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2024-09-04T02:54:57.516243 | 2020-03-02T00:15:09 | 2003.02631 | {
"authors": "Linyan Lu and Zhaohui Yang and Mingzhe Chen and Zelin Zang and\n Mohammad Shikh-Bahaei",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26057",
"submitter": "Zhaohui Yang",
"url": "https://arxiv.org/abs/2003.02631"
} | arxiv-papers | # Machine Learning for Predictive Deployment of UAVs with Multiple Access
Linyan Lu1, Zhaohui Yang1, Mingzhe Chen2, Zelin Zang3, and Mohammad Shikh-
Bahaei1 1Centre for Telecommunications Research, Department of Engineering,
King’s College London, London WC2B 4BG, UK. 2Chinese University of Hong Kong,
Shenzhen, 518172, China,
and also with the Department of Electrical Engineering, Princeton University,
Princeton, NJ, 08544, USA. 3College of Computer Science and Technology,
Zhejiang University of Technology, Hangzhou 310027, China. Emails: lulinyan-
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
In this paper, a machine learning based deployment framework of unmanned
aerial vehicles (UAVs) is studied. In the considered model, UAVs are deployed
as flying base stations (BS) to offload heavy traffic from ground BSs. Due to
time-varying traffic distribution, a long short-term memory (LSTM) based
prediction algorithm is introduced to predict the future cellular traffic. To
predict the user service distribution, a KEG algorithm, which is a joint
K-means and expectation maximization (EM) algorithm based on Gaussian mixture
model (GMM), is proposed for determining the service area of each UAV. Based
on the predicted traffic, the optimal UAV positions are derived and three
multi-access techniques are compared so as to minimize the total transmit
power. Simulation results show that the proposed method can reduce up to 24%
of the total power consumption compared to the conventional method without
traffic prediction. Besides, rate splitting multiple access (RSMA) has the
lower required transmit power compared to frequency domain multiple access
(FDMA) and time domain multiple access (TDMA).
###### Index Terms:
UAV Deployment, LSTM, K-means, EM, GMM, RSMA
## I Introduction
Since user demands for communication services grow dramatically, traditional
base stations (BSs) cannot meet the required demand of the cellular traffic,
which can lead to a bottleneck of cellular communication [1, 2, 3]. Recently,
there has been a growing interest in the study of unmanned aerial vehicle
(UAV) communication due to its excellent attributes of versatility,
maneuverability and flexibility [4]. UAV gradually plays an important role in
wireless communications, which can offer low-cost and efficient wireless
connectivity for devices. UAVs acting as flying BSs is one of the most
important research objects in the UAV communication. Through locations
adjustment, obstacle avoidance, and line-of-sight (LoS) link reinforcement,
UAVs are able to offload data traffic from loaded BSs and increase the
connectivity of wireless network so as to improve the communication
throughput, coverage, and energy efficiency [5].
Therefore, it is a feasible and beneficial option to utilize UAVs to ensure
the connectivity of wireless communication network via meeting the surging
data demands. For efficient and rapid dispatch of UAVs, the prediction of
potential hotspot areas plays a crucial role to help network operators acquire
the information of occurrence and degrees of congestion in advance to reduce
the entire network communication delay [6, 7, 8]. Machine learning techniques
is a useful tool which has the ability to efficiently predict the distribution
of future traffic data [9, 10, 11, 12, 13, 14, 15]. With such predictions, the
target locations of UAVs can be specified beforehand and the deployment can be
more intelligent and on-demand.
There are a number of existing works investigating the applications of UAV
deployment in communication. In UAV deployment, UAV service areas and their
optimal placements are two critical factors [16, 17, 18, 19]. In [16], UAV
latitudes and 2-D locations are optimized based on circle packing theory.
Then, UAV placements and service area boundaries are separately determined for
power efficiency in [17]. To efficiently dispatch UAVs, some intelligent and
practical methods are proposed. A functional split scheme selection of each
UAV is jointed with UAV deployment in [18]. In [19], an adaptive deployment
scheme is introduced to optimize UAV displacement direction, time-dependent
traffic of mobile users can be served in real time.
In UAV communication, machine learning techniques are also applied to improve
system performance [20, 21, 22]. The aerial channel environment is predom
field position (CRF) in [23]. In [20], in order to provide more efficient
downlink service, a learning approach based on the weighted expectation
maximization (WEM) algorithm is used to predict downlink user traffic demand,
meanwhile a traffic load contract using contract theory is introduced.
The main contribution of this work is a predictive UAV deployment scheme for
UAV-assisted networking with ML approaches. Our key contributions include:
* •
The cellular traffic is predicted according to the analysis of previous data
by BP Neural Network model. Then, a joint K-means and EM algorithm of a
Gaussian Mixture Model (KEG) is proposed to divide the entire service area
into clusters as UAV service areas in the temporal and spatial patterns [24].
UAV locations are optimized so as to minimize the total transmit power of the
whole UAV swarm network. Three different access techniques have been compared
in the simulation.
* •
The results show that the proposed UAV deployment framework can reduce the
overall transmit power by over 23.85% compared to the conventional method.
Besides, it is also shown that rate splitting multiple access (RSMA) can
decrease up to 35.5% and 66.4% total power compared to frequency domain
multiple access (FDMA) and time domain multiple access (TDMA), respectively.
## II System Model and Problem Formulation
Given a time-dependent UAV-assisted wireless network, a group of ground users
in the network are distributed in a geographical area $C$. A set $\bm{I}$ of
$I$ UAVs assist a set $\bm{J}$ of $J$ ground BSs to offload amounts of
cellular traffic for congestion alleviation, so the ground users in a time-
variant hotspot area have the air-ground communications with UAVs when ground
BSs are overloaded.
It is assumed that the height of all users and BSs is zero compared to the
UAVs. Besides, each UAV has directional antennas, so the transmissions of
different UAVs will not be interfered with each other. For convenient
elaboration, we specify the ground users served by a UAV as _aerial users_ and
the service area of a UAV $i$ as an _aerial cell_ which can be expressed as
$C_{i}$. In order to consider the communication of all users fairly, the
aerial cells are supposed to completely cover the entire area without any
overlaps.
Because UAVs have limited energy amounts, UAVs should be efficiently deployed
### II-A Air-ground Model
We assume that a ground user $j\in\bm{J}$ located at $(x,y)\in C$ and a UAV
$i\in\bm{I}$ located at $(x_{i},y_{i},h_{i})$ with the aerial cell $C_{i}$, so
the uplink received power of a UAV $i$ and a ground user $j$ is calculated as:
$P_{r,ij}[dB]=P_{ij}[dB]+G_{ij}[dB]-L_{ij}[dB]-r_{ij}[dB],$ (1)
where $P_{ij}$ is the transmit power, $G_{ij}$ is the antenna gain, $L_{ij}$
is the free space path loss, $r_{i}$ is the excessive path loss with a
Gaussian distribution which depends on the category of link. For mathematical
tractability, we give a hypothesis that all of beam alignment of ground-air
links are perfect and UAV antenna gains are same. Thus, $G_{ij}$ can be a
constant number $G$. The free space path loss has a specific formula to be
acquired:
$L_{ij}[dB]=10n\log(\frac{4\pi f_{c}d_{ij}}{c}),$ (2)
where $n\geq 2$ is the path loss exponent, $f_{c}$ is the system carrier
frequency, $c$ is the light speed,
$d_{ij}=[(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+h_{i}^{2})]^{-2}$ is the distance
between a UAV $i$ and a user $j$ [24].
In general, the air-to-ground transmission is separated into two main
cateogories: the line-of-sight link (LoS) and the non-line-of-sight link
(NLoS) . The NLoS link suffers a higher excessive path loss owing to shadowing
and blockage. Figure 2 has shown these two links in the picture.
Figure 1: The LoS Link and the NLoS Link of UAV Transmission
The excessive path loss in these two links can be expressed as
${r_{i}^{LoS}}\sim{\cal N}(\mu_{LoS},\sigma_{LoS}^{2})$ and
${r_{i}^{NLoS}}\sim{\cal N}(\mu_{NLoS},\sigma_{NLoS}^{2})$ respectively. The
probability of the occurance of LoS link is similar to a sigmoid function
[25]:
$p_{ij}^{LoS}=\frac{1}{1+a\exp{[b(a-\theta_{ij})]}},$ (3)
where $a$ and $b$ are environment constant coefficients,
$\theta_{e,ij}=sin^{-1}(h_{i}/d_{ij})$ is the elevation angle of a UAV $i$ and
a user $j$, so the NLoS link existing probability is
$p_{ij}^{NLoS}=1-p_{ij}^{LoS}$ [24]. Therefore, the average excessive path
loss in the uplink transmission is:
$r_{ij}={r_{i}^{LoS}}p_{ij}^{LoS}+{r_{i}^{NLoS}}p_{ij}^{NLoS}$ (4)
To this end, the uplink data rate of a UAV $i$ and a user $j$ can be expressed
as:
$R_{ij}=W_{ij}\log_{2}(\frac{P_{r,ij}}{N_{0}}+1)(bits/s)$ (5)
where $W_{ij}$ is the bandwidth of a UAV $i$ and a user $j$ which can be ,
$N_{0}=n_{0}W_{ij}$ is the power of addictive white Gaussian noise and $n_{0}$
is its average power spectral density. For tractable formulation, each UAV can
offer sufficient overall bandwidth and all transmit bandwidths are assumed to
be a constant value $W$.
### II-B Multi-access Modes
In this paper, three different multi-access techniques will be used. They are
rate splitting multiple access (RSMA), frequency division multiple access
(FDMA) and time division multiple access (TDMA) [26]. We set two users as an
example to give a series of theoretical formulas and set $R_{1}$ and $R_{2}$
are the actual data rates of two users, $g_{1}$ and $g_{2}$ are the channel
gains of two users which include the antenna gain, free space path loss and
the excessive path loss.
RSMA is a multi-access mode with coding and decoding techniques, the equations
for maximizing the sum-rate of users for uplink transmission are shown below
[27, 26, 28, 29, 30, 31, 32, 33, 34]:
$\begin{cases}R_{1}\leq W\log_{2}(1+g_{1}P_{1}/(Wn_{0}))\\\ R_{2}\leq
W\log_{2}(1+g_{2}P_{2}/(Wn_{0}))\\\ R_{1}+R_{2}\leq
W\log_{2}(1+(g_{1}P_{1}+g_{2}P_{2})/(Wn_{0}))\end{cases}$ (6)
FDMA is a technique of BS bandwidth allocation for users to avoid interference
of transmissions in the same area, where $b_{1}$ and $b_{2}$ are the weight
coefficients.
$\begin{cases}R_{1}\leq Wb_{1}\log_{2}(1+g_{1}P_{1}/(Wn_{0}b_{1}))\\\
R_{2}\leq Wb_{2}\log_{2}(1+g_{2}P_{2}/(Wn_{0}b_{2}))\\\
b_{1}+b_{2}=1,b_{1}\geq 0,b_{2}\geq 0\end{cases}$ (7)
Similarly, TDMA assigns the partial time period to users who can use the whole
bandwidth, the equations have been represented.
$\begin{cases}R_{1}\leq Wb_{1}\log_{2}(1+g_{1}P_{1}/(Wn_{0}))\\\ R_{2}\leq
Wb_{2}\log_{2}(1+g_{2}P_{2}/(Wn_{0}))\\\ b_{1}+b_{2}=1,b_{1}\geq 0,b_{2}\geq
0\end{cases}$ (8)
In simulation and analysis section, we will compare these three multi-access
modes for choosing a best one to minimize the total transmit power.
### II-C Uplink Transmit Power Computation
Network operators require to assign UAVs to those traffic congestion areas so
as to offload the heavy traffic from busy BSs. For UAVs, continuous movements
will consume too much transmit power. Thus, we need to analyze the situation
of traffic offloading. A data set is presented as a matrix $\bm{D}$:
$\bm{D}=\\{D^{t}_{d}(x,y)\mid t\in\\{T,...,24T\\},d\in\\{1,...,8\\},(x,y)\in
C\\}$ (9)
where $T$ is the time of an hour, $t$ is the time moment and $d$ is the day.
$D^{t}_{d}(x,y)$ represents the amount of cellular traffic offloaded from a BS
located at $(x,y)$ in a period of $T$ in the day $d$ [24]. In this paper, for
the conveinience of simulation and analysis, we assume that all cellular
traffic of BSs are totally offloaded to UAVs.
Since the positions of mobile users are uncertain and most of mobile users
only move around a single BS in a period of an hour, we assume that all ground
receivers have the same positions as their nearst BSs.
After obtaining the future predictive traffic information through the
implementation of ML methods based on the matrix $\bm{S}$, the required
average data rate within a aerial cell $C_{i}$ in a period of $T$ will be
given. Only when the communication throughput of UAV is not less than the
demanded data rate, the communication can be ensured. Therefore, the
communication condition is formulated as:
$\iint_{C_{i}}R_{i}(x,y)dxdy\geq\frac{1}{T}\iint_{C_{i}}D^{t}_{d}(x,y)dxdy$
(10)
where R(x,y) is the maximum data rate of the overall transmission between a
group of ground users in a ground cell with a BS at $(x,y)$ and a UAV $i$. We
can simplify this equation as
$R_{i}(x,y)\geq D^{t}_{d}(x,y)/T$ (11)
Let us set $D^{t}_{d}(x,y)/T=\alpha(x,y)$, $\alpha(x,y)$ is the minimal
requirement of average data rate. Combining the formula (3.5) and (3.10), the
minimum power for transmission should be provided by a UAV will be:
$P_{i}^{min}(x,y)=(2^{\frac{\alpha(x,y)}{W}}-1)n_{0}WL_{i}(x,y)r_{i}(x,y)/G$
(12)
where both $L_{i}(x,y)$ and $r_{i}(x,y)$ are between the UAV $i$ and the group
of ground users in a ground cell of the BS at $(x,y)$, $L_{i}(x,y)$ is the
path loss in free space, $r_{i}(x,y)$ is the excessive path loss. This
equation provide a target basis for UAV location optimization.
This section proposes a novel predictive scheduling scheme of UAV based on ML.
According to the real data set in City Cellular Traffic Map [35] and the
characteristics of celluar data traffic, for the sake of the efficiency of UAV
deployment, we make some rational assumptions. Because humans have a certain
pace of life with periodic acitivity, the change of cellular data traffic has
a repetitive pattern in daily life [36]. Thus, we assume that the cellular
traffic amount has specific distribution in the same hour in different days
and the data of each hour in the same day is independent. To this end, the
real data set can be classified into 24 independent models and we assume each
single model follows the Guassian mixture model which will be explained in
section 4.3.
The logical procedure diagram of UAV predictive deployment is shown in Figure
3. At first, the acquired real data set is preprocessed to get the cellular
traffic amount of every hour in the first 5 days
$\\{D^{t}_{d}\\}_{d=1,t=T}^{5,24T}$ and the topology information of every BS
$\\{\bm{x}_{n}\\}_{n=1}^{N}=\\{(x_{n},y_{n})\\}_{n=1}^{N}$, where $x_{n}$ is
the relative longitude of $n^{th}$ BS and $y_{n}$ is the relative latitude of
$n^{th}$ BS. Then, a BP neural network model for cellular traffic amount
prediction is come up to predict the cellular amount in $24$ hours in the
$6^{th}$ day. At the same time, a joint K-means and EM algorithm relying on a
GMM for aerial cell classification (ground user clustering) is created, the
point cluster label $\\{l_{n}\\}_{n=1}^{N}$ of every single point $\bm{x}_{n}$
is obtained, the point $\bm{x}_{n}$ with the same label value consists of an
aerial cell. Then, the optimal UAV locations
$\\{\bm{x}_{i}\\}_{i=1}^{K}=\\{(x_{i},y_{i})\\}_{i=1}^{K}$ for minimizing
total transmit power $P_{min}$ is derived according to the system model
introduced in section 3. The purpose of whole process is to achieve an UAV aid
of the on-demand, power-effective, lantency-free network services.
Figure 2: The Logical Procedure Diagram of UAV Predictive Deployment
### II-D Cellular Demand Prediction
In this part, a simple BP neural network model is utilized to predict the
future cellular demand in an hour. Neural network has a variety of categories
and BP neural network is the most basic one. The theory introduction is shown
below.
#### II-D1 The Neuron Model
Refere to biological neural systems, the basic component of neural networks is
neuron model. In a biological neural system, every neuron is connected to
other neurons. A neuron accepts some chemical materials (chemical messages) as
its input emanated from other connected neurons, the electrical potential in
this neuron changes; If the amount of the input is enough, the electrical
potential exceeds to a threshold, this neuron will be activated and send
chemical materials to other neurons.
The famous McCulloch and Pitts model (M-P) of neuron as Figure 4 is based on
the above description, which can be formulated in a mathematical form:
$y=f(\bm{w}\bm{x}-\theta)$ (13)
where $\bm{x}=\\{x_{i}\\}_{i=1}^{n}$ is the input of the neuron,
$\bm{w}=\\{w_{i}\\}_{i=1}^{n}$ is its corresponding weight, $\theta$ is a
threshold of activation, and $f(.)$ is an activation function with multiple
forms. The step function is the most ideal activation function, but it is
discontinuous and not smooth, so it is not convenient to do the operations of
integral and differential. Hence, in most cases, we usually use a canonical
example, the sigmoid function which can squash the large range of input values
into the range from 0 to 1 of outputs. The step function and the sigmoid
function are shown in the Figure 5.
Figure 3: M-P Neuron Model
Figure 4: The Step Function and the Sigmoid Function
When enough training data sets are given, the weights and threshold can be
obtained by learning. The threshold can be seen as a dummy node with fixed
input $1$, so the learning of weights and threshold can be jointed together.
For a single training data set ($x$,$y$), the current output is $\hat{y}$ for
the input $x$. Weight updating can be shown as:
$w_{i}\leftarrow w_{i}+\bigtriangleup w_{i}$ (14) $\bigtriangleup
w_{i}=\eta(y-\hat{y})x_{i}$ (15)
where $\eta\in(0,1)$ is a learning rate. In this way, the weight will be
adjusted until $y=\hat{y}$. This is a single functional neuron, its learning
ability is very limited. Thus, we always add the hidden layers to compose a
multi-layer feedforward nerual network.
#### II-D2 Error Backpropagation
For multi-layer networks, BP(backpropagation) is one of the most successful
learning algorithms. BP algorithm is not only used in the multi-layer feed
forward networks, but also used in other networks such as recurrent neural
networks. The basic structure of the neural networks consists of an input
layer and an output layer with multiple hidden layers between them. For clear
and concise description, the number of hidden layers is set to $1$.
The input layer is $\bm{x}=\\{x_{i}\\}_{i=1}^{d}$, the hidden layer is
$\bm{b}=\\{b_{h}\\}_{h=1}^{q}$, and the output layer is
$\hat{\bm{y}}=\\{\hat{y}\\}_{j=1}^{l}$. The weight between the input layer
node $i$ and the hidden layer node $h$ is $v_{ih}$, The weight between the
hidden layer node $h$ and the output layer node $y$ is $w_{ih}$. The variable
symbols and the BP network model with three layers are shown in Figure 6, the
weights are also represented.
Figure 5: Multilayer Network Sketch
For $h^{th}$ node of hidden layer and $j^{th}$ node of output layer, the
inputs and the outputs are shown below:
$\begin{cases}\alpha_{h}=\sum_{i=1}^{d}v_{i}{}_{h}x_{i}\\\
b_{h}=f(\alpha_{h}-\gamma_{h})\\\ \beta_{j}=\sum_{h=1}^{q}w_{h}{}_{j}b_{h}\\\
\hat{y}_{j}=f(\beta_{j}-\theta_{j})\end{cases}$ (16)
Then the mean square error can be obtained as following, where $1/2$ is for
the convenience of the subsequent calculation.
$E=\frac{1}{2}\sum_{j=1}^{l}(\hat{y}_{j}-y_{j})^{2}$ (17)
Because BP algorithm is based on gradient descent algorithm, parameters are
adjusted according to the negative gradient. Set the gradient of output layer
as an example, the change of weight can be written as
$\bigtriangleup w_{h}{}_{j}=-\eta\frac{\partial E}{\partial w_{h}{}_{j}}$ (18)
Besides, the chain rule can make the formula as
$\frac{\partial E}{\partial w_{h}{}_{j}}=\frac{\partial
E}{\partial\hat{y}_{j}}\cdot\frac{\partial\hat{y}_{j}}{\partial\beta_{j}}\cdot\frac{\partial\beta_{j}}{\partial
w_{h}{}_{j}}$ (19) $\frac{\partial\beta_{j}}{\partial w_{h}{}_{j}}=b_{h}$ (20)
The intermediate variable of gradient can be obtained as
$g_{j}=-\frac{\partial
E}{\partial\hat{y}_{j}}\cdot\frac{\partial\hat{y}_{j}}{\partial\beta_{j}}=\hat{y}_{j}(1-\hat{y}_{j})(y_{j}-\hat{y}_{j})$
(21)
Therefore, the gradient of the hidden layer is similar,
$e_{h}=-\frac{\partial E}{\partial b_{h}}\cdot\frac{\partial
b_{h}}{\partial\alpha_{h}}=b_{h}(1-b_{h})\sum_{j=1}^{l}w_{h}{}_{j}$ (22)
Finally, we can get the update of weights and thresholds for the hidden and
output layer,
$\begin{cases}\bigtriangleup w_{h}{}_{j}=\eta g_{j}b_{h}\\\
\bigtriangleup\theta_{j}=-\eta g_{j}\\\ \bigtriangleup v_{i}{}_{h}=\eta
e_{h}x_{i}\\\ \bigtriangleup\gamma_{h}=-\eta e_{h}\end{cases}$ (23)
Note that the aim of BP algorithm is to minimize the accumulate error for all
training data sets. The workflow of BP algorithm is represented in Algorithm
1. The stop condition is that the iteration has achieved the peak value or the
error $E$ is smaller than a minimum threshold.
Algorithm 1 BP Algorithm [37]
0: A training data set $\bm{S}=\\{(\bm{x}_{k},\bm{y}_{k})\\}^{m}_{k=1}$, a
learning rate $\eta$.
Randomly initiate all weights and thresholds $w$, $\theta$, $v$, $\gamma$
repeat
for all $(\bm{x}_{k},\bm{y}_{k})\in\bm{S}$ do
Calculate the current output $\hat{y}_{k}$ using Equation (4.4)
Calculate intermediate variable of gradient $g$ and $e$ using Equation (4.9)
and (4.10)
Update weights and threshold $w$, $\theta$, $v$, $\gamma$ using Equation
(4.11)
end for
until The stop condition is reached
Weights and thresholds $w$, $\theta$, $v$, $\gamma$
However, BP algorithm has some inadequacies. Overfitting problem is very
obvious because of the powerful presentation of BP networks, early stopping
method and regularization method are often used to prevent this situation. The
limitation of local minimum is another problem, people usually utilize
simulated annealing, genetic algorithm and random gradient descent algorithm
to solve it [37].
### II-E Ground User Clustering
Ground user clustering is a key step in UAV deployement, which also means the
partition of UAV aerial cells. In order to satisfy the fairness and globality
of division, we adopt a KEG algorithm to implement service area
classification. And To show the algorithm practicability, we use the topology
information part of City Cellular Traffic Map [35] as the real data set.
#### II-E1 The K-means Algorithm
The K-means algorithm is a most basic non-hierarchical iterative clustering
algorithm, belonging to unsupervised learning. The goal of this algorithm is
to cluster data points with very low inter-cluster similarity as well as very
high intra-cluster similarity [38]. The similarity often denotes the distances
between data points.
Giving a data set $\bm{X}=\\{\bm{x}_{n}\\}^{N}_{n=1}$ composed of $N$ number
of $M$-dimensional variables $\bm{x}_{n}$. We divide this data set into $K$
clusters, each cluster has a centroid. At first, we set the centroids of
clusters are $K$ vectors $\\{\bm{\mu}_{k}\\}^{K}_{k=1}$ with $M$ dimensions.
These vectors usually randomly take $K$ variables from the given data set
$\bm{X}$ for initialisation. To realize the algorithm goal, each data point is
supposed to be as close as possible to its cluster centroid, and the distances
of each data point and other cluster centroids should be as large as possible.
Thus, we set that point cluster label $\bm{r}_{nk}$ indicates which cluster
the data point $\bm{x}_{n}$ belongs to, this variable can be formulated as:
$\bm{r}_{nk}=\begin{cases}1&\text{if $k=\mathop{\arg\min}_{i}\
\|\bm{x}_{n}-\bm{\mu}_{i}\|$}\\\ 0&\text{otherwise}\end{cases}$ (24)
Then, we assign every data point to its nearest cluster and update the point
cluster label after the distances between each data point and cluster
centroids are calculated. The third one is to update cluster centroids
according to those data point label. The specific K-means algorithm is shown
as Algorithm 2.
Algorithm 2 K-means Algorithm [38]
0: The cluster number $K$, the data point set
$\bm{X}=\\{\bm{x}_{n}\\}^{N}_{n=1}$.
1: Initialize $\\{\bm{\mu}_{k}\\}^{K}_{k=1}$ as $K$ variables chose from
$\bm{X}$ randomly, initialize $\bm{r}_{nk}$ as an $n*k$ all-zero matrix
2: repeat
3: for all $\bm{x}_{n}\in X$ do
4: Allocate each data point $\bm{x}_{n}$ to cluster
$k^{*}=\mathop{\arg\min}_{i}\ \|\bm{x}_{n}-\bm{\mu}_{i}\|$
5: Update point labels $\bm{r}_{nk}(n,k^{*})=1$
6: Calculate cluster centroids
$\bm{\mu}_{k}=\sum\bm{r}_{nk}\bm{x}_{n}/\sum\bm{r}_{nk}$, $k=1,...,K$
7: end for
8: until $\\{\bm{\mu}_{k}\\}^{K}_{k=1}$ will not be changed.
8: The point cluster label $\bm{r}_{nk}$, the cluster centroids
$\\{\bm{\mu}_{k}\\}^{K}_{k=1}$
On the one side, although the K-means algorithm has the capability to cluster
data points, it cannot find the latent variables corresponding the observed
data. On the other side, the principle of the K-means is simple and it is easy
to implement the simulation because of its fast convergence and good clusting
effect, it can be a pratical data initializer for other complex algorithms.
#### II-E2 The EM Algorithm
The EM algorithm has the ability to recognize the important role of latent
variables in a joint distribution. Its goal is to acquire maximum likelihood
results for the models with latent variables [38].
In a mixed distribution model, given a sample data set
$\bm{X}=\\{\bm{x}_{n}\\}^{N}_{n=1}$ with an unknown latent variable set
$\bm{Z}=\\{\bm{z}_{n}\\}^{N}_{n=1}$, we want to find the suitable parameter
set $\bm{\theta}$ to well discribe the joint distribution
$p(\bm{X}|\bm{\theta})=\sum_{\bm{Z}}p(\bm{X},\bm{Z}|\bm{\theta})$ [38].
However, the observed variable set $\bm{X}$ and the latent variable set
$\bm{Z}$ are determined by parameter set $\bm{\theta}$. And we are only given
the incomplete data set $\bm{X}$ without $\bm{Z}$, so we cannot get the
optimal parameter set $\bm{\theta}$. To facilitate analysis, a log likelihood
function is defined as:
${\cal{L}}(\bm{\theta})=\ln{\\{p(\bm{X}|\bm{\theta})\\}}=\ln{\\{\sum_{\bm{Z}}p(\bm{X},\bm{Z}|\bm{\theta})\\}}$
(25)
And a posterior distribution $p(\bm{Z}|\bm{X},\bm{\theta})$ about the latent
variable set $\bm{Z}$ is introduced. Thus, our goal is changed to get the
maximum likelihood function $p(\bm{X}|\bm{\theta})$ with regard to
$\bm{\theta}$. The iteration of EM mainly consists of two step: Expectation
step (E step) and Maximum step (M step). In E step, we evalulate the posterior
probability $p(\bm{Z}|\bm{X},\bm{\theta)}$; In M step, we operate the log
likelihood maximization to update parameter set $\bm{\theta}$ [38]. The
iteration stops until the convergence of the log likelihood function is
checked. The specific algorithm is shown in Algorithm 3.
Algorithm 3 EM Algorithm [38]
0: The observed variable set $\bm{X}$
1: Initialize the parameter set $\bm{\theta}_{old}$,
2: repeat
3: E step Calculate the posterior probability
$p(\bm{Z}|\bm{X},\bm{\theta}_{old})$
4: M step Calculate $\bm{\theta}_{new}=\mathop{\arg\min}_{\bm{\theta}}\
\sum_{\bm{Z}}p(\bm{Z}|\bm{X},\bm{\theta}_{old})\ln{p(\bm{X},\bm{Z}|\bm{\theta})}$
5: Update the parameter set $\bm{\theta}$,
$\bm{\theta}_{old}\leftarrow\bm{\theta}_{new}$
6: until The log likelihood function ${\cal{L}}(\bm{\theta})$ converges.
6: The posterior probability $p(\bm{Z}|\bm{X},\bm{\theta})_{old}$, the
parameter set $\bm{\theta}_{old}$
The EM algorithm can also have good performance in the situation of missing
some observed variable values, the observed variable distribution can be
aquired by marginializing those missing value and taking the whole joint
variable distribution. In this case, the data returned by the sensor which has
some values missing can also be well processed. Therefore, in the scenario of
our ground user clustering, EM algorithm is a useful method to find the latent
variables in data set and classify users in a fair manner.
#### II-E3 The KEG Algorithm
In this paper, the cellular traffic distribution is complex and time-varying,
but a GMM, a linear Gaussian-component superposition model has a remarkable
advantage of abundantly representing data distribution. We model the cellular
traffic distribution by the GMM as:
$p(\bm{X})=\sum_{k=1}^{K}\pi_{k}{\cal{N}}(\bm{X}|\bm{\mu}_{k},\bm{\sigma}_{k})$
(26)
where $\bm{X}=\\{\bm{x}_{n}\\}^{N}_{n=1}$ is the tolopogy information of the
whole area and $\bm{x}_{n}$ is every data point with $M$ dimensions. $p(.)$
represents as a probability function, $K$ is Gaussian component number and
$k\in\\{1,...,K\\}$ denotes a specific one Gaussian component, the mixing
coefficient $\pi$ equals one or zero has $\sum_{k=1}^{K}\pi_{k}=1$,
$\bm{\mu}=\\{\bm{\mu}_{n}\\}^{N}_{n=1}$ is the mean values corresponding to
the cluster centroids with $M$ dimensions,
$\bm{\sigma}=\\{\bm{\sigma}_{n}\\}^{N}_{n=1}$ is the covariance with $M$
dimensions. Besides, in a GMM, the latent variables are discrete.
To this end, we introduce the KEG algorithm based on the K-means algorithm and
the EM algorithm. In the KEG algorithm, the EM part aims to find the discrete
latent variables and the suitable parameters for analyzing data distribution
and clustering data set. Even if a data set is incomplete with missing some
values, the EM can also process the data set in a suitable manner. In the EM
part, the value of the log likelihood function will increase with the number
of iteration rising. When the log likelihood function does not change anymore,
the current parameters are the aims we want to obtain. But since this
algorithm usually needs many iterations to reach the convergent point, the
K-means part is utilized as a data initializer to provide appropriate and
rational initialized values. The integrated KEG algorithm is demonstrated in
Algorithm 4.
Algorithm 4 KEG algorithm
0: The topology data set $\bm{X}=\\{\bm{x}_{n}\\}^{N}_{n=1}$, the clustering
number $K$
1: Set a variable $D$ as the dimension of $\bm{x}_{n}$, initialize the means
$\bm{\mu}_{k}$ as a $1$-by-$D$ matrix by using the K-means algorithm, the
covariance $\bm{\sigma}_{k}$ as a $D$-by-$D$ identical matrix and mixing
coefficients $\pi_{k}=1/K$, $\forall k\in\\{1,...,K\\}$.
2: repeat
3: for all $k\in\\{1,...,K\\}$ do
4: E step: Compute the posterior probability of all $\bm{x_{n}}$
by$\gamma_{nk}=\pi_{k}{\cal{N}}(\bm{x}_{n}|\bm{\mu}_{k},\bm{\sigma}_{k})/\sum_{i=1}^{K}\pi_{i}{\cal{N}}(\bm{x}_{n}|\bm{\mu}_{i},\bm{\sigma}_{i})$
5: For every $\bm{x}_{n}$, allocate it to the cluster
$l_{n}=\mathop{\arg\max}_{k}\gamma_{nk}$
6: M step: Calculate the new parameters
$\bm{\mu}_{k}^{new}=\sum_{n=1}^{N}\gamma_{nk}\bm{x}_{n}/\sum_{n=1}^{N}\gamma_{nk}$
$\bm{\sigma}_{k}^{new}=\sum_{n=1}^{N}\gamma_{nk}(\bm{x}_{n}-\bm{\mu}_{k}^{new})(\bm{x}_{n}-\bm{\mu}_{k}^{new})^{T}/\sum_{n=1}^{N}\gamma_{nk}$$\pi_{k}^{new}=\sum_{n=1}^{N}\gamma_{nk}/\sum_{k=1}^{K}\sum_{n=1}^{N}\gamma_{nk}$
7: end for
8: Calculate the log likelihood using $\bm{\mu}_{k}^{new}$,
$\bm{\sigma}_{k}^{new}$ and $\pi_{k}^{new}$, for
$k\in\\{1,...,K\\}$${\cal{L}}(\bm{\mu},\bm{\sigma},\bm{\pi})=\ln{p(\bm{X}|\bm{\mu},\bm{\sigma},\bm{\pi})}=\sum_{n=1}^{N}\ln{\\{\sum_{k=1}^{K}\pi_{k}{\cal{N}}(\bm{x}_{n}|\bm{\mu}_{k},\bm{\sigma}_{k})\\}}$
9: Update the parameters as $\bm{\mu}_{k}\leftarrow\bm{\mu}_{k}^{new}$,
$\bm{\sigma}_{k}\leftarrow\bm{\sigma}_{k}^{new}$,
$\pi_{k}\leftarrow\pi_{k}^{new}$
10: until The log likelihood function
${\cal{L}}(\bm{\mu},\bm{\sigma},\bm{\pi})$ is converged.
10: The parameters $\\{\pi_{k}$, $\bm{\mu}_{k}$,
$\bm{\sigma}_{k}\\}_{k=1}^{K}$, the cluster labels $\\{l_{n}\\}_{n=1}^{N}$.
When the parameters are obtained, the predicted cellular traffic amount data
is used as the input, only $4^{th}$ step and $5^{th}$ step of Algorithm 4 is
operated, we can get the cluster label $\\{l_{n}\\}_{n=1}^{N}$of the data
points $\\{\bm{x}_{n}\\}^{N}_{n=1}$ that indicates which cluster the data
point belongs to.
### II-F UAV Location Optimization
After determining the aerial cells using the KEG algorithm, our next aim is to
select a optimal location for every UAV so that the minimum transmit power can
be gotten. No matter whether UAVs are in a high altitude platform or a low
altitude platform, we assume that all UAVs are in the same altitude $h$, we
can formulate this problem as:
$\min_{x_{i},y_{i}}\ \
P_{i}=Q\iint_{C_{i}}A_{i}(x,y)d_{i}^{2}(x,y)r_{i}(x,y)dxdy$ (27)
where $Q=(\frac{4\pi f_{c}}{c})^{2}\frac{Wn_{0}}{G}$ is not relevant to the
locations of BS $(x,y)$ in the service area,
$A_{i}(x,y)=2^{\frac{TD^{t}(x,y)}{W}-1}$ is the BS distribution and
$D^{t}(x,y)$ denotes the cellular data amount of BS at $(x,y)$ in a hour in
the predictive day, $d_{i}^{2}(x,y)$ is the distance between the UAV $i$ and
the BS at $(x,y)$, $r_{i}(x,y)$ is also related to $d_{i}^{2}(x,y)$ for the
computation of LoS link probability.
According to the Theorem 1 of paper [17], the function to get $P_{i}^{min}$ is
a convex function, UAV optimal locations can be calculated as:
$x_{i}^{*}=\frac{\iint_{C_{i}}xA_{i}(x,y)dxdy}{\iint_{C_{i}}A_{i}(x,y)dxdy}$
(28)
$y_{i}^{*}=\frac{\iint_{C_{i}}yA_{i}(x,y)dxdy}{\iint_{C_{i}}A_{i}(x,y)dxdy}$
(29)
where there is a condition $h_{i}>>(x-x_{i})^{2}+(y-y_{i})^{2}$ or
$h_{i}<<(x-x_{i})^{2}+(y-y_{i})^{2}$ to be satisfied [17].
At last, we accumulate the transmit power of all operating UAV to get the
minimum total power for transmission:
$P_{min}=\sum_{i}P_{i}^{min}$ (30)
## III Simulation and Analysis
### III-A Illustration of Simulation Process
For simulation, we consider two scenrios, one is the entire area given from
the raw data, the other is the limited area with the relative longitude from
$111.055$ to $111.07$ degrees and the relative altitude from $13.03$ to
$13.05$ degrees. The limited area is the contrast of the entire area. And for
a specific contrast, we classify the entire area and the limited area into $8$
clusters. Some parameters used in the simulation have been given the specific
values which are shown in Table 1 [39]. Moreover, for getting the rational
simulation results, we assume that all BSs have a basic celluar amount $500$
bytes for basic operation, so all the cellular amounts we get from the raw
data add $500$.
TABLE I: Simulation Parameters $f_{c}$ | Carrier frequency | 5GHz
---|---|---
$n_{o}$ | Noise power spectral | -140dBm/Hz
$\mu_{LoS}$ | Excessive path loss for LoS link | 3dB
$\mu_{NLoS}$ | Excessive path loss for NLoS link | 23dB
$W$ | Bandwidth | 1MHz
$h$ | UAV’s altitude | 200m
$G$ | Antenna gain | 10dB
For BP neural network part, we use the _nftool_ built-in box of MATLAB to
implement the cellular traffic amount prediction. We assume that every BS has
the same cellular traffic distribution, we set that the input is the cellular
traffic amount of the first six days and the output is the amount of the
seventh day. The input in the training data set form is
$(\\{\bm{D}^{t}_{d}\\}_{d=1}^{5},\bm{D}^{t}_{d=6})$, where the input layer is
the first $5$ days cellular amount and the output layer is the $6^{th}$ day
cellular amount.
We choose _trainrp_ as the training function and set that the neural network
has 2 hidden layers with $20$ and $10$ neurons respectively, the network
structure is shown in Figure 7. Then we do data training to find the suitable
parameters for our neural network model. The user interface of neural network
training is shown in Figure 8.
Figure 6: Neural Network Structure
Figure 7: Neural Network Training
After training, the input layer $\bm{x}$ is set as
$\\{\bm{D}^{t}_{d}\\}_{d=2}^{6}$, then this output layer is what we want to
obtain. Furthermore, the ratio of training data is $80\%$, valuating data is
$10\%$ and testing data is $10\%$. The data training is for updating weights,
the data evaluation is to detect the overfitting problem and to avoid it as
soon as possible, the data testing is to examine the performance of the neuron
network. In addition, all the cellular data amounts are normalized using min-
normalizaion mothed to make sure every single value is ranged from $0$ to $1$.
Then, for the application of the KEG algorithm, we give two conditions for
finishing the iterations: the first one is the minimum threshold for parameter
error of two neighboring iterations, the second one is the maximum iteration
times. If any of these conditions are met, the algorithm ends.
### III-B The Comparision of Three Multi-access Techniques
In the proposed UAV deployment framework, RSMA as an excellent option of
multiple access technique is adopted in the uplink transmission. FDMA and TDMA
have been developed maturely, but RSMA with excellent robustness and energy
effeciency enjoys great popularity in the new generation of communications.
The Figure 9 is shown below which represents the comparison the performance of
RSMA, FDMA and TDMA. For all three of multi-access techniques, with the
increasing of bandwidth, the system can reduce the total power. But RSMA is
the best one which has the lowest power consumption in all bandwidths. Then
FDMA is the second best and it is better than TDMA. RSMA can reduce 35.3%
power than FDMA and 66.4% than TDMA. Therefore, in this paper, we use RSMA for
uplink transmission.
Figure 8: Comparison of Three Multi-access Modes
### III-C The Simulation Results of the Limited Area and the Entire Area
In order to verify the usefulness and robustness of the KEG algorithm, we use
the data of the limited area and the entire area to do the simulation. The
comparison results of the K-means part are shown in Figure 10 and Figure 11
and the EM part are shown in Figure 12 and Figure 13.
Figure 9: K-means for the Limited Area
Figure 10: EM for the Entire Area
Figure 11: EM for the Limited Area
In the Figure 9, 10, 11 and 12, the scattering points are the locations of BS.
The group of the points with the same color represents a cluster which is an
aerial cell of a UAV. The black crosses denotes the centroids of every
cluster. As we can see, for the K-means part, both the entire area and the
limited area have distinct boundaries of service cell and the centroids locate
at the clusters with the corresponding colors. But for the EM part, the
clusters have the containing and contained states, the centroids may not
appear in the clusters with their own colors. The corresponding situation of
the limited area is much better than that the entire area.
### III-D The Effect of Proposed UAV Scheduling Framework
The main goal of UAV deployment in this paper is to minimize the total
transmit power in the uplink transmission. Thus, based on this goal, we divide
the area into the ground user clusters for aerial cell classification and
determine the optimal locations of UAVs. The Figure 14 shows the schematic
sketch of UAV deployment for the entire area and the limited area.
Figure 12: Deployment Results
As shown in the Figure 14, the number of UAV of the limited area is $8$, but
the one of the entire area changes into $7$ because the system with our
proposed framework judges that $7$ UAVs are enough to supply the offloaded
cellular data amount, this decision is made during the clustering of the
K-means part. Because the initial centroid of the merged category chooses an
unfavorable position leading to a low similarity in the cluster.
Figure 13: Power Comparison for the Limited Area
Figure 14: Power Comparison for the Entire Area
Then we use the total transmit power of four different schemes to evaluate the
performance of the proposed framework. Experimental schemes are a scheme with
KEG and location optimization, a scheme without KEG but with location
optimization, a scheme with KEG but without location optimization, and a
scheme without neither KEG or location optimization.
For the limited area, the power comparison of four schemes is represented in
Figure 15. In general, the scheme with KEG and location optimization consume
the least total transmit power. The performance of the one with no KEG and no
location optimization is worst. The remaining two types of scheme have similar
performance in general. The scheme with KEF and location optimization reduce
24% power consumption than the worst one. For the entire area, four schemes
have been contrasted in Figure 16. Obviously, the scheme without KEG but with
location optimization has the best effect for system performance improvement.
Next, the scheme with KEG but no location optimization is better than the one
with location optimization but no KEG. The scheme with no KEG and no location
is the worst scheme. The scheme with KEF and location optimization reduce
0.47% power consumption than the worst one.
Based on the above simulation results, our proposed framework is not suitable
to apply only several UAVs in the entire area. The number of UAVs is too small
to carry the full range of traffic of an entire city. But still in this
situation, the scheme with KEG and location optimization has some
contributions for reducing the total transmit power consumption. Our UAV
deployment framework has good performance for relatively small area below
dozens of square kilometers, especially for those areas with the approximate
GMM cellular data amount distribution.
## IV Conclusion
In this paper, we have investigated the UAV location optimization in a
downlink system. To effectively optimize the UAV location, we fist predice the
user traffic distribution by a joint K-means and EM algorithm based on GMM.
With the predicted traffic distribution, the optimal locations for UAVs are
accordingly obtained. Simulation results show that RSMA reduces total power
consumption compared to FDMA, and TDMA.
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|
2024-09-04T02:54:57.529715 | 2020-02-25T19:47:19 | 2003.02638 | {
"authors": "Marcus Ebner von Eschenbach, Binyamin Manela, Jan Peters, Armin Biess",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26058",
"submitter": "Marcus Ebner Von Eschenbach",
"url": "https://arxiv.org/abs/2003.02638"
} | arxiv-papers | # Metric-Based Imitation Learning Between Two Dissimilar Anthropomorphic
Robotic Arms
Marcus Ebner von Eschenbach1,2, Binyamin Manela1, Jan Peters2, Armin Biess1
*This work was supported in part by the Helmsley Charitable Trust through the
Agricultural, Biological and Cognitive Robotics Initiative and the Israel
Science Foundation (grant no. 1627/17).1Department of Industrial Engineering
and Management Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel
<EMAIL_ADDRESS>Intelligent Autonomous Systems Group Technical University
Darmstadt, 64289 Darmstadt, Germany Germany<EMAIL_ADDRESS>
###### Abstract
The development of autonomous robotic systems that can learn from human
demonstrations to imitate a desired behavior - rather than being manually
programmed - has huge technological potential. One major challenge in
imitation learning is the correspondence problem: how to establish
corresponding states and actions between the expert and learner, when the
embodiments of the agents are different (morphology, dynamics, degrees of
freedom, etc.). Many existing approaches in imitation learning circumvent the
correspondence problem, for example, kinesthetic teaching or teleoperation,
which are performed on the robot. In this work we explicitly address the
correspondence problem by introducing a distance measure between dissimilar
embodiments. This measure is then used as a loss function for static pose
imitation and as a feedback signal within a model-free deep reinforcement
learning framework for dynamic movement imitation between two anthropomorphic
robotic arms in simulation. We find that the measure is well suited for
describing the similarity between embodiments and for learning imitation
policies by distance minimization.
## I INTRODUCTION
(a)
(b)
Figure 1: Static pose imitation between dissimilar antrophomorphic robotic
arms using a 7-DOF-Panda expert. (a) 4-DOF-Panda learner; (b) 3-DOF-Panda
learner. Dissimilar robots are generated by locking DOFs (expert is shown on
the left, the learner on the right, locked joints in red).
Approaches to imitation learning in robotics have delivered huge success
ranging from helicopter acrobatics [1], high-speed arm skills [2], haptic
control [3, 4], gestures [5], manipulation [6, 7, 8] to legged locomotion [9,
8]. The machine learning algorithms that make imitation learning possible are
well studied and have recently been summarized [10]. Surprisingly, despite all
of these impressive successes in the acquisition of new robot motor skills,
fundamental research questions in imitation learning of central importance
have remained open for decades. Among such core questions is the
correspondence problem: how can one agent (the learner or imitator) produce a
similar behavior - in some aspect - to behavior it perceives in another agent
(the expert or demonstrator), given that the two agents obey different
kinematics and dynamics (body morphology, degrees of freedom (DOFs),
constraints, joints and actuators, torque limits), i.e., occupy different
state spaces [11]?
Existing algorithmic approaches towards imitation learning can be divided into
two groups: behavioral cloning (BC) and inverse reinforcement learning (IRL)
or inverse optimal control (IOC), both can be further subdivided into model-
based and model-free approaches depending on whether the system dynamics is
available or not [10]. BC and IRL make different assumptions about the
correspondence of learner and expert. In BC, a mapping from states to actions
is generated from the demonstrations using supervised learning methods. This
mapping can then be used by the learner to reproduce similar behavior,
provided that the embodiments of expert and learner are alike, otherwise the
method will fail due to lack of correspondence.
Successful implementations of model-based BC algorithms have been obtained for
a hitting-a-ball task with an underactuated robot [12], playing video games
[13] and controlling a UAV through a real forest [14]. Model-free BC
algorithms have been implemented for autonomous RC helicopter flight using
experts demonstrations [1] and for learning tasks, such as tennis swings [15],
ball-paddling [2] and human-robot collaborative motions in tool-handover tasks
[16], autonomous knot-tying using a surgical robot [17] and character
animation [18].
In an IRL framework, the learner infers a reward function for a given task
from expert demonstrations of the task. The underlying assumption is that the
reward function is a parsimonious and portable representation of the task,
which can be transferred and generalized to agents with different embodiments.
Thus, IRL implicitly resolves the correspondence problem but has the
disadvantage of being computationally expensive and requires reinforcement
learning in an inner loop. IRL has been implemented mostly in a model-based
approach for tasks, such as learning to drive a car in a simulator [19] and
path planning [20, 21, 22]. A few model-free IRL algorithms have been proposed
and used to learn policies of robot motion [4, 23], [24].
The correspondence problem results in the following question: what action
sequence in the learner is required to produce behavior that is similar to the
expert, given that learner and expert have different embodiments and given
that a measure of similarity $d$ is defined. If we denote the state-action
pairs of the expert and learner as $(\bm{s}_{t},\bm{a}_{t})$ and
$(\bm{\hat{s}}_{t},\bm{\hat{a}}_{t})$, respectively, with $t=1,2,\dots T$,
then we can formulate the correspondence problem in its simplest form as
follows: for a given set of demonstrations
${\cal{D}}=\\{\bm{s}_{1},\bm{a}_{1},\dots,\bm{a}_{T-1},\bm{s}_{T}\\}_{i=1}^{N}$
find actions $\bm{\hat{a}}_{t},t=1,2,\dots T$, so that $\bm{\hat{s}}_{t}$ is
similar to $\bm{s}_{t}$ for all $t=1,2,\dots T$, where similarity (negative
loss) is defined as
$\sum_{t=1}^{T}d(\bm{s}_{t},\bm{\hat{s}}_{t})~{}\rightarrow~{}\mbox{min}$.
Note that the states depend on the actions via the system dynamics, thus, it
is $\bm{{s}}_{t+1}=\bm{f}(\bm{{s}}_{t},\bm{{a}}_{t})$ and
$\hat{\bm{s}}_{t+1}=\hat{\bm{f}}(\hat{\bm{s}}_{t},\hat{\bm{a}}_{t})$, where
$\bm{f}$ and $\hat{\bm{f}}$ describe the system dynamics of expert and
learner, respectively. Several levels of complexity can be added to this
formulation. (1) Generally, the dynamics for real systems are stochastic,
thus, $\bm{{s}}_{t+1}=\bm{f}(\bm{{s}}_{t},\bm{{a}}_{t})+\bm{\epsilon}_{t}$ and
$\hat{\bm{s}}_{t+1}=\hat{\bm{f}}(\hat{\bm{s}}_{t},\hat{\bm{a}}_{t})+\hat{\bm{\epsilon}}_{t}$,
where $\bm{\epsilon}_{t}$ and $\hat{\bm{\epsilon}}_{t}$ is noise. In this case
the distance can be measured between probability distributions of state
vectors,
$\sum_{t=1}^{T}d(p(\bm{s}_{t}),\hat{p}(\bm{\hat{s}}_{t}))\rightarrow\mbox{min}$,
where $p(\bm{s}_{t})$ and $\hat{p}(\hat{\bm{s}}_{t})$ denote the probability
distribution for expert and learner, respectively (2) The system dynamics of
expert and/or learner are often not known, leading to model-free vs. model-
based approaches. (3) The demonstrations are often given in the form
${\cal{D}}=\\{\bm{s}_{1},\dots,\bm{s}_{T}\\}$, i.e., the actions of the expert
are not available. (4) The states of the expert may be only partially
observable, for instance, if the environment is observed by cameras. The
states must then be inferred from observations $\bm{o}_{t}$ only.
In this work we study imitation tasks between two dissimilar anthropomorphic
robot arms, which are generated by locking degrees of freedom (DOFs) in the
learner. Throughout this work we assume that the dynamics of the learner are
not available to the learning agent, thus, we are in a model-free setting. We
first introduce our definition of an embodiment state and provide a distance
measure to assess the similarity between embodiments. This distance measure is
then used to imitate static poses using neural networks (Fig.1) and as a
feedback signal for movement imitation using reinforcement learning.
## II RELATED WORK
Metric approaches to the correspondence problem in imitation have been
developed in a series of studies [25, 26, 27, 28, 29, 30]. In these studies,
the correspondence problem was formulated in state-action space with separate
metrics for states and actions. Simple global metrics based on the Hamming
norm, Euclidean distance and infinity norm ($L_{p}$-norms) were used to
measure the similarity between expert and learner. Another approach to the
correspondence problem is to explicitly learn the forward dynamics of the
learner. The actions of the learner are then adapted to the given
demonstrations by using the learned forward dynamics. Within this framework,
Englert et al. [12] have used the Kullback-Leibler divergence as similarity
measure to compare trajectory distributions from an expert and a robot
learner. Similarly, Grimes et al. [31, 32] have used Gaussian Mixture Models
to learn a forward model and infer optimal actions of the learner. The model
has been used to transfer human motions to a humanoid robot.
## III METHODS
### III-A Definition of an embodiment
In this study an embodiment consists of a chain of links, which are
represented by frames that are attached to each link. Frames are commonly used
in robotics to describe the orientation and translation (pose) of rigid bodies
with respect to a laboratory frame. Frames are elements of the special
Euclidean group $SE(3)$, which is a non-Euclidean manifold and group (Lie-
group). Frames can be represented as homogeneous matrices defined as
$\displaystyle\bm{T}=\begin{bmatrix}\bm{R}&\bm{p}\\\
\bm{0}^{T}&1\end{bmatrix}\,,$ (1)
where $\bm{R}\in SO(3)$ is a rotation matrix
($\bm{R}\bm{R}^{T}=\bm{R}^{T}\bm{R}=\bm{I},\det\bm{R}=1$) and
$\bm{p}\in\mathbb{R}^{3}$ is a column vector, describing the orientation and
translation of a frame, respectively, with respect to a reference frame. For
simplicity we write $\bm{T}=[\bm{R},\bm{p}]$. The inverse is then defined as
$\bm{T}^{-1}=[\bm{R}^{T},-\bm{R}^{T}\bm{p}]$. The configuration space of an
embodiment consisting of $n$ links with attached frames can be described by an
element of the direct product space $SE(3)^{n}=SE(3)\times
SE(3)\times\dots\times SE(3)$ ($n$ copies). The velocity of a frame is
described by a twist, which encodes the rotational and translational velocity
of the rigid body. A twist is an element of the Lie-algebra $se(3)$, which
defines a vector space. The velocity of the embodiment consisting of $n$
frames is described by an element of the direct product space
$se(3)^{n}=se(3)\times se(3)\times\dots\times se(3)$ ($n$ copies). A twist can
be represented by $4\times 4$ matrices of the form
$\displaystyle{\cal{\bm{V}}}^{b}=\bm{T}^{-1}\dot{\bm{T}}=\begin{bmatrix}[{\bm{\omega}}^{b}]&\bm{v}^{b}\\\
\bm{0}^{T}&0\end{bmatrix}$ (2)
or
$\displaystyle{\cal{\bm{V}}}^{s}=\dot{\bm{T}}\bm{T}^{-1}=\begin{bmatrix}[{\bm{\omega}}^{s}]&\bm{v}^{s}\\\
\bm{0}^{T}&0\end{bmatrix}\;,$ (3)
where (2) defines the body twist and (3) the spatial twist. The notation
$[\cdot]$ denotes a skew symmetric $3\times 3$ matrix composed of the
components of the angular velocity
$\bm{\omega}=[\omega_{1},\omega_{2},\omega_{3}]^{T}$, that is
$\displaystyle[\bm{\omega}]=-[\bm{\omega}]^{T}=\begin{bmatrix}0&-\omega_{3}&\omega_{2}\\\
\omega_{3}&0&-\omega_{1}\\\ -\omega_{2}&\omega_{1}&0\end{bmatrix}\,.$ (4)
Specifically, $[\bm{\omega}^{b}]=\bm{R}^{T}\dot{\bm{R}}\in\mathbb{R}^{3\times
3}$ and $\bm{v}^{b}=\bm{R}^{T}\dot{\bm{p}}\in\mathbb{R}^{3}$ define the
angular velocity and translational velocity of the origin with respect to the
base frame, respectively, both expressed in coordinates of the body frame. A
similar but less intuitive physical interpretation can be given to the spatial
twist. For simplicity we write ${\cal{\bm{V}}}=[\bm{\omega},\bm{v}]$. The
joint of the first link is always attached to the origin of the base frame,
which serves as a reference frame in which all comparisons will be performed.
Each joint rotates around one axis. The forward kinematic map is a map from
joint angles ${\bm{q}=[q_{1},q_{2},\dots,q_{n}]^{T}}$ to frames
$\bm{q}\rightarrow\bm{T}(\bm{q})$, where $n$ denotes the number of DOFs. For
simplicity we first consider a planar manipulator with $n$ DOFs. We assume
that all links have cylindrical shape and constant mass density. We attach a
frame to each link $i$ with its origin at a distance of $r_{i}$ from joint
$i-1$, $i=1,\dots,n,$ and with the $x$-axis that is pointing along the link
direction. The transformation from link frame $i$ to the base frame $0$ can
then be described by a product of matrix exponentials
$\displaystyle\bm{T}_{0i}(q_{1},q_{2},\dots,q_{i})=e^{q_{1}{{\bm{S}}}_{1}}e^{q_{2}{{\bm{S}}}_{2}}\dots
e^{q_{i}{{\bm{S}}}_{i}}\bm{M}_{i}\,,$ (5)
where
$\displaystyle\bm{M}_{i}=\begin{bmatrix}1&0&0&r_{i}\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix}\,,\;\;{{\bm{S}}}_{i}=\begin{bmatrix}[{\bm{n}}_{i}]&-\bm{n}_{i}\times\bm{q}_{i}\cr
0&0\cr\end{bmatrix},$ (6)
for $i=1,\dots,n$. The homogeneous matrix $\bm{M}_{i}$ describes a constant
shift of the frame $i$ by $r_{i}$ along the $x$-axis. The screw $S_{i}\in
se(3)$ is a $4\times 4$ matrix and describes the rotation axis of the revolute
joint $i$. Here $\bm{n}_{i}$ denotes a unit vector in the direction of the
joint axis $i$, $\bm{q}_{i}$ is a vector from the base to any point on the
joint axis $i$ (both expressed in coordinates with respect to the base frame)
and $\times$ denotes the vector cross product. Common choices for $r_{i}$ are
$r_{i}=l_{i}/2$, which corresponds to attaching frames to the center of mass
(COM) of each link (our choice) and $r_{i}=l_{i}$, which corresponds to
attaching frames to the end of each link. Joint angles and joint velocities
${\dot{\bm{q}}=[\dot{q}_{1},\dot{q}_{2},\dots,\dot{q}_{n}]^{T}}$ determine the
twists, thus
$(\bm{q},\dot{\bm{q}})\rightarrow{\cal{\bm{V}}}(\bm{q},\dot{\bm{q}})$. The
(body) twists follow from (2) and (5) as ($i=1,\dots,n$)
$\displaystyle{\cal{\bm{V}}}_{0i}=\bm{T}^{-1}_{0i}(q_{1},q_{2},\dots,q_{i})\cdot\dot{\bm{T}}_{0i}(q_{1},q_{2},\dots,q_{i})\,,$
(7)
which can be determined recursively leading to
$\displaystyle{\cal{\bm{V}}}_{0i}=\mbox{Ad}_{[\bm{M}_{i}e^{\bm{S}_{i}q_{i}}]^{-1}}({\cal{\bm{V}}}_{0i-1})+\bm{S}_{i}\dot{q}_{i}\,,$
(8)
where the adjoint map $\mbox{Ad}_{\bm{T}}({\cal{\bm{V}}})$ for frame
$\bm{T}=[\bm{R},\bm{p}]$ and twist ${\cal{\bm{V}}}=[\bm{\omega},\bm{v}]$ is
defined as
$\mbox{Ad}_{\bm{T}}({\cal{\bm{V}}})=[\bm{R}\bm{\omega},\bm{p}\times\bm{R}\bm{\omega}+\bm{R}\bm{v}]\in
se(3)$ and ${\cal{\bm{V}}}_{00}:=\bm{0}$ [33]. Note that from the (body)
twists ${\cal{\bm{V}}}_{0i}=[\bm{\omega}_{0i},\bm{v}_{0i}]$ the angular
velocity and translational velocity of frame
$\bm{T}_{0i}=[\bm{R}_{0i},\bm{p}_{0i}]$ with respect to the base frame can be
easily obtained by
$\displaystyle\bm{\omega}_{0i}^{s}$ $\displaystyle=$
$\displaystyle\bm{R}_{0i}\bm{\omega}_{0i}\,,$ (9)
$\displaystyle\dot{\bm{p}}_{0i}$ $\displaystyle=$
$\displaystyle\bm{R}_{0i}\bm{v}_{0i}\,.$ (10)
For further details using frames and twists we refer to [34]. After these
derivations we can define the state of an embodiment $\bm{s}$ as the state of
all frames
$\displaystyle\bm{s}$ $\displaystyle=$
$\displaystyle(\bm{s}_{1},\dots,\bm{s}_{i},\dots,\bm{s}_{n})$ (11)
$\displaystyle=$
$\displaystyle(\bm{T}_{01},{\cal{\bm{V}}}_{01},\dots\,,\bm{T}_{0i},{\cal{\bm{V}}}_{0i},\dots\,,\bm{T}_{0n},{\cal{\bm{V}}}_{0n})\,,$
where we surpressed the arguments and
$\bm{s}_{i}=(\bm{T}_{0i},{\cal{\bm{V}}}_{0i})\in\bm{{\cal{S}}}_{i}$ denotes
the state of frame $i$. Note that via the foward kinematic map, which is
assumed to be known in this work, an embodiment state is fully determined by
its joint angles $\bm{q}$ and joint velocities $\dot{\bm{q}}$, i.e.,
$\bm{s}=\bm{s}(\bm{q},\dot{\bm{q}})$. A special case of (11) is obtained when
ignoring rotational information. In this case, an embodiment can be described
by the position and (translational) velocity of a set of candidate points,
defined by the origin of each frame (by setting $\bm{R}=\bm{I}$ in (1), (2)).
The embodiment state is then described by
$\displaystyle\bm{s}$ $\displaystyle=$
$\displaystyle(\bm{x}_{1},\bm{v}_{1},\dots\,,\bm{x}_{i},\bm{v}_{i},\dots\,,\bm{x}_{n},\bm{v}_{n})\,,$
(12)
where $\bm{s}_{i}=(\bm{x}_{i},\bm{v}_{i})$ denotes the state vector of
candidate point $i$. The definition of an embodiment in terms of frames/twists
and candidate points is generic and can be applied to any robot. A
disadvantage of using frames is that they are not elements of a vector space,
but define a non-Euclidean manifold.
### III-B Similarity between embodiments
(a)
(b)
(c)
Figure 2: Pose imitation task between planar manipulators. (a) Demonstrator
pose; (b, c) Learner poses. Learner (b) generates perfect imitation if
similarity is defined by candidate points, e.g., end-effector position.
However, for a similarity measure based on frames, the two embodiments do not
resemble each other. Learner (c) – consisting only of two links – provides
better imitation than learner (b) if similarity is measured between frames
attached to each link.
Similarity of embodiments can be assessed by defining a distance measure $d$
between frames/twists or candidate points, that is
$d(\bm{s}_{i},\hat{\bm{s}}_{j}):\bm{{\cal{S}}}_{i}\times\hat{\bm{{\cal{S}}}}_{j}\mapsto\mathbb{R}^{+}_{0}\,,$
where $\bm{s}_{i}\in\bm{{\cal{S}}}_{i}$ and
$\hat{\bm{s}}_{j}\in\hat{\bm{{\cal{S}}}}_{j}$ result from different state
spaces. We first consider the distance
$\displaystyle d(\bm{T}_{i},\hat{\bm{T}}_{j})$ $\displaystyle=$
$\displaystyle\alpha_{tr}d_{tr}(\bm{T}_{i},\hat{\bm{T}}_{j})+\alpha_{rot}d_{rot}(\bm{T}_{i},\hat{\bm{T}}_{j})$
(13)
between two frames, $\bm{T}_{i}=[\bm{R}_{i},\bm{p}_{i}]$ and
$\hat{\bm{T}}_{j}=[\hat{\bm{R}}_{j},\hat{\bm{p}}_{j}]$. The distance consists
of a translational and rotational part, which are weighted with factors
$\alpha_{tr}$ and $\alpha_{rot}$. The weights can be either constants or
functions of other variables. For the translational part we set the Euclidean
distance between the two frame origins
$\displaystyle
d_{tr}(\bm{T}_{i},\hat{\bm{T}}_{j})=\lVert{\bm{p}_{i}-\hat{\bm{p}}}_{j}\lVert\,.$
(14)
There are various ways to define the rotational distance between frames. We
choose to take the angle between the unit vectors pointing along the $x$-axes
of the frames, i.e., into the directions of the links. Thus, we define
$\displaystyle\beta=\arccos(\bm{e}_{x}^{i}\cdot\hat{\bm{e}}_{x}^{j})$ (15)
leading to values in the interval $[0,\pi]$. This definition results in
numerical problems when performing gradient descent because the derivative of
the $\arccos$-function is
$\displaystyle\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^{2}}}\,.$ (16)
To avoid singularities, a modified rotational distance can be defined by
shifting the negated $\cos\beta$ into the interval $[0,\pi]$ as
$\displaystyle
d_{rot}(\bm{T}_{i},\hat{\bm{T}}_{j})=\frac{\pi}{2}(1-\cos\beta)=\frac{\pi}{2}(1-\bm{e}_{x}^{i}\cdot\hat{\bm{e}}_{x}^{j})\,.$
(17)
Note that the direction of the $x$-axis of frame
$\bm{T}_{0i}=[\bm{R}_{0i},\bm{p}_{0i}]$ with respect to the laboratory frame
can be easily extracted as the first column of the rotation matrix
$\bm{R}_{0i}$. The distance measure introduced in (13), (14) and (17) includes
only the static pose of the embodiment and frames might be considered similar,
even though they move into different directions. To include also motion
information into the distance measure, the twists of the frames need to be
taken into consideration. For dynamic motion imitation we therefore augment
the distance measure between two states
$\bm{s}_{i}=(\bm{T}_{i},{\cal{\bm{V}}}_{i})\in\bm{{\cal{S}}}_{i}$ and
$\hat{\bm{s}}_{j}=(\hat{\bm{T}}_{j},{\hat{\cal{\bm{V}}}}_{j})\in\hat{\bm{{\cal{S}}}}_{j}$
by including the translational and angular velocity (9) and (10), that is
$\displaystyle
d(\bm{s}_{i},\hat{\bm{s}}_{j})=\alpha_{tr}d_{tr}+\alpha_{rot}d_{rot}+\alpha_{v}d_{v}+\alpha_{\omega}d_{\omega}\,,$
(18)
with
$\displaystyle
d_{v}=\lVert\dot{\bm{p}}_{i}-\dot{\hat{\bm{p}}}_{j}\lVert\;\;\;\;\mbox{and}\;\;\;\;d_{\omega}=\lVert\bm{\omega}^{s}_{i}-\hat{\bm{\omega}}^{s}_{j}\lVert\,.$
(19)
Note that the distance measure between two states $\bm{s}_{i}$ and
$\hat{\bm{s}}_{j}$, can be extended over all state spaces by defining the sum
of all mutual distances
$\displaystyle
d(\bm{s},\hat{\bm{s}})=\sum_{i=1}^{n}\sum_{j=1}^{\hat{n}}d(\bm{s}_{i},\hat{\bm{s}}_{j})\,.$
(20)
In the next section a weighted version of (20) will be introduced by
incorporating link correspondences.
### III-C Link Correspondences
To measure similarity between links of two embodiments, we first define how
correspondence between links of different embodiments can be established.
Embodiments may differ in the number and length of links, and thus, a one-to-
one assignment between links is often not possible. To establish
correspondence between links of different embodiments with possibly different
overall size, we first rescale each embodiment by the sum of its link lengths
$L$, resulting in a chain length of $1$. To establish correspondence, we
assign a weight for every possible link-pair combination. Thus, for two
embodiments 1 and 2 with $n$ and $\hat{n}$ links, respectively, link
correspondence can be represented by a correspondence matrix
$\bm{W}\in\mathbb{R}^{n\times\hat{n}}$. Irrelevant combinations result in zero
or close to zero entries and higher values indicate higher relevance. Each row
of the correspondence matrix contains the correspondence weights of one link
of embodiment 1 to all other links of embodiment 2, where the highest value
indicates which link is the most relevant from all links of embodiment 2. The
elements of the correspondence matrix can either be calculated as a function
of embodiment states or pre-calculated for a pair of embodiments, independent
of their current state.
State-Dependent Assignment. For state-dependent calculations of the
correspondence matrix $\bm{W}(\bm{s},\hat{\bm{s}})$, weights are calculated
using the distance measure between frames. The closer the distance, the higher
the weight for this pair of frames. To obtain the correspondence matrix
$\bm{W}(\bm{s},\hat{\bm{s}})$, the mutual distance matrix
$\bm{D}^{\prime}(\bm{s},\hat{\bm{s}})=({D}^{\prime}(\bm{s}_{i},\hat{\bm{s}}_{j}))\,,i=1\dots
n,j=1\dots\hat{n},$ between all links of the two embodiments is computed using
the distance measure in (13). A correspondence matrix can be generated by
replacing the smallest element of each row of
$\bm{D}^{\prime}(\bm{s},\hat{\bm{s}})$ with 1 and all other elements with 0,
resulting in a binary matrix $\bm{W}_{12}(\bm{s},\hat{\bm{s}})$ that assigns
exactly one link of embodiment 1 to each link of embodiment 2 with a weight 1.
The same operation can be applied for each column of
$\bm{D}^{\prime}(\bm{s},\hat{\bm{s}})$, resulting in $\bm{W}_{21}$. Adding
$\bm{W}_{12}$ to $\bm{W}_{21}$ results in a correspondence matrix $\bm{W}$. A
correspondence matrix that only uses the minimum for each row and each column
is very selective and ignores the fact that more than one link of the other
embodiment may be lying in a similar distance and should be taken into
consideration. This effect can be mitigated by applying a softmax function to
the rows and columns of the correspondence matrix, after multiplying with a
constant factor $\xi<0$ to find soft minima instead of maxima and to adjust
the distinctness of the minimum.
### III-D Calculating Distance Between Embodiments
We define distance between embodiments as element-wise multiplication of the
distance function with the correspondence matrix i.e.,
$\bm{D}(\bm{s},\hat{\bm{s}})=\bm{W}(\bm{s},\hat{\bm{s}})\circ\bm{D}^{\prime}(\bm{s},\hat{\bm{s}})$,
where $\circ$ denotes the Hadamard product. Only distances between
corresponding link pairs remain because non-corresponding pairs are weighted
with zero or near-zero values. To obtain one single scalar number, the mean of
all entries of the resulting matrix is taken. Matrix norms, such as the
Frobenius norm, can also be used. For the evaluation of the correspondence
matrix and the distance matrix, suitable weights, $\alpha_{tr}$ and
$\alpha_{rot}$, need to be chosen. Different settings are possible here. The
pseudo-code for calculating the distance measure are shown in Algorithm 1.
Function _distance_measure( $\bm{s}(\bm{q})$, $\hat{\bm{s}}(\hat{\bm{q}})$,
$\alpha_{tr}$, $\alpha_{rot}$)_
Calculate $\\{\bm{T}_{i}\\}_{i=1}^{n},\\{\hat{\bm{T}}_{j}\\}_{j=1}^{\hat{n}}$
from $\bm{q}$, $\hat{\bm{q}}$ using forward kinematics.
$\bm{D}\leftarrow
d(\bm{T},\hat{\bm{T}},\alpha_{tr},\alpha_{rot})\;\bm{foreach}\;(\bm{T},\hat{\bm{T}})$
in $\\{\bm{T}_{i}\\}_{i=1}^{n}\times\\{\hat{\bm{T}}_{j}\\}_{j=1}^{\hat{n}}$
Either calculate correspondence matrix $\bm{W}(\bm{s},\hat{\bm{s}})$ or use
static correspondence matrix $\bm{W}$.
$\bm{D}_{W}\leftarrow\bm{D}\circ\textbf{W}$
$\overline{\textbf{D}_{W}}=\frac{\sum_{i=1}^{m}\sum_{j=1}^{n}D_{W,ij}}{mn}$
return $\overline{\textbf{D}_{W}}$
Algorithm 1 Calculating the Distance Measure
## IV RESULTS
We studied static and dynamic imitation tasks in simulation between two
dissimilar embodiments using the previously derived distance measure. First,
we present results of static pose imitation tasks between planar manipulators
with different links using gradient descent. Second, we examined whether a
neural network can learn the optimal static pose between two planar
manipulators and, furthermore, between two Franka Emika Panda robots for a
given expert pose. In addition, we investigated how well the learner
generalized to poses it has not seen during training. Third, we present
results for a dynamic imitation tasks between two Franka Emika Panda robotic
arms. For this purpose, a simulation environment was built in the physics
simulator Gazebo. In all simulations, we assumed that the learner has no
knowledge of the robot dynamics.
### IV-A Static Pose Imitation Task
In a static pose imitation task, the optimal pose of the learner,
$\hat{\bm{q}}^{\ast}$ is obtained for a given pose of the expert, $\bm{q}$, by
minimization of the distance measure
$\displaystyle\hat{\bm{q}}^{\ast}=\operatorname*{arg\,min}_{\hat{\bm{q}}}\;d(\bm{s}(\bm{q}),\hat{\bm{s}}(\hat{\bm{q}}))\,.$
(21)
Minimizing the distance function is a nonlinear optimization problem for which
generally no analytical solution exists, in particular, for embodiments with a
large number of links. Using mathematical libraries such as TensorFlow, the
gradient of the distance function can be computed and local minima can be
found numerically via gradient descent. Instead of trying to solve the
optimization problem repeatedly for each input, we can learn a mapping from
joint angles of the expert to joint angles of the learner
$\bm{f}_{\bm{\theta}}:\bm{q}\longrightarrow\hat{\bm{q}}$, where
$\bm{q}\in[-\pi,\pi]^{n}$ and $\hat{\bm{q}}\in[-\pi,\pi]^{\hat{n}}$. The
function $\bm{f}_{\bm{\theta}}$ can be approximated by a neural network with
weight parameters $\bm{\theta}$. The distance measure was implemented as a
computation graph in TensorFlow [35].
(a) $\alpha_{tr}=0.5,\;\alpha_{rot}=1.0$
(b) $\alpha_{tr}=3.0,\;\alpha_{rot}=1.0$
(c) $\alpha_{tr}=1.0,\;\alpha_{rot}=1.0$
(d)
(e)
(f)
Figure 3: (a-c) Effects of using different distance weighting factors on pose
imitation between two planar manipulators (expert: blue, learner: orange).
(d-f) Distance function between planar manipulators with two links. (d) State-
dependent weight matrix with $\alpha_{tr}=1.5,\alpha_{rot}=1.0$; (e) State-
dependent weight matrix with distance-dependent weighting factors; (f) State-
independent weight matrix, considering only rotational distance between
corresponding links ($\alpha_{tr}=0$). The distance is plotted over all
possible joint angles of one embodiment, while the other manipulator remains
fixed at $\bm{q}=[1.5,-1.5]$.
Comparing Link Correspondences. Before training the neural network, we
analyzed the behavior of the distance function for a simple toy model. Fig.
3a-c shows an imitation task between planar manipulators with two links. The
distance was measured using a state-dependent correspondence matrix $\bm{W}$,
using varying weight factors $\alpha_{tr}$, $\alpha_{rot}$. Each pose of the
learner was found via gradient descent. The choice of weight factors clearly
has a strong influence on the quality of the result. Balancing between
translational and rotational weights is challenging. One possibility to
overcome this difficulty may be to simply use the translational distance as
the translational weight and to redefine the rotational weight by subtracting
the translational distance from its maximum value and rescaling from $[0,\pi]$
to $[0,2]$. This way, translational and rotational distance are in the same
value range, i.e.,
$\alpha_{tr}=d_{tr},\quad\alpha_{rot}=\frac{2}{\pi}(2-d_{tr}).$ (22)
The maximum distance results from the fact that both embodiments are
normalized to be in a sphere of radius $1$ or diameter $2$, which is the
maximum Euclidean distance between two points in this sphere. Equation (22)
ensures that the sum of $\alpha_{tr}$ and $\alpha_{rot}$ is always $2$,
excluding the transformation factor $2/\pi$. The problem with the above
approaches is that there exist local minima in the distance function, as can
be observed in 3(d) and 3(e) for two-link manipulators. 3(f) on the other
hand, shows that when considering the only the orientational between frames
(setting $\alpha_{tr}=0$) using a static, precalculated correspondence matrix,
only one minimum remains . This approach is less flexible but more robust and
resulted in parallel alignment of corresponding links. Therefore, all
following experiments were conducted using this distance measure.
Pose Imitation Mapping Using a Neural Network. We next implemented a neural
network to map joint angles of the expert to corresponding joint angles of the
learner, leading to a more efficient method for static pose imitation than
conducting a gradient descent search for each state. To generate this
nonlinear map, network parameters
$\bm{\theta^{*}}=\operatorname*{arg\,min}_{\bm{\theta}}\sum_{i=1}^{N}d(\bm{s}(\bm{q}_{i}),\hat{\bm{s}}(f_{\bm{\theta}}(\bm{q}_{i})))\,,$
(23)
need to be determined, which minimize the distance between the states of the
expert and the states of the learner for a given training set
$\\{\bm{q}_{1},\bm{q}_{2},\dots,\bm{q}_{N}\\}$, where the map
$\bm{f}_{\bm{\theta}}(\bm{q}_{i})$ is represented by a neural network and
$\bm{\theta}$ are the network parameters. The training dataset
$\\{\bm{q}_{1},\bm{q}_{2},\dots,\bm{q}_{N}\\}$ can be generated randomly
because it contains only expert angles that do not need to be labeled. The
network structure consists of three hidden layers of size $32$ with LReLU
activation functions. The output layer uses the tanh as activation function,
resulting in output values in $[-1,1]$. These values can then be mapped to
angular values in $[-\pi,\pi]$. Having generated a training dataset and
another dataset for validation, the network was trained using a minibatch-
based stochastic gradient descent method. After dividing the training set into
minibatches, the update step is performed for each of the minibatches.
Afterwards, the whole training set is shuffled and the procedure repeated.
(a)
(b)
(c)
(d)
Figure 4: Imitation between two planar manipulators using a neural network.
(a-b) Expert: 7-DOF, learner: 4-DOF; (c-d) Expert: 4-DOF, learner: 7-DOF.
We trained a neural network to find a mapping from joint angles of a 7-DOF
expert manipulator to angles of a 4-DOF learner manipulator and another
network for the same pair of manipulators but with switched expert/learner
roles. Each time we used a training set of 1024 expert demonstrations,
dividing it into 32 minibatches in each episode. Fig. 4 shows the learned
poses of the trained network for given expert angles that were not included in
the training set.
Pose Imitation Between Three-Dimensional Embodiments. We next applied the
method from the previous section to two Franka Emika Panda robotic arms in
simulation. Dissimilar embodiments were generated by locking individual DOFs
of the learner to $q_{i}=0$. For example, to simulate a four-link-Panda robot
with four DOFs, joint 3, 6, and 7 were locked. To simulate a three-link-Panda
robot, additionally link 2 was locked (see Fig. 1). We first studied static
pose imitation between identical embodiments with all full 7 DOFs enabled
(Video 1 111https://youtu.be/UPZclkFoFXQ). 1(a) shows an example of how the
trained network solves the task between dissimilar embodiments (expert: 7
DOFs, learner: 4 DOFs). The locked joints 6 and 7 (indicated in red) lie at
the very end of the embodiment and therefore do not contribute much to the
overall configuration of the embodiment, in contrast to the locked joint 3 at
the center of the embodiment. In 1(b), the learner only has 3 DOFs. The
learner tries to establish similarity by rotating joint 4 as the second joint
is locked. The results can be seen in Video 2 222https://youtu.be/BmFH6Nr9F1Y.
The experiments have shown that training a neural network with a distance-
based loss function worked reasonably well for static pose imitation. Local
minima in the distance function and over-fitting on the training set posed
some problems. While the former is more difficult to solve, the latter can be
solved by increasing the size of the training set and stopping the training
process at a suitable time. Another possibility to improve the network’s
performance may lie in the structure and training of the network. We used the
same structure for all pose imitation tasks without employing techniques that
decrease the probability of over-fitting, such as dropout or regularization
[36].
### IV-B Dynamic Motion Imitation
In this section, we a apply a reinforcement learning algorithm for motion
imitation by using the online distance measure between embodiments as a
feedback signal. Reinforcement learning has shown great success in learning
motor tasks, for example, [37, 38]. We study the transfer of motions from one
Panda robot with maximally seven DOFs to another one in simulation. As before,
different embodiments are generated by locking DOFs in the learner. We assume
that the dynamics of the robots are unavailable (model-free) and that the
learner is controlled by joint torques $\bm{\tau}$. Consequently, the agent
needs to control the joint positions but also, implicitly, learn the the robot
dynamics.
Simulation Environment. The manufacturer (Franka Emika) of the Panda robot
provided a good integration of the robot into the ROS ecosystem, which we
augmented by the Gazebo physics simulator. Unfortunately, no exact inertia
values for the Panda were provided, which were needed to simulate the
dynamics. The CoR-Lab group from the Universit t Bielefeld published some
estimates of inertia values on their GitHub
repository333https://github.com/corlab/cogimon-gazebo-
models/blob/master/franka/robots/panda_arm.xacro. We used these estimates and
manually adjusted them by using the guide given in the Gazebo manual. The
simulated robot is controlled via joint torque commands. To create
trajectories of the expert, simple PID-controllers were configured for each
joint. Due to the lack of sophisticated controllers and to facilitate the task
for the RL agent, gravity was turned off in the simulation.
RL Environment and Agent. The next task consisted in the implementation of the
reinforcement learning agent and the interface for interaction with the
simulation. The state space consisted of the expert and learner states, thus,
the environment’s state $\bm{s}$ is defined by the tuple
$\bm{s}_{t}=\\{\bm{s}(\bm{q}_{t},\dot{\bm{q}}_{t}),\hat{\bm{s}}(\hat{\bm{q}}_{t},\dot{\hat{\bm{q}}}_{t})\\}$.
The actions are given by the torque commands of the learner,
$\hat{\bm{a}}_{t}=\hat{\bm{\tau}}_{t}$, subject to the torque limits for each
corresponding joint. The control of the expert is not observable by the RL
agent. For training and testing, multiple random trajectories of similar
duration were recorded. The step size was set to $\Delta t=0.1$s and a step
consisted of the following transitions: The agent executed an action in the
environment by sending a torque command to the learner. The torques were then
applied for the duration of the simulation time $\Delta t$. The simulation
then paused and returned the next observed state $s_{t+1}$ together with the
reward $r_{t+1}$, which was calculated from the state as the negative distance
measure between the embodiment states. To train the agent, we used Proximal
Policy Optimization (PPO) [39], which is a state-of-the art actor-critic DRL
algorithm. One big advantage of PPO is its robustness towards hyperparameter
tuning. We employed the GPU-optimized version (PPO2) of the Stable-Baselines
repository, based on OpenAI’s implementations.
Simulation 1: Imitation of a Single Trajectory. We first tested, whether
motion imitation using reinforcement learning is feasible by transferring a
single trajectory between two Panda robots with all 7 DOFs activated, i.e.,
expert and learner had identical embodiments. Both expert and learner started
in their zero pose, in which all joint angles and joint velocities are set to
zero. The trajectory of the expert was recorded off-line by moving each joint
to an arbitrarily chosen goal position. The environment was reset whenever the
trajectory ended. Each trajectory had a total duration of 5 s, which leads to
50 steps per episode. The discount factor was set to $\gamma=0.4$ implying
that the agent acted myopically. This value was chosen because high values led
to very slow training progress. The weights for the frame-distance function
were set to
$\alpha_{tr}=0.0,\;\alpha_{rot}=1.0,\;\alpha_{v}=0.001,\;\alpha_{\omega}=0.01$
in all simulation experiments. Training time was about $7.5$ hours on a
desktop computer using GPU-accelerated computations. The simulation showed
that the learned trajectory resembles the expert’s trajectory very closely and
that imitation of motions is possible using a reinforcement learning framework
with a distance related reward function. The results can be seen in Video 3
444https://youtu.be/dLN314VJTHg.
Simulation 2: Generalization Between Trajectories.
Figure 5: Generalization capabilities of two different agents. Shown is the
distance measure for two 7-DOF-Panda learners while imitating the same,
previously unseen trajectory. One learner (”Specialized“) was trained on a
single trajectory, whereas the other (”Generalized“) was trained on 124
different trajectories.
It was examined next, whether the agent can imitate trajectories it had not
seen before. For this purpose, we trained the agent with different numbers of
training data (one vs. 124 trajectories). Trajectories were recorded as
before, but each time with different final poses. The trajectory of the agent
trained on a single trajectory barely resembles the trajectory of the expert.
The imitation of the agent trained on the larger data set is not perfect, but
resembles the trajectories from the expert more closely (see Video 4
555https://youtu.be/2Q7jiY9DRUg). This improvement shows in a significantly
smaller distance between expert and learner along the trajectory (see Fig. 5).
Simulation 3: Imitation Between Dissimilar Embodiments. In the next
experiment, we studied motion transfer between dissimilar Panda robots.
Towards this goal, we trained the agent again on the same 124 trajectories but
this time the learner had only 4 or 3 DOFs, respectively. As the learning
robot was restricted in its DOFs, some trajectories could could not be
imitated well, resulting in higher values of the distance measure. We found
that the restricted learner moved its links in similar directions as the
expert but the restrictions prevented a more similar imitation. Examples can
be seen in Video 5 666https://youtu.be/Fytw8sz0pG0.
## V CONCLUSIONS
Our main contributions with this work are threefold: First, we have introduced
a definition of embodiment states in terms of frames/twists and candidate
points. Second, we have povided a distance measure between dissimilar
embodiments using correspondences between frames of expert and learner. Third,
we have applied this distance measure to static pose and movement imitation
tasks between manipulators. All tasks have been performed in simulation. In
all experiments we could show that the agent was able to learn the imitation
task, even though no dynamic model has been provided to the learner. The
framework that we have developed is generic and flexible and not limited to
our choice of parameters, distance measures and type of robots. Depending on
the correspondence matrix calculation, the topology of the embodiments is not
crucial. Possibly even free topologies like swarms of flying objects could be
compared and brought into similarity.
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|
2024-09-04T02:54:57.542672 | 2020-03-05T15:21:25 | 2003.02708 | {
"authors": "Shun Fu and Ji Xu",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26059",
"submitter": "Shun Fu",
"url": "https://arxiv.org/abs/2003.02708"
} | arxiv-papers | # The Multi-granularity in Graph Revealed by a Generalized Leading Tree
Shun Fu Ji Xu
###### Abstract
There are hierarchical characteristics in the network and how to effectively
reveal the hierarchical characteristics in the network is a problem in the
research of network structure. If a node is assigned to the community to which
it belongs, how to assign the community to a higher level of community to
which it belongs is a problem. In this paper, the density of data points is
investigated based on the clustering task. By forming the density of data
points, the hierarchical difference of data points is constructed. In
combination with the distance between data points, a density-based leading
tree can be constructed. But in a graph structure, it is a problem to build a
lead tree that reveals the hierarchical relationships of the nodes on the
graph. Based on the method of tree formation based on density, this paper
extends the model of leading tree to the hierarchical structure of graph
nodes, discusses the importance of graph nodes, and forms a leading tree that
can reveal the hierarchical structure of graph nodes and the dependency of
community. Experiments were carried out on real data sets, and a tree
structure was formed in the experiment. This graph leading tree can well
reveal the hierarchical relationships in the graph structure.
###### keywords:
Granular computing , Network representation learning , Graph clustering
††journal: The name of international journal CS
## 1 Introduction
Graph is widely used to model and display the relationships between objects
and objects in the world. On the other hand, the distribution of many data
objects presents multi-level characteristics, such as the organization and
management structure of human beings, the food chain of the earth, the
taxonomy of species, and so on. In the relational network (such as social
network, academic network, traffic network, etc.), the distribution of nodes
also presents multi-level characteristics. Simply taking a single node as an
object to study, the problem will be very complicated. On the other hand,
people can study the objects in the figure at different levels by referring to
the cognitive rules and multi-granularity cognitive methods of human beings
for complex problems. [question] : which nodes in the network are at the
higher level and which are at the lower level? Which nodes make up the
community? And which communities form a larger community? Answering these
questions requires looking at complex relational networks from a multi-
granularity perspective, in which objects (nodes, edges, communities, etc.)
can be subordinate to different granularity levels. Therefore, it is important
to reveal the multi-granularity and multi-level structure of node objects in
the network.
### 1.1 The hierarchical properties in networks
(introduction and related works)
### 1.2 Hierarchical clustering
(introduction and related works)
(a) caption1
(b) caption2
(c) caption3
Figure 1: Caption for subfigures
### 1.3 The leading tree for clustering in Euclidean space
When performing efficient and accurate clustering of general data points in
datasets of any shape in European space, Rodriguez and Laio proposed the
DPClust algorithm [1]. Xu [2] Considered the cluster assignment process of the
DPClust algorithm as a process of finding its parent node for each data point.
Once the search for the parent node is completed, a tree-like subordinate
relationship is formed between the nodes. As shown in Figure 1(a) in the
European space, the tree structure formed by the DPClust algorithm. The
application of Leading Tree in the field of hierarchical clustering makes it
easy to examine the multi-level community relationships in the distribution of
data points at multiple granularities. As shown in Figure 1(b) Leading Tree
maps data to a tree-shaped space. One can easily see that the points in the
upper layer of the tree are more at the core position of the cluster, while
the points in the lower layer correspond to the edge points of a cluster. In
other words, the mapping to the tree structure assigns points in the original
European space to Granular Layers. On the other hand, this tree structure can
be broken down into many subtrees, and small subtrees can be merged into large
subtrees. This corresponds to the multi-level division of clusters in the
original Euclidean space. As shown in Figure 1(c), subtree a corresponds to
class cluster A, and a belongs to a larger subtree b, so in a coarser
granularity, class cluster A is a component of class cluster B.
For the points of Euclidean space, DenPEHC [2] has realized the rapid and
effective multi-level clustering and the clear, intuitive and effective
disclosure of the multi-level dependency of these points through the
construction of DenPEHC guidance tree based on density. But for the data
objects in our widely used Graph space, how can we construct a multi-grained
tree structure to reveal the above relationships?
The construction process of Leading Tree in DenPEHC is carried out for data
points in Euclidean space, which relies on the distance between data points
and the artificially defined density given in data points. So that the point
is subordinate to the point that is closest to it, that is denser. We believe
that in Euclidean space, the process of a data point finding its parent node
can be more generally extended to data objects in any space, such as graph
data objects. The purpose of this paper is to discuss how to extend the idea
of leading tree construction to the data object of graph space so that this
idea of multi-granularity computation can reveal the hierarchical relationship
of data distribution in graph space.
## 2 Contribution
In this paper, we offer contributions by : 1\. Proposing a novel particle
calculation method is introduced to reveal the structure hierarchy of graph
nodes 2\. Provide a new tool and method for Graph Clustering, which can
effectively and intuitively reveal the structure hierarchy of Graph data
objects 3\. The leading tree only applicable to European type space is
expressed in a more general way, and it is extended to graph data objects. 4\.
The measurement standard of the importance of graph nodes is discussed to
inspire the exploration of this problem.
## 3 Related works
### 3.1 Granular computing
Granular computing is an umbrela concept which aims revealing the hierarchical
relationships between objects [3]. Granule is the basic unit of object for
processing in granular computing. The granule can be defined as different
types and they are flexible and scalable. …
### 3.2 Hierarchical Clustering
### 3.3 Graph Clustering
## 4 Problems
### 4.1 A generalized model of leading tree
For any object, or granule, you can define its importance and its distance
from other granule of the same kind. With these two elements, you can generate
a tree structure. This tree structure will reveal the membership and hierarchy
of all granule classes in a set. This tree structure will form a general lead
tree. Not limited to data points in Euclidean space.
For different data objects, or granule, its importance needs to be either
manually defined or machine-learned depending on the needs of the problem. The
distance between two objects also needs to be defined according to the needs
of the problem.
Figure 2: Two elements for GLT generation: tabel of importance and matrix of
distance
Take a set of 9 granule as an example. The distance matrix between them is
shown in the Figure 2. Based on the distance matrix and the importance table,
the parent of a granule can be found. This generates the tree structure on the
right side of the diagram.
With the tree structure, the dependencies between granule are clearly visible.
Here, a granule can be a point in a Euclidean space, a vertex in the Graph, a
person in a collaborating organization, a community of vertices, and so on.
The tree structure in Fig.2 reflects the membership of the nine granule. 1,9
is led by 2. One, two, nine constitute a small grass-roots organization. Then
they are led by 7, along with 5. So 1,2,9,5,7 constitutes a higher-level
organization centered on 7. These organizational levels are abstract concepts,
and if granule is materialized, the meaning of these levels can be
materialized, such as the community membership formed by nodes in the graph,
the management structure of a company, or the wildlife food chain.
## 5 Distance and importance measuring in Graph
As we saw in the previous section, the necessary conditions for building a
generalized granule leading tree are the distance (similarity) measure between
granule and the importance of granule under certain criteria. In this section
we extend the Generalized Leading Tree defined in the previous section
specifically to the Graph problem. Firstly, in the data point set of Euclidean
space, the distance measure can be defined by the cosine distance between the
data point vectors. If the logarithmic data points are clustered, the
importance measure can be the density of a certain point.
In order to extend the general Tree model discussed in the previous section to
the facilities of revealing the Hierarchical/multi - granularity of Property
in Graph, we need to define the distance between the granules and the
importance measurment for granules in Graph. This section lists several
measures and discusses the information focused on under different measures. In
the experimental section, we will reveal the differences in the lead trees
generated under different metrics and discuss the hierarchical nature of the
information expressed through different lead trees.
### 5.1 The importance measure by degrees
The importance of a node can be measured by its degree. Degree is the most
directe sign which counts the link between the vertex to other vertices. In
most senarios, degree reflects the influence capability of one vertex. For
example, in a community, if a vertex has the largest value of degree, there
should be some unique properties in that vertex. Maybe that vertex is the most
social active member in that community.
### 5.2 The importance measure by eigenvector centrality
Eigenvector centrality is a kind of extension of degree—it looks at a
combination of a node’s edges and the edges of that node’s neighbors.
Eigenvector centrality cares if you are a hub, but it also cares how many hubs
you are connected to. It’s calculated as a value from 0 to 1: the closer to
one, the greater the centrality. Eigenvector centrality is useful for
understanding which nodes can get information to many other nodes quickly. If
you know a lot of well-connected people, you could spread a message very
efficiently. If you’ve used Google, then you’re already somewhat familiar with
Eigenvector centrality. Their PageRank algorithm uses an extension of this
formula to decide which webpages get to the top of its search results.
### 5.3 The importance measure by betweenness centrality
Betweenness centrality is a bit different from the other two measures in that
it doesn’t care about the number of edges any one node or set of nodes has.
Betweenness centrality looks at all the shortest paths that pass through a
particular node (see above). To do this, it must first calculate every
possible shortest path in your network, so keep in mind that betweenness
centrality will take longer to calculate than other centrality measures (but
it won’t be an issue in a dataset of this size). Betweenness centrality, which
is also expressed on a scale of 0 to 1, is fairly good at finding nodes that
connect two otherwise disparate parts of a network. If you’re the only thing
connecting two clusters, every communication between those clusters has to
pass through you. In contrast to a hub, this sort of node is often referred to
as a broker.
### 5.4 The importance measure by the length of path
In addition to the measure of importance of vertex, we need to measure the
distance bwteen vertices in graphs for the GLT generation. The most simple way
is the length of path between two vertices. Another way is using the
similarity measurement and minused by one to obtain the distance between
vertices. The similarity measurement can be Jaccard coefficient, SimRank,
PageRank etc.
## 6 The graph leading tree model
For vertex $v$, in the set of vertices that has higher degree than $v$, we
take the closest one (e.g. $u$) as the parent vertex of $v$. By analogy, one
or more trees are formed to form an interpretable semantic tree or semantic
forest that reveals hierarchical semantic concepts on the structure of the
relational network.
## References
* [1] A. Rodriguez, A. Laio, Clustering by fast search and find of density peaks, Science 344 (6191) (2014) 1492–1496 (2014).
* [2] J. Xu, G. Wang, W. Deng, DenPEHC: Density peak based efficient hierarchical clustering, Information Sciences 373 (2016) 200–218 (2016).
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|
2024-09-04T02:54:57.552760 | 2020-03-05T15:56:19 | 2003.02727 | {
"authors": "L. Roa-Leguizam\\'on, H. Torres L\\'opez and A. G. Zamora",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26060",
"submitter": "Hugo Torres",
"url": "https://arxiv.org/abs/2003.02727"
} | arxiv-papers | # On the Segre Invariant for Rank Two Vector Bundles on $\mathbb{P}^{2}$
L. Roa-Leguizamón Instituto de Física y Matemáticas
Universidad Michoacana de San Nicolás de Hidalgo
Edificio C3, Ciudad Universitaria
C.P.58040 Morelia, Mich. México<EMAIL_ADDRESS>, H. Torres-López
CONACyT - U. A. Matemáticas, U. Autónoma de Zacatecas
Calzada Solidaridad entronque Paseo a la Bufa,
C.P. 98000, Zacatecas, Zac. México<EMAIL_ADDRESS>and A. G. Zamora U. A.
Matemáticas, U. Autónoma de Zacatecas
Calzada Solidaridad entronque Paseo a la Bufa,
C.P. 98000, Zacatecas, Zac. México<EMAIL_ADDRESS>
###### Abstract.
We extend the concept of Segre’s Invariant to vector bundles on a surface $X$.
For $X=\mathbb{P}^{2}$ we determine what numbers can appear as the Segre
Invariant of a rank $2$ vector bundle with given Chern’s classes. The
irreducibility of strata with fixed Segre’s invariant is proved and its
dimensions are computed. Finally, we present applications to the Brill-
Noether’s Theory for rank $2$ vector bundles on $\mathbb{P}^{2}.$
###### Key words and phrases:
moduli of vector bundles on surfaces, Segre invariant, stratification of the
moduli space.
This paper was partially supported by CONACyT Grant CB-257079. The first
author acknowledges the financial support of Fondo Institucional de Fomento
Regional para el Desarrollo Científico, Tecnológico y de Innovación, FORDECYT
265667.
## 1\. Introduction
We work over the field of complex numbers $\mathbb{C}$. Given a coherent sheaf
$F$ on a variety $X$ we write $H^{i}(F)$ instead of $H^{i}(X,F)$ and
$h^{i}(F)$ for the corresponding dimension.
Let $C$ be a non-singular irreducible complex projective curve of genus $g$.
Let $E$ be a vector bundle of rank $n$ and degree $d$ over $C$. For any
integer $m$, $1\leq m<n$ the $m-$Segre invariant is defined by
$S_{m}(E):=\min\\{m.\deg\,E-n.\deg\,F\\}$
where the minimum is taken over the subbundles F of $E$ of rank m. This
invariant induces a stratification of the moduli space $M(n,d)$ of stable
vector bundles of rank $n$ and degree $d$ over $C$. This stratification has
been studied by several authors (see for instance [8], [9], [2] and [17]) in
order to get topological and geometric properties of $M(n,d)$.
Given a non-singular, projective surface $X$, the moduli space
$M_{X}(n,c_{1},c_{2})$ of rank $n$ vector bundles with Chern classes $c_{1}$
and $c_{2}$ was constructed by Maruyama [14]. However, relatively little is
known about its geometry and subvarieties. The aim of this paper is to define
the Segre invariant for rank $2$ vector bundles on surfaces, with emphasis in
the case $X=\mathbb{P}^{2}$. Since this invariant defines a semicontinuous
function on the families of vector bundles of rank $2$ on X, we get a
stratification of the moduli space $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ into
locally closed subvarieties $M_{\mathbb{P}^{2}}(2;c_{1},c_{2};s)$ of vector
bundles with fixed Segre’s invariant $s$.
Section 2 introduces the Segre invariant on surfaces and collects a number of
results that will be subsequently used. Section 3 is the core of the paper.
The main issue is to determine what numbers can appear as the Segre invariant
of a vector bundle on $\mathbb{P}^{2}$ with fixed characteristic classes. The
answer is given by:
Theorem 3.1
1. (1)
Let $c_{2}\geq 2$ and $k\in\mathbb{N}$. Then a vector bundle $E\in
M_{\mathbb{P}^{2}}(2,0,c_{2})$ with $S(E)=2k$ exists if and only if
$k^{2}+k\leq c_{2}$. Furthermore, $E$ fits in an exact sequence:
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z}\longrightarrow
0,$
with $Z\subset\mathbb{P}^{2}$ of codimension $2$, and
$\mathcal{O}_{\mathbb{P}^{2}}(-k)\subset{E}$ maximal.
2. (2)
Let $c_{2}\geq 1$ and $k\in\mathbb{N}$. Then a vector bundle $E\in
M_{\mathbb{P}^{2}}(2,-1,c_{2})$ with $S(E)=2k-1$ exists if and only if
$k^{2}\leq c_{2}$. Furthermore, $E$ fits in an exact sequence:
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k-1)\otimes I_{Z}\longrightarrow
0,$
with $Z\subset\mathbb{P}^{2}$ of codimension $2$, and
$\mathcal{O}_{\mathbb{P}^{2}}(-k)\subset{E}$ maximal.
The idea of the proof is to apply Serre’s construction in order to obtain the
desired extension and then to show that the line sub-bundle
$\mathcal{O}_{\mathbb{P}^{2}}(-k)$ is, indeed, maximal. Note that, since
Segre’s invariant is invariant under tensor product with line bundles, it is
sufficient to consider rank 2 vector bundles with $c_{1}(E)=0,-1$.
Theorem 3.1 states in particular that the elements of
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2};s)$ are parameterized by extensions of
stable vector bundles. Thus, the dimension of the stratum is obtained by
“counting parameters” of such extensions. This idea is formalized in Section
4, the precise statement being:
Theorem 4.1
1. (1)
Let $c_{2}\geq 2$, $k\in\mathbb{N}$ such that $k^{2}+k\leq c_{2}$. Then
$M_{\mathbb{P}^{2}}(2;0,c_{2};2k)$ is an irreducible variety of dimension:
$\begin{cases}3c_{2}+k^{2}+3k-2,&\text{if $c_{2}>k^{2}+3k+1$}\\\
4c_{2}-3,&\text{if $c_{2}\leq k^{2}+3k+1$.}\\\ \end{cases}$
2. (2)
Let $c_{2}\geq 1$, $k\in\mathbb{N}$ such that $k^{2}\leq c_{2}$. Then
$M_{\mathbb{P}^{2}}(2;-1,c_{2};-1+2k)$ is an irreducible variety of dimension:
$\begin{cases}3c_{2}+k^{2}+2k-4,&\text{if $c_{2}>k^{2}+2k$}\\\
4c_{2}-4,&\text{if $c_{2}\leq k^{2}+2k$.}\\\ \end{cases}$
Next, observe that the morphism:
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow E,$
leads immediately to the existence of a global section of $E(-k)$. Thus,
Segre’s invariant is naturally related to the existence of vector bundles
admitting at least a section. This is a first connection with Brill-Noether
Theory. Section 5 is devoted to investigate such connection. The main result
is:
Theorem 5.5
1. (1)
Let $r,c_{2},k$ be integer numbers satisfying $r^{2}+2\leq c_{2}$ and $k<r$.
Then, a vector bundle $E\in M_{\mathbb{P}^{2}}(2;2r,c_{2},2k)$ exists such
that $h^{0}(E)\geq(r-k)^{2}+4(r-k)+3$.
2. (2)
Let $r,c_{2}$ be integer numbers satisfying $r^{2}-r+1\leq c_{2}$. Then, a
vector bundle $E\in M_{\mathbb{P}^{2}}(2;2r-1,c_{2},2k-1)$ exists such that
$h^{0}(E)\geq t$, where
$t=\begin{cases}(r-k)^{2}+4(r-k)+3&\textit{ if }k\neq 1,\\\ r^{2}+r-1&\textit{
if }k=1.\end{cases}$
We finish the paper by computing a lower bound for the dimension of these
Brill-Noether’s varieties. As a consequence of our results, the existence of
nonempty Brill-Noether’s locus with negative Brill-Noether number can be
deduced.
## 2\. Segre invariant of vector bundles on Surfaces
We start this section by recalling the main results about vector bundles on
surfaces that shall be use in the sequel. For a further treatment of the
subject see [5] and [13].
Let $X$ be a smooth, irreducible complex projective surface and let $H$ be an
ample divisor on $X$. Let $\mathcal{E}$ be a torsion free coherent sheaf on
$X$ with fixed Chern classes $c_{i}(\mathcal{E})\in H^{2i}(X,\mathbb{Z})$ for
$i=1,2$. The $H-$slope of $\mathcal{E}$ is defined as the rational number
$\mu_{H}(\mathcal{E}):=\frac{c_{1}(\mathcal{E}).H}{rk\,\mathcal{E}},$
where $c_{1}(\mathcal{E}).H$ is the degree of $\mathcal{E}$ with respect to
$H$ and is denoted by $deg_{H}(\mathcal{E})$.
###### Definition 2.1.
_A torsion free coherent sheaf $\mathcal{E}$ is $H$-stable (resp.
$H$-semistable) if for every nonzero subsheaves $\mathcal{F}$ of smaller rank,
we have_
$\mu_{H}(\mathcal{F})<\mu_{H}(\mathcal{E})\text{ (resp. $\leq$). }$
Let $\mathcal{E}$ be a coherent sheaf on $X$. The dual of $\mathcal{E}$ is the
sheaf $\mathcal{E}^{\vee}=\mathcal{H}om(\mathcal{E},\mathcal{O}_{X})$. If the
natural map $\mathcal{E}\longrightarrow\mathcal{E}^{\vee\vee}$ of
$\mathcal{E}$ to its double dual is an isomorphism, we say that $\mathcal{E}$
is reflexive. In particular, any locally free sheaf is reflexive. For more
details related to reflexive sheaves see [11].
###### Remark 2.2.
_Let $E$ be a vector bundle on $X$. Then $E$ is $H$-stable (resp.
$H$-semistable) if for all proper subbundle $F$, we have
$\mu_{H}(F)<\mu_{H}(E)$ (resp. $\leq$). Indeed, let $\mathcal{F}\subset E$ be
a proper subsheaf, hence $\mathcal{F}$ is free torsion and a canonical
embedding exits $\mathcal{F}\rightarrow\mathcal{F}^{\vee\vee}$ which fits in
the following diagram:_
---
$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}^{\vee\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\vee\vee}.}$
_Since $\mathcal{F}^{\vee\vee}$ is a reflexive sheaf and
$c_{1}(\mathcal{F})=c_{1}(\mathcal{F}^{\vee\vee})$, it follows that $E$ is
$H$-stable (resp. $H$-semistable) if for any proper reflexive sheaf
$\mathcal{F}$ we have $\mu_{H}(\mathcal{F})<\mu_{H}(E)$ (resp. $\leq$).
Moreover, since the singular points of reflexive sheaf $\mathcal{F}$ has
codimension greater than two, this implies that $\mathcal{F}$ is a vector
bundle. _
The following lemma allows to establish a relationship between line subbundles
of $E$ and extensions.
###### Lemma 2.3.
[5, Chapter 2. Proposition 5.] Let $\phi:L\longrightarrow E$ be a sub-line
bundle. Then a unique effective divisor $D$ on $X$ exists, such that the map
$\phi$ factors through the inclusion $L\longrightarrow
L\otimes\mathcal{O}_{X}(D)$ and such that $E/(L\otimes\mathcal{O}_{X}(D))$ is
torsion free.
From Lemma 2.3 it follows that if $L\subset E$ is maximal , then $E$ admits an
extension
$0\longrightarrow L\longrightarrow E\longrightarrow L^{\prime}\otimes
I_{Z}\longrightarrow 0,$
where $L^{\prime}\otimes I_{Z}$ is torsion free and $I_{Z}$ denote the ideal
sheaf of a subscheme $Z$ of codimension $2$. Note that
$\displaystyle c_{1}(E)$ $\displaystyle=c_{1}(L)+c_{1}(L^{\prime}),$
$\displaystyle c_{2}(E)$ $\displaystyle=c_{1}(L)\cdot c_{1}(L^{\prime})+l(Z),$
where $l(Z)$ denote the length of $Z$.
Now, we extend the concept of Segre’s invariant for vector bundles on curves
to vector bundles of rank $2$ on surfaces.
###### Definition 2.4.
Let $H$ be an ample divisor on $X$. For a vector bundle $E$ of rank $2$ on $X$
we define the Segre invariant $S_{H}(E)$ by
$S_{H}(E):=2\min\\{\mu_{H}(E)-\mu_{H}(L)\\},$
where the minimum is taken over all line subbundles $L$ of $E$.
Note that Definition 2.4 is equivalent to
$S_{H}(E)=\min\\{c_{1}(E).H-2L.H\\}.$
Segre’s invariant is always a finite number: in case $X=\mathbb{P}^{2}$ it
follows from Serre’s Vanishing Theorem, in general we have:
###### Lemma 2.5.
Let $X$ be a projective smooth surface, $H$ an ample line bundle on $X$ and
$E$ a rank $2$ vector bundle on $X$. Then, the set:
$\\{L.H:L\subset E,L\text{ a line bundle}\\},$
is bounded from above.
###### Proof.
Let $L_{1}$, $L_{2}$ be such that an exact sequence:
$0\to L_{1}\to E\to L_{2}\otimes I_{Z}\to 0$
exists, with $Z$ a zero-cycle on $X$ (see Lemma 2.3 above). Consider $L\subset
E$. If $L\subseteq L_{1}$, then $L.H\leq L_{1}.H$. If not, then
$h^{0}(L_{1}\otimes L^{-1})=0$ and in consequence $h^{0}(L_{2}\otimes
I_{Z}\otimes L^{-1})\neq 0$.
From the stability of $L_{2}\otimes I_{Z}$ it follows that:
$L.H<c_{1}(L_{2}\otimes I_{Z}).H.$
∎
The term invariant is used because $S_{H}(E)=S_{H}(E\otimes L)$ for any line
bundle $L\in Pic(X)$. By Remark 2.2 $E$ is $H-$ stable (resp. $H-$semistable)
if and only if $S_{H}(E)>0$ (resp. $\geq$). We say that $L\subset E$ is
maximal if
$S_{H}(E)=c_{1}(E).H-2c_{1}(L).H.$
The Segre invariant induces a stratification of the moduli space of vector
bundles:
###### Lemma 2.6.
Let $H$ be an ample divisor on $X$ and let $T$ be a variety. Let $\mathcal{E}$
be a vector bundle of rank $2$ on $X\times T$. The function
$\displaystyle S_{H}:T$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{Z}$ $\displaystyle t$ $\displaystyle\longmapsto$
$\displaystyle S_{H}(\mathcal{E}_{t})$
is lower semicontinuous.
###### Proof.
The semicontinuity follows as a slight generalization of the openness property
of stability for which the same proof works (see [14, Theorem 2.8]) ∎
The moduli space of $H$-stable vector bundles with fixed Chern classes $c_{1}$
and $c_{2}$ on $X$ were constructed in the 1970’s by Maruyama (see [15]). We
shall denote the moduli space of $H-$stable vector bundles of rank $n$ and
Chern classes $c_{1}$ and $c_{2}$ on $X$ by $M_{X,H}(n;c_{1},c_{2})$. In case
$X$ is a rational surface the dimension of moduli of rank $2$ vector bundles
is $\dim M_{X,H}(2,c_{1},c_{2})=4c_{2}-c_{1}^{2}-3$.
## 3\. Segre Invariant for rank $2$ vector bundles on $\mathbb{P}^{2}$
In this section we study the Segre invariant for vector bundles of rank $2$ on
$\mathbb{P}^{2}$, in this case a uniquely determined integer number $k_{E}$
exists such that
$c_{1}(E\otimes\mathcal{O}_{\mathbb{P}^{2}}(k_{E}))\in\\{-1,0\\}$. Namely
$k_{E}=\begin{cases}-\frac{c_{1}(E)}{2},&\text{if $c_{1}(E)$ even}\\\
-\frac{c_{1}(E)+1}{2},&\text{if $c_{1}(E)$ odd.}\end{cases}$
Since $S(E)=S(E\otimes L)$ for any line bundle $L$ on $\mathbb{P}^{2}$, in the
remainder of this section we assume that $E$ has degree $c_{1}\in\\{-1,0\\}$
and second Chern class $c_{2}$. Furthermore, by stability we mean stability
with respect to $\mathcal{O}_{\mathbb{P}^{2}}(1)$. In fact as
$Pic\,(\mathbb{P}^{2})\cong\mathbb{Z}$, there is a unique notion of stability
for $\mathbb{P}^{2}$. Let $F$ be a vector bundle on $\mathbb{P}^{2}$, by abuse
of notation we will use $c_{1}(F)$ to denote the degree of $F$ with respect to
$\mathcal{O}_{\mathbb{P}^{2}}(1)$ and we write $S(E)$ to denote the Segre
invariant of the vector bundle $E$ of rank $2$ on $\mathbb{P}^{2}$.
The following theorem is the main result of this paper. It gives necessary and
sufficient conditions for the existence of a stable vector bundle $E$ of rank
$2$, degree $c_{1}=0$ and $S(E)=2k$ (resp. $c_{1}=-1$ and $S(E)=-1+2k$).
###### Theorem 3.1.
1. (1)
Let $c_{2}\geq 2$ and $k\in\mathbb{N}$. Then a vector bundle $E\in
M_{\mathbb{P}^{2}}(2,0,c_{2})$ with $S(E)=2k$ exists if and only if
$k^{2}+k\leq c_{2}$. Furthermore, $E$ fits in an exact sequence:
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z}\longrightarrow
0,$
with $Z\subset\mathbb{P}^{2}$ of codimension $2$ and
$\mathcal{O}_{\mathbb{P}^{2}}(-k)\subset{E}$ maximal.
2. (2)
Let $c_{2}\geq 1$ and $k\in\mathbb{N}$. Then a vector bundle $E\in
M_{\mathbb{P}^{2}}(2,-1,c_{2})$ with $S(E)=2k-1$ exists if and only if
$k^{2}\leq c_{2}$. Furthermore, $E$ fits in an exact sequence:
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k-1)\otimes I_{Z}\longrightarrow
0,$
with $Z\subset\mathbb{P}^{2}$ of codimension $2$ and
$\mathcal{O}_{\mathbb{P}^{2}}(-k)\subset{E}$ maximal.
We need some auxiliary results:
###### Lemma 3.2.
[16, Lemma 1.2.5] Let $E$ be a vector bundle on $\mathbb{P}^{2}$ of rank 2 and
first Chern class $c_{1}=0$ or $c_{1}=-1$. Then $E$ is stable if and only if
$h^{0}(E)=0$.
Serre’s construction provides a method for constructing rank two vector
bundles on a surface $X$ (see [13, Chapter 5] for more details):
###### Theorem 3.3.
[13, Theorem 5.1.] Let $Z\subset X$ be a local complete intersection of
codimension two in the projective non-singular surface $X$, and let $L$ and
$M$ be line bundles on $X$. Then there exists an extension
$0\longrightarrow L\longrightarrow E\longrightarrow M\otimes
I_{Z}\longrightarrow 0$
such that $E$ is locally free if and only if the pair $(L^{-1}\otimes
M\otimes\omega_{X},Z)$ satisfy the Cayley-Bacharach property:
$\displaystyle(CB)\,\,\,$ if $Z^{\prime}\subset Z$ is a sub-scheme with
$l({\tilde{Z}})=l(Z)-1$ and $\displaystyle\text{$s\in H^{0}(L^{-1}\otimes
M\otimes\omega_{X})$ with $s|_{\tilde{Z}}=0$, then $s|_{Z}=0$}.$
###### Lemma 3.4.
Let $Z\subset\mathbb{P}^{2}$ be a zero-cycle in $\mathbb{P}^{2}$ such that
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(d)\otimes I_{Z})=0$. Consider a point
$p\in Supp(Z)$ and write $Z={\tilde{Z}}+p$. Then,
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(d-2)\otimes I_{\tilde{Z}})=0.$
In particular, $(\mathcal{O}_{\mathbb{P}^{2}}(d-2),Z)$ satisfies (CB).
###### Proof.
Suppose that $h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(d-2)\otimes
I_{\tilde{Z}})\neq 0$ and let $C_{1}$ be one of its elements different from
zero. Consider the map
$\displaystyle H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2)\otimes I_{p})$
$\displaystyle\rightarrow$ $\displaystyle
H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(d)\otimes I_{Z}),$ $\displaystyle C$
$\displaystyle\mapsto$ $\displaystyle CC_{1}$
which is lineal and injective. Since
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2)\otimes I_{p})=5$ we obtain a
contradiction. ∎
###### Proof.
Proof of Theorem 3.1
1. (1)
Assume $k^{2}+k\leq c_{2}$. Then a zero cycle $Z$, locally a complete
intersection and of length $l(Z)=c_{2}+k^{2}$ exists such that $Z$ is not
contained in any curve of degree $2k-1$. By the Lemma 3.4, the pair
$(\mathcal{O}_{\mathbb{P}^{2}}(2k-3),Z)$ satisfies the Cayley-Bacharach
property. Therefore (see Theorem 3.3), an extension
(3.1) $0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z}\longrightarrow 0$
exists where $E$ is locally free and has Chern classes $c_{1}(E)=0$ and
$c_{2}(E)=c_{2}$.
Moreover, since $Z$ is not contained in any curve of degree $2k-1$ it follows
that $h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z})=h^{0}(E)=0$.
Therefore, by Lemma 3.2 the vector bundle $E$ is stable.
Finally, we prove that $\mathcal{O}_{\mathbb{P}^{2}}(-k)$ is maximal. Let
$\mathcal{O}_{\mathbb{P}^{2}}(l)$ be a line bundle with $l<k$, from the exact
sequence (3.1) we have
$0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k+l)\rightarrow
E(l)\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(k+l)\otimes I_{Z}\rightarrow 0.$
Since $l<k$ it follows that:
$h^{0}(E(l))=h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(k+l)\otimes I_{Z})\leq
h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes I_{Z})=0.$
This implies that $\mathcal{O}_{\mathbb{P}^{2}}(-l)$ is not a subbundle of $E$
and thus $\mathcal{O}_{\mathbb{P}^{2}}(-k)$ is maximal.
Conversely, assume that a rank $2$ bundle with Chern classes $c_{1}(E)=0$,
$c_{2}(E)=c_{2}<k^{2}+k$ and $S(E)=2k$ exists. Therefore, we have an exact
sequence
(3.2) $0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\rightarrow
E\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z}\rightarrow 0,$
where $Z\subset\mathbb{P}^{2}$ is a local complete intersection of codimension
$2$ with length $l(Z)=c_{2}+k^{2}$ and $\mathcal{O}_{\mathbb{P}^{2}}(-k)$ is
maximal. Hence, $l(Z)<h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1))$ and
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes I_{Z})\neq 0.$
From the exact sequence (3.2) we have:
$0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(-1)\rightarrow
E(k-1)\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes I_{Z}\rightarrow
0,$
which induces the long exact sequence:
$0\rightarrow H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(-1))\rightarrow
H^{0}(E(k-1))\rightarrow H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes
I_{Z})\rightarrow 0$
Thus,
$h^{0}(E(k-1))=h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes I_{Z})\neq 0,$
but this implies that $\mathcal{O}_{\mathbb{P}^{2}}(-k+1)$ is a subbundle of
$E$, contradicting the maximality of $\mathcal{O}_{\mathbb{P}^{2}}(-k)$.
2. (2)
The proof is quite analogous to the proof of $(1)$: taking
$Z\subset\mathbb{P}^{2}$ a local complete intersection of codimension 2 with
length $l(Z)=c_{2}+k^{2}-k$ such that $Z$ is not contained in any curve of
degree $2k-2$ with $k\neq 1$ and noting that
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-2))=2k^{2}-k\leq l(Z)=c_{2}+k^{2}-k$
because $k^{2}\leq c_{2}$. If $k=1$ the pair $(\mathcal{O}(-2),Z)$ satisfies
Cayley- Bacharach and we conclude as in $(1)$.
For the converse, we can proceed analogously to the proof of $(1)$ by noting
that $l(Z)<h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-2))$ and in consequence
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-2)\otimes I_{Z})\neq 0$.
∎
###### Corollary 3.5.
1. (1)
Let $r,c_{2}$ be integer numbers and $k\in\mathbb{N}$ such that $c_{2}\geq
r^{2}+2$. Then a vector bundle $E\in M_{\mathbb{P}^{2}}(2,2r,c_{2})$ with
$S(E)=2k$ exists if and only if $k^{2}+k+r^{2}\leq c_{2}$. Furthermore, $E$
fits in an exact sequence
$\displaystyle 0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(r-k)\rightarrow
E\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(r+k)\rightarrow 0.$
2. (2)
Let $r,c_{2}$ be integer numbers and $k\in\mathbb{N}$ such that $c_{2}\geq
r^{2}+1$. Then a vector bundle $E\in M_{\mathbb{P}^{2}}(2,2r-1,c_{2})$ with
$S(E)=2k-1$ exists if and only if $k^{2}+r^{2}-r\leq c_{2}$. Furthermore, $E$
fits in an exact sequence
$\displaystyle 0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(r-k)\rightarrow
E\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(r+k-1)\rightarrow 0.$
###### Proof.
The proof follows from Theorem 3.1, the fact $S(E(-r))=S(E)$, and the formulas
$\displaystyle c_{1}(E(-r))$ $\displaystyle=$ $\displaystyle c_{1}(E)-2r,$
$\displaystyle c_{2}(E(-r))$ $\displaystyle=$ $\displaystyle
c_{2}(E)-rc_{1}(E)+r^{2}.$
∎
## 4\. A Stratification of the moduli space
$M_{\mathbb{P}^{2}}(2,c_{1},c_{2})$
In this section we use the Segre invariant to induce a stratification of the
moduli space $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ of stable vector bundles of
rank $2$ and Chern classes $c_{1}$ and $c_{2}$ on $\mathbb{P}^{2}$. If
$c_{1}=0$ and $c_{2}$ is odd (resp. $c_{1}=-1$ and $c_{2}$ even) Le Potier
(see for instance [10, Theorem 14.6.2]) has shown that there exists an
universal family $\mathcal{E}$ parameterized by
$M_{\mathbb{P}^{2}}(2;0,c_{2})$ (resp. $M_{\mathbb{P}^{2}}(2;-1,c_{2})$). If
$c_{1}=0$ and $c_{2}$ is even (resp. $c_{1}=-1$ and $c_{2}$ odd) working
locally in the êtale topology we can assume that there is a family
$\mathcal{E}$ parameterized by $M_{\mathbb{P}^{2}}(2;0,c_{2})$ (resp.
$M_{\mathbb{P}^{2}}(2;-1,c_{2})$).
Let $\mathcal{E}$ be a family of rank $2$ vector bundles on $\mathbb{P}^{2}$
parameterized by $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$. By Theorem 2.6, the
function $S:M_{\mathbb{P}^{2}}(2;c_{1},c_{2})\longrightarrow\mathbb{Z}$
induces a stratification of $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ into locally
closed subsets
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2};s):=\\{E\in
M_{\mathbb{P}^{2}}(2;c_{1},c_{2}):S(E)=s\\}$
according to the value of $s$. Without loss of generality we can assume that
if $c_{1}=0$ (resp. $c_{1}=-1$) then $s=2k$ for some $k$, $0<k^{2}+k\leq
c_{2}$ (resp. $s=-1+2k$ for some $k$, $k^{2}\leq c_{2}$).
###### Theorem 4.1.
1. (1)
Let $c_{2}\geq 2$, $k\in\mathbb{N}$ such that $k^{2}+k\leq c_{2}$. Then
$M_{\mathbb{P}^{2}}(2;0,c_{2};2k)$ is an irreducible variety of dimension:
$\begin{cases}3c_{2}+k^{2}+3k-2,&\text{if $c_{2}>k^{2}+3k+1$}\\\
4c_{2}-3,&\text{if $c_{2}\leq k^{2}+3k+1$.}\\\ \end{cases}$
2. (2)
Let $c_{2}\geq 1$, $k\in\mathbb{N}$ such that $k^{2}\leq c_{2}$. Then
$M_{\mathbb{P}^{2}}(2;-1,c_{2};-1+2k)$ is an irreducible variety of dimension
$\begin{cases}3c_{2}+k^{2}+2k-4,&\text{if $c_{2}>k^{2}+2k$}\\\
4c_{2}-4,&\text{if $c_{2}\leq k^{2}+2k$.}\\\ \end{cases}$
###### Proof.
We only prove $(1)$, the proof of $(2)$ being quite analogous. Let $c_{2}\geq
2$, $k\in\mathbb{Z}$ such that $k^{2}+k\leq c_{2}$. Let
$Hilb^{l}(\mathbb{P}^{2})$ be the Hilbert scheme of zero-dimensional
subschemes of length $l=c_{2}+k^{2}$ on $\mathbb{P}^{2}$ and let
$\mathcal{I}_{\mathcal{Z}_{l}}$ be the ideal sheaf of the universal subscheme
$\mathcal{Z}_{l}$ in $\mathbb{P}^{2}\times Hilb^{l}(\mathbb{P}^{2})$. Let
$\mathcal{O}_{\mathbb{P}^{2}}(-k),\mathcal{O}_{\mathbb{P}^{2}}(k)$ be line
bundles on $\mathbb{P}^{2}$. Let $p_{1}$, $p_{2}$ be the projections of
$\mathbb{P}^{2}\times Hilb^{l}(\mathbb{P}^{2})$ on $\mathbb{P}^{2}$ and
$Hilb^{l}(\mathbb{P}^{2})$ respectively. Consider on $\mathbb{P}^{2}\times
Hilb^{l}(\mathbb{P}^{2})$ the sheaf
$\mathcal{H}om(p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes\mathcal{I}_{\mathcal{Z}_{l}},p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-k))$.
Taking higher direct image we obtain on $Hilb^{l}(\mathbb{P}^{2})$ the sheaf:
$R^{1}_{p_{{2}_{*}}}\mathcal{H}om(p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes\mathcal{I}_{\mathcal{Z}_{l}},p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-k).$
From the semicontinuity Theorem [12, Theorem 12.8] we have that the set:
$H:=\\{Z\in
Hilb^{l}(\mathbb{P}^{2}):h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\otimes
I_{Z})<1\\}$
is an open set of $Hilb^{l}(\mathbb{P}^{2})$ which is non-empty by Theorem
3.1. Restricting the sheaf
$R^{1}_{p_{{2}_{*}}}\mathcal{H}om(p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes\mathcal{I}_{\mathcal{Z}_{l}},p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-k))$
to $H$ we have that it is locally free because
$H^{0}(\mathcal{H}om(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes\mathcal{I}_{\mathcal{Z}_{l}},\mathcal{O}_{\mathbb{P}^{2}}(-k)))\cong
Hom(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes\mathcal{I}_{Z},\mathcal{O}_{\mathbb{P}^{2}}(-k))=0,$
and
$\dim\,Ext^{2}(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(-k))=h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\otimes
I_{Z})=0$
for any $Z\in H$. Hence, the fiber over $Z\in H$ is
$Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(-k))$.
Consider on $H$ the sheaf:
$\mathbb{P}\Gamma:=\mathbb{P}R^{1}_{p_{{2}_{*}}}\mathcal{H}om(p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes\mathcal{I}_{\mathcal{Z}_{l}},p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-k)).$
By [7, Lemma 3.2] there exists an exact sequence:
(4.1)
$0\to(id\times\pi)^{*}p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-k)\otimes\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}\Gamma}(1)\to\mathcal{E}\to(id\times\pi)^{*}(p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-k)\otimes\mathcal{I}_{\mathcal{Z}_{l}})\to
0$
on $\mathbb{P}^{2}\times\mathbb{P}\Gamma$ such that for each
$p\in\mathbb{P}\Gamma$ the restriction $\mathcal{E}_{|_{p}}$ of $\mathcal{E}$
to $\mathbb{P}^{2}\times\\{p\\}$ is isomorphic to an extension
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z}\longrightarrow
0.$
Define the set
$U:=\\{p\in\mathbb{P}\Gamma:\text{$\mathcal{E}_{|_{p}}$ is stable and
$S(\mathcal{E}_{p})=2k$}\\}.$
From Theorem 3.1, the lower semicontinuity of the function $S$ and the fact
that stability is an open condition we conclude that the set $U$ is non-empty
and open in $\mathbb{P}\Gamma$. Restricting the sequence (4.1) to
$\mathbb{P}^{2}\times U$ we have, from the universal property of the moduli
space $M_{\mathbb{P}^{2}}(2;0,c_{2})$, a morphism
$f_{s}:U\longrightarrow M_{\mathbb{P}^{2}}(2;0,c_{2})$
where $f_{s}(U)$ is precisely the stratum $M_{\mathbb{P}^{2}}(2;0,c_{2};2k)$.
Hence, $M_{\mathbb{P}^{2}}(2;0,c_{2};2k)$, being the image of an irreducible
variety under a morphism is irreducible.
We can now determine the dimension of
$\dim\,M_{\mathbb{P}^{2}}(2;0,c_{2};2k)=\dim\,U-\dim f_{s}^{-1}(E)$
for general $E$, which is equal to:
$\dim\,M_{\mathbb{P}^{2}}(2;0,c_{2};2k)=\dim\,H+\dim\,Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(-k))-\dim\,\mathbb{P}H^{0}(E(k))-1.$
Since $H$ is an open set of $Hilb^{l}(\mathbb{P}^{2})$ we have:
(4.2) $\displaystyle\dim\,M_{\mathbb{P}^{2}}(2;0,c_{2};2k)=$
$\displaystyle\dim\,Hilb^{l}(\mathbb{P}^{2})+\dim\,Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(-k))$
$\displaystyle-\dim\mathbb{P}H^{0}(E(k))-1,$
We now compute the values of
$\dim\,Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(-k))\text{ and
}\dim\,\mathbb{P}H^{0}(E(k)),$
where $Z\in H$.
Note that by Serre duality $Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(-k))$ is canonically dual to
$Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(-k),\mathcal{O}_{\mathbb{P}^{2}}(k-3)\otimes
I_{Z})$. Since $\mathcal{O}_{\mathbb{P}^{2}}(-k)$ is locally free, then
$Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(-k),\mathcal{O}_{\mathbb{P}^{2}}(k-3)\otimes
I_{Z})\cong H^{1}(\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\otimes I_{Z}).$
By the exact sequence
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\otimes
I_{Z}\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\longrightarrow\mathcal{O}_{Z}\longrightarrow
0$
we have that
$h^{1}(\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\otimes
I_{Z})=-h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-3))+h^{0}(\mathcal{O}_{Z}),$
because $Z\in H$ and therefore:
(4.3) $h^{1}(\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\otimes
I_{Z})=c_{2}-k^{2}+3k-1.$
Now, we compute $h^{0}(E(k))$. Since $E\in M_{\mathbb{P}^{2}}(2;0,c_{2};2k)$
it can be written in an extension
(4.4) $0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z}\longrightarrow 0$
from which we get:
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}\longrightarrow
E(k)\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(2k)\otimes
I_{Z}\longrightarrow 0.$
Therefore,
$h^{0}(E(k))=1+h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k)\otimes I_{Z}).$
It follows that for $Z\in H$ general:
(4.5) $h^{0}(E(k))=\begin{cases}1,&\text{if $c_{2}>k^{2}+3k+1$}\\\
k^{2}+3k-c_{2}+2,&\text{if $c_{2}\leq k^{2}+3k+1.$}\end{cases}$
Replacing (4.3) and (4.5) in (4.2) we have
$\displaystyle\dim\,M_{\mathbb{P}^{2}}(2;0,c_{2};2k)=\begin{cases}3c_{2}+k^{2}+3k-2,&\text{if
$c_{2}>k^{2}+3k+1$}\\\ 4c_{2}-3,&\text{if $c_{2}\leq k^{2}+3k+1$}\\\
\end{cases}$
which proves the theorem. ∎
###### Corollary 4.2.
1. (1)
Let $c_{2},r\in\mathbb{Z}$ and $k\in\mathbb{N}$ such that $r^{2}+k^{2}+k\leq
c_{2}$. Then $M_{\mathbb{P}^{2}}(2;2r,c_{2};2k)$ is an irreducible variety of
dimension
$\begin{cases}3c_{2}-3r^{2}+k^{2}+3k-2,&\text{if }c_{2}>r^{2}+k^{2}+3k+1\\\
4c_{2}-4r^{2}-3,&\text{if }c_{2}\leq r^{2}+k^{2}+3k+1.\\\ \end{cases}$
2. (2)
Let $c_{2}$, $k\in\mathbb{N}$ and $r\in\mathbb{Z}$ such that
$r^{2}-r+k^{2}\leq c_{2}$. Then $M_{\mathbb{P}^{2}}(2;2r-1,c_{2};2k-1)$ is an
irreducible variety of dimension
$\begin{cases}3c_{2}+3r-3r^{2}+k^{2}+2k-4,&\text{if
$c_{2}>r^{2}-r+k^{2}+2k$}\\\ 4c_{2}+4r-4r^{2}-4,&\text{if $c_{2}\leq
r^{2}-r+k^{2}+2k$.}\\\ \end{cases}$
###### Proof.
For $(1)$, since $S(E)=S(E\otimes\mathcal{O}_{\mathbb{P}^{2}}(-r))$, it
follows that the map
$\displaystyle M_{\mathbb{P}^{2}}(2;2r,c_{2};2k)$ $\displaystyle\rightarrow$
$\displaystyle M_{\mathbb{P}^{2}}(2;0,c_{2}-r^{2};2k)$ $\displaystyle E$
$\displaystyle\mapsto$ $\displaystyle
E\otimes\mathcal{O}_{\mathbb{P}^{2}}(-r)$
is an isomorphism. Part (2) follows analogously. ∎
###### Corollary 4.3.
1. (1)
Let $c_{2}\geq r^{2}+2$ and $k\in\mathbb{N}$ the only integer such that
$r^{2}+k^{2}+k\leq c_{2}\leq r^{2}+k^{2}+3k+1$. Then the stratum
$M_{\mathbb{P}^{2}}(2;2r,c_{2};2k)$ has the same dimension as the moduli space
$M_{\mathbb{P}^{2}}(2;2r,c_{2})$.
2. (2)
Let $c_{2}\geq r^{2}-r+1$ and $k\in\mathbb{N}$ the only integer such that
$r^{2}-r+k^{2}\leq c_{2}\leq r^{2}+k^{2}+2k$. Then the stratum
$M_{\mathbb{P}^{2}}(2;2r-1,c_{2};-1+2k)$ has the same dimension as the moduli
space $M_{\mathbb{P}^{2}}(2;2r-1,c_{2})$
###### Remark 4.4.
1. (1)
The uniqueness of the number $k$ satisfying the above inequalities follows by
elementary considerations. It is, indeed, equal to the largest $k$ such that
$r^{2}+k^{2}+k\leq c_{2}$ in (1) and such that $r^{2}-r+k^{2}\leq c_{2}$ in
part (2).
2. (2)
It follows by semicontinuity that the stratum with maximal dimension is an
open set in the moduli space of stable vector bundles.
## 5\. Applications to Brill-Noether Theory
In this section, we use the previous results to study the non-emptiness of
some Brill-Noether loci in the moduli space
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ of stable vector bundles of rank $2$ and
fixed Chern classes $c_{1}$ and $c_{2}$ on $\mathbb{P}^{2}$.
For any $t\geq 0$ the subvariety of $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$
defined as
$W^{t}(2;c_{1},c_{2}):=\\{E\in M_{\mathbb{P}^{2}}(2;c_{1},c_{2}):h^{0}(E)\geq
t\\}$
is called the $t-$Brill-Noether locus of the moduli space
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ (or simply Brill-Noether locus if there is
no confusion).
The following theorem yields information about the variety
$W^{t}(2;c_{1},c_{2})$, in particular shows that $W^{t}(2;c_{1},c_{2})$ is a
determinantal variety and give a formula for the expected dimension. It was
proved by Costa and Miro-Roig in [3] for every smooth projective variety of
dimension $n$.
###### Theorem 5.1.
[3, Corollary 2.8] Let $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ be the moduli space
of stable vector bundles of rank $2$ on $\mathbb{P}^{2}$ with fixed Chern
classes $c_{1}$, $c_{2}$. Then, for any $t\geq 0$, there exists a
determinantal variety
$W^{t}(2;c_{1},c_{2}):=\\{E\in M_{\mathbb{P}^{2}}(2;c_{1},c_{2}):h^{0}(E)\geq
t\\}.$
Moreover, each non-empty irreducible component of $W^{t}(2;c_{1},c_{2})$ has
dimension greater or equal to the Brill-Noether number on $\mathbb{P}^{2}$
$\rho^{t}(2;c_{1},c_{2}):=4c_{2}-c_{1}^{2}-3-t\left(t-\frac{c_{1}^{2}}{2}-\frac{3c_{1}}{2}+c_{2}-2\right)$
and
$W^{t+1}(2;c_{1},c_{2})\subset Sing(W^{t}(2;c_{1},c_{2}))$
whenever $W^{t}(2;c_{1},c_{2})\neq M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$.
The following result allows to establish a relationship between the Brill-
Noether locus $W^{t}(2;c_{1},c_{2})$ and the different strata
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2};s)$.
###### Theorem 5.2.
Let $r,k,c_{2}$ be integers. Assume,
$t=\frac{(r-k+2)(r-k+1)}{2}.$
1. (1)
Let $E\in M_{\mathbb{P}^{2}}(2;2r,c_{2};2k)$. Then $E\notin W^{1}(2;2r,c_{2})$
if $r<k$ and $E\in W^{t}(2;2r,c_{2})$ if $r\geq k$. Moreover, the Brill-
Noether number $\rho^{t}(2,2r,c_{2})<\dim M_{\mathbb{P}^{2}}(2,2r,c_{2};2k)$
for $c_{2}>>0.$
2. (2)
Let $E\in M_{\mathbb{P}^{2}}(2;2r-1,c_{2};-1+2k)$. Then $E\notin
W^{1}(2;2r-1,c_{2})$ if $r<k$ and $E\in W^{t}(2;2r-1,c_{2})$ if $r\geq k$.
Moreover, the Brill-Noether number $\rho^{t}(2,2r-1,c_{2})<\dim
M_{\mathbb{P}^{2}}(2,2r,c_{2};2k-1)$ for $c_{2}>>0.$
###### Proof.
1. (1)
Since $E\in M_{\mathbb{P}^{2}}(2;2r,c_{2};2k)$, it follows that $E(-r)\in
M_{\mathbb{P}^{2}}(2;0,c_{2}-r^{2};2k)$ because $S(E(-r))=S(E).$ By Theorem
3.1 there exists an extension
(5.1) $0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E(-r)\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes
I_{Z}\longrightarrow 0,$
where $Z\subset\mathbb{P}^{2}$ of codimension $2$ has length
$l(Z)=k^{2}+c_{2}-r^{2}$ and $\mathcal{O}_{\mathbb{P}^{2}}(-k)$ is maximal.
Note that $h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes I_{Z})=0$ since
$\mathcal{O}_{\mathbb{P}^{2}}(-k)$ is maximal. From the extension (5.1) we get
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(r-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(r+k)\otimes I_{Z}\longrightarrow
0.$
In consequence, $E\notin W^{1}(2;2r,c_{2})\,\,\text{ if \,\,\, $r<k$}$ and
$h^{0}(E)\geq h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r-k))=t,\,\,\text{if $r\geq
k.$}$
The Brill-Noether number is given by
$\displaystyle\rho^{t}(2;2r,c_{2})=4c_{2}-4r^{2}-3-t(t+c_{2}-2r^{2}-3r-2).$
Therefore, for $c_{2}>>0$ we have:
(5.2) $\displaystyle\rho^{t}(2,2r,c_{2})<3c_{2}-3r^{2}+k^{2}+3k-3=\dim
M_{\mathbb{P}^{2}}(2,2r,c_{2};2k).$
2. (2)
The proof proceeds analogously to the proof of $(1)$.
∎
Suppose that $c_{1}>0$ and the Euler-Poincaré characteristic $\chi=0$, i.e.
$c_{2}=2+\frac{c_{1}^{2}+3c_{1}}{2}$. It is known that the moduli space
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ satisfies Weak Brill-Noether, that is,
there exist $E\in M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ such that $h^{i}(E)=0$
for any $i$ (see [10], Theorem 18.1.1). Note that, by semicontinuity, if $E$
is any sheaf with no cohomology then the cohomology also vanishes for any
general sheaf in $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ (cf. [7], [4]). Using the
Segre Invariant, we can give a different proof that
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ satisfies Weak Brill-Noether. The
advantage in using Segre’s invariant lies in the fact that the open set
satisfying Weak Brill-Noether can be explicitly described as the open stratum
of $M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$.
###### Corollary 5.3.
Suppose that $c_{1}>0$ and $c_{2}=2+\frac{c_{1}^{2}+3c_{1}}{2}$. Then moduli
space $M_{\mathbb{P}^{2}}(2,c_{1},c_{2})$ satisfies Weak-Brill Noether.
###### Proof.
Since $\chi=0$ and $c_{1}>0$, it follows that $h^{2}(E)=0$ for any stable
vector bundle with $c_{1}(E)=c_{1}$. Assume that $c_{1}=2r$ (resp.
$c_{1}=2r-1$) with $r\in\mathbb{N}$. Since the Euler-Poincaré characteristic
$\chi=0$, it follows that $c_{2}=2r^{2}+3r+2$ (resp. $c_{2}=2r^{2}+r+1$). Set
$k=r+1,$ note that $k$ is the largest integer such that $r^{2}+k^{2}+k\leq
c_{2}$ (resp. $k^{2}+r^{2}-r\leq c_{2}$), then by Corollary 4.3 we have that
the stratum $M_{\mathbb{P}^{2}}(2;2r,c_{2};2k)$ (resp.
$M_{\mathbb{P}^{2}}(2;2r-1,c_{2};2k-1)$) is open and by Theorem 5.2
$h^{0}(E)=0$ for any $E\in M_{\mathbb{P}^{2}}(2;2r,c_{2};2k)$ (resp. $E\in
M_{\mathbb{P}^{2}}(2;2r-1,c_{2};2k-1)$). ∎
###### Remark 5.4.
It is also known that under the conditions of Corollary 5.3 the complement of
the maximun stratum is a reduced hypersurface ([6], Theorem 2).
For larger values of $t$ further information can be obtained. For this we use
the existence of special configurations of points:
###### Theorem 5.5.
1. (1)
Let $r,c_{2},k$ be integer numbers satisfying $r^{2}+2\leq c_{2}$ and $k<r$.
Then, a vector bundle $E\in M_{\mathbb{P}^{2}}(2;2r,c_{2},2k)$ exists such
that $h^{0}(E)\geq(r-k)^{2}+4(r-k)+3$.
2. (2)
Let $r,c_{2}$ be integer numbers satisfying $r^{2}-r+1\leq c_{2}$. Then, a
vector bundle $E\in M_{\mathbb{P}^{2}}(2;2r-1,c_{2},2k-1)$ exists such that
$h^{0}(E)\geq t$, where
$t=\begin{cases}(r-k)^{2}+4(r-k)+3&\text{if $k\neq 1$},\\\ r^{2}+r-1&\text{if
$k=1$.}\end{cases}$
###### Proof.
1. (1)
Let $Z^{\prime}$ be a reduced zero-cycle of length
$l(Z^{\prime})=\dim\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1))$ such
that $H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes
I_{Z^{\prime}})=\mathbb{C}.C$ for some curve $C$ of degree $2k-1$. Complete
$Z^{\prime}$ to a zero cycle $\tilde{Z}=Z^{\prime}+Z^{{}^{\prime\prime}}$ such
that $Supp(Z^{{}^{\prime\prime}})\subset C$ and
$l(\tilde{Z})=c_{2}+k^{2}-r^{2}-1$ and set $Z=\tilde{Z}+p$ for $p$ some point
not contained in $C$.
Then it follows from an argument similar to the one used in the proof of the
Lemma 3.4 that the pair $(\mathcal{O}_{\mathbb{P}^{2}}(2k-3),Z)$ satisfies the
Cayley-Bacharach property (see Theorem 3.3). Indeed, if $Z=Z_{0}+p_{0}$ and
$0\neq C_{0}\in H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-3)\otimes I_{Z_{0}})$
exists, then the map:
$H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2)\otimes I_{p_{0}})\to
H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes I_{Z}),$
$C\longmapsto CC_{0},$
must be injective.
Therefore an extension:
(5.3) $0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-k)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(k)\otimes I_{Z}\longrightarrow 0$
exists with $E\in M_{\mathbb{P}^{2}}(2,0,c_{2}-r^{2})$. Since $Z-p\subset C$
we see that any curve of degree $r-k-1$ passing through $p$ gives rise to an
element of $H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r+k)\otimes I_{Z})$. Thus,
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r+k)\otimes I_{Z})\geq
h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r-k+1))-1=\frac{(r-k)^{2}+5(r-k)}{2}+2.$
In this way we obtain that $E(r)\in M_{\mathbb{P}^{2}}(2,2r,c_{2})$ and from
the exact sequence (5.3):
$h^{0}(E(r))=h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r-k))+h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r+k)\otimes
I_{Z})\geq(r-k)^{2}+4(r-k)+3,$
as desired.
2. (2)
The proof for the case $k\neq 1$ follows analogously to the proof of $(1)$.
Taking $C\subset\mathbb{P}^{2}$ be an irreducible curve of degree $2k-2$ and
$Z\subset\mathbb{P}^{2}$ be distinct points with length
$l(Z)=c_{2}+k^{2}-k-r^{2}+r$ such that $Z-\\{p\\}$ is contained in $C$ but $Z$
is not contained in $C$. Note that the pair
$(\mathcal{O}_{\mathbb{P}^{2}}(2k-4),Z)$ satisfies the Cayley- Bacharach
property and
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r+k-1)\otimes I_{Z})\geq
h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r-k+1))-1=\frac{(r-k)^{2}+5(r-k)}{2}+2.$
From the exact sequence
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(r-k)\longrightarrow
E(r)\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(r+k-1)\otimes
I_{Z}\longrightarrow 0,$
we have:
$h^{0}(E(r))=h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r-k))+h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r+k-1)\otimes
I_{Z})\geq(r-k)^{2}+4(r-k)+3.$
We now proceed to prove the case $k=1$. Let $L\subset\mathbb{P}^{2}$ be a line
and let $Z\subset\mathbb{P}^{2}$ be distinct points with length
$l(Z)=c_{2}-r^{2}+r$ such that $Z-\\{p\\}$ is contained in $L$ but $Z$ is not
contained in $L$. Note that the pair $(\mathcal{O}_{\mathbb{P}^{2}}(-2),Z)$
satisfies the Cayley-Bacharach property, thus, there exists an extension
(5.4) $0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-1)\longrightarrow
E\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}\otimes I_{Z}\longrightarrow 0.$
with $c_{1}(E)=-1$ and $c_{2}(E)=c_{2}-r^{2}+r$. Moreover, $E$ is a stable
vector bundle because $Z\neq\emptyset$.
Note that
$h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r)\otimes I_{Z})\geq
h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r-1))-1=\frac{r(r+1)}{2}-1$
and $E(r)$ is a stable vector bundle with $c_{1}(E(r))=2r-1$ and
$c_{2}(E(r))=c_{2}$. From the exact sequence (5.3) we get
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(r-1)\longrightarrow
E(r)\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(r)\otimes I_{Z}\longrightarrow
0,$
and taking cohomology we have
$h^{0}(E(r))=h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r-1))+h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(r)\otimes
I_{Z})\geq r^{2}+r-1.$
∎
Once that Theorem 5.5 has established the non-emptiness of some Brill-Noether
loci it is natural to search for a lower bound for its dimensions.
###### Theorem 5.6.
1. (1)
Let $r,k,c_{2}\in\mathbb{N}$ such that $r\geq 2$, $3k^{2}-4k+r^{2}+2<c_{2}$
and $k<r$. Let $t=(r-k)^{2}+4(r-k)+3$. Then,
$dim\,W^{t}(2;2r,c_{2})\geq\begin{cases}2c_{2}+2k^{2}-2r^{2}+4k-2,&\text{if
$c_{2}>k^{2}+3k+r^{2}+1$}\\\ k^{2}+3c_{2}+k-r^{2}-3,&\text{if $c_{2}\leq
k^{2}+3k+r^{2}+1$.}\end{cases}$
2. (2)
Let $r,k,c_{2}\in\mathbb{N}$ such that $r\geq 2$ and
$3k^{2}-7k+r^{2}-r+5<c_{2}$ and $k<r$.
$t=\begin{cases}(r-k)^{2}+4(r-k)+3,&\text{if $k\neq 1$}\\\ r^{2}+r-1,&\text{if
$k=1$}.\end{cases}$
Then,
$dim\,W^{t}(2;2r-1,c_{2})\geq\begin{cases}2c_{2}+2k^{2}-2r^{2}+2k+2r-4,&\text{if
}c_{2}>k^{2}-2k+r^{2}-r\\\ 3c_{2}+k^{2}-3r^{2}+3r-4,&\text{if }c_{2}\leq
k^{2}-2k+r^{2}-r,\end{cases}$
if $k\neq 1$, and
$dim\,W^{t}(2;2r-1,c_{2})\geq 3c_{2}-3r^{2}+3r+1,$
if $k-1$.
###### Proof.
We only prove $(1)$, the proof of $(2)$ follows by similar arguments.
Let $l=k^{2}+c_{2}-r^{2}$ and let $Hilb^{\tilde{l}}(\mathbb{P}^{2})$ be the
Hilbert scheme of zero-dimensional subschemes of length $\tilde{l}=l-1$. Let
$\mathcal{I}\subset\mathbb{P}H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1))$ be the
subset of irreducible curves of degree $2k-1$ and consider the variety:
$I:=\\{(p,\tilde{Z},C)\in\mathbb{P}^{2}\times
Hilb^{\tilde{l}}(\mathbb{P}^{2})\times\mathcal{I}:p\notin C\,,\tilde{Z}\subset
C\\}.$
For every $C\in\mathcal{I}$ its fiber $p_{3}^{-1}(C)$ under the projection
onto the third factor is the irreducible variety $(\mathbb{P}^{2}-C)\times
S^{\tilde{l}}C$, which have the same dimension for every $C$. Let
$I_{0}\subset I$ be an irreducible component such that
$p_{3}:I_{0}\to\mathcal{I}$ is dominant and $\dim I_{0}=\dim I=k(2k+1)+l$.
Let $p_{12}:I_{0}\to\mathbb{P}^{2}\times Hilb^{\tilde{l}}(\mathbb{P}^{2})$ be
the projection onto the first two factors. Because of the choice of $l$ and
$c_{2}$, we have that $\tilde{l}>(2k-1)^{2}$. Thus $p_{12}$ is injective and
therefore it is a birational morphism. In this way, an open set
$\mathcal{U}\subset p_{12}(I_{0})$ exists such that $\dim
I_{0}=\dim\mathcal{U}$.
Consider the finite morphism
$\displaystyle\phi:\mathbb{P}^{2}\times Hilb^{\tilde{l}}(\mathbb{P}^{2})$
$\displaystyle\longrightarrow Hilb^{l}(\mathbb{P}^{2})$
$\displaystyle(p,\tilde{Z})$ $\displaystyle\longmapsto p+\tilde{Z},$
and the set
(5.5) $H\cap\phi(\mathcal{U}),$
where
$H:=\\{Z\in
Hilb^{l}(\mathbb{P}^{2}):h^{0}(\mathcal{O}_{\mathbb{P}^{2}}(2k-1)\otimes
I_{Z})=0\\}.$
Note that by the proof of the Theorem 3.1 the set (5.5) is non-empty, moreover
$\phi(\mathcal{U})\subset\phi(p_{12}(I))\subset H$. Therefore,
$H\cap\phi(\mathcal{U})=\phi(\mathcal{U})$
Now, we can now proceed analogously to the proof of Theorem 4.1. Consider the
sheaf
$\mathbb{P}\Gamma:=\mathbb{P}R^{1}_{p_{{2}_{*}}}\mathcal{H}om(p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(k+r)\otimes\mathcal{I}_{\mathcal{Z}_{l}},p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(r-k))$
over $\phi(\mathcal{U})$ and the exact sequence
(5.6)
$0\longrightarrow(id\times\pi)^{*}p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(r-k)\otimes\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}\Gamma}(1)\longrightarrow\mathcal{E}\longrightarrow(id\times\pi)^{*}(p_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(k+r)\otimes\mathcal{I}_{\mathcal{Z}_{l}})\longrightarrow
0.$
on $\mathbb{P}^{2}\times\mathbb{P}\Gamma$. Define the set
$U:=\\{p\in\mathbb{P}\Gamma:\text{$\mathcal{E}_{|_{p}}$ is stable and
$S(\mathcal{E}_{p})=2k$}\\}.$
From Theorem 3.1, the lower semicontinuity of the function $S$ and stability
being an open condition we conclude that the set $U$ is non-empty and open in
$\mathbb{P}\Gamma$. Restricting the sequence (5.6) on $\mathbb{P}^{2}\times U$
from the universal property of the moduli space
$M_{\mathbb{P}^{2}}(2;c_{1},c_{2})$ we have a morphism
$f_{t}:U\longrightarrow M_{\mathbb{P}^{2}}(2;2r,c_{2}).$
Note that, by the proof of Theorem 5.5 the set $f_{t}(U)$ is contained in the
locus of Brill-Noether $W^{t}(2;2r,c_{2})$ where $t=(r-k)^{2}+4(r-k)+3$.
Moreover,
$f_{t}(U)\subseteq\overline{f_{t}(U)}\subseteq W^{t}(2;2r,c_{2}).$
We proceed now to determine the dimension of $Im\overline{f_{t}(U)}$ and a
lower bound of the locus $W^{t}(2;2r,c_{2})$,
$\dim\,W^{t}(2;2r,c_{2})\geq\dim\,Im\overline{f_{t}(U)}\geq\dim\,U-dimf_{t}^{-1}(E)$
which is equivalent to
$\displaystyle\dim\,Im\overline{f_{t}(U)}$
$\displaystyle\geq\dim\,\phi{(\overline{p_{12}(I)})}+\dim\,Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k+r)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(r-k))-\dim\,\mathbb{P}H^{0}(E(k-r))-1$
$\displaystyle=\dim\,\phi(p_{12}(I))+\dim\,Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k+r)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(r-k))-\dim\,\mathbb{P}H^{0}(E(k-r))-1$
$\displaystyle\geq\dim\,\phi(\mathcal{U})+\dim\,Ext^{1}(\mathcal{O}_{\mathbb{P}^{2}}(k+r)\otimes
I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(r-k))-\dim\,\mathbb{P}H^{0}(E(k-r))-1.$
From the above and the proof of Theorem 4.1 we conclude that
$\dim\,Im\overline{f_{t}(U)}\geq\begin{cases}c_{2}+k^{2}-r^{2}+4k-2,&\text{if
$c_{2}>k^{2}+r^{2}+3k+1$}\\\ 2c_{2}+k-2r^{2}-3,&\text{if $c_{2}\leq
k^{2}+3k+r^{2}+1$.}\end{cases}$
as claimed.
∎
## References
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* [2] L. Brambila-Paz, H. Lange. — A stratification of the moduli space of vector bundles on curves. Journal für die reine und angewandte Mathematik, 494, 173-187, (1998).
* [3] L. Costa, R.M. Miro-Roig.— Brill-Noether theory for moduli spaces of sheaves on algebraic varieties. Forum Math. 22 (2010), no. 3, 411-432.
* [4] I. Coskun, J. Huizenga. — Weak Brill-Noether for rational surfaces. Local and global methods in algebraic geometry, Contemp. Math., 712, Amer. Math. Soc., Providence, RI, 2018\.
* [5] R. Friedman. — Algebraic surfaces and holomorphic vector bundles. Universitext. Springer-Verlag, New York, 1998. x+328 pp.
* [6] L. Gottsche and A. Hirschowitz.— Weak Brill-Noether for vector bundles on the projective plane. Algebraic geometry (Catania, 1993/Barcelona, 1994), Lecture Notes in Pure and Appl. Math., 200, Dekker, New York, 1998.
* [7] L. Gottsche. — Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces. Mathematische Zeitschrift, 223, 247-260, (1996).
* [8] H. Lange.— Zur Klassifikation von Regelmannigfaltigkeiten. Mathematische Annalen. 262 (4), 447 459, (1983).
* [9] H. Lange, N. Narasimhan. — Maximal subbundles of Rank two vector bundles on curves. Mathematishe Annalen, 266, 55-72, (1983).
* [10] J. Le Potier. — Lectures on vector bundles. Translated by A. Maciocia. Cambridge Studies in Advanced Mathematics, 54. Cambridge University Press, Cambridge, 1997. viii+251 pp.
* [11] R. Hartshorne. — Stable reflexive sheaves. Math. Ann. 254 (1980), no. 2, 121-176.
* [12] R. Hartshorne. — Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp.
* [13] D. Huybrechts, M. Lehn. — The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg Sohn, Braunschweig, 1997. xiv+269 pp.
* [14] M. Maruyama. — Openness of a family of torsion free sheaves. J. Math. Kyoto Univ. 16 (1976), no. 3, 627–637.
* [15] M. Maruyama. — Stable vector bundles on an algebraic surface. Nagoya Math. J. Vol. 58 (1975), 25-68.
* [16] Okonek, Christian; Schneider, Michael; Spindler, Heinz. — .Vector bundles on complex projective spaces. Progress in Mathematics, 3. Birkh user, Boston, Mass., 1980. vii+389 pp.
* [17] B. Russo, M. Teixidor. — On a conjecture of Lange. Journal of Algebraic Geometry, 8, 483-496, (1999).
|
2024-09-04T02:54:57.565027 | 2020-03-05T17:14:27 | 2003.02773 | {
"authors": "Semanti Dutta, B. Sathiapalan, and H. Sonoda",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26061",
"submitter": "Semanti Dutta",
"url": "https://arxiv.org/abs/2003.02773"
} | arxiv-papers | IMSc/2020/02/02
KOBE-TH-20-02
# Wilson Action for the $O(N)$ Model
S. Dutta, B. Sathiapalan
Institute of Mathematical Sciences
CIT Campus, Tharamani
Chennai 600113, India
and
Homi Bhabha National Institute
Training School Complex, Anushakti Nagar
Mumbai 400085, India
and
H. Sonoda
Physics Department, Kobe University
Kobe 657-8501, Japan
###### Abstract
In this paper the fixed-point Wilson action for the critical $O(N)$ model in
$D=4-\epsilon$ dimensions is written down in the $\epsilon$ expansion to order
$\epsilon^{2}$. It is obtained by solving the fixed-point Polchinski Exact
Renormalization Group equation (with anomalous dimension) in powers of
$\epsilon$. This is an example of a theory that has scale and conformal
invariance despite having a finite UV cutoff. The energy-momentum tensor for
this theory is also constructed (at zero momentum) to order $\epsilon^{2}$.
This is done by solving the Ward-Takahashi identity for the fixed point
action. It is verified that the trace of the energy-momentum tensor is
proportional to the violation of scale invariance as given by the exact RG,
i.e., the $\beta$ function. The vanishing of the trace at the fixed point
ensures conformal invariance. Some examples of calculations of correlation
functions are also given.
###### Contents
1. 1 Introduction
2. 2 Background
1. 2.1 Exact Renormalization Group and Fixed Point equation
1. 2.1.1 Exact Renormalization group
2. 2.1.2 Polchinski’s ERG equation
3. 2.1.3 The limit $\Lambda\to 0+$
4. 2.1.4 IR limit of a critical theory
5. 2.1.5 Anomalous dimension in ERG
6. 2.1.6 Dimensionless framework
7. 2.1.7 Fixed-point equation
2. 2.2 Energy Momentum Tensor: Scale Invariance and Conformal Invariance
1. 2.2.1 Energy Momentum Tensor in the Classical Theory
2. 2.2.2 Trace of the Energy Momentum Tensor in the Quantum Theory: Perturbative
3. 2.2.3 Energy Momentum Tensor in Exact RG
3. 3 Wilson-Fisher Fixed Point for the $O(N)$ Model
1. 3.1 Equations for the vertices
2. 3.2 Solving the Equations
1. 3.2.1 $\mathcal{O}(1)$: Retrieving Gaussian theory
2. 3.2.2 $\mathcal{O}(\epsilon)$: Fixed Point value of $m^{2}$
3. 3.2.3 $\mathcal{O}(\epsilon^{2})$: Expression for the six-point vertex
4. 3.2.4 Fixed Point value of $\lambda$: Solution for $U_{4}$ at $\mathcal{O}(\epsilon)$
3. 3.3 Determining Anomalous Dimension
4. 4 Correlation functions
1. 4.1 A more general equation
2. 4.2 Calculation of correlation functions
1. 4.2.1 Two-point function
2. 4.2.2 Four-point function
3. 4.2.3 Six-point function
5. 5 Construction of the energy-momentum tensor at the fixed point
6. 6 Summary and Conclusions:
7. A Fixed Point Action
1. A.1 Evaluation of $U_{4}$
2. A.2 Solving for $\tilde{U}_{4}$
3. A.3 Equation for $\tilde{U}_{2}$
4. A.4 Expression for $\eta$
8. B Asymptotic behaviors of $F(p)$ and $G(p)$
## 1 Introduction
Conformal field theories (CFT) are interesting for a variety of reasons. One
of the most important reason is that a theory critical at a continuous phase
transition is expected to acquire conformal invariance which imposes strong
constraints on the correlation functions[1]. This has motivated the idea of
bootstrap[2]. Particularly in two dimensions these ideas have been very
fruitful [3]. Reviews of later developments and references are given in [4,
5].
The advent of the AdS/CFT correpondence [6, 7, 8, 9] or “holography” between a
boundary CFT and a bulk gravity theory opened up another approach to solving
CFT’s. 111It also opens up the amazing possibility of rewriting quantum
gravity as a quantum field theory in flat space. There is a large amount of
literature on this. See, for example, [10] for a review.
In the AdS/CFT correspondence the radial direction can be interpreted as the
scale of the boundary field theory. Thus, a radial evolution can be thought of
as an RG evolution and has been dubbed “holgraphic RG” [15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28]. The precise connection between the
boundary RG and holographic RG is, however, still an open question.
Recently a connection has been proposed between the Exact Renormalization
Group (ERG) equation [11, 12, 13, 14] and the Holographic Renormalization
Group (Holographic RG) equation. It was shown in [29] that the RG evolution
operator for a Wilson action of a D-dimensional field theory obeying the
Polchinski ERG equation can be formulated as a $D+1$-dimensional functional
integral. The extra dimension, corresponding to the moving scale $\Lambda$ of
the ERG, makes it a “holographic” formulation. Furthermore, a change of field
variables or field redefinition maps the $D+1$ dimensional action for the
functional integral to the action of a free massive scalar field in
$AdS_{D+1}$. It was then shown that the calculation of the two point function
reduces to the familiar calculation using the AdS/CFT correspondence.
This proposal is quite general, and detailed calculations were done for the
Gaussian theory [29]. The scalar field theory action has a free parameter,
i.e., the mass of the scalar field, which is related to the anomalous
dimension of the boundary operator in the AdS/CFT context . This parameter
appears to come out of nowhere. To understand the origin of the anomalous
dimension parameter, an ERG equation with anomalous dimension was analysed in
[30]. The same change of variables mapped this to a scalar field theory in the
AdS space-time, and this time it was easy to see that the mass parameter is
naturally related to the anomalous dimension parameter in the ERG. Normally,
interactions are required for a field to have anomalous dimension. Since the
exact RG for interacting theories is difficult, a Gaussian theory with an
anomalous dimension introduced by hand was studied in [30].
In order to improve our understanding of the connection between ERG and the
AdS/CFT correspondence, it is necessary to have an interacting example — one
needs a non-trivial boundary CFT and a fixed-point Wilson action for this CFT
222Note that the “Wilson action” always has a finite UV cutoff — this is a
point of departure from the usual CFT actions written in the continuum.. Then
the RG evolution of small perturbations to this theory can be studied by ERG.
Using the ideas of [29, 30] this can be mapped to a scalar field theory in
$D+1$-dimensional AdS space. This would make a contact with more detailed
AdS/CFT calculations of higher point correlators. A well studied field theory
is the $\lambda\phi^{4}$ scalar field theory in $4-\epsilon$ dimensions that
has the famous Wilson-Fisher fixed point. When there are $N$ scalar fields,
this is often referred to as the $O(N)$ model. In this paper, as a first step,
we construct a fixed-point Wilson action for this theory to order
$\epsilon^{2}$. It is at this order that the anomalous dimension first shows
up. The action is obtained by solving the fixed-point ERG equation
perturbatively. The fixed-point equation imposes the constraint of scale
invariance.
In fact the theory is also conformally invariant. This follows from the
properties of the energy momentum tensor — if it is traceless the theory is
conformally invariant. Indeed the tracelessness of the energy-momentum tensor
defines what we mean by a CFT [31, 32, 33, 34]. It is thus important to study
the energy-momentum tensor and we construct it in this paper.
The energy-momentum tensor is also important in the context of AdS/CFT: one of
the really interesting aspects of the AdS/CFT correspondence is that the
$D+1$-dimensional bulk theory has dynamical gravity. In addition to the scalar
field, there is the gravitational field that couples to the energy momentum
tensor of the boundary CFT. Thus to extend the ideas of [29, 30] to understand
bulk gravity in AdS/CFT correspondence, from ERG one has to construct the
energy momentum operator.
The energy-momentum tensor for $\phi^{4}$ field theory has been worked out in
the dimensional regularization scheme [33]. The construction of the energy-
momentum tensor from the ERG point of view has been studied in general in [35,
37]. The main idea is to solve the Ward Identity associated with coordinate
transformations. This can be done in perturbation theory. We construct the
leading terms that corresponds to the zero momentum energy momentum tensor.
One can also check that the trace of the energy momentum tensor is
proportional to the number operator. We apply this prescription here and
construct the zero momentum energy momentum tensor to $O(\lambda^{2})$.
This paper is organized as follows: In Section 2 we give a review of ERG and
the fixed-point equation. We also give some background material on the energy-
momentum tensor. In Section 3 we construct the solution to the fixed-point
equation and obtain the fixed-point action. In Section 4 we give a different
approach to obtaining the fixed point equation and also calculate some
correlation functions. In Section 5 the construction of the energy-momentum
tensor is given. We conclude the paper in Section 6.
## 2 Background
### 2.1 Exact Renormalization Group and Fixed Point equation
We review the necessary background in this section. It depends mostly on [44,
54].
#### 2.1.1 Exact Renormalization group
Renormalization means essentially going from one scale $\Lambda_{0}$ to a
lower scale $\Lambda$, where the initial scale $\Lambda_{0}$ is typically
called a bare scale. One will want to see how the physics changes with scale.
What do we mean by physics at $\Lambda_{0}$? It means our theory will not be
sensitive to momentum $p>\Lambda_{0}$. The partition function of the full
theory is given by
$\displaystyle Z=\int\mathcal{D}\phi\,e^{-S[\phi]}$
where
$\displaystyle S=\int_{p}\frac{1}{2}p^{2}\phi^{2}+S_{I}[\phi]$
To make it a partition function at scale $\Lambda_{0}$ we will try to suppress
the kinetic energy term for $\infty<p<\Lambda_{0}$. To execute this we will
put a smooth cutoff in the kinetic energy term to obtain the bare action
$\displaystyle
S_{B}[\phi]\equiv\frac{1}{2}\int_{p}\phi\frac{p^{2}}{K(p^{2}/\Lambda_{0}^{2})}\phi+S_{I,B}[\phi]$
(2.1)
and the bare partition function
$Z_{B}\equiv\int\mathcal{D}\phi\,e^{-S_{B}[\phi]}$ (2.2)
We will choose the cutoff function will follow the condition $K(0)=1$ and
$K(\infty)=0$. In general cutoff functions satisfy stronger properties , but
that will not affect the fixed point values of the couplings [55].
Now we want to go to a lower scale $\Lambda$. For that, observe the following
identity
$\displaystyle\int\mathcal{D}\phi\exp\left[-\frac{1}{2}\int_{p}\phi(-p)\frac{1}{A(p)+B(p)}\phi(p)-S_{I,B}[\phi]\right]$
$\displaystyle=$
$\displaystyle\int\mathcal{D}\phi_{1}\mathcal{D}\phi_{2}\exp\left[-\frac{1}{2}\int_{p}\frac{1}{A(p)}\phi_{1}(-p)\phi_{1}(p)-\frac{1}{2}\int_{p}\frac{1}{B(p)}\phi_{2}(-p)\phi_{2}(p)-S_{I,B}[\phi_{1}+\phi_{2}]\right]$
Using this we can write
$\displaystyle Z_{B}=$
$\displaystyle\int\mathcal{D}\phi_{l}\mathcal{D}\phi_{h}\exp\bigg{\\{}-\frac{1}{2}\int_{p}\frac{p^{2}}{K(p^{2}/\Lambda^{2})}\phi_{l}(-p)\phi_{l}(p)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\int_{p}\frac{p^{2}}{K(p^{2}/\Lambda_{0}^{2})-K(p^{2}/\Lambda^{2})}\phi_{h}(-p)\phi_{h}(p)-S_{I,B}[\phi_{l}+\phi_{h}]\bigg{\\}}$
We can effectively call $\phi_{l}(\phi_{h})$ as low(high) energy field as it
is propagated by low(high) momentum propagator $\Delta_{l}(\Delta_{h})$
defined below
$\displaystyle\Delta_{l}=\frac{K(p^{2}/\Lambda^{2})}{p^{2}},\quad\Delta_{h}=\frac{K(p^{2}/\Lambda^{2})-K(p^{2}/\Lambda_{0}^{2})}{p^{2}}$
(2.3)
So we can write
$\displaystyle Z_{B}=$
$\displaystyle\int\mathcal{D}\phi_{l}\exp\left[-\frac{1}{2}\int_{p}\phi_{l}\Delta_{l}^{-1}\phi_{l}\right]\int\mathcal{D}\phi_{h}\exp\left[-\frac{1}{2}\int_{p}\phi_{h}\Delta_{h}^{-1}\phi_{h}-S_{I,B}[\phi_{l}+\phi_{h}]\right]$
$\displaystyle=$
$\displaystyle\int\mathcal{D}\phi_{l}\exp\left[-\frac{1}{2}\int_{p}\phi_{l}\Delta_{l}^{-1}\phi_{l}\right]\exp\\{-S_{I,\Lambda}[\phi_{l}]\\}$
where
$\displaystyle\exp\\{-S_{I,\Lambda}[\phi_{l}]\\}\equiv\int\mathcal{D}\phi_{h}\exp\bigg{\\{}-\frac{1}{2}\int_{p}\phi_{h}\Delta_{h}^{-1}\phi_{h}-S_{I,B}[\phi_{l}+\phi_{h}]\bigg{\\}}$
(2.4)
$S_{I,\Lambda}$ is the interaction part of an effective low energy field
theory with a UV cutoff $\Lambda$.
Let
$S_{\Lambda}[\phi]\equiv\frac{1}{2}\int_{p}\phi_{l}\Delta_{l}^{-1}\phi_{l}+S_{I,\Lambda}[\phi_{l}]$
(2.5)
be the whole action so that
$Z_{B}=\int\mathcal{D}\phi_{l}\,e^{-S_{\Lambda}[\phi_{l}]}$ (2.6)
Using (2.4), we obtain
$\displaystyle e^{-S_{\Lambda}[\phi]}$
$\displaystyle=\int\mathcal{D}\varphi\,\exp\left[-S_{B}[\varphi]+\frac{1}{2}\int_{p}\frac{p^{2}}{K(p/\Lambda_{0})}\varphi(p)\varphi(-p)-\frac{1}{2}\int_{p}\frac{p^{2}}{K(p/\Lambda)}\phi(p)\phi(-p)\right.$
$\displaystyle\left.\quad-\frac{1}{2}\int_{p}\frac{p^{2}}{K(p/\Lambda_{0})-K(p/\Lambda)}\left(\varphi(p)-\phi(p)\right)\left(\varphi(-p)-\phi(-p)\right)\right]$
(2.7)
where we have written $\phi_{l}$ as $\phi$ and $\phi_{h}$ as $\varphi-\phi$.
This will be useful later.
It is to be noted that one can go back to the bare partition function anytime
. For this reason this scheme is called “exact”, i.e. we lose no physical
information by varying the scale. It is easy to see this explicitly. Using
(2.7), we can calculate the generating functional of $S_{B}$ using
$S_{\Lambda}$ as
$\displaystyle\int\mathcal{D}\phi\,\exp\left(-S_{B}[\phi]-\int_{p}J(-p)\phi(p)\right)$
$\displaystyle=\exp\left[\frac{1}{2}\int_{p}J(p)J(-p)\frac{1}{p^{2}}\left\\{K(p/\Lambda_{0})\left(1-K(p/\Lambda_{0})\right)-\left(\frac{K(p/\Lambda_{0})}{K(p/\Lambda)}\right)^{2}K(p/\Lambda)\left(1-K(p/\Lambda)\right)\right\\}\right]$
$\displaystyle\qquad\times\int\mathcal{D}\phi\,\exp\left(-S_{\Lambda}[\phi]-\int_{p}J(-p)\frac{K(p/\Lambda_{0})}{K(p/\Lambda)}\,\phi(p)\right)$
(2.8)
We observe that the correlation functions of $S_{B}$ are the same as those of
$S_{\Lambda}$ up to the trivial (short-distance) contribution to the two-point
function and up to the momentum-dependent rescaling of the field by
$\frac{K(p/\Lambda_{0})}{K(p/\Lambda)}$ [54]. If we ignore the small
corrections to the two-point functions, we can write
$\prod_{i=1}^{n}\frac{1}{K(p_{i}/\Lambda)}\,\left\langle\phi(p_{1})\cdots\phi(p_{n})\right\rangle_{S_{\Lambda}}=\prod_{i=1}^{n}\frac{1}{K(p_{i}/\Lambda^{\prime})}\,\left\langle\phi(p_{1})\cdots\phi(p_{n})\right\rangle_{S_{\Lambda^{\prime}}}$
(2.9)
#### 2.1.2 Polchinski’s ERG equation
We have given an integral formula (2.4) for $S_{I,\Lambda}$ and (2.7) for
$S_{\Lambda}$. It is easy to derive differential equations from these. From
(2.4), we obtain Polchinski’s ERG equation
$-\Lambda\frac{\partial
S_{I,\Lambda}[\phi]}{\partial\Lambda}=\int_{p}(-)\frac{dK(p/\Lambda)}{dp^{2}}\left(-\frac{\delta
S_{I,\Lambda}[\phi]}{\delta\phi(p)}\frac{\delta
S_{I,\Lambda}[\phi]}{\delta\phi(-p)}+\frac{\delta^{2}S_{I,\Lambda}[\phi]}{\delta\phi(p)\delta\phi(-p)}\right)$
(2.10)
for $S_{I,\Lambda}$. From (2.7) we obtain
$-\Lambda\frac{\partial
S_{\Lambda}[\phi]}{\partial\Lambda}=\int_{p}\left[-2p^{2}\frac{d\ln
K(p/\Lambda)}{dp^{2}}\,\phi(p)\frac{\delta
S_{\Lambda}}{\delta\phi(p)}+\frac{dK(p/\Lambda)}{dp^{2}}\left(-\frac{\delta
S_{\Lambda}}{\delta\phi(p)}\frac{\delta
S_{\Lambda}}{\delta\phi(-p)}+\frac{\delta^{2}S_{\Lambda}}{\delta\phi(p)\delta\phi(-p)}\right)\right]$
(2.11)
for the entire Wilson action.
#### 2.1.3 The limit $\Lambda\to 0+$
In the limit $\Lambda\to 0+$ we expect $S_{\Lambda}[\phi]$ approaches
something related to the partition function. If we substitute
$\lim_{\Lambda\to 0+}K(p/\Lambda)=0$ (2.12)
into (2.7), we get
$\displaystyle\lim_{\Lambda\to
0+}e^{-S_{\Lambda}[\phi]+\frac{1}{2}\int_{p}\frac{p^{2}}{K(p/\Lambda)}\phi(p)\phi(-p)}=\lim_{\Lambda\to
0+}e^{-S_{I,\Lambda}[\phi]}$
$\displaystyle=e^{-\frac{1}{2}\int_{p}\frac{p^{2}}{K(p/\Lambda_{0})}\phi(p)\phi(-p)}\int\mathcal{D}\varphi\,\exp\left[-S_{B}[\varphi]+\int_{p}\frac{p^{2}}{K(p/\Lambda_{0})}\varphi(p)\phi(-p)\right]$
(2.13)
Hence, rewriting $\phi(p)$ by $\frac{K(p/\Lambda_{0})}{p^{2}}J(p)$, we obtain
the generating functional of the bare theory as the $\Lambda\to 0+$ limit of
$S_{I,\Lambda}$:
$\displaystyle Z_{B}[J]$
$\displaystyle\equiv\int\mathcal{D}\varphi\,\exp\left[-S_{B}[\varphi]-\int_{p}\varphi(p)J(-p)\right]$
$\displaystyle=e^{-\frac{1}{2}\int_{p}J(p)J(-p)\frac{K(p/\Lambda_{0})}{p^{2}}}\lim_{\Lambda\to
0+}\exp\left(-S_{I,\Lambda}\left[\frac{K(p/\Lambda_{0})}{p^{2}}J(p)\right]\right)$
(2.14)
#### 2.1.4 IR limit of a critical theory
For the bare theory at criticality, we expect that the correlation functions
$\left\langle\varphi(p_{1})\cdots\varphi(p_{n})\right\rangle_{B}\equiv\int\mathcal{D}\varphi\,\varphi(p_{1})\cdots\varphi(p_{n})\,e^{-S_{B}[\varphi]}$
(2.15)
to become scale invariant in the IR limit, i.e., for small momenta. To be more
precise, we can define the limit
$\mathcal{C}(p_{1},\cdots,p_{n})\equiv\lim_{t\to\infty}e^{\frac{n}{2}\left(-(D+2)+\eta\right)t}\left\langle\varphi(p_{1}e^{-t})\cdots\varphi(p_{n}e^{-t})\right\rangle_{B}$
(2.16)
where $\frac{\eta}{2}$ is the anomalous dimension.
What does this mean for $S_{\Lambda}$ in the limit $\Lambda\to 0+$? As we have
seen above, the interaction part $S_{I,\Lambda}$ becomes the generating
functional of the bare theory in this limit. Since only the IR limit of the
correlation functions are scale invariant, only the low momentum part of
$\lim_{\Lambda\to 0+}S_{I,\Lambda}$ corresponds to the scale invariant theory
defined by the IR limit (2.16).
To understand the IR limit better, we follow Wilson [11] and reformulate the
ERG trasnformation in two steps:
1. 1.
introduction of an anomalous dimension (section 2.1.5) — the anomalous
dimension is an important ingredient of the IR limit. We need to introduce an
anomalous dimension of the field within ERG.
2. 2.
introduction of a dimensionless framework (section 2.1.6) — each time we lower
the cutoff $\Lambda$ we have to rescale space to restore the same momentum
cutoff. This is necessary to realize scale invariance within ERG.
#### 2.1.5 Anomalous dimension in ERG
The cutoff dependent Wilson action $S_{\Lambda}[\phi]$ has two parts:
$S_{\Lambda}[\phi]=\frac{1}{2}\int_{p}\frac{p^{2}}{K(p/\Lambda)}\phi(p)\phi(-p)+S_{I,\Lambda}[\phi]$
(2.17)
The first term is a kinetic term, but this is not the only kinetic term; part
of the interaction quadratic in $\phi$’s also contains the kinetic term. The
normalization of $\phi$ has no physical meaning, and it is natural to
normalize the field so that $S_{I,\Lambda}$ contains no kinetic term.
To do this, we modify the ERG differential equation (2.11) by adding a number
operator [44, 55]:
$\displaystyle-\Lambda\partial_{\Lambda}S_{\Lambda}[\phi]$
$\displaystyle=\int_{p}\left(-2p^{2}\frac{d}{dp^{2}}\ln
K(p/\Lambda)\,\phi(p)\frac{\delta
S_{\Lambda}}{\delta\phi(p)}-\frac{d}{dp^{2}}K(p/\Lambda)\left\\{\frac{\delta^{2}S_{\Lambda}}{\delta\phi(p)\delta\phi(-p)}-\frac{\delta
S_{\Lambda}}{\delta\phi(p)}\frac{\delta
S_{\Lambda}}{\delta\phi(-p)}\right\\}\right)$
$\displaystyle\quad-\frac{\eta_{\Lambda}}{2}\mathcal{N}_{\Lambda}[\phi]$
(2.18)
where the number operator $\mathcal{N}_{\Lambda}[\phi]$ is defined by
$\mathcal{N}_{\Lambda}[\phi]\equiv\int_{p}\left[\phi(p)\frac{\delta
S_{\Lambda}}{\delta\phi(p)}+\frac{K(p/\Lambda)\left(1-K(p/\Lambda)\right)}{p^{2}}\left\\{\frac{\delta^{2}S_{\Lambda}}{\delta\phi(p)\delta\phi(-p)}-\frac{\delta
S_{\Lambda}}{\delta\phi(p)}\frac{\delta
S_{\Lambda}}{\delta\phi(-p)}\right\\}\right]$ (2.19)
This counts the number of fields:
$\left\langle\mathcal{N}_{\Lambda}[\phi]\,\phi(p_{1})\cdots\phi(p_{n})\right\rangle_{S_{\Lambda}}=n\left\langle\phi(p_{1})\cdots\phi(p_{n})\right\rangle_{S_{\Lambda}}$
(2.20)
(Again we are ignoring small corrections to the two-point functions.) Under
(2.18) the correlation functions change as
$\prod_{i=1}^{n}\frac{1}{K(p_{i}/\Lambda)}\,\left\langle\phi(p_{1})\cdots\phi(p_{n})\right\rangle_{S_{\Lambda}}=\left(\frac{Z_{\Lambda}}{Z_{\Lambda^{\prime}}}\right)^{\frac{n}{2}}\prod_{i=1}^{n}\frac{1}{K(p_{i}/\Lambda^{\prime})}\,\left\langle\phi(p_{1})\cdots\phi(p_{n})\right\rangle_{S_{\Lambda^{\prime}}}$
(2.21)
where $Z_{\Lambda}$ is the solution of
$-\Lambda\frac{\partial}{\partial\Lambda}Z_{\Lambda}=\eta_{\Lambda}\,Z_{\Lambda}$
(2.22)
satisfying the initial condition
$Z_{\Lambda_{0}}=1$ (2.23)
We can choose $\eta_{\Lambda}$ so that $S_{\Lambda}$ has the same kinetic term
independent of $\Lambda$. For (2.18), the integral formula (2.7) must be
changed to [54]
$\displaystyle e^{S_{\Lambda}[\phi]}$
$\displaystyle=\int\mathcal{D}\varphi\,e^{S_{0}[\varphi]}$
$\displaystyle\quad\times\exp\left[-\frac{1}{2}\int_{p}\frac{p^{2}}{\frac{1-K(p/\Lambda)}{Z_{\Lambda}K(p/\Lambda)}-\frac{1-K(p/\Lambda_{0})}{K(p/\Lambda_{0})}}\left(\frac{\varphi(p)}{K(p/\Lambda_{0})}-\frac{\phi(p)}{\sqrt{Z_{\Lambda}}\,K(p/\Lambda)}\right)\left(\frac{\varphi(-p)}{K(p/\Lambda_{0})}-\frac{\phi(-p)}{\sqrt{Z_{\Lambda}}\,K(p/\Lambda)}\right)\right]$
(2.24)
This reduces to (2.7) for $Z_{\Lambda}=1$.
#### 2.1.6 Dimensionless framework
To reach the IR limit (2.16) we must look at smaller and smaller momenta as we
lower the cutoff $\Lambda$. We can do this by measuring the momenta in units
of the cutoff $\Lambda$. At the same time we render all the dimensionful
quantities such as $\phi(p)$ dimensionless by using appropriate powers of
$\Lambda$.
We introduce a dimensionless parameter $t$ by
$\Lambda=\mu\,e^{-t}$ (2.25)
where $\mu$ is an arbitary fixed momentum scale. We then define the
dimensionless field with dimensionless momentum by
$\bar{\phi}(p)\equiv\Lambda^{\frac{D+2}{2}}\phi(p\Lambda)$ (2.26)
and define a Wilson action parametrized by $t$:
$\bar{S}_{t}[\bar{\phi}]\equiv S_{\Lambda}[\phi]$ (2.27)
We can now rewrite (2.18) for $\bar{S}_{t}$:
$\displaystyle\partial_{t}\bar{S}_{t}[\bar{\phi}]$
$\displaystyle=\int_{p}\left(-2p^{2}\frac{d}{dp^{2}}\ln
K(p)+p\cdot\partial_{p}+\frac{D+2}{2}\right)\bar{\phi}(p)\cdot\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}(p)}$
$\displaystyle\quad+\int_{p}(-)\frac{d}{dp^{2}}K(p)\,\left\\{\frac{\delta^{2}\bar{S}_{t}}{\delta\bar{\phi}(p)\delta\bar{\phi}(-p)}-\frac{\delta\bar{S}_{t}}{\delta\bar{\phi}(p)}\frac{\delta\bar{S}_{t}}{\delta\bar{\phi}(-p)}\right\\}-\frac{\eta_{t}}{2}\mathcal{N}_{t}[\bar{\phi}]$
(2.28)
where we have replaced $\eta_{\Lambda}$ by $\eta_{t}$, and
$\mathcal{N}_{t}[\bar{\phi}]\equiv\int_{p}\bar{\phi}(p)\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}(p)}+\int_{p}\frac{K(p)\left(1-K(p)\right)}{p^{2}}\left(\frac{\delta^{2}\bar{S}_{t}}{\delta\bar{\phi}(p)\delta\bar{\phi}(-p)}-\frac{\delta\bar{S}_{t}}{\delta\bar{\phi}(p)}\frac{\delta\bar{S}_{t}}{\delta\bar{\phi}(-p)}\right)$
(2.29)
is the number operator for $\bar{S}_{t}$.
Rewriting (2.21) in terms of dimensionless fields, we obtain
$\displaystyle\prod_{i=1}^{n}\frac{1}{K(p_{i})}\,\left\langle\bar{\phi}(p_{1})\cdots\bar{\phi}(p_{n})\right\rangle_{\bar{S}_{t}}$
$\displaystyle=\left(\frac{Z_{t}}{Z_{t^{\prime}}}\right)^{\frac{n}{2}}e^{-\frac{n}{2}\left(D-2\right)(t-t^{\prime})}\prod_{i=1}^{n}\frac{1}{K(p_{i}e^{-(t-t^{\prime})})}\,\left\langle\bar{\phi}(p_{1}e^{-(t-t^{\prime})})\cdots\bar{\phi}(p_{n}e^{-(t-t^{\prime})})\right\rangle_{\bar{S}_{t^{\prime}}}$
(2.30)
where $Z_{t}$ satisfies
$\partial_{t}Z_{t}=\eta_{t}\,Z_{t}$ (2.31)
(The corrections to the two-point functions are ignored.) Comparing (2.30)
with (2.16), the existence of the IR limit implies that
$\lim_{t\to\infty}\eta_{t}=\eta$ (2.32)
and
$\lim_{t\to\infty}\prod_{i=1}^{n}\frac{1}{K(p_{i})}\,\left\langle\bar{\phi}(p_{1})\cdots\bar{\phi}\right\rangle_{\bar{S}_{t}}=\mathcal{C}(p_{1},\cdots,p_{n})$
(2.33)
In other words $\bar{S}_{t}$ approaches a limit as $t\to+\infty$:
$\lim_{t\to+\infty}\bar{S}_{t}=\bar{S}_{\infty}$ (2.34)
We call $\bar{S}_{\infty}$ a fixed point because the right-hand side of (2.28)
vanishes for it:
$\displaystyle 0$ $\displaystyle=\int_{p}\left(-2p^{2}\frac{d}{dp^{2}}\ln
K(p)+p\cdot\partial_{p}+\frac{D+2}{2}\right)\bar{\phi}(p)\cdot\frac{\delta\bar{S}_{\infty}[\bar{\phi}]}{\delta\bar{\phi}(p)}$
$\displaystyle\quad+\int_{p}(-)\frac{d}{dp^{2}}K(p)\,\left\\{\frac{\delta^{2}\bar{S}_{\infty}}{\delta\bar{\phi}(p)\delta\bar{\phi}(-p)}-\frac{\delta\bar{S}_{\infty}}{\delta\bar{\phi}(p)}\frac{\delta\bar{S}_{\infty}}{\delta\bar{\phi}(-p)}\right\\}-\frac{\eta}{2}\mathcal{N}_{\infty}[\bar{\phi}]$
(2.35)
#### 2.1.7 Fixed-point equation
Instead of choosing $\eta$ dependent on $t$, we may choose $\eta$ as a
constant so that there is a non-trivial fixed-point solution
$\bar{S}_{\infty}$ for which the right-hand side of (2.28) vanishes. With a
constant anomalous dimension, the dimensionless ERG equation is given by
$\displaystyle\partial_{t}\bar{S}_{t}[\bar{\phi}]$
$\displaystyle=\int_{p}\left(-2p^{2}\frac{d}{dp^{2}}\ln
K(p)+\frac{D+2}{2}-\frac{\eta}{2}+p\cdot\partial_{p}\right)\bar{\phi}(p)\cdot\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}(p)}$
$\displaystyle\quad+\int_{p}\left(-2\frac{d}{dp^{2}}K(p)-\eta\frac{K(p)\left(1-K(p)\right)}{p^{2}}\right)\frac{1}{2}\left(\frac{\delta^{2}\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}(p)\delta\bar{\phi}(-p)}-\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}(p)}\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}(-p)}\right)$
(2.36)
For the O($N$) model with $N$ fields $\phi^{i}\,(i=1,\cdots,N)$, the ERG
equation becomes
$\displaystyle\partial_{t}\bar{S}_{t}[\bar{\phi}]$
$\displaystyle=\int_{p}\left(-2p^{2}\frac{d}{dp^{2}}\ln
K(p)+\frac{D+2}{2}-\frac{\eta}{2}+p\cdot\partial_{p}\right)\bar{\phi}^{i}(p)\cdot\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}^{i}(p)}$
$\displaystyle\quad+\int_{p}\left(-2\frac{d}{dp^{2}}K(p)-\eta\frac{K(p)\left(1-K(p)\right)}{p^{2}}\right)\frac{1}{2}\left(\frac{\delta^{2}\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}^{i}(p)\delta\bar{\phi}^{i}(-p)}-\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}^{i}(p)}\frac{\delta\bar{S}_{t}[\bar{\phi}]}{\delta\bar{\phi}^{i}(-p)}\right)$
(2.37)
where the repeated indices $i$ are summed over.
### 2.2 Energy Momentum Tensor: Scale Invariance and Conformal Invariance
#### 2.2.1 Energy Momentum Tensor in the Classical Theory
In this paper we will focus on the following Euclidean action whenever a
concrete action is required for a calculation
$S_{E}=\int d^{D}x\sqrt{g}[{1\over
2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+{1\over
2}m^{2}\phi^{2}+\frac{\lambda}{4!}\phi^{4}]$
Using
$\delta g=gg^{\mu\nu}\delta g_{\mu\nu},\quad\delta\sqrt{g}={1\over
2}\sqrt{g}g^{\mu\nu}\delta g_{\mu\nu},\quad\delta
g_{\mu\nu}=-g_{\mu\rho}\delta g^{\rho\sigma}g_{\sigma\nu}$
we get
$\delta S_{E}=-\int d^{D}x~{}{1\over 2}\delta
g_{\mu\nu}\sqrt{g}[\partial^{\mu}\phi\partial^{\nu}\phi-g^{\mu\nu}{\cal
L}]\equiv-\int d^{D}x~{}{1\over 2}\delta g_{\mu\nu}\sqrt{g}T^{\mu\nu}$ (2.38)
where
$T^{\mu\nu}\equiv-\frac{2}{\sqrt{g}}\frac{\delta S}{\delta
g_{\mu\nu}}=\partial^{\mu}\phi\partial^{\nu}\phi-g^{\mu\nu}{\cal L}$ (2.39)
One can check that
$\partial^{\nu}T_{\mu\nu}=-\partial_{\mu}\phi\left[\frac{\partial{\cal
L}}{\partial\phi}-\partial^{\rho}\left(\frac{\partial{\cal
L}}{\partial^{\rho}\phi}\right)\right]=-\partial_{\mu}\phi\frac{\delta
S_{E}}{\delta\phi}$ (2.40)
Thus, classically the energy momentum tensor is conserved on-shell.
Now we rewrite $T_{\mu\nu}$ in a form that will be useful later. Define the
traceless tensor
$t_{\mu\nu}=D\partial_{\mu}\partial_{\nu}-g_{\mu\nu}\Box~{}~{}$ (2.41)
and the transverse tensor
$\sigma_{\mu\nu}=(g_{\mu\nu}\Box-\partial_{\mu}\partial_{\nu})\phi^{2}$ (2.42)
Using the identity
$\partial_{\mu}\phi\partial_{\nu}\phi=\partial_{\mu}\partial_{\nu}{1\over
2}\phi^{2}-\phi\partial_{\mu}\partial_{\nu}\phi$
one can rewrite
$\displaystyle T_{\mu\nu}$
$\displaystyle=\frac{1}{4(D-1)}t_{\mu\nu}\phi^{2}+\frac{D-2}{4(D-1)}(\partial_{\mu}\partial_{\nu}-g_{\mu\nu}\partial^{2})\phi^{2}-\frac{1}{D}\phi
t_{\mu\nu}\phi$
$\displaystyle\quad-\frac{1}{D}g_{\mu\nu}\left[m^{2}\phi^{2}+(4-D)\frac{\lambda}{4!}\phi^{4}+\frac{D-2}{2}E\right]$
(2.43)
The trace which is proportional to $g_{\mu\nu}\frac{\delta S}{\delta
g_{\mu\nu}}$ can be written as $\frac{\partial S}{\partial t}$ when
$g_{\mu\nu}=e^{2t}\delta_{\mu\nu}$ and is the response to scale
transformations.
$T^{\mu}_{\mu}=\frac{(2-D)}{4}\Box\phi^{2}-\left[m^{2}\phi^{2}+(4-D)\frac{\lambda}{4!}\phi^{4}+\frac{D-2}{2}E\right]$
(2.44)
with
$E=\phi\frac{\delta S_{E}}{\delta\phi}$
proportional to the equation of motion. The terms proportional to $m^{2}$ and
$\lambda$ are genuine violations of scale invariance. But the first term can
be gotten rid of by defining the improved energy momentum tensor
$\Theta_{\mu\nu}=T_{\mu\nu}+\frac{D-2}{4(D-1)}\sigma_{\mu\nu}\phi^{2}$ (2.45)
which is still conserved. So in a genuinely classically scale invariant theory
with $m^{2}=0$ and $\lambda=0~{}\mathrm{or}~{}D=4$ one expects
$\Theta^{\mu}_{\mu}=\frac{2-D}{2}E$
#### 2.2.2 Trace of the Energy Momentum Tensor in the Quantum Theory:
Perturbative
When quantum corrections 333We are working in Euclidean space. So “quantum”
fluctuations are actually statistical fluctuations are included the condition
for scale invariance is modified. The trace will be defined as before
proportional to $\frac{\partial S}{\partial t}$. Before we turn to the exact
RG let us see what happens in the usual lowest order perturbation theory. Let
us start at $\Lambda_{0}$ and evolve to $\Lambda$ with $\Lambda$ close to
$\Lambda_{0}$.
$S_{\Lambda_{0}}=\int_{x}\left[{1\over
2}\partial_{\mu}\phi\partial^{\mu}\phi+{1\over
2}m_{0}^{2}\phi^{2}+\lambda_{0}\frac{\phi^{4}}{4!}\right]$ (2.46)
and
$S_{\Lambda}=\int_{x}\left[(1-\delta Z(t)){1\over
2}\partial_{\mu}\phi\partial^{\mu}\phi+{1\over 2}(m_{0}^{2}+\delta
m_{0}(t)^{2})\phi^{2}+(\lambda_{0}+\delta\lambda_{0}(t))\frac{\phi^{4}}{4!}+O(1/\Lambda)\right]$
Here $\delta Z$ is the correction to the kinetic term coming from the two loop
diagram at $\mathcal{O}(\lambda^{2})$, $\delta m_{0}^{2}\approx O(\lambda)$
and $\delta\lambda_{0}\approx\mathcal{O}(\lambda^{2})$ are the corrections
starting at one loop.
We rewrite $S_{\Lambda}$ in a suggestive way by adding and subtracting some
terms proportional to $\delta Z$:
$\displaystyle S_{\Lambda}$ $\displaystyle=\int_{x}\Big{[}{1\over
2}\partial_{\mu}\phi\partial^{\mu}\phi+{1\over 2}\underbrace{(m_{0}^{2}+\delta
m_{0}(t)^{2}+\delta
Zm_{0}^{2})}_{m^{2}(t)=m^{2}_{R}}\phi^{2}+\underbrace{(\lambda_{0}+\delta\lambda_{0}(t)+2\delta
Z\lambda_{0})}_{\lambda(t)=\lambda_{R}}\frac{\phi^{\prime
4}}{4!}+\mathcal{O}(1/\Lambda)\Big{]}$ $\displaystyle\quad-\delta
Z\underbrace{\left[{1\over 2}\partial_{\mu}\phi\partial^{\mu}\phi+{1\over
2}m_{0}^{2}\phi^{2}+2\lambda_{0}\frac{\phi^{4}}{4!}\right]}_{\phi\frac{\partial{\cal
L}}{\partial\phi}}$ (2.47)
If we think of $S_{\Lambda_{0}}$ as the bare action $S_{B}$ and $S_{\Lambda}$
as the renormalized action $S_{R}$ so that $S_{B}=S_{R}+S_{counter-term}$,
then $\lambda_{0}=\lambda_{B}$ and $\lambda(t)=\lambda_{R}$. The relation
between renormalized and bare quantities is
$\lambda_{B}=\frac{\lambda_{R}+\delta\lambda_{R}}{Z^{2}}$
Here $\delta\lambda_{R}$ is the counterterm and is chosen to cancel the
correction $\delta\lambda_{0}$ so $\delta\lambda_{R}=-\delta\lambda_{0}$. Let
us write everything in terms of $\lambda_{B}$:
$\displaystyle\lambda_{B}$
$\displaystyle=\lambda_{R}+\delta\lambda_{R}-2\delta
Z\lambda_{R}\approx\lambda_{R}+\delta\lambda_{R}-2\delta Z\lambda_{0}$
$\displaystyle\lambda_{B}+2\delta Z\lambda_{0}-\delta\lambda_{R}$
$\displaystyle=\lambda_{0}+2\delta
Z\lambda_{0}+\delta\lambda_{0}=\lambda_{R}=\lambda(t)$
Thus for small $t$:
$\lambda(t)=\lambda_{0}+\beta(\lambda_{0})t~{}~{};~{}~{}~{}m^{2}(t)=m^{2}(0)(1+\gamma_{m}t)~{}~{}~{};~{}~{}\delta
Z=-2\gamma t$
Furthermore define
$x=\bar{x}\Lambda^{-1}=\bar{x}\Lambda_{0}e^{t}$
The trace of the energy momentum tensor is given by the dependence on $t$
$\displaystyle-T^{\mu}_{~{}\mu}$ $\displaystyle=\frac{\partial
S_{\Lambda_{0}}}{\partial t}$
$\displaystyle=\Lambda^{-D}\left\\{\int_{\bar{x}}\left[{1\over
2}m_{0}^{2}\gamma_{m}(\lambda_{0})\phi^{2}+\beta(\lambda_{0})\frac{\phi^{4}}{4!}\right]+2\gamma\int_{x}{1\over
2}\phi\frac{\delta S_{\Lambda_{0}}}{\delta\phi(x)}\right.$
$\displaystyle\qquad\left.+D\int_{\bar{x}}[{1\over
2}m_{0}^{2}\phi^{2}+\lambda_{0}\frac{\phi^{4}}{4!}]+(D-2)\int_{\bar{x}}{1\over
2}\partial_{\bar{\mu}}\phi\partial^{\bar{\mu}}\phi+O(1/\Lambda_{0})]\right\\}$
(2.48)
Define dimensionless variables as
$m_{0}^{2}=\bar{m}^{2}\Lambda_{0}^{2}=\bar{m}^{2}e^{2t}\Lambda^{2}$
and
$\lambda_{0}=(\Lambda_{0})^{4-D}\bar{\lambda}_{0}=\bar{\lambda}_{0}e^{(4-D)t}(\Lambda)^{4-D}$
and fields
$\phi=(\Lambda)^{\frac{D-2}{2}}\bar{\phi}=e^{-\frac{D-2}{2}t}\Lambda_{0}^{\frac{D-2}{2}}\bar{\phi}$
Now add and subtract
$(\frac{D-2}{2})\int_{\bar{x}}\bar{\phi}\frac{\delta
S_{\Lambda_{0}}}{\delta\bar{\phi}(x)}$
to get
$-T^{\mu}_{~{}\mu}=\int_{\bar{x}}\underbrace{\Big{[}{1\over
2}\bar{m}^{2}(2+\gamma_{m}(\lambda_{0}))\bar{\phi}^{2}+(\beta(\lambda_{0})-(D-4)\lambda_{0})\frac{\bar{\phi}^{4}}{4!}\Big{]}}_{\textrm{``$\beta$-function''}}+\left(\frac{D-2}{2}+\gamma\right)\int_{\bar{x}}\bar{\phi}\frac{\delta
S_{\Lambda_{0}}}{\delta\bar{\phi}(x)}+O(1/\Lambda_{0})$ (2.49)
LHS can be identified with the trace of the energy momentum tensor in the
quantum theory and can be compared with the corresponding classical expression
in (2.44). The above gives an idea of how the quantum corrections modify
$T_{\mu\nu}$. A detailed calculation of the energy momentum tensor in the
renormalized theory in terms of composite operators and using dimensional
regularization is given in [33]. A systematic and precise treatment is
provided by ERG and is given in [35, 37] and is summarized below.
#### 2.2.3 Energy Momentum Tensor in Exact RG
We summarize the properties of the energy momentum tensor in ERG, given in
[35].
The Ward Identity almost 444up to transverse terms of the form
$\partial_{\mu}\partial_{\nu}-\Box\delta_{\mu\nu}$ that do not contribute
defines the energy momentum. Because of general coordinate invariance
$\delta
x^{\mu}=-\epsilon^{\mu}~{}~{};~{}~{}~{}~{}\phi^{\prime}(x)=\phi(x)+\epsilon^{\mu}\partial_{\mu}\phi(x)$
is equivalent to (Assume that $g_{\mu\nu}=\eta_{\mu\nu}$)
$\delta g_{\mu\nu}=\epsilon_{(\mu,\nu)}$
and
$\int{\cal D}\phi^{\prime}=\int{\cal D}\phi_{g+\delta
g}~{}~{}~{};~{}~{}~{}S[\phi,g+\delta g]=S[\phi^{\prime},g]$
Thus the following identity must hold
$Z[J]=\int{\cal
D}\phi^{\prime}e^{-S[\phi^{\prime}(x)]+\int_{x}J(x)\phi^{\prime}(x)}=\int{\cal
D}\phi_{g+\delta g}e^{-S[\phi(x),g+\delta
g]+\int_{x}J(x)(\phi(x)+\epsilon^{\mu}\partial_{\mu}\phi(x))}$
Then using the definition of the energy momentum tensor, i.e.
$Z[J=0,g+\delta g]=\int{\cal D}\phi_{g+\delta g}e^{-S[\phi,g+\delta
g]}\equiv\int{\cal D}\phi_{g}e^{-S[\phi,g]+{1\over 2}\int\sqrt{g}\delta
g_{\mu\nu}T^{\mu\nu}}$ (2.50)
we get the Ward identity
$-\partial_{\mu}\langle
T^{\mu}_{~{}\nu}(x)\phi(x_{1})...\phi(x_{n})\rangle+\sum_{i=1}^{n}\delta(x-x_{i})\langle\phi(x_{1})....\partial_{\nu}\phi(x_{i})...\phi(x_{n})\rangle=0$
(2.51)
This is a statement of the conservation of $T_{\mu\nu}$ corresponding to the
classical statement (2.40).
In ERG this can be written as a Ward identity for the composite operator
$[T_{\mu\nu}]$
$q^{\mu}[T_{\mu\nu}(q)]=\int_{p}e^{S[\phi]}K(p)(p+q)_{\nu}\frac{\delta}{\delta\phi(p)}([\phi(p+q)]e^{-S[\phi]})$
(2.52)
The equation corresponding to (2.49) and (2.44) is
$T^{\mu}_{\mu}(0)=-\frac{\partial S}{\partial t}-(\frac{D-2}{2}+\gamma){\cal
N}$ (2.53)
where $-\frac{\partial S}{\partial t}$ gives the ERG evolution, with anomalous
dimension, in terms of dimensioness variables - the “$\beta$-function”. It
vanishes at the fixed point. ${\cal N}$ is the number operator. Note that this
equation is obtained for zero momentum or as an integral over space-time in
position space. The classical analog of this is (2.44), which was obtained for
arbitrary momentum.
Note that in equations (2.52) and (2.53), both LHS and RHS are composite
operators. So one strategy will be to evaluate $T_{\mu\nu}$ using these
equations in the bare theory at some scale $\Lambda_{0}$ which will be taken
to be infinity. The bare theory is very simple so the calculations can be done
exactly. Then one can evolve $T_{\mu\nu}$ down to a scale
$\Lambda<<\Lambda_{0}$ order by order using the ERG evolution operator. If we
choose $\lambda$ and $m$ to be on the critical surface we are guaranteed that
at $\Lambda$ the theory flows to the fixed point action. Thus we will have
evaluated the energy momentum tensor at the fixed point.
Another approach is to work directly with the known fixed point action and
solve the Ward identity order by order. In this paper we follow the second
approach.
## 3 Wilson-Fisher Fixed Point for the $O(N)$ Model
We will find the fixed-point Wilson action by putting
$\frac{\partial\bar{S}_{t}}{\partial t}=0$ in (2.1.7). As we will work mostly
with dimensionless variables we will remove the bar sign from the
dimensionless variables unless otherwise mentioned. Also t dependence of
actions and fields being readily implied, the subscript t will be omitted too.
We give the fixed point action $S$ in the following form:
$S=S_{2}+S_{4}+S_{6}$
where $S_{2}$ and $S_{4}$ are given by
$\displaystyle S_{2}$
$\displaystyle=\int\frac{d^{D}p}{(2\pi)^{D}}U_{2}(p){1\over
2}\phi^{I}(p)\phi^{I}(-p)$ (3.54) $\displaystyle S_{4}$
$\displaystyle=\frac{1}{2}\prod_{i=1}^{3}\int\frac{d^{D}p_{i}}{(2\pi)^{D}}U_{4}(p_{1},p_{2};p_{3},p_{4}){1\over
2}\phi^{I}(p_{1})\phi^{I}(p_{2}){1\over 2}\phi^{J}(p_{3})\phi^{J}(p_{4})$
(3.55)
where $p_{1}+p_{2}+p_{3}+p_{4}=0$ is implied. Instead of putting an explicit
delta function and integrating over $p_{4}$ we will simply impose momentum
conservation at every stage. Accordingly $S_{6}$ is given by
$S_{6}=\frac{1}{3!}\prod_{i=1}^{5}\int\frac{d^{D}p_{i}}{(2\pi)^{D}}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6}){1\over
2}\phi^{I}(p_{1})\phi^{I}(p_{2}){1\over
2}\phi^{J}(p_{3})\phi^{J}(p_{4}){1\over 2}\phi^{K}(p_{5})\phi^{K}(p_{6})$
(3.56)
### 3.1 Equations for the vertices
We get the following equations for $U_{2}$,$U_{4}$ and $U_{6}$:
##### Equation for $U_{2}$
$\displaystyle 0=\int$
$\displaystyle\frac{d^{D}p}{(2\pi)^{D}}\Bigg{\\{}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{p^{2}}-K^{\prime}(p^{2})\bigg{)}\frac{1}{8}\bigg{[}4NU_{4}(p_{1},-p_{1};p,-p)+8U_{4}(p_{1},p;-p_{1},-p)\bigg{]}$
$\displaystyle-$
$\displaystyle\frac{1}{2!}2U_{2}(p)U_{2}(p)\delta^{D}(p-p_{1})\Bigg{\\}}+\bigg{(}\frac{-\eta}{2}+1-2\frac{p_{1}^{2}}{K(p_{1}^{2})}K^{\prime}(p_{1}^{2})\bigg{)}U_{2}(p_{1})-\frac{1}{2!}p_{1}\frac{dU_{2}(p_{1})}{dp_{1}}$
##### Equation for $U_{4}$
$\displaystyle 0=$
$\displaystyle\int\frac{d^{D}p}{(2\pi)^{D}}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{p^{2}}-K^{\prime}(p^{2})\bigg{)}\frac{1}{48}$
$\displaystyle\times\bigg{\\{}$ $\displaystyle
6NU_{6}(p_{1},p_{2};p_{3},p_{4};p,-p)+12U_{6}(p_{1},p;p_{2},-p;p_{3},p_{4})+12U_{6}(p_{1},p_{2};p_{3},p;p_{4},-p)\bigg{\\}}$
$\displaystyle-$
$\displaystyle\sum_{j=1}^{4}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{p_{j}^{2}}-K^{\prime}(p_{j}^{2})\bigg{)}U_{2}(p_{j})~{}\frac{2}{8}U_{4}(p_{1},p_{2};p_{3},p_{4})+\sum_{j=1}^{4}\bigg{(}\frac{-\eta}{2}-2\frac{p^{2}}{K(p_{j}^{2})}K^{\prime}(p_{j}^{2})\bigg{)}~{}\frac{1}{8}U_{4}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle+$
$\displaystyle\bigg{[}4-D-\sum_{i=1}^{4}p_{i}\frac{d}{dp_{i}}\bigg{]}\frac{1}{8}U_{4}(p_{1},p_{2};p_{3},p_{4})$
(3.58)
Here $p=p_{a}+p_{b}+p_{n}=-(p_{i}+p_{j}+p_{m})$.
##### Equation for $U_{6}$
$\displaystyle 0=$
$\displaystyle\frac{2}{48}\sum_{6~{}perm~{}of~{}(m,n)}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{(p_{i}+p_{j}+p_{m})^{2})}-K^{\prime}((p_{i}+p_{j}+p_{m})^{2})\bigg{)}U_{4}(p_{i},p_{j};p_{m},p)U_{4}(p_{a},p_{b};p_{n},-p)$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{6}\bigg{(}K^{\prime}(p_{j}^{2})-\frac{-\eta}{2}\frac{K(1-K)}{p_{j}^{2}}\bigg{)}U_{2}(p_{j})\frac{2}{48}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{6}\bigg{(}\frac{-\eta}{2}-2\frac{p^{2}}{K(p_{j}^{2})}K^{\prime}(p_{j}^{2})\bigg{)}\frac{1}{48}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})+\bigg{[}6-2D-\sum_{i=1}^{6}p_{i}\frac{d}{dp_{i}}\bigg{]}\frac{1}{48}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
(3.59)
### 3.2 Solving the Equations
We know that $U_{4}\approx\mathcal{O}(\epsilon)$ and
$U_{6}\approx\mathcal{O}(\epsilon^{2})$ and
$\eta\approx\mathcal{O}(\epsilon^{2})$, where $\epsilon=4-D$.
#### 3.2.1 $\mathcal{O}(1)$: Retrieving Gaussian theory
We start with (3.1) for $U_{2}$. Neglecting $U_{4}$ and $\eta$ and collecting
coefficients of $\phi^{2}$ we get
$0=K^{\prime}(p^{2})U_{2}(p)U_{2}(p)+\bigg{(}1-2\frac{p^{2}}{K(p^{2})}K^{\prime}(p^{2})\bigg{)}U_{2}(p)-p^{2}\frac{dU_{2}(p)}{dp^{2}}$
(3.60)
$U_{2}(p)=\frac{p^{2}}{K(p^{2})}$ solves this equation. This is expected since
the Gaussian theory is expected to be a fixed point - and this ERG was
obtained from Polchinski’s ERG by adding on the kinetic term ${1\over
2}\int\frac{d^{D}p}{(2\pi)^{D}}\phi(p)\frac{p^{2}}{K(p^{2})}\phi(-p)$. Thus
our solution can be written as
$\boldmath
U_{2}(p)=\frac{p^{2}}{K(p^{2})}+\underbrace{U_{2}^{(1)}(p)}_{\mathcal{O}(\epsilon)}+\mathcal{O}(\epsilon^{2})$
(3.61)
#### 3.2.2 $\mathcal{O}(\epsilon)$: Fixed Point value of $m^{2}$
We go back to (3.1) and keep $U_{4}$ which is $\mathcal{O}(\epsilon)$ but drop
$\eta$ which is $\mathcal{O}(\epsilon^{2})$.
$\displaystyle 0=\int$
$\displaystyle\frac{d^{D}p}{(2\pi)^{D}}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{p^{2}}-K^{\prime}(p^{2})\bigg{)}\times$
$\displaystyle\bigg{\\{}$
$\displaystyle\frac{1}{8}\Big{[}4NU_{4}(p_{1},p_{2};p,-p)+8U_{4}(p_{1},p;-p,-p_{1})\Big{]}-\frac{1}{2!}2U_{2}(p)U_{2}(p)\delta^{D}(p-p_{1})\bigg{\\}}$
$\displaystyle+$
$\displaystyle\bigg{(}\frac{-\eta}{2}+1-2\frac{p_{1}^{2}}{K(p_{1}^{2})}K^{\prime}(p_{1}^{2})\bigg{)}U_{2}(p_{1})-\frac{1}{2!}p_{1}\frac{dU_{2}(p_{1})}{dp_{1}}$
We use (3.61) in the above equation and look at the terms of order $\epsilon$.
To leading order we set $U_{4}=\lambda$, which is $\mathcal{O}(\epsilon)$. The
equation for $U_{2}^{(1)}$ is given by
$\displaystyle
0=-\lambda\frac{4N+8}{8}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})+2\frac{p_{1}^{2}}{K(p_{1}^{2})}U_{2}^{(1)}\bigg{(}p_{1})K^{\prime}(p_{1}^{2})+(1-2\frac{p_{1}^{2}}{K(p_{1}^{2})}K^{\prime}(p_{1}^{2})\bigg{)}U_{2}^{(1)}(p_{1})-p_{1}^{2}\frac{dU_{2}^{(1)}(p_{1})}{dp_{1}^{2}}$
To leading order this equation is solved by a constant $U_{2}^{(1)}$, i.e.
$0=-\lambda\frac{4N+8}{8}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})+U_{2}^{(1)}$
(3.63)
Thus
$\boldsymbol{U_{2}^{(1)}=\lambda\frac{N+2}{2}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})}$
(3.64)
Here
$\int\frac{d^{D}p}{(2\pi)^{D}}=\frac{1}{2^{D}\pi^{D/2}\Gamma(D/2)}\int(p^{2})^{\frac{D-2}{2}}dp^{2}$
To get leading results we can set $D=4$:
$U_{2}^{(1)}=\lambda\frac{4N+8}{8}\frac{1}{(4\pi)^{2}}\int_{0}^{\infty}dp^{2}p^{2}K^{\prime}(p^{2})=-\lambda\frac{4N+8}{8}\frac{1}{(4\pi)^{2}}\int_{0}^{\infty}dp^{2}K(p^{2})$
(3.65)
We have used $K(0)=1,K(\infty)=0$. This gives the fixed point value of the
dimensionless mass parameter:
$U_{2}^{(1)}=m_{\star}^{2}=-\lambda\frac{N+2}{2}\frac{1}{(4\pi)^{2}}\int_{0}^{\infty}dp^{2}K(p^{2})$
(3.66)
To evaluate the integral explicitly we need a specific form for $K$. We use
$K(p^{2})=e^{-p^{2}}$. Then the integral is equal to 1.
#### 3.2.3 $\mathcal{O}(\epsilon^{2})$: Expression for the six-point vertex
Let us turn to (3.1) reproduced below:
$\displaystyle 0=-$
$\displaystyle\frac{2}{48}\sum_{6~{}perm~{}of~{}(i,j,m)}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{(p_{i}+p_{j}+p_{m})^{2}}-K^{\prime}((p_{i}+p_{j}+p_{m})^{2})\bigg{)}U_{4}(p_{i},p_{j};p_{m},p)U_{4}(p_{a},p_{b};p_{n},-p)$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{6}\bigg{\\{}\bigg{(}K^{\prime}(p_{j}^{2})-\frac{-\eta}{2}\frac{K(1-K)}{p_{j}^{2}}\bigg{)}2U_{2}(p_{j})+\bigg{(}\frac{-\eta}{2}-2\frac{p^{2}}{K(p_{j}^{2})}K^{\prime}(p_{j}^{2})\bigg{)}\bigg{\\}}\frac{1}{48}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
$\displaystyle+$
$\displaystyle\bigg{[}6-2D-\sum_{i=1}^{6}p_{i}\frac{d}{dp_{i}}\bigg{]}\frac{1}{48}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
(3.67)
where $p=p_{a}+p_{b}+p_{n}=-(p_{i}+p_{j}+p_{m})$.
In this equation we keep terms of $\mathcal{O}(\epsilon^{2})$. Since $\eta$ is
$\mathcal{O}(\epsilon^{2})$, and multiplies terms of
$\mathcal{O}(\epsilon^{2})$, it contributes only at
$\mathcal{O}(\epsilon^{4})$ in this equation, so it can be dropped here.
Furthermore then, if we use the leading order solution for
$U_{2}=\frac{p^{2}}{K(p^{2})}$, the second and third terms cancel each other.
So we are left with
$\displaystyle 0=-$
$\displaystyle\frac{2}{48}\sum_{6~{}perm~{}(i,j,m)}K^{\prime}\big{(}(p_{i}+p_{j}+p_{m})^{2}\big{)}U_{4}(p_{i},p_{j};p_{n},p)U_{4}(p_{a},p_{b};p_{n},-p)$
$\displaystyle+$
$\displaystyle\bigg{[}(6-2D-\sum_{i=1}^{6}p_{i}\frac{d}{dp_{i}})\bigg{]}\frac{1}{48}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
(3.68)
Since $U_{4}=\lambda$ to this order, we obtain
$0=\lambda^{2}\frac{2}{48}\sum_{6~{}perm~{}(i,j,m)}K^{\prime}((p_{i}+p_{j}+p_{m})^{2})+\bigg{[}6-2D-\sum_{i=1}^{6}p_{i}\frac{d}{dp_{i}}\bigg{]}\frac{1}{48}U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
(3.69)
The solution for one permutation is
$U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})=\lambda^{2}\frac{K((p_{1}+p_{2}+p_{3})^{2})-K(0)}{(p_{1}+p_{2}+p_{3})^{2}}$
The full solution is given by
$\displaystyle U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})=-\lambda^{2}$
$\displaystyle\big{\\{}h(p_{1}+p_{2}+p_{3})+h(p_{1}+p_{2}+p_{4})+h(p_{1}+p_{2}+p_{5})$
$\displaystyle+$ $\displaystyle
h(p_{1}+p_{2}+p_{6})+h(p_{1}+p_{3}+p_{4})+h(p_{2}+p_{3}+p_{4})\big{\\}}$
(3.70)
where $h(x)=\frac{K(0)-K(x)}{x^{2}}$.
#### 3.2.4 Fixed Point value of $\lambda$: Solution for $U_{4}$ at
$\mathcal{O}(\epsilon)$
The $U_{4}$ equation is given by (3.1). In this equation $\eta$ can be
neglected as $-\eta\approx\mathcal{O}(\epsilon^{2})$ . Also we put the value
of $U_{2}$ upto order of $\epsilon$ found above. There is a cancellation
between the second and third terms on the R.H.S and we obtain
$\displaystyle\bigg{[}\bigg{(}4-D-\sum_{i=1}^{4}p_{i}\frac{d}{dp_{i}}\bigg{)}-\sum_{j=1}^{4}2K^{\prime}(p_{j}^{2})\frac{\lambda}{16\pi^{2}}\frac{N+2}{2}\bigg{]}\frac{1}{8}U_{4}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle=$
$\displaystyle\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})\frac{1}{48}\bigg{\\{}6NU_{6}(p_{1},p_{2};p_{3},p_{4};p,-p)+12U_{6}(p_{1},p;p_{2},-p;p_{3},p_{4})+12U_{6}(p_{1},p_{2};p_{3},p;p_{4},-p)\bigg{\\}}$
The solution is given in the Appendix (A.1). The fixed point value
$\lambda^{*}$ given below solves the above equation:
$\boldmath\lambda^{*}=(4-D)\frac{16\pi^{2}}{N+8}$ (3.72)
### 3.3 Determining Anomalous Dimension
$U_{2}$ equation at $\mathcal{O}(\epsilon^{2})$
$\displaystyle 0=\int$
$\displaystyle\bigg{\\{}\frac{d^{D}p}{(2\pi)^{D}}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{p^{2}}-K^{\prime}(p^{2})\bigg{)}\left[\frac{\delta^{2}S_{4}}{\delta\phi^{I}(p)\delta\bar{\phi}^{I}(-p)}-\frac{\delta
S_{2}}{\delta\phi^{I}(p)}\frac{\delta
S_{2}}{\delta\phi^{I}(-p)}\right]\bigg{\\}}$ $\displaystyle+\bigg{\\{}$
$\displaystyle-\frac{\eta}{2}-2\frac{p^{2}}{K(p^{2})}K^{\prime}(p^{2})\bigg{\\}}\phi(p).\frac{\delta
S}{\delta\phi(p)}+\mathcal{G}_{dil}^{c}S_{2}$
where we plug in:
$\displaystyle U_{4}(p_{1},p_{2};p_{3},p_{4})$ $\displaystyle=$
$\displaystyle\lambda+\underbrace{\tilde{U}_{4}(p_{1},p_{2};p_{3},p_{4})}_{O(\epsilon^{2})}$
$\displaystyle U_{2}(p)$ $\displaystyle=$
$\displaystyle\frac{p^{2}}{K}-\lambda\frac{N+2}{2}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})+\underbrace{\tilde{U}_{2}(p)}_{O(\epsilon^{2})}$
(3.73)
and keep only $\mathcal{O}(\epsilon^{2})$ terms in the above equation to get
$\displaystyle 0=\int$
$\displaystyle\frac{d^{D}p}{(2\pi)^{D}}\bigg{(}\frac{-\eta}{2}\frac{K(1-K)}{p^{2}}-K^{\prime}(p^{2})\bigg{)}\times$
$\displaystyle\bigg{\\{}$
$\displaystyle\frac{1}{8}\bigg{[}4N\tilde{U}_{4}(p_{1},-p_{1};p,-p)+8\tilde{U}_{4}(p_{1},p;-p,-p_{1})\bigg{]}-\frac{1}{2!}2U_{2}(p)U_{2}(p)\delta^{D}(p-p_{1})]\bigg{\\}}$
$\displaystyle+$
$\displaystyle\bigg{(}\frac{-\eta}{2}+1-2\frac{p_{1}^{2}}{K(p_{1}^{2})}K^{\prime}(p_{1}^{2})\bigg{)}U_{2}(p_{1})-p_{1}^{2}\frac{dU_{2}(p_{1})}{dp_{1}^{2}}$
(3.74)
On simplification it gives
$-\frac{-\eta}{2}\frac{(1-K)}{K}p_{1}^{2}-\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})\frac{1}{8}\bigg{[}4N\tilde{U}_{4}(p_{1},-p_{1};p,-p)+8\tilde{U}_{4}(p_{1},p_{1};-p,-p_{1})\bigg{]}+K^{\prime}(p_{1}^{2})U_{2}(p_{1})U_{2}(p_{1})$
$+\frac{-\eta}{2}\frac{p_{1}^{2}}{K}+\tilde{U}_{2}(p_{1})-p_{1}^{2}\frac{d\tilde{U}_{2}(p_{1})}{dp_{1}^{2}}=0$
(3.75)
In the L.H.S the third term will cancel with part of the second term (shown in
A.3). Also the raison d’etre for introducing $\eta$ is to ensure that
$U_{2}=p^{2}+\mathcal{O}(p^{4})$. So we let
$\tilde{U}_{2}=\mathcal{O}(p^{4})$. So The anomalous dimension is given by
$\boldmath\frac{\eta}{2}=-\frac{d}{dp_{1}^{2}}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})\frac{1}{8}\bigg{[}4N\tilde{U}_{4}^{II}(p_{1},-p_{1};p,-p)+8\tilde{U}_{4}^{II}(p_{1},p;-p_{1},-p)\bigg{]}~{}\Bigg{|}_{p_{1}^{2}=0}$
(3.76)
Here the superscript $II$ is explained in Appendix A and refers to a class of
Feynman diagrams.
$\tilde{U}_{4}$ is determined by solving (3.2.4). So using (3.76) and (A.4)
one can determine $\eta$. This is done in the Appendix (A.4). The result is of
course well known [11]:
$\frac{\eta}{2}=\lambda^{2}\frac{N+2}{4}\frac{1}{(16\pi^{2})^{2}}=\frac{N+2}{(N+8)^{2}}\frac{\epsilon^{2}}{4}$
(3.77)
Collecting results we have (we have put D=4 for $\mathcal{O}(\epsilon^{2})$
terms),
$\displaystyle\boldsymbol{U_{2}(p)=\frac{p^{2}}{K(p^{2})}-\lambda\frac{N+2}{2}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})+\tilde{U}_{2}(p)}$
(3.78)
The expression for $\tilde{U}_{2}(p)$ is given in (A.3) (also in the next
section a neater expression is presented).
$\displaystyle\boldsymbol{U_{4}(p_{1},p_{2};p_{3},p_{4})=}$
$\displaystyle\boldsymbol{(4-D)\frac{16\pi^{2}}{N+8}+\frac{(N+2)}{2}\frac{\lambda^{2}}{16\pi^{2}}\sum_{j=1}^{4}h(p_{j})}$
$\displaystyle\boldsymbol{-}$
$\displaystyle\boldsymbol{\lambda^{2}\bigg{[}(N+4)F(p_{1}+p_{2})+2F(p_{1}+p_{3})+2F(p_{1}+p_{4})\bigg{]}}$
(3.79)
where
$\displaystyle
F(p)=\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}h(q)\bigg{[}h(p+q)-h(q)\bigg{]}$
and
$\displaystyle h(p)=\frac{K(0)-K(p^{2})}{p^{2}}$ $\displaystyle
U_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})=-\lambda^{2}\bigg{\\{}$
$\displaystyle h(p_{1}+p_{2}+p_{3})+h(p_{1}+p_{2}+p_{4})+h(p_{1}+p_{2}+p_{5})$
$\displaystyle+$ $\displaystyle
h(p_{1}+p_{2}+p_{6})+h(p_{1}+p_{3}+p_{4})+h(p_{2}+p_{3}+p_{4})\bigg{\\}}$
(3.80)
and the anomalous dimension is given by
$\displaystyle\boldsymbol{\frac{\eta}{2}=\lambda^{2}\frac{N+2}{4}\frac{1}{(16\pi^{2})^{2}}=\frac{N+2}{(N+8)^{2}}\frac{\epsilon^{2}}{4}}$
(3.81)
To evaluate the integrals we have put $D=4$ and used specific form of
$K(p^{2})=e^{-p^{2}}$.
This completes the solution of the fixed point ERG equation and determination
of the eigenvalue $\eta$ corresponding to anomalous dimension up to
$O(\epsilon^{2})$. In the next section we give a slightly different approach
to obtaining the fixed point action and evaluate correlation functions.
## 4 Correlation functions
### 4.1 A more general equation
In the previous section we set $\frac{\partial S}{\partial t}=0$ and solved
the fixed point equation for the action order by order. One can also solve a
more general equation where the LHS is not set to zero but to $\frac{\partial
S}{\partial t}=\beta_{J}\frac{\partial S}{\partial\lambda_{J}}$. The
parameters can be chosen so that the beta functions are zero. This has the
effect that the equations are modifed at each order by terms of higher order.
The advantage is that the solutions are easier to write down.
We want to obtain the fixed-point Wilson action to order $\lambda^{2}$ in the
following form:
$\displaystyle S[\phi^{I}]$
$\displaystyle=\int_{p}\frac{1}{2}\phi^{I}(p)\phi^{I}(-p)\,\left(\frac{p^{2}}{K(p)}+U_{2}(p)\right)$
$\displaystyle\quad+\frac{1}{2}\int_{p_{1},p_{2},p_{3},p_{4}}\frac{1}{2}\phi^{I}(p_{1})\phi^{I}(p_{2})\frac{1}{2}\phi^{J}(p_{3})\phi^{J}(p_{4})\,\delta\left(\sum_{i=1}^{4}p_{i}\right)\,\bigg{(}\lambda+V_{4}(p_{1},p_{2};p_{3},p_{4})\bigg{)}$
$\displaystyle\quad+\frac{1}{3!}\int_{p_{1},\cdots,p_{6}}\frac{1}{2}\phi^{I}(p_{1})\phi^{I}(p_{2})\frac{1}{2}\phi^{J}(p_{3})\phi^{J}(p_{4})\frac{1}{2}\phi^{K}(p_{5})\phi^{K}(p_{6})\,\delta\left(\sum_{i=1}^{6}p_{i}\right)$
(4.82) $\displaystyle\qquad\qquad\times
V_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
As we will all vertex function in powers of $\lambda$ we have to put the
general expression for $\frac{\partial\lambda}{\partial t}$ i.e
$\displaystyle\frac{\partial\lambda}{\partial
t}=(\epsilon\lambda+\beta_{N}^{(1)}\lambda^{2})$
Where $\beta_{N}^{(1)}$ , the leading term in the beta function, is given by
$\displaystyle\beta_{N}^{(1)}=2(N+8)\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p)\frac{K((0)-K(p)}{p^{2}}\equiv-(N+8)\int_{p}f(p)h(p)$
where $f(p)=-2K^{\prime}(p^{2})$.
If we assume $V_{2}(p)=\lambda
v_{2}^{(1)}(p)+\lambda^{2}v_{2}^{(2)}(p)=\lambda
v_{2}^{(1)}(p)+\Big{(}V_{2}^{I}(p)+V_{2}^{II}(p)\Big{)}$, where
$V_{2}^{I(II)}$ is analog of $\tilde{U}_{2}^{I(II)}$ in A.3, then
$\displaystyle\frac{\partial V_{2}(p)}{\partial
t}=\Big{(}\epsilon\lambda+\beta_{N}^{(1)}\lambda^{2}\Big{)}v_{2}^{(1)}(p)+2\lambda^{2}\epsilon
v_{2}^{(2)}(p)+2\lambda^{3}\beta_{N}^{(1)}v_{2}^{(2)}(p)$
Similarly if
$V_{4}(p_{1},p_{2};p_{3},p_{4})=V_{4}^{I}(p_{1},p_{2};p_{3},p_{4})+V_{4}^{II}(p_{1},p_{2};p_{3},p_{4})$,
where $V_{4}^{I(II)}(p_{1},p_{2};p_{3},p_{4})$ is equivalent to
$\tilde{U}_{4}^{I(II)}(p_{1},p_{2};p_{3},p_{4})$ in A.2.
$\displaystyle\frac{\partial}{\partial
t}\Big{[}\lambda+V_{4}(p_{1},p_{2};p_{3},p_{4})\Big{]}=\Big{(}\epsilon\lambda+\beta_{N}^{(1)}\lambda^{2}\Big{)}+2V_{4}(p_{1},p_{2};p_{3},p_{4})\Big{(}\epsilon+\beta_{N}^{(1)}\lambda\Big{)}$
A. (3.64) is modified to
$\displaystyle\frac{1}{2}\epsilon~{}v_{2}^{(1)}(p)=-\frac{4N+8}{8}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})+v_{2}^{(1)}(p).$
gives
$\displaystyle v_{2}^{(1)}(p)=$
$\displaystyle-\frac{N+2}{2-\epsilon}\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}f(p)$
(4.83) $\displaystyle\equiv$ $\displaystyle-(N+2)v_{2}$
where $v_{2}=\frac{1}{2-\epsilon}\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}f(p)$
B. (A.2) turns into
$\displaystyle\bigg{[}\epsilon+\sum_{j=1}^{4}p_{j}\frac{d}{dp_{j}}\bigg{]}V_{4}^{II}(p_{1},p_{2};p_{3},p_{4}))$
$\displaystyle=$
$\displaystyle-2\lambda^{2}\int_{\bar{p}}K^{\prime}(p^{2})\bigg{[}(N+4)h(p_{1}+p_{2}+p)+2h(p+p_{1}+p_{3})+2h(p+p_{1}+p_{4})-(N+8)h(p)\bigg{]}$
(4.84)
If we write
$V_{4}^{II}(p_{1},p_{2};p_{3},p_{4}))=-\lambda^{2}\Big{\\{}(N+4)F(p_{1}+p_{2})+2F(p_{1}+p_{3})+2F(p_{1}+p_{4})\Big{\\}}$
the equation for $F(p)$ can be written as,
$\displaystyle\big{(}p.\partial
p+\epsilon\big{)}F(p)=\int\frac{d^{D}p}{(2\pi)^{D}}f(q)h(q+p)+\frac{1}{3}\beta^{(1)}$
(4.85)
where
$\displaystyle\frac{1}{3}\beta^{(1)}=-\int\frac{d^{D}p}{(2\pi)^{D}}f(p)h(p)$
The solution , analytic at $p=0$ is,
$\displaystyle
F(q)=\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}h(p)\Big{(}h(q+p)-h(p)\Big{)}$
(4.86)
C. Similarly (A.139a) gets modified to,
$\displaystyle\bigg{[}\epsilon+\sum_{j=1}^{4}p_{j}\frac{d}{dp_{j}}\bigg{]}\frac{1}{8}V_{4}^{I}(p_{1},p_{2};p_{3},p_{4})=\lambda^{2}(N+2)\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})\bigg{\\{}-\frac{1}{8}\sum_{j=1}^{4}h(p_{j})-\frac{1}{4(2-\epsilon)}K^{\prime}(p_{j}^{2})\bigg{\\}}$
(4.87)
whose solution is,
$\displaystyle V_{4}^{I}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle=\lambda^{2}\frac{(N+2)}{2-\epsilon}\int\frac{d^{D}p}{(2\pi)^{D}}(-K^{\prime}(p^{2}))\sum_{j=1}^{4}h(p_{j})$
(4.88)
Also
$\displaystyle\frac{1}{8}\Big{\\{}4NV_{4}^{I}(p_{1},-p_{1};p,-p)+8V_{4}^{I}(p,p_{1};-p,-p_{1})\Big{\\}}$
$\displaystyle=\frac{(N+2)^{2}}{2-\epsilon}\lambda^{2}\int\frac{d^{D}q}{(2\pi)^{D}}(-K^{\prime}(q^{2}))\Big{[}h(p_{1})+h(p)\Big{]}$
D. (3.75) turns into,
$\displaystyle(2-2\epsilon)V_{2}^{I}-2p_{1}^{2}\frac{dV_{2}^{I}(p_{1})}{dp_{1}^{2}}=~{}$
$\displaystyle-\frac{2\lambda^{2}}{2-\epsilon}(N+2)^{2}\bigg{\\{}\int\frac{d^{D}p}{(2\pi)^{D}}(-K^{\prime}(p^{2}))\bigg{\\}}^{2}h(p_{1})-2\big{(}v_{2}^{(1)}\big{)}^{2}K^{\prime}(p_{1}^{2})$
(4.89)
The solution is
$\displaystyle
V_{2}^{I}(p_{1})=-(N+2)^{2}\lambda^{2}\frac{1}{(2-\epsilon)^{2}}\frac{1}{4}\bigg{\\{}\int\frac{d^{D}p}{(2\pi)^{D}}f(p)\bigg{\\}}^{2}h(p_{1})$
E. (A.3) changes to
$\displaystyle\Big{(}-2+2\epsilon\Big{)}V_{2}^{II}(p_{1})+\beta_{N}^{(1)}\lambda^{2}v_{2}^{(1)}(p)+2p_{1}^{2}\frac{dV_{2}^{II}(p_{1})}{dp_{1}^{2}}$
$\displaystyle=$
$\displaystyle-3\lambda^{2}(N+2)\int_{r,p}(-K^{\prime}(p^{2}))h(r)\Big{[}h(p_{1}+p+r)-h(r)\Big{]}+\frac{2}{2-\epsilon}\Big{[}(N+2)^{2}\lambda^{2}\int(-K^{\prime}(q^{2}))\Big{]}\int_{p}(-K^{\prime}(p^{2}))h(p)-\eta~{}p_{1}^{2}$
If we assume
$\displaystyle V_{2}^{II}(p)=-3\lambda^{2}(N+2)G(p)$
Then $G(p)$ satisfies the following equation,
$(p.\partial p-2+2\epsilon)G(p)=\int
f(q)F(p+q)+\frac{2v_{2}}{3}\int_{p}f(p)h(p)+\eta^{(2)}p^{2}$ (4.90)
From (3.76) we get $\eta=3(N+2)\lambda^{2}\eta^{(2)}$ where,
$\displaystyle\eta^{(2)}=-\frac{d}{dp^{2}}\int f(q)F(q+p)\Big{|}_{p=0}$
The solution , analytic at $p=0$ is
$\displaystyle G(p)=\frac{1}{3}\int
h(q)(F(p+q)-F(q))+\frac{1}{\epsilon}\frac{\eta^{(2)}}{2}p^{2}-\frac{1}{2-2\epsilon}\Bigg{(}\int
f(q)F(q)+\frac{2v_{2}}{3}\int_{p}f(p)h(p)\Bigg{)}$ (4.91)
$V_{2}^{I}(p)+V_{2}^{II}(p)$ when calculated in the limit $\epsilon\rightarrow
0$ gives the expression of $\tilde{U}_{2}(p)$ mentioned in the previous
section.
The solutions are given by,
$\displaystyle\boldsymbol{V_{2}(p)}$
$\displaystyle\boldsymbol{=-\lambda(N+2)v_{2}-\lambda^{2}\left(3(N+2)G(p)+(N+2)^{2}\left(v_{2}\right)^{2}h(p)\right)}$
(4.92a) $\displaystyle\boldsymbol{V_{4}(p_{1},p_{2};p_{3},p_{4})}$
$\displaystyle\boldsymbol{=-\lambda^{2}\Big{(}(N+4)F(p_{1}+p_{2})+2F(p_{1}+p_{3})+2F(p_{1}+p_{4})}$
$\displaystyle\qquad\qquad\boldsymbol{-(N+2)v_{2}\sum_{i=1}^{4}h(p_{i})}\Big{)}$
(4.92b) $\displaystyle\boldsymbol{V_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})}$
$\displaystyle\boldsymbol{=-\lambda^{2}}\left(\boldsymbol{h(p_{1}+p_{2}+p_{3})+h(p_{1}+p_{2}+p_{4})+h(p_{1}+p_{2}+p_{5})}\right.$
$\displaystyle\qquad\left.\boldsymbol{+h(p_{1}+p_{2}+p_{6})+h(p_{3}+p_{4}+p_{1})+h(p_{3}+p_{4}+p_{2})}\right)$
(4.92c)
where
$f(p)=-2K^{\prime}(p^{2});~{}h(p)=\frac{K(0)-K(p^{2})}{p^{2}}$
and
$\boldsymbol{v_{2}=\frac{1}{2-\epsilon}\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}f(p)}$
(4.93)
If we take the limit $\epsilon\rightarrow 0$ and $K(p^{2})=e^{-p^{2}}$ we get
$\displaystyle
v_{2}=\frac{1}{2}\int\frac{d^{4}p}{(2\pi)^{4}}e^{-p^{2}}=\frac{1}{2}\frac{1}{16\pi^{2}}$
$F(p)=\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}h(q)\Big{[}h(p+q)-h(q)\Big{]}$
The coupling constant $\lambda$ is given, to order $\epsilon=4-D$, as
$\boldsymbol{\lambda=\frac{\epsilon}{-\beta_{N}^{(1)}}=\frac{(4\pi)^{2}}{N+8}}\,\epsilon$
(4.94)
The anomalous dimension is given, to order $\epsilon^{2}$, as
$\boldsymbol{\eta=\frac{N+2}{2(N+8)^{2}}}\,\boldsymbol{\epsilon^{2}}$ (4.95)
### 4.2 Calculation of correlation functions
In this section we will calculate two-, four-, and six-point correlation
functions. Recall that our Wilson action has a fixed momentum cutoff of order
$1$. If we consider the momenta much larger than the cutoff, the vertices of
the Wilson action gives the correlation functions [36]. We first rescale the
field
$J^{I}(p)\equiv\frac{1}{h(p)}\phi^{I}(p)$ (4.96)
and define
$W[J^{I}]\equiv-S[\phi^{I}]+\frac{1}{2}\int_{p}J^{I}(p)J^{I}(-p)\frac{h(p)}{K(p)}$
(4.97)
For our Wilson action, this is given by
$\displaystyle W[J^{I}]$
$\displaystyle=\int_{p}\frac{1}{2}J^{I}(p)J^{I}(-p)\,h(p)^{2}\left(\frac{1}{h(p)}-V_{2}(p)\right)$
$\displaystyle\quad+\frac{1}{2}\int_{p_{1},p_{2},p_{3},p_{4}}\frac{1}{2}J^{I}(p_{1})J^{I}(p_{2})\frac{1}{2}J^{J}(p_{3})J^{J}(p_{4})\,\delta\left(\sum_{i=1}^{4}p_{i}\right)$
$\displaystyle\qquad\qquad\times\prod_{i=1}^{4}h(p_{i})\,\cdot\left(-\lambda-
V_{4}(p_{1},p_{2};p_{3},p_{4})\right)$
$\displaystyle\quad+\frac{1}{3!}\int_{p_{1},\cdots,p_{6}}\frac{1}{2}J^{I}(p_{1})J^{I}(p_{2})\frac{1}{2}J^{J}(p_{3})J^{J}(p_{4})\frac{1}{2}J^{K}(p_{5})J^{K}(p_{6})\,\delta\left(\sum_{i=1}^{6}p_{i}\right)$
$\displaystyle\qquad\qquad\times\prod_{i=1}^{6}h(p_{i})\,\cdot(-)V_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
(4.98)
In the high momentum limit we obtain the generating functional of the
connected correlation functions
$\mathcal{W}[J^{I}]=\lim_{t\to+\infty}W[J_{t}^{I}]$ (4.99)
where
$J_{t}^{I}(p)\equiv\exp\left(-t\frac{D-2+\eta}{2}\right)J^{I}(pe^{-t})$
(4.100)
In our case we obtain
$\displaystyle W[J_{t}^{I}]$
$\displaystyle=\int_{p}\frac{1}{2}J^{I}(p)J^{I}(-p)\,\exp\left(t(2-\eta)\right)h(pe^{t})^{2}\left(\frac{1}{h(pe^{t})}-V_{2}(pe^{t})\right)$
$\displaystyle\quad+\frac{1}{2}\int_{p_{1},p_{2},p_{3},p_{4}}\frac{1}{2}J^{I}(p_{1})J^{I}(p_{2})\frac{1}{2}J^{J}(p_{3})J^{J}(p_{4})\,\delta\left(\sum_{i=1}^{4}p_{i}\right)$
$\displaystyle\qquad\quad\times\exp\left(t(D+4-2\eta\right)\prod_{i=1}^{4}h(p_{i}e^{t})\,\cdot\left(-\lambda-
V_{4}(p_{1}e^{t},p_{2}e^{t};p_{3}e^{t},p_{4}e^{t})\right)$
$\displaystyle\quad+\frac{1}{3!}\int_{p_{1},\cdots,p_{6}}\frac{1}{2}J^{I}(p_{1})J^{I}(p_{2})\frac{1}{2}J^{J}(p_{3})J^{J}(p_{4})\frac{1}{2}J^{K}(p_{5})J^{K}(p_{6})\,\delta\left(\sum_{i=1}^{6}p_{i}\right)$
$\displaystyle\qquad\quad\times\exp\left(t(2D+6-3\eta\right)\prod_{i=1}^{6}h(p_{i}e^{t})\,\cdot(-)V_{6}(p_{1}e^{t},p_{2}e^{t};p_{3}e^{t},p_{4}e^{t};p_{5}e^{t},p_{6}e^{t})$
(4.101)
In the limit $t\to+\infty$ we obtain
$\displaystyle\mathcal{W}[J^{I}]$
$\displaystyle=\int_{p}\frac{1}{2}J^{I}(p)J^{I}(-p)\,C_{2}(p)$
$\displaystyle\quad+\frac{1}{2}\int_{p_{1},p_{2},p_{3},p_{4}}\frac{1}{2}J^{I}(p_{1})J^{I}(p_{2})\frac{1}{2}J^{J}(p_{3})J^{J}(p_{4})\,\delta\left(\sum_{i=1}^{4}p_{i}\right)\,C_{4}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle\quad+\frac{1}{3!}\int_{p_{1},\cdots,p_{6}}\frac{1}{2}J^{I}(p_{1})J^{I}(p_{2})\frac{1}{2}J^{J}(p_{3})J^{J}(p_{4})\frac{1}{2}J^{K}(p_{5})J^{K}(p_{6})\,\delta\left(\sum_{i=1}^{6}p_{i}\right)$
$\displaystyle\qquad\qquad\times C_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
(4.102)
#### 4.2.1 Two-point function
$\displaystyle C_{2}(p)$
$\displaystyle=\lim_{t\to+\infty}\exp\left(t(2-\eta)\right)h(pe^{t})^{2}\left(\frac{1}{h(pe^{t})}-V_{2}(pe^{t})\right)$
$\displaystyle=\lim_{t\to+\infty}\frac{1}{(p^{2})^{2}}\left[p^{2}(1-\eta\,t)+\lambda^{2}3(N+2)e^{-2t}G(pe^{t})\right]$
(4.103)
Using
$G(pe^{t})\overset{t\to\infty}{\longrightarrow}p^{2}e^{2t}\frac{1}{12(4\pi)^{4}}\ln\left(p^{2}e^{2t}\right)$
(4.104)
we obtain
$\boxed{C_{2}(p)=\frac{1}{p^{2}}\left(1+\frac{\eta}{2}\ln
p^{2}\right)=\frac{1}{p^{2-\eta}}}$ (4.105)
#### 4.2.2 Four-point function
$\displaystyle C_{4}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle=\lim_{t\to+\infty}\exp\left(t(D+4-2\eta)\right)\prod_{i=1}^{4}h(p_{i}e^{t})\,\cdot\left(-\lambda-
V_{4}(p_{1}e^{t},p_{2}e^{t};p_{3}e^{t},p_{4}e^{t})\right)$
$\displaystyle=\prod_{i=1}^{4}\frac{1}{p_{i}^{2}}\lim_{t\to+\infty}\left(1-\epsilon\,t\right)\Big{[}-\lambda$
$\displaystyle\qquad+\lambda^{2}\left((N+4)F\left((p_{1}+p_{2})e^{t}\right)+2F\left((p_{1}+p_{3})e^{t}\right)+2F\left((p_{2}+p_{3})e^{t}\right)\right)\Big{]}$
(4.106)
Using
$F(pe^{t})\overset{t\to+\infty}{\longrightarrow}-\frac{1}{(4\pi)^{2}}\ln\left(pe^{t}\right)$
(4.107)
we obtain
$\prod_{i=1}^{4}p_{i}^{2}\cdot
C_{4}(p_{1},p_{2};p_{3},p_{4})=-\lambda\left(1+\epsilon\frac{1}{N+8}\ln\left\\{(p_{1}+p_{2})^{N+4}(p_{1}+p_{3})^{2}(p_{2}+p_{3})^{2}\right\\}\right)$
(4.108)
#### 4.2.3 Six-point function
Since $V_{6}$ is already of order $\lambda^{2}$, we can take $D=4$ and
$\eta=0$ to obtain
$\displaystyle C_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},p_{6})$
$\displaystyle=\lim_{t\to+\infty}e^{t\left(2D+6-3\eta\right)}\prod_{i=1}^{6}h(p_{i}e^{t})\,(-)V_{6}(p_{1}e^{t},p_{2}e^{t};p_{3}e^{t},p_{4}e^{t};p_{5}e^{t},p_{6}e^{t})$
$\displaystyle=\lim_{t\to+\infty}e^{14t}\prod_{i=1}^{6}\frac{1}{p_{i}^{2}e^{2t}}\cdot\lambda^{2}\left(h\left((p_{1}+p_{2}+p_{3})e^{t}\right)+\cdots\right)$
$\displaystyle=\lambda^{2}\prod_{i=1}^{6}\frac{1}{p_{i}^{2}}\left(\frac{1}{(p_{1}+p_{2}+p_{3})^{2}}+\cdots+\frac{1}{(p_{3}+p_{4}+p_{2})^{2}}\right)$
(4.109)
## 5 Construction of the energy-momentum tensor at the fixed point
Given a fixed-point Wilson action, we wish to construct the energy-momentum
tensor $\Theta_{\mu\nu}(p)$. It is a symmetric tensor implicitly determined by
the Ward identity
$p_{\mu}\Theta_{\mu\nu}(p)=e^{S}\int_{q}K(q)(q+p)_{\nu}\frac{\delta}{\delta\phi^{I}(q)}\left(\left[\phi^{I}(q+p)\right]\,e^{-S}\right)$
(5.110)
where
$\left[\phi^{I}(p)\right]\equiv\frac{1}{K(p)}\left(\phi^{I}(p)-\frac{K(p)\left(1-K(p)\right)}{p^{2}}\frac{\delta
S}{\delta\phi^{I}(-p)}\right)$ (5.111)
is the composite operator corresponding to $\phi^{I}(p)$. The Ward identity
leaves an additive ambiguity of the form
$\left(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu}\right)\mathcal{O}(p)$
where $\mathcal{O}(p)$ is a scalar composite operator. Since $\Theta_{\mu\nu}$
must have zero scale dimension, $\mathcal{O}$ must have scale dimension $-2$.
There is no such $\mathcal{O}$, since the squared mass operator
$\frac{1}{2}\phi^{2}$ acquires a positive anomalous dimension at the fixed
point. Hence, the Ward identity determines $\Theta_{\mu\nu}$ unambiguously. In
fact we are going to calculate $\Theta_{\mu\nu}(p)$ only at $p=0$; we need not
worry about this ambiguity anyway.
It is convenient to expand $\Theta_{\mu\nu}(p)$ in powers of
$\left[\phi^{I}\right]$:
$\displaystyle\Theta_{\mu\nu}(p)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\int_{p_{1},\cdots,p_{2n}}\prod_{i=1}^{n}\frac{1}{2}\left[\phi^{I_{i}}(p_{2i-1})\right]\left[\phi^{I_{i}}(p_{2i})\right]\,\delta\left(\sum_{i=1}^{2n}p_{i}-p\right)$
$\displaystyle\quad\times c_{\mu\nu,2n}(p_{1},p_{2};\cdots;p_{2n-1},p_{2n})$
(5.112)
To order $\lambda^{2}$, we only have three coefficients
$c_{\mu\nu,0},c_{\mu\nu,2},c_{\mu\nu,4}$. Since the field-independent term
($n=0$) is proportional to $\delta(p)$, we cannot determine $c_{\mu\nu,0}$
from the Ward identity. So, we will determine only $c_{\mu\nu,2}$ and
$c_{\mu\nu,4}$.
From (4.1), we obtain
$\displaystyle\left[\phi^{I}(p)\right]=\phi^{I}(p)-h(p)\Big{\\{}V_{2}(p)\phi^{I}(p)$
$\displaystyle\quad+\int_{p_{1},p_{2},p_{3}}\frac{1}{2}\phi^{J}(p_{1})\phi^{J}(p_{2})\phi^{I}(p_{3})\,\delta\left(\sum_{i=1}^{3}p_{i}-p\right)\,\left(\lambda+V_{4}(p_{1},p_{2};p_{3},-p)\right)$
$\displaystyle\quad+\frac{1}{2}\int_{p_{1},\cdots,p_{5}}\frac{1}{2}\phi^{J}(p_{1})\phi^{J}(p_{2})\frac{1}{2}\phi^{K}(p_{3})\phi^{K}(p_{4})\phi^{I}(p_{5})\,\delta\left(\sum_{i=1}^{5}p_{i}-p\right)$
$\displaystyle\qquad\qquad\qquad\times
V_{6}(p_{1},p_{2};p_{3},p_{4};p_{5},-p)\Big{\\}}$ (5.113)
Inverting this we obtain, to order $\lambda^{2}$,
$\displaystyle\phi^{I}(p)=\left[\phi^{I}(p)\right]+h(p)\Big{\\{}V_{2}^{1PI}(p)\left[\phi^{I}(p)\right]$
$\displaystyle\quad+\int_{p_{1},p_{2},p_{3}}\frac{1}{2}\left[\phi^{J}(p_{1})\right]\left[\phi^{J}(p_{2})\right]\left[\phi^{I}(p_{3})\right]\,\delta\left(\sum_{i=1}^{3}p_{i}-p\right)\,\left(\lambda+V_{4}^{1PI}(p_{1},p_{2};p_{3},-p)\right)\Big{\\}}$
(5.114)
where we have defined the 1PI vertices as
$\displaystyle V_{2}^{1PI}(p)$
$\displaystyle=-\lambda(N+2)v_{2}-\lambda^{2}3(N+2)G(p)$ (5.115a)
$\displaystyle V_{4}^{1PI}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle=-\lambda^{2}\left((N+4)F(p_{1}+p_{2})+2F(p_{1}+p_{3})+2F(p_{1}+p_{4})\right)$
(5.115b)
Note that $\phi^{I}$ has no sixth order term expanded in $\left[\phi\right]$’s
to order $\lambda^{2}$.
The rhs of (5.110) gives
$\displaystyle
e^{S}\int_{q}K(q)(q+p)_{\nu}\frac{\delta}{\delta\phi^{I}(q)}\left(\left[\phi^{I}(q+p)\right]e^{-S}\right)$
$\displaystyle=\int_{q}K(q)(q+p)_{\nu}\left(-\left[\phi^{I}(q+p)\right]\frac{\delta
S}{\delta\phi^{I}(q)}+\frac{\delta}{\delta\phi^{I}(q)}\left[\phi^{I}(q+p)\right]\right)$
(5.116)
Expanding this in powers of $\left[\phi\right]$’s, we obtain from (5.110) the
following equations that determine the coefficients $c_{\mu\nu,2}$ and
$c_{\mu\nu,4}$.
$\displaystyle
p_{\mu}c_{\mu\nu,2}(p_{1},p_{2})=-p_{1\nu}p_{2}^{2}-p_{2\nu}p_{1}^{2}$
$\displaystyle\quad+\lambda(N+2)\left(v_{2}p_{\nu}-\int_{q}(q+p)_{\nu}R(q)h(q)h(q+p)\right)$
$\displaystyle\quad+\lambda^{2}(N+2)\Big{[}3\left(p_{1\nu}G(p_{2})+p_{2\nu}G(p_{1})\right)$
$\displaystyle\qquad-(N+2)v_{2}\int_{q}(q+p)_{\nu}R(q)h(q)h(q+p)\left(h(q)+h(q+p)\right)$
$\displaystyle\qquad+\frac{1}{2}\int_{q}\left\\{(q+p)_{\nu}R(q)-q_{\nu}R(q+p)\right\\}h(q)h(q+p)$
$\displaystyle\qquad\qquad\quad\times\left\\{(N+2)F(p)+3F(q+p_{1})+3F(q+p_{2})\right\\}\Big{]}$
(5.117a) and $\displaystyle
p_{\mu}c_{\mu\nu,4}(p_{1},p_{2};p_{3},p_{4})=-\lambda p_{\nu}$
$\displaystyle\quad+\lambda^{2}\Big{\\{}(N+4)\left(F(p_{1}+p_{2})(p_{3}+p_{4})_{\nu}+F(p_{3}+p_{4})(p_{1}+p_{2})_{\nu}\right)$
$\displaystyle\qquad+2p_{1\nu}\left(F(p_{2}+p_{3})+F(p_{2}+p_{4})\right)+2p_{2\nu}\left(F(p_{2}+p_{3})+F(p_{2}+p_{4})\right)$
$\displaystyle\qquad+2p_{3\nu}\left(F(p_{4}+p_{1})+F(p_{4}+p_{2})\right)+2p_{4\nu}\left(F(p_{3}+p_{1})+F(p_{3}+p_{2})\right)\Big{\\}}$
$\displaystyle\quad+\lambda^{2}\frac{1}{2}\int_{q}\left\\{(q+p)_{\nu}R(q)-q_{\nu}R(q+p)\right\\}h(q)h(q+p)$
$\displaystyle\,\times\left\\{(N+4)\left(h(q+p_{1}+p_{2})+h(q+p_{3}+p_{4})\right)+4\left(h(q+p_{1}+p_{3})+h(q+p_{1}+p_{4})\right)\right\\}$
(5.117b)
To determine $c_{\mu\nu,2}(p_{1},p_{2})$ at $p=0$, we substitute
$p_{2}=p-p_{1}$ into the rhs of (5.117a), and expand the result to first order
in $p$. This gives
$\displaystyle c_{\mu\nu,2}(p_{1},-p_{1})$
$\displaystyle=-p_{1}^{2}\delta_{\mu\nu}+2p_{1\mu}p_{1\nu}$
$\displaystyle\quad+\lambda(N+2)\delta_{\mu\nu}\left\\{v_{2}-\int_{q}R(q)\left(h(q)^{2}+\frac{1}{D}h(q)q\cdot\partial_{q}h(q)\right)\right\\}$
$\displaystyle\quad+\lambda^{2}(N+2)\Big{\\{}3\left(\delta_{\mu\nu}G(p_{1})-2p_{1\mu}p_{1\nu}G^{\prime}(p_{1})\right)$
$\displaystyle\qquad+\int_{q}\left(\delta_{\mu\nu}R(q)-2q_{\mu}q_{\nu}R^{\prime}(q)\right)h(q)^{2}\left(-(N+2)v_{2}h(q)+3F(q+p_{1})\right)\Big{\\}}$
(5.118)
Similarly, substituting $p_{4}=p-(p_{1}+p_{2}+p_{3})$ into the rhs of (5.117b)
and expanding the result to first order in $p$, we obtain
$\displaystyle
c_{\mu\nu,4}(p_{1},p_{2};p_{3},-(p_{1}+p_{2}+p_{3}))=-\lambda\delta_{\mu\nu}$
$\displaystyle\quad+\lambda^{2}\Big{\\{}(N+4)\left(\delta_{\mu\nu}F(p_{1}+p_{2})-2(p_{1}+p_{2})_{\mu}(p_{1}+p_{2})_{\nu}F^{\prime}(p_{1}+p_{2})\right)$
$\displaystyle\qquad+2\left(\delta_{\mu\nu}F(p_{1}+p_{3})-2(p_{1}+p_{3})_{\mu}(p_{1}+p_{3})_{\nu}F^{\prime}(p_{1}+p_{3})\right)$
$\displaystyle\qquad+2\left(\delta_{\mu\nu}F(p_{2}+p_{3})-2(p_{2}+p_{3})_{\mu}(p_{2}+p_{3})_{\nu}F^{\prime}(p_{2}+p_{3})\right)$
$\displaystyle\qquad+\int_{q}\left(\delta_{\mu\nu}R(q)-2q_{\mu}q_{\nu}R^{\prime}(q)\right)h(q)^{2}$
$\displaystyle\qquad\quad\times\left((N+4)h(q+p_{1}+p_{2})+2h(q+p_{1}+p_{3})+2h(q+p_{2}+p_{3})\right)\Big{\\}}$
(5.119)
### Check of the trace anomaly
Using the energy-momentum tensor obtained above, we can verify the trace
anomaly
$\Theta(0)=-\left(\frac{D-2}{2}+\frac{1}{2}\eta\right)\mathcal{N}(0)$ (5.120)
where the anomalous dimension is given by (4.95) to order $\epsilon^{2}$.
The trace is easily obtained from (5.118, 5.119) as
$\displaystyle\Theta(0)$
$\displaystyle=\int_{p}\frac{1}{2}\left[\phi^{I}(p)\right]\left[\phi^{I}(-p)\right]\Bigg{[}-(D-2)p^{2}$
$\displaystyle\quad+\lambda(N+2)D\left\\{v_{2}-\int_{q}R(q)\left(h(q)^{2}+\frac{1}{D}h(q)q\cdot\partial_{q}h(q)\right)\right\\}$
$\displaystyle\quad+\lambda^{2}(N+2)\Big{\\{}3(D-p\cdot\partial_{p})G(p)$
$\displaystyle\qquad+\int_{q}\left(D-q\cdot\partial_{q}\right)R(q)\cdot
h(q)^{2}\left(-(N+2)v_{2}+3F(q+p)\right)\Big{\\}}\Bigg{]}$
$\displaystyle\quad+\frac{1}{2}\int_{p_{1},\cdots,p_{4}}\frac{1}{2}\left[\phi^{I}(p_{1})\right]\left[\phi^{I}(p_{2})\right]\frac{1}{2}\left[\phi^{J}(p_{3})\right]\left[\phi^{J}(p_{4})\right]\,\delta\left(\sum_{i=1}^{4}p_{i}\right)$
$\displaystyle\qquad\times\Bigg{[}-\lambda D$
$\displaystyle\qquad\quad+\lambda^{2}\Big{\\{}(N+4)\left(D-p\cdot\partial_{p}\right)F(p)\Big{|}_{p=p_{1}+p_{2}}$
$\displaystyle\qquad\qquad\quad+2\left(D-p\cdot\partial_{p}\right)F(p)\Big{|}_{p=p_{1}+p_{3}}+2\left(D-p\cdot\partial_{p}\right)F(p)\Big{|}_{p=p_{2}+p_{3}}$
$\displaystyle\qquad\qquad+\int_{q}(D-q\cdot\partial_{q})R(q)\cdot h(q)^{2}$
$\displaystyle\qquad\qquad\quad\times\left((N+4)h(q+p_{1}+p_{2})+2h(q+p_{1}+p_{3})+2h(q+p_{2}+p_{3})\right)\Big{\\}}\Bigg{]}$
(5.121)
On the other hand the number operator, defined by
$\mathcal{N}(0)\equiv-e^{S}\int_{q}K(q)\frac{\delta}{\delta\phi^{I}(q)}\left(\left[\phi^{I}(q)\right]e^{-S}\right)\,,$
(5.122)
is calculated as
$\displaystyle\mathcal{N}(0)$
$\displaystyle=\int_{p}\frac{1}{2}\left[\phi^{I}(p)\right]\left[\phi^{I}(-p)\right]\Big{[}2p^{2}+(N+2)\lambda\left(-2v_{2}+\int_{q}R(q)h(q)^{2}\right)$
$\displaystyle\quad+\lambda^{2}(N+2)\left\\{-6G(p)+2(N+2)v_{2}\int_{q}R(q)h(q)^{3}-6\int_{q}R(q)h(q)^{2}F(q+p)\right\\}\Big{]}$
$\displaystyle\quad+\frac{1}{2}\int_{p_{1},\cdots,p_{4}}\frac{1}{2}\left[\phi^{I}(p_{1})\right]\left[\phi^{I}(p_{2})\right]\frac{1}{2}\left[\phi^{J}(p_{3})\right]\left[\phi^{J}(p_{4})\right]\,\delta\left(\sum_{i=1}^{4}p_{i}\right)$
$\displaystyle\qquad\times\Big{[}4\lambda-4\lambda^{2}\left\\{(N+4)F(p_{1}+p_{2})+2F(p_{1}+p_{3})+2F(p_{2}+p_{3})\right\\}$
$\displaystyle\qquad\quad-2\lambda^{2}\int_{q}R(q)h(q)^{2}\big{\\{}(N+4)h(p+p_{1}+p_{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+2h(p+p_{1}+p_{3})+2h(p+p_{2}+p_{3})\big{\\}}\Big{]}$
(5.123)
Using
$f(q)=\left(q\cdot\partial_{q}+2\right)h(q)=(2-q\cdot\partial_{q})R(q)\cdot
h(q)^{2}$ (5.124)
and the equations satisfied by $F$ and $G$
$\displaystyle\left(p\cdot\partial_{p}+\epsilon\right)F(p)$
$\displaystyle=\int_{q}f(q)\cdot\left(h(q+p)-h(q)\right)$ (5.125a)
$\displaystyle\left(p\cdot\partial_{p}-2+2\epsilon\right)G(p)$
$\displaystyle=\frac{2}{3}v_{2}\int_{q}f(q)\cdot
h(q)+\eta\,p^{2}+\int_{q}f(q)\cdot F(q+p)$ (5.125b)
we obtain
$\displaystyle\Theta(0)+\left(\frac{D-2}{2}+\gamma_{N}^{(2)}\lambda^{2}\right)\mathcal{N}(0)$
$\displaystyle=\left(\epsilon\lambda+\beta_{N}^{(1)}\lambda^{2}\right)\Bigg{[}\int_{p}\frac{1}{2}\left[\phi^{I}(p)\right]\left[\phi^{I}(-p)\right]\,(N+2)v_{2}$
$\displaystyle\quad-\frac{1}{2}\int_{p_{1},p_{2},p_{3},p_{4}}\frac{1}{2}\left[\phi^{I}(p_{1})\right]\left[\phi^{I}(p_{2})\right]\frac{1}{2}\left[\phi^{J}(p_{3})\right]\left[\phi^{J}(p_{4})\right]\,\delta\left(\sum_{i=1}^{4}p_{i}\right)\Bigg{]}$
(5.126)
where we have dropped $\epsilon\lambda^{2}G(p)$ and $\epsilon\lambda^{2}F(p)$,
which are terms of order $\epsilon^{3}$. This vanishes at the fixed point,
where
$\epsilon\lambda+\beta_{N}^{(1)}\lambda^{2}=0,$
to order $\epsilon^{2}$.
### Correlation functions
In the previous section we saw how the fixed-point Wilson action gives the
correlation functions. Similarly, the coefficient functions
$c_{\mu\nu,2}(p_{1},p_{2})$ and $c_{\mu\nu,4}(p_{1},p_{2};p_{3},p_{4})$ give
the 1PI correlation functions of the energy-momentum tensor at $p=0$:
$\displaystyle\left\langle\Theta_{\mu\nu}(0)\phi^{I}(p)\phi^{J}(q)\right\rangle^{1PI}$
$\displaystyle=p^{2-\eta}q^{2-\eta}\left\langle\Theta_{\mu\nu}(0)\phi^{I}(p)\phi^{J}(q)\right\rangle$
$\displaystyle=\delta(p+q)\delta^{IJ}\lim_{t\to\infty}e^{(-2+\eta)t}c_{\mu\nu,2}(pe^{t},-pe^{t})$
(5.127)
and
$\displaystyle\left\langle\Theta_{\mu\nu}(0)\phi^{I}(p_{1})\phi^{J}(p_{2})\phi^{K}(p_{3})\phi^{L}(p_{4})\right\rangle^{1PI}$
$\displaystyle\qquad=\prod_{i=1}^{4}p_{i}^{2-\eta}\cdot\left\langle\Theta_{\mu\nu}(0)\phi^{I}(p_{1})\phi^{J}(p_{2})\phi^{K}(p_{3})\phi^{L}(p_{4})\right\rangle$
$\displaystyle\qquad=\delta\left(\sum_{i=1}^{4}p_{i}\right)\lim_{t\to\infty}e^{(-\epsilon+4\eta)t}\left[\delta^{IJ}\delta^{KL}c_{\mu\nu,4}(p_{1}e^{t},p_{2}e^{t};p_{3}e^{t},p_{4}e^{t})\right.$
$\displaystyle\qquad\qquad\left.+\delta^{IK}\delta^{JL}c_{\mu\nu,4}(p_{1}e^{t},p_{3}e^{t};p_{2}e^{t},p_{4}e^{t})+\delta^{IL}\delta^{JK}c_{\mu\nu,4}(p_{1}e^{t},p_{4}e^{t};p_{2}e^{t},p_{3}e^{t})\right]$
(5.128)
We obtain the two-point function as
$\displaystyle\lim_{t\to\infty}e^{(-2+\eta)t}c_{\mu\nu,2}(pe^{t},-pe^{t})$
$\displaystyle=\lim_{t\to\infty}\left\\{\left(1+\eta\,t\right)\left(-p^{2}\delta_{\mu\nu}+2p_{\mu}p_{\nu}\right)\right.$
$\displaystyle\left.\quad+\lambda^{2}(N+2)3e^{-2t}\left(\delta_{\mu\nu}G(pe^{t})-2p_{\mu}p_{\nu}e^{2t}G^{\prime}(pe^{t})\right)\right\\}$
$\displaystyle=p^{-\eta}\left(-p^{2}\delta_{\mu\nu}+2p_{\mu}p_{\nu}\right)$
(5.129)
where we have used the asymptotic form
$G(p)-2p_{\mu}p_{\nu}G^{\prime}(p)\overset{p\to\infty}{\longrightarrow}\frac{1}{12(4\pi)^{4}}\left(p^{2}\delta_{\mu\nu}-2p_{\mu}p_{\nu}\right)\ln
p^{2}$ (5.130)
We obtain the four-point function as
$\displaystyle\lim_{t\to\infty}e^{(-\epsilon+4\eta)t}c_{\mu\nu,4}(p_{1}e^{t},p_{2}e^{t};p_{3}e^{t},p_{4}e^{t})$
$\displaystyle=\lambda\lim_{t\to\infty}(1-\epsilon\,t)\Bigg{[}\delta_{\mu\nu}\big{\\{}-1$
$\displaystyle\qquad\qquad+\lambda\left((N+4)F\left((p_{1}+p_{2})e^{t}\right)+2F\left((p_{1}+p_{3})e^{t}\right)+F\left((p_{2}+p_{3})e^{t}\right)\right)\big{\\}}$
$\displaystyle\,\,-\lambda\left\\{(N+4)\frac{(p_{1}+p_{2})_{\mu}(p_{1}+p_{2})_{\nu}}{(p_{1}+p_{2})^{2}}+\frac{(p_{1}+p_{3})_{\mu}(p_{1}+p_{3})_{\nu}}{(p_{1}+p_{3})^{2}}+\frac{(p_{2}+p_{3})_{\mu}(p_{2}+p_{3})_{\nu}}{(p_{2}+p_{3})^{2}}\right\\}\Bigg{]}$
$\displaystyle=-\lambda\delta_{\mu\nu}\left[1+\frac{\epsilon}{N+8}\ln\left\\{(p_{1}+p_{2})^{N+4}(p_{1}+p_{3})^{2}(p_{2}+p_{3})^{2}\right\\}\right]$
(5.131)
where we have kept only the logarithms of momenta at order $\epsilon^{2}$.
## 6 Summary and Conclusions:
In this paper we have studied some aspects of the $O(N)$ model using the Exact
RG formalism. We have done two things:
1) We have constructed the Wilson action for the $O(N)$ model at the Wilson
Fisher fixed point in $4-\epsilon$ dimensions up to order $\epsilon^{2}$. This
is done by solving the fixed point equation, order by order in $\epsilon$.
Some correlation functions have also been calculated.
2) We have constructed the energy momentum tensor for this theory. This is
done by solving the Ward Identity for diffeomorphism invariance. The
traceless-ness of the energy momentum tensor implies that the Wilson action is
scale and conformal invariant. It is important to note that all this is in the
presence of a finite cutoff $\Lambda$.
As mentioned in the introduction, one of the motivations for this construction
is the use the ideas in [29, 30] and construct the AdS action corresponding to
this CFT. A related problem is to construct the AdS action for sources for
composite operators such as $\phi^{i}\phi^{i}$. Even more interesting would be
to study the massless spin 2 field that would be the source for the energy
momentum tensor. This would give dynamical gravity in the bulk as a
consequence of Exact RG in the boundary by a direct change of variables
similar to what was done for the scalar field in [29, 30].
## Appendix A Fixed Point Action
### A.1 Evaluation of $U_{4}$
We need to solve
$\displaystyle\bigg{[}\bigg{(}4-D-\sum_{i=1}^{4}p_{i}\frac{d}{dp_{i}}\bigg{)}+\sum_{j=1}^{4}2K^{\prime}(p_{j}^{2})U_{2}^{(1)}(p_{j})\bigg{]}\frac{1}{8}U_{4}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle=$
$\displaystyle\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2}))\frac{1}{48}\bigg{\\{}6NU_{6}(p_{1},p_{2};p_{3},p_{4};p,-p)+12U_{6}(p_{1},p;p_{2},-p;p_{3},p_{4})+12U_{6}(p_{1},p_{2};p_{3},p;p_{4},-p)\bigg{\\}}$
$\displaystyle=$
$\displaystyle\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2}))\bigg{\\{}-\frac{(N+2)}{8}\bigg{(}h(p_{1})+h(p_{2})+h(p_{3})+h(p_{4})\bigg{)}$
$\displaystyle-\frac{(N+4)}{4}\bigg{(}h(p+p_{1}+p_{2})+2h(p+p_{1}+p_{3})+2h(p+p_{1}+p_{4})\bigg{)}\bigg{\\}}$
(A.132)
where
$\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2}))\bigg{\\{}-\frac{(N+2)}{8}\bigg{(}h(p_{1})+h(p_{2})+h(p_{3})+h(p_{4})\bigg{)}\bigg{\\}}$
(A.133)
corresponds to the kind of diagrams shown in 1. Here the external loop does
not involve momenta $p_{i}+p_{j}$. We will call it Type I diagrams.
Considering only leading order terms in $p_{j}^{2}$ the contribution from type
I diagram in (A.1) is
$\displaystyle=-\frac{N+2}{8}\frac{\lambda^{2}}{16\pi^{2}}~{}4K^{\prime}(p_{j}^{2})\bigg{|}_{p_{j}=0}$
(A.134)
Now consider the second term in L.H.S of (A.1). In the limit of small external
momenta after putting the value of
$U_{2}^{(1)}(p)=-\frac{N+2}{2}\frac{\lambda}{16\pi^{2}}$( as we are
considering terms of $\mathcal{O}(\epsilon^{2})$ we have put D=4 to find
$U_{2}^{(1)}$) we get
$\displaystyle-$
$\displaystyle\sum_{j=1}^{4}2K^{\prime}(p_{j}^{2})\bigg{|}_{p_{j}\rightarrow
0}\frac{\lambda}{16\pi^{2}}\frac{N+2}{2}\frac{1}{8}V_{4}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle=$
$\displaystyle-4K^{\prime}(p_{j}^{2})\bigg{|}_{p_{j}\rightarrow
0}\frac{\lambda^{2}}{16\pi^{2}}\frac{N+2}{8}$ (A.135)
This cancels exactly with (A.134).
$\phi^{I}$$\phi^{J}$$\phi^{J}$$\phi^{I}$$p$$p_{2}$$p_{3}$$p_{1}$$p_{4}$$-p$
Figure 1: Type I diagram
Similarly in (A.1) the term
$\displaystyle\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})\bigg{\\{}-\frac{(N+4)}{4}\bigg{(}h(p+p_{1}+p_{2})+2h(p+p_{1}+p_{3})+2h(p+p_{1}+p_{4})\bigg{)}\bigg{\\}}$
(A.136)
corresponds to the kind of diagram shown in 2. We will call it Type II
diagram. In the limit $p_{i}\rightarrow 0$ the above term becomes
$\phi^{I}$$\phi^{J}$$\phi^{J}$$\phi^{I}$$\phi^{K}$$\phi^{J}$$\phi^{K}$$p$$-p$$p_{1}$$p_{3}$$p_{2}$$p_{4}$
Figure 2: Type II diagram
$\displaystyle\lambda^{2}~{}\frac{(N+8)}{4}\frac{1}{16\pi^{2}}\int_{0}^{\infty}dp^{2}K^{\prime}(p^{2})\Big{(}K(p^{2})-K(0)\Big{)}$
$\displaystyle=$
$\displaystyle\lambda^{2}\frac{(N+8)}{4}\frac{1}{16\pi^{2}}\int_{0}^{\infty}dp^{2}\bigg{\\{}\frac{1}{2}\frac{d(K^{2})}{dp^{2}}-K(0)K^{\prime}(p^{2})\bigg{\\}}$
Using $K(\infty)=0$ and $K(0)=1$, this integral gives $\frac{1}{2}$. Equating
this contribution with $\epsilon\frac{\lambda}{4!}$ from L.H.S of (A.1) we
obtain
$\frac{1}{8}(4-D)\lambda=\frac{N+8}{8}\frac{\lambda^{2}}{(4\pi)^{2}}$
Thus in addition to the trivial fixed point $\lambda=0$, we have a non trivial
fixed point:
$\boldmath\lambda=(4-D)\frac{16\pi^{2}}{N+8}$ (A.137)
### A.2 Solving for $\tilde{U}_{4}$
$\tilde{U}_{4}$ will have contribution from both type I and II diagram
explained above. We write
$\displaystyle\tilde{U}_{4}=\tilde{U}_{4}^{I}+\tilde{U}_{4}^{II}$
according to contributions from type I(II) diagrams.
(We shall set $D=4$ while evaluating integrations in those terms that are
already of $\mathcal{O}(\epsilon^{2})$.)
##### Type I diagram
In (A.1) the first term on the LHS and the first terms on the RHS (Type I)
cancel only in leading order. In general their difference is
$\lambda^{2}\frac{N+2}{8}\times\frac{1}{(4\pi)^{2}}\int_{0}^{\infty}dp^{2}K^{\prime}(p^{2})\Bigg{[}\sum_{j}\frac{K(p_{j}^{2})-K(0)}{p_{j}^{2}}-K^{\prime}(p_{j}^{2})\Bigg{]}$
Taylor expanding we find
$\lambda^{2}\frac{N+2}{8}\times\frac{1}{(4\pi)^{2}}\int
dp^{2}K^{\prime}(p^{2})K^{\prime\prime}(0){1\over 2}\sum_{j}p_{j}^{2}\equiv
c\sum_{j}p_{j}^{2}$
This is a contribution to $\tilde{U}_{4}(p_{1},p_{2};p_{3},p_{4})$ that we can
call $\Delta U_{4}^{I}(p_{1},p_{2};p_{3},p_{4})$. Consider a type I graph
where the line at one end has $p_{1}$ and lines with momenta
$p_{2},p_{3},p_{4}$ are at the other end. This corresponds to the term
$\lambda^{2}\frac{N+2}{8}\times\frac{1}{(4\pi)^{2}}\int
dp^{2}K^{\prime}(p^{2})K^{\prime\prime}(0){1\over 2}p_{1}^{2}\equiv
cp_{1}^{2}$
when contracted in a loop in order to contribute to $\tilde{U}_{2}$, so that
say $p_{3}=-p_{4}$, we have $p_{2}=-p_{1}$. It contributes to
$\tilde{U}_{2}(p_{1}^{2})$ an amount
$\displaystyle\int dp^{2}K^{\prime}(p^{2}){1\over 2}\Delta
U_{4}^{I}(p_{1},-p_{1},p,-p)=\int dp^{2}K^{\prime}(p^{2}){1\over
2}c(p_{1}^{2})=\Big{[}c\int dp^{2}K^{\prime}(p^{2})\Big{]}p_{1}^{2}\equiv
Ap_{1}^{2}$
This is just a simple wave function renormalization that does not depend on
$p_{1}$. There is no contribution to the mass. The same argument applies to
all the other permutations of the type I terms. A simple wave function
renormalization $\phi^{\prime 2}=(1+A)\phi^{2}$ can ensure the normalization
of the kinetic term.. They do not affect the physics or contribute to $\eta$.
However, type-I term contributes to sub-leading order term of $m^{2}$ or
$U_{2}$.
$\tilde{U}_{4}^{I}$ satisfies the following equation:
$\displaystyle-\sum_{i=1}^{4}p_{i}\frac{d}{dp_{i}}\frac{1}{8}\tilde{U}_{4}^{I}(p_{1},p_{2};p_{3},p_{4})=\lambda^{2}\frac{N+2}{8}\times\frac{1}{(4\pi)^{2}}\int_{0}^{\infty}dp^{2}K^{\prime}(p^{2})\Bigg{[}\sum_{j}\frac{K(p_{j}^{2})-K(0)}{p_{j}^{2}}-K^{\prime}(p_{j}^{2})\Bigg{]}$
(A.138)
The solution is
$\displaystyle\tilde{U}_{4}^{I}(p_{1},p_{2};p_{3},p_{4})=$
$\displaystyle-\lambda^{2}\frac{(N+2)}{2}\frac{1}{16\pi^{2}}\sum_{j=1}^{4}\frac{K(p_{j}^{2})-K(0)}{p_{j}^{2}}$
(A.139a) $\displaystyle=$
$\displaystyle\lambda^{2}\frac{(N+2)}{2}\frac{1}{16\pi^{2}}\sum_{j=1}^{4}h(p_{j})$
(A.139b)
where $K(p)=e^{-p^{2}}$ is assumed.
##### Type II Diagram
In (A.1) if we keep terms upto $\mathcal{O}(\epsilon^{2})$,
$\displaystyle\frac{1}{8}\bigg{[}\sum_{j=1}^{4}p_{j}\frac{d}{dp_{j}}\bigg{]}\tilde{U}_{4}^{II}(p_{1},p_{2};p_{3},p_{4})$
$\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4}\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})\bigg{\\{}(N+4)h(p+p_{1}+p_{2})+2h(p+p_{1}+p_{3})+2h(p+p_{1}+p_{4})-(N+8)h(p)\bigg{\\}}$
(A.140)
where $h(p)=\frac{K(0)-K(p)}{p^{2}}$. It is to be noted in the momentum
independent part $-\epsilon\frac{\lambda}{4!}$ we have written $\epsilon$ in
terms of $\lambda$ using the fixed point value of $\lambda$.
The solution at $\mathcal{O}(\epsilon^{2})$, analytic at zero external
momenta, is given by
$\displaystyle\tilde{U}_{4}^{II}(p_{1},p_{2};p_{3},p_{4})$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{2}\int\frac{d^{D}p}{(2\pi)^{D}}h(p)\Big{[}(N+4)h(p_{1}+p_{2}+p)+2h(p+p_{1}+p_{3})+2h(p+p_{1}+p_{4})-(N+8)h(p)\Big{]}$
(A.141a) $\displaystyle=$
$\displaystyle-\lambda^{2}\Big{[}(N+4)F(p_{1}+p_{2})+2F(p_{1}+p_{3})+2F(p_{1}+p_{4})\Big{]}$
(A.141b)
where
$F(q)=\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}h(p)\Big{(}h(p+q)-h(p)\Big{)}$.
### A.3 Equation for $\tilde{U}_{2}$
From (3.75) we get
$\displaystyle 0=$
$\displaystyle\int\frac{d^{D}p}{(2\pi)^{D}}\Big{(}-K^{\prime}(p^{2})\Big{)}\times$
$\displaystyle\bigg{\\{}$
$\displaystyle\frac{1}{8}\Big{[}4N\tilde{U}_{4}^{I}(p_{1},-p_{1};p,-p)+4N\tilde{U}_{4}^{II}(p_{1},-p_{1};p,-p)+8\tilde{U}_{4}^{I}(p_{1},p;-p_{1},-p)+8\tilde{U}_{4}^{II}(p_{1},p;-p_{1},-p)\Big{]}$
$\displaystyle-$ $\displaystyle
v_{2}^{(1)}(p)v_{2}^{(1)}(p)\delta^{D}(p-p_{1})\bigg{\\}}-\frac{\eta}{2}p_{1}^{2}+\tilde{U}_{2}(p_{1})-p_{1}^{2}\frac{d\tilde{U}_{2}(p_{1})}{dp_{1}^{2}}$
(A.142)
From(A.139a)
$\displaystyle\frac{1}{8}\bigg{\\{}4N\tilde{U}_{4}^{I}(p_{1},-p_{1};p,-p)+8\tilde{U}_{4}^{I}(p_{1},p;-p,-p_{1})\bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(N+2)^{2}\frac{\lambda^{2}}{16\pi^{2}}\bigg{\\{}h(p)+h(p_{1})\bigg{\\}}$
(A.143)
and from (A.141)
$\displaystyle\frac{1}{8}\bigg{\\{}4N\tilde{U}_{4}^{II}(p_{1},-p_{1};p,-p)+8\tilde{U}_{4}^{II}(p_{1},p;-p,-p_{1})\bigg{\\}}$
$\displaystyle=$
$\displaystyle-\frac{3\lambda^{2}}{2}(N+2)\int_{r}\bigg{\\{}h(r)\Big{[}h(r+p_{1}+p)-h(r)\Big{]}\bigg{\\}}$
(A.144)
If we decompose $\tilde{U}_{2}$ in two parts namely $\tilde{U}_{2}^{I}$ and
$\tilde{U}_{2}^{II}$ respectively, in the following way,
1.
$\displaystyle\tilde{U}_{2}^{I}(p_{1})-p_{1}^{2}\frac{d\tilde{U}_{2}^{I}(p_{1})}{dp_{1}^{2}}=\int\frac{d^{D}p}{(2\pi)^{D}}K^{\prime}(p^{2})\frac{1}{2}(N+2)^{2}\frac{\lambda^{2}}{16\pi^{2}}h(p_{1})-\big{(}U_{2}^{(1)}\big{)}^{2}K^{\prime}(p_{1}^{2})$
(A.145)
which gives
$\displaystyle\tilde{U}_{2}^{I}(p_{1})=-\frac{\lambda^{2}}{(16\pi^{2})^{2}}\frac{(N+2)^{2}}{4}h(p_{1})$
(A.146)
2.
$\displaystyle-2\tilde{U}_{2}^{II}(p_{1})+2p_{1}^{2}\frac{d\tilde{U}_{2}^{II}(p_{1})}{dp_{1}^{2}}$
$\displaystyle=$
$\displaystyle-6\lambda^{2}(N+2)\int\frac{d^{D}p}{(2\pi)^{D}}\Big{(}-K^{\prime}(p^{2})\Big{)}F(p_{1}+p)+(N+2)^{2}\frac{\lambda^{2}}{16\pi^{2}}\int\frac{d^{D}p}{(2\pi)^{D}}\Big{(}-K^{\prime}(p^{2})\Big{)}h(p)-\eta~{}p_{1}^{2}$
(A.147)
which gives
$\displaystyle\tilde{U}_{2}^{II}(p_{1})=~{}$ $\displaystyle
p_{1}^{2}\int_{p^{2}=0}^{p_{1}^{2}}dp^{2}\frac{\int\frac{d^{D}q}{(2\pi)^{D}}\Big{\\{}-6\lambda^{2}(N+2)(-K^{\prime}(q^{2}))F(p+q)\Big{\\}}-\eta~{}p^{2}}{2p^{4}}-\frac{(N+2)^{2}}{4}\frac{\lambda^{2}}{(16\pi^{2})^{2}}$
(A.148)
The second term in the expression of $\tilde{U}_{4}^{II}$ is evaluated using
$K(p)=e^{-p^{2}}$.
Hence The full expression of $\tilde{U}_{2}(p_{1})$ is given by
$\displaystyle\tilde{U}_{2}(p_{1})=-$
$\displaystyle\frac{\lambda^{2}}{(16\pi^{2})^{2}}\frac{(N+2)^{2}}{4}h(p_{1})$
$\displaystyle+$ $\displaystyle
p_{1}^{2}\int_{p^{2}=0}^{p_{1}^{2}}dp^{2}\frac{\int\frac{d^{D}q}{(2\pi)^{D}}\Big{\\{}-6\lambda^{2}(N+2)(-K^{\prime}(q^{2}))F(p+q)\Big{\\}}-\eta~{}p^{2}}{2p^{4}}-\frac{(N+2)^{2}}{4}\frac{\lambda^{2}}{(16\pi^{2})^{2}}$
(A.149)
### A.4 Expression for $\eta$
Only Type II diagrams contribute to $\eta$. Because we need the external
momentum to flow through the loop - to get a momentum dependence in $U_{2}$.
This can happen only in Type II terms and that too for certain contractions.
(Calculation of this section requires us to go back to bar denoted variable as
dimensionless variable. So $p$’s from last section are replaced with
$\bar{p}$. )
From (3.76) we have
$\frac{\eta}{2}=-\frac{1}{8}\frac{d}{d\bar{r}^{2}}\int_{\bar{q}}K^{\prime}(\bar{q}^{2})\bigg{\\{}4N\tilde{U}_{4}^{II}(\bar{q},-\bar{q};\bar{r},-\bar{r})+8\tilde{U}_{4}^{II}(\bar{q},\bar{r};-\bar{r},-\bar{q})\bigg{\\}}~{}\Bigg{|}_{\bar{r}^{2}=0}$
(A.150)
We can convert differentiation w.r.t $p_{j}$ into that w.r.t $\Lambda$ , i.e.
$\displaystyle-\sum_{j=1}^{4}\bar{p}_{j}\frac{d}{d\bar{p}_{j}}=\Lambda\frac{d}{d\Lambda}$
So (A.2) gives following expression for $\tilde{U}_{4}^{II}$:
$\displaystyle\frac{1}{8}$
$\displaystyle\tilde{U}_{4}^{II}(\frac{p_{1}}{\Lambda},\frac{p_{2}}{\Lambda};\frac{p_{3}}{\Lambda},\frac{p_{4}}{\Lambda})$
$\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4}\int_{0}^{\ln\Lambda}d\ln\Lambda^{\prime}~{}\int_{\bar{p}}K^{\prime}(\bar{p}^{2})\bigg{[}(N+4)h(\bar{p}+\frac{p_{1}}{\Lambda^{\prime}}+\frac{p_{2}}{\Lambda^{\prime}})+2h(\bar{p}+\frac{p_{1}}{\Lambda^{\prime}}+\frac{p_{3}}{\Lambda^{\prime}})+2h(\bar{p}+\frac{p_{1}}{\Lambda^{\prime}}+\frac{p_{4}}{\Lambda^{\prime}})-(N+8)h(\bar{p})\bigg{]}$
(A.151)
Hence
$\displaystyle\frac{1}{8}\bigg{\\{}4N\tilde{U}_{4}^{II}(\bar{q},-\bar{q};\bar{r},-\bar{r})+8\tilde{U}_{4}^{II}(\bar{q},\bar{r};-\bar{r},-\bar{q})\bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4}\int_{0}^{\ln\Lambda}d\ln\Lambda^{\prime}~{}\int_{\bar{p},\bar{r}}K^{\prime}(\bar{p}^{2})\bigg{\\{}(12N+48)h(\bar{p}+\frac{q}{\Lambda^{\prime}}+\frac{r}{\Lambda^{\prime}})+(12N+48)h(\bar{p}+\frac{q}{\Lambda^{\prime}}-\frac{r}{\Lambda^{\prime}})-24(N+2)h(\bar{p})\bigg{\\}}$
(A.152)
So we need to find the coefficient of $\bar{r}^{2}$ in
$\Big{[}h(\bar{p}+\frac{q}{\Lambda^{\prime}}+\frac{r^{\prime}}{\Lambda^{\prime}})+h(\bar{p}+\frac{q}{\Lambda^{\prime}}-\frac{r^{\prime}}{\Lambda^{\prime}})\Big{]}$
which is calculated as
$\displaystyle{1\over 2}\frac{r^{\mu}r^{\nu}}{\Lambda^{\prime
2}}\frac{d^{2}}{dr^{\prime\mu}dr^{\prime\nu}}\Big{[}h(\bar{p}+\frac{q}{\Lambda^{\prime}}+\frac{r^{\prime}}{\Lambda^{\prime}})+h(\bar{p}+\frac{q}{\Lambda^{\prime}}-\frac{r^{\prime}}{\Lambda^{\prime}})\Big{]}\Bigg{|}_{r^{\prime}=0}$
$\displaystyle=$ $\displaystyle\frac{r^{\mu}r^{\nu}}{\Lambda^{\prime
2}}\bigg{(}\frac{d^{2}}{d\bar{r}^{\prime\mu}d\bar{r}^{\prime\nu}}h(\bar{p}+\frac{q}{\Lambda^{\prime}}+\bar{r}^{\prime}\bigg{)}\Bigg{|}_{\bar{r}^{\prime}=0}$
$\displaystyle=$
$\displaystyle\frac{\bar{r}^{2}}{4}\bigg{(}\frac{d^{2}}{d\bar{r}^{\mu}d\bar{r}_{\mu}}h(\bar{p}+\frac{q}{\Lambda^{\prime}}+\bar{r}\bigg{)}\Bigg{|}_{r=0}$
$\displaystyle=$
$\displaystyle-\frac{\bar{r}^{2}}{4}\frac{d^{2}}{d\bar{r}^{\mu}d\bar{r}_{\mu}}\frac{K(\bar{r}^{2})-1}{\bar{r}^{2}}\Bigg{|}_{\bar{r}=\bar{p}+\frac{q}{\Lambda^{\prime}}}$
$\displaystyle=$
$\displaystyle\bar{r}^{2}K^{\prime\prime}((\bar{p}+\frac{q}{\Lambda^{\prime}})^{2})$
(A.153)
where we have used the facts: in 4 dimensions
$(\frac{d}{dp_{\mu}}\frac{1}{p^{2}})=\delta^{4}(p)$ and $K(0)=1$.
From (A.150),(A.4) and (A.4) we get
$\frac{\eta}{2}=3\lambda^{2}(N+2)\int_{\bar{q}}K^{\prime}(\bar{q}^{2})\int_{0}^{\ln\Lambda}d\ln\Lambda^{\prime}~{}(\frac{\Lambda}{\Lambda^{\prime}})^{2}\int_{\bar{p}}K^{\prime}(\bar{p}^{2})K^{\prime\prime}((\bar{p}+\frac{q}{\Lambda^{\prime}})^{2})$
(A.154)
Evaluation of integral: Let us use
$\bar{q}^{\prime}=\frac{q}{\Lambda^{\prime}}$ and $\Lambda^{\prime}$ as
variables of integration, rather than $\bar{q}=\frac{q}{\Lambda}$ and
$\Lambda^{\prime}$. So change variables:
$\bar{q}=\bar{q}^{\prime}\frac{\Lambda^{\prime}}{\Lambda}~{};~{}~{}~{}\bar{q}^{2}=\bar{q}^{\prime
2}\Big{(}\frac{\Lambda^{\prime}}{\Lambda}\Big{)}^{2}~{};~{}~{}~{}\int
d^{4}\bar{q}=\int
d^{4}\bar{q}\Big{(}\frac{\Lambda^{\prime}}{\Lambda}\Big{)}^{4}$
to get
$\frac{\eta}{2}=-3\lambda^{2}(N+2)\int_{0}^{\ln\Lambda}d\ln\Lambda^{\prime}~{}\int_{\bar{q}^{\prime}}\Big{(}\frac{\Lambda^{\prime}}{\Lambda}\Big{)}^{-2}K^{\prime}(\bar{q}^{\prime
2})\Big{(}\frac{\Lambda^{\prime}}{\Lambda}\Big{)}^{2}\int_{\bar{p}}K^{\prime}(\bar{p}^{2})K^{\prime\prime}((\bar{p}+\frac{q}{\Lambda^{\prime}})^{2})$
Using $K^{\prime}(\bar{q}^{\prime
2})=\frac{dK}{d\Lambda^{\prime}}\frac{d\Lambda^{\prime}}{d\bar{q}^{\prime
2}}=-\frac{\Lambda^{\prime}}{2\bar{q}^{\prime 2}}\frac{dK}{d\Lambda^{\prime}}$
we get
$\frac{\eta}{2}=-3\lambda^{2}(N+2)\int_{0}^{\Lambda}d\Lambda^{\prime}~{}\frac{dK}{d\Lambda^{\prime}}\int_{\bar{q}^{\prime}}\frac{1}{2\bar{q}^{\prime
2}}\int_{\bar{p}}K^{\prime}(\bar{p}^{2})K^{\prime\prime}((\bar{p}+\bar{q}^{\prime})^{2})$
Since $\bar{q}^{\prime}$ is an independent variable we can write this as
$\frac{\eta}{2}=-3\lambda^{2}(N+2)\int_{\bar{q}^{\prime}}\int_{0}^{\Lambda}d\Lambda^{\prime}~{}\frac{dK}{d\Lambda^{\prime}}\frac{1}{2\bar{q}^{\prime
2}}\int_{\bar{p}}K^{\prime}(\bar{p}^{2})K^{\prime\prime}((\bar{p}+\bar{q}^{\prime})^{2})$
The integral over $\bar{p}$ is a function of $\bar{q}^{\prime}$ and not
$\Lambda^{\prime}$. So we can do the $\Lambda^{\prime}$ integral easily. Using
$K(\infty)=0$ we get
$\frac{\eta}{2}=-\frac{3\lambda^{2}}{2}(N+2)\underbrace{\int_{\bar{q}^{\prime}}~{}K(\bar{q}^{\prime
2})\frac{1}{\bar{q}^{\prime
2}}\int_{\bar{p}}K^{\prime}(\bar{p}^{2})K^{\prime\prime}((\bar{p}+\bar{q}^{\prime})^{2})}_{-\frac{\pi^{4}}{6(2\pi)^{8}}}=\frac{1}{4}\lambda^{2}(N+2)\frac{1}{(16\pi^{2})^{2}}$
The integral underbraced above is calculated to give
$-\frac{\pi^{4}}{6(2\pi)^{8}}$ for $K(x)=e^{-x}$. But it can be shown to give
identical result for any smooth $K(x)$ [56]. Using
$\lambda=\frac{16\pi^{2}}{N+8}\epsilon$ we can write the anomalous dimension
as:
$\boldmath\frac{\eta}{2}=\frac{1}{4}\lambda^{2}(N+2)\frac{1}{(16\pi^{2})^{2}}=\frac{N+2}{(N+8)^{2}}\frac{\epsilon^{2}}{4}$
(A.155)
## Appendix B Asymptotic behaviors of $F(p)$ and $G(p)$
The function $F(p)$ is defined by
$\left(p\cdot\partial_{p}+\epsilon\right)F(p)=\int_{q}f(q)\Big{(}h(q+p)-h(q)\Big{)}$
(B.156)
For large $p$, we obtain an equation satisfied by the asymptotic form
$F_{\mathrm{asymp}}(p)$:
$\left(p\cdot\partial_{p}+\epsilon\right)F_{\mathrm{asymp}}(p)=-\int_{q}f(q)h(q)=-\frac{1}{(4\pi)^{2}}+\mathrm{O}(\epsilon)$
(B.157)
This implies
$F_{\mathrm{asymp}}(p)=-\frac{1}{\epsilon}\int_{q}f(q)h(q)+C_{F}(\epsilon)p^{-\epsilon}$
(B.158)
where $C_{F}(\epsilon)$ is independent of $p$. Since $F(p)$ is finite in the
limit $\epsilon\to 0+$, we must find
$C_{F}(\epsilon)=\frac{1}{\epsilon}\frac{1}{(4\pi)^{2}}+\cdots$ (B.159)
Hence, expanding in $\epsilon$, we obtain
$F_{\mathrm{asymp}}(p)=-\frac{1}{(4\pi)^{2}}\ln
p+\mathrm{const}+\mathrm{O}(\epsilon)$ (B.160)
We next consider $G(p)$ satisfying
$\left(p\cdot\partial_{p}-2+2\epsilon\right)G(p)=\int_{q}f(q)F(q+p)+2v_{2}\int_{q}f(q)h(q)+\eta^{(2)}p^{2}$
(B.161)
where
$\eta^{(2)}=-\frac{d}{dp^{2}}\int_{q}f(q)F(q+p)\Big{|}_{p=0}=\frac{1}{6(4\pi)^{4}}+\mathrm{O}(\epsilon)$
(B.162)
The asymptotic form $G_{\mathrm{asymp}}(p)$ satisfies
$\left(p\cdot\partial_{p}-2+2\epsilon\right)G_{\mathrm{asymp}}(p)=\eta^{(2)}p^{2}$
(B.163)
This gives
$G_{\mathrm{asymp}}(p)=\frac{1}{2\epsilon}\eta^{(2)}p^{2}+C_{G}(\epsilon)p^{2-2\epsilon}$
(B.164)
Since $G(p)$ is finite as $\epsilon\to 0+$, we obtain
$C_{G}(\epsilon)=-\frac{1}{\epsilon}\frac{1}{12(4\pi)^{4}}+\cdots$ (B.165)
Hence,
$G_{\mathrm{asymp}}(p)=p^{2}\left(\frac{1}{6(4\pi)^{4}}\ln
p+\mathrm{const}\right)+\mathrm{O}(\epsilon)$ (B.166)
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|
2024-09-04T02:54:57.583754 | 2020-03-05T18:11:11 | 2003.02804 | {
"authors": "Igor V. Tetko, Pavel Karpov, Ruud Van Deursen, Guillaume Godin",
"full_text_license": null,
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26062",
"submitter": "Pavel Karpov Dr",
"url": "https://arxiv.org/abs/2003.02804"
} | arxiv-papers | # State-of-the-Art Augmented NLP Transformer models for direct and single-
step retrosynthesis, which overperforms all published approaches
Igor V. Tetko
Institute of Structural Biology
Helmholtz Zentrum München,
and BigChem GmbH,
Germany, Unterschleißheim
<EMAIL_ADDRESS>
&Pavel Karpov
Institute of Structural Biology
Helmholtz Zentrum München,
and BigChem GmbH,
Germany, Unterschleißheim
<EMAIL_ADDRESS>
&Ruud Van Deursen
Firmenich International SA,
Research&Development Division,
Switzerland, Geneva
<EMAIL_ADDRESS>&Guillaume Godin
Firmenich International SA,
Research&Development Division,
Switzerland, Geneva
<EMAIL_ADDRESS>Helmholtz Zentrum München – Research Center for
Environmental Health (GmbH), Institute of Structural Biology, Ingolstädter
Landstraße 1, D-85764 Neuherberg, Germany
###### Abstract
We investigated the effect of different training scenarios on predicting the
(retro)synthesis of chemical compounds using text-like representation of
chemical reactions (SMILES) and Natural Language Processing neural network
Transformer architecture. We showed that data augmentation, which is a
powerful method used in image processing, eliminated the effect of data
memorization by neural networks and improved their performance for prediction
of new sequences. This effect was observed when augmentation was used
simultaneously for input and the target data simultaneously. The Top-5
accuracy was 84.8% for the prediction of the largest fragment (thus
identifying principal transformation for classical retro-synthesis) for the
USPTO-50k test dataset, and was achieved by a combination of SMILES
augmentation and a beam search algorithm. The same approach provided
significantly better results for the prediction of direct reactions from the
single-step USPTO-MIT test set. Our model achieved 90.6% Top-1 and 96.1% Top-5
accuracy for its challenging mixed set and 97% Top-5 accuracy for the USPTO-
MIT separated set. It also significantly improved results for USPTO-full set
single-step retrosynthesis for both Top-1 and Top-10 accuracies. The
appearance frequency of the most abundantly generated SMILES was well
correlated with the prediction outcome and can be used as a measure of the
quality of reaction prediction.
#### Synposis
Training and application of neural networks with randomized sequences
significantly improves direct and retro-synthetic models and can be used to
estimate quality of reaction predictions
## 1 Introduction
To synthesize an organic compound is to solve a puzzle with many pieces and
potentially several pieces missing. Here, the pieces are single reactions, and
finding their sequential combination to create a final product is the
retrosynthesis task.
The success of the logic of organic synthesis developed by E.J. Corey [1]
triggered the development of computer programs aiming to find appropriate ways
to synthesize a molecule. The first retrosynthesis program LHASA [2] utilizes
a template-based [3, 4] approach. Every template (rule, synthon) in a curated
database of known transformations is sequentially applied to a target
molecule, and then sets of reagents are selected according to a specified
strategy. Reagents, in turn, undergo the same decompositions until a set of
commercially available compounds is found. Retrosynthesis always has multiple
routes – a retrosynthetic tree – ending with different starting materials.
Thus, a practical algorithm for retrosynthesis has to solve not only the rule
acquisition and selection problem but also has capabilities to effectively
navigate this tree [5], taking into account different strategies. These tasks
relate directly to artificial intelligence strategies [6, 7, 8].
Due to the difficulty of maintaining template databases, most projects
dependent on them, including LHASA, did not become widely used tools. The only
major exception is, perhaps, the program Synthia™(previously CHEMATICA [9])
which is a successful commercial product. In the Synthia™program, rules are
automatically extracted from atom-mapped reaction examples [10]. However,
there is an ambiguity in the mapping definition and, more importantly, the
automatic rule does not take into account other undefined possible reactive
centers in a molecule. Applying such transformations may result in molecules
that fail to react as predicted, e.g., ’out-of-scopes’ and special care to
filter out these cases has to be taken [5]. An alternative approach for the
extraction of these rules is to apply a data-driven deep learning technique
that corresponds to a machine learning approach where an algorithm (usually in
the form of a neural network) is trained on the raw data. After the training
finishes, the network contains all the implicitly encoded features (rules) of
the corresponding input via its parameters. Works on reaction prediction
outcomes [11] and retrosynthesis [12, 13] showed the feasibility of a symbolic
approach, where reactions are written as SMILES [14] strings as in a machine
translation. The product is written in the “source language”, whereas the set
of reactants is written in the “target language”. For the “reaction
translation” task both languages, however, are SMILES strings, having the same
alphabet and grammar. The first works on symbolic (retro)synthesis [12, 15]
were carried out with Seq2Seq [16] models following robust and more easy to
train Transformer approaches [17, 18] that bring state-of-the-art results [11,
19]. Meanwhile other approaches based on similarity [20], convolutional [21,
22, 23], and graphs show promising results [24, 25].
The SMILES representation of molecules is ambiguous. Though the
canonicalization procedure exists [26], it has been shown that models benefit
from using a batch of random SMILES (augmentation) during training and
inference [27, 28, 29, 30]. Recently, such augmentation was also applied to
reaction modeling [11, 18, 31, 32]. The augmented (also sometimes called
“random”) SMILES are all valid structures with the exception that the starting
atom and the direction of the graph enumerations are selected randomly.
In this article, we scrutinize the various augmentation regimes and show that
augmentation leads to better performance compared to the standard beam search
inference or evaluation of the model under different temperatures. We clearly
mention that our study is to predict single-step and not multi-step
retrosynthesis, which has been also targeted using Transformer [33, 34]. We
show that by using more complicated data augmentation strategies we decrease
overfitting [35] of neural networks and increase their accuracy to achieve top
performances for both direct and retro-synthesis. We observe that the harder
are the data to train the model, the better it will predict new ones.
Moreover, we introduce a new measure MaxFrag accuracy for the prediction of
the largest fragment (thus identifying principal transformation for classical
retro-synthesis).
## 2 Results and discussion
The baseline dataset contained only canonical SMILES. The other datasets also
contained SMILES, augmented as described in the Methods section, p. Methods.
Four different scenarios were used to augment training set sequences. Namely,
we used augmentation of products only (xN), augmentation of products and
reactants/reagents (xNF), augmentation of products and reactants/reagents
followed by shuffling of the order of reactant/reagents (xNS), and finally
mixed forward/reverse reactions, where each retrosynthesis reaction from xNS
was followed by the inverse (forward synthesis) reaction (xNM). Only the
simplest augmentation xN was used for test sets because no information about
reactant/reagents could be used for the retrosynthesis prediction. At least
one copy of canonical SMILES for each reaction was present in all augmentation
scenarios.
### 2.1 Reaction Synthesis Data
We used a training set filtered from USPTO database [36] containing 50k
reactions classified into 10 reaction types. We used splitting proposed by
[12] and divided it into 40k, 5k and 5k reactions for the training,
validation, and test sets, respectively. As in the previous study [13], after
observing that early stopping using the validation set did not improve model
test accuracy (the model performance for each of the sets was monotonically
increasing with number of iterations, see Fig. A1), we combined the training
and the validation sets into a combined training set. The 5k test reactions
were predicted only once the model training was finished and were not used at
any stage of the model development. In a similar way we joined training and
validation sets of USPTO-MIT [22] dataset for direct reaction prediction. In
order to provide a more straightforward comparison with results of the
previous studies we also reported performances when developing models using
only the respective training sets. Moreover a model with the largest published
USPTO-full set [24] was also developed.
### 2.2 Analysis of canonical datasets
The development of a model with canonical SMILES (x1) as the training set
provided 40.9% accuracy for prediction of the canonical test set. An attempt
to use this model to predict the augmented test set (x5, x10), resulted in
much lower Top-1 predictions of 23.3% and 18.4%, respectively. This result was
to be expected, because the model trained with only canonical sequences was
not able to generalize and predict augmented SMILES, which use different
styles of molecular representation.
### 2.3 Augmentation of products only (xN)
The augmentation of the products (input data), with just one additional
augmented SMILES x2, increased Top-1 accuracy to 43.7% for the test data
composed of canonical sequences. Increasing the number of augmentations in the
training set did not increase the Top-1 prediction accuracy. Thus, the
augmentation of the training set with just one random SMILES contributed the
best performance. This result is in concordance with another study where only
one random SMILES was used to augment data [18].
### 2.4 Analysis of character and exact sequence-based prediction accuracy
To better understand the model training, we also developed several models
where approximately 10% of the dataset did not participate in training but was
used to monitor its prediction performance. Different from the test set, which
tested the performance of models when predicting a new reaction, the
monitoring set tested the ability of the Transformer to predict different
SMILES generated for the same reaction. The Transformer was able to recognize
different representations of the same reaction. For example, when training x1,
the character and exact-sequence-based accuracies when predicting the
monitoring sequences were 96.5% and 34.5%, respectively. The final performance
for the test set, 40.9%, was higher because some reaction products from the
Transformer provided non-canonical SMILES, which were correctly matched after
transformation to canonical ones. When using augmented training sequences
(x10), the accuracies increased to 99.97% and 98.9%, for character and exact
sequence-based accuracy, respectively (see Fig. 1). The Transformer recognized
different representations of SMILES for reactants and reagents of the same
training set reaction, and was able to exactly restore the target products
which were memorized. Demonstrably, it was also able to memorize any random
sequences. To show this, we used a random SMILES sequence (xNR set in Tables
A1, LABEL:tbl:s2 and Fig. A2) instead of the canonical sequences as the target
for prediction. While this task was more difficult and took more epochs to
train, the Transformer was able to perfectly memorize random sequences. Since
the SMILES prediction target was random, the Transformer was not able to learn
canonicalization rules on how to write the target. Despite this fact, it still
calculated a Top-1 prediction accuracy of 26.8% for the test set which was,
however, significantly lower compared to the 42.2% achieved using the x10
dataset with canonical sequences as the target.
Figure 1: Character and exact sequence based accuracies calculated for the
monitoring set. The transformer memorized the target sequences if the target
sequences were all canonical SMILES (red dots). It also reasonably predicted
the sequence composition for randomized target SMILES (cyan rectangle, dashed)
but its performance decreased for prediction of exact full SMILES (cyan
circle). The performance normalized by the percentage of canonical sequences
increased with the number of augmentations, N, since some of the random
sequences were canonical ones.
### 2.5 Augmentation of reactants and reagents
A boost of the Transformer performance was observed when, in addition to
products i.e. the inputs SMILES, we also augmented the target SMILES, i.e.
reactants and reagents. This task was more difficult for the Transformer,
which resulted in a drop in both character and sequence based scores for
monitoring sequences during the training stage. For example, when using the
training dataset with one augmented SMILES, x2F, the character based accuracy
dropped to 91.3%, which was lower than 98.6% calculated with the x2 dataset
composed of canonical product SMILES (Fig. 1). For a larger number of
augmentations, the character-based accuracy converged to a plateau, e.g.
89.96% and 89.67% for the x5F and x20F training sets, respectively. The
character-based accuracy was calculated as the percentage of exact matches
between target and predicted sequences, e.g. “CCCCN” and “NCCCC” have an
accuracy of 80%, despite being the same SMILES but written from different
starting atoms. Thus despite the fact that the Transformer faced a prediction
of random SMILES, it was still able to provide a reasonable prediction of
their character composition.
However, of course, the Transformer was not able to predict the exact random
product SMILES. This resulted in a decrease in sequence-based accuracy based
on the number of augmentations for xNF training datasets (Fig. 1, cyan
circle). Still the Transformer was able to predict some of the sequences,
which corresponded to the subset of canonical sequences in the monitoring set.
Interestingly, the sequence accuracy normalized to the percentage of canonical
SMILES in the monitoring sets increased with the number of augmentations since
some randomly generated sequences were canonical SMILES.
### 2.6 Top-1 performance analysis
For augmentations with 1 or 2 random SMILES, the Top-1 prediction performance
of the models trained with augmentation of reactants and reagents only, xN,
and full reaction augmentation, xNF, were similar. For a larger number of
augmentations the models trained with xNF sets had systematically better
performances than those developed with xN sets (Fig. 2). The training with the
x80F set provided the highest Top-1 performance of 52.3% when this model was
applied to the test set generated with x20 number of augmented sequences.
While it was possible that further increase in the augmentations could still
increase the Top-1 performance, we did not perform such calculations due to
limitations on available computational resources.
Figure 2: Top-1 performance of models developed with different number of
augmentation (shown on x axis) and different augmentation scenarios applied to
both test and training sets (red colour: only products were augmented; cyan
colour: full reactions were augmented). The use of the large number of
augmentations for the test set (solid lines) improved prediction accuracy for
models developed with augmentation of full reactions but did not influence the
performance of models where only input data were augmented.
### 2.7 Shuffling order of reactants
In addition to augmenting the full reaction, we also randomly shuffled the
orders of reactants (see xNS set description in Table A1 and LABEL:tbl:s2).
The effect of this additional data perturbation improved Top-1 performance to
53.1% for the x20S training dataset applied to the test set with the same
number of augmentations (Fig. A3). Further increasing the number of
augmentations resulted in the loss of Top-1 prediction accuracy.
### 2.8 Shuffling and mixing of retrosynthesis and direct reactions
The training of retrosynthesis and direct reactions simultaneously could
create a mixed representation of latent space and further increase the ability
of the Transformer to make generalizations. We tested this hypothesis by
combining direct and reverse reactions in one training set by reversing the
order of product/reactant+reagents and adding a dot to distinguish direct
reactions (see e.g., Table LABEL:tbl:s2, x2M augmentation). Contrary to
previous analysis, which required 20 augmentations of training set sequences
to achieve the highest performance, the mixed dataset achieved it with only 10
augmentations (Fig. A3). Since the mixed dataset also included direct
reactions, it had the same number of 19 augmented SMILES per canonical SMILES
as in the previous analyses. Thus, this number of augmentations was optimal
for the training of the Transformer. A smaller number of augmentations did not
allow it to fully explore the data diversity while a larger number created too
much noise and made it difficult to learn canonization rules, which were
injected by the single canonical sequence. For the x10M training set, the
Transformer calculated 52.8%, which was similar to 53.1% calculated using the
x20S training dataset.
### 2.9 Top-5 performance analysis
This measure provided a relaxed estimation of the performance of the model by
measuring if the correct reaction is listed in the Top-5 predicted reactions.
Actually, it is questionable whether for retrosynthetic models having the
highest Top-1 accuracy is desirable. The goal of a retrosynthetic model is to
obtain several precursor suggestions and not exclusively the ones stated in
the literature. Moreover, multiple reactions for the same product exist. An
example includes the aromatic substitution of an aryl halide (R-X) to an aryl
amine (R-NH2) or aryl hydroxide (R-OH). Models with higher Top-n scores do
suggest other probable reactions (indeed, all reactions amid Top-n have
similar probability) which may correspond to those not reported in the
literature for the analysed example. Thus models with higher Top-N scores but
with similar Top-1 scores could be interesting for a chemist since they do
propose the correct prediction along with similar quality Top-1 reactions.
For each number of augmentations, the Top-5 performance generally increased
with the number of augmented sequences. The highest Top-5 value was
consistently calculated across different scenarios for training sets with 4-5
augmentations only (Fig. 3). The highest accuracy, 78.9%, was calculated for
the mixture dataset using the x5M training set augmentation. This number had
approximately 1% higher accuracy than that calculated using the x5S training
set (Fig. 3).
Figure 3: Top-5 performance of transformer models developed with different
training set augmentation protocol (See Tables A1 & LABEL:tbl:s2) for
prediction of the x20 test set.
### 2.10 Reference USPTO-50k model
For all studies we used a fixed number of epochs N=100. However, we needed to
confirm that this was a sufficient number of epochs and to determine if we
could calculate better results by training for longer. We selected the model
developed with the x5M training set, which provided the highest performance
for Top-5 accuracy, and trained it for an additional 400 iterations (in total
500). This additional training improved Top-1 accuracy to 53.3% while Top-5
performance increased to 79.4% (Table 1111The final reference model was built
using 500 iterations for the x5M training set. Its reference performance was
evaluated using beam size = 5, temperature = 1. The altered parameters are
shown for several other application scenarios. For beam = 1 and x1000
augmentations the model calculated 53.7, 80 and 84.3 for Top-1, Top-5 and
Top-10 predictions, respectively. This augmentation as well as the one with
beam size=10 applied to x100 analysed the same number of predicted sequences.
The best results were shown in bold. Larger beam sizes contributed better
results for larger Top-n predictions.) when using beam=5, e.g. the same as in
previous analyses.
Further improvement was achieved by using a large number of augmentations, and
x100 as the test set. With this setting the model achieved an accuracy of
53.6% and 80.8% for Top-1 and Top-5 predictions, respectively.
Table 1: Analysis of the reference model performance depending on the parameters of the application protocol. Apply model setting | Test set x1 | Test set x20 | Test set x100 |
---|---|---|---|---
Top-1 | Top-5 | Top-1 | Top-5 | Top-1 | Top-5 | Top-10
reference accuracy | 48.5 | 72.5 | 53.3 | 79.4 | 53.6 | 80.8 | 85
temperature, t=1.3 | 49.1 | 67.7 | 52.7 | 77.7 | 53.3 | 78.4 | 83.2
no beam search | 47.7 | 47.7 | 53.3 | 75.3 | 53.8 | 78.8 | 81.7
beam size, beam= 10 | 48.3 | 73.4 | 53.5 | 80 | 53.5 | 81 | 85.7
beam size, beam = 44 | 48.3 | 72.5 | 53.5 | 80 | 53.5 | 80.5 | 85.8
### 2.11 Influence of temperature
In our previous study [13], we observed that using higher temperatures during
beam search increased model accuracy for the Top-1 prediction. It should be
mentioned that no augmentation was used in that study. Under the same
experimental setup with no augmentation, i.e. when predicting test set
composed of only canonical sequences, x1, the Top-1 accuracy of the model
increased from 48.3% to 49.1% and 49.2% when using temperatures 1.3 or 1.5,
respectively. However, the Top-1 and Top-5 performances for the augmented data
(x20) decreased from 53.3% to 52.7% and 52.4%, respectively. For the same test
set the Top-5 accuracies also decreased from 79.4% to 77.7% and 77.4% for both
temperatures, respectively. Thus, while higher temperatures increased the
variability of predictions and thus performance for prediction of canonical
sequences, its effect was negative for the augmented data. In particular, it
resulted in the lower accuracy of Top-5 predictions.
### 2.12 Influence of beam search
In the above studies we consistently used a beam size of 5 for all analyses.
The goal of the beam search was to generate multiple predictions for the same
data and thus to better explore the variability of predictions. For example,
when using the x20 test set and a beam size of 5, we obtained up to 100
individual predictions, which were used to select the most frequently
appearing Top-1 and Top-5 sequences. Increasing the beam size to 10 further
improved Top-1 by 0.2 to 53.5% and Top-5 by 0.6% to 80% for the test set. The
decrease of the beam size to 3 provided a slightly higher Top-1 score of 53.4%
but decreased the Top-5 to 78.5% for the same test set. The use of beam size 1
resulted in a Top-1 accuracy of 53.3% and a reduced Top-5 accuracy of 75.3%
(Table 1). These results were expected: the variation of the beam size
slightly influenced the identification of the highest ranked sequence but its
smaller number reduced exploration of the space of other Top-reactions for
larger $n$.
Both beam search and augmentation increased the number of predicted SMILES
which in turn led to better accuracy of model predictions. Thus both of these
methods could contribute to the generation of multiple predictions to be used
to identify Top-ranked sequences. The maximum size of the beam was restricted
by the size of the target vocabulary (number of characters in the target
SMILES), which was 44 characters for our dataset. Because of the design of the
beam search and because we explicitly excluded duplicated predictions (see
section “Analysis of predicted SMILES” as well as Table A3), the dataset used
for analysis did not generate duplicated sequences for the same beam search.
However, such sequences were indeed generated at different positions of the
beam as different representations of the same SMILES. The number of non-unique
sequences generated within the same beam search increased with the length of
the beam. Interestingly, the use of canonical SMILES as input data contributed
to the largest number of unique SMILES, which were 86%, 82% and 78% for beam
searches of size 5, 10 and 44, respectively. The use of augmented random
SMILES as input contributed smaller numbers of unique sequences, e.g., 42%,
28% and 13% for beam searches of size 5, 10 and 44, respectively. For both
types of SMILES some generated SMILES were erroneous and could not be
correctly converted by RDKit. Such sequences were excluded from analysis. For
large beam sizes, canonical SMILES produced a much bigger percentage of
incorrect SMILES, as compared to the use of random SMILES (see Fig. A4). The
large difference in the results generated when starting from canonical and
random SMILES was also observed for analysis of the percentage of correct
predictions for each beam position. In general, the number of erroneous SMILES
was low, e.g., on average it was less than 1% and 3% for beam search 10, when
using augmented and canonical SMILES as input, respectively (Fig. A4). While
graph-based methods predict exact chemical structures and thus have 0%
syntactically invalid SMILES, a few percentage points of incorrectly predicted
structures by the Transformer model does not make a large difference to these
methods.
The use of canonical SMILES provided (Fig. A4) a higher accuracy for the first
beam position, but its accuracy was much lower for other beams. This was
because the Transformer generated canonical SMILES for the canonical input
sequences (e.g., 91% of valid SMILES produced at the position 1 of the beam
search for input canonical SMILES were canonical ones) and since only one
valid canonical SMILES could be produced, it failed to generate new correct
SMILES. Indeed, during the training phase, the Transformer always had a pair
of canonical SMILES as input and target sequences. Contrary to that, using
augmented SMILES allowed more freedom and allowed it to contribute valid but
not necessarily canonical SMILES (e.g., only 33% of generated SMILES at the
position one of the beam search were canonical ones if augmented SMILES were
used as input).
The decrease in performance of SMILES generated when using canonical SMILES
was one of the main reasons to implement deduplication of data and retain only
the first SMILES for the prediction of reactions (see section “Analysis of
predicted SMILES”). When deduplication was not performed and all SMILES
generated during the beam search were used to rank predictions (compare Tables
A3 and A4), the Top-1 performances of models were most significantly affected
when using only few augmentations, e.g. for the reference model its accuracy
dropped from 48.3% (reference prediction, Table 1) to 47% but did not change
for, e.g. Top-5 performance. In principle, the analysis retaining multiple
predicted sequences was based on more data and thus was more stable.
Therefore, it could be used when several augmentations and/or large values of
Top-n are used for analysis.
As it was mentioned above, both data augmentation and beam search could be
used to generate multiple predictions. For the same number of generated
sequences, 1000 per SMILES, using a beam = 10 search for the x100 set produced
lower accuracy, 53.5% compared to 53.7% using augmented data with the x1000
test set without any beam search. The performance of both methods were the
same and equal to 53.7% when the deduplication procedure was not used.
However, the beam search contributed to better accuracy, i.e., 81% vs 80% and
85.7% vs 84.3% compared to the use of augmentation alone for Top-5 and Top-10,
respectively. Thus, using beam search allowed a better exploration of data
when suggesting several alternative reactions. In any case the augmentation
was a very important part of the beam search and for the best performance,
both of these approaches should be used simultaneously. We also do not exclude
that optimisation of the augmentation may improve its results in the future.
Moreover, data augmentation used alone without a beam search contributed
superior models to the beam search used without any data augmentation.
### 2.13 Accuracy of prediction
For some reaction predictions without the use of augmented sequences or
position at the beam search the majority of predicted sequences were
identical, while for other reactions the Transformer generated as many
different SMILES as possible reactants (see Table A5). While the beam
generation procedure guaranteed that each prediction had exactly the same
sequence of characters, in many cases the Transformer produced multiple non-
canonical instances of the same SMILES. The frequency of the appearance of the
most frequent (after conversion to the canonical representation) SMILES could,
therefore, indicate the confidence of the Transformer in the prediction. Fig.
4 indicated that such frequency (which was calculated on 100x augmented
dataset) correlated well with the accuracy of prediction and could be used as
a confidence score for the chemist. Indeed, the reactions in which the most
frequent SMILES dominated amid all predictions for Top-1 were likely to be
predicted correctly. If the most frequent SMILES had low frequencies, such
predictions were likely to be incorrect ones. For about 20% of the most
frequent predictions, the accuracy of the retrosynthesis prediction was above
80% and for 4% more than 90%. It should be mentioned, that for a practical
implementation which critically depends on the speed, e.g., multistep
synthesis, there is no reason to always run all 100 predictions to get the
confidence estimations. One can always estimate the probability of the most
frequent SMILES and its confidence interval based on a much smaller number of
predictions thus decreasing the number of calculations.
As shown in Fig. 4, the same correlations were observed for two other datasets
USPTO-MIT and USPTO-Full, which are analysed in the following section. The
same approach can be used for Top-n predictions by suggesting one or more
plausible pathways for retrosynthesis. An example of such correlations for
Top-5 MaxFrag accuracy were shown in Fig. A5. Moreover, the same approach also
predicted the accuracy for the direct synthesis as it was demonstrated at Fig.
A6. It should be mentioned that use of data augmentation is not the only
approach to estimate the accuracy of predictions, and other methods based on
the likelihood of the direct reaction prediction were also proposed [18, 34]
and were shown to correlate with the accuracy of the predictions. A comparison
of such methods is beyond the scope of this study.
Figure 4: Accuracy and density (fraction of predictions) of the Transformer
for MaxFrag Top-1 retrosynthesis accuracy as a function of the frequency of
appearance of the Top-1 SMILES in the output of the Transformer for the
respective test sets of the models.
### 2.14 Analysis of prediction accuracy for different scenarios
The accuracy of the reference model was about 5% to 7% (Top-1) and 10% (Top-5)
higher for reactions without stereochemistry than for those with it (Table
2222The classical retro-synthesis accuracy was estimated as the percentage of
correctly predicted largest fragments, i.e., “maximum fragment” (MaxFrag)
accuracy. The best results are shown in bold.). 20% of the reactions in the
test set contained molecules with stereochemistry. An increase in the number
of augmentations of the test set increased the accuracy of both stereo and
non-stereochemical reactions. Stereochemical reactions in the dataset may also
suffer from a larger number of annotation errors or/and can have lower
prediction scores since such data were underrepresented in the training set.
Additionally, for some reactions despite the relative stereochemistry being
conserved it may still define confusing information for the model due to the
reactant satellite effect. The R/S could be also affected by the way the
SMILES was written, e.g. from A to Z or Z to A.
Table 2: Prediction accuracy of the reference model for different subsets of the test set of USPTO-50k using a beam search of size 10. Test set augmentation | Top-1 | Top-5 | Top-10
---|---|---|---
all | stereo (20%) | no stereo (80%) | all | stereo (20%) | no stereo (80%) | all | stereo (20%) | no stereo (80%)
x1 | 48.3 | 44.7 | 49.2 | 73.4 | 67.3 | 74.9 | 77.4 | 71 | 79
x20 | 53.4 | 47.3 | 55 | 80 | 73.3 | 81.9 | 84.2 | 79.2 | 85.4
x100 | 53.5 | 47.1 | 55.1 | 81 | 74.6 | 82.6 | 85.7 | 81.2 | 86.8
MaxFrag, x1 | 53.5 | 48.7 | 54.7 | 79.2 | 72.7 | .80.9 | 81.6 | 75.1 | 83.3
MaxFrag, x20 | 58.5 | 52 | 60.1 | 84.7 | 79 | 86.1 | 88.6 | 83.6 | 89.8
MaxFrag, x100 | 58.5 | 51.2 | 60.3 | 85.4 | 79.4 | 86.9 | 90 | 85.1 | 91.2
### 2.15 Classical Retro-Synthesis accuracy: recognition accuracy for the
largest fragment
The prediction of SMILES for retro-synthesis includes exact prediction of the
reactants. However, the same reaction performed using different reactants can
result in a similar yield. In general the database does not contain all
possible reaction conditions to make a given product. Therefore, a prediction
of only the main (the largest) reactant can be considered more relevant for
retro-synthesis predictions, since we need to first identify the reaction
type. Indeed, a chemist generally writes a retrosynthesis by decomposing a
target molecule into pieces. This classical procedure, focusing only on main
compound transformations, is the minimal information required to get an
efficient retrosynthesis route and at the same time all reactions needed (see
Fig. 5). The selection of reaction conditions of the reactions can be
considered as a subsequent task.
Figure 5: Classical representation of the retrosynthesis of cimetidine
focusing on the principal transformations, as is typically written by
synthetic chemists (adapted from https://de.wikipedia.org/wiki/Cimetidin under
CC BY-SA 3.0 license). The currently used Top-n accuracy measures also include
prediction of other reactants [12, 13, 19, 20, 24, 32], which may not be
necessary for classical methodical retrosynthesis planning.
That is why we decided to consider the recognition of the largest reactant as
a new measure of the model performance: Classical Retro-Synthesis Accuracy,
i.e., the accuracy of prediction of the “Maximal Fragment” (MaxFrag). The
MaxFrag was 85.4% for the Top-5 reaction prediction (Table 2). The MaxFrag is
important to estimate an ability of a system to automatically deduce the
correct reaction class. This strategy is orthogonal to explicitly providing
reaction class information as input to a model [24]. Adding the reaction class
as prior information is equivalent to getting a hint on an exam: this is
impractical and also reduces the chance of proposing alternate feasible
reactions. Using MaxFrag is more accurate and logical than providing a
reaction class as prior information. Besides MaxFrag and Top-n, other scores
were proposed to evaluate the success of retro suggestions/reactions, e.g.,
the matching score by Satoh and Funatsu [37], the ’in-scope’ filter by Segler
et al. [5], and the forward transformer score by Schwaller et al. [34].
However, MaxFrag is the easiest and the interpretable one.
### 2.16 Retrosynthesis data quality and MaxFrag accuracy
The use of the classical retro-synthesis accuracy (MaxFrag Top-n) calculated a
systematic higher accuracy in comparison to the traditional Top-n scores. To
explain this fact, we analysed our datasets and found four types of reactions:
non-reagent reactions, one reagent reactions, multiple reagent reactions, and
unclear reagent reactions. Non-reagent reactions were reactions that did not
work (i.e., A + B -> A). One reagent reactions had only one starting material
for the product (A -> P), multiple reagents had multiple starting materials
for the same end product (A + B -> P), and finally unclear reagents where the
reaction conditions, solvent, salts, and so on, were included as reagents (A +
B + N -> P, where N were chemicals that did not participate to form the
product). Depending on the dataset the proportions of these reaction
categories slightly varied. In the MIT dataset around 0.5% of reactions were
non-reagent reactions, and around 10% of the reactions were unclear reagent
reactions while there were less than 1% of such reactions in the USPTO-50k
dataset. Thus, for the MIT set it would be impossible to fully predict about
10% of reactions for the retrosynthesis, since they contained chemicals “N”
that did not form the reaction, but only conditions, solvent, etc. This more
challenging problem of predicting not only the reactants but also the
reagents, while still keeping diverse precursor suggestions was addressed
elsewhere 34. For the direct synthesis that was not a severe problem since the
Transformer could correctly identify and pick-up the interacting components
(“A” and “B”) and predict the product. However, the use of Top-n for
retrosynthesis is questionable due to the aforementioned problem. The use of
MaxFrag accuracy decreased those effects by focusing on the main reagent. That
is why, in our opinion, the MaxFrag score better reflected the chemistry than
Top-1 accuracy.
Still there is an unsolved challenge with this score due to the possibilities
to synthesize the same products starting from multiple reagents. Both Top-n
and MaxFrag Top-n scores were calculated by using the exact match of the
predicted and target molecules. But, for example, in the reaction R-R1 + NH3
-> R-NH2 multiples choices of R1, i.e. -OH, -Cl, -Br, -I or -F, would be all
correct predictions. The only difference would be the reaction rates and
yields, which are not part of the prediction algorithms. Unfortunately the
currently used scores, including MaxFrag, cannot yet correctly account for
this problem. The problem to some extent could be alleviated by using
Top/MaxFrag-n instead of Top/MaxFrag-1 scores: by considering multiple
reagents generated by the model, we could also get the one provided in the
initial reaction. Thus, the retrosynthesis task is not about getting high
Top-1 accuracy. Any classical organic synthesis book, such as the famous
Larock’s “Comprehensive Organic Transformations” [38] indicates multiple ways
to synthesize chemical compounds and this has to be reflected in the score.
The classical retro-synthesis accuracy measured by MaxFrag is a first attempt
to better handle those data ambiguities during the validation process and we
highly encourage other users to use it. However, in order to enable a
comparison with the previous studies we also reported traditional Top-n
scores.
### 2.17 Analysis of prediction accuracy for different classes
The original dataset USPTO-50k set [12] provides a reaction type label for
every reaction. In total, 10 reaction classes ranging from
protection/deprotection, to carbon-carbon bond and heterocycle formation
present the most common reactions in organic synthesis. The comparison of
accuracy for each class of reactions was presented in Fig. 6. Our best model
showed excellent results, outperforming the state-of-the-art Self-Corrected
Transformer (SCORP) [19]. Functional group interconversion and addition, as
well as carbon-carbon bond formation were the most difficult for the models to
predict. It was not surprising, due to the diverse possibilities for choosing
reactions and corresponding reactants for C-C bond creation compared to more
straightforward oxidation or protection where the set of groups and reactants
is more narrow.
Figure 6: Top-10 accuracy of prediction of different classes of reactions.
### 2.18 Prediction of direct reactions
The same strategy described in this work was applied to predicting direct
reactions from the USPTO-MIT dataset [22]. We used 439k reactions (training
and validation set were joined together) as the training set and predicted 40k
reactions from the test set by training the Transformer with the same
architecture and parameters. The separated and mixed sets were used. In the
separated set reactants and reagents were separated with the “>” sign while in
mixed set all “>” are substituted with “.” and the order of reactants and
reagents was additionally shuffled. The mixed set was more difficult for
training since the Transformer had to identify the reaction center from a
larger number of molecules. However, such a set better reflected a practical
application since separation of data on reactants and reagents in some cases
would not be possible without a knowledge of the target product, and thus it
did provide a hint to the Transformer about the reaction center. We have
removed 316 reactions from the training set where the largest products had
length smaller than 5 characters (no reactions were removed from the test
set). The Transformer was training using the x5N augmentation protocol for the
separated set as well as the x5S and x5M protocols for the mixed set. Since it
would be impractical to predict all reagents and reactants for the
retrosynthesis task, which was used to additionally augment data in the x5M
protocol, only the largest reactant was retained as a target for the reverse
reactions. Augmented test sets were predicted using beam size 10 (Table 3).
For the mixed test set the order of reactants and reagents was shuffled.
Table 3: Prediction accuracy for direct reaction from USPTO-MIT test set using beam size = 10. Training set | Test set x1 | Test set x20 | Test set x100
---|---|---|---
Top-1 | Top-5 | Top-10 | Top-1 | Top-5 | Top-10 | Top-1 | Top-5 | Top-10
x5N (separated) | 91.1 | 96.3 | 96.7 | 91.8 | 96.9 | 97.3 | 91.9 | 97 | 97.4
x5S (mixed) | 90 | 95.8 | 96.2 | 90.4 | 96.4 | 96.9 | 90.4 | 96.5 | 97
x5M (mixed) | 90 | 95.5 | 95.7 | 90.2 | 96.1 | 96.5 | 90.2 | 96.2 | 96.8
As in previous studies, separation of reagent and reactants with “>” symbols
contributed to a model (x5N) with higher prediction scores than for models
with mixed sets (x5S and x5M). The additional augmentation of data using
retrosynthesis reactions (x5M) did not improve the model. This could be due to
the fact that the data for direct reactions were much larger and already
contained sufficient information to develop accurate models. While using the
x100 test set still contributed better prediction accuracy than using x20, the
improvements were in the order of 0.1% or no improvement at all. Thus the
effect of using larger augmentations on model performance reached saturation
for the x100 test set.
### 2.19 Comparison with published models for direct synthesis using USPTO-
MIT set
The USPTO-MIT was used as benchmarking for direct synthesis predictions in
multiple articles. The AT provided the highest gain in performance for
prediction of the more challenging mixed dataset (Table 4333The results of the
models applied to x100 augmented dataset using beam size = 10. Model was
trained on a set of 439k reactions, which combines both the training set of
400k and the validation set of 39k from [22]. The model was trained on the
400k training set to better match performance of previous models.). Since the
model was trained with randomly shuffled augmented data, it was able to
generalise very well and provide excellent predictions for the new mixed data.
In order to provide a more adequate comparison with previous studies we also
developed a model based on exactly the same training data of 400k.
Interestingly, the use of a smaller dataset slightly increased Top-1
performance to 90.6% but decreased Top-5 performance to 96.1. It should be
noted that improvements for direct synthesis look small, i.e. just few
percentages. Indeed, the model performance for the direct synthesis increased
from 88.6 to 90.6 (Top-1) and 96.1 from 94.2 (Top-5) as compared to the single
model reported in ref18. Actually, this is a significant increase in
performance since AT decreased the relative errors by 15% and 30% for both
sets, respectively, if we consider that we can predict direct synthesis with
100%. In reality we approach the experimental accuracy and further decrease of
the errors will be challenging.
Table 4: Comparison of recently published methods for direct synthesis prediction on the USPTO-MIT set. Model | Top-1 | Top-2 | Top-5 | Reference
---|---|---|---|---
separated | mixed | separated | mixed | separated | mixed
Transformer (single) | 90.4 | 88.6 | 93.7 | 92.4 | 95.3 | 94.2 | [18]
Transformer (ensemble) | 91 | | 94.3 | | 95.8 | | [18]
Seq2Seq | 80.3 | | | | 87.5 | | [11]
WLDN | 79.6 | | | | 89.2 | | [32]
GTPN | 83.2 | | | | 86.5 | | [39]
WLDN5 | 85.6 | | | | 93.4 | | [23]
This work (x100, beam 10) | 91.9 | 90.4 | 95.4 | 94.6 | 97 | 96.5 |
AT trained with same training set as in [22] | 92 | 90.6 | 95.4 | 94.4 | 97 | 96.1 |
### 2.20 Comparison with published models for retrosynthesis tasks
#### USPTO-50k:
The proposed augmentation protocol achieved the best published results on the
USPTO-50k dataset (Table 5). In the previous studies with this set the authors
separated data on training, validation and test sets. In all our analyses,
since the validation set was not used for model selection and we did not
observe the model overfitting [35] we joined training and validation sets to
use all data in order to develop better models. While we think this is a fair
comparison (it is up to the developers of the model to decide on how to best
use the available data), we also added results when the model was developed
with only the 40k compounds for USPTO-50k set (Table 5). The accuracies of the
models developed with 40k and 45k sets were very similar for the test set.
Thus, the data augmentation allowed to compensate for the smaller size of the
training set.
Table 5: Comparison of retrosynthesis recently published methods for retrosynthesis prediction on USPTO-50k. Model | Top-1 | Top-2 | Top-5 | Top-10 | Ref | Comments
---|---|---|---|---|---|---
Seq2Seq | 37.4 | | 57.0 | 61.7 | [12] | 40/5/5 split; splitting any reactions with multiple products into multiple single product and removal of trivial products
Transformer (3*6) | 42.7 | 52.5 | 69.8 | - | [13] | 45/5 split: no validation set was used
Transformer (6*8), (self corrected) | 43.7 | | 65.2 | 68.7 | [19] | 40/5/5 split, reagents from reactants are removed
Transformer, augmentation | 44.8 | 57.1 | 57.7 | 79.4 | [32] | same as in [12]
Similarity-based | 37.3 | | 63.3 | 74.1 | [20] | same as in [12]
Graph Logic Network | 52.5 | | 75.6 | 83.7 | [24] | same as in [12, 19]
G2Gs | 48.9 | | 72.5 | 75.5 | [25] | same as in [12]
AT | 53.5 | 69.4 | 81 | 85.7 | | same as in [13]. The results of the reference model applied to x100 augmented dataset using beam size = 10.
AT | 53.2 | 68.1 | 80.5 | 85.2 | | only 40k samples were used as training set to match the other results.
AT MaxFrag | 58.5 | 73 | 85.4 | 90 | | same as in [13]. The classical retro-synthesis accuracy was estimated as accuracy for prediction of the largest fragment (MaxFrag).
AT MaxFrag | 58 | 73.4 | 84.8 | 89.1 | | only 40k samples were used as training set to match the other results.
#### USPTO-MIT:
We also analysed the performance of the model at retrosynthesis of the USPTO-
MIT set. Compared to USPTO-50k this database also contained multiple reagents
and possible catalysts. In our previous analysis (Table 3), we used the
retrosynthesis of the largest fragment as part of the “mix” protocol (x5M),
i.e. the products were used as input data contained with “.” to predict the
largest reactant (as explained in the supplementary materials, in order to
distinguish both direct and retrosynthesis reactions, one of them started with
a dot). The dot in front of the SMILES allowed the Transformer to distinguish
retrosynthesis from the primary studied direct synthesis reaction. But, of
course, the model trained with such data could be also used for
retrosynthesis, provided that input data also started with “.”. We also
developed a new retrosynthesis model for this set by making it more compatible
to USPTO-50k. For this we kept only the 1-2 largest fragments as the targets
for retrosynthesis prediction and trained a new model using the x5S protocol.
Both models were used to predict the 40k test set which was augmented 100
times. The MaxFrag performances of x5S model, 61.9% (Top-1), 84.4% (Top-5) and
86.9% (Top-10) were very similar to those calculated for the USPTO-50k set
(58.5, 85.4 and 90 - see Table 5). The x5M model, which as aforementioned was
a “by-product” of our direct reaction predictions, calculated MaxFrag of
61.1%, 84.4% and 88.2% for the MaxFrag Top1-,Top-5 and Top-10, respectively.
Considering that the USPTO-MIT set contained more diverse reactions than
USPTO-50k, this result clearly demonstrated the excellent performance of the
developed approach and its scalability. The Augmented Transformer (AT) was
able to improve its performance for the Top-1 by extracting knowledge from a
much larger dataset of reactions.
#### USPTO-full:
The final testing was done using a USPTO-full set by Dai et al. [24] The
authors created a large dataset from the entire set of reactions from USPTO
1976-2016. For reactions with multiple products they duplicated them into
multiple ones with one product each. The authors also removed duplications in
reactions as well as those with wrong mapping to obtain train/valid/test
datasets with 800k/100k/100k sizes. Our analysis identified that some
reactions in these sets were still invalid, e.g. contained no products or just
single ions as reactants (e.g., US08163899B2,>>[OH2:11],
US06048982,CC(=O)OCCCCC[I:22]>>[I-:22], US07425593B2,>>[K:12],
US08114877B2,CC[I:13]>>[I-:13]). We eliminated such reactions as well as those
where reactants had less than five atoms in total, since these were unlikely
to be correct reactions. This procedure decreased sizes of the
train/valid/test sets on average by 4% to 769k/96k/96k. The AT trained using
x5M protocol using the 769k training set calculated the higher performance
compared to results from the previous study (Table 6444Model was trained using
a filtered training set of 769k from [20]. Accuracies in parentheses
correspond to those recalculated for the test set by assuming that AT failed
for all 4% of excluded reactions. Results for retrosim and neuralsym
approaches as reported by Dai et al. [24] ). Considering that after the
removal of the 4% erroneous reactions the test dataset was decreased, we also
included recalculated performance for it by assuming the worst case scenario:
that AT and other tested methods failed for all excluded sequences. Even for
this very conservative estimation the AT provided significant improvements
compared to previously reported results. The MaxFrag accuracies for USPTO-full
were lower compared to that of other analysed sets due to the much higher
diversity of this set.
Table 6: Top-k accuracy for retrosynthesis prediction on USPTO-full dataset. | Retrosim [20] | Neuralsym [3] | GLN [24] | AT
---|---|---|---|---
Top-1 | 32.8 | 35.8 | 39.3 | 46.2 (44.4)
Top-2 | | | | 57.2 (54.9)
Top-10 | 56.1 | 60.8 | 63.7 | 73.3 (70.4)
MaxFrag Top-1 | | | | 54
MaxFrag Top-2 | | | | 66.3
MaxFrag Top-5 | | | | 77.3
MaxFrag Top-10 | | | | 80.6
Thus for all analysed data sets the AT provided an outstanding performance by
consistently and significantly overperforming all previously published models
for all statistical performances.
## 3 Conclusions and outlook
This study showed that careful design of the training set was of paramount
importance for the performance of the Transformer. Training the model to learn
different representations of the same reaction by distorting the initial
canonical data eliminated the effect of memorisation and increased the
generalisation performance of models. These ideas are intensively used, e.g.
for image recognition [40], and have been already successfully used in the
context of several chemical problems [27, 28, 29, 30], including reaction
predictions [18, 31], but were limited to the input data. For the first time
we showed that application of augmentation to the target data significantly
boosts the quality of the reaction prediction. We also showed for the first
time that the frequency of predicted SMILES could be used as a confidence
metric for (retro)synthesis prediction and can provide quantitative estimation
of the most probable reactions amid Top-n predicted outcomes. It is very
critical to estimate the quality of the reaction prediction since it could
help to better prioritise multi-step retrosynthesis. The developed methodology
is unique to the use of augmentation techniques, currently unavailable to GCNs
[24], which directly operates with graphs. The estimated accuracy of
prediction can help to distinguish reactions, which are difficult to predict,
from typo and erroneous reaction data, which will be important to clean up the
reaction data and further improve model quality. We also introduced a new
MaxFrag measure, classical retro-synthesis accuracy, which in our opinion
better reflects the requirements for retrosynthesis analysis.
It should be mentioned that use augmentation was first studied by authors of
[18], who introduced Transformer to chemistry and applied it to chemical
reactions by using SMILES instead of the text sequences. The augmentation of
input data, which was done in that article, provided only a minor improvement
of their models. Because of its small impact it was not followed in several
other Transformer-based works, including our own study [13, 19]. In this
article, we brought an original idea on how to augment chemical data, which
provided a significant improvement of the results for all analysed datasets.
## Supporting Information
The Supporting Information contains a description of methods, explanation and
examples of augmentation protocols, illustration of the procedure of ranking
predicted reactions, examples of distributions of predicted SMILES, figures
explaining performances of Transformer models. The training and test set
(including augmented data), model and model predictions are available at .
## Acknowledgments
This study was partially funded by the European Union’s Horizon 2020 research
and innovation program under the Marie Skłodowska-Curie Innovative Training
Network European Industrial Doctorate grant agreement No. 676434, “Big Data in
Chemistry” and ERA-CVD "CardioOncology" project, BMBF 01KL1710 as well as
Intel grant. The article reflects only the author’s view and neither the
European Commission nor the Research Executive Agency (REA) are responsible
for any use that may be made of the information it contains. The authors thank
NVIDIA Corporation for donating Quadro P6000, Titan Xp, and Titan V graphics
cards for this research work. The authors thank Michael Withnall (Apheris AI)
and Alli Michelle Keys (Stanford University) for their comments and English
corrections as well as Marios Theodoropoulos (University of Geneva) for
interesting discussions. We also would like to thank the anonymous reviewers
for their insightful and sometimes even provocative comments answering of
which significantly increased the value of this study.
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Supplementary materials
## Methods
### Model architecture
Following our previous study [13] we used the Transformer [17] architecture to
train all the models. The key component of the Transformer architecture is a
self-attention block equipped with internal memory and attention. During the
training phase the block extracts and structures the incoming data, splitting
it into memory keys and associated values. Thus, the block resembles a
library, where all the books (values) are referred to by an index (keys). On a
new request the model calculates the attention to the known keys and then
extracts knowledge from the values proportionally. The Transformer shows
excellent results not only on (retro) synthesis [13, 11, 19] tasks but also on
ordinary classification and regression QSAR studies [30].
The performance of the Transformer was estimated for the prediction of the
whole training set after each epoch. The five models with the highest fraction
of correctly predicted training set SMILES were stored. As a rule, the stored
models correspond to the latest epochs of training. The weights of five stored
models were averaged to form the final model, which was used to predict
reactions from the test sets.
After several trials, we decided to use a Transformer architecture with 6
layers and 8 heads (6x8), which was used in the original work [11]. We found
that using a smaller architecture with 3 layers and 8 heads (3x8), which was
used in our previous study 1, required more epochs to converge and thus longer
overall training time to achieve the same performance. We restricted training
of the model to 100 epochs to perform model development in a reasonable time
and preserve the possibility to compare different augmentation approaches. For
the final optimal architectures, we further investigated the effect of
training time.
### Influence of the batch size
The speed of calculations using augmented data was linearly increasing with
the dataset size. One epoch using the USPTO-50k set (40k reactions) took 82s
on a Titan V. Training of the USPTO-full augmented set (4.3M reactions) took
9514s, i.e. approximately 120 times longer. The use of a larger batch size (in
our work we formed batches of length ca. 3000 characters, which approximately
corresponded to 12-15 reactions and required about 3.5G of GPU memory for the
given Transformer configuration) could increase the speed of calculations.
However, we noticed that large batches (i.e., we tried a batch of length
30,000 characters on Tesla V100 with 32GB of memory) could result in a
decrease in the speed of convergence. Therefore, for this study we used a
batch with 3000 characters.
### Beam search
When generating new SMILES, the Transformer predicted at each step
probabilities for all characters from its vocabulary. There are two common
approaches to decoding from a linguistic model, such as a Transformer. The
first one, a greedy search, always takes the element (symbol, word) with the
maximum probability at each step. The second one, beam search 6, tracks in
parallel several possible decodings (beam size) and sorts them according to
the sums of logarithms of probabilities of each element. Thus, beam search can
select those decodings where at one step the element to be chosen has no
maximum probability but later symbols have maximum so the overall sum is
greater than in greedy search settings. The beam search with n=5 or n=10 beams
were used to predict the test set for the majority of analyses performed in
this study. As a result of a search using a beam with size n, the Transformer
produced up to n SMILES. Because of the generation procedure these were always
unique sequences. Some of them, however, could be errors or could be different
representations of the same SMILES.
### Augmentation
The datasets used in this study comprised both canonical and so-called
augmented SMILES. Both canonical and augmented SMILES were generated using
RDKit [41]. We introduced this SMILES free augmentation method into RDKit at
the end of 2018 [27, 28]. The augmented SMILES were all valid structures with
an exception that the starting atom and the direction of graph enumerations
were selected by chance. The augmentation increased the diversity of the
training set.
The baseline dataset contained only canonical SMILES. The other datasets also
contained SMILES augmented as summarized. Four different scenarios were used
to augment training set sequences. Sequences were augmented using increasingly
complex datasets as shown in Tables A1 and LABEL:tbl:s2. Namely, we used
augmentation of products only (xN), augmentation of products and
reactant/reagents (xNF), augmentation of products and reactants/reagents
followed by shuffling of the order of reactant/reagents (xNS), and finally
mixed forward/reverse reactions, where each retrosynthesis reaction from xNS
was followed by the inverse (forward synthesis) reaction (xNM). One more
analysis was performed where the Transformer was asked to predict a fixed
random SMILES string.
Only xN were used for augmentations of the test sets because no information
about reactant/reagents could be used for the retrosynthesis prediction.
Table A1: Augmentations of analyzed training datasets. Dataset | Description
---|---
xN | For N=1 the dataset contains canonical SMILES for reactants and products. For N>1 in addition to one canonical SMILES, the dataset contains (N-1) instances of the same reaction with augmented SMILES for the products (input data). The SMILES of reactants were canonical.
xNR | Products are encoded as canonical SMILES, but for reactants only one of possible random SMILES was chosen.
xNF | The first instances of each reaction contained canonical SMILES while other (N-1) instances were augmented for both input (products) and output (reactants) data. The order of SMILES in output data was not changed.
xNS | Same as xNF but the order of SMILES in reactants was randomly shuffled.
xNM | The same as xNS but also contained the same number of inverted (forward synthesis) reactions. The forward reactions started with “.” to distinguish them from retro-synthetic ones.
Table A2: Examples of data augmentations for two reactions. Canonical SMILES are shown in bold. Dataset | Input (product), output (reactants) data | Example
---|---|---
x0 | canonical, canonical | CC(c1ccc(Br)nc1)N(C)C,CC(=O)c1ccc(Br)nc1.CNC O=Cc1cncc(Br)c1,O=C(O)c1cncc(Br)c1
x2 | canonical,canonical random, canonical | CC(c1ccc(Br)nc1)N(C)C,CC(=O)c1ccc(Br)nc1.CNC n1c(Br)ccc(c1)C(N(C)C)C,CC(=O)c1ccc(Br)nc1.CNC O=Cc1cncc(Br)c1,O=C(O)c1cncc(Br)c1 c1(cncc(Br)c1)C=O,O=C(O)c1cncc(Br)c1
x2R | canonical, fixed random random, fixed random | CC(c1ccc(Br)nc1)N(C)C, c1cc(Br)ncc1C(=O)C.CNC n1c(Br)ccc(c1)C(N(C)C)C, c1cc(Br)ncc1C(=O)C.CNC O=Cc1cncc(Br)c1, c1c(cncc1C(=O)O)Br c1(cncc(Br)c1)C=O, c1c(cncc1C(=O)O)Br
x2F | canonical, canonical random, random | CC(c1ccc(Br)nc1)N(C)C, CC(=O)c1ccc(Br)nc1.CNC n1c(Br)ccc(c1)C(N(C)C)C, CC(=O)c1ccc(nc1)Br.CNC O=Cc1cncc(Br)c1, O=C(O)c1cncc(Br)c1 c1(cncc(Br)c1)C=O, c1c(cncc1C(=O)O)Br
x3S | canonical, canonical random, shuffled random, shuffled | CC(c1ccc(Br)nc1)N(C)C,CC(=O)c1ccc(Br)nc1.CNC n1c(Br)ccc(c1)C(N(C)C)C,CNC.CC(=O)c1ccc(nc1)Br CN(C(c1ccc(Br)nc1)C)C,CNC.c1cc(Br)ncc1C(O)C O=Cc1cncc(Br)c1,O=C(O)c1cncc(Br)c1 c1(cncc(Br)c1)C=O,c1c(cncc1C(=O)O)Br n1cc(cc(c1)C=O)Br,OC(=O)c1cncc(c1)Br
x2M | canonical, canonical .canonical, canonical random, shuffled .shuffled. random | CC(c1ccc(Br)nc1)N(C)C,CC(=O)c1ccc(Br)nc1.CNC .CC(=O)c1ccc(Br)nc1.CNC,CC(c1ccc(Br)nc1)N(C)C n1c(Br)ccc(c1)C(N(C)C)C,CNC.CC(=O)c1ccc(nc1)Br .CNC.CC(=O)c1ccc(nc1)Br,n1c(Br)ccc(c1)C(N(C)C)C O=Cc1cncc(Br)c1,O=C(O)c1cncc(Br)c1 .O=C(O)c1cncc(Br)c1,O=Cc1cncc(Br)c1 c1(cncc(Br)c1)C=O,c1c(cncc1C(=O)O)Br .c1c(cncc1C(=O)O)Br,c1(cncc(Br)c1)C=O
## Analysis of predicted SMILES
The beam search was used to infer n=5 (or more) reactant sets from the model
for each entry in the test file. The SMILES predicted within the same beam
search were sorted in the decreasing order of their probabilities. Predictions
containing erroneous SMILES representations, which could not be processed by
RDKit, were discarded. The remaining predictions were converted to canonical
SMILES. In cases where the predicted reaction contained several disconnected
SMILES, they were sorted to have the same representation. If two or more
identical predictions were found for the same input only the first prediction
was kept: in this way we deduplicated reactions predicted for the same input
data. For augmented test datasets, SMILES predicted for the same reaction were
accumulated and those with the largest number of occurrences were selected as
the Top-ranked. If exactly the same number of predictions were found for two
or more SMILES, the weights of the SMILES were set to be inversely
proportional to their relative position in the respective beam search.
Precisely, to rank predictions we used the following formula
$rank(SMILES)=\sum_{n\in[{0,augmentations})}\sum_{i\in[1,beam]}\frac{\delta(SMILES_{n,i},TARGET)}{1.0+0.001*i}$
(1)
where the first sum was over canonical (n=0) and augmented SMILES for the same
input reaction. When the target canonicalized SMILES was equal to the
predicted canonicalized SMILES at position i of the beam search for
augmentation n, $\delta$=1. Otherwise, if predicted and target SMILES did not
coincide, $\delta$=0. The term 0.001*i was used to weight the predicted SMILES
to be inversely proportional to its position in the beam search (see also
Table A3 and A4).
### Top-n performance accuracy
For the analysed input reaction we received a set of generated canonical
SMILES (contributed both by beam search and augmentation procedure), which
were ranked as explained above. If any of these Top-n sequences coincided with
the target canonical SMILES for the analysed reaction, the prediction was
considered to be the correct one. The Top-n accuracy was the ratio of the
number of correct predictions to the total number of sequences in the test
set.
Table A3: Illustration of a procedure used to rank predicted reactions. Step | Input SMILES | Beam1,Beam2,Beam3
---|---|---
Initial prediction | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CC(C),C(C)CC,C(N)N CNN,CCC,CC= CC.CCC,CCC.CC,C#
Canonicalisation, sorting and error detection | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CCC,CCCC,CNN CNN,CCC,error CC.CCC,CC.CCC,error
Elimination of duplicates and erros | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CCC,CCCC,CNN CNN,CCC CC.CCC
Enumeration | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CCC(0),CCCC(1),CNN(2) CNN(0),CCC(1) CC.CCC(0)
Ranks, see Eq. 1. | | CCC = [1] +[ 1/(1+1./1000)] + [0] = 1.999 CNN = [1/(1+2./1000)] + [1] + [0] = 1.998 CC.CCC =[ 0] + [0] + [1] = 1 CCCC = [1/(1+1./1000)] + [0] + [0] = 0.999 The Top-2 ranked predictions are CCC and CNN.
Table A4: Illustration of procedure used to rank predicted reactions when using multiple predictions within the same beam. Step | Input SMILES | Beam1,Beam2,Beam3
---|---|---
Initial prediction | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CC(C),C(C)CC,C(N)N CNN,CCC,CC= CC.CCC,CCC.CC,C#
Canonicalisation, sorting and error detection | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CCC,CCCC,CNN CNN,CCC,error CC.CCC,CC.CCC,error
Elimination of duplicates and erros | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CCC,CCCC,CNN CNN,CCC CC.CCC
Enumeration | SMILES_CAN SMILES_AUG1 SMILES_AUG2 | CCC(0),CCCC(1),CNN(2) CNN(0),CCC(1) CC.CCC(0)
Ranks, see Eq. 1. | | CCC = [1] +[ 1/(1+1./1000)] + [0] = 1.999 CNN = [1/(1+2./1000)] + [1] + [0] = 1.998 CC.CCC =[0] + [0] + [1] + [1/(1+1./1000)] = 1.999 CCCC = [1/(1+1./1000)] + [0] + [0] = 0.999 The Top-2 ranked predictions are CCC and CC.CCC.
The SMILES strings with the largest weights and thus those that appeared most
frequently amidst the first sequences within the beam predictions were
selected as the Top-ranked. The Top-1 and Top-5 SMILES were used to estimate
the prediction performances of models.
### Analysis of stereochemistry-free datasets
About 20% of the reactions in the training and test sets had molecules with
stereochemistry. The stereochemistry was encoded in SMILES with “/”,”\”,”@”
and “@@” characters. However, a number of practical projects have relaxed
stereochemistry requirements. Therefore, we separately reported the
performance of the models for datasets with and without stereo-chemical
information.
### Character and exact sequence performance during training
During the model training, we calculated character-based performance, which
corresponded to the number of exactly predicted characters for the target
SMILES, as well as exact sequence accuracy indicating the number of correctly
predicted exact sequences. Both of these measures were approximations of the
final accuracy, for which the predicted SMILES were first converted to
canonical ones and only after were compared to the target values.
Table A5: Examples of distributions of predicted SMILES. Reaction | Frequency of SMILES | Ratio of the most frequent to all SMILES
---|---|---
CCOC(=O)C1CCCN(C(=O)COc2ccc(-c3ccc(C#N)cc3)cc2)C1>>CCOC(=O)C1CCCNC1. N#Cc1ccc(-c2ccc(OCC(=O)O)cc2)cc1 | 926* 51 7 6 2 1 1 1 1 1 1 1 | 926/999 = 0.93
CCCCC(=O)O>>CCCCC(=O)OC(=O)CCCC | 203 112 107 98 57 19 16 13 12 12 12 12 11 11 11 11 11 11 11 11 11 11 8 8 8 8 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 | 203/999 = 0.2
Star indicates the correctly predicted reaction. For the first reaction the
most frequent SMILES was predicted 926 times or 93% of all predictions. For
this SMILES the Transformer was very confident in the outcome of
retrosynthesis, which it correctly predicted. For the second reaction the
Transformer generated 78 different SMILES and the Top-1 SMILES appeared only
in 20% of all predictions. For this reaction the Transformer failed to predict
the correct SMILES at all.
Unless otherwise noticed, the results presented in the supplementary Figures
were calculated using the USPTO-50k set.
Figure A1: Top-1 accuracies calculated for models developed with different
augmentation scenarios and using 40k sequences as the training set. All models
were applied to the x1 (canonical) test and validation set as they were
defined in 10. As we can see the performance of models is similar for both
training and validation sets and it is monotonically increasing with the
number of iterations. This observation was the main motivation to join
training and validation sets as a single set, which was used for model
development. Figure A2: Monitoring set accuracy (measured as a character
accuracy) of Transformer for prediction of canonical (x10) and random (x10R)
SMILES for USPTO-50k set (see Table 1 and 2 for explanation of used
abbreviations). Figure A3: Top-1 full-sequence retrosynthesis accuracies
calculated for models developed with different augmentation scenarios for
USPTO-50k training set. All models were applied to the x20 test set. Figure
A4: Accuracy of prediction of SMILES generated at the respective position of
the beam search using the largest beam size=44. The results were calculated
for test set prediction using the model trained with 500 iterations on the
USPTO-50k set. The use of canonical SMILES as input produced the highest
accuracy (48.3%) for the first beam, which degraded for other positions of the
beam while use of augmented SMILES provides about 44% correct predictions,
which is slowly decreasing with the increase of the beam position. The number
of erroneous SMILES is increasing with the beam position for both types of
SMILES, but it was significantly higher for predictions when using canonical
SMILES as input. Figure A5: Accuracy and density (fraction of predictions) of
the Transformer for MaxFrag Top-5 retrosynthesis accuracy as a function of the
frequency of appearance of five most frequent SMILESes in the output of the
Transformer (see also Fig. 4 in the article). Due to a small number of samples
and high variability of data the average accuracy is shown for each first left
datapoint for the same or smaller frequencies. For example for USPTO 50k set
the accuracy of 58.9% for frequency 0.6 was calculated by averaging the
MaxFrag accuracies for SMILES with frequencies $\leq$ 0.6. There were 7.1% of
such predictions in the test dataset. Figure A6: Accuracy and density
(fraction of predictions) of the Transformer for MaxFrag Top-1 direct
synthesis accuracy as a function of the frequency of appearance of the Top-1
SMILES in the output of the Transformer for the respective test sets of the
models (see also Fig. 4 in the article).
### Calculation of the decrease of the relative errors
Let us assume that we can (theoretically) get 100% accuracy. Top-5 error of
the previous model for the mixed set was 100-94.2 = 5.8%. The error of our
model is 100 - 96.1 = 3.9%. The relative decrease of the error is
(5.8-3.9)/5.8 = 32.7%.
|
2024-09-04T02:54:57.597595 | 2020-03-05T18:14:33 | 2003.02806 | {
"authors": "Rumen N. Georgiev, Sara O. Toscano, William E. Uspal, Bram Bet, Sela\n Samin, Rene van Roij, and Huseyin Burak Eral",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26063",
"submitter": "Rumen Georgiev",
"url": "https://arxiv.org/abs/2003.02806"
} | arxiv-papers | # Universal motion of mirror-symmetric microparticles in confined Stokes flow
Rumen N. Georgiev and Sara O. Toscano
Process and Energy Department, Delft University of Technology, The Netherlands
William E. Uspal
Department of Mechanical Engineering, University of Hawaií at Manoa, USA
Bram Bet, Sela Samin and René van Roij
Institute for Theoretical Physics, Utrecht University, The Netherlands
Huseyin Burak Eral
Process and Energy Department, Delft University of Technology, The Netherlands
and
Van’t Hoff Laboratory for Physical and Colloid Chemistry, Utrecht University,
The Netherlands
Corresponding author<EMAIL_ADDRESS>
Separation on the microscale is a persistent industrial challenge:
pharmaceutical crystal polymorphs [1, 2], specific strains of yeast cells in
the food industry [3], mammalian cells [4] and microplastic pollutants [5, 6]
all come in different shapes, yet comparable sizes. Advances in microfluidics
have resulted in robust and high throughput methods for micron-scale
segregation. These techniques rely on external force fields [7, 8], sorting
based on fluorescence [9], intricate separator geometries [10, 11, 12, 13, 14,
15, 16, 17, 18] or carriers with non-Newtonian behaviour [19]. An alternative
approach towards microscale separation is to leverage the long-range
hydrodynamic interactions emerging from fluid-structure coupling [20, 21]. By
tuning these interactions particle trajectory can be controlled, thus enabling
separation [22].
A model system common in microfluidic applications, exhibiting such
interactions, is confined Stokes flow in a Hele-Shaw cell. In it, particles or
droplets are sandwiched between a pair of confining walls of a shallow
microfluidic channel and are subjected to creeping flow [23]. Owing to the
shallowness of the cell, the flow is effectively two-dimensional [24]. What is
more, the particle scatters the surrounding fluid, creating a dipolar flow
disturbance, which decays with $1/r^{2}$, where $r$ is the distance from the
particle centre. This flow disturbance strongly couples the particle to its
surroundings. Experimentally, creating and driving particles in shallow
channels has become widely accessible with the advent of microfluidics and
soft lithography [25, 26, 27, 28, 29, 30, 31]. Their easy fabrication and
versatile out-of-equilibrium behaviour make particles in confined Stokes flow
an interesting toy system for the study of flow-mediated separation and self-
assembly [32, 33].
Utilizing long-ranged hydrodynamic interactions (HIs), Beatus et al.
demonstrated how trains of ‘pancake’ droplets flow along a Hele-Shaw cell as
out-of-equilibrium 1D crystals [34, 35]. In a similar experiment, Shen and co-
workers compare the dynamics of clusters comprising 2 or 3 droplets as they
interact near or far away from the side walls of the cell [36]. The presence
of a side wall breaks the symmetry of the system and induces transversal
motion of the cluster. Cross-streamline migration is also present if the
symmetry of an individual particle, rather than that of an ensemble of
particles, is reduced. A particle with two planes of mirror symmetry, such as
a rod [37, 38] or a symmetric disk dimer [22], also moves towards one of the
side walls of a Hele-Shaw cell, provided its long axis is neither normal, nor
parallel to the flow. As one such particle approaches the channel boundary, it
begins to interact with its hydrodynamic image [39], the flow symmetry is
reduced even further and the particle begins to rotate. All three modes of
motion, namely, rotation, streamwise and cross-streamwise translation, are
also present when an asymmetric disk dimer is far away from any side walls as
demonstrated by Uspal, Eral & Doyle [22]. Evidently, screened hydrodynamic
interactions give rise to non-trivial behaviour not only in particle ensembles
[40, 41, 42, 43, 44], but also in single-particle systems with broken symmetry
[45, 46, 47, 48, 49, 50, 51]. A first step towards the development of low-cost
flow separators requires understanding the relation between the geometry of
one such particle and its trajectory in confined Stokes flow.
In this study, we combine theoretical and experimental approaches to
investigate how particle shape can be tailored to induce self-steering under
flow in quasi-2D microchannels. Controlling the motion of a particle in flow
facilitates its separation. To this end, we use optical microscopy to track
the in-plane motion of a variety of particles with a single mirror plane
subjected to creeping flow in a shallow microfluidic channel. The mirror plane
is perpendicular to the top and bottom walls of the channel and bisects the
particle in two identical pieces (white dashes in Fig. 1 (a)-(d)). Through
finite element calculations we link the shape-dependent dynamics of the
particles to the flow disturbances they create as they lag the far-field flow.
Using Stokes linearity and the force-free nature of the particles, we collapse
their re-orientation and cross-streamwise dynamics onto two master curves. We
accomplish this collapse by scaling each particle’s angular and transversal
velocities by two characteristic times. Finally, through minimalistic scaling
relations we link these timescales to a particle’s geometrical parameters
including, but not limited to, area, moment of inertia and length. Our scaling
arguments predict the characteristic times from both experiments and finite
element computations up to a factor on the order of unity. This good agreement
among experiments, simulations and scaling arguments is a strong indication
that the observed dynamics is universal to mirror-symmetric particles in
quasi-2D Stokes flow.
Figure 1: Mirror-symmetric particles in quasi-2D Stokes flow. Stop-flow
lithography [29] produces strongly confined microparticles with various shapes
in a Hele-Shaw cell (a-d). We investigate particles with a single mirror
plane, each consisting of two or three simple building blocks such as disks,
squares or triangles, connected with rigid shafts. These particles are a
useful toy system to study how the geometry of a particle determines its
trajectory. We demonstrate this strong shape dependence by comparing the
trajectories of three particles with $R_{1}/R_{2}=1.5$: from top to bottom a
trimer with $\phi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$, a dimer and a
trimer with $\phi=$68\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (e). The small
arrows denote the orientation of the particles. The trajectories are obtained
via 3D finite element calculations. We assume a planar Poiseuille profile
along the height of the channel and Couette flow in the thin lubrication gaps
with height $h_{\textrm{g}}$ (f). Due to channel symmetry, we only present
half of a Hele-Shaw cell with particle to scale. Upon depth-averaging, we
arrive at the so-called Brinkman flow with steep velocity gradients near the
horizontal walls and constant velocity $u$ along most of the channel width
(g). In this top view the particle is magnified 2.5 times. The streamlines in
all three flow profiles are represented by horizontal blue arrows. Scale bars
are $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Figure 2: Particle-induced
flow disturbances in a Hele-Shaw cell. As the particle thickness
$H_{\mathrm{p}}=H-2h_{\mathrm{g}}$ is comparable to the channel height
$H_{\mathrm{p}}/H\simeq 0.8$, the particle lags the surrounding flow, creating
shape-specific velocity and pressure disturbances (cf. arrows and density
plots in a and b). As the disturbances differ, so too do the hydrodynamic
forces and torque acting on each particle differ. While the streamwise forces
$F_{x}$ on a dimer and a trimer have similar magnitudes (horizontal blue
arrows), the drift forces $F_{y}$ and torques $T_{z}$ acting on them differ
(vertical red arrows and clockwise green arcs, respectively). This shape-
dependence of the forces and torque results in distinct linear and angular
velocities, which manifest themselves in the different trajectories followed
by different particles (cf. c, d and e). The orientation and scaled position
$x/H$ as function of scaled time $t\times u/H$ are strongly dependent on
particle shape. The disturbances to the pressure and velocity fields, as well
as the forces and torques on the particles, are calculated using a 3D finite
element scheme [52]. In all sub-figures the flow is from left to right as
denoted by the white arrow in a. Scale bars are
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
To produce strongly confined polymeric particles with distinct shapes in a
Hele-Shaw cell we use stop-flow lithography (SFL) [29], as depicted in Fig. 1
(a)-(d). In a nutshell, SFL creates particles by projecting the image of a
mask onto a photoreactive fluid. We choose dimeric and trimeric particles,
composed of, respectively, two or three simple shapes connected by rigid
shafts. The building blocks for dimers are either disks, triangles or squares
(Fig. 1 (b), (d) and (a)), while those for trimers are always disks (Fig. 1
(c)). In both cases one of the building blocks is larger with a size ratio
$\kappa\equiv R_{1}/R_{2}$, where $1<\kappa\leq 3$ and $R_{2}$ is the radius
of the circle escribing the smaller shape. This asymmetry in the particle
ensures its rotation even far away from any side walls [53]. The trimers have
an additional geometrical parameter, namely, the angle $\phi$ formed between
the three disks (Fig. 1 (c)). The vertex of $\phi$ is defined as the centre of
the larger disk, while the two rays starting from it point to the centres of
the smaller equally-sized disks. By changing $\phi$ we gain additional control
over the dynamics of the particles (Fig. 1 (e)). The geometry of the particle
profoundly influences its trajectory: particles with identical starting
positions, yet slightly different geometries, follow dramatically different
paths, as demonstrated numerically in Fig. 1 (e).
As the particles are created in situ, we directly track their motion in the
viscous fluid by moving the stage of an optical microscope. We set the system
in motion by applying a small pressure drop across the channel, thus inducing
creeping flow with a Reynolds number $Re\sim 10^{-5}$. This flow regime,
together with the large aspect ratio of the channel $W/H>15$, allows us to
average out the parabolic profile expected along the channel height (Fig. 1
(f)). Thus, the particle is effectively subjected to an in-plane potential
flow with steep velocity gradients near the side walls of the channel and a
constant velocity $u$ for most of its width [54] (Fig. 1 (g)).
Apart from preventing sticking, the fluid layers with thickness
$h_{\mathrm{g}}$ present above and below the particle strongly affect its
motion (Fig. 1 (f) and its inset). As the particle moves along the channel
with a longitudinal velocity $\dot{x}$, it experiences additional drag,
because it shears the lubricating fluid in the gaps. Due to the strong
particle confinement the velocity profile in the gaps is close to linear [38,
52], allowing us to assume Couette flow in the gaps (Fig. 1 (f)). The drag
from the confining walls $F_{x,\mathrm{w}}$ scales with
$2\dot{x}\eta/h_{\mathrm{g}}$ and slows down the particle, where $\eta$ is the
dynamic viscosity of the fluid. Furthermore, it ensures the particle is
confined to the plane of the flow, because any tilt or out-of-plane motion
results in additional force acting on either face of the particle. Thus, the
particle exhibits three degrees of freedom: translation along the length $x$
and width $y$ of the channel and in-plane rotation $\theta$ (Fig. 1 (g)).
The particle lags the flow, perturbing the velocity field, and as a result
pressure builds up on the upstream particle surface. This flow disturbance is
strongly dependent on the particle shape (cf. (a) and (b) in Fig. 2). To
illustrate this phenomenon, we use finite element computations [52] to
calculate the forces and torque acting on two distinctly shaped particles with
$\kappa=1.6$: a dimer and a trimer with
$\phi=$120\text{\,}\mathrm{\SIUnitSymbolDegree}$$. We impose a unidirectional
inlet flow with height-averaged velocity $u$ and prescribe a longitudinal
velocity $\dot{x}=u/2$ to each particle. We orient the particles in such a way
that their mirror axes form an angle
$\theta=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$ with the flow. The particle
heights $H_{\mathrm{p}}$ in both cases are equal and comparable to the channel
height $H$, with $H_{\mathrm{p}}/H\sim 0.8$. While the longitudinal forces
$F_{x}$ acting on the two shapes are identical
($F^{\mathrm{D}}_{x}/F^{\mathrm{T}}_{x}=0.99$), the torques differ – the dimer
experiences a smaller torque $T^{\mathrm{D}}_{x}/T^{\mathrm{T}}_{x}=0.81$. The
superscripts ‘D’ and ‘T’ refer to ‘dimer’ and ‘trimer’. The difference in the
transversal forces $F_{y}$ is even more evident, as its direction also
changes: $F^{\mathrm{D}}_{y}/F^{\mathrm{T}}_{y}=-0.67$. This disparity can be
traced back to the pressure disturbance created by each particle – the larger
the disturbance, the larger the forces.
Figure 3: Universal behaviour of mirror-symmetric particles. Regardless of
their detailed shape, all studied particles follow a universal trajectory.
They exhibit the same quantitative behaviour as long as we take into account
two characteristic times, $\tau$ and $\tau_{y}$, scaling their modes of motion
[52]: exponentially-decaying rotation $\theta(t)$ to orient with the big disk
upstream (top curves) and bell-shaped translation in the lateral direction
$y\left(t\right)-y\left(t_{\perp}\right)$ (bottom curves). The only
geometrical element common to all studied particles is their single plane of
mirror symmetry. In all cases, the error bars denoting experimental
uncertainty are smaller than the symbols and are omitted. The particles and
their motion is sketched in the middle section of the figure. In the legend
disk, square and triangle dimers are dubbed ‘dumbbell’, ‘squares’ and ‘flask’
for brevity. Disk trimers are denoted as ‘tripod’.
The shape-dependence of the disturbances manifests itself in the distinct
dynamics of different particles, as shown in Fig. 1 (e). To demonstrate this
distinction experimentally, we compare the motion of three particles with
different shapes, which have one and the same initial position and
orientation, $x/H$, $y/H$ and $\theta_{0}=7\pi/9$, respectively (Fig. 2 (c),
(d) and (e)). While all three particles rotate to orient their larger building
block upstream, only the dimers experience a significant lateral drift. Nagel
et al. [38] report a similar coupling between longitudinal and transversal
motion for symmetric rods, which drift at a constant velocity as they flow
downstream. However, cross-streamwise motion is orientation-dependent,
resulting in a non-linear cross-stream trajectory when an asymmetric dimer
rotates: as our particles become perpendicular to the flow, their transversal
velocities diminish. Moreover, after acquiring this perpendicular orientation
both particles change the direction of their lateral motion (cf. panel 3 in
Fig. 2 (c) and panel 2 in Fig. 2 (d)). The coupling between rotation and
translation explains why the disk dimer moves further away from its initial
position $\Delta y_{\mathrm{max}}(t\times u/H=60)\sim 1.5H$ compared to the
square dimer, which covers half of that distance in half the time (cf. panel 3
in Fig. 2 (c) and panel 2 in Fig. 2 (d)). Due to its slower rotation, the disk
dimer spends a longer time crossing streamlines before orienting perpendicular
to the flow and starting to move in the opposite direction. This reasoning
does not, however, answer the question why the trimer experiences negligible
drift, even though its rotational velocity is comparable to that of the disk
dimer.
Evidently, the observed coupling among the modes of translation and the
rotation is hallmark of low-symmetry particles in a flow [55]. Mathematically,
we represent this inter-dependence using a resistance tensor
$\textsf{{R}}_{\mathrm{p}}$, a symmetric matrix with size equal to the number
of degrees of freedom a particle exhibits (Supplementary Text 1A). The
resistance tensor relates the hydrodynamic forces and torque a stationary
fluid $u=0$ exerts on a particle, which translates through it with velocities
$\dot{x}$ and $\dot{y}$, while also rotating at a rotational velocity
$\dot{\theta}$ [56, 57]:
$\begin{pmatrix}F_{x}\\\ F_{y}\\\
T_{z}\end{pmatrix}=-\eta\textsf{{R}}_{\mathrm{p}}\cdot\begin{pmatrix}\dot{x}\\\
\dot{y}\\\ \dot{\theta}\end{pmatrix},\text{with}\
\textsf{{R}}_{\mathrm{p}}\sim\begin{pmatrix}l_{xx}&l_{xy}&l_{x\theta}^{2}\\\
l_{yx}&l_{yy}&l_{y\theta}^{2}\\\ l_{\theta x}^{2}&l_{\theta
y}^{2}&l_{\theta\theta}^{3}\end{pmatrix}.$ (1)
We present each component of $\textsf{{R}}_{\mathrm{p}}$ in terms of arbitrary
length scales $l_{ij}$ to demonstrate one of its defining features – much like
Stokes flow itself, the resistance tensor is time-independent and defined
purely by geometry. If the particle possesses only a single mirror plane, all
nine components of $\textsf{{R}}_{\mathrm{p}}$ are generally non-zero,
reflecting the entwined nature of its modes of motion (Supplementary Text 1C).
Conversely, for a rod the $l_{ij}^{2}$ components become zero, since its
coupled translational modes are unaffected by rotation. Particles with an even
higher symmetry such as disks have all three modes independent of each other
and their resistance tensors are diagonal matrices.
Utilizing the concept of the resistance tensor together with Stokes linearity,
we recently derived equations of motion for a force-free mirror-symmetric
particle subjected to confined Stokes flow [58] (Supplementary Text 1B). Both
equations, as presented in [58], seemingly depend on the initial orientation
of the particle $\theta_{0}$. However, once we realize Stokes flow is time-
reversible, $\theta_{0}$ becomes an arbitrary reference angle. For
convenience, we set $\theta_{0}=\pi/2$, resulting in:
$\theta\left(t\right)=2\arctan\left[\exp\left(-\frac{t-t_{\perp}}{\tau}\right)\right]$
(2)
and
$y\left(t\right)=y\left(t_{\perp}\right)+2H\frac{\tau}{\tau_{y}}\left[\mathrm{sech}\left(\frac{t-t_{\perp}}{\tau}\right)-1\right],$
(3)
where $t_{\perp}=t\left(\theta=\pi/2\right)$ denotes the time at which the
particle is perpendicular to the flow. The two timescales, $\tau$ and
$\tau_{y}$, are characteristic for the re-orientation and cross-stream
migration of each particle. Numerically, they can be computed directly from
the resistance tensor [58], and just like $\textsf{{R}}_{\mathrm{p}}$, they
are purely geometrically determined. Furthermore, Eq. 3 captures the coupling
between rotation and translation, because the particle path depends on both
timescales. The generality of these equations of motion points to their
validity for a wide range of particle shapes provided they have at least one
plane of mirror symmetry. The equations also hold for particles that do not
rotate – shapes with more than one mirror plane have an infinitely large
$\tau$ and translate at a constant lateral velocity (Supplementary Text 1C).
To test the validity of these equations, we produce a variety of disk dimers
and track their motion as they rotate from $\theta\sim 0.85\pi$ to $\theta\sim
0.10\pi$. Upon comparing the obtained raw experimental trajectories, we see a
qualitative similarity (Fig. S4). However, as some particles rotate more
slowly than others, the overall paths the particles follow differ considerably
in quantitative terms. We fit Eqs. 2 and 3 to the observed trajectories and
extract the two characteristic times for each particle, as discussed in
Supplementary Text 2. Finally, we transform experimental time to
$\left(t-t_{\perp}\right)/\tau$ for each shape and compare the angle evolution
for the set of dimers (top curve in Fig. 3 (a)). The re-orientation dynamics
of the studied disk dimers do not only agree quantitatively – they seem to be
independent of the exact particle shape as evident from the collapsed
experimental data, which closely follows Eq. 2, as well as 3D finite element
computations. This apparent shape-independence implies that the characteristic
time captures all geometric details of a particle. By condensing them in
$\tau$ and factoring them out, we are left with the general dynamics
determined by the mirror symmetry and described well by our equation for
$\theta\left(t\right)$. This notion is reaffirmed once we take a look at the
lateral motion of the disk dimers (bottom curve in Fig. 3 (a)). Their cross-
streamwise motion also appears shape-independent once we use
$\left(t-t_{\perp}\right)/\tau$ instead of experimental time and scale their
lateral displacement by the channel height and the characteristic times. Even
when the lateral motion of a particle deviates from the one predicted by Eq.
3, the deviation can be traced back to the re-orientation dynamics. Some
dimers stop rotating before their mirror axes align with the flow direction,
leading to a decoupling of rotation and translation. Thus, they begin to
behave as rods with a finite cross stream velocity even at long timescales
[38]. A possible reason for these deviations is interaction with hydrodynamic
images if the particle comes too close to the wall. Additionally, artefacts of
the lithography process such as slight asymmetry in the particle itself or
dust of size comparable to $h_{\mathrm{g}}$, are other possible culprits. We
test these notions by simulating the full trajectory of a dimer whose
experimental behaviour deviates from the theoretically predicted. Since the 3D
finite element results are well-described by the equations of motion and agree
with the experimental trajectories, we conclude that the observed deviations
are indeed experimental artefacts.
Encouraged by the close agreement between theory and experiments in Fig. 3
(a), we broaden our scope to mirror symmetric particles of various shape.
Substituting the disks with pointy building blocks such as squares and
triangles leads to different timescales, but does not affect the general
particle dynamics (Fig. 3 (b)). Increasing the number of building blocks has
the same effect – trimers with different size ratios and inter-disk angle also
behave identically once we isolate the geometrical details condensed in $\tau$
and $\tau_{y}$. This universality, remarkable as it is, is not entirely
unexpected – Eqs. 2 and 3 are derived with the sole assumptions of a force-
and torque-free particle with a mirror plane moving in creeping flow.
Moreover, our findings suggest we should expect this type of dynamics from any
particle that has at least one mirror plane and is subjected to confined
Stokes flow. Our reasoning also raises the question what is the behaviour of
an asymmetric particle, for instance, a trimer where all three disks have
different radii (Fig. S3). One such shape rotates until it acquires a stable
orientation $\theta_{\infty}\neq 0$ as discussed in Supplementary Text 1C.
However, since the flow disturbance it creates is asymmetric, the particle has
a non-zero lateral velocity even after it has ceased re-orienting [58, 48].
Figure 4: Relation of the characteristic timescales to particle geometry. The
(a) rotation and (b) translation timescales needed to fully describe particle
motion via Eqs. 2 and 3 are solely dependent on the geometry of the system.
For identical flow parameters such as depth-averaged flow velocity $u$, gap
thickness $h_{\textrm{g}}$ and channel height $H$, the detailed shape of the
particle determines $\tau$ and $\tau_{y}$. The rotational timescale depends on
the polar moment of inertia $I_{\mathrm{p}}$ of the particle, its area
$S_{\mathrm{p}}$ (yellow particle sketch), its projected length when
perpendicular to the flow $L_{\perp}$ and the distance $r_{\mathrm{arm}}$
spanning from the centroid $\textit{C}_{0}$ to the centre of perimeter
$\textit{C}_{\textrm{p}}$. We obtain the translational timescale via the area
of the particle and its projected lengths $L_{\perp}$ and $L_{\parallel}$ when
its mirror plane is perpendicular or parallel to the flow, respectively. The
vertical error bars represent the standard deviation of the timescales within
an experimental series (Table S1). The horizontal error bars are calculated
from the uncertainty of the confinement $\tilde{h}$. The dashed lines are a
guide to the eye.
Though we have a rigorous description of the general trajectory of a mirror-
symmetric particle, its exact motion still depends on two timescales. Up to
now we obtain $\tau$ and $\tau_{y}$ as fitting parameters in Eqs. 2 and 3.
However, knowing their values a priori opens the door towards tailoring the
shape of a particle to a desired trajectory. One possible way to obtain this
target-specific shape is to survey a large variety of particles, compute their
resistance tensors and estimate $\tau$ and $\tau_{y}$ [58]. As robust as this
method is, it is not particularly insightful as it does not yield an explicit
relation between the timescales and a particle’s geometric parameters. By
considering imbalanced rods, we propose scaling arguments linking the
timescales $\tau$ and $\tau_{y}$ of a particle to its geometry.
We do so by first identifying $\tau$ and $\tau_{y}$ are functions of the
particle velocities $\dot{\theta}$ and $\dot{x}$ at specific orientations:
$\tau_{\mathrm{scaling}}=-1/\dot{\theta}\left(\theta=\pi/2\right)$ and
$\tau_{y\mathrm{,scaling}}=2H/(\dot{x}_{\perp}-\dot{x}_{\parallel})$ as
discussed in Supplementary Text 1D. The subscripts of the streamwise
velocities denote particle orientation:
$\dot{x}_{\perp}=\dot{x}\left(\theta=\pi/2\right)$ and
$\dot{x}_{\parallel}=\dot{x}\left(\theta=0\right)$. To compute the three
velocities, we make use of the force- and torque-free nature of the particle.
At any instant in time, the angular momentum it gains from the in-plane flow
is dissipated as Couette torque from the confining walls above and below its
faces: $T_{\mathrm{f}}+T_{\mathrm{w}}=0$. We write a similar balance for the
streamwise force – the drag from the surrounding fluid and the friction from
the confining walls cancel: $F_{x\mathrm{,f}}+F_{x,\mathrm{w}}=0$. In
Supplementary Text 4 we propose linear scaling expressions for each torque and
force. We solve the two balances for the three velocities and substitute the
solutions in the expressions for the two timescales:
$\tau_{\mathrm{scaling}}\simeq\frac{1}{6\tilde{h}H_{\mathrm{p}}}\times\frac{H}{u}\times\frac{\sqrt{\pi}I_{\mathrm{p}}}{r_{\mathrm{arm}}L_{\parallel}\sqrt{S_{\mathrm{p}}}}$
(4)
and
$\tau_{y,\mathrm{scaling}}\simeq
2H\times\frac{1}{6\tilde{h}H_{\mathrm{p}}}\times\frac{H}{u}\times\frac{\sqrt{\pi
S_{\mathrm{p}}}}{L_{\perp}-L_{\parallel}}\frac{L_{\perp}}{L_{\parallel}}.$ (5)
The proposed scaling relations provide estimates for $\tau$ and $\tau_{y}$ by
simplifying the particle geometry to projections of shape $L_{\perp}$,
$L_{\parallel}$, moment of inertia $I_{\mathrm{p}}$, area $S_{\mathrm{p}}$, as
well as other geometrical parameters. These parameters are illustrated in the
insets of Fig. 4 and detailed in Table 1.
Table 1: Scaling expressions for the longitudinal forces and in-plane torques acting on a particle in confined Stokes flow. The particle moves at velocity $\dot{x}_{i}$ while rotating with frequency $\dot{\theta}$ in a fluid with depth-averaged flow velocity $u$. The subscript $i\equiv\perp\vee\parallel$ denotes orientation. The forces and torques depend on the particle geometry through its area $S_{\mathrm{p}}$, polar moment of inertia $I_{\mathrm{p}}$, thickness $H_{\mathrm{p}}$, confinement $\tilde{h}$ and projected length $L_{i}$. The gap height $h_{\mathrm{g}}=\tilde{h}H$ is made dimensionless with the height of the channel $H$. The two dimensional projected lengths $L_{\perp}$ and $L_{\parallel}$ are sketched in Fig. 1 (g) and Fig. 4 (b). We define $r_{\mathrm{arm}}$ as the distance between the centroid of a particle $C_{0}$ and its centre of perimeter $C_{\mathrm{p}}$ and sketch it in Fig. 4 (a). | | Fluid | | Wall
---|---|---|---|---
$F_{i}\sim$ | | $\displaystyle 12\eta\frac{u}{H^{2}}H_{\mathrm{p}}L_{\parallel}\sqrt{\frac{S}{\pi}}\times\frac{L_{i}}{L_{\perp}}$ | | $\displaystyle-\frac{2\eta}{h}\dot{x}_{i}S_{\mathrm{p}}$
$T\sim$ | | $\displaystyle 12\eta\frac{u}{H^{2}}H_{\mathrm{p}}L_{\parallel}\sqrt{\frac{S}{\pi}}\times r_{\mathrm{arm}}$ | | $\displaystyle-\frac{2\eta}{h}\dot{\theta}I_{\mathrm{p}}$
We verify the scaling models by comparing our experimental timescales to the
ones computed via Eqs. 4 and 5 in Fig. 4. We complement this comparison with
numerical timescales, computed via 3D FEM, and present them in Fig. S18 and
S19. The scaling relation for $\tau_{\mathrm{scaling}}$ overestimates
$\tau_{\mathrm{exp}}$ by a factor of 1.25, while $\tau_{y\mathrm{,scaling}}$
underestimates $\tau_{y\textrm{, exp}}$ by a factor 1.5. This mismatch is to
be expected as the proposed minimalistic scalings strip the particles of any
geometric detail. A possible remedy is the incorporation of mean particle
curvatures, which, however, comes at the expense of model simplicity.
Though we determine the two timescales up to a scaling factor of order 1, Eqs.
4 and 5 accurately predict when $\tau$ or $\tau_{y}$ diverge and when
$\tau_{y}$ becomes negative. In some trivial cases, particles cease to rotate
and $\tau\to\infty$ when they are either too thick ($\tilde{h}\to 0$), too
thin ($H_{\mathrm{p}}\to 0$) or there is no flow ($u\to 0$). The timescale
also diverges when the distance between the centroid and the centre of
perimeter vanishes ($r_{\mathrm{arm}}\to 0$). Particles with more than one
mirror plane – rods, symmetric dimers and disks – all have
$r_{\mathrm{arm}}=0$. Similarly, particles do not cross streamlines when their
two projected lengths match $L_{\perp}=L_{\parallel}$. One such particle is a
trimer with $\kappa=1.5$ and
$\phi\sim$68\text{\,}\mathrm{\SIUnitSymbolDegree}$$, which rotates without
drifting away from the centre-line of the channel, as demonstrated by finite
element computations in Fig. 1 (e). We also observe this phenomenon
experimentally: the trimer with $\kappa=1.84$ and
$\phi=$51\text{\,}\mathrm{\SIUnitSymbolDegree}$$ barely moves in the lateral
direction (Fig. 2 (e)). Its large $\tau_{y}$, dampening its lateral motion, is
due to its comparable projected lengths. Furthermore, $\tau_{y}$ may become
negative for trimers with large $\phi$, as demonstrated in Fig. 1 (e). This
change in drift direction is present experimentally for a trimer with
$\kappa=1.5$ and $\phi\sim$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and is
the reason why we compare $\left|\tau_{y\mathrm{,exp}}\right|$ to
$\left|\tau_{y\mathrm{,model}}\right|$ in Fig. 4 (b).
The applicability of the proposed scaling relations to a wide range of
particles with different geometry and symmetry supports the main conclusion of
our work: in confined Stokes flow, particles with at least one mirror plane
behave identically as long as we scale their trajectories by characteristic
times, directly related to their shape. The proposed scaling can be utilized
to predict trajectories of particles based on minimalistic scaling arguments.
## Conclusion
In summary, by combining experiments, simulations and theory, we investigate
how the trajectory of a confined particle subjected to Stokes flow is
determined by its geometry. We observe that particles with a single mirror
plane exhibit qualitatively similar behaviour: they rotate in-plane to align
their mirror axis with the flow and their larger building block upstream, all
the while crossing streamlines. However, the timescales over which this
dynamics happens are strongly dependent on particle shape. We fit our
experimental trajectories and finite element calculations to theoretical
equations of motion we have recently derived, thus extracting characteristic
rotational and translational times for each particle. By scaling experimental
time by the respective rotational timescale for each experiment, we collapse
the evolution of the orientation for all particles onto a single curve.
Similarly, we obtain a universal bell-shaped path by scaling real time and a
particle’s cross-streamline velocity. Finally, we propose minimalistic scaling
relations linking the characteristic times of a particle to its geometry. We
strip the particles of all geometrical details and treat them as imbalanced
rods, thus reinforcing the idea that it is solely their symmetry that defines
their overall dynamics. Our observations suggest the trajectories are
universal for particles with at least one mirror plane. This finding deepens
our understanding of fluid-structure interactions in confined Stokes flow.
Moreover, it opens new opportunities in lab-on chip and industrial
applications enabling shape-based separation of suspended particles solely
through hydrodynamic interactions.
## Methods
### Experimental setup
Polymeric microparticles are produced and observed with an experimental setup,
similar to the one used by Uspal, Eral and Doyle [22]. Polydimethylsiloxane
(PDMS, Sylgard®184, Dow Corning) microfluidic devices of width
$W=512\pm$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ are fabricated according
to Dendukuri et al. [59]. Disk dimers are tracked in channels with height
$H=30\pm$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Trimers, triangle and
square dimers are tracked in a 33-micron high channels. A UV-crosslinking
oligomer, poly-(ethyleneglycol) diacrylate (PEG-DA $M_{n}=$700\text{\,}$$ ,
$\eta=$95\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ , Sigma-Aldrich), is mixed
with a photoinitiator, hydroxy-2-methylpropiophenone, (Darocur®1173, Sigma-
Aldrich), in a 19:1 volume ratio and the mixture is pumped through the
microfluidic channel. The device, loaded with prepolymer, is mounted on the
stage of a motorized Nikon Ti Eclipse inverted optical microscope. A
photolithographic mask with well-defined shape is inserted as a field stop.
Mask designs are made in Wolfram Mathematica®and post-processed in Dassault
Systémes’ DraftSight®.
### Particle production and tracking
Microparticles are produced by shining a $100\text{\,}\mathrm{ms}$ pulse of UV
light through the mask onto the channel, thus confining photopolymerization to
a discrete part of the prepolymer mixture. Oxygen, diffusing through the
permeable PDMS walls of the device, inhibits polymerization in their vicinity
[59]. This facilitates the formation of two thin lubrication layers,
$h_{\mathrm{g}}=2.5\pm$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, which
separate the particles from the confining walls of the channel. Particles are
produced and observed with a 20X lens. The microparticle is set in motion by
applying a pressure drop $\Delta p\approx$1.5\text{\,}\mathrm{kPa}$$ across
the channel resulting in a depth-average flow velocity
$u=$55\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$$ for
the shallower channel and $u=$70\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$ for the 33-micron high channel. The particle is
tracked by moving the automated microscope stage in a stepwise manner.
The positions and orientations of particles containing disks are extracted
from the acquired time series using a custom-written MATLAB script, which
employs circular Hough transforms to identify the particle shape in each
frame. The script utilizes MATLAB’s Bio-Formats package [60] and the
calcCircle tool. Particles comprising triangles and squares are tracked by
fitting an ellipse to them, calculating the angle and detecting their straight
edges.
### Finite element computations
All computational results are obtained through the Finite element method as
implement in the Creeping Flow module of COMSOL Multiphysics 5.3, which we
couple to MATLAB via LiveLink. Each solution is carried out on a single
computational node fitted with an Intel Xeon E5-2620 v4 @ 2.10GHz CPU and 64
GB memory. Technical details regarding geometry building, meshing and solver
settings are given in Supplementary Text 3 [61, 62, 63].
We use the channel height $H=1$ as a length scale. We set the inlet flow
velocity $u$, the kinematic viscosity of the fluid $\eta$ and its mass density
$\rho$ to unity. To simulate creeping flow at this $Re=1$, we neglect the
inertial term in the momentum equation and solve the Stokes equation with no
external forcing:
$\nabla\cdot\left(-p\textsf{{I}}+\eta\left(\nabla\bm{U}_{\mathrm{f}}+\nabla\bm{U}_{\mathrm{f}}^{\intercal}\right)\right)=0$
$\nabla\cdot\bm{U}_{\mathrm{f}}=0,$
where we solve for $\bm{U}_{\mathrm{f}}$ and $p$, the fluid velocity and
pressure fields. We integrate the total stress over the particle surface to
obtain the forces and torque acting on it at a given position and orientation
with respect to the flow. To compute the force- and torque-free velocities of
the particle at this configuration, we numerically solve the force balance:
$\begin{pmatrix}\dot{x}\\\ \dot{y}\\\
\dot{\theta}\end{pmatrix}=-\frac{1}{\mu}\textsf{{R}}_{\mathrm{p}}^{-1}\cdot\bm{F}_{0},$
where $\bm{F}_{0}$ is the forces and torque acting on a stationary particle in
a flow and $\textsf{{R}}_{\mathrm{p}}$ is the resistance tensor for this
configuration (Supplementary Note 1A, equation (1)). We obtain the trajectory
of a particle through a first order time integration scheme, where we apply
$\left(\dot{x},\dot{y},\dot{\theta}\right)$ over a timestep $t_{\text{step}}$,
which we determine every iteration (Supplementary Text 3).
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* Linkert et al. [2010] M. Linkert, C. T. Rueden, C. Allan, J.-m. Burel, W. Moore, A. Patterson, B. Loranger, J. Moore, C. Neves, D. Macdonald, et al., J. Cell Biol. 189, 777 (2010).
* Amestoy et al. [2001] P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, SIAM J. Matrix Anal. Appl. 23, 15 (2001).
* Amestoy et al. [2006] P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, Parallel Computing 32, 136 (2006).
* Holzbecher and Si [2008] E. Holzbecher and H. Si, in _Proceedings of the COMSOL Conference_ (Hanover, 2008), 1, p. 7, URL https://www.comsol.nl/paper/accuracy-tests-for-comsol-and-delaunay-meshes-5436.
|
2024-09-04T02:54:57.607836 | 2020-03-05T18:16:49 | 2003.02807 | {
"authors": "Shan Jaffry",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26064",
"submitter": "Shan Jaffry",
"url": "https://arxiv.org/abs/2003.02807"
} | arxiv-papers | # Cellular Traffic Prediction with
Recurrent Neural Network
Shan Jaffry DGUT-CNAM Institute
Dongguan University of Technology, Dongguan, China
<EMAIL_ADDRESS>
###### Abstract
Autonomous prediction of traffic demand will be a key function in future
cellular networks. In the past, researchers have used statistical methods such
as Autoregressive integrated moving average (ARIMA) to provide traffic
predictions. However, ARIMA based predictions fail to give an exact and
accurate forecast for dynamic input quantities such as cellular traffic. More
recently, researchers have started to explore deep learning techniques, such
as, recurrent neural networks (RNN) and long-short-term-memory (LSTM) to
autonomously predict future cellular traffic. In this research, we have
designed a LSTM based cellular traffic prediction model. We have compared the
LSTM based prediction with the base line ARIMA model and vanilla feed-forward
neural network (FFNN). The results show that LSTM and FFNN accurately
predicted the future cellular traffic. However, it was found that LSTM train
the prediction model in much shorter time as compared to FFNN. Hence, we
conclude that LSTM models can be effectively even used with small amount of
training data which will allow to timely predict the future cellular traffic.
###### Index Terms:
Cellular traffic prediction, recurrent neural network, LSTM, feed forward
neural network.
## I Introduction
Cellular communication is the most popular and ubiquitous telecommunication
technology. Recently, owing to novel use cases, such as, multimedia video
download, 4K/8K streaming etc., the amount of cellular data traffic has soared
exponentially. It is expected that in the near future, i.e. by 2023, the
monthly mobile data demands will exceed beyond 109 Exabyte (Exa = $10^{18}$)
which currently rests at a modest 20 Exabytes per month consumption [1].
Cellular users, nevertheless, will expect high speed and ubiquitous
connectivity from the network operators. Providing unhindered, ubiquitous, and
high quality of service will be a serious challenge for network operators.
Network operators must update traffic planning tools so they can know in
advance about the state of future traffic demands. Hence, operators will rely
on data-driven self-organizing networks (SON) powered by machine learning (ML)
and artificial intelligence (AI). ML and AI enabled networks can preemptively
take important decisions with limited human intervention. Prediction of
cellular and data traffic patterns will be a key job that SON perform.
Cellular traffic prediction will enable network operators to promptly
distribute resources as per the requirement of competing users. With the
informed network state, operators may also allow resource sharing between
devices [2, 3]. This will also enable high spectral efficiency and will
prevent outages caused due to cell overload. If a network can accurately
predict future traffic loads in specific cells, it may take preventive actions
to avoid outages. For example, network may permit device-to-device
communication to relieve the base station [4].
Figure 1: LTE-A network architecture.
Recent advances in data analytics and the availability of powerful computing
machines have enabled operators to harness the power of Big data to analyze
and predict network operations. Hence, advanced variations of data driven ML
and AI techniques are playing an ever increasing role in all aspects of modern
human lives. In particular, deep learning, a special class of ML and AI
algorithms, can solve enormously complex problems by leveraging the power of
very deep neural network layers [5]. Deep learning algorithms can extract
valuable feature information from the raw data to predict outcomes. Deep
learning has made great strides recently due to advent of user friendly
libraries and programming environment such as Tensorflow, Keras, PyTorch,
Pandas, and Scikit etc. Deep learning algorithms such as recurrent neural
network (RNN), convolutional neural network (CNN) etc. are being extensively
used in application, such as, computer vision [6], health informatics [7],
speech recognition [8], and natural language processing [9] etc. In future, it
is anticipated that majority operations in sixth generation (6G) cellular
network will be solely catered by AI and deep learning algorithms [10].
An AI-enabled SON network will perform long and short term analysis on the
data obtained from the end users and/or network [11]. This self-optimization
will reduce the over all capital expenditures (CAPEX) and operational
expenditure (OPEX) required for network planning and maintenance. For example,
a key issue concerning increasing CAPEX and OPEX for service providers is the
identification and remedy of anomalies that may arise within a cell. To learn
and prevent the cell from going into the anomalous state, it is necessary for
the network to predict the future traffic demands.
Figure 2: Grid 01 Internet Activity for first 5 days.
In the past researcher have proposed to forecast the cellular traffic using
statistical models, such, as Autoregressive integrated moving average (ARIMA)
and its variants [12]. A known limitation of ARIMA is that it reproduce the
time series patterns based on average of the past values. However, ARIMA may
fail to accurately predict traffic patterns in highly dynamic environments
such as cellular network. Nevertheless, ARIMA can give a descent estimate of
future traffic and may serve as a baseline prediction model.
Recently, deep learning based techniques to forecast any time series traffic
is getting more popular. For cellular applications, deep learning techniques
learn the past history of network traffic to train models such as vanilla
feed-forward neural network (FFNN), recurrent neural network (RNN), or long-
short-term-memory (LSTM) etc. In [13], researchers have proposed to use RNN
with multi-task learning to design a spatio-temporal prediction model for
cellular networks. Researchers in [14] have applied neural network models on
cellular traffic to analyze trade activities in urban business districts. A
comparative study between LSTM and ARIMA models was conducted by researchers
in [15].
Inspired by the works presented earlier, in this paper we will use the real
world call data record to forecast future cellular traffic using LSTM. In
particular, we will compare our results with the ARIMA model and vanilla feed
forward neural network (FFNN) models. We will demonstrate that LSTM models
learn the traffic patterns very quickly as compared to FFNN and ARIMA models.
The rest of the paper is organized as follows. The system model is presented
in Section II. The cellular traffic prediction model is presented in Section
III. We discuss the results in Section IV followed by conclusion in Section V.
## II System Model
Figure 1 shows the our system model which comprise of Long Term Evolution -
Advanced (LTE-A) network. The architecture of LTE-A is broadly categorized
into three layers. The core network (CN), the access network, and the end user
equipment (UE) [16].
The wireless communication take place between a UE and evolved NodeB (eNB)
over the access network which is called UMTS terrestrial radio access network
(E-UTRAN) in LTE-A nomenclature. The core network, which is formally known as
evolved packet core (EPC), makes essential the network level decisions. The
EPC further contain several logical entities such as serving gateway (SGW),
packet data network gateway (PGW), and mobility management unit (MMU) etc.
Detailed explanation of these logical entities and LTE-A architecture is out
of scope of current paper. Readers can refer relevant materials, for example
[16]. The call data record (CDR) that we will use in this research was
gathered at the EPC level layer. The execution of LSTM predictive model will
also take place at this layer.
### II-A Data Record Details
The call data record used in this research was published by Telecom Italia for
Big Data Challenge competition [17]. Telecom Italia collected cellular and
internet activities of its subscribers within the city of Milan in Italy. In
the CDR, Milan city is divided into $100\times 100$ square grids. Each grid
has a length of 0.235 Km and an area of 0.055 Km2. The data record has been
collected for 62 days, starting from 1st November 2013 till 1st January 2014.
Data for the single day is stored in a single file which means that there are
62 files in the dataset. Readers can refer to [18] for detailed explanation on
the CDR.
The spatio-temporal CDR contains following fields.
* •
Grid ID.
* •
Time Stamp: Raw timestamp was recorded in milliseconds units with the interval
of 10 minutes.
* •
Country code.
* •
Inbound SMS Activity: Indicates the incoming SMS activity in a particular grid
observed within 10 minute interval.
* •
Outbound SMS Activity: Indicates the outgoing SMS activity in a particular
grid observed within 10 minute interval.
* •
Inbound Call Activity: Indicates the incoming calling in a particular grid
observed within 10 minute interval.
* •
Outbound Call Activity: Indicates the outgoing calling activity in a
particular grid observed within 10 minute interval.
* •
Internet Activity: Indicates the internet usage by cellular users in a
particular grid observed within 10 minute interval.
Figure 3: Single LSTM Cell.
The CDR does not specify activity in terms of particular units. However, an
intuitive interpretations is that the activities are proportional to the
amount of real traffic. For example, the magnitude of Inbound or outbound SMS
activities are high for a greater number of SMS received or sent,
respectively. The data was provided in the raw format. Hence, we will discuss
the data cleansing method in the next step.
### II-B Data cleansing
The CDR, in its raw format, could not be used to extract any meaningful
information. Hence we applied data cleansing and filtering over the CDR. The
timestamps were changed from milliseconds to minutes. There were some missing
fields which we marked as zeros (0). There were multiple entry records for
each timestamp. We summed them to make a single activity record per timestamp.
Figure 2 shows the 24-hour Internet Activity for Grid 01.
In our prediction model, we have only used Internet traffic Activity. However,
the our model can be used to predict activities for SMS and calls without any
modification. We will discuss traffic prediction in the next section.
## III Cellular Traffic Prediction
In this section, we will first briefly describe basics of feed forward and
recurrent neural network (NN) followed by the LSTM based learning model.
### III-A Feed Forward and Recurrent NN
In artificial neural networks, the nodes are connected to form a directed
graph which is ideal for handling temporal sequence predictions. In vanilla
feed forward network (FFNN), information flows only in forward direction. In
FFNN, the input layer feeds the forward looking hidden layer for calculations
and manipulations. The hidden layers forward the information to the output
layer which produce regression of classification predictions.
Figure 4: LSTM Network.
A NN maps inputs to the outputs by learning from the examples provided during
the training phase and can be used for prediction in classification and
regression problems. During the training process, the predictions are compared
against the expected output values (often known as ground truth data) to
calculate the loss function. In the beginning of the training, the loss
function is usually quite high indicating the incorrect prediction by the
model. With back propagation and gradient descent method, the model adjust the
weights and biases corresponding to the input value to minimize the loss
function. A fully trained NN has the minimal loss (also called as error)
between the predicted and the expected output value [19]. After successful
training, the model is validated and a validation error is calculated. A model
is fully trained for prediction when the training and validation errors are
both minimized.
In recurrent neural network (RNN), though the learning process is the same as
FFNN, the architecture is slightly different. RNNs takes the output of one
layer, and feed it as the input to the next layer. Hence, each layer has
information from the past input values. RNN considers the current input as
well as the input received in the previous time steps during training and
prediction. This enables RNN to learn the knowledge from the all previous time
instances to make a well informed prediction for time series data.
However, vanilla RNNs have inherent vanishing and exploding gradient problem
which halts the learning process as gradient either diminishes completely or
explodes to very large value. Hence Long-Short-Term-Memory (LSTM), which is a
variant of RNN was proposed in [20]. LSTMs were designed to avoid the long-
term dependency issue, which is the cause of the vanishing-gradient problem in
Vanilla RNNs [19].
### III-B Learning Through LSTMs
The structure of LSTM units (often known as cells) enable a neural network to
learn long term dependencies. The learning processing is strictly controlled
by multiple gates that allow (or bar) the flow of incoming data from the
previous cell and/or input as shown in Figure 3. The standard LSTM unit is
shown in Figure 3. There are three main gates in any LSTM unit, the forget
gate ($\Gamma_{f}$), the update or input gate ($\Gamma_{i}$), and the output
gate ($\Gamma_{o}$). The cell state for the current unit $C^{t}$ is updated by
the information passed through the update gate ($\Gamma_{i}$). The candidate
value for current cell’s state (i.e. $C_{u}^{t}$) is updated based on the
information from the previous hidden state (i.e. $\text{a}^{t-1}$) and input
$X^{t}$. The update gate decides to allow or bar the flow of this candidate
value to the output state. Finally the output gate $\Gamma_{o}$ allows the
information to pass from the current cell. The forget gate lets the current
cell keep or forget the state value from the previous time step. The
prediction is made as $\hat{y}$ after passing through an activation function
(often sigmoid or softmax).
Figure 5: Traffic prediction with large-sized training set.
The LSTM cells are chained to form one layer of the LSTM network as shown in
Figure 4. Each cells computes operation for one time step and transfer the
output to the next cell. The number of cells in a LSTM network indicates the
amount of observations of data that is being considered before making any
prediction. For our case, the input $X^{t}$ is the internet activity and the
number of observations is the amount of selected time steps T.
The expression for all the gates, cell states, out of the hidden layer, and
the final prediction are given as below:
$\Large\Gamma_{f}^{t}=\sigma(W_{f}[a^{t-1},X^{t}]+b_{f})$ (1)
$\Large\Gamma_{i}^{t}=\sigma(W_{i}[a^{t-1},X^{t}]+b_{i})$ (2)
$\Large C_{u}^{t}=\varphi(W_{c}[a^{t-1},X^{t}]+b_{c})$ (3)
$\Large\Gamma_{o}^{t}=\sigma(W_{o}[a^{t-1},X^{t}]+b_{o})$ (4)
$\Large C^{t}=\Gamma_{f}^{t}\ast C^{t-1}+\Gamma\ast C_{u}^{t}$ (5)
$\Large a^{t}=\Gamma_{o}^{t}\ast\varphi(C^{t})$ (6)
The final output $Y^{t}$ is then calculated as :
$\Large y^{t}=\sigma(W_{y}a^{t}+b_{y})$ (7)
In the equations above, symbol $\sigma$ represent the sigmoid function which
is often known as the squashing function because it limits the output between
0 (gate OFF) and 1 (Gate fully ON). Formally, the sigmoid function is defined
as $\sigma(x)=\Large\frac{1}{1+e^{-x}}$ . Symbol $\varphi$ is another
squashing function and often $\tanh$ or rectified linear unit (relu)
operations are used for $\varphi$. Readers can refer to relevant literature to
gather further information about these functions. [19]. The symbol $\ast$
represents the element-wise multiplication. Finally, $W_{(.)}$ and $b_{(.)}$
are the vectors of weights and biases corresponding to the respective gates,
hidden layer, input, and output layer. The exact values of these weights and
biases are selected by the libraries described in the next sub-section.
### III-C Training and Prediction Software
We have used Matlab for data cleansing and filtering. All the algorithms are
implemented in Python using Keras and Scikit libraries with Tensorflow at the
backend.
Figure 6: Error for traffic prediction with medium-sized training set.
## IV Results and Discussion
In this section we will show the performance comparison of LSTM model with the
base line ARIMA and vanilla feed forward neural network model. We have
compared the performance of each technique with the ground truth test data
from CDR. We have fixed the training epochs to 20 for each cycle. For LSTM
model, we have used two hidden layers to make an even comparison with the FFNN
and ARIMA. The first hidden layer contains 50 LSTM cells followed by a dense
layer with single unit. The FFNN contains two hidden layers with first layer
containing 5 activation units activated by relu operation. The second hidden
layer contains one non-linear activation unit. Training and validation losses
are calculated using mean absolute error.
Figure 7: Traffic prediction with medium-sized training set.
Figure 5 shows the traffic prediction by LSTM, FFNN model, and ARIMA model. We
used 7142 samples for training LSTM and FFNN models. For validation and
testing, we used 893 samples for each case. The LSTM and FFNN both learned the
pattern in less than 5 epochs due to large number of training examples. It can
be observed that LSTM and FFNN predictions match to that of the ground truth
data. The ARIMA model predicts very close to the ground truth but does not
exactly match the traffic pattern.
Figure 8: Traffic prediction with small-sized training set.
We later reduced the training samples to 3571. The training and validation
errors for this case are shown in Figure 6 and the prediction results are
presented in Figure 7. We can observe that LSTM and FFNN still predict very
accurately. The ARIMA baseline model however does not exactly match the ground
truth traffic. It should be noted from Figure 6 that when we reduced the
number of training samples, the training and validation error for LSTM
converges to near zero (0) only after 2 epochs. However, FFNN took at least 10
epochs to fully train the model to enable accurate predictions. Nevertheless,
both models’ errors converged to zero before the 20 epochs limit.
When we further reduced the training samples to 892, we observed that after
training the models for 20 epochs, the FFNN could not predict according to
actual ground truth data. In fact, its performance worsened even to that of
the ARIMA model. The LSTM, on the other hand, very accurately predicted the
traffic activity. This is due to the fact that LSTM trained the network within
20 epochs and the training and validation error converged to zero as shown in
Figure 9. On the other hand, the error for the FFNN remains high even after 20
epochs. Interestingly, FFNN could estimate patterns of future traffic,
however, with very low accuracy.
## V Conclusion
In this paper, we presented cellular data traffic prediction using recurrent
neural network, in particular, with long-short-term-memory model. We
demonstrated that LSTM and vanilla feed-forward neural networks predict more
accurately as compared to the statistical ARIMA model. However, the LSTM
models were shown to be learning more quickly as compared to the FFNN, even
with a small amount of training data sample. As our future work, we are
working to design a LSTM based resource allocation method for 6G networks.
Figure 9: Error for traffic prediction with small-sized training set.
## References
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* [3] ——, “Shared spectrum for mobile-cell’s backhaul and access link,” in _2018 IEEE Global Communications Conference (GLOBECOM)_. IEEE, 2018, pp. 1–6.
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* [14] Y. Zhao, Z. Zhou, X. Wang, T. Liu, Y. Liu, and Z. Yang, “Celltrademap: Delineating trade areas for urban commercial districts with cellular networks,” in _IEEE INFOCOM 2019-IEEE Conference on Computer Communications_. IEEE, 2019, pp. 937–945.
* [15] A. Azari, P. Papapetrou, S. Denic, and G. Peters, “Cellular traffic prediction and classification: a comparative evaluation of lstm and arima,” in _International Conference on Discovery Science_. Springer, 2019, pp. 129–144.
* [16] A. ElNashar, M. A. El-Saidny, and M. Sherif, _Design, deployment and performance of 4G-LTE networks: A practical approach_. John Wiley & Sons, 2014.
* [17] “Telecom Italia, Open Big Data, Milano Grid,” Online, 2014. [Online]. Available: https://dandelion.eu/
* [18] M. S. Parwez, D. B. Rawat, and M. Garuba, “Big data analytics for user-activity analysis and user-anomaly detection in mobile wireless network,” _IEEE Transactions on Industrial Informatics_ , vol. 13, no. 4, pp. 2058–2065, 2017.
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|
2024-09-04T02:54:57.617241 | 2020-03-05T18:25:27 | 2003.02811 | {
"authors": "CMS Collaboration",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26065",
"submitter": "The CMS Collaboration",
"url": "https://arxiv.org/abs/2003.02811"
} | arxiv-papers | FSQ-16-006
FSQ-16-006
# Study of central exclusive $\PGpp\PGpm$ production in proton-proton
collisions at $\sqrt{s}=5.02$ and 13
###### Abstract
Central exclusive and semiexclusive production of $\PGpp\PGpm$ pairs is
measured with the CMS detector in proton-proton collisions at the LHC at
center-of-mass energies of 5.02 and 13. The theoretical description of these
nonperturbative processes, which have not yet been measured in detail at the
LHC, poses a significant challenge to models. The two pions are measured and
identified in the CMS silicon tracker based on specific energy loss, whereas
the absence of other particles is ensured by calorimeter information. The
total and differential cross sections of exclusive and semiexclusive central
$\PGpp\PGpm$ production are measured as functions of invariant mass,
transverse momentum, and rapidity of the $\PGpp\PGpm$ system in the fiducial
region defined as transverse momentum $\pt(\PGp)>0.2\GeV$ and pseudorapidity
$\abs{\eta(\PGp)}<2.4$. The production cross sections for the four resonant
channels 500, $\PGrzP{770}$, 980, and 1270 are extracted using a simple model.
These results represent the first measurement of this process at the LHC
collision energies of 5.02 and 13.
## 0.1 Introduction
The central exclusive production (CEP) process has been studied for a long
time from both theoretical [1, 2, 3, 4, 5, 6, 7] and experimental [8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18] perspectives. In this process, both protons
remain intact in the collision and a central system is produced. The process
is referred to as exclusive when no particles other than the central system
are produced. If one or both protons dissociate into a forward diffractive
system, the process is called semiexclusive production. Various central
systems can be produced in this process, like $\PGpp\PGpm$, $\PKp\PKm$, and
$4\PGp$. In this paper, the $\PGpp\PGpm$ central system is measured. At the
CERN LHC energies, the two dominant mechanisms of $\PGpp\PGpm$ production via
CEP are _double pomeron exchange_ (DPE) and _vector meson photoproduction_
(VMP), which are illustrated by the diagrams shown in Fig. 1. The pomeron
($\mathbb{P}$) is a color singlet object introduced to explain the rise of the
inelastic cross section at high collision energies [19, 20]. The quantum
numbers of the pomeron constrain the possible central systems in DPE
processes, whereas the photon exchange restricts the central system in VMP
processes. By functioning as a quantum number filter, the CEP process is
suitable to study low-mass resonances, which would be difficult to study
otherwise. Furthermore, DPE processes are also suitable to search for
glueballs (bound states of gluons without valence quarks), because they
provide a gluon-rich environment [21, 22]. Another process that could
contribute to the same final state is the two-photon fusion
$\PGg\PGg\to\PGpp\PGpm$, which is expected to have a much smaller cross
section than DPE and VMP processes and gives a negligible contribution [23].
Figure 1: Diagrams of the dominant mechanisms for $\PGpp\PGpm$ production via
CEP in proton-proton collisions: (a) continuum; (b) resonant double pomeron
exchange; and (c) vector meson photoproduction.
The DPE process of pion pair production has two subcategories: continuum and
resonant production. In the case of continuum production, the pion pair is
directly produced; thus the pairs have a nonresonant invariant mass spectrum.
Resonant production means that an intermediate meson resonance is produced
centrally, which manifests itself as a peak in the invariant mass distribution
of the pion pair. Since the pomeron is a Regge trajectory running over states
with quantum numbers $J^{PC}=\\{0^{++},1^{++},2^{++},\dots\\}$ and
$I^{G}=0^{+}$, the resonance is restricted to have $J^{PC}=\\{0^{++}$,
$2^{++}$, $4^{++},\dots\\}$ and $I^{G}=0^{+}$, where $J$ is the total angular
momentum, $I$ is the isospin, $P$ is the parity, $C$ is the charge parity, and
$G=C\,(-1)^{I}$. The known particles [24] satisfying these criteria are the
f0, f2, ${\HepParticle{\PGc}{c0}{}}\Xspace$,
${\HepParticle{\PGc}{c2}{}}\Xspace$, ${\HepParticle{\PGc}{b0}{}}\Xspace$, and
${\HepParticle{\PGc}{b2}{}}\Xspace$ resonances. The cross section for DPE
($\sigma_{\PGpp\PGpm}^{\text{DPE}}$) can be calculated from the amplitude of
continuum ($A_{\PGpp\PGpm}^{\text{DPE,C}}$) and resonant
($A_{\PGpp\PGpm}^{\text{DPE,R}}$) production as
$\sigma_{\PGpp\PGpm}^{\text{DPE}}\propto\abs{A_{\PGpp\PGpm}^{\text{DPE,C}}+A_{\PGpp\PGpm}^{\text{DPE,R}}}^{2}.$
(1)
Interference terms between the continuum and resonant production channels must
be included to describe the observed spectra and to measure the cross sections
for resonances.
In VMP, one of the protons emits a virtual photon, which fluctuates into a
quark-antiquark bound state and scatters from the proton via the pomeron
exchange. The quantum numbers of the possible resonances are constrained by
the quantum numbers of the pomeron and the photon ($J^{PC}=1^{--}$), leading
to mesons with odd spin and the following quantum numbers
$J^{PC}=\\{1^{--},3^{--},\dots\\}$. Resonances satisfying these conditions are
${\HepParticle{\PGr}{}{0}{}{}{}}$, $\PGo$, $\PGf$, , $\PGyP{2S}$, and $\PGU$,
but only the ${\HepParticle{\PGr}{}{0}{}{}{}}\to\PGpp\PGpm$ decay has a
significant branching fraction, since decays in this channel are strongly
suppressed in the case of $\PGf$, , $\PGyP{2S}$, and $\PGU$ according to the
Okubo–Zweig–Iizuka rule [25, 26, 27] and in the case of $\PGo$ because of
G-parity conservation [28].
This paper presents measurements of exclusive and semiexclusive $\PGpp\PGpm$
total and differential cross sections as functions of invariant mass
$m(\PGpp\PGpm)$, transverse momentum $\pt(\PGpp\PGpm)$, and rapidity
$y(\PGpp\PGpm)$ of the pion pair, in a fiducial region defined by single pion
transverse momentum $\pt(\PGp)>0.2\GeV$ and single pion pseudorapidity
$\abs{\eta(\PGp)}<2.4$. Because the outgoing protons are not tagged in this
measurement, there is a residual contribution from semiexclusive production
with all dissociation products outside the $\abs{\eta}>4.9$ range. In the
following, the exclusive and the residual semiexclusive contribution together
will be referred to as central exclusive production. The data were recorded by
CMS with beam conditions ensuring a small probability of multiple $\Pp\Pp$
collisions in the same bunch crossing (pileup) in August 2015 at a center-of-
mass energy of 13with luminosity 258and in November 2015 at 5.02with a
luminosity of 522. The average number of $\Pp\Pp$ collisions in a bunch
crossing was around 0.3–0.5 for the 5.02and around 0.5 for the 13data sets.
## 0.2 The CMS detector
The central feature of the CMS apparatus is a superconducting solenoid of 6m
internal diameter. Within the solenoid volume are a tracker, a lead tungstate
crystal electromagnetic calorimeter (ECAL), and a brass and scintillator
hadron calorimeter (HCAL), each composed of a barrel and two endcap sections,
covering the $\abs{\eta}<3.0$ region. Forward calorimeters extend the $\eta$
coverage provided by the barrel and endcap detectors. Muons are measured in
gas-ionization detectors embedded in the steel flux-return yoke outside the
solenoid.
The silicon tracker measures charged particles within the range
$\abs{\eta}<2.5$. It consists of 1440 silicon pixel and 15 148 silicon strip
detector modules and is located in the 3.8T solenoid field. Three pixel barrel
layers (PXB) are situated at radii of 4.4, 7.3, and 10.2cm; PXB also has two
pixel endcap disks (PXF). The strip tracker consists of the innermost tracker
inner barrel (TIB) and the tracker inner disks (TID), which are surrounded by
the tracker outer barrel (TOB). It is completed by endcaps (TEC) on both
sides. The barrel part of the strip tracker has a total of 10 layers at radii
from 25 to 110cm, whereas the endcap of the strip tracker consists of 12
layers. For charged particles with $\pt<1\GeV$ and $\abs{\eta}<1.4$, the track
resolutions are typically 1–2% in , and 90–300 and 100–350for the transverse
and longitudinal impact parameters, respectively [29]. The tracker provides an
opportunity to identify charged particles with $0.3<p<2\GeV$ based on their
specific ionization in the silicon detector elements [30].
The ECAL consists of 75 848 lead tungstate crystals, which provide coverage in
$\abs{\eta}<1.479$ in the barrel region and $1.479<\abs{\eta}<3.0$ in the two
endcap regions.
The barrel and endcap sections of the HCAL consist of 36 wedges each and cover
the $\abs{\eta}<3.0$ region. In the region $\abs{\eta}<1.74$, the HCAL cells
have widths of 0.087 in $\eta$ and 0.087 radians in azimuth ($\phi$). In the
$\eta$-$\phi$ plane, and for $\abs{\eta}<1.48$, the HCAL cells map onto
$5{\times}5$ ECAL crystal arrays to form calorimeter towers projecting
radially outwards from close to the nominal interaction point. At larger
values of $\abs{\eta}$, the towers are larger and the matching ECAL arrays
contain fewer crystals.
The forward hadron (HF) calorimeter uses steel as an absorber and quartz
fibers as the sensitive material. The two halves of the HF are located at
11.2m from the interaction region, one at each end. Together they provide
coverage in the range $3.0<\abs{\eta}<5.2$. Each HF calorimeter consists of
432 readout towers, containing long and short quartz fibers running parallel
to the beam. The long fibers run the entire depth of the HF calorimeter
(165cm, or approximately 10 interaction lengths), whereas the short fibers
start at a depth of 22cm from the front of the detector. By reading out the
two sets of fibers separately, it is possible to distinguish showers generated
by electrons or photons, which deposit a large fraction of their energy in the
long-fiber calorimeter segment, from those generated by hadrons, which
typically produce, on average, nearly equal signals in both calorimeter
segments.
The triggers used in this analysis are based on signals from the Beam Pick-up
and Timing for eXperiments (BPTX) detectors [31]. The BPTX devices have a time
resolution of less than 0.2ns. They are located around the beam pipe at a
distance of $\pm 175$m from the nominal interaction point, and are designed to
provide precise information on the bunch structure and timing of the proton
beams.
A more detailed description of the CMS detector, together with a definition of
the coordinate system used and the relevant kinematic variables, can be found
in Ref. [32].
## 0.3 Monte Carlo simulations
Two kinds of Monte Carlo (MC) event generators are used in this analysis:
inclusive and exclusive generators. The inclusive generators model the
inclusive diffractive dissociation [33] and nondiffractive interactions, and
are used to estimate the tracking efficiency, multiple reconstruction and
misreconstruction rates. The exclusive generators are used to generate CEP
events and to calculate the vertex correction factors. There are no available
MC event generators that produce exclusive scalar and tensor resonances via
DPE, such as the production of 500, 980, and 1270 mesons.
Event samples are generated with various tunes for diffraction and the
underlying event:
* •
8.205 [34] with CUETP8M1 tune [35] and MBR model [36]: 8 is an inclusive
generator based on the Schuler and Sjöstrand model. It is capable of modeling
a wide variety of physical processes, such as single diffractive (SD), double
diffractive (DD), and central diffractive (CD) dissociation, as well as
nondiffractive (ND) production [33]. The SD, DD, and ND events are generated
with the CUETP8M1 tune. The Minimum Bias Rockefeller (MBR) model of is based
on the renormalized pomeron flux model and it is capable of generating SD, DD,
ND and CD events.
* •
epos [37] with its LHC tune [38]: This inclusive generator is based on the
Regge–Gribov phenomenology [39], and it models SD, DD, CD, and ND processes.
* •
starlight [40]: This event generator models photon-photon and photon-pomeron
interactions in $\Pp\Pp$ and heavy ion collisions. The production of
${\HepParticle{\PGr}{}{0}{}{}{}}$ mesons and their successive decay into two
pions through the VMP process is simulated by starlight. For background
studies, $\PGo$ mesons are also generated with starlight and their decay
simulated by to the $\PGpp\PGpm\PGpz$ final state.
* •
dime mc 1.06 [5]: The dime mc software describes continuum $\PGpp\PGpm$
production through DPE. The generator uses a phenomenological model based on
Regge theory. Events are generated with the Orear-type off-shell meson form
factors with parameters $a_{\text{or}}=0.71\GeV^{-1}$ and
$b_{\text{or}}=0.91\GeV^{-1}$ [5]. Furthermore, two additional MC samples are
generated with an exponential form factor with $b_{\text{exp}}=0.45$ [5] and
$1\GeV^{-2}$ [1] to study the systematic uncertainty in the measured resonance
cross sections arising from uncertainties in the dime mc parametrization.
All of the generated events are processed by a detailed simulation [41] of the
CMS detector.
## 0.4 Event selection
The following triggers were employed:
* •
Zero bias: zero-bias events are selected by using either the BPTX detectors
(13data) or the LHC clock signal and the known LHC bunch structure (5.02data).
Both methods provided zero-bias events.
* •
BPTX XOR: Here XOR stands for the exclusive OR logic, where only one BPTX is
fired, corresponding to an incoming proton bunch from only one direction. This
trigger was used in both 5.02 and 13data sets.
* •
No-BPTX: There is no signal in the BPTX detectors, which means there are no
incoming proton bunches. This trigger was used in both 5.02 and 13data sets.
The present analysis uses events acquired with the zero bias trigger. The BPTX
XOR and No-BPTX triggers select events with no interacting bunches, which are
used to estimate the electronic noise of calorimeters and possible collisions
between beam particles and residual gas molecules in the CMS beampipe (beam-
gas background). The contribution from beam-gas collisions is negligible
because there is no difference in the measured calorimeter tower energy
distributions for the BPTX XOR and No-BPTX triggered events.
In the offline selections, it is required that the event has exactly two
tracks, both of which satisfy $\chi^{2}/\text{ndf}<2$ (where the $\chi^{2}$
value is calculated based on the fitted trajectory and the measured tracker
hits, and ndf is the number of degrees of freedom), $\pt>0.2\GeV$, and
$\abs{\eta}<2.4$ to ensure high track reconstruction efficiency. Only events
with oppositely charged (opposite-sign, OS) tracks are selected for analysis,
whereas events with same-sign (SS) tracks are used in the background
estimation.
Events with a single collision are selected by requiring the two tracks form a
single reconstructed vertex subject to the constraint that
$\abs{z_{1}-z_{2}}<3\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}},$ (2)
where $z_{1}$ and $z_{2}$ are the $z$ coordinates of the closest approach of
the reconstructed tracks to the beamline, and $\sigma_{1}$ and $\sigma_{2}$
are their corresponding uncertainties.
To select exclusive events, all calorimeter towers not matching the
trajectories of the two tracks must have energy deposits below a threshold,
which is defined in Table 0.4. A tower is matched to a track if the
intersection of the extrapolated trajectory with the calorimeter surface is
within three standard deviations in $\eta$ and $\phi$ from the center of the
tower. The threshold values are chosen to have a maximum 1% rejection of
signal events resulting from the electronic noise of the calorimeters. Non-
exclusive events might be also selected because of the lack of coverage in the
eta gap between the HF and central calorimeters; these events are also taken
into account in the background estimation presented later in this paper.
Using all of the above listed event selection criteria, a total of 48 961
events were selected from the 5.02and 20 980 from the 13dataset.
The value of calorimeter thresholds for different calorimeter constituents,
used in the selection of exclusive events. Calorimeter Threshold [] $\eta$
coverage ECAL barrel 0.6 $\abs{\eta}<1.5$ ECAL endcap 3.3 $1.5<\abs{\eta}<3.0$
HCAL barrel 2.0 $\abs{\eta}<1.3$ HCAL endcap 3.8 $1.3<\abs{\eta}<3.0$ HF 4.0
$3.15<\abs{\eta}<5.2$
## 0.5 Data analysis
### 0.5.1 Particle identification
Particle identification is used to select pion pairs by the mean energy loss
($\rd E/\rd x$) of particles in the silicon tracking detectors. The $\rd E/\rd
x$ values shown in the left panel of Fig. 2 are calculated by a second-order
harmonic mean using only the strip detectors [42]:
$\left\langle\frac{\rd E}{\rd
x}\right\rangle=\left(\frac{1}{N}\sum_{i=1}^{N}(\Delta E/\Delta
x)_{i}^{-2}\right)^{-\frac{1}{2}},$ (3)
where $N$ is the number of energy loss measurements, $\Delta E/\Delta x$ is a
single energy loss measurement per path length in one tracker module, and the
sum runs over the strip detectors carrying energy loss measurements. The $-2$
exponent in this formula suppresses high $\Delta E/\Delta x$ values arising
from the highly asymmetric $\Delta E/\Delta x$ Landau distribution, thus
avoiding a bias in the estimate of the average $\rd E/\rd x$ of the track.
Figure 2: Left: The distribution of the logarithm of the mean energy loss and
absolute value of the momentum of tracks from low-multiplicity
($N_{\text{track}}\leq 4$) events collected at $\sqrt{s}=13\TeV$. The
$\PGp$-selection region is shown in the 0.3–2range. All tracks outside this
momentum range are identified as pions. Right: The fit of energy loss
distributions in a given momentum bin with the sum of three Gaussian curves.
Plots are similar for the 5.02data.
The track classification is achieved by fitting the mean energy loss
distributions of tracks from low multiplicity ($N_{\text{track}}\leq 4$)
events with a sum of three Gaussian functions corresponding to pions, kaons,
and protons. An example for such a fit is shown in the right panel of Fig. 2.
In the 0.3–2momentum range pions are selected from the $\pm 3$ standard
deviation region of the corresponding Gaussian peak. This region is shown in
the left panel of Fig. 2. Tracks that have $p<0.3$ or $p>2\GeV$ are assumed to
be pions. The contamination from kaons and protons is estimated using the
data-driven approach described in Section 0.5.3.
### 0.5.2 Corrections
Each event is weighted by several correction factors to compensate for the
detector and reconstruction effects. The multiplying factor is the product of
four independent corrections: tracking, multiple reconstruction, vertex, and
pileup correction.
A tracking correction is used to correct for track reconstruction
inefficiencies:
$C_{\text{tr}}=\frac{1}{\varepsilon_{\text{tr},1}}\,\frac{1}{\varepsilon_{\text{tr},2}},$
(4)
where $\varepsilon_{\text{tr},1}$ ($\varepsilon_{\text{tr},2}$) is the
tracking efficiency in the region where the first (second) particle is
reconstructed. A single charged particle may lead to two reconstructed tracks,
such as spiralling tracks near $\eta\approx 0$ or split tracks in the overlap
region of the tracker barrel and endcap. This effect is corrected using
$\varepsilon_{\text{mrec}}$, which is the probability for this situation to
occur. In this case the correction factor takes the form
$C_{\text{mrec}}=\frac{1}{1-\varepsilon_{\text{mrec},1}}\,\frac{1}{1-\varepsilon_{\text{mrec},2}}.$
(5)
The values of $\varepsilon_{\text{tr}}$ and $\varepsilon_{\text{mrec}}$ are
estimated as a function of $\eta$ and using MC simulations. Their dependence
on the track $\phi$ and the vertex position $z$-coordinate is integrated over.
The simulated events are weighted such that the vertex $z$-coordinate
distribution agrees with collision data.
The vertex correction $C_{\text{vert}}$ accounts for events with an
unreconstructed vertex. It is the reciprocal of the vertex efficiency, which
is calculated using samples produced by the dime mc and starlight generators.
The vertex efficiency has a slight dependence on the invariant mass of the
track pair that is included when applying the vertex correction.
Some real CEP events are rejected because of pileup. To account for these lost
events, a correction factor $C_{\text{pu}}$ for the number of selected events
can be computed. The CEP events are selected from bunch crossings with a
single collision, so by assuming that the number of collisions follows a
Poisson distribution, one can derive $C_{\text{pu}}$:
$C_{\text{pu}}=\frac{N\mu}{N\,\mu\exp{(-\mu)}}=\exp{(\mu)}.$ (6)
Here, $\mu$ is the average number of visible inelastic collisions, in a given
bunch crossing, $N$ is the total number of analyzed events. The value of $\mu$
depends on the instantaneous luminosity associated with individual bunch
crossings, $\mathcal{L}_{\text{bunch}}$, according to the following
expression:
$\mu=\frac{\sigma_{\text{inel,vis}}\mathcal{L}_{\text{bunch}}}{f},$ (7)
where $\sigma_{\text{inel,vis}}$ is the visible inelastic $\Pp\Pp$ cross
section, $f$ is the revolution frequency of protons, and
$\mathcal{L}_{\text{bunch}}$ is the average instantaneous luminosity at the
given bunch crossing position for time periods of 23.3s. The ratio of
$\sigma_{\text{inel,vis}}$ to $f$ is obtained by fitting the fraction of
events with no observed collisions as a function of
$\mathcal{L}_{\text{bunch}}$ with the functional form
$A\exp(-b\,\mathcal{L}_{\text{bunch}})$, where $A$ and $b$ are free parameters
of the fit.
The range of correction factors is summarized in Table 0.5.2.
Correction factors. Type Range Tracking 1.05–1.50 Multiple reconstruction
1.005–1.040 Vertex 1.05–1.33 Pileup 1.3–2.1
Figure 3: The number of extra calorimeter towers over threshold in events
containing an identified pion pair with opposite (left) and same (right)
charge. The known contributions, denoted with the red hatched areas, are used
to estimate the background in the zero bin of the opposite-sign distribution,
which is denoted by the blue hatched area. The error bars correspond to
statistical uncertainties, whereas the error rectangle on the background
denotes the 14% systematic uncertainty in the background normalization. Plots
are similar for 5.02data.
### 0.5.3 Background estimation
The main background contributions to $\PGpp\PGpm$ CEP are the multiparticle
background and the exclusive $\PKp\PKm$/$\Pp\PAp$ production. The
multiparticle background in the selected exclusive sample consists of events
with more than two particles created in the interaction, of which only two are
observed because the additional particles yield energy deposits below the
thresholds, or outside the acceptance. The SD, DD, ND, and CD processes with
more than two centrally produced particles belong to this contribution. A
method based on control regions is used to estimate this multiparticle
background. Control regions are selected in which events have at least two
calorimeter towers above threshold, not matched to the two selected pions,
with all the other selection criteria satisfied. The distribution of the
number of events selected in this way as a function of the number of extra
towers with energy above threshold is shown in Fig. 3. The counts in the bins
with 2, 3, 4, and 5 towers are used to estimate the background. The
normalization factor is calculated using the following assumption:
$\frac{N_{\text{mpart,SS}}(\text{0 extra towers})}{N_{\text{mpart,SS}}(\text{2
--5 extra towers})}=\frac{N_{\text{mpart,OS}}(\text{0 extra
towers})}{N_{\text{mpart,OS}}(\text{2--5 extra towers})},$ (8)
where $N_{\text{mpart,OS/SS}}$ is the number of multiparticle events with two
OS or SS tracks. The validity of this assumption is checked by comparing the
true and predicted number of background events in inclusive MC samples (Table
0.5.3). The observed discrepancy reflects the differences between OS and SS
events and is included as a systematic uncertainty in the estimate of the
total number of multiparticle background events, as discussed in Section
0.5.4. With this formula and the fact that all SS events are multiparticle
events because of charge conservation, it is possible to calculate the value
of $N_{\text{mpart,OS}}(\text{0 towers})$, which is the number of
multiparticle background events. The expected distribution of the
multiparticle background is obtained using OS events with 2–5 extra
calorimeter towers.
This method does not take into account the background contribution from
$\PGo\to\PGpp\PGpm\PGpz$, because this decay cannot be observed in the SS
events. This latter contribution is negligible (0.5%) based on MC simulation
results.
Checking the validity of Eq. (8) by comparing the true and predicted number of
background events in inclusive MC samples. Event generator Difference in
normalization epos $(+11\pm 4)\%$ 8 CUETP8M1 $(-5.5\pm 3)\%$ 8 MBR $(+10\pm
4)\%$
Figure 4: Background distributions as functions of kinematic variables
estimated by data-driven methods. The proton dissociation background is not
shown here, since it is included via scaling of the final cross section
values. The error bars correspond to statistical uncertainties. The results
for the 5.02data set are similar.
Genuine exclusive $\PKp\PKm$ and $\Pp\PAp$ events, where both particles are
misidentified as pions, are included in the previous multiparticle background
estimate. To correct for this contribution, the $\PK/\PGp$ ratios are
calculated in the exclusive events using tracks with $p<1\GeV$. Similarly, the
$\Pp/\PGp$ ratio is calculated in the same sample in the range $1<p<2\GeV$.
The $\PK/\PGp$ and $\Pp/\PGp$ ratios are assumed to be $0.3^{+0.1}_{-0.05}$ in
the region $p>1$ and $p>2\GeV$, respectively [43]. Using this assumption and
the measured ratios, the average $\PK/\PGp$ and $\Pp/\PGp$ ratios are then
calculated over the entire momentum range of the exclusive sample. These
average ratios can then be used to compute the number of $\PKp\PKm$ and
$\Pp\PAp$ events under two extreme scenarios. The first scenario assumes that
the production of a $\PK$ or a $\Pp$ is always accompanied by the production
of its antiparticle, whereas in the second scenario it is assumed that the
production of an individual $\PKp$, $\PKm$, $\Pp$, or $\PAp$ is a totally
independent process. The final estimate of the exclusive $\PKp\PKm$ and
$\Pp\PAp$ background normalization is calculated as the average of the
estimates obtained from assuming these two scenarios. According to these
calculations, there is an 11% residual contribution of exclusive $\PKp\PKm$
and $\Pp\PAp$ events in the sample after the multiparticle background
subtraction. The background distributions of this contribution are calculated
by using two-track OS exclusive events with at least one identified $\PKpm$
(Fig. 4).
The estimated multiparticle and exclusive $\PKp\PKm$/$\Pp\PAp$ background
distributions, as functions of the main kinematic variables, are shown in Fig.
4. These two background contributions are subtracted from the measured
distributions. The background subtracted spectra are divided by the integrated
luminosity to obtain the differential cross sections.
### 0.5.4 Systematic uncertainties
Systematic uncertainties in the measured cross sections originate from various
sources. These include reconstruction effects, particle identification,
correction factors, background estimation, and the luminosity estimation. The
uncertainty assigned to the tracking efficiency in the case of a single track
is 3.9% [29], which corresponds to 7.8% uncertainty for two tracks.
Furthermore, the uncertainty in the multiple reconstruction rate for a single
track is also 3.9%, which propagates to a maximum of 0.4% uncertainty in the
cross section for two tracks, which is neglected in the analysis.
Misreconstructed tracks bias the sample in two ways: either a CEP event is
rejected if a third misreconstructed track is found, or an event is identified
as CEP with a misreconstructed and a genuine track. This source of systematic
uncertainty is estimated to be 1% for a single track, which is the maximal
misreconstruction rate calculated using inclusive MC samples in the kinematic
region ($\pt(\PGp)>0.2\GeV$ and $\abs{\eta(\PGp)}<2.4$) of the analysis. Since
the probability to have two or more misreconstructed tracks in these low-
multiplicity events is negligible, the final uncertainty remains 1%. From the
comparison of the dime mc and starlight simulations, the uncertainty of the
vertex correction is estimated to be 1%.
The systematic uncertainty in the pileup correction factor for a single event
is calculated from only the systematic uncertainties in the luminosity
measurement that do not affect its overall normalization. Indeed, the
normalization-related systematic uncertainties are compensated in the
exponential fit described in Section 0.5.2. The uncertainties that do not
affect the normalization are estimated to be 1.6% and 1.5% for 5.02 [44] and
13[45] data, respectively. These values propagate to a 1% uncertainty in the
pileup correction factor for a single event. After adding up all the selected
events, the pileup uncertainty becomes smaller than 0.1%, which is neglected
in the following.
The measured signal yield is affected by the uncertainty arising from the two
effects associated with calorimeter noise and veto inefficiency caused by the
adopted energy thresholds. A genuine CEP event can be erroneously discarded if
the calorimeter noise appears above the energy thresholds used in the veto.
Conversely a nonCEP event can pass the final selection if the extra particles
pass the veto requirements. In the HF, these uncertainties are estimated by
varying the calorimeter energy thresholds by $\pm 10\%$ [46]. The resulting
uncertainty is estimated to be 3% for both the 5.02 and 13data sets.
Similarly, the ECAL and HCAL thresholds are varied by $\pm 5\%$ [47, 48],
which results in a 1% uncertainty in the corrected yields at both energies.
The systematic uncertainty estimation of the multiparticle background is done
by varying the control region used in the background estimation procedure:
1–2, 2–9, and 5–9 extra towers. The estimate of the systematic uncertainty in
the multiparticle background normalization is 10%. Additionally, a 10%
uncertainty is added to this value quadratically, taking into account the
deviations shown in Table 0.5.3; thus the final uncertainty in the
multiparticle background normalization is 14%. After subtracting this
contribution, this propagates to systematic uncertainties depending on the
invariant mass, transverse momentum and rapidity of the pion pair. The
multiparticle background estimation uncertainty varies between 10–20% below
$1500\MeV$. Over $1500\MeV$ the uncertainty varies between 20–60%, because the
signal versus background ratio is much smaller. The average uncertainty, used
as the systematic uncertainty of the total cross section, is 15%.
The exclusive $\PKp\PKm$ and $\Pp\PAp$ background uncertainty comes from three
sources: (i) multiparticle contamination in the $\rd E/\rd x$ vs. momentum
distribution that modifies the $\PK/\PGp$ and $\Pp/\PGp$ ratios, (ii) the
uncertainty in the $\PK/\PGp$ ratio above 1, and (iii) the uncertainty in the
$\Pp/\PGp$ ratio above 2. The multiparticle contamination is estimated by
checking the difference between two extreme cases: all particle types are
produced independently, or the sample is purely exclusive. The results
correspond to an uncertainty of 70% in the normalization of this background
contribution at both energies. To account for the uncertainty of $\PK/\PGp$
above 1and $\Pp/\PGp$ over 2, the exclusive background normalization is
calculated assuming different values (0.25, 0.30, and 0.40 [43]) for the
$\PK/\PGp$ and $\Pp/\PGp$ ratios in these regions. The uncertainties assigned
to these effects are 16 and 4%, respectively. Thus the total systematic
uncertainty of the exclusive $\PKp\PKm$ and $\Pp\PAp$ background normalization
is 72%. After subtracting this background contribution, this propagates to
systematic uncertainties, which depend on the invariant mass, transverse
momentum, and rapidity of the pion pair. The typical range of this systematic
uncertainty contribution is 5–20%. For the total cross section, this source
contributes to an average uncertainty of 6%.
The sources and average values of systematic uncertainties, used as the
systematic uncertainty of the total cross section. Source Average value
Tracking efficiency 7.8% Misreconstructed tracks 1% Vertex 1% HF energy scale
3% ECAL and HCAL energy scale 1% Multiparticle background 15% Exclusive
$\PKp\PKm$ and $\Pp\PAp$ background 6% Total w/o int. luminosity 18.3% \+
Integrated luminosity 2.3%
All of the systematic uncertainties listed above are the same for the 5.02 and
13data sets. Additionally, the systematic uncertainty in the integrated
luminosity is 2.3% [44, 45]. The average values of the systematic
uncertainties are summarized in Table 0.5.4. The total systematic uncertainty
is obtained by adding the individual contributions in quadrature. All
systematic uncertainty contributions are considered fully correlated across
invariant mass bins.
## 0.6 Results
The differential cross sections are calculated from the selected events as
functions of the invariant mass, transverse momentum, and rapidity of the pion
pair. These are shown in Fig. 5 with the generator-level predictions from the
starlight and dime mc generators, normalized to their cross sections. The MC
generators provide an incomplete description of the available data, since they
do not model the 500, 980, and 1270 resonances as mentioned in Section 0.3.
Figure 5: Differential cross sections as functions of mass (upper row),
transverse momentum (middle row), and rapidity (bottom row), compared with
generator-level simulations for the 5.02 (left) and 13(right) data sets. The
error bars correspond to statistical, whereas the open boxes to systematic
uncertainties.
There is a peak at 800, which corresponds to the $\PGrzP{770}$ resonance.
Since its quantum numbers $I^{G}(J^{PC})=1^{+}(1^{--})$ are forbidden in DPE
processes, the ${\HepParticle{\PGr}{}{0}{}{}{}}$ mesons must be produced in
VMP processes. The sharp drop visible around 1000is expected from previous
measurements [11, 16] and can be attributed to the quantum mechanical
interference of 980 with the continuum contribution. There is a prominent peak
at 1200–1300, which corresponds to the 1270 resonance with
$I^{G}(J^{PC})=0^{+}(2^{++})$ quantum numbers. This resonance is produced via
a DPE process.
Both dime mc and starlight underestimate the measured spectrum as these MC
event generators do not model the forward dissociation of protons.
Figure 6: Fit to the measured cross section with the sum of four interfering
relativistic Breit–Wigner functions convolved with a normal distribution (to
account for the the experimental resolution of the detector) for the 5.02
(left) and 13(right) data sets. The error bars correspond to statistical,
whereas the open boxes correspond to systematic uncertainties.
The total cross section of the CEP process with two pions in the final state
in the kinematic region $\pt(\PGp)>0.2\GeV$ and $\abs{\eta(\PGp)}<2.4$ is
obtained by integrating the observed spectra in this region:
$\displaystyle\sigma_{\Pp\Pp\to\Pp^{\prime}\Pp^{\prime}\PGpp\PGpm}(\sqrt{s}=5.02\TeV)$
$\displaystyle=32.6\pm 0.7\stat\pm 6.0\syst\pm 0.8\lum\,\mu\text{b},$ (9)
$\displaystyle\sigma_{\Pp\Pp\to\Pp^{\prime}\Pp^{\prime}\PGpp\PGpm}(\sqrt{s}=13\TeV)$
$\displaystyle=33.7\pm 1.0\stat\pm 6.2\syst\pm 0.8\lum\,\mu\text{b}.$ (10)
Below, it is demonstrated that the measured invariant $\PGpp\PGpm$ mass
spectrum is well-described by the sum of the continuum distributions obtained
from the dime mc model and four dominant resonances, modeled here by Breit-
Wigner functions. In the fitting procedure the quantum mechanical interference
effect and the detector resolution are also included.
The following fit function is used:
$f(m)=\int
G(m-m^{\prime};\sigma)\Big{[}\abs{A_{\text{RBW}}^{{\HepParticle{\PGr}{}{0}{}{}{}}}(m^{\prime})}^{2}+\abs{A_{\text{RBW}}^{\PfDzP{500}}(m^{\prime})\re^{\mathrm{i}\phi^{\PfDzP{500}}m^{\prime}}+A_{\mathrm{RBW}}^{\PfDzP{980}}(m^{\prime})\re^{\mathrm{i}\phi^{\PfDzP{980}}m^{\prime}}\\\
+A_{\text{RBW}}^{\mathrm{f}_{2}}(m^{\prime})\re^{\mathrm{i}\phi^{\mathrm{f}_{2}}m^{\prime}}+b\,B^{\textsc{dime}}(m^{\prime})}^{2}\Big{]}\rd
m^{\prime}.$ (11)
Here $G(m;\sigma)$ is a Gaussian distribution with variance $\sigma$ and zero
mean, $B^{\textsc{dime}}(m)$ is the nonresonant background estimated from the
dime mc using the Orear-type form factor, and $b$ is a scale factor for the
continuum contribution, and $\phi^{\PfDzP{500}}$, $\phi^{\PfDzP{980}}$, and
$\phi^{\mathrm{f}_{2}}$ are phases that characterize interference effects. The
$A_{\text{RBW}}^{i}(m)$ is the relativistic Breit–Wigner amplitude, which can
be written as [49]:
$\displaystyle A_{\text{RBW}}^{i,J}(m)$
$\displaystyle=A_{i}\frac{\sqrt{mM_{i}\Gamma(m)}}{m^{2}-M_{i}^{2}+iM_{i}\Gamma(m)},$
(12) $\displaystyle\Gamma(m)$
$\displaystyle=\Gamma_{i}\frac{M_{i}}{m}\left[\frac{m^{2}-4m_{\PGp}^{2}}{M_{i}^{2}-4m_{\PGp}^{2}}\right]^{\frac{2J+1}{2}},$
(13)
where $A_{i}$, $M_{i}$, and $\Gamma_{i}$ are the yield, mass, and width of the
resonance, respectively, $m_{\PGp}$ is the mass of charged pions, and $J$ is
the total angular momentum of the resonance. According to Ref. [2], the
magnitude of the interference between the DPE and VMP processes is around 1%,
therefore no interference term is used between
${\HepParticle{\PGr}{}{0}{}{}{}}$ and DPE resonances. The convolution with the
Gaussian distribution models the mass resolution of the detector.
The mass resolution ($\sigma$) is calculated by fitting the distribution of
the difference between generator-level and reconstructed mass from the
starlight and dime mc simulations. Based on these calculations, the mass
resolution is found to vary from 9 to 14in the mass range 500–2000. In the
final fit, an effective mass resolution of 11is used and the systematic
uncertainty associated with this value is taken into account by repeating the
fit with a mass resolution varying from 9 to 14. The resulting systematic
uncertainty is 7–8% for the yield of 980 and around 1–2% for the yields of the
500, 770, and 1270 resonances. The impact of the uncertainty in the
multiparticle (exclusive $\PKp\PKm$ and $\Pp\PAp$) background yield is
included by varying the background normalization in the fit by $\pm 14\%$
($\pm 72\%$).
Cross sections of the resonant processes in the
$\pt(\PGp)>0.2\GeV,\abs{\eta(\PGp)}<2.4$ fiducial region, extracted from the
simple model fit using the sum of the continuum distribution obtained from the
dime mc model and four dominant resonances. The luminosity-related
uncertainties are included in the systematic uncertainties. The starlight
predictions for
$\Pp\Pp\to\Pp^{\prime}\Pp^{\prime}{\HepParticle{\PGr}{}{0}{}{}{}}\to\Pp^{\prime}\Pp^{\prime}\PGpp\PGpm$
processes are 2.3 and 3.0$\,\mu\text{b}$ for 5.02 and 13, respectively, which
is compatible with the fit results. Resonance
$\sigma_{\Pp\Pp\to\Pp^{\prime}\Pp^{\prime}X\to\Pp^{\prime}\Pp^{\prime}\PGpp\PGpm}[\mu\mathrm{b}]$
$\sqrt{s}=5.02\TeV$ $\sqrt{s}=13\TeV$ 500 $2.8\pm 1.4\stat\pm 2.2\syst$
$2.2\pm 0.8\stat\pm 1.3\syst$ $\PGrzP{770}$ $4.7\pm 0.9\stat\pm 1.3\syst$
$4.3\pm 1.3\stat\pm 1.5\syst$ 980 $0.5\pm 0.1\stat\pm 0.1\syst$ $1.1\pm
0.4\stat\pm 0.3\syst$ 1270 $3.6\pm 0.6\stat\pm 0.7\syst$ $4.2\pm 0.9\stat\pm
0.8\syst$
The masses and widths of 770 and 1270 resonances are fixed to the values of
Ref. [24]. The mass and width of 500 and 980 are fixed according to the
results from the most advanced calculations using dispersion relations [50].
The fits are also performed with the mass and width of 500 and 980 varied
according to their uncertainties [24] and the resulting variation in the cross
section of the resonances is added in quadrature to the other systematic
uncertainty contributions. Furthermore the fit is repeated with the two other
dime mc settings and the variation in the cross section is taken as an
additional systematic uncertainty and added in quadrature to the other
uncertainties.
The above simple model fit also provides values for the cross sections of the
resonances; these are obtained by integrating the fitted squared amplitudes
from the dipion threshold ($2m_{\PGp}$) to $M_{i}+5\Gamma_{i}$:
$\sigma^{\text{res}}_{i}=\int_{2m_{\PGp}}^{M_{i}+5\Gamma_{i}}\abs{A^{2}_{\text{RBW},i}(m)}\rd
m.$ (14)
The fits are shown in Fig. 6 and the cross sections are summarized in Table
0.6.
The model of interfering Breit–Wigner resonances with a continuum gives a good
description of the data in the region of resonant peaks (below 1500). The
cross sections for $\PGrzP{770}$ production calculated from the fits are
slightly larger than the predicted values from starlight, which are 2.3 and
3.0$\,\mu\text{b}$ for 5.02 and 13, respectively. The differences can be
attributed to the additional semiexclusive contribution that is not modeled by
starlight. The values of the scale parameter $b$ are $0.7\pm 0.2$ for 5.02and
$1.1\pm 0.3$ for 13, and therefore they are consistent within uncertainties
for the two energies.
## 0.7 Summary
The cross sections for central exclusive pion pair production have been
measured in $\Pp\Pp$ collisions at 5.02 and 13center-of-mass energies.
Exclusive events are selected by vetoing additional energy deposits in the
calorimeters and by requiring two oppositely charged pions identified via
their mean energy loss in the tracker detectors. These events are used
together with correction factors to obtain invariant mass, transverse
momentum, and rapidity distributions of the $\PGpp\PGpm$ system. The measured
total exclusive $\PGpp\PGpm$ production cross section is $32.6\pm 0.7\stat\pm
6.0\syst\pm 0.8\lum$ and $33.7\pm 1.0\stat\pm 6.2\syst\pm
0.8\lum$$\,\mu\text{b}$ for 5.02 and 13, respectively. The observed mass
spectrum exhibits resonant structures, which can be fitted with a simple model
containing four interfering Breit-Wigner functions, corresponding to the 500,
$\PGrzP{770}$, 980, and 1270 resonances, and a continuum contribution modeled
by the dime mc. The exclusive production cross sections are extracted from
this fit. The obtained cross sections of $\PGrzP{770}$ production are higher
than the starlight model prediction, which can be explained by the presence of
semiexclusive production which is not modeled by the starlight generator.
###### Acknowledgements.
We congratulate our colleagues in the CERN accelerator departments for the
excellent performance of the LHC and thank the technical and administrative
staffs at CERN and at other CMS institutes for their contributions to the
success of the CMS effort. In addition, we gratefully acknowledge the
computing centers and personnel of the Worldwide LHC Computing Grid for
delivering so effectively the computing infrastructure essential to our
analyses. Finally, we acknowledge the enduring support for the construction
and operation of the LHC and the CMS detector provided by the following
funding agencies: BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq,
CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST,
and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF
(Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, PUT and ERDF (Estonia); Academy
of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG,
and HGF (Germany); GSRT (Greece); NKFIA (Hungary); DAE and DST (India); IPM
(Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); MES
(Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT,
LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New Zealand); PAEC
(Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom,
RAS, RFBR, and NRC KI (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI, and FEDER
(Spain); MOSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST
(Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK
(Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
Individuals have received support from the Marie-Curie program and the
European Research Council and Horizon 2020 Grant, contract Nos. 675440,
752730, and 765710 (European Union); the Leventis Foundation; the A.P. Sloan
Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science
Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et
dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door
Wetenschap en Technologie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium)
under the “Excellence of Science – EOS” – be.h project n. 30820817; the
Beijing Municipal Science & Technology Commission, No. Z191100007219010; the
Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the
Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy –
EXC 2121 “Quantum Universe” – 390833306; the Lendület (“Momentum”) Program and
the János Bolyai Research Scholarship of the Hungarian Academy of Sciences,
the New National Excellence Program ÚNKP, the NKFIA research grants 123842,
123959, 124845, 124850, 125105, 128713, 128786, and 129058 (Hungary); the
Council of Science and Industrial Research, India; the HOMING PLUS program of
the Foundation for Polish Science, cofinanced from European Union, Regional
Development Fund, the Mobility Plus program of the Ministry of Science and
Higher Education, the National Science Center (Poland), contracts Harmonia
2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and
2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities
Research Program by Qatar National Research Fund; the Ministry of Science and
Education, grant no. 14.W03.31.0026 (Russia); the Tomsk Polytechnic University
Competitiveness Enhancement Program and “Nauka” Project FSWW-2020-0008
(Russia); the Programa Estatal de Fomento de la Investigación Científica y
Técnica de Excelencia María de Maeztu, grant MDM-2015-0509 and the Programa
Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programs
cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for
Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn
Academic into Its 2nd Century Project Advancement Project (Thailand); the
Kavli Foundation; the Nvidia Corporation; the SuperMicro Corporation; the
Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA).
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## .8 The CMS Collaboration
Yerevan Physics Institute, Yerevan, Armenia
A.M. Sirunyan${}^{\textrm{\textdagger}}$, A. Tumasyan Institut für
Hochenergiephysik, Wien, Austria
W. Adam, F. Ambrogi, T. Bergauer, J. Brandstetter, M. Dragicevic, J. Erö, A.
Escalante Del Valle, M. Flechl, R. Frühwirth1, M. Jeitler1, N. Krammer, I.
Krätschmer, D. Liko, T. Madlener, I. Mikulec, N. Rad, J. Schieck1, R.
Schöfbeck, M. Spanring, D. Spitzbart, W. Waltenberger, J. Wittmann, C.-E.
Wulz1, M. Zarucki Institute for Nuclear Problems, Minsk, Belarus
V. Drugakov, V. Mossolov, J. Suarez Gonzalez Universiteit Antwerpen,
Antwerpen, Belgium
M.R. Darwish, E.A. De Wolf, D. Di Croce, X. Janssen, J. Lauwers, A. Lelek, M.
Pieters, H. Rejeb Sfar, H. Van Haevermaet, P. Van Mechelen, S. Van Putte, N.
Van Remortel Vrije Universiteit Brussel, Brussel, Belgium
F. Blekman, E.S. Bols, S.S. Chhibra, J. D’Hondt, J. De Clercq, D. Lontkovskyi,
S. Lowette, I. Marchesini, S. Moortgat, L. Moreels, Q. Python, K. Skovpen, S.
Tavernier, W. Van Doninck, P. Van Mulders, I. Van Parijs Université Libre de
Bruxelles, Bruxelles, Belgium
D. Beghin, B. Bilin, H. Brun, B. Clerbaux, G. De Lentdecker, H. Delannoy, B.
Dorney, L. Favart, A. Grebenyuk, A.K. Kalsi, J. Luetic, A. Popov, N. Postiau,
E. Starling, L. Thomas, C. Vander Velde, P. Vanlaer, D. Vannerom, Q. Wang
Ghent University, Ghent, Belgium
T. Cornelis, D. Dobur, I. Khvastunov2, C. Roskas, D. Trocino, M. Tytgat, W.
Verbeke, B. Vermassen, M. Vit, N. Zaganidis Université Catholique de Louvain,
Louvain-la-Neuve, Belgium
O. Bondu, G. Bruno, C. Caputo, P. David, C. Delaere, M. Delcourt, A.
Giammanco, G. Krintiras, V. Lemaitre, A. Magitteri, K. Piotrzkowski, J.
Prisciandaro, A. Saggio, M. Vidal Marono, P. Vischia, J. Zobec Centro
Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil
F.L. Alves, G.A. Alves, G. Correia Silva, C. Hensel, A. Moraes, P. Rebello
Teles Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
E. Belchior Batista Das Chagas, W. Carvalho, J. Chinellato3, E. Coelho, E.M.
Da Costa, G.G. Da Silveira4, D. De Jesus Damiao, C. De Oliveira Martins, S.
Fonseca De Souza, L.M. Huertas Guativa, H. Malbouisson, J. Martins5, D. Matos
Figueiredo, M. Medina Jaime6, M. Melo De Almeida, C. Mora Herrera, L. Mundim,
H. Nogima, W.L. Prado Da Silva, L.J. Sanchez Rosas, A. Santoro, A. Sznajder,
M. Thiel, E.J. Tonelli Manganote3, F. Torres Da Silva De Araujo, A. Vilela
Pereira Universidade Estadual Paulista a, Universidade Federal do ABC b, São
Paulo, Brazil
S. Ahujaa, C.A. Bernardesa, L. Calligarisa, T.R. Fernandez Perez Tomeia, E.M.
Gregoresb, D.S. Lemos, P.G. Mercadanteb, S.F. Novaesa, SandraS. Padulaa
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of
Sciences, Sofia, Bulgaria
A. Aleksandrov, G. Antchev, R. Hadjiiska, P. Iaydjiev, A. Marinov, M. Misheva,
M. Rodozov, M. Shopova, G. Sultanov University of Sofia, Sofia, Bulgaria
A. Dimitrov, L. Litov, B. Pavlov, P. Petkov Beihang University, Beijing, China
W. Fang7, X. Gao7, L. Yuan Department of Physics, Tsinghua University,
Beijing, China
Z. Hu, Y. Wang Institute of High Energy Physics, Beijing, China
M. Ahmad, G.M. Chen, H.S. Chen, M. Chen, C.H. Jiang, D. Leggat, H. Liao, Z.
Liu, S.M. Shaheen8, A. Spiezia, J. Tao, E. Yazgan, H. Zhang, S. Zhang8, J.
Zhao State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing, China
A. Agapitos, Y. Ban, G. Chen, A. Levin, J. Li, L. Li, Q. Li, Y. Mao, S.J.
Qian, D. Wang Universidad de Los Andes, Bogota, Colombia
C. Avila, A. Cabrera, L.F. Chaparro Sierra, C. Florez, C.F. González
Hernández, M.A. Segura Delgado University of Split, Faculty of Electrical
Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia
D. Giljanović, N. Godinovic, D. Lelas, I. Puljak, T. Sculac University of
Split, Faculty of Science, Split, Croatia
Z. Antunovic, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia
V. Brigljevic, S. Ceci, D. Ferencek, K. Kadija, B. Mesic, M. Roguljic, A.
Starodumov9, T. Susa University of Cyprus, Nicosia, Cyprus
M.W. Ather, A. Attikis, E. Erodotou, A. Ioannou, M. Kolosova, S. Konstantinou,
G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A. Razis, H.
Rykaczewski, D. Tsiakkouri Charles University, Prague, Czech Republic
M. Finger10, M. Finger Jr.10, A. Kveton, J. Tomsa Escuela Politecnica
Nacional, Quito, Ecuador
E. Ayala Universidad San Francisco de Quito, Quito, Ecuador
E. Carrera Jarrin Academy of Scientific Research and Technology of the Arab
Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt
M.A. Mahmoud11,12, Y. Mohammed11 National Institute of Chemical Physics and
Biophysics, Tallinn, Estonia
S. Bhowmik, A. Carvalho Antunes De Oliveira, R.K. Dewanjee, K. Ehataht, M.
Kadastik, M. Raidal, C. Veelken Department of Physics, University of Helsinki,
Helsinki, Finland
P. Eerola, L. Forthomme, H. Kirschenmann, K. Osterberg, J. Pekkanen, M.
Voutilainen Helsinki Institute of Physics, Helsinki, Finland
F. Garcia, J. Havukainen, J.K. Heikkilä, T. Järvinen, V. Karimäki, R.
Kinnunen, T. Lampén, K. Lassila-Perini, S. Laurila, S. Lehti, T. Lindén, P.
Luukka, T. Mäenpää, H. Siikonen, E. Tuominen, J. Tuominiemi Lappeenranta
University of Technology, Lappeenranta, Finland
T. Tuuva IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
M. Besancon, F. Couderc, M. Dejardin, D. Denegri, B. Fabbro, J.L. Faure, F.
Ferri, S. Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry,
C. Leloup, E. Locci, J. Malcles, J. Rander, A. Rosowsky, M.Ö. Sahin, A. Savoy-
Navarro13, M. Titov Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole
Polytechnique, Institut Polytechnique de Paris
C. Amendola, F. Beaudette, P. Busson, C. Charlot, B. Diab, R. Granier de
Cassagnac, I. Kucher, A. Lobanov, C. Martin Perez, M. Nguyen, C. Ochando, P.
Paganini, J. Rembser, R. Salerno, J.B. Sauvan, Y. Sirois, A. Zabi, A. Zghiche
Université de Strasbourg, CNRS, IPHC UMR 7178, Strasbourg, France
J.-L. Agram14, J. Andrea, D. Bloch, G. Bourgatte, J.-M. Brom, E.C. Chabert, C.
Collard, E. Conte14, J.-C. Fontaine14, D. Gelé, U. Goerlach, M. Jansová, A.-C.
Le Bihan, N. Tonon, P. Van Hove Centre de Calcul de l’Institut National de
Physique Nucleaire et de Physique des Particules, CNRS/IN2P3, Villeurbanne,
France
S. Gadrat Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3,
Institut de Physique Nucléaire de Lyon, Villeurbanne, France
S. Beauceron, C. Bernet, G. Boudoul, C. Camen, N. Chanon, R. Chierici, D.
Contardo, P. Depasse, H. El Mamouni, J. Fay, S. Gascon, M. Gouzevitch, B.
Ille, Sa. Jain, F. Lagarde, I.B. Laktineh, H. Lattaud, M. Lethuillier, L.
Mirabito, S. Perries, V. Sordini, G. Touquet, M. Vander Donckt, S. Viret
Georgian Technical University, Tbilisi, Georgia
T. Toriashvili15 Tbilisi State University, Tbilisi, Georgia
Z. Tsamalaidze10 RWTH Aachen University, I. Physikalisches Institut, Aachen,
Germany
C. Autermann, L. Feld, M.K. Kiesel, K. Klein, M. Lipinski, D. Meuser, A.
Pauls, M. Preuten, M.P. Rauch, C. Schomakers, J. Schulz, M. Teroerde, B.
Wittmer RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
A. Albert, M. Erdmann, S. Erdweg, T. Esch, B. Fischer, R. Fischer, S. Ghosh,
T. Hebbeker, K. Hoepfner, H. Keller, L. Mastrolorenzo, M. Merschmeyer, A.
Meyer, P. Millet, G. Mocellin, S. Mondal, S. Mukherjee, D. Noll, A. Novak, T.
Pook, A. Pozdnyakov, T. Quast, M. Radziej, Y. Rath, H. Reithler, M. Rieger, A.
Schmidt, S.C. Schuler, A. Sharma, S. Thüer, S. Wiedenbeck RWTH Aachen
University, III. Physikalisches Institut B, Aachen, Germany
G. Flügge, W. Haj Ahmad16, O. Hlushchenko, T. Kress, T. Müller, A. Nehrkorn,
A. Nowack, C. Pistone, O. Pooth, D. Roy, H. Sert, A. Stahl17 Deutsches
Elektronen-Synchrotron, Hamburg, Germany
M. Aldaya Martin, C. Asawatangtrakuldee, P. Asmuss, I. Babounikau, H.
Bakhshiansohi, K. Beernaert, O. Behnke, U. Behrens, A. Bermúdez Martínez, D.
Bertsche, A.A. Bin Anuar, K. Borras18, V. Botta, A. Campbell, A. Cardini, P.
Connor, S. Consuegra Rodríguez, C. Contreras-Campana, V. Danilov, A. De Wit,
M.M. Defranchis, C. Diez Pardos, D. Domínguez Damiani, G. Eckerlin, D.
Eckstein, T. Eichhorn, A. Elwood, E. Eren, E. Gallo19, A. Geiser, J.M. Grados
Luyando, A. Grohsjean, M. Guthoff, M. Haranko, A. Harb, N.Z. Jomhari, H. Jung,
A. Kasem18, M. Kasemann, J. Keaveney, C. Kleinwort, J. Knolle, D. Krücker, W.
Lange, T. Lenz, J. Leonard, J. Lidrych, K. Lipka, W. Lohmann20, R. Mankel,
I.-A. Melzer-Pellmann, A.B. Meyer, M. Meyer, M. Missiroli, G. Mittag, J.
Mnich, A. Mussgiller, V. Myronenko, D. Pérez Adán, S.K. Pflitsch, D. Pitzl, A.
Raspereza, A. Saibel, M. Savitskyi, V. Scheurer, P. Schütze, C.
Schwanenberger, R. Shevchenko, A. Singh, H. Tholen, O. Turkot, A. Vagnerini,
M. Van De Klundert, G.P. Van Onsem, R. Walsh, Y. Wen, K. Wichmann, C. Wissing,
O. Zenaiev, R. Zlebcik University of Hamburg, Hamburg, Germany
R. Aggleton, S. Bein, L. Benato, A. Benecke, V. Blobel, T. Dreyer, A.
Ebrahimi, A. Fröhlich, C. Garbers, E. Garutti, D. Gonzalez, P. Gunnellini, J.
Haller, A. Hinzmann, A. Karavdina, G. Kasieczka, R. Klanner, R. Kogler, N.
Kovalchuk, S. Kurz, V. Kutzner, J. Lange, T. Lange, A. Malara, D. Marconi, J.
Multhaup, M. Niedziela, C.E.N. Niemeyer, D. Nowatschin, A. Perieanu, A.
Reimers, O. Rieger, C. Scharf, P. Schleper, S. Schumann, J. Schwandt, J.
Sonneveld, H. Stadie, G. Steinbrück, F.M. Stober, M. Stöver, B. Vormwald, I.
Zoi Karlsruher Institut fuer Technologie, Karlsruhe, Germany
M. Akbiyik, C. Barth, M. Baselga, S. Baur, T. Berger, E. Butz, R. Caspart, T.
Chwalek, W. De Boer, A. Dierlamm, K. El Morabit, N. Faltermann, M. Giffels, P.
Goldenzweig, A. Gottmann, M.A. Harrendorf, F. Hartmann17, U. Husemann, S.
Kudella, S. Mitra, M.U. Mozer, Th. Müller, M. Musich, A. Nürnberg, G. Quast,
K. Rabbertz, M. Schröder, I. Shvetsov, H.J. Simonis, R. Ulrich, M. Weber, C.
Wöhrmann, R. Wolf Institute of Nuclear and Particle Physics (INPP), NCSR
Demokritos, Aghia Paraskevi, Greece
G. Anagnostou, P. Asenov, G. Daskalakis, T. Geralis, A. Kyriakis, D. Loukas,
G. Paspalaki National and Kapodistrian University of Athens, Athens, Greece
M. Diamantopoulou, G. Karathanasis, P. Kontaxakis, A. Panagiotou, I.
Papavergou, N. Saoulidou, A. Stakia, K. Theofilatos, K. Vellidis National
Technical University of Athens, Athens, Greece
G. Bakas, K. Kousouris, I. Papakrivopoulos, G. Tsipolitis University of
Ioánnina, Ioánnina, Greece
I. Evangelou, C. Foudas, P. Gianneios, P. Katsoulis, P. Kokkas, S. Mallios, K.
Manitara, N. Manthos, I. Papadopoulos, J. Strologas, F.A. Triantis, D.
Tsitsonis MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary
M. Bartók21, M. Csanad, P. Major, K. Mandal, A. Mehta, M.I. Nagy, G. Pasztor,
O. Surányi, G.I. Veres Wigner Research Centre for Physics, Budapest, Hungary
G. Bencze, C. Hajdu, D. Horvath22, F. Sikler, T.Á. Vámi, V. Veszpremi, G.
Vesztergombi${}^{\textrm{\textdagger}}$ Institute of Nuclear Research ATOMKI,
Debrecen, Hungary
N. Beni, S. Czellar, J. Karancsi21, A. Makovec, J. Molnar, Z. Szillasi
Institute of Physics, University of Debrecen, Debrecen, Hungary
P. Raics, D. Teyssier, Z.L. Trocsanyi, B. Ujvari Eszterhazy Karoly University,
Karoly Robert Campus, Gyongyos, Hungary
T.F. Csorgo, W.J. Metzger, F. Nemes, T. Novak Indian Institute of Science
(IISc), Bangalore, India
S. Choudhury, J.R. Komaragiri, P.C. Tiwari National Institute of Science
Education and Research, HBNI, Bhubaneswar, India
S. Bahinipati24, C. Kar, G. Kole, P. Mal, V.K. Muraleedharan Nair Bindhu, A.
Nayak25, S. Roy Chowdhury, D.K. Sahoo24, S.K. Swain Panjab University,
Chandigarh, India
S. Bansal, S.B. Beri, V. Bhatnagar, S. Chauhan, R. Chawla, N. Dhingra, R.
Gupta, A. Kaur, M. Kaur, S. Kaur, P. Kumari, M. Lohan, M. Meena, K. Sandeep,
S. Sharma, J.B. Singh, A.K. Virdi, G. Walia University of Delhi, Delhi, India
A. Bhardwaj, B.C. Choudhary, R.B. Garg, M. Gola, S. Keshri, Ashok Kumar, S.
Malhotra, M. Naimuddin, P. Priyanka, K. Ranjan, Aashaq Shah, R. Sharma Saha
Institute of Nuclear Physics, HBNI, Kolkata, India
R. Bhardwaj26, M. Bharti26, R. Bhattacharya, S. Bhattacharya, U. Bhawandeep26,
D. Bhowmik, S. Dey, S. Dutta, S. Ghosh, M. Maity27, K. Mondal, S. Nandan, A.
Purohit, P.K. Rout, A. Roy, G. Saha, S. Sarkar, T. Sarkar27, M. Sharan, B.
Singh26, S. Thakur26 Indian Institute of Technology Madras, Madras, India
P.K. Behera, P. Kalbhor, A. Muhammad, P.R. Pujahari, A. Sharma, A.K. Sikdar
Bhabha Atomic Research Centre, Mumbai, India
R. Chudasama, D. Dutta, V. Jha, V. Kumar, D.K. Mishra, P.K. Netrakanti, L.M.
Pant, P. Shukla Tata Institute of Fundamental Research-A, Mumbai, India
T. Aziz, M.A. Bhat, S. Dugad, G.B. Mohanty, N. Sur, RavindraKumar Verma Tata
Institute of Fundamental Research-B, Mumbai, India
S. Banerjee, S. Bhattacharya, S. Chatterjee, P. Das, M. Guchait, S. Karmakar,
S. Kumar, G. Majumder, K. Mazumdar, N. Sahoo, S. Sawant Indian Institute of
Science Education and Research (IISER), Pune, India
S. Chauhan, S. Dube, V. Hegde, A. Kapoor, K. Kothekar, S. Pandey, A. Rane, A.
Rastogi, S. Sharma Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran
S. Chenarani28, E. Eskandari Tadavani, S.M. Etesami28, M. Khakzad, M.
Mohammadi Najafabadi, M. Naseri, F. Rezaei Hosseinabadi University College
Dublin, Dublin, Ireland
M. Felcini, M. Grunewald INFN Sezione di Bari a, Università di Bari b,
Politecnico di Bari c, Bari, Italy
M. Abbresciaa,b, C. Calabriaa,b, A. Colaleoa, D. Creanzaa,c, L. Cristellaa,b,
N. De Filippisa,c, M. De Palmaa,b, A. Di Florioa,b, L. Fiorea, A. Gelmia,b, G.
Iasellia,c, M. Incea,b, S. Lezkia,b, G. Maggia,c, M. Maggia, G. Minielloa,b,
S. Mya,b, S. Nuzzoa,b, A. Pompilia,b, G. Pugliesea,c, R. Radognaa, A.
Ranieria, G. Selvaggia,b, L. Silvestrisa, R. Vendittia, P. Verwilligena INFN
Sezione di Bologna a, Università di Bologna b, Bologna, Italy
G. Abbiendia, C. Battilanaa,b, D. Bonacorsia,b, L. Borgonovia,b, S. Braibant-
Giacomellia,b, R. Campaninia,b, P. Capiluppia,b, A. Castroa,b, F.R. Cavalloa,
C. Cioccaa, G. Codispotia,b, M. Cuffiania,b, G.M. Dallavallea, F. Fabbria, A.
Fanfania,b, E. Fontanesi, P. Giacomellia, C. Grandia, L. Guiduccia,b, F.
Iemmia,b, S. Lo Meoa,29, S. Marcellinia, G. Masettia, F.L. Navarriaa,b, A.
Perrottaa, F. Primaveraa,b, A.M. Rossia,b, T. Rovellia,b, G.P. Sirolia,b, N.
Tosia INFN Sezione di Catania a, Università di Catania b, Catania, Italy
S. Albergoa,b,30, S. Costaa,b, A. Di Mattiaa, R. Potenzaa,b, A. Tricomia,b,30,
C. Tuvea,b INFN Sezione di Firenze a, Università di Firenze b, Firenze, Italy
G. Barbaglia, R. Ceccarellia,b, K. Chatterjeea,b, V. Ciullia,b, C. Civininia,
R. D’Alessandroa,b, E. Focardia,b, G. Latino, P. Lenzia,b, M. Meschinia, S.
Paolettia, L. Russoa,31, G. Sguazzonia, D. Stroma, L. Viliania INFN Laboratori
Nazionali di Frascati, Frascati, Italy
L. Benussi, S. Bianco, D. Piccolo INFN Sezione di Genova a, Università di
Genova b, Genova, Italy
M. Bozzoa,b, F. Ferroa, R. Mulargiaa,b, E. Robuttia, S. Tosia,b INFN Sezione
di Milano-Bicocca a, Università di Milano-Bicocca b, Milano, Italy
A. Benagliaa, A. Beschia,b, F. Brivioa,b, V. Cirioloa,b,17, S. Di Guidaa,b,17,
M.E. Dinardoa,b, P. Dinia, S. Fiorendia,b, S. Gennaia, A. Ghezzia,b, P.
Govonia,b, L. Guzzia,b, M. Malbertia, S. Malvezzia, D. Menascea, F. Montia,b,
L. Moronia, G. Ortonaa,b, M. Paganonia,b, D. Pedrinia, S. Ragazzia,b, T.
Tabarelli de Fatisa,b, D. Zuoloa,b INFN Sezione di Napoli a, Università di
Napoli ’Federico II’ b, Napoli, Italy, Università della Basilicata c, Potenza,
Italy, Università G. Marconi d, Roma, Italy
S. Buontempoa, N. Cavalloa,c, A. De Iorioa,b, A. Di Crescenzoa,b, F.
Fabozzia,c, F. Fiengaa, G. Galatia, A.O.M. Iorioa,b, L. Listaa,b, S.
Meolaa,d,17, P. Paoluccia,17, B. Rossia, C. Sciaccaa,b, E. Voevodinaa,b INFN
Sezione di Padova a, Università di Padova b, Padova, Italy, Università di
Trento c, Trento, Italy
P. Azzia, N. Bacchettaa, D. Biselloa,b, A. Bolettia,b, A. Bragagnolo, R.
Carlina,b, P. Checchiaa, P. De Castro Manzanoa, T. Dorigoa, U. Dossellia, F.
Gasparinia,b, U. Gasparinia,b, A. Gozzelinoa, S.Y. Hoh, P. Lujan, M.
Margonia,b, A.T. Meneguzzoa,b, J. Pazzinia,b, M. Presillab, P. Ronchesea,b, R.
Rossina,b, F. Simonettoa,b, A. Tiko, M. Tosia,b, M. Zanettia,b, P. Zottoa,b,
G. Zumerlea,b INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy
A. Braghieria, P. Montagnaa,b, S.P. Rattia,b, V. Rea, M. Ressegottia,b, C.
Riccardia,b, P. Salvinia, I. Vaia,b, P. Vituloa,b INFN Sezione di Perugia a,
Università di Perugia b, Perugia, Italy
M. Biasinia,b, G.M. Bileia, C. Cecchia,b, D. Ciangottinia,b, L. Fanòa,b, P.
Laricciaa,b, R. Leonardia,b, E. Manonia, G. Mantovania,b, V. Mariania,b, M.
Menichellia, A. Rossia,b, A. Santocchiaa,b, D. Spigaa INFN Sezione di Pisa a,
Università di Pisa b, Scuola Normale Superiore di Pisa c, Pisa, Italy
K. Androsova, P. Azzurria, G. Bagliesia, V. Bertacchia,c, L. Bianchinia, T.
Boccalia, R. Castaldia, M.A. Cioccia,b, R. Dell’Orsoa, G. Fedia, F. Fioria,c,
L. Gianninia,c, A. Giassia, M.T. Grippoa, F. Ligabuea,c, E. Mancaa,c, G.
Mandorlia,c, A. Messineoa,b, F. Pallaa, A. Rizzia,b, G. Rolandi32, A.
Scribanoa, P. Spagnoloa, R. Tenchinia, G. Tonellia,b, N. Turinia, A. Venturia,
P.G. Verdinia INFN Sezione di Roma a, Sapienza Università di Roma b, Rome,
Italy
F. Cavallaria, M. Cipriania,b, D. Del Rea,b, E. Di Marcoa,b, M. Diemoza, E.
Longoa,b, B. Marzocchia,b, P. Meridiania, G. Organtinia,b, F. Pandolfia, R.
Paramattia,b, C. Quarantaa,b, S. Rahatloua,b, C. Rovellia, F.
Santanastasioa,b, L. Soffia,b INFN Sezione di Torino a, Università di Torino
b, Torino, Italy, Università del Piemonte Orientale c, Novara, Italy
N. Amapanea,b, R. Arcidiaconoa,c, S. Argiroa,b, M. Arneodoa,c, N. Bartosika,
R. Bellana,b, C. Biinoa, A. Cappatia,b, N. Cartigliaa, S. Comettia, M.
Costaa,b, R. Covarellia,b, N. Demariaa, B. Kiania,b, C. Mariottia, S.
Masellia, E. Migliorea,b, V. Monacoa,b, E. Monteila,b, M. Montenoa, M.M.
Obertinoa,b, L. Pachera,b, N. Pastronea, M. Pelliccionia, G.L. Pinna
Angionia,b, A. Romeroa,b, M. Ruspaa,c, R. Sacchia,b, R. Salvaticoa,b, K.
Shchelinaa,b, V. Solaa, A. Solanoa,b, D. Soldia,b, A. Staianoa INFN Sezione di
Trieste a, Università di Trieste b, Trieste, Italy
S. Belfortea, V. Candelisea,b, M. Casarsaa, F. Cossuttia, A. Da Rolda,b, G.
Della Riccaa,b, F. Vazzolera,b, A. Zanettia Kyungpook National University,
Daegu, Korea
B. Kim, D.H. Kim, G.N. Kim, M.S. Kim, J. Lee, S.W. Lee, C.S. Moon, Y.D. Oh,
S.I. Pak, S. Sekmen, D.C. Son, Y.C. Yang Chonnam National University,
Institute for Universe and Elementary Particles, Kwangju, Korea
H. Kim, D.H. Moon, G. Oh Hanyang University, Seoul, Korea
B. Francois, T.J. Kim, J. Park Korea University, Seoul, Korea
S. Cho, S. Choi, Y. Go, D. Gyun, S. Ha, B. Hong, K. Lee, K.S. Lee, J. Lim, J.
Park, S.K. Park, Y. Roh Kyung Hee University, Department of Physics
J. Goh Sejong University, Seoul, Korea
H.S. Kim Seoul National University, Seoul, Korea
J. Almond, J.H. Bhyun, J. Choi, S. Jeon, J. Kim, J.S. Kim, H. Lee, K. Lee, S.
Lee, K. Nam, S.B. Oh, B.C. Radburn-Smith, S.h. Seo, U.K. Yang, H.D. Yoo, I.
Yoon, G.B. Yu University of Seoul, Seoul, Korea
D. Jeon, H. Kim, J.H. Kim, J.S.H. Lee, I.C. Park, I. Watson Sungkyunkwan
University, Suwon, Korea
Y. Choi, C. Hwang, Y. Jeong, J. Lee, Y. Lee, I. Yu Riga Technical University,
Riga, Latvia
V. Veckalns33 Vilnius University, Vilnius, Lithuania
V. Dudenas, A. Juodagalvis, J. Vaitkus National Centre for Particle Physics,
Universiti Malaya, Kuala Lumpur, Malaysia
Z.A. Ibrahim, F. Mohamad Idris34, W.A.T. Wan Abdullah, M.N. Yusli, Z. Zolkapli
Universidad de Sonora (UNISON), Hermosillo, Mexico
J.F. Benitez, A. Castaneda Hernandez, J.A. Murillo Quijada, L. Valencia Palomo
Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico
H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-De La Cruz35, R. Lopez-
Fernandez, A. Sanchez-Hernandez Universidad Iberoamericana, Mexico City,
Mexico
S. Carrillo Moreno, C. Oropeza Barrera, M. Ramirez-Garcia, F. Vazquez Valencia
Benemerita Universidad Autonoma de Puebla, Puebla, Mexico
J. Eysermans, I. Pedraza, H.A. Salazar Ibarguen, C. Uribe Estrada Universidad
Autónoma de San Luis Potosí, San Luis Potosí, Mexico
A. Morelos Pineda University of Montenegro, Podgorica, Montenegro
N. Raicevic University of Auckland, Auckland, New Zealand
D. Krofcheck University of Canterbury, Christchurch, New Zealand
S. Bheesette, P.H. Butler National Centre for Physics, Quaid-I-Azam
University, Islamabad, Pakistan
A. Ahmad, M. Ahmad, Q. Hassan, H.R. Hoorani, W.A. Khan, M.A. Shah, M. Shoaib,
M. Waqas AGH University of Science and Technology Faculty of Computer Science,
Electronics and Telecommunications, Krakow, Poland
V. Avati, L. Grzanka, M. Malawski National Centre for Nuclear Research,
Swierk, Poland
H. Bialkowska, M. Bluj, B. Boimska, M. Górski, M. Kazana, M. Szleper, P.
Zalewski Institute of Experimental Physics, Faculty of Physics, University of
Warsaw, Warsaw, Poland
K. Bunkowski, A. Byszuk36, K. Doroba, A. Kalinowski, M. Konecki, J.
Krolikowski, M. Misiura, M. Olszewski, A. Pyskir, M. Walczak Laboratório de
Instrumentação e Física Experimental de Partículas, Lisboa, Portugal
M. Araujo, P. Bargassa, D. Bastos, A. Di Francesco, P. Faccioli, B. Galinhas,
M. Gallinaro, J. Hollar, N. Leonardo, J. Seixas, G. Strong, O. Toldaiev, J.
Varela Joint Institute for Nuclear Research, Dubna, Russia
S. Afanasiev, P. Bunin, M. Gavrilenko, I. Golutvin, I. Gorbunov, A. Kamenev,
V. Karjavine, A. Lanev, A. Malakhov, V. Matveev37,38, P. Moisenz, V. Palichik,
V. Perelygin, M. Savina, S. Shmatov, S. Shulha, N. Skatchkov, V. Smirnov, N.
Voytishin, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St.
Petersburg), Russia
L. Chtchipounov, V. Golovtsov, Y. Ivanov, V. Kim39, E. Kuznetsova40, P.
Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, D. Sosnov, V. Sulimov, L.
Uvarov, A. Vorobyev Institute for Nuclear Research, Moscow, Russia
Yu. Andreev, A. Dermenev, S. Gninenko, N. Golubev, A. Karneyeu, M. Kirsanov,
N. Krasnikov, A. Pashenkov, D. Tlisov, A. Toropin Institute for Theoretical
and Experimental Physics named by A.I. Alikhanov of NRC ‘Kurchatov Institute’,
Moscow, Russia
V. Epshteyn, V. Gavrilov, N. Lychkovskaya, A. Nikitenko41, V. Popov, I.
Pozdnyakov, G. Safronov, A. Spiridonov, A. Stepennov, M. Toms, E. Vlasov, A.
Zhokin Moscow Institute of Physics and Technology, Moscow, Russia
T. Aushev National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
M. Chadeeva42, D. Philippov, E. Popova, V. Rusinov P.N. Lebedev Physical
Institute, Moscow, Russia
V. Andreev, M. Azarkin, I. Dremin38, M. Kirakosyan, A. Terkulov Skobeltsyn
Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow,
Russia
A. Belyaev, E. Boos, A. Ershov, A. Gribushin, L. Khein, V. Klyukhin, O.
Kodolova, I. Lokhtin, O. Lukina, S. Obraztsov, S. Petrushanko, V. Savrin, A.
Snigirev Novosibirsk State University (NSU), Novosibirsk, Russia
A. Barnyakov43, V. Blinov43, T. Dimova43, L. Kardapoltsev43, Y. Skovpen43
Institute for High Energy Physics of National Research Centre ‘Kurchatov
Institute’, Protvino, Russia
I. Azhgirey, I. Bayshev, S. Bitioukov, V. Kachanov, D. Konstantinov, P.
Mandrik, V. Petrov, R. Ryutin, S. Slabospitskii, A. Sobol, S. Troshin, N.
Tyurin, A. Uzunian, A. Volkov National Research Tomsk Polytechnic University,
Tomsk, Russia
A. Babaev, A. Iuzhakov, V. Okhotnikov Tomsk State University, Tomsk, Russia
V. Borchsh, V. Ivanchenko, E. Tcherniaev University of Belgrade: Faculty of
Physics and VINCA Institute of Nuclear Sciences
P. Adzic44, P. Cirkovic, D. Devetak, M. Dordevic, P. Milenovic, J. Milosevic,
M. Stojanovic Centro de Investigaciones Energéticas Medioambientales y
Tecnológicas (CIEMAT), Madrid, Spain
M. Aguilar-Benitez, J. Alcaraz Maestre, A. Álvarez Fernández, I. Bachiller, M.
Barrio Luna, J.A. Brochero Cifuentes, C.A. Carrillo Montoya, M. Cepeda, M.
Cerrada, N. Colino, B. De La Cruz, A. Delgado Peris, C. Fernandez Bedoya, J.P.
Fernández Ramos, J. Flix, M.C. Fouz, O. Gonzalez Lopez, S. Goy Lopez, J.M.
Hernandez, M.I. Josa, D. Moran, Á. Navarro Tobar, A. Pérez-Calero Yzquierdo,
J. Puerta Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, M.S. Soares, A.
Triossi, C. Willmott Universidad Autónoma de Madrid, Madrid, Spain
C. Albajar, J.F. de Trocóniz Universidad de Oviedo, Instituto Universitario de
Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Oviedo, Spain
B. Alvarez Gonzalez, J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras,
I. Gonzalez Caballero, J.R. González Fernández, E. Palencia Cortezon, V.
Rodríguez Bouza, S. Sanchez Cruz Instituto de Física de Cantabria (IFCA),
CSIC-Universidad de Cantabria, Santander, Spain
I.J. Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M.
Fernandez, P.J. Fernández Manteca, A. García Alonso, G. Gomez, C. Martinez
Rivero, P. Martinez Ruiz del Arbol, F. Matorras, J. Piedra Gomez, C. Prieels,
T. Rodrigo, A. Ruiz-Jimeno, L. Scodellaro, N. Trevisani, I. Vila, J.M. Vizan
Garcia University of Colombo, Colombo, Sri Lanka
D.U.J. Sonnadara University of Ruhuna, Department of Physics, Matara, Sri
Lanka
W.G.D. Dharmaratna, N. Wickramage CERN, European Organization for Nuclear
Research, Geneva, Switzerland
D. Abbaneo, B. Akgun, E. Auffray, G. Auzinger, J. Baechler, P. Baillon, A.H.
Ball, D. Barney, J. Bendavid, M. Bianco, A. Bocci, E. Bossini, C. Botta, E.
Brondolin, T. Camporesi, A. Caratelli, G. Cerminara, E. Chapon, G. Cucciati,
D. d’Enterria, A. Dabrowski, N. Daci, V. Daponte, A. David, A. De Roeck, N.
Deelen, M. Deile, M. Dobson, M. Dünser, N. Dupont, A. Elliott-Peisert, F.
Fallavollita45, D. Fasanella, G. Franzoni, J. Fulcher, W. Funk, S. Giani, D.
Gigi, A. Gilbert, K. Gill, F. Glege, M. Gruchala, M. Guilbaud, D. Gulhan, J.
Hegeman, C. Heidegger, Y. Iiyama, V. Innocente, A. Jafari, P. Janot, O.
Karacheban20, J. Kaspar, J. Kieseler, M. Krammer1, C. Lange, P. Lecoq, C.
Lourenço, L. Malgeri, M. Mannelli, A. Massironi, F. Meijers, J.A. Merlin, S.
Mersi, E. Meschi, F. Moortgat, M. Mulders, J. Ngadiuba, S. Nourbakhsh, S.
Orfanelli, L. Orsini, F. Pantaleo17, L. Pape, E. Perez, M. Peruzzi, A.
Petrilli, G. Petrucciani, A. Pfeiffer, M. Pierini, F.M. Pitters, M. Quinto, D.
Rabady, A. Racz, M. Rovere, H. Sakulin, C. Schäfer, C. Schwick, M. Selvaggi,
A. Sharma, P. Silva, W. Snoeys, P. Sphicas46, J. Steggemann, V.R. Tavolaro, D.
Treille, A. Tsirou, A. Vartak, M. Verzetti, W.D. Zeuner Paul Scherrer
Institut, Villigen, Switzerland
L. Caminada47, K. Deiters, W. Erdmann, R. Horisberger, Q. Ingram, H.C.
Kaestli, D. Kotlinski, U. Langenegger, T. Rohe, S.A. Wiederkehr ETH Zurich -
Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland
M. Backhaus, P. Berger, N. Chernyavskaya, G. Dissertori, M. Dittmar, M.
Donegà, C. Dorfer, T.A. Gómez Espinosa, C. Grab, D. Hits, T. Klijnsma, W.
Lustermann, R.A. Manzoni, M. Marionneau, M.T. Meinhard, F. Micheli, P.
Musella, F. Nessi-Tedaldi, F. Pauss, G. Perrin, L. Perrozzi, S. Pigazzini, M.
Reichmann, C. Reissel, T. Reitenspiess, D. Ruini, D.A. Sanz Becerra, M.
Schönenberger, L. Shchutska, M.L. Vesterbacka Olsson, R. Wallny, D.H. Zhu
Universität Zürich, Zurich, Switzerland
T.K. Aarrestad, C. Amsler48, D. Brzhechko, M.F. Canelli, A. De Cosa, R. Del
Burgo, S. Donato, C. Galloni, B. Kilminster, S. Leontsinis, V.M. Mikuni, I.
Neutelings, G. Rauco, P. Robmann, D. Salerno, K. Schweiger, C. Seitz, Y.
Takahashi, S. Wertz, A. Zucchetta National Central University, Chung-Li,
Taiwan
T.H. Doan, C.M. Kuo, W. Lin, S.S. Yu National Taiwan University (NTU), Taipei,
Taiwan
P. Chang, Y. Chao, K.F. Chen, P.H. Chen, W.-S. Hou, Y.y. Li, R.-S. Lu, E.
Paganis, A. Psallidas, A. Steen Chulalongkorn University, Faculty of Science,
Department of Physics, Bangkok, Thailand
B. Asavapibhop, N. Srimanobhas, N. Suwonjandee Çukurova University, Physics
Department, Science and Art Faculty, Adana, Turkey
M.N. Bakirci49, A. Bat, F. Boran, S. Damarseckin50, Z.S. Demiroglu, F. Dolek,
C. Dozen, I. Dumanoglu, E. Eskut, G. Gokbulut, EmineGurpinar Guler51, Y.
Guler, I. Hos52, C. Isik, E.E. Kangal53, O. Kara, A. Kayis Topaksu, U.
Kiminsu, M. Oglakci, G. Onengut, K. Ozdemir54, A.E. Simsek, D. Sunar Cerci55,
U.G. Tok, S. Turkcapar, I.S. Zorbakir, C. Zorbilmez Middle East Technical
University, Physics Department, Ankara, Turkey
B. Isildak56, G. Karapinar57, M. Yalvac Bogazici University, Istanbul, Turkey
I.O. Atakisi, E. Gülmez, M. Kaya58, O. Kaya59, B. Kaynak, Ö. Özçelik, S.
Ozkorucuklu60, S. Tekten, E.A. Yetkin61 Istanbul Technical University,
Istanbul, Turkey
A. Cakir, K. Cankocak, Y. Komurcu, S. Sen62 Institute for Scintillation
Materials of National Academy of Science of Ukraine, Kharkov, Ukraine
B. Grynyov National Scientific Center, Kharkov Institute of Physics and
Technology, Kharkov, Ukraine
L. Levchuk University of Bristol, Bristol, United Kingdom
F. Ball, E. Bhal, S. Bologna, J.J. Brooke, D. Burns, E. Clement, D. Cussans,
O. Davignon, H. Flacher, J. Goldstein, G.P. Heath, H.F. Heath, L. Kreczko, S.
Paramesvaran, B. Penning, T. Sakuma, S. Seif El Nasr-Storey, D. Smith, V.J.
Smith, J. Taylor, A. Titterton Rutherford Appleton Laboratory, Didcot, United
Kingdom
K.W. Bell, A. Belyaev63, C. Brew, R.M. Brown, D. Cieri, D.J.A. Cockerill, J.A.
Coughlan, K. Harder, S. Harper, J. Linacre, K. Manolopoulos, D.M. Newbold, E.
Olaiya, D. Petyt, T. Reis, T. Schuh, C.H. Shepherd-Themistocleous, A. Thea,
I.R. Tomalin, T. Williams, W.J. Womersley Imperial College, London, United
Kingdom
R. Bainbridge, P. Bloch, J. Borg, S. Breeze, O. Buchmuller, A. Bundock,
GurpreetSingh CHAHAL64, D. Colling, P. Dauncey, G. Davies, M. Della Negra, R.
Di Maria, P. Everaerts, G. Hall, G. Iles, T. James, M. Komm, C. Laner, L.
Lyons, A.-M. Magnan, S. Malik, A. Martelli, V. Milosevic, J. Nash65, V.
Palladino, M. Pesaresi, D.M. Raymond, A. Richards, A. Rose, E. Scott, C. Seez,
A. Shtipliyski, M. Stoye, T. Strebler, S. Summers, A. Tapper, K. Uchida, T.
Virdee17, N. Wardle, D. Winterbottom, J. Wright, A.G. Zecchinelli, S.C. Zenz
Brunel University, Uxbridge, United Kingdom
J.E. Cole, P.R. Hobson, A. Khan, P. Kyberd, C.K. Mackay, A. Morton, I.D. Reid,
L. Teodorescu, S. Zahid Baylor University, Waco, USA
K. Call, J. Dittmann, K. Hatakeyama, C. Madrid, B. McMaster, N. Pastika, C.
Smith Catholic University of America, Washington, DC, USA
R. Bartek, A. Dominguez, R. Uniyal The University of Alabama, Tuscaloosa, USA
A. Buccilli, S.I. Cooper, C. Henderson, P. Rumerio, C. West Boston University,
Boston, USA
D. Arcaro, T. Bose, Z. Demiragli, D. Gastler, S. Girgis, D. Pinna, C.
Richardson, J. Rohlf, D. Sperka, I. Suarez, L. Sulak, D. Zou Brown University,
Providence, USA
G. Benelli, B. Burkle, X. Coubez, D. Cutts, M. Hadley, J. Hakala, U. Heintz,
J.M. Hogan66, K.H.M. Kwok, E. Laird, G. Landsberg, J. Lee, Z. Mao, M. Narain,
S. Sagir67, R. Syarif, E. Usai, D. Yu University of California, Davis, Davis,
USA
R. Band, C. Brainerd, R. Breedon, M. Calderon De La Barca Sanchez, M. Chertok,
J. Conway, R. Conway, P.T. Cox, R. Erbacher, C. Flores, G. Funk, F. Jensen, W.
Ko, O. Kukral, R. Lander, M. Mulhearn, D. Pellett, J. Pilot, M. Shi, D. Stolp,
D. Taylor, K. Tos, M. Tripathi, Z. Wang, F. Zhang University of California,
Los Angeles, USA
M. Bachtis, C. Bravo, R. Cousins, A. Dasgupta, A. Florent, J. Hauser, M.
Ignatenko, N. Mccoll, S. Regnard, D. Saltzberg, C. Schnaible, V. Valuev
University of California, Riverside, Riverside, USA
K. Burt, R. Clare, J.W. Gary, S.M.A. Ghiasi Shirazi, G. Hanson, G.
Karapostoli, E. Kennedy, O.R. Long, M. Olmedo Negrete, M.I. Paneva, W. Si, L.
Wang, H. Wei, S. Wimpenny, B.R. Yates, Y. Zhang University of California, San
Diego, La Jolla, USA
J.G. Branson, P. Chang, S. Cittolin, M. Derdzinski, R. Gerosa, D. Gilbert, B.
Hashemi, D. Klein, V. Krutelyov, J. Letts, M. Masciovecchio, S. May, S. Padhi,
M. Pieri, V. Sharma, M. Tadel, F. Würthwein, A. Yagil, G. Zevi Della Porta
University of California, Santa Barbara - Department of Physics, Santa
Barbara, USA
N. Amin, R. Bhandari, C. Campagnari, M. Citron, V. Dutta, M. Franco Sevilla,
L. Gouskos, J. Incandela, B. Marsh, H. Mei, A. Ovcharova, H. Qu, J. Richman,
U. Sarica, D. Stuart, S. Wang, J. Yoo California Institute of Technology,
Pasadena, USA
D. Anderson, A. Bornheim, J.M. Lawhorn, N. Lu, H.B. Newman, T.Q. Nguyen, J.
Pata, M. Spiropulu, J.R. Vlimant, S. Xie, Z. Zhang, R.Y. Zhu Carnegie Mellon
University, Pittsburgh, USA
M.B. Andrews, T. Ferguson, T. Mudholkar, M. Paulini, M. Sun, I. Vorobiev, M.
Weinberg University of Colorado Boulder, Boulder, USA
J.P. Cumalat, W.T. Ford, A. Johnson, E. MacDonald, T. Mulholland, R. Patel, A.
Perloff, K. Stenson, K.A. Ulmer, S.R. Wagner Cornell University, Ithaca, USA
J. Alexander, J. Chaves, Y. Cheng, J. Chu, A. Datta, A. Frankenthal, K.
Mcdermott, N. Mirman, J.R. Patterson, D. Quach, A. Rinkevicius68, A. Ryd, S.M.
Tan, Z. Tao, J. Thom, P. Wittich, M. Zientek Fermi National Accelerator
Laboratory, Batavia, USA
S. Abdullin, M. Albrow, M. Alyari, G. Apollinari, A. Apresyan, A. Apyan, S.
Banerjee, L.A.T. Bauerdick, A. Beretvas, J. Berryhill, P.C. Bhat, K. Burkett,
J.N. Butler, A. Canepa, G.B. Cerati, H.W.K. Cheung, F. Chlebana, M. Cremonesi,
J. Duarte, V.D. Elvira, J. Freeman, Z. Gecse, E. Gottschalk, L. Gray, D.
Green, S. Grünendahl, O. Gutsche, AllisonReinsvold Hall, J. Hanlon, R.M.
Harris, S. Hasegawa, R. Heller, J. Hirschauer, B. Jayatilaka, S. Jindariani,
M. Johnson, U. Joshi, B. Klima, M.J. Kortelainen, B. Kreis, S. Lammel, J.
Lewis, D. Lincoln, R. Lipton, M. Liu, T. Liu, J. Lykken, K. Maeshima, J.M.
Marraffino, D. Mason, P. McBride, P. Merkel, S. Mrenna, S. Nahn, V. O’Dell, V.
Papadimitriou, K. Pedro, C. Pena, G. Rakness, F. Ravera, L. Ristori, B.
Schneider, E. Sexton-Kennedy, N. Smith, A. Soha, W.J. Spalding, L. Spiegel, S.
Stoynev, J. Strait, N. Strobbe, L. Taylor, S. Tkaczyk, N.V. Tran, L. Uplegger,
E.W. Vaandering, C. Vernieri, M. Verzocchi, R. Vidal, M. Wang, H.A. Weber
University of Florida, Gainesville, USA
D. Acosta, P. Avery, P. Bortignon, D. Bourilkov, A. Brinkerhoff, L. Cadamuro,
A. Carnes, V. Cherepanov, D. Curry, F. Errico, R.D. Field, S.V. Gleyzer, B.M.
Joshi, M. Kim, J. Konigsberg, A. Korytov, K.H. Lo, P. Ma, K. Matchev, N.
Menendez, G. Mitselmakher, D. Rosenzweig, K. Shi, J. Wang, S. Wang, X. Zuo
Florida International University, Miami, USA
Y.R. Joshi Florida State University, Tallahassee, USA
T. Adams, A. Askew, S. Hagopian, V. Hagopian, K.F. Johnson, R. Khurana, T.
Kolberg, G. Martinez, T. Perry, H. Prosper, C. Schiber, R. Yohay, J. Zhang
Florida Institute of Technology, Melbourne, USA
M.M. Baarmand, V. Bhopatkar, S. Butalla, M. Hohlmann, D. Noonan, M. Rahmani,
M. Saunders, F. Yumiceva University of Illinois at Chicago (UIC), Chicago, USA
M.R. Adams, L. Apanasevich, D. Berry, R.R. Betts, R. Cavanaugh, X. Chen, S.
Dittmer, O. Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, K. Jung, C.
Mills, T. Roy, M.B. Tonjes, N. Varelas, H. Wang, X. Wang, Z. Wu The University
of Iowa, Iowa City, USA
M. Alhusseini, B. Bilki51, W. Clarida, K. Dilsiz69, S. Durgut, R.P.
Gandrajula, M. Haytmyradov, V. Khristenko, O.K. Köseyan, J.-P. Merlo, A.
Mestvirishvili70, A. Moeller, J. Nachtman, H. Ogul71, Y. Onel, F. Ozok72, A.
Penzo, C. Snyder, E. Tiras, J. Wetzel Johns Hopkins University, Baltimore, USA
B. Blumenfeld, A. Cocoros, N. Eminizer, D. Fehling, L. Feng, A.V. Gritsan,
W.T. Hung, P. Maksimovic, J. Roskes, M. Swartz, M. Xiao The University of
Kansas, Lawrence, USA
C. Baldenegro Barrera, P. Baringer, A. Bean, S. Boren, J. Bowen, A. Bylinkin,
T. Isidori, S. Khalil, J. King, A. Kropivnitskaya, C. Lindsey, D. Majumder, W.
Mcbrayer, N. Minafra, M. Murray, C. Rogan, C. Royon, S. Sanders, E. Schmitz,
J.D. Tapia Takaki, Q. Wang, J. Williams Kansas State University, Manhattan,
USA
S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y. Maravin, D.R. Mendis, T. Mitchell,
A. Modak, A. Mohammadi Lawrence Livermore National Laboratory, Livermore, USA
F. Rebassoo, D. Wright University of Maryland, College Park, USA
A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng, N.J. Hadley, S. Jabeen,
G.Y. Jeng, R.G. Kellogg, J. Kunkle, A.C. Mignerey, S. Nabili, F. Ricci-Tam, M.
Seidel, Y.H. Shin, A. Skuja, S.C. Tonwar, K. Wong Massachusetts Institute of
Technology, Cambridge, USA
D. Abercrombie, B. Allen, A. Baty, R. Bi, S. Brandt, W. Busza, I.A. Cali, M.
D’Alfonso, G. Gomez Ceballos, M. Goncharov, P. Harris, D. Hsu, M. Hu, M.
Klute, D. Kovalskyi, Y.-J. Lee, P.D. Luckey, B. Maier, A.C. Marini, C. Mcginn,
C. Mironov, S. Narayanan, X. Niu, C. Paus, D. Rankin, C. Roland, G. Roland, Z.
Shi, G.S.F. Stephans, K. Sumorok, K. Tatar, D. Velicanu, J. Wang, T.W. Wang,
B. Wyslouch University of Minnesota, Minneapolis, USA
A.C. Benvenuti${}^{\textrm{\textdagger}}$, R.M. Chatterjee, A. Evans, S. Guts,
P. Hansen, J. Hiltbrand, Sh. Jain, S. Kalafut, Y. Kubota, Z. Lesko, J. Mans,
R. Rusack, M.A. Wadud University of Mississippi, Oxford, USA
J.G. Acosta, S. Oliveros University of Nebraska-Lincoln, Lincoln, USA
K. Bloom, D.R. Claes, C. Fangmeier, L. Finco, F. Golf, R. Gonzalez Suarez, R.
Kamalieddin, I. Kravchenko, J.E. Siado, G.R. Snow, B. Stieger State University
of New York at Buffalo, Buffalo, USA
C. Harrington, I. Iashvili, A. Kharchilava, C. Mclean, D. Nguyen, A. Parker,
S. Rappoccio, B. Roozbahani Northeastern University, Boston, USA
G. Alverson, E. Barberis, C. Freer, Y. Haddad, A. Hortiangtham, G. Madigan,
D.M. Morse, T. Orimoto, L. Skinnari, A. Tishelman-Charny, T. Wamorkar, B.
Wang, A. Wisecarver, D. Wood Northwestern University, Evanston, USA
S. Bhattacharya, J. Bueghly, T. Gunter, K.A. Hahn, N. Odell, M.H. Schmitt, K.
Sung, M. Trovato, M. Velasco University of Notre Dame, Notre Dame, USA
R. Bucci, N. Dev, R. Goldouzian, M. Hildreth, K. Hurtado Anampa, C. Jessop,
D.J. Karmgard, K. Lannon, W. Li, N. Loukas, N. Marinelli, I. Mcalister, F.
Meng, C. Mueller, Y. Musienko37, M. Planer, R. Ruchti, P. Siddireddy, G.
Smith, S. Taroni, M. Wayne, A. Wightman, M. Wolf, A. Woodard The Ohio State
University, Columbus, USA
J. Alimena, B. Bylsma, L.S. Durkin, S. Flowers, B. Francis, C. Hill, W. Ji, A.
Lefeld, T.Y. Ling, B.L. Winer Princeton University, Princeton, USA
S. Cooperstein, G. Dezoort, P. Elmer, J. Hardenbrook, N. Haubrich, S.
Higginbotham, A. Kalogeropoulos, S. Kwan, D. Lange, M.T. Lucchini, J. Luo, D.
Marlow, K. Mei, I. Ojalvo, J. Olsen, C. Palmer, P. Piroué, J. Salfeld-Nebgen,
D. Stickland, C. Tully, Z. Wang University of Puerto Rico, Mayaguez, USA
S. Malik, S. Norberg Purdue University, West Lafayette, USA
A. Barker, V.E. Barnes, S. Das, L. Gutay, M. Jones, A.W. Jung, A. Khatiwada,
B. Mahakud, D.H. Miller, G. Negro, N. Neumeister, C.C. Peng, S. Piperov, H.
Qiu, J.F. Schulte, J. Sun, F. Wang, R. Xiao, W. Xie Purdue University
Northwest, Hammond, USA
T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA
K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Kilpatrick, Arun Kumar, W. Li, B.P.
Padley, R. Redjimi, J. Roberts, J. Rorie, W. Shi, A.G. Stahl Leiton, Z. Tu, A.
Zhang University of Rochester, Rochester, USA
A. Bodek, P. de Barbaro, R. Demina, Y.t. Duh, J.L. Dulemba, C. Fallon, M.
Galanti, A. Garcia-Bellido, J. Han, O. Hindrichs, A. Khukhunaishvili, E.
Ranken, P. Tan, R. Taus The Rockefeller University, New York, USA
R. Ciesielski Rutgers, The State University of New Jersey, Piscataway, USA
B. Chiarito, J.P. Chou, A. Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart,
M. Heindl, E. Hughes, S. Kaplan, S. Kyriacou, I. Laflotte, A. Lath, R.
Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, D. Sheffield,
S. Somalwar, R. Stone, S. Thomas, P. Thomassen University of Tennessee,
Knoxville, USA
H. Acharya, A.G. Delannoy, J. Heideman, G. Riley, S. Spanier Texas A&M
University, College Station, USA
O. Bouhali73, A. Celik, M. Dalchenko, M. De Mattia, A. Delgado, S. Dildick, R.
Eusebi, J. Gilmore, T. Huang, T. Kamon74, S. Luo, D. Marley, R. Mueller, D.
Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock,
USA
N. Akchurin, J. Damgov, F. De Guio, S. Kunori, K. Lamichhane, S.W. Lee, T.
Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang, A.
Whitbeck Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij University of
Virginia, Charlottesville, USA
M.W. Arenton, P. Barria, B. Cox, G. Cummings, R. Hirosky, M. Joyce, A.
Ledovskoy, C. Neu, B. Tannenwald, Y. Wang, E. Wolfe, F. Xia Wayne State
University, Detroit, USA
R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa, S. Zaleski University
of Wisconsin - Madison, Madison, WI, USA
J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De Bruyn, L. Dodd, B.
Gomber75, M. Herndon, A. Hervé, U. Hussain, P. Klabbers, A. Lanaro, A.
Loeliger, K. Long, R. Loveless, J. Madhusudanan Sreekala, T. Ruggles, A.
Savin, V. Sharma, W.H. Smith, D. Teague, S. Trembath-reichert, N. Woods †:
Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at UFMS, Nova Andradina, Brazil
6: Also at Universidade Federal de Pelotas, Pelotas, Brazil
7: Also at Université Libre de Bruxelles, Bruxelles, Belgium
8: Also at University of Chinese Academy of Sciences, Beijing, China
9: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
10: Also at Joint Institute for Nuclear Research, Dubna, Russia
11: Also at Fayoum University, El-Fayoum, Egypt
12: Now at British University in Egypt, Cairo, Egypt
13: Also at Purdue University, West Lafayette, USA
14: Also at Université de Haute Alsace, Mulhouse, France
15: Also at Tbilisi State University, Tbilisi, Georgia
16: Also at Erzincan Binali Yildirim University, Erzincan, Turkey
17: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
18: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
19: Also at University of Hamburg, Hamburg, Germany
20: Also at Brandenburg University of Technology, Cottbus, Germany
21: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary,
Debrecen, Hungary
22: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
23: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary, Budapest, Hungary
24: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India
25: Also at Institute of Physics, Bhubaneswar, India
26: Also at Shoolini University, Solan, India
27: Also at University of Visva-Bharati, Santiniketan, India
28: Also at Isfahan University of Technology, Isfahan, Iran
29: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
30: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
31: Also at Università degli Studi di Siena, Siena, Italy
32: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
33: Also at Riga Technical University, Riga, Latvia, Riga, Latvia
34: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
35: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
36: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
37: Also at Institute for Nuclear Research, Moscow, Russia
38: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
39: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
40: Also at University of Florida, Gainesville, USA
41: Also at Imperial College, London, United Kingdom
42: Also at P.N. Lebedev Physical Institute, Moscow, Russia
43: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
44: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
45: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy,
Pavia, Italy
46: Also at National and Kapodistrian University of Athens, Athens, Greece
47: Also at Universität Zürich, Zurich, Switzerland
48: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria,
Vienna, Austria
49: Also at Gaziosmanpasa University, Tokat, Turkey
50: Also at Şırnak University, Sirnak, Turkey
51: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey
52: Also at Istanbul Aydin University, Application and Research Center for
Advanced Studies (App. & Res. Cent. for Advanced Studies), Istanbul, Turkey
53: Also at Mersin University, Mersin, Turkey
54: Also at Piri Reis University, Istanbul, Turkey
55: Also at Adiyaman University, Adiyaman, Turkey
56: Also at Ozyegin University, Istanbul, Turkey
57: Also at Izmir Institute of Technology, Izmir, Turkey
58: Also at Marmara University, Istanbul, Turkey
59: Also at Kafkas University, Kars, Turkey
60: Also at Istanbul University, Istanbul, Turkey
61: Also at Istanbul Bilgi University, Istanbul, Turkey
62: Also at Hacettepe University, Ankara, Turkey
63: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
64: Also at IPPP Durham University, Durham, United Kingdom
65: Also at Monash University, Faculty of Science, Clayton, Australia
66: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA
67: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
68: Also at Vilnius University, Vilnius, Lithuania
69: Also at Bingol University, Bingol, Turkey
70: Also at Georgian Technical University, Tbilisi, Georgia
71: Also at Sinop University, Sinop, Turkey
72: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
73: Also at Texas A&M University at Qatar, Doha, Qatar
74: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea
75: Also at University of Hyderabad, Hyderabad, India
|
2024-09-04T02:54:57.644390 | 2020-03-05T19:30:54 | 2003.02881 | {
"authors": "Dirk Keppens",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26066",
"submitter": "Dirk Keppens",
"url": "https://arxiv.org/abs/2003.02881"
} | arxiv-papers | # On the history of ring geometry
(with a thematical overview of literature)
Dirk Keppens
Faculty of Engineering Technology, KU Leuven
Gebr. Desmetstraat 1
B-9000 Ghent BELGIUM
<EMAIL_ADDRESS>
###### Abstract
In this survey paper we give an historical and at the same time thematical
overview of the development of “ring geometry” from its origin to the current
state of the art. A comprehensive up-to-date list of literature is added with
articles that treat ring geometry within the scope of incidence geometry.
In questo documento di ricerca forniamo una panoramica storica e allo stesso
tempo tematica dello sviluppo della “geometria sopra un anello” dalla sua
origine allo stato attuale. È aggiunto una lista di letteratura aggiornata
completa di articoli che trattano la geometria degli anelli nel contesto della
geometria dell’incidenza.
In diesem Forschungsartikel geben wir einen historischen und gleichzeitig
thematischen Überblick über die Entwicklung der “Ringgeometrie” von ihrem
Ursprung bis zum aktuellen Stand der Technik, mit einer Liste aktualisierter
Literatur einschließlich Artikeln zur Ringgeometrie im Kontext der
Inzidenzgeometrie.
Dans ce document de recherche, nous fournissons un aperçu historique et à la
fois thématique du développement de la “géométrie sur un anneau”, de son
origine à l’état actuel des connaissances. Nous ajoutons une liste de la
littérature actualisée comprenant des articles traitant la géométrie sur un
anneau dans le contexte de la géométrie de l’incidence.
Keywords: Ring geometry, projective ring plane, Hjelmslev geometry,
Klingenberg geometry, Barbilian plane, neighbor relation, bibliography
AMS Classification: Primary 51C05; Secondary 00A15, 01A60, 01A61, 51-03,
51A05, 51B05, 51E26, 51H10, 51F15
## 1 Introduction
The current version of the Mathematics Subject Classification (MSC), a
classification scheme used by the two major mathematical reviewing databases,
Mathematical Reviews (MathSciNet) and Zentralblatt MATH (zbMATH), provides
code 51C05 for all research papers dealing with Ring Geometry. This rather
small category contains articles about geometries that are not only provided
with an incidence relation but also with a neighbor relation (or its negation,
a non–neighbor or remoteness relation). Included are all geometries obtained
from rings that are not division rings. Ring geometry, i.e. the theory of
geometries equipped with a neighbor relation in general and of geometries over
rings in particular, is a rather young discipline. Its origin lies in the
beginning of the 20th century and its importance has grown steadily. For that
reason it has also got a full–fledged place as a chapter in the “Handbook of
Incidence Geometry” [A24]. In the past decades several mathematicians have
contributed to ring geometry. Newly discovered connections with coding theory,
with the theory of Tits–buildings and with quantum information theory, have
opened new horizons. In this survey paper we present an historical and
thematical overview with attention to many aspects. An out-of-date list with
articles on the subject, up to 1989, was composed by Törner and Veldkamp in
[A23] as an addition and completion of two even older lists [A02] and [A12]
written by respectively Artmann, Drake et al. and Jungnickel. Up to now such
list of literature is not available for the period after 1990, except for a
survey paper by the author [A13], dealing exclusively with plane projective
geometries over finite rings. In the present work we fill that gap and we add
a new updated list of existing literature, ordered thematically (and
containing the relevant material from the preceding lists). Articles on
algebraic geometry and on differential geometry over rings are not included.
We also do not aim for completeness when it concerns metric aspects, neither
for geometries on modular lattices nor for geometric algebra over rings. We
think that the bibliography might be useful for researchers who want to attack
the future challenges of ring geometry.
## 2 The first traces of ring geometry: dual numbers and Johannes Hjelmslev
The first traces of ring geometry date back to the beginning of the twentieth
century. The Danish geometer Johannes Trolle Hjelmslev (1873–1950) who was
born as Johannes Petersen but who changed his name in 1904, graduated in
mathematics from the University of Copenhagen and received his PhD degree in
1897. Hjelmslev can be viewed as one of the early founders of ring geometry.
In a series of four lectures, held at the University of Hamburg in July 1922
and published one year later in the Abhandlungen aus dem Mathematischen
Seminar [A10], he presented an axiomatic framework for geometry that better
reflected the properties observed in the real world. The basic observation
made by Hjelmslev was that, if one draws lines “close” to each other (meaning
that the sharp angle they define is very small), then it is hard to identify
the intersection point, and it looks as if the lines have a little segment in
common. Dually, if two points are close to each other, they belong to a line
segment that can be part of several joining lines. Hjelmslev called this Die
natürliche Geometrie, the “natural geometry”. In fact, Hjelmslev put forward
this idea already some years earlier, in 1916, in an article [A09] on what he
called Die Geometrie der Wirklichkeit, the “geometry of reality”.
In order to obtain a model for his geometry, Hjelmslev made use of the ring of
dual numbers of the form $a+b\varepsilon$ with $a$ and $b$ both real and
$\varepsilon^{2}=0$.
Dual numbers were already well–known before Hjelmslev. William Clifford
defined them for the first time in 1873 [A06] and they were used as a
convenient tool in mechanics by Aleksander Kotelnikov [A18] and by Eduard
Study in his famous work “Geometrie der Dynamen” [A22].
According to Benz [A05] the first traces of ring geometry can be observed
already in Study’s work and in that of some of his contemporaries like Josef
Grünwald, Pilo Predella and Corrado Segre. In their papers dual numbers are
treated from a geometrical viewpoint, by connecting the geometry of oriented
lines (spears) in the real euclidean space with a spherical geometry over the
ring of dual numbers [A08, A19, A21].
## 3 The pioneers of ring geometry: Barbilian and Klingenberg
Twenty years after Hjelmslev, Dan Barbilian (1895–1961), besides a
mathematician at the University of Bucharest, also one of the greatest
Romanian poets (with pseudonym Ion Barbu), took up the line again. In two
papers [A03, A04] (extended versions of a lecture hold in Baden–Baden in 1940)
he gave the first axiomatic foundation for plane projective ring geometry. He
started by investigating the conditions which must be imposed upon an
arbitrary associative ring in order that the corresponding geometry may have
“useful” properties. It turns out that the rings must have a unit element and
that all left singular elements must be two–sided singular or equivalently
$ab=1$ implies $ba=1$. Barbilian called these rings “Zweiseitig singuläre
Ringe” or $Z$–rings. Today such rings are also known as Dedekind–finite rings.
They include of course all commutative rings but also all finite rings (even
non–commutative) and many other classes of rings (e.g. matrix rings over a
field). Starting with $C$–rings (“Kategorische Ringe”), being $Z$–rings with
some additional property, Barbilian defined a kind of projective plane.
Conversely, he formulated a set of axioms for a plane geometry with incidence
and non–neighborship (“klare Lage”) and proved that this leads to a $C$–ring.
Barbilian’s work showed some shortcomings, but nevertheless it was of great
importance for the development of geometry over arbitrary rings as we will
discuss further (see also section 11).
Another monument among the ring geometers was the German mathematician Wilhelm
Klingenberg (1924–2010). He is best known for his work on differential
geometry, but in a series of papers [A14, A15, A16, A17], published in the
mid–fifties, he also laid the foundation for affine, metric (euclidean) and
projective geometries with neighbor relation (“mit Nachbarelementen”). His
central idea is the existence of a natural map from the geometry under
consideration onto an “underlying” ordinary geometry, a consequence of the
assumption that the neighbor relation is transitive. Therefore Klingenberg
called such geometries “Geometrien mit Homomorphismus”. He not only defined
them axiomatically, but he also constructed explicit examples using rings. The
assumption of a transitive neighbor relation is interlaced with the fact that
the rings must be local ones. A not necessarily commutative local ring $R$ is
a ring with a unique maximal left (or equivalently a unique maximal right)
ideal $R_{0}$. The quotient ring $R/R_{0}$ is a (skew)field, coordinatizing an
ordinary geometry which is the natural epimorphic image of the ring geometry
over $R$.
The work of Klingenberg was the source of two mainstreams in ring geometry in
the decades afterwards: Klingenberg geometry and Hjelmslev geometry.
Geometries with a transitive neighbor relation or equivalently with a
canonical map onto an ordinary geometry are now named Klingenberg geometries.
Their epimorphic image is obtained by considering the equivalence classes of
neighboring elements as “thick” elements. Non–neighboring elements behave like
distinct elements in the epimorphic image (e.g. two non–neighboring points are
incident with exactly one line in a projective Klingenberg plane since two
distinct points in its epimorphic projective plane are incident with a unique
line). No further assumptions are made about the neighboring elements.
Klingenberg geometries comprise the smaller, but more intensively studied
class of Hjelmslev geometries for which additional axioms must hold when it
comes to neighboring elements (e.g. two neighboring points can be connected by
at least two distinct lines in a projective Hjelmslev plane). This idea relies
on the natural geometry of Hjelmslev (see section 2). In the case of a
Klingenberg plane over a local ring $R$, the Hjelmslev condition means that
the ring must be a two–sided chain ring (a ring is a right chain ring if for
any $a$ and $b$ in $R$ either $a\in bR$ or $b\in aR$, similar for left chain
ring) with the additional property that its maximal ideal contains only zero
divisors. Such rings are called PH–rings (projective Hjelmslev rings or
$H$–rings). Finite chain rings are always $H$–rings and, equivalently, local
principal ideal rings. Klingenberg himself considered both Klingenberg planes
(over local rings) and Hjelmslev planes (over Hjelmslev rings) in [A16].
To be complete, we must mention also two isolated, rather unnoticed,
contributions to early geometry over rings. J.W. Archbold [A01] wrote on
projective geometry over group algebras $K[G]$ and worked out a small finite
example taking $K=$ GF(2) and $G$ a group of order 2, while Rimhak Ree in
[A20] considered projective spaces as the modular lattices $L(R,M)$ of all
sub–modules of a module $M$ over a ring $R$, in particular over a full matrix
ring.
## 4 The Belgian contribution to early ring geometry: projective geometry
over rings of matrices
The ideas of Barbilian concerning projective geometries over rings instead of
fields, got some followers in the sixties and early seventies. Among them we
find a lot of Belgian contributors who studied projective geometries over
rings of matrices. Since such rings are not local, the geometries are not
Klingenberg nor Hjelmslev geometries.
Julien Depunt studied in [B01] the projective line and in [B03, B04]
projective planes over the ring of ternions. This ring contains the elements
$a_{1}\varepsilon_{1}+a_{2}\varepsilon_{2}+a_{3}\varepsilon_{3}$ with
$a_{i}\in\mathbb{C}$ and $\varepsilon_{i}^{2}=\varepsilon_{i}$ ($i=1,2$),
$\varepsilon_{1}\varepsilon_{3}=\varepsilon_{3}=\varepsilon_{3}\varepsilon_{2}$
and all other products equal to $0$. The ring of ternions is isomorphic to the
ring of upper triangular matrices with elements in the complex field. The main
result of Depunt concerns the embedding of the plane over the ternions into
the 5–dimensional complex projective space.
Inspired by this work, Cléry Vanhelleputte could generalize the results for
planes over the full matrix ring of $2\times 2$–matrices with elements in an
arbitrary commutative field. His doctoral thesis was published in [B14] and
fitted in the research program under Julien Bilo (1914–2006) who was head of
the Geometry Department at the University of Ghent at that time.
A few years later, in 1969, Joseph Thas wrote his PhD thesis under the
supervision of Bilo about the projective line over the ring of $3\times
3$–matrices with elements in an algebraically closed field [B11], see also
[B10]. Soon afterwards Thas published [B12, B13] on projective ring geometry
over matrix rings, in particular over finite rings $M_{n}(GF(q))$ of $n\times
n$–matrices with entries in a Galois field of order $q$. Especially [B12] was
important as it contains some concepts (ovals, arcs, caps, etc.) and
combinatorial theorems which extend known results from classical Galois
geometry over finite fields and which appear for the first time for ring
geometries. Later Thas grew out to an authority in finite geometry, but his
interest shifted from finite ring geometry to Galois geometry and other
topics, in particular finite generalized quadrangles.
Thas was a colleague of Willem Mielants during many years. It was he who
guided Hendrik Van Maldeghem (see section 14) and the author towards ring
geometry as PhD supervisor for both of us.
Paul De Winne was the last student of Bilo studying ring geometry, see [B05].
His doctoral thesis was published in [B06].
Around the same time, Xavier Hubaut, another Belgian mathematician from the
Université Libre de Bruxelles, introduced the projective line over a general
associative algebra and investigated the structure of its group of
projectivities in [B08, B09]. His colleague Franz Bingen could generalize the
results to $n$–dimensional projective spaces over semi–primary rings in [B02].
Mind that the geometries over matrix rings studied by all these people, may
not be confused with the “geometries of matrices” initiated by the Chinese
mathematician Loo–Keng Hua in the mid–forties [B07]. Hua’s geometries of
matrices are not ring geometries in the narrow sense, despite the suggestive
name (although a connection is possible with projective lines over rings, see
section 13).
## 5 The foundations of plane affine ring geometry: from Benz to Leissner and
beyond
Barbilian, as well as all members of the Belgian School, dealt exclusively
with projective ring geometry. The first traces of affine geometries over
rings can be found in a paper from 1942 by Cornelius Everett, concerning
affine planes over rings without zero divisors [C16].
Walter Benz (1931–2017), a renowned German geometer, especially for his work
on circle geometries, considered in [C04, C05] affine planes over a special
kind of commutative rings and in [C06] plane affine (and metric–affine)
geometries over arbitrary rings with unit. A big part of this interesting
paper discusses the relation between algebraic properties of the ring and
geometric properties of the plane.
Peter Ashley Lawrence obtained a PhD under the supervision of Benz. His thesis
“Affine mappings in the geometries of algebras” considers ring geometries
associated with modules and algebras over a commutative ring and was published
in [C21]. Two other publications [C02, C03] by Hans–Joachim Arnold can be
classified within this section. They contain a simultaneous generalization of
affine geometry over rings and generalized affine spaces in the sense of
Emanuel Sperner [C41]. Arnold starts with the axiomatic definition of an
affine–line geometry which can be coordinatized by a vectorial groupoid and he
adds some extra axioms to turn it into a module over a unitary ring. Also
[C18] and [C34] both fit into this approach.
Affine planes over local rings, today also known as desarguesian affine
Klingenberg planes, appear as particular examples in the papers of Benz, but
this was not for the first time. Klingenberg already considered them on the
side of projective ring planes and in the more general context of “Affine
Ebenen mit Nachbarelementen” in the articles [A14, A15, A16, A17].
A reasonable part of the research concerning affine ring geometry focusses on
affine Hjelmslev planes (AH–planes). An important paper [C28] in that respect
was written by Heinz Lüneburg (1935–2009) in 1962. It contains the main
results of his doctoral thesis entitled “Affine Hjelmslev–Ebenen mit
transitiver Translationsgruppe”, dealing with translation AH–planes as
generalizations of ordinary translation planes. Werner Seier investigated in
more detail desarguesian and translation AH–planes in [C36, C37, C38, C39,
C40]. The central theme in his work is a characterization of desarguesian
AH–planes by an affine variant of Desargues’ theorem and the transitivity of
the set of translations in the plane. Some other papers on affine Hjelmslev
planes are not mentioned here and we postpone them to sections 7 and 8 where
they will be discussed in connection with projective Hjelmslev planes.
Without minimizing the importance of the papers cited above, we can state that
the real breakthrough of affine ring geometry in full generality was not
achieved yet at this point. Therefore one has to wait for Werner Leissner, a
student of Benz. Leissner wrote two papers in the mid–seventies on, what he
called, “affine Barbilian planes” (today the name “Leissner planes” would be
more convenient). In [C22] such planes are defined axiomatically as affine
structures in which two parallelodromy axioms hold (suitable substitutes for
the affine Desargues theorem) and they are coordinatized by an arbitrary
$Z$–ring $R$ together with a special subset $B$ of $R\times R$ (a Barbilian
domain). In [C23] the converse is proved: any affine ring plane over a
$Z$–ring is an affine Barbilian plane. Leissner considered even more general
affine Barbilian structures in [C24].
The papers of Leissner were a source of inspiration for further
generalizations by several people. Victoria Groze and Angela Vasiu considered
affine structures over arbitrary rings in [C17], Francisc Radó defined affine
Barbilian structures in [C33] by weakening Leissner’s axioms, Armentrout et
al. investigated in [C01] generalized affine planes which can be coordinatized
by near-rings and Pickert studied tactical configurations over quasirings in
[C32].
Other contributors to the theory of plane affine geometries with neighbor
relation and parallellism were Gernot Dorn [C13], Kvetoslav Burian [C08, C09,
C10], Frantisek Machala [C29, C30, C31], Stefan Schmidt and Ralph Steinitz
[C35], Franco Eugeni [C14, C15] and Angela Vasiu [C42, C43, C44]. The
Barbilian domains defined by Leissner also formed a study object in
themselves. Several authors have contributed to this, see [C07, C20, C25, C26,
C27]. For a recent and fairly complete overview of affine planes over finite
rings we refer to the survey paper [C19] of the author. We also mention here
two papers [C11, C12] by Basra Çelik on finite hyperbolic Klingenberg planes
as they are not far away from finite affine Klingenberg planes.
## 6 Metric geometry over rings and the school of Bachmann
The goal of this section is just to give a glimpse of metric ring geometry. We
do not aim for completeness when it concerns metric aspects, because they are
only indirectly related to incidence geometry. The literature about this
subject is extensive and we have selected only a few representative papers in
which much more references can be found. The basis of “modern” metric geometry
is provided by the work of several German mathematicians, including Bachmann,
Lingenberg and Schröder.
The pivotal figure is Friedrich Bachmann (1909–1982). In the second edition of
his famous book “Aufbau der Geometrie aus dem Spiegelungsbegriff” [D01] he
considers plane metric geometries in which points, lines, incidence and
orthogonality are defined by means of a group of reflections. A group $G$ with
a subset $S$ of involutory elements which are invariant under inner
automorphisms of $G$ and which generate $G$, determines a group plane
$E=E(G,S)$ as follows: the elements of $S$ are the lines of $E$ and the
involutory elements of $S^{2}$ are the points. Two lines that commute are
perpendicular. A point and a line that commute are incident. If the group
plane $E$ satisfies the conditions (1) a point and a line determine a unique
perpendicular, and (2) the product of three lines that are either concurrent
or have a common perpendicular is again a line, then the pair $(G,S)$ is
called a Hjelmslev group and the associated group plane $E$ is called a metric
(non–elliptic) Hjelmslev plane.
If the uniqueness of the line incident with two distinct points is not
required, then metric Hjelmslev planes stripped of their metric structure,
reduce to incidence Hjelmslev planes. In section 2 we have already discussed
the role of Hjelmslev as founder of (incidence) ring geometry. In some later
work, the “Algemeine Kongruenzlehre” [D06], Hjelmslev used special
transformations (reflections) to define orthogonality. Hence, the rudiments of
metric ring geometry also go back to Hjelmslev and have been worked out in
more detail by Bachmann. There is also a minor contribution by Klingenberg who
treats metric aspects in ring geometries in [A15].
Rolf Lingenberg (1929–1978), a student of Bachmann, continues his work in
[D12]. The main result is an algebraic characterization of classical metric
group planes $E(G,S)$ with an additional axiom, as planes associated with a
metric vectorspace $(V,q)$ with $q$ a quadratic form.
Eberhard Schröder in [D18, D19] considers metric planes of different kind
starting from a pappian affine plane over a field and using twodimensional
algebras for the introduction of the metric notions of angle, distance and
orthogonality. His work is strongly related with circle geometries studied
intensively by Benz (see section 13). A good overview of classical metric
geometry (but not including ring geometries) can be found in Chapter 17
written by Schröder in the Handbook of Incidence Geometry [D20].
There are numerous contributions by Bachmann and his school to the theory of
Hjelmslev groups, connecting geometric and algebraic properties of groups and
planes. From Bachmann himself we mention the important additions [D02, D03,
D04] to his book [D01]. In the seventies and eighties his successors, most of
them from the University of Kiel (Germany), have extended the knowledge.
Finite Hjelmslev groups were characterized by Rolf Stölting in his PhD thesis
“Endliche Hjelmslev-Gruppen und Erweiterungen von Hjelmslev-Gruppen”, also
published in [D21]. In [D22] Hjelmslev groups are constructed from a module
$M$ over a commutative ring $R$ endowed with a bilinear form.
Edzard Salow introduced singular Hjelmslev groups (in which the product of
three points is always a point) in his doctoral thesis “Beiträge zur Theorie
der Hjelmslev-Gruppen: Homomorphismen und Singuläre Hjelmslev-Gruppen”
published in [D15]. The main result is the construction of a coordinate ring
$R$ for the group plane of a singular Hjelmslev group, proving that these
metric planes are indeed ring geometries. The process of algebraization of
metric Hjelmslev planes is investigated also in [D16]. It is proved there that
a Hjelmslev group with some additional axioms can be embedded in the
orthogonal group of a metric module $(R^{3},f)$ with $R$ a commutative ring
with unit for which $2$ and any non zero–divisor is invertible and with $f$ a
symmetric bilinear form of the free module $R^{3}$.
In [D17] Salow studies another class of metric ring planes using a commutative
algebra over a ring and the concept of an angle. In an early paper of Benz
[C04] the metric notion of angle also played an important role and in [C06] he
devotes a paragraph to metric geometry, using an elliptic form over an
arbitrary commutative ring.
Similar work can be found in [D13, D14] in which Wolfgang Nolte proves that a
class of metric planes $E(G,S)$ with additional axioms can be embedded into a
projective Hjelmslev plane over a local ring and that $G$ is isomorphic to a
subgroup of an orthogonal group. Other results in this direction were obtained
by Frieder Knüppel, in [D07, D08, D09, D10, D27], Gerald Fischbach [D05],
Michael Kunze [D11] and Rolf and Horst Struve [D23, D24, D25, D26].
The influence of Bachmann is also clear from the large amount of doctoral
theses produced in Germany on Hjelmslev groups and metric geometry, e.g. R.
Schnabel (1974), M. Kunze (1975), H. Struve (1979), R. Struve (1979), M.
Gegenwart (1987), W. Vonck (1988) and A. Bach (1998).
## 7 The florescence of Hjelmslev geometry in the era of Drake, Artmann and
Törner
The first period of florescence of ring geometry (especially Hjelmslev
geometry) regarded as incidence geometry, started in the late sixties and
reached his culmination point in the seventies. This is reflected in a large
number of publications. Two mathematicians who were very productive in this
area and left their mark, were David Allyn Drake (1937–) from the University
of Florida, Gainesville (USA) and Benno Artmann (1933–2010) from the
University of Giessen (Germany).
Drake obtained his PhD in 1967 with a thesis entitled “Neighborhood
collineations and affine projective extensions of Hjelmslev planes” under the
supervision of Erwin Kleinfeld. Kleinfeld himself, an authority in algebra,
with a lot of publications on non–associative alternative rings, published
only one, though interesting paper [E25] about ring geometry. It was the first
publication concerning finite Hjelmslev planes. Among other things he
introduced a two parameter set $(s,t)$ of non–zero integers such that for each
flag $(P,\ell)$ in a finite projective Hjelmslev plane, there are exactly $t$
points on $\ell$ neighboring with $P$ and exactly $s$ points on $\ell$ not
neighboring with $P$. It was proved that $s\leq t^{2}$ or $t=1$. If $s=t^{2}$,
the plane is called uniform. In that case all point neighborhoods have the
structure of ordinary affine planes. Robert Craig proved in [E11] that any
finite projective plane can be extended to a uniform projective Hjelmslev
plane.
The notion of uniformity (and its generalization to $n$–uniformity) has played
a crucial role in the work of Drake. It was also related to another issue: the
extension of an affine Hjelmslev plane to a projective Hjelmslev plane. An
ordinary affine plane can always be extended to a projective plane, but for
Hjelmslev planes the situation is much more complicated. Drake has written
several papers about this problem in the period from 1968 to 1975. In [E14] he
proves that any uniform affine Hjelmslev plane has at least one (uniform)
projective extension and he gives an example of a desarguesian uniform affine
Hjelmslev plane with a non–desarguesian projective extension. In [E15]
uniformity is generalized to $n$–uniformity inductively (a PH–plane is
$n$–uniform if the point neighborhoods are $(n-1)$-uniform AH–planes).
Strongly $n$–uniform planes are characterized by a local property, which leads
to the theorem: an $n$–uniform PH–plane is strongly $n$–uniform if and only if
its dual is $n$–uniform. Drake also proved that every finite desarguesian
PH–plane is strongly $n$–uniform.
A further study of $n$–uniform Hjelmslev planes (projective and affine) was
made in [E16, E17, E18, E19, E20, E21, E22] where even more general affine
geometries with neighbor relation appear. Drake could also prove that there do
exist affine Hjelmslev planes which cannot be extended to projective Hjelmslev
planes.
Artmann was a contemporary of Drake and he wrote his doctoral thesis
“Automorphismen und Koordinaten bei ebenen Verbänden” in 1965 under the
supervision of Günther Pickert. In his early work on Hjelmslev geometry we can
observe a strong relation with the theory of modular lattices. In [E01] he
gives a sufficient condition for a modular lattice to define a projective
Hjelmslev plane and in [E03] he proves that any uniform PH–plane can be
derived from a modular lattice.
Like Drake, Artmann also studies refinements of the neighbor relation
(“verfeinerten Nachbarschaftsrelationen”) and the affine–projective extension
question. In [E04] he proves that a uniform affine Hjelmslev plane can be
extended to at least two non–isomorphic projective Hjelmslev planes. A new
concept introduced by him in [E02] is that of a projective Hjelmslev plane of
level $n$ (“$n$–ter Stufe”) based on the refinement of the neighbor relation.
This definition was extended by Drake to the affine case in [E20]. Artmann
proves that desarguesian PH–planes over a Hjelmslev ring $R$ are of level $n$
if and only if the maximal ideal of $R$ is nilpotent of index $n$, see [E05,
E06].
Another theorem proved by Artmann [E07] states that for any projective plane
$\mathcal{P}$ and any integer $n>0$ there exists a PH–plane of level $n$ with
$\mathcal{P}$ as epimorphic image. Moreover, given a sequence of PH–planes
$\ldots\rightarrow H_{i}\rightarrow H_{i-1}\rightarrow\ldots\rightarrow
H_{1}=\mathcal{P}$ with $H_{i}$ of level $i$, the inverse limit is a
projective plane. Arno Cronheim constructed in [E12] in a purely algebraic
way, using formal power series over a cartesian group, a chain of Hjelmslev
planes whose inverse limit is a projective plane. Cronheim also obtained a
complete classification of all finite uniform desarguesian projective
Hjelmslev planes in [E13]. They are either planes over a ring of twisted dual
numbers over GF($q$) (a non–commutative generalization of the classical dual
numbers) or over a truncated Witt ring $W_{2}(q)$ of length 2.
Both Drake and Artmann had a great influence on the mathematical research in
the domain of Hjelmslev geometry (even when Artmann’s interest shifted to
other subjects after a few years). One of Drake’s students was Phyrne Bacon.
She wrote both her Master’s thesis “On Hjelmslev planes with small invariants”
and her PhD thesis “Coordinatized Hjelmslev planes” on Hjelmslev geometry,
resulting in two papers [E08] and [E09]. She proved that a finite Hjelmslev
plane is strongly $n$–uniform if and only if it is of level $n$ which unifies
the two notions introduced by Drake and Artmann respectively. Later, Bacon’s
attention shifted to the more general Klingenberg geometries (see section 9).
Artmann was the supervisor of Manfred Dugas and Günther Törner who both made
important contributions to Hjelmslev geometry. The PhD thesis of Dugas
“Charakterisierungen endlicher desarguescher uniformer Hjelmslev-Ebenen”
contains many new ideas, including a coordinatization method (see section 9).
In [E23] Dugas proves that a finite translation AH–plane can be extended to a
PH–plane if it is or a desarguesian PH–plane or an ordinary translation
projective plane, while in [E24] he gives a necessary and sufficient condition
for a projective Hjelmslev plane to be derivable from a lattice. In particular
the PH–planes of level $n$ are always lattice–derivable.
Törner wrote his Master’s thesis on “Hjelmslev–Ringe und die Geometrie der
Nachbarschaftsbereiche in der zugehörigen Ebenen” and obtained his PhD with
“Eine Klassifizierung von Hjelmslev-Ringen und Hjelmslev-Ebenen” in the same
year as Dugas, under the supervision of Artmann and Pickert. His research in
the domain of ring geometry focusses on two main themes: the structure of
(finite) Hjelmslev planes and the ideal structure of chain rings. Among his
publications we mention here [E36] which contains the main results from his
thesis: a classification of PH–planes based on congruence relations. He proves
that the set of all congruence relations of a finite PH–plane is linearly
ordered under inclusion and consequently, the canonical epimorphism onto the
associated projective plane admits an essentially unique factorization into
indecomposable epimorphisms. The plane $\mathcal{H}$ is of “type $n$” or
“height $n$” if the canonical epimorphism $\varphi$ from $\mathcal{H}$ onto
the projective plane $\overline{\mathcal{H}}$ has a maximal factorization
$\mathcal{H}=\mathcal{H}_{n}\rightarrow\mathcal{H}_{n-1}\rightarrow\ldots\rightarrow\mathcal{H}_{1}=\overline{\mathcal{H}}$.
In [E40, E41] Törner investigates the equivalence of finite $n$–uniform planes
or planes of level $n$ as defined by Drake and Artmann to planes of type $n$.
In [E39] he proves that $n$–uniform projective Hjelmslev planes are strongly
$n$–uniform. Some of the results were later extended to the infinite case in
[E43]. In [E40] much attention goes also to affine Hjelmslev planes, in
particular translation AH–planes over near–rings. In Törner’s work
homomorphisms play an important role as can also be seen from [E37, E38].
In the desarguesian case (Hjelmslev planes over a chain ring) the structure of
the plane is intrinsically connected with the ideal structure of the ring. The
structure of chain rings and valuation rings was investigated by Törner partly
in collaboration with Hans–Heinrich Brungs. One of their papers [E10] concerns
the embedding of right chain rings into chain rings (related with the problem
of embedding desarguesian affine Hjelmslev planes into projective ones), see
also [E32, E33, E35, E42].
With a postdoctoral scholarship Artmann stayed for a short time at the
McMaster University in Ontario (Canada). There he inspired J.W. (Mike) Lorimer
who would later become one of the leading figures in topological Hjelmslev
geometry (see section 10). In [E31] Lorimer and Lane study desarguesian
Hjelmslev planes. They prove that an affine Hjelmslev plane is desarguesian if
and only if it can be coordinatized by an AH–ring and that not every
desarguesian AH–plane can be extended to a desarguesian PH–plane. Morphisms
between affine Hjelmslev planes are the main subject in [E26, E27, E30] while
[E28, E29] deal with the structure of Hjelmslev rings.
## 8 The continuation of the Hjelmslev epoch under Drake, Jungnickel and Sane
In his publications on $n$–uniform planes, we can observe already that Drake
had a particular interest in finite Hjelmslev planes. This is continued when
his attention goes more and more to the problem of existence and non–existence
of finite Hjelmslev planes with given parameters. In a series of papers [F05,
F06, F07, F08, F13, F16, F20], some of them with co–author, he attacked this
problem and he linked finite PH–planes to nets in [F10, F19].
Meanwhile, also finite Klingenberg planes came to the attention [F09]. Drake
and Lenz considered a parameter set for finite PK–planes in [F15] together
with new examples of finite PH–planes.
Structure theorems for finite chain rings (needed for finite desarguesian
Klingenberg planes) were proved by Edwin Clark et al. in [F03, F04] and
independently by Arnold Neumaier in [F33] and Al–Khamees [F01]. The
classification of all chain rings is still an open problem but partial results
are known. Galois rings GR($q^{n},p^{n}$) with $q^{n}$ elements and
characteristic $p^{n}$, with $q=p^{r}$, play a crucial role.
Beside Drake another player came to the forefront, Dieter Jungnickel, who was
active at the Universities of Giessen and Augsburg (Germany). He was a student
of Hanfried Lenz and became an expert in the theory of designs. In 1976 he
wrote his Diplomarbeit at the University of Berlin (Germany) on “Klingenberg
and Hjelmslev planes” and with the dissertation “Konstruktion transitiver
Inzidenzstrukturen mit Differenzenverfahren” he obtained his doctoral degree.
His most important contribution to the theory of Hjelmslev planes (and the
more general class of Klingenberg planes) concerns “regularity” [F14, F23,
F24, F26, F29]. A PK– or PH–plane is regular if it has an abelian automorphism
group $G=Z\oplus N$, where $G$ acts regularly (sharply transitively) on the
point set and on the line set and $N$ acts regularly on each neighborhood. It
is proved in [F21] that any finite PK–plane over a commutative local ring is
regular. Regularity is also interpreted in terms of difference sets and
auxiliary matrices, leading to new families of finite Hjelmslev and
Klingenberg planes. An interesting result, connecting PH–planes with designs,
is: the projective uniform Hjelmslev planes of order $q$ (with $q>2$) are
precisely the symmetric divisible partial designs on two classes with
parameters
$v=b=q^{2}(q^{2}+q+1),k=r=q(q+1),s=q^{2},t=q^{2}+q+1,\lambda_{1}=q,\lambda_{2}=1$.
For $q=2$ counterexamples exist, see [F27]. The concepts of regularity and
uniformity were also considered in $K$–structures, a further generalization of
Klingenberg planes (see [F09, F11, F12, F22, F25, F28, F30, F31]). Nino
Civolani [F02] considers free extensions of partial Klingenberg planes.
Jungnickel’s work was continued by Sharad Sane. Sane studied at the Indian
Institute of Technology Bombay, Mumbai (India) and obtained his PhD with the
dissertation “Studies in Partial Designs and Projective Hjelmslev Planes”
under the supervision of Balmohan Vishnu Limaye. In [F32] Sane and Limaye
demonstrate that $n$–uniform PH–planes are a kind of divisible partial designs
and by taking advantage of this property, they can give an alternative proof
for the fact that $n$–uniform planes are strongly $n$–uniform, as was proved
before in another way by Törner [E39]. In some other papers [F17, F18, F34,
F35, F36, F37, F38] Sane contributes to the theory of finite Hjelmslev and
Klingenberg planes.
## 9 The coordinatization of Hjelmslev and Klingenberg planes: a versatile
story
The coordinatization of affine and projective planes is one of the most
powerful tools in the study of such geometries. It permits to reformulate
geometric properties into algebraic ones (and vice versa), leading to a better
insight, including the construction of many non–desarguesian examples. This
coordinatization goes back to Marshall Hall Jr. (1910–1990). His important
paper [G37] published in 1943 is still one of the most cited. The basic
concept is a Hall ternary ring, also called PTR (planar ternary ring), an
algebraic structure $(R,T)$ with $R$ a non–empty set containing two distinct
elements $0$ and $1$ and with $T$ a ternary operation on $R$ such that
$y=T(x,m,k)$ means that the point with coordinates $(x,y)$ lies on the line
with coordinates $[m,k]$. With $(R,T)$ one can associate two loops $(R,+)$ and
$(R,\circ)$ when defining $a+b:=T(a,1,b)$ and $a\circ b:=T(a,b,0)$. The
properties of the plane (formulated in terms of the validity of Desargues’
configuration or in terms of transitivity of the automorphism group) are
reflected in the richness of the coordinatizing algebraic structure. If the
theorem of Desargues is always valid or equivalently if the plane is
$(P,\ell)$–transitive for any choice of the point $P$ and the line $\ell$,
then it turns out that $(R,+)$ and $(R\setminus\\{0\\},\circ)$ are both
groups, hence $(R,+,\circ)$ is a division ring or skewfield. Conversely, any
skewfield gives rise to a desarguesian projective plane. Slight variations on
Hall’s coordinatization method were made by Daniel Hughes [G38, G39] and by
Günter Pickert [G55]. Independently the russian mathematician Lev Anatolevich
Skornyakov described in 1949 a similar coordinatization method in [G58]. His
work [G59] was important for the distribution of the knowledge on projective
planes in the russian speaking mathematical community.
In the seventies and the eighties several attempts were made to coordinatize
in a similar way affine and projective Hjelmslev and Klingenberg planes. In
1967 the russian geometer V.K. Cyganova worked out a first successful
coordinatization for the more restrictive class of affine Hjelmslev planes
[G21]. She used the concept of an $H$–ternar, an algebraic structure with two
ternary operations, generalizing a Hall ternary ring (one of the main
differences being the existence of zero divisors). Since her paper was written
in russian, it remained unfortunately unaccessible for many people.
Independently from Cyganova, J.W. Lorimer introduced in 1971 in his PhD thesis
“Hjelmslev Planes and Topological Hjelmslev Planes” generalized ternary rings
(very similar to $H$–ternars) as the coordinatizing structures of affine
Hjelmslev planes. Three years later, Phyrne Bacon streamlined the work of
Cyganova and Lorimer in her thesis “Coordinatized Hjelmslev planes”, and she
introduced the name biternary ring (in appendix A of her thesis she gives a
comprehensive list of annotations including some mistakes and imperfections in
the work of her predecessors). The interaction between the geometric
properties of an affine Hjelmslev plane and the algebraic properties of its
coordinatizing biternary ring was examined in more detail by Lorimer in [G46],
by Cyganova in [G20, G22, G23, G24, G25, G26, G27], by Emelchenkov in [G33,
G35, G36] and by Shatokhin in [G56, G57].
To be complete, we also have to mention a paper by Drake [G28] in which he
obtains a kind of coordinatization for a special class of affine Hjelmslev
planes (radial planes) by means of a module.
After the coordinatization of affine Hjelmslev planes, a similar theory for
the more general class of affine Klingenberg planes has been worked out by
several authors. Bacon generalized her biternary rings. Drake, her supervisor,
encouraged her to publish in mathematical journals, but she was stubborn and,
apart from one single publication [G05], she refused. Her voluminous work,
totaling about 1000 pages, is contained in four books [G04], edited in own
management. For that reason it was often overlooked and seldom recognized as
an acceptable reference. In [G05] the “triangle theorem” is proved: a PK–plane
$\mathcal{P}$ possessing a nondegenerate triangle with sides
$\ell_{1},\ell_{2}$ and $\ell_{3}$ such that each derived AK–plane
$\mathcal{A}_{i}=\mathcal{P}\setminus g_{i}$ is desarguesian, is itself a
desarguesian PK–plane.
The Czech mathematician Frantisek Machala from the University of Olomouc
introduced in [G49] affine local ternary rings as an alternative for the
coordinatization of affine Klingenberg planes. Much later he could prove the
equivalence between his coordinatization method and the one of Bacon. He also
proved that any “incomplete” biternary ring (with one ternary and one partial
ternary operator) can be extended to a biternary ring with two (full) ternary
operators [G52]. In [G11] it is shown that this biternary ring extension is
unique.
In the case of ordinary planes each (desarguesian) affine plane can be
extended to a (desarguesian) projective plane. This does not hold any longer
for Hjelmslev planes (see section 7). This observation has also a serious
impact on the coordinatization of projective Hjelmslev planes: it does not
follow immediately from the affine coordinatization. The projective case was
first attacked by the russian mathematician E.P. Emelchenkov in 1972 in his
PhD thesis ”Ternars and automorphisms of Hjelmslev planes” (in russian) and in
[G34]. Due to the language barrier, his work was not accessible for many
researchers and for that reason, like Cyganova’s work, it was somewhat
overlooked.
Coordinatization methods for the more general case of projective Klingenberg
planes were worked out by both Machala and Bacon. Their approach is totally
different. Machala’s method is based on the concept of an extended local
ternary ring, an algebraic structure $(R,R^{\prime},T)$ with two disjoint sets
of coordinates $R$ and $R^{\prime}$ and one ternary operation $T$ on $R\cup
R^{\prime}$ (see [G47, G48, G50]). This coordinatization was not very
successful because it is not obvious to see any interaction between properties
of the extended local ternary ring and geometric properties of the
coordinatized plane.
The approach of Bacon is based on the fact that a PK–plane can be covered by
three AK–planes corresponding to three biternary rings. This yields a
coordinatizing structure for a projective Klingenberg plane as a triplet of
biternary rings, called sexternary ring [G04]. In Bacon’s voluminous work, the
interaction between geometric properties and the algebraic structure is
examined in depth. Unfortunately, a big part of these results remained hidden
for the reason mentioned above. In [G31] Manfred Dugas used a similar
coordinatizing structure, with six ternary operations.
Independently from the people mentioned above, the author introduced in 1987
in his PhD thesis “Klingenberg incidence structures, a contribution to ring
geometry” (in Dutch) (see also [G41, G42, G43]) planar sexternary rings
(PSR’s) with one full and five partial ternary operators. His coordinatization
method for PK–planes was inspired by the Hughes variant of the Hall ternary
ring. A small deficiency in his method was detected later (as pointed out by
Baker and Lorimer in [G11]). As a consequence of this shortcoming, the
coordinatization of a PK–plane by a PSR wasn’t fully compatible with the
coordinatization of a derived AK–plane by the biternary ring obtained from the
PSR. To overcome this anomaly, Baker and Lorimer (op.cit.) developed a new
coordinate ring, called incomplete sexternary ring (in the spirit of Dugas) as
a substitute for the planar sexternary ring. They even proved that such a
structure can be extended (in a unique way) to a sexternary ring with six full
ternary operators.
The coordinatization of projective planes is a handy instrument for the
construction of non–desarguesian examples in an algebraic manner. A lot of new
planes were found using quasifields, nearfields or alternative division rings.
Because of the bigger complexity of sexternary rings it seems that much less
examples of non–desarguesian PK–planes were obtained in this manner.
Nevertheless examples of non–desarguesian AK– and PK–planes obtained from
algebraic structures, are known. The oldest examples are the Moulton affine
Hjelmslev planes, given by Baker in [G06]. A projective version is contructed
in [G43] by the author. Klingenberg planes over local nearrings and Hjelmslev
planes over Hjelmslev–nearrings were studied by Emanuel Kolb in [G44, G45].
Much attention has gone to Moufang planes which can be coordinatized by local
alternative rings. Moufang–Hjelmslev planes first appear in a paper of Dugas
[G29]. He proves that all finite uniform Moufang PH–planes are desarguesian. A
stronger version of that theorem was later proved in [G30] (the uniformity
condition could be dropped if the order of the plane is bigger than 2).
Similar results were found by Baker, Lane and Lorimer for Moufang Klingenberg
planes, see [G07, G09, G10]. They prove that the class of Moufang PK–planes
coincides with the class of planes over local alternative rings and that a
finite Moufang PK–plane in which any two points have at least one joining
line, is a desarguesian projective Hjelmslev plane. Also a stronger version of
Bacon’s triangle theorem was proved in [G08]: a PK–plane with a non–degenerate
triangle for which the three derived AK–planes are translation AK–planes (and
with epimorphic image distinct from PG(2,2)), is Moufang.
More recently, a group of mathematicians around Basri Çelik and Süleyman
Çiftçi, from the University of Uludag, Bursa (Turkey), published several
papers concerning a particular class of Moufang–Klingenberg planes [G01, G02,
G14, G03, G15, G16, G17, G18, G19]. Their results, all variations on the same
theme, overlap with work of Andrea Blunck [G12, G13]. The role of Pappus’
theorem (its validity in a desarguesian plane implies the commutativity of the
coordinatizing ring) was investigated by Nolte and Maurer in [G53, G54].
## 10 Order and topology in Hjelmslev geometry: Machala and Lorimer
The theory of ordered incidence structures can be traced back mainly to Pasch,
Vorlesungen über neuere Geometrie from 1882. An excellent survey paper on the
axiomatics of ordered incidence geometry is [H27].
Ordered affine and projective Hjelmslev planes were studied by a group of
Canadian mathematicians, starting in the seventies. At least three
dissertations were written in that period at the McMaster University of
Hamilton, Ontario (Canada) under supervision of Norman Lane who was also a
world–class canoeist, competing in two Olympic games (bronze medal in 1948 in
London), before he started his academic career.
In 1975, Lynda Ann Thomas, wrote her Master’s thesis on “Ordered desarguesian
affine Hjelmslev planes” in which she proved that any ordered AH–ring gives
rise to an ordered desarguesian affine Hjelmslev plane and vice versa. This
result was published a few years later in [H30].
James Laxton, another student of Lane, treated in his Master’s thesis “Ordered
non–desarguesian affine Hjelmslev planes”. Catherine Baker, also a student of
Lane, wrote her Master’s thesis on “Affine Hjelmslev and generalized affine
Hjelmslev planes” and her doctoral thesis in 1978 on “Ordered Hjelmslev
planes”. In that thesis she investigates in detail the relationship between
ordered AH–planes and the coordinatizing ordered biternary rings, extending
results of Laxton and Thomas. Baker published several papers about ordered
(affine and projective) Hjelmslev planes (also with co–author): [H01, H02,
H03, H04, H05, H06].
It would be disrespectful if we should attribute all the results on ordered
ring geometries to the “Canadian School” only. Independently, a theory of
orderings for Klingenberg planes was worked out by Machala. He published many
papers on ordered Klingenberg planes [H18, H19, H20, H21, H22, H23, H24, H25]
and one overview work [H26]. The work of Baker et al. resembles in many
aspects Machala’s, but there are some differences. For a comparison between
both approaches, one may consult [H06].
The study of topological Klingenberg and Hjelmslev planes remained an
exclusive Canadian affair. The most prominent student of Lane, was undoubtedly
J.W. (Mike) Lorimer. In his doctoral thesis on “Hjelmslev planes and
topological Hjelmslev planes”, he not only introduced a coordinatization (see
the previous section) but he also laid the foundation for topological
Hjelmslev geometry. His work generalizes that of Salzmann [H28] and Skornyakov
[H29] on topological projective planes. In a series of publications [H07, H08,
H09, H10, H11, H12, H13, H14, H15, H16, H17] he further developed the theory
in close connection with the coordinatization problem. Among the most
important theorems proved by Lorimer, we mention the following
characterization theorem: the only locally compact connected pappian
projective Hjelmslev planes are the ones over the rings $\mathbb{K}[x]/\langle
x^{n}\rangle$ with $\mathbb{K}$ the field of real or complex numbers.
## 11 The revival of ring geometry in the eighties and nineties under
Veldkamp and Faulkner
In the seventies ring geometry was restricted almost exclusively to Hjelmslev
and Klingenberg geometry (in the desarguesian case to geometries over local
rings and Hjelmslev rings). The Dutch mathematician Ferdinand Douwe Veldkamp
(1931-1999), who is well–known for his work on geometries associated with
exceptional Lie groups and in particular polar spaces, reverted back to the
pioneering work of Barbilian where geometries over the broader class of
$Z$–rings were considered. It was Veldkamp’s aim to give an axiom system for
projective planes (and higherdimensional spaces) over arbitrary rings with
unit (without the imperfection in Barbilian’s attempt [A03, A04]). From
conversations with his colleague van der Kallen at the University of Utrecht
(an expert in $K$–theory), it became clear that the best setting for this
project is provided by rings of stable rank two. A ring $R$ has stable rank
two if the following property holds: if $a,b\in R$ and $Ra+Rb=R$ then there
exists a $r$ in $R$ such that $a+rb$ is invertible in $R$. The class of stable
rank two rings comprises all local rings. Hence, the projective ring planes
introduced by Veldkamp include the desarguesian Klingenberg and Hjelmslev
planes. A ring of stable rank two is always a $Z$–ring.
Veldkamp first worked out the theory for planes in [I23] with some special
cases in [I24] and later for spaces of higher dimension (see section 12). The
ring planes defined by Veldkamp are also known today as desarguesian Veldkamp
planes.
John Robert Faulkner, an authority in the domain of non–associative algebra
and geometry, further extended the theory of Veldkamp planes in the
non–desarguesian direction by introducing alternative (non–associative) rings
of stable rank two in [I01]. He then proved in [I02] that a Veldkamp plane has
the Moufang property (i.e. $(P,\ell)$–transitivity holds for all $P,\ell$ with
$P$ incident with $\ell$) if and only if it is a plane
$\mathcal{P}(\mathbb{A})$ over an alternative ring $\mathbb{A}$ of stable rank
two. Inspired by the work of his predecessors, Faulkner introduced in [I03]
Faulkner planes as a very general class of plane incidence structures with
neighbor (or remoteness) relation. They comprise the Veldkamp planes and the
planes introduced by Barbilian. A (connected) Faulkner plane for which the
group of $(P,\ell)$–transvections (automorphisms fixing all objects incident
with $P$ or $\ell$) is transitive on the set of points not neighboring with
$\ell$, is called a transvection plane. The coordinatization of transvection
Faulkner planes by a not necessarily associative alternative ring with the
property that $ab=1$ implies $ba=1$ involves a rather technical procedure
based on group theory and a lot of new concepts such as covering planes and
tangent bundle planes. A transvection Faulkner plane for which the tangent
bundle plane is also a transvection plane is called a Lie transvection
Faulkner plane. To every such plane an alternative two–sided units ring can be
attached and conversely with an alternative two–sided units ring $R$ a
corresponding Lie transvection Faulkner plane can be constructed. However,
this plane is not determined unambiguously when the ring does not have stable
rank 2. This high price has to be paid for the generalization from Veldkamp
planes to Faulkner planes. In [I04] Faulkner gives a geometric construction of
Barbilian planes coordinatized by composition algebras (including the Moufang
plane) using Jordan algebras. His book [I07] is completely devoted to the role
of such algebras in projective geometry.
Faulkner was surrounded by students at the University of Virginia,
Charlottesville (USA), who all were involved with the study of ring
geometries. Terese Deltz Magnus obtained her PhD in 1991 on “Geometries over
non–division rings” and she could generalize Faulkner’s axioms and results for
geometries of higher dimension (see section 12). Eve Torrence also graduated
under Faulkner’s supervision with “The coordinatization of a
hexagonal–Barbilian plane by a quadratic Jordan algebra”, a generalization of
the classical notion of generalized hexagon. Another student was Catherine
Moore d’Ortona who studied homomorphisms between projective ring planes in her
PhD thesis “Homomorphisms of remotely projective planes”, published in [I16].
Finally Karen Klintworth wrote her PhD thesis on “Affine remoteness planes”.
Faulkner himself considered a slightly more general axiomatization of Faulkner
planes in [I06]. There he chooses for the remoteness relation, the negation of
the neighbor relation. Much of the results obtained in [I03] are extended but
the coordinate rings that appear are no longer always alternative. It is
proved that $\mathcal{P}$ is a transvection plane if and only if $\mathcal{P}$
is isomorphic to $P(G,N)$, the plane associated with a group $G$ of Steinberg
type parametrized by the ring $R$ and with $N$ a certain subgroup of $G$.
Necessary and sufficient conditions are given for $R$ to be alternative,
associative or commutative. In [I06] also projective remoteness planes with
reflections (hence metric planes) have been considered. The content of this
paper is closely related to some work of Knüppel and Salow in [D10] (see
section 6). It also contains a part on affine ring planes and elementary basis
sets which are closely related to Barbilian domains as introduced by Leissner
(see section 5).
In the slipstream of Veldkamp’s paper on projective ring planes, several
slightly modified axiom systems have been described, leading to other classes
of projective ring geometries. We have already mentioned Frieder Knüppel in
the section on metric ring geometry, but some of his papers rather join the
spirit of this section. In [I14] Knüppel considers ring geometries over
associative rings based on four axioms adapted from Veldkamp (based on
remoteness rather than neighborship). A coordinatization theorem is stated
without proof. In [I15] he studies homomorphisms between such planes.
Renata Spanicciati defines near–Barbilian planes and strong near–Barbilian
planes in [I22] by adapting some of the axioms of Veldkamp. The neighbour
relation between points turns out to be the identity, and the neighbour
relation between lines becomes an equivalence relation. In [I11] Guy Hanssens
and Hendrik Van Maldeghem prove that any near–Barbilian plane is strong
near–Barbilian. Kálman Péntek gives a necessary and sufficient condition for a
Veldkamp plane to be a direct product of a finite number of Veldkamp planes in
[I17, I18, I19].
The study of homomorphisms between ring geometries was also a central theme in
several papers of Veldkamp (some of them in joint work with Joseph Ferrar)
[I08, I09, I10, I25, I26]. The geometric homomorphisms of distinct kind
(incidence preserving, neighbor–preserving, distant–preserving) are
characterized in terms of algebraic morphisms between the underlying rings.
Veldkamp’s results were generalized by the author in the non–desarguesian case
for homomorphisms between projective Klingenberg planes using the
coordinatizing planar sexternary rings (see section 9) in [I12, I13].
Thorsten Pfeiffer could generalize a well–known theorem for planes over fields
to planes over rings: a desarguesian ring plane $\mathcal{P}(R)$ is pappian
(Pappus’ theorem is valid) if and only if $R$ is commutative.
## 12 Projective and affine Hjelmslev spaces and spaces over arbitrary rings
Hitherto we only discussed plane ring geometries. The theory of higher
dimensional projective spaces over rings has been developed by different
people. Projective spaces over local rings appear for the first time in the
work of Klingenberg [A17]. Today we call them Klingenberg spaces (PK–spaces).
The first study of PK–spaces after Klingenberg, is due to Hans–Heinrich Lück,
a student of Lüneburg. His PhD thesis, published as an article [J34] in 1970
under the somewhat misleading title “Projektive Hjelmslevräume”, contains an
axiomatic characterization of a class of incidence structures which permit a
coordinatization by local rings. Hence, the paper deals with projective
Klingenberg spaces rather than with Hjelmslev spaces. Lück proves that in an
axiomatically defined PK–space of dimension at least three, the theorem of
Desargues holds and that it must be isomorphic to a space derived from a
module over a local ring. Hence all projective Klingenberg spaces of dimension
$\geq 3$ are desarguesian, a situation analogous to the case of classical
projective spaces.
Independently from Lück, Machala defined and studied projective Klingenberg
spaces (Projektive Räume mit Homomorphismus) of finite or infinite dimension
in [J35, J36, J37, J38, J39]. He proved that the planes in a PK–space are
PK–planes and that PK–spaces of dimension at least three come from modules
over a local ring (cfr. Lück). Machala also investigated homomorphisms between
PK–spaces and the fundamental theorem (isomorphisms between spaces can be
represented by semilinear mappings between the underlying modules). A paper by
Jukl [J13] is in conformity with this.
An axiomatic approach for the more restricted class of projective Hjelmslev
spaces (PH–spaces) was initiated by John Lamb Jr. from the University of Texas
at Austin (USA) in his PhD thesis, entitled “The Structure of Hjelmslev space,
a generalization of projective space” (1969) but its content (related to
lattice theory) was not published.
Much more widespread is the work of Karl Mathiak, who was very productive in
this field. He defined a class of special projective Hjelmslev spaces starting
from a vectorspace over a (skew)field endowed with a valuation (“Bewertete
Vektorräume”). The structure of the ideals in the corresponding valuation ring
plays a central role in his approach. The theory is thoroughly worked out in a
series of papers, published over a period of twenty years between 1967 and
1987, see [J41, J42, J43, J44, J45, J46, J47, J48, J49, J50].
His compatriot Alexander Kreuzer introduced an axiom system for arbitrary
PH–spaces in his doctoral thesis “Projektive Hjelmslevräume” which was
published afterwards as an article in [J18] with some preliminary work in
[J17]. This study is continued in [J19, J20].
For the affine case we have to go to Canada again. In four papers, Tibor
Bisztriczky together with J.W. Lorimer, worked out two axiom systems for
affine Klingenberg spaces [J02, J03, J04, J05]. Neither of their axiom systems
assumes the existence of an overlying projective Klingenberg space or the
existence of an underlying ordinary affine space. Machala in [J38] also
defined affine Klingenberg spaces, but not separate from PK–spaces (similar to
ordinary affine spaces obtained from projective spaces by deleting a
hyperplane).
The most general study of projective ring spaces (over not necessarily local
rings) was undertaken by Ferdinand Veldkamp. We have already indicated his
interest in section 11. In [J62, J63], Veldkamp gives a self–dual axiom system
for projective “Barbilian spaces” of finite dimension using the basic concepts
of points, hyperplanes, incidence and neighbor relation. We call them now
Veldkamp spaces. The main result is that Veldkamp spaces of dimension $\geq 3$
are spaces over rings of stable rank 2.
Theresa Magnus defines Faulkner spaces as even more general geometries by
extending the theory of Faulkner planes [J40]. She proves that any Faulkner
space of dimension $n\geq 3$ is coordinatized by a unique associative
two–sided units ring $R$ and that the group generated by all transvections is
a group of Steinberg type over $R$. A Faulkner space over the ring
$\mathbb{Z}$ of integers is constructed, providing an example of a Faulkner
geometry which is not a Veldkamp space, since $\mathbb{Z}$ has stable rank 3.
Under the Veldkamp spaces we also find the projective spaces over matrix rings
over GF($q$), studied by Thas [B12] in the early days of ring geometry (see
section 4). Other contributions to finite ring spaces are due to Kapralova who
considered projective spaces over the ring of dual numbers over a Galois field
in [J14] and to Ivan Landjev and Peter Vandendriesche [J24, J25].
A well-developed theory for affine spaces over rings is also due to Veldkamp.
In [J64] he defines Barbilian domains in free modules of rank $n$ and
introduces $n$–dimensional affine ring geometries. A geometrical
interpretation of Barbilian domains is given by Sprenger in [J61].
Other attempts for setting up a theory of higher dimensional affine ring
geometries (incidence structures with parallellism) are scattered in the
literature. The definitions and the methods used are very diverse.
Contributions in this field are due Permutti and Pizzarello [J54, J55], Miron
[J51], Leissner, Severin and Wolf [J31, J32], Ostrowski [J53], Schmidt and
Weller [J58], Kreis [J15, J16], Seier [J59, J60], Bach [J01] and others.
One of the problems for higher dimensional geometries over rings that has got
much attention is that of morphisms and the fundamental theorem. For classical
projective geometries $P(V)$ induced by a vectorspace over a field this
theorem states that any bijective incidence preserving map (projectivity)
between projective spaces $P(V)$ and $P(W)$ can be algebraically characterized
by a semilinear map from $V$ to $W$. The first generalization of the
fundamental theorem to ring geometries was obtained by the Indian
mathematicians Ojanguren and Sridharan who could prove it in case of
module–induced geometries $P(M)$ with $M$ a free module of finite rank $\geq
3$ over a commutative ring [J52]. Generalizations to other classes of rings
were proved later by Sarath and Varadarajan in [J57] and by James in [J12] and
Faure in [J06].
For the sake of completeness we also refer to module–induced geometries as
defined by Marcus Greferath and Stefan Schmidt [J08, J09, J10, J11]. Instead
of the usual definition of the pointset as the set of all submodules generated
by a unimodular element in a free module (cfr. Veldkamp), they take all
submodules of rank one, leading to the bizarre situation of points properly
contained in bigger points. In [J07] an extension of the fundamental theorem
is proved for such module–induced geometries. Close relative to this, there is
an abundance of articles by the school of the Georgian mathematician Alexander
Lashkhi. They all contain variations on the same theme: an extension of the
fundamental theorem for affine and projective geometries related with modules
over rings, from the lattice–theoretic point of view. We have not included the
whole collection of papers by Lashki and his students. Some of them have been
published multiple times in different journals (in Russian and in English).
The literature list only mentions a few representative ones: [J21, J22, J23,
J26, J27, J28, J29, J30, J56].
## 13 Projective lines and circle geometries over rings: Blunck, Havlicek and
Keppens
The “smallest” projective geometries that can be considered, are the
projective lines. From the viewpoint of incidence geometry not much can be
said about these rather poor structures. But combining the study of projective
lines with those of their automorphisms (the general linear group) puts them
into a new light. The theory has common ground with what is usually called
geometric algebra. Indeed, the projective line $P(R)$ over any ring $R$ can be
defined in terms of the free left $R$–module $R^{2}$ as follows: it is the
orbit of a starter point $R(1,0)$ under the action of the general linear group
$GL_{2}(R)$ on $R^{2}$. Since geometric algebra over rings only fits sideways
in the section of incidence geometry, we did not make an effort to be complete
in the literature list for this item. Nevertheless we mention a number of
relevant references in which more information can be found, e.g. [K46].
Central themes that keep returning are the notions of cross–ratio and harmonic
quadruples and the fundamental theorem, also known as Von Staudt’s theorem. In
the case of classical projective lines over a field or a skewfield, this
theorem characterizes mappings of the projective line which preserve
harmonicity as projectivities.
Among the first publications on projective ring lines (and we do not consider
here papers only dealing with linear groups over rings) belong some articles
by the Indian mathematicians Nirmala and Balmohan Limaye. They prove a
generalization of Von Staudt’s theorem for some special classes of commutative
and non–commutative rings in [K38, K39, K40, K41, K42]. A little bit earlier,
in 1968, Melchior, a student of Benz, wrote his PhD thesis, entitled “Die
projektive Gerade über einem lokalen Ring: ihre lineare Gruppe und ihre
Geometrie”. Other contributions to this item appeared in [K02, K04, K05, K10,
K16, K22, K23, K24, K25, K30, K31, K34, K35, K37, K45, K47, K49]. Some papers
by Bilo and Depunt [B01], Hubaut [B08, B09] and Thas [B10, B11] also deal with
projective lines over rings (see section 4). They were followed by Havlicek et
al. in [K26, K27, K28].
Projective lines over rings are also intrinsically related with circle
geometries. This relation was established for the first time by Benz in his
famous book “Vorlesungen über Geometrie der Algebren” [K06] from 1973. He
presented a unified treatment of plane circle geometries, now called Benz
planes, using the projective line over a commutative ring which is a
two–dimensional $\mathbb{K}$–algebra over a field $\mathbb{K}$. His definition
was extended by Andrea Blunck, a student of Benz, and Armin Herzer who
introduced more general chain geometries $\Sigma(\mathbb{K},R)$ with $R$ a
(not necessarily two–dimensional) $\mathbb{K}$–algebra. A chain geometry is an
incidence structure whose point set is the set of points of the projective
line over $R$ and the $GL_{2}(R)$ orbit of $P(\mathbb{K})$ is the set of
chains. The plane circle geometries of Möbius, Laguerre and Minkowski type are
particular chain geometries for $R=\mathbb{L}$ (a quadratic field extension of
$\mathbb{K}$), $R=\mathbb{K}+\mathbb{K}\varepsilon$ with $\varepsilon^{2}=0$
(dual numbers) or $R=\mathbb{K}+\mathbb{K}t$ with $t^{2}=t$ (double numbers)
respectively. For an overview of chain geometry we refer to [K19]. Chain
geometries were also treated by Schaeffer, a student of Benz, in his doctoral
thesis entitled “Zum Automorphismenproblem in affinen Geometrien und
Kettengeometrien über Ringen. The study of chain geometries is continued by
Blunck who introduced generalized chain geometries by considering projective
lines over non–associative, alternative rings. In her PhD thesis
“Doppelverhältnisse und lokale Alternativringe” (1990) and in [K07] she
extended the notion of cross–ratio and investigated chain geometries (in
relation to projective lines over non–associative rings) in [K08, K11, K12].
Around the same time the author in [K32, K33] defined Klingenberg–Benz planes
axiomatically, i.e. plane circle geometries with neighbor relation which admit
a natural epimorphism onto a classical Benz plane. Using the projective line
over three kinds of quadratic ring extensions of a local ring (instead of a
field), he was able to construct algebraic models of such geometries. Also
Konrad Lang in [K36] studied independently of us a class of Hjelmslev–Möbius
planes. Blunck and Stroppel extended our definition of Klingenberg–Benz planes
to Klingenberg chain spaces in [K21] and Blunck also proved that a Klingenberg
chain space can be embedded into a projective Klingenberg space, such that the
points are identified with points of a quadric and the chains with plane
sections [K09]. In [K50] Seier constructed analogously a class of chain
geometries $\Sigma(H,L)$ with $H$ a Hjelmslev–ring and $L$ a ring extension of
$H$ and in [K51] he defined a Möbius plane with neighbor relation of a
different kind than the one defined by us.
A basic notion concerning the projective line over a ring $R$ is its distant
relation: two points are called distant if they can be represented by the
elements of a two-element basis of $R^{2}$. The distant graph has as vertices
the points of the projective line and as edges the pairs of distant points.
The distant graph is connected precisely when $GL_{2}(R)$ is generated by the
elementary linear group $E_{2}(R)$. This aspect of projective ring lines was
studied in more detail by Blunck, Havlicek, Matraś and some others in [K01,
K13, K14, K15, K29, K43, K44]. In [K17, K18] the interaction between ring
geometry and the geometry of matrices in the sense of Hua (see [B07]) is
investigated in more detail.
## 14 Ring geometries and buildings: Van Maldeghem and co.
The theory of buildings was invented by the Belgium-born French mathematician
Jacques Tits. Roughly speaking, buildings are incidence geometries on which
groups act. Tits received many awards for his fundamental and path–breaking
mathematical ideas, including the Abel Prize in 2008. One of his achievements
is the classification of affine buildings of rank at least 4. They are known
to be “classical”, i.e. they arise from algebraic groups over a local field.
In the rank three case (where affine buildings are of three possible types
$\tilde{A}_{2}$, $\tilde{C}_{2}$ or $\tilde{G}_{2}$) many non–classical
counterexamples are known.
In his PhD thesis “Niet–klassieke driehoeksgebouwen” on triangle buildings
(affine buildings of type $\tilde{A}_{2}$) Hendrik Van Maldeghem observed that
a special kind of ring geometry is present as the suitably defined geometry at
distance $n$ from any given vertex of the building (the so-called $n$–th
floor). This was described among other things in [L18, L19]. A little bit
later Hanssens and Van Maldeghem could prove that those ring geometries are in
fact projective Hjelmslev planes of level $n$, see [L07]. In [L08] they give a
universal construction for level $n$ Hjelmslev planes (see also [L06] for the
$2$–uniform case) and as a corollary any level $n$ projective Hjelmslev plane
is isomorphic to the $n$–th floor of a triangle building. This result links
the theory of PH–planes to that of triangle buildings, a rather unexpected but
fascinating fact. In the same spirit Van Maldeghem investigated another class
of rank three affine buildings, of type $\tilde{C}_{2}$, and proved that the
$n$–th floor turns out to be another type of ring geometry which can be seen
as a generalization of an ordinary generalized quadrangle. He called it
“Hjelmslev–quadrangle” of level $n$ (see [L20, L21]). In joint work with
Hanssens a complete characterization of $\tilde{C}_{2}$–buildings by Hjelmslev
quadrangles was obtained [L09, L10].
The author defined “Klingenberg–quadrangles” as another generalization of
ordinary generalized quadrangles in [L12]. The connection between
Klingenberg–quadrangles and Hjelmslev–quadrangles is explained in [L20].
The relation of polar spaces of higher rank to generalized quadrangles is
comparable with the relation of projective spaces to projective planes.
Generalized quadrangles are polar spaces of rank two. Projective ring spaces
of dimension at least three have got as much attention in the literature as
projective ring planes. This is not the case so far for general
“Klingenberg–polar spaces” or “polar spaces over rings” if the rank is bigger
than two. Only one paper by James [L11] is known to us. Certainly this topic
offers perspectives for future research.
The discovery by Van Maldeghem of the connection between buildings and ring
geometry has a precedent. Twenty years earlier, in 1968, Veldkamp in joint
work with Tonny Springer considered a geometry over the split octonions (over
the complex number field) in [L16]. This geometry is a kind of analogue of the
non–desarguesian projective plane over the alternative division ring of
(non–split) octonions $\mathbb{O}$ but in which two distinct lines may have
more than one point in common and dually. It was called Moufang–Hjelmslev
plane, but this name is misleading since it is not a projective Hjelmslev
plane (the neighbor relation is not transitive) and hence it is completely
distinct from the Moufang PH–plane (over an alternative local ring) studied
elsewhere (see section 9). In two subsequent papers [L25, L26] more results on
Hjelmslev–Moufang planes are obtained, concerning projective groups. The
geometry of Veldkamp and Springer is the same as the one constructed by Tits
[L17] starting from the split algebraic group of type $E_{6}$. In this
geometry each line has the structure of a polar space and two lines can meet
in more than one point (namely, in a maximal singular subspace of the
corresponding polar spaces).
John Faulkner considered Hjelmslev–Moufang planes by allowing an arbitrary
ground field instead of $\mathbb{C}$ in [L04, L05] and also Robert Bix studied
generalized Moufang planes in [L02, L03].
The relationship between Hjelmslev planes and buildings was further exploited
by Van Maldeghem and Van Steen to give a characterization of some rank three
buildings by automorphism groups [L22, L23, L24].
In the margins of the study of buildings some other questions have emerged.
One of them concerns embeddings. Embeddings of point–line geometries into
projective spaces are well–known in the literature. The embedding question for
ring geometries, in particular for projective Hjelmslev planes, is first
attacked by Artmann [L01] who shows that the PH–plane over the ring of plural
numbers $\mathbb{F}[t]/t^{n}$ ($\mathbb{F}$ a field), can be embedded in the
$(3n-1)$–dimensional projective space over $\mathbb{F}$. In [L13] the author
and Van Maldeghem prove a nice characterization theorem for embeddable
Klingenberg planes: if $\mathcal{P}$ is a projective Klingenberg plane that is
fully embedded in the projective space PG(5,$\mathbb{K}$) for some skewfield
$\mathbb{K}$, then $\mathcal{P}$ is either a desarguesian Klingenberg plane
over a ring of twisted dual numbers or a subgeometry of an ordinary projective
plane. The embedding of the projective plane over a matrix ring with entries
in GF($q$) into a projective space over GF($q$) was also observed by Thas in
[B12, B13].
Veronesean sets are closely connected with embeddings. In [L14] Schillewaert
and Van Maldeghem define geometries with an additional axiom by which the
Hjelmslev–Moufang plane (in the sense of Springer–Veldkamp) and its relatives
fit into the framework using the modern notion of parapolar spaces. In [L15]
they provide a common characterisation of projective planes over two-
dimensional quadratic algebras (over an arbitrary field) in terms of
associated Veronesean sets. Anneleen De Schepper and Van Maldeghem [L27] have
considered Veronese representations of Hjelmslev planes over quadratic
alternative algebras as part of a more general study of Veronese varieties and
Mazzocca–Melone sets.
## 15 Ring geometries in coding theory: Honold, Kiermaier and Landjev
One of the fastest growing disciplines in mathematics is coding theory. Since
its introduction by Claude Shannon in 1948, the number of publications about
codes has exploded, in particular due to its importance in cryptography, data
transmission and data storage. Initially mostly linear codes over finite
fields were studied, but after the publication in 1994 of the paper [M07] by
Hammons _et al._ , a new era has begun. In that paper it is proved that all
(non–linear) binary Kerdock-, Preparata-, Goethals- and Delsarte-Goethals-
codes are images of $\mathbb{Z}_{4}$–linear codes under the Gray map. This
discovery was quite peculiar and the paper got in 1995 the Information Theory
Paper Award from the IEEE Information Theory Society. It was the start of a
search for new codes by considering linear codes over the ring
$\mathbb{Z}_{4}$ and over more general finite rings (see e.g. [M01]). Also
some papers by Aleksandr Nechaev [M47, M48] have led the research in that
direction.
We will not give a survey of all results obtained up to now for codes over
finite rings, because even this niche has become too wide. We restrict
ourselves here (and in the literature list) to the publications in which the
direct relation between codes over rings and ring geometries is exhibited.
Indeed, linear codes over finite chain rings can be associated with finite
projective Hjelmslev geometries, the hyperplanes corresponding to the
codewords. This correspondence offers opportunities for investigating the
structure and the construction of ring–linear codes by pure geometrical
methods.
The technique was first applied in [M14] by Thomas Honold, now working at the
Zhejiang Universität in Hangzhou (China), and Ivan Landjev from the New
Bulgarian University of Sofia (Bulgaria). They prove that certain MacDonald
codes can be represented by linear codes over the ring of twisted dual numbers
on a finite field, using multisets of points in Hjelmslev spaces. In [M15]
they prove that all Reed–Muller codes are linearly representable over the ring
of dual numbers over $\mathbb{Z}_{2}$. In [M16] a general theory of linear
codes over finite chain rings has been developed as a natural generalization
of the theory of linear codes over finite fields and the correspondence with
Hjelmslev spaces is investigated. In [M17] and [M18] an update of that paper
is given. Geometric arguments are also used explicitly in [M19] for the
construction of particular linear codes over chain rings of order four,
generalizing a result obtained by Michael Kiermaier and Johannes Zwanzger in
[M34, M35]. Keisuke Shiromoto and Leo Storme defined in [M49] a Griesmer type
bound for linear codes over finite quasi-Frobenius rings and they give a
geometrical characterization of linear codes meeting the bound, viz. a one-to-
one correspondence between these codes and minihypers in projective Hjelmslev
spaces.
Kiermaier was a student of Honold and wrote his Master’s thesis on “Arcs und
Codes über endlichen Kettenringen” in 2006 and obtained his PhD in 2012 at the
University of Bayreuth (Germany) with the thesis “Geometrische Konstruktionen
linearer Codes über Galois-Ringen der Charakteristik 4 von hoher homogener
Minimaldistanz”.
The study of codes over chain rings from the viewpoint of Hjelmslev geometry
has also led to the generalization of several special point sets (arcs, ovals,
blocking sets, caps), already well–known in classical geometry over fields.
The thus far most studied sets in Hjelmslev planes are arcs. A $(k,n)$-arc
refers to a set of $k$ points which meet every line in at most $n$ points.
This definition was given by Honold and Kiermaier in [M43]. General upper
bounds on the cardinality of such arcs were found as well as the maximum
possible size. For a chain ring $R$ of length 2 with Jacobson radical $R_{0}$,
such that $|R/R_{0}|=q$, the maximum size of a $(k,2)$-arc in the projective
Hjelmslev plane over $R$ is $q^{2}$ if $q$ is odd and $q^{2}+q+1$ if $q$ is
even (see [M20]). In [M21] the existence of maximal $(q^{2}+q+1,2)$-arcs (i.e.
hyperovals) is proved for $q$ even and in [M12] the existence of maximal
$(q^{2},2)$-arcs for $q$ odd is proved. The results on maximal arcs are also
used to construct interesting codes with a linear representation over a chain
ring. Examples, non–existence results and upper bounds for the length of arcs
are also present in [M02, M08, M10, M11, M13, M23, M32, M33, M36, M50].
Caps in finite projective Hjelmslev spaces over chain rings of nilpotency
index 2 are defined by Honold and Landjev in [M22]. A geometric construction
for caps in the threedimensional space is given, using ovoids in the
epimorphic space PG(3,$q$) as well as an algebraic construction using
Teichmüller sets. Blocking sets in Hjelmslev planes and their relation with
codes is the subject of [M03, M40, M42] while [M39] and [M54] deal with
spreads.
Another aspect that appears in the literature is the correspondence between
two–weight codes and strongly regular graphs. In [M04] regular projective two-
weight codes over finite Frobenius rings are introduced and it is shown that
such codes give rise to a strongly regular graph. In [M04, M41] two-weight
codes are constructed using ring geometries and they yield infinite families
of strongly regular graphs with non-trivial parameters.
Ovals in an ordinary projective plane of order $q$ are just $q+1$ arcs and
every conic is an oval. By a celebrated theorem of Segre every $(q+1)$–arc in
PG(2,$q$) with $q$ odd, is a conic. The study of ovals, conics and unitals in
finite projective Klingenberg and Hjelmslev planes was initiated by the author
in his PhD thesis but only a part of it was published, e.g. in [M31]. The
author also defined polarities in projective Klingenberg planes and spaces and
he investigated their sets of absolute points which give rise to ovals or
ovoids in some cases. A comprehensive study of polarities in $n$–uniform
Hjelmslev planes and spaces over the ring GF($q$)[$t$]/$t^{n}$ appeared in
[M28, M29, M30]. Also the papers [M37, M38] enlight some aspects of ovals and
conics in ring planes.
A combinatorial study of conics in finite desarguesian Hjelmslev planes was
made by Ratislav Jurga and Viliam Chvál. Their papers contain formulas for the
number of interior and exterior points, tangents and secants of a conic [M05,
M25, M26, M27]. A related problem is the projective equivalence of quadrics in
projective Klingenberg spaces. This was first formulated in [M06] and analyzed
in detail by O.A. Starikova et al. in [M24, M44, M51, M52, M53].
Recently, a special class of codes (toric codes) is found to be related to
affine ring geometries (Leissner planes) by Little in [M45, M46], revealing
still another correspondence between ring geometry and coding theory.
## 16 Ring geometry in quantum information theory: Saniga and Planat
Besides the fact that ring geometries play a substantial role in coding
theory, there is another domain in which they arise unexpectedly, namely in
quantum information theory. In this important branch of quantum physics it is
studied how information can be stored and retrieved from a quantum mechanical
system. In 2006 Metod Saniga from the Astronomical Institute at the Slovak
Academy of Sciences and Michel Planat from the Université de Franche–Comté
(France) discovered a connection between finite ring geometry and quantum
information theory [N09]. It is not clear yet whether or not the
correspondence goes further than just equality in numbers of objects, but
anyhow it is remarkable that ring geometries seem to play a role in quantum
physics.
The notion of mutually unbiased bases (MUBs) has turned into a cornerstone of
the modern theory. Saniga and Planat observed that the basic combinatorial
properties of a complete set of MUBs of a $q$–dimensional Hilbert space
$\mathcal{H}_{q}$ with $q=p^{r}$, $p$ being a prime and $r$ a positive
integer, are qualitatively mimicked by the configuration of points lying on a
proper conic in a projective Hjelmslev plane defined over a Galois ring of
characteristic $p^{2}$ and rank $r$. The $q$ vectors of a basis of
$\mathcal{H}_{q}$ correspond to the $q$ points in a neighbour class and the
$q+1$ MUBs answer to the total number of pairwise disjoint neighbour classes
on the conic. In a series of subsequently published papers, other
combinatorial correspondences between concepts from quantum theory and ring
geometries over finite rings (in particular projective ring lines) are
observed. One of these similarities concerns the structure of the generalized
Pauli group associated with a specific single $d$-dimensional qudit (qudits
are generalizations tot $d$–level quantum systems of qubits which are the
basic units in quantum information systems of level two). See [N01, N02, N03,
N04, N06, N08, N11, N12, N13, N14, N15, N16, N17, N18] and the survey papers
[N05, N10] for an overview on this topic. The study of the relation with
finite geometry (not only restricted to ring geometry but also in connection
with small generalized polygons and polar spaces) is still ongoing, see e.g.
[N07].
## 17 Literature on ring geometry and geometry over rings
In the separate bibliographic list we only mention publications in scientific
journals (or books). Other references, e.g. Master’s and PhD theses which are
quoted explicitly in the foregoing text, are not repeated here, except in case
they were also published. Articles belonging to conference proceedings are
also omitted if there exist copies with almost identical content, published in
other journals. We have grouped the references in themes, corresponding to the
sections in the paper. Some papers can be classified in multiple sections. In
that case they are listed in the section in which they are referred to for the
first time. We do not claim that the bibliography is complete but we have
tried to compose an accurate list which complements the existing obsolete
lists, especially for the period after 1990. The author will be grateful for
additions, completions and corrections to this list.
## A First traces and pioneers of ring geometry
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* [A02] Artmann B., Dorn D., Drake D. and Törner G., Hjelmslev’sche Inzidenzgeometrie und verwandte Gebiete. Literaturverzeichnis, J. Geom. 7, 175–191 (1976)
* [A03] Barbilian D., Zur Axiomatik der projektiven ebenen Ringgeometrien I, Jahresber. Deutsch. Math.-Verein. 50, 179–229 (1940)
* [A04] Barbilian D., Zur Axiomatik der projektiven ebenen Ringgeometrien II, Jahresber. Deutsch. Math.-Verein. 51, 34–76 (1941)
* [A05] Benz W., On Study’s Übertragungsprinzip, J. Geom. 64, 1–7 (1999)
* [A06] Clifford W. K., Preliminary Sketch of Bi-Quaternions, Proc. Lond. Math. Soc. 4, 381–395 (1873)
* [A07] Dembowski P. Finite Geometries, reprint of the 1968 Edition, Springer-Verlag, Berlin Heidelberg, 379 pp. (1997)
* [A08] Grünwald J., Über duale Zahlen und ihre Anwendung in der Geometrie, Monatsh. Math. 17, 81–136 (1906)
* [A09] Hjelmslev J., Die Geometrie der Wirklichkeit, Acta Math. 40, 35–66 (1916)
* [A10] Hjelmslev J., Die natürliche Geometrie, Abh. Math. Sem. Univ. Hamburg 2, 1–36 (1923)
* [A11] Iordănescu R., The geometrical Barbilian’s work from a modern point of view, Balkan J. Geom. Appl. 1, 31–36 (1996)
* [A12] Jungnickel D., Hjelmslev’sche Inzidenzgeometrie und verwandte Gebiete. Literaturverzeichnis II, J. Geom. 16, 138–147 (1981)
* [A13] Keppens D., 50 years of finite geometry, the “geometries over finite rings” part, Innov. Incidence Geom. 15, 123–143 (2017)
* [A14] Klingenberg W., Projektive und affine Ebenen mit Nachbarelementen, Math. Z. 60, 384–406 (1954)
* [A15] Klingenberg W., Euklidische Ebenen mit Nachbarelementen, Math. Z. 61, 1–25 (1954)
* [A16] Klingenberg W., Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg 20, 97–111 (1955)
* [A17] Klingenberg W., Projektive Geometrien mit Homomorphismus, Math. Ann. 132, 180–200 (1956)
* [A18] Kotelnikov A. P., Screw Calculus and Some Applications to Geometry and Mechanics (in Russian), Annals of the Imperial University of Kazan, Russia (1895)
* [A19] Predella P., Saggio di Geometria non-Archimedea (Nota II), Batt. G. 3, 161–171 (1912)
* [A20] Ree R., On projective geometry over full matrix rings, Trans. Amer. Math. Soc. 6, 144–150 (1955)
* [A21] Segre C., Le geometrie proiettive nei campi di numeri duali, Nota I e II, Atti R. Acc. Scienze Torino 47, 114–133 and 164–185 (1911–1912) (in: Corrado Segre, Opere, a cura della Unione Matematica Italiana, Volume II, Edizione Cremonese, Roma, 396–431 (1958))
* [A22] Study E., Geometrie der Dynamen, Leipzig, Germany, 603 pp. (1903)
* [A23] Törner G. and Veldkamp F.D., Literature on geometry over rings, J. Geom. 42, 180–200 (1991)
* [A24] Veldkamp F.D., Geometry over rings, chapter 19 in Handbook of Incidence Geometry: Buildings and Foundations. Edited by F. Buekenhout, Elsevier, Amsterdam, 1420 pp. (1995)
## B The Belgian contribution to early ring geometry
* [B01] Bilo J. and Depunt J., Over de lineaire transformaties van de ternionenrechte (in Dutch), Med. Koninkl. Vlaamse Acad. Wet., Lett. schone Kunsten België, Kl. Wet. 24 (8), 1–36 (1962)
* [B02] Bingen F., Géométrie projective sur un anneau semi–primaire, Acad. Roy. Belg. Bull. Cl. Sci. 52, 13–24 (1966)
* [B03] Depunt J., Sur la géométrie ternionienne dans le plan, Bull. Soc. Math. Belg. 11, 123–133 (1959)
* [B04] Depunt J., Grondslagen van de analytische projectieve ternionenmeetkunde van het platte vlak (in Dutch), Verh. Kon. Vl. Ac. voor Wet., Lett. en Sch. K. van België, Kl. der Wet. 63, 1–99 (1960)
* [B05] De Winne P., An extension of the notion “cross-ratio of an ordered 4-tuple of points of the projective line” to an ordered ($n+3$)-tuple of points (resp. hyperplanes) of the $n$-dimensional space over an arbitrary ring with identity, part I: the $n$-dimensional projective space $S_{n}$ over an arbitrary ring with identity, Simon Stevin, 47, 139–159 (1974)
* [B06] De Winne P., Een studie betreffende het projectieve vlak over de totale matrixalgebra $M_{2}((K)$ der $2\times 2$–matrices met elementen in een algebraïsch afgesloten veld (in Dutch), Verh. Kon. Vl. Ac. voor Wet., Lett. en Sch. K. van België, Kl. der Wet., 144, 1–80 (1978)
* [B07] Hua L.-K., Geometries of Matrices I. Generalizations of Von Staudt’s theorem, Trans. Amer. Math. Soc. 57, 441–481 (1945)
* [B08] Hubaut X., Construction d’une droite projective sur une algèbre associative, Acad. Roy. Belg. Bull. Cl. Sci., 50, 618–623 (1964).
* [B09] Hubaut X., Algèbres projectives, Bull. Soc. Math. Belg. 17, 495–502 (1965)
* [B10] Thas J.A., Dubbelverhouding van een geordend puntenviertal op de projectieve rechte over een associatieve algebra met éénelement (in Dutch), Simon Stevin, 42 (3), 97–111 (1969)
* [B11] Thas J.A., Een studie betreffende de projectieve rechte over de totale matrix algebra $M_{3}(K)$ der $3\times 3$–matrices met elementen in een algebraïsch afgesloten veld $K$ (in Dutch), Verh. Kon. Vl. Ac. voor Wet., Lett. en Sch. K. van België, Kl. der Wet., 112, 1–151 (1969)
* [B12] Thas J.A., The $m$-dimensional projective space $S_{m}(M_{n}(GF(q)))$ over the total matrix algebra $M_{n}(GF(q))$ of the $n\times n$-matrices with elements in the Galois field $GF(q)$, Rend. Mat. 4, 459–532 (1971)
* [B13] Thas J.A., Deduction of properties, valid in the projective space $S_{3n-1}(K)$, using the projective plane over the total matrix algebra $M_{n}(K)$, Simon Stevin, 46, 3–16 (1972)
* [B14] Vanhelleputte C., Een studie betreffende de projectieve meetkunde over de ring der $(2\times 2)$-matrices met elementen in een commutatief lichaam, Verhdl. Vlaamse Acad. Wet., Lett. schone Kunsten België, Kl. Wet. (in Dutch) 92, 1–93 (1966)
## C The foundations of plane affine ring geometry
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* [C02] Arnold H-J., Die Geometrie der Ringe im Rahmen allgemeiner affiner Strukturen, Hamburger Mathematische Einzelschriften, Göttingen 4, 86 pp. (1971)
* [C03] Arnold H-J., A way to the geometry of rings, J. Geom. 1, 155–167 (1971)
* [C04] Benz W., Süssche Gruppen in affinen Ebenen mit Nachbarelementen und allgemeineren Strukturen, Abh. Math. Sem. Univ. Hamburg 26, 83–101 (1963)
* [C05] Benz W., $\Omega$–Geometrie und Geometrie von Hjelmslev, Math. Ann. 164, 118–123 (1966)
* [C06] Benz W., Ebene Geometrie über einem Ring, Math. Nachr. 59, 163–193 (1974)
* [C07] Benz W., On Barbilian domains over commutative rings, J. Geom. 12, 146–151 (1979)
* [C08] Burian K., Affine H-structures, Comment. Math. Univ. Carolin. 13, 629–635 (1972)
* [C09] Burian K., Affine parallel $H$–structures (in Czech), Sb. Prací Ped. Fak. v Ostrav$\breve{\rm e}$ Ser. A 9, 3–5 (1974)
* [C10] Burian K., Translation $H$–structures, Sb. Prací Ped. Fak. v Ostrav$\breve{\rm e}$ Ser. A Mat. Fyz. 21, 15–25 (1986)
* [C11] Çelik B., On the hyperbolic Klingenberg plane classes constructed by deleting subplanes, J. Inequal. Appl. 357, 1–6 (2013)
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* [C13] Dorn G., Affine Geometrie über Matrizenringen, Mitt. Math. Sem. Giessen 109, 120 pp. (1974)
* [C14] Eugeni F. and Galiè E., Sui piani costruiti su anelli, Dipartimento M.E.T., Università di Teramo, Italy, 143–162 (1991)
* [C15] Eugeni F. and Maturo A., Generalized affine planes, J. Inform. Optim. Sci. 12, 431–439 (1991)
* [C16] Everett C.J., Affine geometries of vector spaces over rings, Duke Math. J. 9, 873–878 (1942).
* [C17] Groze V. and Vasiu A., Affine structures over an arbitrary ring, Studia Univ. Babes-Bolyai Math. 25, 28–31 (1980)
* [C18] Kaerlein G., Zur Isomorphie vektorieller Gruppen und affiner Liniengeometrien, J. Geom. 16, 1–4 (1981)
* [C19] Keppens D., Affine planes over finite rings, a summary, Aequationes Math. 91, 979–993 (2017)
* [C20] Lantz D., Uniqueness of Barbilian domains, J. Geom. 15, 21–27 (1981)
* [C21] Lawrence P.A., Affine mappings in the geometries of algebras, J. Geom. 2, 115–143 (1972)
* [C22] Leissner W., Affine Barbilian-Ebenen. I, J. Geom. 6, 31–57 (1975)
* [C23] Leissner W., Affine Barbilian-Ebenen. II, J. Geom. 6, 105–129 (1975)
* [C24] Leissner W., Parallelodromie–Ebenen, J. Geom. 8, 117–135 (1976)
* [C25] Leissner W., Barbilianbereiche, in Beiträge zur Geometrischen Algebra, Birkhäuser, Basel–Stuttgart, 219–224 (1977)
* [C26] Leissner W., Rings of stable rank 2 are Barbilian rings, Result. Math. 20, 530–537 (1991)
* [C27] Lorimer J. W., What is a collineation of the integer plane?, Amer. Math. Mon. 103, 687–691 (1996)
* [C28] Lüneburg H., Affine Hjelmslev–Ebenen mit transitiver Translationsgruppe, Math. Z. 79, 260–288 (1962)
* [C29] Machala F., Affine Klingenbergsche strukturen, J. Geom. 11 , 16–34 (1978)
* [C30] Machala F., Desarguessche affine Ebenen mit Homomorphismus, Geom. Dedicata 3, 493–509 (1975)
* [C31] Machala F., Affine planes with homomorphism (in Czech), Kni$\breve{z}$nice, Odb. V$\breve{e}$d. Spisu Vys. U$\breve{c}$. Tech. Brn$\breve{e}$ 56, 79–84 (1975)
* [C32] Pickert G., Taktische Konfigurationen über Quasiringen, Aequationes Math. 58, 31–40 (1999)
* [C33] Radó F., Affine Barbilian structures, J. Geom. 14, 75–102 (1980)
* [C34] Schleicher R., Die Eindeutigkeit der Koordinatisierung von affinen Liniengeometrien durch freie Moduln, J. Geom. 22, 143–148 (1984)
* [C35] Schmidt S. and Steinitz R., The coordinatization of affine planes by rings, Geom. Dedicata 62, 299–317 (1996)
* [C36] Seier W., Der kleine Satz von Desargues in affinen Hjelmslev-Ebenen, Geom. Dedicata 3, 215–219 (1974)
* [C37] Seier W., Über Translationen in affinen Hjelmslev–Ebenen, Abh. Math. Sem. Univ. Hamburg 43, 224–228 (1975)
* [C38] Seier W., Die Quasitranslationen desarguesscher affiner Hjelmslev-Ebenen, Math. Z. 177, 181–186 (1981)
* [C39] Seier W., Streckungstransitive affine Hjelmslev–Ebenen, Geom. Dedicata 11, 329–336 (1981)
* [C40] Seier W., Eine Bemerkung zum grossen Satz von Desargues in affinen Hjelmslev–Ebenen, J. Geom. 20, 181–191 (1983)
* [C41] Sperner E., Affine Räume mit schwacher Inzidenz und zugehörige algebraische Strukturen, J. Reine und angewandte Math. 204, 205–215 (1960)
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* [C43] Vasiu A., The coordinatisation of a class of $\mathcal{B}$–structures (in Romanian), Stud. Univ. Babes-Bolyai, Math. 31, 35–40 (1986)
* [C44] Vasiu A., On a class of planes with neighbouring elements, Prepr., Babes-Bolyai Univ., Fac. Math., Res. Semin. 10, 59–70 (1986)
## D Metric geometry over rings
* [D01] Bachmann F., Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd ed.Die Grundlehren der mathematischen Wissenschaften, Band 96. Springer-Verlag, Berlin-New York, 374 pp. (1973)
* [D02] Bachmann F., Hjelmslev–Gruppen, Neudruck. Mit einem Kapitel und einem Anhang von R. Stölting. Arbeitsgemeinschaft über geometrische Fragen, Universität Kiel, Kiel, 175 pp. (1974)
* [D03] Bachmann F., Eine neuere Entwicklung in der ebenen metrischen Geometrie, Ber. Math.-Statist. Sekt. Forsch. Graz 92–95, 22 pp. (1978)
* [D04] Bachmann F., Ebene Spiegelungsgeometrie. Eine Vorlesung über Hjelmslev-Gruppen, Bibliographisches Institut, Mannheim, 340 pp. (1989)
* [D05] Fischbach G., Ein Darstellungssatz für Translations-Hjelmslev-Ebenen mit Spiegelungen, J. Geom. 46, 45–54 (1993)
* [D06] Hjelmslev J., Einleitung in die allgemeine Kongruenzlehre, 1. Mitt. Danske Vid. Selsk. mat.-fys. Medd. 8 (1929); 2. Mitt. 10 (1929); 3. Mitt. 19 (1942); 4. und 5. Mitt. 22 (1945); 6. Mitt. 25 (1949).
* [D07] Knüppel F., Äquiforme Ebenen über kommutativen Ringen und singuläre Prä-Hjelmslev-Gruppen, Abh. Math. Semin. Univ. Hamburg 53, 229–257 (1983)
* [D08] Knüppel F. and Kunze M., Neighbor relation and neighbor homomorphism of Hjelmslev groups, Canad. J. Math. 31, 680–699 (1979)
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* [D10] Knüppel F. and Salow E., Plane elliptic geometry over rings, Pacific J. Math. 123, 337–384 (1986)
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* [D12] Lingenberg R., Metric planes and metric vector spaces. Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 209 pp. (1979)
* [D13] Nolte W., Minkowskische Hjelmslevgruppen über lokalen Ringen, Result. Math. 12, 376–385 (1987)
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* [D21] Stölting R., Über endliche Hjelmslev-Gruppen, Math. Z. 135, 249–255 (1973/74)
* [D22] Stölting R., Ebene metrische Geometrien über projektiven Moduln, Abh. Math. Sem. Univ. Hamburg 50, 166–177 (1980)
* [D23] Struve H. and Struve R., Hjelmslevgruppen, in denen sich die Punkte gegen Geraden austauschen lassen, Geom. Dedicata 13, 399–417 (1983)
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* [D25] Struve H. and Struve R., Coeuklidische Hjelmslevgruppen, J. Geom. 34, 181–186 (1989)
* [D26] Struve R., Algebraisierung singulärer Hjelmslevgruppen, Geom. Dedicata 13, 309–323 (1982)
* [D27] von Benda H. and Knüppel F., Hjelmslev-Gruppen über lokalen Ringen, Geom. Dedicata 5, 195–206 (1976)
## E The florescence of Hjelmslev geometry
* [E01] Artmann B., Hjelmslev planes derived from modular lattices, Canad. J. Math. 21, 76–83 (1969)
* [E02] Artmann B., Hjelmslev–Ebenen mit verfeinerten Nachbarschaftsrelationen, Math. Z. 112, 163–180 (1969)
* [E03] Artmann B., Uniforme Hjelmslev–Ebenen und modulare Verbände, Math. Z. 111, 15–45 (1969)
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* [E05] Artmann B., Existenz und projektive Limiten von Hjelmslev–Ebenen $n$-ter Stufe, Atti Convegno Geom. combinat. Appl. Perugia, 27–41 (1971)
* [E06] Artmann B., Desarguessche Hjelmslev-Ebenen $n$–ter Stufe, Mitt. Math. Sem. Giessen 91, 1–19 (1971)
* [E07] Artmann B., Geometric aspects of primary lattices, Pacific J. Math. 43, 15–25 (1972)
* [E08] Bacon P., Strongly $n$–uniform and level $n$ Hjelmslev planes, Math. Z. 127, 1–9 (1972)
* [E09] Bacon P., On the extension of projectively uniform affine Hjelmslev planes, Abh. Math. Semin. Univ. Hamburg 41, 185–189 (1974)
* [E10] Brungs H. and Törner G., Embedding right chain rings in chain rings, Canad. J. Math. 30, 1079–1086 (1978)
* [E11] Craig R., Extensions of finite projective planes. I. Uniform Hjelmslev planes, Canad. J. Math. 16, 261–266 (1964)
* [E12] Cronheim A., Cartesian groups, formal power series and Hjelmslev planes, Arch. Math. 27, 209–220 (1976)
* [E13] Cronheim A., Dual numbers, Witt vectors, and Hjelmslev planes, Geom. Dedicata 7, 287–302 (1978)
* [E14] Drake D.A., Projective extensions of uniform affine Hjelmslev planes, Math. Z. 105, 196–207 (1968)
* [E15] Drake D.A., On $n$–uniform Hjelmslev planes, J. Comb. Theory Ser. A 9, 267–288 (1970)
* [E16] Drake D.A., The translation groups of $n$–uniform translation Hjelmlev planes, Pacific J. Math. 38, 365–375 (1971)
* [E17] Drake D.A., The structure of $n$–uniform translation Hjelmslev planes, Trans. Amer. Math. Soc. 175, 249–282 (1973)
* [E18] Drake D.A., Near affine Hjelmslev planes, J. Comb. Theory Ser. A 16, 34–50 (1974)
* [E19] Drake D.A., Existence of parallelisms and projective extensions for strongly $n$–uniform near affine Hjelmslev planes, Geom. Dedicata 3, 191–214 (1974)
* [E20] Drake D.A., Affine Hjelmslev-Ebenen mit verfeinerten Nachbarschaftsrelationen, Math. Z. 143, 15–25 (1975)
* [E21] Drake D.A., All $n$–uniform quasitranslation Hjelmslev planes are strongly $n$–uniform, Proc. Amer. Math. Soc. 51, 494–498 (1975)
* [E22] Drake D.A., Squeezing the accordion in a strongly $n$–uniform Hjelmslev plane, Math. Z. 185, 151–166 (1984)
* [E23] Dugas M., Eine Kennzeichnung der endlichen desarguesschen Hjelmslev-Ebenen, Geom. Dedicata 3, 295–324 (1974)
* [E24] Dugas M., Der Zusammenhang zwischen Hjelmslev-Ebenen und H-Verbänden, Geom. Dedicata 3, 295–324 (1974)
* [E25] Kleinfeld E., Finite Hjelmslev planes, Illinois J. Math. 3, 403–407 (1959)
* [E26] Lorimer J.W., The fundamental theorem of desarguesian affine Hjelmslev planes, Mitt. Math. Sem. Giessen 119, 6–14 (1975)
* [E27] Lorimer J.W., Morphisms and the fundamental theorem of affine Hjelmslev planes, Mathematical Report 64, McMaster University, Hamilton, Ontario, Canada (1973)
* [E28] Lorimer J. W., Structure theorems for commutative Hjelmslev rings with nilpotent radicals, C. R. Math. Rep. Acad. Sci. Canada 6, 123–127 (1984)
* [E29] Lorimer J. W., Affine Hjelmslev rings and planes, Annals of Discr. Math. 37, 265–276 (1988)
* [E30] Lorimer J.W. and Lane N.D., Morphisms of affine Hjelmslev planes, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 56, 880–885 (1974)
* [E31] Lorimer J.W. and Lane N.D., Desarguesian affine Hjelmslev planes, J. für die reine und angew. Math. 1, 336–352 (1975)
* [E32] Machala F., Über projektive Erweiterung affiner Klingenbergscher Ebenen, Czech. Math. J. 29, 116–129 (1979)
* [E33] Machala F., Über eine Klasse affiner Klingenbergscher Ebenen, die projektiv erweiterbar sind, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 19, 65–74 (1980)
* [E34] Seier W., Zentrale und axiale Kollineationen in projektiven Hjelmslev–Ebenen, J. Geom. 17, 35–45 (1981)
* [E35] Skornjakov, L. A., Rings chain–like from the left (in Russian) Izv. Vyssh. Uchebn. Zaved. Matematika 4, 114–117 (1966)
* [E36] Törner G., Eine klassifizierung von Hjelmslev–ringen und Hjelmslev–Ebenen, Mitt. Math. Sem. Giessen, 107, 1–77 (1974)
* [E37] Törner G., Über Homomorphismen projektiver Hjelmslev–Ebenen, J. Geom. 5, 1–13 (1974)
* [E38] Törner G., Homomorphismen von affinen Hjelmslev–Ebenen, Math. Z. 141, 159–167 (1975)
* [E39] Törner G., $n$-uniforme projektive Hjelmslev-Ebenen sind stark $n$-uniform, Geom. Dedicata 6, 291–295 (1977)
* [E40] Törner G., Über den Stufenaufbau von Hjelmslev–Ebenen, Mitt. Math. Sem. Giessen, 126, 1–43 (1977)
* [E41] Törner G., $(r^{n-1},r)$ Hjelmslev–Ebenen des Typs $n$, Math. Z. 154, 189–201 (1977)
* [E42] Törner G., Über ein Problem von Klingenberg, Arch. Math. 28, 253–254 (1977)
* [E43] Törner G., Faktorisierungen von Epimorphismen projektiver Ebenen, Geom. Dedicata 18, 281–291 (1985)
## F The continuation of the Hjelmslev epoch
* [F01] Al-Khamees Y., The enumeration of finite principal completely primary rings, Abh. Math. Sem. Univ. Hamburg 51, 226–231 (1981)
* [F02] Civolani N., Hyperfree extensions of partial Klingenberg planes, Geom. Dedicata 9, 467–475 (1980)
* [F03] Clark W.E. and Liang J., Enumeration of finite commutative chain rings, J. Algebra 27, 445–453 (1973)
* [F04] Clark W.E. and Drake D.A., Finite chain rings, Abh. Math. Sem. Univ. Hamburg 39, 147–153 (1973)
* [F05] Drake D.A., Nonexistence results for finite Hjelmslev planes, Abh. Math. Sem. Univ. Hamburg 40, 100–110 (1974)
* [F06] Drake D.A., More new integer pairs for finite Hjelmslev planes, Illinois J. Math. 19, 618–627 (1975)
* [F07] Drake D.A., Charakterisierungen der Hjelmslev-Ebenen mit Invarianten (4,2), Arch. Math. 27, 436–440 (1976)
* [F08] Drake D.A., Constructions of Hjelmslev planes, J. Geom. 10, 179–193 (1977)
* [F09] Drake D.A., The use of auxiliary sets of matrices in the construction of Hjelmslev and Klingenberg structures, Lecture Notes in Pure and Appl. Math. 82, 129–153 (1983)
* [F10] Drake D.A. and Hale M., Group constructible $(t,k)$–nets and Hjelmslev planes, J. Algebra 48, 301–331 (1977)
* [F11] Drake D.A. and Jungnickel D., Klingenberg structures and partial designs I: Congruence relations and solutions, J. Stat. Plann. Inference 1, 265–287 (1977)
* [F12] Drake D.A. and Jungnickel D., Klingenberg structures and partial designs. II: Regularity and uniformity, Pacific J. Math. 76, 389–415 (1978)
* [F13] Drake D.A. and Jungnickel D., Das Existenzproblem für projektive (8,5)-Hjelmslevebenen, Abh. Math. Sem. Univ. Hamburg 50, 118–126 (1980)
* [F14] Drake D.A. and Jungnickel D., Finite Hjelmslev planes and Klingenberg epimorphisms, in: Rings and geometry, Proc. NATO Adv. Study Inst., Istanbul/Turkey 1984, NATO ASI Ser., Ser. C 160, 153–231 (1985)
* [F15] Drake D.A. and Lenz H., Finite Klingenberg planes, Abh. Math. Sem. Univ. Hamburg 44, 70–83 (1975)
* [F16] Drake D.A. and Lenz H., Finite Hjelmslev planes with new integer invariants, Bull. Amer. Math. Soc. 82, 265–267 (1976)
* [F17] Drake D.A. and Sane S., Maximal intersecting families of finite sets and $n$–uniform Hjelmslev planes, Proc. Amer. Math. Soc. 86, 358–362 (1982)
* [F18] Drake D.A. and Sane S., Auxiliary sets of matrices with new step parameter sequences, Linear Algebra Appl. 46, 131–153 (1982)
* [F19] Drake D.A. and Shult E., Construction of Hjelmslev planes from $(t,k)$–nets, Geom. Dedicata 5, 377–392 (1976)
* [F20] Drake D.A. and Törner G., Die Invarianten einer Klasse projektiver Hjelmslev–Ebenen, J. Geom. 7, 157–174 (1976)
* [F21] Hale M. and Jungnickel D., A generalization of Singer’s theorem, Proc. Amer. Math. Soc. 71, 280–284 (1978)
* [F22] Jungnickel D., Verallgemeinerte Klingenberg–Ebenen, Mitt. Math. Sem. Giessen 120, 1–10 (1976)
* [F23] Jungnickel D., Hjelmslevebenen mit regulärer abelscher Kollineationsgruppe, Beiträge zur geometrischen Algebra (Proc. Sympos. Duisburg 1977), 157–165 (1977)
* [F24] Jungnickel D., Regular Hjelmslev planes. II, Trans. Amer. Math. Soc. 241, 321–330 (1978)
* [F25] Jungnickel D., On the congruence relations of regular Klingenberg structures, J. Combin. Inform. System Sci. 3, 49–57 (1978)
* [F26] Jungnickel D., Regular Hjelmslev planes, J. Comb. Theory Ser. A 26, 20–37 (1979)
* [F27] Jungnickel D., On an assertion of Dembowski, J. Geom. 12, 168–174 (1979)
* [F28] Jungnickel D., Construction of regular proper CK–planes, J. Combin. Inform. System Sci. 4, 14–18 (1979)
* [F29] Jungnickel D., On balanced regular Hjelmslev planes, Geom. Dedicata 8, 445–462 (1979)
* [F30] Jungnickel D., Some new combinatorial results on finite Klingenberg structures, Utilitas Math. 16, 249–269 (1979)
* [F31] Jungnickel D., A class of uniform Klingenberg matrices, Ars Combin. 10, 91–94 (1980)
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* [F34] Sane S., Some new invariant pairs $(t,3)$ for projective Hjelmslev planes, J. Geom. 15, 64–73 (1981)
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* [F39] Sedlar V., Incidence matrices of finite projective uniform Hjelmslev planes (in Czech) Sb. Praci Ped. Fak. v Ostrave Ser. A 17, 25–38 (1982)
## G The coordinatization of Klingenberg and Hjelmslev planes
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* [G06] Baker, C., Moulton affine Hjelmslev planes Canad. Math. Bull. 21, 135–142 (1978)
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* [G08] Baker, C., Lane N. and Lorimer J.W., An affine characterization of Moufang projective Klingenberg planes, Results Math. 17, 27–36 (1990)
* [G09] Baker, C., Lane N. and Lorimer J.W., The Artin-Zorn theorem for finite punctually cohesive projective Klingenberg planes, Ars Comb. Ser. B 29, 143–149 (1990)
* [G10] Baker, C., Lane N. and Lorimer J.W., A coordinatization for Moufang Klingenberg planes, Simon Stevin 65, 3–22 (1991)
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* [G50] Machala F., Projektive Ebenen mit Homomorphismus und erweiterte lokale Ternärringe, Math. Slovaca 29, 227–237 (1979)
* [G51] Machala F., Epimorphismen von lokalen Ternärringen, Czech. Math. J. 33, 70–75 (1983)
* [G52] Machala F., Biternärringe und affine lokale Ternärringe, Acta. Univ. Palack. Olomuc. Fac. Rerum. Natur. Math. 27 , 25–37 (1988)
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## H Order and topology in ring geometry
* [H01] Baker C., Ordered affine Hjelmslev planes, J. Geom. 23, 1–13 (1984)
* [H02] Baker C., Preordered uniform Hjelmslev planes, J. Geom. 24, 14–17 (1985)
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* [H07] Lorimer J.W., Topological Hjelmslev planes, Geom. Dedicata 7, 185–207 (1978)
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## I The revival of ring geometry (Veldkamp, Faulkner)
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## J Projective and affine Hjelmslev spaces and spaces over rings
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## K Projective lines and circle geometries over rings
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## L Ring geometries and buildings
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## M Ring geometries and coding theory
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|
2024-09-04T02:54:57.666922 | 2020-03-05T19:57:48 | 2003.02895 | {
"authors": "Monica Alexander, Kivan Polimis, Emilio Zagheni",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26067",
"submitter": "Monica Alexander",
"url": "https://arxiv.org/abs/2003.02895"
} | arxiv-papers | # Combining social media and survey data to nowcast migrant stocks in the
United States
Monica Alexander111University of Toronto<EMAIL_ADDRESS>
Kivan Polimis222University of Washington
Emilio Zagheni333Max Planck Institute for Demographic Research
###### Abstract
Measuring and forecasting migration patterns, and how they change over time,
has important implications for understanding broader population trends, for
designing policy effectively and for allocating resources. However, data on
migration and mobility are often lacking, and those that do exist are not
available in a timely manner. Social media data offer new opportunities to
provide more up-to-date demographic estimates and to complement more
traditional data sources. Facebook, for example, can be thought of as a large
digital census that is regularly updated. However, its users are not
representative of the underlying population. This paper proposes a statistical
framework to combine social media data with traditional survey data to produce
timely ‘nowcasts’ of migrant stocks by state in the United States. The model
incorporates bias adjustment of the Facebook data, and a pooled principal
component time series approach, to account for correlations across age, time
and space. We illustrate the results for migrants from Mexico, India and
Germany, and show that the model outperforms alternatives that rely solely on
either social media or survey data.
## 1 Introduction
Accurate, reliable and timely estimates of migration indicators, such as flows
and stocks, are crucial for understanding population dynamics and demographic
change, for designing effective economic, social and health policies, and for
supporting migrants and their families. However, data on migration from
traditional sources, such as censuses, surveys or administrative registers,
are often insufficient. Even when these sources exist, the data available may
lack the granularity of information required to understand migration trends,
or are not released in a manner that is timely enough to monitor changes in
trends. As migration flows can change substantially over a short period of
time — such as in response to a natural disaster or war and conflict — relying
on out-dated data is often not sufficient.
Timely and reliable information about migration stocks is important not only
to understand migration patterns. It is key also to monitor fertility,
population health and mortality. Even when accurate data on births and deaths
exist, often demographers are faced with large uncertainty in population
counts, which, in various disaggregations, form the denominators for standard
demographic rates. Most of the uncertainty is driven by lack of appropriate
information on how migration stocks change over time and space.
As a consequence of data availability issues, we need to consider how non-
traditional data can be leveraged to complement existing sources in order to
improve estimates and predictions of migration indicators over time. Previous
work has explored the use of data such as call detail records (Blumenstock,
2012; Pestre et al., 2020), air traffic data (Gabrielli et al., 2019), tax
file records (Engels and Healy, 1981) and other sources like billing addresses
or school enrollment (Foulkes and Newbold, 2008) to estimate migration.
Additionally, an increasingly large body of work has investigated the use of
social media data, from websites such as Twitter (Zagheni et al., 2014),
Facebook (Zagheni et al., 2017) and LinkedIn (State et al., 2014). Provided
that the data can be obtained in a reliable, timely, and ethical way,
information about the users of social media websites is potentially an
incredibly rich demographic data source. Data on these populations are
essentially collected in real time, and while individual-level information is
usually restricted, many of the social media websites provide a certain amount
of aggregate-level information through their advertising platforms (Cesare et
al., 2018). In particular, Facebook’s Advertising platform allows information
to be extracted on the relative size of groups by key demographics such as
age, sex, location of residence and country of origin, thereby potentially
acting as a measure of the relative size of migrant groups in a particular
country.
While these data have clear potential for use in demographic research, with
respect to timeliness and the size of the sample being considered, there are
some notable issues that need to be overcome. In particular, for any given
population subgroup of interest, the corresponding users of Facebook or any
other social media platform are unlikely to be a representative sample. An
additional challenge is to use these data in a way that meaningfully combines
new ‘signal’ or information about migration trends with existing knowledge on
probable migration trends from historical data sources.
This paper proposes a statistical framework to combine social media data from
Facebook, with traditional survey data from the ACS, in order to produce
timely ‘nowcasts’ of migrant stocks by state in the United States. The
framework consists of a Bayesian hierarchical model which incorporates bias
adjustment of the Facebook data, a demographic time series approach to account
for historical past trends, and a geographic pooling component which allows
information about the age structure of migrants to be shared across space. The
model also accounts for the different types of uncertainty that are likely to
be present in Facebook and traditional survey data. The resulting model
produces estimates and short-term projections of migrant stocks by US state of
destination and country of origin, and is shown to out-perform valid modeling
alternatives.
The remainder of the paper is structured as follows. First, we briefly discuss
previous demographic research which incorporates social media. Then we outline
the data sources used, and in particular how the Facebook data were collected.
Section 4 discusses the model set-up, assumptions and computation. We then
present results for Mexican, Indian and German migrants by US state, and
validate model performance against reasonable alternatives. Finally, the
strengths and limitations of the model are discussed, together with avenues
for future research.
## 2 Background
The lack of good-quality data on migration is a global problem, with data
sparsity issues prevalent in both developed and developing countries (Landau
and Achiume, 2017). This has prompted scholars to investigate the use of other
types of data to monitor migration trends. In particular, with the rise of
social media use around the world, new data that have potential for
demographic research have emerged.
Scholars began using social media and web data to estimate and track
demographic indicators over time in the early 2010s. The earliest papers
illustrated how geo-located data from email services and web-based
applications such as Twitter, Google Latitude, Foursquare or Yahoo! can be
used (Ferrari et al., 2011; Noulas et al., 2011; Zagheni and Weber, 2012).
Initial research focused on evaluating spatial mobility of populations at a
city or regional level. For example, Ferrari et al. (2011) used Twitter data
to study patterns of urban movement in New York. In the first effort to tackle
global trends, Zagheni and Weber (2012) linked the geographic locations of IP
addresses of Yahoo! emails to the user’s self-reported demographic data to
estimate age- and sex-specific migration flows in a large number of countries
around the world.
Recent efforts have focused on using data from social media and networking
websites such as Twitter, LinkedIn and, more recently, Facebook and Instagram.
These websites provide public access to an Application Programming Interface
(API), which makes it possible to send requests and receive responses from
these websites for data like tweet hashtag counts, the number of jobs in a
certain industry, or number of cell phone users in a particular area.
Researchers have utilized these APIs to extract publicly available demographic
and location data for use in social research, in particular to study outcomes
such as migration (Yildiz et al., 2017; Zagheni et al., 2017), fertility
(Rampazzo et al., 2018), gender equality (Fatehkia et al., 2018; Garcia et
al., 2018), and health (Araujo et al., 2017a). For instance, Garcia et al.
(2018) used Facebook data to create an index of the internet gender divide in
217 countries, showing that this indicator encapsulated gender equality
indices in education, health and economic opportunity. Yildiz et al. (2017)
used a combination of geo-located tweets and image recognition software to
obtain estimates of internal migration in England.
In work relevant to this paper, Zagheni et al. (2017) presented a proof of
concept for estimating migration stocks in the United State by age, sex and
state, using Facebook’s Advertising Platform. More recently, Alexander et al.
(2019) used the same type of data to track changes in migrants over time, in
the context of estimating out-migration from Puerto Rico following Hurricane
Maria in September 2017.
The main gap in the literature is related to the lack of a suitable
statistical model for combining ‘traditional’ data sources on migrants — in
the form of censuses, nationally representative surveys, or other vital
statistics — with migration information from social media data. The goal of
this paper is thus to develop a probabilistic framework that allows
representative and historical time series to be combined, in a sound
statistical framework, with non-representative – but timely – sources from
social media.
## 3 Data
### 3.1 Facebook Advertising data
Facebook for Business has developed a targeted advertising platform, called
Ads Manager that provides a graphical user interface to allow advertisers to
micro-target specific audiences. Demographic characteristics that can be
targeted include information directly reported by Facebook users, such as age
or sex, and information indirectly inferred from using Facebook’s platform or
affiliated websites, such as location and behavioral interests. Before
launching an advertisement, an advertiser can select a variety of
characteristics (e.g., Australians living in California, who are female, and
aged 30-35) and get an estimate of the ‘potential reach’ (monthly active
users) to this subgroup. These estimates can be obtained, in a programmatic
way, for a variety of different expatriate (‘expat’) groups by age, sex, and
education.
We use the estimates of potential reach by expat group, age and sex to track
sizes of migration stocks over time. These estimates can be obtained before
the launch of an advertisement, and as such are obtained free of charge. We
use the Ads Manager backend application, Facebook’s Marketing API, to extract
estimates of potential reach over time programmatically with the Python module
pySocialWatcher (Araujo et al., 2017b). With pySocialWatcher, we collected
data across 11 age groups (10 UN age groups from 15-19 to 60-65; an 11th group
for the entire available Facebook population of 13-65 was also used), three
gender groups (female, male, and total population) and multiple education
categories. Data was collected using Amazon Web Services (AWS) EC2 Instance
servers.
As part of a broader project on using social media in demographic research, we
started data collection in January 2017. For each wave of data collection we
obtained state-level estimates of all Facebook users (by age, sex, and gender)
as well as state-level estimates of 90 expat groups. We have been collecting a
new wave of data every 2-3 months. 444The waves used in this paper are: Wave
1: January 2017; Wave 2: April 2017; Wave 3: June 2017; Wave 4: October 2017;
Wave 5: January 2018; Wave 6: March 2018
### 3.2 American Community Survey
The American Community Survey (ACS) is an annual survey of the U.S. Census
Bureau, designed to supplement the decennial census. Based on the long-form
version of the census, the ACS collects information on topics including
population, housing, employment and education from a nationally representative
sample. Data on migrant stocks can be readily obtained from the ACS. In
particular, in every year of the ACS, the survey has contained a question
asking the birthplace of the person; if it is inside the United States, the
state is recorded, and if it is outside the United States, the country is
recorded. This birthplace variable is recorded as a three digit code to
indicate the US state or country of birth. In addition to the birthplace
variable, the ACS has information on current state of residence. Thus, we can
tabulate the number of migrants from a particular country living in a
particular state by looking at the combination of these two variables. From a
modeling perspective, we are interested in the proportion of migrants from a
particular origin of the total population by five-year age groups
($15-19,20-24,\dots,50-54$) in each state.
We calculated the migrant stock proportions using the 1-year ACS for each year
between 2001 and 2017 using micro-data available through the Integrated Public
Use Microdata (IPUMS) US project (Ruggles et al., 2000). Standard errors
around the calculated proportions based on sampling variation were calculated
based on ACS accuracy guidelines (US Census Bureau, 2020) and using the Delta
method.
## 4 Model
We are considering two data sources of migration trends in the US: data from
Facebook’s Advertising Platform, and the ACS. The overall goal of the modeling
strategy is to combine information from both these sources to produce
estimates of current and future migrant stocks. To do this, the model should
have three main characteristics. Firstly, we want to adjust for biases in
Facebook data to effectively use up-to-date information on migration patterns
from this source. Secondly, we want to be able to incorporate longer time
series of information from the ACS. Finally, the data should be combined in a
probabilistic way, in order to objectively weigh information from both
sources. We propose a Bayesian hierarchical model which achieves these goals.
In this section we describe the model in detail.
For a particular migrant group, define $\mu_{xts}$ to be the (‘true’)
proportion of migrants of the total populations in age group $x$ at time $t$
and in state $s$. This quantity $\mu_{xts}$ is the main parameter of interest
to be estimated. We have observations of this proportion, which will be
denoted $p_{xts}$. The observed proportions are either from Facebook
($p_{xts}^{FB}$) or from the ACS ($p^{ACS}_{xts}$). The $p_{xts}$s are
observed, and it is assumed that these are somehow related to the underlying
true proportions, $\mu_{xts}$, with some associated error. We use the term
‘true’ in a statistical sense, referring to a latent variable of interest.
### 4.1 Facebook bias adjustment
The first goal is to adjust the Facebook data to account for the non-
representativeness of the Facebook user population. Previous research from
Zagheni et al. (2017) showed that, while the bias in the Facebook migrant data
is substantial, it is also relatively systematic by age and migrant group and
can be modelled.
Following their approach, we introduce a regression model which relates the
proportions of migrants in Facbook, $p_{xts}^{FB}$, to the proportions in the
ACS in a similar time period, $p_{xts}^{ACS}$, plus a series of age and state
variables. In particular, for a particular migrant group, express
$p_{xts}^{ACS}$, on a log scale, as
$\log p_{xts}^{ACS}=\alpha_{0}+\alpha_{1}\log p_{xts}^{FB}+\mathbf{\beta
X}+\varepsilon_{FB}$ (1)
where $\mathbf{X}$ is a covariate matrix containing an indicator variable for
each age group ($15-19,20-24,\dots,50-54$) and each of the 50 states plus
Washington D.C. This means that we estimate a fixed effect for each age group
and state. In addition, we assume that the error is i.i.d and that
$\varepsilon_{FB}\sim N(0,\sigma^{2}_{FB}).$
Estimates of the coefficients $\alpha_{0}$, $\alpha_{1}$ and the vector of
$\beta$’s are obtained using the first wave of the Facebook data and the 2016
ACS data. Once obtained, these coefficient estimates are then used to adjust
subsequent waves of Facebook data, i.e. we calculate
$\log p_{xts}^{*}=\hat{\alpha}_{0}+\hat{\alpha}_{1}\log
p_{xts}^{FB}+\mathbf{\hat{\beta}X}$
where $\log p_{xts}^{*}$ is a ‘bias-adjusted’ version of the Facebook data.
This is taken to be our ‘best guess’ of what the migrant stocks in group $xts$
are, based on the Facebook data alone. Note that an estimate of
$\sigma^{2}_{FB}$ is also obtained, that is, the variance of the error terms,
which becomes important in the final model (see section 4.3 below).
### 4.2 Time series modeling of ACS using principal components
In addition to using data from Facebook, we also want to incorporate the
relatively long historical time series of information on migrant stocks
obtained from the ACS. A reasonable short term forecast based on ACS should
model historical trends and project them forward.
There are many different time series models that could be used in this
context. Perhaps the simplest approach would be to project forward a moving
average of the time series for each age group and state combination.
Alternatively, we could use a classical Box-Jenkins approach and model the
time series of migrant stocks in each age group and state separately using an
appropriately specified ARIMA model. However, these methods would not place
any constraints on the age structure of migration. Given this demographic
context, we expect that the age distribution of migration displays strong
patterns and changes in a relatively regular way over time. This is because of
regularities in the age at migration as well as historical trends which
include different waves of migrants, who also age over time. As such, we chose
to incorporate this prior knowledge into our model through a principal
components approach.
Principal component-based models have a long history in demographic modeling,
with the most well known example being the Lee-Carter mortality model (Lee and
Carter, 1992). The idea is that a set of age-specific demographic rates
observed over time can be expressed as a combination of a series of
characteristic patterns (or principal components). The Lee-Carter approach
uses the mean age-specific mortality schedule and first principal component,
which is interpreted as age-specific contributions to mortality change over
time. This model can easily be extended to include higher-order principal
components, which various researchers have done.
Apart from the Lee-Carter model and variants (e.g. Li et al. (2004), Lee
(2000), Renshaw and Haberman (2006)), principal component models have been
recently used to estimate and forecast mortality (e.g. Alexander et al.
(2017)), fertility (e.g. Schmertmann et al. (2014)) and overall population
(Wiśniowski et al. (2015)). Here, we extend this idea to parsimoniously
estimate and project migration stocks by age and state.
#### 4.2.1 Model overview
Age-specific migration schedules are decomposed into independent age and time
components. The time component is then projected forward as a time series,
taking auto-correlated error into account. We propose a log-linear model for
$p_{xts}$:
$\log p_{xts}^{ACS}=\beta_{ts,1}Z_{x,1}+\beta_{ts,2}Z_{x,2}+\varepsilon_{xts}$
(2)
where $Z_{x,1}$ and $Z_{x,2}$ are the first and second ‘principal components’,
$\beta_{ts,1}$ and $\beta_{ts,2}$ are state and time-specific coefficients, to
be estimated, and $\varepsilon_{xts}$ is an error term. The principal
components are obtained via Singular Value Decomposition (SVD), as outlined in
the next section. To obtain estimates of $\beta_{ts,1}$ and $\beta_{ts,2}$, we
impose some smoothing over time and pooling of information across space, as
outlined in section 4.2.3. Finally, as discussed in Section 4.2.4 we place a
time series model on the error term, $\varepsilon_{xts}$, to account for auto-
correlation.
#### 4.2.2 Obtaining the principal components
The principal component terms $Z_{x,1}$ and $Z_{x,2}$ aim to capture the main
sources of systematic variation in migration patterns across age. They are
obtained by first creating a matrix of (logged) historical age-specific
migration schedules based on ACS data from 2001 to 2016. Singular Value
Decomposition (SVD) is then performed on this matrix to obtain principal
components of the age-specific migration. In particular, let $\bf{X}$ be a
$N\times G$ matrix of log-migration stock rates, where $N$ is the number of
state-years and $G$ is the number of age-groups. In this case, we had $N=51$
states + DC $\times 16$ years $=816$ observations of $G=9$ age-groups
($15-19,20-24,\dots,50-54$). The SVD of $\bf X$ is
$\bf X=\bf{UDV^{\prime}},$ (3)
where $\bf U$ is a $N\times N$ matrix, $\bf D$ is a $N\times G$ matrix and
$\bf V$ is a $G\times G$ matrix. The first two columns of $\bf V$ (the first
two right-singular values of $\bf X$) are $Z_{x,1}$ and $Z_{x,2}$.
For example, Fig. 1 shows the resulting $Z_{.,1}$ and $Z_{.,2}$ for the
Mexican migrant group in the US. These were obtained via the following steps:
1. 1.
Calculate $p_{xts}^{ACS}$, i.e. the proportion of migrants in age group $x$,
year $t$ and state $s$ for each age group, year and state in ACS 2001-2016.
2. 2.
Create $\bf{X}$ where each element is $\log p_{xts}^{ACS}$, every row is a
state-year and every column is an age group.
3. 3.
Perform SVD on $\bf X$ and extract the first two columns555We used the ‘svd’
function in R. of $\bf V$.
Figure 1: Principal Components for Mexico
The principal components shown in Fig. 1 can be interpreted as a baseline
migration age schedule ($Z_{.,1}$) and age-specific contributions to change
over time ($Z_{.,2}$). In the model, the coefficient on $Z_{.,1}$
($\beta_{ts,1}$) moves the overall level of Mexican migrants up or down,
depending on the year and state. The coefficient on $Z_{.,2}$ allows the age
distribution to shift to older or younger ages. For $Z_{.,2}$, the sign
changes from negative to positive at age 35. This means that the larger and
more positive the value of $\beta_{ts,2}$, the older the migrant age
distribution.666Note that the interpretation of $Z_{.,1}$ and $Z_{.,2}$ is
similar to the interpretation of the $a_{x}$ and $b_{x}$ terms in the usual
Lee-Carter model.
#### 4.2.3 Sharing information across time and space
The model specified in Eqn. 2 requires the estimation of two coefficients,
$\beta_{ts,1}$ and $\beta_{ts,2}$ for each time $t$ and state $s$. One option
would be to estimate each of these coefficients separately for every year and
state. However, we would like to incorporate the knowledge that trends in
migration over time are likely to exhibit relatively regular patterns. In
addition, for the coefficient on the second principal component — which allows
for the age distribution of migrants to shift to the left or right, we would
like to share information about the patterns in migration across geographic
space.
The coefficient on the first principal component, $\beta_{ts,1}$, is modeled
as a random walk, i.e.
$\beta_{ts,1}\sim N(\beta_{t-1s,1},\sigma_{\beta_{1}}^{2})$
This allows for information about the level of migration within each state to
be smoothed over time. The random walk structure allows for the estimate in
the current time period, $\beta_{ts,1}$, to be partially informed by the
previous period.
For the coefficient on the second principal component, we place the following
hierarchical structure on the $\beta$’s:
$\beta_{ts,2}\sim N(\Phi_{t},\sigma_{\beta}^{2})$ $\Phi_{t}\sim
N(\Phi_{t-1},\sigma_{\Phi}^{2})$
where $p=1,2$. The $\Phi_{t}$ term represents essentially a national mean; as
such the $\beta_{ts,2}$’s are a draw from a national distribution with some
mean and variance. In this way, information about how the age distribution is
ageing over time is shared across states. The more information about migration
there is available for a particular state (i.e., the larger the migrant
population), the less the estimate of $\beta_{ts,2}$ is influenced by the
overall mean. Conversely, states with smaller migrant populations where the
trends over time are less clear from the data are partially informed by
patterns in larger states.
Note that the geographical hierarchical structure is not present on the first
coefficient, as this represents an overall level of migration. Pooling
information across space about the level of migration would artificially
increase migrant proportions in smaller states.
#### 4.2.4 Auto-correlated error
The final piece of the time series model is the error term
$\varepsilon_{xts}$. This term is included in the model to allow for extra
variation in migration age schedules that is not otherwise picked up by the
principal components. We expect the extra variation to be autocorrelated, and
as such we model the error term as an AR(1) process:
$\varepsilon_{xts}\sim
N(\rho_{xs}\varepsilon_{xt-1s},\sigma_{\varepsilon}^{2})$
where $\rho_{xs}\in[0,1]$.
#### 4.2.5 Projection
The model described above is fit to ACS data from 2001-2016. However,
estimates in more recent years can easily be obtained by projecting the time
series aspects of this model forward. In particular, for time $t+1$:
* •
Obtain an estimate for $\beta_{t+1s,1}$ from $\beta_{t+1s,1}\sim
N(\beta_{ts,1},\sigma^{2}_{\beta})$.
* •
Obtain an estimate for $\beta_{t+1s,2}$ from $\beta_{t+1s}\sim
N(\Phi_{t+1},\sigma^{2}_{\Phi})$ and $\Phi_{t+1}\sim
N(\Phi_{t},\sigma_{\Phi}^{2})$.
* •
Obtain an estimate for $\varepsilon_{xt+1s}$ from $\varepsilon_{xt+1s}\sim
N(\rho_{xs}\varepsilon_{xts},\sigma_{\varepsilon}^{2}).$
* •
Calculate $\log p_{xt+1s}^{ACS}$ based on Eqn. 2.
### 4.3 Bringing it all together
Sections 4.1 and 4.2 described two ways to obtain current ‘nowcasts’ of
migrant stocks. One option would be to take the most recent data obtained from
Facebook, adjust using the bias-adjustment model, and take the resulting
estimate as our nowcast. Another option would be to project forward the ACS
model to the time period of interest. Ideally, we would like to incorporate
both sources into our final estimate. Perhaps an option would be to just take
an average of the two resulting estimates. However, we would like to weigh the
estimates from both sources more objectively, taking different sorts of
uncertainty into consideration.
Our solution is to combine both models into one framework, and use the results
from both methods as data points for our ‘best estimate’ nowcast. This is
illustrated in Fig. 2. Facebook inputs are calibrated with the ACS via the
adjustment model. ACS data are used to obtained principal components based on
past migration data. The modeling structure allows for information exchange
over time and across geographic space. The key piece of the combined model,
which has yet to be explained, is the data model (or likelihood), which allows
data from the different sources to have different associated error.
Figure 2: Modeling framework
#### 4.3.1 Data model
As outlined above, we observe migrant proportions $p_{xts}$ from either
Facebook or the ACS. The data model assumes
$\log p_{xts}\sim N(\log\mu_{xts},\sigma_{p}^{2})$
i.e. the log of the observed proportion is assumed to have mean
$\log\mu_{xts}$ and variance $\sigma_{p}^{2}$, where $\sigma_{p}^{2}$ depends
on the data source:
$\sigma_{p}^{2}=\begin{cases}\sigma_{s}^{2},&\text{if ACS}\\\
\sigma_{s}^{2}+\sigma_{FB}^{2}+\sigma_{ns}^{2},&\text{if
Facebook}\end{cases}\\\ $
Here, $\sigma^{2}_{s}$ refers to sampling error, and is assumed to be present
in both ACS and Facebook data. For the ACS data, sampling errors are
calculated based on guidelines from the US Census Bureau (2020). For Facebook
data, the sampling error is calculated assuming the binomial approximation to
the Normal distribution and calculating
$\sigma^{2}_{s}=\frac{p_{xts}\cdot(1-p_{xts})}{N_{xts}^{FB}}$
where $N_{xts}^{FB}$ is the total size of the Facebook population in subgroup
$x,t,s$.
For the Facebook data there are two additional error terms. $\sigma_{FB}^{2}$
refers to the error associated with our bias-adjustment model (Eqn. 1) and is
estimated within this model. This captures the fact that our adjustment model
is imperfect and that extra variation remains. Additionally, we allow for a
non-sampling error with $\sigma_{ns}^{2}$, which aims at capturing additional
uncertainty like variation in the way potential reach is estimated across
waves.
For a given population size, the sampling error is going to be of similar size
for ACS and Facebook data. As such, the error term associated with the
Facebook data, which is the sum of three terms, will always be bigger than for
ACS. In practice, this means that estimates from the model will follow (i.e.
give more weight to) the ACS data.
#### 4.3.2 Summary of full model
The full model is summarized below. Equation 4 is the data model. Equations
5-9 relate to the ACS time series model. Equation 12 is related to the
Facebook regression model. Equations 10 and 11 allow the observation of the
proportion of interest to come from a different source (Facebook or ACS),
which has a different associated variance. Note that $\mu_{xts}$ is estimated
on a yearly basis, but it is assumed that $j$ waves of Facebook data are
collected within any one year.
$\displaystyle\log p_{xts}$ $\displaystyle\sim$ $\displaystyle
N(\log\mu_{xts},\sigma^{2})$ (4) $\displaystyle\log\mu_{xts}$ $\displaystyle=$
$\displaystyle\beta_{ts,1}Z_{x,1}+\beta_{ts,2}Z_{x,2}+\varepsilon_{xts}$ (5)
$\displaystyle\beta_{ts,1}$ $\displaystyle\sim$ $\displaystyle
N(\beta_{t-1s,1},\sigma_{\beta_{1}}^{2})$ (6) $\displaystyle\beta_{ts,2}$
$\displaystyle\sim$ $\displaystyle N(\Phi_{t,2},\sigma_{\beta}^{2})$ (7)
$\displaystyle\Phi_{t}$ $\displaystyle\sim$ $\displaystyle
N(\Phi_{t-1},\sigma_{\Phi}^{2})$ (8) $\displaystyle\varepsilon_{xts}$
$\displaystyle\sim$ $\displaystyle
N(\rho_{xs}\varepsilon_{xt-1s},\sigma_{\varepsilon}^{2})$ (9) $\displaystyle
p_{xts}$ $\displaystyle=$ $\displaystyle\begin{cases}p_{xts}^{ACS},&\text{if
}2001\leq t\leq 2016\\\ p_{xtsj}^{*},&\text{if }t\geq 2017\end{cases}$ (10)
$\displaystyle\sigma^{2}$ $\displaystyle=$
$\displaystyle\begin{cases}\sigma_{s}^{2},&\text{if ACS}\\\
\sigma_{s}^{2}+\sigma_{FB}^{2}+\sigma_{ns}^{2},&\text{if Facebook}\\\
\end{cases}$ (11) $\displaystyle p_{xtsj}^{*}$ $\displaystyle\sim$
$\displaystyle N({\alpha_{0}}+{\alpha_{1}}\cdot p_{xtsj}^{\text{
Facebook}}+X{\Gamma},\sigma^{2}_{FB})$ (12)
#### 4.3.3 Priors
Weakly-informative priors were placed on the coefficients in the Facebook
bias-adjustment model, as well as the principal component coefficients in the
initial periods:
$\displaystyle\alpha_{0}$ $\displaystyle\sim$ $\displaystyle N(0,100)$
$\displaystyle\alpha_{1}$ $\displaystyle\sim$ $\displaystyle N(0,100)$
$\displaystyle\Gamma_{0}$ $\displaystyle\sim$ $\displaystyle N(0,100)$
$\displaystyle\beta_{1,s,1}$ $\displaystyle\sim$ $\displaystyle N(0,100)$
$\displaystyle\Phi_{1}$ $\displaystyle\sim$ $\displaystyle N(0,100)$
In addition, we put weakly-informative half-Normal priors on the two standard
deviation terms to be estimated:
$\displaystyle\sigma_{FB}$ $\displaystyle\sim$ $\displaystyle N_{+}(0,1)$
$\displaystyle\sigma_{ns}$ $\displaystyle\sim$ $\displaystyle N_{+}(0,1).$
#### 4.3.4 Computation
The model was fitted in a Bayesian framework using the statistical software R.
Samples were taken from the posterior distributions of the parameters via a
Markov Chain Monte Carlo (MCMC) algorithm. This was performed using JAGS
software (Plummer et al., 2003). Standard diagnostic checks using trace plots
and the Gelman and Rubin diagnostic were used to check convergence (Gelman et
al., 2013).
Best estimates of all parameters of interest were taken to be the median of
the relevant posterior samples. The 95% Bayesian credible intervals were
calculated by finding the 2.5% and 97.5% quantiles of the posterior samples.
All code and data are available on GitHub: https://github.com/MJAlexander/fb-
migration-bayes
## 5 Results
We illustrate the model on male migrants from three different countries:
Mexico, India, and Germany. These three migrant groups represent three
different scenarios of levels and trends over time, as illustrated by the
trends in the ACS data shown in Figures 3(a) and 3(b).
Firstly, Mexican migrants make up a relatively large share of the overall
population, but the proportion has been generally declining since around 2007.
The age distribution at the national level peaks in the 40-44 year old age
group. Secondly, Indian migrants make up a moderate proportion of the total
population, but this share is increasing over time. The age distribution peaks
at younger ages (30-34), compared to Mexicans. Finally, German migrants make
up a low and declining share of the population. In contrast to the other
migrant groups, the age distribution of German migrants at the national level
is relatively flat, increasing slightly across age.
(a) By proportion of the total population
(b) By proportion of each age group, 2016
Figure 3: German, Indian and Mexican migrants
### 5.1 Bias adjustment of Facebook data
We firstly illustrate the results of the bias-adjustment step of the Facebook
data. Figure 4 shows, for each US state and five-year age group where data are
available, the proportion of migrants in each age group for the ACS data in
2016 (black dots), the un-adjusted Facebook data (blue dots), and the
estimated bias-adjusted Facebook data (red line and associated shaded area)
for Mexican migrants. Similar plots for migrants from India and Germany are
shown in Appendix A. The interpretation is that if the bias-adjustment step is
working reasonably well, the red line would be close to the black dots. In
general, this appears to be the case.
For all three migrant groups, the raw Facebook data are generally lower than
the ACS data, but the bias-adjustment model adjusts these values upwards. In
general, across the three migrant groups and across states, the shape of the
age distributions in the Facebook and ACS data are similar, with more
substantial under-representation in Facebook in the older age groups. These
systematic differences mean that the model works well to adjust the raw
Facebook data based on age and state effects.
Figure 4: Bias adjustment of Facebook data for Mexican migrants
### 5.2 Nowcasts by age group and state
Now we move on to short term projections by age and state. Figure 5 shows the
estimated age distribution in 2008 (red) and projected distribution in 2018
(blue) for Mexican migrants. Similar plots for India and Germany can be found
in Appendix LABEL:appendix_age_facets.
For Mexico (Figure 5), the relatively high proportions in the border states
and on the West coast are apparent, with the highest proportions in
California, Texas, Nevada and Arizona. Additionally, the age distribution of
Mexican migrants is generally aging over time (shifting to the right), which
is consistent with relatively constant stocks.
Figure 5: Estimated and projected age distributions of Mexican migrants by
state, 2008 and 2018
#### 5.2.1 Projected time series
Figures 6 to 8 zoom in on two states for each migrant group and show how the
Facebook data are used to project forward the time series to the most recent
two years (2017 and 2018). The full estimated and projected time series from
2001 to 2018 is shown. In the figures, each facet is a five-year age group.
The red dots and associated shaded area represent the ACS data and sampling
standard errors; these data are broadly available from 2001 to 2016, although
some observations are missing (if sample sizes in the ACS were too small to
capture information about migrants in that particular state and age group).
The blue dots represent the (adjusted) Facebook observations, which are
available in years 2017 and 2018. The black line and associated shaded area is
the model estimate and 95% uncertainty intervals.
Mexican males in California (Figure 6(a)) represent by far the highest
proportions of any of the migrant origin/ state combinations considered. The
proportion is as high as 0.25 in some age groups, for example 25-29 year olds
in 2001 and 40-44 year olds in 2018. As a consequence, the sampling error
around the ACS data for this migrant group is relatively small and the model
estimates closely follow these data. For the most recent two years, where only
Facebook data are available, note that the model estimates do not follow the
data as closely and the uncertainty around the model estimates increases. This
reflects the fact that there are more sources of error associated with the
Facebook data.
In Georgia (Figure 6(b)) the levels of Mexican migrants are around half as
high as in California. Due to smaller sample sizes in Georgia, the standard
errors around the ACS data are much larger, and as such the model estimates do
not follow the data as closely. However, the trends for Mexican migrants in
California, and Georgia are broadly the same: decreases in the younger age
groups, and increases in the older age groups, representing an aging stock of
migrants.
For Indian males in California and Georgia (Figure 7), the proportions are
much lower than for the Mexican migrant population, peaking at around 3-4% of
the population in the 30-39 year old age groups. The proportions are
increasing over time, however, particularly in the 25-44 age bracket. Finally,
for German male migrants in California and Georgia (Figure 8), we see low and
constant migrant proportions. The uncertainty around the ACS data is already
relatively high, and so there is not so much of an increase in uncertainty in
the final two years.
(a) California
(b) Georgia
Figure 6: Mexican male migrants by age group, California and Georgia,
2001-2018
(a) California
(b) Georgia
Figure 7: Indian male migrants by age group, California and Georgia, 2001-2018
(a) California
(b) Georgia
Figure 8: German male migrants by age group, California and Georgia, 2001-2018
## 6 Validation
We evaluated the performance of the Bayesian model compared to other
reasonable forecasting alternatives. To do this, we ran the model on data from
2001 to 2016, and forecast migration stocks in 2017. We then compared these
forecasts to the actual ACS data in 2017. We compared the accuracy of the
Bayesian model forecast to forecasts produced by three other models:
1. 1.
Three-year moving average of the ACS data. This is one of the simplest options
available and does not require the Facebook data or any statistical modeling.
2. 2.
Facebook data only: Estimates are based just on the available Facebook data in
2017, after it has been adjusted for biases.
3. 3.
ACS time series model: Here, we ran the Bayesian hierarchical time series
model described in section 4 above, but just using data from the ACS (no
Facebook).
In order to assess model performance, we compare the root mean squared error
(RMSE):
$RMSE=\sqrt{\frac{\sum_{n}\left(\hat{p}_{g,2017}-p^{ACS}_{g2017}\right)^{2}}{N}}$
(13)
where $\hat{p}_{g,2017}$ is the estimated proportion of migrants from a
particular group $g$, $p^{ACS}_{g2017}$ is the equivalent proportion from the
ACS and $N$ is the size of the group. Here, the $g$ can refer to any
combination of age group, state and migrant origin.
Table 1 shows the overall RMSE for the four models for Mexican, Indian and
German migrants. The main result is that in each of the three migrant groups,
the Bayesian model presented (which combines the ACS and bias-adjusted
Facebook data and thus is referred to as the ’combined model’), produces the
lowest RMSE and thus the most accurate forecasts. The overall results also
illustrate that the Bayesian hierarchical time series model produces
substantially more accurate forecasts compared to a simple moving average or
the bias-adjusted Facebook data alone, producing RMSEs that are up to an order
of magnitude smaller. This gain in accuracy is much larger than the gain
moving from ACS-only to the combined model, although there is still a gain in
each case.
Model | Mexico | India | Germany
---|---|---|---
Moving average | 0.01280 | 0.0480 | 0.0129
Facebook | 0.01210 | 0.00584 | 0.0142
ACS | 0.00995 | 0.00453 | 0.00263
Combined | 0.00970 | 0.00356 | 0.00261
Table 1: Overall RMSE by model and migrant origin
Figure 9 illustrates the RMSE by age group and model type for each of the
three migrant groups. Similar plots by state can be found in Appendix C. For
Mexico, there is generally an incremental decline in the RMSE moving from the
Facebook-only model, to moving average, to ACS, to the combined model. The
RMSE is highest in the 30-34 year old age group, which is also where the
proportion of migrants is highest (see Figure 6). For India (Figure 9b)), the
RMSE is particularly high from the Facebook-only model in the 25-34 age
bracket. As a consequence, the combined model RMSEs in those age groups are
higher in the combined model versus the ACS model. For Germany, the gain in
accuracy in moving to the hierarchical time series set-up is much larger. This
is most likely related to the fact that the proportions of German migrants are
in general a lot lower than for Mexico or India, and so there are noticeable
gains in pooling information across state, age and time.
(a) Mexico
(b) India
(c) Germany
Figure 9: RMSE by age group and model
To summarize, the validation exercise comparing the one-year-out predictions
from a range of models to the migrant proportions reported in the ACS
illustrate both (i) the strength of the proposed Bayesian hierarchical time
series model as a general framework, and (ii) the additional information
obtained from including up-to-date Facebook data compared to just historical
ACS data alone.
## 7 Discussion
As the size and frequency of migration movements continues to increase
worldwide, new sources of data are being considered in order to better
understand both historical and future migration trends. There is a growing
body of work considering the feasibility of using social media data to achieve
such goals, from platforms such as Facebook, Twitter and LinkedIn. While the
granularity of social media use varies widely (from individual geo-tagged
tweets, to aggregated advertising demographic data, as is in this paper), the
common challenges of using such data remain: firstly, to adequately adjust for
known biases in the social media data, primarily as a consequence of the non-
representativeness of the population of social media users; and secondly, to
meaningfully combine information from social media data with information from
more traditional data sources, such as surveys or censuses.
In this paper we presented a statistical framework to achieve these goals in
the context of producing short-term projections of migrant stocks in the
United States. The model includes a bias-adjustment process of the Facebook
data, and a ‘principal components time series’ model, which allows for the
projection of trends in stocks into the future, considering both Facebook and
ACS data. The model allows for different types of uncertainty around the
different data sources, and shares information on migration trends over time
and pools across geographic space. Illustrative results were presented for
three separate migrant groups: Mexicans, Indians and Germans. The results of
the validation exercise, comparing projections with 2017 ACS data, suggest
that the proposed model improves prediction of short-term trends when compared
to viable alternatives.
The validation exercise illustrated the substantial gain in accuracy achieved
when moving to the Bayesian hierarchical time series model, regardless of
whether or not the Facebook data were included. While the benefits of
including the Facebook data in this particular case were relatively marginal,
more generally the Facebook data has the advantage of being up-to-date and
essentially available in real time. Thus, in a situation of a ‘shock’, such as
a natural disaster or other event, the collection of Facebook data allows for
a more immediate estimate of the effects of that shock on migration. The
combination of these data with past trends allows for the identification of
surprising increases or decreases, that are out of the expected bounds based
on historical patterns.
There are several limitations of the proposed model, which naturally lead into
avenues for future work. Firstly, the bias-adjustment model assumes that the
systematic bias in the Facebook data (by age and state) is constant over time.
In reality, it is reasonable to believe that the biases in the Facebook data
are changing over time, as the composition of the underlying Facebook
population changes. The relationship between the age/location composition of
the Facebook and the actual population (as measured by the ACS) could be
investigated in future work. Secondly, the bias-adjustment model also assumes
that the non-sampling error is constant over Facebook’s ‘waves’ of data
collection — that is, sources of error that include changes in how the
population of reach is calculated, or other computational reasons, are assumed
to be constant. In practice, and in other work using these data (Alexander et
al. 2019), we have observed that this is probably not the case, and needs to
be further investigated to better understand non-migration-related
fluctuations over time. While we only consider two data sources in this paper,
the general statistical framework could easily be extended to include
information from other sources. Future work could also include taking
advantage of the rich demographic and socioeconomic data available through the
Facebook Advertising Platform, including information on education and
occupation.
While this work focused on a model for the estimation of migrant stocks, the
philosophy of combining social media data with more traditional data sources
in one statistical framework – allowing for different sources of uncertainty –
can be readily extended to model other demographic indicators. Indeed, the
underlying time series model was itself an extension of principal component
techniques that have been previously been used in demography to study
mortality and fertility. We show the strength of combining more traditional
demographic modeling techniques and data with newer sources of data to gain
insights into underlying population processes.
## Appendix A Bias adjustment plots for migrants from India and Germany
Figure 10: Bias adjustment of Facebook data for Indian migrants Figure 11:
Bias adjustment of Facebook data for German migrants
## Appendix B Age distributions in 2008 and 2018 for Indian and Germany
migrants by state
For India (Figure 12) we see that the proportions across age groups have
generally increased over the decade, with relatively high proportions
concentrated in the northeast region of the country. Unlike Mexico, the age
distribution is fairly constant, with the highest proportions generally being
in the 30-34 age group. The German male migrant populations (Figure 13) by
state show relatively flat, low and unchanging levels over the decade
2008–2018.
Figure 12: Estimated and projected age distributions of Indian migrants by
state, 2008 and 2018 Figure 13: Estimated and projected age distributions of
German migrants by state, 2008 and 2018
## Appendix C Validation results by state
The figures below show the RMSE for each model, by state of destination in the
US, for Mexican, Indian and German migrants. The figures show that, in
general, the combined model out-performs the other three alternative models.
For Mexico (Figure 14), this is particularly the case for the states with
relatively high proportions of Mexican migrants. Results are more variable for
Indian migrants (Figure 15), with the combined model performing relatively
poorly in the West coast states but well in the mid-West states. For German
migrants (Figure 16), the geographic pooling appears to vastly improve the
projection accuracy.
Figure 14: RSME by state and model type for Mexican migrants in 2018 Figure
15: RSME by state and model type for Indian migrants in 2018 Figure 16: RSME
by state and model type for German migrants in 2018
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|
2024-09-04T02:54:57.680225 | 2020-03-05T20:07:53 | 2003.02900 | {
"authors": "Dirk Keppens",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26068",
"submitter": "Dirk Keppens",
"url": "https://arxiv.org/abs/2003.02900"
} | arxiv-papers | # 50 years of Finite Geometry,
the “geometries over finite rings” part
Dirk Keppens
Faculty of Engineering Technology, KU Leuven
Gebr. Desmetstraat 1
B-9000 Ghent BELGIUM
<EMAIL_ADDRESS>
###### Abstract
Whereas for a substantial part, “Finite Geometry” during the past 50 years has
focussed on geometries over finite fields, geometries over finite rings that
are not division rings have got less attention. Nevertheless, several
important classes of finite rings give rise to interesting geometries.
In this paper we bring together some results, scattered over the literature,
concerning finite rings and plane projective geometry over such rings. It
doesn’t contain new material, but by collecting stuff in one place, we hope to
stimulate further research in this area for at least another 50 years of
Finite Geometry.
Keywords: Ring geometry, finite geometry, finite ring, projective plane
AMS Classification: 51C05,51E26,13M05,16P10,16Y30,16Y60
## 1 Introduction
Geometries over rings that are not division rings have been studied for a long
time. The first systematic study was done by Dan Barbilian [11], besides a
mathematician also one of the greatest Romanian poets (with pseudonym Ion
Barbu). He introduced plane projective geometries over a class of associative
rings with unit, called Z–rings (abbreviation for Zweiseitig singuläre Ringe)
which today are also known as Dedekind-finite rings. These are rings with the
property that $ab=1$ implies $ba=1$ and they include of course all commutative
rings but also all finite rings (even non–commutative).
Wilhelm Klingenberg introduced in [52] projective planes and 3–spaces over
local rings. A ring $R$ is local if it possesses a unique maximal right ideal
(which turns out to be the Jacobson radical $J(R)$). For a local ring $R$ the
quotient ring $R/J(R)$ is a division ring (= skewfield or field) and the
natural homomorphism of $R$ onto $\mathbb{K}=R/J(R)$ induces an epimorphism of
the plane $P_{2}(R)$ over $R$ onto the ordinary desarguesian projective plane
PG(2,$\mathbb{K}$). Nowadays planes over local rings are called (desarguesian)
Klingenberg planes (see also [7]). In the finite case such planes have the
finite projective plane PG(2,$q$) over the Galois field GF($q$) as epimorphic
image.
In three other papers [49], [50] and [51], Klingenberg studied projective
planes over local rings with some additional properties, called $H$–rings
(short for Hjelmslev rings). In these rings the left and right ideals form a
chain and the maximal ideal contains only zero divisors. If one drops that
last condition, one gets chain rings. In the finite case any chain ring is an
$H$–ring. Planes over $H$–rings are now called (desarguesian) Hjelmslev planes
after the Danish mathematician Johannes Hjelmslev (born as Johannes Petersen)
who was the first one to consider plane geometries in which two distinct lines
may have more than one point in common [41]. Among the finite $H$–rings are
the Galois rings GR($p^{nr},p^{n}$) of cardinality $p^{nr}$ and characteristic
$p^{n}$ which are natural generalizations of Galois fields.
In the early seventies another class of rings came under the attention: full
matrix rings over fields. Strongly inspired by the work of the italian “father
of Galois geometry” Beniamino Segre on geometries over finite fields (e.g.
[77]), J.A. Thas defined projective planes (and higher dimensional spaces)
over full matrix rings with elements in a field and investigated combinatorial
properties in the finite planes over the matrix rings $M_{n}(GF(q))$ of
$n\times n$–matrices over Galois fields [80]. We will refer to these planes
further as Thas planes.
In the eighties F.D. Veldkamp was very productive in the area of projective
ring planes and their generalizations. He gave in [84] and [86] an axiomatic
description of projective planes and higher dimensional geometries over the
large class of rings of stable rank 2, a notion coming from algebraic
$K$–theory.
A ring $R$ has stable rank 2 if for any $a,b\in R$ with $Ra+Rb=R$ there exists
$r\in R$ such that $a+rb$ is a unit. The class of rings of stable rank 2
includes the class of semilocal rings (hence also all finite rings, local
rings, chain rings, $H$–rings and matrix rings over a division ring) and a
ring of stable rank 2 is always Dedekind–finite (hence a Z–ring in the sense
of Barbilian). Projective planes over rings of stable rank 2 are called
(desarguesian) Veldkamp planes. Among these are Klingenberg planes, Hjelmslev
planes, Thas planes and also the projective planes over semiprimary rings
(i.e. rings with nilpotent Jacobson radical and with $R/J(R)$ semisimple)
treated by Bingen in [13].
In almost all papers on projective geometry over rings no special attention is
paid to the finite case. Mostly, theorems deal with rings in general (with no
specification for finite or infinite). In this paper we restrict ourselves to
the finite case. First we bring together some results on finite rings with
special attention for local rings. Then we have a closer look at projective
plane geometries over finite rings. In the last section we deal with some
generalizations of rings (semirings, nearrings and alternative rings) and
projective plane geometries over such algebraic structures.
## 2 Finite rings
In this section the word “ring” always refers to an associative ring with unit
$1\not=0$, but with multiplication not necessarily commutative.
Finite fields or Galois fields are well-known algebraic structures. Finite
fields of order $q$ only exist if $q$ is a prime power ($q=p^{r}$) and for
each such $q$ their is a unique (up to isomorphism) field of that order which
is denoted by $\mathbb{F}_{q}$ or by GF($q$). The prime number $p$ is the
characteristic of the field.
It is natural to look at generalizations of finite fields to finite rings, but
the situation is much more complicated. First there exist finite
non–commutative rings unlike the situation in finite division rings where the
famous theorem of Wedderburn forces any finite skewfield to be a field. Also
the order of a finite ring doesn’t uniquely determine that ring (for example
there are four non–isomorphic rings of order four, including one field). A
complete classification of finite rings seems to be a “mission impossible”
(even if one restricts to the commutative case).
The paper of Raghavendran [72] on rings of prime power order was the starting
point for the study of the structure of finite rings. Also the work of Wilson
[88] and [89] was of great importance. A recent survey on results obtained so
far with an extensive bibliography can be found in Nechaev [68].
Local rings, first defined by Krull in [56], play a central role in the
structure theory of (finite) rings. Recall that a ring $R$ is called local if
it possesses a unique maximal right ideal (or equivalently a unique maximal
left ideal). This is stronger than asking that $R$ has a unique maximal
two–sided ideal (e.g. the ring $M_{n}(\mathbb{Z}_{p^{n}})$ of $n\times
n$–matrices over $\mathbb{Z}/p^{n}\mathbb{Z}$ has a unique maximal two–sided
ideal but is not local). The unique maximal right or left ideal in a local
ring turns out to be the Jacobson radical $J(R)$. Other characterizations of
local rings are possible. E.g. $R$ is local iff the set of non–units forms a
right (or left) proper ideal in $R$. Also $R$ is local if and only if for all
$r\in R$ either $r$ or $1-r$ is invertible. Finally $R$ is local iff $R/J(R)$
is a division ring. Other characterizations in terms of zero divisors are
given in [76]. In the finite case one can say even more: $R$ is local iff
$R\setminus J(R)$ is the set of units of $R$ or equivalently $J(R)$ is the set
of nilpotent elements of $R$.
The following theorem gives parameters for finite local rings.
###### Theorem 2.1.
(Raghavendran [72]) Let $R$ be a finite local ring. Then there exist unique
numbers $p$, $n$, $r$ and $k$ such that $|R|=p^{nr}$, $|J(R)|=p^{(n-1)r}$ and
the characteristic of $R$ is $p^{k}$ with $1\leq k\leq n$. The number $p^{r}$
is the order of the Galois field $R/J(R)$ and the number $n$ is the index of
nilpotency of $J(R)$. If $k=n$ then $R$ is commutative.
There is also a more recent result which conversely characterizes local rings
among finite rings just by a couple of parameters.
###### Theorem 2.2.
(Behboodi and Beyranvand [12], González [36]) Let $R$ be a finite ring and let
$Z(R)$ be the set of zero–divisors of $R$. Then $R$ is local if and only if
$|R|=p^{m}$ and $|Z(R)|=p^{n}$ for some prime number $p$ and integers $1\leq
n<m$. Moreover, when $R$ is local with these parameters, then the order of
$R/J(R)=R/Z(R)=p^{r}$ with $r=m-n$.
The structure of commutative finite local rings was first studied by Ganske
and McDonald [33]. Classification theorems are proved for fixed orders or
fixed characteristic in [16], [17], [24] and [72]. In the non–commutative case
Wirt [90] has contributed to the theory.
By the following important structure theorem the classification problem of
commutative finite rings can be reduced to that of finite local rings.
###### Theorem 2.3.
(McDonald [67]) Let $R$ be a finite commutative ring. Then $R$ decomposes (up
to order of summands) uniquely as a direct sum of finite local rings.
Another decomposition theorem, also valid in the non–commutative case, shows
once more the importance of finite local rings.
###### Theorem 2.4.
(McDonald [67] and Wirt [90]) Let $R$ be a finite ring. Then $R$ decomposes as
$S+N$ with $S$ a direct sum of full matrix rings over finite local rings and
$N$ a subring of the Jacobson radical $J(R)$.
Next we look at principal ideal rings. A ring is called a right principal
ideal ring if any right ideal $I$ is right principal, i.e. generated by one
element ($I=aR$). Similar definition for left principal ideal ring. If a ring
is a left and right principal ideal ring, it is called a principal ideal ring
(PIR). A right principal ideal ring is always a right noetherian ring, since
any right ideal is finitely generated. It is also a right Bézout ring since
any finitely generated right ideal is principal. In fact the right PIR’s are
just the rings which are both right noetherian and right Bézout. Similar
results are true for left principal ideal rings and PIR’s.
The structure of finite principal ideal rings was first studied by Fisher in
[32]. For finite rings the notions of left PIR, right PIR and PIR are
equivalent (see [68]).
Another important class of rings are the chain rings. A ring is called a right
chain ring if for any $a$ and $b$ in $R$ either $a\in bR$ or $b\in aR$. For a
right chain ring the lattice of right ideals is totally ordered by inclusion
and it follows that $R$ is a local ring. Analogous definitions and results for
left chain rings. A ring which is a left and right chain ring is called a
chain ring. In the infinite case there are examples of right chain rings which
are not chain rings (see [63], [79] and [15]), but in the finite case there is
a left–right equivalence. Every ideal of a chain ring is a power of the unique
maximal ideal. A (left or right) chain ring with the additional property that
any non–unit is a two–sided zero divisor is called a (left or right) $H$–ring
or affine Hjelmslev ring (two–sided chain rings are also known as projective
Hjelmslev rings). Finite chain rings are always left and right $H$–rings. For
a comprehensive study of $H$–rings linked to the behaviour of ideals, we refer
to [81].
The following theorem shows that finite chain rings are nothing but finite
local principal ideal rings!
###### Theorem 2.5.
(Clarke and Drake [18] and Lorimer [62]) Let $R$ be a finite ring. Then the
following conditions are equivalent: (a) $R$ is a local PIR (b) $R$ is a local
ring with principal maximal ideal (b) $R$ is a left chain ring (c) $R$ is a
right chain ring (d) $R$ is a chain ring
A valuation ring in a division ring $\mathbb{D}$ is a proper subring $R$ with
the property that $x$ or $x^{-1}$ $\in R$ for each nonzero $x\in\mathbb{D}$. A
ring is a valuation ring if and only if it is a left and right chain domain
(i.e. a chain ring without zero divisors) (for a proof, see [60]. Since any
finite domain is a finite field, there do not exist finite valuation rings.
A ring $R$ is called E–ring if and only if it possesses an ideal $I$ so that
all ideals of $R$ are of the form $I^{n}$. In the infinite case $E$–rings can
be characterized as $H$–rings with nilpotent radical and also as proper
homomorphic images of discrete valuation rings (see [4], [5], [60] and [61]).
In the finite case the notions of $H$–ring and $E$–ring coincide.
The simplest and most investigated finite chain rings are the Galois rings,
first defined by Krull [56] as “Grundringe” and later rediscovered by Janusz
[46] and Raghavendran [72]. A Galois ring is a commutative local PIR such that
$J(R)=(p)$ with $p=1+1+\ldots+1$ ($p$ terms) for some prime $p$. These rings
are very close to Galois fields. In the past ten years, finite chain rings and
in particular Galois rings got a lot of attention in connection with coding
theory (see e.g. [42] and [43]).
As for Galois fields one has
###### Theorem 2.6.
(Raghavendran [72] and McDonald [67]) For any prime $p\in\mathbb{N}$ and for
any $n,r\in\mathbb{N}$ there exists a unique (up to isomorphism) Galois ring
$R$ consisting of $q^{n}$ (with $q=p^{r}$) elements and with characteristic
$p^{n}$
The unique Galois ring in the preceding theorem is denoted by
GR($q^{n},p^{n}$) (or sometimes also by GR($p^{n},r$)). For $p=q$ we have
GR($p^{n},p^{n}$) = $\mathbb{Z}_{p^{n}}$, the ring of integers module $p^{n}$,
and for $n=1$ we obtain the Galois field GF($q$)=GR($q,p$). All Galois rings
can be constructed in the form $R=\mathbb{Z}_{p^{n}}[x]/(f(x))$ where $f(x)$
is a monic polynomial of degree $r$ which is irreducible modulo $p$ and hence
GR($q^{n},p^{n})$, $q=p^{r}$, can be seen as Galois extensions of degree $r$
of its subring $\mathbb{Z}_{p^{n}}$.
The properties of Galois rings are well known, e.g. the structure of the group
of units, the automorphism group, the possible subrings etc. Many results can
be found in [14]. The classification of all chain rings is still an open
problem but partial results are known. Galois rings occur in the construction
of finite chain rings as can be seen from next theorem.
###### Theorem 2.7.
(Clark and Liang [19], Wirt [90], Neumaier [69]) Let $R$ be a finite chain
ring with parameters $p,n,r$ and $k$ as in theorem 2.1. Then there exist
integers $t$ and $s$ such that $R$ is isomorphic to
$S[x,\sigma]/(g(x),p^{k-1}x^{t})$ with $S$ = GR($q^{k},p^{k}$) and
$S[x,\sigma]$ the Ore–skew polynomial ring over $S$, i.e. with usual addition
and the multiplication $xa=\sigma(a)x$ for $\sigma\in Aut\;S$, and with
$g(x)\in$ $S[x,\sigma]$ an Eisenstein polynomial of degree $s$,
$g(x)=x^{s}-p(a_{0}+a_{1}x+\ldots+a_{s-1}x^{s-1})$ with $a_{0}$ a unit in $S$
and $n=(k-1)s+t$ ($1\leq t\leq s\leq n$)
The integer $s$ in the theorem above is called the ramification index of $R$.
It is the smallest integer such that the ideal $(p)$ is equal to $J(R)^{s}$.
For a given set of parameters $p,n,r,k,s$ and $t$ one could ask for the number
of non–isomorphic finite chain rings. In general this problem is still open,
but partial results are known (see e.g. [1], [3], [74], [68] ). In some cases
the parameters determine completely the ring:
###### Theorem 2.8.
(Clark and Liang [19] and Arkhipov [3]) Let $R$ be a finite chain ring with
parameters $p,n,r,k,t$ and $s$ as in theorem 2.7.
(a) If $k=1$ (hence $R$ has minimal characteristic $p$), then $R$ is uniquely
determined (up to isomorphism) and $R\cong$ GF($q$)$[x,\sigma]/(x^{n})$ (a
truncated skew polynomial ring)
(b) If $k=n$ (hence $R$ has maximal characteristic $p^{r}$), then $R$ is
uniquely determined (up to isomorphism) and $R\cong$ GR($q^{n},p^{n}$) (always
commutative)
Some more results are known for finite chain rings with characteristic $p^{k}$
with $1<k<n$ (see [44]). In [91] still another description of finite
(commutative) chain rings is given as certain homomorphic images of the
polynomial ring $\mathbb{Z}_{p^{r}}[x,y]$.
An important special subclass of chain rings is the one for which the Jacobson
radical $J$ of $R$ has index of nilpotency $2$, so $J^{2}=0$. In this case
$n=2$ and the two cases in theorem 2.8 are the only possible ones. Finite
chain rings with $J\not=0$ and $J^{2}=0$ are called uniform. The
classification of finite uniform chain rings follows from theorem 2.8 but was
also proved directly by Cronheim.
###### Theorem 2.9.
(Cronheim [25]) Every finite uniform chain ring with $R/J\cong$ GF($q$) is
either a ring of (twisted) dual numbers, or a truncated Witt ring of length 2,
over the field GF($q$).
Rings of (twisted) dual numbers are the rings $\mathbb{D}(q,\sigma)$ =
GF($q$)$[x,\sigma]/(x^{2})$ (twisted for $\sigma\not=1$) (corresponding to
case ($a$) with $n=2$ in theorem 2.8).
$\mathbb{D}(q,\sigma)$ can also be represented as the subring of matrices
$\left(\begin{array}[]{ll}a&b\\\ 0&a^{\sigma}\end{array}\right)$ in the full
matrix ring $M_{2}(q)$ of $2\times 2$ matrices with elements in GF($q$).
$W_{2}(q)$, the truncated Witt ring of length 2 over GF($q$) is defined on the
set GF($q$) $\times$ GF($q$), $q=p^{k}$, as follows:
addition:
$(x_{0},x_{1})+(y_{0},y_{1})=(x_{0}+y_{0},x_{1}+y_{1}+\dfrac{x_{0}^{p}+y_{0}^{p}-(x_{0}+y_{0})^{p}}{p})$
and multiplication:
$(x_{0},x_{1})\cdot(y_{0},y_{1})=(x_{0}y_{0},x_{0}^{p}y_{1}+x_{1}y_{0}^{p})$
It can be proved that $W_{2}(q)$ is isomorphic to the Galois ring
GR($q^{2},p^{2}$) (this is case ($b$) with $n=2$ in theorem 2.8).
## 3 Finite ring planes
In this section we deal with geometries over finite rings and we restrict
ourselves to the case of plane projective geometries. The projective line,
higher dimensional projective geometries, affine and metric geometries, planar
circle geometries (Benz–planes), chain geometries and polar geometries over
finite rings will not be considered here.
A big part of finite geometry, called Galois geometry, is related to finite
fields, see e.g. the work of Hirschfeld [40]. Since the pioneering work of B.
Segre, the finite desarguesian (pappian) projective plane PG($2,q$) and its
interesting point sets (arcs, ovals, blocking sets, unitals, $\ldots$) have
been studied extensively. For planes over finite rings still a lot of work has
to be done. As already mentioned in the introductory section, plane geometries
over some important classes of rings have been defined in a suitable way,
starting somewhere in 1940 by Barbilian (some isolated cases over particular
rings were even known longer ago).
Before we look at planes over finite rings, we first recall the definition of
a projective plane over an arbitrary ring (not necessarily finite).
Let $R$ be an arbitrary ring (associative and with unit element). Denote the
set of twosided invertible elements of $R$ by $R^{\star}$. Following [11],
[26] or [55], we can construct a plane projective geometry PG($2,R$) over $R$
as follows:
points are the left unimodular triples $(x,y,z)\in R\times R\times R$ up to a
right scalar in $R^{\star}$ (where $(x,y,z)$ left unimodular means that there
exist $a,b,c\in R$ such that $ax+by+cz=1$ or equivalently $Rx+Ry+Rz=R$)
lines are the right unimodular triples $[u,v,w]\in R\times R\times R$ up to a
left scalar in $R^{\star}$ (where $[u,v,w]$ right unimodular means that there
exist $a,b,c\in R$ such that $ua+vb+wc=1$ or equivalently $xR+yR+zR=R$)
incidence I (between points and lines) is defined as follows: $(x,y,z)$ I
$[u,v,w]$ if and only if $ux+vy+wz=0$
neighborship $\sim$ (between points and lines) is defined by:
$(x,y,z)\sim[u,v,w]$ if and only if $ux+vy+wz\in R\setminus R^{\star}$
It is clear that incidence always implies neighborship, so $p$ I $L$ implies
$p\sim L$ for any point $p$ and any line $L$.
The incidence structure (with neighbor relation) obtained in this way is
called right projective plane over $R$. In the same way one can define the
left projective plane over $R$ which is clearly isomorphic to the right
projective plane over the opposite ring $R^{\circ}$. Therefore we will drop
from now on the specification “right” or “left”.
Although the denomination “projective plane” is used here, the projective
plane over a ring (which is not a division ring) isn’t a projective plane in
the usual sense, as two distinct points may be incident with none or with more
than one line and dually.
In addition to the neighbor relation for point–line pairs, also a neighbor
relation between points (between lines respectively) can be considered in
PG($2,R$): points $(x,y,z)$ and $(x^{\prime},y^{\prime},z^{\prime})$ are
neighboring iff $\left[\begin{array}[]{lll}x&y&z\\\
x^{\prime}&y^{\prime}&z^{\prime}\end{array}\right]$ cannot be extended to an
invertible $3\times 3$–matrix over $R$ (and similar for lines).
Dealing with projective planes over rings it is natural to assume that
non–neighboring elements behave the same as distinct elements in an ordinary
projective plane over a division ring. To get that situation, it is necessary
to restrict to the class of rings for which every one–sided unit is a
two–sided unit, which was first observed by Barbilian [11]. Indeed, assume
that $r$ is right–invertible (with right inverse $a$), but not left
invertible. Consider the lines $[1,0,0]$ and $[r,0,0]$ (remark that $[r,0,0]$
is right unimodular since $r\cdot a+0\cdot b+0\cdot c=1$). These lines are
distinct as otherwise there would exist a left scalar $l\in R^{\star}$ for
which $[1,0,0]=l\cdot[r,0,0]$, so $1=l\cdot r$ in contradiction with the
assumption that $r$ has not a left inverse. Now these two distinct lines are
incident with the non–neighboring points $(0,1,0)$ and $(0,0,1)$.
The restriction to rings in which any left (or right) invertible element is a
two–sided unit (or equivalently $a\cdot b=1$ implies $b\cdot a=1$), the
so–called Dedekind–finite rings, only comes up when one deals with infinite
rings. In the finite case (but also for many important classes of infinite
rings) invertible elements are always two–sided invertible. In this context it
is also interesting to mention that all reversible rings (i.e. $a\cdot b=0$
implies $b\cdot a=0$) are Dedekind–finite.
Next we are interested in the connection between properties of the ring $R$
and the projective plane PG($2,R$). Most of the following results are
reformulations for the finite case of theorems that can be found in Veldkamp
[84] and [85] .
In the projective plane over a Dedekind–finite ring, there is a unique line
incident with two given non–neighboring points (and dually). One might ask
whether the neighbor relation is completely determined by the incidence
relation in the sense that two points are non–neighboring if and only if there
is unique line incident with them (and dually for lines). This is not always
the case, but it does if every non–invertible element in $R$ is a right and
left zero–divisor.
A (left or right) artinian ring is a ring in which any non–empty set of (left
or right) ideals that is partially ordered by inclusion, has a minimal
element. In a (left or right) artinian ring any non–invertible element is a
(left or right) zero divisor. As finite rings are always left and right
artinian we get that the neighbor relation is completely determined by the
incidence relation in planes PG($2,R$) over finite rings. In [23] a proof is
given of the property that in a finite ring any left zero divisor is also a
right zero divisor.
###### Theorem 3.1.
(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then two distinct points are neighboring if and only if they are
incident with either no or at least two lines. Two distinct lines are
neighboring if and only if they are incident with either no or at least two
points.
A projective ring plane is called linearly connected (Veldkamp [84]), neighbor
cohesive (Drake and Jungnickel [27]) or punctally cohesive (Baker et al. [9])
if any two distinct points are incident with at least one line. For planes
over rings of stable rank 2 it is proved in Veldkamp that two points are
incident with at least one line if and only if $R$ has the following property:
for any two $r_{1},r_{2}\in R$ there exists $a\in R$ such that
$Rr_{1}+Rr_{2}=R(r_{2}+ar_{1})$. This is fulfilled for $R$ a left Bézout ring,
i.e. a ring for which any finitely generated left ideal is a principal ideal.
Dually two lines are incident with at least one point iff $R$ is a right
Bézout ring. For finite rings the Bézout conditions amount to the condition
that $R$ is a principal ideal ring (recall that for a finite ring the notions
left principal, right principal and principal coincide). So we can reformulate
the theorem for finite rings as follows:
###### Theorem 3.2.
(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then any two points are incident with at least one line (the plane
is linearly connected) and dually if and only if $R$ is a principal ideal
ring.
The possibility of more than one line incident with two neighboring points
(and dually) corresponds to the presence of zero divisors in the ring. So any
two distinct points are incident with exactly one line (and dually) if and
only if $R$ is a Bézout domain. In the finite case this becomes: if $R$ is a
principal ideal domain, hence if $R$ is a finite field. Hence:
###### Theorem 3.3.
(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then any two distinct points are incident with exactly one line and
dually if and only if $R$ is a finite field (i.e. PG($2,R$) is a pappian
projective plane).
Next we look at the special case of (finite) local rings. For such rings the
definition of the projective plane PG($2,R$) and his neighbor relations, can
be adapted (in an equivalent way) a little. E.g. two points $(x,y,z)$ and
$(x^{\prime},y^{\prime},z^{\prime})$ are neighbors if and only if
$(x^{\prime},y^{\prime},z^{\prime})$ $-$ $(x,y,z)\lambda\in J\times J\times J$
for some $\lambda\in R\setminus J$ with $J$ the maximal ideal of $R$ and
similarly for lines.
###### Theorem 3.4.
(Veldkamp [84]) Let $R$ be a (finite) ring and PG($2,R$) the projective plane
over $R$. Then the neighbor relation $\approx$ between points (between lines
resp.) is transitive if and only if $R$ is a (finite) local ring.
For local rings $R$ there is a canonical epimorphism $\varphi$ from $R$ onto
the division ring $\mathbb{K}=R/J$. This epimorphism induces an epimorphism
$\pi$ of the projective plane PG($2,R$) onto the (ordinary) projective plane
PG($2,\mathbb{K}$) by putting $\pi(x,y,z)=(\phi(x),\phi(y),\phi(z))$ and
$\pi[u,v,w]=[\phi(u),\phi(v),\phi(w)]$ and the neighbor relation can be
expressed by means of $\pi$ : $p\sim L$ if and only if $\pi(p)$ I $\pi(L)$ and
similarly $p\sim q$ iff $\pi(p)=\pi(q)$ and $L\sim M$ iff $\pi(L)=\pi(M)$.
The projective plane over a local ring, also known as a (desarguesian)
projective Klingenberg plane, therefore is strongly connected with an ordinary
desarguesian projective plane. One could say that the points (and lines) of an
ordinary projective plane are blown up to clusters of neighboring points
(lines) to produce a projective Klingenberg plane. In the finite case the
epimorphic image of PG($2,R$) is the plane PG($2,q$) over the Galois field
GF($q$).
Combining theorems 3.1, 3.2 and 3.4 yields the following:
###### Theorem 3.5.
(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then the neighbor relation $\approx$ between points (between lines
resp.) is transitive and two neighboring points are incident with at least two
lines and dually, if and only if $R$ is a finite local principal ideal ring.
From section 2 we know that finite local principal ideal rings are synonym for
finite chain rings or finite $H$–rings. Recall that projective planes over
$H$–rings are called (desarguesian) projective Hjelmslev planes.
We now summarize the possibilities for projective planes over a finite ring.
###### Corollary 3.6.
Let $R$ be a finite ring and PG($2,R$) the projective plane over $R$. Then
only four cases are possible:
(a) $R$ has no zero divisors and hence is a field and PG($2,R$) is an ordinary
pappian projective plane (two distinct points are incident with exactly one
line and dually)
(b) $R$ is a local principal ideal ring (hence a chain ring = $H$–ring) and
PG($2,R$) is a desarguesian projective Hjelmslev plane
(c) $R$ is local but not a principal ideal ring and PG($2,R$) is a
desarguesian projective Klingenberg plane (but not a Hjelmslev plane)
(d) $R$ is semilocal (but not local) and PG($2,R$) has non–transitive neighbor
relations
The fourth class (d) is the wildest as it contains all finite rings which are
not local (but necessarily semilocal due to the finiteness, i.e. with a finite
number of maximal ideals). Important examples of rings belonging to this class
are the full matrix rings $M_{n}(q)$ over GF($q$). Projective planes over full
matrix rings were first mentioned by Ree [73] and further studied by J.A. Thas
[80] who also gave an interpretation of PG($2,M_{n}(q)$) in terms of the
projective space PG($3n-1,q$). Other examples are the rings $\mathbb{Z}_{m}$
with $m\not=p^{r}$ (see [29]). Of special interest are also the rings of
double numbers $\mathbb{B}(q)=$GF($q$) + GF($q$)$\,t$ with $t^{2}=t$. They
possess exactly two maximal ideals. In [75] projective planes over
$\mathbb{B}(q)$ are studied.
Examples of finite local rings that are not chain rings (c) are provided by
the rings GF($q$)[$x,y$]/$\langle x^{n},xy,y^{n}\rangle$ ($n>1$). The
corresponding planes are finite desarguesian Klingenberg planes that are not
Hjelmslev planes.
Class (b) contains many interesting examples, including the Galois rings
GR($q^{n},p^{n}$) ($q=p^{r}$) and the rings $\mathbb{A}(p^{r},n)=$
GF($p^{r}$)$[x]/x^{n}$ (called quasi–Galois rings in [14]). The rings
$\mathbb{A}(p^{r},n)$ can also be interpreted as matrix rings, consisting of
all matrices $(a_{ij})$ with elements belonging to GF($q$) and
$a_{i,j+i-1}=a_{1j}$ and $a_{ij}=0$ for $i>j$. For $n=2$ the ring of dual
numbers $\mathbb{D}(q)$ over GF($q$) is included. Projective planes over dual
numbers were considered yet a century ago by Corrado Segre [78].
Class (a) finally consists of all the Galois fields GF($q$) with the
associated projective planes PG($2,q$).
For finite (not necessarily desarguesian) projective Klingenberg and Hjelmslev
planes a unique set of parameters (the order) can be given (see [48] and
[28]): for any flag $(p,L)$ there are exactly $t$ points on $L$ neighboring
with $p$ and exactly $s$ points on $L$ not neighboring with $p$. Moreover: the
number of points = the number of lines = $s^{2}+st+t^{2}$, any line is
incident with $s+t$ points, any point is incident with $s+t$ lines, any point
has $t^{2}$ neighbors, any line has $t^{2}$ neighbors, $t|s$ and
$r=\frac{s}{t}$ is the order of the projective plane that is the canonical
epimorphic image of the Klingenberg plane, and $s\leq t^{2}$ or $t=1$.
For a finite desarguesian projective Klingenberg plane this yields
###### Theorem 3.7.
(Drake and Jungnickel [27] ) Let $R$ be a finite local ring. Then the
projective Klingenberg plane (Hjelmslev plane in some cases) PG($2,R$) has
parameters $s=|R|=q^{n}$ and $t=|J|=q^{n-1}$ with $q=p^{r}$ a prime power.
To conclude this section we consider rings of order 4. This is the smallest
order for which there exist rings that are not division rings. There are four
non–isomorphic rings (with unit) of order 4. The first is the Galois ring
GF($4$) that gives rise to the projective plane PG($2,4$). The second is the
chain ring $\mathbb{Z}_{4}\cong GR(4,4)\cong W_{2}(2)$ with characteristic 4,
the coordinate ring of a projective Hjelmslev plane. The third is the chain
ring $\mathbb{D}(2)\cong\mathbb{A}(2,2)$ of characteristic 2 and with GF($2$)
as a subfield (dual numbers over GF($2$)) that gives rise to another
projective Hjelmslev plane (it is proved in [47] that the plane over
$\mathbb{D}(2)$ is embeddable in PG($5,q$) while the plane over
$\mathbb{Z}_{4}$ is not). Finally the fourth is the non–local ring
$\mathbb{B}(2)\cong GF(2)[t]/(t^{2}-t)$ of characteristic 2 (double numbers
over GF($2$)), associated to a Veldkamp plane with non–transitive neighbor
relation.
## 4 Finite ring–like structures and ring–like planes
Besides finite rings, even more general finite ring–like algebraic structures
deserve a closer look in relationship to geometry. Till now very few research
has been done in this area (except for finite field–like algebraic
structures).
In the literature several generalizations of rings can be found. Among the
most important are: non–associative rings, nearrings and semirings. For all
these structures there are finite examples and a generalization of the concept
“local ring” exists, which opens perspectives for Klingenberg–like geometries
associated to these generalized rings.
### 4.1 Semirings
A semiring is a structure $(S,+,\cdot)$ with $(S,+)$ a commutative semigroup
with identity element $0$, $(S,\cdot)$ a (not necessarily commutative)
semigroup with identity element $1$ ($\not=0$), in which the left and right
distributivity of multiplication over addition hold and in which $0$ is
absorbing for multiplication $a\cdot 0=0\cdot a=0$. Hence semirings differ
from rings by the fact that elements do not always have an inverse for the
addition (the additive group of a ring is replaced by a semigroup).
Semirings were first introduced by Vandiver [83] in 1935 and in the past years
there was an enormous amount of publications on the subject (see e.g. the work
of Glazec [34] for a survey), mainly in relation to computer science and
automata theory but they also are interesting algebraic objects on their own,
see [38] and [35].
Examples of finite semirings are $B(n,i)$ on the set $\\{0,1,\ldots,n-1\\}$
($n\geq 2$ and $0\leq i\leq n-1$) with addition $\oplus$ defined by $a\oplus
b=a+b$ if $0\leq a+b<n$ and $a\oplus b=c$ if $a+b\geq n$ with $c$ the unique
number such that $c\equiv a+b$ (mod $n-i$) and $0\leq c\leq n-1$.
Multiplication $\odot$ is defined in a similar way. In particular $B(n,0)$ is
the ring $\mathbb{Z}_{n}$ of integers modulo $n$ and $B(2,1)$ is known as the
boolean semiring $\mathbb{B}$. For other values of $n$ and $i$ one obtains
semirings that are not rings.
In semirings zero divisors and zero sums are of interest ($a$ is a zero sum of
$S$ if there exists an element $b\not=0$ in $S$ such that $a+b=0$). In [39] it
is proved that if $S$ is a finite commutative semiring then either every zero
sum is a zero divisor or $S$ is a ring. As a corollary one has that a finite
commutative semiring without zero divisors (a semidomain) either is zero sum
free ($a+b=0$ always implies $a=b=0$) or is a finite domain (and hence a
field).
A semiring is called a semifield if $(S^{\ast},\cdot)$, with
$S^{\ast}=S\setminus\\{0\\}$, is a group (in a semifield the multiplication
need not to be commutative, so the term semi division ring would be better).
We have to warn for confusion between the semifields considered here and the
semifields defined in the context of non–desarguesian projective planes by
e.g. Albert, Dickson, Knuth and others and which are also known under the name
division algebras or distributive quasifields. Those are generalizations of
division rings by dropping the need for associativity of the multiplication,
so $(S,+,\cdot)$ is in that context a semifield if $(S,+)$ is a group (which
turns out to be commutative) and $(S^{\ast},\cdot)$ is a loop and the two
distributivity laws of multiplication over addition hold (for a survey on
those semifields, see e.g. [57])
To my knowledge no research has been done yet on geometries over (finite)
semirings that are not rings. So there may be opportunities. In particular a
generalization to Klingenberg–like planes (over local semirings) seems
possible though not trivial as the concept of ideals in semirings is much more
complicated. As a starting point for ideals in semirings and the concept of
local semiring see e.g. [6] and [37].
### 4.2 Nearrings
A left nearring is a structure $(N,+,\cdot)$ with $(N,+)$ a (not necessarily)
commutative group with identity element $0$, $(N,\cdot)$ a (not necessarily
commutative) semigroup with identity element $1$ ($\not=0$), in which the left
distributivity of multiplication over addition holds: $a\cdot(b+c)=a\cdot
b+a\cdot c$. Similar for right nearring. Nearrings differ from rings by the
fact that addition isn’t necessarily commutative and there is only
distributivity on one side. Nearrings which are distributive on both sides are
rings (the commutativity of the addition then follows automatically). Most of
the material on nearrings can be found in the work of Pilz [70] and [71].
A (left or right) nearring is called a (left or right) nearfield if
$(S^{\ast},\cdot)$ is a group, with $S^{\ast}=S\setminus\\{0\\}$. So
nearfields are division rings with only distributivity on one side. Nearfields
were first discovered by Dickson in 1905 and are useful in constructing
examples of non–desarguesian projective planes. All finite nearfields are
classified by Zassenhaus. They are either Dickson–nearfields or they belong to
one of seven exceptional classes.
Little research has been done on geometry over nearrings that are not
nearfields (except for the class of planar nearrings which give rise to
balanced incomplete block designs, but these nearrings don’t possess a
multiplicative identity element and therefore are less usable in the context
of projective geometry). So another suggestion for research could be a
treatment of plane projective geometries over (finite) nearrings that are not
rings. The special case of Klingenberg planes (over local nearrings) and
Hjelmslev planes (over $H$–nearrings) was already initiated in two general
papers by E. Kolb, see [53] and [54] and by Törner in [82]. Local nearrings
were introduced by Maxson in [65] and partially classified in [66]. Among the
results is the fact that the additive group of a finite local nearring always
is a $p$–group and the existence of a natural epimorphism from a local
nearring onto a local nearfield. Other results on finite nearrings can be
found in [2], [20], [21], [22], [45], [58], [59], [64] and [87].
### 4.3 Alternative rings
Finally, we mention some results on non–associative rings. A non–associative
ring is a structure $(A,+,\cdot)$ which satisfies all axioms for an
(associative) ring with multiplicative identity element, except for the
associativity of the multiplication. An alternative ring is a non–associative
ring such that $a\cdot(a\cdot b)=a^{2}\cdot b$ and $(a\cdot b)\cdot b=a\cdot
b^{2}$ for all $a,b\in A$. Alternativity is a weaker condition than
associativity. If any element in an alternative ring $A$ is a unit, then $A$
is called an alternative division ring. Alternative division rings are used to
construct a class of non–desarguesian projective planes, called Moufang
planes. By the theorem of Artin–Zorn every finite alternative division ring is
a field, hence finite Moufang projective planes are desarguesian (and
pappian).
Generalizations to alternative rings that are not division rings are due to
Baker, Lorimer and Lane. In [10] Moufang projective Klingenberg planes are
defined as projective Klingenberg planes that are $(p,L)$–transitive for all
flags and it is proved that they can be coordinatized by a local alternative
ring. In [8] several characterizations of local alternative rings are given
and an analogue for non–associative chain rings and $H$–rings is defined
properly. In the finite case it is proved that the concepts of alternative
$H$–ring, left (or right) alternative chain ring and local alternative
principal ideal ring are equivalent. Moreover the theorem of Artin–Zorn is
expanded : any finite alternative chain ring (or $H$–ring) is associative [9].
Leaving out the condition of being local, leads to more general alternative
rings. It is hard to define projective ring planes over such rings in a
suitable way. Faulkner has done it for alternative stable rank 2 rings in [30]
(generalizing Veldkamp’s results for associative stable rank 2 rings) and for
alternative rings in which any one–sided unit is two–sided in [31]
(generalizing the planes of Barbilian).
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|
2024-09-04T02:54:57.693651 | 2020-03-05T22:55:45 | 2003.02955 | {
"authors": "Seid Muhie Yimam and Gopalakrishnan Venkatesh and John Sie Yuen Lee\n and Chris Biemann",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26069",
"submitter": "Seid Muhie Yimam",
"url": "https://arxiv.org/abs/2003.02955"
} | arxiv-papers | # Automatic Compilation of Resources for Academic Writing
and Evaluating with Informal Word Identification and Paraphrasing System
###### Abstract
We present the first approach to automatically building resources for academic
writing. The aim is to build a writing aid system that automatically edits a
text so that it better adheres to the academic style of writing. On top of
existing academic resources, such as the Corpus of Contemporary American
English (COCA) academic Word List, the New Academic Word List, and the
Academic Collocation List, we also explore how to dynamically build such
resources that would be used to automatically identify informal or non-
academic words or phrases. The resources are compiled using different generic
approaches that can be extended for different domains and languages. We
describe the evaluation of resources with a system implementation. The system
consists of an informal word identification (IWI), academic candidate
paraphrase generation, and paraphrase ranking components. To generate
candidates and rank them in context, we have used the PPDB and WordNet
paraphrase resources. We use the _Concepts in Context_ (CoInCO) ”All-Words”
lexical substitution dataset both for the informal word identification and
paraphrase generation experiments. Our informal word identification component
achieves an F-1 score of 82%, significantly outperforming a stratified
classifier baseline. The main contribution of this work is a _domain-
independent_ methodology to build targeted resources for writing aids.
Keywords: academic writing, academic word, academic phrase, informal word
identification, academic text paraphrasing
languageresourceLanguage Resources
Automatic Compilation of Resources for Academic Writing
and Evaluating with Informal Word Identification and Paraphrasing System
Seid Muhie Yimam1, Gopalakrishnan Venkatesh2, John Sie Yuen Lee3 and Chris
Biemann1
---
Universität Hamburg, Germany1, International Institute of Information
Technology, Bangalore, India2,
City University of Hong Kong, Hong Kong SAR3
<EMAIL_ADDRESS>
Abstract content
## 1 Introduction
We present the first approach to building resources for an academic writing
aid system automatically. Academic writing aid systems help in automatically
editing a text so that it better adheres to the academic style of writing,
particularly by choosing a better academic word in a given domain. In the
context of academic paraphrasing tasks, the resources are mainly words or
phrases, that are more appropriate to use in an academic writing style.
Moreover, the academic resources might vary from domain to domain as some
words or phrases are extensively used in one domain over the other.
The first step in building an academic writing aid tool is to collect
resources that determines whether a given phrase follows the style of writing
in academia. This involves analyzing a given sentence and determining if the
lexemes of the sentences are well-selected academic words and phrases or not.
To evaluate the resources compiled, we have to build a system, analogous to
the lexical substitution and text simplification tasks, for example, [Szarvas
et al., 2013, Štajner and Saggion, 2018], that consists of informal word
identification, academic candidate generation, and candidate paraphrase
ranking components (see Figure 1). While it is possible to follow the same
approaches as the lexical substitution and text simplification approaches for
academic text rewriting tasks, the main challenge for the academic
paraphrasing task is the collection of resources for academic texts.
The following are the main objectives of building academic resources:
1. 1.
Identify suitable academic and non-academic datasets that are to be used to
build academic resources.
2. 2.
Design a generic, _domain-independent_ , approach to extract academic
resources.
3. 3.
Evaluate the quality of the collected resources and use these resources for
informal word identification (IWI) and academic paraphrasing systems.
The informal word identification (IWI) component automatically identifies
informal words (see Section 4.2) that are going to be replaced with academic
paraphrases. The candidate generation and ranking component determine the best
academic candidate paraphrase to replace the informal words.
The ultimate goal of this research work is to integrate the informal word
identification, candidate generation, and paraphrase ranking components into
writing aid tools, for example to word processors or text composing software
like latex packages, to automatically assist users in academic text composing.
In this work, we have targeted the following research questions 1) How to
build academic resources (words or phrases), which are used to replace
informal or less academic expressions in academic texts? 2) How to build a
system that can be used to evaluate the collected resources?
In Section 2, a brief review of related works is presented. In Section 3, we
discuss how to build academic resources using reference corpora and evaluate
the quality of the resource. In Section 4, we present the approaches that are
used to build an informal word identification and paraphrasing system for
academic rewriting. Setups of the academic paraphrasing systems and the
experimental results are discussed in Section 5. Analysis of system results
and conclusion of the research are presented in Section 6 and Section 7
respectively.
## 2 Previous Work
In this section, we review previous work in lexical substitution, a closely
related task, and discuss how the academic text rewriting system potentially
differs.
In essence, our system is similar to lexical substitution (LS) and text
simplification tasks, in such a way that both focus on the rewriting of an
original text towards a given goal. Lexical substitution system mainly focuses
on rewriting texts by replacing some of the words or phrases without altering
the original meaning [Szarvas et al., 2013, Štajner and Saggion, 2018]. The
work by ?) targeted text simplification based on the sequence-to-sequence deep
neural network model, where its entailment and paraphrasing capabilities are
improved via multi-task learning.
While the complex word identification (CWI) task focuses on identifying
lexical units that pose difficulties to understand the sentence [Yimam et al.,
2017b, Yimam et al., 2017a, Yimam et al., 2018, Paetzold and Specia, 2016],
our informal word identification (IWI) component focuses on identifying words
that are not fitting or adhering to the academic style of writing.
The work by ?) focuses on the lexical substitution task, particularly for
medical documents. They have relied on Distributional Thesaurus (DT), computed
on medical texts to generate synonyms for target words.
Existing resources for academic writing are limited to a precompiled list of
words such as the Corpus Of Contemporary American English (COCA) [Gardner and
Davies, 2013] and the New Academic Word List 1.0 (NAWL) [Browne et al., 2013]
vocabulary lists. Regarding phrases (multi-word expressions) for academic
writing, the only available resources are the academic bi-grams compiled by
Pearson111Academic collocation list:
https://pearsonpte.com/organizations/resea.
However, these resources are 1) limited to a certain domain and target writers
(mostly L2 learners and students), 2) their vocabulary is fixed, thus
requiring manual work for an extension, and 3) the resources are limited to
uni-gram and bi-gram lists. In this work, we build academic resources that are
more generic, which can be built from existing reference corpora. In addition
to uni-gram and bi-gram resources, we also design a system that can produce
resources up to a length of four words (quad-grams).
As far as we know, the only system available to academic writing is the work
of ?), which addresses a different aspect, which is a sentence restructuring
based on nominalizing verbal expressions.
## 3 Building Academic Resources
In this section, we will first discuss the existing academic resources, how
they are built and their limitations. Then, we will present our approach that
describes the process of building academic resources from different reference
corpora. Finally, we will discuss the quality of the collected resources
against two evaluation measures, namely comparing with the existing resources
and manually evaluating the academic fitness of resources.
### 3.1 Existing Resources for Academic Writing
In this subsection, we will present the existing academic word lists and
phrases, which will be used to evaluate the quality of the dataset we build
from reference corpora.
#### 3.1.1 Academic Vocabulary
There are some efforts in building a list of vocabularies or words for
academic writing. Some of them are created by analyzing text from academic
writing corpora such as journals, theses works, and essays. One such resource
is the Corpus Of Contemporary American English (COCA) [Gardner and Davies,
2013] vocabulary list, which contains about 3,000 words (in lemmas) that are
derived from a 120 million word sub-corpus of the 560 million words.
Similarly, the New Academic Word List 1.0 (NAWL) [Browne et al., 2013] was
also built in the same way as the COCA list as a reference resource for second
language learners of English, which is selected from an academic corpus of 288
million words.
#### 3.1.2 Academic Phrases
Academic phrases are a list of collocated words (multi-word expressions),
which are mostly used for academic writing. The list from ?) comprises of
2,468 bi-gram collocations. The list is compiled from the written curricular
component of the Pearson International Corpus of Academic English (PICAE)
comprising of over 25 million words. However, the academic phrases, like the
academic word lists, are mostly used as a guideline (study material) to
practice academic writing.
### 3.2 Academic and Non-Academic Reference Corpora
The existing resources that are presented in Section 3.1 are prepared mostly
as references or study guidelines for academic writers. However, to build
automatic writing support, it is required to have more comprehensive and
larger resources that can also be updated dynamically. In addition to single
word and bi-gram lists, it would be also beneficial if the resource includes
longer sequences of words. Hence, we have further extended the academic phrase
list that includes up to four-gram phrases. The resource helps the academic
paraphrasing or rewriting system in 1) identifying words or phrases in a text
that are less academic and 2) providing alternative academic words or phrases
that are more relevant to the contexts presented.
Figure 1: Frequencies of the highest occurring tri-grams collected from the
reference corpora based on our approach.
To this end, we have compiled a list of academic phrases that are extracted
from the ACL Anthology Reference Corpus (ACLAC) [Bird et al., 2008]. This
corpus contains 22,878 scholarly publications (articles) about Computational
Linguistics. To understand the syntactic difference of an academic corpus from
a non-academic corpus, we have used the Amazon Review Full Score Dataset
[Zhang et al., 2015] as our non-academic reference. The non-academic dataset
is constructed by randomly taking 600,000 training samples and 130,000 testing
samples for each review score from 1 to 5 [Zhang et al., 2015]. In this paper,
a review refers to the review text from the training sample.
Resource | Size | Coverage (%)
---|---|---
COCA | 3,015 | 95.39
NAWL | 963 | 99.90
Academic phrases | 2,468 | 79.34
Table 1: Coverage of the existing resources for academic writing in our
reference ACLAC corpus.
The above two corpora can be considered to be a good fit as it shows a high
match with the existing academic vocabulary or phrase list, as shown in Table
1. From Table 1, we can see that 95% of the academic words from COCA and
99.90% of the academic words from NAWL are represented in the ACLAC corpus.
Similarly, around 80% of the bi-grams from the academic phrases (PICAE) are
contained in the ACLAC corpus.
### 3.3 Approach to Build the New Academic Resource
On analyzing the corpora, we noticed that the non-academic corpus is much
larger (in terms of the number of words) than the academic corpus. Therefore,
we downsampled the non-academic text (to have comparable resources in terms of
size) and ensured that the total number of words in both of the corpora are
comparable. As a part of the pre-processing step, we clean the corpus
(removing special characters) and lower case each word. We have considered a
total of 991,798 reviews, which results in 75,184,498 tokens.
Using the NLTK’s222https://www.nltk.org/ Bi-, Tri- and Quad-Gram multi-word
expression finder, we have extracted phrases from the two corpora (ACLAC and
Amazon Review Full Score Dataset) and also compute the frequency distributions
of these phrases across both the corpora as it can bee seen in Figure 1. The
phrases extracted from both corpora can be used to assess naively the
distribution across the two domains.
However, we have followed two different widely adopted approaches to extract
representative phrases in a corpus, which is specifically known as keyphrases.
The first approach is called Term Frequency-Inverse Document Frequency (TF-
IDF), which is one of the most important statistics that show the relative
importance of a term in a document in comparison to the corpus. The importance
increases proportionally to the number of times a word appears in the document
while its weight is lowered when the term occurs in many documents. We used
the scikit-learn333https://scikit-learn.org/ implementation of TF-IDF to
compute the scores of the different n-grams and thereby select the phrases
that have maximum TF-IDF scores as keyphrases. In the ACLAC corpus, we have
considered an article as one document while for the Amazon Review dataset, a
review is considered as a single document.
In the second approach, we explore keyphrase extraction techniques based on
part-of-speech sequences. We have employed _EmbedRank_ , an unsupervised
keyphrase extraction tool trained with sentence embeddings [Bennani-Smires et
al., 2018]. We consider only those phrases that consist of zero or more
adjectives followed by one or multiple nouns [Wan and Xiao, 2008]. While using
the official implementation444https://github.com/swisscom/ai-research-
keyphrase-extraction, we also explored the possibility of using the _Spacy_
555https://spacy.io/ POS tagger for keyphrase extraction in our corpora, which
has a permissive license to redistribute our resource generation system as an
open-source project.
As per the heuristic approach followed in the COCA word list compilation, we
only retain those phrases that occur at least 50% more frequently in the
academic portion of the corpora than would otherwise be expected. In other
words, the ratio of the academic frequency of a term (in the ACLAC dataset) to
the non-academic frequency (in the Amazon Review Full Score Dataset) should be
1.50 or higher [Gardner and Davies, 2013]. Using a similar approach, we have
also created the non-academic resources, which are also used to evaluate the
quality of the academic resources in the human evaluation experiment (cf.
Section 3.5)
### 3.4 Newly Collected Academic Resources
Based on the two keyphrase extraction approaches discussed in Section 3.3 (TF-
IDF and EmbedRank based keyphrase extractions), we have compiled a total of
6,836 academic phrases (5,275 from EmbedRank and 1,900 from the TF-IDF
approach). From Table 2, we can see that most of the academic keyphrases are
extracted using the EmbedRank approach.
Newly Collected resources
---
Approach | Uni-gram | Bi-gram | Tri-gram | Quad-gram
EmbedRank | 1,267 | 3,848 | 156 | 4
TF-IDF | 1,090 | 690 | 109 | 11
From Existing Resources
COCA | 3,016 | 0 | 0 | 0
NAWAL | 960 | 0 | 0 | 0
PICAE | 0 | 2,468 | 0 | 0
Table 2: Academic word and phrases lists from the existing as well as from
newly collected resources.
### 3.5 Manual Evaluation of Resources
From the automatically compiled list of resources (words and phrases), we have
randomly sampled 520 words and phrases comprising of 155 uni-grams, 100 bi-
grams and 5 tri-grams from each of the compiled academic and non-academic
phrase list. We then distributed the word and phrase lists to a total of 9
annotators (Ph.D. and postdoctoral researchers) and requested the participants
to label each entry as academic or non-academic. The sampled words and phrases
are evaluated by two sets of annotators and the annotators were able to label
the entries with an inter-annotator agreement of 68.22%.
### 3.6 Results and Discussions on the Collected Resources
While analyzing the COCA list, we noticed that it contains a few stop words
such as both and above. Hence, while relying on TF-IDF, we have considered
extracting academic resources in different scenarios. First, we remove stop
words as a part of the preprocessing step and in the second approach we have
used the whole corpus as it is.
The system proposed by us relies on the relative frequencies in the reference
corpora which can be computed independently of the language used. Thus the
compilation of such an academic resource (through keyphrase extraction) can be
considered language agnostic.
While performing the human evaluation, the annotators were asked to classify
whether the given phrase is academic or not. The evaluation would have been
more rigorous if they had to classify the phrases given the context in which
the term had occurred. The annotators have at times labeled an entry as both
academic and non-academic. Consider the word attention, it was used both in an
academic (ACLAC) and non-academic (Amazon Review Full Score Dataset) context,
for example as ”LSTM with attention” and ”the kid’s attention to the game”
respectively.
## 4 Evaluating the Resources for Academic Rewriting System
### 4.1 Academic Words
We define a word as academic or formal if it is in one of the following lists
of academic phrases 1) keyphrases (up to four-grams) compiled by our system
(cf. Section 3.3 – comprises of 6,836 phrases) 2) the COCA list [Davies, 2012]
3) the New Academic Word List [Browne et al., 2013]666
http://www.newgeneralservicelist.org/nawl-new-academic-word-list.
Some example academic words are shown in Table 3. The academic word lists are
also extended to phrases or multi-word expressions. Pearson has published a
set of academic bi-grams777Academic collocation list:
https://pearsonpte.com/organizations/resea. Words like _best_ , _almost_ , and
_way_ are not by themselves _academic_ , but they can be combined with other
words to form academic expressions such as _best described_ , _almost
identical_ , and _appropriate way_.
### 4.2 Informal Words
The naive approach is to attempt to rewrite every non-academic word, using our
definition above. That is a misplaced goal, however, since even the average
document in the BAWE corpus [Alsop and Nesi, 2009] contains a considerable
number of words outside the list, including function words and other words
commonly used in all English documents.
We define a word as informal if it is a non-academic term that can be
paraphrased by an academic term. If the term is academic, or it is non-
academic but does not have an academic paraphrase, it is termed as formal.
### 4.3 Architecture
Figure 2: Architecture of the system.
As shown in Figure 2, our proposed system consists of four components, which
is analogous to the lexical simplification systems [Paetzold and Specia,
2017]. The components of our system constituted informal word identification
(IWI), paraphrase generation, candidate selection, and paraphrase ranking.
#### 4.3.1 Informal Word Identification
The informal word identification (IWI) component identifies each word as
_informal_ , or not. The system attempts to paraphrase only the informal words
in the rest of the pipeline.
Similar to CWI [Yimam et al., 2017b, Yimam et al., 2017a, Yimam et al., 2018,
Paetzold and Specia, 2016], IWI is more accurate when placed in context. The
word _big_ , for example, may need to be paraphrased to _major_ in the context
of ” _This article makes two big contributions._ ” It should not be
paraphrased, however, when it is part of the expression _big data_.
#### 4.3.2 Paraphrase Generation, Selection, and Ranking
Given an informal word, this step generates a list of substitution candidates.
While there are different approaches to generate candidates for target words,
such as using existing paraphrase resources like WordNet and Distributional
thesaurus (see ?)), we depend solely on the CoInCo [Kremer et al., 2014],
WordNet [Miller, 1995], and the paraphrase database (PPDB) [Pavlick et al.,
2015] resources to generate candidates.
Once the candidates are generated, all of the candidates, which must be
academic words are retained for the paraphrase ranking component. Given a list
of academic substitution candidates, the paraphrase ranking component finds
the one that fits best in the context. The detailed approach is presented in
Section 4.4.
Academic words | report, state, claim…
---|---
Non-academic words | say, declare, mention, allege…
Table 3: Example of academic and non-academic words based on our academic
resources.
### 4.4 Datasets for IWI and the Paraphrasing Components
For this evaluation, we derive our dataset from a lexical substitution dataset
called the Concepts in Context (CoInCo) [Kremer et al., 2014]. The CoInCo
dataset is an _All-Words_ lexical substitution dataset, where all words that
could be substituted are manually annotated. The corpus is sampled from
newswire and fiction genres of the Manually Annotated Sub-Corpus (MASC)
corpus888http://www.anc.org/data/masc/. While the targets (words that are
going to be substituted) are used to build the informal word identification
dataset, the candidates are further processed to perform the academic
paraphrase ranking task.
A total of 1,608 training and 866 test sentences are compiled out of 2474
sentences from the CoInCo dataset. Statistics on the IWI dataset are shown in
Table 5.
#### 4.4.1 Building the IWI Dataset
We automatically generated an IWI dataset from CoInCo as follows. For each
non-academic target word, we determine if its substitution candidates include
at least one academic word. If so, it is labeled as informal; otherwise, it is
labeled as formal. All academic target words and all words without
substitution candidates are labeled as formal. An example is given in Example
4.1 and Table 3.
CoInCo annotation | Pacific First Financial Corp said[paraphrases: report, state, detect] shareholders
---|---
IWI dataset | Pacific[N] First[N] Financial[N] Corp[N] said[Y] shareholders
Table 4: Transformation of the CoInCo dataset into IWI dataset, with respect
to the academic word list in Table 3
###### Example 4.1.
Sentence: Pacific First Financial Corp said shareholders … CoInCo annotation:
Target word: said. Paraphrases: report, state, claim, allege, announce,
mention, declare IWI dataset ([I]–informal, [F]–formal): Pacific[F] First[F]
Financial[F] Corp[F] said[I] shareholders[N]
Dataset | # Tokens | #Types
---|---|---
| I | F | I | F
IWI training | 6,783 | 3,358 | 2,266 | 1,509
IWI test | 3,666 | 1,822 | 1,577 | 994
Table 5: Statistics on the IWI dataset. _#Tokens_ shows the total number of
tokens (formal (_F_) and informal (_I_)) while _#Types_ shows the unique
occurrences of tokens in the IWI training and test sets. _I_ stands for
informal and _F_ for formal tokens and types resp.
#### 4.4.2 Paraphrase Candidates
To generate non-academic to academic word pairs for paraphrasing, we used the
paraphrases (word pairs) in CoInCo, WordNet, and PDPB as the starting point.
For the CoInCo dataset, we have only included those word pairs where: 1) the
target word is non-academic, 2) the substitution candidate is academic, 3) the
target word has a higher word frequency than the substitute candidate in our
academic resources. Since the academic resource is not exhaustive, some proper
academic terms may be mistakenly considered as non-academic. This requirement
aims to prevent these words from being substituted.
For example, from the sentence in Example 4.1, we obtained the word pairs
say:report, say:state, and say:claim. We have collected a total of 23,476 word
pairs from the CoInCo Training Set.
The dataset is prepared with 4 candidates for each informal target, where 2
candidates are academic and 2 candidates are non-academic. When we do not have
appropriate candidates we extract further candidates from WordNet [Miller,
1995] and PPDB [Pavlick et al., 2015]. Table 6 shows the statistics of target
words extracted from the CoInCo dataset, where 59% of the informal words have
possible candidate paraphrases.
### 4.5 Academic Paraphrase Corpus
In general, any existing paraphrase or lexical substitution corpus can be
converted into an academic paraphrase corpus with the following steps:
1) Discard all academic target words since they do not need to be paraphrased.
2) Remove all non-academic substitution candidates for the remaining (non-
academic) target words.
If no candidate is left after step (2), also remove that target word.
# target words | Paraphrase coverage
---|---
Original | Our corpus | in (%)
5,480 | 3,250 | 59.30
Table 6: Statistics on our evaluation dataset. The last column shows the
percentage of non-academic words in the corpus for which paraphrases can be
obtained.
### 4.6 Informal Word Identification Models
We trained three Support Vector Machine (SVM) classifiers, using Radial Basis
Function kernel, from scikit-learn999https://scikit-learn.org/ with different
feature sets. We use the following features:
Word frequency: We use word frequencies 1) in the Beautiful
Data101010https://norvig.com/ngrams/ which are derived from the Google Web
Trillion Word Corpus, 2) in the general COCA list, and 3) in the ACL anthology
corpus [Bird et al., 2008].
Word Embedding: We have used GloVe [Pennington et al., 2014] word embedding to
compute the cosine similarity between the word and the sentence111111Embedding
for the sentence is calculated by averaging the embedding of words in the
sentence. We also explore the option of using Euclidean distance between the
word and the sentence as a feature while training the classifier.
Part of Speech Tag (POS): The POS tag of the word obtained from the
TreeTagger121212https://www.cis.uni-muenchen.de/~schmid/tools/TreeTagger/.
Word level features: We use the word length and the number of vowels as
features for training the classifier.
### 4.7 Paraphrase Ranking Models
In order to rank the best candidates for academic rewriting, we have followed
the learning-to-rank machine learning approach, where candidates are ranked
based on their relevance score. The number of annotators selected the given
candidate is considered as a relevance score. The TF-Ranking deep learning
model provided by _TensorFlow Ranking_
131313https://github.com/tensorflow/ranking library [Pasumarthi et al., 2019]
is used to build the paraphrase ranking model.
## 5 Experiments
### 5.1 Informal Word Identification
We trained the IWI classifier on the CoInCo Train Set using SVM. Similar to
most of the CWI evaluation metrics, we evaluate the performance of the system
on the following evaluation metrics:
Precision: The number of correct informal targets, out of all targets proposed
by the system.
Recall: The number of correct informal targets, out of all informal words that
should be paraphrased.
F-Measure: The harmonic average of precision and recall.
Table 7 shows IWI precision and recalls on the CoInCo Test Set. We use a
simple stratified randomization algorithm from scikit-learn as a baseline
system. The proposed algorithm (SVM classifier) achieves a better performance
overall in the F-Score of 0.8204. As it can be seen in Table 7, the following
features work better for the IWI task: frequencies, cosine similarity, and
Euclidean distance.
Method | Precision | Recall | F-score
---|---|---|---
Baseline | 0.6679 | 0.6787 | 0.6733
SVM Fe1 | 0.7584 | 0.8933 | 0.8204
SVM Fe2 | 0.7650 | 0.8748 | 0.8162
SVM Fe3 | 0.7552 | 0.8912 | 0.8176
Table 7: Precision and recall on the informal word identification task. The
baseline system has been setup using the Stratified classifier from scikit-
learn: The stratified classifier in scikit-learn generates predictions by
respecting the training set’s class distribution. Fe1 = (Frequencies, cosine
similarity), Fe2 = Fe1 + (Euclidean distance), Fe3 = All features
### 5.2 Academic Paraphrasing
We evaluate the system performance on automatically generating academic
paraphrases and ranking them. Following standard evaluation metrics in lexical
simplification, we report on the Mean Reciprocal Rank
(MRR)141414https://en.wikipedia.org/wiki/Mean_reciprocal_rank metric.
The model from TF-Ranking [Pasumarthi et al., 2019] library has been trained
to re-rank the candidates on the CoInCo test set. The model was trained using
the _Adagrad_ optimizer with a learning rate of 0.05. Experiments were
performed on various loss functions (_pairwise_logistic_loss_ and
_softmax_loss_) and different _step_ 151515Steps are the number of training
iterations executed. (50, 100 and 200) values. Table 8 shows the experimental
results.
Parameters | Ranking metric
---|---
Loss | Steps | MRR
Logistic | 50 | 0.8861
100 | 0.8926
200 | 0.8895
Softmax | 50 | 0.8893
100 | 0.8895
200 | 0.8914
Table 8: Academic paraphrasing performance on the CoInCo Test Set using the
MRR ranking metric.
## 6 Analysis of Results
For the informal word identification task, our models have a slightly lower
precision as our dataset is not balanced (we have more informal words than
formal words, as shown in Table 5).
From an error analysis, we find out that even if the term is academic in
general, its usage in the test dataset is inclined to be informal. For
example, in the sentence ”It was last February, after the winter break, that
we moved in together.”, break is labeled as academic but should be labeled as
informal. This issue could be solved by further enhancing the dataset by
employing human annotators during the resource compilation process.
Similarly, some of the errors from the system’s prediction are to be
attributed to the annotation process of the test set. For example, in the
sentence ”They included support for marine reserves and money for fisheries
management reform.”, reserves is annotated as informal while the system
identified it as formal.
In general, while bootstrapping the academic resource compilation and the
informal word identification tasks, a minimal intervention of human annotators
would enhance the overall system. Furthermore, integration of a BERT or other
contextualized embedding model [Devlin et al., 2019] could also help to
improve the performance of the system. Contextualized word embeddings provide
word vector representations based on their context. As the vector
representation of words varies as per the context, they implicitly provide a
model for word sense disambiguation (WSD).
## 7 Conclusion and Future Direction
In the realm of academic text writing, we explored how to compile academic
resources, automatically identify informal words (words that are less formal
for academic writing), and provide better substitutes. We have used a generic
approach to compile the academic resources, which can be easily transferred to
domains or languages as it only requires text corpus. The academic text
rewriting system, analogous to lexical substitution systems, consists of
informal word identification, candidate generation, candidate selection, and
ranking components. As far as we know, this is the first experiment towards
the development of academic writing support for academia, while there might be
commercial cases (for example Grammarly161616https://www.grammarly.com/) that
we do not know how the systems operate.
We envision this system to be embedded into open source academic writing aid
tools where the academic sources are used to detect informal terms and propose
academic substitutes. For the resource compilation process, it would be nice
to extend the EmbedRank approach to extract keyphrases beyond the adjective
and noun POS tag patterns, especially to cover verbs used in academic
contexts.
Source code and resources of this the paper are released
publicly171717https://github.com/uhh-lt/par4Acad on the Github repository
under permissive licenses (ASL 2.0, CC-BY).
## Acknowledgments
This work was partially funded by a HKSAR UGC Teaching & Learning Grant
(Meeting the Challenge of Teaching and Learning Language in the University:
Enhancing Linguistic Competence and Performance in English and Chinese) in the
2016-19 Triennium.
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|
2024-09-04T02:54:57.704651 | 2020-03-06T00:39:37 | 2003.02979 | {
"authors": "Hengyuan Hu, Adam Lerer, Alex Peysakhovich, Jakob Foerster",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26070",
"submitter": "Hengyuan Hu",
"url": "https://arxiv.org/abs/2003.02979"
} | arxiv-papers | # “Other-Play” for Zero-Shot Coordination
Hengyuan Hu Adam Lerer Alex Peysakhovich Jakob Foerster
###### Abstract
We consider the problem of zero-shot coordination - constructing AI agents
that can coordinate with novel partners they have not seen before (e.g.
humans). Standard Multi-Agent Reinforcement Learning (MARL) methods typically
focus on the self-play (SP) setting where agents construct strategies by
playing the game with themselves repeatedly. Unfortunately, applying SP
naively to the zero-shot coordination problem can produce agents that
establish highly specialized conventions that do not carry over to novel
partners they have not been trained with. We introduce a novel learning
algorithm called _other-play_ (OP), that enhances self-play by looking for
more robust strategies, exploiting the presence of known symmetries in the
underlying problem. We characterize OP theoretically as well as
experimentally. We study the cooperative card game Hanabi and show that OP
agents achieve higher scores when paired with independently trained agents. In
preliminary results we also show that our OP agents obtains higher average
scores when paired with human players, compared to state-of-the-art SP agents.
Machine Learning, ICML
## 1 Introduction
A central challenge for AI is constructing agents that can coordinate and
cooperate with partners they have not seen before (Kleiman-Weiner et al.,
2016; Lerer & Peysakhovich, 2017; Carroll et al., 2019; Shum et al., 2019).
This is particularly important in applications such as cooperative game
playing, communication, or autonomous driving (Foerster et al., 2016;
Lazaridou et al., 2016; Sukhbaatar et al., 2016; Resnick et al., 2018). In
this paper we consider the question of zero-shot coordination where agents are
placed into a cooperative situation with a novel partner and must quickly
coordinate if they wish to earn high payoffs.
Our setting is a partially observed cooperative Markov game (MG) which is
commonly known among both agents. The agents are able to construct strategies
separately in the training phase but cannot coordinate on the strategies that
they construct. They must then play these strategies when paired together one
time. We refer to this as zero-shot coordination.
A popular way of constructing strategies for MG with unknown opponents is
self-play (or “self-training”), (Tesauro, 1994). Here the agent controls both
players during training and iteratively improves both players’ strategies. The
agent then uses this strategy at test time. If it converges, self-play finds a
Nash equilibrium of the game and yields superhuman AI agents in two-player
zero-sum games such as Chess, Go and Poker (Campbell et al., 2002; Silver et
al., 2017; Brown & Sandholm, 2018). However, in complex environments self-play
agents typically construct ‘inhuman’ strategies (Carroll et al., 2019). This
may be a benefit for zero-sum games, but is less useful when it is important
to coordinate with, not trick, one’s partner.
Our main contribution is “other-play” (OP), an algorithm for constructing good
strategies for the zero-shot coordination setting. We assume that with every
MG we are provided with a set of symmetries, i.e. arbitrary relabelings of the
state/action space that leave trajectories unchanged up to relabeling. One
source of miscoordination in zero-shot settings is that agents have no good
way to break the symmetries (e.g. should we drive on the left or the right?).
In most MDPs, there are classes of strategies that require more or less
coordinated symmetry breaking. OP’s goal is to find a strategy that is
maximally robust to partners breaking symmetries in different ways while still
playing in the same class. OP works as follows: it uses RL to maximize reward
when matched with agents playing the same policy under a random relabeling of
states and actions under the known symmetries.
Figure 1: The _lever coordination game_ illustrates the counter intuitive
outcome of zero-shot coordination.
To show the intuition behind OP consider the following game: you need to
coordinate with an unknown stranger by independently choosing one from a set
of 10 different levers (Figure 1a). If both of you pick the same lever a
reward of 1 point is paid out, otherwise you leave the game empty-handed.
Clearly, without any prior coordination the only option is to pick one of the
levers at random, leading to an expected reward of $1/10=0.1$.
Next we consider a game that instead only pays $0.9$ for one of the levers,
keeping all other levers unchanged (Figure 1b). How does this change the
coordination problem?
From the point of view of the MG the $1$ payoff levers have no labels and so
are symmetric. Since agents cannot coordinate on how to break symmetries,
picking one of the $1.0$ levers leads to $0.11$ expected return. By contrast,
OP suggests the choice of the $0.9$ lever.
We note that this example illustrates another facet of OP: it is an
equilibrium in meta-strategies. That is, neither agent wishes to deviate from
OP as a reasoning strategy if the other agent is using it.
Note that OP does not use any action labels. Instead OP uses only features of
the problem description to coordinate. Furthermore, note that the OP policy in
this setting is the only policy that would _never_ be chosen by the types of
algorithms that try to use self-play to optimize team performance, e.g. VDN
(Sunehag et al., 2018) or SAD (Hu & Foerster, 2019).
The main contributions of this work are: 1) we introduce OP as a way of
solving the zero-shot coordination problem, 2) we show that OP is the highest
payoff meta-equilibrium for the zero-shot coordination problem, 3) we show how
to implement OP using deep reinforcement learning (deep RL) based methods, and
4) we evaluate OP in the cooperative card game Hanabi (Bard et al., 2020).
## 2 Related Work
### 2.1 Self-Play in Cooperative Settings
There is a large body of research on constructing agents to do well in
positive-sum games. Self-play, if it converges, converges to an equilibrium of
the game and so in purely cooperative games SP agents will be able to
coordinate. Here the main problem is that SP may reach inefficient equilibria
and so there is a large literature on pushing self-play toward higher payoff
equilibria using various algorithmic innovations (Babes et al., 2008; Devlin
et al., 2011; Devlin & Kudenko, 2016; Peysakhovich & Lerer, 2018). However,
the setting where agents play with the same agents they have been trained with
(aka. centralized training with decentralized execution) is quite different
from the zero-shot coordination one which we study.
### 2.2 Cooperation and Coordination
A closely related problem to zero-shot coordination is ad-hoc teamwork (Stone
et al., 2010; Barrett et al., 2011). For example: a robot agent joining
another existing group of agents to play soccer (Barrett et al., 2011). Ad-hoc
teamwork differs from the zero-shot coordination problem in that it is
typically framed as a learning problem of learning the policies/capabilities
of other agents during interaction whereas the pure zero-shot coordination
scenario is one where there is no time to update a fixed policy that is
constructed during training. These problems are closely linked and
incorporating ideas from this literature into algorithms like OP is an
interesting question for future research. However, another difference is that
zero-shot agents only need to coordinate well with teams of agents that are
optimized for the zero-shot setting, rather than arbitrary teams self-play of
agents.
There is recent work looking at the situation where the one RL agent, trained
separately, must join a group of new AI agents or humans (Lerer &
Peysakhovich, 2018; Tucker et al., 2020; Carroll et al., 2019). These papers
focus on using small amounts of observed behavior from partnered test time
agents to either guide self-play to selecting the equilibrium (or “social
convention”) of the existing agents (Lerer & Peysakhovich, 2018; Tucker et
al., 2020) or allow building a human model which can be used to learn an
approximate best response using RL (Carroll et al., 2019). This setting is
related, but zero-shot coordination gives no behavioral data to either agent
to guide self-play or allow building a model of the other agent. Instead,
zero-shot makes the assumption that test-time agents being themselves are
optimized for the zero-shot setting (rather than the SP setting).
### 2.3 Game Theory and Tacit Coordination
Within behavorial game theory a large body of work considers coordination
based on “focal points” or other shared grounding such as the famous “you lost
your friend in New York City, where are you going to meet?” coordination
problem (Schelling, 1980; Mehta et al., 1994). However, such focal points
typically come from the fact that these coordination problems are not just
abstract but grounded in _exogenous_ features, _action labels_ , that are
meaningful due to a prior shared context. The zero-shot coordination setting
thus is a special form of the tacit coordination problem in which there are no
shared exogenous features between the different agents and OP can be thought
of as a way to coordinate in this setting.
There is also a large theoretical literature on learning and evolving
coordination (Nowak, 2006). However, as with the self-play literature, it
focuses on long run outcomes within a single group of agents learning or
evolving together and does not typically focus on the question of engineering
agents as we do.
### 2.4 Predicting Human Decision Making
Clearly, if we were able to accurately predict how our human counterparts are
going to act in any given situation, the zero-shot coordination with human
counterparts would reduce to learning a best response to those predicted
actions. There is a large body of work using formal models to predict and
understand human decision making (Camerer, 2011) and recent work that
incorporates machine learning into this question (Wright & Leyton-Brown, 2010;
Hartford et al., 2016; Peysakhovich & Naecker, 2017; Kleinberg et al., 2017;
Fudenberg & Liang, 2019). However, the majority of this research focuses on
extremely simple settings such as small normal form games (Wright & Leyton-
Brown, 2010; Hartford et al., 2016; Fudenberg & Liang, 2019) or single
decision problems (Peysakhovich & Naecker, 2017; Kleinberg et al., 2017)
rather than complex cooperative settings with partial observability.
### 2.5 Domain Randomization
Our work is also related to the idea of domain randomization (Tobin et al.,
2017). In RL and supervised learning domain randomization tries to make the
realized model invariant to some feature of the environment. For example, an
object detector should be invariant to the exact camera angle from which a
view of an object is captured. OP applies a similar idea: a policy should be
invariant to how an agent’s partner breaks symmetries in the underlying game.
### 2.6 Exploiting Symmetries in Other Contexts
In the single agent context, it is harder to plan in MDPs that have more
states. The idea of abstraction is to use underlying symmetries to ‘compress’
a large MDP into a simpler one, solve for the optimal strategy in the
abstraction, and then lift the strategy to the original MDP. One set of such
methods are MDP homomorphisms (van der Pol et al., 2020; Ravindran & Barto,
2004). These, like OP, use underlying symmetries but their goal is different:
they want to find payoff maximizing policies for single agent decision
problems, while OP seeks to find robust policies for zero-shot coordination.
Note that as the lever game illustrates robust policies are not necessarily
the payoff maximizing ones. In addition, these methods do not solve the
problem of equilibrium selection among ‘symmetric’ policies in games, because
the symmetry in the MDP just becomes a symmetry in the homomorphism.
A similar technique (compress, solve, then lift) is also used for finding Nash
equilibria in large games like poker (Gilpin & Sandholm, 2007). In this case
the abstraction treats ‘isomorphic’ states equally and thus reduces the
effective number of states in the game. Again, the goal is different - poker
abstractions are trying to find Nash equilibrium strategies in the original
game while OP uses symmetries to select among a set of possible equilibria.
## 3 Zero-Shot Coordination
In this paper we study fully cooperative Markov games. To construct this
environment we start out with a Dec-POMDP (Nair et al., 2003) with states
$s_{t}\in\mathcal{S}$. There are $i=1,\cdots N$ agents who each choose
actions, $a^{i}_{t}\in\mathcal{A}$ at each time step.
The game is partially observable, $o^{i}_{t}\sim O(o|i,s_{t})$ being each
agent’s stochastic observation function. At time $t$ each agent has an action-
observation history
$\tau^{i}_{t}=\\{o^{i}_{0},a^{i}_{0},r^{i}_{0},\cdots,o^{i}_{t}\\}$ and
selects action $a_{i}^{t}$ using stochastic policies of the form
$\pi^{i}_{\theta}(a^{i}|\tau^{i}_{t})$. The transition function,
$P(s^{\prime}|s,\mathbf{a})$, conditions on the joint action, $\mathbf{a}$.
The game is fully cooperative, agents share the reward $r_{t}$ which is
conditioned on the joint action and the state. Thus, the goal is to maximize
the expected return $J=\mathbb{E}_{\tau}R(\tau)$, where
$R(\tau)=\sum_{t}\gamma^{t}r_{t}$ is calculated using the discount factor
$\gamma$.
Most work on cooperative MARL focuses on a setting where agents are trained
together, although they must execute their policies independently at least at
test time , _e.g._ (Lowe et al., 2017; Foerster et al., 2018a, b). The goal is
to construct learning rules, i.e. functions that map Markov games to (joint)
policies that select policies for each agent that together maximize expected
discounted return. Because agents are trained together, these policies may be
arbitrarily complex.
We are instead interested in achieving high returns with partners that were
not trained together with our agent. Instead, we will frame the problem as
follows: suppose that multiple independent AI designers will construct agents
that have to interact in various but ex-ante unknown Dec-POMDPs without being
able to coordinate beforehand, what learning rule should these designers agree
on? To make this even more concrete, consider the case of independent
autonomous vehicles made by multiple firms which have to interact in novel
traffic situations on a daily basis.
Figure 2:
The first key concept we introduce is the class of equivalence mappings,
$\Phi$, for a given Dec-POMDP.
Each element of $\Phi$ is a bijection of each of $\mathcal{S}$, $O$, and
$\mathcal{A}$ onto itself, such that it leaves the Dec-POMDP unchanged:
$\displaystyle\phi\in\Phi\iff$ $\displaystyle
P(\phi(s^{\prime})|\phi(s),\phi(a))=P(s^{\prime}|s,a)$ $\displaystyle\land$
$\displaystyle R(\phi(s^{\prime}),\phi(a),\phi(s))=R(s^{\prime},a,s)$
$\displaystyle\land$ $\displaystyle O(\phi(o)|\phi(s),\phi(a),i)=O(o|s,a,i)$
$\displaystyle\text{where equalities apply }\forall{s^{\prime},s,a^{\prime}}$
In other words, $\Phi$ describes the symmetries in the underlying Dec-POMDP.
We note that our notation is heavily overloaded since each $\phi$ can act on
actions, states and the observation function, so $\phi$ shorthand for
$\phi=\\{\phi_{\mathcal{S}},\phi_{\mathcal{A}},\phi_{O}\\}$.
Next, we extend $\phi$ to also act on trajectories:
$\displaystyle\phi(\tau^{i}_{t})=\\{\phi(o^{i}_{0}),\phi(a^{i}_{0}),\phi(r_{0}),\cdots,\phi(o^{i}_{t})\\}.$
At this point an example might be helpful: consider a gridworld with a robot,
shown in Figure 2, that can move in the 4 cardinal directions. In our example
the goal is in the middle of the room, which leaves two axis of symmetry: We
can invert either the x-axis, the y-axis or both, as long as we make the
corresponding changes to the action space, for example mapping “up” to “down”
and vice versa when inverting the y-axis.
In a similar way, we can extend $\phi$ to act on policies $\pi$, as follows:
$\displaystyle\pi^{\prime}=\phi(\pi)\iff\pi^{\prime}(\phi(a)|\phi(\tau))=\pi(a|\tau)\vspace{1mm},\vspace{1mm}\forall\tau,a$
These symmetries are the “payoff irrelevant” parts of the Dec-POMDP. They come
from the fact that the actions and states in the Dec-POMDP do not come with
labels and so taking a policy and permuting it with respect to these
symmetries does not change the outcome of interest: the trajectory and the
reward.
It is precisely these symmetries that can cause problems for self-play trained
agents. Since agents are trained together, they can coordinate on how to break
symmetries. However, there is no guarantee that multiple SP agents trained
separately will break symmetries in the same way. In this case when they are
paired together their policies may fail spectacularly.
The goal of OP, then, will be to build policies which are maximally robust to
this failure mode.
## 4 Other Play
We consider the $2$ agent case for ease of notation with $\pi^{1},\pi^{2}$
denoting each agent’s component of the policy and $\mathbf{\pi}$ denoting the
joint policy.
First, consider self-play (SP) learning rule. This is the learning rule which
tries to optimize the following objective:
$\displaystyle\mathbf{\pi}^{*}=\arg\max_{\mathbf{\pi}}J(\pi^{1},\pi^{2})$ (1)
When the Dec-POMDP is tabular we can solve this via various methods. When it
is not, deep reinforcement learning (deep RL) can be used to apply function
approximation and gradient based optimization of this objective function.
Though there is a large literature focusing on various issues in multi-agent
optimization (Busoniu et al., 2006; Hernandez-Leal et al., 2019), our paper is
agnostic to the precise method used.
These policies can be arbitrary and in complicated Dec-POMDPs multiple maxima
to Equation 1 will often exist. These multiple policies can (and, as we will
see in our experiments, often will) use coordinated symmetry breaking to
receive high payoffs. Therefore, 2 matched, separately trained, SP agents will
not necessarily receive the same payoff with each other as they receive with
themselves.
To alleviate this issue, we need to make the optimization problem more robust
to the symmetry breaking.
Let us consider the point of view of constructing a strategy for agent $1$
where agent $2$ will be the unknown novel partner. The _other-play_ (OP)
objective function for agent $1$ maximizes expected return when randomly
matched with a symmetry-equivalent policy of agent $2$ rather than with a
particular one. In other words, we perform a version of self-play where agents
are not assumed to be able to coordinate on exactly how to break symmetries.
$\displaystyle\mathbf{\pi}^{*}=\arg\max_{\mathbf{\pi}}\mathbb{E}_{\phi\sim\Phi}\hskip
2.84526pt{J(\pi^{1},\phi(\pi^{2}))}$ (2)
Here the expectation is taken with respect to a uniform distribution on
$\Phi$. We call this expected return $J_{OP}$. We will now consider what
policies maximize $J_{OP}$.
###### Lemma 1.
$\displaystyle
J(\pi_{A},\pi_{B})=J(\phi(\pi_{A}^{1}),\phi(\pi_{B}^{2})),\hskip
5.69054pt\forall\phi\in\Phi,\pi_{A},\pi_{B}$
This Lemma follows directly from the fact that the MDP is invariant to any
$\phi\in\Phi$.
###### Lemma 2.
$\displaystyle\\{\phi\cdot\phi^{\prime}:\phi^{\prime}\in\Phi\\}=\Phi\hskip
2.84526pt,\hskip 2.84526pt\forall\phi\in\Phi$
This Lemma follows from the fact that $\Phi$ is a bijection.
###### Proposition 1.
The expected OP return of $\pi$ is equal to the expected return of each player
independently playing a policy $\pi_{\Phi}^{i}$ which is the uniform mixture
of $\phi(\pi^{i})$ for all $\phi\in\Phi$.
###### Proof.
$\displaystyle J_{OP}(\pi)$ $\displaystyle=\mathbb{E}_{\phi\sim\Phi}\hskip
2.84526pt{J(\pi^{1},\phi(\pi^{2}))}$ (3)
$\displaystyle=\mathbb{E}_{\phi_{1}\sim\Phi,\phi_{2}\sim\Phi}\hskip
2.84526pt{J(\phi_{1}(\pi^{1}),\phi_{1}(\phi_{2}(\pi^{2})))}$ (4)
$\displaystyle=\mathbb{E}_{\phi_{1}\sim\Phi,\phi_{2}\sim\Phi}\hskip
2.84526pt{J(\phi_{1}(\pi^{1}),\phi_{2}(\pi^{2}))}$ (5)
$\displaystyle=J(\pi_{\Phi})$ (6)
(4) follows from Lemma 1, (5) follows from Lemma 2.
∎
###### Corollary 1.
The distribution $\pi^{*}_{OP}$ produced by OP will be the uniform mixture
$\pi_{\Phi}$ with the highest return $J(\pi_{\Phi})$.
Let $\mathcal{L}_{i}$ be the set of learning rules which input a Dec-POMDP and
output a policy for agent $i$.
A meta-equilibrium is a learning rule for each agent such that neither agent
can improve their expected payoff by unilaterally deviating to a different
learning rule.
###### Proposition 2.
If agent $i$ uses OP as their learning rule then OP is a payoff maximizing
learning rule for the agent’s partner. Furthermore both agents using OP is the
best possible meta-equilibrium.
###### Proof.
Since the Dec-POMDP has no labels for actions and states, $\mathcal{L}_{i}$
must choose all $\phi(\pi)$ with equal probability. Among these possible
outputs, $\pi^{*}_{OP}$ maximizes the return by Corollary 1. ∎
## 5 Implementing Other Play via Deep RL
We now turn to optimizing the OP objective function. In many applications of
interest the Dec-POMDP is not tabular. Thus, deep RL algorithms use function
approximation for the state space and attempt to find local maxima of Equation
1 using self-play reinforcement learning.
We show how to adapt this method to optimize the other-play objective
(Equation 2). This amounts to applying a very specific kind of asymmetric
domain randomization (Tobin et al., 2017) during training.
During each episode of MARL training, for each agent $i$ a random permutation
$\phi_{i}\in\Phi$ is chosen uniformly iid from $\Phi$, and agent $i$ observes
and acts on $\phi(\mathcal{S},O,\mathcal{A}$). Importantly, the agents act in
different permutations of the same environment.
This environment randomization is equivalent to other-play, because the MDP
remains constant under $\phi_{i}$ while the effect of agent $i$’s policy on
the environment is $\phi_{i}(\pi_{i})$. The fixed points of independent
optimization of $\pi$ under this learning rule will be joint policies where
each $\pi_{i}$ is a BR to the uniform mixture of permutations of partner
policies, i.e. precisely the permutation-invariant equilibria that are the
solutions of other-play.
We note that OP is fundamentally compatible with any type of optimization
strategy and can be applied whenever there are symmetries in the underlying
MDP.
## 6 Experiments
We evaluate OP in two different settings. In each setting we will compare
standard SP against OP. We will perform comparisons of agents trained together
to agents trained separately that are placed into a zero-shot coordination
test game.
### 6.1 Lever Game
We begin with the “lever game” mentioned in the introduction. This environment
is tabular, there are only $10$ actions possible per player. Here, during
training, we use simple joint action learning, compute the true gradient with
respect to the current policy and update. We show training time (i.e. expected
reward with itself) and test time (zero-shot) coordination performance for
both SP (optimizing equation 1) and OP (optimizing equation 2). The code is
available as a notebook online here and can be executed online without
downloading: https://bit.ly/2vYkfI7.
Figure 3 shows the results. As expected, OP agents coordinate on the unique
option of $0.9$ points both during the training phase and at test time. As a
consequence, OP agents can carry out successful zero-shot coordination when
paired with other OP agents.
In contrast, SP agents achieve higher rewards of $1.0$ points during the
training phase but entirely fail to coordinate with other, independently
trained, SP agents.
Figure 3: Train and test performance of self-play and other-play algorithms on
the lever coordination game. Shown is the mean, shading is the standard error
of the mean (s.e.m.), across 30 different seeds.
### 6.2 Hanabi with AI Agents
We now turn to a much more complex environment. We construct agents for the
cooperative card game Hanabi, which has recently been established a benchmark
environment for multi-agent decision making in partially observable settings
(Bard et al., 2020).
Hanabi is a cooperative card game with the interesting twist that players
cannot see their own cards and hence have to rely on receiving information
from the other player (who can see their hand). In Hanabi, there are two main
ways of exchanging information: first of all, players can take costly “hint”
actions that point out subsets of cards based on rank or color. For example,
hinting for “blue” reveals the color of all blue cards. Secondly, observing
the actions themselves can be informative, in particular when players have
pre-established conventions. The goal in Hanabi is to play cards in a legal
order completing stacks of cards, one for each color. There are 5 color and 5
cards and the maximum points is 25. Players will lost a life token if they
play a card out of order. Once they exhaust the deck or lose all 3 lives
(“bomb out”), the game will terminate.
As noted, the vast majority of research on Hanabi has been in the self-play
setting, in which a group of agents is trained to jointly obtain the highest
possible score. To apply OP in Hanabi we note that assuming no side
information, a permutation of the colors of the cards leaves the game
unchanged. We use this as our class of symmetries.
### 6.3 MARL Training Details
OP can be applied on top of any SP algorithm. In Hanabi the Simplified Action
Decoder (SAD) (Hu & Foerster, 2019) method achieves SOTA performance for RL
agents. We use SAD as a base algorithm onto which we add OP. We use the open-
sourced implementation of SAD as well as most of its hyper-parameters but with
two major modifications. First, we use 2 GPUs for simulation instead of 1 as
in the original paper. This doubles the data generation speed and has a
profound effect on reducing the wall clock time required to achieve
competitive performance. Second, we introduce extra hyper-parameters that
control the network architecture to add diversity to the model capacity in
order to better demonstrate the effectiveness of OP. Specifically, the network
can have either 1 or 2 fully connected layers before 2 LSTM layers and can
have an optional residual connection to by-pass the LSTM layers. For SP, we
re-train the base SAD and the SAD + AUX variant proposed in Hu & Foerster
(2019). SAD + AUX is specifically engineered for Hanabi by adding an auxiliary
task to predict whether a card is playable, discardable, or unknown. We train
agents with the aforementioned 4 different network architectures. We run each
hyper-parameter configuration with 3 different seeds and thus 12 models are
produced for each category of {SAD, SAD + AUX, SAD + OP, SAD + AUX + OP}.
### 6.4 Evaluation
We evaluate the models within the same category by pairing different models
together to play the game, a process we refer to as _cross-play_. Clearly, if
independent training runs (“seeds”) from the same training method fail to
coordinate with each other at test time it is unlikely they will coordinate
with agents optimized by through a different process, let alone humans. As
such, cross-play is a cheap proxy to evaluate whether a training method has
potential for zero-shot coordination with human players. Figure 4 shows the
scores obtained between all pairs of agents. Table 1 shows the average within-
pair and cross-play scores. We see that SAD coordinates with itself but fails
to coordinate with any other SAD agent. SAD with OP, however, significantly
improves the cross-play. The effect is especially profound when the model has
limited representation power. The top left corner of the graph, which
corresponds to the simplest models that have only 1 fully connected layers, 2
LSTM layers and no residual connection, shows almost perfect cooperation
scores. With the network growing more complicated, different strategies start
to emerge and the cross-play performance drops. Auxiliary task implicitly
improves cross-play scores by encouraging all agents to act basing on grounded
information and confident predictions. Nonetheless, adding OP to SAD + AUX
further improves performance and achieves the highest cross-play payoffs.
Figure 4: Cross-Play Matrix. Visualization of paired evaluation of different agents trained under the same method. y-axis represents the agent index of player 1 (first mover) and x-axis represents that of player 2. Agents 0-2: 1-layer FC, without residual connection; Agents 3-5: 1-layer FC, with residual connection; Agents 6-8: 2-layer FC, without residual connection; Agents 9-11: 2-layer FC, with residual connection. All agents have 2-layer LSTMs after the FC layers. Each block in the grid is obtained by evaluating the pair on 10K games with different seeds. Please refer to Table 1 for numeric results. Method | Cross-Play | Cross-Play(*) | Self-Play
---|---|---|---
SAD | 2.52 $\pm$ 0.34 | 3.02 $\pm$ 0.39 | 23.97 $\pm$ 0.04
SAD + OP | 15.32 $\pm$ 0.65 | 18.28 $\pm$ 0.36 | 23.93 $\pm$ 0.02
SAD + AUX | 17.65 $\pm$ 0.69 | 21.09 $\pm$ 0.18 | 24.09 $\pm$ 0.03
SAD + AUX + OP | 22.07 $\pm$ 0.11 | 22.49 $\pm$ 0.18 | 24.06 $\pm$ 0.02
Table 1: Cross-Play Performance. The average performance of pairs of agents
that are train with the same method but different network architecture and/or
seeds. Please refer to Figure 4 for visualization of performance for each
individual pair. Cross-Play score is non-diagonal mean of each grid. Cross-
Play(*) is the cross-play score after removing the worst model from the grid.
Self-Play score is the score attained when agents play with the partner they
are trained with.
We can further study the policies resulting from these learning algorithms.
Figure 5 picks the agent with highest cross-play performance in each category
(top row) as well as their worst possible partner (bottom row) and presents
$P(a^{i}_{t}\mid a^{j}_{t-1})$ over a subset of actions averaged over time-
steps in 1000 episodes generated through self-play. In other words, we ask, do
the agents respond very differently to possible actions of their partner? A
large difference indicates that what an agent would do in a situation is very
different from what their partner would do: a recipe for miscoordination!
We see that two paired SAD agents have very different policies and thus
miscoordinate a lot. They also learn “inhuman” conventions that are hard for
human to understand. For example, the agent hints Color5 to indicate
discarding the 1st card while its partner interprets that as playing the 2nd
card. OP eliminates these type of conventions. From the plot and our
experience of playing with the SAD + OP agent, we find that it tends to use
color hints to indicate either that the partner should save the card, or to
disambiguate with a subsequent rank hint. This is not a typical strategy
played by seasoned human players but is easy understand and thus makes the
agent easier to cooperate with. However, due to the way we implement OP in
Hanabi, it is still possible to form secretive conventions such as using all
color hints to indicate a specific move. For example, the worst partner of SAD
+ OP uses all color hints to indicate playing the 5th card.
Figure 5: $P(a^{i}_{t}\mid a^{j}_{t-1})$ Matrices. Each subplot can be used as
a proxy to roughly decipher conventions of an agent. The y-axis means the
action taken at time-step $t$ and x-axis means the percentage of each action
as response at time-step $t+1$. Here we only show a quarter of the matrix that
corresponds to the interaction between color/rank hint and play/discard
positions. C1-C5 and R1-R5 mean hinting 5 different colors and ranks
respectively while D1-D5 and P1-P5 mean discarding and playing 1st-5th cards
of the hand. For each plot, we take an agent and run 1000 episodes of self-
play to compute statistics. The agents that achieved the highest cross-play
scores in Figure 4 are used to generate the top row and their worst partners
are chosen to render the bottom row.
### 6.5 Hanabi with Humans
So far we have focused on AI agents that play other AI agents and have shown
that OP is a meta-equilibrium with respect to learning rules in the zero-shot
setting.
We now ask: do human strategies in Hanabi also have an OP-like quality? In
other words, do OP agents perform well with humans?
To begin to answer this question we recruited 20 individuals from a board game
club. These individuals were familiar with Hanabi but not expert players. We
asked each individual to play a game of Hanabi with two bots, in random order,
using the user interface open-sourced by (Lerer et al., 2019). We note that we
did not provide the participants with any information about the bots, either
regarding their strategy or the method through which they were trained.
For testing we selected our best SAD + AUX + OP agent based on the cross-play
performance (henceforth OP bot). We also have individuals play with the SOTA
self-play agent from (Hu & Foerster, 2019) (henceforth SP bot). We download
models from their GitHub repo and pick the model based on cross-play scores.
For reference, the SP model used here gets 23.99 in self-play and 20.99 in
cross-play with other agents where the only difference among them is seed.
Since in Hanabi the exact deck being used can make a huge difference (for
example, some hands are unwinnable), to reduce the variance of our results we
play each seed by two different players, one for our OP agent, and one for the
control. Importantly, to prevent any adaptation advantages, we alternate the
order between which bot came first across different participants.
Humans achieved an average score of 15.75 (s.e.m. 1.23) with the OP bot and
“bombed out” 45% of games. Thus the OP bot, which has high cross-play scores
with other OP bots is also able to play with humans. Note, in our counting
convention players keep the current score when the bomb out, which be believe
is more appropriate for the zero-shot setting.
By comparison, humans paired with the SP bot achieve an average score of 9.15
(s.e.m. 1.18) and an 85% bomb rate. To the best of our knowledge, the only
other research involving human-AI collaboration in Hanabi is (Eger et al.,
2017). Here, a hand-coded AI agent, designed to play well with humans is used
and achieves an average of around 15.0 points when paired with humans. Beyond
the average scores and “bomb-out” rates, we thus also have access to pairwise
comparisons for the two bots when playing two different people on the same
deck.
Figure 6: Shown are all scores obtained by human testing. Each blue dot is one
seed, when humans are matched with the SP bot (x-axis) and the OP bot
(y-axis).
These preliminary numbers confirm that our OP bot significantly outperformed
the SOTA self-play bot from (Hu & Foerster, 2019) when paired with humans. In
particular, OP won 15 out of the 20 per-seed comparisons and tied in 2 cases,
losing to the control group for 3 seeds (p=0.004111exact binomial test of the
null hypothesis being that P(OP higher score) $\leq$ P(control higher
score).).
Of course, these results do not suggest that OP will work in every zero-shot
coordination where AI agents need to cooperate with humans. However, they are
encouraging and suggest that OP is a fruitful research direction for the
important problem of human-AI coordination.
## 7 Other Attempts
While attempting to make progress on the zero-shot coordination problem in
Hanabi we tried a variety of approaches. Here we discuss some other approaches
that seemed promising but did not yield agents that were able to coordinate
with agents they were not trained with. While this does not necessarily mean
these approaches are doomed to failure, we report these results as information
for other researchers interested in this problem.
In particular we tried multi-agent RL adaptations of cognitive hierarchies
(Stahl, 1993), k-level reasoning (Costa-Gomes et al., 2001) and training a
population of agents. Our original inspiration was that both cognitive
hierarchies and k-level reasoning should reduce the tendency towards arbitrary
symmetry breaking and have been shown to produce human like decision making in
other settings (Wright & Leyton-Brown, 2010). Similarly, population based
approaches are gaining popularity for regularizing communication protocols in
the field of emergent communication, see e.g. (Fitzgerald, 2019; Tieleman et
al., 2018; Lowe et al., 2019). We found that none of these approaches produced
high cross-play performance in Hanabi, which we now consider a necessary
condition for high zero-shot performance with humans. In hind-sight,
considering that all of these approaches would necessarily fail in the matrix
game example, this is not at all surprising. Still, to help future researchers
learn from our endeavours, we are adding all results to the supplementary
material and will open-source the corresponding agents.
## 8 Conclusion
We have shown that a simple expansion of self-play which we call _other-play_
can construct agents that are better able to zero-shot coordinate with
partners they have not seen before. We have proven theoretical properties of
the OP strategy, shown how to implement it with deep RL, and shown in
experiments with the cooperative card game Hanabi that OP can construct robust
agents that can play with other AIs as well as with humans.
We do not claim that OP is a silver bullet for all zero-shot coordination
problems. In particular, because OP is a modification of the SP algorithm, it
can be combined with many of the algorithmic innovations that have been
developed to improve SP in various games (Lanctot et al., 2017; Foerster et
al., 2018a; Lowe et al., 2017; Foerster et al., 2017). Thus, we believe that
this represents an exciting research direction for those interested in moving
deep RL beyond two-player, zero-sum environments to ones involving
coordination and cooperation. Currently we are assuming that the symmetries
$\Phi$ are given to the algorithm. However, in principle, discovering the
symmetries of an MDP is another optimization problem, which opens interesting
avenues for future work.
## Acknowledgements
We would like to thank Noam Brown for encouraging discussions and Pratik
Ringshia for help with the UI for human testing. We would also like to thank
our human participants for offering to help.
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## Appendix A Details on Other Attempts
k | Cross-Play | Self-Play
---|---|---
1 | 1.06 $\pm$ 0.04 | 1.04 $\pm$ 0.06
2 | 0.95 $\pm$ 0.18 | 0.99 $\pm$ 0.33
3 | 1.49 $\pm$ 0.11 | 2.63 $\pm$ 0.34
4 | 2.48 $\pm$ 0.22 | 5.89 $\pm$ 0.75
5 | 2.04 $\pm$ 0.35 | 7.22 $\pm$ 0.56
Table 2: Cognitive Hierarchies Performance. We train CH for 5 levels with 3
seeds. The cross-play and self-play results are computed by averaging scores
of intra-level pairing of agents trained with different seeds. Cross-play
score is averaged over 6 pairs and self-play score is averaged over 3 pairs
for each cell.
In this section we provide more details on other attempts mentioned
previously.
The core idea behind cognitive hierarchies (CH) (Stahl, 1993) and k-level
reasoning (Costa-Gomes et al., 2001) is to train a sequence of $K$ agents of
different capabilities. The final agents may hopefully learn strategies that
cross-play well through such explicit route of evolution. In our
implementation of CH, the first agent $a^{(0)}$ in the sequence is a random
agent that pick actions uniformly regardless of the state. The $k$th agent
$a^{(k)}$ is trained to be the “best response” to the pool of agents
$\\{a^{(0)},...,a^{(k-1)}\\}$. Intuitively, this means that the first agent
will learn to play based only on the hinted facts, i.e. ground information, to
play Hanabi because $a^{(0)}$’s actions contain no intentions nor conventions.
$a^{(2)}$ can then learn to give more useful hints and the subsequent agents
may learn more complicated behaviors. In k-level, the $k$th agent $a^{(k)}$
only learns to best respond $a^{(k-1)}$ and other aspects remain the same.
Because the agents are trained with a random agent $a^{(0)}$ and they will
“bomb out” inevitably, we alter the reward scheme so that the agents receive
reward 0, instead of the negative of current score, when they lose all life
tokens.
The performance of CH is shown in Table 2. The most prominent phenomenon is
that CH converges quite slowly, due to the fact that it needs to cooperate
with a pool of different yet primitive policies. For each level, we train the
models until convergence and 5 levels normally take several days to complete.
For reference, it roughly takes less than 20 hours for SAD and our other-play
agents to reach 23 points in self-play under the same settings and hardware.
The prohibitive cost in time and computation make CH unsuitable for
complicated tasks like MARL in Hanabi. Moreover, even though the self-play
score is low, we can already see a clear performance gap between self-play and
cross-play, making it safe to assume that CH will not work well in zero-shot
coordination.
k | Cross-Play | Self-Play
---|---|---
1 | 0.73 $\pm$ 0.18 | 0.95 $\pm$ 0.36
2 | 0.50 $\pm$ 0.02 | 0.54 $\pm$ 0.13
3 | 2.99 $\pm$ 0.11 | 3.24 $\pm$ 0.24
4 | 1.71 $\pm$ 0.12 | 2.57 $\pm$ 0.11
5 | 6.14 $\pm$ 0.58 | 7.27 $\pm$ 1.48
6 | 2.08 $\pm$ 0.22 | 4.52 $\pm$ 1.05
7 | 6.28 $\pm$ 0.92 | 8.82 $\pm$ 2.17
8 | 1.82 $\pm$ 0.25 | 4.89 $\pm$ 1.95
9 | 6.87 $\pm$ 0.91 | 10.26 $\pm$ 2.52
10 | 2.05 $\pm$ 0.28 | 6.54 $\pm$ 2.65
Table 3: K-Level Performance. We train K-Level with $K=10$ and 3 seeds. The
cross-play and self-play scores are computed by intra-level pairing of agents
trained with different seeds.
In Table 3, we show results of K-Level method trained for 10 levels. Despite
the gap between cross-play and self-play being smaller, this method suffers
from non-monotonic improvements between levels and the same high cost in time
and sample complexity.
| Population1 | Population2
---|---|---
Population1 | 23.46 $\pm$ 0.01 | 19.97 $\pm$ 0.06
Population2 | 20.16 $\pm$ 0.06 | 23.44 $\pm$ 0.01
Table 4: Performance of Population Based Method. Each cell is computed by
pairing all agents from one population with those from the other population
and then average the scores. Diagonal can be seen as self-play score and non-
diagonal can be seen as cross-play score under a population setting.
Figure 7: $P(a^{i}_{t}\mid a^{j}_{t-1})$ matrices of one model from each
population. The semantic of the visualization is identical to Figure 5.
Different from CH and K-Level where agents are trained sequentially,
population based approaches (Fitzgerald, 2019; Tieleman et al., 2018; Lowe et
al., 2019) train agents simultaneously by pairing distinct agents together to
generate samples. We briefly experimented with a simple population setting
where we initialize $N$ different agents with their private replay buffers.
They are uniformly paired together with each other to generate samples and
write their observation action sequences into the their own buffer. Each agent
is optimized independently at each training step. We train 2 populations with
different seeds. Each of them contains 4 agents initialized differently. The
numerical results are shown in Table 4. This method can achieve decent cross-
play scores. It is worth noting that the diversity between the hyper-
parameters of the two populations is much smaller than that of the experiments
shown in Figure 4 so that they are not directly comparable. However, a closer
look at their respective policy through $P(a^{i}_{t}\mid a^{j}_{t-1})$
matrices reveals the problem. The way they use color hints not only differs
greatly from each other but also breaks the color symmetry of the game, which
is the exact problem other-play tries to solve. Qualitatively they are hard
for human to play with. They manage to achieve good cross-play scores because
the agents seldom use color hints.
|
2024-09-04T02:54:57.717177 | 2020-03-06T02:30:15 | 2003.03006 | {
"authors": "Lijiang Geng, Guanyu Hu",
"full_text_license": null,
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26071",
"submitter": "Guanyu Hu",
"url": "https://arxiv.org/abs/2003.03006"
} | arxiv-papers | # Bayesian Spatial Homogeneity Pursuit for Survival Data with an Application
to the SEER Respiratory Cancer Data
Lijiang Geng
Department of Statistics, University of Connecticut, CT 06269, USA Guanyu Hu
Department of Statistics, University of Missouri - Columbia, MO 65211, USA
###### Abstract
In this work, we propose a new Bayesian spatial homogeneity pursuit method for
survival data under the proportional hazards model to detect spatially
clustered patterns in baseline hazard and regression coefficients. Specially,
regression coefficients and baseline hazard are assumed to have spatial
homogeneity pattern over space. To capture such homogeneity, we develop a
geographically weighted Chinese restaurant process prior to simultaneously
estimate coefficients and baseline hazards and their uncertainty measures. An
efficient Markov chain Monte Carlo (MCMC) algorithm is designed for our
proposed methods. Performance is evaluated using simulated data, and further
applied to a real data analysis of respiratory cancer in the state of
Louisiana.
Keywords: Geographically Weighted Chinese Restaurant Process, MCMC, Piecewise
Constant Baseline Hazard, Spatial Clustering
## 1 Introduction
Clinical data on individuals are often collected from different geographical
regions and then aggregated and analyzed in public health studies. The most
popular dataset is the Surveillance, Epidemiology, and End Results (SEER)
program (SEER, 2016) data which routinely collects population-based cancer
patient data from 20 registries across the United States. This data provides
prognostic and demographic factors of cancer patients. In this paper, we focus
our study on Louisiana respiratory cancer data which was analyzed in Mu et al.
(2020). Analysis of such data conducted on a higher level often assume that
covariate effects are constant over the entire spatial domain. This is a
rather strong assumption, as all intrinsic heterogeneities in data are
ignored. For example, if one was to study the hazard for patients with lung
cancer, it is expected that the true hazard is not the same in areas where
there is little air pollution and severe air pollution, even for patients with
similar characteristics. From Tobler’s first law of geography (Tobler, 1970),
it is reasonable to consider similarities between nearby locations in survival
data due to environmental circumstances in geographically close regions. In
this paper, we will recover the spatial homogeneity pattern of respiratory
cancer survival rates among different counties in state of Louisiana.
Existing approaches that account for such patterns in survival data can be put
into two major categories. The first one is to incorporate spatial random
effects in survival models such as the accelerated failure time (AFT) model
and the proportional hazards model (Banerjee et al., 2003; Banerjee and Dey,
2005; Zhou et al., 2008; Zhang and Lawson, 2011; Henderson et al., 2012), such
that spatial variations are accounted for by different intercepts for
different regions, while parameters for covariates are held constant. Another
important approach, instead of assuming all covariate effects are constant,
allows parameters to be spatially varying in parametric, nonparametric, and
semiparametric models (Hu et al., 2020; Hu and Huffer, 2020; Xue et al.,
2020).
Despite their flexibility, the aforementioned spatially varying coefficients
models can be unnecessarily large. Imposing certain constraints on nearby
regions so that they have the same parameter values provides an efficient way
of reducing the model size without sacrificing too much of its flexibility.
While similar endeavor have been made to cluster spatial survival responses
(Huang et al., 2007; Bhatt and Tiwari, 2014), the clustering of covariate
effects and baseline hazards have yet to be studied for survival data.
Two challenges are to be tackled for clustering of coefficients and baseline
hazards for spatial survival models. First, the spatial structure needs to be
appropriately incorporated into the clustering process. Contiguousness
constraints should be added so that truly similar neighbors are driven to the
same cluster. The constraints, however, should not be overly emphasized, as
two distant regions may still share similar geographical and demographical
characteristics and thus parameters. Existing methods, such as in Lee et al.
(2017, 2019) and Li and Sang (2019), do not allow for globally discontiguous
clusters, which is a serious limitation. Second, the true number of clusters
is unknown, and needs to be estimated. With the probabilistic Bayesian
framework, simultaneous estimation of the number of clusters and the
clustering configuration for each region is achieved by complicated search
algorithms (e.g., reversible jump MCMC, Green, 1995) in variable dimensional
parameter spaces. Such algorithms assign a prior to the number of clusters
that needs to be updated in every MCMC iteration, which made them difficult to
implement or automate, and suffer from mixing issues as well as lack of
scalability. Nonparametric Bayesian approaches, such as the Chinese restaurant
process (CRP; Pitman, 1995), provide another approach to allow for
uncertainties in the number of clusters. Its extension, the distance dependent
CRP (ddCRP; Blei and Frazier, 2011), considers spatial information, and makes
a flexible class of distributions over partitions that allows for dependencies
between their elements. The CRP framework, however, has been shown to be
inconsistent in its estimation of number of clusters (Miller and Harrison,
2013). Lu et al. (2018) proposed the powered CRP that suppresses the small
tail clusters. Similar to the traditional CRP, however, it does not consider
distance information, and therefore is not well-suited when spatial
homogeneity is to be detected.
To address these challenges, in this work, we consider a spatial proportional
hazards model, and propose a geographically weighted Chinese restaurant
process (gwCRP) to capture the spatial homogeneity of both the regression
coefficients and baseline hazards over subareas under piecewise constant
hazards models framework (Friedman et al., 1982). Our main contributions in
this paper are three folds. First, we develop a new nonparametric Bayesian
method for spatial clustering which combines the ideas of geographical weights
and Dirichlet mixture models to leverage geographical information. Compared
with existing methods, our proposed approach is able to capture both locally
spatially contiguous clusters and globally discontiguous clusters. Second, an
efficient Markov chain Monte Carlo (MCMC) algorithm is proposed for our
proposed model without reversible jumps to simultaneously estimate the number
of clusters and clustering configuration. In addition, we apply our method to
the analysis of the Surveillance, Epidemiology, and End Results (SEER) Program
data in the state of Louisiana among different counties, which provide
important information to study spatial survival rates.
The remainder of the paper is organized as follows. In Section 2, we develop a
homogeneity pursuit of survival data in the piecewise constant proportional
hazard framework with gwCRP prior. In Section 3, a collapsed Gibbs sampler
algorithm and post MCMC inference are discussed. The extensive simulation
studies are carried out in Section 4. For illustration, our proposed
methodology is applied to respiratory cancer survival data in Section 5.
Finally, we conclude this paper with a brief discussion in Section 6.
## 2 Methodology
### 2.1 Spatial Piecewise Constant Hazards Models
Let $T_{\ell i}$ denote the survival time for patient $\ell$ at location
$s_{i}$, with $\delta_{\ell i}=1$ representing the event and $\delta_{\ell
i}=0$ indicating censored, and $X_{\ell}(s_{i})$ denotes the vector of
covariates corresponding to $T_{\ell i}$ for $i=1,2,...,n$, and
$\ell=1,2,...,n_{i}$, where $n_{i}$ denotes the number of the patients at
location $s_{i}$. In this paper, $s_{1},s_{2},\ldots,s_{n}$ are areal units
which is defined in Banerjee et al. (2014). Let $\bm{D}=\\{(T_{\ell
i},\delta_{\ell i},X_{\ell}(s_{i})),i=1,2,...,n,\ell=1,2,...,n_{i}\\}$. denote
the observed data. We consider a proportional hazards model (Cox, 1972) with
piecewise constant baseline hazard. We partition $[0,\infty)$ into $J$
intervals ($0=a_{0}<a_{1}<\dots<a_{J}=\infty$), then the hazard function is
given by
$\lambda(t|X_{\ell}(s_{i}))=\lambda_{0}(t)\exp(X_{\ell}(s_{i})^{\top}{\mbox{\boldmath$\beta$}}),\quad$
(1)
with piecewise constant baseline hazard function $\lambda_{0}(t)=\lambda_{j}$
for $a_{j-1}\leq t<a_{j},~{}j=1,\ldots,J$.
For the piecewise constant hazard function mentioned in (1), the baseline
hazards $\lambda_{1},\dots,\lambda_{J}$ and regression coefficients $\beta$
are constants over different regions. Due to observed environmental factors,
spatially varying patterns in baseline hazards and regression coefficients of
hazard function need to be considered. The piecewise constant hazard function
with spatially varying pattern is therefore given by
$\lambda(t|X_{\ell}(s_{i}))=\lambda_{0(s_{i})}(t)\exp(X_{\ell}(s_{i})^{\top}{\mbox{\boldmath$\beta$}}(s_{i})),\quad$
(2)
where $\lambda_{0(s_{i})}(t)=\lambda_{j}(s_{i})$ for $a_{j-1}\leq
t<a_{j},~{}j=1,\ldots,J$. Under this model,
${\mbox{\boldmath$\lambda$}}(s_{i})=(\lambda_{1}(s_{i}),\dots,\lambda_{J}(s_{i}))^{\top}$
and ${\mbox{\boldmath$\beta$}}(s_{i})$ represent the location-specific
baseline hazards and regression coefficients.
After some algebra, the logarithm of likelihood function for observed survival
data $\bm{D}$ is obtained as
$\begin{split}&\log\mathcal{L}({\mbox{\boldmath$\beta$}}(s_{i}),{\mbox{\boldmath$\lambda$}}(s_{i}),i=1,\ldots,n\mid\bm{D})\\\
&=\sum_{i=1}^{n}\left\\{\sum_{j=1}^{J}d_{ji}\log\lambda_{j}(s_{i})+\sum_{\ell=1}^{n_{i}}\delta_{\ell
i}X_{\ell}(s_{i})^{\top}{\mbox{\boldmath$\beta$}}(s_{i})-\sum_{j=1}^{J}\lambda_{j}(s_{i})\left[\sum_{\ell=1}^{n_{i}}\Delta_{j}(T_{\ell
i})\exp(X_{\ell}(s_{i})^{\top}{\mbox{\boldmath$\beta$}}(s_{i}))\right]\right\\},\end{split}$
(3)
where $d_{ji}=\sum_{\ell=1}^{n_{i}}\delta_{\ell
i}{\bf{{1}}}_{[a_{j-1},a_{j})}(T_{\ell i})$, which represents the number of
people at location $s_{i}$ who experience the event during the time period
from $a_{j-1}$ to $a_{j}$, and $\Delta_{j}(t)=t-a_{j-1}\text{ for }a_{j-1}\leq
t<a_{j}.$
For one particular location $s_{i}$, let
$\bm{\eta}(s_{i})=\log{\mbox{\boldmath$\lambda$}}(s_{i})$ and define
${\mbox{\boldmath$\theta$}}(s_{i})=({\mbox{\boldmath$\beta$}}(s_{i})^{\top},\bm{\eta}(s_{i})^{\top})^{\top}$
the collection of parameters, then the maximized likelihood estimate (MLE)
$\widehat{{\mbox{\boldmath$\theta$}}}(s_{i})$ can be obtained by solving the
score function, which is the derivative of the logarithm of likelihood
function in (3), and the estimated variance-covariance matrix of MLE is
$\widehat{\Sigma}_{i}=(-H)^{-1}$, respectively, where $(-H)$ denotes the
negative Hessian matrix. Based on the MLEs and estimated variance-covariance
matrices, we have the following approximation of the likelihood.
###### Proposition 1.
We assume the regularity conditions A-D in Friedman et al. (1982). As
$n_{i}\rightarrow\infty,\,i=1\ldots,n$, the data likelihood
$\mathcal{L}({\mbox{\boldmath$\beta$}}(s_{i}),{\mbox{\boldmath$\lambda$}}(s_{i}),i=1,\ldots,n\mid\bm{D})$
is approximated as
$\mathcal{L}({\mbox{\boldmath$\beta$}}(s_{i}),{\mbox{\boldmath$\lambda$}}(s_{i}),i=1,\ldots,n\mid\bm{D})\approx\prod_{i=1}^{n}\text{MVN}(\widehat{{\mbox{\boldmath$\theta$}}}(s_{i})|{\mbox{\boldmath$\theta$}}(s_{i}),\widehat{\Sigma}_{i}),$
(4)
where MVN stands for the multivariate normal distribution.
The derivations of $\widehat{\Sigma}_{i}$ and the proof of Proposition 1 are
given in Section A and Section B of Supporting information. Instead of using
the log likelihood in (3), our following model is based on normal
approximation in Proposition 1 for computational convenience.
Based on the normal approximation given in Proposition 1, a natural way which
follows Gelfand et al. (2003) for spatially varying pattern of baseline
hazards and regression is to give a Gaussian process prior to
${{\mbox{\boldmath$\theta$}}}(s_{i}),\,i=1,\ldots,n$. The Gaussian process for
${{\mbox{\boldmath$\theta$}}}(s_{i}),\,i=1,\ldots,n$ is defined as
${\mbox{\boldmath$\theta$}}\sim\text{MVN}(\bm{1}_{n\times
1}\otimes\bm{\mu},\bm{H}(\phi)\otimes\Sigma),$ (5)
where
${\mbox{\boldmath$\theta$}}=({\mbox{\boldmath$\theta$}}(s_{1})^{\top},\ldots,{\mbox{\boldmath$\theta$}}(s_{n})^{\top})^{\top}$,
$\bm{\mu}$ is a $p+J$ dimensional vector, $\bm{H}(\phi)$ is a $n\times n$
spatial correlation matrix depending on the distance matrix with parameter
$\phi$, $\Sigma$ is a $(p+J)\times(p+J)$ covariance matrix, and $\otimes$
denotes Kronecker product. The $(i,j)$-th entry of $\bm{H}(\phi)$ is
$\exp(-\phi|s_{i}-s_{j}|)$, where $|s_{i}-s_{j}|$ is the distance between
$s_{i}$ and $s_{j}$, and $\phi>0$ is the range parameter for spatial
correlation. For the Gaussian process prior, the parameters of closer
locations have stronger correlations.
For many spatial survival data, some regions will share same covariate effects
or baseline hazards with their nearby regions. In addition, some regions will
share similar parameters regardless of their geographical distances, due to
the similarities of regions’ demographical information such as income
distribution (Ma et al., 2020; Hu et al., 2020), food environment index, air
pollution (Zhao et al., 2020), and etc.. A spatially varying pattern for
${{\mbox{\boldmath$\theta$}}}(s_{i}),\,i=1,\ldots,n$ is not always valid.
Based on the homogeneity pattern, we focus on the clustering of spatially-
varying parameters. In our setting, we assume that the $n$ parameter vectors
can be clustered into $k$ groups, i.e.,
${\mbox{\boldmath$\theta$}}(s_{i})={\mbox{\boldmath$\theta$}}_{z_{i}}$ where
$z_{i}\in\\{1,2,\ldots,k\\}$.
### 2.2 Geographically Weighted Chinese Restaurant Process
A latent clustering structure can be introduced to accommodate the spatial
heterogeneity on parameters of sub-areas. Under the frequentist framework, the
clustering problem could be solved in a two-stage approach: first obtain the
estimate of number of clusters, $\widehat{k}$, then detect the optimal
clustering assignment among all possible clusterings of $n$ elements into
$\widehat{k}$ clusters. However, in this approach, the performance of the
estimation of cluster assignments highly relies on the estimated number of
clusters, it may ignore uncertainty in the first stage and cause redundant
cluster assignments. Bayesian nonparametric method is a natural remedy to
simultaneously estimate the number of clusters and cluster assignments. The
Chinese restaurant process (CRP; Pitman, 1995; Neal, 2000) offers choices to
allow uncertainty in the number of clusters by assigning a prior distribution
on $(z_{1},z_{2},\ldots,z_{n})$. In CRP, $z_{i},~{}i=2,\ldots,n$ are defined
through the following conditional distribution (also called a Pólya urn
scheme, Blackwell et al., 1973).
$\displaystyle P(z_{i}=c\mid
z_{1},\ldots,z_{i-1})\propto\begin{cases}\absolutevalue{c},&\text{at an
existing cluster labeled}\,c,\\\ \alpha,&\text{at a new cluster}.\end{cases}$
(6)
Here $\absolutevalue{c}$ refers to the size of cluster labeled $c$, and
$\alpha$ is the concentration parameter of the underlying Dirichlet process.
Based on the Pólya urn scheme shown in (6), the customers will have no
preference for sitting with different customers. For the spatial survival
data, nearby regions will share similar environmental effects such as P.M.
2.5, water quality, etc.. These similar effects will lead the nearby sub-
regions to share similar parameters. In order to consider similar effects
caused by geographical distance, we modify the traditional CRP to
geographically weighted CRP (gwCRP) so that the customer will have higher
probability sitting with their familiar customers which are geographically
nearby. We have the conditional distribution of $\bm{\theta}(s_{i})$ given
${\mbox{\boldmath$\theta$}}(s_{1}),\ldots,{\mbox{\boldmath$\theta$}}(s_{i-1})$
based on following definition.
###### Definition 1.
If $G_{0}$ is a continuous distribution and $i>1$, the distribution of
$\bm{\theta}(s_{i})$ given $\bm{\theta}(s_{1}),\ldots,\bm{\theta}(s_{i-1})$ is
proportional to
$\displaystyle
f(\bm{\theta}(s_{i})\mid{\mbox{\boldmath$\theta$}}(s_{1}),\ldots,{\mbox{\boldmath$\theta$}}(s_{i-1}))\propto\sum_{r=1}^{K^{*}}\sum_{j=1}^{i-1}w_{ij}\bm{1}(\bm{\theta}(s_{j})=\bm{\theta}^{*}_{r})\delta_{\bm{\theta}^{*}_{r}}(\bm{\theta}(s_{i}))+\alpha
G_{0}(\bm{\theta}(s_{i})),$ (7)
where $f(\cdot)$ is the distribution density function, $K^{*}$ denote the
number of clusters excluding the $i$-th observation,
$\bm{\theta}^{*}_{1},\ldots,\bm{\theta}^{*}_{K^{*}}$ are $K^{*}$ distinguished
values of $\bm{\theta}_{1},\ldots,\bm{\theta}_{i-1}$, $w_{ij}$ is geographical
weight which is calculated by the distance between $s_{i}$ and $s_{j}$, and
$\delta(\cdot)$ is the Dirac measure.
Based on the Definition 1, we have similar Pólya urn scheme called gwCRP for
conditional distribution in (7) with CRP.
###### Proposition 2.
A Pólya urn scheme of gwCRP is defined as
$\displaystyle P(z_{i}=c\mid
z_{1},\ldots,z_{i-1})\propto\begin{cases}\absolutevalue{c^{*}},&\text{at an
existing cluster labeled}\,c,\\\ \alpha,&\text{at a new cluster}.\end{cases}$
(8)
$\absolutevalue{c^{*}}=\sum_{j=1}^{i-1}w_{ij}{\bf{{1}}}(z_{j}=c)$, where
$w_{ij}$ is the geographical weight.
Compared the existing geographically weighted regression literatures, our
weights are obtained by graph distance between different areas. Following Xue
et al. (2020), we denote a graph as $G$, with set of vertices
$V(G)=\\{v_{1},\ldots,v_{n}\\}$, and set of edges
$E(G)=\\{e_{1},\ldots,e_{m}\\}$. The graph distance between two vertices
$v_{i}$ and $v_{j}$ is defined as:
$d_{v_{i}v_{j}}=\begin{cases}|V(e)|,&\text{if }e\text{ is the shortest path
connecting }v_{i}\text{ and }v_{j},\\\ \infty,&\text{if }v_{i}\text{ and
}\,v_{j}\text{ are not connected},\end{cases}$ (9)
where $|V(e)|$ represents the cardinality of edges in $e$. For the county
level data, we construct the graph $G$ based on adjacency matrix among
different counties. We treat $n$ counties as $n$ vertices of this graph and
$v_{i}$ and $v_{j}$ are connected when the corresponding counties share the
boundary. Based on the graph distance calculated by (9), we calculate the
geographical weights by:
$w_{ij}=\begin{cases}1,&\text{ if }d_{v_{i}v_{j}}\leq 1,\\\
\exp(-d_{v_{i}v_{j}}\times h),&\text{ if }1<d_{v_{i}v_{j}},\end{cases}$ (10)
where $d_{v_{i}v_{j}}$ is the graph distance between areas $i$ and $j$. For
the weighting function in (10), we give the largest weight ($w_{ij}\equiv 1$)
for the areas sharing the same boundaries, which follows the first law of
geography (Tobler, 1970). For simplicity, we refer to gwCRP introduced above
as $\text{gwCRP}(\alpha,h)$, where $\alpha$ is the concentration parameter for
Dirichlet distribution and $h$ is the spatial smoothness parameter.
###### Remark 1.
Based on the Pólya urn scheme defined in (8) and geographical weighting scheme
defined in (10), we find that (i) when $h=0$, the gwCRP reduces to traditional
CRP, which leads to over-clustering problem in estimating of the number of
clusters; (ii) when $h\rightarrow\infty$, a new customer just only choose the
table representing spatially contiguous regions. This will also lead to the
same over-clustering problem as CRP.
### 2.3 gwCRP for Piecewise Constant Hazards Models
Adapting gwCRP to the piecewise constant hazards models, our model and prior
can be expressed hierarchically as:
$\displaystyle\begin{split}&\log\mathcal{L}({\mbox{\boldmath$\beta$}}_{z_{i}},{\mbox{\boldmath$\lambda$}}_{z_{i}},i=1,\ldots,n\mid\bm{D})\\\
&=\sum_{i=1}^{n}\left\\{\sum_{j=1}^{J}d_{ji}\log\lambda_{jz_{i}}+\sum_{\ell=1}^{n_{i}}\delta_{\ell
i}X_{\ell}(s_{i})^{\top}{\mbox{\boldmath$\beta$}}_{z_{i}}-\sum_{j=1}^{J}\lambda_{jz_{i}}\left[\sum_{\ell=1}^{n_{i}}\Delta_{j}(T_{\ell
i})\exp(X_{\ell}(s_{i})^{\top}{\mbox{\boldmath$\beta$}}_{z_{i}})\right]\right\\},\\\
&z_{i}\mid\bm{\pi},k\sim\text{Multinomial}(\pi_{1},\cdots,\pi_{k}),\\\
&\bm{\pi}\sim\text{gwCRP}(\alpha,h),\\\
&\bm{\theta}_{r}\sim\mbox{MVN}(0,\Sigma_{0}),\quad r=1,\ldots,k,\end{split}$
(11)
where
${\mbox{\boldmath$\theta$}}_{r}=(\beta_{1r},\ldots,\beta_{pr},\log\lambda_{1r},\ldots,\log\lambda_{Jr})^{\top}$
is a $p+J$ dimensional vector. And let $k\rightarrow\infty$, and $\Sigma_{0}$
be hyperparameter for base distribution of
${\mbox{\boldmath$\theta$}}_{1},\ldots,{\mbox{\boldmath$\theta$}}_{r}$. We
choose $\Sigma_{0}=100\bm{I}$ in all the simulations and real data analysis
providing noninformative priors. The concentration parameter $\alpha$ controls
the probability of introducing a new cluster which is similar with CRP.
Different values of $h$ lead to different weighting scale for different sub-
regions. In our following simulations and real data analysis, we fix
$\alpha=1$ and tune $h$ with different values.
## 3 Bayesian Inference
In this section, we will introduce the MCMC sampling algorithm, post MCMC
inference method, and Bayesian model selection criterion.
Our goal is to sample from the posterior distribution of the unknown
parameters $k$, $\bm{z}=(z_{1},...,z_{n})\in\\{1,...,k\\}^{n}$,
$\bm{\beta}=(\bm{\beta}_{1},\ldots,\bm{\beta}_{k})$, and
$\bm{\lambda}=(\bm{\lambda}_{1},\ldots,\bm{\lambda}_{k})$. Based on
Proposition 1 and Proposition 2, we can efficiently cycles through the full
conditional distributions of $z_{i}|z_{-i}$ for $i=1,2,\ldots,n$ and
$\bm{\beta}^{\top},\log\bm{\lambda}^{\top}$, where
$z_{-i}=\bm{z}\setminus{\\{z_{i}}\\}$. The marginalization over $k$ can avoid
complicated reversible jump MCMC algorithms or even allocation samplers. The
full conditionals of $z_{1},\ldots,z_{n}$ are given in Proposition 3. The
details of sampling algorithm is given in Section C of Supporting information.
###### Proposition 3.
The full conditional distributions $P(z_{i}=c\mid
z_{-i},\widehat{\bm{\theta}},{\mbox{\boldmath$\theta$}})$ of
$z_{1},\ldots,z_{n}$ is given as
$\propto\left\\{\begin{array}[]{ll}&\left(\sum_{j\neq
i}w_{ij}{\bf{{1}}}(z_{j}=c)\right)(2\pi)^{-\frac{p}{2}}|\widehat{\Sigma}_{i}|^{-\frac{1}{2}}\exp\left\\{-\frac{1}{2}\left((\widehat{\bm{\theta}}(s_{i})-{\mbox{\boldmath$\theta$}}_{c})^{\top}\widehat{\Sigma}_{i}^{-1}(\widehat{\bm{\theta}}(s_{i})-{\mbox{\boldmath$\theta$}}_{c})\right)\right\\}\text{
at existing $c$}\\\
&\alpha(2\pi)^{-\frac{p}{2}}|\widehat{\Sigma}_{i}|^{-\frac{1}{2}}|\Sigma_{0}|^{-\frac{1}{2}}|\widehat{\Sigma}_{i}^{-1}+\Sigma_{0}^{-1}|^{-\frac{1}{2}}\exp\left\\{-\frac{1}{2}\left(\widehat{\bm{\theta}}(s_{i})^{\top}(\widehat{\Sigma}_{i}+\Sigma_{0})^{-1}\widehat{\bm{\theta}}(s_{i})\right)\right\\}\text{
if $c$ is a new cluster}\\\ \end{array}\right.$
where $\widehat{\Sigma}_{i}$ is the estimated variance-covariance matrix of
MLE $\widehat{\bm{\theta}}(s_{i})$ for $i=1,\ldots,n$, and $\Sigma_{0}$ is the
variance hyperparameter for the base distribution of
${\mbox{\boldmath$\theta$}}_{1},\ldots,{\mbox{\boldmath$\theta$}}_{r}$.
We carry out posterior inference on the group memberships $z_{1},\ldots,z_{n}$
by using Dahl’s method (Dahl, 2006), which proceeds as follows
1. 1.
Define membership matrices
$B^{(l)}=(B(i,j))_{i,j\in\left\\{1,\ldots,n\right\\}}=(\bm{1}(z_{i}^{(l)}=z_{j}^{(l)}))_{n\times
n}$, where $l=1,\ldots,B$ indexes the number of retained MCMC draws after
burn-in, and $\bm{1}(\cdot)$ is the indicator function.
2. 2.
Calculate the average membership matrix
$\overline{B}=\frac{1}{B}\sum_{l=1}^{L}B^{(l)}$, where the summation is
element-wise.
3. 3.
Identify the most _representative_ posterior sample as the one that is closest
to $\overline{B}$ with respect to the element-wise Euclidean distance
$\sum_{i=1}^{n}\sum_{j=1}^{n}(B^{(l)}(i,j)-\overline{B}(i,j))^{2}$ among the
retained $l=1,\ldots,B$ posterior samples.
Therefore, the posterior estimates of cluster memberships $z_{1},\ldots,z_{n}$
and model parameters $\bm{\theta}$ can be obtained based on the draw
identified by Dahl’s method.
We recast the choice of decaying parameter $h$ as a model selection problem.
We use the Logarithm of the Pseudo-Marginal Likelihood (LPML; Ibrahim et al.,
2001) based on conditional predictive ordinate (CPO; Gelfand et al., 1992;
Geisser, 1993; Gelfand and Dey, 1994) to select $h$. The LPML is defined as
$\text{LPML}=\sum_{i=1}^{N}\text{log}(\text{CPO}_{i}),$ (12)
where $\text{CPO}_{i}$ is the i-th conditional predictive ordinate. Following
Chen et al. (2000), a Monte Carlo estimate of the CPO, within the Bayesian
framework, can be obtained as
$\widehat{\text{CPO}}_{i}^{-1}=\frac{1}{B}\sum_{b=1}^{B}\frac{1}{f(D_{i}|{\mbox{\boldmath$\theta$}}_{z_{i}}^{b})},$
(13)
where $B$ is the total number of Monte Carlo iterations,
${\mbox{\boldmath$\theta$}}_{z_{i}}^{b}$ is the $b$-th posterior sample, and
$f(\cdot)$ is the likelihood function define in (3). An estimate of the LPML
can subsequently be calculated as:
$\widehat{\text{LPML}}=\sum_{i=1}^{N}\text{log}(\widehat{\text{CPO}}_{i}).$
(14)
A model with a larger LPML value is preferred.
## 4 Simulation
### 4.1 Simulation Setting and Evaluation Metrics
In this section, we present simulation studies under four different designs to
illustrate the performance of our proposed gwCRP method and compare with
traditional CRP, in terms of both clustering configuration and estimation of
regression coefficients and piecewise constant baseline hazards under
proportional hazards model. Survival datasets that resemble the SEER
respiratory cancer data for Louisiana are generated. The censoring rate is
around 30%. We design four different geographical clustering patterns in
Louisiana state, which are shown in Figure 1. Designs I and III have three
true clusters, and Designs II and IV have two true clusters. In addition,
Designs II and III both have one cluster consisting of two disjoint areas
since, in practice, it is still possible for two distant counties to belong to
the same cluster. Design IV has two clusters both consisting of disjoint
areas.
Figure 1: Geographical clustering patterns in Louisiana state of simulation
designs (This figure appears in color in the electronic version of this
article, and any mention of color refers to that version.)
For each design, 100 replicate datasets are generated under proportional
hazards model with piecewise constant baseline hazard. In each replicate, we
generate survival data of 60 subjects for each county, including three
regression covariates from $N(0,1)$ i.i.d., survival time and censoring. We
set three pieces for the baseline hazards with cutting points 1.5 and 6 for
all designs, and over four designs, we have three true clusters at maximum,
and the true regression coefficients and baseline hazards used are chosen from
${\mbox{\boldmath$\beta$}}_{1}=(1,0.5,1),{\mbox{\boldmath$\lambda$}}_{1}=(0.045,0.036,0.045)$,
${\mbox{\boldmath$\beta$}}_{2}=(1.5,1,1),{\mbox{\boldmath$\lambda$}}_{2}=(0.045,0.036,0.036)$,
and
${\mbox{\boldmath$\beta$}}_{3}=(2,0.5,1.5),{\mbox{\boldmath$\lambda$}}_{3}=(0.036,0.045,0.0495)$.
Censoring times are generated independently by taking the minimum of 150 and
random values from Exp(0.01) with expectation 100. For each replicate, we set
$\alpha=1$ and run different values of $h$, from 0 to 2 with grid 0.2, and
from 3 to 10 with grid 1, and select the optimal $h$ via LPML. A total of 2000
MCMC iterations are run for each replicate, with the first 500 iterations as
burn-in.
To compare the performance of clustering of gwCRP under different values of
$h$, both estimation of the number of clusters and the matchability of
clustering configurations are reported. In our simulation, we use mean Rand
Index (Rand, 1971) which is obtained by using R-package fossil (Vavrek, 2011)
to measure the clustering performance.
In addition to clustering performance, we further evaluate the estimation
performance of covariates coefficients and baseline hazards, which is assessed
by average of bias (AB) and average of mean squared error (AMSE) defined as
follows. Let $\bm{z}=(z_{1},\ldots,z_{n})$ be the true clustering label
vector, ${\mbox{\boldmath$\theta$}}_{r}(s_{i})$ be the true parameter value of
cluster $r$, $\kappa_{r}=\sum_{i=1}^{64}{\bf{{1}}}(z_{i}=r)$ be the number of
counties in cluster $r$, $r=1,\ldots,k$, $\sum_{r=1}^{k}\kappa_{r}=n$, and for
simulated data set $t$, let
$\widehat{{\mbox{\boldmath$\theta$}}}_{(t)}(s_{i})$ be Dahl’s method estimate
at location $s_{i}$ for $t$-th replicates. Then AB is calculated as
$\text{AB}=\frac{1}{k}\sum_{r=1}^{k}\frac{1}{\kappa_{r}}\sum_{i|z_{i}=r}\frac{1}{100}\sum_{t=1}^{100}(\widehat{{\mbox{\boldmath$\theta$}}}_{(t)}(s_{i})-{\mbox{\boldmath$\theta$}}_{r}(s_{i})),$
and AMSE is calculated as
$\text{AMSE}=\frac{1}{k}\sum_{r=1}^{k}\frac{1}{\kappa_{r}}\sum_{i|z_{i}=r}\frac{1}{100}\sum_{t=1}^{100}(\widehat{{\mbox{\boldmath$\theta$}}}_{(t)}(s_{i})-{\mbox{\boldmath$\theta$}}_{r}(s_{i}))^{2},$
which calculates mean squared errors for each cluster first, and then average
across clusters.
### 4.2 Simulation Results
Figure 2 shows the histogram of $k$ estimates and boxplots of Rand Index under
different $h$ and the optimal selected by LPML for four simulation designs. We
see that when $h=0$, the proposed gwCRP method is identical to the traditional
CRP method, and in this case, CRP always tends to over-cluster and often
yields smaller Rand Index than the results under $h>0$. Another important
trend is that, as $h$ increases, the estimated number of clusters decreases
first and then increases, and the Rand Index increases first and then
decreases as $h$ becomes too large. As we discussed in Remark 1, this is
because when $h$ increases from 0, the spatial patterns in the data is
captured by the proposed gwCRP method. However, as $h\rightarrow\infty$, the
geographical weights $w_{ij}$ for spatial-discontiguous counties become 0,
which means only adjacent counties can be classified into the same cluster,
therefore leading to over-clustering phenomenon again. It is also discovered
that the clustering perfomance under optimal $h$ selected by LPML is very
well, with the probability of selecting true number of clusters always greater
than 0.75, and Rand Index larger than or similar to the highest results
attained by some fixed value of $h$.
Figure 2: Histogram of estimates of $k$ and boxplot of Rand Index under
different $h$ and LPML selection for simulation designs. The average h
selected by LPML is 1.296 in Design 1, 1.412 in Design 2, 1.366 in Design 3,
1.602 in Design 4.
Table 1 summarizes the AB and AMSE results of estimating parameters of gwCRP
under different $h$ for different designs. For simplicity of summary results,
here the AMSE of $\beta$ is the average of AMSE of
$\beta_{1},\beta_{2},\beta_{3}$ since they have similar scales, and the value
of $\log{\mbox{\boldmath$\lambda$}}$ is the average of AMSE of
$\log\lambda_{1},\log\lambda_{2},\log\lambda_{3}$, respectively.
For Designs II, III and IV, the absolute value of AB decrease as $h$ increase
from 0 to moderate values, and increase again as $h$ increase to relatively
large values. For all four designs, the absolute values of AB for $\lambda$’s
of optimal $h$ selected by LPML always are the smallest, and the absolute
values for both $\beta$’s and $\lambda$’s of optimal $h$ selected by LPML are
always smaller than the values of traditional CRP. The patterns in AMSE are
more clear when comparing different methods, that traditional CRP has the
largest AMSE and AMSE decrease as $h$ increase from 0 to moderate values, and
increase again as $h$ increase to relatively large values. The results of
optimal $h$ selected by LPML also has the best performance in estimation.
Table 1: AB and AMSE for parameter estimation under different $h$ and LPML selection for different simulation designs Method | Parameter | Design I | Design II | Design III | Design IV
---|---|---|---|---|---
| | AB | AMSE | AB | AMSE | AB | AMSE | AB | AMSE
gwCRP $h=0.6$ | $\beta$ | -0.0063 | 0.0069 | 0.0038 | 0.0067 | -0.0065 | 0.0083 | -0.0027 | 0.0060
| $\lambda$ | 0.0815 | 0.0193 | 0.0732 | 0.0186 | 0.0797 | 0.0219 | 0.0827 | 0.0190
gwCRP $h=1.2$ | $\beta$ | -0.0078 | 0.0065 | -0.0002 | 0.0058 | -0.0061 | 0.0087 | -0.0017 | 0.0049
| $\lambda$ | 0.0818 | 0.0194 | 0.0775 | 0.0171 | 0.0794 | 0.0217 | 0.0815 | 0.0172
gwCRP $h=2$ | $\beta$ | -0.0096 | 0.0068 | -0.0006 | 0.0055 | -0.0056 | 0.0085 | -0.0051 | 0.0049
| $\lambda$ | 0.0849 | 0.0190 | 0.0814 | 0.0158 | 0.0798 | 0.0216 | 0.0873 | 0.0191
gwCRP $h=6$ | $\beta$ | -0.0061 | 0.0129 | 0.0030 | 0.0072 | -0.0299 | 0.0204 | -0.0064 | 0.0074
| $\lambda$ | 0.0805 | 0.0281 | 0.0770 | 0.0217 | 0.1005 | 0.0296 | 0.0852 | 0.0195
gwCRP Optimal | $\beta$ | -0.0039 | 0.0059 | 0.0042 | 0.0055 | 0.0074 | 0.0067 | -0.0005 | 0.0035
| $\lambda$ | 0.0661 | 0.0177 | 0.0711 | 0.0145 | 0.0732 | 0.0203 | 0.0777 | 0.0177
CRP | $\beta$ | -0.0046 | 0.0086 | 0.0018 | 0.0092 | -0.0056 | 0.0089 | 0.0003 | 0.0082
| $\lambda$ | 0.0760 | 0.0228 | 0.0717 | 0.0233 | 0.0787 | 0.0239 | 0.0742 | 0.0223
A sensitivity analysis regarding $\alpha$ and the weighting function is
conducted. $\alpha=0.5,1,2,5$ and the weighting function
$w_{ij}=\exp(-d_{v_{i}v_{j}}^{2}\times
h^{2}){\bf{{1}}}\\{d_{v_{i}v_{j}}>1\\}+{\bf{{1}}}\\{d_{v_{i}v_{j}}\leq 1\\}$
which has a faster decay to 0 are ran, and all the results are presented in
Section D of Supporting information. The results show that the results of
optimal $h$ selected by LPML are insensitive to the choice of $\alpha$ and
weighting function.
In a brief conclusion based on our simulation studies, gwCRP models have
better performance than CRP both for clustering and parameter estimation. Our
proposed model selection criterion, LPML, can nearly select the best
performing $h$ value for both clustering and parameter estimation.
## 5 SEER Respiratory Cancer Data
### 5.1 Data Description
In this section, we apply our proposed model to analyze respiratory cancer
data in Louisiana state, which is downloaded from the Surveillance,
Epidemiology, and End Results (SEER) Program.We analyzed the survival time of
respiratory cancer patients using the SEER public use data (SEER 1973-2016
Public-Use). We refer to Mu et al. (2020) for the detailed data clean
description. After cleaning, there are 16213 observations left, and the
censoring rate is 30.44%. We select Age, Gender, Cancer grade and Historical
stage of cancer for our analysis, and give the summary of survival times and
covariates in Table 2. The median survival times for patients in each county
are provided in Section E of Supporting information .
Table 2: Demographics for the studied dataset. For continuous variables, the mean and standard deviation (SD) are reported. For binary variables, the frequency and percentage of each class are reported. | Mean(SD) / Frequency (Percentage)
---|---
Survival Time | 22.43 (31.90)
Event | 12.63 (18.32)
Censor | 44.85 (43.06)
Diagnostic Age | 66.55 (11.66)
Sex |
Female | 6548 ($40.39\%$)
Male | 9665 ($59.61\%$)
Cancer Grade |
the class of lower grades | 5307 ($32.73\%$)
the class of III or IV | 10906 ($67.27\%$)
Historical Stage 111Distant stage means that a tumor has spread to areas of the body distant or remote from the primary tumor |
not distant | 9005 ($55.54\%$)
distant | 7208 ($44.46\%$)
We first fit the Cox model of patients for each county using the covariates
selected. The regression coefficients are visualized in Section E of
Supporting information. From results shown in Supporting information, it is
seen that some counties have similar characteristics, no limited to only
adjacent counties, indicating possibilities of globally discontiguous
clusters.
### 5.2 Data Analysis
To select the optimal number of pieces for the baseline hazard and $h$, we run
$J=2$ with the cutpoints $(0,9.01)$, $J=3$ with cutpoints $(0,3.01,9.01)$,
$J=4$ with cutpoints $(0,1.01,4.01,9.01)$, and $J=5$ with cutpoints
$(0,1.01,3.01,5.01,9.01)$. The cutpoints are set by dividing the start point
$0$ and the median survival time $9.01$ by quantiles evenly to ensure there
are events at each piece for each county. For each $J$, we run $h$ from 0 to
10 with grid 0.1, and for each combination of $J$ and $h$, 5000 MCMC
iterations are run and drop the first 2000 as burn-in. The optimal values
selected by LPML is $J=4$ and $h=9.0$, under which the corresponding estimate
of number of clusters is two, while the traditional CRP classifies the
counties into five clusters. The trace plots of different chains of posterior
samples of the estimates for selected counties are presented in the Section E
of Supporting information to show the convergence of the MCMC. The plots of
clustering patterns of CRP and gwCRP Optimal are shown in Figure 3, from which
it is seen that the gwCRP captures the globally discontiguous clusters very
well. The estimates and $95\%$ Credible Intervals of regression covariates
coefficients and baseline hazards obtained by gwCRP Optimal are given in Table
3, from which we see that, though the baseline hazards are similar, the
regression covariates coefficients are quite different across different
clusters. We see that our proposed method successfully detects both spatially
contiguous cluster and discontinuous cluster simultaneously. The parameter
estimates for Age are positive in all counties, indicating that older patients
on average are more likely to have the event than younger patients. For the
counties in cluster 1, the diagnostic ages has higher hazards effects than
other counties. However, for the counties in Cluster 2, male, later cancer
stage will have higher hazards effects than other counties. The historical
distance stage effects are very similar in two clusters which indicates that
the subjects with tumor spreading will have higher hazards.
Figure 3: Clustering patterns of counties in Louisiana state under CRP (when J=4) and gwCRP Optimal (when J=4, h=9.0) methods (This figure appears in color in the electronic version of this article, and any mention of color refers to that version.) Table 3: Estimates results of regression coefficients and baseline hazards obtained by gwCRP Optimal ($J=4,~{}h=9.0$). The $95\%$ Credible Interval for estimates of Cluster One is calculated by the $95\%$ HPD Interval of County 13, and The $95\%$ Credible Interval for estimates of Cluster Two is calculated by the $95\%$ HPD Interval of County 33, where the counties were selected by the minimum Euclidean distance from the posterior mean to the average estimate. Parameter | Cluster 1 | Cluster 2
---|---|---
Estimate | $95\%$ Credible Interval | Estimate | $95\%$ Credible Interval
$\beta_{\text{Age}}$ | 0.1847 | (0.1693, 0.2158) | 0.0728 | (0.0138, 0.2056)
$\beta_{\text{Sex}}$ | 0.1239 | (0.0912, 0.1732) | 0.3411 | (0.1118, 0.5189)
$\beta_{\text{Grade}}$ | 0.5075 | (0.4366, 0.5291) | 0.8290 | (0.4583, 0.9578)
$\beta_{\text{Hist-Stage}}$ | 1.3271 | (1.2926, 1.3824) | 1.4434 | (1.3150, 1.7101)
$\lambda_{1}$ | 1.0690 | (0.9938, 1.0912) | 1.0359 | (0.9060, 1.2330)
$\lambda_{2}$ | 1.0716 | (0.9909, 1.1172) | 1.0877 | (0.8432, 1.3035)
$\lambda_{3}$ | 0.9843 | (0.9787, 1.1040) | 1.0912 | (0.8535, 1.2899)
$\lambda_{4}$ | 1.0040 | (0.9583, 1.0513) | 1.0209 | (0.8724, 1.2091)
## 6 Discussion
In this paper, we proposed a geographically weighted Chinese restaurant
process to capture spatial homogeneity of regression coefficients and baseline
hazards based on piecewise constant hazard model. An efficient MCMC algorithm
is proposed for our methods without complicated reversible jump algorithm.
Extensive simulation results are carried out to show that our proposed method
has better clustering performance than the traditional CRP in spatial
homogeneity pursuit for survival data. Simulation studies also show that our
proposed methods have promising results in coefficients and baseline hazard
estimation. An application to analysis of SEER data provides an interesting
illustration of our proposed methods.
Furthermore, four topics beyond the scope of this paper are worth further
investigation. In this paper, our proposed algorithm is based on two step
estimation under piecewise constant proportional hazard model assumption.
Proposing an efficient sampling algorithm without Laplace approximation is an
important future work. Furthermore, we fixed the number of pieces of baseline
hazards in both simulation studies and real data analysis. Imposing adaptive
number of pieces model in baseline hazards is devoted for future research. In
addition, variable selection approaches based on hierarchical CRP (Griffiths
et al., 2004) is also worth being investigated. Allowing different covariates
and baseline hazard share different clustering processes is also an important
future work.
## Acknowledgement
The authors would like to thank the editor, the associate editors and two
reviewers for their valuable comments which help improve the presentation of
this paper.
## Data Availability Statement
The data that support the findings of this paper are available from the
corresponding author upon reasonable request.
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## Supporting Information
Web Appendices, Tables, and Figures referenced in Sections 2-5 and R scripts
for simulationsand real data examples are available with this paper at the
Biometrics website on WileyOnline Library. The R code for the computations of
this paper is available at https://github.com/lj-geng/GWCRP.
|
2024-09-04T02:54:57.730130 | 2020-03-06T04:51:37 | 2003.03029 | {
"authors": "Vitaly Bergelson and Andreu Ferr\\'e Moragues",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26072",
"submitter": "Andreu Ferr\\'e Moragues",
"url": "https://arxiv.org/abs/2003.03029"
} | arxiv-papers | # An ergodic correspondence principle, invariant means and applications
Vitaly Bergelson Department of Mathematics, Ohio State University, Columbus,
OH 43210, USA<EMAIL_ADDRESS>and Andreu Ferré Moragues
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
<EMAIL_ADDRESS>
###### Abstract.
A theorem due to Hindman states that if $E$ is a subset of $\mathbb{N}$ with
$d^{*}(E)>0$, where $d^{*}$ denotes the upper Banach density, then for any
$\varepsilon>0$ there exists $N\in\mathbb{N}$ such that
$d^{*}\left(\bigcup_{i=1}^{N}(E-i)\right)>1-\varepsilon$. Curiously, this
result does not hold if one replaces the upper Banach density $d^{*}$ with the
upper density $\bar{d}$. Originally proved combinatorially, Hindman’s theorem
allows for a quick and easy proof using an _ergodic_ version of Furstenberg’s
correspondence principle. In this paper, we establish a variant of the ergodic
Furstenberg’s correspondence principle for general amenable (semi)-groups and
obtain some new applications, which include a refinement and a generalization
of Hindman’s theorem and a characterization of countable amenable minimally
almost periodic groups.
## 1\. Introduction
Many results in additive combinatorics are of the form: If
$E\subseteq\mathbb{N}=\\{1,2,\dots,\\}$ is a "large" set, then $E$ is "highly
organized". For example, the celebrated Szemerédi theorem [Sz] states that if
$E$ has positive upper density,
$\bar{d}(E):=\limsup_{N\to\infty}\frac{|E\cap\\{1,\dots,N\\}|}{N}>0$, then $E$
is "AP-rich", meaning that $E$ contains arbitrarily long arithmetic
progressions. An equivalent form of Szemerédi’s theorem is the following: if
$E\subseteq\mathbb{N}$ has positive upper Banach density, i.e.
$d^{*}(E):=\limsup_{N-M\to\infty}\frac{|E\cap\\{M,\dots,N-1\\}|}{N-M}>0$, then
$E$ is AP-rich 111Indeed, one can show that both the $d^{*}$ and $\bar{d}$
versions of Szemerédi’s Theorem are equivalent to the original ”finitistic”
version in [Sz]. See also Theorem 1.5 in [B1]. In [F3], Furstenberg obtained a
proof of Szemerédi’s theorem via the ergodic Szemerédi theorem (EST), which
states that for any probability measure preserving system
$(X,\mathcal{B},\mu,T)$, any $A\in\mathcal{B}$ with $\mu(A)>0$, and any
$k\in\mathbb{N}$, there exists $n\in\mathbb{N}$ such that
$\mu(A\cap T^{-n}A\cap\dots\cap T^{-kn}A)>0.$
Furstenberg’s approach (see [F3]) to deriving Szemerédi’s theorem from his EST
can be described as follows. Suppose that $\bar{d}(E)>0$. Viewing the $0$-$1$
valued sequence $\xi(m)=\mathbb{1}_{E}(m)$, $m\in\mathbb{Z}$, as an element of
the symbolic space $\\{0,1\\}^{\mathbb{Z}}$, and denoting by $T$ the shift
transformation $Tx(n)=x(n+1)$ for all $n\in\mathbb{Z}$, Furstenberg
establishes the existence of a $T$-invariant measure $\mu$ on
$\\{0,1\\}^{\mathbb{Z}}$ as follows. First, by a diagonalization argument, we
can find a common subsequence $(N_{k})$ so that the following limit exists for
a countable dense subset of $C(\\{0,1\\}^{\mathbb{Z}})$
(1.1) $L(f):=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}f(T^{n}\xi).$
Applying a standard approximation argument, we see that formula (1.1) holds
for all $f\in C(\\{0,1\\}^{\mathbb{Z}})$. Now, $L$ is a positive, normalized
functional, and so by Riesz’s representation theorem, there is a Borel
probability measure $\mu$ on $\\{0,1\\}^{\mathbb{Z}}$ such that
$L(f)=\int_{\\{0,1\\}^{\mathbb{Z}}}f\ d\mu$. Let
$X:=\overline{\\{T^{n}\xi:n\in\mathbb{Z}\\}}$ be the orbit closure of $\xi$.
Observe that $\mu$ is supported on $X$. Letting
$A:=X\cap\\{x\in\\{0,1\\}^{\mathbb{Z}}:x(0)=1\\}$, we get
$\mu(A)=\int_{\\{0,1\\}^{\mathbb{Z}}}\varphi\ d\mu=L(\varphi)=\bar{d}(E)>0$,
for $\varphi(x):=x(0)$. (Note that the fact $L(\varphi)=\bar{d}(E)$ follows
from (1.1)).
By the EST there exists some $n\in\mathbb{N}$ such that $\mu(A\cap
T^{-n}A\cap\dots\cap T^{-kn}A)>0$. Then, for any $x\in A\cap
T^{-n}A\cap\dots\cap T^{-kn}A$ we have $x(0)=1,x(n)=1,\dots,x(kn)=1$. Since
$x\in X$, we can choose $l\in\mathbb{N}$ such that $T^{l}\xi$ and $x$ are as
close as we wish. This implies that for some $l\in\mathbb{N}$,
$\mathbb{1}_{E}(l)=\mathbb{1}_{E}(l+n)=\dots=\mathbb{1}_{E}(l+kn)=1$ and hence
$E$ contains the arithmetic progression $\\{l,l+n,\dots,l+kn\\}$ of length
$k+1$.
One can check that the functional $L$ satisfies the identity
(1.2)
$\lim_{k\to\infty}\frac{|E\cap(E-n)\cap\dots\cap(E-kn)\cap[1,N_{k}]|}{N_{k}}=\\\
L(\varphi\cdot T^{n}\varphi\cdot\dotso\cdot
T^{kn}\varphi)=\int_{\\{0,1\\}^{\mathbb{Z}}}\mathbb{1}_{A}(x)\cdot\mathbb{1}_{A}(T^{n}x)\cdot\dotso\cdot\mathbb{1}_{A}(T^{kn}x)\
d\mu.\\\ \textrm{ (see \cite[cite]{[\@@bibref{}{fsz}{}{}]} p. 210)}$
Now, from (1.2) we can derive the inequality
(1.3) $\bar{d}(E\cap(E-n)\cap\dots\cap(E-kn))\geq\mu(A\cap
T^{-n}A\cap\dots\cap T^{-kn}A).$
The foregoing discussion leads to the more general principle:
###### Theorem 1.1 (Furstenberg’s correspondence principle for $\bar{d}$.
(cf. [B1], Theorem 1.1)).
Let $E\subseteq\mathbb{N}$ with $\bar{d}(E)>0$. Then, there is an invertible
measure preserving system $(X,\mathcal{B},\mu,T)$ and a set $A\in\mathcal{B}$
with $\mu(A)=\bar{d}(E)>0$ satisfying
(1.4) $\bar{d}(E\cap(E-h_{1})\cap\dotso\cap(E-h_{r}))\geq\mu(A\cap
T^{-h_{1}}A\cap\dotso\cap T^{-h_{r}}A)$
for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$.
###### Remark 1.2.
All proofs of the above result known to the authors also give a version of
(1.4) for unions:
(1.5) $\bar{d}(E\cup(E-h_{1})\cup\dots\cup(E-h_{r}))\geq\mu(A\cup
T^{-h_{1}}A\cup\dots\cup T^{-h_{r}}A)$
for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$. See the
discussion and ramifications of this fact below.
A priori, one could expect that, given $E$ with $\bar{d}(E)>0$, it is possible
to judiciously choose the system $(X,\mathcal{B},\mu,T)$ to be ergodic in the
above construction. It turns out that, this is not always the case. To see
this, we will invoke the following interesting result of Hindman:
###### Theorem 1.3 (Hindman’s covering theorem [H2]).
Let $E\subseteq\mathbb{N}$ be a set with $d^{*}(E)>0$. Then, for every
$\varepsilon>0$ there is some $N\in\mathbb{N}$ such that
(1.6) $d^{*}\left(\bigcup_{i=1}^{N}(E-i)\right)>1-\varepsilon.$
Curiously enough, Theorem 1.3 fails to be true if one replaces $d^{*}$ with
$\bar{d}$. Consider, for example, the following set $E\subseteq\mathbb{N}$
provided by Hindman in [H2]:
(1.7) $E:=\bigcup_{n\in\mathbb{N}}[2^{2n},2^{2n+1}).$
Then $\bar{d}(E)=\frac{2}{3}$, and one can check that, moreover,
$\bar{d}(\bigcup_{i=0}^{N}(E-i))=\frac{2}{3}$ for all $N\in\mathbb{N}$. It
follows that for this set $E$ any measure preserving system
$(X,\mathcal{B},\mu,T)$ satisfying (1.5) cannot be ergodic. The reason is that
for an ergodic system $(X,\mathcal{B},\mu,T)$, if $\mu(A)>0$, then
$\lim_{N\to\infty}\mu\left(\bigcup_{i=1}^{N}T^{-i}A\right)=1$. Assuming the
inequality (1.5) is valid for the system in question, we would have
(1.8)
$\bar{d}\left(\bigcup_{i=1}^{N}(E-i)\right)\geq\mu\left(\bigcup_{i=1}^{N}T^{-i}A\right),$
and this cannot hold in our example since the left hand side is bounded away
from $1$.
However, by appropriately amplifying the proof of Theorem 1.1 discussed above,
one can obtain the following ergodic variant thereof. This amplification will
be carried out in the appropriate generality in Sections 2 and 6.
###### Theorem 1.4 (Ergodic Furstenberg’s correspondence principle. (cf.
[BHK], Proposition 3.1)).
Let $E\subseteq\mathbb{N}$ be such that $d^{*}(E)>0$. Then, there is an
ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ and a set
$A\in\mathcal{B}$ with $\mu(A)=d^{*}(E)>0$ satisfying
(1.9) $d^{*}(E\cap(E-h_{1})\cap\dotso\cap(E-h_{r}))\geq\mu(A\cap
T^{-h_{1}}A\cap\dotso\cap T^{-h_{r}}A)$
for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$.
Similarly to the situation with Theorem 1.1, one can show that a version of
(1.9) holds for unions, i.e.
(1.10) $d^{*}(E\cup(E-h_{1})\cup\dots\cup(E-h_{r}))\geq\mu(A\cup
T^{-h_{1}}A\cup\dots\cup T^{-h_{r}}A)$
for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$. This
observation was made and utilized in [BBF] (cf. [BBF] Prop. 2.3).
The functions $\bar{d}$ and $d^{*}$ have very similar properties. For example,
$\bar{d}$ and $d^{*}$ satisfy $d^{*}(\mathbb{N})=1$ and
$\bar{d}(\mathbb{N})=1$ and are shift-invariant (i.e.
$\bar{d}(E-n)=\bar{d}(E)$ for all $n\in\mathbb{N}$ and $d^{*}(E-n)=d^{*}(E)$
for all $n\in\mathbb{N}$), which allows one to view $\mathbb{N}$ as a
generalized probability space with either $\bar{d}$ or $d^{*}$ serving as an
(admittedly vague) substitute for the probability measure. Moreover, either of
Theorems 1.1 and 1.4 can be used to derive Szemerédi’s theorem from EST.
While in certain investigations (see [HF1], [HF2], [Fra], [GLR]), it is of
interest to consider the non-ergodic measure preserving systems corresponding
to sets satisfying $\bar{d}(E)>0$ (or more generally, sets with the property
$\bar{d}_{(I_{N})}(E):=\limsup_{N\to\infty}\frac{|E\cap I_{N}|}{|I_{N}|}>0$,
where $(I_{N})$ is a sequence of intervals with increasing length), in some
other situations, $d^{*}$ allows for stronger/sharper results. One such
application was obtained in [BHK]. Also, observe that Theorem 1.4 immediately
implies (via (1.10)) Theorem 1.3. The ergodic approach to Theorem 1.3 has two
additional advantages. First, it will allow us to characterize sequences
$(n_{k})_{k\in\mathbb{N}}$ with the "Hindman property", i.e., sequences
$(n_{k})_{k\in\mathbb{N}}$ such that for any $E\subseteq\mathbb{Z}$ with
$d^{*}(E)>0$ one has that for all $\varepsilon>0$ there is some
$N\in\mathbb{N}$ such that
(1.11) $d^{*}\left(\bigcup_{k=1}^{N}(E-n_{k})\right)>1-\varepsilon,$
(see (1.6)). Second, the robustness of the ergodic approach will enable us to
formulate and prove with ease a Hindman-like result for any amenable group. (A
different proof of this result was obtained in [BBF] through combinatorial
methods).
While the fact that Furstenberg’s correspondence principle works equally well
for unions was not observed/utilized in the early papers in ergodic Ramsey
theory, in hindsight this observation is quite natural if one takes into
account the algebraic nature of this principle. The versatility of
Furstenberg’s correspondence principle can be perhaps best of all perceived
via Gelfand’s representation theorem. The possibility of using Gelfand’s
representation theorem was mentioned in [F4] and was explicitly implemented in
[B1], [BF], [BCRZ], and [B2]. See also the appendix to Section I in [F2]. We
would like to point out that the correspondence principle was not born with
the ergodic theoretical proof of Szemerédi’s theorem; indeed, a form of it
appears already in Furstenberg’s thesis [F1], where it was used as a tool to
reconstruct a stationary process from its past. More concretely, the seminal
idea which was put to action in [F1] was to replace the approximate measure
space $\mathbb{Z}$ together with the density preserving transformation
$x\mapsto x+1$ by a genuine measure preserving system, namely: the orbit
closure of the sequence $(\mathbb{1}_{E}(n))_{n\in\mathbb{Z}}$ in
$\\{0,1\\}^{\mathbb{Z}}$, where $\mathbb{1}_{E}$ corresponds to the given time
series.
It follows from Gelfand’s representation theorem that any commutative, unital,
countably generated $C^{*}$-algebra $\mathcal{A}$ is topologically and
algebraically isomorphic to the algebra of continuous functions $C(X)$, where
$X$ is a compact metric space. In our situation, such a $C^{*}$-algebra can be
naturally generated by the family $(\mathbb{1}_{E-n})_{n\in\mathbb{Z}}$, where
$E\subseteq\mathbb{Z}$ satisfies
$d_{(I_{N})}(E):=\lim_{N\to\infty}\frac{|E\cap I_{N}|}{|I_{N}|}>0$ for some
sequence of intervals $(I_{N})$ with $|I_{N}|\to\infty$. Let us denote this
$C^{*}$-algebra by $\mathcal{A}_{E}$.
One can then refine $(I_{N})$ to obtain a subsequence $(I_{N_{k}})$ so that
for any set $F$ in the Boolean algebra $\mathcal{B}_{E}$ generated by the
family $(E-n)_{n\in\mathbb{Z}}$, $d_{(I_{N_{k}})}(F)$ is well defined. In
other words, $d_{(I_{N_{k}})}(\cdot)$ is a _shift-invariant density_ on
$\mathcal{B}_{E}$. In turn, $d_{(I_{N_{k}})}(\cdot)$ induces a shift-invariant
mean on $\mathcal{A}_{E}$, i.e., a positive functional
$L:\mathcal{A}_{E}\rightarrow\mathbb{C}$ such that $L(1)=1$ and for any
$F\in\mathcal{B}_{E}$, $L(\mathbb{1}_{F})=d_{(I_{N_{k}})}(F)$. By Gelfand’s
representation theorem, there exists a compact metric space $X$ such that
$\mathcal{A}_{E}$ is algebraically and topologically isomorphic to $C(X)$. Let
$\tilde{L}:C(X)\rightarrow\mathbb{C}$ be the linear functional on $C(X)$
induced by $L$. By Riesz’s representation theorem, the functional $\tilde{L}$
is given by a Borel probability measure on $X$. Let
$\Gamma:\mathcal{A}_{E}\rightarrow C(X)$ denote the Gelfand isomorphism. Then,
for all $\varphi\in\mathcal{A}_{E}$ we have
$L(\varphi)=\tilde{L}(\Gamma(\varphi))=\int_{X}\Gamma(\varphi)\ d\mu.$
Since $\mathbb{1}_{E}$ is an idempotent, and since $\Gamma$ is, in particular,
an algebraic isomorphism, the image, $\Gamma(\mathbb{1}_{E})$, is again an
idempotent, and hence of the form $\mathbb{1}_{A}$, where
$A\in\textrm{Borel}(X)$ and satisfies
$\mu(A)=\tilde{L}(\mathbb{1}_{A})=\tilde{L}(\Gamma(\mathbb{1}_{E}))=d_{(I_{N})}(E)$.
The $L$-invariant shift operator given by $\varphi(n)\mapsto\varphi(n+1)$,
$\varphi\in\mathcal{A}_{E}$ induces a $\mathbb{Z}$-action on $C(X)$, which is,
by a theorem of Banach, induced by a measure preserving homeomorphism
$T:X\rightarrow X$. Let $\varphi=\mathbb{1}_{E-h_{1}\cap\dots\cap
E-h_{r}}=\prod_{i=1}^{r}\mathbb{1}_{E-h_{i}}$,
$h_{1},\dots,h_{r}\in\mathbb{Z}$. Since $\mathcal{A}_{E}$ is an algebra, it is
clear that $\varphi\in\mathcal{A}_{E}$. This leads us to the equality
(1.12) $d_{(I_{N_{k}})}(E\cap E-h_{1}\cap\dots\cap
E-h_{r})=L(\mathbb{1}_{E\cap E-h_{1}\cap\dots\cap E-h_{r}})=\mu(A\cap
T^{-h_{1}}A\cap\dots\cap T^{-h_{r}}A).$
Linearity and the inclusion-exclusion principle imply that functions of the
form $\varphi=\mathbb{1}_{E\cup E-h_{1}\cup\dots\cup E-h_{r}}$ are also in
$\mathcal{A}_{E}$. Since $\Gamma$ is an algebraic isomorphism, we see that a
formula similar to (1.12) holds for unions as well:
(1.13) $d_{(I_{N_{k}})}(E\cup E-h_{1}\cup\dots\cup
E-h_{r})=L(\mathbb{1}_{E\cup E-h_{1}\cup\dots\cup E-h_{r}})=\mu(A\cup
T^{-h_{1}}A\cup\dots\cup T^{-h_{r}}A).$
As we already mentioned above, all known proofs of Furstenberg’s
correspondence principle are such that they allow for (1.5) and (1.10). We
explained above how one can get (1.5) with the help of Gelfand’s transform,
but, as it will be seen in Sections 2 and 6 (where we will prove and juxtapose
general versions of Theorems 1.1 and 1.4 in the setup of amenable groups), the
other methods also have an algebraic aspect which is adequate for our
purposes. In any case, whatever method is used, it still has to be properly
modified and amplified to allow for ergodicity. The approach that we choose in
Section 2 uses symbolic dynamics and goes along the lines of the proof of the
correspondence principle in [F3], which is sketched above. As we will see, it
allows to conveniently localize the place in the proof where the amplification
leading to ergodicity has to be made.
At this point, we would like to make a comment which explains why utilizing
$d^{*}$ allows one to establish the ergodic version of Furstenberg’s
correspondence principle. The key advantage of dealing with $d^{*}$ is that,
in the course of proving the correspondence principle, one can keep changing
the sequence of intervals along which the averaging scheme is applied, which
is essential for applicability of the ergodic decomposition. On the other
hand, the proof of the $\bar{d}$ version of Furstenberg’s correspondence
priniciple is based on refining the averaging scheme given by the sequence of
intervals along which $\bar{d}_{(I_{N})}(E)>0$. Theorems 1.1 and 1.4 will be
generalized, juxtaposed and exploited in Section 2.
The structure of the paper is as follows. In Section 2 we give a proof of a
general version of Theorem 1.4, which works for any combination of unions,
intersections and complements of shifts of $E$ for countable amenable groups.
Moreover, we also give a proof of a general version of Theorem 1.1, comparing
the two methods of proof, and pinpointing what exactly in the proof allows us
to have ergodicity.
In Section 3 we give a quick proof of a general form of Hindman’s theorem 1.3
for countable cancellative amenable semigroups in Section 3. We also
generalize the example (1.7) to countable abelian groups and to finitely
generated virtually nilpotent groups. In Section 3 we prove a few more
combinatorial results which make use of the "ergodic" nature of $d^{*}$.
Section 4 is devoted to the characterization of sequences $(n_{k})$ for which
(1.11) holds and is followed by Section 5, where we give a characterization of
countable amenable WM groups via the “Hindman property” with the help of a
general version of Theorem 1.4 proved in Section 2. A group $G$ is called
_weakly mixing_ or WM if any measure preserving ergodic action of $G$ on a
probability space is automatically weakly mixing 222Such groups also appear in
the literature under the name _minimally almost periodic groups_.. For example
$A(\mathbb{N})$, the group of finite even permutations of $\mathbb{N}$ is a
countable amenable WM group. It is worth noting that in [BCRZ] a different
characterization of WM groups is obtained via another result due to Hindman,
namely his "finite sums" theorem [H1].
Lastly, in Section 6 we establish a general form of an ergodic Furstenberg
correspondence principle for discrete amenable semigroups and derive as a
corollary a general form of Hindman’s theorem.
###### Remark 1.5.
Throughout this paper, all the measures used are normalized so that
$\mu(X)=1$.
## 2\. Furstenberg’s correspondence principle for amenable groups: $\bar{d}$
and $d^{*}$ versions
The goal of this section is two-fold. First, we will formulate and prove
"amenable" versions of Theorems 1.1 and 1.4 (Theorems 2.3 and 2.8) that (i)
encompass not only intersections but unions of sets and their complements, and
(ii) are valid for general discrete countable amenable groups. Second, we will
pinpoint the distinction between $\bar{d}$ and $d^{*}$ which allows for the
stronger, ergodic version of Furstenberg’s correspondence principle (see
Theorem 2.8 below).
A definition of amenability which is convenient for our purposes uses the
notion of Følner sequence. A sequence $(F_{N})$ of finite non-empty subsets of
a countable group $G$ is a _(left) Følner sequence_ if
$\lim_{N\to\infty}\frac{|F_{N}\Delta gF_{N}|}{|F_{N}|}=0.$
for all $g\in G$. A countable group $G$ is _amenable_ if it admits a (left)
Følner sequence 333One can show that every amenable group $G$ admits also
right- and indeed two-sided analogues of left Følner sequences (see Corollary
5.3 in [N]). Throughout this paper we deal only with left Følner sequences and
will routinely omit the adjective ”left”..
To facilitate the discussion, we will present first the versions of Theorems
1.1 and 1.4 (see Theorems 2.3 and 2.8) for general countable amenable groups.
The proofs are in essence the same as those of Theorems 1.1 and 1.4, but we
will need this generality for the applications in the forthcoming sections.
We will then juxtapose the proofs of Theorems 2.3 and 2.8 which will allow us
to explain what exactly in the proof of Theorem 2.8 leads to the ergodicity of
the system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$.
To formulate Theorems 2.3 and 2.8 we need a few definitions:
###### Definition 2.1.
Let $E\subseteq G$ and let $(F_{N})$ be a Følner sequence. We denote by
$\bar{d}_{(F_{N})}(E)$ the upper density of the set $E$ along $(F_{N})$. This
notion of largeness is given by the formula
$\bar{d}_{(F_{N})}(E):=\limsup_{N\to\infty}\frac{|F_{N}\cap E|}{|F_{N}|}$
We are now in a position to define upper Banach density in this more general
context:
###### Definition 2.2.
Let $E\subseteq G$. We denote by $d^{*}(E)$ the upper Banach density of the
set $E$, which is given by
$d^{*}(E):=\sup\\{\bar{d}_{(F_{N})}(E):(F_{N})\textrm{ is a F\o lner
sequence}\\}.$
We begin with a short proof (based on the idea of the proof of Theorem 1.1 in
[F3] and [F4] 555We could also use for our goals the proofs of the amenable
version of Theorem 1.1 given in [BMc] (Theorem 2.1) and [B2] (Theorem 4.11).
(see also [FKO])) of a generalization of Theorem 1.1 for countable amenable
groups. Note that Theorem 1.1 corresponds to the special case $G:=\mathbb{Z}$
and $F_{N}:=[a_{N},b_{N}]$ with $b_{N}-a_{N}\to\infty$.
In what follows we will use the notation $A^{1}=A$ and $A^{0}=A^{c}$.
###### Theorem 2.3 (Furstenberg Correspondence Principle for countable
amenable groups, $\bar{d}_{(F_{N})}$ version).
Let $(F_{N})$ be a Følner sequence in a countable amenable group $G$ and let
$E$ be a subset of $G$ with $\bar{d}_{(F_{N})}(E)>0$. Then there exist a
probability measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
and a set $A\in\mathcal{B}$ with $\mu(A)=\bar{d}_{(F_{N})}(E)$ such that
(2.1)
$\mu((T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}})\leq\bar{d}_{(F_{N})}(g_{1}^{-1}E^{w_{1}}\star\dots\star
g_{k}^{-1}E^{w_{k}}),$
for all $k\in\mathbb{N}$, all $\\{g_{1},\dots,g_{k}\\}\subset G$ and all
$(w_{1},\dots,w_{k})\in\\{0,1\\}^{k}$, and where each of the stars denotes
either union or intersection with the understanding that
* (i)
for all $1\leq i\leq k-1$, the operation represented by $\star$ which stands
between $E^{w_{i}}$ and $E^{w_{i+1}}$ is the same as the operation appearing
between $A^{w_{i}}$ and $A^{w_{i+1}}$.
* (ii)
the choices of parentheses which are needed to make the expressions on both
sides of formula (2.1) well defined also match. 666For example, we have
$\mu(((T_{g_{1}})^{-1}A\cup(T_{g_{2}})^{-1}A^{c})\cap(T_{g_{3}})^{-1}A)\leq\bar{d}_{(F_{N})}((g_{1}^{-1}E\cup
g_{2}^{-1}E^{c})\cap g_{3}^{-1}E)\textrm{ and }$
$\mu((T_{g_{1}})^{-1}A\cup((T_{g_{2}})^{-1}A^{c}\cap(T_{g_{3}})^{-1}A))\leq\bar{d}_{(F_{N})}(g_{1}^{-1}E\cup(g_{2}^{-1}E^{c}\cap
g_{3}^{-1}E)).$
###### Proof.
Let $X=\\{0,1\\}^{G}$ (viewed as a compact metric space with the usual product
topology). Let $(T_{g})_{g\in G}$ be the action of $G$ on $X$ by
homeomorphisms defined by the formula $(T_{g}x)_{g_{0}}=x_{gg_{0}}$ for all
$g,g_{0}\in G$. Define $\omega\in X$ by setting $\omega(g)=1$ if $g\in E$ and
$\omega(g)=0$ otherwise. Put $A=\\{x\in X:x(e)=1\\}$ (Here and elsewhere, $e$
denotes the neutral element of the group $G$). Note that $A$ is a clopen set
in $X$ (and hence $\mathbb{1}_{A}$ is a continuous function). Moreover, we
have that $T_{g}\omega\in A$ if and only if $g\in E$. Let $(F_{N_{k}})$ be a
subsequence such that
$\bar{d}_{(F_{N})}(E)=\lim_{k\to\infty}\frac{|E\cap F_{N_{k}}|}{|F_{N_{k}}|}.$
Let $\mu$ be any weak* limit point of the sequence of measures
$\frac{1}{|F_{N_{k}}|}\sum_{g\in F_{N_{k}}}\delta_{T_{g}\omega}.$
Moreover, since $C(Y)$ is separable, there exists a further subsequence of
$(F_{N_{k}})$, which we will, in order not to overload the notation, still
denote by $(F_{N_{k}})$, such that, in the weak* topology,
$\mu=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}\delta_{T_{g}\omega}$. Clearly, $\mu$ is a $G$-invariant probability
measure on $X$. We claim that $\mu(A)=\bar{d}_{(F_{N})}(E)$. Indeed, since
$\mathbb{1}_{A}$ is a continuous function, we can write
(2.2) $\mu(A)=\int_{X}\mathbb{1}_{A}\
d\mu=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}\mathbb{1}_{A}(T_{g}\omega)=d_{(F_{N_{k}})}(E)=\bar{d}_{(F_{N})}(E).$
Now let $g_{1},\dots,g_{k}\in G$ and consider the set
$(T_{g_{1}})^{-1}A^{w_{1}}\star\dotso\star(T_{g_{k}})^{-1}A^{w_{k}}$, where
the stars denote an arbitrary fixed choice of either union or intersection.
This is a clopen set in $X$, so its indicator function is, again, continuous.
We have
$\mu\left((T_{g_{1}})^{-1}A^{w_{1}}\star\dotso\star(T_{g_{k}})^{-1}A^{w_{k}}\right)=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}\mathbb{1}_{(T_{g_{1}})^{-1}A^{w_{1}}\star\dotso\star(T_{g_{k}})^{-1}A^{w_{k}}}(T_{g}\omega)$
$=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}|(g_{1}^{-1}E^{w_{1}}\star\dotso\star
g_{k}^{-1}E^{w_{k}})\cap
F_{N_{k}}|\leq\bar{d}_{(F_{N})}(g_{1}^{-1}E^{w_{1}}\star\dotso\star
g_{k}^{-1}E^{w_{k}}).$
We are done. ∎
###### Remark 2.4.
Observe that when $G:=\mathbb{Z}$ and $F_{N}:=[a_{N},b_{N}]$ (where
$b_{N}-a_{N}\to\infty$) inequality (2.1) implies the inequalities (1.4) and
(1.5). As we saw in the Introduction with the help of Theorem 1.3, for some
sets $E\subseteq\mathbb{Z}$ no system $(X,\mathcal{B},\mu,T)$ satisfying
inequality (1.5) can be ergodic. The general version of Theorem 1.1 which also
involves complements of sets, allows us to arrive at the same conclusion
without invoking Theorem 1.3. Indeed, we see that (for $G:=\mathbb{Z}$ and
$F_{N}:=[1,N]$) inequality (2.1) implies
(2.3) $\bar{d}(E^{c}\cap(E-h))\geq\mu(A^{c}\cap T^{-h}A)$
for all $h\in\mathbb{Z}$. Take $E$ as in the Introduction (see (1.7)). Then,
$\mu(A)=\bar{d}(E)=\frac{2}{3}$, so $\mu(A^{c})>0$. However, one can easily
check (see also Section 3), that $\bar{d}(E^{c}\cap(E-h))=0$ for all
$h\in\mathbb{Z}$. This contradicts inequality (2.3).
###### Remark 2.5.
Let $G$ be a countable amenable group. Call a set $E$ a _Hindman set_ if there
exists some Følner sequence $(F_{N})$ such that
$0<\bar{d}_{(F_{N})}(E)<1\textrm{ and }\bar{d}_{(F_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)<\frac{3}{4}$
for all finite sets $F\subseteq G$. One can show (see Proposition 3.3) that
any countable abelian group has a Hindman set.
It is interesting to observe that if our countable amenable group $G$ admits a
Hindman set, then any measure preserving system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ satisfying inequalities (2.1) cannot be
ergodic. This can be seen in two ways. First, using the special case of
inequality (2.3) for unions, i.e.
(2.4) $\bar{d}_{(F_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)\geq\mu\left(\bigcup_{g\in F}g^{-1}A\right)\textrm{ for all
finite sets }F\subseteq G,$
and arguing, as in the Introduction, that inequality (2.4) together with the
fact that $\bar{d}_{(F_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)=\bar{d}_{(F_{N})}(E)<1$ contradict ergodicity.
Alternatively, we can use another special case of inequality (2.3), namely:
$\bar{d}_{(F_{N})}(E^{c}\cap g^{-1}E)\geq\mu(A^{c}\cap g^{-1}A)\textrm{ for
all }g\in G,$
together with the fact that if $E$ is a Hindman set then
$\bar{d}_{(F_{N})}(E^{c}\cap g^{-1}E)=0$ for all $g\in G$. (This discussion
will be completed in Section 3).
We move now to an ergodic version of Theorem 1.4 for general countable
amenable groups. The ergodic amplification of Theorem 2.3 hinges on two
additional tools: the ergodic decomposition for amenable group actions, and a
result about quasi-generic points (see Proposition 2.7 below).
###### Definition 2.6.
Let $X$ be a compact metric space on which $G$ acts by homeomorphisms. Let
$\mu$ be a $G$-invariant measure. We say that $x_{0}\in X$ is _quasi-generic_
for $\mu$ if there exists a Følner sequence $(F_{N})$ such that
$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}f(T_{g}x_{0})=\int_{X}f\
d\mu$
for every $f\in C(X)$.
The following Proposition is an amenable version of Proposition 3.9 in [F5].
We include the proof for the reader’s convenience.
###### Proposition 2.7.
Let $(T_{g})_{g\in G}$ be an action of $G$ by homeomorphisms on a compact
metric space $X$. Let $x_{0}\in X$, and let $Y:=\overline{\\{T_{g}x_{0}:g\in
G\\}}$. Suppose that $\mu\in\mathcal{M}(Y)$ is an ergodic $G$-invariant
measure. Then $x_{0}$ is quasi-generic for $\mu$.
###### Proof.
Since $\mu$ is an ergodic measure, it follows by the mean ergodic theorem
(which can be proved in the same way as the classical mean ergodic theorem for
isometries of Hilbert spaces, see, for example, Theorem 4.15 in [B2]), that
for any Følner sequence $(F_{N})$, and any $f\in L^{2}(\mu)$
$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}f(T_{g}x)=\int_{Y}f\
d\mu,$
where the convergence is in the $L^{2}(\mu)$-norm. Thus, there exists some
subsequence $(F_{N_{k}})$ along which we have pointwise convergence for all
$f$ in a countable dense subset of $C(Y)$, which in turn, by a simple triangle
inequality argument, implies that, for any $f\in C(Y)$ we have
$\lim_{N\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}f(T_{g}x)=\int_{X}f\ d\mu,$
for a.e. $x\in Y$ (and so a.e. $x\in Y$ is quasi-generic for $\mu$).
Let $x_{1}\in Y$ be quasi-generic for $\mu$ along the Følner sequence
$(F_{N_{k}})$. Take a countable dense set $\\{f_{k}:k\in\mathbb{N}\\}$ in
$C(Y)$ such that
(2.5) $\left|\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}f_{j}(T_{g}x_{1})-\int_{Y}f_{j}\ d\mu\right|<\frac{1}{k}$
for all $k\in\mathbb{N}$ and all $j=1,\dots,k$. Since the functions $f_{j}$
are continuous, we can pick $(g_{k})\subseteq G$ such that the inequality
(2.5) holds if we replace $x_{1}$ by $T_{g_{k}}x_{0}$, which after a change of
variables in the sum yields
$\left|\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}g_{k}}f_{j}(T_{g}x_{0})-\int_{Y}f_{j}\ d\mu\right|<\frac{1}{k},$
which implies that
$\lim_{k\to\infty}\frac{1}{|F_{N_{k}}g_{k}|}\sum_{g\in
F_{N_{k}}g_{k}}f(T_{g}x_{0})=\int_{Y}f\ d\mu$
for all $f\in C(Y)$. In other words, $x_{0}$ is a quasi-generic point for
$\mu$ with respect to the Følner sequence $(F_{N_{k}}g_{k})$. Indeed, observe
that for all $g\in G$ we have that $|gF_{N_{k}}g_{k}\Delta
F_{N_{k}}g_{k}|=|gF_{N_{k}}\Delta F_{N_{k}}|$, which implies that
$(F_{N_{k}}g_{k})$ is still a Følner sequence ∎
We are now ready to formulate and prove the amenable ergodic Furstenberg’s
Correspondence Principle, adapting arguments from both [BHK] and [BBF]. We
note that Theorem 1.4 corresponds to the special case $G:=\mathbb{Z}$ in
Theorem 2.8.
###### Theorem 2.8 (Ergodic Furstenberg Correspondence Principle for
countable amenable groups; $d^{*}$ version).
Let $E$ be a subset of a countable amenable group $G$ with positive upper
Banach density. Then there exists an ergodic probability measure preserving
system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ such that
(2.6)
$\mu((T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}})\leq
d^{*}(g_{1}^{-1}E^{w_{1}}\star\dots\star g_{k}^{-1}E^{w_{k}})$
for all $k\in\mathbb{N}$, all $\\{g_{1},\dots,g_{k}\\}\subset G$ and all
$(w_{1},\dots,w_{k})\in\\{0,1\\}^{k}$, and where each of the stars denotes
either union or intersection with the understanding that
* (i)
for all $1\leq i\leq k-1$, the operation represented by $\star$ which stands
between $E^{w_{i}}$ and $E^{w_{i+1}}$ is the same as the operation appearing
between $A^{w_{i}}$ and $A^{w_{i+1}}$.
* (ii)
the choices of parentheses which are needed to make the expressions on both
sides of formula (2.1) well defined also match.
###### Proof.
We start as in the proof of Theorem 2.3. Let $X=\\{0,1\\}^{G}$ and
$(T_{g})_{g\in G}$ be the action of $G$ on $X$ by homeomorphisms defined by
the formula $(T_{g}x)_{g_{0}}=x_{gg_{0}}$ for all $g,g_{0}\in G$. Define, as
before, $\omega\in X$ by setting $\omega(g)=1$ if $g\in E$ and $\omega(g)=0$
otherwise.
Let $Y=\overline{\\{T_{g}\omega:g\in G\\}}$. Put $A=\\{x\in Y:x(e)=1\\}$.
Then, $A$ is a clopen set in $Y$. Moreover, we have that $T_{g}\omega\in A$ if
and only if $g\in E$. Let $(F_{N})$ be a Følner sequence such that
$d^{*}(E)=\bar{d}_{(F_{N})}(E)$. Let $\nu$ be any weak* limit point of the
sequence of measures
$\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\delta_{T_{g}\omega}.$
Then, as in Theorem 2.3, $\nu$ is a $G$-invariant probability measure on $Y$
such that $\nu(A)=d^{*}(E)$.
By the ergodic decomposition theorem (see Theorem 4.2 in [V]), there is a
probability measure $\lambda$ on the set of ergodic normalized measures
$\mathcal{M}_{G}(X)$ such that
(2.7) $\nu(C)=\int_{\mathcal{M}_{G}(X)}\mu_{z}(C)\ d\lambda(z)$
for all $C\in\mathcal{B}$. It follows from the equality (2.7) that there
exists some $z$ such that $\mu_{z}(A)\geq\nu(A)=d^{*}(E)$.
We show that the measure $\mu_{z}$, which we will now denote by $\mu$, works.
Let $g_{1},\dots,g_{k}\in G$ and observe that the set
$(T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}}$ is a
clopen set in $Y$, so its indicator function is continuous.
By Proposition 2.7, there exists a Følner sequence $(G_{N})$, with respect to
which the point $\omega$ is quasi-generic for the measure $\mu$. This implies
that
(2.8)
$\mu((T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}})=\lim_{N\to\infty}\frac{1}{|G_{N}|}\sum_{g\in
G_{N}}\mathbb{1}_{T_{g_{1}}^{-1}A^{w_{1}}\star\dots\star
T_{g_{k}}^{-1}A^{w_{k}}}(T_{g}\omega)=\\\
\lim_{N\to\infty}\frac{1}{|G_{N}|}\left|(g_{1}^{-1}E^{w_{1}}\star\dots\star
g_{k}^{-1}E^{w_{k}})\cap G_{N}\right|\leq
d^{*}(g_{1}^{-1}E^{w_{1}}\star\dots\star g_{k}^{-1}E^{w_{k}}).$
In particular, letting $k=1,w_{1}=1$ and $g_{1}=e$ in inequality (2.8) we
obtain $\mu(A)\leq d^{*}(E)$. Recalling the previous inequality
$\mu(A)\geq\nu(A)=d^{*}(E)$ we see that $\mu(A)=d^{*}(E)$, so we are done. ∎
We conclude this section with some comments on why the utilization of $d^{*}$
allows us to achieve in Theorem 2.8 the goal of ergodicity (whereas, as we saw
above, there are sets $E$ for which Theorem 2.3 cannot guarantee it). In both
proofs, we start with a Følner sequence $(F_{N})$ wich satisfies
$\bar{d}_{(F_{N})}(E)>0$ (in Theorem 2.3) or $d^{*}(E)=\bar{d}_{(F_{N})}(E)$
(in Theorem 2.8). Then we consider weak* limits of the sequences of measures
$\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\delta_{T_{g}\omega}$
($\omega=(\mathbb{1}_{E}(g))_{g\in G}$) along a subsequence of $(F_{N})$. The
advantage of $d^{*}$ comes in handy when, after invoking the ergodic
decomposition in the proof of Theorem 2.8, we start using shifts
$(F_{N_{k}}g_{k})$ of a relevant subsequence $(F_{N_{k}})$ in order to use
quasi-genericity of $\omega$, as guaranteed by Proposition 2.7. Unlike the
proof of Theorem 2.3 where we are restricted to a given Følner sequence
$(F_{N})$ and its subsequences, in the proof of Theorem 2.8, we are
conveniently passing to a different Følner sequence without affecting the
value of $d^{*}$ for expressions of the form
$g_{1}^{-1}E^{w_{1}}\star\dotso\star g_{k}^{-1}E^{w_{k}}$ 777This property of
$d^{*}$, when applied to unions, is also behind Hindman’s proof of Theorem
1.3..
## 3\. Hindman’s Theorem via ergodic theory and some other consequences of
the ergodic version of Furstenberg’s correspondence principle
In this section we will give a short proof of a natural generalization of
Hindman’s theorem to the context of countable amenable groups. Strictly
speaking, Hindman’s theorem, as formulated in the introduction (Theorem 1.3)
deals with $(\mathbb{N},+)$, which is a semigroup rather than a group, but the
results of this section are easily adjusted so that they hold for countable
cancellative amenable semigroups. See the explanatory remark at the end of the
section (see also Section 6 for a general semigroup version). We also deduce
other ergodic-flavored corollaries which provide additional evidence to our
claim that $d^{*}$ is better suited for applications.
We begin with showing that a general version of Hindman’s theorem (see Theorem
1.6 in the introduction) is an immediate corollary of Theorem 2.8:
###### Theorem 3.1.
Let $G$ be a countable amenable group, and let $E$ be a subset of $G$ with
$d^{*}(E)>0$. Then, for every $\varepsilon>0$, there exists $k\in\mathbb{N}$
and $g_{1},\dots,g_{k}\in G$ such that $d^{*}(g_{1}^{-1}E\cup\dots\cup
g_{k}^{-1}E)>1-\varepsilon$.
###### Proof.
By Theorem 2.8 there exists an ergodic system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ such that
$\mu(A)=d^{*}(E)$ and for which
$d^{*}(h_{1}^{-1}E\cup\dots\cup
h_{k}^{-1}E)\geq\mu((T_{h_{1}})^{-1}A\cup\dots\cup(T_{h_{k}})^{-1}A)$
for all $k\in\mathbb{N}$ and $h_{1},\dots,h_{k}\in G$. By ergodicity of the
action $(T_{g})_{g\in G}$ we have
$\mu\left(\bigcup_{g\in G}(T_{g})^{-1}A\right)=1,$
and so since $G$ is countable, the result follows. ∎
Another corollary that can be obtained from Theorem 2.8, is an ergodicity-like
statement for the group $G$. Namely:
###### Corollary 3.2.
Let $G$ be a countable amenable group, and let $E\subseteq G$ be such that
$d^{*}(E)\in(0,1)$. Then there exists some $g\in G$ such that $d^{*}(E^{c}\cap
g^{-1}E)>0$.
###### Proof.
Let $E\subseteq G$ be such that $d^{*}(E)\in(0,1)$. By Theorem 2.8 we can find
an ergodic measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
and a set $A\in\mathcal{B}$ such that $\mu(A)=d^{*}(E)$ and for all $h\in G$,
$d^{*}(E^{c}\cap h^{-1}E)\geq\mu(A^{c}\cap(T_{h})^{-1}A).$
Since $\mu(A)>0$ and $\mu(A^{c})>0$ and the action $(T_{g})_{g\in G}$ is
ergodic, there is some $g\in G$ such that $\mu(A^{c}\cap(T_{g})^{-1}A)>0$, so
we are done. ∎
We proceed to show that an example similar to (1.7) exists in the context of
countable abelian groups.
###### Proposition 3.3.
Let $G$ be a countably infinite abelian group. Let $(F_{N})\subseteq G$ be a
Følner sequence. Then, there exists a set $E\subseteq G$ such that
(3.1) $\bar{d}_{(F_{N})}(E)>0,\textrm{ and
}\bar{d}_{(F_{N})}\left(\bigcup_{g\in F}g^{-1}E\right)<\frac{3}{4},$
for all finite sets $F\subseteq G$.
###### Proof.
We will show first that the assertion of the proposition holds for a
particular Følner sequence, and then upgrade it to an arbitrary Følner
sequence.
Assume first that $G$ is finitely generated and $\\{g_{1},\dots,g_{k}\\}$
generate $G$. Then, one of the generators of $G$, say $g_{1}$ has infinite
order. Consider the Følner sequence
$G_{N}:=\\{g_{1}^{a_{1}}\dotso g_{k}^{a_{N}}:0\leq a_{i}\leq 2^{2N}\textrm{
for }1\leq i\leq N\\}$. Define
$A_{N}=\\{g_{1}^{a_{1}}\dotso g_{k}^{a_{N}}:2^{2N-1}\leq a_{1}\leq
2^{2N}\textrm{ and }0\leq a_{i}\leq 2^{2N}\textrm{ otherwise }\\}$ and put
$E=\bigcup_{N\in\mathbb{N}}A_{N}$. It is not hard to check that the set $E$
satisfies (3.3).
Assume now that $G$ is infinitely generated, and let
$\\{g_{n}:n\in\mathbb{N}\\}$ be a set of generators for the group $G$. We
distinguish two cases. In the first case, one of the generators, say $g_{1}$
has infinite order. Consider the Følner sequence
$G_{N}:=\\{g_{1}^{a_{1}}g_{2}^{a_{2}}\cdot\dotso\cdot g_{N}^{a_{N}}:0\leq
a_{i}\leq 2^{2N},\textrm{ for }1\leq i\leq N\\},$
and set
(3.2) $A_{N}:=\\{g_{1}^{a_{1}}g_{2}^{a_{2}}\cdot\dotso\cdot
g_{N}^{a_{N}}:2^{N}\leq a_{1}\leq 2^{2N},\textrm{ and }0\leq a_{i}\leq
2^{2N}\textrm{ for }2\leq i\leq N\\}.$
Letting $E:=\bigcup_{N\in\mathbb{N}}A_{N}$, we get
$\bar{d}_{(G_{N})}(E)=\bar{d}_{(G_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)<\frac{3}{4},$
for all finite sets $F\subseteq G$.
Now assume that all elements of $G$ have finite order and that the enumeration
of $(g_{n})$ is such that $\textrm{ord}(g_{n+1})\geq\textrm{ord}(g_{n})$ for
all $n\in\mathbb{N}$. Consider the Følner sequence
$G_{N}:=\\{g_{1}^{a_{1}}\cdot\dotso\cdot g_{2^{2N}}^{a_{N}}:0\leq
a_{i}\leq\textrm{ord}(g_{i})\textrm{ for all }1\leq i\leq 2^{2N}\\}$
and the sets
$A_{N}:=\\{g_{1}^{a_{1}}\cdot\dotso\cdot g_{2^{2N}}^{a_{N}}:0\leq a_{i}\leq
b_{i}\textrm{ for all }1\leq i\leq 2^{2N},\ \textrm{with }i\textrm{ even;
}a_{i}=0\textrm{ with }i\textrm{ odd }\\},$
where $b_{i}$ is chosen so that $\frac{\left|(\bigcup_{j=1}^{i}A_{j})\cap
G_{i}\right|}{|G_{i}|}\in(\frac{1}{4},\frac{1}{2})$. Letting
$E:=\bigcup_{N\in\mathbb{N}}A_{N}$, we get that
$\bar{d}_{(G_{N})}(E)\geq\frac{1}{4},\quad\textrm{ and
}\quad\bar{d}_{(G_{N})}\left(\bigcup_{g\in F}g^{-1}E\right)<\frac{3}{4},$
for all finite subsets $F\subseteq G$.
Let now $(F_{N})$ be an arbitrary Følner sequence. We indicate next how to
construct the set $E$ that satisfies (3.1) for $(F_{N})$. To do this, we need
the following general fact, which follows from Lemma 4.1 in [DHZ] (we thank
Tomasz Downarowicz for providing this information).
###### Fact 3.4.
Let $\varepsilon>0$. We say that a family of finite sets
$\\{A_{n}:n\in\mathbb{N}\\}$ is $\varepsilon$-disjoint if each set $A_{n}$
admits a subset $A_{n}^{\prime}$ such that
$|A_{n}^{\prime}|\geq(1-\varepsilon)|A_{n}|$, and such that the new family
$\\{A_{n}^{\prime}:n\in\mathbb{N}\\}$ is disjoint. Fix a Følner sequence
$(F_{N})$ and let $(\varepsilon_{N})$ be a sequence of positive numbers with
$\varepsilon_{N}\to 0$ as $N\to\infty$. Given another Følner sequence
$(G_{N})$, we can find a Følner sequence $(G_{N}^{\prime})$ that is equivalent
to $(G_{N})$ (i.e., satisfying $\frac{|G_{N}\Delta
G_{N}^{\prime}|}{|G_{N}|}\to 0$ as $N\to\infty$) such that for each
$N\in\mathbb{N}$, $G_{N}^{\prime}$ is a union of $\varepsilon_{N}$-disjoint
subsets of the form $\\{F_{N}g:g\in G\\}$.
It is not hard to see that Fact 3.4 implies that the set $E$ in question can
be constructed with the help of the same argument utilized above for the
special Følner sequence $(G_{N})$. ∎
###### Remark 3.5.
For given $0<a\leq b<1$, one can construct sets $E\subseteq\mathbb{Z}$ with
$\bar{d}(E)=a$ such that $\bar{d}(\bigcup_{i=1}^{N}(E-i))=b$, for all large
enough $N$. This can be done as follows. Start with a sequence of disjoint
intervals $[a_{n},b_{n}]$ such that the set
$A:=\bigcup_{n\in\mathbb{N}}[a_{n},b_{n}]$ satisfies $\bar{d}(A)=b$ and
$\bar{d}(\bigcup_{i=1}^{N}(A-i))=b$ for all $N\in\mathbb{N}$ (this can be done
by imitating Hindman’s construction for $b=\frac{2}{3}$). Now let
$\beta=\frac{b}{a}$ and let $E:=A\cap\\{\lfloor
n\beta\rfloor:n\in\mathbb{N}\\}$. One easily checks that $E$ satisfies the
required properties.
###### Remark 3.6.
It is worth noting that the set $E$ constructed in Proposition 3.3 also
satisfies (as in Hindman’s original example)
(3.3) $\bar{d}_{(F_{N})}(E^{c}\cap g^{-1}E)=0$
for all $g\in G$.
It is of interest to know whether the phenomenon exhibited in Proposition 3.3
takes place in non-commutative amenable groups. We cannot show this in
complete generality, but for virtually nilpotent groups, we have the following
theorem:
###### Theorem 3.7.
Let $G$ be a finitely generated, countable virtually nilpotent group. Let
$(F_{N})$ be a Følner sequence. Then, there exists a set $E\subseteq G$ with
(3.4) $\bar{d}_{(F_{N})}(E)>0\textrm{ and
}\bar{d}_{(F_{N})}\left(\bigcup_{g\in B}g^{-1}E\right)<\frac{3}{4},$
for all finite subsets $B\subseteq G$.
###### Sketch of the proof.
First, by Fact 3.4, it suffices to show the result for a particular Følner
sequence $(F_{N})$ of our choosing. Let $F:=\\{x_{1},\dots,x_{k}\\}$ be a set
of generators for $G$. Then, the sequence
$\left(\bigcup_{j=1}^{n}F^{j}\right)$ (i.e. words of length at most $n$
generated by $F$) is a Følner sequence of polynomial growth.
Let $F_{N}:=\bigcup_{j=1}^{2^{2N}}F^{j}$, and let $A_{N}$ the set of words in
$F$ of length between $2^{2N-1}$ and $2^{2N}$. Then the set
$E:=\bigcup_{N\in\mathbb{N}}A_{N}$ satisfies (3.4). ∎
We will describe now another interesting application of Theorem 2.8
###### Definition 3.8.
Let $E\subseteq G$. We denote by $\mathcal{A}_{E}$ the algebra of sets
generated by shifts of the set $E$, i.e. sets of the form $\\{g^{-1}E:g\in
G\\}$ with the help of the operations of union, intersection and complement.
###### Theorem 3.9.
Let $E\subseteq G$ be such that $d^{*}(E)>0$. Then, there exists a Følner
sequence $(G_{N})_{N\in\mathbb{N}}$ in $G$, such that for all Følner sequences
$(F_{N})_{N\in\mathbb{N}}$, for all $E_{1},E_{2}\in\mathcal{A}_{E}$, we have
(3.5) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}d_{(G_{K})}(E_{1}\cap
g^{-1}E_{2})=d_{(G_{K})}(E_{1})d_{(G_{K})}(E_{2}).$
Moreover, if we let $E_{1}=E_{2}=E$ in (3.5), we get
(3.6) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}d_{(G_{K})}(E\cap
g^{-1}E)=d_{(G_{K})}(E)^{2}=d^{*}(E)^{2}.$
###### Proof.
Let $E\subseteq G$ with $d^{*}(E)>0$ and $E_{1},E_{2}\in\mathcal{A}_{E}$. By
Theorem 2.8 there exists an ergodic measure preserving system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ satisfying (2.6). Let $A_{1},A_{2}\in\mathcal{B}$ be the
sets corresponding to the sets $E_{1},E_{2}$. By the proof of Theorem 2.8, the
functions $\mathbb{1}_{A_{1}}$ and $\mathbb{1}_{A_{2}}$ are continuous. Let
$(G_{K})$ be a Følner sequence such that
$\mu=\textrm{w*-}\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\delta_{T_{g}\omega}$, where $\omega=(\mathbb{1}_{E}(g))_{g\in G}$.
Thus, we can write
(3.7) $\mu(A_{1}\cap g^{-1}A_{2})=\lim_{N\to\infty}\frac{1}{|G_{N}|}\sum_{g\in
G_{N}}\mathbb{1}_{A_{1}\cap g^{-1}A_{2}}(T_{g}\omega)=d_{(G_{K})}(E_{1}\cap
g^{-1}E_{2}).$
Taking an average over $g\in G$ in (3.7) along any left Følner sequence
$(F_{N})$ in $G$ immediately yields (3.5), given that the action
$(T_{g})_{g\in G}$ is ergodic. By construction of $\mu$, we have that
$\mu(A)=d_{(G_{K})}(E)=d^{*}(E)$ whence (3.6) also follows. ∎
We remark that (3.5) and (3.6) do not hold for arbitrary sequences $(G_{N})$:
let $E$ be as in (1.7), take $E_{1}=E_{2}=E$ and put $G_{N}=[1,2^{2N}]$.
Nonetheless, we have the following Proposition:
###### Proposition 3.10.
Let $E\subseteq G$. Let $(G_{N})$ be a Følner sequence in $G$. Then, there
exists a Følner subsequence $(G_{N_{k}})$ such that for all Følner sequences
$(F_{N})$ and all sets $F\in\mathcal{A}_{E}$ we have
(3.8) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}d_{(G_{N_{k}})}(F\cap g^{-1}F)\geq d_{(G_{N_{k}})}(F)^{2}.$
If we let $F=E$ in (3.8) we have the following variant of (3.8):
(3.9) $\liminf_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\bar{d}_{(G_{N})}(E\cap g^{-1}E)\geq\bar{d}_{(G_{N})}(E)^{2}.$
###### Proof.
Given that $G$ is countable, the family of sets $\mathcal{A}_{E}$ is
countable. Thus, via a diagonal procedure, we can take a subsequence
$(G_{N_{k}})$ of our given Følner sequence $(G_{N})$ so that
(3.10) $\mu:=\textrm{w*-}\lim_{k\to\infty}\frac{1}{|G_{N_{k}}|}\sum_{g\in
G_{N_{k}}}\delta_{T_{g}\omega}$
exists, where $\omega=(\mathbb{1}_{E}(g))_{g\in G}$.
Letting $A:=\\{x:x(e)=1\\}$ we see that $F(x)=x(0)=\mathbb{1}_{A}(x)\in C(X)$,
so the function $F_{1}$ representing the set $E$ is also in $C(X)$. Thus, we
can write
(3.11) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}d_{(G_{N_{k}})}(F\cap
g^{-1}F)=\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\int F_{1}\cdot
T_{g}F_{1}\ d\mu.$
Using the ergodic decomposition for $\mu$, we see that the last term in
Equation (3.11) can be rewritten as
(3.12) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\int\left(\int_{X}F_{1}\cdot T_{g}F_{1}\ d\mu_{t}\right)d\lambda(t),$
where each $\mu_{t}$ is an ergodic measure. Thus, by von Neumann’s Mean
Ergodic Theorem and Jensen’s inequality, we have
(3.13) $\int\left(\int_{X}F_{1}\
d\mu_{t}\right)^{2}d\lambda(t)\geq\left(\int_{X}F_{1}\ d\mu\right)^{2}.$
By construction, the right hand side in (3.13) is equal to
$d_{(G_{N_{k}})}(F)^{2}$. To prove (3.9), we use Theorem 2.3 to obtain a
measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set
$A\in\mathcal{B}$ with $\mu(A)=\bar{d}_{(G_{N})}(E)$ satisfying inequality
(2.1). In particular, this means that for all $g\in G$ we have
(3.14) $\bar{d}_{(G_{N})}(E\cap g^{-1}E)\geq\mu(A\cap(T_{g})^{-1}A).$
Using the same argument that leads to inequality (3.13), we get that for any
Følner sequence $(F_{N})$
(3.15) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\mu(A\cap(T_{g})^{-1}A)\geq\mu(A)^{2}.$
Combining (3.14) and (3.15) we obtain
$\liminf_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\bar{d}_{(G_{N})}(E\cap
g^{-1}E)\geq\mu^{2}(A)=\bar{d}_{(G_{N})}(E)^{2},$
as desired. ∎
###### Remark 3.11.
As was mentioned at the beginning of the section, the results contained herein
can be effortlessly carried over to countable cancellative amenable
semigroups. Indeed, if $S$ is such a semigroup, then one can embed it into
$G:=\\{st^{-1}:s,t\in S\\}$, which is now going to be a countable amenable
group (see Proposition 1.17 in [P]). It is straightforward to check that a
Følner sequence in $S$ becomes a Følner sequence in $G$, and this is all that
is needed to push the results to this more general context.
## 4\. Ergodic sequences and Hindman’s theorem
The goal of this section is to extend Hindman’s covering theorem (Theorem 1.3)
to unions of the form $\bigcup_{n=1}^{N}(E-k_{n})$. In particular, we will use
Ergodic Theory to characterize (and provide numerous examples of) the
sequences $(k_{n})$ with the property that for any $E\subseteq\mathbb{Z}$ with
$d^{*}(E)>0$ one has $d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\to 1$ as
$N\to\infty$. The ergodic approach can easily be extended to amenable groups.
We will discuss this after the proof of Theorem 4.8.
###### Definition 4.1.
We say that a sequence of integers $(k_{n})_{n\in\mathbb{N}}$ has the
_combinatorial sweeping out property_ if for every $E\subseteq\mathbb{Z}$ with
$d^{*}(E)>0$ we have
(4.1)
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\xrightarrow[N\to\infty]{}1$
The class of sequences satisfying (4.1) is quite wide. For example, as we will
see below, ergodic sequences have the combinatorial sweeping out property.
###### Definition 4.2.
We say that a sequence of positive integers $(k_{n})$ is an _ergodic sequence_
if for every ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ we have
(4.2) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap
T^{-k_{n}}B)=\mu(A)\mu(B).$
for all $A,B\in\mathcal{B}$.
###### Remark 4.3.
One can show (see Theorem 4.9 below) that a sequence is ergodic if and only if
for all $f\in L^{2}(\mu)$ we have
(4.3) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f=\int_{X}f\ d\mu,$
where the convergence is with respect to the $L^{2}(\mu)$ norm.
The following definition deals with an ergodic counterpart of the notion of
combinatorial sweeping out:
###### Definition 4.4.
We say that a sequence of integers $(k_{n})_{n\in\mathbb{N}}$ has the _ergodic
sweeping out property_ if for every ergodic measure preserving system
$(X,\mathcal{B},\mu,T)$ and for every $A\in\mathcal{B}$ with $\mu(A)>0$ we
have
(4.4) $\mu\left(\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A\right)=1.$
As we will see in Theorem 4.8, the notions of ergodic sweeping out and
combinatorial sweeping out, in fact, coincide, so one can use the term
sweeping out unambiguously. But first we will show, as promised, that ergodic
sequences have the combinatorial sweeping out property.
###### Proposition 4.5.
Let $(k_{n})$ be an ergodic sequence. Then $(k_{n})$ has the combinatorial
sweeping out property.
###### Proof.
We first note that if $(k_{n})$ is an ergodic sequence,
$(X,\mathcal{B},\mu,T)$ is an ergodic measure preserving system and
$A\in\mathcal{B}$ is such that $\mu(A)>0$, then
(4.5) $\mu\left(\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A\right)=1.$
(Otherwise taking $B=X\setminus\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A$ we would
get a contradiction with (4.2).)
Now let $E\subseteq\mathbb{Z}$ be such that $d^{*}(E)>0$ and take
$\varepsilon>0$. By Theorem 2.8 there exists an ergodic measure preserving
system $(X,\mathcal{B},\mu,T)$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ satisfying
(4.6)
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\geq\mu\left(\bigcup_{n=1}^{N}T^{-k_{n}}A\right)$
for all $N\in\mathbb{N}$. Using (4.5) and continuity of $\mu$ we see that
$(k_{n})$ satisfies (1.8). ∎
We would like to note that in the proof of Proposition 4.5 the ergodic
Furstenberg correspondence principle, Theorem 2.8, was utilized. Theorem 2.8
will also play an instrumental role in the proof of the equivalence of ergodic
sweeping out and combinatorial sweeping out (see Theorem 4.8).
There are sequences with the combinatorial sweeping out property that are not
ergodic. A rather cheap example is provided by the sequence $k_{n}:=[\log n]$.
While this sequence does not satisfy (4.2), it takes on all nonnegative
integer values and hence is sweeping out. A more interesting example is given
by the sequence $k_{n}:=[n^{2}+\log n]$. By [BKQW], $(k_{n})$ is not ergodic.
However, one can show that for any ergodic measure preserving system
$(X,\mathcal{B},\mu,T)$ and any sets $A,B\in\mathcal{B}$ with $\mu(A)>0$ and
$\mu(B)>0$ there is some $n\in\mathbb{N}$ such that $\mu(A\cap
T^{-([n^{2}+\log n])}B)>0$. This implies that
$\mu\left(\bigcup_{n\in\mathbb{N}}T^{-[n^{2}+\log n]}A\right)=1$. This fact
together with Theorem 4.8 below imply that $(k_{n})$ is sweeping out.
###### Definition 4.6.
We say that an invertible measure preserving system $(X,\mathcal{B},\mu,T)$
has a topological model if there exists a measure-theoretically isomorphic
system $(\hat{X},\hat{\mathcal{B}},\hat{\mu},\hat{T})$, where $\hat{X}$ is a
compact metric space and $T$ is a homeomorphism from $\hat{X}$ to itself.
###### Theorem 4.7 (Jewett-Krieger Theorem, [J], [Kr]).
Every ergodic invertible measure preserving system $(X,\mathcal{B},\mu,T)$ has
a
uniquely ergodic 888A measure preserving system $(X,\mathcal{B},\mu,T)$ is
called _uniquely ergodic_ if $X$ is a compact metric space, $T:X\rightarrow X$
is a homeomorphism and $\mu$ is the unique $T$-invariant normalized Borel
measure on $X$. topological model.
The following theorem characterizes sequences which are "good" for Hindman’s
covering theorem (see Theorem 1.3) and establishes the equivalence between
measurable sweeping out and combinatorial sweeping out:
###### Theorem 4.8.
A sequence of integers has the combinatorial sweeping out property if and only
if it has the ergodic sweeping out property.
###### Proof.
In one direction, assume that $(k_{n})$ has the ergodic sweeping out property.
Let $E\subseteq\mathbb{Z}$ with $d^{*}(E)>0$. Let $(X,\mathcal{B},\mu,T)$ be a
measure preserving system and $A\in\mathcal{B}$ which are guaranteed by
Theorem 2.8 and satisfy the following special case of (2.6):
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\geq\mu\left(\bigcup_{n=1}^{N}T^{-k_{n}}A\right)\textrm{
for all }N\in\mathbb{N}.$
Since $\mu(A)=d^{*}(E)>0$ and the system $(X,\mathcal{B},\mu,T)$ is ergodic,
equality (4.4) holds. Continuity of the measure $\mu$ and equality (4.4)
together imply that
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\xrightarrow[N\to\infty]{}1$
and so $(k_{n})$ is combinatorially sweeping out.
In the other direction, we will show the contrapositive. Assume that there
exists an ergodic system $(X,\mathcal{B},\mu,T)$ and a set $A\in\mathcal{B}$
with $\mu(A)>0$ such that
$\mu\left(\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A\right)=1-\delta$ for some
$\delta>0$. Without loss of generality, one can assume that
$(X,\mathcal{B},\mu,T)$ is invertible. Indeed, otherwise we can work with the
invertible extension of $(X,\mathcal{B},\mu,T)$.
In view of the Jewett-Krieger theorem (Theorem 4.7), we can also assume that
$(X,T)$ is a uniquely ergodic topological dynamical system.
Since $X$ is a compact metric space, the probability measure $\mu$ is regular.
Let $K_{1}\subseteq A$ be a compact set such that $\mu(K_{1})>0$ and
$K_{2}\subseteq X\setminus\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A$ a compact
subset such that $\mu(K_{2})\geq\frac{\delta}{2}$. Since $T$ is ergodic, von
Neumann’s mean ergodic theorem implies
$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{n}f=\int_{X}f\ d\mu,$
where $f=\mathbb{1}_{K_{1}}$ and the convergence is in the $L^{2}(\mu)$-norm.
Thus, there is a subsequence $(N_{k})$ and $X_{0}\subseteq X$ with
$\mu(X_{0})=1$ such that for all $x\in X_{0}$ we have
$\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}f(T^{n}x)=\int_{X}f\ d\mu.$
Let $x_{0}\in X_{0}$ and consider the set
$E:=\\{n\in\mathbb{Z}:T^{n}x_{0}\in K_{1}\\}.$
Note that by the choice of $x_{0}$ and $E$
$\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}\mathbb{1}_{E}(n)=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}\mathbb{1}_{K_{1}}(T^{n}x_{0})=\int_{X}\mathbb{1}_{K_{1}}\
d\mu=\mu(K_{1})>0$
which implies that $d^{*}(E)>0$. We claim that for all $N\in\mathbb{N}$, we
have
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\leq 1-\frac{\delta}{2}.$
Indeed, let $N\in\mathbb{N}$. By our choice of $K_{2}$, the set
$T^{-k_{1}}K_{1}\cup\dots\cup T^{-k_{N}}K_{1}$ is a compact set disjoint from
$K_{2}$, so by Urysohn’s lemma there is a continuous function
$f:X\rightarrow[0,1]$ such that $f(x)=1$ if $x\in T^{-k_{1}}K_{1}\cup\dots\cup
T^{-k_{N}}K_{1}$ and $f(x)=0$ if $x\in K_{2}$. Thus,
(4.7)
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)=\limsup_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}\mathbb{1}_{T^{-k_{1}}K_{1}\cup\dots\cup
T^{-k_{N}}K_{1}}(T^{n}x_{0})\\\
\leq\lim_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}f(T^{n}x_{0})=\int_{X}f\
d\mu\leq 1-\frac{\delta}{2}.$
(Note that we used the fact that for uniquely ergodic systems,
$\lim_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}f(T^{n}x_{0})$ exists for any
continuous function $f$ and any $x_{0}\in X$). ∎
While we have chosen to stick with $\mathbb{Z}$ for the sake of clarity, it is
worth mentioning that one can establish a version of Theorem 4.8 for general
countable amenable groups. Indeed, it is not hard to define in total analogy
with Definition 4.1 and 4.2 the notions of combinatorial and measurable
sweeping out for general countable amenable groups. To carry out the amenable
generalization of Theorem 4.8, one has to invoke a general form of Jewett-
Krieger’s theorem due to Rosenthal (see [R]). Note that Definition 4.2
naturally extends to $\mathbb{Z}^{d}$ and even to any amenable group, and can
be used to provide numerous examples of ergodic (and hence sweeping out)
sequences. One can show, for example, that every subset of positive density of
a minimally almost periodic group is ergodic.
We give now some examples of ergodic sequences both in $\mathbb{Z}$ and
$\mathbb{Z}^{d}$. (See [BLes] and [BKQW].)
* (1)
$\\{[bn^{c}]:n\in\mathbb{N}\\}$, where $c\notin\mathbb{Q}$, $c>1$ and $b\neq
0$.
* (2)
$\\{[bn^{c}+dn^{a}]:n\in\mathbb{N}\\}$, where $b,d\neq 0$,
$b/d\notin\mathbb{Q}$, $c\geq 1$, $a>0$ and $a\neq c$.
* (3)
$\\{[bn^{c}(\log n)^{d}]:n\in\mathbb{N}\\}$, where $b\neq 0$,
$c\notin\mathbb{Q}$, $c>1$ and $d$ is any number
* (4)
$\\{[bn^{c}(\log n)^{d}]:n\in\mathbb{N}\\}$, where $b\neq 0$,
$c\in\mathbb{Q}$, $c>1$ and $d\neq 0$.
* (5)
$\\{[bn^{c}+d(\log n)^{a}]:n\in\mathbb{N}\\}$, where $b,d\neq 0$, $c\geq 1$
and $a>1$.
Another class of examples of ergodic sequences is provided by sequences of the
form $[g(n)]$, where $g$ is any _tempered function_ 999Let $k$ be a non-
negative integer. A real-valued function $g$ which is $(k+1)$ times
continuously differentiable on $[x_{0},\infty)$, where $x_{0}\geq 0$, is
called a _tempered function of order_ $k$ if (a) $g^{(k+1)}(x)$ tends
monotonically to zero as $x\to\infty$, and (b)
$\lim_{x\to\infty}x|g^{(k+1)}(x)|=+\infty$. (see [BK], Theorem 7.1)
The paper [BKS] provides a class of examples of ergodic sequences involving
primes for $\mathbb{Z}^{d}$ actions. It is clear that these sequences will be
sweeping out for $\mathbb{Z}^{d}$ due to a straightforward generalization of
Lemma 4.5. Namely, sequences of the form
$\\{([\xi_{1}(p_{n})],\dots,[\xi_{d}(p_{n})]):n\in\mathbb{N}\\}$, where
$p_{n}$ denotes the $n$-th prime with the standard order, and where
$\xi_{1},\dots,\xi_{d}$ are functions in a Hardy field with subpolynomial
growth such that either
(4.8)
$\lim_{x\to\infty}\frac{\xi(x)}{x^{l+1}}=\lim_{x\to\infty}\frac{x^{l}}{\xi(x)}=0\textrm{
for some }l\in\mathbb{N},\textrm{ or
}\lim_{x\to\infty}\frac{\xi(x)}{x}=\lim_{x\to\infty}\frac{\log x}{\xi(x)}=0,$
and such that any combination of the form $\sum_{i=1}^{d}b_{i}\xi_{i}$ also
satisfies (4.8) for all
$(b_{1},\dots,b_{d})\in\mathbb{R}^{d}\setminus\\{\vec{0}\\}$. A particular
case of this would be sequences of the form
$\\{([p_{n}^{c_{1}}],\dots,[p_{n}^{c_{d}}]):n\in\mathbb{N}\\}$, where $p_{n}$
denotes the $n$-th prime, and where $c_{1},\dots,c_{d}$ are distinct positive
real numbers such that $c_{i}\notin\mathbb{N}$ for all $i$. (This special case
was obtained in [BKMST]).
We conclude with showing that strong and weak convergence lead to the same
definition of an ergodic sequence. We did not need this fact for this section,
but we believe it is of independent interest. Note that Theorem 4.9 can be
viewed as a generalization of the well-known fact that the $L^{2}$ version of
the mean ergodic theorem follows from its weak convergence version.
###### Theorem 4.9.
For a sequence $(k_{n})\subseteq\mathbb{Z}$, the following are equivalent:
* (1)
For every ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ and for
all $A,B\in\mathcal{B}$
(4.9) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap
T^{-k_{n}}B)=\mu(A)\mu(B).$
* (2)
For every ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ and for
all $f\in L^{2}(\mu)$
(4.10) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f=\int_{X}f\ d\mu.$
where the convergence is with respect to the $L^{2}(\mu)$ norm.
###### Proof.
Strong convergence implies weak convergence, so one direction is trivial. In
the other direction, notice that for any $x\in\mathbb{T}$, (4.9) implies that
$(k_{n}x)_{n\in\mathbb{N}}$ is uniformly distributed in
$\overline{\\{nx:n\in\mathbb{Z}\\}}\subseteq\mathbb{T}$. To see this we argue
as follows. First, observe that, by a standard approximation argument, (4.9)
implies that for all $f,g\in L^{2}(\mu)$
(4.11) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{X}f\cdot
T^{k_{n}}\bar{g}\ d\mu=\int_{X}f\ d\mu\int_{X}\bar{g}\ d\mu.$
Let $X=\mathbb{T}$, $x\in\mathbb{T}\setminus\\{0\\}$ and put $T:X\rightarrow
X$ the rotation by $x$, i.e. $Tz=z+x$ for all $z\in\mathbb{T}$. Let $\chi$ be
a non-trivial character of the compact abelian group
$G_{x}:=\overline{\\{nx:n\in\mathbb{Z}\\}}$ (note that either
$G_{x}=\mathbb{T}$ or it is a finite subgroup of $\mathbb{T}$), and set
$f(y)=\bar{\chi}(y)$ and $g(y)=f(y)=\bar{\chi}(y)$. With these choices,
equation (4.11) becomes
(4.12)
$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{X}\bar{\chi}(y)\chi(y+k_{n}x)\
d\nu(y)=\int_{X}\chi(y)\ d\nu(y)\int_{X}\bar{\chi}(y)\ d\nu(y),$
where $\nu$ is the Haar measure on $G_{x}$. Now, simplifying (4.12) we get
(4.13) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\chi(k_{n}x)=0,$
so $(k_{n}x)$ is equidistributed on $G_{x}$, as desired. Then, for any $f\in
L^{2}(\mu)$, we have
(4.14) $\left|\left|\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f-\int_{X}f\
d\mu\right|\right|^{2}=\left|\left|\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f\right|\right|^{2}-\left|\int_{X}f\
d\mu\right|^{2}$
after expanding the inner product and recombining. Next we are going to invoke
the classical Herglotz’s theorem, which states that the positive definite
sequence $a(n)=\langle T^{n}f,f\rangle$ has a representation
$a(n)=\int_{\mathbb{T}}e^{2\pi inx}\ d\nu_{f}(x)$, for some finite positive
measure $\nu_{f}$ on $\mathbb{T}$. Thus, the right hand side of (4.14) becomes
(4.15) $\frac{1}{N^{2}}\sum_{n,m=1}^{N}\langle
T^{k_{n}-k_{m}}f,f\rangle-\left|\int_{X}f\
d\mu\right|^{2}=\frac{1}{N^{2}}\sum_{n,m=1}^{N}\int_{\mathbb{T}}e^{2\pi
i(k_{n}-k_{m})x}\ d\nu_{f}(x)-\left|\int_{X}f\ d\mu\right|^{2}=\\\
\int_{\mathbb{T}}\left|\frac{1}{N}\sum_{n=1}^{N}e^{2\pi ik_{n}x}\right|^{2}\
d\nu_{f}(x)-\left|\int_{X}f\ d\mu\right|^{2}.$
Using (4.13) for $\chi(x)=e^{2\pi ix}$, and invoking ergodicity of $T$ we get
(4.16)
$\lim_{N\to\infty}\int_{\mathbb{T}}\left|\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
ik_{n}x}\right|^{2}\ d\nu_{f}(x)-\left|\int_{X}f\
d\mu\right|^{2}=\nu_{f}(\\{0\\})-\left|\int_{X}f\ d\mu\right|^{2}=0.$
(In the last step we used the fact that
$\nu_{f}(\\{0\\})=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{\mathbb{T}}e^{2\pi
inx}\ d\nu_{f}(x)=\\\ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\langle
T^{n}f,f\rangle=\left|\int_{X}f\ d\mu\right|^{2}$.) ∎
###### Remark 4.10.
Theorem 4.9 can be extended to the context of locally compact abelian groups.
We omit the details.
## 5\. A characterization of countable amenable weakly mixing groups
In this section, we will establish a characterization of countable weakly
mixing groups with the help of the amenable version of Hindman’s covering
theorem (Theorem 3.1). Recall that a group $G$ is _weakly mixing_ (or
minimally almost periodic) if any ergodic measure preserving action of $G$ on
a probability space $\mathds{X}=(X,\mathcal{B},\mu)$ is automatically weakly
mixing i.e., the diagonal action of $G$ on $\mathds{X}\times\mathds{X}$ is
ergodic. (In this case, the diagonal action on an arbitrary finite product
$\mathds{X}\times\dots\times\mathds{X}$ is also ergodic).
We begin with the following proposition:
###### Proposition 5.1.
Let $G$ be a countable amenable WM group. Let $E\subseteq G$ with $d^{*}(E)>0$
and let $d\in\mathbb{N}$. Then, for all $\varepsilon>0$, there exist
$g_{1},\dots,g_{k}\in G$ such that
(5.1)
$d^{*}\left(\bigcup_{i=1}^{k}(g_{i},\dots,g_{i})^{-1}\underbrace{(E\times\dots\times
E)}_{d\ \textrm{times}}\right)>1-\varepsilon.$
###### Proof.
By Theorem 2.8 there exists an ergodic measure preserving system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ satisfying (2.6). Since $G$ is a WM group, the measure
$\mu\otimes\mu$ will be ergodic for the diagonal action on $X\times X$.
Let $(G_{N})$ be a Følner sequence such that
$\mu=\textrm{w*-}\lim_{N\to\infty}\frac{1}{|G_{N}|}\sum_{g\in
G_{N}}\delta_{T_{g}\omega}$, where $\omega=(\mathbb{1}_{E}(g))_{g\in G}$, as
in Section 2. Notice that, for each $k\in\mathbb{N}$, we have
(5.2) $d^{*}\left(\bigcup_{i=1}^{k}(g_{i},\dots,g_{i})^{-1}(E\times\dots\times
E)\right)\geq d_{(G_{N}\times\dots\times
G_{N})}\left(\bigcup_{i=1}^{k}(g_{i},\dots,g_{i})^{-1}(E\times\dots\times
E)\right),$
for all $g_{1},\dots,g_{k}\in G$. Applying the inclusion-exclusion principle
we can change the unions in (5.2) for intersections. The density of the
typical intersection can be computed with a product of densities (note that
$d_{(G_{N}\times G_{N})}(E_{1}\cap E_{2}\times F_{1}\cap
F_{2})=d_{(G_{N})}(E_{1}\cap E_{2})d_{(G_{N})}(F_{1}\cap F_{2})$). Finally,
using the definition of $\mu$ we see that the quantity on the right hand side
in (5.2) equals
$\mu\otimes\dots\otimes\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}}\times\dots\times
T_{g_{i}})(A\times\dots\times A)\right)$
Since $G$ is a WM group, the measure $\mu\otimes\dots\otimes\mu$ is ergodic
for the diagonal $G$-action on the product space $X\times\dots\times X$. Since
$\mu(A)>0$ we have $(\mu\otimes\dots\otimes\mu)(A\times\dots\times A)>0$,
whence
$(\mu\otimes\dots\otimes\mu)\left(\bigcup_{g\in G}(T_{g}\times\dots\times
T_{g})^{-1}(A\times\dots\times A)\right)=1$, so the result follows by
continuity of $\mu$. ∎
We will show next that the covering property (5.1) characterizes weakly mixing
groups:
###### Theorem 5.2.
Let $G$ be a countable amenable group that is not WM. Then there exists a set
$E\subseteq G$ with $d^{*}(E)\in(0,1)$ such that for all $r\in\mathbb{N}$ and
any finite subset $\\{h_{1},\dots,h_{r}\\}$ of $G$ we have
(5.3) $d^{*}\left(\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times E)\right)\leq
C<1,$
where the constant $C$ in (5.3) is independent of $\\{h_{1},\dots,h_{r}\\}$.
###### Proof.
Before constructing the set $E$, we need to do some preparatory work. First,
we observe that by a Corollary of Lemma 3.3 in [BBF], (see equation (5) in
[BBF]) for any $B\subseteq G$, and for any Følner sequence $(F_{n})\subseteq
G$, there is a sequence $(t_{n})$ such that
$d^{*}(B)=d_{F_{n}t_{n}}(B).$
Hence, it suffices to show that for any pair of Følner sequences
$(I_{N})\subseteq G$ and $(J_{N})\subseteq G$ there exists $0<C<1$ such that
(5.4) $\bar{d}_{(I_{N}\times
J_{N})}\left(\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times E)\right)\leq C<1.$
for all $r\in\mathbb{N}$ and all finite sets $\\{h_{1},\dots,h_{r}\\}$. Note
that this step is essential because general Følner sequences of $G\times G$
need not be of the form $(I_{N}\times J_{N})$.
Next, we observe that since $G$ is not WM it must admit a non-trivial finite
dimensional representation $\pi:G\rightarrow U(k)$, where $U(k)$ is the
unitary group of $k\times k$ complex matrices (see for example [S], Theorem
3.4). Thus, $H=\\{\pi(g):g\in G\\}$ is a non-trivial subgroup of $U(k)$.
Let $X=\overline{H}$, the closure of $H$ in $U(k)$. Then, $X$ is a compact
metric group with the topology inherited from $\mathbb{C}^{k^{2}}$. As such,
it carries a unique normalized Haar measure (fully supported on $X$) which we
denote by $\mu$.
Moreover, $G$ acts on $X$ by translations as follows: let $R_{g}x:=\pi(g)\cdot
x$. Note that the action $(R_{g})_{g\in G}$ is minimal because $\pi(G)=H$ and
$H$ is dense in $X$. Clearly, $(X,\textrm{Borel}(X),\mu,(R_{g})_{g\in G})$ is
a uniquely ergodic measure preserving system.
Let $d$ be a bi-invariant metric on $X$ (such a metric exists by the Birkhoff-
Kakutani Theorem, see [Bi] and [K]). Let $g_{0}\in X$ be such that
$d(e,g_{0})=b=\max\\{d(e,g):g\in G\\}>0$ (since $X$ is compact and non-
trivial, $0<b<\infty$). Let $U:=B(e,\frac{b}{16})$ (the open ball of radius
$\frac{b}{16}$ centered at $e$). Clearly $\mu(U)>0$ since $\mu$ is fully
supported on $X$. Let $U^{\prime}:=B(e,r)$ where $0<r<\frac{b}{16}$ is such
that $\mu(\partial U^{\prime})=0$. Such an $r$ must exist (see for example
Lemma 2.21 in [BCRZ]). At this point we are ready to define the set $E$.
Namely, choose an arbitrary $x_{0}\in X$ and put
(5.5) $E:=\\{g\in G:R_{g}x_{0}\in U^{\prime}\\}.$
Since $(X,\mu,(R_{g})_{g\in G})$ is uniquely ergodic and $\mu(\partial
U^{\prime})=0$, the function $\mathbb{1}_{U^{\prime}}$ can be approximated by
continuous functions. From these two facts, it follows that
$\textrm{w*-}\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\delta_{T_{g}x_{0}}$ converges to $\mu$.
Next, observe that for all $g\in X$, the open set
$W:=B(g_{0},\frac{b}{16})\times B(e,\frac{b}{16})$ satisfies
(5.6) $(g,g)^{-1}(U\times U)\cap W=\emptyset,$
because otherwise, by the triangle inequality we would get that $d(e,g)\leq
d(e,g_{1}u)+d(g_{1}u_{1},g_{1}u_{2})+d(g_{1}u_{2},g_{0})<2\cdot\frac{b}{16}+d(u_{1},u_{2})<\frac{b}{8}+\frac{b}{8}<b$,
where $g_{1}\in G$ and $u_{1},u_{2}\in U$ are such that
$d(e,g_{1}u_{1})<\frac{b}{16}$ and $d(g_{0},g_{1}u_{2})<\frac{b}{16}$, a
contradiction with our choice of $e,g_{0}$.
It follows from (5.6) and from the fact that $G$ acts minimally on $X$ that
$\overline{\Delta\cdot(U\times U)}\neq X\times X$, where $\Delta$ is the
diagonal in $X^{2}$. Thus,
$(\mu\otimes\mu)(\Delta\cdot(U\times U)):=C<1.$
Note also that since $\mu$ is fully supported on $X$, $\mu\otimes\mu$ is fully
supported on $X\times X$, and so we have $(\mu\otimes\mu)(\Delta\cdot(U\times
U)>0$. Let $(I_{N}),(J_{N})$ be two Følner sequences in $G$, and
$\\{h_{1},\dots,h_{r}\\}$ a finite set of elements of $G$. Then,
(5.7) $\bar{d}_{(I_{N}\times
J_{N})}\left(\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times
E)\right)=\limsup_{N\to\infty}\frac{1}{|I_{N}||J_{N}|}\sum_{g\in
F_{N}}\sum_{h\in J_{N}}\mathbb{1}_{\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times
E)}(g,h).$
By (5.5), the last term in equation (5.7) is equal to
(5.8) $\limsup_{N\to\infty}\frac{1}{|I_{N}||J_{N}|}\sum_{g\in F_{N}}\sum_{h\in
J_{N}}\mathbb{1}_{\bigcup_{i=1}^{r}(T_{h_{i}}\times
T_{h_{i}})^{-1}(U^{\prime}\times U^{\prime})}(T_{g}x_{0},T_{h}x_{0}).$
We now use the inclusion-exclusion principle to separate the variables in each
summand. Let $\prod_{i=1}^{r}\mathbb{1}_{T_{h_{i}}U^{\prime}}$, $1\leq t\leq
r$, be a typical term obtained through this process. By unique ergodicity of
the action $(R_{g})_{g\in G}$ we see that for any Følner sequence $(F_{N})$
$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\mathbb{1}_{\bigcap_{i=1}^{t}T_{h_{i}}U^{\prime}}(T_{g}x_{0})=\mu\left(\bigcap_{i=1}^{t}T_{h_{i}}U^{\prime}\right).$
Indeed, since $U^{\prime}$ is an open set with $\mu(\partial U^{\prime})=0$,
we also have that $\mu(\partial(T_{g}U^{\prime}))=0$ for all $g\in G$ given
that $T_{g}(\partial U^{\prime})=\partial(T_{g}U^{\prime})$ as $T_{g}$ is a
measure preserving homeomorphism. Consequently, functions of the form
$\prod_{i=1}^{t}\mathbb{1}_{T_{g_{i}}U^{\prime}}$, $1\leq t\leq r$, can be
approximated by continuous functions. Thus, the limit in equation (5.8) in
fact exists and (after performing the inclusion-exclusion principle backwards)
is equal to
(5.9) $(\mu\otimes\mu)\left(\bigcup_{i=1}^{r}(T_{h_{i}}\times
T_{h_{i}})^{-1}(U^{\prime}\times
U^{\prime})\right)\leq(\mu\otimes\mu)(\Delta\cdot(U\times U))=C<1,$
which completes the proof. ∎
## 6\. A general form of Hindman’s covering theorem
One may wonder if a version of Hindman’s covering theorem (Theorem 3.1) is
valid for discrete amenable semigroups which are not necessarily countable or
cancellative. In this section we will show that this is indeed the case.
Recall that a discrete semigroup $G$ is left amenable if there exists a left
invariant mean $m:\ell^{\infty}(G)\rightarrow\mathbb{C}$ 101010We say that
$m\in\ell^{\infty}(G)^{*}$ is a left _invariant mean_ if it is a continuous
linear functional from $\ell^{\infty}(G)$ to $\mathbb{C}$ such that (i) for
every $f\in\ell^{\infty}(G)$ and for every $g\in G$ we have $m({}_{g}f)=m(f)$,
where ${}_{g}f(x):=f(gx)$ for all $x\in G$, (ii) $m(f)\geq 0$ for any bounded
function $f:G\rightarrow[0,\infty)$, and (iii) $m(\mathbb{1}_{G})=1$.. (Note
that, for discrete countable groups, this is equivalent to the definition of
left amenability given in Section 2.)
In the context of means, a notion of largeness presents itself. Let us denote
by $\mathfrak{M}(G)$ the space of left invariant means on $G$. We say a subset
$E$ of a discrete left amenable semigroup $G$ is large if
$m(\mathbb{1}_{E})>0$ for some mean $m\in\mathfrak{M}(G)$. This leads to the
following definition of a notion of upper Banach density that is valid in all
amenable semigroups (see Definition 2.7 [BG]):
###### Definition 6.1.
Let $G$ be an amenable semigroup, and let
$E\subseteq G$. The _upper Banach density_ of $E$ is
$d^{*}(E):=\sup\\{m(\mathbb{1}_{E}):m\in\mathfrak{M}(G).\\}$
Notice that since $\mathfrak{M}(G)$ is weak* compact there is some
$m\in\mathfrak{M}(G)$ that achieves the supremum in Definition 6.1. If $G$ is
a discrete countable amenable group and $E\subseteq G$, then the upper Banach
density of $E$ as given in Definition 2.2 agrees with the one in Definition
6.1. Moreover, in this case, we have
$d^{*}(E)=\max\\{m(\mathbb{1}_{E}):m\in\mathfrak{M}(G)\\}$. We are now in a
position to formulate a general version of Hindman’s theorem.
###### Theorem 6.2.
Let $G$ be an amenable semigroup, and let $E$ be a subset of $G$ with
$d^{*}(E)>0$. Then, for every $\varepsilon>0$, there exists $k\in\mathbb{N}$
and $g_{1},\dots,g_{k}\in G$ such that $d^{*}(g_{1}^{-1}E\cup\dots\cup
g_{k}^{-1}E)>1-\varepsilon$.
The proof of Theorem 6.2 requires a few preliminary results which will be
given next. Recall that the space of _weakly almost periodic_ functions on
$G$, denoted by $\textrm{WAP}(G)$ is comprised of functions
$f\in\ell^{\infty}(G)$ such that the weak closure of their shifts, i.e.
$\overline{\\{_{g}f:g\in G\\}}$ is weakly compact (i.e., compact with respect
to the weak topology induced by functionals on $\ell^{\infty}(G)$) .
###### Theorem 6.3 (Ryll-Nardzewski, cf. [P], page 86).
There is a unique left invariant mean on $\textrm{WAP}(G)$.
We will be using the fact (due to Eberlein, [E]) that for a discrete semigroup
$G$, functions of the form $f(g)=\langle h_{1},T_{g}h_{2}\rangle$, for
$h_{1},h_{2}\in L^{2}(\mu)$ are in $\textrm{WAP}(G)$ (see, for example,
Theorem 3.1 in [Bu]).
###### Lemma 6.4.
Let $G$ be a discrete semigroup and let $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
be an ergodic measure preserving system. Let $m$ be the unique left invariant
mean on $\textrm{WAP}(G)$ and $f_{1},f_{2}\in L^{2}(\mu)$. Then,
$m(\langle f_{1},T_{g}f_{2}\rangle)=\int_{X}f_{1}\ d\mu\int_{X}\bar{f}_{2}\
d\mu$
###### Proof.
Let $F_{2}\in L^{2}(\mu)$ be defined via
$\langle F_{1},F_{2}\rangle=m(\langle F_{1},T_{g}f_{2}\rangle)$
for all $F_{1}\in L^{2}(\mu)$. We will show that $F_{2}=\int_{X}\bar{f}_{2}\
d\mu$, which implies the result. First observe that since the action
$(T_{g})_{g\in G}$ is ergodic, the only invariant functions are the constants.
Next, we show that $T_{h}F_{2}=F_{2}$ for all $h\in G$, which implies $F_{2}$
is a constant by ergodicity. Indeed,
$\langle
F_{1},T_{h}F_{2}\rangle=\langle(T_{h})^{*}F_{1},F_{2}\rangle=m(\langle(T_{h})^{*}F_{1},T_{g}f_{2}\rangle=m(\langle
F_{1},(T_{hg})f_{2}\rangle)=$ $m(\langle F_{1},T_{g}f_{2}\rangle)=\langle
F_{1},F_{2}\rangle.$
Hence, $F_{2}$ is a constant, so $F_{2}=\langle F_{2},\mathbb{1}_{X}\rangle$.
Thus, for all $f_{1}\in L^{2}(\mu)$ we have
$\int_{X}f_{1}\cdot\bar{F}_{2}\ d\mu=\langle f_{1},F_{2}\rangle=\langle
f_{1},\langle F_{2},\mathbb{1}_{X}\rangle\rangle=\langle\langle
f_{1},\mathbb{1}_{X}\rangle,F_{2}\rangle=$ $m(\langle\langle
f_{1},\mathbb{1}_{X}\rangle,(T_{g})f_{2}\rangle)=m\left(\int_{X}f_{1}\
d\mu\int_{X}T_{g}\bar{f}_{2}\ d\mu\right)=\int_{X}f_{1}\
d\mu\int_{X}\bar{f}_{2}\ d\mu,$
so we are done. ∎
###### Remark 6.5.
It follows from Lemma 6.4 that for any (not necessarily ergodic) measure
preserving action of a discrete semigroup $G$ on a probability space
$(X,\mathcal{B},\mu)$ and any $A\in\mathcal{B}$ we have
$m(\mu(A\cap(T_{g})^{-1}A))\geq\mu^{2}(A).$
Indeed, invoking a general form of the ergodic decomposition we have
$m(\mu(A\cap(T_{g})^{-1}A))=m\left(\int_{\Omega}\mu_{t}(A\cap(T_{g})^{-1}A)\
d\nu(t)\right),$
where $\mu_{t}$ is an ergodic measure for $\nu$-a.e. $t\in\Omega$. Then, since
$m$ is linear, Lemma 6.4 implies
$m\left(\int_{\Omega}\mu_{t}(A\cap(T_{g})^{-1}A)\
d\nu(t)\right)=\int_{\Omega}m(\mu_{t}(A\cap(T_{g})^{-1}A))\
d\nu(t)=\int_{\Omega}\mu_{t}(A)^{2}\ d\nu(t).$
Applying Jensen’s inequality we get
$\int_{\Omega}\mu_{t}(A)^{2}\ d\nu(t)\geq\left(\int_{\Omega}\mu_{t}(A)\
d\nu(t)\right)^{2}=\mu^{2}(A),$
as desired.
###### Lemma 6.6.
Let $G$ be a discrete semigroup and let $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
an ergodic measure preserving system. Let $A,B\in\mathcal{B}$ with $\mu(A)>0$
and $\mu(B)>0$. Then,
(6.1) $R_{A,B}:=\left\\{g\in
G:\mu(B\cap(T_{g})^{-1}A)>\frac{\mu(A)\mu(B)}{2}\right\\}\neq\emptyset.$
###### Proof.
We proceed by contradiction. Assume that for all $g\in G$ we have
$\mu(B\cap(T_{g})^{-1}A)\leq\frac{\mu(A)\mu(B)}{2}$. Let $m$ be the unique
left invariant mean on $\textrm{WAP}(G)$. Let $A,B\in\mathcal{B}$ with
$\mu(A)>0$ and $\mu(B)>0$. By Lemma 6.4, we have
(6.2) $m(\mu(B\cap(T_{g})^{-1}A))=\mu(A)\mu(B).$
Equation (6.2) contradicts the assumption that, for all $g\in G$,
$\mu(B\cap(T_{g})^{-1}A)\leq\frac{\mu(A)\mu(B)}{2}$, (given that $m(f)\geq 0$
if $f\geq 0$) so we are done. ∎
###### Lemma 6.7.
Let $G$ be a discrete semigroup and $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ an
ergodic measure preserving system. Let $A\in\mathcal{B}$ with $\mu(A)>0$.
Then, for all $\varepsilon>0$, there are $g_{1},\dots,g_{k}\in G$ such that
$\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}})^{-1}A\right)>1-\varepsilon.$
###### Proof.
Let $A\in\mathcal{B}$ with $\mu(A)>0$. We claim that
(6.3) $\sup\left\\{\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right):B\subseteq
G\textrm{ and }|B|\leq|\mathbb{N}|\right\\}=1.$
We proceed by contradiction. Let us assume that this is not the case. Then
there exists $0<\delta<1$ such that
(6.4) $\sup\left\\{\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right):B\subseteq
G\textrm{ and }|B|\leq|\mathbb{N}|\right\\}=1-\delta.$
Let $0<\varepsilon^{\prime}<\frac{\mu(A)\delta}{2}$, and choose $B\subseteq G$
with $|B|\leq|\mathbb{N}|$ such that
(6.5) $\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right)\geq
1-\delta-\varepsilon^{\prime}.$
By assumption, there is some $C\subseteq X\setminus\bigcup_{g\in
B}(T_{g})^{-1}A$ with
$\mu(C)\geq\delta$. Now, by Lemma 6.6, we can find $g_{0}\in G$ such that
$\mu(C\cap(T_{g_{0}})^{-1}A)\geq\frac{\mu(A)\mu(C)}{2}$. (Notice that, in
particular, it follows that $g_{0}\notin B$). We have
(6.6) $\mu\left(\bigcup_{g\in B\cup\\{g_{0}\\}}(T_{g})^{-1}A\right)\geq
1-\delta-\varepsilon^{\prime}+\mu(C\cap(T_{g_{0}})^{-1}A)\geq
1-\delta-\varepsilon^{\prime}+\frac{\mu(A)\delta}{2}>1-\delta,$
by our choice of $\varepsilon^{\prime}$ and $g_{0}$. This contradicts the
definition of supremum, since clearly $B\cup\\{g_{0}\\}$ is still a countable
subset of $G$. Thus, (6.3) holds.
To complete the proof, let $\varepsilon>0$ and choose $B\subseteq G$ with
$|B|\leq|\mathbb{N}|$ such that
$\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right)>1-\frac{\varepsilon}{2}.$
Since $\mu$ is continuous and $B$ is countable, there exist
$g_{1},\dots,g_{k}\in B$ such that
$\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}})^{-1}A\right)>1-\frac{\varepsilon}{2}-\frac{\varepsilon}{2},$
so we are done. ∎
Before proving an ergodic version of Furstenberg’s correspondence principle
for means, we formulate two additional results. We start with a version of
Theorem 2.3 for means, which we will juxtapose with its ergodic counterpart,
Theorem 6.11, below.
###### Theorem 6.8 (Furstenberg correspondence principle for means (cf.
[BLei] and [BMc])).
Let $G$ be a discrete amenable semigroup. Let $m\in\mathfrak{M}(G)$,
$E\subseteq G$ with $m(\mathbb{1}_{E})>0$. Then there exists a probability
measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ such that $X$
is compact and Hausdorff, $\mathcal{B}=\textrm{Borel}(X)$ and $(T_{g})_{g\in
G}$ is a $G$-action on $X$ by continuous self-maps of $X$. Finally, $A$ is a
set in $\mathcal{B}$ for which $\mu(A)=m(\mathbb{1}_{B})$, and $\mu$ is such
that for all $k\in\mathbb{N}$, $g_{1},\dots,g_{k}\in G$ we have
(6.7)
$m\left(\mathbb{1}_{E}\prod_{i=1}^{k}\mathbb{1}_{g_{i}^{-1}E}\right)=\mu(A\cap
T_{g_{1}}^{-1}A\cap\dots\cap T_{g_{k}}^{-1}A).$
###### Proof.
The proof for discrete amenable groups provided in [BLei] extends verbatim to
our context without any major modification. ∎
###### Remark 6.9.
The proof of Theorem 6.8 in [BLei] can be easily adjusted to include unions
and complements as in Theorem 2.3.
The other result we need can be found in [P]:
###### Theorem 6.10 (Proposition 0.1 [P]).
The space of left invariant means $\mathfrak{M}(G)$ is a weak*-compact, convex
spanning subset of $\ell^{\infty}(G)^{*}$.
We are now in position to formulate and prove a general version of the ergodic
Furstenberg correspondence principle.
###### Theorem 6.11 (Ergodic Furstenberg correspondence principle for means).
Let $E\subseteq G$ be such that $m(\mathbb{1}_{E})>0$ for some mean
$m\in\mathfrak{M}(G)$. Then there exists a mean $\tilde{m}\in\mathfrak{M}(G)$,
and an ergodic measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in
G})$ such that for all $k\in\mathbb{N}$ we have
(6.8) $\tilde{m}(\mathbb{1}_{E^{w_{0}}\star
g_{1}^{-1}E^{w_{1}}\star\dotso\star
g_{k}^{-1}E^{w_{k}}})=\mu(A^{w_{0}}\star(T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}}),$
where $A\in\mathcal{B}$ is such that
$0<\mu(A)=m(\mathbb{1}_{E})\leq\tilde{m}(\mathbb{1}_{E})$, and each of the
stars denotes either union or intersection with the understanding that
* (i)
for all $1\leq i\leq k-1$, the operation represented by $\star$ which stands
between $E^{w_{i}}$ and $E^{w_{i+1}}$ is the same as the operation appearing
between $A^{w_{i}}$ and $A^{w_{i+1}}$.
* (ii)
the choices of parentheses which are needed to make the expressions on both
sides of formula (2.1) well defined also match.
###### Proof.
First, we remark that, as in the proof of Theorem 6.8, any invariant mean on
$G$ is given by a $G$-invariant probability measure on $\beta G$, the Stone-
Čech compactification of $G$, via the isomorphism $(\ell^{\infty}G)^{*}\cong
C(\beta G)^{*}$121212See [GJ] for the background on Stone-Čech
compactifications.. It is easy to see that this isomorphism is behind the
formula (6.8).
By Choquet’s theorem (which we can apply in view of Theorem 6.10), we can
write
(6.9) $m=\int_{\textrm{Ext}(\mathfrak{M}(G))}m_{t}\ d\lambda(t),$
for some probability measure $\lambda$ supported on
$\textrm{Ext}(\mathfrak{M}(G))$, the set of extreme points of
$\mathfrak{M}(G)$.
Notice that extreme points of $\mathfrak{M}(G)$ get mapped to extreme points
of the set of probability measures on $\beta G$ via the isomorphism
$(\ell^{\infty}(G))^{*}\cong C(\beta G)^{*}$. It is well known that the
measures that are extreme points in the set of $G$-invariant probability
measures on $\beta G$ are in fact ergodic.
It follows from formula (6.9), that since $m(\mathbb{1}_{E})>0$, we have that
$m_{t}(\mathbb{1}_{E})>0$ for a set of $t$ of positive $\lambda$-measure.
Thus we can choose $\tilde{m}$, an extreme point of $\mathfrak{M}(G)$, for
which $\tilde{m}(\mathbb{1}_{E})\geq m(\mathbb{1}_{E})>0$. Using the
aforementioned isomorphism, we obtain an ergodic measure $\mu$ for which (6.8)
holds. ∎
###### Remark 6.12.
It is worth noticing that the ergodicity of the system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ in Theorem 6.11 stems from the fact
that passed from the mean $m$ to $\tilde{m}$. If one restricts oneself to only
using a given mean $m$, as in Theorem 6.8, ergodicity cannot be guaranteed. We
also see in the statement of Theorem 6.11 that moving form $m$ to $\tilde{m}$
does not affect combinatorial applications, as we can require
$\tilde{m}(\mathbb{1}_{E})\geq m(\mathbb{1}_{E})$. This situation is similar
to the one discussed in the comments at the end of Section 2.
We are now ready for a proof of Theorem 6.2:
###### Proof of Theorem 6.2.
Let $E\subseteq G$ with $d^{*}(E)>0$. Then, there is some
$m_{0}\in\mathfrak{M}(G)$ such that $m_{0}(\mathbb{1}_{E})>0$. Let
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure preserving system
satisfying the equality (6.8) for some set $A\in\mathcal{B}$ with $\mu(A)>0$.
(See Theorem 6.11 above). Let $E\subseteq G$ with $d^{*}(E)>0$. Then, there is
some $m_{0}\in\mathfrak{M}(G)$ such that $m_{0}(\mathbb{1}_{E})>0$. Let
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure preserving system
satisfying the equality (6.8) for some set $A\in\mathcal{B}$ with $\mu(A)>0$.
(See Theorem 6.11 above).
Let $E\subseteq G$ with $d^{*}(E)>0$. Then, there is some
$m_{0}\in\mathfrak{M}(G)$ such that $m_{0}(\mathbb{1}_{E})>0$. Let
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure preserving system
satisfying the equality (6.8) for some set $A\in\mathcal{B}$ with $\mu(A)>0$.
(See Theorem 6.11 above).
Let $\varepsilon>0$. By Lemma 6.7, we can find $g_{1},\dots,g_{k}\in G$ such
that
$\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}})^{-1}A\right)>1-\varepsilon,$
since $\mu(A)>0$. Equality (6.8) then implies that for some
$m\in\mathfrak{M}(G)$ we have
$m(\mathbb{1}_{\bigcup_{i=1}^{k}g_{i}^{-1}E})>1-\varepsilon,$
whence the result follows from the definition of $d^{*}$. ∎
We conclude this section with brief remarks on yet another generalization of
Hindman’s covering theorem. Assume that $G$ is a locally compact amenable
group (i.e. that $G$ has a left invariant _topological_ mean, see [Gre] for
more details). The Furstenberg’s correspondence principle which was proved in
[BCRZ] (see also [BF]) can be "upgraded" to an ergodic Furstenberg
correspondence principle similar to Theorem 2.8 and Theorem 6.8. Based on this
enhancement one can prove a version of Hindman’s Theorem for locally compact
groups. Before providing the formulation we need two definitions.
###### Definition 6.13.
Let $G$ be a locally compact amenable group and let $E\subseteq G$. We define
the upper Banach density of $E$ as follows:
(6.10) $d^{*}(E)=\sup\\{m(\mathbb{1}_{E}):m\textrm{ is a left-invariant
topological mean on }L^{1}(G,\mu)\\},$
where $\mu$ is a Haar measure on $G$.
###### Definition 6.14.
Let $G$ be a locally compact amenable group. We say that a set $E\subseteq G$
is _substantial_ if $E\supseteq UW$, where $U$ is a non-empty open subset in
$G$ containing $\textrm{id}_{G}$ and $W$ is a measurable set with
$d^{*}(W)>0$.
###### Theorem 6.15.
Let $G$ be a locally compact amenable group. Let $E\subseteq G$ be a
substantial set. Let $\varepsilon>0$. Then there exist $g_{1},\dots,g_{k}\in
G$ such that
$d^{*}\left(\bigcup_{i=1}^{k}g_{i}^{-1}E\right)>1-\varepsilon.$
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|
2024-09-04T02:54:57.745485 | 2020-03-06T05:13:47 | 2003.03035 | {
"authors": "Masaaki Fujii and Akihiko Takahashi",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26073",
"submitter": "Masaaki Fujii",
"url": "https://arxiv.org/abs/2003.03035"
} | arxiv-papers | # A Mean Field Game Approach to Equilibrium Pricing
with Market Clearing Condition 111 All the contents expressed in this research
are solely those of the author and do not represent any views or opinions of
any institutions. The author is not responsible or liable in any manner for
any losses and/or damages caused by the use of any contents in this research.
Masaaki Fujii222Quantitative Finance Course, Graduate School of Economics, The
University of Tokyo. , Akihiko Takahashi333Quantitative Finance Course,
Graduate School of Economics, The University of Tokyo.
( First version: 6 March, 2020
This version: 3 October, 2020 )
###### Abstract
In this work, we study an equilibrium-based continuous asset pricing problem
which seeks to form a price process endogenously by requiring it to balance
the flow of sales-and-purchase orders in the exchange market, where a large
number of agents $1\leq i\leq N$ are interacting through the market price.
Adopting a mean field game (MFG) approach, we find a special form of forward-
backward stochastic differential equations of McKean-Vlasov type with common
noise whose solution provides a good approximate of the market price. We show
the convergence of the net order flow to zero in the large $N$-limit and get
the order of convergence in $N$ under some conditions. We also extend the
model to a setup with multiple populations where the agents within each
population share the same cost and coefficient functions but they can be
different population by population.
Keywords : FBSDE of McKean-Vlasov type, common noise, general equilibrium
## 1 Introduction
One of the most important problems in the financial economics is to understand
how the asset price processes are formed through the interaction among a large
number of rational competitive agents. In this paper, using a stylized model
of security exchange, we try to explicitly form an approximate market price
process which balances the flow of sales-and-purchase orders from a large
number of rational financial institutions. If we directly force the price
process to balance the net order flow, the strategies of the agents become
strongly coupled and the problem is hardly solvable. In fact, it is even
unclear how to make the cost functions of the agents well-defined, since the
market price results in a very complicated recursive functional of strategies
of all the agents that makes it difficult to guarantee the convexity of the
cost functions. In order to circumvent this problem, we make use of the recent
developments of mean field games.
Since its inception brought by the pioneering works of Lasry & Lions [20, 21,
22] and Huang, Malhame & Caines [18], mean field game has rapidly developed
into one of the most actively studied topics in the field of probability
theory, applied mathematics, engineering, finance and economics. The greatest
strength of the mean field game approach is to render notoriously difficult
problems of stochastic differential games among many agents tractable by
transforming it to a simpler form of stochastic control problem. There exist
two approaches to the mean field games, one is analytic approach using partial
differential equations (PDEs), and the others is probabilistic approach based
on forward-backward stochastic differential equations (FBSDEs). For details of
analytic approach and its applications, the interested readers may consult the
works of Bensousssan, Frehse & Yam [3], Gomes, Nurbekyan & Pimentel [14],
Achdou et.al. [1], Gomes, Pimentel & Voskanyan [15] and also Kolokoltsov &
Malafeyev [19]. On the other hand, the probabilistic approach was developed by
the series of works of Carmona & Delarue [4, 5, 6] and the recent two volumes
of monograph [7, 8] provide its full mathematical details and many references
for a wide array of applications of mean field games.
Interestingly, from the perspective of equilibrium asset pricing, the number
of applications of mean field games is quite limited. In most of the existing
literature, the authors have given a response function of the price process
exogenously and searched an approximate Nash equilibrium among agents. See,
for example, applications to optimal trading as well as liquidation of
portfolio, exploitation of exhaustible resources and related issues among many
agents [10, 11, 12, 23], or an application to electricity pricing with smart
grids [2, 9]. In the work [17], the authors treat explicitly the balance of
demand and supply in the oil market, but the demand is exogenously given as a
function of the oil price. One notable exception is the work of Gomes & Saude
[16], in which the authors explicitly force demand and supply to balance and
endogenously construct the market clearing electricity price. They use the
analytic approach and the resultant equilibrium price process becomes
deterministic due to the absence of common noise.
In the current paper, we extend the work [16] by adopting the probabilistic
approach. In order to understand the price processes, in particular those of
financial assets, including systemic signals which impacts all the agents is
crucially important. We find an interesting form of FBSDEs of McKean-Vlasov
type with common noise as a limit problem. Although it involves dependence in
conditional law, it only appears as a conditional expectation. This allows us
to adopt the well-known Peng-Wu’s continuation method [24] to prove the
existence of a unique strong solution. The resultant candidate of the market
price process is derived completely endogenously by the optimal trading
strategies of the agents facing systemic information (including securities’
coupon stream) as well as idiosyncratic noise. Another benefit of
probabilistic approach is that it allows us to quantify the relation between
the actual game with finite number of agents and its large population limit.
In a similar manner to the standard mean field games in proving
$\varepsilon$-Nash equilibrium, we show that the solution of the mean-field
limit problem actually provides asymptotic market clearing in the large-$N$
limit. Under additional integrability conditions, Glivenko-Cantelli
convergence theorem in the Wasserstein distance even provides a specific order
of convergence in terms of the number of agents $N$. We also discuss the
extension of the model to the situation with multiple populations where the
agents share the same cost and coefficient functions within each population
but they can be different population by population. This will provide an
important tool to study the price formation in the presence of different type
of agents such as Buy-side and Sell-side institutions, for example.
The organization of the paper is as follows: After explaining the notation in
Section 2, we give an intuitive derivation of the limit problem from the game
of finite number of agents in Section 3, which motivates the readers to study
the special type of FBSDEs of MKV-type. The solvability of the FBSDE is
studied in Section 4. Using the derived regularity of the solution, we prove
the asymptotic market clearing in Section 5. In Section 6, we discuss the
extension of the model to the setup with multiple populations. Finally, in
Section 7, we give concluding remarks. We discuss further extensions of the
model and future directions of research.
## 2 Notation
We introduce (N+1) complete probability spaces:
$\displaystyle(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0})\quad{\rm{and}}\quad(\overline{\Omega}^{i},\overline{{\cal
F}}^{i},\overline{\mathbb{P}}^{i})_{i=1}^{N}~{},$
endowed with filtrations $\overline{\mathbb{F}}^{i}:=(\overline{{\cal
F}}_{t}^{i})_{t\geq 0}$, $i\in\\{0,\cdots,N\\}$. Here,
$\overline{\mathbb{F}}^{0}$ is the completion of the filtration generated by
$d^{0}$-dimensional Brownian motion $\boldsymbol{W}^{0}$ (hence right-
continuous) and, for each $i\in\\{1,\cdots,N\\}$, $\overline{\mathbb{F}}^{i}$
is the complete and right-continuous augmentation of the filtration generated
by $d$-dimensional Brownian motions $\boldsymbol{W}^{i}$ as well as a
$\boldsymbol{W}^{i}$-independent $n$-dimensional square-integrable random
variables $(\xi^{i})$. $(\xi^{i})_{i=1}^{N}$ are supposed to have the same
law. We also introduce the product probability spaces
$\Omega^{i}=\overline{\Omega}^{0}\times\overline{\Omega}^{i},\quad{\cal
F}^{i},\quad\mathbb{F}^{i}=({\cal F}_{t}^{i})_{t\geq
0},\quad\mathbb{P}^{i}~{},i\in\\{1,\cdots,N\\}$
where $({\cal F}^{i},\mathbb{P}^{i})$ is the completion of $(\overline{{\cal
F}}^{0}\otimes\overline{{\cal
F}}^{i},\overline{\mathbb{P}}^{0}\otimes\overline{\mathbb{P}}^{i})$ and
$\mathbb{F}^{i}$ is the complete and right-continuous augmentation of
$(\overline{{\cal F}}_{t}^{0}\otimes\overline{{\cal F}}_{t}^{i})_{t\geq 0}$.
In the same way, we define the complete probability space $(\Omega,{\cal
F},\mathbb{P})$ endowed with $\mathbb{F}=({\cal F}_{t})_{t\geq 0}$ satisfying
the usual conditions as a product of $(\overline{\Omega}^{i},\overline{{\cal
F}}^{i},\overline{\mathbb{P}}^{i};\overline{\mathbb{F}}^{i})_{i=0}^{N}$.
Throughout the work, the symbol $L$ and $L_{\varpi}$ denote given positive
constants, the symbol $C$ a general positive constant which may change line by
line. When we want to emphasize that $C$ depends only on some specific
variables, say $a$ and $b$, we use the symbol $C(a,b)$. For a given constant
$T>0$, we use the following notation for frequently encountered spaces:
$\bullet~{}$$\mathbb{L}^{2}({\cal G};\mathbb{R}^{d})$ denotes the set of
$\mathbb{R}^{d}$-valued ${\cal G}$-measurable square integrable random
variables.
$\bullet~{}$$\mathbb{S}^{2}(\mathbb{G};\mathbb{R}^{d})$ is the set of
$\mathbb{R}^{d}$-valued $\mathbb{G}$-adapted continuous processes
$\boldsymbol{X}$ satisfying
$\displaystyle||X||_{\mathbb{S}^{2}}:=\mathbb{E}\bigl{[}\sup_{t\in[0,T]}|X_{t}|^{2}\bigr{]}^{\frac{1}{2}}<\infty~{}.$
$\bullet~{}$$\mathbb{H}^{2}(\mathbb{G};\mathbb{R}^{d})$ is the set of
$\mathbb{R}^{d}$-valued $\mathbb{G}$-progressively measurable processes
$\boldsymbol{Z}$ satisfying
$\displaystyle||Z||_{\mathbb{H}^{2}}:=\mathbb{E}\Bigl{[}\Bigl{(}\int_{0}^{T}|Z_{t}|^{2}dt\Bigr{)}\Bigr{]}^{\frac{1}{2}}<\infty~{}.$
$\bullet~{}$${\cal L}(X)$ denotes the law of a random variable $X$.
$\bullet~{}{\cal P}(\mathbb{R}^{d})$ is the set of probability measures on
$(\mathbb{R}^{d},{\cal B}(\mathbb{R}^{d}))$.
$\bullet~{}{\cal P}_{p}(\mathbb{R}^{d})$ with $p\geq 1$ is the subset of
${\cal P}(\mathbb{R}^{d})$ with finite $p$-th moment; i.e., the set of
$\mu\in{\cal P}(\mathbb{R}^{d})$ satisfying
$\displaystyle
M_{p}(\mu):=\Bigl{(}\int_{\mathbb{R}^{d}}|x|^{p}\mu(dx)\Bigr{)}^{\frac{1}{p}}<\infty~{}.$
We always assign ${\cal P}_{p}(\mathbb{R}^{d})$ with $(p\geq 1)$ the
$p$-Wasserstein distance $W_{p}$, which makes ${\cal P}_{p}(\mathbb{R}^{d})$ a
complete separable metric space. As an important property, for any
$\mu,\nu\in{\cal P}_{p}(\mathbb{R}^{d})$, we have
$\displaystyle
W_{p}(\mu,\nu)={\inf}\Bigl{\\{}\mathbb{E}[|X-Y|^{p}]^{\frac{1}{p}};{\cal
L}(X)=\mu,{\cal L}(Y)=\nu\Bigr{\\}}~{},$ ( 2.1)
where “$\inf$” is taken over all random variables with laws equal to $\mu$ and
$\nu$, respectively. For more details, see Chapter 5 in [7]. We frequently
omit the arguments such as $(\mathbb{G},\mathbb{R}^{d})$ in the above
definitions when there is no confusion from the context.
## 3 Intuitive Derivation of the Mean Field Problem
In this section, in order to introduce the special form of forward-backward
stochastic differential equations of McKean-Vlasov type to be studied in this
paper, we give a heuristic derivation of the mean-field limit problem from the
corresponding equilibrium problem with finite number of agents. As a
motivating example, we consider the equilibrium-based pricing problem of $n$
types of securities, which are continuously traded in the security exchange in
the presence of a large number of participating agents indexed by
$i\in\\{1,\cdots,N\\}$. Every agent is supposed to have many small clients who
can only trade directly to the agent via over-the-counter markets (OTC) and
have no access to the security exchange.
We suppose that each agent $i\in\\{1,\cdots,N\\}$ tries to solve the problem
$\inf_{\boldsymbol{\alpha^{i}}\in\mathbb{A}^{i}}J^{i}(\boldsymbol{\alpha}^{i})$
( 3.1)
with some cost functional
$\displaystyle
J^{i}(\boldsymbol{\alpha}^{i}):=\mathbb{E}\Bigl{[}\int_{0}^{T}f(t,X_{t}^{i},\alpha_{t}^{i},\varpi_{t},c_{t}^{0},c_{t}^{i})dt+g(X_{T}^{i},\varpi_{T},c_{T}^{0},c_{T}^{i})\Bigr{]}~{},$
subject to the dynamic constraint:
$\displaystyle
dX_{t}^{i}=\Bigl{(}\alpha_{t}^{i}+l(t,\varpi_{t},c_{t}^{0},c_{t}^{i})\Bigr{)}dt+\sigma_{0}(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{0}+\sigma(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{i}$
with $X_{0}^{i}=\xi^{i}\in\mathbb{L}^{2}(\overline{{\cal
F}}_{0}^{i};\mathbb{R}^{n})$. Here, $(X_{t}^{i})_{t\geq 0}$ is an
$\mathbb{R}^{n}$-valued process denoting the time-$t$ position of the $n$
securities for the agent $i$ with the initial position $\xi^{i}$.
$(c_{t}^{0})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$ denotes the
coupon payments from the securities or the market news commonly available to
all the agents, while $(c_{t}^{i})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{i};\mathbb{R}^{n})$ denotes some
independent factors affecting only on the agent $i$. Moreover,
$(c_{t}^{i})_{t\geq 0}$ are also assumed to have the common law for all $1\leq
i\leq N$. We further suppose $c_{T}^{0}$ and $c_{T}^{i}$ are square integrable
to handle the terminal cost $g$. Each agent controls $(\alpha_{t}^{i})_{t\geq
0}$ denoting the trading speed though the security exchange. The remaining
terms $(l,\sigma_{0},\sigma)$ denote the order flow to the agent from his/her
clients through over-the-counter (OTC) markets. $(\varpi_{t})_{t\geq 0}$ is
the market price of the $n$ securities. The space of admissible strategies
$\mathbb{A}^{i}$ of the agent $i$ is the set of processes
$(\alpha^{i}_{t})_{t\geq 0}$ adapted to the complete right-continuous
augmentation of the filtration $\bigl{(}\sigma\\{\varpi_{s}:s\leq
t\\})\vee{\cal F}_{t}^{i}\bigr{)}_{t\geq 0}$ satisfying
$\mathbb{E}\int_{0}^{T}|\alpha_{t}^{i}|^{2}dt<\infty~{}.$
In contrast to the standard optimization problems with a given market price
process, we want to understand the fundamental mechanism of financial market
which determines the market price by the equilibrium condition. The
equilibrium price $(\varpi_{t})_{t\geq 0}$ adapted to the filtration
$\mathbb{F}$ is determined endogenously so that the optimal strategies of the
agents $(\widehat{\alpha}^{i}_{t})_{i=1}^{N}$ satisfy the market clearing
condition for every $t\in[0,T]$, $\mathbb{P}$-a.s.
$\displaystyle\sum_{i=1}^{N}\widehat{\alpha}_{t}^{i}=0~{},$ ( 3.2)
which denotes the balance point of demand and supply at the security exchange.
Although we have already made simplistic assumptions such that the cost
functions as well as the coefficient functions of all the agents are common,
the problem is still hardly solvable. Due to the clearing condition
$(\ref{orig-clearing})$, we cannot adopt an open-loop equilibrium approach. In
particular, $(\varpi_{t})_{t\geq 0}$ becomes a complicated functional of the
agents’ trading strategies and hence the problem for each agent is highly
recursive with respect to $(\alpha^{i}_{t})_{t\geq 0,1\leq i\leq N}$. It is
even unclear how to guarantee the cost function well-defined by making it
convex with respect to the controls.
In order to obtain some insight, let us consider a much simpler situation. It
is natural to suppose that the impact to the market price
$(\varpi_{t})_{t\in[0,T]}$ from the individual agent becomes negligibly small
when $N$ is sufficiently large. Moreover, $(\varpi_{t})_{t\in[0,T]}$ is likely
to be given by $\overline{\mathbb{F}}^{0}$-progressively measurable process
since the effects from the idiosyncratic parts from many agents are expected
to be canceled out. If this is the case, the problem for each agent reduces to
the standard stochastic optimal control problem in a given random environment
$(\varpi_{t},c_{t}^{0},c_{t}^{i})_{t\in[0,T]}$ with $\mathbb{F}^{i}$-adapted
trading strategy $(\alpha^{i}_{t})_{t\in[0,T]}$. Let us first investigate this
simple problem in details. We introduce the cost functions:
$f:[0,T]\times(\mathbb{R}^{n})^{5}\rightarrow\mathbb{R}$,
$g:(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$,
$\overline{f}:[0,T]\times(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$ and
$\overline{g}:(\mathbb{R}^{n})^{3}\rightarrow\mathbb{R}$, which are measurable
functions such that
$\displaystyle
f(t,x,\alpha,\varpi,c^{0},c):=\langle\varpi,\alpha\rangle+\frac{1}{2}\langle\alpha,\Lambda\alpha\rangle+\overline{f}(t,x,\varpi,c^{0},c),$
$\displaystyle
g(x,\varpi,c^{0},c):=-\delta\langle\varpi,x\rangle+\overline{g}(x,c^{0},c)~{}.$
###### Assumption 3.1.
(MFG-a)
(i) $\Lambda$ is a positive definite $n\times n$ symmetric matrix with
$\underline{\lambda}I_{n\times n}\leq\Lambda\leq\overline{\lambda}I_{n\times
n}$ in the sense of 2nd-order form where $\underline{\lambda}$ and
$\overline{\lambda}$ are some constants satisfying
$0<\underline{\lambda}\leq\overline{\lambda}$.
(ii) For any $(t,x,\varpi,c^{0},c)$,
$\displaystyle|\overline{f}(t,x,\varpi,c^{0},c)|+|\overline{g}(x,c^{0},c)|\leq
L(1+|x|^{2}+|\varpi|^{2}+|c^{0}|^{2}+|c|^{2})~{}.$
(iii) $\overline{f}$ and $\overline{g}$ are continuously differentiable in $x$
and satisfy, for any $(t,x,x^{\prime},\varpi,c^{0},c)$,
$|\partial_{x}\overline{f}(t,x^{\prime},\varpi,c^{0},c)-\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}(x^{\prime},c^{0},c)-\partial_{x}\overline{g}(x,c^{0},c)|\leq
L|x^{\prime}-x|~{},$
and
$|\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}(x,c^{0},c)|\leq
L(1+|x|+|\varpi|+|c^{0}|+|c|)$.
(iv)The functions $\overline{f}$ and $\overline{g}$ are convex in $x$ in the
sense that for any $(t,x,x^{\prime},\varpi,c^{0},c)$,
$\displaystyle\overline{f}(t,x^{\prime},\varpi,c^{0},c)-\overline{f}(t,x,\varpi,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)\rangle\geq\frac{\gamma^{f}}{2}|x^{\prime}-x|^{2}~{},$
$\displaystyle\overline{g}(x^{\prime},c^{0},c)-\overline{g}(x,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{g}(x,c^{0},c)\rangle\geq\frac{\gamma^{g}}{2}|x^{\prime}-x|^{2}~{},$
with some constants $\gamma^{f},\gamma^{g}\geq 0$.
(v) $l,\sigma_{0},\sigma$ are the measurable functions defined on
$[0,T]\times(\mathbb{R}^{n})^{3}$ and are $\mathbb{R}^{n},\mathbb{R}^{n\times
d^{0}}$ and $\mathbb{R}^{n\times d}$-valued, respectively. Moreover they
satisfy the linear growth condition:
$\displaystyle|(l,\sigma_{0},\sigma)(t,\varpi,c^{0},c)|\leq
L(1+|\varpi|+|c^{0}|+|c|)$
for any $(t,\varpi,c^{0},c)$.
(vi) $\delta\in[0,1)$ is a given constant.
The first term $\langle\varpi,\alpha\rangle$ of $f$ denotes the direct cost
incurred by the sales and purchase of the securities and the second term
$\frac{1}{2}\langle\alpha,\Lambda\alpha\rangle$ is some fee to be paid to the
exchange depending on the trading speed, or may be interpreted as some
internal cost. The first term of $g$ denotes the mark-to-market value at the
closing time with some discount factor $\delta<1$.444We shall see that the
condition $\delta<1$ is necessary to obtain well-defined terminal condition
for the limit problem. $\overline{f}$ and $\overline{g}$ denote the running as
well as the terminal cost which are affected by the market price, coupon
streams, or the news.
###### Remark 3.1.
If we think $c^{0}$ as a coupon stream of the securities, one may consider for
example,
$\displaystyle\overline{f}(t,x,\varpi,c^{0},c)=-\langle
c^{0},x\rangle+\overline{f}^{\prime}(t,x,\varpi,c)$
as a running cost with an appropriate measurable function
$\overline{f}^{\prime}$. For securities with a given maturity $T$ with
exogenously specified payoff $c^{0}$, such as bonds and futures, it is natural
to consider
$\displaystyle g(x,c^{0})=\overline{g}(x,c^{0})=-\langle c^{0},x\rangle$
as the terminal cost.
For this problem, the (reduced) Hamiltonian is given by
$\displaystyle H(t,x,y,\alpha,\varpi,c^{0},c)=\langle
y,\alpha+l(t,\varpi,c^{0},c)\rangle+f(t,x,\alpha,\varpi,c^{0},c)~{}.$
Since
$\partial_{\alpha}H(t,x,y,\alpha,\varpi,c^{0},c)=y+\varpi+\Lambda\alpha$, the
minimizer of the Hamiltonian is
$\displaystyle\widehat{\alpha}(y,\varpi):=-\overline{\Lambda}(y+\varpi)$ (
3.3)
where $\overline{\Lambda}:=\Lambda^{-1}$. The adjoint FBSDE associated with
the stochastic maximal principle for each agent $1\leq i\leq N$ is thus given
by,
$\displaystyle
dX_{t}^{i}=\Bigl{(}\widehat{\alpha}(Y_{t}^{i},\varpi_{t})+l(t,\varpi_{t},c_{t}^{0},c_{t}^{i})\Bigr{)}dt+\sigma_{0}(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{0}+\sigma(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{i}~{},$
$\displaystyle
dY_{t}^{i}=-\partial_{x}\overline{f}(t,X_{t}^{i},\varpi_{t},c_{t}^{0},c_{t}^{i})dt+Z_{t}^{i,0}dW_{t}^{0}+Z_{t}^{i}dW_{t}^{i}~{},$
( 3.4)
with $X_{0}^{i}=\xi^{i}$ and
$Y_{T}^{i}=\partial_{x}g(X_{T}^{i},\varpi_{T},c_{T}^{0},c_{T}^{i})$.
###### Theorem 3.1.
Under Assumption (MFG-a) and a given
$(\varpi_{t})_{t\in[0,T]}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$,
the problem $(\ref{agent-P})$ for each agent is uniquely characterized by the
FBSDE $(\ref{agent-FBSDE})$ which is strongly solvable with a unique solution
$(X^{i},Y^{i},Z^{i,0},Z^{i})\in\mathbb{S}^{2}(\mathbb{F}^{i};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{i};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{i};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{i};\mathbb{R}^{n\times d})$.
###### Proof.
Since the cost functions are jointly convex with $(x,\alpha)$ and strictly
convex in $\alpha$, the problem is the special situation investigated in
Section 1.4.4 in [8]. Note that, in our case, the diffusion terms
$\sigma_{0},\sigma$ are independent of $(X^{i},\alpha^{i})$. The proof is the
direct result of Theorem 1.60 in the same reference. ∎
Using the above solution, the optimal strategy of each agent is given by
$\displaystyle\widehat{\alpha}^{i}_{t}=-\overline{\Lambda}(Y_{t}^{i}+\varpi_{t}),\quad
t\in[0,T]~{}.$
Let us check the market clearing condition. In the current situation,
$(\ref{orig-clearing})$ is equivalent to
$\displaystyle\varpi_{t}=-\frac{1}{N}\sum_{i=1}^{N}Y_{t}^{i}$
which is of course inconsistent with the our simplifying assumption that
requires $(\varpi_{t})_{t\geq 0}$ to be an $\overline{\mathbb{F}}^{0}$-adapted
process. However, in the current setup, for any $t\in[0,T]$,
$(Y^{i}_{t})_{i=1}^{N}$ are exchangeable random variables due to the
construction of the probability space, common coefficient functions, and the
fact that $(\xi^{i})_{i=1}^{N}$ as well as $(c_{t}^{i},t\in[0,T])_{i=1}^{N}$
are assumed to be i.i.d. Thus De Finetti’s theory of exchangeable sequence of
random variables tells,
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}Y_{t}^{i}=\mathbb{E}\Bigl{[}Y_{t}^{1}|\bigcap_{k\geq
1}\sigma\\{Y_{t}^{j},j\geq k\\}\Bigr{]}\quad{\rm a.s.}$
See for example Theorem 2.1 in [8]. It also seems natural to expect that the
tail $\sigma$-field is reduced to $\overline{{\cal F}}_{t}^{0}$. Therefore we
can expect that, in the large-$N$ limit, the market price of the securities
may be given by $\varpi_{t}=-\mathbb{E}[Y_{t}^{1}|\overline{{\cal
F}}_{t}^{0}]$.
The above observation motivates us to consider the following FBSDE:
$\displaystyle
dX_{t}=\Bigl{(}\widehat{\alpha}\bigl{(}Y_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]\bigr{)}+l\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}\Bigr{)}dt$
$\displaystyle\qquad\quad+\sigma_{0}\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{0}+\sigma\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{1}~{},$ $\displaystyle
dY_{t}=-\partial_{x}\overline{f}\bigl{(}t,X_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dt+Z_{t}^{0}dW_{t}^{0}+Z_{t}dW_{t}^{1}~{},$
with $X_{0}=\xi$ with $\displaystyle
Y_{T}=\frac{\delta}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})$. To
simplify the notation, we have omitted the superscript $1$ from $Y^{1}$,
$X^{1}$, $\xi^{1}$ and $c^{1}$. Let us remark on the terminal condition.
$Y_{T}=\partial_{x}g(X_{T},-\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}],c_{T}^{0},c_{T})$ is not yet fully specified. Taking the
conditional expectation in the both sides gives
$\displaystyle\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}]=\delta\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}]+\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}~{},$
which implies $\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}]=\frac{1}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}$. Substituting this expression for
$\mathbb{E}[Y_{T}|\overline{{\cal F}}_{T}^{0}]$ in $\partial_{x}g$, we get the
above specification of the terminal condition.
This is the FBSDE we are going to study in the following. It is of McKean-
Vlasov type with common noise, and similar to the FBSDEs relevant for the
extended mean field games. In the following, we are going to prove the
existence of a unique solution to the above FBSDE under appropriate conditions
and then show that $-\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}]$ is
actually a good approximate of the market price by investigating how
accurately it achieves the market clearing condition $(\ref{orig-clearing})$
when $N$ increases.
## 4 Solvability of the mean-field FBSDE
We now investigate the solvability of the FBSDE derived in the last section
$\displaystyle
dX_{t}=\Bigl{(}\widehat{\alpha}\bigl{(}Y_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]\bigr{)}+l\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}\Bigr{)}dt$
$\displaystyle\qquad\quad+\sigma_{0}\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{0}+\sigma\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{1}~{},$ $\displaystyle
dY_{t}=-\partial_{x}\overline{f}\bigl{(}t,X_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dt+Z_{t}^{0}dW_{t}^{0}+Z_{t}dW_{t}^{1}~{},$
( 4.1)
with $X_{0}=\xi$ with $\displaystyle
Y_{T}=\frac{\delta}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})$.
$\widehat{\alpha}$ is defined as in $(\ref{def-ha})$. $(c^{0}_{t})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$ and
$(c_{t})_{t\geq 0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{1};\mathbb{R}^{n})$
with square integrable $c_{T}^{0},c_{T}$ are given as inputs. Let us remind
the notation to write $\xi=\xi^{1}$ and $c=c^{1}$.
### 4.1 Unique existence for small $T$
###### Assumption 4.1.
(MFG-b)
For any $(t,x,c^{0},c)\in[0,T]\times(\mathbb{R}^{n})^{3}$ and any
$\varpi,\varpi^{\prime}\in\mathbb{R}^{n}$, the coefficient functions
$l,\sigma_{0},\sigma$ and $\overline{f}$ satisfy
$\displaystyle|(l,\sigma_{0},\sigma)(t,\varpi,c^{0},c)-(l,\sigma_{0},\sigma)(t,\varpi^{\prime},c^{0},c)|+|\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)-\partial_{x}\overline{f}(t,x,\varpi^{\prime},c^{0},c)|\leq
L_{\varpi}|\varpi-\varpi^{\prime}|~{}.$
Due to the Lipschitz continuity and the absence of $(Z^{0},Z)$ in the
diffusion coefficients of the forward SDE, we have the following short-term
existence result.
###### Theorem 4.1.
Under Assumptions (MFG-a,b), there exists some constant $\tau>0$ which depends
only on $(L,L_{\varpi},\underline{\lambda},\delta)$ such that for any
$T\leq\tau$, there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE $(\ref{fbsde-single-p})$.
###### Proof.
Although there exist terms involving $\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]$, one can adopt the standard technique for the Lipschitz FBSDE.
See, for example, the proof of Theorem 1.45 [8]. ∎
### 4.2 Unique existence for general $T$
In order to obtain existence result for general $T$, we are going to apply the
technique developed by Peng & Wu [24]. In the case of the standard
optimization problem, the joint convexity in the state and control variables
combined with strict convexity in the control variable are enough to obtain
the unique existence. Interestingly however, we need a strict convexity also
in the state variable $X$ in our problem. As we shall see, this is because the
term $-\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}]$ which appears due to the
clearing condition weakens the convexity.
###### Assumption 4.2.
(MFG-c1)
(i) The functions $\sigma_{0}$ and $\sigma$ are independent of the argument
$\varpi$.
(ii) For any $t\in[0,T]$, any random variables
$x,x^{\prime},c^{0},c\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and any
sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the function $l$ satisfies the
monotone condition with some positive constant $\gamma^{l}>0$:
$\displaystyle\mathbb{E}\Bigl{[}\langle l(t,\mathbb{E}[x|{\cal
G}],c^{0},c)-l(t,\mathbb{E}[x^{\prime}|{\cal
G}],c^{0},c),x-x^{\prime}\rangle\Bigr{]}\geq\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[x-x^{\prime}|{\cal
G}]^{2}\bigr{]}~{}.$
(iii) There exists a strictly positive constant $\gamma$ satisfying
$0<\gamma\leq\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\wedge\gamma^{g}$.
Moreover, for any $x,x^{\prime},c^{0},c\in\mathbb{L}^{2}({\cal
F};\mathbb{R}^{n})$ and any sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the
function $\overline{g}$ satisfies
$\displaystyle\gamma^{g}\mathbb{E}[|x-x^{\prime}|^{2}]+\frac{\delta}{1-\delta}\mathbb{E}\Bigl{[}\langle\mathbb{E}\bigl{[}\partial_{x}\overline{g}(x,c^{0},c)-\partial_{x}\overline{g}(x^{\prime},c^{0},c)|{\cal
G}\bigr{]},x-x^{\prime}\rangle\Bigr{]}\geq\gamma\mathbb{E}[|x-x^{\prime}|^{2}]~{}.$
###### Remark 4.1.
If $l$ and $\partial_{x}\overline{g}$ have separable forms such as
$h(x)+h^{c}(c^{0},c)$ with some functions $h$ and $h^{c}$, then the conditions
(ii) and (iii) are satisfied when the function $h$ is monotone. Economically
speaking, the condition (ii) implies that the demand from the individual OTC
clients of each agent toward the security decreases when its market price
rises.
The next theorem is the first main existence result.
###### Theorem 4.2.
Under Assumptions (MFG-a,b,c1), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE $(\ref{fbsde-single-p})$.
###### Proof.
In order to simplify the notation, let us define the functionals $B,F$ and $G$
for any $y,x,c^{0},c\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ by
$\displaystyle
B(t,y,c^{0},c):=\Bigl{(}-\overline{\Lambda}(y-\mathbb{E}[y|\overline{{\cal
F}}_{t}^{0}])+l(t,-\mathbb{E}[y|\overline{{\cal
F}}_{t}^{0}],c^{0},c)\Bigr{)},$ $\displaystyle
F(t,x,y,c^{0},c):=-\partial_{x}\overline{f}\bigl{(}t,x,-\mathbb{E}[y|\overline{{\cal
F}}_{t}^{0}],c^{0},c\bigr{)},$ $\displaystyle
G(x,c^{0},c):=\frac{\delta}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(x,c^{0},c)|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}(x,c^{0},c)~{}.$ ( 4.2)
With the convention $\Delta y:=y-y^{\prime}$, $\Delta x:=x-x^{\prime}$, one
can easily confirms
$\displaystyle\mathbb{E}\bigl{[}\langle
B(t,y,c^{0},c)-B(t,y^{\prime},c^{0},c),\Delta
y\rangle\bigr{]}\leq-\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\Delta
y|\overline{{\cal F}}_{t}^{0}]^{2}\bigr{]}~{},$
$\displaystyle\mathbb{E}\bigl{[}\langle
F(t,x,y,c^{0},c)-F(t,x^{\prime},y^{\prime},c^{0},c),\Delta
x\rangle\bigr{]}\leq-\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\mathbb{E}[|\Delta
x|^{2}]+\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\Delta y|\overline{{\cal
F}}_{t}^{0}]^{2}\bigr{]}~{},$ $\displaystyle\mathbb{E}\bigl{[}\langle
G(x,c^{0},c)-G(x^{\prime},c^{0},c),\Delta
x\rangle\bigr{]}\geq\gamma\mathbb{E}[|\Delta x|^{2}],$ ( 4.3)
where the first estimate follows from (MFG-c1)(ii) and Jensen’s inequality,
the second from (MFG-a)(iv), (MFG-b) and Cauchy-Schwarz inequality. The third
one is the direct consequence of (MFG-c)(iii).
We first make the following hypothesis: there exists some constant
$\varrho\in[0,1)$ such that, for any $(I_{t}^{b})_{t\geq 0}$,
$(I_{t}^{f})_{t\geq 0}$ in $\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})$ and
any $\eta\in\mathbb{L}^{2}({\cal F}_{T}^{1};\mathbb{R}^{n})$, there exists a
unique solution
$(x^{\varrho},y^{\varrho},z^{0,\varrho},z^{\varrho})\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE:
$\displaystyle dx_{t}^{\varrho}=\bigl{(}\varrho
B(t,y_{t}^{\varrho},c_{t}^{0},c_{t})+I_{t}^{b}\bigr{)}dt+\sigma_{0}(t,c_{t}^{0},c_{t})dW_{t}^{0}+\sigma(t,c_{t}^{0},c_{t})dW_{t}^{1}~{},$
$\displaystyle dy_{t}^{\varrho}=-\bigl{(}(1-\varrho)\gamma
x_{t}^{\varrho}-\varrho
F(t,x_{t}^{\varrho},y_{t}^{\varrho},c_{t}^{0},c_{t})+I_{t}^{f}\bigr{)}dt+z_{t}^{0,\varrho}dW_{t}^{0}+z_{t}^{\varrho}dW_{t}^{1}~{},$
( 4.4)
with $x_{0}^{\varrho}=\xi$ and $y_{T}^{\varrho}=\varrho
G(x_{T}^{\varrho},c_{T}^{0},c_{T})+(1-\varrho)x_{T}^{\varrho}+\eta$. Note that
when $\varrho=0$ we have a decoupled set of SDE and BSDE and hence the
hypothesis trivially holds. Our goal is to extend the $\varrho$ up to $1$ by
following Peng-Wu’s continuation method [24]. Now, for an arbitrary set of
inputs
$(x,y,z^{0},z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})^{2}\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ and
constant $\zeta\in(0,1)$, consider
$\displaystyle dX_{t}=\bigl{[}\varrho B(t,Y_{t},c_{t}^{0},c_{t})+\zeta
B(t,y_{t},c_{t}^{0},c_{t})+I_{t}^{b}\bigr{]}dt+\sigma_{0}(t,c_{t}^{0},c_{t})dW_{t}^{0}+\sigma(t,c_{t}^{0},c_{t})dW_{t}^{1}~{},$
$\displaystyle dY_{t}=-\bigl{[}(1-\varrho)\gamma X_{t}-\varrho
F(t,X_{t},Y_{t},c_{t}^{0},c_{t})+\zeta(-\gamma
x_{t}-F(t,x_{t},y_{t},c_{t}^{0},c_{t}))+I_{t}^{f}\bigr{]}dt$
$\displaystyle\qquad\quad+Z_{t}^{0}dW_{t}^{0}+Z_{t}dW_{t}^{1}~{},$ ( 4.5)
with $X_{0}=\xi$ and $Y_{T}=\varrho
G(X_{T},c_{T}^{0},c_{T})+(1-\varrho)X_{T}+\zeta(G(x_{T},c_{T}^{0},c_{T})-x_{T})+\eta$.
The existence of the solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\times\mathbb{S}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$
is guaranteed by the previous hypothesis. We are going to prove the map
$(x,y,z^{0},z)\mapsto(X,Y,Z^{0},Z)$ defined above becomes strict contraction
when $\zeta>0$ is chosen small enough.
For two set of inputs $(x,y,z^{0},z)$ and
$(x^{\prime},y^{\prime},z^{0\prime},z^{\prime})$, let us denote the
corresponding solutions to $(\ref{shifted-1})$ by $(X,Y,Z^{0},Z)$ and
$(X^{\prime},Y^{\prime},Z^{0\prime},Z^{\prime})$, respectively. We put $\Delta
X_{t}:=X_{t}-X_{t}^{\prime}$, $\Delta Y_{t}:=Y_{t}-Y_{t}^{\prime}$ and
similarly for the others. Applying Itô’s formula to $\langle\Delta
X_{t},\Delta Y_{t}\rangle$ and using the estimates $(\ref{peng-wu-
condition})$, we obtain
$\displaystyle\mathbb{E}\bigl{[}\langle\Delta X_{T},\Delta
Y_{T}\rangle\bigr{]}\leq-\gamma\mathbb{E}\int_{0}^{T}|\Delta X_{t}|^{2}dt$
$\displaystyle\hskip 85.35826pt+\zeta C\mathbb{E}\int_{0}^{T}\Bigl{[}|\Delta
Y_{t}|(|\Delta y_{t}|+\mathbb{E}[\Delta y_{t}|\overline{{\cal
F}}_{t}^{0}])+|\Delta X_{t}|(|\Delta x_{t}|+\mathbb{E}[|\Delta
y_{t}|\overline{{\cal F}}_{t}^{0}])\Bigr{]}dt~{},$
$\displaystyle\mathbb{E}\bigl{[}\langle\Delta X_{T},\Delta
Y_{T}\rangle\bigr{]}\geq(\varrho\gamma+(1-\varrho))\mathbb{E}[|\Delta
X_{T}|^{2}]-\zeta C\mathbb{E}\bigl{[}|\Delta X_{T}|(|\Delta
x_{T}|+\mathbb{E}[|\Delta x_{T}||\overline{{\cal F}}_{T}^{0}])\bigr{]}~{},$
with some $\varrho$-independent constant $C$. Let us set
$\gamma_{c}:=\min(1,\gamma)>0$. Then one easily confirms
$0<\gamma_{c}\leq\varrho\gamma+(1-\varrho)$ for any $\varrho\in[0,1)$. Then
the above estimates yields
$\displaystyle\gamma_{c}\mathbb{E}\Bigl{[}|\Delta
X_{T}|^{2}+\int_{0}^{T}|\Delta X_{t}|^{2}dt\Bigr{]}$ $\displaystyle\leq$
$\displaystyle\zeta C\mathbb{E}\bigl{[}|\Delta X_{T}|(|\Delta
x_{T}|+\mathbb{E}[|\Delta x_{T}||\overline{{\cal F}}_{T}^{0}])\bigr{]}$
$\displaystyle+\zeta C\mathbb{E}\int_{0}^{T}\Bigl{[}|\Delta Y_{t}|(|\Delta
y_{t}|+\mathbb{E}[\Delta y_{t}|\overline{{\cal F}}_{t}^{0}])+|\Delta
X_{t}|(|\Delta x_{t}|+\mathbb{E}[|\Delta y_{t}|\overline{{\cal
F}}_{t}^{0}])\Bigr{]}dt~{}.$
Using Young’s inequality and and a new constant $C$, we get
$\displaystyle\mathbb{E}[|\Delta X_{T}|^{2}]+\mathbb{E}\int_{0}^{T}|\Delta
X_{t}|^{2}dtd\leq\zeta C\mathbb{E}\int_{0}^{T}\bigl{(}|\Delta
Y_{t}|^{2}+(|\Delta x_{t}|^{2}+|\Delta y_{t}|^{2})\bigr{)}dt+\zeta
C\mathbb{E}[|\Delta x_{T}|^{2}]~{}.$ ( 4.6)
Treating $X,X^{\prime}$ as inputs, the standard estimates for the Lipschitz
BSDEs (see, for example, Theorem 4.2.3 in [25]) gives
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
Y_{t}|^{2}+\int_{0}^{T}(|\Delta Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$
$\displaystyle\qquad\quad\leq C\mathbb{E}\Bigl{[}|\Delta
X_{T}|^{2}+\int_{0}^{T}|\Delta X_{t}|^{2}dt\Bigr{]}+\zeta
C\mathbb{E}\Bigl{[}|\Delta x_{T}|^{2}+\int_{0}^{T}(|\Delta x_{t}|^{2}+|\Delta
y_{t}|^{2})dt\Bigr{]}~{}.$
Combining with $(\ref{eq-pw-1})$ and choosing $\zeta>0$ small, we obtain
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
Y_{t}|^{2}+\int_{0}^{T}(|\Delta Z_{t}^{0}|^{2}+|\Delta
Z_{t}|^{2})dt\Bigr{]}\leq\zeta C\mathbb{E}\Bigl{[}|\Delta
x_{T}|^{2}+\int_{0}^{T}(|\Delta x_{t}|^{2}+|\Delta y_{t}|^{2})dt\Bigr{]}~{}.$
( 4.7)
By the similar procedures, we also have
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}\Bigr{]}\leq\zeta C\mathbb{E}\Bigl{[}|\Delta
x_{T}|^{2}+\int_{0}^{T}(|\Delta x_{t}|^{2}+|\Delta y_{t}|^{2})dt\Bigr{]}~{}.$
( 4.8)
From $(\ref{eq-pw-2})$ and $(\ref{eq-pw-3})$, we obtain
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}+\sup_{t\in[0,T]}|\Delta Y_{t}|^{2}+\int_{0}^{T}(|\Delta
Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$ $\displaystyle\leq\zeta
C\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta x_{t}|^{2}+\sup_{t\in[0,T]}|\Delta
y_{t}|^{2}+\int_{0}^{T}(|\Delta z_{t}^{0}|^{2}+|\Delta z_{t}|^{2})dt\Bigr{]}.$
Thus there exists $\zeta>0$, being independent of the size of $\varrho$, that
makes the map $(x,y,z^{0},z)\mapsto(X,Y,Z^{0},Z)$ strict contraction.
Therefore the initial hypothesis holds true for $(\varrho+\zeta)$, which
establishes the existence. The uniqueness follows from the next proposition. ∎
###### Proposition 4.1.
Given two set of inputs $(\xi,c^{0},c),(\xi^{\prime},c^{0\prime},c^{\prime})$,
coefficients $(\delta,\Lambda),(\delta^{\prime},\Lambda^{\prime})$ and the
coefficient functions
$(l,\sigma_{0},\sigma,\overline{f},\overline{g}),(l^{\prime},\sigma_{0}^{\prime},\sigma^{\prime},\overline{f}^{\prime},\overline{g}^{\prime})$
satisfying Assumptions (MFG-a,b,c1), let us denote the corresponding solutions
to $(\ref{fbsde-single-p})$ by $(X,Y,Z^{0},Z)$ and
$(X^{\prime},Y^{\prime},Z^{0\prime},Z^{\prime})$, respectively. We also define
the functionals $(B,F,G)$and $(B^{\prime},F^{\prime},G^{\prime})$ by
$(\ref{BG-notation})$ with corresponding coefficients, respectively. Then, we
have the following stability result:
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}+\sup_{t\in[0,T]}|\Delta Y_{t}|^{2}+\int_{0}^{T}(|\Delta
Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$ $\displaystyle\qquad\leq
C\mathbb{E}\Bigl{[}|\Delta\xi|^{2}+|\overline{G}|^{2}+\int_{0}^{T}\Bigl{(}|\overline{F}(t)|^{2}+|\overline{B}(t)|^{2}+|\overline{\sigma}_{0}(t)|^{2}+|\overline{\sigma}(t)|^{2}\Bigr{)}dt\Bigr{]}~{},$
where $C$ is a constant depending only on $T$ as well as the Lipschitz
constants of the system, and
$\displaystyle\overline{B}(t):=B(t,Y_{t}^{\prime},c^{0}_{t},c_{t})-B^{\prime}(t,Y_{t}^{\prime},c_{t}^{0\prime},c_{t}^{\prime}),$
$\displaystyle\overline{F}(t):=F(t,X_{t}^{\prime},Y_{t}^{\prime},c_{t}^{0},c_{t})-F^{\prime}(t,X_{t}^{\prime},Y_{t}^{\prime},c_{t}^{0\prime},c_{t}^{\prime}),$
$\displaystyle(\overline{\sigma}_{0},\overline{\sigma})(t)=(\sigma_{0}(t,c_{t}^{0},c_{t})-\sigma_{0}^{\prime}(t,c_{t}^{0\prime},c_{t}^{\prime}),\sigma(t,c_{t}^{0},c_{t})-\sigma^{\prime}(t,c_{t}^{0\prime},c_{t}^{\prime})),$
$\displaystyle\overline{G}:=G(X_{T}^{\prime},c_{T}^{0},c_{T})-G^{\prime}(X_{T}^{\prime},c_{T}^{0\prime},c_{T}^{\prime})~{},$
and $\Delta\xi:=\xi-\xi^{\prime}$, $\Delta X_{t}:=X_{t}-X_{t}^{\prime}$ and
similarly for the other variables.
###### Proof.
Let us put $\Delta
B(t):=B(t,Y_{t},c_{t}^{0},c_{t})-B(t,Y_{t}^{\prime},c_{t}^{0},c_{t})$, $\Delta
F(t):=F(t,X_{t},Y_{t},c_{t}^{0},c_{t})-F(t,X_{t}^{\prime},Y_{t}^{\prime},c_{t}^{0},c_{t})$
and $\Delta G:=G(X_{T},c_{T}^{0},c_{T})-G(X_{T}^{\prime},c_{T}^{0},c_{T})$. We
get by Itô’s formula that
$\displaystyle\mathbb{E}\bigl{[}\langle\Delta X_{T},\Delta
G+\overline{G}\rangle\bigr{]}=\mathbb{E}\Bigl{[}\langle\Delta\xi,\Delta
Y_{0}\rangle+\int_{0}^{T}\Bigl{(}\langle\overline{F}(t),\Delta
X_{t}\rangle+\langle\overline{B}(t),\Delta Y_{t}\rangle$
$\displaystyle\qquad+\langle\overline{\sigma}_{0}(t),\Delta
Z_{t}^{0}\rangle+\langle\overline{\sigma}(t),\Delta
Z_{t}\rangle+\bigl{(}\langle\Delta F(t),\Delta X_{t}\rangle+\langle\Delta
B(t),\Delta Y_{t}\rangle\bigr{)}\Bigr{)}dt\Bigr{]}~{}.$
Using $(\ref{peng-wu-condition})$, we obtain
$\displaystyle\gamma\mathbb{E}\Bigl{[}|\Delta X_{T}|^{2}+\int_{0}^{T}|\Delta
X_{t}|^{2}dt\Bigr{]}\leq\mathbb{E}\Bigl{[}\langle\Delta\xi,\Delta
Y_{0}\rangle-\langle\Delta X_{T},\overline{G}\rangle$
$\displaystyle\qquad+\int_{0}^{T}\Bigl{(}\langle\overline{F}(t),\Delta
X_{t}\rangle+\langle\overline{B}(t),\Delta
Y_{t}\rangle+\langle\overline{\sigma}_{0}(t),\Delta
Z_{t}^{0}\rangle+\langle\overline{\sigma}(t),\Delta
Z_{t}\rangle\Bigr{)}dt\Bigr{]}~{}.$ ( 4.9)
On the other hand, the standard estimates for Lipschitz SDEs and BSDEs give
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
Y_{t}|^{2}+\int_{0}^{T}(|\Delta Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$
$\displaystyle\qquad\leq
C\mathbb{E}\Bigl{[}|\overline{G}|^{2}+\int_{0}^{T}|\overline{F}(t)|^{2}dt\Bigr{]}+C\mathbb{E}\Bigl{[}|\Delta
X_{T}|^{2}+\int_{0}^{T}|\Delta X_{t}|^{2}dt\Bigr{]}~{},$ ( 4.10)
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta X_{t}|^{2}\Bigr{]}\leq
C\mathbb{E}\Bigl{[}|\Delta\xi|^{2}+\int_{0}^{T}\bigl{[}|\overline{B}(t)|^{2}+|\overline{\sigma}_{0}(t)|^{2}+|\overline{\sigma}(t)|^{2}\bigr{]}dt\Bigr{]}+C\mathbb{E}\int_{0}^{T}|\Delta
Y_{t}|^{2}dt~{}.$
Combining the above inequalities $(\ref{stability-1})$ and
$(\ref{stability-y})$ gives
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}+\sup_{t\in[0,T]}|\Delta Y_{t}|^{2}+\int_{0}^{T}(|\Delta
Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$ $\displaystyle\leq
C\mathbb{E}\Bigl{[}|\Delta\xi|^{2}+|\overline{G}|^{2}+\int_{0}^{T}\bigl{[}|\overline{F}(t)|^{2}+|\overline{B}(t)|^{2}+|\overline{\sigma}_{0}(t)|^{2}+|\overline{\sigma}(t)|^{2}\bigr{]}dt\Bigr{]}$
$\displaystyle+C\mathbb{E}\Bigl{[}\langle\Delta\xi,\Delta
Y_{0}\rangle-\langle\Delta
X_{T},\overline{G}\rangle+\int_{0}^{T}\bigl{[}\langle\overline{F}(t),\Delta
X_{t}\rangle+\langle\overline{B}(t),\Delta
Y_{t}\rangle+\langle\overline{\sigma}_{0}(t),\Delta
Z_{t}^{0}\rangle+\langle\overline{\sigma}(t),\Delta
Z_{t}\rangle\bigr{]}dt\Bigr{]}~{}.$
Now simple application of Young’s inequality establishes the claim. ∎
###### Corollary 4.1.
Under Assumptions (MFG-a,b,c1), the solution $(X,Y,Z^{0},Z)$ to the FBSDE
$(\ref{fbsde-single-p})$ satisfies the following estimate:
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|X_{t}|^{2}+\sup_{t\in[0,T]}|Y_{t}|^{2}+\int_{0}^{T}(|Z_{t}^{0}|^{2}+|Z_{t}|^{2})dt\Bigr{]}\leq
C\mathbb{E}\Bigl{[}|\xi|^{2}+|\partial_{x}\overline{g}(0,c_{T}^{0},c_{T})|^{2}$
$\displaystyle\hskip
113.81102pt+\int_{0}^{T}\Bigl{(}|\partial_{x}\overline{f}(t,0,0,c_{t}^{0},c_{t})|^{2}+|l(t,0,c_{t}^{0},c_{t})|^{2}+|(\sigma_{0},\sigma)(t,c_{t}^{0},c_{t})|^{2}\Bigr{)}dt\Bigr{]}~{},$
where $C$ is a constant depending only on $T,\delta$ and Lipschitz constants
of the system.
###### Proof.
By quick inspection of the proof for Proposition 4.1, one can confirm that as
long as there exists a solution
$(X^{\prime},Y^{\prime},Z^{0\prime},Z^{\prime})\in\mathbb{S}^{2}\times\mathbb{S}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$,
their coefficients need not satisfy Assumption (MFG-a,b,c1). In particular, by
putting $\xi^{\prime}$ and
$(l^{\prime},\sigma_{0}^{\prime},\sigma^{\prime},\overline{f}^{\prime},\overline{g}^{\prime})$
all zero, we have a trivial solution
$(X^{\prime},Y^{\prime},Z^{0^{\prime}},Z^{\prime})=(0,0,0,0)$. The desired
estimate now follows from Proposition 4.1. ∎
#### Securities of maturity $T$ with exogenously specified payoff
If we consider the exchange markets of bonds and futures, or other financial
derivatives with maturity $T$, those securities cease to exist at $T$ after
paying exogenously specified amount of cash $c_{T}^{0}$. In this case, it is
natural to consider with $\delta=0$ and
$\displaystyle g(x,c^{0})=\overline{g}(x,c^{0}):=-\langle c^{0},x\rangle~{},$
( 4.11)
since there is no reason to put penalty on the outstanding volume at $T$. In
this case, the terminal function $g$ in $(\ref{terminal-special})$ does not
have the strict convexity. Fortunately, even in this case, we can prove the
unique existence as well as the stability result of the same form.
###### Assumption 4.3.
(MFG-c2)
(i) The functions $\sigma_{0}$ and $\sigma$ are independent of the argument
$\varpi$.
(ii) For any $t\in[0,T]$, any random variables
$x,x^{\prime},c^{0},c\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and any
sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the function $l$ satisfies the
monotone condition with some positive constant $\gamma^{l}>0$:
$\displaystyle\mathbb{E}\Bigl{[}\langle l(t,\mathbb{E}[x|{\cal
G}],c^{0},c)-l(t,\mathbb{E}[x^{\prime}|{\cal
G}],c^{0},c),x-x^{\prime}\rangle\Bigr{]}\geq\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[x-x^{\prime}|{\cal
G}]^{2}\bigr{]}~{}.$
(iii) $\gamma:=\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}$ is strictly
positive and the terminal function $g$ is given by $(\ref{terminal-special})$
with $\delta=0$.
###### Theorem 4.3.
Under Assumptions (MFG-a,b,c2), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE $(\ref{fbsde-single-p})$. Moreover, the same form of stability and
$\mathbb{L}^{2}$ estimates given in Proposition 4.1 and Corollary 4.1 hold.
###### Proof.
Note that, in this case, the terminal condition for the BSDE is independent of
$X_{T}$. Thus, as in Theorem 2.3 [24], we put
$y_{T}^{\varrho}=Y_{T}=-c_{T}^{0}$ in $(\ref{shifted-0})$ and
$(\ref{shifted-1})$, respectively. Using the fact that $\langle\Delta
X_{T},\Delta Y_{T}\rangle=0$, one can follow the same arguments to get the
desired result. The proof of the stability result can also be done in almost
exactly the same way. ∎
## 5 Asymptotic Market Clearing
We are now ready to investigate if our FBSDE $(\ref{fbsde-single-p})$ actually
provides a good approximate of the market price and if so, how accurate it is.
By Theorem 3.1, if we use $(-\mathbb{E}\bigl{[}Y_{t}|\overline{{\cal
F}}_{t}^{0}\bigr{]})_{t\in[0,T]}$ as the input $(\varpi_{t})_{t\in[0,T]}$,
where $(Y_{t})_{t\in[0,T]}$ is the unique solution to the FBSDE $(\ref{fbsde-
single-p})$ with the convention $\xi=\xi^{1}$ and $c=c^{1}$, the optimal
strategy of the individual agent is given by
$\displaystyle\widehat{\alpha}^{i}_{\rm{mf}}(t):=\widehat{\alpha}(Y_{t}^{i},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])=-\overline{\Lambda}(Y_{t}^{i}-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])$ ( 5.1)
where $(Y^{i}_{t})_{t\in[0,T]}$ is the solution to $(\ref{agent-FBSDE})$ with
$(\varpi_{t}=-\mathbb{E}\bigl{[}Y_{t}|\overline{{\cal
F}}_{t}^{0}\bigr{]})_{t\in[0,T]}$.
###### Theorem 5.1.
If the conditions for Theorem 4.1, Theorem 4.2 or Theorem 4.3 are satisfied
then we have
$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)\Bigr{|}^{2}dt=0~{}.$
Moreover if there exists some constant $\Gamma$ such that
$\sup_{t\in[0,T]}\mathbb{E}\bigl{[}|Y_{t}|^{q}\bigr{]}^{\frac{1}{q}}\leq\Gamma<\infty$
for some $q>4$, then there exists some constant $C$ independent of $N$ such
that
$\displaystyle\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)\Bigr{|}^{2}dt\leq
C\Gamma^{2}\epsilon_{N},$ ( 5.2)
where
$\epsilon_{N}:=N^{-2/\max(n,4)}\bigl{(}1+\log(N)\mathbb{1}_{\\{n=4\\}}\bigr{)}$.
###### Proof.
Let us consider the following set of FBSDEs with $1\leq i\leq N$ on the
filtered probability space $(\Omega,{\cal F},\mathbb{P};\mathbb{F})$
constructed in Section 2.
$\displaystyle d\underline{X}_{t}^{i}$ $\displaystyle=$
$\displaystyle\Bigl{(}-\overline{\Lambda}(\underline{Y}_{t}^{i}-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}])+l(t,-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})\Bigr{)}dt$
$\displaystyle\qquad+\sigma_{0}(t,-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})dW_{t}^{0}+\sigma(t,-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})dW_{t}^{i},$ $\displaystyle
d\underline{Y}_{t}^{i}$ $\displaystyle=$
$\displaystyle-\partial_{x}\overline{f}(t,\underline{X}_{t}^{i},-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})dt+\underline{Z}_{t}^{i,0}dW_{t}^{0}+\underline{Z}_{t}^{i}dW_{t}^{i},$
with $\underline{X}_{0}^{i}=\xi^{i}$ and
$\underline{Y}_{T}^{i}=\delta/(1-\delta)\mathbb{E}[\partial_{x}\overline{g}(\underline{X}_{T}^{i},c_{T}^{0},c_{T}^{i})|\overline{{\cal
F}}_{T}^{0}]+\partial_{x}\overline{g}(\underline{X}_{T}^{i},c_{T}^{0},c_{T}^{i})$.
Thanks to the existence of unique strong solution, Yamada-Watanabe Theorem for
FBSDEs (see, Theorem 1.33 [8]), there exists some measurable function $\Phi$
such that for every $1\leq i\leq N$,
$\displaystyle(\underline{X}^{i}_{t},\underline{Y}^{i}_{t})_{t\in[0,T]}=\Phi\Bigl{(}(c_{t}^{0})_{t\in[0,T]},(W_{t}^{0})_{t\in[0,T]},\xi^{i},(c_{t}^{i})_{t\in[0,T]},(W_{t}^{i})_{t\in[0,T]}\Bigr{)}~{}.$
Hence, conditionally on $\overline{{\cal F}}^{0}$, the set of proceses
$(\underline{X}^{i}_{t},\underline{Y}^{i}_{t})_{t\in[0,T]}$ with $1\leq i\leq
N$ are independently and identically distributed. In particular, we have
$\mathbb{P}$-a.s.
$\displaystyle\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}]=\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}],\quad\forall
t\in[0,T],$
$\displaystyle\mathbb{E}[\partial_{x}\overline{g}(\underline{X}_{T}^{i},c_{T}^{0},c_{T}^{i})|\overline{{\cal
F}}_{T}^{0}]=\mathbb{E}[\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}]~{}.$ ( 5.3)
Note that, under the convention $\xi^{1}=\xi$ and $c^{1}=c$, we actually have
$(\underline{X}^{1},\underline{Y}^{1})=(X,Y)$. From $(\ref{law-identity})$, we
conclude that
$(X^{i}_{t},Y^{i}_{t},Z^{i,0}_{t},Z^{i}_{t})_{t\in[0,T]}=(\underline{X}^{i}_{t},\underline{Y}^{i}_{t},\underline{Z}^{i,0}_{t},\underline{Z}_{t}^{i})_{t\in[0,T]}$
in
$\mathbb{S}^{2}(\mathbb{F}^{i})\times\mathbb{S}^{2}(\mathbb{F}^{i})\times\mathbb{H}^{2}(\mathbb{F}^{i})\times\mathbb{H}^{2}(\mathbb{F}^{i})$.
Therefore,
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)=-\overline{\Lambda}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\underline{Y}_{t}^{i}-\mathbb{E}[\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}~{}.$ ( 5.4)
We can easily check that
$\displaystyle\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0})\Bigr{)}^{2}\Bigr{|}\overline{{\cal
F}}_{t}^{0}\Bigr{]}\leq\frac{2}{N}\sum_{i=1}^{N}\mathbb{E}\bigl{[}|\underline{Y}_{t}^{i}|^{2}|\overline{{\cal
F}}_{t}^{0}\bigr{]}+2\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}|^{2}|\overline{{\cal
F}}_{t}^{0}\bigr{]}=4\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}|^{2}|\overline{{\cal
F}}_{t}^{0}\bigr{]}~{}.$
Since $(\underline{Y}_{t}^{i})_{1\leq i\leq N}$ are $\overline{{\cal
F}}^{0}_{t}$-conditionally independently and identically distributed and also
$\underline{Y}^{1}\in\mathbb{S}^{2}$, the same arguments leading to $(2.14)$
in [8] imply that the pointwise convergence holds:
$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0})\Bigr{)}^{2}\Bigr{]}=0~{}.$ ( 5.5)
We are now going to show that the set of functions, $(f_{N})_{N\in\mathbb{N}}$
defined by
$\displaystyle[0,T]\ni t\mapsto
f_{N}(t):=\mathbb{E}\bigl{[}W_{2}\bigl{(}\overline{\mu}_{t},\mu_{t}\bigr{)}^{2}\bigr{]}\in\mathbb{R}$
with
$\overline{\mu}_{t}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}}$
and $\mu_{t}:={\cal L}(\underline{Y}_{t}^{1}|\overline{{\cal F}}_{t}^{0})$ are
precompact in the set ${\cal C}([0,T];\mathbb{R})$ endowed with the topology
of uniform convergence. In fact, uniformly in $N$,
$\displaystyle\sup_{t\in[0,T]}|f_{N}(t)|\leq
4\sup_{t\in[0,T]}\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}|^{2}\bigr{]}\leq
C<\infty$ ( 5.6)
where $C$ is given by the estimate in Corollary 4.1. Moreover, for any $0\leq
t,s\leq T$, Cauchy-Schwarz, $(\ref{fN-inequality})$ and the triangular
inequalities give
$\displaystyle|f_{N}(t)-f_{N}(s)|\leq\mathbb{E}\Bigl{[}\Bigl{(}W_{2}(\overline{\mu}_{t},\mu_{t})+W_{2}(\overline{\mu}_{s},\mu_{2})\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\mathbb{E}\Bigl{[}\Bigl{(}W_{2}(\overline{\mu}_{t},\mu_{t})-W_{2}(\overline{\mu}_{s},\mu_{2})\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}$
$\displaystyle\quad\leq
C\mathbb{E}\Bigl{[}\Bigl{(}W_{2}(\overline{\mu}_{t},\mu_{t})-W_{2}(\overline{\mu}_{s},\mu_{s})\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\leq
C\mathbb{E}\Bigl{[}W_{2}(\overline{\mu}_{t},\overline{\mu}_{s})^{2}+W_{2}(\mu_{t},\mu_{s})^{2}\Bigr{]}^{\frac{1}{2}}~{}.$
$\displaystyle\quad\leq
C\mathbb{E}\Bigl{[}\frac{1}{N}\sum_{i=1}^{N}|\underline{Y}_{t}^{i}-\underline{Y}_{s}^{i}|^{2}+|\underline{Y}_{t}^{1}-\underline{Y}_{s}^{1}|^{2}\Bigr{]}^{\frac{1}{2}}$
$\displaystyle\quad\leq
C\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}-\underline{Y}_{s}^{1}|^{2}\bigr{]}^{\frac{1}{2}}~{},$
uniformly in $N$, where we have used the fact that $(\underline{Y}^{i})_{i\geq
1}$ are conditionally i.i.d at the last inequality. Since
$(\underline{Y}_{t}^{1})_{t\in[0,T]}$ is a continuous process, the above
estimate tells that $(f_{N})_{N\in\mathbb{N}}$ is equicontinuous, which is
also uniformly equicontinuous since we are working on the finite interval.
Now, Arzela-Ascoli theorem implies the desired precompactness.
Combining with the pointwise convergence $(\ref{pointwise-conv})$, we thus
conclude
$\displaystyle\lim_{N\rightarrow\infty}\sup_{t\in[0,T]}\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0})\Bigr{)}^{2}\Bigr{]}=0~{}.$ ( 5.7)
From the definition of Wasserstein distance $(\ref{def-W})$, we have
$\displaystyle\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\underline{Y}_{t}^{i}-\mathbb{E}[\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0}]\Bigr{|}\leq
W_{1}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal F}}_{t}^{0})\Bigr{)}~{},$
and hence, from $(\ref{alpha-m})$,
$\displaystyle\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)\Bigr{|}^{2}dt\leq
C\sup_{t\in[0,T]}\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal F}}_{t}^{0})\Bigr{)}^{2}\Bigr{]}~{}.$
( 5.8)
The first conclusion now follows from $(\ref{law-conv})$. The latter claims
directly follows from the expression $(\ref{law-conv-2})$ and the (Fourth
Step) in the proof of Theorem 2.12 in [8]. ∎
Theorem 5.1 justifies our intuitive understanding and a special type of FBSDEs
$(\ref{fbsde-single-p})$ derived in Section 3 as a reasonable model to
approximate the market clearing price. When there exists higher integrability,
Glivenko-Cantelli convergence theorem in the Wasserstein distance even
provides a specific order $\epsilon_{N}$ of convergence in terms of the number
of agents $N$ $(\ref{Glivenko-Cantelli})$. See Theorem 5.8 and Remark 5.9 in
[7] for more details.
###### Remark 5.1.
Consider the situation treated in Theorem 4.3, for example, a market model of
a Futures contract. If the contract pays unit amount of the underlying asset
per contract whose value is exogenously given by $c_{T}^{0}$, our mean-field
limit model $(\ref{fbsde-single-p})$ gives $Y_{T}=-c_{T}^{0}$. This means that
the modeled Futures price satisfies
$\varpi_{T}=-\mathbb{E}[Y_{T}|\overline{{\cal F}}_{T}^{0}]=c_{T}^{0}$, which
guarantees the convergence of the modeled price to the value of the underlying
asset at the maturity $T$. This is a crucially important feature that any
market model of this type of securities must satisfy.
## 6 Extension to Multiple Populations
The main limitation of the last model is that there exists only one type of
agents who share the common cost functions as well as the coefficient
functions for their state dynamics. Interestingly, it is rather
straightforward to extend the model to the situation with multiple
populations, where the agents in each population share the same cost and
coefficient functions but they can be different population by population. From
the perspective of the practical applications, this is a big advantage since
we can analyze, for example, the interactions between the Sell-side and Buy-
side institutions for financial applications, or consumers and producers for
economic applications. For general issues of mean field games as well as mean
field type control problems in the presence of multiple populations without
common noise, see Fujii [13]. Although there exists a common noise in the
current model, the conditional law only enters as a form of expectation.
Therefore, as long as the system of FBSDEs is Lipschitz continuous, there
exists a unique strong solution at least for small $T$. For general $T$,
although it is rather difficult to find appropriate set of assumptions, it is
still possible for some simple cases. In this section, our main task is to
find an appropriate limit model that extends $(\ref{fbsde-single-p})$ for
multiple populations and the sufficient conditions that make appropriate
monotone conditions hold, which guarantees the existence of unique solution.
In the following, we shall treat $m$ populations indexed by
$p\in\\{1,\cdots,m\\}$. For each $p$, $N_{p}\geq 1$ agents are assumed to
belong to the population. We denote by $(p,i)$ the ith agent in the population
$p$. First, let us enlarge the probability space constructed in Section 2. In
addition to $(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$, we introduce
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{P}}^{p,i})$ with
$1\leq i\leq N_{p}$ and $1\leq p\leq m$, each of which is generated by
$(\xi^{p,i},\boldsymbol{W}^{p,i})$ with $d$-dimensional Brownian motion
$\boldsymbol{W}^{p,i}$ and a $\boldsymbol{W}^{p,i}$-independent
$\mathbb{R}^{n}$-valued square integrable random variable $\xi^{p,i}$. For
each $p$, $(\xi^{p,i})_{i=1}^{N_{p}}$ are assumed to have the common law. We
define $(\Omega^{p,i},{\cal F}^{p,i},\mathbb{P}^{p,i};\mathbb{F}^{p,i})$ as
the product of $(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$ and
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{P}}^{p,i})$. Finally
$(\Omega,{\cal F},\mathbb{P};\mathbb{F})$ is defined as a product of all the
spaces $(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$ and
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{F}}^{p,i})$, $1\leq
i\leq N_{p},1\leq p\leq m$, and $(\Omega^{i},{\cal
F}^{i},\mathbb{P}^{i};\mathbb{F}^{i})$ as a product of
$(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$ and
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{F}}^{p,i})$ with
$1\leq p\leq m$. Every probability space is assumed to be complete and every
filtration is assumed to be complete and right-continuously augmented to
satisfy the usual conditions.
As we have done in Section 3, we first assume that the market price of $n$
securities is given exogenously by
$\varpi_{t}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$. Under
this setup, we consider the control problem for each $(p,i)$ agent defined by
$\inf_{\boldsymbol{\alpha}^{p,i}\in\mathbb{A}^{p,i}}J^{p,i}(\boldsymbol{\alpha}^{p,i})~{},$
( 6.1)
with
$\displaystyle
J^{p,i}(\boldsymbol{\alpha}^{p,i}):=\mathbb{E}\Bigl{[}\int_{0}^{T}f_{p}(t,X_{t}^{p,i},\alpha_{t}^{p,i},\varpi_{t},c_{t}^{0},c_{t}^{p,i})dt+g_{p}(X_{T}^{p,i},\varpi_{T},c_{T}^{0},c_{T}^{p,i})\Bigr{]}~{},$
subject to the dynamic constraint:
$\displaystyle
dX_{t}^{p,i}=\Bigl{(}\alpha_{t}^{p,i}+l_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})\Bigr{)}dt+\sigma_{p,0}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{0}+\sigma_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{p,i}$
with $X_{0}^{p,i}=\xi^{p,i}$. As before we assume $(c_{t}^{0})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$ and
$(c_{t}^{p,i})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{p,i};\mathbb{R}^{n})$. In addition,
within each population $p$, the random sources $(c^{p,i}_{t})_{t\geq 0}$ are
assumed to have a common law $1\leq i\leq N_{p}$. Admissible strategies
$\mathbb{A}^{p,i}$ is the space
$\mathbb{H}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n})$. The measurable functions
$f_{p}:[0,T]\times(\mathbb{R}^{n})^{5}\rightarrow\mathbb{R}$,
$g_{p}:(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$,
$\overline{f}_{p}:[0,T]\times(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$ and
$\overline{g}_{p}:(\mathbb{R}^{n})^{3}\rightarrow\mathbb{R}$ are given by
$\displaystyle
f_{p}(t,x,\alpha,\varpi,c^{0},c):=\langle\varpi,\alpha\rangle+\frac{1}{2}\langle\alpha,\Lambda_{p}\alpha\rangle+\overline{f}_{p}(t,x,\varpi,c^{0},c)~{},$
$\displaystyle
g_{p}(x,\varpi,c^{0},c):=-\delta\langle\varpi,x\rangle+\overline{g}_{p}(x,c^{0},c)~{}.$
###### Assumption 6.1.
(MFG-A) We assume the following conditions uniformly in
$p\in\\{1,\cdots,m\\}$.
(i) $\Lambda_{p}$ is a positive definite $n\times n$ symmetric matrix with
$\underline{\lambda}I_{n\times
n}\leq\Lambda_{p}\leq\overline{\lambda}I_{n\times n}$ in the sense of 2nd-
order form where $\underline{\lambda}$ and $\overline{\lambda}$ are some
constants satisfying $0<\underline{\lambda}\leq\overline{\lambda}$.
(ii) For any $(t,x,\varpi,c^{0},c)$,
$\displaystyle|\overline{f}_{p}(t,x,\varpi,c^{0},c)|+|\overline{g}_{p}(x,c^{0},c)|\leq
L(1+|x|^{2}+|\varpi|^{2}+|c^{0}|^{2}+|c|^{2})~{}.$
(iii) $\overline{f}_{p}$ and $\overline{g}_{p}$ are continuously
differentiable in $x$ and satisfy, for any $(t,x,x^{\prime},\varpi,c^{0},c)$,
$|\partial_{x}\overline{f}_{p}(t,x^{\prime},\varpi,c^{0},c)-\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}_{p}(x^{\prime},c^{0},c)-\partial_{x}\overline{g}_{p}(x,c^{0},c)|\leq
L|x^{\prime}-x|~{},$
and
$|\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}_{p}(x,c^{0},c)|\leq
L(1+|x|+|\varpi|+|c^{0}|+|c|)$.
(iv)The functions $\overline{f}_{p}$ and $\overline{g}_{p}$ are convex in $x$
in the sense that for any $(t,x,x^{\prime},\varpi,c^{0},c)$,
$\displaystyle\overline{f}_{p}(t,x^{\prime},\varpi,c^{0},c)-\overline{f}_{p}(t,x,\varpi,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)\rangle\geq\frac{\gamma^{f}}{2}|x^{\prime}-x|^{2}~{},$
$\displaystyle\overline{g}_{p}(x^{\prime},c^{0},c)-\overline{g}_{p}(x,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{g}_{p}(x,c^{0},c)\rangle\geq\frac{\gamma^{g}}{2}|x^{\prime}-x|^{2}~{},$
with some constants $\gamma^{f},\gamma^{g}\geq 0$.
(v) $l_{p},\sigma_{p,0},\sigma_{p}$ are the measurable functions defined on
$[0,T]\times(\mathbb{R}^{n})^{3}$ and are $\mathbb{R}^{n},\mathbb{R}^{n\times
d^{0}}$ and $\mathbb{R}^{n\times d}$-valued, respectively. Moreover they
satisfy the linear growth condition:
$\displaystyle|(l_{p},\sigma_{p,0},\sigma_{p})(t,\varpi,c^{0},c)|\leq
L(1+|\varpi|+|c^{0}|+|c|)$
for any $(t,\varpi,c^{0},c)$.
(vi) $\delta\in[0,1)$ is a given constant.
Under Assumption (MFG-A), Theorem 3.1 guarantees that the control problem
$(\ref{control-pi})$ for each agent $(p,i)$ is uniquely characterized by
$\displaystyle
dX_{t}^{p,i}=\Bigl{(}\widehat{\alpha}_{p}(Y_{t}^{p,i},\varpi_{t})+l_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})\Bigr{)}dt+\sigma_{p,0}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{0}+\sigma_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{p,i},$
$\displaystyle
dY_{t}^{p,i}=-\partial_{x}\overline{f}_{p}(t,X_{t}^{p,i},\varpi_{t},c_{t}^{0},c_{t}^{p,i})dt+Z_{t}^{p,i,0}dW_{t}^{0}+Z_{t}^{p,i}dW_{t}^{p,i},$
( 6.2)
with $X_{0}^{p,i}=\xi^{p,i}$ and
$Y_{T}^{p,i}=-\delta\varpi_{T}+\partial_{x}\overline{g}_{p}(X_{T}^{p,i},c_{T}^{0},c_{T}^{p,i})$.
We have defined
$\widehat{\alpha}_{p}(y,\varpi):=-\overline{\Lambda}_{p}(y+\varpi)$ and
$\overline{\Lambda}_{p}:=(\Lambda_{p})^{-1}$ as before. There exists a unique
strong solution
$(X^{p,i}_{t},Y^{p,i}_{t},Z^{p,i,0}_{t},Z^{p,i}_{t})_{t\in[0,T]}\in\mathbb{S}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n\times d})$, and the
optimal trading strategy for the agent $(p,i)$ is given by
$\displaystyle\widehat{\alpha}_{t}^{p,i}=\widehat{\alpha}_{p}(Y_{t}^{p,i},\varpi_{t})~{},\forall
t\in[0,T].$
Let us check the market clearing condition under this setup. In order to
balance the demand and supply of securities at the exchange, we need to have
$\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}(Y^{p,i}_{t},\varpi_{t})=0$.
This requires the market price to satisfy
$\displaystyle\varpi_{t}=-\Bigl{(}\sum_{p=1}^{m}n_{p}\overline{\Lambda}_{p}\Bigr{)}^{-1}\sum_{p=1}^{m}n_{p}\overline{\Lambda}_{p}\Bigl{(}\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}Y_{t}^{p,i}\Bigr{)}~{},$
where $N=\sum_{p=1}^{m}N_{p}$ and $n_{p}:=N_{p}/N$. At the moment, this is
inconsistent to the initial assumption that requires $(\varpi_{t})_{t\geq 0}$
to be $\overline{\mathbb{F}}^{0}$-adapted. However, since for each $1\leq
p\leq m$, $(Y^{p,i}_{t})_{i=1}^{N_{p}}$ are $\overline{{\cal
F}}^{0}$-conditionally independently and identically distributed, we may
follow the same arguments used in Section 3. If we take $N\rightarrow\infty$
while keeping the relative size of populations $n_{p}$ constant, we can expect
to obtain
$\displaystyle\varpi_{t}=-\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal
F}}_{t}^{0}]$ ( 6.3)
in the large population limit where
$\hat{\Lambda}_{p}:=n_{p}\overline{\Lambda}_{p},\qquad\hat{\Xi}:=\Bigl{(}\sum_{p=1}^{m}\hat{\Lambda}_{p}\Bigr{)}^{-1}~{}.$
###### Remark 6.1.
When $\Lambda_{p}=\Lambda$ for every population $p$, one can easily check that
$(\ref{market-price-guess})$ becomes
$\varpi_{t}=-\sum_{p=1}^{m}n_{p}\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal
F}}_{t}^{0}]~{}.$
Since $Y$ of the adjoint equation represents the marginal cost i.e., the first
order derivative of the value function with respect to the state variable $x$,
the above expression of $\varpi$ implies that the market price may be given by
the population-weighted average of the marginal benefit (-cost) across the
entire populations.
### 6.1 Limit problem with multiple populations
By the observation we have just made, we are motivated to study the following
limit problem with $1\leq p\leq m$:
$\displaystyle dX_{t}^{p}$ $\displaystyle=$
$\displaystyle\Bigl{(}\widehat{\alpha}_{p}\bigl{(}Y_{t}^{p},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])\bigr{)}+l_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}\Bigr{)}dt$
$\displaystyle+\sigma_{p,0}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}dW_{t}^{0}+\sigma_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}dW_{t}^{p,1},$ $\displaystyle
dY_{t}^{p}$ $\displaystyle=$
$\displaystyle-\partial_{x}\overline{f}_{p}\bigl{(}t,X_{t}^{p},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}dt+Z_{t}^{p,0}dW_{t}^{0}+Z_{t}^{p}dW_{t}^{p,1}~{},$
( 6.4)
with $X_{0}^{p}=\xi^{p}$ and
$\displaystyle
Y_{T}^{p}=\frac{\delta}{1-\delta}\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}\bigl{[}\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})~{}.$
We put as before $\xi^{p}:=\xi^{p,1}$ and $c^{p}:=c^{p,1}$ to lighten the
notation. Here,
$\displaystyle\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]):=-\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}],\quad\widehat{\alpha}_{p}(y,\varpi):=-\overline{\Lambda}_{p}(y+\varpi)$
and hence $(\ref{fbsde-multiple-p})$ is actually an $m$-coupled system of
FBSDEs of McKean-Vlasov type. One can derive the terminal condition from
$Y_{T}^{p}=-\delta\varpi(\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}])+\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})~{},$
( 6.5)
by summing over $1\leq p\leq m$ after taking conditional expectation given
$\overline{{\cal F}}_{T}^{0}$. In the following, we use the notation
$(X_{t},Y_{t},Z_{t}^{0},Z_{t})_{t\in[0,T]}=\Bigl{(}(X_{t}^{p})_{p=1}^{m},(Y_{t}^{p})_{p=1}^{m},(Z_{t}^{p,0})_{p=1}^{m},(Z_{t}^{p})_{p=1}^{m}\Bigr{)}_{t\in[0,T]}~{}.$
( 6.6)
### 6.2 Solvability for small $T$
For small $T$, Lipschitz continuity suffices to guarantee the existence of a
unique solution.
###### Assumption 6.2.
(MFG-B)
Uniformly in $p\in\\{1,\cdots,m\\}$, for any
$(t,x,c^{0},c)\in[0,T]\times(\mathbb{R}^{n})^{3}$ and any
$\varpi,\varpi^{\prime}\in\mathbb{R}^{n}$, the coefficient functions
$l_{p},\sigma_{p,0},\sigma_{p}$ and $\overline{f}_{p}$ satisfy
$\displaystyle|(l_{p},\sigma_{p,0},\sigma_{p})(t,\varpi,c^{0},c)-(l_{p},\sigma_{p,0},\sigma_{p})(t,\varpi^{\prime},c^{0},c)|$
$\displaystyle\qquad\qquad+|\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)-\partial_{x}\overline{f}_{p}(t,x,\varpi^{\prime},c^{0},c)|\leq
L_{\varpi}|\varpi-\varpi^{\prime}|~{}.$
The following theorem follows exactly in the same way as Theorem 4.1.
###### Theorem 6.1.
Under Assumptions (MFG-A,B), there exists some constant $\tau>0$ which depends
only on $(L,L_{\varpi},\delta,n_{p},\Lambda_{p})$ such that for any
$T\leq\tau$, there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d^{0}})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d})^{m}\bigr{)}$ to the FBSDE $(\ref{fbsde-multiple-p})$.
###### Remark 6.2.
Note that the above system of FBSDEs becomes a linear-quadratic form by
choosing $(l_{p},\sigma_{p,0},\sigma_{p},\overline{f}_{p},\overline{g}_{p})$
appropriately. In this case, the problem reduces to solving ordinary
differential equations of Riccati type. Therefore, the existence of a solution
for a given $T$ can be tested, at lest numerically, by checking the absence of
a “blow up” in its solution.
### 6.3 Solvability for general $T$
We now move on to the existence result of a unique solution for general $T$.
It is very difficult to find general existence criteria for fully-coupled
multi-dimensional FBSDEs. A the moment, in order to apply well-known Peng-Wu’s
method, let us put the following simplifying assumptions.
###### Assumption 6.3.
(MFG-C1)
(i) For every $1\leq p\leq m$, the functions $\sigma_{p,0}$ and $\sigma_{p}$
are independent of the argument $\varpi$.
(ii) $\Lambda_{p}$=$\Lambda$ and $n_{p}=1/m$ for every $p$.
(iii) For any $t\in[0,T]$, any random variables
$x^{p},x^{p\prime},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and
any sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the functions
$(l_{p})_{p=1}^{m}$ satisfy with some positive constant $\gamma^{l}>0$,
$\displaystyle\sum_{p=1}^{m}\mathbb{E}\Bigl{[}\bigl{\langle}l_{p}\bigl{(}t,\mathbb{E}[\overline{x}|{\cal
G}],c^{0},c^{0}\bigr{)}-l_{p}\bigl{(}t,\mathbb{E}[\overline{x}^{\prime}|{\cal
G}],c^{0},c^{p}\bigr{)},x^{p}-x^{p\prime}\bigr{\rangle}\Bigr{]}\geq
m\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\overline{x}-\overline{x}^{\prime}|{\cal
G}]^{2}\bigr{]},$
where $\overline{x}:=\frac{1}{m}\sum_{p=1}^{m}x^{p}$ and similarly for
$\overline{x}^{\prime}$.
(iv) There exists a strictly positive constant $\gamma$ satisfying
$0<\gamma\leq\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\wedge\gamma^{g}$.
Moreover, the functions $(\overline{g}_{p})_{p=1}^{m}$ satisfy for any
$x^{p},x^{p\prime},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and
any sub-$\sigma$-field ${\cal G}\subset{\cal F}$,
$\displaystyle\frac{\delta}{1-\delta}m^{-1}\mathbb{E}\Bigl{[}\bigl{\langle}\sum_{p=1}^{m}\mathbb{E}[\partial_{x}\overline{g}_{p}(x^{p},c^{0},c^{p})-\partial_{x}\overline{g}_{p}(x^{p\prime},c^{0},c^{p})|{\cal
G}],\sum_{p=1}^{m}(x^{p}-x^{p\prime})\bigr{\rangle}\Bigr{]}$
$\displaystyle\qquad\quad+\gamma^{g}\sum_{p=1}^{m}\mathbb{E}[|x^{p}-x^{p\prime}|^{2}]\geq\gamma\sum_{p=1}^{m}\mathbb{E}[|x^{p}-x^{p\prime}|^{2}]~{}.$
###### Remark 6.3.
The conditions (iii) and (iv) in the above assumption are rather restrictive.
The condition (iii) is satisfied, for example, if $l_{p}$ has a separable form
$l_{p}=h(x)+h_{p}(c^{0}_{t},c_{t}^{p})$ with some function $h$, which is
common to every population and strictly monotone. (iv) is also satisfied by
requiring similar structure. Or, since $\partial_{x}\overline{g}_{p}$ is
Lipschitz continuous in $x$, the absolute value of the first term is bounded
by
$\frac{\delta}{1-\delta}\max((L_{p})_{p=1}^{m})\sum_{p=1}^{m}\mathbb{E}|x^{p}-x^{p\prime}|^{2}$,
where the $L_{p}$ is the Lipschitz constant for
$\partial_{x}\overline{g}_{p}$. Thus the condition (iv) is satisfied if
$\delta\max((L_{p})_{p=1}^{m})$ is sufficiently small.
The next result is the counterpart of Theorem 4.2.
###### Theorem 6.2.
Under Assumptions (MFG-A,B,C1), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d^{0}})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d})^{m}\bigr{)}$ to the FBSDE $(\ref{fbsde-multiple-p})$. Moreover, the same
form of stability and $\mathbb{L}^{2}$ estimates given in Proposition 4.1 and
Corollary 4.1 hold.
###### Proof.
Under Assumption (MFG-C1), $(\ref{fbsde-multiple-p})$ can be written as
$\displaystyle
dX_{t}^{p}=\Bigl{\\{}-\overline{\Lambda}\Bigl{(}Y_{t}^{p}-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}+l_{p}\Bigl{(}t,-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{p}\Bigr{)}\Bigr{\\}}dt$
$\displaystyle\qquad\qquad+\sigma_{p,0}(t,c_{t}^{0},c_{t}^{p})dW_{t}^{0}+\sigma_{p}(t,c_{t}^{0},c_{t}^{p})dW_{t}^{p,1},$
$\displaystyle
dY_{t}^{p}=-\partial_{x}\overline{f}_{p}\Bigl{(}t,X_{t}^{p},-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{p}\Bigr{)}dt+Z_{t}^{p,0}dW_{t}^{0}+Z_{t}^{p}dW_{t}^{p,1},$
with $X_{0}^{p}=\xi^{p}$ and
$Y_{T}^{p}=\frac{\delta}{1-\delta}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}\bigl{[}\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})~{}.$
For each $p$, let us define the functionals $B_{p},F_{p}$ and $G_{p}$ for any
$y^{p},x^{p},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ with
$y:=(y^{p})_{p=1}^{m}$, $x:=(x^{p})_{p=1}^{m}$ and $c:=(c^{p})_{p=1}^{m}$ by
$\displaystyle
B_{p}(t,y,c^{0},c^{p}):=-\overline{\Lambda}\Bigl{(}y^{p}-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[y^{p}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}+l_{p}\Bigl{(}t,-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[y^{p}|\overline{{\cal
F}}_{t}^{0}],c^{0},c^{p}\Bigr{)}$ $\displaystyle
F_{p}(t,x^{p},y,c^{0},c^{p}):=-\partial_{x}\overline{f}\Bigl{(}t,x^{p},-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[y^{p}|\overline{{\cal
F}}_{t}^{0}],c^{0},c^{p}\Bigr{)},$ $\displaystyle
G_{p}(x,c^{0},c):=\frac{\delta}{1-\delta}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\partial_{x}\overline{g}_{p}(x^{p},c^{0},c^{p})|\overline{{\cal
F}}_{T}^{0}]+\partial_{x}\overline{g}_{p}(x^{p},c^{0},c^{p})~{},$
and set $B(t,y,c^{0},c):=(B_{p}(t,y,c^{0},c^{p}))_{p=1}^{m}$,
$F(t,x,y,c^{0},c):=(F_{p}(t,x^{p},y,c^{0},c^{p}))_{p=1}^{m}$ and
$G(x,c^{0},c):=(G_{p}(x,c^{0},c))_{p=1}^{m}$. With $\Delta y:=y-y^{\prime}$
and $\Delta x:=x-x^{\prime}$, we have from (MFG-C1)(iii),
$\displaystyle\mathbb{E}\Bigl{[}\langle
B(t,y,c^{0},c)-B(t,y^{\prime},c^{0},c),\Delta
y\rangle\Bigr{]}:=\sum_{p=1}^{m}\mathbb{E}\Bigl{[}\langle
B_{p}(t,y,c^{0},c)-B_{p}(t,y^{\prime},c^{0},c),\Delta y^{p}\rangle\Bigr{]}$
$\displaystyle\leq-\sum_{p=1}^{m}\mathbb{E}[\langle\Delta
y^{p},\overline{\Lambda}\Delta
y^{p}\rangle]+\frac{1}{m}\mathbb{E}\Bigl{[}\bigl{\langle}\sum_{p=1}^{m}\mathbb{E}[\Delta
y^{p}|\overline{{\cal F}}_{t}^{0}],\overline{\Lambda}\sum_{p=1}^{m}\Delta
y^{p}\bigr{\rangle}\Bigr{]}-m\gamma^{l}\mathbb{E}\Bigl{[}\Bigl{(}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\Delta
y^{p}|\overline{{\cal F}}_{t}^{0}]\Bigr{)}^{2}\Bigr{]}$
$\displaystyle\leq-m\gamma^{l}\mathbb{E}\Bigl{[}\Bigl{(}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\Delta
y^{p}|\overline{{\cal F}}_{t}^{0}]\Bigr{)}^{2}\Bigr{]}.$ ( 6.7)
There exists a orthogonal matrix $P$ such that $P^{\top}\overline{\Lambda}P$
becomes diagonal. Then working on the new basis $\hat{y}^{p}=P^{\top}\Delta
y^{p}$, $1\leq p\leq m$, the last inequality of $(\ref{DelB-ineq})$ can be
checked component by component $1\leq i\leq n$ by the fact
$(\sum_{p=1}^{m}\hat{y}^{p}_{i})^{2}\leq
m\sum_{p=1}^{m}|\hat{y}^{p}_{i}|^{2}$. Second, from (MFG-A)(iv), (MFG-B) and
Cauchy-Schwarz inequality,
$\displaystyle\mathbb{E}\Bigl{[}\langle
F(t,x,y,c^{0},c)-F(t,x^{\prime},y^{\prime},c^{0},c),\Delta x\rangle\Bigr{]}$
$\displaystyle\hskip
85.35826pt\leq-\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\mathbb{E}[|\Delta
x|^{2}]+m\gamma^{l}\mathbb{E}\Bigl{[}\Bigl{(}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\Delta
y_{t}^{p}|\overline{{\cal F}}_{t}^{0}]\Bigr{)}^{2}\Bigr{]}.$ ( 6.8)
Finally, from (MFG-A, C1)(iv), we immediately get
$\displaystyle\mathbb{E}\Bigl{[}\langle
G(x,c^{0},c)-G(x^{\prime},c^{0},c),\Delta
x\rangle\Bigr{]}\geq\gamma\mathbb{E}[|\Delta x|^{2}]~{}.$
Now we have established the monotone conditions corresponding to $(\ref{peng-
wu-condition})$ for the current model. We can now repeat the same procedures
in the proof of Theorem 4.2 and Proposition 4.1. ∎
Let us give the results for the securities of maturity $T$ with exogenously
specified payoff.
###### Assumption 6.4.
(MFG-C2)
(i) For every $1\leq p\leq m$, the functions $\sigma_{p,0}$ and $\sigma_{p}$
are independent of the argument $\varpi$.
(ii) $\Lambda_{p}$=$\Lambda$ and $n_{p}=1/m$ for every $p$.
(iii) For any $t\in[0,T]$, any random variables
$x^{p},x^{p\prime},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and
any sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the functions
$(l_{p})_{p=1}^{m}$ satisfy with some positive constant $\gamma^{l}>0$,
$\displaystyle\sum_{p=1}^{m}\mathbb{E}\Bigl{[}\bigl{\langle}l_{p}\bigl{(}t,\mathbb{E}[\overline{x}|{\cal
G}],c^{0},c^{0}\bigr{)}-l_{p}\bigl{(}t,\mathbb{E}[\overline{x}^{\prime}|{\cal
G}],c^{0},c^{p}\bigr{)},x^{p}-x^{p\prime}\bigr{\rangle}\Bigr{]}\geq
m\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\overline{x}-\overline{x}^{\prime}|{\cal
G}]^{2}\bigr{]},$
where $\overline{x}:=\frac{1}{m}\sum_{p=1}^{m}x^{p}$ and similarly for
$\overline{x}^{\prime}$.
(iv) $\gamma:=\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}$ is strictly
positive. Moreover, $\delta=0$ and the terminal function $g_{p}$ is given by
$\displaystyle g_{p}(x,c^{0})=\overline{g}_{p}(x,c^{0}):=-\langle
c^{0},x\rangle$ ( 6.9)
for every $1\leq p\leq m$.
###### Theorem 6.3.
Under Assumptions (MFG-A,B,C2), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d^{0}})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d})^{m}\bigr{)}$ to the FBSDE $(\ref{fbsde-multiple-p})$. Moreover, the same
form of the stability and $\mathbb{L}^{2}$ estimates given in Proposition 4.1
and Corollary 4.1 holds.
###### Proof.
Using the inequalities $(\ref{DelB-ineq})$ and $(\ref{DelF-ineq})$ with
$\sum_{p=1}^{m}\langle\Delta X_{T}^{p},\Delta Y_{T}^{p}\rangle=0$, we can
follow the same arguments in the proof of Theorem 4.3. ∎
### 6.4 Asymptotic market clearing for multi-population model
At the last part of this section, we investigate the asymptotic market
clearing in the presence of multiple populations. As in Section 5, we define
$(\varpi_{t})_{t\in[0,T]}$ using the solution to the system of the mean-field
FBSDEs:
$\displaystyle\varpi_{t}=\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]):=-\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}\bigl{[}Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}\bigr{]}$
where $(Y_{t}^{p})_{p=1}^{m}$ is the solution of $(\ref{fbsde-multiple-p})$.
In order to test the accuracy of the above $(\varpi_{t})_{t\in[0,T]}$ as a
market clearing price, we solve the individual agent problem $(\ref{control-
pi})$ with this $\varpi$ as an input. The corresponding individual problem
$(\ref{control-pi})$ for the agent $(p,i)$ is given by the unique strong
solution $(X^{p,i},Y^{p,i},Z^{p,i,0},Z^{p,i})$ of $(\ref{fbsde-pi})$. The
optimal strategy for the agent $(p,i)$ is then given by
$\displaystyle\widehat{\alpha}_{\rm{mf}}^{p,i}(t):=-\overline{\Lambda}_{p}\Bigl{(}Y_{t}^{p,i}-\hat{\Xi}\sum_{q=1}^{m}\hat{\Lambda}_{q}\mathbb{E}\bigl{[}Y_{t}^{q}|\overline{{\cal
F}}_{t}^{0}\bigr{]}\Bigr{)}~{},\forall t\in[0,T]~{}.$
###### Theorem 6.4.
If the conditions for Theorem 6.1, Theorem 6.2 or Theorem 6.3 are satisfied
then we have
$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)\Bigr{|}^{2}dt=0~{},$
where $N:=\sum_{p=1}^{m}N_{p}$ and the limit is taken while keeping
$(n_{p}:=N_{p}/N)_{1\leq p\leq m}$ constant. Moreover if there exists some
constant $\Gamma$ such that
$\sup_{t\in[0,T]}\mathbb{E}\bigl{[}|Y_{t}|^{q}\bigr{]}^{\frac{1}{q}}\leq\Gamma<\infty$
for some $q>4$, then there exists some constant $C$ independent of $N$ such
that
$\displaystyle\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)\Bigr{|}^{2}dt\leq
C\Gamma^{2}\epsilon_{N},$
where
$\epsilon_{N}:=N^{-2/\max(n,4)}\bigl{(}1+\log(N)\mathbb{1}_{\\{n=4\\}}\bigr{)}$.
###### Proof.
By the definition of $\widehat{\alpha}^{p,i}_{\rm{mf}}$, we have
$\displaystyle\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)$
$\displaystyle=$
$\displaystyle-\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\overline{\Lambda}_{p}\Bigl{(}Y_{t}^{p,i}-\hat{\Xi}\sum_{q=1}^{m}\hat{\Lambda}_{q}\mathbb{E}[Y_{t}^{q}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}$ ( 6.10) $\displaystyle=$
$\displaystyle-\sum_{p=1}^{m}\hat{\Lambda}_{p}\Bigl{(}\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}Y_{t}^{p,i}-\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}~{}.$
On the other hand, we have for each $1\leq p\leq m$, $1\leq i\leq N_{p}$,
$\displaystyle dX_{t}^{p,i}$ $\displaystyle=$
$\displaystyle\Bigl{(}\widehat{\alpha}_{p}\bigl{(}Y_{t}^{p,i},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])\bigr{)}+l_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}\Bigr{)}dt$
$\displaystyle+\sigma_{p,0}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}dW_{t}^{0}+\sigma_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}dW_{t}^{p,i},$ $\displaystyle
dY_{t}^{p,i}$ $\displaystyle=$
$\displaystyle-\partial_{x}\overline{f}_{p}\bigl{(}t,X_{t}^{p,i},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}dt+Z_{t}^{p,i,0}dW_{t}^{0}+Z_{t}^{p,i}dW_{t}^{p,i}~{},$
with $X_{0}^{p,i}=\xi^{p,i}$,
$\displaystyle Y_{T}^{p,i}=-\delta\varpi(\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}])+\partial_{x}\overline{g}_{p}(X_{T}^{p,i},c_{T}^{0},c_{T}^{p,i})~{}.$
By the unique strong solvability, Yamada-Watanabe theorem implies that there
exists some function $\Phi_{p}$ for each $1\leq p\leq m$ such that for every
$1\leq i\leq N_{p}$,
$\displaystyle(Y_{t}^{p,i})_{t\in[0,T]}=\Phi_{p}\Bigl{(}c^{0},(W_{t}^{0})_{t\in[0,T]},(\mathbb{E}[Y_{t}^{q}|\overline{{\cal
F}}_{t}^{0}]_{t\in[0,T]})_{1\leq q\leq
m},\xi^{p,i},(c_{t}^{p,i})_{t\in[0,T]},(W^{p,i}_{t})_{t\in[0,T]}\Bigr{)}.$
Hence $(Y_{t}^{p,i})_{t\in[0,T],1\leq i\leq N_{p}}$ are independently and
identically distributed conditionally on $\overline{{\cal F}}^{0}$. In
particular, we have $\mathbb{E}[Y_{t}^{p,i}|\overline{{\cal
F}}_{t}^{0}]=\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal F}}_{t}^{0}]$.
We now compare
$(X^{p,1}_{t},Y^{p,1}_{t},Z^{p,1,0}_{t},Z^{p,1}_{t})_{t\in[0,T]}$ with
$(X^{p}_{t},Y^{p}_{t},Z^{p,0}_{t},Z^{p}_{t})_{t\in[0,T]}$ by treating
$\varpi(\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}])$ as external inputs.
Note that the terminal condition of the latter satisfies the relation
$(\ref{terminal-pre})$. Then the standard stability result of the Lipschitz
FBSDEs implies $(Y^{p,1}_{t})_{t\in[0,T]}=(Y^{p}_{t})_{t\in[0,T]}$ in
$\mathbb{S}^{2}(\mathbb{F}^{p,1};\mathbb{R}^{n})$. As a result we have
obtained $\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}]=\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal F}}_{t}^{0}]$. Using the
expression $(\ref{clearing-mp})$, we obtain
$\displaystyle\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)$
$\displaystyle=$
$\displaystyle-\sum_{p=1}^{m}\hat{\Lambda}_{p}\Bigl{(}\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}Y_{t}^{p,i}-\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}~{}.$
We can now repeat the last part of the proof for Theorem 5.1. ∎
## 7 Concluding Remarks and Further Extensions
In this work, we have studied endogenous formation of market clearing price
using a stylized model of a security exchange. We have derived a special type
of FBSDE of McKean-Vlasov type with common noise whose solution provides a
good approximate of the equilibrium price. In addition to the existence of
strong unique solution to the FBSDE, we have proved that the modeled price
asymptotically clear the market in the large $N$-limit. We also gave the order
of convergence $\epsilon_{N}$ when the solution of the FBSDE possesses higher
order of integrability. In the following, let us list up of a further
extension of our technique and some interesting topics for future projects:
$\bullet~{}$Dependence on the conditional law of the state: For applications
to energy and commodity markets, or economic models with producers and
consumers, one may want to study the cost functions
$(\overline{f},\overline{g})$ depending on the empirical distribution of the
sate $X$ of the agents such as
${\overline{f}}\Bigl{(}t,X_{t}^{i},\frac{1}{N}\sum_{j=1}^{N}\delta_{X_{t}^{j}},\varpi_{t},c^{0}_{t},c^{i}_{t}\Bigr{)}$.
Under the setup with conditional independence, the cost function for the limit
problem is naturally given by ${\overline{f}}\Bigl{(}t,X_{t},{\cal
L}(X_{t}|\overline{{\cal F}}_{t}^{0}),\varpi_{t},c^{0}_{t},c_{t}\Bigr{)}$.
Even in this case, the resultant FBSDE $(\ref{fbsde-single-p})$ is solvable,
at least for small $T$, if
$(\partial_{x}\overline{f},\partial_{x}\overline{g})$ are Lipschitz continuous
in the measure argument with respect to $W_{2}$-distance. Under the stronger
assumption guaranteeing the monotone conditions $(\ref{peng-wu-condition})$,
one can even achieve the existence of unique solution in general $T$. As long
as the source of common noise is solely from the filtration
$\overline{\mathbb{F}}^{0}$ generated by $\boldsymbol{W}^{0}$, we can avoid
subtleties regarding the admissibility (so-called $H$-hypothesis). See Remark
2.10 in [8] as a useful summary for this issue.
$\bullet~{}$Explicit solution: If we chose $\overline{f},\overline{g}$ as
quadratic functions and $l,\sigma_{0},\sigma$ as affine functions, we obtain a
linear-quadratic mean field game with common noise. In this case, an explicit
solution may be available where the coefficients functions are given as the
solutions to differential equations of Riccati type.
$\bullet~{}$Property of market price process: It seems interesting to study
the properties of the market clearing price theoretically and numerically. For
example, if $n=d^{0}$ the equivalent martingale measure (EMM) can be uniquely
determined. Based on the payoff distribution $c^{0}$ and the cost functions of
the agents $(\overline{f},\overline{g})$, one may study how the market price
process under the EMM behaves, for example, the relation between the skew of
its implied volatility and the risk-averseness of the agents.
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|
2024-09-04T02:54:57.763763 | 2020-03-06T08:29:37 | 2003.03072 | {
"authors": "Chan Hee Song, Dawn Lawrie, Tim Finin, James Mayfield",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26074",
"submitter": "Chan Hee Song",
"url": "https://arxiv.org/abs/2003.03072"
} | arxiv-papers | # Improving Neural Named Entity Recognition with Gazetteers
Chan Hee Song
University of Notre Dame
<EMAIL_ADDRESS>Dawn Lawrie
HLTCOE, JHU
<EMAIL_ADDRESS>Tim Finin
UMBC, HLTCOE
<EMAIL_ADDRESS>James Mayfield
HLTCOE, JHU
<EMAIL_ADDRESS>
###### Abstract
The goal of this work is to improve the performance of a neural named entity
recognition system by adding input features that indicate a word is part of a
name included in a gazetteer. This article describes how to generate
gazetteers from the Wikidata knowledge graph as well as how to integrate the
information into a neural NER system. Experiments reveal that the approach
yields performance gains in two distinct languages: a high-resource, word-
based language, English and a high-resource, character-based language,
Chinese. Experiments were also performed in a low-resource language, Russian
on a newly annotated Russian NER corpus from Reddit tagged with four core
types and twelve extended types. This article reports a baseline score. It is
a longer version of a paper in the 33rd FLAIRS conference (?).
## 1 Introduction
Named Entity Recognition (NER) is an important task in natural language
understanding that entails spotting mentions of conceptual entities in text
and classifying them according to a given set of categories. It is
particularly useful for downstream tasks such as information retrieval,
question answering, and knowledge graph population. Developing a well-
performing, robust NER system can facilitate more sophisticated queries that
involve entity types in information retrieval and more complete extraction of
information for knowledge graph population.
Various approaches exist to automated named entity recognition. Older
statistical methods use conditional random fields (?), perceptrons (?), and
support vector machines (?). More recent approaches have applied deep neural
models, beginning with ? (?). Further advances came from the addition of a
BiLSTM model with CRF decoding (?), which led to the current state-of-the-art
model.
Our NER architecture combines recent advances in transfer learning (?) and a
BiLSTM-CRF model, producing a BERT-BiLSTM-CRF model. In our model, BERT
generates an embedding for each word. This embedding is fed into a multi-layer
BiLSTM, which is often jointly trained with a pre-trained encoder at training
time. This fine-tunes the encoder to the NER task. At test time, the BiLSTM
outputs are decoded using a CRF. Other approaches show similar results, such
as a BiLSTM-CRF that uses a character-level CNN added to BiLSTM-CRF (?; ?). We
adopt a BERT-based model as the baseline system for comparison.
In the context of natural language understanding, a gazetteer is simply a
collection of common entity names typically organized by their entity type.
These have been widely used in natural language processing systems since the
early 1990s, such as a large list of English place names provided by the MUC-5
Message Understanding Conference (?) to support its TIPSTER information
extraction task. Initially these lists were employed to help recognize and
process mentions of entities that were places or geo-political regions, hence
the name gazetteer. Their use quickly evolved to cover more entity types and
subtypes, such as cities, people, organizations, political parties and
religions.
Statistical approaches have benefited by using gazetteers as an additional
source of information, often because the amount of labeled data for training
an NER system tends to be small. A lack of training data is of particular
concern when using neural architectures, which generally require large amounts
of training data to perform well. Gazetteers are much easier to produce than
labeled training data and can be mined from existing sources. Therefore, it is
important to know whether this rich source of information can be effectively
integrated into a neural model.
This paper first focuses on generating gazetteers from Wikidata, presenting a
simple way to gather a large quantity of annotated entities from Wikidata. It
then describes how to integrate the gazetteers with a neural architecture by
generating features from gazetteers alongside the features from BERT as input
to the BiLSTM. We aim to provide an additional external knowledge base to
neural systems similar to the way people use external knowledge to determine
what is an entity and to what category it belongs.
Adding gazetteer features (often called lexical features) to neural systems
has been shown to improve performance on well-studied datasets like English
OntoNotes and CONLL-2003 NER using a closed-world neural system (_i.e.,_
BiLSTM-CRF) (?). We extended this approach and validated that gazetteer
features are still beneficial to datasets in a more diverse set of languages
and with models that use a pre-trained encoder.
For generality, we applied and evaluated our approaches on datasets in three
languages: a high-resource, word-based language, English; a high-resource,
character-based language, Chinese; and a lower-resource, high morphological
language, Russian. We will first present how our gazetteer is generated from
publicly available data source, Wikidata. Then we will analyze our
experimental results.
Type | Description | Examples
---|---|---
PER | Person | Enrico Rastelli, Beyoncé
ORG | Organization | International Jugglers Association
COMM | Commercial Org. | Penguin Magic, Papermoon Diner
POL | Political Organization | Green Party, United Auto Workers
GPE | Geo-political Entity | Morocco, Carlisle, Côte d’Ivoire
LOC | Natural Location | Kartchner Caverns, Lake Erie
FAC | Facility | Stieff Silver Building, Hoover Dam
GOVT | Government Building | White House, 10 Downing St.
AIR | Airport | Ninoy Aquino International, BWI
EVNT | Named Event | WWII, Operation Desert Storm
VEH | Vehicle | HMS Titanic, Boeing 747
COMP | Computer Hard/Software | Nvidia GeForce RTX 2080 Ti, Reunion
MIL | Military Equip. | AK-47, Fat Man, cudgel
MIL_G | Generic Military Equip. | tank, aircraft carrier, rifle
MIL_N | Named Military Equip. | USS Nimitz, 13"/35 caliber gun
CHEM | Chemical | Iron, NaCl, hydrochloric acid
MISC | Other named entity | Dark Star, Lord of the Rings
Table 1: We worked with types where training data was available in several
languages, including four core types (in bold) and twelve additional ones.
## 2 Related Work
Our work builds on the neural approach to NER, which was introduced when ? (?)
used Long Short-Term Memory (LSTM), achieving just above average performance
for English and improvement for German. LSTM was proposed by ? (?), expanded
by ? (?), and reached its modern form with ? (?). Recent NER systems have
adopted a forward-backward LSTM or BiLSTM, mainly using the BiLSTM-CRF
architecture first proposed by ? (?), and now widely studied and augmented.
For example, ? (?) and ? (?) augmented the BiLSTM-CRF architecture with a
character-level CNN to add additional features to the architecture.
Adding lexical features to the system has been studied widely, mainly by
matching words in the dataset to words in pre-gathered gazetteers. ? (?) uses
gazetteers during embedding generation; ? (?) uses gazetteers to generate a
one-hot encoded match of the words in the data to those in the gazetteers; and
? (?) generates gazetteer embeddings from Wikipedia. ? (?) presents an
architecture incorporating gazetteer information for Chinese, which is a
language that often has a greater number of false positive matches because it
is logographic.
However, our approach provides a simple augmentation to existing neural models
and demonstrates that Chinese can benefit from gazetteer matches. We take the
? (?)’s approach to matching the gazetteer because of its simplicity and
universality in application to many different neural models. We also show that
it is applicable to neural models with a deep pre-trained encoder.
Transfer learning architectures have shown significant improvement in various
natural language processing tasks such as understanding, inference, question
answering, and machine translation. BERT (?), uses stacked bi-directional
transformer layers trained on masked word prediction and next sentence
prediction tasks. BERT is trained on over 3.3 billion words gathered mainly
from Wikipedia and Google Books. By adding a final output layer, BERT can be
adapted to many different natural language processing tasks. In this work, we
apply BERT to NER and use BiLSTM-CRF as the output layer of BERT. Our approach
embodies a simple architecture that does not require a dataset-specific
architecture or feature engineering.
## 3 Gazetteer Creation
We describe the knowledge source we use to create our gazetteers and outline
the process we used to automatically produce cleaned gazetteers for the entity
types of interest.
Figure 1: Statistics for canonical names for Wikidata entities for each type.
Additional lists hold aliases and, for Russian, inflected forms.
### 3.1 Wikidata Knowledge Graph
Our gazetteers were created by extracting canonical names (e.g., Manchester
United F.C.) and aliases (e.g., Red Devil, Man U) of entities of a given type
(e.g., ORG) from Wikidata (?). Wikidata is a large, collaboratively edited
knowledge graph with information drawn from and used by a number of Wikimedia
projects, including 310 Wikipedia sites in different languages. Its goal is to
integrate entities and knowable facts about them for use in Wikimedia sites in
a language-independent manner. Wikidata is multilingual, with all of its
strings tagged with a two-letter ISO 639-1 language code.
Wikidata currently has more than 900 million statements about 77 million
items, supported by an ontology with nearly 2.4 million fine-grained types and
more than 7,250 properties. An example item is the entity Q7186 shown in
Figure 2. Items have a canonical name, short description and set of aliases in
one or more languages. Property statements encode relations between items or
between an item and a literal value and can have metadata including qualifiers
(e.g., a period of time during which the property held), provenance
information (e.g., the URL of an attesting source), and a rank (e.g., to
distinguish a preferred value from alternative or deprecated ones).
The data is exposed as RDF triples and can be queried using Wikimedia APIs or
SPARQL queries sent to a public query service. We used the public SPARQL
service to get both canonical names (e.g., Johns Hopkins University) and
aliases (e.g., JHU, Johns Hopkins, Hopkins) in each of the languages studied
for a given entity type (e.g., ORG). In addition, the Wikimedia community has
developed many tools for searching for items, exploring the ontology, and
updating entries.
### 3.2 Gazetteer Generation
The gazetteer is generated by searching Wikidata via SPARQL queries sent to
the public query server to retrieve both canonical names (e.g., Johns Hopkins
University) and aliases (e.g., JHU, Johns Hopkins, Hopkins)) in each of the
languages studied. The first step was to construct a mapping from our
project’s 16 target types shown in Table 1 to Wikidata’s fine-grained type
system (?). Our types included four common core types (person, organization,
geopolitical entity (GPE), location) and twelve additional types (airport,
chemical, commercial organization, computer hardware/software, event,
facility, government building, military equipment, money, political
organization, title, vehicle).
The mapping for some types was simple: person corresponds to Wikidata’s Q5 and
vehicle to Q42889. Others had a complex mapping that eliminate Wikidata
subtypes that seemed too specialized (e.g., lunar craters and ice rumples from
Wikidata’s geographic object) or allow us to retrieve more entity names given
the public server’s one-minute query timeout.
Figure 2: Wikidata entities have a unique ID, a canonical name, aliases and a
short description in one or more languages along with any number of statements
representing properties or relations and including qualifiers and provenance
metadata. Figure 3: New training data is generated by replacing existing
annotated names with gazetteer names of the same type.
The initial name lists were filtered by type-dependant regular expressions to
delete names we thought to be unhelpful (e.g., Francis of Assisi as a person
because historical figures are unlikely to be mentioned in our targeted
genres), remove Wikipedia artifacts (e.g., parentheticals), and eliminate
punctuation, names that were too short or too long, and duplicate names.
Although one could say that these changes bias the gazetteers, there is no
reason not engineer a gazetteer in a way that is most helpful for the data.
Wikidata is still being used in an automated way since we are relying on
available labels.
Text | Jack | is | on | Hong | Kong | International | Airport | in | Lantau | Island | , | Hong | Kong
---|---|---|---|---|---|---|---|---|---|---|---|---|---
PER | B-PER | O | O | O | O | O | O | O | O | O | O | O | O
LOC | O | O | O | O | O | O | O | O | B-LOC | I-LOC | O | O | O
GPE | O | O | O | B-GPE | I-GPE | O | O | O | O | O | O | B-GPE | I-GPE
ORG | O | O | O | O | O | I-ORG | I-ORG | O | O | O | O | O | O
Figure 4: This example shows gazetteer matches for the sentence "Jack is on
Hong Kong International Airport in Lantau Island, Hong Kong" showing both full
and partial matches.
We produced additional lists for Russian using a custom script that generates
type-sensitive inflected and familiar forms of canonical names and aliases.
For an extreme example, the Russian name for the person Vladimir Vladimirovich
Putin ( Владимир Владимирович Путин) produces more than 100 variations. The
result is a collection of 96 gazetteer files with total 15.7M entity names,
4.2M for English, 2.1M for Russian and 584K for Chinese with an additional
8.7M Russian names produced by our morphological scripts. We kept the
gazetteers for canonical names, aliases, and inflected forms separate to
facilitate experimentation.
We also worked with data downloaded directly from Wikidata, which uses a JSON
serialization in which only the immediate types of each entity is provided.
The entity Museum of Modern Art, for example, is identified as an instance of
an art museum, an art institution and a copyright holder’s organisation. To
decide which, if any, of our target types an entity belongs, we constructed a
dictionary that maps relevant types to our 16 target types. For our example,
this turns out to be three types: ORG, LOC, and FAC.
Although doing the mapping sounds daunting given an ontology with more than
two million types, it is simplified by exploiting the fact that most of these
types do not have any immediate instances. We developed SPARQL queries that
identified for each of our 16 types all of their subtypes that had one or more
immediate instances. The ORG type, for example has 15,904 subtypes but only
5,962 have immediate instances; the LOC type has only 1,181 subtypes with
immediate instances. The resulting dictionary was thus relatively small
without losing any information and was used to quickly recognize entities of
interest in the Wikidata dumps as well as identify their target types.
## 4 Exploiting Gazetteers
### 4.1 Gazetteer Features
To use a gazetteer as a feature in the NER system, words in the dataset are
matched with a gazetteer and turned into one-hot vectors for each entity type.
Those one-hot vectors are then concatenated with word embeddings generated
from other sources. For example, a word embedding of size 768 from BERT is
concatenated with the gazetteer one-hot vectors sized to the number of
entities $x$. Although each gazetteer represents an entity type, no attempt is
made to communicate that type to the Bi-LSTM layer.
We use the BIO (Beginning-Inside-Outside) tagging scheme where B-<type> tags
the first token of an entity, I-<type> the subsequent ones, and O tagging non-
entity tokens. For gazetteer matches, the gazetteer uses two matching schemes:
full match and partial match.
* •
Full match: an $n$-gram in the dataset matches fully with a gazetteer entry.
If there are multiple matches in same entity category, the longest match is
preferred.
* •
Partial match: an $n$-gram in the dataset matches partially with the
gazetteer. Only partial matches of length greater than one are accepted,
except for the PER type, due to frequency of one-word person names.
As an example, consider a gazetteer that contains {Jack:PER, Lantau
Island:LOC, Hong Kong Government:GPE, JFK International Airport:ORG}, Figure 4
shows how full matches and partial matches are handled. Jack is fully matched
with PER, so it is tagged B-PER. Lantau Island is also fully matched and
tagged LOC. Hong Kong partially matches Hong Kong Government, and since the
match length is greater than one, it is considered a match and thus tagged as
GPE. Hong Kong International Airport is also partially matched with gazetteer
entry JFK International Airport, so International Airport is tagged with ORG.
As seen in the example match, tags of the gazetteer entries are assigned
during a partial match. For character-level tokenized text, like Chinese, we
forgo partial matching because it produces too many false matches. However for
all other language, we both utilize full match and partial match as shown in
the experiments.
After matching, matches are one-hot encoded with each tag type assigned a
separate one-hot vector. Therefore, for each token in the text, it gets
assigned a number of tag types of one-hot vectors. These one-hot vectors are
concatenated to the other features, which are fed into the BiLSTM.
### 4.2 Generating Augmented Training Data
We experimented with a second application of gazetteers that uses them to
generate additional training data. In this approach, we select sentences from
our initial human-annotated training data, replace one or more of the
annotated entities with a randomly selected gazetteer entity of the same type,
and retrain the system. However, this approach did not produce statistically
significant improvements.
We developed a script that takes as input a BIO-tagged file, a type, and
gazetteer for that type, and produces a modified version of the file with
entity instances of the type replace with a random entity selected from the
gazetteer. Additional arguments control whether all instances of a given
entity in the input BIO file are replaced with the same gazetteer entity and
specify the random seed, to support repeatable experiments. Our current
experiments were run by replacing entities for all types and to allowing a
given input entity to be replaced with different gazetteer entities each time
it appears. Figure 3 shows an example with an original annotated sentence from
OntoNotes on the left, and a new, generated training instance on the right.
## 5 Architecture
Hyperparameters
---
BiLSTM layers | $1$
BiLSTM hidden size | $256$
BiLSTM dropout | $.5$
Optimizer | adafactor
Gradient clipping | $1.0$
Learning rate scheduler | cosine decay
BERT layers used | $-4,-3$, $-2,-1$
Weight decay | $.005$
Mini batch size | $8$
Table 2: Default hyperparameters used in the baseline model
We use a common baseline Bi-LSTM-CRF model like many sequence to sequence
closed-world NER systems (?), which includes a stacked bi-directional
recurrent neural network with long short-term memory units and a conditional
random field decoder and is similar to ? (?) without the character-level CNN.
We combine this system with BERT (?), which is a stack of bi-directional
transformer encoders. We keep the BERT frozen during training and testing,
feeding the text into BERT and concatenating its final four layers as an input
to our Bi-LSTM-CRF. In addition, the features generated from gazetteers are
concatenated with the outputs from BERT and fed into the Bi-LSTM-CRF. Table 2
shows the hyperparameters used for our experiments. We did not perform a
hyperparameter search.
## 6 Experimental Data Sets
To demonstrate the effectiveness of our new approaches to NER, datasets
labeled with names are required. For English and Chinese, there are
established datasets. We chose OntoNotes v5.0 (?) because it has a large
number of labeled entities. Table 3 contains the statistics for these
datasets. Russian, on the other hand, has little labeled data for NER. We
chose to create our own dataset over Russian informal text. This section
describes the construction of this collection, as well as its tag set and
statistics.
Figure 5: Architecture used in our models, with baseline components in blue
and additional gazetteer features in green.
The Russian Reddit collection comes from Russian comments collected over 433
threads on the Reddit online discussion forum (?). Reddit organizes threads
around a submission that is posted to a channel. The first step in building
the collection was to identify Russian threads. Annotators examined threads
with at least ten comments in a majority of Cyrillic characters to determine
whether the thread was written in Russian. We eliminated images and movies
from the thread seeds, as well as seeds from sites primarily devoted to image
content; such seeds typically contain few named entities in their comments.
Over 30,000 threads met these criteria. Threads were prioritized based on the
source of the material in the submission, where newswire and blogs were
preferred.
Annotators examined around 800 of these threads and identified the language of
the comments, 433 of which were in Russian. These comments were automatically
sentence-segmented using CoreNLP (?), so that named entity tagging could be
performed by annotators at the sentence level. The Dragonfly annotation tool
(?) was used to record the entity tags through an in-house Mechanical Turk-
like interface.
One goal of this collection was to have a wider variety of entity types so
that future research could investigate types that have varying frequencies of
attestation. Beyond the common core types, types were chosen that were
sufficiently attested in the data. In addition, we desired to have a few
subtypes of the common types to be able to experiment with this hierarchical
relationship.
To assure quality annotations, either sentences were doubly-annotated and a
third annotator reviewed disagreements or the sentence was singly-annotated
and a second annotator reviewed the annotations. The inter-annotator agreement
on the doubly annotated text was 53%. The annotators agreed on whether a token
was part of a name 63% of the time. Since agreement was measured at the token
level, both a name’s tag and span had to match exactly. Finally the collection
was split 80-10-10 into train, development, and test respectively. Table 3
shows the size of the collection, which was labeled with 16 types as shown in
Table 1. The frequency of each type is shown in Table 4. The data set is
available from https://github.com/hltcoe/rus-reddit-ner-dataset.
Dataset | Type | Train | Test | Dev
---|---|---|---|---
English OntoNotes | Sentences | 82.1k | 9.0k | 12.7k
| Tokens | 1644.2k | 172.1k | 251.0k
| Entities | 70.3k | 6.9k | 10.9k
Chinese OntoNotes | Sentences | 37.5k | 4.3k | 6.2k
| Tokens | 1241.1k | 149.7k | 178.4k
| Entities | 37.9k | 4.5k | 5.4k
Russian Reddit | Sentences | 22.8k | 3.2k | 3.1k
| Tokens | 281.7k | 39.3k | 37.9k
| Core Ent. | 8.1k | 1.1k | 1.0k
| Extended Ent. | 11.2k | 1.5k | 1.4k
Table 3: Statistics of dataset sizes Tag Type | Frequency by Collection
---|---
| English | Chinese | Russian
| OntoNotes | OntoNotes | Reddit
PER | 27.4k | 14.1k | 3.3k
ORG | 30.0k | 10.1k | 1.1k
COMM | - | - | 409
POL | - | - | 174
GPE | 28.2k | 20.2k | 5.5k
LOC | 2.7k | 2.7k | 451
FAC | - | - | 50
GOVT | - | - | 36
AIR | - | - | 5
EVNT | - | - | 152
VEH | - | - | 63
COMP | - | - | 273
MIL_G | - | - | 260
MIL_N | - | - | 139
CHEM | - | - | 21
MISC | - | - | 1.5k
Table 4: Statistics of datasets by tag type
## 7 Results using Gazetteer Features
We use our models for NER tasks on the English OntoNotes, Chinese OntoNotes
Russian Reddit datasets. For each, we run our baseline models and the models
with added gazetteer features at least ten times, depending on the size of the
collection. Smaller collections were run a greater number of times because of
the greater variability in the output. Performance is reported as precision
(P), recall (R), and their harmonic mean (F1). Statistics of the dataset are
shown in Tables 3 and 4. The statistics for gazetteer coverage for individual
datasets are shown in Table 6. We use only the four core types for English and
Chinese because our gazetteer tag types do not include the extended OntoNotes
types. However, we experiment with both core and extended types for the
Russian dataset. For each experiment, we train for fixed epochs and choose the
model that shows the minimum loss on the development set.
### 7.1 English and Chinese OntoNotes
We use the English OntoNotes v5.0 dataset compiled for the CoNLL-2013 shared
task (?) and follow the standard train/dev/test split as presented in (?). We
use pre-trained Cased BERT-Base with 12-layer, 768-hidden, 12-heads, 110M
parameters available on the Google Github. The experiment is run for 10 trials
and trained for 30 epochs. The model with the minimum dev set loss is selected
and run on the test set. Table 5 shows our experiment results. We compute the
p-value of the distribution using a $t$-test. We show that adding gazetteer
features increases $0.52$ F1 score, an improvement that is statistically
significant ($p<0.001$). We attribute this to an even coverage of the
percentage of entities across train, dev, and test sets as seen in Table 6, as
well as a high coverage (high 80s) for GPE entities, the entity type with the
largest F1 gain.
We use the Chinese OntoNotes v5.0 dataset with four core types compiled for
CoNLL-2013 and follow the standard train/dev/test split as before. We use the
pre-trained Chinese BERT-Base for simplified and traditional Chinese which has
12-layer, 768-hidden, 12-heads, 110M parameters. The experiment is run for 10
trials and trained for 30 epochs. The model with minimum loss on dev set is
selected for testing. The gazetteer feature leads to a statically significant
improvement ($p=0.003$), which we attribute this to high GPE coverage and even
coverage across dataset splits. However, the absolute increase in F1 score is
around 0.3, which is lower than English dataset. We believe Chinese showed
less improvement due to our decision to forgo partial matches due to the high
frequency of partial matched n-grams stemming from the language’s logographic
nature.
Dataset | Model | P | R | F1
---|---|---|---|---
English | Baseline | 92.46 | 91.77 | 92.11 (SD: $0.10$)
Gazetteer | 92.82 | 92.44 | 92.63 (SD: $0.12$)
+Aliases | 92.69 | 92.50 | 92.59 (SD: $0.11$)
Chinese | Baseline | 83.40 | 84.63 | 84.01 (SD: $0.16$)
Gazetteer | 83.91 | 84.72 | 84.31 (SD: $0.23$)
+Aliases | 83.84 | 84.76 | 84.30 (SD: $0.25$)
Table 5: Performance of BERT-BiLSTM-CRF baseline and $+$ gazetteer features on
English and Chinese OntoNotes, SD stands for standard deviation
### 7.2 Russian Reddit Dataset
We use the Russian Reddit dataset to evaluate the performance of Russian NER.
We use pre-trained Multilingual Cased BERT-Base with 12-layer, 768-hidden,
12-heads, 110M parameters. We use the same baseline BERT-BiLSTM-CRF model with
gazetteer feature added. For the Russian Reddit dataset, the experiment is run
for 20 trials with 30 epochs. The model with minimum loss on dev is selected
for testing. The different number of trials is due to the smaller size of this
dataset, as is shown in Table 3. We report experiments with both core types
and extended types using the Russian Reddit dataset. Table 7 shows Russian
dataset experiments with gazetteers and different tag types.
Dataset | Type | Train | Test | Dev
---|---|---|---|---
English OntoNotes | PER | 38.2% | 44.3% | 37.5%
ORG | 19.3% | 17.2% | 19.0%
GPE | 88.7% | 86.8% | 87.2%
LOC | 26.3% | 23.7% | 30.0%
Chinese OntoNotes | PER | 24.0% | 21.2% | 21.6%
ORG | 18.0% | 17.4% | 23.4%
GPE | 76.2% | 75.4% | 77.2%
LOC | 18.1% | 17.4% | 14.1%
Russian Reddit | PER | 12.5% | 15.4% | 11.4%
ORG | 16.9% | 26.6% | 9.8%
COMM | 31.4% | 33.3% | 5.3%
GPE | 23.4% | 23.1% | 20.2%
LOC | 7.4% | 0% | 5.8%
FAC | 4.3% | 50.0% | 0%
GOVT | 7.7% | 0% | 0%
AIR | 0% | 0% | 0%
EVNT | 3.4% | 0% | 0%
VEH | 6.1% | 11.1% | 20.0%
COMP | 24.0% | 17.6% | 0%
MIL_G | 0% | 0% | 0%
MIL_N | 0% | 0% | 0%
CHEM | 0% | 0% | 14.3%
MISC | 0% | 0% | 0%
Table 6: Statistics for the entity types and subtypes for each of the three
collections. Our OntoNotes data only covered the core types while our Russian
Reddit included additional types and subtypes.
While the mean of the trials is slightly higher for those with gazetteer
features, none of the results shows statistical significance. We attribute
this to (a) lower coverage of our gazetteer for those in the dataset; and (b)
uneven gazetteer coverage throughout train, dev, and test sets as is seen in
Table 6. That table reports additional results from using inflected and
familiar forms of entity canonical names and aliases, as described in Section
2. However, our takeaway here is that adding gazetteer features does not hurt
the performance of the neural systems, but only improves it when the gazetteer
has high coverage, as can be seen in the English and Chinese experiments.
## 8 Results for Training Data Augmentation
Using our gazetteers to produce additional training data produced mixed
results and generally was inconsistent in improving performance for our models
using BERT for either the four core types or the extended set. We ran early
experiments using FastText embeddings (?) and found that using our gazetteers
with Russian inflections for PER improved performance for most types. However,
we did not see the same gains when using BERT.
Tags-Model | P | R | F1
---|---|---|---
C-Baseline | 80.21 | 72.12 | 75.95 (SD:0.43)
C-Gazetteer | 79.81 | 72.03 | 75.72 (SD:0.44)
C-Inflected | 79.75 | 72.01 | 75.68 (SD:0.42)
C-Alias | 79.68 | 72.05 | 75.67 (SD:0.48)
E-Baseline | 73.36 | 56.88 | 64.08 (SD:0.44)
E-Gazetteer | 73.33 | 57.05 | 64.17 (SD:0.58)
E-Inflected | 73.31 | 57.01 | 64.14 (SD:0.51)
E-Alias | 73.08 | 57.08 | 64.10 (SD:0.48)
Table 7: Performance of BERT-BiLSTM-CRF baseline and model with gazetteer
features on Russian Reddit datasets for the Core (C) and Extended (E) tag sets
This may be due to several reasons. First, both core and extended types are
quite broad. Replacing the annotated ORG Harvard University with the gazetteer
ORG Disneyland in the sentence "Professor Pinker teaches at Harvard
University" seems anomalous to us and probably also to our model. Second, our
experiments were done with relatively small amounts of annotated training
data, especially for Russian. While drawing on gazetteer data may help
introduce new patterns not present in the training data, such as ORGs
beginning with "Association of", the chances of this helping when evaluated
with the relatively small test partition is low. Third, entity names were
extracted from Wikidata without regard to their utility, including both very
prominent entities (e.g., the LOC Atlantic Ocean) and very obscure ones (e.g.,
Avalonia, a microcontinent in the Paleozoic era).
We plan to further explore this use case by replacing annotated entities with
gazetteer entities that are in a same finer-grained Wikidata type. We can
readily identify all of the Wikidata types to which a reasonably prominent
entity (e.g., Harvard University) belongs. We will select several hundred of
these types as targets for gazetteer entity replacement (e.g., Q2385804 –
education institution) that are similar to an annotated entity. We can then
associate gazetteer entities with these target types, replacing an entity such
as "Harvard University" with an entity that is more similar, e.g., "Swarthmore
College" or "Loyola Academy"). We also plan to limit obscure entities using a
measure of prominence derived from Wikidata metrics, such as their number of
incoming and outgoing links.
## 9 Conclusion and Future Work
We present a simple way to generate a gazetteer, and show how it can be used
in a neural NER systems. We also present a new Russian NER corpus gathered
from Reddit. We show that with enough coverage on the dataset, gazetteer
features improve neural NER systems, even systems using deep pre-trained
models such as BERT. We hypothesize that paying attention to how tuned the
signal is between the gazetteer and the training set greatly impacts ho much
the neural system learns to pay attention to the gazetteer. Modifying the
gazetteer based on the training data is a path we plan to explored. In
general, we believe gazetteer features should be a standard addition to any
NER system and show that even with low coverage, the gazetteer features do not
hurt the performance of neural NER systems.
While our gazetteer data augmentation did not show consistent improvement, we
believe that future work in more sophisticated and contextualized replacement
scheme will benefit low-resource languages such as Russian. In addition the
noisiness of the gazetteers may have a great impact on performance since the
NER system may learn not to trust a gazetteer that does not assist with
tagging a sufficient number of time. Future work will identify techniques to
produce gazetteers that are trustworthy relative to the training data to see
if such gazetteers can be shown to be more helpful.
Gazetteers and associated software is available from
https://github.com/hltcoe/gazetteer-collection.
## Acknowledgments
We thank Johns Hopkins University, the Human Language Technology Center of
Excellence and the 2019 SCALE workshop for their hospitality and for
facilitating an excellent research environment.
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|
2024-09-04T02:54:57.773516 | 2020-03-06T08:58:25 | 2003.03085 | {
"authors": "Hanaa Zitane, Ali Boutoulout, Delfim F. M. Torres",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26075",
"submitter": "Delfim F. M. Torres",
"url": "https://arxiv.org/abs/2003.03085"
} | arxiv-papers | ###### Abstract
We investigate the stability and stabilization concepts for infinite
dimensional time fractional differential linear systems in Hilbert spaces with
Caputo derivatives. Firstly, based on a family of operators generated by
strongly continuous semigroups and on a probability density function, we
provide sufficient and necessary conditions for the exponential stability of
the considered class of systems. Then, by assuming that the system dynamics is
symmetric and uniformly elliptic and by using the properties of the
Mittag–Leffler function, we provide sufficient conditions that ensure strong
stability. Finally, we characterize an explicit feedback control that
guarantees the strong stabilization of a controlled Caputo time fractional
linear system through a decomposition approach. Some examples are presented
that illustrate the effectiveness of our results.
###### keywords:
fractional differential equations; fractional diffusion systems; Caputo
derivative; stability and stabilization in Hilbert spaces; decomposition
method.
xx 1 5 Submitted: 31 Dec 2019; Revised: 21 and 28 Feb 2020; Accepted: 02 March
2020. Published in: Mathematics 2020, Volume 8, Issue 3, 353.
DOI:black10.3390/math8030353 The Stability and Stabilization of Infinite
Dimensional Caputo-Time Fractional Differential Linear Systems Hanaa Zitane
1,†, Ali Boutoulout 2,† and Delfim F. M. Torres 3,†* Hanaa Zitane, Ali
Boutoulout and Delfim F. M. Torres Correspondence<EMAIL_ADDRESS>Tel.:
+351-234-370-668 (D.F.M.T.) The authors contributed equally to this work.
26A33; 93D15.
## 1 Introduction
Fractional order calculus is a natural generalization of classical integer
order calculus. It deals with integrals and derivatives of an arbitrary real
or complex order. Fractional order calculus has become very popular, in recent
years, due to its demonstrated applications in many fields of applied sciences
and engineering, such as the spread of contaminants in underground water, the
charge transport in amorphous semiconductors, and diffusion of pollution in
the atmosphere Rahimy (2010); Kilbas et al. (2006); Diethelm (2010). Because
it generalizes and includes in the limit the integer order calculus, the
fractional calculus has the potential to accomplish much more than what
integer order calculus achieves Hilfer (2000). In particular, it has proved to
be a powerful tool to describe long-term memory and hereditary properties of
various dynamical complex processes Sabatier et al. (2007), diffusion
processes, such as those found in batteries Gabano and Poinot (2011) and
electrochemical and control processes Ichise et al. (1971), to model and
control epidemics Rosa and Torres (2019); Silva and Torres (2019) and
mechanical properties of viscoelastic systems and damping materials, such as
stress and strain Bagley and Calico (1991).
One can find in the literature several different fractional calculus. Here we
use the fractional calculus of Caputo, which was introduced by Michele Caputo
in his 1967 paper Caputo (2008). Such calculus has appeared, in a natural way,
for representing observed phenomena in laboratory experiments and field
observations, where the mathematical theory was checked with experimental
data. Indeed, the operator introduced by Caputo in 1967, and used by us in the
present work, represents an observed linear dissipative mechanism phenomena
with a time derivative of order 0.15 entering the stress-strain relation
Caputo (2008). More recently, a variational analysis with Caputo operators has
been developed, which provides further mathematical substance to the use of
Caputo fractional operators Malinowska and Torres (2012); Almeida et al.
(2015).
In the analysis and design of control systems, the stability issue has always
an important role Mahmoud and Karaki (2018); Rocha et al. (2018). For a
dynamical system, an equilibrium state is said to be stable if such system
remains close to this state for small disturbances, and for an unstable system
the question is how to stabilize it, especially by a feedback control law
Sontag (2012). The stabilization concept for integer order systems and related
problems has been considered in several works, see, e.g., Pritchard and
Zabczyk (1981); Curtain and Zwart (1995); Triggiani (1975); Balakrishnan
(1981) and references cited therein. In Pritchard and Zabczyk (1981), the
relationship between the asymptotic behavior of a system, the spectrum
properties of its dynamics, and the existence of a Lyapunov functional is
provided. Several techniques are considered to study different kinds of
stabilization, for example, the exponential stabilization is studied via a
decomposition method Triggiani (1975) while the strong stabilization is
developed using the Riccati approach Balakrishnan (1981).
Similarly as classical dynamical systems, stability analysis is a central task
in the study of fractional dynamical systems, which has attracted increasing
interest of many researchers Silva and Torres (2019); Wojtak et al. (2018).
For finite dimensional systems, the stability concept for fractional
differential systems equipped with the Caputo derivative is investigated in
many works Zhang et al. (2011). In Matignon (1996), Matignon studies the
asymptotic behavior for linear fractional differential systems with the Caputo
derivative, where their dynamics $A$ is a constant coefficient matrix. In this
case the stability is guaranteed if the eigenvalues of the dynamics matrix
$A$, $\lambda\in\sigma(A)$, satisfy $|arg(\lambda)|>\dfrac{\pi\alpha}{2}$
Matignon (1996). Since then, many scholars have carried out further studies on
the stability for different classes of fractional linear systems Qian et al.
(2010); Li et al. (2010). In Qian et al. (2010), stability theorems for
fractional differential systems, which include linear systems, time-delayed
systems, and perturbed systems, are established, while in Li et al. (2010),
Ge, Chen and Kou provide results on the Mittag–Leffler stability and propose a
Lyapunov direct method, which covers the power law stability and the
exponential stability. See also Matar and Abu Skhail (2018), where the
Mittag–Leffler and the class-K function stability of fractional differential
equations of order $\alpha\in(1,2)$ are investigated. In 2018, the notion of
regional stability was introduced for fractional systems in Ge et al. (2018),
where the authors study the Mittag–Leffler stability and the stabilization of
systems with Caputo derivatives, but only on a sub-region of its geometrical
domain. More recently, fractional output stabilization problems for
distributed systems in the Riemann–Liouville sense were studied Zitane et al.
(2020, in press, 2019), where feedback controls, which ensure exponential,
strong, and weak stabilization of the state fractional spatial derivatives,
with real and complex orders, are characterized.
An analysis of the literature shows that existing results on stability of
fractional systems are essentially limited to finite-dimensional fractional
order linear systems, while results on infinite-dimensional spaces are a
rarity. In contrast, here we investigate global stability and stabilization of
infinite dimensional fractional dynamical linear systems in the Hilbert space
$L^{2}(\Omega)$ with Caputo derivatives of fractional order $0<\alpha<1$. In
particular, we characterize exponential and strong stability for fractional
Caputo systems on infinite-dimensional spaces.
The remainder of this paper is organized as follows. In Section 2, some basic
knowledge of fractional calculus and some preliminary results, which will be
used throughout the paper, are given. In Section 3, we prove results on the
global asymptotic and exponential stability of Caputo-time fractional
differential linear systems. In contrast with available results in the
literature, which are restricted to systems of integer order or to fractional
systems in the finite dimensional state space $\mathbb{R}^{n}$, here we study
a completely different class of systems: we investigate fractional linear
systems where the state space is the Hillbert space $L^{2}(\Omega)$. We also
characterize the stabilization of a controlled Caputo diffusion linear system
via a decomposition method. Section 4 presents the main conclusions of the
work and some interesting open questions that deserve further investigations.
## 2 Preliminaries and Notation
In this section, we introduce several definitions and results of fractional
calculus that are used in the sequel.
[Kilbas et al. (2006)] Let $0<\alpha<1$ and $T>0$. The Caputo derivative of
fractional order $\alpha$ for an absolutely continuous function $y(\cdot)$ on
$[0,T]$ can be defined as follows:
${}^{C}D_{t}^{\alpha}y(t)=\dfrac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha}\dfrac{d}{ds}y(s)\,\mathrm{d}s,$
where $\Gamma(1-\alpha)$ is the Euler Gamma function.
[Zhou and Jiao (2010)] For any given function $g\in L^{2}(0,T,L^{2}(\Omega))$,
we say that function $y\in C(0,T,L^{2}(\Omega))$ is a mild solution of the
system
$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}y(t)=Ay(t)+g(t)&t\in]0,+\infty[\\\
y(0)=y_{0}&y_{0}\in L^{2}(\Omega)\end{array}\right.$ (1)
if it satisfies
$y(t)=S_{\alpha}(t)y_{0}+\int_{0}^{t}(t-s)^{\alpha-1}K_{\alpha}(t-s)g(s)\,\mathrm{d}s,$
(2)
where
$S_{\alpha}(t)=\int_{0}^{+\infty}\Psi_{\alpha}(\theta)S(t^{\alpha}\theta)\,\mathrm{d}\theta$
(3)
and
$K_{\alpha}(t)=\alpha\int_{0}^{+\infty}\theta\Psi_{\alpha}(\theta)S(t^{\alpha}\theta)\,\mathrm{d}\theta$
(4)
with
$\Psi_{\alpha}(\theta)=\frac{1}{\alpha}\theta^{-1-\frac{1}{\alpha}}T_{\alpha}(\theta^{-\frac{1}{\alpha}}),$
(5)
$(S(t))_{t\geq 0}$ the strongly continuous semigroup generated by operator
$A$, and $T_{\alpha}$ the probability density function defined on $(0,\infty)$
by
$T_{\alpha}=\frac{1}{\pi}\sum_{n=1}^{+\infty}(-1)^{n}\theta^{\alpha
n-1}\frac{\Gamma(n\alpha+1)}{n!}\sin(n\pi\alpha).$
[Mainardi et al. (2007)] The probability density function $T_{\alpha}$ defined
on $(0,\infty)$ satisfies
$T_{\alpha}(\theta)\geq 0,\quad\theta\in(0,\infty),\quad\text{ and
}\int_{0}^{+\infty}T_{\alpha}(\theta)\,\mathrm{d}\theta=1.$
[Erdélyi et al. (1981)] The Mittag–Leffler function of one parameter is
defined as
$E_{\eta}(z)=\sum_{n=0}^{+\infty}\frac{z^{n}}{\Gamma(\eta
n+1)}\quad\text{with}\quad Re(\eta)>0,\quad z~{}\in\mathbb{C}.$
[Erdélyi et al. (1981)] The Mittag–Leffler function of two parameters is
defined as
$E_{\eta,\beta}(z)=\sum_{n=0}^{+\infty}\frac{z^{n}}{\Gamma(\eta
n+\beta)}\quad\text{with}\quad Re(\eta)>0,\quad\beta>0,\quad z\in\mathbb{C}.$
The Mittag–Leffler function appears naturally in the solution of fractional
differential equations and in various applications: see Erdélyi et al. (1981)
and references therein. The exponential function is a special case of the
Mittag–Leffler function Joshi et al. (2020): for $\beta=1$ one has
$E_{\eta,1}(z)=E_{\eta}(z)$ and $E_{1,1}(z)=e^{z}$.
[Mainardi (2014)] The Mittag–Leffler function $E_{\alpha}(-t^{\alpha})$ is
completely monotonic: for all $0<\alpha<1$, for all $n\in\mathbb{N}$ and
$t>0$, one has
$(-1)^{n}\frac{d^{n}}{dt^{n}}E_{\alpha}(-t^{\alpha})\geq 0.$
[Schneider (1996)] The generalized Mittag–Leffler function
$E_{\eta,\beta}(-x)$, $x\geq 0$, is completely monotonic for $\eta,\beta>0$ if
and only if $\eta\in(0,1]$ and $\beta\geq\eta$.
[Podlubny (1999)] Let $\beta>0$, $0<\eta<2$, and $\mu$ be an arbitrary real
number such that $\frac{\pi\eta}{2}<\mu<\min\\{\pi,\pi\eta\\}$. Then, the
following asymptotic expressions hold:
* •
if $|arg(z)|\leq\mu$ and $|z|>0$, then
$|E_{\eta,\beta}(z)|\leq
M_{1}(1+|z|)^{(1-\beta)/\eta}e^{Re(z^{\frac{1}{{\eta}}})}+\frac{M_{2}}{1+|z|};$
(6)
* •
if $\mu<|arg(z)|\leq\pi$ and $|z|\geq 0$, then
$|E_{\eta,\beta}(z)|\leq\frac{M_{2}}{1+|z|},$ (7)
where $M_{1}$ and $M_{2}$ are positive constants.
## 3 Main Results
Our main goal is to study the stability and provide stabilization for a class
of abstract Caputo-time fractional differential linear systems.
### 3.1 Stability of Time Fractional Differential Systems
Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$, $n=1,2,3,\ldots$,
and let us consider the following abstract time fractional order differential
system:
$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}z(t)=Az(t),&t\in\,]0,+\infty[,\\\
z(0)=z_{0},&z_{0}\in L^{2}(\Omega),\end{array}\right.$ (8)
where ${}^{C}D_{t}^{\alpha}$ is the left-sided Caputo fractional derivative of
order $0<\alpha<1$, the second order operator $A:D(A)\subset
L^{2}(\Omega)\longrightarrow L^{2}(\Omega)$ is linear, with dense domain and
such that the coefficients do not depend on time $t$, and such that it is also
the infinitesimal generator of the $C_{0}$-semi-group $(S(t))_{t\geq 0}$ on
the Hilbert state space $L^{2}(\Omega)$ endowed with its usual inner product
$<\cdot,\cdot>$ and the corresponding norm $\Arrowvert\cdot\Arrowvert$. The
unique mild solution of system (8) can be written, from Lemma 2, as
$z(t)=S_{\alpha}(t)z_{0},$
where $S_{\alpha}(t)$ is defined by (3).
We begin by proving the following lemma, which will be used thereafter.
Let $A$ be the infinitesimal generator of a $C_{0}$-semi-group $(S(t))_{t\geq
0}$ on the Hilbert space $L^{2}(\Omega)$. Assume that there exists a function
$h(\cdot)\in L^{2}(0,+\infty;\mathbb{R}^{+})$ satisfying
$\Arrowvert S_{\alpha}(t+s)z\Arrowvert\leq h(t)\Arrowvert
S_{\alpha}(s)z\Arrowvert,\quad\forall\,t,s\geq 0,\quad\forall\,z\in
L^{2}(\Omega).$ (9)
Then the operators $(S_{\alpha}(t))_{t\geq 0}$ are uniformly bounded.
###### Proof.
To prove that $(S_{\alpha}(t))_{t\geq 0}$ are bounded, we have to show that
$\forall z\in L^{2}(\Omega)~{}\underset{t\geq 0}{\sup}\Arrowvert
S_{\alpha}(t)z\Arrowvert<\infty.$ (10)
By _reductio ad absurdum_ , let us suppose that (10) does not hold, which
means that there exists a sequence $(t_{s}+\tau_{n})$, $t_{s}>0$ and
$\tau_{n}\longrightarrow+\infty$, satisfying
$\Arrowvert
S_{\alpha}(t_{s}+\tau_{n})z\Arrowvert\longrightarrow+\infty~{}as~{}n\longrightarrow+\infty.$
(11)
From relation
$\displaystyle\int_{0}^{+\infty}\Arrowvert
S_{\alpha}(s+\tau_{n})z\Arrowvert^{2}\,\mathrm{d}s=\displaystyle\int_{\tau_{n}}^{+\infty}\Arrowvert
S_{\alpha}(s)z\Arrowvert^{2}\,\mathrm{d}s,~{}~{}0\leq s<+\infty,$
it follows that the right-hand side goes to $0$ as $n\longrightarrow+\infty$.
Using Fatou’s Lemma yields
$\underset{n\longrightarrow+\infty}{\lim~{}\inf}~{}\Arrowvert
S_{\alpha}(s+\tau_{n})z\Arrowvert=0~{}~{}\forall~{}s>0.$
Therefore, for some $s_{0}<t_{s}$, we may find a subsequence $\tau_{n_{k}}$
such that
$\underset{k\longrightarrow+\infty}{\lim}\Arrowvert
S_{\alpha}(s_{0}+\tau_{n_{k}})z\Arrowvert=0.$
By virtue of condition (9), one obtains
$\Arrowvert S_{\alpha}(t_{s}+\tau_{n_{k}})z\Arrowvert\leq
h(t_{s}-s_{0})\Arrowvert
S_{\alpha}(s_{0}+\tau_{n_{k}})z\Arrowvert\underset{k\longrightarrow+\infty}{\longrightarrow
0},$
which contradicts (11). The intended conclusion follows from the uniform
boundedness principle. ∎
Let $z_{0}\in L^{2}(\Omega)$. System (8) is said to be exponentially stable if
there exist two strictly positive constants, $M>0$ and $\omega>0$, such that
$\Arrowvert z(t)\Arrowvert\leq Me^{-\omega t}\Arrowvert
z_{0}\Arrowvert,\quad\forall t\geq 0.$
The next theorem provides necessary and sufficient conditions for exponential
stability of the abstract fractional order differential system (8).
Suppose that the operators $(S_{\alpha}(t))_{t\geq 0}$ fulfill assumption (9)
and
$\forall z\in L^{2}(\Omega)~{}~{}~{}~{}~{}~{}\Arrowvert
S_{\alpha}(t+s)z\Arrowvert\leq\Arrowvert
S_{\alpha}(t)z\Arrowvert\cdot\Arrowvert S_{\alpha}(s)z\Arrowvert,\quad\forall
t,s\geq 0.$ (12)
Then, system (8) is exponentially stable if, and only if, for every $z\in
L^{2}(\Omega)$ there exists a positive constant $\delta<\infty$ such that
$\displaystyle\int_{0}^{+\infty}\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}\,\mathrm{d}t<\delta.$ (13)
###### Proof.
One has
$\begin{split}t\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}&=\displaystyle\int_{0}^{t}\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}\,\mathrm{d}s\\\
&=\displaystyle\int_{0}^{t}\Arrowvert
S_{\alpha}(t-s+s)z\Arrowvert^{2}\,\mathrm{d}s.\end{split}$
Combining assumption (9), Lemma 3.1, and condition (13), one gets
$\begin{split}t\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}&\leq\displaystyle\int_{0}^{t}\Arrowvert
S_{\alpha}(s)z\Arrowvert^{2}\Arrowvert
S_{\alpha}(t-s)z\Arrowvert^{2}\,\mathrm{d}s\\\ &\leq N\delta\Arrowvert
z\Arrowvert^{2}\end{split}$
for some $N>0$. Therefore, for $t$ sufficiently large, it follows that
$\Arrowvert S_{\alpha}(t)\Arrowvert<1.$
Then, there exists $t_{1}>0$ such that
$\ln\Arrowvert S_{\alpha}(t)\Arrowvert<0,\quad\forall t\geq t_{1}.$
Thus,
$\omega_{0}=\underset{t\geq 0}{\inf}~{}\dfrac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}<0.$
Now, let us show that
$\omega_{0}=\underset{t\longrightarrow+\infty}{\lim}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}.$ (14)
Let $t_{s}>0$ be a fixed number and
$N^{{}^{\prime}}=\underset{t\in[0,t_{s}]}{\sup}{\Arrowvert
S_{\alpha}(t)\Arrowvert}$. Thus, for each $t>t_{s}$, there exists
$m\in\mathbb{N}$ such that $mt_{s}\leq t\leq(m+1)t_{s}$. From (12), it follows
that
$\begin{split}\Arrowvert S_{\alpha}(t)\Arrowvert&=\Arrowvert
S_{\alpha}(mt_{s}+(t-mt_{s}))\Arrowvert\\\ &\leq\Arrowvert
S_{\alpha}(mt_{s})\Arrowvert\Arrowvert
S_{\alpha}(t-mt_{s})\Arrowvert,\end{split}$
which yields
$\frac{\ln\Arrowvert S_{\alpha}(t)\Arrowvert}{t}\leq\frac{\ln\Arrowvert
S_{\alpha}(mt_{s})\Arrowvert}{t}+\frac{\ln\Arrowvert
S_{\alpha}(t-mt_{s})\Arrowvert}{t}.$
Using again (12), it results that
$\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}\leq\frac{mt_{s}}{t}\frac{\ln\Arrowvert
S_{\alpha}(t_{s})\Arrowvert}{t_{s}}+\frac{\ln\Arrowvert
N^{{}^{\prime}}\Arrowvert}{t}.$
Since $\dfrac{mt_{s}}{t}\leq 1$ and $t_{s}$ is arbitrary, one obtains
$\underset{t\longrightarrow+\infty}{\lim~{}\sup}~{}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}\leq\underset{t>0}{\inf}~{}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}\leq\underset{t\longrightarrow+\infty}{\lim~{}\inf}~{}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}.$
Consequently, (14) holds. Hence, we conclude that for all
$\omega\in\,]0,-\omega_{0}[$, there exists $M>0$ such that
$\forall z\in L^{2}(\Omega)\quad\Arrowvert S_{\alpha}(t)z\Arrowvert\leq
Me^{-\omega t}\Arrowvert z\Arrowvert,~{}~{}\forall t\geq 0,$
which means that system (8) is exponentially stable. The converse is obvious.
∎
When $\alpha=1$, the conditions (9) and (12) are verified, and we retrieve
from our Theorem 3.1 the results established in Curtain and Zwart (1995);
Pritchard and Zabczyk (1981) about the exponential stability of system (8) on
$\Omega$, which is equivalent to
$\int_{0}^{+\infty}\Arrowvert
S(t)z\Arrowvert^{2}\,\mathrm{d}t<\infty,\quad\forall z\in L^{2}(\Omega).$
Let $z_{0}\in L^{2}(\Omega)$. System (8) is said to be strongly stable if its
corresponding solution $z(t)$ satisfies
$\Arrowvert z(t)\Arrowvert\longrightarrow
0~{}~{}as~{}~{}t\longrightarrow+\infty.$
In our next theorem, we provide sufficient conditions that guaranty the strong
stability of the fractional order differential system (8). The result
generalizes the asymptotic result established by Matignon for finite
dimensional state spaces, where the dynamics of the system $A$ is considered
to be a matrix with constant coefficients in $\mathbb{R}^{n}$ Matignon (1996).
In contrast, here we tackle the stability for a different class of systems.
Precisely, we consider fractional systems where the system dynamics $A$ is a
linear operator generating a strongly continuous semigroup in the infinite
dimensional state space $L^{2}(\Omega)$.
Let $(\lambda_{p})_{p\geq 1}$ and $(\phi_{p})_{p\geq 1}$ be the eigenvalues
and the corresponding eigenfunctions of operator $A$ on $L^{2}(\Omega)$. If
$A$ is a symmetric uniformly elliptic operator, then system (8) is strongly
stable on $\Omega$.
###### Proof.
Since $A$ is a symmetric uniformly elliptic operator, it follows that system
(8) admits a weak solution defined by
$z(t)=\sum_{p=1}^{+\infty}E_{\alpha}(\lambda_{p}t^{\alpha})\langle
z_{0},\phi_{p}\rangle\phi_{p}\quad\forall\,z_{0}\in L^{2}(\Omega),$
where $(\lambda_{p})_{p\geq 1}$ satisfy
$0>\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{j}\geq\cdots,\lim\limits_{j\longrightarrow\infty}=-\infty,$
and $(\phi_{p})_{p\geq 1}$ forms an orthonormal basis in $L^{2}(\Omega)$
Sakamoto and Yamamoto (2011); Courant and Hilbert (1953). Using the fact that
function $E_{\alpha}(-t^{\alpha})$ is completely monotonic, for all
$\alpha\in(0,1)$ and $t>0$ (Lemma 2), yields
$\begin{split}\Arrowvert
z(t)\Arrowvert&=\left\Arrowvert\displaystyle\sum\limits_{{p=1}}^{+\infty}E_{\alpha}(\lambda_{p}t^{\alpha})\langle
z_{0},\phi_{p}\rangle\phi_{p}\right\Arrowvert\\\
&\leq|E_{\alpha}(\lambda_{1}t^{\alpha})|\Arrowvert
z_{0}\Arrowvert.\end{split}$
Moreover, from Lemma 2, it follows that
$\Arrowvert z(t)\Arrowvert\leq\frac{M_{2}}{1-\lambda_{1}t^{\alpha}}\Arrowvert
z_{0}\Arrowvert\longrightarrow 0\quad\text{ as }\quad t\longrightarrow+\infty$
for some $M_{2}>0$. Hence, system (8) is strongly stable on $\Omega$. ∎
Let us consider, on $\Omega=]0,1[$, the following one-dimensional fractional
diffusion system defined by
$\left\\{\begin{array}[]{lll}{}^{C}D_{t}^{0.5}z(x,t)=\dfrac{\partial^{2}z}{\partial
x^{2}}(x,t),&x\in\Omega,\quad t\in\,]0,+\infty[,\\\ z(0,t)=z(1,t)=0,&\forall
t>0,\\\ z(x,0)=z_{0},&x\in\Omega,\end{array}\right.$ (15)
where the second order operator $A=\dfrac{\partial^{2}}{\partial x^{2}}$ has
its spectrum given by the eigenvalues $\lambda_{p}=-(p\pi)^{2}$, $p\geq 1$,
and the corresponding eigenfunctions are
$\phi_{p}(x)=\sqrt{\frac{2}{1+(p\pi)^{2}}}\sin(p\pi x)$, $p\geq 1$. Operator
$A$ generates a $C_{0}$-semi-group $(S(t))_{t\geq 0}$ defined by
$S(t)z_{0}=\sum_{p=1}^{+\infty}e^{\lambda_{p}t}\langle
z_{0},\phi_{p}\rangle\phi_{p}.$
Moreover, the solution of system (15) is given by
$S_{0.5}(t)z_{0}=\sum_{p=1}^{+\infty}E_{0.5}(\lambda_{p}t^{0.5})\langle
z_{0},\phi_{p}\rangle\phi_{p}.$
One has that operator $A$ is symmetric and uniformly elliptic. Consequently,
from our Theorem 3.1, we deduce that system (15) is strongly stable on
$\Omega$. This is illustrated numerically in Figure 1 for $z(x,0)=\sin(\pi
x)$, $t=0.1$, $t=0.15$, $t=0.2$, and $t=1.0$.
Figure 1: The state of system (15) for $z(x,0)=\sin(\pi x)$, $t=0.1$,
$t=0.15$, $t=0.2$, and $t=1.0$, illustrating the fact that (15) is strongly
stable on $\Omega=]0,1[$.
### 3.2 Stabilization of Time Fractional Differential Systems
Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$, $n=1,2,3,\ldots$.
We consider the following Caputo-time fractional differential linear system:
$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}z(t)=Az(t)+Bu(t),&t\in]0,+\infty[,\quad
0<\alpha<1,\\\ z(0)=z_{0},&z_{0}\in L^{2}(\Omega),\end{array}\right.$ (16)
with the same assumptions on $A$ as in Section 3.1 and where $B$ is a bounded
linear operator from $U$ into $L^{2}(\Omega)$, where $U$ is the space of
controls, assumed to be a Hilbert space. By Lemma 2, the unique mild solution
$z(\cdot)$ of system (16) is defined by
$z(t)=S_{\alpha}(t)z_{0}+\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}K_{\alpha}(t-s)Bu(s)\,\mathrm{d}s,$
(17)
where $S_{\alpha}(t)$ and $K_{\alpha}(t)$ are given, respectively, by (3) and
(4) .
System (16) is said to be exponentially (respectively strongly) stabilizable
if there exists a bounded operator $K\in\mathcal{L}(L^{2}(\Omega),U)$ such
that the system
$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}z(t)=(A+BK)z(t),&t\in\,]0,+\infty[,\\\
z(0)=z_{0},&z_{0}\in L^{2}(\Omega),\end{array}\right.$ (18)
is exponentially (respectively strongly) stable on $\Omega$.
It is clear that the exponential stabilization of system (16) implies the
strong stabilization of (16). Note that the concept is general: when
$\alpha=1$, we obtain the classical definitions of stability and
stabilization.
Let $(S^{k}(t))_{t\geq 0}$ be the strongly continuous semi-group generated by
$A+BK$, where $K\in\mathcal{L}(L^{2}(\Omega),U)$ is the feedback operator. The
unique mild solution of system (16) can be written as
$z(t)=S^{k}_{\alpha}(t)z_{0}$
with
$S^{k}_{\alpha}(t)=\displaystyle\int_{0}^{+\infty}\Psi_{\alpha}(\theta)S^{k}(t^{\alpha}\theta)\,\mathrm{d}\theta,$
where $\Psi_{\alpha}(\theta)$ is defined by (5).
Let $A+BK$ generate a strongly continuous semi-group $(S^{k}(t))_{t\geq 0}$ on
$L^{2}(\Omega)$. If the operator $(S^{k}_{\alpha}(t))_{t\geq 0}$ satisfies
conditions (9) and (12) and if
$\forall z\in
L^{2}(\Omega)~{}~{}~{}~{}~{}~{}\displaystyle\int_{0}^{+\infty}\Arrowvert
S^{k}_{\alpha}(t)z\Arrowvert^{2}\,\mathrm{d}t<~{}\infty$
holds, then system (16) is exponentially stabilizable on $\Omega$.
###### Proof.
The proof is similar to the proof of Theorem 3.1. ∎
Let $(\lambda^{k}_{p})_{p\geq 1}$ and $(\phi^{k}_{p})_{p\geq 1}$ be the
eigenvalues and the corresponding eigenfunctions of operator $A+BK$ on
$L^{2}(\Omega)$. If $A+BK$ is a symmetric uniformly elliptic operator, then
system (16) is strongly stabilizable on $\Omega$.
###### Proof.
The proof is similar to the proof of Theorem 3.1. ∎
Let us consider, on $\Omega=]0,1[$, the following fractional differential
system of order $\alpha=0.2$:
$\begin{cases}{}^{C}D_{t}^{0.2}z(x,t)=\displaystyle\frac{1}{100}\dfrac{\partial^{2}z}{\partial
x^{2}}(x,t)+\displaystyle\frac{1}{2}z(x,t)+BKz(x,t),&(x,t)\in\Omega\times]0,+\infty[,\\\
z(0,t)=z(1,t)=0,&\forall t>0,\\\ z(x,0)=z_{0},&x\in\Omega,\end{cases}$ (19)
with the linear bounded operator $B=I$ and where we take $K=-B^{*}=-I$. The
operator
$A+BK=\frac{1}{100}\dfrac{\partial^{2}}{\partial x^{2}}-\frac{1}{2},$
with spectrum given by the eigenvalues
$\lambda^{k}_{p}=-\frac{1}{2}-\frac{1}{100}(p\pi)^{2}$, $p\geq 1$, and the
corresponding eigenfunctions
$\phi^{k}_{p}(x)=\sqrt{\frac{2}{1+(p\pi)^{2}}}\cos(p\pi x)$, $p\geq 1$,
generates a $C_{0}$-semi-group $(S^{k}(t))_{t\geq 0}$ defined by
$S^{k}(t)z_{0}=\displaystyle\sum_{p=1}^{+\infty}e^{\lambda^{k}_{p}t}\langle
z_{0},\phi^{k}_{p}\rangle\phi^{k}_{p}.$
Furthermore, the solution of system (19) can be written as
$z(t)=S^{k}_{0.2}(t)z_{0}=\displaystyle\sum_{p=1}^{+\infty}E_{0.2}(\lambda^{k}_{p}t^{0.2})\langle
z_{0},\phi^{k}_{p}\rangle\phi^{k}_{p}.$
It is clear that $A+BK$ is a symmetric and uniformly elliptic operator. Hence,
from Theorem 3.2, we deduce that system (19) is strongly stabilizable on
$\Omega$, i.e., the system
$\begin{cases}{}^{C}D_{t}^{0.2}z(x,t)=\displaystyle\frac{1}{100}\dfrac{\partial^{2}z}{\partial
x^{2}}(x,t)+\displaystyle\frac{1}{2}z(x,t)+Bu(t),&(x,t)\in\Omega\times]0,+\infty[,\\\
z(0,t)=z(1,t)=0,&\forall t>0,\\\ z(x,0)=z_{0},&x\in\Omega,\end{cases}$
is strongly stabilizable by the feedback control $u(t)=-B^{*}z(t)$. Figure 2
shows, for $z(x,0)=x(x-1)$, that the state $z(x,t)$ of system (19) is unstable
at $t=0$. Moreover, we see that the state evolves close to 0 at $t=10$.
Numerically, the state is stabilized by $u(t)=-B^{*}z(t)$ with an error equal
to $1.75\times 10^{-04}$.
Figure 2: The state of system (19) for $z(x,0)=x(x-1)$, $t=0$, and $t=10$,
illustrating the fact that (19) is unstable at $t=0$ but it is stabilized at
$t=10$ on $\Omega=]0,1[$.
### 3.3 Decomposition Method
Now, we study the stabilization of system (16) using the decomposition method,
which consists in decomposing the state space and the system using the
spectral properties of operator $A$.
Let $\xi>0$ be fixed and assume that there are at most finitely many
nonnegative eigenvalues of $A$ and each with finite dimensional eigenspace. In
other words, assume there exists $l\in\mathbb{N}$ such that
$\sigma(A)=\sigma_{u}(A)\cup\sigma_{s}(A),$ (20)
where $\sigma_{u}(A)=\sigma(A)\cap\\{\lambda_{p},~{}~{}~{}p=1,2,\ldots,l\\}$,
$\sigma_{s}(A)=\sigma(A)\cap\\{\lambda_{p},~{}~{}~{}p=l+1,l+2\ldots\\}$ with
$\lambda_{l}\geq 0$ and $\lambda_{l+1}\leq-\xi$. Because the sequence
$(\phi_{p})_{p\geq 1}$ forms a complete and orthonormal basis in
$H=L^{2}(\Omega)$, it follows that the state space $H$ can be decomposed as
$H=H_{u}\oplus H_{s},$ (21)
where $H_{u}=PH=\mathrm{span}\\{\phi_{1},\phi_{2},\ldots,\phi_{l}\\}$ and
$H_{s}=(I-P)H=\mathrm{span}\\{\phi_{l+1},\phi_{l+2},\ldots\\}$ with
$P\in\mathcal{L}(H)$ the projection operator Kato (1966). Hence, system (16)
can be decomposed into the following two sub-systems:
$\begin{cases}{}^{C}D_{t}^{\alpha}z_{u}(t)=A_{u}z_{u}(t)+PBu(t),\\\
z_{0u}=Pz_{0},\end{cases}$ (22)
and
$\begin{cases}{}^{C}D_{t}^{\alpha}z_{s}(t)=A_{s}z_{s}(t)+(I-P)Bu(t),\\\
z_{0s}=(I-P)z_{0},\end{cases}$ (23)
where $A_{s}$ and $A_{u}$ are the restrictions of $A$ on $H_{s}$ and $H_{u}$,
respectively, and are such that $\sigma(A_{s})=\sigma_{s}(A)$,
$\sigma(A_{u})=\sigma_{u}(A)$, and $A_{u}$ is a bounded operator on $H_{u}$.
Our next result asserts that stabilization of system (16) is equivalent to the
one of system (22).
Let the spectrum $\sigma(A)$ of $A$ satisfy the above spectrum decomposition
assumptions (20) for some $\xi>0$ and $A_{s}$ be a symmetric uniformly
elliptic operator. If system (22) is strongly stabilizable by the control
$u(t)=D_{u}z_{u}(t)$ (24)
with $D_{u}\in\mathcal{L}(H,U)$ such that
$\Arrowvert z_{u}(t)\Arrowvert\leq C~{}t^{-\mu},\quad\mu,C>0,$ (25)
then system (16) is strongly stabilizable using the feedback control
$v(t)=D_{u}z_{u}(t)$.
###### Proof.
Using the fact that system (22) is strongly stabilizable by control (24), and
inequality (25) yields
$\Arrowvert z_{u}(t)\Arrowvert\longrightarrow 0\text{ as
}t\longrightarrow+\infty$ (26)
and
$\Arrowvert u(t)\Arrowvert\leq C\Arrowvert D_{u}\Arrowvert t^{-\mu},$ (27)
the unique weak solution of system (23) can be written in the space $H_{s}$ as
$z_{s}(t)=\sum_{p=l+1}^{+\infty}E_{\alpha}(\lambda_{p}t^{\alpha})\langle
z_{0s},\phi_{p}\rangle\phi_{p}+\sum_{p=l+1}^{+\infty}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}(\lambda_{p}(t-s)^{\alpha})\langle(I-P)Bu(s),\phi_{p}\rangle\phi_{p}\,\mathrm{d}s$
since $A_{s}$ is a symmetric uniformly elliptic operator Sakamoto and Yamamoto
(2011). Using the spectrum decomposition relation (20), Lemma 2, and Lemma 2,
one has that
$E_{\alpha}(\lambda_{p}t^{\alpha})\leq E_{\alpha}(-\xi t^{\alpha})\quad\text{
for all }p\geq l+1$ (28)
and
$E_{\alpha,\alpha}(\lambda_{p}(t-s)^{\alpha})\leq
E_{\alpha,\alpha}(-\xi(t-s)^{\alpha})\quad\text{ for all }p\geq l+1.$ (29)
Then, feeding system (23) by the same control $u(t)=D_{u}z_{u}(t)$ and using
(27)–(29), it follows that
$\begin{split}\Arrowvert z_{s}(t)\Arrowvert&\leq E_{\alpha}(-\xi
t^{\alpha})\Arrowvert z_{0s}\Arrowvert+C\Arrowvert D_{u}\Arrowvert\Arrowvert
I-P\Arrowvert\Arrowvert
B\Arrowvert\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}s^{-\mu}E_{\alpha,\alpha}(-\xi(t-s)^{\alpha})\,\mathrm{d}s\\\
&\leq E_{\alpha}(-\xi t^{\alpha})\Arrowvert z_{0s}\Arrowvert+C\Arrowvert
D_{u}\Arrowvert\Arrowvert I-P\Arrowvert\Arrowvert
B\Arrowvert\displaystyle\sum\limits_{{n=1}}^{+\infty}\displaystyle\int_{0}^{t}\dfrac{(-\xi)^{n}(t-s)^{\alpha
n+\alpha-1}s^{-\mu}\,\mathrm{d}s}{\Gamma(\alpha n+\alpha)}\\\ &\leq
E_{\alpha}(-\xi t^{\alpha})\Arrowvert z_{0s}\Arrowvert+C\Arrowvert
D_{u}\Arrowvert\Arrowvert I-P\Arrowvert\Arrowvert
B\Arrowvert\displaystyle\sum\limits_{{n=1}}^{+\infty}\dfrac{(-\xi)^{n}t^{\alpha
n+\alpha-\mu}}{\Gamma(\alpha n+\alpha-\mu-1)\Gamma(1-\mu)^{-1}}\\\ &\leq
E_{\alpha}(-\xi t^{\alpha})\Arrowvert
z_{0s}\Arrowvert+C\Gamma(1-\mu)\Arrowvert D_{u}\Arrowvert\Arrowvert
I-P\Arrowvert\Arrowvert B\Arrowvert t^{\alpha-\mu}E_{\alpha,\alpha-\mu+1}(-\xi
t^{\alpha}).\end{split}$
Lemma 2 implies that
$\Arrowvert z_{s}(t)\Arrowvert\leq\frac{M_{1}}{1+\xi t^{\alpha}}\Arrowvert
z_{s0}\Arrowvert+C\Gamma(1-\mu)\Arrowvert D_{u}\Arrowvert\Arrowvert
I-P\Arrowvert\Arrowvert B\Arrowvert\frac{M_{2}t^{\alpha-\mu}}{1+\xi
t^{\alpha}}$
for some $M_{1},M_{2}>0$. Therefore,
$\Arrowvert z_{s}(t)\Arrowvert\longrightarrow 0\text{ as
}t\longrightarrow+\infty.$ (30)
On the other hand, we have that
$\Arrowvert z(t)\Arrowvert=\Arrowvert
z_{s}(t)+z_{u}(t)\Arrowvert\leq\Arrowvert z_{s}(t)\Arrowvert+\Arrowvert
z_{u}(t)\Arrowvert.$ (31)
Combining (26), (30), and (31), we deduce the strong stabilization of system
(16). ∎
## 4 Conclusions and Future Work
We investigated the stability problem of infinite dimensional time fractional
differential linear systems under Caputo derivatives of order
$\alpha\in(0,1)$, where the state space is the Hillbert space $L^{2}(\Omega)$.
We proved necessary and sufficient conditions for exponential stability and
obtained a characterization for the asymptotic stability, which is guaranteed
if the system dynamics is symmetric and uniformly elliptic. Moreover, some
stabilization criteria were also proved. Finally, we investigated the strong
stabilization of the system via a decomposition method, where an explicit
feedback control is obtained. Illustrative examples were given, showing the
effectiveness of the theoretical results. As future work, we intend to extend
our work to the class of infinite dimensional time fractional differential
nonlinear systems. Various other questions are still open and deserve further
investigations, such as, studying boundary stability and gradient stability
for time fractional differential linear systems or considering the more recent
notion of $\Lambda$-fractional derivative Lazopoulos and Lazopoulos (2019),
and thus obtaining a geometrical interpretation.
Each author equally contributed to this paper, read and approved the final
manuscript.
This research was funded by Moulay Ismail University (H.Z.); by Hassan II
Academy of Science and Technology, project N 630/2016 (A.B.); and by The
Portuguese Foundation for Science and Technology, R&D unit CIDMA, within
project UIDB/04106/2020 (D.F.M.T.).
###### Acknowledgements.
This research is part of first author’s Ph.D. project, which is carried out at
Moulay Ismail University, Meknes, and has began during a one-month visit of
Zitane to the R&D Unit CIDMA, Department of Mathematics, University of Aveiro,
Portugal, June 2019. The hospitality of the host institution is here
gratefully acknowledged. The authors are strongly grateful to three anonymous
referees for their suggestions and invaluable comments. The authors declare no
conflict of interest. References
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|
2024-09-04T02:54:57.783309 | 2020-03-06T09:06:34 | 2003.03089 | {
"authors": "Sergii Morozov, Stefano Vezzoli, Ali Hossain Khan, Iwan Moreels,\n Riccardo Sapienza",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26076",
"submitter": "Sergii Morozov",
"url": "https://arxiv.org/abs/2003.03089"
} | arxiv-papers | Present address:] Centre for Nano Optics, University of Southern Denmark,
Campusvej 55, Odense M, DK-5230, Denmark.
# Objective-free excitation of quantum emitters
with a laser-written micro parabolic mirror
Sergii Morozov [ Stefano Vezzoli The Blackett Laboratory, Department of
Physics, Imperial College London,
London SW7 2BW, United Kingdom Ali Hossain Khan Iwan Moreels Department of
Chemistry, Ghent University, Krijgslaan 281-S3, Gent 9000, Belgium Riccardo
Sapienza<EMAIL_ADDRESS>The Blackett Laboratory, Department of
Physics, Imperial College London,
London SW7 2BW, United Kingdom
###### Abstract
The efficient excitation of quantum sources such as quantum dots or single
molecules requires high NA optics which is often a challenge in cryogenics, or
in ultrafast optics. Here we propose a 3.2 $\mu$m wide parabolic mirror, with
a 0.8 $\mu$m focal length, fabricated by direct laser writing on CdSe/CdS
colloidal quantum dots, capable of focusing the excitation light to a sub-
wavelength spot and to extract the generated emission by collimating it into a
narrow beam. This mirror is fabricated via in-situ volumetric optical
lithography, which can be aligned to individual emitters, and it can be easily
adapted to other geometries beyond the paraboloid. This compact solid-state
transducer from far-field to the emitter has important applications in
objective-free quantum technologies.
††preprint: APP20-LE-00202
## I Introduction
Highly confined optical fields are required for excitation of individual
emitters and for efficient generation of single photons. Confining light to
sub-wavelength volumes to decrease the background signal or to increase the
fluency, is a common strategy in many optical domains, such as confocal
microscopy, super-resolution microscopy, optical lithography, optical
tweezers, ion and atom trapping, high density data storage, material
processing. Lens-based objectives are traditionally used to focus light to a
diffraction limited spot, a size which depends on the numerical aperture (NA)
Abbe (1883). High NA objectives are bulky and have to be operated at short
working distances which complicates their practical applications, especially
at cryogenic temperatures, while their chromatic aberrations and temporal
dispersion hamper their use with ultra-short pulses.
Objectives can be replaced by reflective architectures which provide inherent
achromaticity and nonparaxial focusing due to the high NA. The simplest
reflective objective is a parabolic mirror, which concentrates an incident
beam around the geometrical focal point. The resulting focal spot has a size
which is on a par with or better than that of high quality lens objectives
Lieb and Meixner (2001); Stadler _et al._ (2008). Parabolic mirrors have
found multiple applications in confocal microscopy Drechsler _et al._ (2001),
cryostat based single molecule spectroscopy Durand _et al._ (1999), scanning
optical near-field microscopy Sackrow _et al._ (2008), Raman microscopy Zhang
_et al._ (2009), solar cells Kosten _et al._ (2013), light-emitting diodes
Tanriseven, Maaskant, and Corbett (2008), nonlinear optics Penjweini _et al._
(2019). Recently, deep parabolic mirrors covering $4\pi$ solid angle gained
attention in quantum optics because of their efficient focusing, ability of
trapping individual quantum emitters, and of extracting a collimated beam of
single photons Sondermann and Leuchs (2015); Salakhutdinov _et al._ (2016).
However, future practical applications require miniaturisation and integration
of parabolic mirrors.
When dealing with integrated photonics, different strategies to focus light
into quantum emitters and collect their radiation have been explored.
Nanofocusing has been achieved with structured metamaterials, as for example
using a metalense based on plasmonic Fresnel plates Ma and Liu (2010),
hypergratings Thongrattanasiri and Podolskiy (2009), and plasmonic metamirrors
Ding _et al._ (2019). Plasmonic antennas have been shown to confine optical
fields to zeptoliter volumes delivering nano resolution for plasmonic direct
writing lithography Wang _et al._ (2016), albeit often hampered by ohmic
losses, difficulty in precise nanometric alignment with an emitter, as well as
complicated and expensive fabrication techniques.
Deterministic integration of micro optical components and individual quantum
emitters can be achieved either by pre- or post-fabrication alignment,
combined with lithographic fabrication. By lithographic techniques, various
compact optical systems have been fabricated to control and manipulate light
at the nanoscale, as for example waveguides Shi _et al._ (2016); Colautti
_et al._ (2020), polarisation rotators Schumann _et al._ (2014), microdisc
resonators Schell _et al._ (2013), objectives Fischbach _et al._ (2017);
Gissibl _et al._ (2016), dielectric pillar antennas Au _et al._ (2019),
pillar microcavities Dousse _et al._ (2008). High-index solid immersion
lenses have been used to improve the coupling to quantum light sources,
decreasing the laser excitation spot by a factor $1/n$ and magnifying the
photoluminescence image of an emitter by a factor $n$ Sapienza _et al._
(2015); Sartison _et al._ (2017); Schmidt _et al._ (2019). Microscale
parabolic antennas have been shown to be an ideal design for directing light
from quantum emitters Schell _et al._ (2014); Morozov _et al._ (2018),
however the focusing abilities of such compact structures have not been
explored.
In this letter, we report a compact parabolic mirror for sub-wavelength
excitation of quantum emitters placed in its focal spot. The mirror also
directs the generated photons into a low divergent beam along the parabola
symmetry axis. The parabolic mirror is fabricated by in-situ optical
volumetric lithography, which produces paraboloid structures in a single laser
exposure step and results in high optical quality surfaces. We experimentally
demonstrate that this mirror can focus light to a spot which is comparable to
one of a high NA oil immersion objective. With a focal length of 0.8 $\mu$m,
and a predicted focal spot of 120 nm ($\sim\lambda_{ex}/4$), the micro-mirror
acts as an ideal optical transducer from the emitter to free-space.
## II Results and Discussion
### II.1 Focusing with a micro parabolic mirror
A parabolic mirror illuminated with a collimated laser beam concentrates the
excitation energy in its focal point. If the NA is large enough, the mirror
has a sub-wavelength focal spot formed with minimal aberrations Lieb and
Meixner (2001), and broadband response in the visible and near infrared range
of electromagnetic spectrum. While refractive optics is intrinsically limited
by the frequency-dependent dielectric constant of its constituent, this is not
the case for the parabolic mirror which is intrinsically achromatic (i.e.
different wavelengths focus always in the same plane). Instead, the different
refractive index experienced by wavelength will affect the focal spot size
(see SI Fig.S2).
We consider a micro parabolic mirror with focal length $f=0.8~{}\mu$m and
aperture diameter $d=3.2~{}\mu$m covering a $2\pi$ solid angle. In such a
geometry, the radius $R$ of the dish is double of its height $H$, that is
$R=2H=2f$. The dish is filled with a polymer ($n_{p}=1.49$) and illuminated
from a glass substrate ($n_{g}=1.52$) with a linearly polarised plane wave at
$\lambda_{ex}=442$ nm as sketched in Fig.1(a). Due to the small size of the
mirror, diffraction effects are expected, which can be well captured in
finite-difference time-domain (FDTD) numerical simulations. The map of total
electric field intensity $|\bm{E}|^{2}$ through the focal spot is plotted in
Fig.1(b-d). The focal plane in Fig.1(b) demonstrates that the maximum
confinement is achieved in the $y$ direction, which is orthogonal to the
excitation polarisation ($x$ here).
The intensity cross-sections through the focal spot in $x$, $y$, and $z$
directions are plotted in Fig.1(e), where the maximum of normalised total
electric field intensity in $z$ direction is close to the position of
geometrical focal point, just 30 nm shifted into the glass substrate. We
ascribe this to the small refractive index mismatch of the parabolic mirror
filling $n_{p}$ and the glass substrate $n_{g}$ (see SI Fig.S1). The obtained
intensity distributions around the focal point have a full width at half
maximum (FWHM) in lateral ($x$ and $y$) and axial ($z$) directions of $\Delta
x=222$ nm, $\Delta y=120$ nm and $\Delta z=247$ nm. The focal spot size scales
with the refractive index of the parabolic mirror filling, and thus a much
tighter focusing can be reached in comparison with an air-filled parabolic
mirror (see SI Fig.S2). Hence, the sub-wavelength focus of
$\sim\lambda_{ex}/4$ can be achieved for a focal length of only 0.8 $\mu$m,
being unaffected by size-effects and diffraction. Mirrors with focal length
shorter than that drastically decrease the intensity in the focal point (see
SI Fig.S3), and loose the ability to collimate the generated emission Morozov
_et al._ (2018).
Another important characteristic of high-NA parabolic mirrors is their ability
to convert the incident light polarisation from transverse ($x$ here) to
longitudinal ($z$ here), allowing for the generation of optical fields with a
strong electric field component along the optical axis Drechsler _et al._
(2001); Debus _et al._ (2003). The numerical simulations confirm the
polarisation conversion, indicating a $|\bm{E_{x}}|^{2}$ component is the
dominant contribution to the intensity in focal region, while
$|\bm{E_{y}}|^{2}$ and $|\bm{E_{z}}|^{2}$ components have 24 and 4 times lower
magnitude (see SI Fig.S4). Hence, the micro parabolic mirror is able to excite
also quantum emitters with electric dipoles oriented perpendicularly to the
sample plane, with about $20$% efficiency.
Figure 1: Micro parabolic mirror for focusing light at the nanoscale. (a) In
geometrical optics, a parabolic mirror concentrates the collimated
illumination in its focal point $F$. (b-d) FDTD simulations of intensity
distribution in a micro parabolic mirror with focal length of 0.8 $\mu$m
illuminated with a collimated $x$-polarised plane wave at $\lambda=442$ nm.
The intensity distributions of total electric field intensity are shown in
$x-y$ focal plane in (b), as well as in orthogonal $x-z$ and $y-z$ planes
through the focal point in (c) and (d), respectively. The outline of parabolic
mirror is shown by orange line. (e) Sections of the intensity profiles from
panels (b-d) demonstrate the sub-wavelength focal spot of the micro parabolic
mirror. Figure 2: Fabrication of micro parabolic mirrors by in-situ
volumetric lithography. (a) The laser beam was focused below the sample plane
to polymerise a parabolic micro structure over a quantum dot layer (not to
scale). (b) The unexposed photoresist was washed away resulting in a polymeric
parabolic dish on glass substrate. (c) The sample was coated with 80 nm gold
layer to form a micro parabolic mirror with quantum dots in its focal plane.
(d) Deviation of a micro parabolic mirror surface from a perfect paraboloid
shape was obtained by AFM scanning. The black dashed circle represents the
mirror aperture. (e) Confocal scan of the parabolic mirror focal plane reveals
a quasi homogeneous photoluminescence intensity distribution, while the
photoluminescence of quantum dots outside aperture (dashed white circle) is
completely quenched by the deposited gold layer.
### II.2 Fabrication
We fabricated micro parabolic mirrors over a fluorescent layer composed of
colloidal giant shell CdSe/CdS quantum dots emitting at $\lambda_{em}=650$ nm
Christodoulou _et al._ (2014). The giant shell configuration of quantum dots
is highly photostable, and helped us to eliminate such undesirable artifacts
as photobleaching. The quantum dots are used here to map the intensity
distribution in the focal plane of the micro parabolic mirror, as their
fluorescence is proportional to the excitation light intensity, provided that
it is well below the saturation intensity. A layer of quantum dots was
deposited by spin-coating over a glass coverslip prior to the fabrication of
the mirror dish. In the fabrication, we followed the one-photon direct laser
writing which we have described in Morozov _et al._ (2018). Such one-photon
photopolymerization has a power threshold Delrot _et al._ (2018), thus a
clear boundary in the polymerization is formed. Our lithography technique is
based on the polymerization of structures with the volumetric intensity
profile of a focused Gaussian beam (the blue hourglass-shaped profile in
Fig.2). First, we localised the quantum dots by means of scanning confocal
microscopy retrieving the position of the sample plane. Next, we controllably
defocused the confocal system using a piezo stage in order to expose the
photoresist to a part of hourglass-shaped intensity iso-surface (Fig.2(a)).
This outer part of intensity iso-surface follows a paraboloid shape allowing
for the polymerisation of parabolic structures in a single laser exposure
step. After washing off the unexposed photoresist, a parabolic polymer
structure reveals over the quantum dot layer (Fig.2(b)). In the final step, we
sputtered an 80 nm gold layer over the sample to form a metal parabolic mirror
(Fig.2(c)). The fabrication technique results in a smooth mirror surface of
high optical quality as demonstrated in Fig.2(d): the deviation of a micro
parabolic mirror shape from a perfect paraboloid surface was within $\pm 70$
nm, extracted from a 3D fit of an atomic force microscope (AFM) scan. A
confocal scan in Fig.2(e) shows the focal plane of a fabricated parabolic
mirror, where the fluorescence signal originates from the quantum dot layer.
The fluorescence signal within the parabolic mirror aperture is distributed
quasi homogeneously without a dominant intensity spot ($I_{avg}=392$ cnts/px,
$I_{std}=96$ cnts/px), while the quantum dot layer fluorescence outside the
mirror aperture is completely quenched by the deposited gold layer.
### II.3 Excitation of quantum emitters
The fabricated micro parabolic mirror with a quantum dot fluorescence layer in
its focal plane was illuminated with a collimated laser beam at
$\lambda_{ex}=442$ nm to demonstrate its focusing properties. The resulting
fluorescence intensity profile of the parabolic mirror focal plane (Fig.3(a))
was imaged on a CCD camera using an objective (Nikon Plan Apochromat 100x, NA
= 1.45). The obtained intensity distribution in Fig.3(a) is very different
from the confocal intensity map presented in Fig.2(e), and a dominant
intensity peak is clearly visible in the position of the micro parabolic
mirror focal spot. The residual background in Fig.3(b) comes from fluorescence
signal of the quantum dots in the focal plane of the micro parabolic mirror
excited by the collimated laser beam. In addition, the intensity profile is
modulated by interference rings around the central bright focal spot, which we
attribute to the reflection of photoluminescence between parabolic surface and
the focal plane (see SI Fig.S7).
In order to characterise the fluorescence intensity distribution in the focal
plane of the parabolic mirror (Fig.3(a)), we extract an intensity cross
section through the focal spot as shown by green line in Fig.3(b). The
experimental intensity cross section is characterised by a FWHM of
$\Delta_{exp}=554\pm 38$ nm, which is comparable with the simulated focal spot
size $\Delta_{sim}=347$ nm. This simulated focal spot shown by red line in
Fig.3(b) was obtained by a convolution of the parabolic mirror excitation
profile presented in Fig.1 with the point spread function $\Delta_{PSF}=224$
nm of the imaging system with NA=1.45 at $\lambda_{em}=650$ nm, estimated from
the Abbe diffraction limit (see SI Fig.S5). The difference in FWHM between the
experimental and simulated values of focal spot sizes is due to the presence
of the quantum dot layer, which can scatter both the exciting beam and the
emitted light, thus slightly widening the collected spot.
To compare the performance of the parabolic mirror we also directly excited
the quantum dots layer at the position of the micro parabolic mirror focal
spot with the high NA objective. In Fig.3(b) we present a cross section (blue
line) of the photoluminescence intensity excited in this way (see also SI
Fig.S6). This profile has the same FWHM of $\Delta_{exp}=555\pm 38$ nm as in
the case of focusing with the micro parabolic mirror. While one could conclude
that the micro parabolic mirror focusing power is comparable to that of the
1.45 NA objective, we point out that the measured spot size comes from (i)
excitation through the mirror with an expected average focal spot size of
$\Delta_{par}$=171 nm (Fig.1), and (ii) imaging through an objective with
finite resolution. Although we expect a slightly smaller focal spot size for a
1.45 NA objective ($\Delta_{obj}$=152 nm), its measured focal spot would be
very similar to the micro parabolic mirror (see SI). Therefore, we conclude
that the results shown in Fig.3(b) are compatible with the expected focusing
power of the micro parabolic mirror, and $\Delta_{exp}=555$ nm is only an
upper bound for the size of the focal spot.
Figure 3: Excitation of quantum emitters with the micro parabolic mirror. (a)
A collimated laser beam at $\lambda_{ex}=442$ nm causes fluorescence of all
quantum dots in the focal plane, while the parabolic mirror reflects the laser
beam exciting quantum dots only in its focal spot resulting in a bright
intensity spot in the centre of mirror aperture (dashed white circle). (b) The
green intensity cross section of the micro parabolic focal spot was acquired
along the dotted white line in panel (a). The blue cross section was obtained
by focusing with the high NA objective in the position of parabolic mirror
focal spot. The red line shows the simulated focal spot intensity cross
section.
### II.4 Directing quantum emission
The photons emitted by the quantum dots in the focal spot are collimated by
the micro parabolic mirror. We demonstrate this by imaging the back focal
plane of microscope objective to obtain the Fourier space momentum
distribution of quantum dot emission. Such a Fourier space image of the micro
parabolic mirror focal plane intensity profile from Fig.3(a) is presented in
Fig.4(a). The emission of the quantum dots excited with the micro parabolic
mirror is directed into a narrow beam along the optical axis, which is
completely contained in an NA=0.5 solid angle. The fluorescence beam
collimation is characterised in Fig.4(b), which presents the intensity cross
sections of the micro parabolic mirror radiation pattern. This result
summarizes the full operational principle of the parabolic mirror: first it
efficiently couples a directional plane wave onto the emitter, similarly to
what achieved with a conventional objective Morozov _et al._ (2018), and then
it directs the generated photoluminescence back to the far-field.
The simulation of an isolated single horizontal dipole in the focal spot of
the micro parabolic mirror confirms the unidirectional radiation pattern and
is plotted as a red curve in Fig.4(b), with a half power beam width of
$\theta_{1/2}^{sim}=8^{\circ}$ at $\lambda_{em}=650$ nm. The green curve in
Fig.4(b) is the radiation pattern cross section obtained from Fig.4(a), and
the micro parabolic mirror excitation with a collimated laser beam in the
widefield configuration yields in the collimated fluorescence beam with
$\theta_{1/2}^{exp}=25^{\circ}$ (after the background subtraction, see SI
Fig.S8). The difference in the half power beam width originates from the layer
of quantum dots distributed quasi-homogeneously in the sample plane. The
emission of the quantum dots out of the focal spot of the micro parabolic
mirror is directional as well, however it is not aligned with the mirror
optical axis. The parabolic mirror reflects their emission at different
angles, and thus broadens the radiation pattern in Fig.4. The background would
be absent in case of a single quantum dot in the focal point of the micro
parabolic mirror as we have shown in ref. Morozov _et al._ (2018). In the
current experimental conditions the background contribution can be reduced by
confocal excitation with the high NA objective in the position of focal spot
of micro parabolic antenna resulting in the collimated emission with
$\theta_{1/2}^{exp}=19^{\circ}$ (blue curve in Fig.4(b)).
Figure 4: The fluorescence of quantum dots in the focal spot of the micro
parabolic mirror is collimated. (a) A Fourier space image of quantum dots in
the focal spot of the micro parabolic mirror presented in Fig.3(a). The main
lobe of the radiation pattern is completely within NA=0.5. (b) Radiation
pattern in polar coordinates demonstrate the low beam divergence. The
radiation pattern obtained using the micro parabolic antenna excitation is
shown by green curve acquired along the dotted white line in the panel (a).
The blue line represents the radiation pattern obtained using confocal
excitation of the quantum dots in the focal spot of the same micro parabolic
mirror. The red line is the simulation of an isolated single $x$-oriented
dipole in the focal spot of the micro parabolic mirror.
## III Conclusion
In conclusion, we scaled down to microscale the concept of a reflective
objective based on the micro parabolic mirror. We fabricated such a compact
parabolic mirror with a focal length of only $0.8$ $\mu$m, capable of focusing
the excitation light to sub-wavelength spot and to extract the fluorescence
from nanoscale emitters. The fabrication process is fast, single-shot, and
leads to a low surface roughness. The volumetric lithography provides fast
fabrication of microstructures of high optical quality in sub second single
laser shots over large areas, which can be also aligned with individual
emitters. This is an ideal approach to objective-free microscopy, especially
for cryogenic and vacuum conditions, and could become a powerful tool in
nanoscale quantum optics.
The presented design of the micro parabolic mirror covers 2$\pi$ solid angle
around an emitter in the focal point, which limits the focal spot size as well
as the photon extraction efficiency. The mirror directs photons out of the
sample plane, while more sophisticated designs could be exploited for
applications where photons need to be guided in plane. The in-situ volumetric
lithography allows for fabrication of structures beyond the parabolic shape at
the microscale. More complex spatial intensity distribution could be achieved
by using wavefront-shaping techniques, such as spatial light modulators (SLMs)
or digital micromirror devices (DMDs) Jenness _et al._ (2010); Tian and Wang
(2020).
## Supporting Information
See the Supporting Information for the effect of refractive index mismatch
between parabolic mirror filling and glass substrate; scaling of the focal
spot size with the refractive index of parabolic dish filling; evolution of
the focal spot dimensions with further decreasing of parabolic mirror focal
length; polarization of the parabolic mirror focal spot; simulation of
experimental intensity distribution in the focal point; interference rings in
the focal plane of parabolic mirror; measurements and simulations of radiation
pattern.
###### Acknowledgements.
S.M. and R.S. acknowledge funding by EPSRC (EP/P033369 and EP/M013812/1).
A.H.K. and I.M. acknowledge funding from the European Research Council (ERC)
under the European Unions Horizon 2020 research and innovation program (714876
PHOCONA).
## Data availability
The data that support the findings of this study are openly available in
Figshare at http://doi.org/10.6084/m9.figshare.12357356.
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|
2024-09-04T02:54:57.793390 | 2020-03-06T10:32:03 | 2003.03125 | {
"authors": "Radha Kopparti, Tillman Weyde",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26077",
"submitter": "Radha Kopparti",
"url": "https://arxiv.org/abs/2003.03125"
} | arxiv-papers | # Bayesian Weight Priors for Learning Identity Relations
Radha Kopparti
Department of Computer Science
City, University of London
London, United Kingdom
<EMAIL_ADDRESS>
&Tillman Weyde
Department of Computer Science
City, University of London
London, United Kigndom
<EMAIL_ADDRESS>
# Weight Priors for Learning Identity Relations
Radha Kopparti
Department of Computer Science
City, University of London
London, United Kingdom
<EMAIL_ADDRESS>
&Tillman Weyde
Department of Computer Science
City, University of London
London, United Kigndom
<EMAIL_ADDRESS>
###### Abstract
Learning abstract and systematic relations has been an open issue in neural
network learning for over 30 years. It has been shown recently that neural
networks do not learn relations based on identity and are unable to generalize
well to unseen data. The Relation Based Pattern (RBP) approach has been
proposed as a solution for this problem. In this work, we extend RBP by
realizing it as a Bayesian prior on network weights to model the identity
relations. This weight prior leads to a modified regularization term in
otherwise standard network learning. In our experiments, we show that the
Bayesian weight priors lead to perfect generalization when learning identity
based relations and do not impede general neural network learning. We believe
that the approach of creating an inductive bias with weight priors can be
extended easily to other forms of relations and will be beneficial for many
other learning tasks.
## 1 Introduction
Humans, including infants. are very effective at learning patterns of
relations based on identity from sensory input and systematically applying
them to new stimuli, even after a very brief exposure, while current neural
networks in their standard form fail to detect these relations in unseen data
[Marcus et al., 1999]. It has become evident that there are still relevant
limitations to systematic generalization in current neural network
architectures [Lake and Baroni, 2018, Marcus, 2018]. Neural networks are seen
to be good at memorizing the numerical patterns seen in the training set but
often not to extrapolate this representation outside the training set [Liska
et al., 2018]. It was found that standard neural networks do not seem to learn
identity relations, i.e. the equality of two vectors [Weyde and Kopparti,
2018], which are fundamental for many higher level tasks.
A well known study in this direction was conducted by Marcus et al. [1999],
where a recurrent neural network failed to distinguish abstract patterns,
based on equality or identity relations between the input stimuli, although
seven-month-old infants showed the ability to distinguish them after a few
minutes of exposure. This was followed by an lively exchange on rule learning
by neural networks and in human language acquisition in general, where results
by Elman [1999], Altmann and Dienes [1999] and Shultz and Bale [2001] could
not be reproduced by Vilcu and Hadley [2001, 2005], and Shultz and Bale [2006]
disputed claims by Vilcu and Hadley [2005]. Other approaches, such as those by
Shastri and Chang [1999] and Dominey and Ramus [2000], Alhama and Zuidema
[2016], use specialized network architectures or different problem
formulations or evaluation methods.
Recently, the Relation Based Patterns (RBP) approach has been introduced as a
way to create a suitable inductive bias for the problem of learning identity
relations on binary vectors Weyde and Kopparti [2018, 2019]. The task is to
classify whether two halves of the input vectors are equal. i.e.
$u_{i}=u_{n+i}\forall i\in\\{1,\dots,n\\}$ for an input vector with $2n$
dimensions. The RBP model introduced in Weyde and Kopparti [2018, 2019] is
based on the comparison of input neurons that correspond to each other in a
relation, e.g. the corresponding dimensions in a pair of binary vectors. For
the comparison they introduce Differentiator-Rectifier (DR) units, which
calculate the absolute difference of two inputs: $f(x,y)=|x-y|$. For each
dimension of the input vectors, a DR unit is introduced. This simplifies the
learning problem because the summation of activations of the DR units is
sufficient for generalisable identity detection. The results show no
restriction on the learning of other tasks in practice.
There are different ways of integrating DR units into neural networks in RBP:
Early and Mid Fusion. In _Early Fusion_ , DR units are concatenated to input
units, and in _Mid Fusion_ they are concatenated to the hidden layer. In both
cases, the existing input and hidden units are unchanged.
Adding units with hard-wired connections and unusual activation function
limits the flexibility of the RBP approach. In this work, we introduce a
modified RBP structure that can be formulated as a Bayesian prior to model the
weight structure of Mid Fusion RBP in a standard feed forward network setting.
## 2 A Bayesian Approach to Relation Based Patterns
In the weight prior approach, we replace each DR units with two standard
neurons and model the fixed weights with a default weight matrix $D$. This
matrix contains the weights that enable a dimension-wise comparison of the
inputs.
A diagram of the structure is given in Figure 1(a) where $\alpha$ and $\beta$
are the two input values being compared. A small example of this default
matrix is shown in Figure 1b). The incoming connections from two corresponding
inputs to a neuron in the hidden layer to be compared have values of 1 and
$-1$. For the same pair we use another hidden neuron with inverted signs of
the weights, as in rows 1 and 2. With a ReLU activation function, this means
that the value of one of the hidden neurons will be positive, if one input
neurons have different activations. We therefore need at least $n$ hidden
units in the first hidden layer for a comparison of n two $n/2$-dimensional
vectors. All other incoming connection weights are set to 0, including the
bias. This ensures that in all cases where corresponding inputs are not equal,
there will be a positive activation in one of the neurons.
a) | b) $\displaystyle D={\text{\boldmath$\begin{pmatrix}+1&0&0&-1&0&0\\\ -1&0&0&+1&0&0\\\ 0&+1&0&0&-1&0\\\ 0&-1&0&0&+1&0\\\ 0&0&+1&0&0&-1\\\ 0&0&-1&0&0&+1\\\ 0&0&0&0&0&0\\\ &\vdots&&&\vdots&\\\ \end{pmatrix}$}}$
---|---
Figure 1: a) For each input dimension there are two hidden layer neurons, e.g.
$h_{1},h_{2}$ for dimension $1$. $\alpha$ and $\beta$ indicate two input
vectors of dimensionality $n$. b) The default weight matrix $D$ for vector dim
n=3. Each row corresponds to the incoming weights of a hidden neuron, e.g. the
first row to $h_{1}$ in figure 1 a). If there are more hidden neurons than
pairs of input vectors, the additional rows contain only zeros.
The matrix $D$ is then used to define default values of the network weights,
i.e. we impose a loss based on the difference between $D$ and the actual
weight matrix $W$, that connects the input neurons to the first hidden layer.
This loss term $l_{RBP_{1/2}}$ for L1 or L2 loss are defined as:
$l_{RBP_{1}}=\sum_{i=1}^{k}|W_{k}-D_{k}|,\hskip
28.45274ptl_{RBP_{2}}=\sum_{i=1}^{k}(W_{k}-D_{k})^{2},$ (1)
where $k$ is the number of elements in $W$. These loss functions correspond to
Bayesian priors on the weights with the mean defined by the values of $D$. The
$L2$ loss corresponds to a Gaussian and $L1$ loss to a Laplacian prior, such
that backpropagation maximizes the posterior likelihood of the weights given
the data Williams [1995]. The overall training loss $l_{t}$ is defined as
$l_{t}=l_{c}+\lambda\times l_{RBP}$ (2)
where $l_{c}$ is the cross entropy and $\lambda$ is the regularization
parameter, corresponding to the inverse of the variance of the prior,
effectively regulating the strength of the RBP regularization. We call these
methods of embedding the RBP into a standard network ERBP L1 and ERBP L2
respectively.
## 3 Generalizing and Learning of Identity Relations
For the task of learning identity relations, we generate synthetic data and
use a standard feed-forward neural network. The input vector is binary and the
target values are $[0,1]$ for unequal and $[1,0]$ for equal vector halves as
described in section 1.
We use a grid search over hyper-parameters: the number of epochs [10,20,30],
the number of neurons per hidden layer was varied as [10,20,30]. For the
Bayesian weight prior, we varied the regularization parameter $\lambda$ with
values [0.01,0.03,0.1,0.3,1,3,10,30]. We ran a total of 10 simulations using
the SGD and Adam optimizers, training for 20 epochs. We used a single hidden
layer and a batch size of 1 unless indicated otherwise. The networks have been
implemented using the PyTorch library111http://pytorch.org.
The train/test split was set to 75% / 25% for all tasks. We tested with vector
dimensionalities $n=3,10,30$. We generate all vectors with equal halves and
take a random sample of all those with unequal halves to balance the classes.
We downsample the size of the dataset to 1000 when it is greater.
### 3.1 Identity Relations
Identity is an abstract relation in the sense that it is independent of the
actual values of the individual arguments, it just depends on their combined
configuration. In this task, pairs of vectors are presented to a feed-forward
network and the task is to distinguish whether the two vectors are equal or
not. We evaluated the performance of the network in different configurations
on a held-out test set and the results are tabulated in Table 1 for different
vector dimensions. As observed by Weyde and Kopparti [2018], standard networks
do not improve much over random guessing, while ERBP L1 and ERBP L2, as well
as Mid Fusion, almost always achieve perfect generalizsation with sufficiently
strong $\lambda$ (see next section for details).
Table 1: Test set classification accuracy (in %) and standard deviation over 10 simulations (in brackets) using different models for Identity Learning (vector dimensions $n=3,10,30$). The networks were trained with the Adam optimizer for 20 epochs. Type | Standard | Early Fusion | Mid Fusion | ERBP L1 | ERBP L2
---|---|---|---|---|---
$n=3$ | 55 (1.91) | 65 (1.34) | 100 (0.04) | 100 (0.00) | 100 (0.00)
$n=10$ | 51 (1.67) | 65 (1.32) | 100 (0.08) | 100 (0.04) | 100 (0.02)
$n=30$ | 50 (1.52) | 65 (1.27) | 100 (0.07) | 100 (0.05) | 100 (0.04)
### 3.2 Parameter Variations
We study the effect of several parameters: number of hidden layers, choice of
optimizer, regularization factor and weight initialization on identity
relation learning tasks.
Network Depth: We tested identity learning with deeper neural network models,
using $h={2,3,4,5}$ hidden layers. The results are tabulated in Table 2,
showing only minor improvements in the network performance for deeper
networks. However, ERBP L1 and L2 generalization is consistent and independent
of the network depth.
Table 2: Test set classification accuracy (in %) with standard deviation (in brackets) for identity learning ($n=3$) using deeper networks. The networks were trained with the Adam optimizer for 20 epochs. Hidden | No | Early | Mid | ERBP L1 | ERBP L2
---|---|---|---|---|---
layers | RBP | Fusion | Fusion | |
h = 2 | 55 (1.65) | 65 (1.26) | 100 (0.02) | 100 (0.00) | 100 (0.00)
h = 3 | 55 (1.67) | 67 (1.14) | 100 (0.03) | 100 (0.00) | 100 (0.00)
h = 4 | 58 (1.63) | 68 (1.25) | 100 (0.02) | 100 (0.00) | 100 (0.00)
h = 5 | 59 (1.68) | 72 (1.23) | 100 (0.02) | 100 (0.00) | 100 (0.00)
Table 3: Accuracy (in %) and standard deviation of identity learning ($n=3$) using Adam and SGD optimizer for ERBP L1 and L2. SGD can also lead to 100% accuracy on the test set, but needs higher $\lambda$ values. Type | ERBP L1 | ERBP L2
---|---|---
Adam | 100 (0.00) | 100 (0.00)
SGD | 98 (0.06) | 96 (0.05)
Optimiser: We used both Stochastic Gradient Descent (SGD) and the Adam
optimizer [Diederik P and Jimmy Lei, 2014] for training the ERBP L1 and L2. We
observed faster convergence and greater improvement in the overall accuracy
with the Adam compared to SGD. We observed similar results for both ERBP L1
and L2. Table 3 below summarizes the results of identity learning for both the
optimizers with the regularization parameter $\lambda$ set to 1. We observe
that the SGD does not reach full generalization in this setting, however it
does so at higher values of $\lambda$.
Regularization Factor: We varied the regularization factor $\lambda$ in the
loss function of the ERBP models. We observed that a factor of 3 or above
reliably leads to perfect generalization in identity learning task. Figure 2,
shows how the effect depends on the size the regularization factor $\lambda$
using L1 and L2 loss functions for learning identity relations.
Figure 2: Test set classification accuracy (in %) of the network with ERBP L1
and L2 when varying the regularization parameter $\lambda$ (shown in
logarithmic scale) for identity learning with ($n=3$) and using the Adam
optimizer.
### 3.3 Jointly learning identity relations and bit patterns
In order to test, whether the RBP and ERBP impedes other learning tasks, we
also tested learning with some non-relational patterns that are based on
values of specific neurons. The simplest form is that we have one bit that
determines the target class. This case is learned with 100% generalization
performance for a network with 2 additional outputs for the pattern
classification.
Furthermore, we tested the learning of classification based on all even or odd
elements in the vector being $0$. In this case, we also get perfect
generalization for $\lambda$ up to 10. For $\lambda=30$ we see first
deterioration of the accuracy, but there is a wide range of values where both
the identity relations and the even/odd classification generalize perfectly.
## 4 Conclusions
Identity based relations are a fundamental form of relational learning. In
this work, we re-visit the problem of learning identity relations using
standard neural networks and show that creating a weight prior on the network
weights leads to generalisable solutions of learning identity based relations.
This also did not affect learning of non-relational patterns in our
preliminary experiments, although more thorough testing remains to be done
here. We believe that addressing these issues and coming up with effective
solutions is necessary for more higher level relational learning tasks and
also is relevant to address problems in general neural network learning. In
future, we would like to extend this work towards learning other complex
relational learning tasks and come with effective ways of creating an
inductive bias using weight priors within the standard neural network
architectures.
## References
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* Marcus [2018] Gary Marcus. Deep learning : a critical appraisal. _arXiv:1801.00631_ , 2018.
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* Weyde and Kopparti [2018] Tillman Weyde and Radha Kopparti. Feed-forward neural networks need inductive bias to learn equality relations. _https://arxiv.org/abs/1812.01662_ , 2018.
* Elman [1999] Jeffrey Elman. Generalization, rules, and neural networks: A simulation of marcus et. al. Technical report, 1999. URL {https://crl.ucsd.edu/~elman/Papers/MVRVsimulation.html}.
* Altmann and Dienes [1999] Gerry Altmann and Zoltan Dienes. Technical comment on rule learning by seven-month-old infants and neural networks. _In Science_ , 284(5416):875–875, 1999.
* Shultz and Bale [2001] Thomas R. Shultz and Alan C. Bale. Neural network simulation of infant familiarization to artificial sentences: Rule-like behavior without explicit rules and variables. _Infancy_ , 2(4):501–536, 2001.
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* Vilcu and Hadley [2005] Marius Vilcu and Robert F Hadley. Two apparent ‘counterexamples’ to marcus: A closer look. _Minds and Machines_ , 15(3-4):359–382, 2005\.
* Shultz and Bale [2006] Thomas R Shultz and Alan C Bale. Neural networks discover a near-identity relation to distinguish simple syntactic forms. _Minds and Machines_ , 16(2):107–139, 2006.
* Shastri and Chang [1999] Shastri and Chang. A spatiotemporal connectionist model of algebraic rule-learning. _International Computer Science Institute_ , pages TR–99–011, 1999\.
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* Alhama and Zuidema [2016] Raquel G. Alhama and Willem Zuidema. Pre-wiring and pre-training: What does a neural network need to learn truly general identity rules. _CoCo at NIPS_ , 2016.
* Weyde and Kopparti [2019] Tillman Weyde and Radha Kopparti. Modelling identity rules with neural networks. _Journal or Applied Logics_ , 6(4):745–769, 2019\.
* Williams [1995] Peter M Williams. Bayesian regularization and pruning using a laplace prior. _Neural computation_ , 7(1):117–143, 1995.
* Diederik P and Jimmy Lei [2014] Kingma Diederik P and Ba Jimmy Lei. Adam : A method for stochastic optimization. 2014\. URL https://arxiv.org/abs/1412.6980.
|
2024-09-04T02:54:57.801137 | 2020-03-06T10:54:46 | 2003.03130 | {
"authors": "Sebastian J. M\\\"uller (1), Franziska Weigl (2), Carina Bezold (1), Ana\n Sancho (2,3), Christian B\\\"acher (1), Krystyna Albrecht (2) and Stephan Gekle\n (1) ((1) Theoretical Physics VI, Biofluid Simulation and Modeling, University\n of Bayreuth, (2) Department of Functional Materials in Medicine and Dentistry\n and Bavarian Polymer Institute (BPI), University of W\\\"urzburg, (3)\n Department of Automatic Control and Systems Engineering, University of the\n Basque Country UPV/EHU, San Sebastian, Spain)",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26078",
"submitter": "Sebastian Johannes M\\\"uller",
"url": "https://arxiv.org/abs/2003.03130"
} | arxiv-papers | Published 2020 in: Biomechanics and Modeling in Mechanobiology
DOI: 10.1007/s10237-020-01397-2
Unfortunately, the journal version misses an important contributor.
The correct author list is the one given in this document.
A hyperelastic model for simulating cells in flow
Sebastian J. Müller 1, Franziska Weigl 2, Carina Bezold 1, Ana Sancho 2,3,
Christian Bächer 1, Krystyna Albrecht 2 and Stephan Gekle 1
1 Theoretical Physics VI, Biofluid Simulation and Modeling, University of
Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany
2 Department of Functional Materials in Medicine and Dentistry and Bavarian
Polymer Institute (BPI), University of Würzburg, Pleicherwall 2, 97070
Würzburg, Germany
3 Department of Automatic Control and Systems Engineering, University of the
Basque Country UPV/EHU, San Sebastian, Spain
###### Abstract
In the emerging field of 3D bioprinting, cell damage due to large deformations
is considered a main cause for cell death and loss of functionality inside the
printed construct. Those deformations, in turn, strongly depend on the
mechano-elastic response of the cell to the hydrodynamic stresses experienced
during printing. In this work, we present a numerical model to simulate the
deformation of biological cells in arbitrary three-dimensional flows. We
consider cells as an elastic continuum according to the hyperelastic
Mooney–Rivlin model. We then employ force calculations on a tetrahedralized
volume mesh.
To calibrate our model, we perform a series of FluidFM® compression
experiments with REF52 cells demonstrating that all three parameters of the
Mooney–Rivlin model are required for a good description of the experimental
data at very large deformations up to $80\text{\,}\mathrm{\char 37\relax}$. In
addition, we validate the model by comparing to previous AFM experiments on
bovine endothelial cells and artificial hydrogel particles. To investigate
cell deformation in flow, we incorporate our model into Lattice Boltzmann
simulations via an Immersed-Boundary algorithm. In linear shear flows, our
model shows excellent agreement with analytical calculations and previous
simulation data.
Keywords: Hyperelasticity, Cell deformation, Mooney–Rivlin, Atomic force
Microscopy, Shear flow, Lattice-Boltzmann
## 1 Introduction
The dynamic behavior of flowing cells is central to the functioning of
organisms and forms the base for a variety of biomedical applications.
Technological systems that make use of the elastic behavior of cells are, for
example, cell sorting [1], real-time deformability cytometry [2, 3] or probing
techniques for cytoskeletal mechanics [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15]. In most, but not all, of these applications cell deformations typically
remain rather small. A specific example where large deformations become
important is 3D bioprinting. Bioprinting is a technology which, analogously to
common 3D printing, pushes a suspension of cells in highly viscous hydrogels—a
so-called bioink—through a fine nozzle to create three-dimensional tissue
structures. A major challenge in this process lies in the control of large
cell deformations and cell damage during printing. Those deformations arise
from hydrodynamic stresses in the printer nozzle and ultimately affect the
viability and functionality of the cells in the printed construct [16, 17, 18,
19, 20]. How exactly these hydrodynamic forces correlate with cell
deformation, however, strongly depends on the elastic behavior of the cell and
its interaction with the flowing liquid. Theoretical and computational
modeling efforts in this area have thus far been restricted to pure fluid
simulations without actually incorporating the cells [21, 22, 17] or simple 2D
geometries [23, 24]. The complexity of cell mechanics and the diversity of
possible applications make theoretical modeling of cell mechanics in flow a
challenge which, to start with, requires reliable experimental data for large
cell deformations.
The most appropriate tool to measure cellular response at large deformations
is atomic force microscopy (AFM) [25, 26, 27, 28, 8, 29, 30, 31, 32, 33, 34].
AFM cantilevers with pyramidal tips, colloidal probes, or flat geometries are
used to indent or compress cells. Therefore, a common approach to characterize
the elasticity of cells utilizes the Hertzian theory, which describes the
contact between two linear elastic solids [35, p. 90-104], but is limited to
the range of small deformations [36]. Experimental measurements with medium-
to-large deformations typically show significant deviations from the Hertz
prediction, e. g., for cells or hydrogel particles [37]. Instead of linear
elasticity, a suitable description of cell mechanics for bioprinting
applications requires more advanced hyperelastic material properties. While
for simple anucleate fluid-filled cells such as, e.g., red blood cells,
theoretical models abound [38, 39, 40, 41, 42], the availability of models for
cells including a complex cytoskeleton is rather limited. In axisymmetric
geometries, Caille et al. [43] and Mokbel et al. [44] used an axisymmetric
finite element model with neo-Hookean hyperelasticity to model AFM and
microchannel experiments on biological cells. In shear flow, approximate
analytical treatments are possible [45, 46, 47, 48]. Computationally, Gao and
Hu [46] carried out 2D simulations while in 3D Lykov et al. [49] utilized a
DPD technique based on a bead-spring model. Furthermore, Villone et al. [50,
51] presented an arbitrary Lagrangian-Eulerian approach for elastic particles
in viscoelastic fluids. Finally, Rosti et al. [52] and Saadat et al. [53]
considered viscoelastic and neo-Hookean finite element models, respectively,
in shear flow.
In this work, we introduce and calibrate a computational model for fully
three-dimensional simulations of cells in arbitrary flows. Our approach uses a
Lattice-Boltzmann solver for the fluid and a direct force formulation for the
elastic equations. In contrast to earlier works [43, 47, 44, 52, 53] our model
uses a three-parameter Mooney–Rivlin elastic energy functional. To demonstrate
the need for this more complex elastic model, we carry out extensive FluidFM®
indentation experiments for REF52 (rat embryonic fibroblast) cells at large
cell deformation up to $80\text{\,}\mathrm{\char 37\relax}$ [54]. In addition,
our model compares favorably with previous AFM experiments on bovine
endothelial cells [43] as well as artificial hydrogel particles [37]. Our
model provides a much more realistic force–deformation behavior compared to
the small-deformation Hertz approximation, but is still simple and fast enough
to allow the simulation of dense cell suspensions in reasonable time.
Particularly, our approach is less computationally demanding than conventional
finite-element methods which usually require large matrix operations.
Furthermore, it is easily extensible and allows, e.g., the inclusion of a cell
nucleus by the choice of different elastic moduli for different parts of the
volume.
We finally present simulations of our cell model in different flow scenarios
using an Immersed-Boundary algorithm to couple our model with Lattice
Boltzmann fluid calculations. In a plane Couette (linear shear) flow, we
investigate the shear stress dependency of single cell deformation, which we
compare to the average cell deformation in suspensions with higher volume
fractions, and show that our results in the neo-Hookean limit are in
accordance with earlier elastic cell models [47, 52, 53].
## 2 Theory
In general, hyperelastic models are used to describe materials that respond
elastically to large deformations [55, p. 93]. Many cell types can be
subjected to large reversible shape changes. This section provides a brief
overview of the hyperelastic Mooney–Rivlin model implemented in this work.
The displacement of a point is given by
$\displaystyle u_{i}=y_{i}-x_{i}\>,$ (1)
where $x_{i}$ ($i=1,2,3$) refers to the undeformed configuration (material
frame) and $y_{i}$ to the deformed coordinates (spatial frame). We define the
deformation gradient tensor and its inverse as [55, p. 14,18]
$\displaystyle\mathsf{F}_{ij}=\frac{\partial{y_{i}}}{\partial{x_{j}}}=\frac{\partial{u_{i}}}{\partial{x_{j}}}+\delta_{ij}\quad\mathrm{and}\quad\mathsf{F}_{ij}^{-1}=\frac{\partial{x_{i}}}{\partial{y_{j}}}\>.$
(2)
Together with the right Cauchy-Green deformation tensor,
$\mathsf{C}=\mathsf{F}^{\intercal}\mathsf{F}$ (material description), we can
define the following invariants which are needed for the strain energy density
calculation below:
$\displaystyle J$ $\displaystyle=\det\mathsf{F}$ (3) $\displaystyle I$
$\displaystyle=T_{\mathsf{C}}J^{-2/3}$ (4) $\displaystyle K$
$\displaystyle=\tfrac{1}{2}\left(T_{\mathsf{C}}^{2}-T_{\mathsf{C}^{2}}\right)J^{-4/3}$
(5)
Here,
$\displaystyle T_{\mathsf{C}}=\mathrm{tr}\,\mathsf{C}\quad\mathrm{and}\quad
T_{\mathsf{C}^{2}}=\mathrm{tr}\,\left(\mathsf{C}^{2}\right)$ (6)
are the trace of the right Cauchy-Green deformation tensor and its square,
respectively. The nonlinear strain energy density of the Mooney–Rivlin model
is given by [56, 57]
$\displaystyle
U=\left[\frac{\mu_{1}}{2}\left(I-3\right)+\frac{\mu_{2}}{2}\left(K-3\right)+\frac{\kappa}{2}\left(J-1\right)^{2}\right]\>,$
(7)
where $\mu_{1}$, $\mu_{2}$, and $\kappa$ are material properties. They
correspond—for consistency with linear elasticity in the range of small
deformations—to the shear modulus $\mu=\mu_{1}+\mu_{2}$ and bulk modulus
$\kappa$ of the material and are therefore related to the Young’s modulus $E$
and the Poisson ratio $\nu$ via [55, p. 74]
$\displaystyle\mu=\frac{E}{2\left(1+\nu\right)}\quad\mathrm{and}\quad\kappa=\frac{E}{3\left(1-2\nu\right)}\>.$
(8)
Through the choice $\mu_{2}=0$ in (7), we recover the simpler and frequently
used [47, 53] neo-Hookean strain energy density:
$\displaystyle
U_{\mathrm{NH}}=\left[\frac{\mu}{2}\left(I-3\right)+\frac{\kappa}{2}\left(J-1\right)^{2}\right]$
(9)
As we show later, this can be a sufficient description for some cell types. To
control the strength of the second term and quickly switch between neo-Hookean
and Mooney–Rivlin strain energy density calculation, we introduce a factor
$w\in\left[0,1\right]$ and set
$\displaystyle\mu_{1}=w\mu\quad\mathrm{and}\quad\mu_{2}=(1-w)\mu\>$ (10)
such that $w=1$, which equals setting $\mu_{2}=0$ in (7), corresponds to the
purely neo-Hookean description in (9), while $w<1$ increases the influence of
the $\mu_{2}$-term and thus leads to a more pronounced strain hardening as
shown in figure S-6 of the Supporting Information.
## 3 Tetrahedralized cell model
In this section we apply the hyperelastic theory of section 2 to a
tetrahedralized mesh as shown in figure 1.
### 3.1 Calculation of elastic forces
We consider a mesh consisting of tetrahedral elements as depicted in figure 1.
The superscript $\alpha$ refers to the four vertices of the tetrahedron. The
elastic force acting on vertex $\alpha$ in direction $i$ is obtained from (7)
by differentiating the strain energy density $U$ with respect to the vertex
displacement as
$\displaystyle f_{i}^{\alpha}=$ $\displaystyle-
V_{0}\frac{\partial{U}}{\partial{u_{i}^{\alpha}}}\>,$ (11)
where $V_{0}$ is the reference volume of the tetrahedron. In contrast to
Saadat et al. [53], the numerical calculation of the force in our model does
not rely on the integration of the stress tensor, but on a differentiation
where the calculation of all resulting terms involves only simple arithmetics.
Applying the chain rule for differentiation yields:
$\displaystyle f_{i}^{\alpha}=-V_{0}\Bigg{[}$
$\displaystyle\left(\frac{\partial{U}}{\partial{I}}\frac{\partial{I}}{\partial{T_{\mathsf{C}}}}+\frac{\partial{U}}{\partial{K}}\frac{\partial{K}}{\partial{T_{\mathsf{C}}}}\right)\frac{\partial{T_{\mathsf{C}}}}{\partial{\mathsf{F}_{kl}}}$
$\displaystyle+\left(\frac{\partial{U}}{\partial{I}}\frac{\partial{I}}{\partial{J}}+\frac{\partial{U}}{\partial{K}}\frac{\partial{K}}{\partial{J}}+\frac{\partial{U}}{\partial{J}}\right)\frac{\partial{J}}{\partial{\mathsf{F}_{kl}}}$
$\displaystyle+\frac{\partial{U}}{\partial{K}}\frac{\partial{K}}{\partial{T_{\mathsf{C}^{2}}}}\frac{\partial{T_{\mathsf{C}^{2}}}}{\partial{\mathsf{F}_{kl}}}\Bigg{]}\frac{\partial{\mathsf{F}_{kl}}}{\partial{u_{i}^{\alpha}}}$
(12)
The evaluation of (3.1) requires the calculation of the deformation gradient
tensor $\mathsf{F}$, which is achieved by linear interpolation of the
coordinates and displacements inside each tetrahedral mesh element as detailed
in the next section. We note that our elastic force calculation is purely
local making it straightforward to employ different elastic models in
different regions of the cell and/or to combine it with elastic shell models.
This flexibility can be used to describe, e.g., the cell nucleus [43] or an
actin cortex [58] surrounding the cell interior.
### 3.2 Interpolation of the displacement field
Following standard methods, e.g. Bower [55], we start by interpolating a point
$x_{i}$ inside a single tetrahedron using the vertex positions
$x_{i}^{\alpha}$ ($\alpha=1,2,3,4$). The interpolation uses an inscribed,
dimensionless coordinate system, denoted by
$\left(\xi_{1},\xi_{2},\xi_{3}\right)$ with $0\leq\xi_{i}\leq 1$111Bower [55,
p. 481,483] erroneously states a range of $-1\leq\xi_{i}\leq 1$ for the
tetrahedral element., as depicted in figure 1a. One vertex defines the origin
while the remaining three indicate the coordinate axes. A set of shape
functions, i. e., interpolation functions,
$N^{\alpha}\left(\xi_{1},\xi_{2},\xi_{3}\right)$ is employed to interpolate
positions inside the tetrahedron volume. An arbitrary point $x_{i}$ inside the
element is interpolated as
$\displaystyle
x_{i}=\sum\limits_{\alpha=1}^{4}N^{\alpha}\left(\xi_{1},\xi_{2},\xi_{3}\right)x_{i}^{\alpha}\>,$
(13)
where the shape functions are defined as [55, p. 483]:
$\displaystyle N^{1}\left(\xi_{1},\xi_{2},\xi_{3}\right)$
$\displaystyle=\xi_{1}$ (14) $\displaystyle
N^{2}\left(\xi_{1},\xi_{2},\xi_{3}\right)$ $\displaystyle=\xi_{2}$ (15)
$\displaystyle N^{3}\left(\xi_{1},\xi_{2},\xi_{3}\right)$
$\displaystyle=\xi_{3}$ (16) $\displaystyle
N^{4}\left(\xi_{1},\xi_{2},\xi_{3}\right)$
$\displaystyle=1-\xi_{1}-\xi_{2}-\xi_{3}$ (17)
According to (1), the displacement of vertex $\alpha$ in $i$-direction is
given by
$\displaystyle u_{i}^{\alpha}=y_{i}^{\alpha}-x_{i}^{\alpha}\>.$ (18)
Therefore similar to (13), the displacement at an arbitrary point in the
volume can also be expressed in terms of the shape functions and the vertex
displacements as
$\displaystyle
u_{i}=\sum\limits_{\alpha=1}^{4}N^{\alpha}\left(\xi_{1},\xi_{2},\xi_{3}\right)u_{i}^{\alpha}\>.$
(19)
The calculation of the deformation gradient tensor according to (2) requires
the spatial derivative of the displacement:
$\displaystyle\mathsf{F}_{ij}-\delta_{ij}=\frac{\partial{u_{i}}}{\partial{x_{j}}}$
$\displaystyle=\frac{\partial{u_{i}}}{\partial{\xi_{k}}}\frac{\partial{\xi_{k}}}{\partial{x_{j}}}=\mathsf{A}_{ik}\mathsf{B}_{kj}$
(20)
By inserting (19) into (20) and evaluating the shape functions, the components
of the matrix $\mathsf{A}$ are easily determined to be the difference of the
displacements between the origin (vertex 4) and the remaining vertices 1, 2
and 3:
$\displaystyle\mathsf{A}_{ik}=u_{i}^{k}-u_{i}^{4}$ (21)
Note that due to the linear interpolation $\mathsf{A}_{ik}$ is constant inside
a given tetrahedron. The matrix $\mathsf{B}=\mathsf{\mathsf{J}}^{-1}$ is the
inverse of the Jacobian matrix, obtained similarly to (21) as
$\displaystyle\mathsf{J}_{ik}$
$\displaystyle=\frac{\partial{x_{i}}}{\partial{\xi_{k}}}=x_{i}^{k}-x_{i}^{4}\>.$
(22)
Since $x_{i}$ refers to the reference coordinates, the calculation of the
matrices $\mathsf{\mathsf{J}}$ and $\mathsf{B}$ has to be performed only once
at the beginning of a simulation. With the interpolation of the displacement
in each tetrahedron, we can write all derivatives occurring in (3.1), as
listed in the following:
$\frac{\partial{U}}{\partial{I}}=\frac{\mu_{1}}{2}$
$\frac{\partial{I}}{\partial{T_{\mathsf{C}}}}=J^{-\frac{2}{3}}$
$\frac{\partial{U}}{\partial{K}}=\frac{\mu_{2}}{2}$
$\frac{\partial{K}}{\partial{T_{\mathsf{C}}}}=T_{\mathsf{C}}J^{-\frac{4}{3}}$
$\frac{\partial{T_{\mathsf{C}}}}{\partial{\mathsf{F}_{il}}}=2\mathsf{F}_{il}$
$\frac{\partial{I}}{\partial{J}}=-\frac{2}{3}T_{\mathsf{C}}J^{-\frac{5}{3}}$
$\frac{\partial{K}}{\partial{J}}=-\frac{2}{3}\left(T_{\mathsf{C}}^{2}-T_{\mathsf{C}^{2}}\right)J^{-\frac{7}{3}}$
$\frac{\partial{U}}{\partial{J}}=\kappa\left(J-1\right)$
$\frac{\partial{J}}{\partial{\mathsf{F}_{il}}}=J\mathsf{F}_{li}^{-1}$
$\frac{\partial{K}}{\partial{T_{\mathsf{C}^{2}}}}=-\tfrac{1}{2}J^{-\frac{4}{3}}$
$\frac{\partial{T_{\mathsf{C}^{2}}}}{\partial{\mathsf{F}_{il}}}=4\mathsf{F}_{ik}\mathsf{C}_{kl}$
$\frac{\partial{\mathsf{F}_{kl}}}{\partial{u_{i}^{\alpha}}}=\delta_{ki}\mathsf{B}_{ml}\left(\delta_{m\alpha}-\delta_{4\alpha}\right)$
(a) (b) (c)
Figure 1: (a) The four noded tetrahedron as mesh element within a local
dimensionless coordinate system $\left\\{\xi_{1},\xi_{2},\xi_{3}\right\\}$.
(b) The spherical cell model with its triangulated surface and (c) its inner
tetrahedralized mesh
### 3.3 Taylor deformation parameter
As a measure for the cell deformation, we use the Taylor deformation parameter
[59, 60, 61, 53]
$\displaystyle D=\frac{a_{3}-a_{1}}{a_{3}+a_{1}}\>,$ (23)
where $a_{1}$ and $a_{3}$ are respectively the minor and major semi axis of an
ellipsoid corresponding to the inertia tensor of the cell. The Taylor
deformation is a good measure for approximately elliptic cell deformations, as
they occur in shear flow (cf. section 6).
To calculate $D$, first the components of the inertia tensor
$\displaystyle\mathsf{\Theta}_{ij}=\int\limits_{V}x_{k}x_{k}\delta_{ij}-x_{i}x_{j}{\mathrm{d}{V}}\>,$
(24)
where $\vec{x}$ is a vector inside the volume $V$, are calculated using our
discretized cell with $N_{\mathrm{tet}}$ tetrahedra as
$\displaystyle\mathsf{\Theta}_{ij}=\sum\limits_{l=1}^{N_{\mathrm{tet}}}V_{l}\left(r_{k}^{l}r_{k}^{l}\delta_{ij}-r_{i}^{l}r_{j}^{l}\right)\>.$
(25)
The vector $\vec{r}^{\,l}$ denotes the center of mass of the $l^{\mathrm{th}}$
tetrahedron and $V_{l}$ is its current volume. The eigenvalues
$\theta_{1}>\theta_{2}>\theta_{3}$ of $\mathsf{\Theta}$ can be used to fit the
semi axes $a_{1}<a_{2}<a_{3}$ of the corresponding ellipsoid:
$\displaystyle a_{1}$
$\displaystyle=\frac{5}{2M}\left(-\theta_{1}+\theta_{2}+\theta_{3}\right)$
$\displaystyle a_{2}$
$\displaystyle=\frac{5}{2M}\left(\theta_{1}-\theta_{2}+\theta_{3}\right)$
$\displaystyle a_{3}$
$\displaystyle=\frac{5}{2M}\left(\theta_{1}+\theta_{2}-\theta_{3}\right)$ (26)
The prefactor contains the mass $M$ of the ellipsoid (considering uniform mass
density) and drops out in the calculation of $D$.
## 4 Comparison of the numerical model to FluidFM® measurements on REF52
cells
In this section, we validate compression simulations of our cell model with
FluidFM® compression experiments of REF52 cells stably expressing paxillin-YFP
[54]. These experiments provide as an output the required force to produce a
certain deformation of the cell, which can be directly compared to our model.
We start with a detailed description of the experiments and show the
suitability of our model to describe the elastic behavior of REF52 cells
afterwards.
### 4.1 FluidFM® indentation measurements
We perform a series of compression measurements of REF52 cells with a Flex FPM
(Nanosurf GmbH, Germany) system that combines the AFM with the FluidFM®
technology (Cytosurge AG, Switzerland). In contrast to conventional AFM
techniques, FluidFM® uses flat cantilevers that possess a microchannel
connected to a pressure system. By applying a suction pressure, cells can be
aspirated and retained at the aperture of the cantilever’s tip. A more
detailed description of the setup and its functionality is already reported in
[31]. All experiments are based on a cantilever with an aperture of
$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter and a nominal spring
constant of $2\text{\,}\mathrm{N}\text{\,}{\mathrm{m}}^{-1}$. In order to
measure the cellular deformation, a cell was sucked onto the tip and
compressed between the cantilever and the substrate until a setpoint of
$100\text{\,}\mathrm{nN}$ was reached. Immediately before the experiment, the
cells were detached by using Accutase (Sigma Aldrich) and were therefore in
suspension at the time of indentation. In this way, it can be ensured that
only a single cell is deformed during each measurement.
An example micrograph of the experiment before compression is shown in figure
2. Analogously to AFM, primary data in form of cantilever position (in
$\mathrm{m}$) and deflection (in $\mathrm{V}$) has to be converted to force
and deformation through the deflection sensitivity (in
$\mathrm{m}\text{\,}{\mathrm{V}}^{-1}$) and the cantilevers’ spring constant.
The cellular deformation further requires the determination of the contact
point, which we choose as the cantilever position where the measured force
starts to increase. The undeformed cell size is obtained as mean from a
horizontal and vertical diameter measurement using the software imageJ.
Figure 2: Example micrograph showing the FluidFM® cantilever and a cell viewed
from the top. Scale bar is $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$
### 4.2 Simulation setup
The experimental setup of the previous section is easily transferred and
implemented for our cell model: the undeformed spherical cell rests on a fixed
plate while a second plate approaches from above to compress the cell as
depicted in figure 3 (a and b). In section 5.2 below we will also use a
slightly modified version where a sphere indents the cell as shown in figure 3
(c and d). A repulsive force prevents the cell vertices from penetrating the
plates or the spherical indenter. The elastic restoring forces (cf. section 3)
acting against this imposed compression are transmitted throughout the whole
mesh, deforming the cell.
We use meshes consisting of $2000$ to $5000$ vertices and about $10\,000$ to
$30\,000$ tetrahedra to build up a spherical structure. More details of the
mesh and its generation (section S-2.4) as well as the algorithm (section S-3)
are provided in the SI.
(a) (b) (c) (d)
Figure 3: (a and b) Cell compression simulations: The cell is compressed
between a lower, resting, and an upper, moving, plate. (c and d) Colloidal
probe cell indentation simulations: The cell rests on a plate, while being
indented with a sphere
### 4.3 Results
In our FluidFM® experiment series with REF52 cells, the cell radii lie between
$7.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$10.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with an overall average of
$8.6(7)\text{\,}\mathrm{\SIUnitSymbolMicro m}$. In figure 4 we depict the
force as function of the non-dimensionalized deformation, i. e., the absolute
compression divided by the cell diameter. The experimental data curves share
general characteristics: The force increases slowly in the range of small
deformations up to roughly $40\text{\,}\mathrm{\char 37\relax}$, while a
rapidly increasing force is observed for larger deformations. Although the
variation of the cell radius in the different measurements is already taken
into account in the deformation, the point of the force upturn differs
significantly which indicates a certain variability in the elastic parameters
of the individual cells.
We use the compression simulation setup as detailed in section 4.2 to
calculate force–deformation curves of our cell model. The Poisson ratio is
chosen as $\nu=0.48$. In section S-2.7 of the Supporting Information we show
that variations of the $\nu$ do not strongly affect the results. A best fit
approach is used to determine the Young’s modulus and the ratio of shear
moduli $w$ and leads to very good agreement between model prediction and
experimental data as shown in figure 4 as well as section S-1 of the SI. While
the general range of force values is controlled using the Young’s modulus, the
Mooney–Rivlin ratio $w$ especially defines the point of the force upturn. We
find Young’s moduli in the range $110\text{\,}\mathrm{Pa}$ to
$160\text{\,}\mathrm{Pa}$ and $w=0.25$, $0.5$, and $1$. For very small
deformations our hyperelastic model produces the same results as would be
expected from a linear elastic model according to the Hertz theory. See the SI
(section S-2.5) for further details on the calculation of the
force–deformation according to the Hertzian theory. For large deformations,
the force rapidly increases due to its nonlinear character, showing strain-
hardening behavior and huge deviations from the Hertz theory. Overall, we find
an excellent match between simulation and our FluidFM® measurements with REF52
cells.
Figure 4: Our numerical model in comparison to our FluidFM® measurements on
REF52 cells. The labels give the two fit parameters $E$ and $w$. We find
Young’s moduli in the range of $110\text{\,}\mathrm{Pa}$ to
$160\text{\,}\mathrm{Pa}$. The Hertz theory is shown for a Young’s modulus of
$180\text{\,}\mathrm{Pa}$
## 5 Comparison of our numerical model to other micromechanical setups
In this section, we compare our simulations to axisymmetric calculations using
the commercial software Abaqus and validate our cell model with further
experimental data for bovine endothelial cells from [43] and very recent data
for hydrogel particles from [37].
### 5.1 Validation with axisymmetric simulations
To validate our model numerically, we compare our simulated force–deformation
curves to calculations using the commercial software Abaqus [62] (version
6.14).
In Abaqus, we use a rotationally symmetric setup consisting of a two-
dimensional semicircle, which is compressed between two planes, similar to our
simulation setup in section 4.2 and the finite element model utilized in [43].
The semicircle has a radius $r=$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, a
Young’s modulus of $E=$2.25\text{\,}\mathrm{kPa}$$ and a Poisson ratio of
$\nu=0.48$. We choose a triangular mesh and the built-in implementation of the
hyperelastic neo-Hookean model. In figure 5 we see very good agreement between
the results of the two different numerical methods.
Figure 5: Comparison of force–deformation curves obtained from our model (red
line) with the linear elastic Hertz theory (black line) and the two-
dimensional simulation with Abaqus (red squares), showing good agreement
between our three-dimensional and the axisymmetric model
### 5.2 Validation with AFM experiments
To compare with the AFM experiments of Caille et al. [43], we simulate a cell
with radius $15\text{\,}\mathrm{\SIUnitSymbolMicro m}$ using the setup of
section 4.2. For the hydrogel particle indentation [37] we use the setup
depicted in figure 3 (c and d) with a particle radius of
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a radius of the colloidal probe
of $26.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The Poisson ratio is chosen
as $0.48$ in all simulations and the Young’s modulus is determined using a
best fit to the experimental data points. Since the neo-Hookean description
appears to be sufficient for these data sets, we further set $w=1$.
In figure 6a, we show the experimental data for suspended, round, bovine
endothelial cells of five separate measurements from [43] together with the
prediction of the Hertz theory for a Young’s modulus of
$1000\text{\,}\mathrm{Pa}$. Fitting our data with Young’s moduli in the range
of $550\text{\,}\mathrm{Pa}$ to $2400\text{\,}\mathrm{Pa}$, we find good
agreement between our calculations and the experimental data. We note that
Caille et al. [43] observed similarly good agreement for their axisymmetric
incompressible neo-Hookean FEM simulations which, however, cannot be coupled
to external flows in contrast to the approach presented here. The same
procedure is applied to the colloidal probe indentation data of hydrogel
particles from [37], showing in figure 6b the experimental data and the
prediction of the Hertz theory from [37]. We find excellent agreement between
our model calculations for Young’s moduli in the range of
$$580$\pm$100\text{\,}\mathrm{Pa}$$ and the experimental data. For both
systems, figure 6 shows large deviations between the Hertzian theory and the
experimental data for medium-to-large deformations. Our model provides a
significant improvement in this range.
(a)
(b)
Figure 6: (a) Our numerical model in comparison to experimental measurements
of bovine endothelial cells from [43]. The black line depicts the prediction
of the Hertz theory for a Young’s modulus of $1000\text{\,}\mathrm{Pa}$. (b)
Our numerical model in comparison to experimental measurements of hydrogel
particles from [37]. The indicated range corresponds to the experimentally
found range of $\pm$100\text{\,}\mathrm{Pa}$$ for the Young’s modulus
according to the depicted Hertz model
## 6 Application in shear flow
We now apply our model to study the behavior of cells in a plane Couette
(linear shear) flow setup and compare the steady cell deformation to other
numerical and analytical cell models of Gao et al. [47], Rosti et al. [52] and
Saadat et al. [53]. A sketch of the simulation setup is shown in figure 7. For
simplicity, we choose $w=1$ to reduce the Mooney–Rivlin description (7) to two
free parameters $\mu$ and $\kappa$ (or $E$ and $\nu$), obtaining a
compressible neo-Hookean form. We use the Lattice Boltzmann implementation of
the open source software package ESPResSo [63, 64]. Coupling between fluid and
cell is achieved via the immersed-boundary algorithm [65, 53] which we
implemented into ESPResSo [66, 58]. We note here that, in contrast to Saadat
et al. [53], we do not subtract the fluid stress within the particle interior.
This leads to a small viscous response of the cell material in addition to its
elasticity. To obtain (approximately) the limit of a purely elastic particle,
we exploit a recently developed method by Lehmann et al. [67] to discriminate
between the cell interior and exterior during the simulation. Using this
technique, we can tune the ratio between inner and outer viscosity $\lambda$
with $\lambda\to 0$ representing a purely elastic particle. For simplicity, we
will nevertheless set $\lambda=1$ in the following, except where otherwise
noted. Details of the method are provided in the SI (section S-4.1). As
measure for the deformation, we investigate the Taylor parameter $D$ (23) of
our initially spherical cell model in shear flow at different shear rates
$\dot{\gamma}$.
### 6.1 Single cell simulation
The first simulation setup, a single cell in infinite shear flow, is realized
by choosing a simulation box of the dimensions $10\times 15\times 5$ ($x\times
y\times z$) in units of the cell radius. The infinite shear flow is
approximated by applying a tangential velocity $u_{\mathrm{wall}}$ on the
$x$-$z$-planes at $y=0$ in negative and at $y=15$ in positive $x$-direction,
as depicted in figure 7. The tangential wall velocity is calculated using the
distance $H$ of the parallel planes and the constant shear rate $\dot{\gamma}$
via
$\displaystyle u_{\mathrm{wall}}=\tfrac{1}{2}H\dot{\gamma}\>.$ (27)
The box is periodic in $x$ and $z$. A single cell is placed at the center of
the simulation box corresponding to a volume fraction of $\phi=0.0003$. We
choose the following parameters: fluid mass density
$\varrho=${10}^{3}\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$$, dynamic
viscosity $\eta=${10}^{-3}\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$$, and shear
rate $\dot{\gamma}=$4\text{\,}{\mathrm{s}}^{-1}$$. The capillary number is
defined by [46]
$\displaystyle\mathrm{Ca}=\frac{\eta\dot{\gamma}}{\mu}\>,$ (28)
and is used to set the shear modulus $\mu$ of our cell relative to the fluid
shear stress $\eta\dot{\gamma}$. Simulation snapshots of the steady state
deformation of a single cell in shear flow are depicted in dependency of the
capillary number in figure 8a. We compare the Taylor deformation parameter $D$
to previous approximate analytical calculations of Gao et al. [47] for a
three-dimensional elastic solid in infinite shear flow in figure 8b and see
reasonable agreement for our standard case of $\lambda=1$. Reducing the inner
viscosity by setting $\lambda=0.05$, i.e. close to the limit of a purely
elastic solid, the agreement is nearly perfect. Finally, we demonstrate that
the elastic particle exhibits a tank-treading motion in section S-4.2.
A possibly even more intuitive way to measure cell deformation is the net
strain of the cell which we define as
$\displaystyle\Delta\epsilon=\frac{\left(d_{\mathrm{max}}-d_{\mathrm{ref}}\right)}{d_{\mathrm{ref}}}\>.$
(29)
It describes the relative stretching of the cell using the maximum elongation
$d_{\mathrm{max}}$, i. e., the maximum distance of two cell vertices, and its
reference diameter $d_{\mathrm{ref}}=2R$. A strain of $\Delta\epsilon=$1$$
thus corresponds to an elongation of the cell by an additional
$100\text{\,}\mathrm{\char 37\relax}$ of its original size. In figure 8c, we
depict the $\Delta\epsilon$ as function of $\mathrm{Ca}$. For small capillary
numbers, i. e., small shear stresses, a linear stress-strain dependency is
observed. Above $\mathrm{Ca}\approx 0.3$, the strain-hardening, nonlinear
behavior of the neo-Hookean model can be seen. By stretching the cell up to
$280\text{\,}\mathrm{\char 37\relax}$ of its initial size, this plot
demonstrates again the capability of our model to smoothly treat large
deformations.
Figure 7: Schematic of the single cell in shear flow. The cell sits in the
center of the box and shows an approximately elliptic deformation as well as
tank-treading, i. e., a rotation of the membrane around the steady shape in
the $x$-$y$-plane
(a)
(b)
(c)
Figure 8: (a) Converged shapes of a single cell in a $10\times 15\times 5$
($x\times y\times z$) simulation box (in units of the cell radius) with a
shear flow in $x$-direction as function of the capillary number $\mathrm{Ca}$.
(b) Comparison of our model predictions for a single cell in shear flow to the
analytical 3D calculations in figure 7 of Gao et al. [47] in the range of
$\mathrm{Ca}\in\left[0.01,2.0\right]$. (c) The relative stretch
$\Delta\epsilon$ of our cell model as function of the capillary number
$\mathrm{Ca}$. A linear behavior is found for small capillary numbers up to
$\mathrm{Ca}=0.3$, while increasing stress is required for larger deformations
due to the strain-hardening quality of the neo-Hookean model. Lines are a
guide to the eye
### 6.2 Multiple cell simulations
The second simulation setup, implemented to investigate the multiple particle
aspect of our model, consists of $4$ ($8$) cells in a $5\times 8\times 4$
simulation box (in units of the cell radius), corresponding to a volume
fraction of $\phi=0.11$ ($\phi=0.22$) occupied by cells. The cells are
inserted at random initial positions in the box and the flow parameters are
the same as in the first setup (cf. section 6.1).
Figure 9a shows simulation snapshots of the cells in suspensions with volume
fraction $\phi=0.11$ and $\phi=0.22$ for $\mathrm{Ca}=0.2$. The Taylor
deformation of the suspensions, depicted in figure 9b, is calculated as an
average over all cells and over time after an initial transient timespan. We
find good agreement when comparing the averaged cell deformation in suspension
with Rosti et al. [52], Saadat et al. [53].
(a)
(b)
Figure 9: (a) Multiple cells in a $5\times 8\times 4$ ($x\times y\times z$)
simulation box (in units of the cell radius) with a confined shear flow in
$x$-direction for a capillary number of $\mathrm{Ca}=0.2$ and $4$ cells
corresponding to a volume fraction of $\phi=0.11$, and $8$ cells corresponding
to $\phi=0.22$. (b) Averaged deformation of multiple cell simulations with
$\phi=0.11$ and $\phi=0.22$ in comparison to data from figure 3 of Rosti et
al. [52] and figure 13 of Saadat et al. [53]
## 7 Conclusion
We presented a simple but accurate numerical model for cells and other
microscopic particles for the use in computational fluid-particle dynamics
simulations.
The elastic behavior of the cells is modeled by applying Mooney–Rivlin strain
energy calculations on a uniformly tetrahedralized spherical mesh. We
performed a series of FluidFM® compression experiments with REF52 cells as an
example for cells used in bioprinting processes and found excellent agreement
between our numerical model and the measurements if all three parameters of
the Mooney–Rivlin model are used. In addition, we showed that the model
compares very favorably to force versus deformation data from previous AFM
compression experiments on bovine endothelial cells [43] as well as colloidal
probe AFM indentation of artificial hydrogel particles [37]. At large
deformations, a clear improvement compared to Hertzian contact theory has been
observed.
By coupling our model to Lattice Boltzmann fluid calculations via the
Immersed-Boundary method, the cell deformation in linear shear flow as
function of the capillary number was found in good agreement with analytical
calculations by Gao et al. [47] on isolated cells as well as previous
simulations of neo-Hookean and viscoelastic solids [52, 53] at various volume
fractions.
The presented method together with the precise determination of model
parameters by FluidFM® /AFM experiments may provide an improved set of tools
to predict cell deformation - and ultimately cell viability - in strong
hydrodynamic flows as occurring, e.g., in bioprinting applications.
## Acknowledgements
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) — Project number 326998133 — TRR 225 “Biofabrication” (subproject
B07). We gratefully acknowledge computing time provided by the SuperMUC system
of the Leibniz Rechenzentrum, Garching. We further acknowledge support through
the computational resources provided by the Bavarian Polymer Institute.
Christian Bächer thanks the Studienstiftung des deutschen Volkes for financial
support and acknowledges support by the study program “Biological Physics” of
the Elite Network of Bavaria. Furthermore, we thank the laboratory of
professor Alexander Bershadsky at Weizmann Insitute of Science in Isreal for
providing the REF52 cells stably expressing paxillin-YFP.
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See pages - of Supplementary_Information.pdf
|
2024-09-04T02:54:57.813735 | 2020-03-06T11:00:12 | 2003.03133 | {
"authors": "Kshitij Tiwari, Ville Kyrki, Allen Cheung, Naohide Yamamoto",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26079",
"submitter": "Kshitij Tiwari",
"url": "https://arxiv.org/abs/2003.03133"
} | arxiv-papers | # DeFINE: Delayed Feedback based Immersive Navigation Environment for Studying
Goal-Directed Human Navigation
###### Abstract
With the advent of consumer-grade products for presenting an immersive virtual
environment (VE), there is a growing interest in utilizing VEs for testing
human navigation behavior. However, preparing a VE still requires a high level
of technical expertise in computer graphics and virtual reality, posing a
significant hurdle to embracing the emerging technology. To address this
issue, this paper presents Delayed Feedback based Immersive Navigation
Environment (DeFINE), a framework that allows for easy creation and
administration of navigation tasks within customizable VEs via intuitive
graphical user interfaces and simple settings files. Importantly, DeFINE has a
built-in capability to provide performance feedback to participants during an
experiment, a feature that is critically missing in other similar frameworks.
To show the usability of DeFINE from both experimentalists’ and participants’
perspectives, a demonstration was made in which participants navigated to a
hidden goal location with feedback that differentially weighted speed and
accuracy of their responses. In addition, the participants evaluated DeFINE in
terms of its ease of use, required workload, and proneness to induce
cybersickness. The demonstration exemplified typical experimental
manipulations DeFINE accommodates and what types of data it can collect for
characterizing participants’ task performance. With its out-of-the-box
functionality and potential customizability due to open-source licensing,
DeFINE makes VEs more accessible to many researchers.
###### keywords:
Virtual reality, Software, Closed-loop, Locomotion, Gamification, Unity
Kshitij Tiwari* and Ville KyrkiAllen CheungNaohide Yamamoto* Department of
Electrical Engineering and Automation, Aalto UniversityQueensland Brain
Institute, The University of QueenslandSchool of Psychology and Counselling,
Queensland University of Technology (QUT)
Orcid IDs: Kshitij Tiwari https://orcid.org/0000-0003-1789-7961
Ville Kyrki https://orcid.org/0000-0002-5230-5549
Allen Cheung https://orcid.org/0000-0001-9770-217X
Naohide Yamamoto https://orcid.org/0000-0001-9734-7470
The authors would like to thank Onur Sari and Ville Sinkkonen for their
contributions to the development of the framework presented in this article.
*Corresponding authors—KT: Center of Ubiquitous Computing, Faculty of Information Technology and Electrical Engineering, University of Oulu, Finland (email: [email protected]); NY: School of Psychology and Counselling, Queensland University of Technology, Brisbane, Australia (email: [email protected]).
Behavioral researchers are increasingly becoming interested in understanding
the underlying mechanisms for goal-directed navigation in humans (Spiers
Maguire, 2006; Cornwell ., 2008; Pezzulo ., 2014). Whilst it would be ideal to
gather data on navigation in a well-controlled real-world setting, not every
researcher has access to such a setup. However, with the advances in virtual
reality (VR), it is becoming easier to set up economical game-like
environments to test hypotheses about human navigational behaviors.
Although VR technologies are becoming more accessible, they are primarily for
playing computer games and not for carrying out behavioral experiments. Thus,
to use them for investigating human navigation, researchers need to have
working knowledge of graphics design and game engines like Unity
(https://unity.com/) and Panda3D (https://www.panda3d.org/), which may not be
the case even when they are proficient in general scientific programming. To
address this issue, attempts have been made at developing easy-to-use VR
frameworks for researchers who are novice at computer graphics and game
creation. Notable examples include Python-based Experiment Programming Library
(PyEPL; Geller ., 2007), Maze Suite (Ayaz ., 2008, 2011), PandaEPL (Solway .,
2013), Experiments in Virtual Environments (EVE; Grübel ., 2017), Virtual
Reality Experiments (VREX; Vasser ., 2017), VRmaze (Machado ., 2019), Unity
Experiment Framework (UXF; Brookes ., 2020), Route Learning and Navigation
Test Battery (Wiener ., 2020), NavWell (Commins ., 2020), BiomotionLab Toolkit
for Unity Experiments (bmlTUX; Bebko Troje, 2020), and Landmarks (Starrett .,
2020).
These existing frameworks are similar in that they offer some or all of the
following functions for conducting behavioral experiments in VR: modeling and
rendering virtual environments (VEs), designing the structure of an experiment
(e.g., setting the number and order of trials), executing the trials,
recording data (e.g., participants’ navigation trajectories), and performing
preset analyses of the recorded data. A major difference between the
frameworks is in the extent to which they are designed for specific purposes.
For example, Maze Suite is specialized for creating and running experiments in
standard mazes (i.e., mazes created by dividing an enclosed space into
connected paths by walls). By limiting its scope, Maze Suite achieves a high
level of ease of use—that is, everything can be done in graphical user
interfaces (GUIs) and no coding is necessary on the part of end-users. At the
other end of the spectrum, PyEPL and PandaEPL are Python libraries for
programming behavioral experiments in general (PyEPL) and spatial navigation
experiments in particular (PandaEPL). Because they are not compiled software
packages but programming libraries, researchers can create any experiments
using PyEPL and PandaEPL, as far as their programming proficiency goes. The
other frameworks lie in between, balancing ease-of-use and flexibility to
varying degrees by preparing some ready-made functionality while providing
users with the way of doing their own coding for fine customization. Some of
them are still relatively specific to certain types of experiments (e.g., VREX
focuses on indoor environments that consist of connected rooms). The others
offer generic modules for experimental design and data recording, leaving
high-level features of experiments including stimulus presentation up to
users’ programming (e.g., UXF).
Despite the different purposes these frameworks are designed to achieve, one
aspect shared by them is that they focus on the stimulus–response relationship
in examining navigational behavior. That is, participants are presented with a
stimulus with which they carry out a navigation response (e.g., walking to a
goal location indicated by visual cues), and during and at the completion of
the response, they do not receive any feedback on their performance. Although
such a research design is appropriate and even required for investigating
certain aspects of navigational behavior (Philbeck Loomis, 1997; Loomis Knapp,
2003), it makes it impossible to examine how participants modulate their
subsequent response to the stimulus using feedback (Brand, 2008).
Importantly, when goal-directed navigation is performed in real-world
settings, navigators often receive feedback with which they can adjust their
behavior. For example, when they walk in the dark to reach a door and fail to
touch its knob, the lack of tactile sensation serves as feedback and informs
them that they still have a few more steps to go. In this instance, the
navigational behavior should be characterized as consisting of a closed
stimulus–response–feedback loop, instead of an open stimulus–response loop as
in the foregoing frameworks. Indeed, studies have been conducted to capture
human navigation by the stimulus–response–feedback loop. For instance, Carton
. (2016) have found evidence that humans tend to adapt their trajectory
planning horizon (i.e., how far in the future they plan their locomotion
trajectory) when they detect potential collision situations. In this case,
impending collisions constitute a stimulus that triggers a response, which is
witnessed in the form of a shortened planning horizon. The successful
avoidance of collision or a failure thereof functions as feedback that can
subsequently be used to tune the length of planning horizons. In this manner,
experiments on human navigation come in a variety of designs, utilizing both
stimulus–response and stimulus–response–feedback loops. To fully accommodate
this diversity of the experiments, there is a need for a new VR framework that
allows for incorporation of feedback into the experimental design. To this
end, we developed the Delayed Feedback based Immersive Navigation Environment
(DeFINE).
## 1 DeFINE: Delayed Feedback based Immersive Navigation Environment
DeFINE is freely available to download via GitHub at
https://github.com/ktiwari9/define-VR. DeFINE is based on the Unity game
engine, and hence, relies heavily on C# as a programming language. All the
low-level implementation is already taken care of to minimize the workload of
end-users who will use DeFINE in its default settings or with minimal
customization as required by their experimental design. DeFINE aims
specifically to provide an easy-to-use experimental framework that is based on
the stimulus–response–feedback architecture, which can be used to study goal-
directed spatial navigation in an immersive three-dimensional space. In order
to reduce the burden of researchers when setting up an experiment, DeFINE
allows them to make simple alterations of experimental parameters through GUIs
and by changing settings files that are in a straightforward JSON format. We
provide a short video that succinctly explains how to use the basic
functionality of DeFINE “out-of-the-box” (https://youtu.be/OVYiSHygye0). To
allow for further customization of the experiment and the framework itself,
DeFINE is being released open-source under the MIT license. Thus, interested
researchers can modify its code directly. To facilitate this, DeFINE’s
codebase is made modular, making it possible to alter a particular pre-
programmed functionality by adjusting the code in a specific module only, and
also to incorporate a new functionality into the framework by creating a new
module and placing it in an appropriate folder. The file hierarchy is kept
intuitive for this purpose. Another video we provide offers a quick tutorial
of how to change various elements of DeFINE using the Unity software
(https://youtu.be/smIp5n9kyAM). A detailed user manual is also available
(https://github.com/ktiwari9/define-VR/blob/master/user_manual.pdf).
Currently, DeFINE can be integrated into Unity tasks built for Windows
personal computers. It is assumed that DeFINE will be used with a head-mounted
display (HMD) such as HTC Vive and Oculus Rift. For example, DeFINE is
designed to utilize the HMD’s motion tracking sensors for implementing various
methods of participants’ locomotion within VEs (see the locomotion methods
section below for details). In addition to the HMD worn by the participants,
DeFINE simultaneously presents a VE to a desktop display so that
experimentalists can monitor the progress of an experiment. Further details
about hardware and software requirements for DeFINE are available in the user
manual.
The main capabilities and options of DeFINE are detailed below in the
following order: (1) the generic experimental structure, (2) time- and
accuracy-based feedback, (3) the GUI, (4) a diverse suite of locomotion
methods, (5) static and dynamic goals, (6) performance leader-board, and (7)
intra-VR surveys.
### 1.1 Experiment Structure
Human behavioral experiments are often defined by a trial–block–session
architecture which allows experimentalists to repeat a task multiple times to
acquire requisite data (Figure 1). Just like the UXF (Brookes ., 2020), DeFINE
adopts this architecture. A trial is an instance of the experiment where
participants are presented with a stimulus and their response is recorded. At
the end of the trial, the participants receive feedback. To clarify that the
feedback is given after the response is made, as opposed to during the trial
as the response unfolds, this feedback is referred to as “delayed” in DeFINE.
Trials are often repeated multiple times for various purposes (e.g., to
measure variability of responses to the same stimulus, to decipher the
learning effects over trials, or to train the participants on the task),
constituting a block. At the end of the block, the participants are assumed to
be familiar with the environment and to have formulated a behavior of choice
for the task at hand. In order to evaluate the quality of this behavior, the
experimentalists may choose to make some modifications to the environment
before proceeding with the next block. The experiment can consist of a single
or multiple blocks, and when there are multiple blocks, a single iteration of
the task over the blocks is called a session.
Figure 1: Structure of typical human spatial navigation experiments.
Participants are presented with a stimulus, to which they provide a response
and are given feedback after the trial has been fully executed. This feedback
is then used to modify performance in the next trial. Usually, multiple trials
are carried out under the same condition, which constitute a block. Once the
participants are finished with the block, they progress to the next block
which may involve modifications of the condition, and their navigation
performance is evaluated again over multiple trials. A single iteration of the
task is called a session. This trial–block–session architecture as well as the
diagram itself are modeled after those presented by Brookes . (2020). Figure
2: The architecture of the Delayed Feedback based Immersive Navigation
Environment (DeFINE). The left panel shows the low-level functionality
provided by DeFINE while the high-level experimentalist-defined
functionalities are shown on the right. The surveys/questionnaires are
optional, and hence, shown using dashed lines. Modules shown in purple are
parsed during a trial, those in red during a block and those in green during a
session. The logo of DeFINE shown at the top was inspired by that of UXF
(Brookes ., 2020), symbolizing that DeFINE is built as a navigation-focused
extension of UXF.
DeFINE has primarily two levels of abstraction as shown by the two columns
(demarcated by orange dotted lines) in Figure 2. All modules mentioned on the
left are those implemented at the low-level abstraction that come pre-
programmed with DeFINE, while those on the right are implemented at a higher
level for easy and quick modifications. Such an arrangement streamlines
customization of an experiment to suit different research studies because the
modules at the low level of abstraction are common across many experiments and
thus, the experimentalists can rely on DeFINE’s built-in functionality, which
in turn allows them to focus their effort on customizing high-level modules
that tend to be unique to each experiment. All modules shown by dotted black
lines represent optional modules that can be utilized as seen fit by the
experimentalists. In keeping with the trial–block–session architecture as
described in Figure 1, all modules clustered under the green polygon (“Begin
Session”) represent a session, those under the red polygon (“Begin Block”)
represent blocks, and those under the purple polygon (“Begin Trial”) represent
trials. All other modules (under either column of abstraction) are preset
before the session starts.
The flow of an experiment starts from the top of Figure 2. DeFINE takes care
of initializing and rendering an immersive environment and setting up a
session, blocks, and trials, but the experimentalists are required to
configure the VE (e.g., setting its size) and specify the experimental design
(e.g., the number of trials per block) by modifying the settings files in
order to make DeFINE behave in the way they need for their experiment. Details
of the settings files are described later. If the experimentalists want
questionnaires to be filled within the VE, these questionnaires need to be
created online, and the respective links to the questionnaires need to be
added to the environment settings. Setting up the questionnaires is followed
by inputting a participant’s information and selecting preset settings that
are appropriate for the experiment. Once the participant information has been
entered, the experimentalists can start the experimental session that DeFINE
will generate. If there are block-specific settings, those are applied by
DeFINE before starting the first trial. The trials follow the
stimulus–response–feedback structure, by showing the participant a stimulus in
the environment, to which the participant responds, and after the
experimentalist-defined end condition has been met, the trial ends and
feedback is shown to the participant. A new trial is started automatically
after the feedback has been given, until the program reaches the end of the
block. At the end of the block, the participant will be shown the
questionnaires, if any were specified during the initial setup. During the
trials the experimentalists can take down notes, or manually mark the session
as bad to indicate that it should not be taken into account for data analysis.
At any point of the session the experimentalists can abort the session, which
will also mark the session as bad in the stored session notes. If there are
more blocks remaining in the session, the next block is started after DeFINE
has applied its specific settings to the environment. After the final block of
trials has been completed, and the questionnaires following it have been
answered, the session ends and the participant and experimentalists are
returned to the startup screen. For each of the trials executed with DeFINE,
the participant’s trajectory during the trial, the status and changes of
environment variables, the feedback score obtained at the end of the trial
(see the next section for details), trial start time, trial end time, total
time taken for the trial, and straight-line distance to the goal when the
participant ends the response are logged.
### 1.2 Feedback
As opposed to other closed-loop systems where real-time feedback may be made
available to participants as they carry out a response, DeFINE provides
delayed feedback at the end of each trial. This is because continuous feedback
will most likely go against the purpose of typical goal-directed navigation
experiments. Usually, these experiments are to test participants’ ability to
estimate their location relative to a goal by using sensory and other spatial
cues in an environment (Loomis Philbeck, 2008). If the participants were given
external feedback about whether they were moving in the right direction for
every step they took, this feedback would essentially be a non-spatial cue
that would directly aid them in their location estimation. In extreme cases,
with continuous feedback, the participants could perform the task by moving in
an arbitrary direction without processing the sensory and spatial cues and
seeing if that would result in positive feedback. Such a strategy would lead
them to take myopic unstructured paths to the goal, causing non-optimal
navigation performance. Thus, by default, DeFINE is designed to give
performance feedback only after the trial is completed. However, it is
possible for experimentalists to modify DeFINE’s source code in Unity and have
it provide real-time feedback, if they so choose.
Feedback on a goal-directed navigation behavior can be given in a number of
different forms, but DeFINE adopts a reward/cost function that evaluates
participants’ performance and provides feedback as gains and losses of scores.
It has been shown that feedback of this type is very effective in affecting
participants’ behavior and decision making under a variety of conditions
(Brand, 2008; Hossain List, 2012; Yechiam Hochman, 2014). The reward and cost
in the context of navigation can also be defined in various ways, and it is up
to experimentalists’ discretion how they formulate the reward/cost function in
DeFINE, but one straightforward method would be to define them by using speed
and accuracy of navigation. That is, the quicker the participants are in
performing a trial, the greater the reward (or the smaller the cost); and the
more accurate they are in reaching a goal, the greater the reward (or the
smaller the cost). By default, DeFINE implements a reward/cost function of
this form. Specifically:
$R=\beta_{1}\exp(-\alpha_{1}t)+\beta_{2}\exp(-\alpha_{2}d)$ (1)
In Equation (1), $t$ refers to the time taken for navigating towards a goal in
a trial, and $d$ refers to the residual straight-line distance to the goal
from the location at which participants end the trial. $\beta_{1}\text{ and
}\beta_{2}$ are weights used for combining the time and distance into the
decaying reward function, which penalizes both the time taken and the residual
distance to the goal (i.e., rewarding shorter times and smaller distances with
higher scores). $\alpha_{1}\text{ and }\alpha_{2}$ are factors for scaling the
effects of time and distance. If experimentalists choose to use this function
in their experiments, they can assign values of their choice to these
parameters simply by specifying them in a settings file (details shown later).
If they are to calculate the reward/cost scores using their own equation, they
can do so by modifying a relevant section of DeFINE’s codes in Unity. It
should further be noted that, by changing the relevant codes and implementing
their own equation, the experimentalists can use any kinds of feedback that do
not take the form of a cost/reward function. For example, it is possible to
simply present how far away participants are from the goal at the end of each
trial.
### 1.3 Graphical User Interface (GUI)
Utilizing the GUI of UXF (Brookes ., 2020), DeFINE allows experimentalists to
log participant information including, but not limited to, name (or
participant identification), age, gender, and educational qualification
(Figure 3(a)). Should other personal particulars be required, they can be
easily added to the framework by modifying relevant settings files. As an
extension to UXF’s original GUI, DeFINE allows the experimentalists to quickly
set up the environment of choice with a desired locomotion method (see the
next section for details about the locomotion methods). This unique feature
can also be scaled and automated to handle multiple combinations of
environments and locomotion methods via DeFINE’s auto-pilot mode. In this
mode, the experimentalists can provide DeFINE with preset instructions so that
it loads specific combinations of the environment and the locomotion method in
a specified order. This way, a sequence of participants can be tested
automatically, doing away with the need to individually set up an appropriate
combination of the environment and the locomotion method for every
participant. For example, if an experimental design requires that each
participant is shown a different environment, a sequence of environments can
be explicitly listed in settings files which will then be autonomously parsed
when executing the auto-pilot mode. If each participant is to do trials with a
different locomotion method, explicit participant-locomotion method
combinations can be listed in the settings files in a similar fashion.
As a significant extension to the predecessor, UXF, DeFINE also provides
functionalities to study the role of lighting conditions and auditory cues in
spatial navigation. At any point during a trial, experimentalists can toggle
the lights and sound of a VE on or off by clicking on the dedicated buttons on
the user interface, shown in Figure 3(b). The change of the status of these
environmental variables are logged along with the information about
participants’ performance in a navigation task (e.g., their position within
the environment at a given time point).
(a) Drop-down menus to specify experimental parameters (adapted from UXF;
Brookes ., 2020).
(b) The buttons for toggling sound and lights.
Figure 3: DeFINE’s intuitive GUI for experimentalists to set up an experiment.
### 1.4 Locomotion Methods
In order to provide a locomotion suite for participants to perform goal-
directed navigation in VR, DeFINE comes equipped with a variety of locomotion
methods.
#### 1.4.1 Teleoperated locomotion
In order to allow teleoperated locomotion, DeFINE is compatible with both a
keyboard based and a VR controller based teleoperated methods. A typical use
case may involve head direction sensors in an HMD being used to update
participants’ headings in a VE while the keyboard or the VR controller being
used to linearly traverse at preset velocity (which is to be specified in a
settings file by experimentalists). The necessary key-bindings and further
details are available in the user manual (https://github.com/ktiwari9/define-
VR/blob/master/user_manual.pdf).
#### 1.4.2 Arm-swing locomotion
Arm-swing locomotion is an implementation of walking-in-place locomotion. In
this method, the participants walk in place, including swinging their arms in
a manner consistent with their pace of walking. It has been shown that such
arm swings are effective in having the participants experience a naturalistic
sense of locomotion without actually moving in real space (Kunz ., 2009;
Yamamoto ., 2018). This locomotion method uses the physical movement of the VR
controller(s), held by the participants, to determine forward speed in the VE.
The movement speed is calculated from the positional difference of the tracked
controller(s) between two consecutive frames. This calculation can be done
either by requiring movement of two controllers (typically, one in each hand)
or by using one controller (or either of the two controllers) that moves more
than a given threshold amount between the frames. When the use of the two
controllers is required, the forward speed in the environment is set to be
zero, unless both of the controllers exceed the threshold value.
#### 1.4.3 Head-bob locomotion
Head-bob locomotion is another implementation of walking-in-place locomotion.
In order to move forward in the VE, the participants need to walk in place,
and as they do, their head, and in particular, the HMD, bobs slightly
vertically. This locomotion method uses this vertical bobbing to determine the
forward velocity. DeFINE tracks the vertical direction of the bobbing and its
starting position. Once the direction changes, the participants are considered
to have stepped, if the vertical height difference between two successive
flexion points exceeds a threshold value specified in the settings of the
locomotion method. The detected physical step is then translated into a step
in the VE so that the participants walk forward in the VE at preset velocity.
Due to the fact that the HMD is in front of the participants’ face, turning
their head up or down causes the HMD to move vertically. In order to avoid
reading these vertical movements as steps of the participants, DeFINE also
tracks the participants’ rotational head movements about the pitch axis and
ignores any “bobs” that are accompanied with the rotational head movements
that exceed a specified threshold value.
#### 1.4.4 Physical walking
Physical walking is the only locomotion method in which the participants
physically move around in the real world. The movement of the participants is
tracked by using the HMD’s motion tracking sensors and the participants’
position in the VE is updated accordingly. Owing to the limited size of a
physical area in which the participants’ movement can be tracked (which is
typically around $10\times 10$ m), the size of the VE is going to be limited.
To alleviate this limitation, sometimes modified physical walking methods such
as redirected walking are adapted (Paludan ., 2016; Nilsson ., 2018). In these
methods, the rotations and translations of the participants are slightly
altered between physical and virtual worlds in order to steer the participants
away from the edges of the available physical area. However, DeFINE does not
utilize these methods because they can induce disruption to mental and neural
spatial representations as well as to navigational behavior by causing a
mismatch between intended (and physically carried out) movements and
consequent virtual movements (Du ., 2020; Tcheang ., 2011).
In DeFINE, a visible grid barrier, shown in Figure 4, is displayed in the HMD
when the participants approach the limits of a configured area in which they
can safely move around. The grid barrier serves two purposes. First and
foremost, it prevents the participants from going out of the physical safe
area, ensuring their safety. It is advisable that a navigation task in DeFINE
be well confined to an area smaller than the safe area so that the
participants will never encounter the barriers in the first place. If they do
view the barrier, it essentially functions as an extra landmark that informs
about the boundary of an environment, which can induce significant bias in
their navigational behavior (Cheung, 2014; Mou Zhou, 2013; Bird ., 2010).
Second, the barrier makes it possible to extend a navigable virtual space
beyond the physical safe area, in case it is necessary for an experiment. To
do this, the participants hold a trigger button in the VR controller, which
locks the VE in place. While the VE is locked, the participants’ physical
rotation in real space is not reflected in their virtual heading. Thus, the
participants appear to keep facing the same direction in the VE, despite
physically turning to face away from the edge of the safe area. The grid
barrier remains visible and in correct orientation in respect to the physical
safe area, allowing the participants to reorient themselves before continuing.
In order to minimize the motion sickness caused by the VE remaining still
during the participants’ physical rotation, DeFINE blurs the VE during the
rotation. Unlike similar approaches used in the literature (Williams ., 2007),
DeFINE does not require the participants to rotate a fixed amount as long as
they steer clear of the physical boundary.
Although this method of extending the virtual space can be practical, it must
be used with caution because by physically rotating in the locked VE, the
participants will most likely be forced to go through a mental process of
dissociating real and virtual spaces once and realigning them after the
physical rotation. It is very probable that this process will have significant
impact on the participants’ mental and neural spatial representations, and in
turn, on their subsequent navigational behavior.
Figure 4: The visible blue grid marking the edge of a configured safe area.
#### 1.4.5 Teleportation
Teleportation locomotion differs from all of the other locomotion methods in
that the participants never move through space, but instead teleport directly
to their desired location some distance away. Before and after teleporting,
the direction of the participants’ body and head remain unchanged relative to
the environment. In DeFINE, the participants teleport by holding the trigger
of a VR controller, which brings up the teleportation marker, as seen in
Figure 5. Then the participants place the marker on the desired teleportation
target location by aiming the controller at the location. Once the
participants release the trigger, they are teleported to the marked location,
given that the location is on the horizontal X-Z plane and clear of all
collision regions around the objects in the environment. A valid target
location is indicated by the blue color of the marker, whereas invalid
locations turn the marker red. Although teleportation is not a naturalistic
method of locomotion, its use is increasingly common in VEs including those
for spatial navigation research (Cherep ., 2020; Kelly ., 2020).
Figure 5: Teleportation locomotion with the teleportation marker visible.
### 1.5 Goal Demarcation
To accommodate a variety of experiments, DeFINE offers two possible ways of
demarcating the goal location for a navigation task: firstly, presenting
static objects at goal locations (e.g., arrows, exclamation marks, or other
similar objects) and secondly, showing dynamic objects like a buzzing fly that
can give imprecise indication of the goal location. An example of the dynamic
goal markers is available in the demonstration section below.
### 1.6 Leader-board
Gamification of learning has shown increased participant engagement be it for
online programs (Looyestyn ., 2017) or education (Barata ., 2013). Thus, as an
optional feature, DeFINE is equipped with a leader-board which provides a
ranking based on scores obtained using Equation (1) or other equivalent
equations implemented by experimentalists (Figure 6). DeFINE keeps track of
the scores and displays ten best scores in the leader-board. A new high-score
is indicated with red font in the leader-board, while a score that was not
high enough to get to the leader-board is shown at the bottom to illustrate
the difference between the latest score and pre-existing scores. If
participants are to carry out some practice or training trials first, it may
not be appropriate to compare their scores against the pre-existing scores
before they become fully familiarized with an experimental task. In that case,
it is possible to show a provisional ranking which is not integrated with the
leader-board. For clarity, this is labeled with a red Practice tag in the
board. Once the practice phase is finished, the actual scores of the
participants are integrated into the leader-board that includes their own
previous high scores.
(a) A new high-score is shown in red in the leader-board.
(b) The latest score, which does not make it to the top ten, is shown at the
bottom.
Figure 6: Leader-boards inform participants about their performance in a given
trial relative to their own and other participants’ previous trials.
While having a leader-board can motivate participants, it can also cause the
conditions of an experiment to be different between the participants. As
earlier participants obtain their place in the leader-board, they keep
replacing lower scores on it. As such, it gets systematically more difficult
for later participants to score high enough to make it to the top-ten scores
of all time. Having a leader-board that is seemingly unreachable might provide
a different motivation to the later participants than having an easily
reachable one would. In order to ensure that each participant can have an
equal experimental condition, DeFINE offers two options. First, because the
leader-board is an optional feature, experimentalists can choose to remove it
entirely. Second, they can use a fake leader-board that behaves like a normal
leader-board during the session of one participant, except that the changes to
the board are not in fact stored to a log file. Once the next participant
begins a session, the board reverts to its original condition, giving
subsequent participants the same competitive challenge.
### 1.7 Surveys
Often in behavioral studies, experimentalists would like participants to fill
surveys for quality assurance or other purposes. Some of the most commonly
used surveys for such studies using VR include the simulator sickness
questionnaire (SSQ; Kennedy ., 1993) and the NASA task load index (NASA TLX;
Hart, 2006). The SSQ studies the onset of simulator sickness symptoms like
nausea or headache owing to being immersed in VR. It contains 27 questions and
the participants answer each of them using a scale ranging from none (0) to
severe (3). The NASA TLX is a survey for evaluating the workload of a given
task utilizing six questions. Administering these and any other surveys has
been made conveniently possible in DeFINE (Figure 7). The surveys are visible
in an HMD to the participants and also on a desktop display to the
experimentalists. While questions that have preset choices can be answered
directly by the participants using the VR controller(s), questions that
require free-form responses are to be typed in by the experimentalists on
behalf of the participants.
While DeFINE’s survey system allows the experimentalists to administer surveys
while keeping the participants immersed in VR, other systems typically require
them to take off an HMD to answer the surveys. Thus, if an experiment involves
multiple sessions or blocks, each of which contains surveys, the participants
need to be re-immersed in VR every time they remove the HMD and put it back on
(Schatz ., 2017). This can be very cumbersome and make the participants feel
uncomfortable, possibly inducing cybersickness. An alternative could be that
the experimentalists orally ask questions and fill in the surveys on behalf of
the participants, but this can feel intrusive to the participants and reduce
the sense of immersion in VR because the participants have to directly
communicate with the experimentalists who do not belong to the virtual world
(Bowman ., 2002). DeFINE remedies these issues by displaying the surveys in
the HMD. To our knowledge, only Grübel . (2017) and Regal . (2018) implemented
a similar system previously.
Figure 7: Filling in surveys whilst being immersed in VR using DeFINE.
## 2 Demonstration
To demonstrate various built-in functionalities of DeFINE and its overall
usability for human navigation experiments, we asked actual participants to
perform a simple goal-directed navigation task in DeFINE. In each trial, the
participants navigated to a hidden goal that was indicated by a dynamic visual
cue and received a feedback score on the leader-board. The participants’
navigation response was characterized by the total duration of the response,
the residual distance to the goal at the end of the trial, the feedback score,
and detailed time-course plots of their locomotion trajectory, exemplifying
the types of data DeFINE can record. To implement the trial–block–session
architecture, two blocks of trials were used, between which the environment
and the dynamic cue were modified. Two ways of deriving the feedback scores,
which differentially weighted the response duration and the residual distance,
were defined through DeFINE’s settings files. The participants also answered
the SSQ and NASA TLX. This was to give a demonstration of DeFINE’s intra-VR
survey feature, and also to examine whether the participants tolerated the use
of DeFINE in the current experiment, which was of typical scale as a
behavioral experiment.
Figure 8: The default experimentalist view of an environment containing walls
and floor with non-repeating textures. A countdown timer and score window
helps participants keep track of their performance. Light and sound can be
toggled by experimentalists. The firefly serves as a noisy visual cue to guide
the participants to a goal. The light/audio toggle switches (top right) and
experimentalist button descriptors (top left) are only visible to the
experimentalists on a desktop running DeFINE, and are not made visible to the
participants in an HMD. Except for this difference, the participant and
experimentalist views are the same. The arrows specifying X, Y, and Z axes in
the lower-left corner are shown in this figure only; they appear neither in
the participant nor experimentalist view. See text for further details.
### 2.1 Method
#### 2.1.1 Participants
Twenty-four participants (15 males, 8 females, and 1 other) took part in this
experiment. Twenty-three of them were students of Aalto University, and one
was from the vocational university Metropolia, Finland. The mean age of the
participants was $24.8\pm 2.9$ years. The participants’ educational background
ranged from having graduated from high school to having a master’s degree. All
participants gave written informed consent to participate in the experiment
and received a movie ticket in return for their participation. The protocol of
the experiment was approved by the Aalto University Research Ethics Committee.
#### 2.1.2 Design and materials
The participants were asked to navigate from start to goal positions within a
virtual room of $10\leavevmode\nobreak\ \times 10$ m with non-repeating
textures as shown in Figure 8. The walls were $4$-m tall. The participants
navigated using the controller teleoperation method. Specifically, they turned
the head to face the direction they wanted to go and moved in that direction
by pressing a button of a VR controller. The participants’ eye height in the
virtual room was set at $1.36$ m, which approximately corresponded to their
actual eye height while seated in a real room. The participants first went
through a block containing $15$ trials in which the room walls were visible.
Subsequently, they performed the same navigation task in the second block in
which the walls were removed and the floor texture was extended to the
horizon. This block also consisted of $15$ trials. The starting and goal
positions were fixed across the trials as well as across the blocks. Relative
to the center of the room, in a left-handed coordinate system, the starting
position was at ($4.5$ m, $4.5$ m) and the goal position was at ($-3$ m, $-1$
m). At the starting position, the participants directly faced the hidden goal
position with an orientation of $-135^{\circ}$ about the vertical Y-axis. As a
dynamic goal marker, a firefly buzzed around the goal position in such a way
that its randomly fluctuating flying trajectory had its center directly over
the goal position. Specifically, in each frame, the fly’s position along the
horizontal X-Z plane was randomly sampled within the radii of 0.75 and 1.5 m
from the goal in the first and second blocks, respectively. The height of the
fly along the Y-axis was randomly sampled in the range of
$0.75$–$1.25\leavevmode\nobreak\ $m. To make the fly move smoothly, its
position was incremented with a step size of $5$ mm. For a graphical
presentation of X, Y, and Z axes, see Figure 8.
In this manner, the fly represented a noisy visual cue to guide the
participants to the goal position. That is, the exact goal position was never
revealed to the participants, and instead they were told that the goal was
somewhere inside the area delimited by the fly’s trajectory. Hence, the goal
position was provided imprecisely to the participants via the noisy visual
cue, and also the feedback score.
The participants were assigned to one of two groups, both of which received
scores for their performance that depended on both navigation speed and
accuracy, but with different weights. Time group received feedback that put
more importance on speed while the feedback in accuracy group was weighted in
favor of being as close as possible to the goal. The feedback provided to the
participants was computed using Equation (1) with the constants shown in Table
1. The score was presented to the participants at the end of each trial.
Table 1: Constants of the reward function for two participant groups. Constant | Time group value | Accuracy group value
---|---|---
$\alpha_{1}$ | $-0.05$ | 0.2
$\alpha_{2}$ | 0.2 | 1
$\beta_{1}$ | $-2$ | 0.5
$\beta_{2}$ | 6.2 | 3.4
The participants were informed that they would be graded according to the time
elapsed and residual distance to the actual goal. However, they were not told
about the existence of the two groups or which group they belonged to.
Instead, the participants of both groups were told that the scores obtained
for the trials would be based on both speed and accuracy. In order to make the
scores easier for the participants to follow, they were scaled up by a factor
of 300 and their minimum value was set at 0 (i.e., no negative scores).
DeFINE was run on a high-performance Windows 10 personal computer with an
Intel Core i5 processor, 32 gigabytes of random-access memory, and an Nvidia
GTX 1070 graphics card. The HTC Vive Pro with a wireless extension was used
for a VR HMD. DeFINE was used for presenting the VE and recording data for
each frame (approximately 90 frames per second). Specifically, the following
log files were created for each participant per session:
##### Environment settings
This file specified the size of the virtual room, the fly’s trajectory
(buzzing speed, minimum and maximum heights of flight, and a buzzing radius),
and whether to remove or retain the bounding walls during the second block.
Additionally, the links to survey forms were added here.
##### Locomotion settings
This log file specified the locomotion method used and its presets like
traversal gain and rotation speed.
##### Scenario settings
This log file specified how many trials were to be presented in first and
second blocks, the longest possible duration of a trial (with the maximum of
120 s), and start and goal locations.
##### Participant particulars
The participant information as collected via the GUI (i.e., identification,
age, gender, and highest qualification achieved) was recorded.
##### Movement logs
This log file recorded each participant’s X and Z positions and rotation about
the Y axis with time stamps. Owing to flexibility provided by DeFINE to toggle
lights and sounds even during a trial, the status of these parameters was also
logged every frame in these logs. A new log was created per trial along with
trial numbers.
##### Trial results
This log file assimilated component-wise and cumulative rewards per
participant along with distance covered and time elapsed during a trial.
##### Notes
The experimentalists’ notes during the experiment were recorded in this file.
For instance, if some participants felt dizzy and opted for early termination,
the particular session can be marked as bad and further details can be stored
as notes for later use.
#### 2.1.3 Procedure
The participants sat in the middle of a room that was clear of any obstacles.
At the outset of the experiment, the participants were asked to fill the SSQ
to log their state of health before being immersed in VR. Their age, gender,
and the highest qualification achieved were also recorded using DeFINE’s GUI
(Figure 3(a)). An experimentalist then put an HMD on the participants’ head
(over the spectacles, as and when necessary) and handed them hand-held VR
controllers. The participants were run individually.
As soon as the participants had verbally confirmed to be ready, the first
block was started by the experimentalist. The participants began a trial by
leaving the start position, using the controller teleoperation method to
navigate, and ended their locomotion by pressing a key on a VR controller when
they thought they had reached the goal position. The goal was positioned
diagonally across the other side of the room and remained unchanged across
trials. The participants then received a score from the trial in a leader-
board (Figure 6). The fake leader-board feature was used so that all
participants performed the navigation task with the same competitive
challenge. The board was filled with made-up scores at the beginning, which
were set low to give everyone a reasonable chance of making it to the top ten.
Pilot testing was conducted to empirically determine how low should be low
enough for this purpose. The leader-board was displayed for 10 s (or until the
participants pressed the “End Trial” key on the VR controller), and the room
was automatically shown from the start position again for the next trial
thereby resetting the scene to the exact same configuration for each trial.
The participants completed the trials at their own pace, until reaching the
end of the block. The participants were allowed to have a short break between
trials. When necessary, the participants were able to skip a trial by pressing
a controller key. At the end of the first block, the participants filled the
NASA TLX in the DeFINE’s form system (i.e., without taking off the HMD) using
a 7-point Likert scale. Upon having filled the form, the participants started
the second block at their own input, prompted on the HMD. Once again the
participants performed the trials at their own pace, until filling the NASA
TLX one more time at the end of the second block. Filling the form completed
the VR part of the experiment.
After taking off the HMD and the controllers, the participants filled the SSQ
again to evaluate their simulation sickness after the exposure to the
immersive VR. In addition, the participants were invited to provide feedback
about the experiment and DeFINE by indicating the degree of agreement with
each of the following five statements in a 5-point Likert scale: “Instructions
were easy to understand”; “I understood what the score depended on”; “moving
in the VE was easy”; “the walls in the practice phase were helpful”; and
“filling a form in the VE was easy”.
### 2.2 Results
Two participants from each group misunderstood task instructions and simply
chased the fly rather than navigating towards the goal it indicated. This was
determined through real-time observation of their locomotion patterns on the
experimentalist’s desktop (Figure 8). Although it was not intended, this gave
another demonstration of the utility of the experimentalist view that allowed
the experimentalist to observe participants’ responses as they took place
during trials. Due to this behavior, data from the four participants were
excluded from analysis. The data presented in this section represent the
results of the remaining $10$ participants per group, accounting for $20$
participants in total. In addition, in $0.7\%$ of all trials, the participants
accidentally pressed the button to end the trial immediately after it had
begun. These trials were also discarded for the analysis presented herewith.
For each trial, the total elapsed time, the residual distance to the goal, and
the score were derived as dependent measures from log files. The entire
trajectory of a participant’s locomotion was also reconstructed from recorded
position data. For each dependent measure, data points that were more than
three standard deviations away from each participant’s mean of each block were
defined as outliers and removed from analysis. This resulted in removal of
1.17% of trials on average.
Table 2: Means and standard deviations of the total elapsed time, the residual distance to the goal, and the score as a function of participant group and block. | Time group | | Accuracy group
---|---|---|---
| First block | Second block | | First block | Second block
Time (s) | 11.13 (5.93) | 7.81 (4.44) | | 10.97 (5.25) | 13.33 (10.77)
Distance (m) | 0.43 (0.18) | 0.67 (0.32) | | 0.63 (0.34) | 0.80 (0.36)
Score | 664.97 (239.02) | 718.87 (255.85) | | 624.77 (141.36) | 542.41 (160.26)
Note. Standard deviations are shown in parentheses.
#### 2.2.1 Time, distance, and score
Table 2 shows descriptive statistics of the dependent measures as a function
of participant groups and blocks. Overall, participants in the time group
performed trials more quickly in the second block than in the first block, and
those in the accuracy group showed the opposite pattern. It is likely that
this resulted from the types of feedback the participants received in each
group—that is, the swiftness of a response was more heavily rewarded in the
time group, whereas the closeness to the goal was given more emphasis in the
accuracy group. In terms of the residual distance to the goal, both groups
performed worse in the second block, reflecting the fact that the navigation
task was more challenging in the second block because of the lack of walls and
decreased precision of the dynamic goal marker. Capturing these outcomes, the
scores increased in the time group and decreased in the accuracy group between
the first and second blocks.
Figure 9: Mean times participants spent for performing each trial (top row),
mean residual distances to the goal (middle row), and mean feedback scores the
participants received (bottom row) as a function of participant group, block,
and trial number. Error bars represent $\pm 1$ standard error of the mean.
Figure 9 shows the elapsed time, residual distance, and score in each trial,
providing a more detailed picture of the participants’ performance. In terms
of speed, the two groups performed similarly in the first block, but they
differed in the second block. Specifically, the time group maintained
approximately the same speed throughout the block, performing the trials
consistently quicker than the accuracy group. This pattern suggests that the
feedback scores affected the participants’ navigation differently in the two
groups. On the other hand, with the reward parameters used in this experiment
(Table 1), the effects of the scores were less clear on the accuracy of
performance. The participants in the accuracy group showed no visible
improvement of accuracy in later trials, even though they received scores that
rewarded accurate performance. This might indicate that the parameters did not
favor accuracy enough to elicit observable change in behavior within a block.
Between the blocks, it is also possible that the increase of task difficulty
overrode the effects of the scores on navigation performance, as suggested by
the overall lower scores in the second block.
To statistically examine the time, distance, and score data, they were
analyzed separately by mixed analyses of variance (ANOVAs) in which block
(first and second) was a within-participant factor and group (time and
accuracy) was a between-participant factor. Because the data plotted in Figure
9 suggest that the first trial of the first block yielded rather different
results (particularly in time and score), the ANOVAs were run with and without
this trial. The two sets of ANOVAs showed the same outcomes, and thus those
including the first trial are reported here. The main effect of block on
distance was significant, $F(1,18)=10.52,p=0.005,{\eta_{G}}^{2}=0.11$,
reflecting the overall decrease of accuracy in the second block. All the other
main effects and interactions were not significant,
$F\text{s}(1,18)<3.09,p\text{s}>0.096,{\eta_{G}}^{2}\text{s}<0.075$.
#### 2.2.2 Trajectory
Figure 10: The residual distance to the goal as a function of participant
group and time in a trial (first 30 s only). Panels A and B show sample
participants’ 30 trials (black lines), one participant from each group. The
red lines represent the mean time courses of the residual distance derived
from the 30 trials. To calculate the means, trials that lasted shorter than 30
s were extended by using the end values of each such trial. The insets display
the corresponding trajectories of each participant from the start (cyan
circle) to the goal (red cross). Panels C and D show the mean time courses of
each participant (black lines) separately for the two groups. The red lines
indicate group means. In calculating them, short time series were dealt with
in the same way as in Panels A and B.
In addition to the time, distance, and score measures that aggregate
participants’ performance in a trial, DeFINE can provide dynamic information
about their movement within the trial. To illustrate this feature, the time
courses of the residual distance to the goal as well as sample participants’
trajectories are plotted in Figure 10. By visualizing the trajectories in this
manner, researchers can gain additional insights into the way participants
navigate in their experiments. For example, by comparing panels A and B of
Figure 10, it can be seen that the trajectories of the sample participant from
the time group were more dispersed than those of the sample participant from
the accuracy group. Such an observation is easily possible in this
visualization, but it is not readily available from the aggregate measures.
#### 2.2.3 Simulation sickness questionnaire (SSQ)
Responses to the SSQ are summarized in Table 3. As shown in the table, the
participants scored very low not only before but also after exposure to the
VE. Because the scores were very low overall, we used total raw scores for
analysis, instead of deriving weighted scores for each sub-scale of the SSQ
(Kennedy ., 1993). The total raw SSQ scores were analyzed by a mixed ANOVA
with exposure (before and after) as a within-participant factor and group
(time and accuracy) as a between-participant factor. This ANOVA yielded no
significant outcomes,
$F\text{s}(1,18)<0.22,p\text{s}>0.64,{{\eta_{G}}^{2}}\text{s}<0.010$,
suggesting that the SSQ scores did not differ between pre- and post-exposure
to the VE as well as between the time and accuracy groups. These results
indicate that the use of DeFINE did not induce any major symptoms of
cybersickness.
Because non-significant results in an ANOVA do not necessarily constitute
positive evidence for null hypotheses, we also conducted a Bayes factor
analysis to gauge the extent to which the data actually supported the claim
that neither exposure nor group had an effect on the SSQ scores (Rouder .,
2012). When the null model was compared against the full model that included
the main effects of exposure and group as well as the interaction between the
two, it yielded a Bayes factor of 13.9. This constitutes positive evidence for
the null hypothesis (Kass Raftery, 1995), supporting the conclusion that doing
the navigation task in DeFINE (and which group each participant was in) did
not cause cybersickness above and beyond what the participants had prior to
navigating in the VE.
Table 3: Means and standard deviations of raw scores of the simulator sickness
questionnaire (SSQ).
| Pre-exposure | | Post-exposure
---|---|---|---
| Time group | Accuracy group | | Time group | Accuracy group
General discomfort | 0.1 (0.32) | 0.2 (0.63) | | 0.2 (0.42) | 0.3 (0.48)
Fatigue | 0.4 (0.52) | 0.7 (0.95) | | 0.2 (0.42) | 0.6 (0.70)
Boredom | 0.3 (0.67) | 0.1 (0.32) | | 0.3 (0.67) | 0.1 (0.32)
Drowsiness | 0.5 (0.71) | 0.2 (0.42) | | 0 (0) | 0.2 (0.63)
Headache | 0.1 (0.32) | 0 (0) | | 0.1 (0.32) | 0.2 (0.42)
Eyestrain | 0.5 (0.53) | 0.5 (0.97) | | 0.7 (0.67) | 0.5 (0.97)
Difficulty focusing | 0.2 (0.42) | 0.4 (0.70) | | 0.1 (0.32) | 0.3 (0.48)
Salivation increase/decrease | 0.1 (0.32) | 0.2 (0.42) | | 0.1 (0.32) | 0 (0)
Sweating | 0 (0) | 0.1 (0.32) | | 0.2 (0.42) | 0.2 (0.63)
Nausea | 0 (0) | 0 (0) | | 0.2 (0.42) | 0.1 (0.32)
Difficulty concentrating | 0.3 (0.48) | 0.3 (0.67) | | 0.1 (0.32) | 0.3 (0.48)
Mental depression | 0.2 (0.42) | 0.2 (0.42) | | 0.2 (0.42) | 0.2 (0.42)
Fullness of the head | 0.3 (0.48) | 0.4 (0.52) | | 0.3 (0.48) | 0.2 (0.42)
Blurred vision | 0.1 (0.32) | 0.3 (0.48) | | 0.2 (0.42) | 0.4 (0.52)
Dizziness with eyes open/closed | 0.1 (0.32) | 0.2 (0.63) | | 0.2 (0.42) | 0.3 (0.48)
Vertigo | 0 (0) | 0 (0) | | 0.1 (0.32) | 0.1 (0.32)
Visual flashbacks | 0 (0) | 0 (0) | | 0.3 (0.67) | 0.1 (0.33)
Faintness | 0.1 (0.32) | 0 (0) | | 0 (0) | 0.1 (0.32)
Breathing awareness | 0.4 (0.52) | 0 (0) | | 0.3 (0.48) | 0.1 (0.32)
Stomach awareness | 0 (0) | 0.1 (0.32) | | 0.1 (0.32) | 0.1 (0.32)
Loss of appetite | 0 (0) | 0 (0) | | 0 (0) | 0.1 (0.32)
Increase of appetite | 0.1 (0.32) | 0.3 (0.48) | | 0.1 (0.32) | 0.4 (0.52)
Desire to move bowels | 0.1 (0.32) | 0 (0) | | 0.1 (0.32) | 0 (0)
Confusion | 0 (0) | 0.4 (0.70) | | 0.1 (0.32) | 0.1 (0.32)
Burping | 0 (0) | 0 (0) | | 0 (0) | 0 (0)
Vomiting | 0 (0) | 0 (0) | | 0 (0) | 0.1 (0.32)
Others | 0 (0) | 0 (0) | | 0 (0) | 0 (0)
Total | 3.9 (3.51) | 4.6 (5.66) | | 4.2 (3.74) | 5.1 (4.12)
Note. Standard deviations are shown in parentheses. The possible range of the
total score was from 0 to 81.
Table 4: Means and standard deviations of scores of the NASA task load index (NASA TLX). | Time group | | Accuracy group
---|---|---|---
| First block | Second block | | First block | Second block
Mental demand | 3.0 (1.33) | 3.7 (1.64) | | 2.3 (1.25) | 3.0 (1.41)
Physical demand | 1.7 (0.82) | 2.2 (1.14) | | 1.7 (1.25) | 2.7 (1.77)
Temporal demand | 3.5 (1.35) | 4.1 (1.79) | | 2.9 (1.60) | 2.9 (1.85)
Effort | 3.4 (1.58) | 3.6 (1.71) | | 2.8 (1.14) | 3.2 (1.32)
Performance | 3.5 (1.43) | 2.6 (1.35) | | 3.9 (1.60) | 4.0 (1.70)
Frustration level | 2.6 (1.71) | 3.2 (1.62) | | 1.9 (1.20) | 2.5 (1.43)
Total | 17.7 (6.38) | 19.4 (7.29) | | 15.5 (5.15) | 18.3 (7.89)
Note. Standard deviations are shown in parentheses. The possible range of the
total score was from 6 to 42.
#### 2.2.4 NASA task load index (NASA TLX)
Responses to each item of the NASA TLX ranged from one to seven, with smaller
scores indicating lower task load. As shown in Table 4, participants generally
indicated that doing the navigation task in DeFINE required medium workload.
There was some variation of the scores between groups, blocks, and questions.
For example, the scores of the temporal demand question suggest that the time
group felt stronger time pressure than the accuracy group, which is consistent
with the feedback function that put emphasis on speedy response in the time
group. In addition, scores in the second block tended to be higher than those
in the first block, which corresponds to the fact that the task was made more
difficult in the second block. In line with these observations, a mixed ANOVA
with block (first and second) and question (six questions of the NASA TLX) as
within-participant factors and group (time and accuracy) as a between-
participant factor yielded a significant interaction between question and
group, $F(5,90)=2.96,p=0.035,{\eta_{G}}^{2}=0.049,\epsilon=0.66$ (this ANOVA
was corrected for non-sphericity with the Greenhouse-Geisser method when
appropriate). The interaction between question and block as well as the main
effect of question were also significant,
$F(5,90)=3.31,p=0.008,{\eta_{G}}^{2}=0.019$ and
$F(5,90)=7.14,p<0.001,{\eta_{G}}^{2}=0.11,\epsilon=0.66$, respectively. The
main effect of block was not significant,
$F(1,18)=3.58,p=0.075,{\eta_{G}}^{2}=0.018$. The interaction between block and
group and the main effect of group were virtually non-existent,
$F\text{s}(1,18)<0.36,p\text{s}>0.55,{{\eta_{G}}^{2}}\text{s}<0.010$,
suggesting that overall, the two groups tolerated the workload of using DeFINE
in a similar way.
#### 2.2.5 Participant feedback on the experiment and DeFINE
Scores of the participant feedback survey at the end of the experiment are
summarized in Table 5. Larger scores denote stronger agreement with the
statements. Overall, participants gave high scores, indicating that DeFINE
provided an easy-to-use interface for doing the navigation experiment. The
scores were analyzed by a mixed ANOVA in which statement (five statements in
the survey) was a within-participant factor and group (time and accuracy) was
a between-participant factor. The main effect of statement was significant,
$F(4,72)=6.51,p<0.001,{\eta_{G}}^{2}=0.20$, which suggests that scores were
reliably lower in the statement about the usefulness of walls than in the
other statements. The interaction between statement and group and the main
effect of group were not significant,
$F(4,72)=1.73,p=0.15,{\eta_{G}}^{2}=0.062$ and
$F(1,18)=0.18,p=0.67,{\eta_{G}}^{2}=0.003$, respectively, suggesting that
there was no overall difference between the groups in the way they responded
to the feedback survey.
Table 5: Means and standard deviations of scores of the participant feedback survey. | Time group | Accuracy group
---|---|---
Clarity of instructions | 4.2 (0.63) | 4.1 (0.99)
Score interpretation | 4.5 (0.71) | 4.0 (1.05)
Ease of movement | 4.3 (0.67) | 4.2 (0.79)
Usefulness of walls | 2.9 (1.37) | 3.3 (0.82)
Ease of filling forms in DeFINE | 3.4 (0.97) | 4.2 (1.03)
Total | 19.3 (2.63) | 19.8 (2.62)
Note. Standard deviations are shown in parentheses. The possible range of the
total score was from 5 to 25.
### 2.3 Discussion
The purpose of this demonstration was to showcase the key functionalities of
DeFINE—namely, its ability to set up an experiment in a trial–block–session
structure, run goal-directed navigation trials in a stimulus–response–feedback
loop, and collect both moment-by-moment and aggregate measures of
participants’ task performance. In this experiment, the measures included the
duration of a trial, the residual distance to the goal, the feedback score
given to the participants, and the time course of the participants’ positions
within the trial. It may be worth noting that among these measures, those that
capture temporal information about the participants’ locomotion (i.e., the
trial duration and dynamic position data) can be of particular use in future
studies because past studies of goal-directed navigation in small-scale space
tended to put emphasis on accuracy or precision of responses with little
regard for how quickly they were carried out (e.g., Chen ., 2017; Chrastil
Warren, 2014; Harris Wolbers, 2012; Yamamoto, Meléndez Menzies, 2014;
Yamamoto, Philbeck ., 2014). However, it is important to consider the speed of
the responses in evaluating their accuracy because there can be a trade-off
relationship between them (Bogacz ., 2010). DeFINE allows researchers to
examine the speed and accuracy of navigation either in conjunction as in the
current experiment or in isolation by setting the parameters of the reward
function accordingly (e.g., $\beta_{1}=0$ makes the reward function
exclusively focused on the accuracy).
A defining feature of DeFINE is that with its default settings, it provides
feedback on participants’ performance in each trial. Results from this
experiment suggested that by using differential weights on speed and accuracy
of navigation in calculating feedback scores, DeFINE has potential for
eliciting different responses from the participants. Although the behavioral
measures of navigation were largely non-significant in the statistical
analyses, the effects of the feedback scores, particularly those on navigation
speed, were implied in Figure 9. The time group improved the speed of
responses in early trials and kept the same speed during the rest of the
experiment because this helped increase and then maintain the feedback scores.
On the other hand, the accuracy group appeared to care less for making a
speedy response toward the end of the experiment, as the slower speed had
little influence on the feedback scores in this group. It is likely that the
feedback scores were more effective in affecting the speed because of the
specific way in which the current experiment was designed—that is, the
participants were self-aware of the speed of their response, but the accuracy
was never explicitly revealed to them, making it harder for the participants
to improve the accuracy. Importantly, this pattern is a result of one
particular installation of DeFINE, and its architecture flexibly enables
researchers to set up a suitable balance between the speed and accuracy
according to the objectives of their studies. For example, by giving heavier
weights to accuracy in the feedback function, researchers can make feedback
scores more directly informative about how well participants are reaching a
goal. Similarly, by demarcating the goal location more specifically by using
different goal markers (e.g., a static marker or a dynamic marker with less
variability) and environmental features (e.g., walls that provide spatial
cues), researchers can run experiments in which focus is entirely on speed
(i.e., accuracy is a given) or subtle changes in accuracy are scrutinized.
This experiment also examined the participants’ experience in using DeFINE.
Results from the SSQ indicated that DeFINE caused no major symptoms of
cybersickness. Considering that the participants repeatedly experienced
sensory conflicts between their vision and body-based (i.e., vestibular and
proprioceptive) senses due to the use of the teleoperation method, the absence
of cybersickness is notable (Bos ., 2008). The NASA TLX showed that the
participants found doing the navigation task in DeFINE moderately challenging
but not unreasonably taxing. In the feedback survey, the participants gave a
positive evaluation to DeFINE itself and the design of the experiment.
Generally, these results did not differ between the two groups of the
participants, suggesting that DeFINE provided a versatile platform that
accommodates different types of experiments.
In sum, this demonstration showed that by using DeFINE in its default
settings, we were able to run a human navigation experiment of typical scale,
involving 24 participants and consisting of a single continuous session with
two blocks of 15 trials each. It exemplified an assortment of data types that
DeFINE can collect, which allow for detailed characterization of navigational
behavior. Both objective and subjective measures of the participants’
experience indicated that they found DeFINE easy to use and the navigation
task it implemented well tolerable in terms of cybersickness and task
workload, irrespective of the ways in which they carried out the navigation
task (i.e., whether they were implicitly driven to perform it more quickly or
accurately via feedback scores). Together, these results validated DeFINE’s
capability and potential as a tool for investigating goal-directed navigation
in humans under a variety of conditions.
## 3 Conclusions
This paper presented the Delayed Feedback based Immersive Navigation
Environment (DeFINE) for studying goal-directed navigation in humans using VR.
Although similar frameworks have already been developed (Brookes ., 2020;
Vasser ., 2017; Commins ., 2020; Machado ., 2019; Wiener ., 2020; Geller .,
2007; Ayaz ., 2008, 2011; Solway ., 2013; Grübel ., 2017; Bebko Troje, 2020;
Starrett ., 2020), they are based on an open-loop stimulus–response
architecture that omits performance feedback to participants. DeFINE
distinguishes itself from the previous frameworks by implementing the closed-
loop stimulus–response–feedback architecture as its core element (Figures 1
and 2). The feedback is delayed by default in order to suit the needs of
typical navigation experiments, but if necessary, it can be made real-time
through relatively simple changes in DeFINE’s code so that the
stimulus–response–feedback loop is even more tightly closed.
As discussed in the introduction, the VR frameworks for navigation research
mainly differ in whether they are geared toward ease of use by limiting their
scope or wide applicability by providing general-purpose tools that demand
technical skills of end-users. In this spectrum, DeFINE aims to position
itself toward the ease-of-use end by focusing primarily on goal-directed
navigation tasks and also by making it possible to set up an experiment mostly
through GUIs and simple settings files (demonstrated in the video clips
available online). However, this does not mean that coding is absolutely
unnecessary or impossible in DeFINE. Indeed, some customization that goes
beyond the GUIs and settings files is expected, and for this purpose the
software is made open-source and its codebase is modularized.
The demonstration experiment showed the utility of DeFINE as a platform for
navigation research and its general friendliness to participants in VR
experiments. Additionally, this experiment demonstrated DeFINE’s potential as
a tool for testing hypotheses about the temporal aspects of navigational
behavior. The optional feature of seamlessly administering surveys within an
HMD enhances the immersion of the participants in VR, thereby improving the
quality of data collected via DeFINE. The optional leader-board enables
investigation of the effect of gamification on spatial navigation. Previous
studies have shown its impact in other domains of learning (Barata ., 2013;
Looyestyn ., 2017), but it is yet to be thoroughly explored for navigation-
related applications (Coutrot ., 2018; Coughlan ., 2019). These out-of-the-box
features of DeFINE, together with its customizability via the Unity software,
open up many new possibilities for human navigation research.
## 4 Open Practices Statements
The software used in the experiment reported in this article—the Delayed
Feedback based Immersive Navigation Environment (DeFINE)—is available at
https://github.com/ktiwari9/define-VR. The data and other materials for the
experiment are available upon request. The experiment was not preregistered.
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|
2024-09-04T02:54:57.841423 | 2020-03-06T13:09:56 | 2003.03175 | {
"authors": "Jens Hesse",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26080",
"submitter": "Jens Hesse",
"url": "https://arxiv.org/abs/2003.03175"
} | arxiv-papers | # Central leaves on Shimura varieties with parahoric reduction
Jens Hesse Technische Universität Darmstadt<EMAIL_ADDRESS>
###### Abstract
We investigate the geometry of the special fiber of the integral model of a
Shimura variety with parahoric level at a given prime place.
To be more precise, we deal with the definition of central leaves in this
situation, their local closedness, and the relationship between the folations
for varying parahoric level. This is connected to the verification of axioms
for integral models formulated by He and Rapoport.
###### Contents
1. 1 Background
1. 1.1 Shimura data of Hodge type
2. 1.2 Bruhat-Tits buildings
3. 1.3 Alteration of the Hodge embedding
4. 1.4 Siegel integral models
5. 1.5 Construction of the integral model
6. 1.6 Maps between Shimura varieties
7. 1.7 Local structure of the integral model
1. 1.7.1 Generizations and irreducible components
2. 1.7.2 Normalization and completion
8. 1.8 Hodge tensors and (lack of) moduli interpretation
1. 1.8.1 The story for the $\mathbb{C}$-valued points
2. 1.8.2 The story for the integral model
2. 2 Central leaves in the case of parahoric reduction
1. 2.1 Definition of central leaves
1. 2.1.1 An alternative characterization of the central leaves
2. 2.2 Local closedness of central leaves
1. 2.2.1 Topological lemmas
2. 2.2.2 Local closedness
3. 2.3 Quasi-isogenies of $p$-divisible groups
4. 2.4 Almost product structure
5. 2.5 Change of parahoric level
1. 2.5.1 The change-of-parahoric morphism
2. 2.5.2 Newton-Igusa variety and change-of-parahoric
Introduction
Shimura varieties are objects of arithmetic geometry (namely varieties over
number fields) that naturally arise in the search for generalized, non-abelian
reciprocity laws (i.e., in the Langlands program) and as moduli spaces of
abelian varieties (with certain extra structures on them). One way of
approaching these objects is to try to understand their mod-$p$ reduction
(which has to be carefully defined first). Insofar as a moduli interpretation
in the above sense exists and continues to exist likewise for the mod-$p$
reduction111There need not be a _literal_ moduli interpretation, but in any
event the stratifications in question derive from a close connection to moduli
problems., it allows us to stratify the moduli space according to several
invariants of the abelian varieties parametrized, e.g., the isomorphism
classes of their $p$-torsion. (An important observation is that these
stratifications genuinely live in the characteristic $p$ world, making use of
Frobenius endomorphisms and so on.) This, very roughly, is the general theme
everything in this article revolves around.
More precisely, we will be dealing with Shimura varieties of Hodge type and
parahoric level structure, at some fixed prime $v\mid p$ of the number field
over which the Shimura variety is defined. Under some reasonably mild
assumptions, cf. 1.16, Kisin and Pappas [KP15] constructed a canonical
integral model for such a Shimura variety. We try to understand some aspects
of the geometry of the special fiber of said integral model, namely the
central leaves (roughly the patches where the isomorphism class of the
$p$-divisible group associated with the abelian variety is constant); their
being locally closed and how they vary as the (parahoric) level is varied.
Let us now go into more detail.
On the integral model $\mathscr{S}_{K}$ ($K$ parahoric level) we have a
“universal” abelian scheme (the quotation marks indicating that it is not
really universal for some moduli problem on $\mathscr{S}_{K}$, but it comes
from a universal abelian scheme via pullback) and we have various kinds of
Hodge tensors. We also have a “universal” isogeny chain of abelian schemes
tightly connected to the “universal” abelian scheme. We define the _naive
central leaves_ on the special fiber $\overline{\mathscr{S}}_{K}$ to be the
loci where the isomorphism type of the geometric fibers of the $p$-divisible
group associated with the “universal” abelian scheme (the “universal”
$p$-divisible group) is constant (alternatively, we may use the “universal”
isogeny chain). We arrive at the non-naive version by taking into account the
Hodge tensors. This is the content of section 2.1.
Next we show
###### Theorem A:
(Corollary 2.12) The central leaves are locally closed.
using a somewhat simpler construction than the one given in [HK17]. By
foundational work of Oort [Oor04] we already know the naive central leaves to
be locally closed. We show that the central leaves are open and closed inside
the naive central leaves. Some basic topological considerations allow us to
treat this question in perfected formal neighborhoods, allowing us to phrase
it as a question about $p$-divisible groups with crystalline Tate tensors; a
question which was answered by Hamacher [Ham17]. This forms section 2.2.
Then we consider (tensor-respecting) self-quasi-isogenies of $p$-divisible
groups (with tensors). The main take-away here is that, if we consider the
geometric fibers of the “universal” isogeny chain of $p$-divisible groups in
the above setting, then
###### Theorem B:
(Example 2.21) The self-quasi-isogenies are independent of the level.
In section 2.4 we recall the almost product structure (under an additional
technical assumption 2.23 concerning the Rapoport-Zink uniformization map,
which is satisfied e.g. if our reductive group over $\mathbb{Q}_{p}$ is
residually split222A reductive group $G/\mathbb{Q}_{p}$ is _residually split_
if it has the same rank as its base change to the maximal unramified extension
of $\mathbb{Q}_{p}$. The name “residually split” derives from the fact that in
this case the maximal reductive quotients of parahoric group schemes
associated with $G$ are split reductive. [Zho18]), which expresses a variant
of the Igusa variety (the _Newton-Igusa variety_ , with quasi-isogenies
instead of isomorphisms) as a product of the Igusa variety and the Rapoport-
Zink space.
We apply this to the question of how the central leaves behave under change of
the parahoric level. We begin by constructing the change-of-parahoric
morphisms on the Shimura varieties, Igusa varieties, Rapoport-Zink spaces and
Newton-Igusa varieties. The almost product isomorphism is compatible with
these morphisms (Lemma 2.38).
From this we can derive
###### Theorem C:
(Corollary 2.42) The map between Igusa varieties for varying parahoric level
is an isomorphism.
This implies in particular
###### Theorem D:
(Corollary 2.43) The change-of-parahoric map between central leaves is
surjective.
This is the most difficult part of the axioms on integral models He and
Rapoport give in [HR17]. Our proof is independent of the one by Rong Zhou in
[Zho18], even though we use some of the results he gave in the first version
of the cited preprint, which did not contain a proof of the surjectivity.
Positively answering another conjecture by He and Rapoport [HR17, Rmk. 3.4],
we show
###### Theorem E:
(Corollary 2.48) The change-of-parahoric map between central leaves is the
composition of a flat universal homeomorphism of finite type and a finite
étale morphism.
### Acknowledgements
This article essentially is an extract of my doctoral thesis [Hes20] (another
extract333In particular, there is a large overlap between the “Background”
sections of the two articles., dealing with the EKOR stratification, is
[Hes20a]). I thank Torsten Wedhorn for suggesting the topic of the
dissertation, his support and patiently answering my questions. Moreover I
thank Eva Viehmann and Paul Hamacher for their hospitality and helpful
discussions during a month-long stay in Munich at the TU München. I am also
grateful to Timo Richarz and Timo Henkel for numerous helpful discussions.
This research was supported by the Deutsche Forschungsgemeinschaft (DFG),
project number WE 2380/5.
## 1 Background
### 1.1 Shimura data of Hodge type
This article deals with aspects of the geometry of Shimura varieties (of Hodge
type), which are the (systems of) varieties associated with Shimura data (of
Hodge type).
###### Definition 1.1.
A _Shimura datum of Hodge type_ is a pair $(G,X)$, where $G$ is a reductive
algebraic group over $\mathbb{Q}$ and
$X\subseteq\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})$
is a $G(\mathbb{R})$-conjugacy class
($\mathbb{S}:=\operatorname{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}$
being the Deligne torus) subject to the following conditions:
1. (1)
For $h\in X$, the induced Hodge structure
$\mathbb{S}\xrightarrow{h}G_{\mathbb{R}}\xrightarrow{\mathrm{Ad}}\operatorname{GL}(\operatorname{Lie}(G_{\mathbb{R}}))$
satisfies
$\operatorname{Lie}(G_{\mathbb{C}})=\operatorname{Lie}(G_{\mathbb{C}})^{-1,1}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{0,0}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{1,-1}$.
2. (2)
$\operatorname{int}(h(i))\colon G^{\mathrm{ad}}_{\mathbb{R}}\to
G^{\mathrm{ad}}_{\mathbb{R}}$ is a Cartan involution, i.e., $\\{g\in
G^{\mathrm{ad}}(\mathbb{C})\;|\;gh(i)=h(i)\overline{g}\\}$ is compact. Another
way of phrasing this condition: Every finite-dimensional real representation
$V$ of $G^{\mathrm{ad}}_{\mathbb{R}}$ carries a
$G^{\mathrm{ad}}_{\mathbb{R}}$-invariant bilinear form $\varphi$ such that
$(u,v)\mapsto\varphi(u,h(i)v)$ is symmetric and positive definite. It is
enough to show that this holds for one _faithful_ finite-dimensional real
representation $V$.
3. (3)
$G^{\mathrm{ad}}$ _cannot_ be non-trivially written as $G^{\mathrm{ad}}\cong
H\times I$ over $\mathbb{Q}$ with $\mathbb{S}\to
G_{\mathbb{R}}\xrightarrow{\mathrm{proj}}H_{\mathbb{R}}$ trivial.
4. (4)
There exists an embedding
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$, where
$(\operatorname{GSp}(V),S^{\pm})$ is the Shimura datum associated with a
finite-dimensional symplectic $\mathbb{Q}$-vector space $V$ (see below). That
is, we have an embedding $G\hookrightarrow\operatorname{GSp}(V)$ of
$\mathbb{Q}$-group schemes such that the induced map
$\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})\hookrightarrow\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},\operatorname{GSp}(V_{\mathbb{R}}))$
restricts to a map $X\hookrightarrow S^{\pm}$.
###### Example 1.2.
Let $W$ be a finite-dimensional $\mathbb{R}$-vector space.
$\mathbb{R}$-group homomorphisms $\mathbb{S}\to\operatorname{GL}(W)$ then
correspond to Hodge decompositions of $W$, i.e., to decompositions
$W_{\mathbb{C}}=\oplus_{(p,q)\in\mathbb{Z}^{2}}W_{\mathbb{C}}^{p,q}$, such
that $W_{\mathbb{C}}^{p,q}$ is the complex conjugate of $W_{\mathbb{C}}^{q,p}$
for all $(p,q)\in\mathbb{Z}^{2}$. Under this correspondence,
$h\colon\mathbb{S}\to\operatorname{GL}(W)$ corresponds to the Hodge
decomposition $W_{\mathbb{C}}^{p,q}=\\{w\in W_{\mathbb{C}}\;|\;\forall
z\in\mathbb{S}(\mathbb{R})=\mathbb{C}^{\times}\colon
h(z)w=z^{-p}\bar{z}^{-q}w\\}$. Hodge decompositions of $W$ of type
$(-1,0)+(0,-1)$ correspond to complex structures on $W$: If
$h\colon\mathbb{S}\to\operatorname{GL}(W)$ yields such a Hodge decomposition,
then $h(i)$ gives an $\mathbb{R}$-endomorphism $J$ of $W$ with $J\circ
J=-\operatorname{id}_{W}$.
Let $V=(V,\psi)$ be a finite-dimensional symplectic $\mathbb{Q}$-vector space.
We say that a complex structure $J$ on $V_{\mathbb{R}}$ is positive (resp.
negative) if $\psi_{J}:=\psi_{\mathbb{R}}(\\_,J\\_)$ is a positive definite
(resp. negative definite) symmetric bilinear form on $V_{\mathbb{R}}$. Define
$S^{+}$ (resp. $S^{-}$) to be the set of positive (resp. negative) complex
structures on $(V_{\mathbb{R}},\psi_{\mathbb{R}})$ and $S^{\pm}:=S^{+}\sqcup
S^{-}$.
We can make this more concrete: A symplectic basis of
$(V_{\mathbb{R}},\psi_{\mathbb{R}})$ is a basis $e_{1},\dotsc,\allowbreak
e_{g},e_{-g},\dotsc,\allowbreak e_{-1}$, such that $\psi_{\mathbb{R}}$ is of
the form
$\begin{pmatrix}&\tilde{I}_{g}\\\ -\tilde{I}_{g}&\end{pmatrix}$
with respect to this basis, where $\tilde{I}_{g}=\begin{pmatrix}&&1\\\
&\iddots&\\\ 1&&\end{pmatrix}$ is the antidiagonal identity
matrix.444Occasionally (in particular when doing concrete matrix
calculations), it is more convenient to number the basis vectors
$1,\dotsc,g,-1,\dotsc,-g$ instead of $1,\dotsc,g,-g,\dotsc,-1$. Then the
standard symplectic form is given by $\left(\begin{smallmatrix}&I_{g}\\\
-I_{g}&\end{smallmatrix}\right)$, $I_{g}$ being the $g\times g$ identity
matrix.
Let $J$ be the endomorphism of $V_{\mathbb{R}}$ of the form
$\begin{pmatrix}&-\tilde{I}_{g}\\\ \tilde{I}_{g}&\end{pmatrix}$
with respect to this basis. Then $J\in S^{+}$ and what we have described is a
surjective map
$\\{\text{symplectic bases of
}(V_{\mathbb{R}},\psi_{\mathbb{R}})\\}\twoheadrightarrow S^{+}.$
In particular we see that
$\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=\psi_{\mathbb{R}}\\}$
(by virtue of acting simply transitively on the symplectic bases) acts
transitively on
$S^{+}\cong\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})/\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J)$
(where we define
$\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J):=\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})\cap
O(V_{\mathbb{R}},\psi_{J})=U((V_{\mathbb{R}},J),\psi_{J})$ for a fixed choice
of $J\in S^{+}$) and therefore the general symplectic group
$\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=c\cdot\psi_{\mathbb{R}}\text{
for some }c\in\mathbb{R}^{\times}\\}$ acts transitively on $S^{\pm}$ (note
that the element of the form $e_{\pm i}\mapsto e_{\mp i}$ of
$\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}})$ for any given choice of
symplectic basis $\left(e_{i}\right)_{i}$ permutes $S^{+}$ and $S^{-}$).
###### Definition 1.3.
Condition (1) of Definition 1.1 implies that the action of
$\mathbb{G}_{m,\mathbb{R}}$ (embedded in $\mathbb{S}$ in the natural way) on
$\operatorname{Lie}(G_{\mathbb{R}})$ is trivial, so that $h$ induces a
homomorphism
${w\colon\mathbb{G}_{m,\mathbb{R}}\to\operatorname{Cent}(G_{\mathbb{R}})}$.
This homomorphism is independent of the choice of $h\in X$ and is called the
_weight homomorphism_ of $(G,X)$.
Moreover, we denote by $\\{\mu\\}$ the the $G(\mathbb{C})$-conjugacy class of
the cocharacter
$\mu_{h}:=h\circ(\operatorname{id}_{\mathbb{G}_{m,\mathbb{C}}},1)\colon\mathbb{G}_{m,\mathbb{C}}\to\mathbb{G}_{m,\mathbb{C}}^{2}\cong\mathbb{S}_{\mathbb{C}}\to
G_{\mathbb{C}}$, where $h$ is as above. Obviously, the conjugacy class
$\\{\mu\\}$ is independent of the particular choice of $h\in X$.
###### Remark 1.4.
Let $L/\mathbb{Q}$ be a field extension such that $G_{L}$ contains a split
maximal torus $T$. Let $W:=\operatorname{Norm}_{G(L)}(T)/T$ be the Weyl group.
Then the natural map
$W\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},T)\to
G(L)\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},G_{L})$
is bijective.
Since the left hand side remains unchanged if we go from $L=\bar{\mathbb{Q}}$
(where as usual $\bar{\mathbb{Q}}$ denotes an algebraic closure of
$\mathbb{Q}$) to $L=\mathbb{C}$, we see that $\\{\mu\\}$ contains a
cocharacter defined over $\bar{\mathbb{Q}}$ and that we may then also consider
$\\{\mu\\}$ as a $G(\bar{\mathbb{Q}})$-conjugacy class.
###### Definition 1.5.
The _reflex field_ $\mathbf{E}=\mathbf{E}(G,X)$ of $(G,X)$ is the field of
definition of $\\{\mu\\}$, i.e., the fixed field in $\bar{\mathbb{Q}}$ of
$\\{\gamma\in\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\;|\;\gamma(\\{\mu\\})=\\{\mu\\}\\}$.
###### Example 1.6.
The reflex field of the Shimura datum
$(\operatorname{GSp}_{2g,\mathbb{Q}},S^{\pm})$ of Example 1.2 is $\mathbb{Q}$.
To wit, one of the cocharacters in the conjugacy class $\\{\mu\\}$ is
$\mu(z)=\left(\begin{smallmatrix}z&&&&&\\\ &\ddots&&&&\\\ &&z&&&\\\ &&&1&&\\\
&&&&\ddots&\\\ &&&&&1\end{smallmatrix}\right).$
###### Notation 1.7.
We denote the ring of (rational) adeles by
$\mathbb{A}:=\mathbb{A}_{\mathbb{Q}}$, the subring of finite adeles by
$\mathbb{A}_{f}:=\mathbb{A}_{\mathbb{Q},f}$ and the subring of finite adeles
away from some fixed prime $p$ by $\mathbb{A}_{f}^{p}$.
###### Definition and Remark 1.8.
Let $K\subseteq G(\mathbb{A}_{f})$ be a compact open subgroup. The _Shimura
variety of level $K$ associated with $(G,X)$_ is the double coset space
$\operatorname{Sh}_{K}(G,X):=G(\mathbb{Q})\backslash(X\times(G(\mathbb{A}_{f})/K)).$
A priori, this is just a set, but if $K$ is sufficiently small (i.e., “neat”
in the sense of [Bor69, Pin90]), $\operatorname{Sh}_{K}(G,X)$ can be
canonically written as a finite disjoint union of hermitian symmetric
domains.555If $K$ fails to be sufficiently small, one might very reasonably
argue that our definition of the Shimura variety of level $K$ really is the
definition of the _coarse_ Shimura variety and that one should be working with
stacks instead. Since we will only be interested in sufficiently small level,
this is inconsequential for us. In particular, this gives
$\operatorname{Sh}_{K}(G,X)$ the structure of a complex manifold.
In fact, by the theorem of Baily-Borel, this complex manifold attains the
structure of a quasi-projective complex variety in a canonical way.
By work of Deligne, Milne and Borovoi, this variety is defined already (and
again in a canonical way) over the reflex field $\mathbf{E}$. So in
particular, it is defined over a number field independent of $K$. This is
important when varying $K$ and it is the reason why we consider the whole
Shimura variety instead of its connected components over $\mathbb{C}$ on their
own. It is possible for the Shimura variety to have multiple connected
components over $\mathbb{C}$ while being connected over $\mathbf{E}$.
More detailed explanations may be found in [Mil05].
### 1.2 Bruhat-Tits buildings
Let $K$ be a complete discrete valuation field with ring of integers
$\mathcal{O}$, uniformizer $\varpi$ and perfect residue field
$\kappa:=\mathcal{O}/\varpi$.
###### Notation 1.9.
For a (connected) reductive group $G$ over $K$, we denote by
$\mathcal{B}(G,K)$ the extended (or enlarged) and by
$\mathcal{B}^{\mathrm{red}}(G,K)$ the reduced (i.e., non-extended) Bruhat-Tits
building of $G$ over $K$ [BT84]. Moreover,
$\mathcal{B}^{\mathrm{abstract}}(G,K)$ denotes the underlying abstract
simplicial complex.
###### Remark 1.10.
Let $V$ be a finite-dimensional $K$-vector space.
As described in [KP15, 1.1.9] (originally in [BT84a]), the points of
$\mathcal{B}(\operatorname{GL}(V),K)$ correspond to graded periodic lattice
chains $(\mathcal{L},c)$, i.e.,
* •
$\emptyset\neq\mathcal{L}$ is a totally ordered set of full
$\mathcal{O}$-lattices in $V$ stable under scalar multiplication (i.e.,
$\Lambda\in\mathcal{L}\iff\varpi\Lambda\in\mathcal{L}$),
* •
$c\colon\mathcal{L}\to\mathbb{R}$ is a strictly decreasing function such that
$c(\varpi^{n}\Lambda)=c(\Lambda)+n$.
###### Remark 1.11.
Fix such an $\mathcal{L}$ and let $\Lambda^{0}\in\mathcal{L}$. Then every
homothety class of lattices has a unique representative $\Lambda$ such that
$\Lambda\subseteq\Lambda^{0}$ and $\Lambda\not\subseteq\varpi\Lambda^{0}$.
Consider such representatives $\Lambda^{i}$ for all of the distinct homothety
classes of lattices that make up $\mathcal{L}$. Because $\mathcal{L}$ is
totally ordered and $\Lambda^{i}\not\subseteq\varpi\Lambda^{0}$, it follows
that $\Lambda^{i}\supseteq\varpi\Lambda^{0}$ for all $i$ and that
$\left\\{\Lambda^{i}/\varpi\Lambda^{0}\right\\}_{i}$ is a flag of non-trivial
linear subspaces of $\Lambda^{0}/\varpi\Lambda^{0}\cong\kappa^{n}$, where
$n:=\dim V$. Consequently, the number $r$ of homothety classes is in
$\\{1,\dotsc,n\\}$; it is called the _period length_ (or _rank_) of
$\mathcal{L}$. Numbering the $\Lambda^{i}$ in descending order we hence obtain
$r$ lattices $\Lambda^{0},\Lambda^{1},\dotsc,\Lambda^{r-1}$ such that
$\Lambda^{0}\supsetneqq\Lambda^{1}\supsetneqq\dotsb\supsetneqq\Lambda^{r-1}\supsetneqq\varpi\Lambda^{0}$
(1.12)
and $\mathcal{L}$ is given by the the strictly descending sequence of lattices
$\Lambda^{qr+i}=\varpi^{q}\Lambda^{i},\quad q\in\mathbb{Z},\;0\leq i<r.$
###### Remark 1.13.
Let $V$ be a finite-dimensional symplectic $K$-vector space.
$\mathcal{B}(\operatorname{GSp}(V),K)$ embeds into the subset of
$\mathcal{B}(\operatorname{GL}(V),K)$ consisting of those $(\mathcal{L},c)$
such that $\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}$.
Passing to the underlying abstract simplicial complexes means forgetting about
the grading $c$ and
$\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)=\\{\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GL}(V),K)\;|\;\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}\\}.$
If $\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)$
and $\\{\Lambda^{i}\\}_{i}$ is as in Remark 1.11, then there is an involution
$t\colon\mathbb{Z}\to\mathbb{Z}$ with
$\left(\Lambda^{i}\right)^{\vee}=\Lambda^{t(i)}$, $t(i+qr)=t(i)-qr$, and
$i<j\implies t(i)>t(j)$. So $-a:=t(0)>t(1)>\dotsb>t(r)=-a-r$, which implies
$t(i)=-i-a$. Thus $i_{0}-t(i_{0})=2i_{0}+a\in\\{0,1\\}$ for some unique
$i_{0}\in\mathbb{Z}$. Hence, upon renumbering the $\Lambda^{i}$, we may assume
that $a\in\\{0,1\\}$.
We therefore have
$\displaystyle\varpi\Lambda^{0}\subsetneqq\Lambda^{r-1}\subsetneqq\Lambda^{r-2}\subsetneqq\dotsb\subsetneqq\Lambda^{0}\subseteq\left(\Lambda^{0}\right)^{\vee}=\Lambda^{-a}\subsetneqq\left(\Lambda^{1}\right)^{\vee}=\Lambda^{-1-a}$
$\displaystyle\subsetneqq\dotsb\subsetneqq\left(\Lambda^{r-1}\right)^{\vee}=\Lambda^{-r+1-a}\subseteq\Lambda^{-r}=\varpi^{-1}\Lambda^{0}.$
### 1.3 Alteration of the Hodge embedding
###### Notation 1.14.
Let $E$ be a finite field extension of $\mathbb{Q}_{p}$.
Denote by $\breve{E}$ the completion of the maximal unramified extension of
$E$ (hence $\breve{E}=E\cdot\breve{\mathbb{Q}}_{p}$).
###### Remark 1.15.
If $E/\mathbb{Q}_{p}$ is unramified, then ${{\cal
O}_{\breve{E}}}=W(\bar{\mathbb{F}}_{p})$, $\bar{\mathbb{F}}_{p}$ denoting an
algebraic closure of $\mathbb{F}_{p}$ and
$W\colon\mathrm{Ring}\to\mathrm{Ring}$ being the ($p$-adic) Witt vectors
functor. This generalizes to the ramified case using _ramified Witt vectors_
instead, see e.g. [Haz78, Chap. IV, (18.6.13)] or [Ahs11, Chapter 1].
Let $(G,X)$ be a Shimura datum of Hodge type, let
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$ be an embedding as in
Definition 1.1 (4), and let $x\in\mathcal{B}(G,\mathbb{Q}_{p})$ be a point in
the Bruhat-Tits building of $G$ over $\mathbb{Q}_{p}$.
We consider the associated Bruhat-Tits scheme ${\cal G}_{x}$, i.e., the affine
smooth model of $G_{\mathbb{Q}_{p}}$ over $\mathbb{Z}_{p}$ such that ${\cal
G}_{x}(\breve{\mathbb{Z}}_{p})\subseteq G(\breve{\mathbb{Q}}_{p})$ is the
stabilizer of the facet of $x$ in ${\cal
B}(G,\breve{\mathbb{Q}}_{p})\overset{\text{\cite[cite]{[\@@bibref{}{landvogt}{}{},
Prop.\leavevmode\nobreak\
2.1.3]}}}{=}\mathcal{B}(G,\mathbb{Q}^{\mathrm{ur}}_{p})$. Let $K_{p}:={\cal
G}_{x}(\mathbb{Z}_{p})\subseteq G(\mathbb{Q}_{p})$ and let $K^{p}\subseteq
G(\mathbb{A}_{f}^{p})$ be a sufficiently small open compact subgroup. Define
$K:=K_{p}K^{p}\subseteq G(\mathbb{A}_{f})$.
###### Assumptions 1.16.
From now on, we will always make the following assumptions:
* •
$\mathcal{G}_{x}=\mathcal{G}_{x}^{\circ}$ is connected.
* •
$G$ splits over a tamely ramified extension of $\mathbb{Q}_{p}$.
* •
$p\nmid\\#\pi_{1}(G^{\mathrm{der}})$.
###### Notation 1.17.
In order not to make notation overly cumbersome, we usually denote the base
change $G_{\mathbb{Q}_{p}}$ of $G$ to $\mathbb{Q}_{p}$ by $G$ again. (Later,
we will almost exclusively be dealing with $G_{\mathbb{Q}_{p}}$.)
Under the above assumptions, Kisin and Pappas construct in [KP15, section 1.2]
(building on [Lan00]) a toral $G(\breve{\mathbb{Q}}_{p})$\- and
$\operatorname{Gal}(\breve{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$-equivariant
embedding
$\iota\colon\mathcal{B}(G,\breve{\mathbb{Q}}_{p})\to\mathcal{B}(\operatorname{GL}(V),\breve{\mathbb{Q}}_{p}),$
restricting666In this case, this is obvious from Galois-equivariance, since
$\mathcal{B}(G,K)=\mathcal{B}(G,K^{\prime})^{\operatorname{Gal}(K^{\prime}/K)}$
if $K^{\prime}/K$ is an unramified extension. to a map
$\iota\colon\mathcal{B}(G,\mathbb{Q}_{p})\to\mathcal{B}(\operatorname{GL}(V),\mathbb{Q}_{p})$.
$\iota$ being a toral embedding means the following: $\iota$ is isometric
after a suitable normalization of the norm on
$\mathcal{B}(G,\breve{\mathbb{Q}}_{p})$; moreover it is the canonical
extension of a map
${\mathcal{B}^{\mathrm{red}}(G,\breve{\mathbb{Q}}_{p})\to\mathcal{B}^{\mathrm{red}}(\operatorname{GL}(V),\breve{\mathbb{Q}}_{p})}$
such that for each maximal $\breve{\mathbb{Q}}_{p}$-split torus $S\subseteq G$
there exists a maximal $\breve{\mathbb{Q}}_{p}$-split torus
$T\subseteq\operatorname{GL}(V)$ such that
$G\hookrightarrow\operatorname{GL}(V)$ maps $S$ into $T$ and $\iota$ restricts
to a map between the reduced apartments associated with $(G,S)$ and
$(\operatorname{GL}(V),T)$, respectively, compatible with translations.
This embedding $\iota$ depends on some choices: By assumption, there exists a
tamely ramified Galois extension $\tilde{K}/\mathbb{Q}_{p}$ with finite
inertia group such that $G_{\tilde{K}}$ is split reductive. Let
$H\to\operatorname{Spec}\mathbb{Z}_{p}$ be the split Chevalley form of $G$
over $\mathbb{Z}_{p}$. Then $\iota$ depends on the choice of
* •
an isomorphism $G_{\tilde{K}}\cong H_{\tilde{K}}$,
* •
a pinning
$\left(T,M,f,R,\Delta,\left(X_{\alpha}\right)_{\alpha\in\Delta}\right)$ of $H$
(cf. [SGA3, Exp. XXIII]777Since $\operatorname{Spec}\mathbb{Z}_{p}$ is
connected, some technicalities from _loc. cit._ disappear here.), which also
entails the following: Let $B$ be the Borel subgroup of $H$ corresponding to
$\Delta$. When we talk about roots, it is with respect to $T$, and when we
talk about positive roots and so on, it is with respect to $(T,B)$. We also
fix a hyperspecial vertex $x_{o}$ in $\mathcal{B}(H,\mathbb{Q}_{p})$ with
stabilizer $H(\mathbb{Z}_{p})$.
* •
for every $\mathbb{Q}_{p}$-irreducible summand888Note that by reductivity and
$\operatorname{char}(\mathbb{Q}_{p})=0$, every finite-dimensional
representation of $G_{\mathbb{Q}_{p}}$ is completely reducible. $V_{i}$ of the
representation $G_{\mathbb{Q}_{p}}\to\operatorname{GL}(V)_{\mathbb{Q}_{p}}$ a
lattice $\Lambda_{i}=U(\mathfrak{n}^{-})v_{i}\subseteq V_{i}$, where
$v_{i}\neq 0$ is a highest weight vector of $V_{i}$ and $\mathfrak{n}^{-}$ is
the (strictly) negative root space inside the Lie algebra of $H$ over
$\mathbb{Z}_{p}$,
* •
a grading $c_{\Lambda_{i}}+t_{i}$ of the lattice chain
$\\{p^{n}\Lambda_{i}\\}_{n\in\mathbb{Z}}$ given by real numbers
$t_{i}\in\mathbb{R}$. Here
${(c_{\Lambda_{i}}+t_{i})(p^{n}\Lambda_{i})}:=n+t_{i}$.
By [KP15, Lemma 2.3.3], we can (and do) arrange these choices in such a way
that $\iota$ factors
$\mathcal{B}(G,\breve{\mathbb{Q}}_{p})\xrightarrow{j}\mathcal{B}(\operatorname{GSp}(V),\breve{\mathbb{Q}}_{p})\to\mathcal{B}(\operatorname{GL}(V),\breve{\mathbb{Q}}_{p}),$
where the last map is the canonical toral embedding (whose definition will be
clear from Remark 1.13). Again, $j$ restricts to a map
$j\colon\mathcal{B}(G,\mathbb{Q}_{p})\hookrightarrow\mathcal{B}(\operatorname{GSp}(V),\mathbb{Q}_{p}).$
(1.18)
Let $y=(\mathcal{L},c)$ be the image of $x\in\mathcal{B}(G,\mathbb{Q}_{p})$
under the map (1.18) and let $\left(\Lambda^{i}\right)_{i},r,a$ be as in
Remark 1.13. We define
$N_{p}:=\operatorname{Stab}_{\operatorname{GSp}(V)(\mathbb{Q}_{p})}(\mathcal{L})$.
Consider the symplectic $\mathbb{Q}$-vector space
$V^{\S}:=\bigoplus_{i=-(r-1)-a}^{r-1}V$
(direct sum of symplectic spaces, i.e., if $\psi$ denotes the symplectic form
on $V$, then the symplectic form $\psi^{\S}$ on $V^{\S}$ is given by
$\bigoplus_{i=-(r-1)-a}^{r-1}\psi$) and the lattice in
$V^{\S}_{\mathbb{Q}_{p}}$
$\Lambda^{\S}:=\bigoplus_{i=-(r-1)-a}^{r-1}\Lambda^{i}.$
By replacing $\Lambda^{\S}$ by a homothetic lattice, we may assume that
$\Lambda^{\S}\subseteq\left(\Lambda^{\S}\right)^{\vee}$ (hence
$\left(\Lambda^{\S}\right)^{\vee\vee}=\Lambda^{\S}$).
We have a diagonal embedding
$\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$
and (with calligraphic script meaning that we talk about Bruhat-Tits group
schemes)
$\mathcal{GSP}(V)_{y}=\operatorname{Stab}_{\operatorname{GSp}(V)}(\mathcal{L})=\bigcap\operatorname{Stab}_{\operatorname{GSp}(V)}(\Lambda^{i})\subseteq\operatorname{GSp}(\Lambda^{\S})\subseteq\operatorname{GL}(\Lambda^{\S})\subseteq\mathcal{GL}(V)_{\Lambda^{\S}},$
so that upon replacing our original embedding
$G\hookrightarrow\operatorname{GSp}(V)$ by the embedding
$G\hookrightarrow\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$,
we can assume that the parahoric subgroup on the Siegel side is given as
$\operatorname{Stab}_{\operatorname{GL}(V^{\S})(\mathbb{Q}_{p})}(\Lambda^{\S})\cap\operatorname{GSp}(V^{\S})(\mathbb{Q}_{p}),$
i.e., that our level is essentially given by a single lattice, albeit one that
is (in general) not self-dual. This is [KP15, 2.3.15].
### 1.4 Siegel integral models
With notation as above let
$\displaystyle N_{p}$
$\displaystyle:=\operatorname{Stab}_{\operatorname{GSp}(V)(\mathbb{Q}_{p})}(\mathcal{L})\quad\text{(as
before)},$ $\displaystyle J_{p}$
$\displaystyle:=\operatorname{Stab}_{\operatorname{GL}(V^{\S})(\mathbb{Q}_{p})}(\Lambda^{\S})\cap\operatorname{GSp}(V^{\S})(\mathbb{Q}_{p}).$
Let $N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$ and
$J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ be sufficiently
small open compact subgroups, and $N:=N_{p}N^{p}$, $J:=J_{p}J^{p}$.
In this subsection, we are going to describe integral models of
$\operatorname{Sh}_{N}(\operatorname{GSp}(V),S^{\pm})$ and of
$\operatorname{Sh}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ over
$\mathbb{Z}_{(p)}$ and relate the two.
###### Remark 1.19.
By [RZ96, Definition 6.9], the integral model
$\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})$ is given by the moduli
problem $(\mathbb{Z}_{(p)}\text{-scheme})\ni
S\mapsto\left\\{(A,\bar{\lambda},\eta^{p})\right\\}/{\scriptstyle\cong}$,
where:
1. (a)
$A=\left(A_{\Lambda}\right)_{\Lambda\in\mathcal{L}}$ is an $\mathcal{L}$-set
of abelian schemes, i.e.,
* •
for every $\Lambda\in\mathcal{L}$, an abelian $S$-scheme up to
$\mathbb{Z}_{(p)}$-isogeny $A_{\Lambda}$ (i.e., $A_{\Lambda}$ is an object of
the category $(\text{abelian }S\text{-schemes})\otimes\mathbb{Z}_{(p)}$, where
the category $\mathcal{A}\otimes R$ for $\mathcal{A}$ an preadditive category
and $R$ a ring has the same objects as $\mathcal{A}$ and
$\operatorname{Hom}_{\mathcal{A}\otimes
R}(X,Y)=\operatorname{Hom}(X,Y)\otimes_{\mathbb{Z}}R$ for all objects $X,Y$),
* •
for every inclusion $\Lambda_{1}\subseteq\Lambda_{2}$ a
$\mathbb{Z}_{(p)}$-isogeny $\rho_{\Lambda_{2},\Lambda_{1}}\colon
A_{\Lambda_{1}}\to A_{\Lambda_{2}}$,
* •
$\rho_{\Lambda_{3},\Lambda_{1}}=\rho_{\Lambda_{3},\Lambda_{2}}\circ\rho_{\Lambda_{2},\Lambda_{1}}$
if $\Lambda_{1}\subseteq\Lambda_{2}\subseteq\Lambda_{3}$ in $\mathcal{L}$,
* •
the height of $\rho_{\Lambda_{2},\Lambda_{1}}$ is
$\log_{p}|\Lambda_{2}/\Lambda_{1}|$. Here $\rho_{\Lambda_{2},\Lambda_{1}}$
gives rise to a well-defined homomorphism of $p$-divisible groups, and what we
mean is that the kernel of this homomorphism (which is a finite locally free
commutative group scheme, which we also refer to simply as the kernel of
$\rho_{\Lambda_{2},\Lambda_{1}}$) is to have order
$|\Lambda_{2}/\Lambda_{1}|$.
* •
For every $\Lambda\in\mathcal{L}$, there is an isomorphism (called
_periodicity isomorphism_) $\theta_{\Lambda}\colon A_{\Lambda}\to
A_{p\Lambda}$ such that
$\rho_{\Lambda,p\Lambda}\circ\theta_{\Lambda}=[p]\colon A_{\Lambda}\to
A_{\Lambda}$ is the multiplication-by-$p$ isogeny.
2. (b)
$\bar{\lambda}\colon A\to\tilde{A}$ is a $\mathbb{Q}$-homogeneous principal
polarization, i.e., a $\underline{\mathbb{Q}^{\times}}$-orbit of a principal
polarization $\lambda\colon A\to\tilde{A}$. Here $\tilde{A}$ is the
$\mathcal{L}$-set of abelian schemes over $S$ up to prime-to-$p$ isogeny given
by $\tilde{A}_{\Lambda}:=(A_{\Lambda^{\vee}})^{\vee}$. And being a
polarization $\lambda$ means being a quasi-isogeny of $\mathcal{L}$-sets
$\lambda\colon A\to\tilde{A}$ such that
$A_{\Lambda}\xrightarrow{\lambda_{\Lambda}}\tilde{A}_{\Lambda}=(A_{\Lambda^{\vee}})^{\vee}\xrightarrow{\varrho_{\Lambda^{\vee},\Lambda}^{\vee}}(A_{\Lambda})^{\vee}$
is a polarization of $A_{\Lambda}$ for all $\Lambda$. If $\lambda_{\Lambda}$
can be chosen to be an isomorphism up to prime-to-$p$ isogeny for all
$\Lambda$, then we speak of a principal polarization. In that case, when
referring to $\lambda_{\Lambda}$, we mean a $\lambda_{\Lambda}$ which is an
isomorphism up to prime-to-$p$ isogeny.
3. (c)
$\eta^{p}$ is a level-$N^{p}$-structure, i.e. (if $S$ is connected), it is a
$\pi_{1}(S,s)$-invariant $N^{p}$-orbit of symplectic similitudes
$\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to H_{1}(A_{s},\mathbb{A}_{f}^{p})$
(where $s$ is some geometric basepoint and $H_{1}(A_{s},\mathbb{A}_{f}^{p})$
with its $\pi_{1}(S,s)$-action corresponds to the Tate
$\mathbb{A}_{f}^{p}$-module of $A$ (cf. [RZ96, 6.8]), which is a smooth
$\mathbb{A}_{f}^{p}$-sheaf). Note that this forces the abelian schemes
$A_{\Lambda}$ to be $(\dim_{\mathbb{Q}}V)$-dimensional.
###### Definition 1.20.
Set $\Lambda^{\S}_{\mathbb{Z}_{(p)}}:=\Lambda^{\S}_{\mathbb{Z}_{p}}\cap
V^{\S}_{\mathbb{Q}}=\prod_{i=-(r-1)-a}^{r-1}\Lambda_{\mathbb{Z}_{(p)}}^{i}$.
We choose a lattice $\Lambda^{\S}_{\mathbb{Z}}\subseteq V^{\S}$ such that
$\Lambda^{\S}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}=\Lambda^{\S}_{\mathbb{Z}_{(p)}}$
and $\Lambda^{\S}_{\mathbb{Z}}\subseteq(\Lambda^{\S}_{\mathbb{Z}})^{\vee}$.
###### Remark 1.21.
Set
$d:=\bigl{|}\left(\Lambda_{\mathbb{Z}}^{\S}\right)^{\vee}/\Lambda_{\mathbb{Z}}^{\S}\bigr{|}$.
By [Kis10, 2.3.3, 3.2.4], the integral model
$\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ is given by the
moduli problem $(\mathbb{Z}_{(p)}\text{-schemes})\ni
S\mapsto\left\\{(A^{\S},\lambda^{\S},\epsilon^{p})\right\\}/{\scriptstyle\cong}$,
where
1. (a)
$A^{\S}$ is an abelian scheme over $S$ up to $\mathbb{Z}_{(p)}$-isogeny,
2. (b)
$\lambda^{\S}\colon A^{\S}\to\left(A^{\S}\right)^{\vee}$ is a polarization of
degree $d$ (i.e., the polarization of the (well-defined) associated
$p$-divisible group has degree $d$),
3. (c)
$\epsilon^{p}$ is a level-$J^{p}$-structure, i.e. (if $S$ is connected), it is
a $\pi_{1}(S,s)$-invariant $J^{p}$-orbit of symplectic similitudes
$\epsilon^{p}\colon V^{\S}_{\mathbb{A}_{f}^{p}}\to
H_{1}(A^{\S}_{s},\mathbb{A}_{f}^{p})$. Note that this forces the abelian
schemes $A^{\S}$ to be $(\dim_{\mathbb{Q}}V^{\S})$-dimensional.
This completes the descriptions of the moduli problems, and we turn to the
question of the relationship between the two. Consider (for appropriate
$N^{p},J^{p}$; see below) the morphism
$\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$
given on $S$-valued points by sending $(A,\bar{\lambda},\eta^{p})$ to
$(A^{\S},\lambda^{\S},\epsilon^{p})$, where
1. (a)
$\displaystyle A^{\S}:=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$,
2. (b)
$\displaystyle\lambda^{\S}:=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{\Lambda^{i}}\right)$,
3. (c)
$\epsilon^{p}$ is the product $\prod_{i=-(r-1)-a}^{r-1}\eta^{p}$, to be
interpreted as the product over $\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to
H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong
H_{1}(A_{s},\mathbb{A}_{f}^{p})$, where the isomorphism
$H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong
H_{1}(A_{s},\mathbb{A}_{f}^{p})$ is by definition the identity for some fixed
$i=i_{0}$ and otherwise induced by the transition map
$\rho_{\Lambda^{i},\Lambda^{i_{0}}}$. We need that $N^{p}$ is mapped into
$J^{p}$ by $\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$
for this to make sense.
###### Lemma 1.22.
Let $S$ be a scheme, $\ell\neq p$ prime numbers. If $\ell$ does not appear as
a residue characteristic of $S$, then the Tate module functors
$\displaystyle H_{1}(\\_,\mathbb{Z}_{\ell})$
$\displaystyle\colon(\text{abelian }S\text{-schemes})\to(\text{\~{A}©tale
}\mathbb{Z}_{\ell}\text{-local systems on }S),$ $\displaystyle
H_{1}(\\_,\mathbb{Q}_{\ell})$ $\displaystyle\colon(\text{abelian
}S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{Q}_{\ell}\text{-local systems
on }S)$
(cf. [Gro74, III, 5.4 and 6.2] for precise definitions) are faithful.
If only $p$ and $0$ appear as residue characteristics of $S$, then the Tate
module functor
$H_{1}(\\_,\mathbb{A}_{f}^{p})\colon(\text{abelian
}S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{A}_{f}^{p}\text{-local systems
on }S)$
is faithful.
###### Proof:
First note that the statements about $H_{1}(\\_,\mathbb{Q}_{\ell})$ and
$H_{1}(\\_,\mathbb{A}_{f}^{p})$ follows from the statement about
$H_{1}(\\_,\mathbb{Z}_{\ell})$, which is why it is enough to only look at
$H_{1}(\\_,\mathbb{Z}_{\ell})$.
A homomorphism of abelian $S$-schemes $f\colon A\to B$ vanishes if and only if
it vanishes over every (geometric) fiber of $S$: Indeed, if it vanishes
fiberwise, then it is flat by the fiber criterion for flatness. Applying that
criterion again we see that the closed immersion and fiberwise isomorphism
$\ker(f)\hookrightarrow A$ is flat, which means that is an isomorphism.
This way we are reduced to the case where $R$ is an (algebraically closed)
field of characteristic different from $\ell$. In this setting the
faithfulness is well-known (the salient point being that the $\ell$-primary
torsion is dense). □
###### Lemma 1.23.
Let $H$ be a totally disconnected locally compact999By (our) definition,
locally compact implies Hausdorff. group (i.e., a locally profinite group) and
let $N\subseteq H$ be a compact subgroup. Then
$N=\bigcap_{\begin{subarray}{c}N\subseteq J\\\ J\subseteq H\text{ open compact
subgrp.}\end{subarray}}J.$
Note that this is (a variant of) a well-known theorem by van Dantzig if
$N=\\{1\\}$ [Dan36].
###### Proof:
We make use of the following fact [AT08, Prop. 3.1.7]: A Hausdorff space is
locally compact and totally disconnected if and only if the open compact sets
form a basis of the topology. (Van Dantzig’s theorem is the group version of
this, which talks only about a neighborhood basis of the identity and open
compact _subgroups_.)
First we show that $N$ is contained in some open compact subset $K\subseteq
H$. For every $x\in N$ choose a compact open neighborhood $x\in K_{x}\subseteq
H$. This is possible by the fact cited above. Then there is a finite subset
$I\subseteq N$ such that $N\subseteq\bigcup_{x\in I}K_{x}=:K$.
Next, for every $x\in N$ choose an open neighborhood of the identity $U_{x}$
such that $xU_{x}K\subseteq K$. With $N\subseteq U:=\bigcup_{x\in N}xU_{x}$ we
obtain $UK\subseteq K$. Replacing $U$ by $U\cap U^{-1}$, we may moreover
assume it is symmetric. The subgroup generated by $U$ is open (hence closed)
and contained in $K$, hence is an open compact subgroup.
Thus $N$ even is contained in an open compact sub _group_ ; in other words, we
may assume that $H$ is compact, i.e., is a profinite group.
Then $H/N$ is compact101010Hausdorff quotient spaces of compact spaces are
compact again, but for “locally compact” the analogous statement is not true
in general! and totally disconnected111111Take $x,y\in H$ such that $xN\neq
yN$. We show that any subspace $S\subseteq H/N$ containing both $xN$ and $yN$
is disconnected. Let $U\subseteq H/N$ be a neighborhood of $xN$ not containing
$yN$. Let $x\in V\subseteq\pi^{-1}(U)$ be open and compact, where $\pi\colon
H\to H/N$ is the projection. Then $yN\notin\pi(V)\subseteq H/N$ is open and
compact (hence closed) and we have $S=(\pi(V)\cap S)\sqcup S\setminus\pi(V)$
where both $\pi(V)\cap S$ and $S\setminus\pi(V)$ are open in $S$. This shows
that $S$ is disconnected. (i.e., is a Stone space). By the fact cited above,
$H/N\supseteq\\{1\\}=\bigcap_{L\subseteq H/N\text{ open compact subset}}L.$
Observe that the quotient map $H\to H/N$ is proper to deduce
$N=\bigcap_{\begin{subarray}{c}N\subseteq M\\\ M\subseteq H\text{ open compact
subset}\end{subarray}}M.$
Say $M$ is an open and compact subset of $H$ containing $N$. As we have shown
above, there is an open compact subgroup $J\subseteq H$ in between $N$ and
$M$, and this is all we need to complete the proof. □
###### Proposition 1.24.
For every compact open subgroup
$N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$
$\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$
is a well-defined morphism for all compact open subgroups $N^{p}\subseteq
J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ and is a closed
immersion for all sufficiently small compact open subgroups $N^{p}\subseteq
J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$.
###### Proof:
The fact that it’s well-defined is clear from the construction.
To show the second statement, as in [Del71, Prop. 1.15], it is enough to show
that
$\mathscr{S}_{N_{p}N^{p}}(\operatorname{GSp}(V),S^{\pm})\to\varprojlim_{J^{p}}\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$
is a closed immersion, i.e., a proper monomorphism.
We begin by proving that it is a monomorphism, i.e., injective on $S$-valued
points ($S$ arbitrary $\mathbb{Z}_{(p)}$-scheme). So, say
$(A_{1},\lambda_{1},\eta_{1}^{p})$ and $(A_{2},\lambda_{2},\eta_{2}^{p})$ both
map to $(A^{\S},\lambda^{\S},\epsilon_{J^{p}}^{p})$. That means precisely that
there is an isomorphism of abelian $S$-schemes up to
$\mathbb{Z}_{(p)}$-isogeny
$\phi\colon\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\cong}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}$
such that
$\phi^{\vee}\circ\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{2,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{2,\Lambda^{i}}\right)\circ\phi=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{1,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{1,\Lambda^{i}}\right)$
and
$H_{1}(\phi,\mathbb{A}_{f}^{p})\circ\epsilon_{1,J^{p}}^{p}=\epsilon_{2,J^{p}}^{p}\mod{J^{p}}.$
We claim that $\phi$ comes from isomorphisms
$\phi_{i}\colon A_{1,\Lambda^{i}}\xrightarrow{\cong}A_{2,\Lambda^{i}}.$
Certainly there is but one candidate for $\phi_{i}$: define $\phi_{i}$ to be
the composition
$A_{1,\Lambda^{i}}\xrightarrow{\mathrm{incl}}\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\phi}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}\xrightarrow{\mathrm{proj}}A_{2,\Lambda^{i}}.$
Our claim then is that
$\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}.$
Apply $H^{1}(\\_,\mathbb{A}_{f}^{p})$ on both sides. For the left hand side,
we have
$H_{1}(\phi,\mathbb{A}_{f}^{p})=\epsilon_{2,J^{p}}^{p}\circ\left(\epsilon_{1,J^{p}}^{p}\right)^{-1}\mod{J^{p}}.$
and the right hand side of this equation is block diagonal. So
$H_{1}(\phi,\mathbb{A}_{f}^{p})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{A}_{f}^{p})\mod{J^{p}}.$
Since (by Lemma 1.23)
$N^{p}=\bigcap_{\begin{subarray}{c}N_{\ell}\subseteq J_{\ell}\\\
J_{\ell}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{Q}_{\ell})\text{ cpt. open
subgrp.}\end{subarray}}J_{\ell},$
it follows that (with $\ell\neq p$)
$H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})\mod{N_{\ell}},$
hence (since $N_{\ell}$ acts block-diagonally) that
$H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})$.
Since $H_{1}(\\_,\mathbb{Q}_{\ell})$ is faithful (Lemma 1.22), this implies
$\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}$, as desired.
Next, consider the extension by zero of
$\left(H_{1}(\rho_{1/2,\Lambda^{j},\Lambda^{i}},\mathbb{A}_{f}^{p})\right)_{i,j}$
(where for “$1/2$” either “$1$” or “$2$” can be plugged in) to a map
$H_{1}(A^{\S},\mathbb{A}_{f}^{p})\to H_{1}(A^{\S},\mathbb{A}_{f}^{p})$. Under
the isomorphism given by the $J^{p}$-level structure this corresponds, up to
the $J^{p}$-action, to the map $V^{\S}_{\mathbb{A}_{f}^{p}}\to
V^{\S}_{\mathbb{A}_{f}^{p}}$ given by mapping the $i$’th copy of
$V_{\mathbb{A}_{f}^{p}}$ identically to the $j$’th copy and the rest to zero.
Thus $\rho_{1/2,i,j}$ yield the same up to $J^{p}$ after applying
$H_{1}(\\_,\mathbb{A}_{f}^{p})$, hence they are equal in the
$\mathbb{Z}_{(p)}$-isogeny category.
Consequently, $\chi$ is a monomorphism.
For properness, we will use the valuative criterion. Let $R$ be a discrete
valuation ring with field of fractions $K$ and assume that a $K$-point
$A^{\S}=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$ with its additional
structures coming from $(A_{\Lambda^{i}})_{i}$ extends to an $R$-point
$\mathcal{A}^{\S}$. Consider the map $A^{\S}\to A_{\Lambda^{i_{0}}}\to A^{\S}$
where the first map is a projection and the second an inclusion. By the Néron
mapping property, this extends to a map $\mathcal{A}^{\S}\to\mathcal{A}^{\S}$.
Define $\mathcal{A}_{\Lambda^{i_{0}}}$ to be the image of this map.
The Néron mapping property also allows us to extend the transition isogenies
$\rho_{\Lambda^{i_{0}},\Lambda^{j_{0}}}\colon\allowbreak{A_{\Lambda^{j_{0}}}\to
A_{\Lambda^{i_{0}}}}$, $i_{0}\leq j_{0}$, the periodicity isomorphisms, and
the polarization.
Since $\pi_{1}(\operatorname{Spec}K)$ surjects onto
$\pi_{1}(\operatorname{Spec}R)$ (see [Stacks, Tag 0BQM]), extending the level
structure away from $p$ is trivial. □
### 1.5 Construction of the integral model
Let $\mathbf{E}$ be the reflex field of $(G,X)$ and $v\mid p$ a place of
$\mathbf{E}$.
As the first step towards the construction of the integral model, we define
$\mathscr{S}_{K}^{-}(G,X)\to\operatorname{Spec}\mathcal{O}_{\mathbf{E},(v)}$
to be the closure of $\operatorname{Sh}_{K}(G,X)$ in
$\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$
(or, equivalently by Proposition 1.24, in
$\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$).
By this we mean the topological closure with the reduced subscheme structure
or, equivalently (since $\operatorname{Sh}_{K}(G,X)$ is reduced), the flat
closure (in the sense of [EGA4, § 2.8]).
Then we define
$\mathscr{S}_{K}(G,X)\to\operatorname{Spec}\mathcal{O}_{\mathbf{E},(v)}$ to be
the normalization of
$\mathscr{S}_{K}^{-}(G,X)\to\operatorname{Spec}\mathcal{O}_{\mathbf{E},(v)}$.
This can be regarded as “the obvious definition”, but it is entirely non-
obvious and due to Kisin and Pappas that this really behaves as one would
expect from a canonical integral model. These expectations, properly explained
in [KP15], mainly concern the satisfaction of an extension property which is a
weaker version of the valuative criterion for properness, and the existence of
a local model diagram that essentially shows that the integral model is
étale-locally isomorphic to the local model described by Pappas and Zhu
[PZ13].
Define $E:=\mathbf{E}_{v}$, the $v$-adic completion of $\mathbf{E}$, and
denote by $\kappa$ the residue field of $E$, a finite extension of
$\mathbb{F}_{p}$. By abuse of notation, we also denote the base change of
$\mathscr{S}_{K}(G,X)$ to $\mathcal{O}_{E}$ by $\mathscr{S}_{K}(G,X)$ again.
By $\overline{\mathscr{S}}_{K}(G,X)$ we denote the base change of
$\mathscr{S}_{K}(G,X)$ to $\kappa$.
### 1.6 Maps between Shimura varieties
This section is based on [Zho18], where more detailed explanations may be
found. We let $K_{p}^{\prime}=\mathcal{G}^{\prime}(\mathbb{Z}_{p})$ be another
parahoric where $\mathcal{G}^{\prime}$ is a Bruhat-Tits group scheme with
$\mathcal{G}^{\prime}=(\mathcal{G}^{\prime})^{\circ}$. We assume that
$K_{p}\subseteq K_{p}^{\prime}$, i.e., that the facet (in the Bruhat-Tits
building) associated with $K_{p}^{\prime}$ is contained in the closure of the
one associated with $K_{p}$.
###### Theorem 1.25.
[Zho18, Theorem 7.1] For sufficiently small $K^{p}$ there exists a morphism
$\pi_{K_{p}K^{p},K_{p}^{\prime}K^{p}}\colon\mathscr{S}_{K_{p}K^{p}}(G,X)\to\mathscr{S}_{K_{p}^{\prime}K^{p}}(G,X).$
In moving towards a proof, let us first note that
$\operatorname{GSp}(V,\psi)\to\operatorname{GSp}(V^{\S},\psi^{\S})$ factors
through
$M:=\left\\{\left(g_{i}\right)_{i}\in\prod_{i=-(r-1)-a}^{r-1}\operatorname{GSp}(V,\psi)\;|\;c(g_{-(r-1)-a})=\dotsb=c(g_{r-1})\right\\},$
where $c\colon\operatorname{GSp}(V,\psi)\to\mathbb{G}_{m}$ is the
multiplicator homomorphism.
There is a natural $X_{M}$ that makes $(M,X_{M})$ into a Shimura datum. Define
$H_{p}:=\operatorname{Stab}_{M(\mathbb{Q}_{p})}(\Lambda^{\S}),\quad
J_{p}:=\operatorname{Stab}_{\operatorname{GSp}(V^{\S}_{\mathbb{Q}_{p}},\psi^{\S}_{\mathbb{Q}_{p}})}(\Lambda^{\S}).$
Then for sufficiently small $H^{p},J^{p}$, we obtain a closed immersion
$i\colon\operatorname{Sh}_{H_{p}H^{p}}(M,X_{M})\hookrightarrow\operatorname{Sh}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm}).$
$\operatorname{Sh}_{H_{p}H^{p}}$ has a moduli interpretation over
$\mathbb{Z}_{(p)}$ (essentially: $\\#\\{-(r-1)-a,\dotsc,r-1\\}=2(r-1)+a+1$
abelian schemes up to prime-to-$p$ isogeny endowed with certain polarizations
and prime-to-$p$ level structure).
###### Proposition 1.26.
[Zho18, Prop. 7.2] If $J^{p}$ is sufficiently small, then $i$ extends to a
closed immersion
$i\colon\mathscr{S}_{H_{p}H^{p}}(M,X_{M})\to\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm}).$
Now consider the embedding
$i\colon\mathcal{B}(G,\mathbb{Q}_{p})\to\mathcal{B}(\operatorname{GSp}(V,\psi),\mathbb{Q}_{p})$.
We still have $K_{p}=\mathcal{G}(\mathbb{Z}_{p})$ with
$\mathcal{G}=\mathcal{G}_{x}$, $x\in\mathcal{B}(G,\mathbb{Q}_{p})$. Let
$\mathfrak{g}$ be the minimal facet in
$\mathcal{B}(\operatorname{GSp}(V,\psi),\mathbb{Q}_{p})$ containing $i(x)$. So
$\mathfrak{g}$ corresponds to some lattice chain
$\Lambda^{0}\supseteq\dotsb\supseteq\Lambda^{r-1}$ as above. We have
$\operatorname{Sh}_{K_{p}K^{p}}(G,X)\xrightarrow{\text{cl.
imm.}}\operatorname{Sh}_{H_{p}H^{p}}(M,X_{M})_{\mathbf{E}}\xrightarrow{\text{cl.
imm.}}\operatorname{Sh}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm})_{\mathbf{E}}$
and $\mathscr{S}^{-}_{K}(G,X)$ is defined to be the closure of
$\operatorname{Sh}_{K_{p}K^{p}}(G,X)$ in
$\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$.
###### Corollary 1.27.
[Zho18, Cor. 7.3] $\mathscr{S}^{-}_{K}(G,X)$ also can be described as the
closure of $\operatorname{Sh}_{K_{p}K^{p}}(G,X)$ in
$\mathscr{S}_{H_{p}H^{p}}(M,X_{M})_{\mathcal{O}_{\mathbf{E},(v)}}$.
Let $\mathfrak{f}^{\prime}$ be a facet of
$\mathcal{B}(\operatorname{GSp}(V^{\S},\psi^{\S}),\mathbb{Q}_{p})$ in the
closure of $\mathfrak{g}$. Then $\mathfrak{f}^{\prime}$ corresponds to a “sub-
lattice chain” $\Lambda^{i_{1}}\supseteq\dotsb\supseteq\Lambda^{i_{s}}$,
$\left\\{i_{1},\dotsc,i_{s}\right\\}\subseteq\\{0,\dotsc,r-1\\}$. Defining
$M^{\prime},H_{p}^{\prime}$ as above, we naturally obtain morphisms $M\to
M^{\prime}$ and
$\omega_{H_{p}H^{p},H_{p}^{\prime}\left.H^{\prime}\right.^{p}}\colon\mathscr{S}_{H_{p}H^{p}}(M,X_{M})\to\mathscr{S}_{H^{\prime}_{p}\left.H^{\prime}\right.^{p}}(M^{\prime},X_{M^{\prime}})$
for suitable levels $H^{p},\left.H^{\prime}\right.^{p}$ away from $p$.
###### Sketch of Proof (of Theorem 1.25):
Let $\mathfrak{f}$ be the facet of $K_{p}$ (that is to say $\mathfrak{f}$ is
the minimal facet satisfying $x\in\mathfrak{f}$), $\mathfrak{f}^{\prime}$ that
of $K_{p}^{\prime}$. Then
$\mathfrak{f}\subseteq\overline{\mathfrak{f}^{\prime}}$.
Let $x^{\prime}\in\mathfrak{f}^{\prime}$ be so close to $x$ that
$\mathfrak{g}\subseteq\overline{\mathfrak{g}^{\prime}}$ in
$\mathcal{B}(\operatorname{GSp}(V,\psi),\mathbb{Q}_{p})$, where $\mathfrak{g}$
and $\mathfrak{g}^{\prime}$ denote the minimal facets containing $i(x)$ and
$i(x^{\prime})$ respectively.
The constructions from above yield:
$\mathscr{S}_{K_{p}K^{p}}^{-}(G,X)$$\mathscr{S}_{K^{\prime}_{p}\left.K^{\prime}\right.^{p}}^{-}(G,X)$$\mathscr{S}_{H_{p}^{\prime}\left.H^{\prime}\right.^{p}}(M^{\prime},X_{M^{\prime}})_{\mathcal{O}_{\mathbf{E},(v)}}$$\mathscr{S}_{H_{p}\left.H\right.^{p}}(M,X_{M})_{\mathcal{O}_{\mathbf{E},(v)}}$closed
imm.closed imm.$\omega_{H_{p}H^{p},H_{p}^{\prime}\left.H^{\prime}\right.^{p}}$
(1.28)
On the generic fiber we can complete this to a commutative square.
By Corollary 1.27, this implies that (1.28) also completes to a commutative
square. Now normalize. □
### 1.7 Local structure of the integral model
#### 1.7.1 Generizations and irreducible components
Let $\mathscr{X}\to\operatorname{Spec}\mathcal{O}_{\breve{E}}$ be a flat
scheme locally of finite type; denote the special fiber by
$X\to\operatorname{Spec}\bar{\mathbb{F}}_{p}$ and the generic fiber by
$\mathcal{X}\to\operatorname{Spec}\breve{E}$. We assume that $\mathcal{X}$ is
locally integral (e.g. smooth).
For example, we can consider
$(\mathscr{X},X,\mathcal{X})=(\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}},\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}\otimes_{{{\cal
O}_{\breve{E}}}}\bar{\mathbb{F}}_{p},\allowbreak{\operatorname{Sh}_{K}(G,X)\otimes_{E}\breve{E}})$.
Let $\bar{x}\in X(\bar{\mathbb{F}}_{p})$.
###### Lemma 1.29.
There is a generization $x$ of $\bar{x}$ which lies in the generic fiber
$\mathcal{X}$, and is a closed point in there, i.e., $x\in\mathcal{X}(L)$ for
a finite extension $L/\breve{E}$.
###### Definition 1.30.
We shall call such a point $x$ a _closed point generization_ of $\bar{x}$ for
short.
###### Proof:
Due to flatness (going-down) there is _some_ generization in the generic
fiber; call it $x_{0}$.
By [Stacks, Tag 053U] the following set is dense (and in particular non-empty)
in the closure of $\\{x_{0}\\}$ in $\mathcal{X}$:
$\left\\{x\in\mathscr{X}\;|\;x\text{ is a specialization of }x_{0}\text{ and a
closed point generization of }\bar{x}\right\\}.$
□
###### Lemma 1.31.
Notation as in the preceding lemma.
The specialization $x\leadsto\bar{x}$ can be realized by an ${\cal
O}_{L}$-valued point of $\mathscr{X}$.
###### Proof:
First off, by [EGA2, 7.1.9], it can be realized by a morphism
$\operatorname{Spec}R=\\{\eta,s\\}\to\mathscr{X}$ of ${{\cal
O}_{\breve{E}}}$-schemes, where $R$ is a discrete valuation ring such that
$L\cong\kappa(\eta)=\operatorname{Quot}(R)$ as field extensions of
$\kappa(x)$.
We hence get local homomorphisms of local rings ${{\cal
O}_{\breve{E}}}\to{\cal O}_{\mathscr{X},\bar{x}}\to R$.
Thus the discrete valuation on $L$ defined by $R$ extends the discrete
valuation on $\breve{E}$. But there is but one such extension and its
valuation ring is ${\cal O}_{L}$ (by definition). □
###### Lemma 1.32.
Mapping $x$ to the unique irreducible component of $\mathscr{X}$ that contains
$x$ establishes a surjection from the set of closed point generizations $x$ of
$\bar{x}$ to the set of irreducible components of $\mathscr{X}$ containing
$\bar{x}$.
###### Proof:
If $x_{0}\in\mathcal{X}$ is a generization of $\bar{x}$, then $x_{0}$ lies in
a unique irreducible component of $\mathscr{X}$ because $\mathcal{X}$ is
locally irreducible. Hence the map described above is well-defined.
Now for surjectivity: Given an irreducible component $C$ of $\mathscr{X}$
containing $\bar{x}$, let $x_{0}\in C$ be the generic point. Then $x_{0}$ must
be in the generic fiber (else we would be able to find a generization in the
generic fiber by going-down). Now go through the proof of Lemma 1.29 with this
particular choice of $x_{0}$. □
#### 1.7.2 Normalization and completion
For reference, we collect some facts concerning the passage to normalization
and completion and in particular how it applies to integral models of Shimura
varieties.
###### Fact 1.33.
$\mathscr{S}^{-}_{K}(G,X)\to\operatorname{Spec}{{\cal O}_{\breve{E}}}$ is
quasi-projective, so in particular of finite type.
Hence $\mathscr{S}^{-}_{K}(G,X)$ and $\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}$ are excellent.
As a consequence the normalization
$\mathscr{S}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}\xrightarrow{\nu}\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}$
is finite. (This really is a normalization because normalization and
completion behave well together in the excellent case (to get from ${\cal
O}_{E^{\mathrm{ur}}}$ to ${{\cal O}_{\breve{E}}}$ and from ${\cal
O}_{\mathbf{E},(v)}$ to ${\cal O}_{E}$) and because normalization commutes
with base change along filtered colimits of smooth morphisms (to get from
${\cal O}_{E}$ to ${\cal O}_{E^{\mathrm{ur}}}$)).
We will always denote by $(\;)^{\sim}$ the integral closure of a ring in its
total ring of fractions.
$\bar{x}$ still denotes an $\bar{\mathbb{F}}_{p}$-valued point of
$\mathscr{S}^{-}_{K}(G,X)$. Let
$\nu^{-1}(\bar{x})=\\{\bar{y}=\bar{y}_{1},\dotsc,\bar{y}_{n}\\}$.
Let $\mathscr{S}^{-}:=\mathscr{S}^{-}_{K}(G,X)_{{{\cal O}_{\breve{E}}}}$ and
$\mathscr{S}:=\mathscr{S}_{K}(G,X)_{{{\cal O}_{\breve{E}}}}$.
###### Fact 1.34.
$\left(\nu_{*}{\cal
O}_{\mathscr{S}^{-}_{K}(G,X)}\right)_{\bar{x}}^{\wedge}=\prod_{j=1}^{n}\widehat{\cal
O}_{\mathscr{S},\bar{y}_{j}}$.
###### Fact 1.35.
By [Stacks, Tag 0C3B]: $\left(\nu_{*}{\cal
O}_{\mathscr{S}^{-}_{K}(G,X)}\right)_{\bar{x}}=\widetilde{\cal
O}_{\mathscr{S}^{-},\bar{x}}$.
###### Fact 1.36.
By [Stacks, Tag 035P]:
$\widetilde{A}=\prod_{\mathfrak{q}\in\operatorname{Min}(A)}(A/\mathfrak{q})^{\sim}$,
if $A$ is a reduced ring and $\\#\operatorname{Min}(A)<\infty$.
###### Fact 1.37.
$(\;)^{\sim}$ and $(\;)^{\wedge}$ commute in the case of excellence (see e.g.
[EGA4, 7.6.1, 7.8.3.1(vii)]).
For instance, $\left.{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\wedge}\right.^{\sim}=\left.{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\sim}\right.^{\wedge}$ (on the right hand side,
we have a completion of a ${\cal O}_{\mathscr{S}^{-},\bar{x}}$-module).
###### Fact 1.38.
In the case of excellence, the completion of a normal domain is a normal
domain (see e.g. [GW10, 12.50]).
Thus we have
$\displaystyle\prod_{\mathfrak{q}\in\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q})^{\sim}$
$\displaystyle\cong\left.\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\wedge}\right.^{\sim}\cong\left.\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\sim}\right.^{\wedge}$ (1.39)
$\displaystyle\cong\left(\nu_{*}{\cal
O}_{\mathscr{S}^{-}_{K}(G,X)}\right)_{\bar{x}}^{\wedge}\cong\prod_{j=1}^{n}\widehat{\cal
O}_{\mathscr{S},\bar{y}_{j}}$
and the rings $\widehat{\cal O}_{\mathscr{S},\bar{y}_{j}}$ are normal domains.
Hence we obtain a bijection
$\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})\xleftrightarrow{\;1:1\;}\nu^{-1}(\bar{x})$
such that there exists a numbering $\mathfrak{q}_{1},\dotsc,\mathfrak{q}_{n}$
of the elements of $\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})$ such that (1.39) restricts to an isomorphism
$\widehat{\cal O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}_{j}\cong\widehat{\cal
O}_{\mathscr{S},\bar{y}_{j}}$
(also see [EGA4, 7.6.2]).
Also:
$\displaystyle\left.\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\sim}\right.^{\wedge}$
$\displaystyle\cong\left(\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}({\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q})^{\sim}\right)^{\wedge}$
$\displaystyle\cong\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}\left.\left({\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\right)^{\sim}\right.^{\wedge}$
$\displaystyle\cong\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}\left.\left({\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\right)^{\wedge}\right.^{\sim}$
$\displaystyle\overset{\mathclap{\text{Artin-
Rees}}}{\cong}\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}\left(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\widehat{\mathfrak{q}}\right)^{\sim},\quad\widehat{\mathfrak{q}}=\mathfrak{q}\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}.$
Now by [KP15, 2.1.2, 4.2.2] for all
$\mathfrak{q}\in\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})$ we have that
$\left(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\right)^{\sim}=\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\cong
R_{G,\bar{x},\mathfrak{q}}=R_{G}$
is normal and $\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\cong\widehat{\cal
O}_{\mathscr{S},\bar{y}}$ for an appropriate choice of $\mathfrak{q}$ (i.e.,
of $x$). (The notation $R_{G}$ is from [KP15], where it is defined as the
formal local ring of the local model.)
### 1.8 Hodge tensors and (lack of) moduli interpretation
We discuss the moduli interpretation of Shimura varieties of Hodge type and
the partial extension of this interpretation to the integral model as
constructed in Section 1.5. Let $(G,X)$ be a Shimura datum of Hodge type, let
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$ be an embedding as in
Definition 1.1 (4).
#### 1.8.1 The story for the $\mathbb{C}$-valued points
###### Lemma 1.40.
[Del82, Prop. 3.1] There exist numbers $n,r_{i},s_{i}\in\mathbb{Z}_{\geq 0}$
and tensors $t_{i}\in V^{\otimes r_{i}}\otimes(V^{*})^{\otimes s_{i}}$, $1\leq
i\leq n$, such that $G$ is the subgroup of $\operatorname{GSp}(V)$ fixing the
$t_{i}$.
###### Remark 1.41.
(See [Mil05, Section 7].) For $K$ a compact open subgroup of
$G(\mathbb{A}_{f})$, the set $\operatorname{Sh}_{K}(G,X)(\mathbb{C})$ of
Definition 1.8 has the following moduli interpretation: isomorphism classes of
triples $((W,h),\left(u_{i}\right)_{0\leq i\leq n},\eta K)$, where
1. (a)
$(W,h)$ rational Hodge structure of type $(-1,0)+(0,-1)$,
2. (b)
$\pm u_{0}$ polarization of $(W,h)$ (i.e., either $u_{0}$ or $-u_{0}$ is a
polarization),
3. (c)
$u_{i}\in V^{\otimes r_{i}}\otimes(V^{*})^{\otimes s_{i}}$ for $1\leq i\leq
n$,
4. (d)
$\eta K$ is a $K$-orbit of isomorphisms
$V\otimes\mathbb{A}_{f}\xrightarrow{\sim}W\otimes\mathbb{A}_{f}$, mapping
$\psi$ to a $\mathbb{A}_{f}^{\times}$-multiple of $u_{0}$, and every $t_{i}$
to $u_{i}$ ($i\geq 1$),
such that there exists an isomorphism $a\colon W\xrightarrow{\sim}V$ mapping
$u_{0}$ to a $\mathbb{Q}^{\times}$-multiple of $\psi$, every $u_{i}$ to
$t_{i}$ ($i\geq 1$), and $h$ to an element of $X$ (so
${}^{a}h:=\operatorname{GL}(a)\circ h\colon\mathbb{S}\to\operatorname{GL}(V)$
is in $X$).
###### Sketch of Proof:
Given $((W,h),\left(u_{i}\right)_{0\leq i\leq n},\eta K)$ and $a$ as above, we
consider the pair ${(^{a},a\circ\eta)}$. By assumption, ${}^{a}h\in X$ and
$a\circ\eta$ is a symplectic similitude fixing the $t_{i}$; hence
$(^{a}h,a\circ\eta)\in X\times G(\mathbb{A}_{f})$. The double quotient now
results precisely from the ambiguity in the choices of $a$ and $\eta\in\eta
K$. □
###### Remark 1.42.
Denote by $\mathbb{Q}(r)$ the rational Hodge structure of type $(-r,-r)$ with
underlying vector space $\mathbb{Q}$, and denote the multiplier character
$\operatorname{GSp}(V)\to\mathbb{G}_{m}$ as well as its restriction to $G$ by
$c$. Then for $h\in X$, $c\circ h\cong\mathbb{Q}(1)$ as rational Hodge
structure, and the symplectic form gives an isomorphism $V\cong
V^{*}\otimes\mathbb{Q}(1)$.
With this, we may also interpret a tensor $t\in V^{\otimes
r}\otimes(V^{*})^{\otimes s}$ as a multilinear map $V^{r+s}\to\mathbb{Q}$,
and, if $t$ is fixed by $G$, as a morphism
$(V,h)^{\otimes(r+s)}\to\mathbb{Q}(r)$ of Hodge structures. Provided $t\neq
0$, this implies $r+s=2r$, i.e. $r=s$.
###### Remark 1.43.
Since $A\mapsto H_{1}(A(\mathbb{C}),\mathbb{Q})$ yields an equivalence between
the category of complex abelian varieties up to isogeny and the category of
polarizable rational Hodge structures of type $(-1,0)+(0,-1)$, we may also
take $\operatorname{Sh}_{K}(G,X)(\mathbb{C})$ to be a moduli problem of
abelian varieties with extra structure: consider isomorphism classes of
triples $(A,\left(u_{i}\right)_{0\leq i\leq n},\eta K)$, where
1. (a)
$A$ is a complex abelian variety up to isogeny,
2. (b)
$\pm u_{0}$ is a polarization of the rational Hodge structure
$V:=H_{1}(A(\mathbb{C}),\mathbb{Q})$,
3. (c)
$u_{1},\dotsc,u_{n}$ is as in Remark 1.41,
4. (d)
$\eta K$ is a $K$-orbit of $\mathbb{A}_{f}$-linear isomorphisms
$V\otimes\mathbb{A}_{f}\to
V_{f}(A):=T_{f}(A)\otimes_{\mathbb{Z}}\mathbb{Q}=\left(\varprojlim
A(k)[n]\right)\otimes_{\mathbb{Z}}\mathbb{Q}$, mapping $\psi$ to a
$\mathbb{A}_{f}^{\times}$-multiple of $u_{0}$ and every $t_{i}$ to $u_{i}$
($i\geq 1$),
such that there exists an isomorphism $a\colon
H_{1}(A(\mathbb{C}),\mathbb{Q})\xrightarrow{\sim}V$ mapping $u_{0}$ to a
$\mathbb{Q}^{\times}$-multiple of $\psi$, $u_{i}$ to $t_{i}$ ($i\geq 1$) and
$h$ to an element of $X$.
###### Remark 1.44.
When rephrasing the moduli problem appropriately, the Shimura variety as an
$\mathbf{E}$-variety is given by such a moduli problem, see [Mil05, Section
14].
In particular, we have a universal abelian scheme
$\mathcal{A}\to\operatorname{Sh}_{K}(G,X)$.
#### 1.8.2 The story for the integral model
Integral models of Hodge type Shimura varieties in general seem not to afford
nice straightforward moduli interpretations. Still, the moduli interpretation
of the $\mathbb{C}$-valued points extends to _some_ degree. First, we do have
a generalization of Lemma 1.40 as follows.
###### Proposition 1.45.
[Kis10, Prop. 1.3.2] Let $R$ be a discrete valuation ring of mixed
characteristic, $M$ a finite free $R$-module, and
$\mathcal{G}\subseteq\operatorname{GL}(M)$ a closed $R$-flat subgroup with
reductive generic fiber. Then $\mathcal{G}$ is defined by a finite collection
of tensors $\left(s_{\alpha}\right)_{\alpha}\subset M^{\otimes}$.
Here $M^{\otimes}$ is the direct sum of all the $R$-modules which can be
formed from $M$ by taking duals and (finite) tensor products.121212Kisin in
_loc. cit._ also allows for taking symmetric and exterior powers. As Deligne
pointed out, this is unnecessary. [Del11]
We return to the setting of Section 1.3; in particular we consider a Bruhat-
Tits scheme $\mathcal{G}:=\mathcal{G}_{x}$.
By the proposition just stated, we find a collection of tensors
$\left(s_{\alpha}\right)_{\alpha}\subset(\Lambda^{\S}_{\mathbb{Z}_{(p)}})^{\otimes}$
whose pointwise stabilizer is the Zariski closure $G_{\mathbb{Z}_{(p)}}$ of
$G$ in $\operatorname{GL}(\Lambda^{\S}_{\mathbb{Z}_{(p)}})$.
###### Remark 1.46.
We can for example consider, for every $j\in\\{-(r-1)-a,\dotsc,r-1\\}$, the
projection
$\operatorname{pr}_{j}\colon\Lambda_{\mathbb{Z}_{(p)}}^{\S}\to\Lambda_{\mathbb{Z}_{(p)}}^{j}$.
Since the $G$-action on
$\Lambda_{\mathbb{Z}_{(p)}}^{\S}=\prod_{i=-(r-1)-a}^{r-1}\Lambda_{\mathbb{Z}_{(p)}}^{i}$
is diagonal, the $\operatorname{pr}_{j}$ are fixed, i.e., we may count the
$\left(\operatorname{pr}_{j}\right)_{j}$ among the
$\left(s_{\alpha}\right)_{\alpha}$.
###### Lemma 1.47.
$G_{\mathbb{Z}_{(p)}}\otimes_{\mathbb{Z}_{(p)}}\mathbb{Z}_{p}={\cal G}$.
###### Proof:
First, $G_{\mathbb{Z}_{(p)}}\otimes_{\mathbb{Z}_{(p)}}\mathbb{Z}_{p}$ is the
Zariski closure $G_{\mathbb{Z}_{p}}$ of $G$ in
$\operatorname{GL}(\Lambda^{\S}_{\mathbb{Z}_{p}})$ and
$G_{\mathbb{Z}_{p}}\otimes\mathbb{Q}_{p}=G$: This follows from [GW10, Lemma
14.6].
According to [Stacks, Tag 056B], $G_{\mathbb{Z}_{p}}$ is the topological
closure of $G$ in $\operatorname{GL}(\Lambda^{\S}_{\mathbb{Z}_{p}})$ endowed
with the reduced subscheme structure. The special fiber therefore is reduced
as well, i.e. ($\mathbb{F}_{p}$ being perfect), geometrically reduced, i.e.
(being a group scheme), smooth. $G_{\mathbb{Z}_{p}}$ is flat over
$\mathbb{Z}_{p}$ by [GW10, Prop. 14.14]. $G_{\mathbb{Z}_{p}}$ obviously is of
finite presentation over $\mathbb{Z}_{p}$. Hence, $G_{\mathbb{Z}_{p}}$ is
smooth over $\mathbb{Z}_{p}$. It also is affine and is the stabilizer of
$\Lambda^{\S}_{\mathbb{Z}_{p}}$ (or rather,
$G_{\mathbb{Z}_{p}}(\mathbb{Z}_{p})\subseteq G(\mathbb{Q}_{p})$ is (and
analogously for $G_{\mathbb{Z}_{p}}(\breve{\mathbb{Z}}_{p})\subseteq
G(\breve{\mathbb{Q}}_{p})$)). □
###### Remarks 1.48.
1. (1)
Obtaining an embedding of the Bruhat-Tits group scheme to which we can apply
Proposition 1.45 is the main reason for altering the Hodge embedding in the
way outlined in Section 1.3.
2. (2)
We can also interpret the $s_{\alpha}$ as tensors in
$((\Lambda^{\S}_{\mathbb{Z}_{(p)}})^{*})^{\otimes}$ and that
$G_{\mathbb{Z}_{(p)}}$ also is the Zariski closure of $G$ in
$\operatorname{GL}((\Lambda^{\S}_{\mathbb{Z}_{(p)}})^{*})$ (using the
contragredient representation).
##### Hodge tensors: generic fiber.
###### Notation 1.49.
Recall the universal abelian scheme
$h\colon\mathcal{A}\to\operatorname{Sh}_{K}(G,X)$ from Remark 1.44.
We consider the local system
$\mathcal{L}_{B}:=R^{1}h_{*}^{\mathrm{an}}\underline{\mathbb{Q}}$ on
$\operatorname{Sh}_{K}(G,X)_{\mathbb{C}}^{\mathrm{an}}$ and the flat vector
bundle $\mathcal{V}_{\mathrm{dR}}:=R^{1}h_{*}\Omega^{\bullet}$ with Gauß-Manin
connection $\nabla$.
###### Lemma 1.50.
(See [CS17, Lemma 2.3.1].) $\mathcal{L}_{B}$ can be identified with the local
system of $\mathbb{Q}$-vector spaces on
$\operatorname{Sh}_{K}(G,X)^{\mathrm{an}}$ given by the
$G(\mathbb{Q})$-representation $V$ and the $G(\mathbb{Q})$-torsor
$p\colon X\times(G(\mathbb{A}_{f})/K)\to
G(\mathbb{Q})\backslash(X\times(G(\mathbb{A}_{f})/K)=\operatorname{Sh}_{K}(G,X)(\mathbb{C}).$
Said torsor is isomorphic to the $G(\mathbb{Q})$-torsor
$I:=\underline{\mathrm{Isom}}((V,s_{\alpha}),({\cal L}_{B},s_{\alpha,B}))$
that maps an open subset $U\subseteq\operatorname{Sh}_{K}(\mathbb{C})$ to
$I(U)=\\{\beta\colon V\times U\xrightarrow{\sim}\left.{\cal
L}_{B}\right|_{U}\;|\;\beta(s_{\alpha})=s_{\alpha,B}\\},$
where the $\left(s_{\alpha,B}\right)_{\alpha}\subset\mathcal{L}_{B}^{\otimes}$
are the global sections corresponding to the $G(\mathbb{Q})$-invariant tensors
$\left(s_{\alpha}\right)_{\alpha}$.
###### Remark 1.51.
By de Rham’s theorem,
$\mathcal{V}_{\mathrm{dR},\mathbb{C}}^{\mathrm{an}}\cong\mathcal{L}_{B}\otimes_{\mathbb{Q}}\mathcal{O}_{\operatorname{Sh}_{K}(G,X)^{\mathrm{an}}}.$
In particular the global sections
$\left(s_{\alpha,B}\right)_{\alpha}\subset\mathcal{L}_{B}^{\otimes}$ yield
flat global sections
$\left(s_{\alpha,\mathrm{dR}}\right)_{\alpha}\subset(\mathcal{V}_{\mathrm{dR},\mathbb{C}}^{\mathrm{an}})^{\otimes}$.
All such sections arise from sections in
$\mathcal{V}_{\mathrm{dR},\mathbb{C}}^{\otimes}$ (which we will denote the
same), see [Kis10, 2.2].
###### Remark 1.52.
Let $k/E$ be a field extension embeddable into $\mathbb{C}$, and choose an
embedding $\mathbb{Q}_{p}\hookrightarrow\mathbb{C}$ and an $E$-embedding
$\sigma\colon\bar{k}\hookrightarrow\mathbb{C}$. Let
$x\in\operatorname{Sh}_{K}(G,X)(k)$. By $p$-adic Hodge theory, the embedding
$\sigma$ gives rise to isomorphisms
$H^{1}_{\mathrm{dR}}(\mathcal{A}_{x}/k)\otimes_{k}\mathbb{C}\xrightarrow{\sim}H^{1}_{B}(\mathcal{A}_{x}(\mathbb{C}),\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}\xrightarrow{\sim}H^{1}_{\mathrm{\acute{e}t}}(\mathcal{A}_{x,\bar{k}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C},$
and by proper base change $(\mathcal{L}_{B})_{x}:=x^{-1}\mathcal{L}_{B}\cong
H^{1}_{B}(\mathcal{A}_{x}(\mathbb{C}),\mathbb{Q})$.
###### Notation 1.53.
We denote by $s_{\alpha,B,x}$ the fiber of $s_{\alpha,B}$ at $x$, and by
$s_{\alpha,\mathrm{dR},x}$ and $s_{\alpha,\mathrm{\acute{e}t},x}$,
respectively, the images of $s_{\alpha,B,x}$ under the above isomorphisms. For
the Betti–étale comparison, we don’t need to go all the way to $\mathbb{C}$
and have $s_{\alpha,\mathrm{\acute{e}t},x}\in
H^{1}_{\mathrm{\acute{e}t}}(\mathcal{A}_{x,\bar{k}},\mathbb{Q}_{p})$.
###### Remark 1.54.
$(s_{\alpha,\mathrm{dR},x},s_{\alpha,\mathrm{\acute{e}t},x})$ is an absolute
Hodge cycle in the sense of [Del82], for every $\alpha$.
###### Lemma 1.55.
[Kis10, Lemma 2.2.1] The $\operatorname{Gal}(\bar{k}/k)$-action on
$H^{1}_{\mathrm{\acute{e}t}}(\mathcal{A}_{x,\bar{k}},\mathbb{Q}_{p})$ fixes
every $s_{\alpha,\mathrm{\acute{e}t},x}$ and factors through
$G(\mathbb{Q}_{p})$. Moreover, $s_{\alpha,\mathrm{dR},x}\in
H^{1}_{\mathrm{dR}}(\mathcal{A}_{x}/k)$.
In particular, $(s_{\alpha,\mathrm{dR},x},s_{\alpha,\mathrm{\acute{e}t},x})$
is independent of the choices made above.
###### Corollary 1.56.
[Kis10, Lemma 2.2.2] $s_{\alpha,\mathrm{dR}}$ is defined over $E$ for all
$\alpha$, i.e.,
$s_{\alpha,\mathrm{dR}}\in\mathcal{V}_{\mathrm{dR},E}^{\otimes}$.
##### Hodge tensors: special fiber.
###### Notation 1.57.
Denote by $\mathscr{A}_{\mathrm{Siegel}}$ the universal abelian scheme over
$\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$, cf. Remark 1.21, and
by $\mathscr{A}$ the pullback of $\mathscr{A}_{\mathrm{Siegel}}$ to
$\mathscr{S}_{K}(G,X)$. Then the pullback of $\mathscr{A}$ to
$\operatorname{Sh}_{K}(G,X)_{E}$ agrees with the pullback of $\mathcal{A}$ of
Remark 1.44.
We will occasionally call $\mathscr{A}$ the _“universal” abelian scheme_ (with
quotation marks).
Let $\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$ and let
$\mathbb{D}_{\bar{x}}:=\mathbb{D}(\mathscr{A}_{\bar{x}}[p^{\infty}])(W)$ be
the Dieudonné module of the associated fiber of the “universal” abelian
scheme, $W=W(\bar{\mathbb{F}}_{p})=\breve{\mathbb{Z}}_{p}$. Choose a closed
point generization $x\in\operatorname{Sh}_{K}(G,X)(L)$, $L/E$ finite field
extension, in the sense of Definition 1.30. Note that $L$ can be embedded into
$\mathbb{C}_{p}\cong\mathbb{C}$. Then
$H^{1}_{\mathrm{\acute{e}t}}(\mathscr{A}_{x,\bar{E}},\mathbb{Z}_{p})\cong(T_{p}\mathscr{A}_{x,\bar{E}})^{*}\cong(T_{p}\mathscr{A}_{x,\bar{E}}^{*})(-1)\cong(\Lambda^{\S})^{*},$
allowing us to identify the tensors $s_{\alpha}$ with tensors
$s_{\alpha,\mathrm{\acute{e}t},x}\in
H^{1}(\mathscr{A}_{x,\bar{E}},\mathbb{Z}_{p})^{\otimes}$. Again, these tensors
can be shown to be $\operatorname{Gal}(\bar{E}/L)$-invariant. Now we have the
$p$-adic comparison isomorphism
$H_{\mathrm{\acute{e}t}}^{1}(A_{x,\bar{E}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}B_{\mathrm{cris}}\cong
H^{1}_{\mathrm{cris}}(\mathscr{A}_{\bar{x}}/W)\otimes_{W}B_{\mathrm{cris}}=\mathbb{D}(\mathscr{A}_{\bar{x}}[p^{\infty}])(W)\otimes_{W}B_{\mathrm{cris}},$
(1.58)
and by [KP15, 3.3.8], via this isomorphism, the
$s_{\alpha,\mathrm{\acute{e}t},x}$ also correspond to tensors
$s_{\alpha,0}:=s_{\alpha,0,\bar{x}}$ in $\mathbb{D}_{\bar{x}}^{\otimes}$. In
fact, we get an isomorphism
$(\Lambda^{\S})^{*}\otimes_{\mathbb{Z}_{p}}\breve{\mathbb{Z}}_{p}\cong\mathbb{D}_{\bar{x}}$
(1.59)
identifying $s_{\alpha}\otimes 1$ with $s_{\alpha,0,\bar{x}}$.
##### Globalizing crystalline tensors.
Above we constructed crystalline tensors fiberwise. This will not suffice to
understand the local geometry of the special fiber of the integral model.
Therefore we now “globalize” the tensors.
Let $\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$. As in Section
1.7.2, we write
$R_{G}:=R_{G,\bar{x}}:=\widehat{\mathcal{O}}_{\mathscr{S}_{K}(G,X)_{{\cal
O}_{\breve{E}}},\bar{x}}$. (So $\operatorname{Spf}R_{G}$ is the $\bar{x}$-adic
completion of $\mathscr{S}_{K}(G,X)_{{\cal O}_{\breve{E}}}$.)
###### Remark 1.60.
In [KP15, 3.2.12, 3.2.14] a construction of a “universal” deformation
$p$-divisible group
$\mathscr{G}_{R_{G}}\to\operatorname{Spec}R_{G}$ (1.61)
is given, characterized by its Dieudonné display, to wit
$\mathrm{DDisp}(\mathscr{G}_{R_{G}})=\mathbb{D}_{\bar{x}}\otimes_{W}\hat{W}(R_{G})$
endowed with a natural Dieudonné display structure.
$\mathscr{G}_{R_{G}}$ can be identified with the pullback of the $p$-divisible
group of the “universal” abelian scheme $\mathscr{A}$.
###### Remark 1.62.
$R_{G}$ is a complete local normal noetherian ring with perfect residue field.
###### Remark 1.63.
By [Lau14, Theorem B], the Dieudonné crystal associated with
$\mathrm{DDisp}(\mathscr{G}_{R_{G}})$ coincides with the Dieudonné crystal
$\mathbb{D}(\mathscr{G}_{R_{G}})$ of $\mathscr{G}_{R_{G}}$.
In fact,
$\mathrm{DDisp}(\mathscr{G}_{R_{G}})=\mathbb{D}(\mathscr{G}_{R_{G}})^{\vee}(\hat{W}(R_{G}))$
endowed with a natural Dieudonné display structure, cf. [KP15, 3.1.7].
In particular we get tensors
$t_{\alpha,\bar{x}}^{\mathrm{def}}\in\mathbb{D}(\mathscr{G}_{R_{G}})^{\vee}(\hat{W}(R_{G}))^{\otimes}$
corresponding to $s_{\alpha,0,\bar{x}}\otimes
1\in(\mathbb{D}_{\bar{x}}\otimes_{W}\hat{W}(R_{G}))^{\otimes}$, and
accordingly tensors $t_{\alpha,\bar{x}}^{\mathrm{def}}\otimes
1\in\mathbb{D}(\mathscr{G}_{R_{G}})^{\vee}(W(R_{G}))^{\otimes}$.
###### Definition 1.64.
(See [Ham17, Definition 2.8].) Let $\mathscr{G}\to S$ be a $p$-divisible group
over a formally smooth $\mathbb{F}_{p}$-scheme $S$. Denote by
$\mathbb{D}(\mathscr{G})$ its contravariant Dieudonné crystal of
$\mathscr{G}$ [BBM82, Def. 3.3.6], and by
$\mathds{1}:=\mathbb{D}(\underline{\mathbb{Q}_{p}/\mathbb{Z}_{p}}_{S})$ the
unit object in the tensor category of locally free
$\mathcal{O}_{S/\mathbb{Z}_{p},\mathrm{CRIS}}$-modules. Note that
$\mathbb{D}(\mathscr{G})$ comes with a Frobenius morphism
$\mathbb{D}(\mathscr{G})^{(p)}\to\mathbb{D}(\mathscr{G})$, and similar for
$\mathds{1}$.
1. (1)
A _tensor_ $t$ of $\mathbb{D}(\mathscr{G})$ is a morphism
$\mathds{1}\to\mathbb{D}(\mathscr{G})^{\otimes}$ of locally free
$\mathcal{O}_{S/\mathbb{Z}_{p},\mathrm{CRIS}}$-modules.131313$(\;)^{\otimes}$
here (and also in point (2)) is defined as it is defined for $R$-modules in
Prop. 1.45.
2. (2)
A tensor $t$ of $\mathbb{D}(\mathscr{G})$ is called a _crystalline Tate tensor
on $\mathscr{G}$_ if it induces a morphism of $F$-isocrystals
$\mathds{1}\to\mathbb{D}(\mathscr{G})[\tfrac{1}{p}]^{\otimes}$.
###### Remarks 1.65.
1. (1)
To give a tensor as in the preceding definition therefore means the following:
For every $(U,T,i,\delta)$ with
1. (a)
an $S$-scheme $U$,
2. (b)
a $\mathbb{Z}_{p}$-scheme $T$ on which $p$ is locally nilpotent,
3. (c)
a closed $\mathbb{Z}_{p}$-immersion $i\colon U\hookrightarrow T$,
4. (d)
a pd-structure $\delta$ on the ideal in $\mathcal{O}_{T}$ defining the
immersion $i$, compatible with the canonical pd-structure on
$p\mathbb{Z}_{p}$,
functorially to give a morphism
$\mathds{1}(U,T,i,\delta)=\mathcal{O}_{T}(T)\to\mathbb{D}(\mathscr{G})(U,T,i,\delta)^{\otimes}$
of $\mathcal{O}_{T}(T)$-modules, i.e., functorially to give elements of
$\mathbb{D}(\mathscr{G})(U,T,i,\delta)^{\otimes}$.
2. (2)
Assume $S=\operatorname{Spec}A$, where $A$ is an $\mathbb{F}_{p}$-algebra
which has a $p$-basis (e.g., $A$ perfect) _or_ which satisfies [Jon95,
(1.3.1.1)], the latter signifying the existence of an ideal $I\subseteq A$
such that
* •
$A$ is noetherian and $I$-adically complete,
* •
$A$ is formally smooth as a topological $\mathbb{F}_{p}$-algebra with the
$I$-adic topology,
* •
$A/I$ contains a field with a finite $p$-basis and is finitely generated as an
algebra over this field.
Also fix a _lift_ $\tilde{A}$ of $A$ in the sense of [Jon95, Def. 1.2.1],
i.e., a $p$-adically complete $\mathbb{Z}_{p}$-flat ring $\tilde{A}$ together
with an isomorphism $\tilde{A}/p\tilde{A}\cong A$ and a ring endomorphism
$\sigma\colon\tilde{A}\to\tilde{A}$ such that $\sigma(a)\equiv
a^{p}\mod{p\tilde{A}}$. Note that if $A$ is perfect, then $\tilde{A}:=W(A)$
with the usual Frobenius lift works.
Then by [BM90, Prop. 1.3.3] and [Jon95, Cor. 2.2.3] the category of crystals
of quasi-coherent
$\mathcal{O}_{\operatorname{Spec}(A)/\mathbb{Z}_{p},\mathrm{CRIS}}$-modules is
equivalent to the category of $p$-adically complete $\tilde{A}$-modules $M$
endowed with an integrable topologically quasi-nilpotent connection
$\nabla\colon M\to M\otimes_{\tilde{A}}\hat{\Omega}_{\tilde{A}}^{1}$.
3. (3)
In the setting of (2), say $(M,\nabla)$ is the image of
$\mathbb{D}(\mathscr{G})$ under the equivalence. Then
$M=\mathbb{D}(\mathscr{G})(\tilde{A})$, and to give a tensor
$\mathds{1}\to\mathbb{D}(\mathscr{G})^{\otimes}$ means to give a horizontal
section of $M^{\otimes}$.
###### Proposition 1.66.
[HK17, Prop. 3.3.1, Cor. 3.3.7] Denote by
$\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}}$ and
$\mathscr{G}_{\hat{\mathscr{S}}}$ the pullbacks of the $p$-divisible group
$\mathscr{A}[p^{\infty}]$ to the perfection141414This being the inverse
perfection of
$\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{E}}\bar{\mathbb{F}}_{p}$ in the
terminology of [BG18, Section 5].
$\overline{\mathscr{S}}^{\mathrm{perf}}:=(\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{E}}\bar{\mathbb{F}}_{p})^{\mathrm{perf}}$
and the $p$-adic completion $\hat{\mathscr{S}}$ of
$\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{E}}{{\cal O}_{\breve{E}}}$,
respectively.
For a $p$-divisible group $\mathscr{X}$ over a $p$-adic formal scheme
$\mathfrak{S}$ denote by $P(\mathscr{X})$ the locally free
$W(\mathcal{O}_{\mathfrak{S}})$-module given by
$P(\mathscr{X})(\operatorname{Spf}A)=\mathbb{D}(\mathscr{X}_{A})^{\vee}(W(A))$
for every open affine formal subscheme
$\operatorname{Spf}A\subseteq\mathfrak{S}$.
Then there exist tensors $\left(t_{\alpha}\right)_{\alpha}\subset
P(\mathscr{G}_{\hat{\mathscr{S}}})^{\otimes}$ whose pullback to
$P(\mathscr{G}_{R_{G},\bar{x}})^{\otimes}$ coincides with
$t_{\alpha,\bar{x}}^{\mathrm{def}}\otimes 1$ for all
$\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$.
Via pullback, these $t_{\alpha}$ yield crystalline Tate tensors
$(\bar{t}_{\alpha})_{\alpha}$ on
$\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}}$.
Moreover, if $x\in\mathscr{S}_{K}(G,X)(\mathcal{O}_{L})$, $L/\breve{E}$ finite
field extension, is a closed point generization of
$\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$, then
$s_{\alpha,\mathrm{\acute{e}t},x}\in(T_{p}\mathscr{A}_{x,\bar{E}})^{\otimes}$
gets identified with
$\bar{t}_{\alpha,\bar{x}}\in\mathbb{D}(\mathscr{A}_{\bar{x}}[p^{\infty}])(W(\bar{\mathbb{F}}_{p}))$
by the $p$-adic comparison isomorphism (1.58).
## 2 Central leaves in the case of parahoric reduction
We begin by giving some history. The foliation given by the central leaves was
introduced by Oort [Oor04, Oor09] in the setting of a $p$-divisible group over
a characteristic $p$ scheme (cf. Remark 2.2 below). As already indicated in
the introduction, the idea is to consider for every isomorphism class of
$p$-divisible groups over an algebraically closed field of characteristic $p$
the locus where the isomorphism class of the geometric fibers of the
$p$-divisible group is the given one. It is not at all readily apparent why
this would be a reasonable notion in geometrical or topological terms. The key
tool here is given by the slope filtrations introduced by Zink [Zin01a], and a
key insight is that there is a number $N\geq 0$ such that a $p$-divisible
group over an algebraically closed field of characteristic $p$ is in the
prescribed isomorphism class if and only if the analogous statement holds for
the $p^{N}$-torsion subgroups. For Shimura varieties of Hodge type and good
reduction, the central leaves were studied by Mantovan [Man04, Man05] (PEL
cases), Vasiu [Vas08] and Zhang [Zha15] (a good survey), among others.
He and Rapoport formulated a set of axioms that integral models for Shimura
varieties with parahoric level are supposed to satisfy in order for them to
merit the label “canonical”. Among these is having a notion of well-behaved
central leaves. This is what we investigate in the Hodge type situation. As
already mentioned, Zhou [Zho18] concurrently and independently also worked on
this and obtained very similar results in a different way, and we use some of
his results from the first version of the preprint, which didn’t yet address
the change-of-parahoric map between central leaves. Central leaves in the
parahoric level case also are subject in work of Hamacher and Kim [Ham17,
Kim19, HK17], but they do not deal with changing the parahoric level.
We still fix a Shimura datum $(G,X)$ of Hodge type, a parahoric subgroup
$K_{p}\subseteq G(\mathbb{Q}_{p})$ (associated with a Bruhat-Tits group scheme
$\mathcal{G}\to\operatorname{Spec}\mathbb{Z}_{p}$) and a sufficiently small
open compact subgroup $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$. We also keep up
our standard assumptions 1.16.
### 2.1 Definition of central leaves
We use the notation for Hodge tensors, in particular
$\left(s_{\alpha}\right)_{\alpha}$ and $\left(s_{\alpha,0}\right)_{\alpha}$,
established in Section 1.8.2.
###### Definition 2.1.
Two points
$\bar{x}_{1},\bar{x}_{2}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$ lie in
the same _central leaf_ if there is an isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ of Dieudonné modules
that takes $s_{\alpha,0,\bar{x}_{1}}$ to $s_{\alpha,0,\bar{x}_{2}}$.
They lie in the same _naive central leaf_ if there is an arbitrary isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ of Dieudonné modules
(i.e., an isomorphism
$\mathscr{A}_{\bar{x}_{1}}[p^{\infty}]\cong\mathscr{A}_{\bar{x}_{1}}[p^{\infty}]$
of $p$-divisible groups).
###### Remark 2.2.
More generally, one can make analogous definitions when given any abelian
scheme $\mathcal{A}\to S$ over any $\mathbb{F}_{p}$-scheme $S$ (such that for
every point we are given tensors on the Dieudonné module of the corresponding
abelian variety). Given two arbitrary points, one compares them after going to
a common algebraically closed extension of the residue fields, cf. [Oor04].
###### Notation 2.3.
Recall that we denote the completion of the maximal unramified extension
$\mathbb{Q}_{p}^{\mathrm{ur}}$ of $\mathbb{Q}_{p}$ by
$\breve{\mathbb{Q}}_{p}$, and its ring of integers accordingly by
$\breve{\mathbb{Z}}_{p}$. We set
$\breve{K}:=\breve{K}_{p}:=\mathcal{G}(\breve{\mathbb{Z}}_{p})$ and define
$\breve{K}_{\sigma}$ to be the graph of the Frobenius
$\sigma\colon\breve{K}\to\breve{K}$. So dividing out the action of
$\breve{K}_{\sigma}$ (which is mostly how $\breve{K}_{\sigma}$ will make an
appearance) means dividing out the action of $\breve{K}$ by
$\sigma$-conjugation.
###### Definition 2.4.
The central leaves are the fibers of the map
$\Upsilon=\Upsilon_{K}\colon\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})\to
G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$
given as follows: For $\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$
there is an isomorphism $\beta\colon
V_{\mathbb{Z}_{p}}^{*}\otimes_{\mathbb{Z}_{p}}\breve{\mathbb{Z}}_{p}\cong\mathbb{D}_{\bar{x}}$
(equation (1.59)) sending $s_{\alpha}\otimes 1$ to $s_{\alpha,0,\bar{x}}$. We
hence can interpret the Frobenius on $\mathbb{D}_{\bar{x}}$ as an element of
$G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$, where dividing out
$\breve{K}_{\sigma}$ rids us of the ambiguity introduced by the choice of the
isomorphism $\beta$.
###### Notation 2.5.
We write
$\displaystyle G(\breve{\mathbb{Q}}_{p})$ $\displaystyle\to{}$ $\displaystyle
C(G):=G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$ $\displaystyle\to
B(G):=G(\breve{\mathbb{Q}}_{p})/G(\breve{\mathbb{Q}}_{p})_{\sigma},$
$\displaystyle b$ $\displaystyle\mapsto{}$
$\displaystyle[\mkern-2.0mu[b]\mkern-2.0mu]$ $\displaystyle\mapsto[b].$
#### 2.1.1 An alternative characterization of the central leaves
Consider two points
$\bar{x}_{1},\bar{x}_{2}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$ in the
same central leaf, i.e., such that there is an isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ of Dieudonné modules
that takes $s_{\alpha,0,\bar{x}_{1}}$ to $s_{\alpha,0,\bar{x}_{2}}$.
$\bar{x}_{j}$ for $j=1,2$ has an associated isogeny chain of abelian schemes
in $\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})$, cf. Remark 1.19, and
$\mathbb{D}_{\bar{x}_{j}}=\mathbb{D}(\prod_{i=-(r-1)-a}^{r-1}A_{j,i})=\prod_{i=-(r-1)-a}^{r-1}\mathbb{D}(A_{j,i})$,
where $(A_{j,i})_{i}$ is the isogeny chain associated with $\bar{x}_{j}$ under
$\mathscr{S}_{K}(G,X)\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$,
cf. Remark 1.21 and Proposition 1.24. Note that
$A_{j}=\prod_{i=-(r-1)-a}^{r-1}A_{j,i}$ (plus extra structure) is the point
associated with $\bar{x}_{j}$ under
$\mathscr{S}_{K}(G,X)\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$.
###### Lemma 2.6.
Any tensor-preserving isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ yields, for all $i$,
isomorphisms ${\mathbb{D}(A_{1,i})\cong\mathbb{D}(A_{2,i})}$.
###### Proof:
This follows immediately from Remark 1.46. □
We may thus rephrase our definition of central leaves as follows.
###### Lemma 2.7.
Let $\bar{x}_{1},\bar{x}_{2}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$
and, as above, denote by $A_{j,i}$ the abelian varieties in the associated
isogeny chains, $j=1,2$.
$\bar{x}_{1}$ and $\bar{x}_{2}$ lie in the same central leaf if and only if
there is an isomorphism between the associated rational Dieudonné modules of
$A_{j,i}$ (which is common to all $A_{j,i}$, $j$ fixed, $i$ variable)
respecting the tensors and identifying the lattices $\mathbb{D}(A_{1,i})$ and
$\mathbb{D}(A_{2,i})$ for all $i$.
“Respecting the tensors” means the following: We get an induced identification
of the rational Dieudonné modules of $A_{1}$ and $A_{2}$. This identification
has to preserve the tensors.
### 2.2 Local closedness of central leaves
#### 2.2.1 Topological lemmas
We begin with some purely topological preliminaries which we shall use to
globalize statements about formal neighborhoods.
###### Lemma 2.8.
Let $X$ be a topological space with a subset $A\subseteq X$, and for $x\in X$
let $U_{x}\subseteq X$ be the set of generizations of $x$. Consider the
statements
1. (i)
$A\cap U_{x}\subseteq U_{x}$ closed (resp. open) for all $x\in X$.
2. (ii)
$A\cap U_{x}\subseteq U_{x}$ closed (resp. open) for all closed points $x\in
X$.
3. (iii)
$A\subseteq X$ stable under specialization (resp. generization).
We have (i)$\implies$(ii),(iii), and if every $x\in X$ has a specialization in
$x$ that is a closed point, we also have (ii)$\implies$(iii).
###### Proof:
Denote the closure of $A$ in $X$ by $\operatorname{cl}_{X}(A)$.
For “(i)$\implies$(iii)” (with “closed” and “specialization”, respectively)
let $x\in A$ and $s\in\operatorname{cl}_{X}(\\{x\\})$. Then $s,x\in U_{s}$,
and $s\in\operatorname{cl}_{X}(\\{x\\})\cap
U_{s}=\operatorname{cl}_{U_{s}}(\\{x\\})$. By assumption $A\cap U_{s}\subseteq
U_{s}$ is stable under specialization, hence $s\in A\cap U_{s}\subseteq A$.
The rest is proven along similar lines. □
###### Example 2.9.
$\mathbb{N}\subseteq\mathbb{A}^{1}_{\mathbb{C}}$ satisfies property (i) from
the lemma but is not closed (hence not constructible).
###### Lemma 2.10.
Let $X$ be a sober topological space and let $A\subseteq X$ be stable under
both generization and specialization.
Then $A$ is a union of irreducible components of $X$.
If additionally $X$ only has finitely many irreducible components, $A$ even is
a union of connected components and is open and closed.
###### Proof:
Consider the unique generic points $\left\\{\eta_{i}\right\\}_{i\in I}$ of the
irreducible components of $X$. By assumption, we have, firstly, that if $A$
contains $\eta_{i}$ then it contains the entire irreducible component
$\overline{\\{\eta_{i}\\}}$, and, secondly, that if $A$ contains any one point
of $\overline{\\{\eta_{i}\\}}$, then it also contains $\eta_{i}$. Similarly,
if $A$ contains any one irreducible component $\overline{\\{\eta_{i}\\}}$, it
also contains all irreducible components that meet
$\overline{\\{\eta_{i}\\}}$. □
#### 2.2.2 Local closedness
To show the local closedness of central leaves, we can content ourselves with
a construction in the spirit of Proposition 1.66 but somewhat simpler. Namely,
we pull back (1.61) to obtain
$\displaystyle\mathscr{G}_{\bar{x}}\to\operatorname{Spec}\bar{R}_{G}$
$\displaystyle:=\operatorname{Spec}R_{G}\otimes_{{{\cal
O}_{\breve{E}}}}\bar{\mathbb{F}}_{p}$
$\displaystyle=\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}},\bar{x}}\otimes\bar{\mathbb{F}}_{p})=\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}).$
The Dieudonné display of $\mathscr{G}_{\bar{x}}$ then is
$\mathrm{DDisp}(\mathscr{G}_{\bar{x}})=\mathbb{D}(\mathscr{G}_{\bar{x}})^{\vee}(\hat{W}(\bar{R}_{G}))=\mathbb{D}_{\bar{x}}\otimes_{\breve{\mathbb{Z}}_{p}}\hat{W}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}).$
In particular we can consider the tensors $\left(s_{\alpha,0}\otimes
1\right)_{\alpha}$ on this Dieudonné display and on
$\mathbb{D}(\mathscr{G}_{\bar{x}})^{\vee}(W(\bar{R}_{G}))$.
Thus we get crystalline Tate tensors
$\left(u_{\alpha,\bar{x}}\right)_{\alpha}$ on
$\mathbb{D}(\mathscr{G}_{\bar{x},\mathrm{perf}})$, where
$\mathscr{G}_{\bar{x},\mathrm{perf}}$ is the pullback of
$\mathscr{G}_{\bar{x}}$ to $\bar{R}_{G}^{\mathrm{perf}}$.
###### Theorem 2.11.
The central leaves on $\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}$ are
open and closed in the naive central leaves.
###### Proof:
We consider the $p$-divisible group $\mathscr{G}_{\bar{x},\mathrm{perf}}$ over
$\widehat{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}^{\mathrm{perf}}$ together
with its crystalline Tate tensors. Note that perfection makes no difference
for topological considerations.
On the naive central leaves on $\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}})=\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}^{\mathrm{perf}})$ the
$p$-divisible group is geometrically fiberwise constant. By a lemma of
Hamacher [Ham17, 2.12], the tensors then are geometrically fiberwise constant
as well.
Using the fact that $\widehat{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$ is noetherian, we obtain
that the central leaf on $\widehat{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$ is closed and open in the
corresponding naive central leaf.
$\operatorname{Spec}{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$ has the quotient topology
with respect to the fpqc covering $\operatorname{Spec}\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}\to\operatorname{Spec}{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$, hence we also get the same
result after leaving away completion.
By Lemma 2.10 we conclude that we also get the same result for the central
leaves and naive central leaves on
$\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}$. □
###### Corollary 2.12.
The central leaves on $\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}$ are
locally closed.
###### Proof:
Because Oort [Oor04] has proven that the naive central leaves on
${\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}}$ are locally closed, this
is an immediate consequence of the preceding theorem. □
###### Corollary 2.13.
In fact, Oort has shown that the naive central leaf is closed in the naive
Newton stratum. So our argument even goes to show that the central leaves are
closed in their respective Newton strata (naive and non-naive).
### 2.3 Quasi-isogenies of $p$-divisible groups
Let $b\in
G(\breve{\mathbb{Q}}_{p})=\mathcal{G}(\breve{\mathbb{Q}}_{p})\subseteq\operatorname{GL}(\Lambda^{\S})(\breve{\mathbb{Q}}_{p})$
and denote by $b^{\S}$ the image in
$\operatorname{GL}(V^{\S})(\breve{\mathbb{Q}}_{p})$.
Let $J_{b}$ denote the $\mathbb{Q}_{p}$-reductive group given on $R$-valued
points, $R\in(\mathbb{Q}_{p}\text{-alg})$, by
$J_{b}(R):=\left\\{g\in
G(R\otimes_{\mathbb{Q}_{p}}\breve{\mathbb{Q}}_{p})=\operatorname{Res}_{\breve{\mathbb{Q}}_{p}/\mathbb{Q}_{p}}(G)(R)\;|\;gb\sigma(g)^{-1}=b\right\\}$
(cf. [RZ96, Prop. 1.12] and [Kim19, Prop. 2.2.6]).
$J_{b}(\mathbb{Q}_{p})$ then naturally attains the structure of a locally
profinite group. We make it into a formal group scheme
$\underline{J_{b}(\mathbb{Q}_{p})}$ over
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$; with
$\underline{J_{b}(\mathbb{Q}_{p})}(U)=\mathrm{Map}_{\mathrm{cont}}(U,J_{b}(\mathbb{Q}_{p}))$
for formal test schemes
$U\to\operatorname{Spf}\breve{\mathbb{Z}}_{p}$.151515Locally: If
$G=\varprojlim_{n}G_{n}$ is a profinite set (or even profinite group), then
$\underline{G}:=\varprojlim_{n}\underline{G_{n}}$ with $\underline{G_{n}}$
constant formal (group) scheme.
Let $\mathbb{X}_{b}=\mathbb{X}_{b,K}=\mathbb{X}_{b^{\S}}$ be a polarized
$p$-divisible group over $\bar{\mathbb{F}}_{p}$ with a distinguished
isomorphism between its Dieudonné (symplectic) module and
$(\Lambda^{\S})_{\breve{\mathbb{Z}}_{p}}$ under which the Frobenius is
identified with $b$. The existence of such a $p$-divisible group depends on
$b$, but we _assume_ that $\mathbb{X}_{b}$ exists from now on (see also 2.35
(2) below). By Dieudonné theory we obtain a bijection
$\displaystyle J_{b^{\S}}(\mathbb{Q}_{p})$
$\displaystyle:=\left\\{g\in\operatorname{GL}(\Lambda^{\S}\otimes\breve{\mathbb{Q}}_{p})\;|\;gb^{\S}\sigma(g)^{-1}=b^{\S}\right\\}$
$\displaystyle\cong\\{\text{(self-)quasi-isogenies of }\mathbb{X}_{b}\text{
(without polarization)}\\}$
(and similar with polarization (replace $\operatorname{GL}$ by
$\mathrm{(G)Sp}$)), and $J_{b}(\mathbb{Q}_{p})\subseteq
J_{b^{\S}}(\mathbb{Q}_{p})$ under this bijection corresponds to the _tensor-
preserving_ quasi-isogenies of $\mathbb{X}_{b}$.
We need to understand more than this; at least we will need to understand the
tensor-preserving quasi-isogenies of $(\mathbb{X}_{b})_{\Omega}$ for all
algebraically closed fields of characteristic $p$. Using internal hom
$p$-divisible groups first defined by Chai and Oort, Caraiani and Scholze
[CS17, Prop. 4.2.11] worked out the structure of the quasi-isogeny group
(albeit in a more special setting than ours; Kim [Kim19] generalized it to our
setting).
First of all, following [Kim19, section 3.2], we denote by
$\operatorname{Qisg}(\mathbb{X}_{b})$ the formal group scheme over
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$ given by
$\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}\to\mathrm{Grp},\quad
R\mapsto\\{\text{quasi-isogenies of }(\mathbb{X}_{b})_{R/p}\\},$
where $\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ is the category of
$\breve{\mathbb{Z}}_{p}$-algebras on which $p$ is nilpotent (anti-equivalent
to the category of affine schemes over
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$161616That is, affine schemes
together with a natural transformation of functors
$\mathrm{Ring}\to\mathrm{Set}$ from it to
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$.).
###### Definition 2.14.
Let $R$ be a topological ring.
1. (1)
$R$ is called _adic_ , if there is an ideal $I$ (called _ideal of definition_)
such that $\left\\{I^{n}\right\\}_{n\in\mathbb{N}}$ is a basis of
neighborhoods (or, equivalently, basis of _open_ neighborhoods) of $0$.
2. (2)
$R$ is called _f-adic_ (or _Huber ring_), if there exists an open subring
$A_{0}\subseteq A$ that is adic with finitely generated ideal of definition.
Such a ring is called a _ring of definition_.
###### Definition 2.15.
Let $R$ be a ring of characteristic $p$.
1. (1)
$R$ is _semiperfect_ , if the Frobenius endomorphism $\Phi\colon R\to R$ is
surjective.
2. (2)
$R$ is _f-semiperfect_ , if it is semiperfect and the perfection
$R^{\flat}:=\varprojlim_{\Phi}R$ (with the inverse limit topology) is f-adic.
There is a functorial construction which assigns to every semiperfect ring $R$
its universal $p$-adically complete pd-thickening $A_{\mathrm{cris}}(R)$
[SW13, Prop. 4.1.3]. Set
$B_{\mathrm{cris}}^{+}(R):=A_{\mathrm{cris}}(R)[\frac{1}{p}]$. We have
$\mathbb{D}((\mathbb{X}_{b})_{R})(A_{\mathrm{cris}}(R))\cong\mathbb{D}(\mathbb{X}_{b})(\breve{\mathbb{Z}}_{p})\otimes_{\breve{\mathbb{Z}}_{p}}A_{\mathrm{cris}}(R)\cong\Lambda^{\S}\otimes_{\mathbb{Z}_{p}}A_{\mathrm{cris}}(R)$
by construction.
###### Assumption 2.16.
We will assume from now on that $[b]\in B(G)$ is _neutral acceptable_ in the
sense of [RV14, Def. 2.3], i.e., $[b]\in B(G,\\{\mu\\})$. (Of course this is a
completely harmless assumption with regard to the setting of Shimura
varieties.)
There exists an internal tensor-preserving quasi-endomorphism $p$-divisible
group:
###### Lemma 2.17.
[Kim19, Lemma 3.1.3] There is a $p$-divisible group $\mathcal{H}_{b}^{G}$ such
that for any f-semiperfect $\bar{\mathbb{F}}_{p}$-algebra $R$ there is a
natural $\mathbb{Q}_{p}$-linear isomorphism
$\tilde{\mathcal{H}}_{b}^{G}(R)\cong\operatorname{End}_{(s_{\alpha})}((\mathbb{X}_{b})_{R})\bigl{[}\tfrac{1}{p}\bigr{]},$
where $\tilde{\mathcal{H}}_{b}^{G}$ is the universal cover of
$\mathcal{H}_{b}^{G}$, and where on the right hand side we have the set of
$\gamma\in\operatorname{End}((\mathbb{X}_{b})_{R})[\frac{1}{p}]$ such that the
endomorphism of $\Lambda^{\S}\otimes_{\mathbb{Z}_{p}}B_{\mathrm{cris}}^{+}(R)$
induced by $\gamma$ preserves the tensors $s_{\alpha}\otimes 1$.
What we really need is an internal tensor-preserving quasi-isogeny
$p$-divisible group. Still following [Kim19], to this end we consider the
group sheaf $\operatorname{Qisg}(\mathbb{X}_{b})$ on
$\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ given by
$R\mapsto\\{\text{self-quasi-isogenies of }(\mathbb{X}_{b})_{R}\\}$. This can
be realized as a closed formal subscheme of $\tilde{\mathcal{H}}_{b}^{2}$
(where
$\mathcal{H}_{b}:=\mathcal{H}_{b^{\S}}=\mathcal{H}_{b^{\S}}^{\operatorname{GL}(V^{\S})}$
(no tensors)).
The closed formal subscheme
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})\subseteq\operatorname{Qisg}(\mathbb{X}_{b})$
over $\operatorname{Spf}\breve{\mathbb{Z}}_{p}$ then is defined by
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})=\operatorname{Qisg}(\mathbb{X}_{b})\times_{\tilde{\mathcal{H}}_{b}^{2}}(\tilde{\mathcal{H}}_{b}^{G})^{2}.$
###### Proposition 2.18.
[Kim19, Prop 3.2.4] We have a natural map
$\underline{J_{b}(\mathbb{Q}_{p})}\to\operatorname{Qisg}_{G}(\mathbb{X}_{b})$,
which has a natural retraction
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})\to\underline{J_{b}(\mathbb{Q}_{p})}$
with all fibers isomorphic to
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x_{1}^{p^{-\infty}},\dotsc,x_{d}^{p^{-\infty}}]\mkern-2.0mu]$
as formal schemes, where $d=\langle 2\rho,\nu_{[b]}\rangle$ with $2\rho$ the
sum of all the positive roots of $G_{\bar{\mathbb{Q}}_{p}}$ and $\nu_{[b]}$
the dominant171717This requires distinguishing a Weyl chamber as the positive
one; this arises from the choice of a convenient maximal torus in [Kim19, just
before Rmk. 2.1.4]. representative of the
$G(\breve{\mathbb{Q}}_{p})$-conjugacy class of $\nu_{b}\in
X_{*}(G_{\breve{\mathbb{Q}}_{p}})_{\mathbb{Q}}$.181818So $b\mapsto\nu_{[b]}$
is the _Newton map_ in the parlance of [Kot97, 256].
###### Corollary 2.19.
Let $R\in\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ be such that there is
no non-trivial continuous homomorphism of $\breve{\mathbb{Z}}_{p}$-algebras
$\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x^{p^{-\infty}}]\mkern-2.0mu]\to R$. Put
differently, there is no $r\in R$ having a compatible system of $p$-power
roots such that all power series with $\breve{\mathbb{Z}}_{p}$-coefficients in
$r$ and its $p$-power roots converge.
Then
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})(R)\cong\underline{J_{b}(\mathbb{Q}_{p})}(R).$
###### Proof:
With $d$ as in the proposition, the assumption implies that there is no non-
trivial continuous homomorphism of $\breve{\mathbb{Z}}_{p}$-algebras
$\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x_{1}^{p^{-\infty}},\dotsc,x_{d}^{p^{-\infty}}]\mkern-2.0mu]\to
R$.
Define
$\operatorname{Qisg}_{G}^{\circ}(\mathbb{X}_{b}):=\ker(\operatorname{Qisg}_{G}(\mathbb{X}_{b})\to\underline{J_{b}(\mathbb{Q}_{p})})\cong\operatorname{Spf}\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x_{1}^{p^{-\infty}},\dotsc,x_{d}^{p^{-\infty}}]\mkern-2.0mu]$
and consider the split exact sequence
$0\to\operatorname{Qisg}_{G}^{\circ}(\mathbb{X}_{b})\to\operatorname{Qisg}_{G}(\mathbb{X}_{b})\to\underline{J_{b}(\mathbb{Q}_{p})}\to
0.$
□
###### Example 2.20.
Say $R\in\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ has characteristic $p$,
i.e., $R$ is a $\bar{\mathbb{F}}_{p}$-algebra and the $p$-adic topology is the
discrete topology. Let $r$ be an element of $R$ having a compatible system of
$p$-power roots such that all power series with
$\breve{\mathbb{Z}}_{p}$-coefficients in $r$ and its $p$-power roots converge.
Then $r$ must be nilpotent. Therefore, if
$R\in\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ has characteristic $p$ and
is reduced (e.g., is a field), then the conditions of Corollary 2.19 are
satisfied.
In particular we deduce:
###### Example 2.21.
Let $L/\bar{\mathbb{F}}_{p}$ be a field extension. Then
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})(L)\cong J_{b}(\mathbb{Q}_{p}).$
### 2.4 Almost product structure
###### Notation 2.22.
We shorten notation by writing
$\mathscr{S}:=\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{\mathbf{E},(v)}}{{\cal
O}_{\breve{E}}}\quad\text{and}\quad\overline{\mathscr{S}}:=\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},$
as well as
$\mathscr{S}^{\S}:=\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\pm,\S})\otimes_{\mathbb{Z}_{(p)}}\breve{\mathbb{Z}}_{p}\quad\text{and}\quad\overline{\mathscr{S}}^{\S}:={\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\pm,\S})\otimes_{\mathbb{Z}_{(p)}}\bar{\mathbb{F}}_{p}}.$
Moreover, we denote the central leaf associated with $b$ by
$\overline{C}^{b}:=\overline{C}^{b}_{K}:=\Upsilon_{K}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])\subseteq\overline{\mathscr{S}}$,
and by $\overline{C}^{b^{\S}}\subseteq\overline{\mathscr{S}}^{\S}$ the
corresponding central leaf in $\overline{\mathscr{S}}^{\S}$.
###### Assumption 2.23.
From now on, we assume that Axiom A from [HK17] holds.
###### Remark 2.24.
To explain Axiom A: Consider the _affine Deligne-Lusztig variety_
$X_{\mu}(b)_{K}(\bar{\mathbb{F}}_{p}):=\left\\{g\breve{K}_{p}\in
G(\breve{\mathbb{Q}}_{p})/\breve{K}_{p}\;|\;g^{-1}b\sigma(g)\in\breve{K}_{p}\mathrm{Adm}(\\{\mu\\})\breve{K}_{p}\right\\}$
(here $\mathrm{Adm}(\\{\mu\\})$ is the _admissible subset_ as defined in
[PRS13]).
Choose a point $\bar{x}\in\overline{\mathscr{S}}(\bar{\mathbb{F}}_{p})$ with a
quasi-isogeny $j\colon\mathbb{X}_{b}\to\mathscr{A}_{\bar{x}}[p^{\infty}]$
compatible with the extra structure.
We have a map $i^{\S}_{(\bar{x},j)}\colon
X_{\mu^{\S}}(b^{\S})(\bar{\mathbb{F}}_{p})\to\overline{\mathscr{S}}^{\S}(\bar{\mathbb{F}}_{p})$
given as follows: For $g\breve{J}_{p}\in
X_{\mu^{\S}}(b^{\S})(\bar{\mathbb{F}}_{p})$, there is a $p$-divisible group
$g\mathbb{X}_{b}$ isogenous to $\mathbb{X}_{b}$ with Frobenius on the
Dieudonné module given by $g^{-1}b\sigma(g)$. The quasi-isogeny
$g\mathbb{X}_{b}\to\mathbb{X}_{b}\to\mathscr{A}_{\bar{x}}[p^{\infty}]$ lifts
to a quasi-isogeny of abelian schemes191919Idea: A lift of a quasi-isogeny
$y\colon A[p^{\infty}]\to X$, $A$ abelian scheme, $X$ $p$-divisible group, is
given by $A\to A/\ker(y)$.. We also get a polarization and a level structure
away from $p$ on the new abelian scheme, which is the image of
$g\breve{J}_{b}$ under $i^{\S}_{(\bar{x},j)}$.
The content of Axiom A (= Assumption 2.23) now is that
$X_{\sigma(\mu)}(b)_{K}(\bar{\mathbb{F}}_{p})\hookrightarrow
X_{\mu^{\S}}(b^{\S})(\bar{\mathbb{F}}_{p})\xrightarrow{i^{\S}_{(\bar{x},j)}}\overline{\mathscr{S}}^{\S}(\bar{\mathbb{F}}_{p})$
is to factor through $\overline{\mathscr{S}}^{-}(\bar{\mathbb{F}}_{p})$ such
that there is a unique lift $i\colon
X_{\sigma(\mu)}(b)_{K}(\bar{\mathbb{F}}_{p})\to\overline{\mathscr{S}}(\bar{\mathbb{F}}_{p})$
with $s_{\alpha,0,i(g\breve{K}_{p})}=s_{\alpha,0,\bar{x}}$ for all
$g\breve{K}_{p}\in X_{\sigma(\mu)}(b)_{K}(\bar{\mathbb{F}}_{p})$.
###### Remark 2.25.
Assumption 2.23 holds in the hyperspecial case [Kis17, Thm. 1.4.4], and it
holds if $G$ is residually split [Zho18, Prop. 6.4].
###### Remark 2.26.
With Assumption 2.23 in place, the central leaf $\overline{C}^{b}$ of $b$ is
non-empty by [HK17, Rmk. 4.3.1].
We review some constructions from the paper [HK17]. By
$\overline{\mathrm{Ig}}^{b^{\S}}$ we denote the (special fiber of the) Igusa
variety over $\overline{C}^{b^{\S}}$, cf. [CS17, Prop. 1.12]. It parameterizes
isomorphisms between the standard $p$-divisible group and the universal one,
and it is a perfect scheme over $\bar{\mathbb{F}}_{p}$. Define
$\overline{\mathrm{Ig}}^{b,\diamond}:=\overline{\mathrm{Ig}}^{b^{\S}}\times_{\overline{\mathscr{S}}^{\S,\mathrm{perf}}}\overline{\mathscr{S}}^{\mathrm{perf}}$.
Furthermore, $\mathfrak{Ig}^{b^{\S}},\mathfrak{Ig}^{b,\diamond}$ are the
unique flat lifts202020Locally: Let $R$ be a perfect
$\bar{\mathbb{F}}_{p}$-algebra. Then $W_{\mathcal{O}_{E}}(R)$ (ramified Witt
vectors, cf. Remark 1.15) is the unique $\varpi$-adically complete ${{\cal
O}_{\breve{E}}}$-flat algebra lifting it (cf. [Ahs11, Prop. 1.3.3]). of
$\overline{\mathrm{Ig}}^{b^{\S}}$ and $\overline{\mathrm{Ig}}^{b,\diamond}$
over $\operatorname{Spf}{{\cal O}_{\breve{E}}}$, given by the formula
$\mathfrak{Ig}(R):=\overline{\mathrm{Ig}}(R/\varpi)\quad\text{for all
}R\in\operatorname{Nilp}_{{{\cal O}_{\breve{E}}}}$ (2.27)
in both cases, where $\varpi$ is a uniformizer of ${{\cal O}_{\breve{E}}}$.
By definition, we have an isomorphism
$j\colon\mathbb{X}_{b}\times\overline{\mathrm{Ig}}^{b,\diamond}\xrightarrow{\cong}\mathscr{A}_{\overline{C}^{b}}[p^{\infty}]\times_{\overline{C}^{b}}\overline{\mathrm{Ig}}^{b,\diamond}$
and $\overline{\mathrm{Ig}}^{b}$ is by definition the locus of geometric
points of $\overline{\mathrm{Ig}}^{b,\diamond}$, where $j$ respects the
crystalline Tate tensors. This is a closed union of connected components of
$\overline{\mathrm{Ig}}^{b,\diamond}$ [HK17, Def./Lemma 5.1.1].
$\operatorname{Qisg}(\mathbb{X}_{b})$ acts on
$\overline{\mathrm{Ig}}^{b^{\S}}$ and $\mathfrak{Ig}^{b^{\S}}$,
$\overline{\mathrm{Ig}}^{b}\to\overline{\mathrm{Ig}}^{b^{\S}}$ is a closed
immersion with
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})_{\bar{\mathbb{F}}_{p}}$-stable image,
and $\mathfrak{Ig}^{b}\to\mathfrak{Ig}^{b^{\S}}$ is a closed immersion with
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})$-stable image [HK17, Prop. 5.1.2, Cor
5.1.3].
Let $\mathfrak{M}^{b^{\S}}\to\operatorname{Spf}\breve{\mathbb{Z}}_{p}$ be the
Rapoport-Zink space given by
$\mathfrak{M}^{b^{\S}}(R)=\left\\{(\mathscr{X},\rho)\;\middle|\;\begin{tabular}[]{@{}l@{}}$\mathscr{X}/R$
polarized $p$-divisible group and\\\ $\rho\colon\mathbb{X}_{b}\otimes
R/p\to\mathscr{X}\otimes R/p$ quasi-isogeny resp.
polarizations\end{tabular}\right\\}.$
Choose a point $\bar{x}\in\overline{\mathscr{S}}(\bar{\mathbb{F}}_{p})$ with a
quasi-isogeny $j\colon\mathbb{X}_{b}\to\mathscr{A}_{\bar{x}}[p^{\infty}]$
compatible with the extra structure.
$\mathfrak{M}^{b^{\S}}$ comes with the Rapoport-Zink uniformization map
$\Theta^{\S}\colon\mathfrak{M}^{b^{\S}}\to\mathscr{S}^{\S}$ depending on this
choice [RZ96, (6.3)]: Given $(\mathscr{X},\rho)$ as above, $\rho$ lifts to a
quasi-isogeny $\tilde{\rho}\colon\mathbb{X}_{b}\otimes R\to\mathscr{X}$, and
there is an abelian scheme $\mathscr{Y}$ with $p$-divisible group
$\mathscr{X}$ and a unique lift $\tilde{\mathscr{A}}_{\bar{x}}\otimes
R\to\mathscr{Y}$ of $\tilde{\rho}$ for a chosen lift
$\tilde{\mathscr{A}}_{\bar{x}}$ of $\mathscr{A}_{\bar{x}}$ to
$\breve{\mathbb{Z}}_{p}$. We also get a polarization and a level structure
away from $p$ on $\mathscr{Y}$, and $\mathscr{Y}$ with these extra structures
is the image of $(\mathscr{X},\rho)$ under $\Theta^{\S}$.
###### Remark 2.28.
The affine Deligne-Lusztig variety is the perfection of the special fiber of
the Rapoport-Zink space [Zhu17, Prop. 0.4]. Under this isomorphism,
$\Theta^{\S}$ corresponds to $i^{\S}$ from Remark 2.24.
Define
$\mathfrak{M}^{b,\diamond}:=\mathfrak{M}^{b^{\S}}\times_{\mathscr{S}^{\S}}\mathscr{S}$
and define $\Theta^{\diamond}\colon\mathfrak{M}^{b,\diamond}\to\mathscr{S}$ to
be the base change of $\Theta^{\S}$.
Let $\mathscr{X}^{\diamond}\to\mathfrak{M}^{b,\diamond}$ be the pullback of
the universal $p$-divisible group over $\mathfrak{M}^{b^{\S}}$. Then we have
two families of tensors on $\mathbb{D}(\mathscr{X}^{\diamond})[\tfrac{1}{p}]$
(or more precisely, two families of maps
$\mathfrak{M}^{b,\diamond}(\bar{\mathbb{F}}_{p})\ni\bar{y}\mapsto$ tensor on
$\mathbb{D}(\mathscr{X}^{\diamond}_{\bar{y}})[\tfrac{1}{p}]$):
1. (1)
$\left(t_{\alpha}^{\diamond}\right)_{\alpha}$ obtained from
$\left(s_{\alpha}\right)_{\alpha}$ via
$\mathbb{D}(\mathbb{X}_{b}\otimes\mathfrak{M}^{b,\diamond})[\tfrac{1}{p}]\cong\mathbb{D}(\mathscr{X}^{\diamond})[\tfrac{1}{p}]$,
and
2. (2)
for every $\bar{y}\in\mathfrak{M}^{b,\diamond}(\bar{\mathbb{F}}_{p})$,
$\left(u_{\alpha,\bar{y}}^{\diamond}\right)_{\alpha}$ with
$u_{\alpha,\bar{y}}^{\diamond}:=(\Theta^{\diamond})^{*}s_{\alpha,0,\Theta^{\diamond}(\bar{y})}$.
$\mathfrak{M}^{b}$ is defined to be the formal subscheme of
$\mathfrak{M}^{b,\diamond}$ corresponding to the locus where these agree.
Also, $\Theta\colon\mathfrak{M}^{b}\to\mathscr{S}$ is defined to be the
restriction of $\Theta^{\diamond}$. Moreover, $\mathfrak{M}^{b}$ has a natural
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})$-action.
By [CS17, § 4], there is an isomorphism
$\mathfrak{Ig}^{b^{\S}}\times\mathfrak{M}^{b^{\S}}\xrightarrow{\sim}\mathfrak{X}^{b^{\S}}$
(2.29)
with the _Newton-Igusa variety_
$\mathfrak{X}^{b^{\S}}(R)=\left\\{(A,\lambda,\eta^{p};\psi)\;\middle|\;\begin{tabular}[]{@{}l@{}}$(A,\lambda,\eta^{p})\in\mathscr{S}^{\S}(R)$,\\\
$\psi\colon(\mathbb{X}_{b},\lambda_{\mathbb{X}_{b}})\otimes
R/p\to(A[p^{\infty}],\lambda)\otimes R/p$ quasi-
isogeny\end{tabular}\right\\}.$
###### Remark 2.30.
Let us quickly recall how the isomorphism (2.29) works (suppressing
polarizations in the notation for simplicity). Let
$(\mathcal{A},\xi)\in\mathfrak{Ig}^{b^{\S}}(R)$ and
$(\mathscr{X},\rho)\in\mathfrak{M}^{b^{\S}}(R)$ be given. Consider the
composition
$\mathscr{X}\xrightarrow{\rho^{-1}}\mathbb{X}_{b}\otimes
R\xrightarrow{\xi}\mathcal{A}[p^{\infty}],$ (2.31)
where we denote a lift $\mathbb{X}_{b}\otimes
R/p\xrightarrow{\rho}\mathscr{X}\otimes R/p$ by $\rho$ again. Lift (2.31) to a
quasi-isogeny of abelian schemes $\mathcal{A}^{\prime}\to\mathcal{A}$. Then
$(\mathcal{A}^{\prime},\rho)\in\mathfrak{X}^{b^{\S}}(R)$ is our image point.
Define $\mathfrak{X}^{b}\subseteq\mathfrak{X}^{b^{\S}}$ to be the image of
$\mathfrak{Ig}^{b}\times\mathfrak{M}^{b}$ under the isomorphism (2.29). This
comes with a natural $\operatorname{Qisg}_{G}(\mathbb{X}_{b})$-action; cf.
[HK17, section 5.2].
We have canonical maps
$\pi_{\infty}^{\S}\colon\mathfrak{X}^{b^{\S}}\to\mathscr{S}^{\S}$ and
$\pi_{\infty}\colon\mathfrak{X}^{b}\to\mathscr{S}$.
###### Remark 2.32.
By [HK17, Thm. 5.2.6 (1)],
$\pi_{\infty,\bar{\mathbb{F}}_{p}}^{\mathrm{perf}}\colon\mathfrak{X}^{b,\mathrm{perf}}_{\bar{\mathbb{F}}_{p}}\to\overline{\mathscr{S}}^{\mathrm{perf}}$
represents the moduli problem
$\displaystyle(\text{perfect affine
}\overline{\mathscr{S}}^{\mathrm{perf}}\text{-schemes})$
$\displaystyle\to\mathrm{Set},$
$\displaystyle(\operatorname{Spec}(R)\xrightarrow{Q}\overline{\mathscr{S}}^{\mathrm{perf}})$
$\displaystyle\mapsto\begin{Bmatrix}\psi\colon(\mathbb{X}_{b})_{R}\to(\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}})_{Q}\text{
quasi-isogeny}\\\ \text{compatible with crystalline Tate-
tensors}\end{Bmatrix}$
with $\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}}$ and crystalline
Tate tensors on the right hand side as in Proposition 1.66.
### 2.5 Change of parahoric level
Now we consider the question of how the central leaves behave when the level
at $p$ is changed from $K_{p}$ to a larger $K_{p}^{\prime}$, where $K_{p}$ and
$K_{p}^{\prime}$ are associated with points
$x,x^{\prime}\in\mathcal{B}(G,\mathbb{Q}_{p})$ as described in Chapter 1.
Define $K:=K_{p}K^{p}$, as usual, and $K^{\prime}:=K_{p}^{\prime}K^{p}$. Let
$b\in G(\breve{\mathbb{Q}}_{p})$.
Let $(\mathcal{L},c)$ and $(\mathcal{L}^{\prime},c^{\prime})$ be the graded
lattice chains that are the images of $x$ and $x^{\prime}$, respectively,
under the embedding of buildings
$\iota\colon\mathcal{B}(G,\mathbb{Q}_{p})\hookrightarrow\mathcal{B}(\operatorname{GSp}(V),\mathbb{Q}_{p})$
described in Section 1.2. Note that the gradings $c$ and $c^{\prime}$ play no
role for our purposes.
###### Remarks 2.33.
1. (1)
Observe that $K_{p}$ and $K_{p}^{\prime}$ depend only on the minimal facet
$\mathfrak{f}$ and $\mathfrak{f}^{\prime}$ with $x\in\mathfrak{f}$ and
$x^{\prime}\in\mathfrak{f}^{\prime}$, respectively. The inclusion
$K_{p}\subseteq K_{p}^{\prime}$ means precisely that
$\mathfrak{f}^{\prime}\subseteq\overline{\mathfrak{f}}$. We can move
$x^{\prime}$ arbitrarily close to $x$ without altering $K_{p}^{\prime}$. The
minimal facets $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ of
$\mathcal{B}(\operatorname{GSp}(V),\mathbb{Q}_{p})$ containing $\iota(x)$ and
$\iota(x^{\prime})$, respectively, do _not_ depend only on
$\mathfrak{f},\mathfrak{f}^{\prime}$. Still, for $x^{\prime}$ sufficiently
close to $x$, we may assume that
$\mathfrak{g}^{\prime}\subseteq\overline{\mathfrak{g}}$ (if it were not so, we
would not be able to move $\iota(x^{\prime})$ arbitrarily close to $\iota(x)$
— but recall that $\iota$ is continuous), i.e., that $\mathcal{L}^{\prime}$ is
a thinned out version of $\mathcal{L}$. Say $\mathcal{L}$ is given by
$\Lambda^{0}\supsetneqq\Lambda^{1}\supsetneqq\dotsb\supsetneqq\Lambda^{r-1}\supsetneqq
p\Lambda^{0}$
(as in (1.12)); then $\mathcal{L}^{\prime}$ is given by
$(\Lambda^{i_{j}})_{j}$ for some $0\leq i_{1}<i_{2}<\dotsb<i_{s}<r$.
2. (2)
In proofs, we can often reduce to the case where “$\mathcal{L}$ and
$\mathcal{L}^{\prime}$ differ by one element”, which of course is supposed to
mean that $s=r-2$ in this notation.
###### Notation 2.34.
We define
$N_{p},N_{p}^{\prime},J_{p},J_{p}^{\prime},\Lambda^{\S},\left.\Lambda^{\prime}\right.^{\S},V^{\S},\left.V^{\prime}\right.^{\S}$
as in Section 1.4.
#### 2.5.1 The change-of-parahoric morphism
As explained in Section 1.6, we obtain (for $K^{p}$ sufficiently small and
some compact open subgroup
$N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$) a commutative
diagram
$\mathscr{S}_{K_{p}K^{p}}(G,X)$$\mathscr{S}_{K^{\prime}_{p}\left.K\right.^{p}}(G,X)$$\mathscr{S}_{N_{p}N^{p}}(\operatorname{GSp}(V),S^{\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$$\mathscr{S}_{N_{p}^{\prime}\left.N\right.^{p}}(\operatorname{GSp}(V),S^{\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$finitefinite$\pi_{K_{p}K^{p},K_{p}^{\prime}\left.K\right.^{p}}$$\pi_{N_{p}N^{p},N_{p}^{\prime}\left.N\right.^{p}}$
###### Notation 2.35.
1. (1)
The change-of-parahoric map
$\pi_{K,K^{\prime}}:=\pi_{K_{p}K^{p},K_{p}^{\prime}\left.K\right.^{p}}$
restricts to a change-of-parahoric map
$\Upsilon_{K_{p}K^{p}}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])\to\Upsilon_{K_{p}^{\prime}K^{p}}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])$
between leaves, which we denote by
$\left.\pi_{K,K^{\prime}}\right|_{\Upsilon_{K}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])}$
or simply by $\pi_{K,K^{\prime}}$ again.
2. (2)
We choose a base point
$\bar{x}_{b}\in\overline{C}^{b}_{K}(\bar{\mathbb{F}}_{p})$ and take its image
under $\pi_{K,K^{\prime}}$ as a base point
$\bar{x}_{b}^{\prime}\in\overline{C}^{b}_{K^{\prime}}(\bar{\mathbb{F}}_{p})$.
By $\mathbb{X}_{b,K}$ and $\mathbb{X}_{b,K^{\prime}}$, respectively, we denote
the corresponding polarized $p$-divisible groups.
3. (3)
Denote by $\mathscr{A}_{K}$ the universal polarized abelian scheme over
$\overline{\mathscr{S}}_{K}$. By slight abuse of notation, we will also use
the same notation for its pullback to
$\overline{\mathscr{S}}_{K}^{\mathrm{perf}}$.
###### Remark 2.36.
We have morphisms $\mathbb{X}_{b,K}\to\mathbb{X}_{b,K^{\prime}}$ and, more
generally, $\mathscr{A}_{K}\to\mathscr{A}_{K^{\prime}}$ (lying over
$\overline{\mathscr{S}}_{K}\to\overline{\mathscr{S}}_{K^{\prime}}$). This
follows from the “thinning out” interpretation, Remark 2.33 (1).
#### 2.5.2 Newton-Igusa variety and change-of-parahoric
###### Lemma 2.37.
There are natural change-of-parahoric morphisms
$\mathfrak{Ig}^{b}_{K}\to\mathfrak{Ig}^{b}_{K^{\prime}},\quad\mathfrak{M}^{b}_{K}\to\mathfrak{M}^{b}_{K^{\prime}},\quad\mathfrak{X}^{b}_{K}\to\mathfrak{X}^{b}_{K^{\prime}}.$
lying over the change-of-parahoric morphism
$\mathscr{S}_{K}\to\mathscr{S}_{K^{\prime}}$ and compatible with the Siegel
embeddings.
###### Proof:
Consider the universal isomorphism
$j\colon\mathbb{X}_{b,K}\times\overline{\mathrm{Ig}}_{K}^{b,\diamond}\xrightarrow{\cong}\mathscr{A}_{K,\overline{C}^{b}_{K}}[p^{\infty}]\times_{\overline{C}^{b}_{K}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}=\mathscr{A}_{K}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$
and the diagram
$\mathbb{X}_{b,K}\times\overline{\mathrm{Ig}}^{b,\diamond}_{K}$$\mathscr{A}_{K}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$$\mathbb{X}_{b,K^{\prime}}\times\overline{\mathrm{Ig}}^{b,\diamond}_{K}$$\mathscr{A}_{K^{\prime}}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K^{\prime}}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$$\cong$$\cong$
The dashed arrow exists on $\overline{\mathrm{Ig}}^{b}_{K}$, i.e.,
$\mathbb{X}_{b,K^{\prime}}\times\overline{\mathrm{Ig}}^{b}_{K}\xrightarrow{\cong}\mathscr{A}_{K^{\prime}}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K^{\prime}}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b}$
exists. The reason is that the tensors are respected on
$\overline{\mathrm{Ig}}_{K}^{b}$ which in particular forces
$\mathbb{X}_{b,K}\times\overline{\mathrm{Ig}}^{b,\diamond}_{K}\xrightarrow{\cong}\mathscr{A}_{K}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$
to be “diagonal”, similar to the proof of Proposition 1.24.
This isomorphism
$\mathbb{X}_{b,K^{\prime}}\times\overline{\mathrm{Ig}}^{b}_{K}\xrightarrow{\cong}\mathscr{A}_{K^{\prime}}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K^{\prime}}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b}$
now corresponds to a morphism
$\overline{\mathrm{Ig}}_{K}^{b}\to\overline{\mathrm{Ig}}_{K^{\prime}}^{b}$. By
construction (2.27) of $\mathfrak{Ig}^{b}$, we also get a corresponding
morphism $\mathfrak{Ig}_{K}^{b}\to\mathfrak{Ig}_{K^{\prime}}^{b}$.
For $\mathfrak{X}$ and $\mathfrak{M}$, it is very similar. □
###### Lemma 2.38.
The isomorphism $\mathfrak{Ig}^{b}\times\mathfrak{M}^{b}\cong\mathfrak{X}^{b}$
is compatible with change-of-parahoric, where the change-of-parahoric map for
$\mathfrak{Ig}^{b}\times\mathfrak{M}^{b}$ is by definition the product of
those for $\mathfrak{Ig}^{b}$ and $\mathfrak{M}^{b}$, respectively.
###### Proof:
All of $\mathfrak{Ig}^{b},\mathfrak{M}^{b},\mathfrak{X}^{b}$ are embedded into
the corresponding Siegel versions as closed formal subschemes, compatible with
change-of-parahoric. Hence we may assume that we are in the Siegel case.
In that case, the isomorphism has a very concrete description, cf. Remark
2.30, from which the lemma is clear. □
###### Definition 2.39.
By $\overline{\mathscr{S}}_{K}^{b}:=\overline{\mathscr{S}}_{K}^{[b]}$ we
denote the Newton stratum associated with ${[b]\in B(G)}$, i.e.,
$\bar{S}_{K,[b]}$ in the notation of [HR17].
###### Lemma 2.40.
Let $\Omega/\mathbb{F}_{p}$ be an algebraically closed field.
The action of $J_{b}(\mathbb{Q}_{p})$ on the fibers of
$\mathfrak{X}_{K}^{b}(\Omega)\to\overline{\mathscr{S}}^{b}_{K}(\Omega)$ is
simply transitive and the change-of-parahoric map
$\mathfrak{X}_{K}^{b}(\Omega)\to\mathfrak{X}_{K^{\prime}}^{b}(\Omega)$
is $J_{b}(\mathbb{Q}_{p})$-equivariant.
###### Proof:
The moduli description of Remark 2.32 gives us that the action is simply
transitive upon noting that
$\operatorname{Qisg}_{G}(\mathbb{X}_{b,K})(\Omega)\overset{\ref{QisgJbField}}{\cong}J_{b}(\mathbb{Q}_{p})\overset{\ref{QisgJbField}}{\cong}\operatorname{Qisg}_{G}(\mathbb{X}_{b,K^{\prime}})(\Omega).$
Equivariance follows from the description of the action of
$J_{b}(\mathbb{Q}_{p})$ and of the change-of-parahoric map in terms of lattice
chains in Dieudonné theory: Given a point
$Q\colon\operatorname{Spec}\Omega\to\overline{\mathscr{S}}_{K}^{b}$, we
consider the map between the fiber of $Q$ under
$\mathfrak{X}^{b}_{K}(\Omega)\to\overline{\mathscr{S}}^{b}_{K}(\Omega)$ and
the fiber of the image of $Q$ under
$\mathfrak{X}^{b}_{K^{\prime}}(\Omega)\to\overline{\mathscr{S}}^{b}_{K^{\prime}}(\Omega)$.
We identify the lattice chain associated with $Q$ with a standard lattice
chain such that the Frobenius is identified with a $b^{\prime}\in
G(\breve{\mathbb{Q}}_{p})$, $b^{\prime}\equiv b\mod\breve{K}_{p,\sigma}$.
Elements of $J_{b}(\mathbb{Q}_{p})$ then act naturally on the common rational
Dieudonné module. The action on the fiber of $Q$ is given by altering the
quasi-isogenies appearing in the description of that set by the quasi-isogeny
obtained this way. Passing from $K$ to $K^{\prime}$, i.e., from $Q$ to the
image of $Q$, means leaving out parts of the lattice chain and enlarging
$\breve{K}_{p}$ (so one still has the same $b^{\prime}$). □
###### Proposition 2.41.
The change-of-parahoric map
$\mathfrak{X}^{b}_{K,\bar{\mathbb{F}}_{p}}\to\mathfrak{X}^{b}_{K^{\prime},\bar{\mathbb{F}}_{p}}$
is surjective.
###### Proof:
We check that
$\mathfrak{X}^{b}_{K}(\Omega)\to\mathfrak{X}^{b}_{K^{\prime}}(\Omega)$
is surjective for every algebraically closed field $\Omega/\mathbb{F}_{p}$.
Considering the diagram
$\mathfrak{X}^{b}_{K}(\Omega)$$\mathfrak{X}^{b}_{K^{\prime}}(\Omega)$$\overline{\mathscr{S}}_{K}^{b}(\Omega)$$\overline{\mathscr{S}}_{K^{\prime}}^{b}(\Omega)$
this follows from the preceding lemma and the fact212121This follows simply
from the surjectiveness of
$\overline{\mathscr{S}}_{K}\to\overline{\mathscr{S}}_{K^{\prime}}$ and the
commutativity of the diagram
$\overline{\mathscr{S}}_{K}$$\overline{\mathscr{S}}_{K^{\prime}}$$B(G)$, non-
horizontal maps being the Newton maps. that the change-of-parahoric map
between Newton strata is surjective. □
###### Corollary 2.42.
The map
$\overline{\mathrm{Ig}}_{K}^{b}\to\overline{\mathrm{Ig}}_{K^{\prime}}^{b}$ is
an isomorphism.
###### Proof:
Lemma 2.38 and Proposition 2.41 imply that it is surjective.
Also, it is the restriction to closed subschemes of the isomorphism
$\overline{\mathrm{Ig}}_{K}^{b^{\S}}\to\overline{\mathrm{Ig}}_{K^{\prime}}^{b^{\S}}$.
Since all involved schemes are reduced, the corollary follows. □
###### Corollary 2.43.
The change-of-parahoric morphism between central leaves is surjective.
###### Proof:
We have a diagram
$\overline{\mathrm{Ig}}_{K}^{b}$$\overline{\mathrm{Ig}}_{K^{\prime}}^{b}$$\Upsilon^{-1}_{K}(b)$$\Upsilon^{-1}_{K^{\prime}}(b)$$\cong$
where we already know that all maps but the lower horizontal one are
surjective. □
###### Corollary 2.44.
The separable rank of $\Upsilon^{-1}_{K}(b)\to\Upsilon^{-1}_{K^{\prime}}(b)$,
i.e., the number of geometric points of the fibers, is finite and constant.
###### Proof:
We have
$\Upsilon^{-1}_{K}(b)\cong\overline{\mathrm{Ig}}_{K}^{b}/\operatorname{Aut}(\mathbb{X}_{b,K})\cong\overline{\mathrm{Ig}}_{K^{\prime}}^{b}/\operatorname{Aut}(\mathbb{X}_{b,K})\quad\text{and}\quad\Upsilon^{-1}_{K^{\prime}}(b)\cong\overline{\mathrm{Ig}}_{K^{\prime}}^{b}/\operatorname{Aut}(\mathbb{X}_{b,K^{\prime}}),$
so that all fibers are isomorphic to
$\operatorname{Aut}(\mathbb{X}_{b,K^{\prime}})/\operatorname{Aut}(\mathbb{X}_{b,K})$.
□
###### Corollary 2.45.
The change-of-parahoric map between central leaves is finite.
###### Proof:
We have just seen it to be quasi-finite.
Corollary 2.13, combined with the properness of the change-of-parahoric itself
and therefore of the change-of-parahoric map restricted to Newton strata
(similar argument as in footnote 21), implies the properness of the change-of-
parahoric map between central leaves. □
###### Remark 2.46.
By [Kim19, Cor. 5.3.1], leaves are equidimensional smooth and the dimension of
a leaf depends only on the Newton stratum it is in; in particular, if we
consider a change-of-parahoric map between leaves,
$\pi_{K,K^{\prime}}\colon\Upsilon^{-1}_{K}(y)\to\Upsilon_{K^{\prime}}^{-1}(y^{\prime}),$
then $\dim\Upsilon^{-1}_{K}(y)=\dim\Upsilon_{K^{\prime}}^{-1}(y^{\prime})$.
###### Corollary 2.47.
The change-of-parahoric map between central leaves is finite locally free.
###### Proof:
Combine Corollary 2.45 with the preceding remark and [GW10, Cor.
14.128]222222Note that “$y\in Y$” may be replaced by “$y\in f(X)$” in the
statement of [GW10, Cor. 14.128].. □
###### Corollary 2.48.
The change-of-parahoric morphism between central leaves is the composition of
a flat universal homeomorphism of finite type and a finite étale morphism.
###### Proof:
This follows from what has been established about the morphism by using
[Mes72, Lemma 4.8]. □
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|
2024-09-04T02:54:57.861480 | 2020-03-06T13:23:44 | 2003.03184 | {
"authors": "ALICE Collaboration",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26081",
"submitter": "ALICE Publications",
"url": "https://arxiv.org/abs/2003.03184"
} | arxiv-papers | 2020 025 03 March
(Anti-)Deuteron production in pp collisions at $\sqrt{s}=13$ TeV
ALICE Collaboration††thanks: See Appendix A for the list of collaboration
members ALICE Collaboration
The study of (anti-)deuteron production in pp collisions has proven to be a
powerful tool to investigate the formation mechanism of loosely bound states
in high energy hadronic collisions. In this paper the production of
(anti-)deuterons is studied as a function of the charged particle multiplicity
in inelastic pp collisions at $\sqrt{s}=13$ TeV using the ALICE experiment.
Thanks to the large number of accumulated minimum bias events, it has been
possible to measure (anti-)deuteron production in pp collisions up to the same
charged particle multiplicity ($\mbox{$\mathrm{d}N_{ch}/\mathrm{d}\eta$}\sim
26$) as measured in p–Pb collisions at similar centre-of-mass energies. Within
the uncertainties, the deuteron yield in pp collisions resembles the one in
p–Pb interactions, suggesting a common formation mechanism behind the
production of light nuclei in hadronic interactions. In this context the
measurements are compared with the expectations of coalescence and Statistical
Hadronisation Models (SHM).
## 1 Introduction
High energy collisions at the Large Hadron Collider (LHC) create a suitable
environment for the production of light (anti-)nuclei. In ultra-relativistic
heavy-ion collisions light (anti-)nuclei are abundantly produced [1, 2, 3],
but in elementary pp collisions their production is lower [4, 5, 1, 6]. As a
consequence, there are only few detailed measurements of (anti-)nuclei
production rate in pp collisions. However, with the recently collected large
data sample it is now possible to perform more differential measurements of
light (anti-)nuclei production as a function of multiplicity and transverse
momentum. In this paper, we present the detailed study of the multiplicity
dependence of (anti-)deuteron production in pp collisions at $\sqrt{s}=$ 13
TeV, the highest collision energy so far delivered at the LHC.
The production mechanism of light (anti-)nuclei in high energy hadronic
collisions is not completely understood. However, two groups of models have
turned out to be particularly useful, namely Statistical Hadronisation Models
(SHM) and coalescence models. The SHMs, which assume particle production
according to the thermal equilibrium expectation, have been very successful in
explaining the yields of light (anti-)nuclei along with other hadrons in Pb–Pb
collisions [7], suggesting a common chemical freeze-out temperature for light
(anti-)nuclei and other hadron species. The ratio between the
$p_{\mathrm{T}}$-integrated yields of deuterons and protons (d/p ratio) in
Pb–Pb collisions remains constant as a function of centrality, but rises in pp
and p–Pb collisions with increasing multiplicity, finally reaching the value
observed in Pb–Pb [1, 8, 9]. The constant d/p ratio in Pb–Pb collisions as a
function of centrality is consistent with thermal production, suggesting that
the chemical freeze-out temperature in Pb–Pb collisions does not vary with
centrality [10]. Assuming thermal production in pp collisions as well, the
lower d/p ratio would indicate a lower freeze-out temperature [10]. On the
other hand, the ratio between the $p_{\mathrm{T}}$-integrated yields of
protons and pions (p/$\pi$ ratio) does not show a significant difference
between pp and Pb–Pb collisions [11, 12]. Also, for p–Pb collisions the
freeze-out temperature obtained with SHMs using only light-flavoured particles
is constant with multiplicity and its value is similar to that obtained in
Pb–Pb collisions [13]. Thus, the increase of the d/p ratio with multiplicity
for smaller systems cannot be explained within the scope of the grand-
canonical SHM as is done in case of Pb–Pb. It is also not consistent with a
simple SHM that the d/p and p/$\pi$ ratios behave differently as a function of
multiplicity even though numerator and denominator differ in both cases by one
unit of baryon number. Nonetheless, a process similar to the canonical
suppression of strange particles might be worth considering also for baryons.
A recent calculation within the SHM approach with exact conservation of baryon
number, electric charge, and strangeness focuses on this aspect [14].
In coalescence models (anti-)nuclei are formed by nucleons close in phase-
space [15]. In this approach, the coalescence parameter $B_{2}$ quantitatively
describes the production of (anti-)deuterons. $B_{2}$ is defined as
$B_{2}\left(p_{\mathrm{T}}^{p}\right)=E_{d}\frac{\mathrm{d}^{3}N_{d}}{\mathrm{d}p_{d}^{3}}\bigg{/}\left(E_{p}\frac{\mathrm{d}^{3}N_{p}}{\mathrm{d}p_{p}^{3}}\right)^{2}=\frac{1}{2\pi
p_{\mathrm{T}}^{d}}\frac{\mathrm{d}^{2}N_{d}}{\mathrm{d}y\mathrm{d}p^{d}_{\mathrm{T}}}\;\bigg{/}\left(\frac{1}{2\pi
p_{\mathrm{T}}^{p}}\frac{\mathrm{d}^{2}N_{p}}{\mathrm{d}y\mathrm{d}p_{\mathrm{T}}^{p}}\right)^{2},$
(1)
where $E$ is the energy, $p$ is the momentum, $p_{\mathrm{T}}$ is the
transverse momentum and $y$ is the rapidity. The labels $p$ and $d$ are used
to denote properties related to protons and deuterons, respectively. The
invariant spectra of the (anti-)protons are evaluated at half of the
transverse momentum of the deuterons, so that
$p_{\mathrm{T}}^{p}=p_{\mathrm{T}}^{d}/2$. Neutron spectra are assumed to be
equivalent to proton spectra, since neutrons and protons belong to the same
isospin doublet. Since the coalescence process is expected to occur at the
late stage of the collision, the parameter $B_{2}$ is related to the emission
volume. In a simple coalescence approach, which describes the uncorrelated
particle emission from a point-like source, $B_{2}$ is expected to be
independent of $p_{\mathrm{T}}$ and multiplicity. However, it has been
observed that $B_{2}$ at a given transverse momentum decreases as a function
of multiplicity, suggesting that the nuclear emission volume increases with
multiplicity [2, 16, 9]. In Pb–Pb collisions the $B_{2}$ parameter as a
function of $p_{T}$ shows an increasing trend, which is usually attributed to
the position-momentum correlations caused by radial flow or hard scatterings
[17, 18]. Such an increase of $B_{2}$ as a function of $p_{\mathrm{T}}$ has in
fact also been observed in pp collisions at $\sqrt{s}=7$ TeV [6]. However, if
pp collisions are studied in separate intervals of multiplicity, $B_{2}$ is
found to be almost constant as a function of $p_{\mathrm{T}}$ [8]. Similarly,
$B_{2}$ does not depend on $p_{\mathrm{T}}$ in multiplicity selected p–Pb
collisions [9]. Moreover, the highest multiplicities reached in pp collisions
are comparable with those obtained in p–Pb collisions and not too far from
peripheral Pb–Pb collisions. Therefore, the measure of $B_{2}$ as a function
of $p_{\mathrm{T}}$ for finer multiplicity intervals in pp collisions at
$\sqrt{s}=$ 13 TeV gives the opportunity to compare different collision
systems and to evaluate the dependence on the system size.
The paper is organized as follows. Section 2 discusses the details of the
ALICE detector. Section 3 describes the data sample used for the analysis and
the corresponding event and track selection criteria. Section 4 presents the
data analysis steps in detail, such as raw yield extraction and various
corrections, as well as the systematic uncertainty estimation. In Section 5,
the results are presented and discussed. Finally, conclusions are given in
Section 6.
## 2 The ALICE detector
A detailed description of the ALICE detectors can be found in [19] and
references therein. For the present analysis the main sub-detectors used are
the V0, the Inner Tracking System (ITS), the Time Projection Chamber (TPC) and
the Time-of-Flight (TOF), which are all located inside a 0.5 T solenoidal
magnetic field.
The V0 detector [20] is formed by two arrays of scintillation counters placed
around the beampipe on either side of the interaction point: one covering the
pseudorapidity range $2.8<\eta<5.1$ (V0A) and the other one covering
$-3.7<\eta<-1.7$ (V0C). The collision multiplicity is estimated using the
counts in the V0 detector, which is also used as trigger detector. More
details will be given in Section 3.
The ITS [21], designed to provide high resolution track points in the
proximity of the interaction region, is composed of three subsystems of
silicon detectors placed around the interaction region with a cylindrical
symmetry. The Silicon Pixel Detector (SPD) is the subsystem closest to the
beampipe and is made of two layers of pixel detectors. The third and the
fourth layers consist of Silicon Drift Detectors (SDD), while the outermost
two layers are equipped with double-sided Silicon Strip Detectors (SSD). The
inner radius of the SPD, 3.9 cm, is essentially given by the radius of the
beam pipe, while the inner field cage of the TPC limits the radial span of the
entire ITS to be 43 cm. The ITS covers the pseudorapidity range $|\eta|<0.9$
and it is hermetic in azimuth.
The same pseudorapidity range is covered by the TPC [22], which is the main
tracking detector, consisting of a hollow cylinder whose axis coincides with
the nominal beam axis. The active volume, filled with a Ne/CO2/N2 gas mixture
(Ar/CO2/N2 in 2016), at atmospheric pressure, has an inner radius of about 85
cm, an outer radius of about 250 cm, and an overall length along the beam
direction of 500 cm. The gas is ionised by charged particles traversing the
detector and the ionisation electrons drift, under the influence of a constant
electric field of $\sim$ 400 V/cm, towards the endplates, where their position
and arrival time are measured. The trajectory of a charged particle is
estimated using up to 159 combined measurements (clusters) of drift times and
radial positions of the ionisation electrons. The charged-particle tracks are
then formed by combining the hits in the ITS and the reconstructed clusters in
the TPC. The TPC is used for particle identification by measuring the specific
energy loss ($\mathrm{d}E/\mathrm{d}x$) in the TPC gas.
The TOF system [23] covers the full azimuth for the pseudorapidity interval
$|\eta|<0.9$. The detector is based on the Multi-gap Resistive Plate Chambers
(MRPCs) technology and it is located, with a cylindrical symmetry, at an
average distance of 380 cm from the beam axis. The particle identification is
based on the difference between the measured time-of-flight and its expected
value, computed for each mass hypothesis from track momentum and length. The
overall resolution on the time-of-flight of particles is about 80 ps.
A precise starting signal for the TOF system can be also provided by the T0
detector, consisting of two arrays of Cherenkov counters, T0A and T0C, which
cover the pseudorapidity regions $4.61<\eta<4.92$ and $3.28<\eta<2.97$,
respectively [24]. Alternatively, the start time can be provided by the TOF
itself or the bunch-crossing time can be used, as described in [24].
## 3 Data sample
The data samples used in this work consist of approximately 950 million
minimum bias pp events collected during the LHC proton runs in 2016 and 2017.
The data were collected using a minimum-bias trigger requiring at least one
hit in both the V0 detectors. Moreover, the timing information of the V0
scintillators is used for the offline rejection of events triggered by
interactions of the beam with the residual gas in the LHC vacuum pipe. To
ensure the best possible performance of the detector, events with more than
one reconstructed primary interaction vertex (pile-up events) were rejected.
The production of primary (anti-)deuterons is measured around mid-rapidity. In
particular, the spectra are provided within a rapidity window of $|y|<0.5$. To
ensure that all tracks have the maximal length, only those in the
pseudorapidity interval $|\eta|<0.8$ are selected. In order to guarantee good
track momentum and $\mathrm{d}E/\mathrm{d}x$ resolution in the relevant
$p_{\mathrm{T}}$ ranges, the selected tracks are required to have at least 70
reconstructed points in the TPC and two points in the ITS. In addition, at
least one of the ITS points has to be measured by the SPD in order to assure
for the selected tracks a resolution better than 300 $\mu$m on the distance of
closest approach to the primary vertex in the plane perpendicular (DCAxy) and
parallel (DCAz) to the beam axis [19]. Furthermore, it is required that the
$\chi^{2}$ per TPC reconstructed point is less than 4 and tracks originating
from kink topologies of weak decays are rejected.
Data are divided into ten multiplicity classes, identified by a roman number
from I to X, going from the highest to the lowest multiplicity. However, in
this analysis classes IV and V are merged into a single class to achieve a
better statistical precision. The multiplicity classes are determined from the
sum of the V0 signal amplitudes and defined in terms of percentiles of the
INEL$>0$ pp cross section, where INEL $>$ 0 events are defined as collisions
with at least one charged particle in the pseudo-rapidity region $|\eta|<$1
[25]. The mean charged particle multiplicity
$\left<\mbox{$\mathrm{d}N_{ch}/\mathrm{d}\eta$}\right>$ for each class is
reported in Table 2.
## 4 Data analysis
### 4.1 Raw yield extraction
The identification of (anti-)deuterons is performed with two different
methods, depending on their transverse momentum. For $p_{\mathrm{T}}<$ 1
GeV/c, the identification is done using a measurement of the
$\mathrm{d}E/\mathrm{d}x$ in the TPC only. In particular, for each
$p_{\mathrm{T}}$ interval the number of (anti-)deuterons is extracted through
a fit with a Gaussian with two exponential tails to the $n_{\sigma}$
distribution. Here, $n_{\sigma}$ is the difference between the measured TPC
$\mathrm{d}E/\mathrm{d}x$ and the expected one for (anti-)deuterons divided by
the TPC $\mathrm{d}E/\mathrm{d}x$ resolution. However, for
$p_{\mathrm{T}}\geq$ 1 GeV/c it is more difficult to separate (anti-)deuterons
from other charged particles with this technique. Therefore, the particle
identification in this kinematic region is performed using the TOF detector.
The squared mass of the particle is computed as
$m^{2}=p^{2}\left(t_{\mathrm{TOF}}^{2}/L^{2}-1/c^{2}\right)$, where $t_{TOF}$
is the measured time-of-flight, $L$ is the length of the track and $p$ is the
momentum of the particle. In order to reduce the background, only the
candidates with a d$E$/d$x$ measured in the TPC compatible within 3$\sigma$
with the expected value for a (anti-)deuteron are selected. The squared-mass-
distributions are fitted with a Gaussian function with an exponential tail for
the signal. A significant background is present for $p_{\mathrm{T}}\geq$ 1.8
GeV/c and is modelled with two exponential functions. In the range where the
background is negligible, the raw yield is extracted by directly counting the
candidates. Otherwise, the squared-mass distribution is fitted with the
described model, using an extended-maximum-likelihood approach. The
(anti-)deuteron yield is then obtained by a fit parameter.
### 4.2 Efficiency and acceptance correction
A correction for the tracking efficiency and the detector acceptance must be
applied to obtain the real yield. The correction is evaluated from Monte Carlo
(MC) simulated events. The events are generated using the standard generator
PYTHIA8 (Monash 2013)[26]. However, PYTHIA8 does not handle the production of
nuclei. Therefore, in each event it is necessary to inject (anti-)deuterons.
In each pp collision one deuteron or one anti-deuteron is injected, randomly
chosen from a flat rapidity distribution in the range $|y|<$ 1 and a flat
$p_{\mathrm{T}}$ distribution in the range $p_{\mathrm{T}}\in[0,10]$ GeV/c.
The correction is defined as the ratio between the number of reconstructed
(anti-)deuterons in the rapidity range $|y|<0.5$ and in the pseudorapidity
interval $|\eta|<0.8$ and the number of generated ones in $|y|<0.5$. The
correction is computed separately for deuterons and anti-deuterons and for the
TPC and TOF analyses.
Another correction is related to the trigger efficiency. All the selected
events are required to have at least one charged particle in the acceptance,
i.e. in the pseudo-rapidity region $|\eta|<$1 (INEL $>$ 0) [25]. Due to the
imperfection of the trigger, some INEL $>$ 0 events are wrongly rejected
(event loss). Consequently, all the (anti-)deuterons produced in the
erroneously rejected events are lost as well (signal loss). Therefore, it is
necessary to correct the spectra for the event and the signal losses. Event
loss is more relevant at low multiplicity and almost negligible at high
multiplicity ($\sim$ 12% for multiplicity class X and $<$ 1‰ for multiplicity
class I). The corrections are computed from MC simulations, because both the
number of rejected events and the number of (anti-)deuterons produced in those
same events are known. However, it is not possible to count the number of lost
(anti-)deuterons directly, because the artificial injection of one
(anti-)deuteron per event will bias the number of lost candidates that can be
extracted from this MC data set. Instead, the number of lost pions, kaons and
protons are extracted from a different MC data set and then these values are
extrapolated to the deuteron mass. The standard transport code used in ALICE
simulations is GEANT3. However, it is known from other ALICE analyses on
nuclei that GEANT4 provides a more realistic transport of (anti-)nuclei. The
GEANT3 response is hence scaled to the GEANT4 one to take into account this
effect. Moreover, the spectra obtained with TOF are further corrected to take
into account the TPC-TOF matching efficiency using a data-driven approach.
This correction was evaluated for the analysis of the (anti-)deuteron
production in the p–Pb data sample collected in 2013 [9]. In that year not all
the modules of the Transition Radiation Detector (TRD), which is located
between the TPC and the TOF, were already installed. In this way it was
possible to compute the effects of the presence of the TRD, comparing the
(anti-)deuteron yields in the regions where the TRD modules were present and
in those where they were not yet installed. This correction was also verified
with Run 2 data, by comparing the yields extracted with the TPC with those
extracted with the TOF in the $p_{\mathrm{T}}$ region where both the
techniques can be used.
### 4.3 Subtraction of secondary deuterons
Secondary deuterons are produced in the interaction of particles with the
detector material and their contribution must be subtracted from the total
measured deuteron yield. However, the production of secondary anti-deuterons
is extremely rare due to baryon number conservation. Hence, the correction is
applied only to the deuteron spectra. The fraction of primary deuterons is
evaluated via a fit to the DCAxy distribution of the data, as described in
[1]. The template for primary deuterons is obtained from the measured DCAxy of
anti-deuterons. The template from secondary deuterons is instead obtained from
MC simulations. The production of secondary deuterons is more relevant at low
$p_{\mathrm{T}}$ (at $p_{\mathrm{T}}=$ 0.7 GeV/c the fraction of secondary
deuterons is $\sim$ 40%) and decreases exponentially with the transverse
momentum ($<$ 5% for $p_{\mathrm{T}}=$ 1.4 GeV/c). The only other possible
contribution to secondary deuterons that is known is the decay
${}^{3}_{\Lambda}\mathrm{H}\rightarrow\mathrm{d}+\mathrm{p}+\pi$. However,
${}^{3}_{\Lambda}\mathrm{H}$ production has not yet been observed in pp
collisions and its production yield is therefore lower than that of 3He, which
is less than a thousandth of the deuteron production rate [6].
### 4.4 Systematic uncertainties
A list of all the sources of systematic uncertainty is shown in Table 1. The
values are reported for the multiplicity classes I and X, for the lowest and
highest $p_{\mathrm{T}}$ values.
The track selection criteria are a source of systematic uncertainty. In this
category we include all the contributions related to the single-track
selection: DCA, number of clusters in the TPC and, for the TOF analysis, the
width of the $\mathrm{d}E/\mathrm{d}x$ selection applied in the TPC. These
uncertainties are evaluated by varying the relevant selections, as done in
[8]. At low $p_{\mathrm{T}}$ ($p_{\mathrm{T}}<1$ GeV/c) the contribution is 2%
for deuterons due to the DCAz and DCAxy selections, which influence the
estimation of the fraction of primary deuterons, while for anti-deuterons this
systematic uncertainty is around 1%. It increases with $p_{\mathrm{T}}$ and
the growth is more pronounced for low multiplicity. The systematic uncertainty
on the signal extraction is evaluated by directly counting the (anti-)deuteron
candidates. It is obtained by varying the interval in which the direct
counting is performed. Its contribution is $\sim$ 1% at low $p_{\mathrm{T}}$
and increases with $p_{\mathrm{T}}$. Another source of systematic uncertainty
is given by the incomplete knowledge of the material budget of the detector in
the Monte Carlo simulations. The effect is evaluated by comparing different MC
simulations in which the material budget was increased and decreased by 4.5%.
This value corresponds to the uncertainty on the determination of the material
budget by measuring photon conversions. This particular systematic uncertainty
is below 1%. The imperfect knowledge of the hadronic interaction cross section
of (anti-)deuterons with the material contributes to the systematic
uncertainty as well. Its effect is evaluated with the same data-driven
approach used to investigate the TOF-matching efficiency, as described in
section 4.2. Half of the correction, corresponding to the 1$\sigma$ confidence
interval, is taken as its uncertainty contributing 4% to the systematic
uncertainty for deuterons and 7.5% for anti-deuterons. Similarly, an
uncertainty related to the ITS-TPC matching is considered. It is evaluated
from the difference between the ITS-TPC matching efficiencies in data and MC
and its contribution is less than 2.5%. Finally, a source of systematic
uncertainties results from the signal loss correction. It is assumed to be
half of the difference between the signal-loss correction (described in
section 4.2) and 1. It is strongly dependent on the event multiplicity: it is
negligible at high multiplicity (multiplicity classes from I to VII) and
contributes up to 6% in the lowest multiplicity class (class X). Where
present, it decreases with $p_{\mathrm{T}}$.
Table 1: Summary of the main contributions to the systematic uncertainties for the extreme multiplicity classes I and X. Values in brackets are referred to anti-deuterons. If they are not present, the systematic uncertainty is common for deuterons and anti-deuterons. More details about the sources of the uncertainties can be found in the text. Source | d ($\bar{\mathrm{d}}$)
---|---
Multiplicity | | Class I | | Class X
$p_{\mathrm{T}}$ (GeV/$c$) | | 0.7 | 3.8 | | 0.7 | 2.6
Track selection | | 2% (1%) | 2% (3%) | | 2% (1%) | 5% (6%)
Signal extraction | | 1% | 7% (7%) | | 1% | 5% (5%)
Material budget | | $<1\%$ | $<1\%$ | | $<1\%$ | $<1\%$
TPC-TOF matching | | 4% (7.5%) | 4% (7.5%) | | 4% (7.5%) | 4% (7.5%)
ITS-TPC matching | | 1% | 2.5% | | 1% | 2.5%
Signal Loss | | - | - | | 6% | 3%
Total | | 5% (8%) | 9% (11%) | | 8% (10%) | 10% (12%)
Figure 1: Transverse-momentum spectra of deuterons (top) and anti-deuterons
(bottom) measured in pp collisions at $\sqrt{s}~{}=~{}$13 TeV in different
multiplicity classes (circles) and in INEL$>$0 events (squares). The mean
charged-particle multiplicity for classes I and X are reported in the figures
and all the values for the multiplicity classes can be found in Table 2. For
the analyses in multiplicity classes, the multiplicity increases moving from
the bottom of the figure upwards. The statistical uncertainties are
represented by vertical bars while the systematic uncertainties are
represented by boxes. The dashed lines are individual fits with a Lévy-Tsallis
function [27].
## 5 Results and Discussion
The transverse momentum spectra of deuterons and anti-deuterons in different
multiplicity classes as well as INEL$>$0 pp collisions are reported in Figure
1. The spectra normalised to inelastic pp collisions (INEL) are included in
the data provided with this paper. The mean charged-particle multiplicity
$\left<\mbox{$\mathrm{d}N_{ch}/\mathrm{d}\eta$}\right>$ for each class is
reported in Table 2. The spectra exhibit a slight hardening with increasing
multiplicity: the slope of the spectra becomes less steep and the mean
transverse momentum $\left<p_{\mathrm{T}}\right>$ moves towards higher values.
This effect is similar to that observed in Pb–Pb collisions, where it is
explained with the presence of increasing radial flow with centrality [1, 28].
However, in pp collisions the intensity of the hardening is not as dramatic.
The ratio between the spectra of anti-deuterons and deuterons for all the
multiplicity classes under study is reported in Figure 2. The ratio is
compatible within uncertainties with unity in all multiplicity classes.
To calculate the integrated yield ($\mathrm{d}N/\mathrm{d}y$) and the mean
$p_{\mathrm{T}}$ the spectra have been fitted with the Lévy-Tsallis function
[27, 29, 30]:
$\frac{\mathrm{d}^{2}N}{\mathrm{d}y\,\mathrm{d}p_{\mathrm{T}}}=\frac{\mathrm{d}N}{\mathrm{d}y}\frac{p_{\mathrm{T}}\left(n-1\right)\left(n-2\right)}{nC[nC+m\left(n-2\right)]}\left(1+\frac{m_{\mathrm{T}}-m}{nC}\right)^{-n},$
(2)
where $m$ is the particle rest mass (i.e. the mass of the deuteron),
$m_{\mathrm{T}}=\sqrt{m^{2}+p_{\mathrm{T}}^{2}}$ is the transverse mass, while
$n$, $\mathrm{d}N/\mathrm{d}y$ and $C$ are free fit parameters. The Lévy-
Tsallis function is used to extrapolate the spectra in the unmeasured regions
of $p_{\mathrm{T}}$. One contribution to the systematic uncertainty is
obtained by shifting the data points to the upper border of their systematic
uncertainty and to the corresponding lower border. The difference between
these values and the reference one is taken as an uncertainty which amounts to
$\sim$ 11%. Another contribution to the systematic uncertainty is estimated by
using alternative fit functions such as simple exponentials depending on
$p_{\mathrm{T}}$ and $m_{\mathrm{T}}$, as well as a Boltzmann function, and is
found to be $\sim$ 3%.The two contributions are summed in quadrature. The
extrapolation amounts to 25% of the total yield in the highest multiplicity
class, where the widest $p_{\mathrm{T}}$ range is measured, and increases up
to 35% in the lowest multiplicity class.
The statistical uncertainty on the integrated yield is obtained by moving the
data points randomly within their statistical uncertainties, using a Gaussian
probability distribution centered at the measured data point, with a standard
deviation corresponding to the statistical uncertainty. In the unmeasured
regions at low and high $p_{\mathrm{T}}$, the value of the fit function at a
given $p_{\mathrm{T}}$ is considered. In this case the statistical uncertainty
is estimated using a Monte Carlo method to propagate the uncertainties on the
fit parameters. Following the same procedure, the $\langle
p_{\mathrm{T}}\rangle$ and its statistical and systematic uncertainties are
computed. The resulting mean $p_{\mathrm{T}}$ and $\mathrm{d}N/\mathrm{d}y$,
as well as the parameters of the individual Lévy-Tsallis fits, are listed in
Table 2.
Figure 2: Ratio between the transverse momentum spectra of anti-deuterons and
deuterons in different multiplicity classes. The statistical uncertainties are
represented by vertical bars while the systematic uncertainties are
represented by boxes. Table 2: Summary of the relevant information about the
multiplicity classes and the fits to the measured transverse momentum spectra
of anti-deuterons. $\langle\mathrm{d}N_{ch}/\mathrm{d}\eta\rangle$ is the mean
pseudorapidity density of the primary charged particles [25]. $n$ and $C$ are
the parameters of the Lévy-Tsallis fit function [27].
$\mathrm{d}N/\mathrm{d}y$ is the integrated yield, with statistical
uncertainties, multiplicity-uncorrelated and multiplicity-correlated
systematic uncertainties (see the text for details). $\langle
p_{\mathrm{T}}\rangle$ is the mean transverse momentum.
Multiplicity | $\langle\mathrm{d}N_{ch}/\mathrm{d}\eta\rangle$ | $n$ | $C$ (GeV) | $\mathrm{d}N/\mathrm{d}y\left(\times 10^{-4}\right)$ | $\langle p_{\mathrm{T}}\rangle$ (GeV/c)
---|---|---|---|---|---
class
I | 26.02 $\pm$ 0.35 | 7 $\pm$ 3 | 0.37 $\pm$ 0.05 | 16.0 $\pm$ 0.4 $\pm$ 0.5 $\pm$ 1.8 | 1.57 $\pm$ 0.08 $\pm$ 0.05 $\pm$ 0.03
II | 20.02 $\pm$ 0.27 | 7 $\pm$ 3 | 0.32 $\pm$ 0.04 | 12.2 $\pm$ 0.2 $\pm$ 0.4 $\pm$ 1.4 | 1.43 $\pm$ 0.04 $\pm$ 0.04 $\pm$ 0.02
III | 16.17 $\pm$ 0.22 | 6 $\pm$ 2 | 0.27 $\pm$ 0.03 | 9.4 $\pm$ 0.1 $\pm$ 0\. 3 $\pm$ 1.1 | 1.31 $\pm$ 0.03 $\pm$ 0.03 $\pm$ 0.04
IV + V | 12.91 $\pm$ 0.13 | 8 $\pm$ 3 | 0.27 $\pm$ 0.03 | 7.13 $\pm$ 0.08 $\pm$ 0.20 $\pm$ 0.79 | 1.21 $\pm$ 0.02 $\pm$ 0.01 $\pm$ 0.03
VI | 10.02 $\pm$ 0.14 | 7 $\pm$ 2 | 0.23 $\pm$ 0.03 | 5.34 $\pm$ 0.07 $\pm$ 0.20 $\pm$ 0.59 | 1.12 $\pm$ 0.02 $\pm$ 0.01 $\pm$ 0.03
VII | 7.95 $\pm$ 0.11 | 6 $\pm$ 2 | 0.19 $\pm$ 0.03 | 3.99 $\pm$ 0.07 $\pm$ 0.20 $\pm$ 0.44 | 1.06 $\pm$ 0.02 $\pm$ 0.01 $\pm$ 0.03
VIII | 6.32 $\pm$ 0.09 | 17 $\pm$ 13 | 0.23 $\pm$ 0.03 | 2.73 $\pm$ 0.04 $\pm$ 0.06 $\pm$ 0.30 | 0.98 $\pm$ 0.01 $\pm$ 0.01 $\pm$ 0.03
IX | 4.50 $\pm$ 0.07 | 10 $\pm$ 5 | 0.19 $\pm$ 0.03 | 1.64 $\pm$ 0.03 $\pm$ 0.06 $\pm$ 0.19 | 0.92 $\pm$ 0.01 $\pm$ 0.01 $\pm$ 0.03
X | 2.55 $\pm$ 0.04 | 10 $\pm$ 5 | 0.15 $\pm$ 0.02 | 0.59 $\pm$ 0.02 $\pm$ 0.04 $\pm$ 0.07 | 0.82 $\pm$ 0.01 $\pm$ 0.02 $\pm$ 0.02
The coalescence parameter as a function of the transverse momentum is shown in
Figure 3. The transverse momentum spectra needed for the $B_{2}$ computation
are taken from Ref. [31]. The $B_{2}$ values for INEL$>$0 collisions show a
significant deviation from a transverse momentum independent coalescence
parameter as expected by the simplest implementation of the coalescence model.
However, it has been shown [8] that the the multiplicity-integrated
coalescence parameter is distorted because deuterons are biased more towards
higher multiplicity than protons, and consequently have harder
$p_{\mathrm{T}}$ spectra than expected from inclusive protons. The coalescence
parameter evaluated in fine multiplicity classes is consistent with a flat
behaviour, in agreement with the expectation of the simple coalescence model.
Figure 3: Coalescence parameter $B_{2}$ for anti-deuterons for different
multiplicity classes (circles) and for INEL$>$0 collisions (squares). For the
analyses in multiplicity classes, the multiplicity decreases moving from the
bottom of the figure upwards. The statistical uncertainties are represented by
vertical bars while the systematic uncertainties are represented by boxes.
$B_{2}$ is shown as a function of $p_{\mathrm{T}}/A$, being $A~{}=~{}2$ the
mass number of the deuteron.
The evolution of the coalescence parameter as a function of the charged
particle multiplicity is sensitive to the production mechanism of deuterons.
Recent formulations of the coalescence model [32, 33] implement an interplay
between the size of the collision system and the size of the light nuclei
produced via coalescence.
Figure 4 shows how the $B_{2}$, for a fixed transverse momentum interval,
evolves in different systems as a function of the charged particle
multiplicity. $B_{2}$ is shown at $p_{\mathrm{T}}~{}=~{}0.75$ GeV$/c$, which
was measured in all the analyses. However, the trend is the same for other
$p_{\mathrm{T}}$ values. The measurements are compared with the model
descriptions detailed in [33]. The two descriptions use different
parameterisations for the size of the source. Parameterisation A uses the
ALICE measurements of system radii $R$ from HBT studies as a function of
multiplicity[34]. These values are fitted with the function:
$R=a\;\langle\mathrm{d}N/\mathrm{d}\eta\rangle^{1/3}+b,$ (3)
where $a$ and $b$ are free parameters. In Parameterisation B the free
parameters $a$ and $b$ in Eq. 3 are fixed to reproduce the $B_{2}$ of
deuterons in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV in the
centrality class 0–10%. The first parameterisation (dashed red line) describes
well the measured $B_{2}$ in pp and p–Pb collisions, while it overestimates
the measurements in Pb–Pb collisions. However, as outlined by the authors in
[33], a more refined parameterisation of the HBT radius evolution through
different systems might reduce the observed discrepancy. The parameterisation
of the source size fixed to the $B_{2}$ measurement in central Pb–Pb
collisions already departs from the measurements in peripheral Pb–Pb
collisions and it underestimates the coalescence parameter for small colliding
systems.
Figure 4: Coalescence parameter $B_{2}$ at $p_{\mathrm{T}}/A=$ 0.75 GeV/c as a
function of multiplicity in pp collisions at $\sqrt{s}~{}=~{}13$ TeV (anti-
deuterons) and in $\sqrt{s}~{}=~{}7$ TeV [8] (average of deuterons and anti-
deuterons), in p–Pb collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV [9]
(deuterons) and in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}2.76$ TeV
[1] (deuterons). The statistical uncertainties are represented by vertical
bars while the systematic uncertainties are represented by boxes. The two
lines are theoretical predictions based on two different parameterisations of
the HBT radius, see text for details.
Figure 5 shows the ratio of the $p_{\mathrm{T}}$-integrated yields of
deuterons and protons for different multiplicities in different collisions
systems and at different energies. The ratio increases monotonically with
multiplicity for pp and p–Pb collisions and eventually saturates for Pb–Pb
collisions. The experimental data are compared with a SHM prediction. In this
implementation of the model, called the Canonical Statistical Model (CSM),
exact conservation of baryon number ($B$), charge ($Q$), and strangeness ($S$)
is enforced using the recently developed THERMAL-FIST package [14]. The
calculations with the CSM are performed using 155 MeV for the chemical freeze-
out temperature, $B$ = $Q$ = $S$ = 0 and two different values of the
correlation volume, which is expressed in terms of rapidity units
$\mathrm{d}V/\mathrm{d}y$, corresponding to one and three units of rapidity,
respectively. The model qualitatively reproduces the trend observed in data.
This might suggest that for small collision systems the light (anti-) nuclei
production could be canonically suppressed and that a canonical correlation
volume might exist. The correlation volume required to describe the
measurements is larger than one unit of rapidity. However, such a canonical
suppression should also affect the p/$\pi$ ratio in a similar way and this is
not observed in the experimental measurements [11, 35].
A full coalescence calculation, taking into account the interplay between the
system size and the width of the wave function of the produced
(anti-)deuterons, is also able to describe the measured trend of the d/p ratio
[36] and it describes the data consistently better than CSM for all system
sizes.
Figure 5: Ratio between the $p_{\mathrm{T}}$-integrated yields of deuterons
and protons (sum of protons and anti-protons) for different multiplicities in
pp collisions at $\sqrt{s}~{}=~{}13$ TeV (anti-deuterons) and in
$\sqrt{s}~{}=~{}7$ TeV [8] (deuterons), in p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV [9] (deuterons) and in Pb–Pb
collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}2.76$ TeV [1] (deuterons). The
statistical uncertainties are represented by vertical bars while the
systematic uncertainties are represented by boxes. The two black lines are the
theoretical predictions of the Thermal-FIST statistical model [14] for two
sizes of the correlation volume $V_{C}$, while the magenta line represents the
expectation from a coalescence model [36].
## 6 Conclusions
The results on (anti-)deuteron production presented in this paper display a
smooth evolution with multiplicity across different reaction systems, in
agreement with the measurements of other light-flavoured hadrons. This
suggests that a common physics process might be able to describe the nuclei
production in all hadronic collision systems. Coalescence and statistical
hadronisation models are able to describe qualitatively the observed trend in
the d/p ratio and $B_{2}$ as a function of the charged particle multiplicity.
However, with the precision of the current measurements it is not possible to
distinguish which mechanism drives the (anti-)deuteron production. On the
other hand, it is not clear whether the CSM would be able to describe
simultaneously the d/p and the p/$\pi$ ratios with the same chemical freeze-
out conditions.
No substantial differences are seen in the dependence of nuclei production on
the charged multiplicity in pp and p–Pb collisions and with the Pb–Pb data
sample collected in Run 2 it will be also possible to perform a direct
comparison with peripheral Pb–Pb collisions. With the enhanced luminosity in
Run 3, it will be possible to measure pp collisions with multiplicities
similar to those observed in mid-central Pb–Pb collisions. It will be
interesting to see whether ALICE can confirm this dependence when measuring
nuclei production in pp and Pb–Pb collisions at the same multiplicity.
## Acknowledgements
The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Ministry of Education of China (MOEC) , Ministry of Science & Technology of China (MSTC) and National Natural Science Foundation of China (NSFC), China; Ministry of Science and Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the VILLUM FONDEN and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à l’Energie Atomique (CEA), Institut National de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS) and Région des Pays de la Loire, France; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE), Department of Science and Technology, Government of India (DST), University Grants Commission, Government of India (UGC) and Council of Scientific and Industrial Research (CSIR), India; Indonesian Institute of Science, Indonesia; Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi and Istituto Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative Science and Technology , Nagasaki Institute of Applied Science (IIST), Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Commission on Science and Technology for Sustainable Development in the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru; Ministry of Science and Higher Education and National Science Centre, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics and Ministry of Research and Innovation and Institute of Atomic Physics, Romania; Joint Institute for Nuclear Research (JINR), Ministry of Education and Science of the Russian Federation, National Research Centre Kurchatov Institute, Russian Science Foundation and Russian Foundation for Basic Research, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSDTA) and Office of the Higher Education Commission under NRU project of Thailand, Thailand; Turkish Atomic Energy Agency (TAEK), Turkey; National Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities Council (STFC), United Kingdom; National Science Foundation of the United States of America (NSF) and United States Department of Energy, Office of Nuclear Physics (DOE NP), United States of America.
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## Appendix A The ALICE Collaboration
S. Acharyaorg141&D. Adamováorg94&A. Adlerorg74&J. Adolfssonorg80&M.M.
Aggarwalorg99&G. Aglieri Rinellaorg33&M. Agnelloorg30&N. Agrawalorg10,org53&Z.
Ahammedorg141&S. Ahmadorg16&S.U. Ahnorg76&A. Akindinovorg91&M. Al-
Turanyorg106&S.N. Alamorg141&D.S.D. Albuquerqueorg122&D. Aleksandrovorg87&B.
Alessandroorg58&H.M. Alfandaorg6&R. Alfaro Molinaorg71&B. Aliorg16&Y.
Aliorg14&A. Aliciorg10,org26,org53&A. Alkinorg2&J. Almeorg21&T. Altorg68&L.
Altenkamperorg21&I. Altsybeevorg112&M.N. Anaamorg6&C. Andreiorg47&D.
Andreouorg33&H.A. Andrewsorg110&A. Andronicorg144&M. Angelettiorg33&V.
Anguelovorg103&C. Ansonorg15&T. Antičićorg107&F. Antinoriorg56&P.
Antonioliorg53&R. Anwarorg125&N. Apadulaorg79&L. Aphecetcheorg114&H.
Appelshäuserorg68&S. Arcelliorg26&R. Arnaldiorg58&M. Arratiaorg79&I.C.
Arseneorg20&M. Arslandokorg103&A. Augustinusorg33&R. Averbeckorg106&S.
Azizorg61&M.D. Azmiorg16&A. Badalàorg55&Y.W. Baekorg40&S. Bagnascoorg58&X.
Baiorg106&R. Bailhacheorg68&R. Balaorg100&A. Baldisseriorg137&M. Ballorg42&S.
Balouzaorg104&R. Barberaorg27&L. Barioglioorg25&G.G. Barnaföldiorg145&L.S.
Barnbyorg93&V. Barretorg134&P. Bartaliniorg6&K. Barthorg33&E. Bartschorg68&F.
Baruffaldiorg28&N. Bastidorg134&S. Basuorg143&G. Batigneorg114&B.
Batyunyaorg75&D. Bauriorg48&J.L. Bazo Albaorg111&I.G. Beardenorg88&C.
Beddaorg63&N.K. Beheraorg60&I. Belikovorg136&A.D.C. Bell Hechavarriaorg144&F.
Belliniorg33&R. Bellwiedorg125&V. Belyaevorg92&G. Bencediorg145&S.
Beoleorg25&A. Bercuciorg47&Y. Berdnikovorg97&D. Berenyiorg145&R.A.
Bertensorg130&D. Berzanoorg58&M.G. Besoiuorg67&L. Betevorg33&A.
Bhasinorg100&I.R. Bhatorg100&M.A. Bhatorg3&H. Bhattorg48&B.
Bhattacharjeeorg41&A. Bianchiorg25&L. Bianchiorg25&N. Bianchiorg51&J.
Bielčíkorg36&J. Bielčíkováorg94&A. Bilandzicorg104,org117&G. Biroorg145&R.
Biswasorg3&S. Biswasorg3&J.T. Blairorg119&D. Blauorg87&C. Blumeorg68&G.
Bocaorg139&F. Bockorg33,org95&A. Bogdanovorg92&S. Boiorg23&L.
Boldizsárorg145&A. Bolozdynyaorg92&M. Bombaraorg37&G. Bonomiorg140&H.
Borelorg137&A. Borissovorg92,org144&H. Bossiorg146&E. Bottaorg25&L.
Bratrudorg68&P. Braun-Munzingerorg106&M. Bregantorg121&M. Brozorg36&E.J.
Bruckenorg43&E. Brunaorg58&G.E. Brunoorg105&M.D. Bucklandorg127&D.
Budnikovorg108&H. Bueschingorg68&S. Bufalinoorg30&O. Bugnonorg114&P.
Buhlerorg113&P. Buncicorg33&Z. Butheleziorg72,org131&J.B. Buttorg14&J.T.
Buxtonorg96&S.A. Bysiakorg118&D. Caffarriorg89&A. Calivaorg106&E. Calvo
Villarorg111&R.S. Camachoorg44&P. Cameriniorg24&A.A. Caponorg113&F.
Carnesecchiorg10,org26&R. Caronorg137&J. Castillo Castellanosorg137&A.J.
Castroorg130&E.A.R. Casulaorg54&F. Catalanoorg30&C. Ceballos Sanchezorg52&P.
Chakrabortyorg48&S. Chandraorg141&W. Changorg6&S. Chapelandorg33&M.
Chartierorg127&S. Chattopadhyayorg141&S. Chattopadhyayorg109&A.
Chauvinorg23&C. Cheshkovorg135&B. Cheynisorg135&V. Chibante Barrosoorg33&D.D.
Chinellatoorg122&S. Choorg60&P. Chochulaorg33&T. Chowdhuryorg134&P.
Christakoglouorg89&C.H. Christensenorg88&P. Christiansenorg80&T.
Chujoorg133&C. Cicaloorg54&L. Cifarelliorg10,org26&F. Cindoloorg53&J.
Cleymansorg124&F. Colamariaorg52&D. Colellaorg52&A. Colluorg79&M.
Colocciorg26&M. Concasorg58orgI&G. Conesa Balbastreorg78&Z. Conesa del
Valleorg61&G. Continorg24,org127&J.G. Contrerasorg36&T.M. Cormierorg95&Y.
Corrales Moralesorg25&P. Corteseorg31&M.R. Cosentinoorg123&F. Costaorg33&S.
Costanzaorg139&P. Crochetorg134&E. Cuautleorg69&P. Cuiorg6&L.
Cunqueiroorg95&D. Dabrowskiorg142&T. Dahmsorg104,org117&A. Daineseorg56&F.P.A.
Damasorg114,org137&M.C. Danischorg103&A. Danuorg67&D. Dasorg109&I.
Dasorg109&P. Dasorg85&P. Dasorg3&S. Dasorg3&A. Dashorg85&S. Dashorg48&S.
Deorg85&A. De Caroorg29&G. de Cataldoorg52&J. de Cuvelandorg38&A. De
Falcoorg23&D. De Gruttolaorg10&N. De Marcoorg58&S. De Pasqualeorg29&S.
Deborg49&B. Debjaniorg3&H.F. Degenhardtorg121&K.R. Dejaorg142&A.
Delofforg84&S. Delsantoorg25,org131&D. Devetakorg106&P. Dhankherorg48&D. Di
Bariorg32&A. Di Mauroorg33&R.A. Diazorg8&T. Dietelorg124&P.
Dillensegerorg68&Y. Dingorg6&R. Diviàorg33&D.U. Dixitorg19&Ø.
Djuvslandorg21&U. Dmitrievaorg62&A. Dobrinorg33,org67&B. Dönigusorg68&O.
Dordicorg20&A.K. Dubeyorg141&A. Dublaorg106&S. Dudiorg99&M. Dukhishyamorg85&P.
Dupieuxorg134&R.J. Ehlersorg146&V.N. Eikelandorg21&D. Eliaorg52&H.
Engelorg74&E. Eppleorg146&B. Erazmusorg114&F. Erhardtorg98&A.
Erokhinorg112&M.R. Ersdalorg21&B. Espagnonorg61&G. Eulisseorg33&D.
Evansorg110&S. Evdokimovorg90&L. Fabbiettiorg104,org117&M. Fagginorg28&J.
Faivreorg78&F. Fanorg6&A. Fantoniorg51&M. Faselorg95&P. Fecchioorg30&A.
Felicielloorg58&G. Feofilovorg112&A. Fernández Téllezorg44&A. Ferreroorg137&A.
Ferrettiorg25&A. Festantiorg33&V.J.G. Feuillardorg103&J. Figielorg118&S.
Filchaginorg108&D. Finogeevorg62&F.M. Fiondaorg21&G. Fiorenzaorg52&F.
Flororg125&S. Foertschorg72&P. Fokaorg106&S. Fokinorg87&E. Fragiacomoorg59&U.
Frankenfeldorg106&U. Fuchsorg33&C. Furgetorg78&A. Fursorg62&M. Fusco
Girardorg29&J.J. Gaardhøjeorg88&M. Gagliardiorg25&A.M. Gagoorg111&A.
Galorg136&C.D. Galvanorg120&P. Ganotiorg83&C. Garabatosorg106&E. Garcia-
Solisorg11&K. Gargorg27&C. Gargiuloorg33&A. Garibliorg86&K. Garnerorg144&P.
Gasikorg104,org117&E.F. Gaugerorg119&M.B. Gay Ducatiorg70&M. Germainorg114&J.
Ghoshorg109&P. Ghoshorg141&S.K. Ghoshorg3&P. Gianottiorg51&P.
Giubellinoorg58,org106&P. Giubilatoorg28&P. Glässelorg103&D.M. Goméz
Coralorg71&A. Gomez Ramirezorg74&V. Gonzalezorg106&P. González-Zamoraorg44&S.
Gorbunovorg38&L. Görlichorg118&S. Gotovacorg34&V. Grabskiorg71&L.K.
Graczykowskiorg142&K.L. Grahamorg110&L. Greinerorg79&A. Grelliorg63&C.
Grigorasorg33&V. Grigorievorg92&A. Grigoryanorg1&S. Grigoryanorg75&O.S.
Groettvikorg21&F. Grosaorg30&J.F. Grosse-Oetringhausorg33&R. Grossoorg106&R.
Guernaneorg78&M. Guittiereorg114&K. Gulbrandsenorg88&T. Gunjiorg132&A.
Guptaorg100&R. Guptaorg100&I.B. Guzmanorg44&R. Haakeorg146&M.K. Habiborg106&C.
Hadjidakisorg61&H. Hamagakiorg81&G. Hamarorg145&M. Hamidorg6&R.
Hanniganorg119&M.R. Haqueorg63,org85&A. Harlenderovaorg106&J.W.
Harrisorg146&A. Hartonorg11&J.A. Hasenbichlerorg33&H. Hassanorg95&D.
Hatzifotiadouorg10,org53&P. Hauerorg42&S. Hayashiorg132&S.T.
Heckelorg68,org104&E. Hellbärorg68&H. Helstruporg35&A. Herghelegiuorg47&T.
Hermanorg36&E.G. Hernandezorg44&G. Herrera Corralorg9&F. Herrmannorg144&K.F.
Hetlandorg35&T.E. Hildenorg43&H. Hillemannsorg33&C. Hillsorg127&B.
Hippolyteorg136&B. Hohlwegerorg104&D. Horakorg36&A. Hornungorg68&S.
Hornungorg106&R. Hosokawaorg15,org133&P. Hristovorg33&C. Huangorg61&C.
Hughesorg130&P. Huhnorg68&T.J. Humanicorg96&H. Hushnudorg109&L.A.
Husovaorg144&N. Hussainorg41&S.A. Hussainorg14&D. Hutterorg38&J.P.
Iddonorg33,org127&R. Ilkaevorg108&M. Inabaorg133&G.M. Innocentiorg33&M.
Ippolitovorg87&A. Isakovorg94&M.S. Islamorg109&M. Ivanovorg106&V.
Ivanovorg97&V. Izucheevorg90&B. Jacakorg79&N. Jacazioorg53&P.M. Jacobsorg79&S.
Jadlovskaorg116&J. Jadlovskyorg116&S. Jaelaniorg63&C. Jahnkeorg121&M.J.
Jakubowskaorg142&M.A. Janikorg142&T. Jansonorg74&M. Jercicorg98&O.
Jevonsorg110&M. Jinorg125&F. Jonasorg95,org144&P.G. Jonesorg110&J.
Jungorg68&M. Jungorg68&A. Juskoorg110&P. Kalinakorg64&A. Kalweitorg33&V.
Kaplinorg92&S. Karorg6&A. Karasu Uysalorg77&O. Karavichevorg62&T.
Karavichevaorg62&P. Karczmarczykorg33&E. Karpechevorg62&A. Kazantsevorg87&U.
Kebschullorg74&R. Keidelorg46&M. Keilorg33&B. Ketzerorg42&Z.
Khabanovaorg89&A.M. Khanorg6&S. Khanorg16&S.A. Khanorg141&A.
Khanzadeevorg97&Y. Kharlovorg90&A. Khatunorg16&A. Khuntiaorg118&B.
Kilengorg35&B. Kimorg60&B. Kimorg133&D. Kimorg147&D.J. Kimorg126&E.J.
Kimorg73&H. Kimorg17,org147&J. Kimorg147&J.S. Kimorg40&J. Kimorg103&J.
Kimorg147&J. Kimorg73&M. Kimorg103&S. Kimorg18&T. Kimorg147&T. Kimorg147&S.
Kirschorg38,org68&I. Kiselorg38&S. Kiselevorg91&A. Kisielorg142&J.L.
Klayorg5&C. Kleinorg68&J. Kleinorg58&S. Kleinorg79&C. Klein-Bösingorg144&M.
Kleinerorg68&A. Klugeorg33&M.L. Knichelorg33&A.G. Knospeorg125&C.
Kobdajorg115&M.K. Köhlerorg103&T. Kolleggerorg106&A. Kondratyevorg75&N.
Kondratyevaorg92&E. Kondratyukorg90&J. Konigorg68&P.J. Konopkaorg33&L.
Koskaorg116&O. Kovalenkoorg84&V. Kovalenkoorg112&M. Kowalskiorg118&I.
Králikorg64&A. Kravčákováorg37&L. Kreisorg106&B. Krimphofforg68&M.
Krivdaorg64,org110&F. Krizekorg94&K. Krizkova Gajdosovaorg36&M. Krügerorg68&E.
Kryshenorg97&M. Krzewickiorg38&A.M. Kuberaorg96&V. Kučeraorg60&C.
Kuhnorg136&P.G. Kuijerorg89&L. Kumarorg99&S. Kumarorg48&S. Kunduorg85&P.
Kurashviliorg84&A. Kurepinorg62&A.B. Kurepinorg62&A. Kuryakinorg108&S.
Kushpilorg94&J. Kvapilorg110&M.J. Kweonorg60&J.Y. Kwonorg60&Y. Kwonorg147&S.L.
La Pointeorg38&P. La Roccaorg27&Y.S. Laiorg79&R. Langoyorg129&K.
Lapidusorg33&A. Lardeuxorg20&P. Larionovorg51&E. Laudiorg33&R. Lavickaorg36&T.
Lazarevaorg112&R. Leaorg24&L. Leardiniorg103&J. Leeorg133&S. Leeorg147&F.
Lehasorg89&S. Lehnerorg113&J. Lehrbachorg38&R.C. Lemmonorg93&I. León
Monzónorg120&E.D. Lesserorg19&M. Lettrichorg33&P. Lévaiorg145&X. Liorg12&X.L.
Liorg6&J. Lienorg129&R. Lietavaorg110&B. Limorg17&V. Lindenstruthorg38&S.W.
Lindsayorg127&C. Lippmannorg106&M.A. Lisaorg96&V. Litichevskyiorg43&A.
Liuorg19&S. Liuorg96&W.J. Llopeorg143&I.M. Lofnesorg21&V. Loginovorg92&C.
Loizidesorg95&P. Loncarorg34&X. Lopezorg134&E. López Torresorg8&J.R.
Luhderorg144&M. Lunardonorg28&G. Luparelloorg59&Y. Maorg39&A.
Maevskayaorg62&M. Magerorg33&S.M. Mahmoodorg20&T. Mahmoudorg42&A.
Maireorg136&R.D. Majkaorg146&M. Malaevorg97&Q.W. Malikorg20&L.
Malininaorg75orgII&D. Mal’Kevichorg91&P. Malzacherorg106&G. Mandaglioorg55&V.
Mankoorg87&F. Mansoorg134&V. Manzariorg52&Y. Maoorg6&M. Marchisoneorg135&J.
Marešorg66&G.V. Margagliottiorg24&A. Margottiorg53&J. Marguttiorg63&A.
Marínorg106&C. Markertorg119&M. Marquardorg68&N.A. Martinorg103&P.
Martinengoorg33&J.L. Martinezorg125&M.I. Martínezorg44&G. Martínez
Garcíaorg114&M. Martinez Pedreiraorg33&S. Masciocchiorg106&M. Maseraorg25&A.
Masoniorg54&L. Massacrierorg61&E. Massonorg114&A. Mastroserioorg52,org138&A.M.
Mathisorg104,org117&O. Matonohaorg80&P.F.T. Matuokaorg121&A. Matyjaorg118&C.
Mayerorg118&M. Mazzilliorg52&M.A. Mazzoniorg57&A.F. Mechlerorg68&F.
Meddiorg22&Y. Melikyanorg62,org92&A. Menchaca-Rochaorg71&C. Mengkeorg6&E.
Meninnoorg29,org113&M. Meresorg13&S. Mhlangaorg124&Y. Miakeorg133&L.
Michelettiorg25&D.L. Mihaylovorg104&K. Mikhaylovorg75,org91&A.
Mischkeorg63org*&A.N. Mishraorg69&D. Miśkowiecorg106&A. Modakorg3&N.
Mohammadiorg33&A.P. Mohantyorg63&B. Mohantyorg85&M. Mohisin Khanorg16orgIII&C.
Mordasiniorg104&D.A. Moreira De Godoyorg144&L.A.P. Morenoorg44&I.
Morozovorg62&A. Morschorg33&T. Mrnjavacorg33&V. Mucciforaorg51&E.
Mudnicorg34&D. Mühlheimorg144&S. Muhuriorg141&J.D. Mulliganorg79&M.G.
Munhozorg121&R.H. Munzerorg68&H. Murakamiorg132&S. Murrayorg124&L.
Musaorg33&J. Musinskyorg64&C.J. Myersorg125&J.W. Myrchaorg142&B. Naikorg48&R.
Nairorg84&B.K. Nandiorg48&R. Naniaorg10,org53&E. Nappiorg52&M.U.
Naruorg14&A.F. Nassirpourorg80&C. Nattrassorg130&R. Nayakorg48&T.K.
Nayakorg85&S. Nazarenkoorg108&A. Neaguorg20&R.A. Negrao De Oliveiraorg68&L.
Nellenorg69&S.V. Nesboorg35&G. Neskovicorg38&D. Nesterovorg112&L.T.
Neumannorg142&B.S. Nielsenorg88&S. Nikolaevorg87&S. Nikulinorg87&V.
Nikulinorg97&F. Noferiniorg10,org53&P. Nomokonovorg75&J. Normanorg78,org127&N.
Novitzkyorg133&P. Nowakowskiorg142&A. Nyaninorg87&J. Nystrandorg21&M.
Oginoorg81&A. Ohlsonorg80,org103&J. Oleniaczorg142&A.C. Oliveira Da
Silvaorg121,org130&M.H. Oliverorg146&C. Oppedisanoorg58&R. Oravaorg43&A. Ortiz
Velasquezorg69&A. Oskarssonorg80&J. Otwinowskiorg118&K. Oyamaorg81&Y.
Pachmayerorg103&V. Pacikorg88&D. Paganoorg140&G. Paićorg69&J. Panorg143&A.K.
Pandeyorg48&S. Panebiancoorg137&P. Pareekorg49,org141&J. Parkorg60&J.E.
Parkkilaorg126&S. Parmarorg99&S.P. Pathakorg125&R.N. Patraorg141&B.
Paulorg23,org58&H. Peiorg6&T. Peitzmannorg63&X. Pengorg6&L.G. Pereiraorg70&H.
Pereira Da Costaorg137&D. Peresunkoorg87&G.M. Perezorg8&E. Perez
Lezamaorg68&V. Peskovorg68&Y. Pestovorg4&V. Petráčekorg36&M.
Petroviciorg47&R.P. Pezziorg70&S. Pianoorg59&M. Piknaorg13&P. Pillotorg114&O.
Pinazzaorg33,org53&L. Pinskyorg125&C. Pintoorg27&S. Pisanoorg10,org51&D.
Pistoneorg55&M. Płoskońorg79&M. Planinicorg98&F. Pliquettorg68&J.
Plutaorg142&S. Pochybovaorg145org*&M.G. Poghosyanorg95&B. Polichtchoukorg90&N.
Poljakorg98&A. Poporg47&H. Poppenborgorg144&S. Porteboeuf-Houssaisorg134&V.
Pozdniakovorg75&S.K. Prasadorg3&R. Preghenellaorg53&F. Prinoorg58&C.A.
Pruneauorg143&I. Pshenichnovorg62&M. Puccioorg25,org33&J. Putschkeorg143&R.E.
Quishpeorg125&S. Ragoniorg110&S. Rahaorg3&S. Rajputorg100&J. Rakorg126&A.
Rakotozafindrabeorg137&L. Ramelloorg31&F. Ramiorg136&R. Raniwalaorg101&S.
Raniwalaorg101&S.S. Räsänenorg43&R. Rathorg49&V. Ratzaorg42&I.
Ravasengaorg30,org89&K.F. Readorg95,org130&K. Redlichorg84orgIV&A.
Rehmanorg21&P. Reicheltorg68&F. Reidtorg33&X. Renorg6&R. Renfordtorg68&Z.
Rescakovaorg37&J.-P. Revolorg10&K. Reygersorg103&V. Riabovorg97&T.
Richertorg80,org88&M. Richterorg20&P. Riedlerorg33&W. Rieglerorg33&F.
Riggiorg27&C. Risteaorg67&S.P. Rodeorg49&M. Rodríguez Cahuantziorg44&K.
Røedorg20&R. Rogalevorg90&E. Rogochayaorg75&D. Rohrorg33&D. Röhrichorg21&P.S.
Rokitaorg142&F. Ronchettiorg51&E.D. Rosasorg69&K. Roslonorg142&A.
Rossiorg28,org56&A. Rotondiorg139&A. Royorg49&P. Royorg109&O.V. Ruedaorg80&R.
Ruiorg24&B. Rumyantsevorg75&A. Rustamovorg86&E. Ryabinkinorg87&Y.
Ryabovorg97&A. Rybickiorg118&H. Rytkonenorg126&O.A.M. Saarimakiorg43&S.
Sadhuorg141&S. Sadovskyorg90&K. Šafaříkorg36&S.K. Sahaorg141&B. Sahooorg48&P.
Sahooorg48,org49&R. Sahooorg49&S. Sahooorg65&P.K. Sahuorg65&J. Sainiorg141&S.
Sakaiorg133&S. Sambyalorg100&V. Samsonovorg92,org97&D. Sarkarorg143&N.
Sarkarorg141&P. Sarmaorg41&V.M. Sartiorg104&M.H.P. Sasorg63&E.
Scapparoneorg53&B. Schaeferorg95&J. Schambachorg119&H.S. Scheidorg68&C.
Schiauaorg47&R. Schickerorg103&A. Schmahorg103&C. Schmidtorg106&H.R.
Schmidtorg102&M.O. Schmidtorg103&M. Schmidtorg102&N.V. Schmidtorg68,org95&A.R.
Schmierorg130&J. Schukraftorg88&Y. Schutzorg33,org136&K. Schwarzorg106&K.
Schwedaorg106&G. Scioliorg26&E. Scomparinorg58&M. Šefčíkorg37&J.E.
Segerorg15&Y. Sekiguchiorg132&D. Sekihataorg132&I. Selyuzhenkovorg92,org106&S.
Senyukovorg136&D. Serebryakovorg62&E. Serradillaorg71&A. Sevcencoorg67&A.
Shabanovorg62&A. Shabetaiorg114&R. Shahoyanorg33&W. Shaikhorg109&A.
Shangaraevorg90&A. Sharmaorg99&A. Sharmaorg100&H. Sharmaorg118&M.
Sharmaorg100&N. Sharmaorg99&A.I. Sheikhorg141&K. Shigakiorg45&M.
Shimomuraorg82&S. Shirinkinorg91&Q. Shouorg39&Y. Sibiriakorg87&S.
Siddhantaorg54&T. Siemiarczukorg84&D. Silvermyrorg80&G. Simatovicorg89&G.
Simonettiorg33,org104&R. Singhorg85&R. Singhorg100&R. Singhorg49&V.K.
Singhorg141&V. Singhalorg141&T. Sinhaorg109&B. Sitarorg13&M. Sittaorg31&T.B.
Skaaliorg20&M. Slupeckiorg126&N. Smirnovorg146&R.J.M. Snellingsorg63&T.W.
Snellmanorg43,org126&C. Sonccoorg111&J. Songorg60,org125&A.
Songmoolnakorg115&F. Soramelorg28&S. Sorensenorg130&I. Sputowskaorg118&J.
Stachelorg103&I. Stanorg67&P. Stankusorg95&P.J. Steffanicorg130&E.
Stenlundorg80&D. Stoccoorg114&M.M. Storetvedtorg35&L.D. Strittoorg29&A.A.P.
Suaideorg121&T. Sugitateorg45&C. Suireorg61&M. Suleymanovorg14&M.
Suljicorg33&R. Sultanovorg91&M. Šumberaorg94&S. Sumowidagdoorg50&S.
Swainorg65&A. Szaboorg13&I. Szarkaorg13&U. Tabassamorg14&G.
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Acostaorg19&S.R. Torresorg9,org120&A. Trifiroorg55&S. Tripathyorg49&T.
Tripathyorg48&S. Trogoloorg28&G. Trombettaorg32&L. Tropporg37&V.
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Tsujiorg132&A. Tumkinorg108&R. Turrisiorg56&T.S. Tveterorg20&K.
Ullalandorg21&E.N. Umakaorg125&A. Urasorg135&G.L. Usaiorg23&A.
Utrobicicorg98&M. Valaorg37&N. Valleorg139&S. Valleroorg58&N. van der
Kolkorg63&L.V.R. van Doremalenorg63&M. van Leeuwenorg63&P. Vande Vyvreorg33&D.
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## Affiliation notes
org*Deceased
orgIDipartimento DET del Politecnico di Torino, Turin, Italy
orgIIM.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of
Nuclear, Physics, Moscow, Russia
orgIIIDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India
orgIVInstitute of Theoretical Physics, University of Wroclaw, Poland
## Collaboration Institutes
org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia
org2Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine
org3Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India
org4Budker Institute for Nuclear Physics, Novosibirsk, Russia
org5California Polytechnic State University, San Luis Obispo, California,
United States
org6Central China Normal University, Wuhan, China
org7Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France
org8Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba
org9Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico
org10Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi’, Rome, Italy
org11Chicago State University, Chicago, Illinois, United States
org12China Institute of Atomic Energy, Beijing, China
org13Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovakia
org14COMSATS University Islamabad, Islamabad, Pakistan
org15Creighton University, Omaha, Nebraska, United States
org16Department of Physics, Aligarh Muslim University, Aligarh, India
org17Department of Physics, Pusan National University, Pusan, Republic of
Korea
org18Department of Physics, Sejong University, Seoul, Republic of Korea
org19Department of Physics, University of California, Berkeley, California,
United States
org20Department of Physics, University of Oslo, Oslo, Norway
org21Department of Physics and Technology, University of Bergen, Bergen,
Norway
org22Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN,
Rome, Italy
org23Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
org24Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
org25Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy
org26Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy
org27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy
org29Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
org30Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy
org31Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy
org32Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy
org33European Organization for Nuclear Research (CERN), Geneva, Switzerland
org34Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia
org35Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway
org36Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
org37Faculty of Science, P.J. Šafárik University, Košice, Slovakia
org38Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany
org39Fudan University, Shanghai, China
org40Gangneung-Wonju National University, Gangneung, Republic of Korea
org41Gauhati University, Department of Physics, Guwahati, India
org42Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany
org43Helsinki Institute of Physics (HIP), Helsinki, Finland
org44High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico
org45Hiroshima University, Hiroshima, Japan
org46Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation
(ZTT), Worms, Germany
org47Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania
org48Indian Institute of Technology Bombay (IIT), Mumbai, India
org49Indian Institute of Technology Indore, Indore, India
org50Indonesian Institute of Sciences, Jakarta, Indonesia
org51INFN, Laboratori Nazionali di Frascati, Frascati, Italy
org52INFN, Sezione di Bari, Bari, Italy
org53INFN, Sezione di Bologna, Bologna, Italy
org54INFN, Sezione di Cagliari, Cagliari, Italy
org55INFN, Sezione di Catania, Catania, Italy
org56INFN, Sezione di Padova, Padova, Italy
org57INFN, Sezione di Roma, Rome, Italy
org58INFN, Sezione di Torino, Turin, Italy
org59INFN, Sezione di Trieste, Trieste, Italy
org60Inha University, Incheon, Republic of Korea
org61Institut de Physique Nucléaire d’Orsay (IPNO), Institut National de
Physique Nucléaire et de Physique des Particules (IN2P3/CNRS), Université de
Paris-Sud, Université Paris-Saclay, Orsay, France
org62Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
org63Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht,
Netherlands
org64Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia
org65Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India
org66Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic
org67Institute of Space Science (ISS), Bucharest, Romania
org68Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
org69Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico
org70Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil
org71Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico
org72iThemba LABS, National Research Foundation, Somerset West, South Africa
org73Jeonbuk National University, Jeonju, Republic of Korea
org74Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany
org75Joint Institute for Nuclear Research (JINR), Dubna, Russia
org76Korea Institute of Science and Technology Information, Daejeon, Republic
of Korea
org77KTO Karatay University, Konya, Turkey
org78Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France
org79Lawrence Berkeley National Laboratory, Berkeley, California, United
States
org80Lund University Department of Physics, Division of Particle Physics,
Lund, Sweden
org81Nagasaki Institute of Applied Science, Nagasaki, Japan
org82Nara Women’s University (NWU), Nara, Japan
org83National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece
org84National Centre for Nuclear Research, Warsaw, Poland
org85National Institute of Science Education and Research, Homi Bhabha
National Institute, Jatni, India
org86National Nuclear Research Center, Baku, Azerbaijan
org87National Research Centre Kurchatov Institute, Moscow, Russia
org88Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
org89Nikhef, National institute for subatomic physics, Amsterdam, Netherlands
org90NRC Kurchatov Institute IHEP, Protvino, Russia
org91NRC «Kurchatov Institute» - ITEP, Moscow, Russia
org92NRNU Moscow Engineering Physics Institute, Moscow, Russia
org93Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom
org94Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy,
Czech Republic
org95Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
org96Ohio State University, Columbus, Ohio, United States
org97Petersburg Nuclear Physics Institute, Gatchina, Russia
org98Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia
org99Physics Department, Panjab University, Chandigarh, India
org100Physics Department, University of Jammu, Jammu, India
org101Physics Department, University of Rajasthan, Jaipur, India
org102Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany
org103Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany
org104Physik Department, Technische Universität München, Munich, Germany
org105Politecnico di Bari, Bari, Italy
org106Research Division and ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany
org107Rudjer Bošković Institute, Zagreb, Croatia
org108Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
org109Saha Institute of Nuclear Physics, Homi Bhabha National Institute,
Kolkata, India
org110School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
org111Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru
org112St. Petersburg State University, St. Petersburg, Russia
org113Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria
org114SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes,
France
org115Suranaree University of Technology, Nakhon Ratchasima, Thailand
org116Technical University of Košice, Košice, Slovakia
org117Technische Universität München, Excellence Cluster ’Universe’, Munich,
Germany
org118The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland
org119The University of Texas at Austin, Austin, Texas, United States
org120Universidad Autónoma de Sinaloa, Culiacán, Mexico
org121Universidade de São Paulo (USP), São Paulo, Brazil
org122Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
org123Universidade Federal do ABC, Santo Andre, Brazil
org124University of Cape Town, Cape Town, South Africa
org125University of Houston, Houston, Texas, United States
org126University of Jyväskylä, Jyväskylä, Finland
org127University of Liverpool, Liverpool, United Kingdom
org128University of Science and Techonology of China, Hefei, China
org129University of South-Eastern Norway, Tonsberg, Norway
org130University of Tennessee, Knoxville, Tennessee, United States
org131University of the Witwatersrand, Johannesburg, South Africa
org132University of Tokyo, Tokyo, Japan
org133University of Tsukuba, Tsukuba, Japan
org134Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
org135Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, Lyon, France
org136Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg,
France, Strasbourg, France
org137Université Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU,
Départment de Physique Nucléaire (DPhN), Saclay, France
org138Università degli Studi di Foggia, Foggia, Italy
org139Università degli Studi di Pavia, Pavia, Italy
org140Università di Brescia, Brescia, Italy
org141Variable Energy Cyclotron Centre, Homi Bhabha National Institute,
Kolkata, India
org142Warsaw University of Technology, Warsaw, Poland
org143Wayne State University, Detroit, Michigan, United States
org144Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany
org145Wigner Research Centre for Physics, Budapest, Hungary
org146Yale University, New Haven, Connecticut, United States
org147Yonsei University, Seoul, Republic of Korea
|
2024-09-04T02:54:57.876967 | 2020-03-06T13:52:46 | 2003.03206 | {
"authors": "Yuanhang Zhang, Shuang Yang, Jingyun Xiao, Shiguang Shan, Xilin Chen",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26082",
"submitter": "Yuanhang Zhang",
"url": "https://arxiv.org/abs/2003.03206"
} | arxiv-papers | # Can We Read Speech Beyond the Lips?
Rethinking RoI Selection for Deep Visual Speech Recognition
Yuanhang Zhang1,2, Shuang Yang1, Jingyun Xiao1,2, Shiguang Shan1,2, Xilin
Chen1,2
1 Key Laboratory of Intelligent Information Processing of Chinese Academy of
Sciences (CAS), Institute of Computing Technology, CAS, Beijing 100190, China
2 University of Chinese Academy of Sciences, Beijing 100049, China This work
was done by Yuanhang Zhang during his internship at Institute of Computing
Technology, Chinese Academy of Sciences.
###### Abstract
Recent advances in deep learning have heightened interest among researchers in
the field of visual speech recognition (VSR). Currently, most existing methods
equate VSR with automatic lip reading, which attempts to recognise speech by
analysing lip motion. However, human experience and psychological studies
suggest that we do not always fix our gaze at each other’s lips during a face-
to-face conversation, but rather scan the whole face repetitively. This
inspires us to revisit a fundamental yet somehow overlooked problem: can VSR
models benefit from reading extraoral facial regions, i.e. beyond the lips? In
this paper, we perform a comprehensive study to evaluate the effects of
different facial regions with state-of-the-art VSR models, including the
mouth, the whole face, the upper face, and even the cheeks. Experiments are
conducted on both word-level and sentence-level benchmarks with different
characteristics. We find that despite the complex variations of the data,
incorporating information from extraoral facial regions, even the upper face,
consistently benefits VSR performance. Furthermore, we introduce a simple yet
effective method based on Cutout to learn more discriminative features for
face-based VSR, hoping to maximise the utility of information encoded in
different facial regions. Our experiments show obvious improvements over
existing state-of-the-art methods that use only the lip region as inputs, a
result we believe would probably provide the VSR community with some new and
exciting insights.
## I Introduction
Visual speech recognition (VSR) is the task of recognising speech by analysing
video sequences of people speaking. A robust VSR system has a variety of
useful applications, such as silent speech interfaces [29], audio-visual
speech recognition (AVSR) in noisy environments [1], face liveness detection,
and so on. Its performance has progressed significantly over the past few
years, thanks to several successful deep learning architectures, such as
convolutional neural networks (CNNs) and recurrent neural networks (RNNs). It
also benefits from the emergence of large-scale, in-the-wild audiovisual
datasets, from which deep neural networks can automatically learn strong
representations that outperform previous hand-crafted features.
Traditionally, the term “visual speech recognition” is used almost
interchangeably with “lip reading” within the VSR community, since it is
usually believed that lip shapes and lip motion most likely contain almost all
the information correlated with speech. The information in other facial
regions is considered by default weak, and not helpful for VSR in practical
use, due to the diversity of the speaker’s pose and other variations in the
facial region that are unrelated to speech production. Accordingly, as part of
the dataset creation pipeline, almost all researchers crop regions-of-interest
(RoIs) around the mouth after obtaining face bounding boxes and landmarks. By
observing only the cropped RoI, the model is expected to focus on fine-grained
discrimination of “clean” motion signals within the most relevant lip region,
and not be distracted by other parts of the face, whose utilities are less
obvious. Some common practices for RoI selection in previous work are depicted
in Fig. 1.
Figure 1: Common practices for RoI selection in VSR. To date, there is no
clear consensus on a best practice, resulting in very different RoIs in
different works (see Sec. II-A). Top row: examples of frames from talking face
videos. Bottom row: some examples of cropped RoIs in prior work, from left to
right: [3, 4, 9, 17, 27, 1, 35].
However, this convention of explicit mouth RoI cropping inevitably brings
about many questions. Firstly, lip motion is not the only visual signal we can
rely on to decode speech. Research on the advantage of performing
speechreading with the whole face has a long history [28, 6]. In particular,
movements of articulatory muscles such as the orbicularis oris (which is near
the lips) lead to skin movements, often reflected by the cheeks being pushed
and pulled, as well as changes in the visibility of the nasolabial folds. The
now widely adopted term “speechreading”, used in place of “lip reading”
implies precisely the contribution of extraoral facial regions, such as the
tongue, teeth, and cheeks, to the speech perception task. Evidence from
psychology studies, and our experience in human communication also suggest
that we in fact do not focus on the speaker’s lip all the time throughout a
conversation, even in very noisy environments. Instead, we scan different
regions of the person’s face periodically [31].
Secondly, if we do use a mouth RoI, there are many factors that need to be
considered: how much of the face should the RoI cover? Will increased spatial
context improve performance by supplying more information, or hinder
performance by distracting the network with information unrelated to speech?
Should we apply additional corrections to handle pose variation? Answers to
these questions are not evident or straight, and there has been no consensus
or a universal guideline for choosing RoIs until now. In fact, RoIs are often
chosen based on intuition and the researchers’ experience. The choices can be
very different for different datasets and different methods, making transfer
learning and cross-dataset evaluation difficult. Meanwhile, this specialised
RoI selection step separates VSR from other face-related tasks, hindering
further joint analysis. Using mouth inputs alone makes VSR an isolated
problem, while including other facial regions opens up the possibility of
various research in affective analysis and visual speech understanding. For
example, joint modeling of non-verbal behaviour and verbal behaviour has been
shown to be beneficial for learning adaptive word representations [32].
Finally, data encountered in real-world applications may not have the same
luxury that some academic datasets enjoy, where data is biased towards frontal
or near-frontal views, and high-resolution mouth crops are easily obtainable.
Models trained on such well-behaved, cropped data may not perform as well when
presented with the various challenges in practice. This severely limits the
potential usage scenarios of VSR systems.
Motivated by these observations, we conduct a comparative study on input RoI
selection, to (a) quantitatively estimate the contribution of different facial
regions to the VSR task, and (b) determine whether state-of-the-art VSR
networks, when presented with complex in-the-wild data, still benefit from
additional clues within extraoral regions. Previous studies have explored
different RoIs in the lower face [17, 24], and demonstrated the importance of
suitable RoI coverage and selection. However, these attempts are limited to
the lower face, and relatively small datasets with narrow vocabularies. For
this study, we approach the problem with state-of-the-art, deep VSR models
trained on large-scale “in-the-wild” VSR datasets, which depict many real-
world variations, such as pose, lighting, scale, background clutter, makeup,
expression, different speaking manners, etc. This is a much fairer reflection
of real-world scenarios. Besides, we propose Cutout as an effective approach
to encourage the model to utilise all facial regions. This allows researchers
to sidestep the ambiguous choice of selecting appropriate RoIs, enhances the
visual features, and increases the model’s robustness to mild occlusion within
the mouth or other facial regions.
## II Related Work
### II-A Visual Speech Recognition
Visual speech recognition (VSR), commonly referred to as automatic lip
reading, is a classical problem in computer vision and has received increased
interest over recent years. The combination of deep learning approaches and
large-scale audiovisual datasets has been highly successful, achieving
remarkable word recognition rates and even surpassing human performance. The
deep network approach has become increasingly common and mainstream [9, 4,
35]. These methods have retraced the success of the CNN-LSTM-DNN (CLDNN)
paradigm [22] in the automated speech recognition (ASR) community. However,
though many works have reported state-of-the-art results on multiple
challenging datasets, few have investigated the influence of input RoI
selection, which can be a nuisance due to variations in mouth shapes, face
geometry, and pose, as well as the relatively unknown effect of spatial
context and extraoral regions. Unlike face recognition, where face
frontalisation is a recognised need for pose invariance, and has been
thoroughly investigated as an important part of the pipeline, VSR researchers
tend to specify RoIs based on their own experience, and in a dataset-specific
manner.
In some controlled-environment datasets, e.g. OuluVS [37] and GRID [11], the
subjects remain relatively still while speaking. Lombard GRID [2] uses a head-
mounted camera which the speakers face directly at for the frontal view,
essentially fully removing head motion. The Lip Reading in the Wild dataset
[9] is of short duration with small or no face scale variation within clips,
and loose registration is enforced by aligning nose centers. A fixed, mouth-
centered (and sometimes affine-transformed) rectangular RoI is preferable on
these datasets, because the faces are usually stable, frontal or near-frontal,
and of uniform sizes. In contrast, OuluVS2 [3] is a multi-view dataset, and
RoIs are processed separately for each camera viewpoint. Chung and Zisserman
[10] are the first to investigate large-scale deep lip reading in profile, and
use an extended bounding box covering the whole face to account for
significant pose variations. Yang et al. [35] determine the RoI size based on
the distance between the nose and the mouth center and the width of the
speaker’s lips, which also effectively handles pose variations.
There have been some previous attempts to address the RoI selection problem
explicitly. The most relevant work is [17], which perform experiments with
rectangular lower face regions with different spatial context and resolution
in a connected digits recognition task. [24] experiments with optical flow
features in non-rectangular RoIs after removing head motion, and obtain
results better than rectangular RoIs. However, these work are limited by the
amount of data used and model capacity, and only investigate the utility of
the lower face. We adopt face inputs, and mimic real-world scenarios by
experimenting on large-scale, in-the-wild VSR benchmarks that are a magnitude
larger than those used in the above two papers.
### II-B Human Perception Studies
Our intuition that we do not fix our gaze at the speaker’s lip region when
communicating with others is supported by a number of psychology studies on
human gaze behaviour during visual and audio-visual speech perception [30, 31,
18]. It has been reported that human gaze patterns usually involve repetitive
transitioning between the eyes and the mouth, even at high noise levels and
when audio is absent [30]. Interestingly, [18] suggests that in a visual-only
scenario, speechreading accuracy was related to the difficulty of the
presented sentences and individual proficiency, but not to the proportion of
gaze time at a specific part of the face. The roles of the upper face, which
is less intuitive for VSR, have also been studied [13]. Studies in audiovisual
speech perception show that head and eyebrow motion can help discriminate
prosodic contrasts [12], which may be helpful for the VSR task. Moreover, in
tonal languages like Mandarin Chinese, movements in the neck, head, and mouth
might be related to the lexical tones of syllables [7]. In the context of
deep-learning based VSR, recently [38] has shown that using video information
from the cropped lip region together with pinyin (Chinese syllables)
information yields $0.85\%$ reduction in tone prediction error, supporting
this hypothesis.
## III Exploring the Influence of RoIs on VSR
In this section, we first introduce the deep VSR architectures used in this
work, and four manually selected RoIs (some of which include extraoral
regions) to be experimented on. Next, we introduce Cutout as a simple strategy
to enhance face-based VSR. Finally, we introduce a few visualisation methods
we use to diagnose the resulting models.
### III-A Model Architecture
3D-ResNet18. The 3D-ResNet$18$ backbone which holds the current state-of-the-
art on the LRW dataset [27, 26] is used for all word-level experiments. It
consists of a spatiotemporal convolution layer and a $18$-layer 2D residual
network which gradually reduces spatial dimensionality, and yields a
$512$-dimensional average-pooled vector for each frame. We use a $2$-layer
bidirectional Gated Recurrent Unit (Bi-GRU) with $1024$ hidden units as the
recurrent backend, and do not experiment with further regularisation such as
Dropout and recurrent batch normalisation [26], as it deviates from the main
objective of this paper.
LipNet. For sentence-level VSR, we use the LipNet architecture, which achieves
state-of-the-art performance on the GRID corpus [4]. With only three
spatiotemporal convolutional layers in the frontend, this lightweight model
should be less prone to overfitting on such a small-scale dataset. We also use
temporal augmentation and Dropout as recommended by the authors.
### III-B Description of Manually Selected RoIs
For fixed-size mouth crops, there are two dominant approaches: one uses fixed
bounding box coordinates, and the other uses mouth-centered crops. To make
this study comprehensive and self-complete, we experiment with both choices
(although eventually we found no difference; see Table I).
Original frame Landmark detection Aligned face
Figure 2: Illustration of the sub-face RoIs defined in this paper. We train
baselines on the whole face (blue), the upper face (purple), the cheeks
(orange), and the mouth (red).
For face based models, we first crop and align the faces using detected eye
and nose landmarks. This is because these parts undergo less non-rigid
deformation, and this allows us to capture more significant motion in the
lower face including the cheeks, the mouth and the jaw. At a low computational
cost, this allows for more stable face tracks, and helps the network capture
information in different spatial patches more consistently. However, it should
be noted that face motion consists of rigid head movement (yaw, pitch, and
roll) and non-rigid movement (e.g. frowning, raising eyebrows, and skin
movement). Aligning the face retains yaw and pitch, but totally removes roll
rotation. To this end, we also experiment with whole face and upper face
cropped directly by fixed coordinates on LRW, since the faces are of uniform
sizes and loosely aligned. The input sizes we choose are those commonly
adopted in prior work, $112\times 112$ and $100\times 100$. We do not
elaborate on the effects of spatial downsampling, since lower RoI resolution
does not always yield poorer performance, as long as it remains above an
acceptable level (e.g. $50\times 50$ pixels) [5, 15, 35]. Besides, in any case
datasets collected in-the-wild already contain inherent scale and image
quality variations.
To crop the cheeks without revealing too much information in other regions, we
crop a rectangular tile from the aligned face, which has no in-plane rotation.
The vertical center of the RoI is set as the mean $y$ coordinate of the 18th,
27th, 57th, and 59th landmark after transformation.
Finally, we do not experiment with the jaw and the lower neck. Although these
parts are directly related to the vibrations of the vocal tract, these regions
are not always visible, and such an RoI will be difficult to define.
### III-C Enhancing Face Based VSR with Cutout
Our motivation of using face as input is that it covers all other three sub-
RoIs, which has the possibility of providing a strong “holistic”
representation. If the network is able to remain robust against the speech-
irrelevant variations that human faces present, including pose, lighting,
makeup, etc., we can benefit from additional information in other facial
regions all at once. However, a network has only limited capacity. On a fixed
budget, the goals of achieving invariance to nuisance factors and keeping
stronger discrimination capabilities are inherently complimentary, and the
recognition performance may suffer if either is not sufficiently realised.
Indeed, our experiments will later show that the face models have already
achieved performance comparable to lip based models. However, when we inspect
the convolutional responses of vanilla face-based models (see Fig. 5) we find
that the network tends to focus on the lip region and does not sufficiently
utilise other parts of the face. Inspired by the observation that the vanilla
face-based models able to achieve un-expected good performance, we try to
enhance further the performance by asking the model to learn more
discriminative features by utilising signals spread across the whole face. An
intuitive idea is to create a strong patch-based ensemble, which has been
shown feasible for facial expression recognition [19]. However, for VSR it
would be computationally expensive and impractical for deployment, since we
are dealing with spatiotemporal data. Moreover, the redundancy between patches
will burden optimisation.
Therefore, we propose to apply Cutout [14], a regularisation technique which
has been popular with image CNNs. It augments the dataset with partially
occluded versions of the samples in the dataset, and has already been
successfully applied to image classification and object detection [39]. Our
motivation is that this “adversarial erasing” process should help the model
pay more attention to less significant motion signals related to speech in
extraoral regions.
During training, a patch within the face region is randomly zeroed. Note that
Cutout is applied to identical spatial positions across the video, since we
expect the same facial region to be in roughly the same position after decent
alignment. Over the course of the training process the model will encounter
many masked versions of the same video, and eventually learn discriminative
features for all parts of the face, which the model should be able to combine
to its advantage during inference. Cutout fits seamlessly into the training
process, and can be performed efficiently as a data augmentation step at no
additional cost.
### III-D Interpreting Model Behaviour
To explore what our models are paying attention to throughout the video, we
apply three visualisation techniques that can provide insights as to what the
network focuses on for the task of visual speech recognition – similar to the
use of gaze trackers in human psychology experiments.
Feature maps. Feature maps are filter responses to input images and outputs
from previous layers, and can provide insight into intermediate
representations learned by the model. Looking at responses from increasingly
deeper layers can give us a sense of how the model combines low-level features
into higher-level concepts that are useful for recognition.
Saliency maps. We use guided backpropagation saliency visualisation [25],
which has been previously utilised to interpret lip reading [4, 9] and image
classification models. Specifically, let $\mathbf{x}\in\mathbb{R}^{T\times
H\times W\times C}$ be an input video, $V$ be the alphabet or vocabulary, and
$\phi$ be the non-linear function that underlies the deep recognition network.
For word-level classification, the network outputs a score for the $i$-th
class in the vocabulary, $p(i\mid\mathbf{x})=\phi(\mathbf{x})_{i}$, where
$1\leq i\leq|V|$. We compute its gradient with respect to the input
$\mathbf{x}$ using guided backpropagation. Likewise, for sentence-level VSR,
we compute the likelihood of the greedily decoded output
$\mathbf{y}\in(V\cup\\{\text{\textvisiblespace}\\})^{*}$, which is
$\prod_{t}p(\mathbf{y}_{t}\mid\mathbf{x})$, and derive it against the input
video sequence $\mathbf{x}$ to obtain the saliency maps. Eventually we obtain
a tensor the same size as the input, which depicts spatial regions the model
bases its prediction on for different timesteps.
Spatiotemporal masking. This approach, adapted from the patch masking method
for visualising the receptive fields of 2D ConvNets [36] has been used to
visualise important space-time video patches in audio-visual speech
enhancement models [16]. In our experiments, we mask each frame at the
identical position for spatially aligned faces using a small $7\times 7$ patch
in a sliding window fashion, and measure how overall accuracy is affected by
computing performance drop, e.g. $\Delta_{\text{accuracy}}$. This process
results in a heatmap depicting the contribution of different facial regions to
recognition.
## IV Experiments and Results
We train and evaluate on three VSR benchmarks, which cover tonal and atonal
languages as well as in-the-wild and scripted speech: the Lip Reading in the
Wild (LRW) dataset, the recent LRW-1000 dataset, and the GRID audiovisual
corpus. In this section, we first briefly introduce the three datasets we use,
and present some implementation details. Next, we compare recognition
performance using the four manually selected RoIs in Sec. III-B. We highlight
the benefits of incorporating extraoral regions, in particular by using
aligned entire faces. Finally, we present results of our best performing model
which combine Cutout with face inputs, and make a few useful remarks.
### IV-A Datasets
LRW. LRW [9] is a challenging “in-the-wild” English word-level lip reading
dataset derived from BBC news collections, with $500$ classes and over
$500,000$ instances, of which $25,000$ are reserved for testing.
LRW-1000. LRW-1000 [35] is a $1000$-class, large-scale, naturally distributed
Mandarin Chinese word-level lip reading dataset, also derived from TV
broadcasts. With over $700,000$ word instances, it is even more challenging,
with significant pose, scale, background clutter, word length, and inter-
speaker variations. Note that the whole face was not provided with LRW-1000 in
the initial release. We will release face tracks for LRW-1000111For now, we
train and evaluate our models with the original word alignment and landmark
annotations in [35] for fair comparison. Note that the original paper used
ResNet-$34$, while this work uses ResNet-$18$ which is currently more popular
due to fewer parameters and better performance. along with annotations that
have undergone another round of manual cleaning as well as corresponding
baselines in due course.
GRID. The GRID audiovisual corpus [11], released in 2006, is a popular
benchmark for sentence-level VSR. It consists of video recordings from $34$
speakers222One speaker’s data is unavailable due to technical reasons.,
yielding $33,000$ utterances. All sentences follow a fixed grammar.
### IV-B Implementation Details
Data preprocessing. We detect faces and facial landmarks with the open-source
SeetaFace2 toolkit [23], and align faces with similarity transformations using
upper face landmarks [8], which are smoothed using a temporal Gaussian kernel
of width $3$. For LRW and LRW-1000, the faces are resized to $122\times 122$
and randomly cropped to $112\times 112$ during training. For GRID, the faces
are resized to $100\times 100$.
Training details. All models are implemented with PyTorch and trained on $2$
to $3$ NVIDIA Titan X GPUs, each with $12$GB memory. We use the Adam optimizer
with default hyperparameters. For word-level VSR, we use three-stage training
described in [27]. Weight decay is set to $0.0001$, and learning rate is
initialised to $0.0003$ for stage I / II and $0.001$ for stage III, decreasing
on log scale from the $5$th or the $10$th epoch onwards, depending on whether
Cutout is applied. We apply identical spatial augmentation to every frame in
the form of random cropping and random horizontal flipping during training,
and take a central crop for testing. Also, for LRW-1000 training, we
initialise front-end weights from the corresponding best model on LRW,
following [35]. For sentence-level VSR, we use a fixed learning rate of
$0.0001$ and use no weight decay. We also apply random horizontal flipping,
but no random cropping is used because of relatively accurate and stable
landmarking results.
Data partitioning, and evaluation metrics. We use the train / validation /
test partitioning provided with LRW and LRW-1000. For GRID, we use $255$
random sentences from each speaker for evaluation, and the remainder for
training, the same as previous state-of-the-art [4, 34]. The evaluation
metrics for word-level and sentence-level tasks are classification accuracy
and Word Error Rate (WER), respectively, where WER is defined as
$\texttt{WER}=\frac{\text{\\# of substitutions, deletions, and
insertions}}{\text{length of ground truth transcript}}.$
Hence lower WERs are preferred. Note that the two metrics are equivalent if
one views word recognition as a one-word sentence recognition task, in the
sense that $\texttt{Acc}=1-\texttt{WER}$.
### IV-C Experiments on Manually Selected RoIs
Baseline results on the manually defined RoIs are shown in Table I, Table II,
and III. We analyze the results step by step for the rest of this subsection.
TABLE I: Evaluation on the LRW dataset with different RoI choices. The second
group uses directly cropped (upper) faces while the third applies face
alignment.
Region | Resolution | Accuracy | Description
---|---|---|---
Mouth | $88\times 88$ | $83.30\%$ | Fixed bounding box [21, 26]
Mouth | $88\times 88$ | $83.30\%$ | Mouth-centered [9]
Face | $112\times 112$ | $83.46\%$ | Nose-centered, $7/8$ original size
Upper face | $112\times 112$ | $42.28\%$ | Resized upper half from above
Face | $112\times 112$ | $83.10\%$ | Aligned with eye & nose landmarks [8]
Cheeks | $112\times 112$ | $62.49\%$ | Resized $40\times 112$ crop from above
Upper face | $112\times 112$ | $48.33\%$ | Resized upper half
Face (CBAM) | $112\times 112$ | $83.14\%$ |
Face (Cutout) | $112\times 112$ | $\bf 85.02\%$ |
TABLE II: Evaluation on LRW-1000 with different RoI choices.
Region | Resolution | Accuracy | Description
---|---|---|---
Mouth | $88\times 88$ | $38.64\%$ | Mouth-centered, no roll (as in [35])
Face | $112\times 112$ | $41.71\%$ | Aligned with eye & nose landmarks
Upper face | $112\times 112$ | $15.84\%$ | Resized upper half
Cheeks | $112\times 112$ | $32.50\%$ | Resized $40\times 112$ crop from above
Face (Cutout) | $112\times 112$ | $\bf 45.24\%$ |
Upper face | $112\times 112$ | $13.58\%$ | Front-end loaded from LRW and fixed
TABLE III: Evaluation on the GRID corpus with different RoI choices. A
$5$-gram, character-level language model is used during beam search. “CER”
stands for Character Error Rate. The lower the error rates, the better.
Region | Resolution | WER | CER | Description
---|---|---|---|---
Mouth | $100\times 50$ | $4.8\%$[4] | $1.9\%$ | Affine warped
Mouth | $100\times 50$ | $4.7\%$ | $1.9\%$ | Above (reproduced)
Face | $100\times 100$ | $3.1\%$ | $1.3\%$ | Aligned with eye & nose landmarks
Upper face | $100\times 50$ | $14.4\%$ | $7.4\%$ | Upper half of above
Cheeks | $100\times 50$ | $6.8\%$ | $3.1\%$ | Resized $36\times 100$ crop from above
Face (Cutout) | $100\times 100$ | $\bf 2.9\%$ | $\bf 1.2\%$ |
Effectiveness of including extraoral regions. Experiment results clearly show
that upper face and cheeks carry useful information, since recognition rates
are far above chance. Counterintuitively, the upper face achieves nearly half
the accuracy of face and mouth models. To ensure that the model has learned
useful discriminative features instead of some unknown inherent bias within
the dataset, we conduct an additional experiment, transferring from LRW to
LRW-1000 while keeping the front-end fixed. Intuitively, if the front-end has
learned spurious clues that are irrelevant to the task itself, it should
behave poorly on a dataset it has not been trained on. However, despite
significant differences between the two datasets in terms of quality and
language, classification accuracy is still far above chance after we fixed the
front-end, with only $2.26\%$ absolute performance loss.
Seeing how the upper face and cheeks convey useful information for VSR, by
feeding the model with the entire face, we would expect it to benefit from the
additional spatial context, which is indeed the case. Using the entire face
instead of only the mouth region yields $1.6\%$ WER reduction on GRID,
$3.07\%$ improvement on LRW-1000, and $0.16\%$ improvement on LRW (when
directly cropped faces are used). The slight performance regression on LRW
with aligned faces will be discussed next.
Making a case for face alignment. For LRW, there are two sets of face-based
experiments, one with face alignment, and the other using faces directly
cropped by fixed coordinates. We observe a small performance degradation on
LRW when we align faces to our canonical template ($0.2\%$ to $0.3\%$ drop
relative to mouth and $7/8$ original resolution direct crops). Since
recognition performance using the upper face actually benefits from alignment,
it can be argued that the performance drop is most likely due to slightly
lower mouth resolution (about $70\times 70$), and not removal of roll during
the alignment process. This is not desirable, but acceptable, and there may be
room for improvement if we adopt higher resolution inputs and improve
landmarking quality. Therefore, we consider aligning the faces into stable
face tracks to be beneficial, and use aligned faces for the remaining
experiments. By any means, this is also necessary for structured facial region
erasing with Cutout.
Failure modes. As an illustrative example, we further compare confusions made
by the model under each crop setting in Table IV and Table V for the two
English datasets. Overall, short words with less context and words that invoke
weak extraoral motions perform worst. We observe that the words best predicted
by the upper face are long words, such as “Westminster” and “Temperatures”,
and cheeks are good at making predictions for words that invoke significant
extraoral motion. The face based model, which achieves comparable but slightly
inferior performance compared to the mouth based model, fails to discriminate
words with only subtle differences, such as “benefits” and “benefit”.
Recognising such words correctly requires analysis of tongue movement, which
is hindered by the lowered overall resolution. However, on the GRID corpus,
the face-based model seems to have identified idiosyncratic speaking styles,
allowing it to make correct predictions for even short words and letters that
are not easy to distinguish, and eventually obtain results better than cropped
mouths.
TABLE IV: Top predicted words, worst predicted words and pairs exhibiting
highest confusion in LRW under each crop setting.
Accuracy / Network
---
Mouth | Face | Cheeks | Upper Face
AGREEMENT (1) | ACCUSED (1) | AFTERNOON (0.98) | WESTMINSTER (0.96)
ALLEGATIONS (1) | AGREEMENT (1) | WEEKEND (0.98) | TEMPERATURES (0.92)
BEFORE (1) | BEFORE (1) | WELFARE (0.98) | AFTERNOON (0.88)
PERHAPS (1) | CAMPAIGN (1) | WESTMINSTER (0.98) | SUNSHINE (0.88)
PRIME (1) | FOLLOWING (1) | INFORMATION (0.96) | DESCRIBED (0.86)
Accuracy / Network
---
Mouth | Face | Cheeks | Upper Face
ASKED (0.58) | WORLD (0.6) | REALLY (0.32) | GREAT (0.18)
BRITAIN (0.58) | ANSWER (0.58) | COULD (0.3) | OTHER (0.18)
MATTER (0.58) | BECAUSE (0.58) | GETTING (0.3) | UNTIL (0.18)
SPEND (0.58) | COURT (0.58) | MAKES (0.3) | WHICH (0.18)
TAKEN (0.58) | PERSON (0.58) | MATTER (0.3) | BRING (0.16)
Target / Estimated (# Errors) / Network
---
Mouth | Face | Cheeks | Upper Face
SPEND / SPENT (11) | BENEFITS / BENEFIT (14) | CLAIMS / GAMES (11) | MILLION / BILLION (11)
PRESS / PRICE (11) | PRESS / PRICE (12) | CHALLENGE / CHANGE (10) | BENEFITS / BENEFIT (10)
WORST / WORDS (10) | LIVING / GIVING (12) | SYRIAN / SYRIA (10) | EVERYONE / EVERYBODY (10)
PRICE / PRESS (10) | WORST / WORDS (11) | GROUND / AROUND (9) | TAKEN / SECOND (9)
SERIES / SERIOUS (9) | THEIR / THERE (10) | INDUSTRY / HISTORY (9) | TERMS / TIMES (9)
TABLE V: Examples of predictions on GRID under each crop setting. Prediction
errors are highlighted in red. GT: Ground Truth; UF: Upper Face; C: Cheeks; M:
Mouth; F: (Aligned) Face.
GT | lay white in u four now
---|---
UF | lay white in u four now
C | lay white at o four now
M | lay white at o four now
F | lay white in u four now
GT | lay white in q five again
---|---
UF | set white at t five again
C | lay white at q five again
M | lay white at q five again
F | lay white in q five again
Superiority of using entire faces for cross-dataset transfer. A byproduct of
using the entire face rather than specifying sub-RoIs is that the visual
front-end can be transferred across datasets easily with no explicit sub-RoI
cropping after face detection and registration, which is already very routine
in face recognition. Empirically, we found that transferring across face crops
is much better than transferring across mouth crops from LRW to LRW-1000. When
both fine-tuned for $9$ epochs, mouth-based models suffer from source and
target domain discrepancies, resulting in fluctuating validation loss and
accuracy (best value $2.8482/38.40\%$), whereas the face-based model
transferred stably, and converged to a better result (best value
$2.7202/40.35\%$).
### IV-D Experiments on Enhancing Face Inputs with Cutout
Results of combining Cutout with aligned faces can also be found in previous
tables. This strategy is extremely powerful, enabling state-of-the-art
performance on all three benchmarks. In particular, Cutout significantly
reduces overfitting on LRW to the point where the model starts to underfit, as
can be seen from the training curves in Fig. 3. This is radically different
from models trained on faces or mouths without Cutout, where the network
eventually reaches near $100\%$ accuracy on the training data, and proves that
this harsh regularisation policy is indeed useful for VSR. Below we elaborate
on a few interesting observations.
Figure 3: Accuracy curves on LRW as training progresses (only the first 75k
steps of the temporal convolution backend stage is shown). Compared to vanilla
mouth and face based models which easily achieve nearly $100\%$ accuracy on
the training set, Cutout (denoted by red curves) significantly reduces
overfitting.
Effect of the Cutout patch size. The size of the masked out region is an
important hyperparameter for Cutout. We experiment with four different sizes:
$70\times 70$, $56\times 56$, $42\times 42$, and $28\times 28$, which are
$5/8$, $1/2$, $3/8$, and $1/4$ the scale of the whole face, respectively.
Experiments on LRW show that among those a $56\times 56$ patch is most
effective (see Fig. 4). This is probably because it is approximately the
average size of the mouth, and the possibility of the entire mouth being
blocked allows for more efficient utilisation of extraoral regions. Since we
adopt the same canonical template, we use $1/2$-size masks for all datasets
(i.e. $50\times 50$ for GRID).
$W/4$$3W/8$$W/2$$5W/8$$82.5$$83.0$$83.5$$84.0$$84.5$$85.0$$85.5$$86.0$ValTestBaseline
(val)Baseline (test)Cutout patch sizeAccuracy (%)
Figure 4: Ablation results on LRW with different Cutout patch sizes. We
achieve best validation accuracy with patches half the size of the input.
Figure 5: Visualisations of model behaviour on LRW, LRW-1000, and GRID with
Cutout applied. In particular, note how models trained with Cutout preserve
fine-grained features, and yields clearly visible saliency around the cheeks.
(a) ResNet layer 3 feature maps for “hundred”; (b) Saliency maps for “bin
blue”; (c) Saliency maps for “er ling yi qi” (twenty-seventeen).
Visualising key facial regions. Fig. 5 provide some saliency visualisations
which show that models with Cutout can learn more discriminative features.
Fig. 5(a) is generated by extracting feature maps from the third ResNet layer,
and performing max-pooling across channels, and Fig. 5(b)(c) (GRID and
LRW-1000) by computing back-prop saliency. The derived maps are colored by
intensity and overlaid onto the original frames. Each face thumbnail
corresponds to a time step, and the two rows can be compared side-by-side to
see the effects of introducing Cutout, especially regions highlighted with a
dotted red box. For example, for LRW the convolutional responses are no longer
confined to the lips, and for GRID there is stronger saliency in the cheeks,
the eyebrows, and other facial muscles. The third row shows that after
transferring from LRW to LRW-1000, saliency in extraoral regions persist,
which means that the learned facial features generalize well and are
relatively robust.
We also identify regions that are important to the network’s predictions by
spatiotemporal masking on the entire test set of LRW, and results are depicted
in Fig. 6. It can be seen that for both models, the area that leads to the
most significant performance degradation when masked is still the lip region,
which agrees with common intuition. In addition, the model trained with Cutout
is also affected by occlusion in extraoral regions such as the cheeks and the
upper face, showing that the model has learned strong visual features that
encode these weak signals to complement lip motion. More importantly, while
the plain model suffers up to $40\%$ performance drop even when only a
$7\times 7$ patch is occluded, the drop remains below $2\%$ for the model
trained with Cutout. This observation again strongly supports the usefulness
of extraoral regions.
Figure 6: Important facial regions determined by spatiotemporal masking. Left:
without Cutout. Middle: an aligned face for reference. Right: with Cutout.
Regions that result in more accuracy drop when occluded are colored brighter,
and more crucial for model prediction.
Comparison with attention-based RoI selection. The attention mechanism,
inspired by human vision, has been widely used for implicit RoI selection and
fine-grained image recognition. Here we compare Cutout with a Convolutional
Block Attention Module (CBAM) [33] augmented baseline on LRW, where we plug
CBAM into ResNet blocks. Here, we use $5\times 5$ instead of $7\times 7$
kernels for the spatial attention modules to accommodate to the smaller
feature maps. Although CBAM is a powerful spatial-channel attention mechanism
which achieved remarkable performance on ImageNet classification and robust
remote heart rate estimation [20], results show that the attention-augmented
model is only marginally better than the baseline on LRW. We believe this is
because the subtle movements in extraoral regions are too weak to be captured
with the attention mechanism, and the model is still biased towards lip
motion.
Performance across pose. We are also interested in how well the Cutout-
augmented model can handle pose variations. We turn to LRW-1000, where the
data is divided into different difficulty levels according to yaw rotation.
From Table VI, we can see that the model trained with Cutout outperforms the
no-augmentation face-based baseline by about $2\%$ on both the easy (yaw $\geq
20^{\circ}$) and the medium ($\geq 40^{\circ}$) subset, but degrades slightly
on the hard subset ($\geq 60^{\circ}$). We believe this is mainly because
effective areas are smaller in the hard setting, where there are more frames
in profile view. The effective regions are more likely to be erased or
occluded when there is significant yaw rotation.
TABLE VI: Performance w.r.t Pose on LRW-1000 Methods | Easy | Medium | Hard | All
---|---|---|---|---
Mouth (ResNet34) [35] | $24.89\%$ | $20.76\%$ | $15.9\%$ | $38.19\%$
Mouth | $25.95\%$ | $21.55\%$ | $18.36\%$ | $38.64\%$
Face | 28.87% | 28.45% | $\mathbf{27.21\%}$ | 41.71%
Face (Cutout) | $\mathbf{31.74\%}$ | $\mathbf{30.04\%}$ | 26.89% | $\mathbf{45.24\%}$
## V Conclusion
In this paper, we have investigated a previously overlooked problem in VSR:
the use of extraoral information. We demonstrate that extraoral regions in the
face, such as the upper face and the cheeks, can also be included to boost
performance. We show that using simple Cutout augmentation with aligned face
inputs can yield stronger features, and vastly improve recognition performance
by forcing the model to learn the less obvious extraoral cues from data.
Beyond VSR, our findings also have clear implications for other speech-related
vision tasks, such as realistic talking face generation, face spoofing
detection and audio-visual speech enhancement. Next steps include extending
the method to sentence-level VSR where more contextual clues are available,
increasing input resolution, and eliminating the need for explicit face
alignment.
## VI Acknowledgments
We would like to thank Chenhao Wang and Mingshuang Luo’s extensive help with
data processing. This work is supported in part by the National Key R&D
Program of China (No. 2017YFA0700804), and Natural Science Foundation of China
(No. 61702486, 61876171).
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|
2024-09-04T02:54:57.888713 | 2020-03-06T14:01:01 | 2003.03220 | {
"authors": "Ozan \\c{C}atal, Samuel Wauthier, Tim Verbelen, Cedric De Boom, Bart\n Dhoedt",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26083",
"submitter": "Ozan \\c{C}atal",
"url": "https://arxiv.org/abs/2003.03220"
} | arxiv-papers | # Deep Active Inference
for Autonomous Robot Navigation
Ozan Çatal, Samuel Wauthier, Tim Verbelen, Cedric De Boom, & Bart Dhoedt
IDLab, Department of Information Technology
Ghent University –imec
Ghent, Belgium
<EMAIL_ADDRESS>
###### Abstract
Active inference is a theory that underpins the way biological agent’s
perceive and act in the real world. At its core, active inference is based on
the principle that the brain is an approximate Bayesian inference engine,
building an internal generative model to drive agents towards minimal
surprise. Although this theory has shown interesting results with grounding in
cognitive neuroscience, its application remains limited to simulations with
small, predefined sensor and state spaces.
In this paper, we leverage recent advances in deep learning to build more
complex generative models that can work without a predefined states space.
State representations are learned end-to-end from real-world, high-dimensional
sensory data such as camera frames. We also show that these generative models
can be used to engage in active inference. To the best of our knowledge this
is the first application of deep active inference for a real-world robot
navigation task.
## 1 Introduction
Active inference and the free energy principle underpins the way our brain –
and natural agents in general – work. The core idea is that the brain
entertains a (generative) model of the world which allows it to learn cause
and effect and to predict future sensory observations. It does so by
constantly minimising its prediction error or “surprise”, either by updating
the generative model, or by inferring actions that will lead to less
surprising states. As such, the brain acts as an approximate Bayesian
inference engine, constantly striving for homeostasis.
There is ample evidence (Friston, 2012; Friston et al., 2013a; 2014) that
different regions of the brain actively engage in variational free energy
minimisation. Theoretical grounds indicate that even the simplest of life
forms act in a free energy minimising way (Friston, 2013).
Although there is a large body of work on active inference for artificial
agents (Friston et al., 2006; 2009; 2017; 2013b; Cullen et al., 2018),
experiments are typically done in a simulated environment with predefined and
simple state and sensor spaces. Recently, research has been done on using deep
neural networks as an implementation of the active inference generative model,
resulting in the umbrella term “deep active inference”. However, so far all of
these approaches were only tested on fairly simple, simulated environments
(Ueltzhöffer, 2018; Millidge, 2019; Çatal et al., 2019). In this paper, we
apply deep active inference on a robot navigation task, with high-dimensional
camera observations and deploy it on a mobile robot platform. To the best of
our knowledge, this is the first time that active inference is applied on a
real-world robot navigation task.
In the remainder of this paper we will first introduce the active inference
theory in Section 2. Next, we show how we implement active inference using
deep neural networks in Section 3, and discuss initial experiments in Section
4.
## 2 Active Inference
Active inference is a process theory of the brain that utilises the concept of
free energy (Friston, 2013) to describe the behaviour of various agents. It
stipulates that all agents act in order to minimise their own uncertainty of
the world. This uncertainty is expressed as Bayesian Surprise, or
alternatively the variational free energy. In this context this is
characterised by the difference between what an agent imagines about the world
and what it has perceived about the world (Friston, 2010). More concretely,
the agent builds a generative model
$P(\tilde{{\bm{o}}},\tilde{{\bm{s}}},\tilde{{\bm{a}}})$, linking together the
agents internal belief states ${\bm{s}}$ with the perceived actions ${\bm{a}}$
and observations ${\bm{o}}$ in the form of a joint distribution. We use a
tilde to denote a sequence of variables through time. This generative model
can be factorised as in Equation 1.
$P(\tilde{{\bm{o}}},\tilde{{\bm{s}}},\tilde{{\bm{a}}})=P(\tilde{{\bm{a}}})P({\bm{s}}_{0})\prod_{t=1}^{T}P({\bm{o}}_{t}|{\bm{s}}_{t})P({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1})$
(1)
The free energy or Bayesian surprise is then defined as:
$\begin{split}F&=\mathbb{E}_{Q}[\log Q(\tilde{{\bm{s}}})-\log
P(\tilde{{\bm{o}}},\tilde{{\bm{s}}},\tilde{{\bm{a}}})]\\\
&=D_{\mathrm{KL}}(Q(\tilde{{\bm{s}}})\|P(\tilde{{\bm{s}}},\tilde{{\bm{a}}}|\tilde{{\bm{o}}}))-\log
P(\tilde{{\bm{o}}})\\\
&=D_{\mathrm{KL}}(Q(\tilde{{\bm{s}}})\|P(\tilde{{\bm{s}}},\tilde{{\bm{a}}}))-\mathbb{E}_{Q}[\log
P(\tilde{{\bm{o}}}|\tilde{{\bm{s}}})]\end{split}$ (2)
Here, $Q(\tilde{{\bm{s}}})$ is an approximate posterior distribution. The
second equality shows that free energy is equivalent to the (negative)
evidence lower bound (ELBO) (Kingma & Welling, 2013; Rezende et al., 2014).
The final equation frames the problem of free energy minimisation as
explaining the world from the agents beliefs whilst minimising the complexity
of accurate explanations (Friston et al., 2016).
Crucially, in active inference agents will act according to the belief that
they will keep minimising surprise in the future. This means agents will infer
policies that yield minimal expected free energy in the future, with a policy
$\pi$ being the sequence of future actions ${\bm{a}}_{t:t+H}$ starting at
current time step $t$ with a time horizon $H$. This principle is formalised in
Equation 3 with $\sigma$ being the softmax function with precision parameter
$\gamma$.
$\begin{split}P(\pi)&=\sigma(-\gamma G(\pi))\\\
G(\pi)&=\sum_{\tau=t}^{t+H}G(\pi,\tau)\end{split}$ (3)
Expanding the expected free energy functional $G(\pi,\tau)$ we get Equation 4.
Using the factorisation of the generative model from Equation 1 we approximate
$Q({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)\approx
P({\bm{o}}_{\tau}|{\bm{s}}_{\tau})Q({\bm{s}}_{\tau}|\pi)$.
$\begin{split}G(\pi,\tau)&=\mathbb{E}_{Q({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)}[\log
Q({\bm{s}}_{\tau}|\pi)-\log P({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)]\\\
&=\mathbb{E}_{Q({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)}[\log
Q({\bm{s}}_{\tau}|\pi)-\log P({\bm{o}}_{\tau}|{\bm{s}}_{\tau},\pi)-\log
P({\bm{s}}_{\tau}|\pi)]\\\
&=D_{\mathrm{KL}}(Q({\bm{s}}_{\tau}|\pi)\|P({\bm{s}}_{\tau}))+\mathbb{E}_{Q({\bm{s}}_{\tau})}[H(P({\bm{o}}_{\tau}|{\bm{s}}_{\tau}))]\end{split}$
(4)
Note that, in the final equality, we substitute $P({\bm{s}}_{\tau}|\pi)$ by
$P({\bm{s}}_{\tau})$, a global prior distribution on the so-called “preferred”
states of the agent. This reflects the fact that the agent has prior
expectations about the states it will reach. Hence, minimising expected free
energy entails both realising preferences, while minimising the ambiguity of
the visited states.
## 3 Deep active inference
Figure 1: The various components of the agent rolled out trough time. We
minimise the variational free energy by minimising both the negative log
likelihood of observations and the KL divergence between the state transition
model and the observation model. The inferred hidden state is characterised as
a multivariate Gaussian distribution.
In current treatments of active inference the state spaces are typically
completely fixed upfront (Friston et al., 2009; Millidge, 2019) or partially
(Ueltzhöffer, 2018). However, this does not scale well for more complex tasks
as it is often difficult to design meaningful state spaces for such problems.
Therefore we allow for the agent to learn by itself what the exact
parameterisation of its belief space should be. We enable this by using deep
neural networks to generate the various necessary probability distributions
for our agent.
We approximate the variational posterior distribution for a _single_ timestep
$Q({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1},{\bm{o}}_{t})$ with a network
$q_{\phi}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1},{\bm{o}}_{t})$. Similarly
we approximate the likelihood model $P({\bm{o}}_{t}|{\bm{s}}_{t})$ with the
network $p_{\xi}({\bm{o}}_{t}|{\bm{s}}_{t})$ and the prior
$P({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1})$ with the network
$p_{\theta}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1})$. Each of the networks
output a multivariate normal distribution with a diagonal covariance matrix
using the reparameterisation trick (Kingma & Welling, 2013). These neural
networks cooperate in a way similar to a VAE, where the fixed standard normal
prior is replaced with the learnable prior $p_{\theta}$, the decoder by
$p_{\xi}$ and finally the encoder by $q_{\phi}$, as visualised in Figure 1.
These networks are trained end-to-end using the free energy formula from the
previous section as an objective.
$\forall t:\underset{\phi,\theta,\xi}{\text{minimise}}:-\log
p_{\xi}({\bm{o}}_{t}|{\bm{s}}_{t})+D_{\mathrm{KL}}(q_{\phi}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1},{\bm{o}}_{t})\|p_{\theta}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1}))$
(5)
As in a conventional VAE (Kingma & Welling, 2013) the negative log likelihood
(NLL) term in the objective punishes reconstruction error forcing the model to
learn relevant information on the belief state to be captured in the posterior
output, while the KL term pulls the prior output towards the posterior output,
forcing the prior and posterior to agree on the content of the belief state in
a way that still allows the likelihood model to reconstruct the current
observation.
We can now use the learned models to engage in active inference, and infer
which action the agent has to take next. This is done by generating imagined
trajectories for different policies using $p_{\theta}$ and $p_{\xi}$,
calculating the expected free energy $G$ and selecting the action of the
policy that yields the lowest $G$. These policies to evaluate can be
predefined, or generated through random shooting, using cross-entropy method
(Boer et al., 2005) or by building a search tree.
## 4 Experiments
We validate our deep active inference approach on a real world robotics
navigation task. First, we collect a dataset consisting of two hours worth of
real world action-observation sequences by driving a Kuka Youbot base platform
up and down the aisles of a warehouse lab. Camera observations are recorded
with a front mounted Intel Realsense RGB-D camera, without taking into account
the depth information. The x, y and angular velocities are recorded as actions
at a recording frequency of 10Hz. The models are trained on a subsampled
version of the data resulting in a train set with data points every 200ms.
Next, we instantiate neural networks $q_{\phi}$ and $p_{\xi}$ as a
convolutional encoder and decoder network, and $p_{\theta}$ using an LSTM.
These are trained with Adam optimizer using the objective function from
Equation 5 for 1M iterations. We use a minibatch size of 128 and a sequence
length of 10 timesteps. A detailed overview of all hyperparameters is given in
appendix.
We utilise the same approach as in Çatal et al. (2020) for our imaginary
trajectories and planning. The agent has access to three base policies to pick
from: drive straight, turn left and turn right. Actions from these policies
are propagated to the learned models at different time horizons $H=10,25$ or
$55$. For each resulting imaginary trajectory, the expected free energy $G$ is
calculated. Finally the trajectory with lowest $G$ is picked, and the first
action of the chosen policy is executed, after which the imaginary planning
restarts. The robot’s preferences are given by demonstration, using the state
distribution of the robot while driving in the middle of the aisle. This
should encourage the robot to navigate in the aisles.
At each trial the robot is placed at a random starting position and random
orientation and tasked to navigate to the preferred position. Figure 2
presents a single experiment as an illustrative example. Figure 2(a) shows the
reconstructed preferred observation from the given preferred state, while
Figure 2(b) shows the trial’s start state from an actual observation. Figure
2(c) shows the imagined results of either following the policy “always turn
right”, “always go straight” or “always turn left”. Figure 2(d) is the result
of utilising the planning method explained above. Additional examples can be
found in the supplementary material.
The robot indeed turns and keeps driving in the middle of the aisle, until it
reaches the end and then turns around 111A movie demonstrating the results is
available at https://tinyurl.com/smvyk53. When one perturbs the robot by
pushing it, it will again recover and continue to the middle of the aisle.
(a) Preferred state.
(b) Start state
(c) Imaginary future trajectories for different policies, i.e. going straight
ahead (top), turning right (middle), turning left (bottom).
(d) Actually followed trajectory.
Figure 2: Experimental results: Figure (a) shows the target observation in
imagined (reconstructed) space. (b) The start observation of the trial. Figure
(c) shows different imaginary planning results, whilst (d) shows the actually
followed trajectory.
## 5 Conclusion
In this paper we present how we can implement a generative model for active
inference using deep neural networks. We show that we are able to successfully
execute a simple navigation task on a real world robot with our approach. As
future work we want to allow the robot to continuously learn from past
autonomous behaviour, effectively “filling the gaps” in its generative model.
Also how to define the “preferred state” distributions and which policies to
evaluate remains an open research challenge for more complex tasks and
environments.
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## Appendix A Neural architecture
| Layer | Neurons/Filters | activation function
---|---|---|---
Posterior | Convolutional | 8 | Leaky ReLU
Convolutional | 16 | Leaky ReLU
Convolutional | 32 | Leaky ReLU
Convolutional | 64 | Leaky ReLU
Convolutional | 128 | Leaky ReLU
Concat | N.A. | N.A.
Linear | 2 x 128 states | Softplus
Likelihood | Linear | 128 x 8 x 8 | Leaky ReLU
Convolutional | 128 | Leaky ReLU
Convolutional | 64 | Leaky ReLU
Convolutional | 32 | Leaky ReLU
Convolutional | 16 | Leaky ReLU
Convolutional | 8 | LeakyReLU
Prior | LSTM cell | 400 | Leaky ReLU
Linear | 2 x128 states | Softplus
Table 1: Neural network architectures. All convolutional layers have a 3x3
kernel. The convolutional layers in the Likelihood model have a stride and
padding of 1 to ensure that they preserve the input shape. Upsampling is done
by nearest neighbour interpolation. The concat step concatenates the processed
image pipeline with the vector inputs ${\bm{a}}$ and ${\bm{s}}$.
## Appendix B Hyperparameters
| Parameter | Value
---|---|---
Learning | learning rate | 0.0001
batch size | 128
train iterations | 1M
sequence length | 10
Planning | $\gamma$ | 100
D (Çatal et al., 2020) | 1
K (Çatal et al., 2020) | 10, 25, 55
N (Çatal et al., 2020) | 5
$\rho$ (Çatal et al., 2020) | 0.001
Table 2: Overview of the model hyperparemeters.
## Appendix C Detailed Planning example
A movie demonstrating the results is available at https://tinyurl.com/smvyk53.
Figure 3: Trial preferred state Figure 4: Short term planning
Figure 5: Middle long term planning
Figure 6: Long term planning
|
2024-09-04T02:54:57.902017 | 2020-03-06T14:35:20 | 2003.03239 | {
"authors": "Mutian He, Yangqiu Song, Kun Xu, Dong Yu",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26084",
"submitter": "Mutian He",
"url": "https://arxiv.org/abs/2003.03239"
} | arxiv-papers | # On the Role of Conceptualization in
Commonsense Knowledge Graph Construction
Mutian He1, Yangqiu Song1, Kun Xu2, Dong Yu 2
1Hong Kong University of Science and Technology
2Tencent
{mhear<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
Commonsense knowledge graphs (CKGs) like Atomic and ASER are substantially
different from conventional KGs as they consist of much larger number of nodes
formed by loosely-structured text, which, though, enables them to handle
highly diverse queries in natural language related to commonsense, leads to
unique challenges for automatic KG construction methods. Besides identifying
relations absent from the KG between nodes, such methods are also expected to
explore absent nodes represented by text, in which different real-world
things, or entities, may appear. To deal with the innumerable entities
involved with commonsense in the real world, we introduce to CKG construction
methods _conceptualization_ , i.e., to view entities mentioned in text as
instances of specific concepts or vice versa. We build synthetic triples by
conceptualization, and further formulate the task as triple classification,
handled by a discriminatory model with knowledge transferred from pretrained
language models and fine-tuned by negative sampling. Experiments demonstrate
that our methods can effectively identify plausible triples and expand the KG
by triples of both new nodes and edges of high diversity and novelty.
## 1 Introduction
Commonsense knowledge, such as knowing that a trophy could not fit in a
suitcase because the trophy is too big, is implicitly acknowledged among human
beings through real-life experience rather than systematic learning. As a
result, artificial intelligence meets difficulties in capturing such
commonsense. To deal with the issue, commonsense knowledge graphs (CKGs) like
ConceptNet (Speer et al., 2017), Atomic (Sap et al., 2019), ASER (Zhang et
al., 2019a), etc. have been proposed. Such graphs are aimed at collecting and
solidifying the implicit commonsense in the form of triples $\langle
h,r,t\rangle$, with the head node _h_ and the tail node _t_ connected by a
relation (i.e., edge) _r_. However, a key difference between traditional
knowledge graphs (like WordNet, Freebase, etc.) and CKGs is that commonsense
is often difficult to represent as two strictly formed nodes compared by a
specific relation. Instead, recent approaches represent a node with loosely-
structured text, either annotated by humans or retrieved from text corpora.
Nodes are then linked by one of some predefined types of relations, such as
the triple $\langle$_h_ : I’m hungry, _r_ : Result, _t_ : I have
lunch$\rangle$ in ASER.111Words in ASER are lemmatized, but in this paper we
always show the original text for easier understanding.
| ASER | Atomic
---|---|---
#Nodes | 194.0M | 309.5K
#Triples | 64.4M | 877.1K
#Relation Types | 15 | 9
Average Degree | 0.66 | 5.67
Entity Coverage | 52.33% | 6.98%
Average Distinct Entity | 0.026 | 0.082
Table 1: Statistics for recently proposed commonsense knowledge graphs.
Entity Coverage is calculated by the proportion of top 1% most frequent
entities in Probase, that are mentioned by nodes in each CKG. Average Distinct
Entity is given by average number of distinct Probase entities per node in
each CKG. Core version of ASER is used for these two results.
Such CKGs, storing an exceptionally large number of triples, as shown in Table
1, are capable of representing a much broader range of knowledge and handling
flexible queries related to commonsense. However, the complexity of real world
commonsense is still immense. Particularly, with innumerable eventualities
involved with commonsense in the real world, it is of high cost for a CKG to
cover all of them in nodes; and even when covered, to acquire the
corresponding relation between each pair of nodes is of quadratic difficulty.
Such a situation is particularly demonstrated by the sparsity of edges in
current CKGs in Table 1: even automatic extraction methods as in ASER, fail to
capture edges between most nodes. Therefore, alternative KG construction
methods are in need.
Figure 1: A sample for conceptualization in CKG. Given a commonsense reasoning
problem like (a), even though the corresponding triple (c) is not in the KG, a
triple (b) which is present in the KG could be used through abstraction. This
is done by identifying real world entities of trophy and brown suitcase in the
text of (c), and then substituting them using IsA relations. Following the
same idea, (c) can be produced from (b) to be included in the CKG via
instantiation.
As nodes are represented by text, utilizing semantic information becomes
critical for CKG construction. Such semantic information can be leveraged by
large pretrained language models like BERT (Devlin et al., 2018) and GPT-2
(Radford et al., 2019): These models capture rich semantic knowledge from
unsupervised pretraining, which can be transferred to a wide range of
downstream tasks to reach impressive results. Therefore, efforts have been
made to extract knowledge from such models for KG construction. COMeT
(Bosselut et al., 2019) is a notable attempt which fine-tuned GPT on CKG
triples to predict tails from heads and relations. However, it is often
observed that such generative neural models suffer from the diversity and
novelty issue: They tend to overproduce high-probability samples similar to
those in the training set, while failing to cover a broader range of possible
triples absent in a CKG with diverse entities involved. In contrast,
discriminatory methods incorporate language models to KG completion tasks such
as link prediction and triple classification to evaluate whether a triple is
valid, and could be extended to arbitrary text for triples on various KGs
(Malaviya et al., 2019; Davison et al., 2019; Yao et al., 2019). However it
would be computationally infeasible to identify new plausible triples on
recent large CKGs if we aimlessly explore new nodes and evaluate each possible
triple, without leveraging existent nodes and triples as in generative
methods. Therefore, all methods above have certain shortcomings.
Particularly, we observe that current methods miss the variation of real world
entities, which is a critical factor for the diversity of nodes. For example,
a CKG may cover the node I eat apple and its relevant edges, but the node I
eat banana might be missing or the edges are incomplete. As shown in Table 1,
a large portion of the most common real world entities in Probase are never
mentioned in the recent CKGs like ASER, not to say edges related to the
entities. More, as demonstrated by the low number of average distinct entity
per node, directly expanding the scale of CKGs would not be cost-effective to
cover diverse entities.
To relieve the issue, we posit the importance of a specific element of human
commonsense, conceptualization, which, though found useful for certain natural
language understanding tasks (Song et al., 2011; Wang et al., 2015; Hua et
al., 2015), has not been investigated in depth in this area. As observed by
psychologists, “concepts are the glue that holds our mental world together”
(Murphy, 2004). Human beings are able to make reasonable inferences by
utilizing the IsA relationship between real-world concepts and instances. For
example, without knowing what a _floppy disk_ is, given that it is a memory
device, people may infer that it may store data and be readable by a computer,
etc. From this viewpoint, instead of directly building triples with countless
entities, a CKG can be broadly expanded to handle various queries, as shown in
Figure 1, by such substitution of instances on text with the corresponding
concepts (i.e., abstraction), or vice versa (i.e., instantiation), given an
extra CKG of IsA relations.
However, such conceptualization is never strict induction or deduction that is
guaranteed to be true. As shown in Figure 2, it is still a challenging task to
determine whether a triple built from conceptualization is reasonable, and
requires both the context within the triple and a broader range of
commonsense. Such discriminatory problem could be viewed as a particular case
of the well-studied task of KG completion. Therefore, we propose to formulate
our problem as a triple classification task, one of the standard tasks for KG
completion (Socher et al., 2013). The difference is that, instead of
considering arbitrary substitution of the head or tail with existent nodes, we
apply conceptualization as described in Section 2.1, and train our model by
negative sampling as discussed in Section 2.2. We leverage the rich semantic
information with large pretrained language models by fine-tuning them as
discriminators. In this way, the models are expected to take triples with
arbitrary node as inputs and evaluate whether the triple is reasonable.
Figure 2: A sample from Atomic for discriminating conceptualization. Some
conceptualizations, like replacing milk with beverage in the tail node is
valid, while with dairy is invalid in the context, as with commonsense one
would often want something to drink after eating cookies, while the general
concept of dairy is not relevant in such a scenario, and dairy products are
not all drinkable.
To conclude, our contributions are three-fold:
1. 1.
We introduce conceptualization to CKG construction to explore broader range of
triples.
2. 2.
We formulate the conceptualization-based CKG construction as a triple
classification task to be performed on synthetic triples.
3. 3.
We propose a method for the task by fine-tuning pretrained language models
with negative sampling.
Our code and pipeline is available at github.com/mutiann/ccc.
## 2 Methodologies
Our methodologies of CKG construction are based on the idea that given a set
of ground truth triples as seeds, new triples can be built from them by
abstraction or instantiation of entities mentioned in the head or tail node,
i.e., substituting a mentioned entity with a more general or more specific
one, using the particular commonsense of _IsA_ relations. Therefore, we need a
CKG, K, viewed as seeds, and a conceptualization KG, C, both denoting a set of
triples $\langle h,r,t\rangle$, while in C, _r_ is always _IsA_.
### 2.1 Conceptualization
Since there are diverse ways to conceptualize an entity from commonsense, C
must sufficiently cover various real-world entities connected by IsA
relations. Therefore, we choose Probase (Wu et al., 2012), which is a large-
scale KG that consists of 17.9M nodes and 87.6M edges, extracted from various
corpora, and is proved to be suitable for conceptualization (Song et al.,
2011).
A single entity may have various ways of being abstracted or instantiated,
with different typicality. For example, either Linux or BeOS is an operating
system, and a pen is either a writing tool or an enclosure for animals, though
for both examples the two choices are not equally common. Since Probase is
extracted from real corpora, the frequencies of text showing the triple
$\langle h,IsA,t\rangle$in the source corpora can demonstrate how common the
relation is. Such frequencies $f$ are given by Probase along with the triple,
forming 4-tuples of $\langle h,IsA,t,f\rangle$. The frequency information in
Probase allows us to balance between IsA relations of different typicality and
filter out noise of rare relations in the graph.
Figure 3: A sample of identifying entities. All noun phrases are identified as
possible entities on which substitutions are proposed. The phrases include
basketball, basketball team, team, player, professional player, but except
professional , which is tagged as an adjective here.
With C prepared, conceptualization can then be performed on any mentioned
real-world entity in the head or tail nodes in each triple, which could be a
single noun or noun phrases, serving as subjects, objects or even modifiers,
as shown in Figure 3. What leads to more complexity is that although nodes in
C are all real-world entities in some context, the word or phrase could be
used in different manners within a triple of interest. Therefore, for raw text
as in Atomic, we choose to perform dependency parsing on each node by spaCy
(Honnibal and Montani, 2017). Then we need to filter out all nouns or noun
phrases that also present in C as possible candidates.
The method to identify entities is given in Algorithm 1: We iterate through
each noun $w$ as the root of the entity, and choose all continuous sequences
of words within the range of the subtree corresponding to $w$ in the
dependency tree, to ensure that all entities rooted by $w$ (possibly with
different modifiers) are collected. We then query the possible abstraction and
instantiation of each candidate with Probase, and, for any result, add the
substituted text and the corresponding frequency into the list of results. If
the text in the CKG are given in the original form (i.e., not lemmatized,
unlike ASER), we further use a set of rules to inflect the substitution $s$
returned, and to modify the determiner (if any) in the returned text so as to
avoid any false statistical clues of grammatical mistakes introduced by the
substitution.
Input: W $[w_{1},w_{2},...,w_{n}]$: a node represented by a sequence of words
P $[p_{1},p_{2},...,p_{n}]$: POS tags of words in W
D: Dependency tree for W
C {$\langle x,IsA,y,f\rangle$}: Probase of IsA relations
Result: S $[W^{\prime}_{1},W^{\prime}_{2},...]$: List of substituted word
sequences
F $[f_{1},f_{2},...]$: List of frequencies for each substitution
$\;\;S\leftarrow[]$
$F\leftarrow[]$
for _$k\in[1,n]$_ do
if _$p_{k}\in\\{\mathrm{noun,propn}\\}$_ then
$\;\;T\leftarrow$subtree of $w_{k}$ in $D$
$L\leftarrow\mathrm{min}_{x\in T}$ {index of $x$ in $W$}
$R\leftarrow\mathrm{max}_{x\in T}$ {index of $x$ in $W$}
foreach _$(l,r),L\leq l\leq k\leq r\leq R$_ do
$\;\;E\leftarrow[w_{l},...,w_{r}]$
$\;\;A\leftarrow\\{(S,f)|(E,IsA,S,f)\in C\\}$
$I\leftarrow\\{(S,f)|(S,IsA,E,f)\in C\\}$
foreach _$(s,f)\in A\cup I$_ do
$\;\;S$.add($[w_{1},...,w_{l-1}]+s+[w_{r+1},...,w_{k}]$)
$F$.add(f)
end foreach
end foreach
end if
end for
Algorithm 1 IdentifyConceptualization
### 2.2 Discriminator
We are aimed at building a model capable of evaluating whether a triple is
valid, possibly with its head or tail conceptualized. However, as in the well-
studied field of KG completion, we fall into the difficult situation that we
need to undertake the evaluation using only positive ground truth present in
K, except that in our case not only unseen edges but also nodes need to be
considered. For such KG completion tasks, commonly it is assumed that it is
unknown whether triples not present in K are valid, while the method of
negative sampling is applied. In this way, synthetic triples are built from
substitution (a.k.a. corruption) of the head or tail of present ones, often
using random nodes in the KG. Then the triples are viewed as more likely to be
invalid than the original ones, and labelled as members of a negative set
$D_{-}$. In combination with triples in K as the positive set $D_{+}$, the
model can be trained by such pseudo-labelled data in a self-supervised manner
and evaluated by the classification accuracy under triples in K held out from
$D_{+}$ and the corresponding negative samples generated in the same way
(Bordes et al., 2013; Socher et al., 2013; Nickel et al., 2011).
Although there could be false-negatives, it is demonstrated that models
trained in this way can successfully identify valid triples (though possibly
labelled negative) missing from the KG. More, it is discovered that instead of
uniform sampling, using negative samples similar to valid ones leads to better
performance (Cai and Wang, 2018; Wang et al., 2018; Zhang et al., 2019b).
Therefore, we further propose to sample the substitution of a node from its
conceptualized versions, which might be missing in the original KG. This fits
into previous KG completion methods if we view the conceptualized new nodes as
isolated nodes. We expect that in this way the model could better evaluate
conceptualization of triples.
To generate negative samples, two different settings are applied.
1. 1.
Node Substitution (NS): The common corruption method, as in Bordes et al.
(2013), that substitutes the head or tail (each with 0.5 probability) with a
random node from the KG. For a CKG like Atomic in which head nodes and tails
nodes, as well as tail nodes from triples with different relations, can often
be easily distinguished from each other222For instance, the head node in
Atomic always starts with Person X, and tail nodes for relation type xAttr
(attribute of Person X) are often adjectives., we follow Socher et al. (2013)
to pick random heads only from other heads, and random tails only from other
tails appearing in triples with the same relation.
2. 2.
Entity Conceptualization (EC): To enable the model to identify false triples
with inappropriate conceptualization, we randomly choose the head or tail
(each with 0.5 probability) and corrupt the node as in Section 2.1 by
substituting an entity in the node with its abstraction or instantiation. This
method ensures that the substituted nodes are often plausible. Then we use the
triples with the head or tail substituted as negative samples. Particularly,
we make use of the frequencies returned by Algorithm 1 as weights (or
unnormalized probabilities), based on which we sample from possible
conceptualized nodes, as shown in Algorithm 2. In this way, we strike a
balance between the diversity and the typicality of _IsA_ relations used.
Input: N: A node, represented by a sequence of words
P: POS tags of words in N
D: Dependency tree for N
C: Probase
Result: $N^{\prime}$: corrupted node
$\;\;S$, $F$ $\leftarrow$ IdentifyConceptualization($N$, $P$, $D$, $C$)
$W$ $\leftarrow$ $F$ / $\sum F$
$k\sim Categorical(W)$
$N^{\prime}$ $\leftarrow$ $S_{k}$
Algorithm 2 BuildSampleEC
Building negative samples with both settings, we expect that the model will be
capable of discriminating whether a triple is valid or not, when the triple is
possibly corrupted in either way. To reduce noise in training and evaluation,
negative samples are filtered to ensure that they are different from any
positive samples, which matches the filtered setting from Bordes et al.
(2013).
To make the best use of semantic information from the textual description of
nodes, we apply the widely used transformer-based pretrained language models
like BERT as the discriminator. Particularly, the structure of our task
matches the next sentence prediction (NSP) in Devlin et al. (2018). As a
result, we follow a similar setting that takes pairs of sentences separated by
a special [SEP] token and marked by different token type IDs as inputs, with
_h_ the first sentence and the concatenation of _r_ and _t_ the second
sentence. Binary classification is then performed by a fully-connected layer
taking the final representation corresponding to the [CLS] token, as shown in
Figure 4. All parameters in the model, except those in the final fully-
connected layer for binary classification, can be initialized from the
pretrained model. Then the model is fine-tuned using the positive and negative
samples mentioned above, with a 1:1 frequency during training, using binary
cross-entropy loss below, based on the output $s$, which is a scalar after a
logistic sigmoid activation, indicating the confidence on the input being
valid:
$L=-\sum_{(x,y)\in D_{+}\cup D_{-}}(y\log s+(1-y)\log(1-s)).$ (1)
Figure 4: Architecture of the BERT-based discriminator model. Raw text are fed
into the model to predict the binary label y. All except the last fully-
connected layer are pretrained but not frozen.
## 3 Experiments
### 3.1 Datasets
Two different datasets, Atomic and ASER, which are typical CKGs using open-
form text as nodes, are used in our experiments. Earlier CKGs such as
ConceptNet (Speer et al., 2017) are not discussed, as in ConceptNet, unlike
the more recent CKGs, only simple text, mostly noun phrases, is used as nodes,
and previous work can already reach close-to-human results (Saito et al.,
2018).
#### 3.1.1 Atomic
Atomic is a CKG of 877K triples on cause-and-effect relations between everyday
activities (Sap et al., 2019). Within Atomic, head nodes, or base events, are
extracted from text corpora, with the form of some person’s action or state
(e.g., Person X is on the basketball team). The dataset further categorizes
the cause-and-effect relations into nine types, covering the intention,
prerequisites and impacts with regard to the agent Person X and other people.
The tail entities are then written in open-form text by crowd-sourced workers
under these categories. In this way, a broad range of eventualities and their
relations are covered by the dataset. We follow the original data split of
Atomic in our experiments.
#### 3.1.2 ASER
ASER (Zhang et al., 2019a) is a large-scale CKG of 194M nodes representing
verb-centric eventualities matching certain patterns (e.g., s-v-o for I love
dogs and s-v-v-o for I want to eat an apple) extracted from various corpora.
Relations of 15 types between nodes are then extracted as well, identified by
matching the text with language patterns such as “$E_{1}$, because $E_{2}$”
for Reason and “$E_{1}$, $E_{2}$ instead” for ChosenAlternative. A total
194.0M nodes and 64.4M triples are extracted in ASER. In our experiments, we
use the core release of ASER with 27.6M nodes and 10.4M edges. Triples of the
CoOccurance type and isolated nodes are further removed to create a smaller
and cleaner KG with 1.4M nodes and 1.1M edges. Triples are then randomly split
into train, dev, and test set at 8:1:1.
### 3.2 Settings
To build our CKG Construction by Conceptualization (CCC) discriminator, we
follow the scheme for fine-tuning BERT on downstream tasks (Devlin et al.,
2018), and use the pretrained 12-layer BERT-base model on GTX1080 GPUs with
8GB memory. To evaluate the impact of the two different ways of producing
negative samples given in Section 2.2, and to trade off between model
capabilities of discriminating triples under general cases and specifically
identifying inappropriate conceptualization, we perform experiments with
different percentages of negative samples built by conceptualization, i.e.,
the EC setting. Specifically, models with 50%, 75%, and 87.5% negative samples
created by EC (and the rest by NS) are trained and reported. For evaluation,
negative samples are generated on the dev and test samples as well by both
methods, forming the EC and NS dev and test sets with 1:1 positive and
negative samples. Under EC, those triples with nodes containing no entities to
be conceptualized (e.g., I am fine, for which Algorithm 1 returns empty
results) are ignored. Nevertheless, 79.65% and 83.56% of triples in the dev
and test sets are collected in the EC set for Atomic and ASER respectively,
showing that a majority of triples can be conceptualized. Test results of the
models at best EC dev accuracy are reported.
### 3.3 Baselines
We train COMeT333Available at github.com/atcbosselut/comet-commonsense and KG-
BERT444Available at github.com/yao8839836/kg-bert on the two datasets as our
baselines. Particularly, our model will degenerate into KG-BERT with 0% EC
samples, as the NS setting is what KG-BERT trained under. Since COMeT itself
is not a discriminative model, we use the perplexity per token555Unlike the
case in Malaviya et al. (2019), conceptualization would not significantly
change the length of a triple, so we only use the Normalized setting. as its
score given to each triple, and use the dev set to find a classification
threshold with best accuracy.
| ASER | Atomic
---|---|---
| EC | NS | EC | NS
COMeT | 0.6388 | 0.5869 | 0.6927 | 0.5730
KG-BERT | 0.7091 | 0.8018 | 0.7669 | 0.6575
CCC-50 | 0.8716 | 0.7775 | 0.9016 | 0.7840
CCC-75 | 0.8995 | 0.7250 | 0.9221 | 0.7446
CCC-87.5 | 0.9156 | 0.6635 | 0.9355 | 0.6980
CCC-75-scratch | 0.8284 | 0.5587 | 0.8579 | 0.5003
CCC-75-RoBERTa | 0.8999 | 0.6938 | 0.9305 | 0.7350
Table 2: Results of accuracy on EC and NS test set of baselines and our
models on different datasets. CCC denotes our model, with the number attached
representing percentage of EC training.
### 3.4 Results
#### 3.4.1 Triple Classification
Test accuracies of triple classification on both methods are given in Table 2.
As shown by the results, COMeT lacks discriminative power, which is consistent
with the results in Malaviya et al. (2019). KG-BERT, which has been
successfully applied on traditional KGs, produce satisfactory results on CKG
as well, while our methods perform better than both baseline methods by a
large margin on the EC tests. Hence it is demonstrated that introducing
conceptualization during training is effective to create a model capable of
identifying false conceptualization. Particularly, the percentage of EC
samples in training is critical for a trade-off between EC and NS tasks:
Increased EC percentage will lead to better EC results, but the NS results
will drop. The Atomic CCC models reach better results on NS than KG-BERT,
which is possibly due to the fact that Atomic nodes are mostly about everyday
activities, in contrast to ASER which covers a broader range of topics.
Therefore by EC training a more diverse set of nodes could be seen by the
model in training, and could be helpful for the model to generalize in the NS
test.
#### 3.4.2 Ablation Studies
We perform ablation studies to examine the importance of pretraining and model
selection. With the model trained from scratch on our task without using
pretrained parameters, the performance significantly drops, as shown in the
CCC-75-scratch results in Table 2. We also attempted to use RoBERTa, an
alternative pretrained language model that makes improvements on BERT training
and has demonstrated better performance on downstream tasks (Liu et al.,
2019). However, the results using the pretrained RoBERTa-base model
(CCC-75-RoBERTa) are generally on par with our model using BERT. This could be
possibly explained by that BERT is sufficient in our current settings, that
RoBERTa uses a larger batch size while on our GPU the batch size is more
limited, and that the NSP pretraining task is used in BERT but is absent in
RoBERTa, as NSP exactly matches the input scheme of our task.
| ASER | Atomic
---|---|---
| COMeT | CCC-75 | COMeT | CCC-75
N/Seed | 10 | 8.28 | 10 | 5.02
Dist-N | 24.68% | 96.57% | 6.49% | 51.26%
Dist-1 | 1.62% | 6.30% | 0.63% | 8.34%
Dist-2 | 10.56% | 15.45% | 2.87% | 49.76%
N/T N | 88.48% | 98.60% | 10.30% | 94.65%
N/U N | 93.38% | 99.18% | 69.17% | 96.66%
Dist-N-Norm | 10.74% | 84.16% | 4.35% | 46.36%
N/T N-Norm | 9.37% | 86.01% | 5.12% | 87.95%
N/U N-Norm | 65.72% | 95.02% | 58.71% | 93.01%
Table 3: Results for diversity and novelty on generations, larger for better results. All rows except N/seed are given in percentage. | head | tail
---|---|---
Seed | another promises him a scholarship | his parents own a successful business
COMeT | – | he never gets it
he does not get it
he never gets one
he could not pay it
he does not receive it
CCC-75 | another promises him a grant | –
another promises him an award
– | his parents own a successful shop
his parents own a successful bank
his parents own a successful hotel
Table 4: Samples for ASER generations given the seed. In this sample the head
and tail are connected by the relation of Concession, i.e., although.
#### 3.4.3 Generations
We generate triples using the test set from ASER and Atomic as seeds by both
COMeT with 10-beam search and the CCC-75 model by conceptualization, and apply
various metrics on the results, as shown in Table 3. Both methods may produce
a large number of triples, as given by the number of generations per seed,
N/Seed. For diversity, we report Dist-1 (number of distinct words per node),
Dist-2 (number of distinct bigrams per node), and Dist-N (number of distinct
nodes per node). Due to the different number of generated triplets, the
results are all normalized by number of nodes.666Results from Bosselut et al.
(2019) of test perplexity are reproduced in our experiments. Differences on
diversity metrics are due to the fact that we use 10-beam search instead for
fair comparison.
Novelty is measured by N/T N, the proportion of nodes among all produced nodes
that are novel, i.e. not present in the training set, and N/U N, the
proportion of novel distinct nodes, among all distinct nodes. Moreover, since
generative methods may produce nodes of essentially the same meaning with
slight changes in forms, we also have the produced nodes normalized, by
removing structural words like determiners, auxiliary verbs, pronouns, etc. We
then report the results for the metrics above but applied to generations after
such normalization, denoted as Dist-N-Norm, N/T N-Norm, and N/U N-Norm
respectively. Furthermore, samples of generations by both models are shown in
Table 4.
It is clearly demonstrated in the diversity results that a majority of
generations by COMeT are similar to each other given a certain head node and
relation, as the number of distinct nodes, words and bigrams are all
relatively low. The novelty results further show that generated nodes are
often similar to the seen ones in the training set as well. It can be
particularly observed that, though the original diversity and novelty metrics
appear to be acceptable, which is consistent with Bosselut et al. (2019),
results drop sharply when the generations are normalized. This indicates that
COMeT may produce slightly different nodes paraphrasing each other, as shown
in Table 4 where four of five generated tails are similar to each other
(saying he does not get it), while this is not the case for CCC, as generated
nodes are mostly discussing different entities, and the results will often be
diverse and novel.
## 4 Related Work
Automatic construction of structured KGs is a well-studied task, and a number
of learning-based methods have been proposed, including KG embedding methods
based on translational distances (Bordes et al., 2013; Lin et al., 2015; Ji et
al., 2015; Wang et al., 2014; Shang et al., 2019) and semantic matching
(Nickel et al., 2011; Socher et al., 2013; Yang et al., 2014; Trouillon et
al., 2016), typically trained by negative sampling techniques and applied on
tasks like link prediction and triple classification. Furthermore, graph
neural networks can be used to better capture structural information
(Schlichtkrull et al., 2018), GANs are applied to improve negative sampling
(Cai and Wang, 2018; Wang et al., 2018; Zhang et al., 2019b) by mining more
difficult examples, and textual information from the node can be
leveraged(Wang and Li, 2016; Xie et al., 2016; Xiao et al., 2017; An et al.,
2018).
Textual information is more critical on CKGs with nodes carrying complicated
eventualities, often in open-form text. Therefore, Li et al. (2016) proposed
to score ConceptNet triples by neural sequence models taking text inputs so as
to discover new triples, while Saito et al. (2018) and Sap et al. (2019)
further proposed to generate tail nodes by a sequence-to-sequence LSTM model
with head and relation as inputs. Recently powerful large pretrained models
like BERT and GPT-2 have been proposed (Devlin et al., 2018; Radford et al.,
2019), from which, it has been observed by Trinh and Le (2018) and Radford et
al. (2019) that rich knowledge including commonsense can be extracted.
Therefore, different ways for KG construction have been introduced on such
models as downstream tasks: In KG-BERT, BERT was fine-tuned for KG completion
tasks like link prediction and triple classification (Yao et al., 2019); COMeT
used GPT-based models to generate tails (Bosselut et al., 2019); LAMA directly
predicted masked words in triples on various KGs by BERT (Petroni et al.,
2019); Davison et al. (2019) considered both generation of new tails and
scoring given triples; Malaviya et al. (2019) utilized both structural and
semantic information for CKG construction on link prediction tasks.
## 5 Conclusion
We introduce conceptualization to commonsense knowledge graph construction and
propose a novel method for the task by generating new triples with
conceptualization and examine them by a discriminator transferred from
pretrained language models. Future studies will be focused on strategies of
conceptualization and its role in natural languages and commonsense by deep
learning approaches.
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|
2024-09-04T02:54:57.912432 | 2020-03-06T14:45:46 | 2003.03243 | {
"authors": "M. Redies, F. R. Lux, J.-P. Hanke, P. M. Buhl, S. Bl\\\"ugel, Y.\n Mokrousov",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26085",
"submitter": "Matthias Redies",
"url": "https://arxiv.org/abs/2003.03243"
} | arxiv-papers | # Mixed topology ring states for Hall effect and orbital magnetism
in skyrmions of Weyl semimetals
M. Redies<EMAIL_ADDRESS>Peter Grünberg Institut and Institute for
Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
Department of Physics, RWTH Aachen University, 52056 Aachen, Germany F. R.
Lux Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, 52425 Jülich, Germany Department of
Physics, RWTH Aachen University, 52056 Aachen, Germany J.-P. Hanke Institute
of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany P. M.
Buhl Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz,
Germany S. Blügel Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany Y.
Mokrousov<EMAIL_ADDRESS>Peter Grünberg Institut and Institute for
Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz,
Germany
###### Abstract
As skyrmion lattices are attracting increasing attention owing to their
properties driven by real-space topology, properties of magnetic Weyl
semimetals with complex $k$-space topology are drawing increasing attention.
We consider Hall transport properties and orbital magnetism of skyrmion
lattices imprinted in topological semimetals, by employing a minimal model of
a mixed Weyl semimetal which, as a function of the magnetization direction,
exhibits two Chern insulator phases separated by a Weyl state. We find that
while the orbital magnetization is topologically robust and Hall transport
properties exhibit a behavior consistent with that expected for the recently
discovered chiral Hall effect Lux _et al._ (2020), their evolution in the
region of the Chern insulator gap is largely determined by the properties of
the so-called mixed topology ring states, emerging in domain walls that
separate the skyrmion core from the ferromagnetic background. In particular,
we show that these localized ring states possess a robust orbital chirality
which reverses sign as a function of the skyrmion radius, thereby mediating a
smooth switching dynamics of the orbital magnetization. We speculate that
while the emergent ring states can possibly play a role in the physics of
Majorana states, probing their properties experimentally can provide insights
into the details of skyrmionic spin structures.
## I Introduction
Topological chiral spin textures such as magnetic skyrmions have established
themselves as an exciting platform for the realization of novel physical
effects and for innovative ideas in the realm of practical applications of
magnetic systems Lai _et al._ (2017); Zhang _et al._ (2015a); Tomasello _et
al._ (2015); Zhang _et al._ (2020a); Zázvorka _et al._ (2019); Bourianoff
_et al._ (2018); Pinna _et al._ (2019); Zhang _et al._ (2015b); Luo _et
al._ (2018); Zhang _et al._ (2020a). On the other side of the topology scale,
magnetic materials which exhibit non-trivial $k$-space topology in the
electronic structure $-$ such as quantum anomalous Hall insulators Deng _et
al._ (2020); Chang _et al._ (2013), or antiferromagnetic topological
insulators Otrokov _et al._ (2019); Niu _et al._ (2020) $-$ are at the heart
of current research in solid state physics and spintronics. Bringing together
the benefits of skyrmions, such as their efficient dynamics and stability,
with the advantages of $k$-topological materials, normally associated with
dissipationless transport and topological protection, appears to be an
exciting avenue to pursue. To date, however, skyrmions have been mostly
realized in metallic ferromagnetic materials that do not exhibit a distinct
non-trivial $k$-space topology, such as metallic ferromagnetic materials Miao
_et al._ (2014); Münzer _et al._ (2010). And while the interest in skyrmions
realized in insulating materials is rising Zhang _et al._ (2020b); Kong and
Zang (2013); Onose _et al._ (2012), the emergence of global Chern insulating
states in skyrmion lattices has been recently shown theoretically Lado and
Fernández-Rossier (2015); Hamamoto _et al._ (2015); Göbel _et al._ (2018,
2017, 2019).
At the same time, another flavor of $k$-space topology exhibited by magnetic
Weyl semimetals is gaining increasing attention, and the number of specific
material candidates which exhibit distinct topological band crossings in their
electronic structure is constantly growing Wang _et al._ (2017); Liu _et
al._ (2019). In three dimensions, Weyl semimetals host bands crossings known
as Weyl nodes, that appear in pairs and carry an opposite non-zero topological
charge. Recently, it was suggested that in two dimensions (2D) it is natural
to interpret the emergence of topological band crossings in the context of
mixed Weyl points, emphasizing that their topology and properties can be
classified most optimally by including the magnetization direction
$\hat{\mathbf{m}}$ and the mixed components of the Berry curvature tensor into
the topological analysis Hanke _et al._ (2017a); Niu _et al._ (2019). The 2D
mixed Weyl semimetals thus behave in many aspects similarly to the 3D Weyl
semimetals in $(\mathbf{k},\hat{\mathbf{m}})$-space. Recently, it was realized
that the presence of Weyl points in spin textures in 2D or 3D Weyl semimetals
can have a drastic effect on their magneto-electric properties, orbital
magnetism and dynamics, see e.g. Refs. [Araki, 2020; Araki and Nomura, 2018;
Lux _et al._ , 2018; Hanke _et al._ , 2017a; Niu _et al._ , 2019].
In this work, by performing explicit tight-binding calculations of skyrmion
lattices imprinted into a 2D Weyl semimetal, we attempt to understand the role
that the complex mixed topology can have on the Hall transport properties and
orbital magnetism in these systems. Our main finding is the demonstration that
properties of skyrmions of mixed Weyl semimetals, whose chemical potential
resides in the vicinity of Weyl points, are largely determined by so-called
ring states, which are localized at the skyrmion boundary and carry an orbital
moment of a specific orbital chirality. While we discuss how the transport
properties and orbital magnetization in these systems can be used to get
insights into the details of spin distribution, we also consider a range of
phenomena where ring states can be utilized for shaping skyrmion dynamics and
for mediating the emergence of novel topological phases. This article is
structured as follows. In Sec. II, we describe the tight-binding model and
parameters used in this paper. The details of the setup and transport
calculations are given in Sec. III. In Sec. IV we present and discuss the
results of our calculations, providing an outlook in Sec. V.
## II Model
In order to assess the transport properties of 2D skyrmions in mixed Weyl
semimetals we choose the tight-binding model such that the underlying
electronic structure of the ferromagnetic host exhibits a mixed Weyl point in
an extended phase space of the magnetization direction and $k$-space. A
similar model has been used to study the topological properties of
ferromagnetic mixed Weyl semimetals in the past Hanke _et al._ (2017a). The
tight-binding Hamiltonian of this model on a honeycomb lattice with two
structurally inequivalent atoms per unit cell and two spin-split orbitals per
site reads:
$\begin{split}H&=\,\lambda\sum_{i\epsilon\zeta}\left(\hat{\mathbf{m}_{i}}\cdot\boldsymbol{\sigma}\right)c_{i\epsilon}^{\dagger}c_{i\zeta}\\\
-t\sum_{\left<ij\right>\epsilon}c_{i\epsilon}^{\dagger}c_{j\epsilon}&+it_{\text{so}}\sum_{\left<ij\right>\epsilon\zeta}\hat{\mathbf{e}}_{z}\cdot\left(\boldsymbol{\sigma}\times\hat{\mathbf{d}}_{ij}\right)c_{i\epsilon}^{\dagger}c_{j\zeta}\end{split}$
(1)
The properties of this model of magnetic graphene with Rashba spin-orbit
interaction have been extensively studied in the past for collinear
ferromagnetic and antiferromagnetic cases Castro _et al._ (2008); Matte _et
al._ (2009); Qiao _et al._ (2010). In Eq. (1), $t$ is the magnitude of the
nearest neighbour hopping, $t_{\text{so}}$ is the magnitude of Rasba-like
spin-orbit coupling Bychkov and Rashba (1984) , and $\lambda$ characterizes
the magnitude of the Stoner exchange splitting Stoner (1936, 1938). In the
above expression the indices $i$ and $j$ run over nearest-neighbor atoms,
while $\epsilon$ and $\zeta$ mark the spin channel. Further,
$\hat{\mathbf{e}}_{z}$ is the unit vector in the $z$-direction (out of the two
dimensional plane), while $\hat{\mathbf{d}}_{ij}$ is the unit vector
connecting atom sites $i$ and $j$. The operators $c_{i\epsilon}^{\dagger}$ and
$c_{i\epsilon}$ are the creation and annihilation operators of an electron on
site $i$ with spin $\epsilon$. The vector of Pauli matrices is denoted as
$\boldsymbol{\sigma}$, while $\hat{\mathbf{m}_{i}}$ is the unit vector of the
magnetization direction at atom $i$, which generally varies in real space when
a spin texture is present.
Throughout this work, we choose $t=-1.0$ eV, $t_{so}=~{}0.4$ eV and
$\lambda=1.4$ eV in order to realize a mixed Weyl point in the electronic
structure for the ferromagnetic case when all $\hat{\mathbf{m}_{i}}$ are
aligned along a single direction $\hat{\mathbf{m}}$. The bandstructure of the
model for the out-of-plane direction of the ferromagnetic magnetization with
these parameters is shown in Fig. 1(a). The emergence of the mixed Weyl point
upon varying the magnetization direction is shown in Fig. 1(c). The
topologically non-trivial character of this crossing point can be shown by
computing the flux of the Berry curvature tensor with components in $k$\- and
$\theta$-space, which quantifies the mixed topological charge of the mixed
Weyl point to be $+2$ for $\theta=\pi/2$, and $-2$ for $\theta=-\pi/2$ Niu
_et al._ (2019).
Figure 1: (a) The bandstructure of the ferromagnetic model with an out-of-
plane magnetization. The green dashed line marks the region $[-0.47,-0.46]$
eV. (b) The bandstructure of a skyrmion with the parameterization described in
the text. The red-shaded area in (a) and (b) indicates the band gap region of
about 0.6 eV of the ferromagnetic model. (c) A section of the bandstructure is
shown for different angles $\theta$ that the magnetization in the
ferromagnetic model makes with the $z$-axis. (d) The magnetization is
characterized by the azimuthal angle $\theta$ and the polar angle $\phi$. The
skyrmion is placed in the $xy$-plane. (e,f) The Hall conductance (e) and
orbital magnetization (f) of the ferromagnetic model with an out-of-plane
magnetization (along $+z$) as a function of band filling. Red shaded areas
coincide with the band gap marked in (a). All plots except for (b) and (d)
are for $\phi=0$.
## III Computational approach
Similarly to our previous work on transport properties of chiral bobbers
Redies _et al._ (2019) we calculate the Hall conductance and orbital
magnetization of the system by employing the $k$-space Berry curvature
formalism, with the only non-vanishing component of the Berry curvature tensor
in $k$-space being
$\Omega_{xy}^{n}=-2\imaginary\innerproduct{\frac{\partial
u_{n\bm{k}}}{\partial k_{x}}}{\frac{\partial u_{n\bm{k}}}{\partial k_{y}}},$
(2)
where $u_{n\bm{k}}$ is the lattice-periodic part of the Bloch wave function of
the band $n$ for a Bloch vector $\bm{k}$. The intrinsic Hall conductance (in
units of $e^{2}/h$) is then evaluated as the Brillouin zone integral of the
Berry curvature of all occupied states Blügel _et al._ (2017)
$\sigma_{xy}=\sum_{n}^{occ}\int\limits_{\rm
BZ}\Omega_{xy}^{n}(\bm{k})d\bm{k}.$ (3)
For an insulator the Hall conductance is quantized and it is proportional to
the value of an integer Chern number
$\mathcal{C}:=\int_{S}\Omega_{xy}dS/2\pi=-\sigma_{xy}/2\pi$. We calculate the
out-of-plane component of the orbital magnetization (OM) in the system,
$M_{orb}$, employing the modern theory of orbital magnetism Thonhauser _et
al._ (2005); Xiao _et al._ (2005); Thonhauser (2011), according to which
$\begin{split}M_{orb}=&M^{LC}_{orb}+M^{IC}_{orb}=\frac{e}{2\hbar
c}\imaginary\int\limits_{\rm BZ}\frac{d\bm{k}}{(2\pi)^{2}}\\\
&\sum_{n}^{occ.}\matrixelement{\frac{\partial u_{n\bm{k}}}{\partial
k_{x}}}{\times\left(H_{\bm{k}}+\varepsilon_{n\bm{k}}-2\mu\right)}{\frac{\partial
u_{n\bm{k}}}{\partial k_{y}}},\end{split}$ (4)
where $M^{LC}_{orb}$ and $M^{IC}_{orb}$ are the local circulation and
itinerant circulation parts of the OM, respectively, $\varepsilon_{n\bm{k}}$
is the energy of the $n$-th Bloch state at $\bm{k}$, and $H_{\bm{k}}$ is the
lattice-periodic part of the Hamiltionian
$H_{\bm{k}}:=e^{-i\bm{k}\bm{r}}He^{i\bm{k}\bm{r}}$. In order to compute the
derivatives of the Bloch states in the tight-binding basis we employ first-
order perturbation theory:
$\ket{\frac{\partial u_{n\bm{k}}}{\partial k_{i}}}=\sum_{m\neq
n}\frac{\matrixelement{u_{m\bm{k}}}{\nicefrac{{\partial H_{\bm{k}}}}{{\partial
k_{i}}}}{u_{n\bm{k}}}}{\varepsilon_{n\bm{k}}-\varepsilon_{m\bm{k}}+i\eta}\ket{u_{m\bm{k}}},$
(5)
which results in the well-known gauge-invariant expression for the Berry
curvature and OM Yao _et al._ (2004); Thonhauser (2011). In order to avoid
divergences for (nearly) degenerate states a broadening $\eta=10^{-8}\mbox{
Hartree}$ is introduced.
The calculated Hall conductance (HC) and OM of the model as a function of band
filling for the out-of-plane magnetization are shown in Fig. 1(e) and (f),
respectively. As evident from these plots, at half filling the system is a
Chern insulator with the Chern number of $-2$. Therefore, the variation of the
OM in the topologically non-trivial gap is linear with the chemical potential
$\mu$, according to the relation valid for insulators Thonhauser (2011):
$\frac{dM_{orb}}{d\mu}=\frac{e}{(2\pi)^{2}\hbar c}\,\sigma_{xy},$ (6)
where $e$ is the electron charge and $c$ is the speed of light. The emergence
of the metallic point in the spectrum of the model is due to the change in the
Chern number from $-2$ to $+2$ at half filling upon reversing the direction of
$\hat{\mathbf{m}}$.
Based on the model (1) we imprint the skyrmion lattice by varying the
direction of $\hat{\mathbf{m}}_{i}$ in real space according to the
parametrization of the skyrmions discussed below. We consider a hexagonal
lattice of skyrmions where in the center of each unit cell we place a Néel
skyrmion. Skyrmions of this type can be stabilized experimentally in the
presence of a small external magnetic field, the effect of which on the
electronic properties we do not take into account. The chemical unit cell
contains two atoms, which are located at $(0,\pm a,0)^{T}$. The lattice
vectors of the chemical unit cell are $(l_{\text{chem}},0,0)^{T}$ and
$l_{\text{chem}}\cdot(\frac{1}{2},\frac{\sqrt{3}}{2},0)^{T}$, where
$l_{\text{chem}}=\sqrt{3}a$ and $a$ is the distance between the inequivalent
atoms in the chemical unit cell. The unit cell of the skyrmion lattice is
hexagonal and contains 1568 atoms. The lattice vectors of the supercell are
$(l_{\text{mag}},0,0)^{T}$ and
$l_{\text{mag}}\cdot(\frac{1}{2},\frac{\sqrt{3}}{2},0)^{T}$, where
$l_{\text{mag}}=28\,l_{\text{chem}}$. This corresponds to 28 atoms in between
the centers of neighboring skyrmions and 1568 atoms in the unit cell. We use
an adaptive integration scheme to evaluate the integrals in Eqs. (3) and (4).
This leads to the convergence of the results presented in Figs. 3 and 4 with
6144 $k$-points, and results presented in Fig. 1(e,f) with 16042 $k$-points.
The full source code is available as open source Redies (2020).
## IV Results
### IV.1 Emergence of mixed topology ring states in skyrmions of mixed Weyl
semimetals
Here, we analyze the electronic structure of a skyrmion lattice of a mixed
Weyl semimetal at half filling, and in the following the zero of energy is
associated with the position of Fermi energy for the case when exactly two
electronic states per structural unit cell are occupied. For the ferromagnetic
system this corresponds to the position of the metallic Weyl point in the
spectrum when the magnetization is in-plane, see Fig. 1(c). Correspondingly,
in the limit of very large skyrmions, when the regions with homogeneous out-of
plane magnetization in the skyrmion center and in-between the skyrmions are
large, we expect an emergence of electronic states around the zero energy.
This state is expected to be localized to the region where the magnetization
lies in-plane, i.e. within the domain wall separating the skyrmion center with
the outside region.
Figure 2: Emergence of mixed topology ring states (MTRS) in a skyrmion
lattice of a mixed Weyl semimetal with skyrmion radius of
$r_{\text{max}}=\frac{1}{2}\,l_{mag}$. (a-c) The local density of states
(LDOS) in real space integrated over the energy range from $-$0.3 to $+$0.3
eV, corresponding to the gap of the ferromagnetic system in Fig. 1(a,b). The
magnitude of the integrated LDOS is indicated in the color bar. The arrows to
the right indicate the spin direction along the orange line in the
$(xy)$-plane. Here the dark red (blue) arrows indicate a
$\mathbf{m}=(-)\hat{e}_{z}$ magnetization direction. In (a) $d=0.2$, in (b)
$d=0.5$ and in (c) $d=0.8$. The LDOS marks the formation of MTRS. (d) The LDOS
integrated in the energy range $[-0.47,-0.46]\mbox{ eV}$, away from the gap,
marked green in Fig 1(a). The magnetic structure is identical to (c). In this
energy region the MTRS are absent. In the center of the skyrmion
$\mathbf{m}=+\mathbf{e}_{z}$, while $\mathbf{m}=-\mathbf{e}_{z}$ at the edges
of the unit cell.
We verify this by performing explicit calculations of the electronic structure
of the skyrmion lattice with the model described above. The Néel skyrmions of
radius $R$ (with the maximal value of it being $R_{max}=l_{\rm mag}/2$), are
parameterized by the angle $\alpha$ that a spin at the distance $r$ from the
center of the skyrmion makes with the $z$-axis, while the parameter $d$ is
introduced to tune the width of the domain wall region where the spins are
lying in-plane so that larger $d$ results in a larger region of “flat”
magnetization (see examples in Fig 2(a-c) for different values of $d$). Note
that the rate of change of the magnetization outside of the in-plane regions
does not change. We use this parametrization to artificially tune the internal
width of the states which emerge in the “flat” region. Below we consider the
case of $R=R_{max}/2$ when introducing the parameterization in terms of $r$
and $d$ as follows:
$\alpha(r,d)=\pi\frac{\beta(r/R_{max},d)-\beta(1,d)}{\beta(0,d)-\beta(1,d)}$,
with
$\beta(x,d)=\operatorname{atan}\left(R_{max}(6x-3d-3)\right)+\operatorname{atan}\left(R_{max}(6x+3d-3)\right)$.
As we show below, this parameterization allows us to manipulate the spatial
spread of states associated with the transition between the two out-of-plane
domains.
In order to analyze the spatial localization of the electronic states, we
compute the space-resolved local density of states (LDOS), presenting the
results in Fig. 2. Namely, we look at the LDOS of states which are appearing
in the electronic structure of the skyrmion in the gap of the out-of-plane
ferromagnetic system between $-$0.3 and $+$0.3 eV, see Fig. 1(a,b). In Fig. 2
(a-c) the spatial distribution of the skyrmion’s LDOS, which has been
integrated in this energy region, is shown for different widths of the domain
wall as controlled by the parameter $d$. Quite remarkably, our calculations
show that, irrespective of the domain wall width, the LDOS is vanishing
exactly outside of the domain wall $-$ i.e., in the center of the skyrmion,
and in the region between the skyrmions. This is in sharp contrast to all
other states of the skyrmion lattice which lie outside of the gap in the
“metallic” region: as an example, in Fig. 2(d) the LDOS of the system, which
has been integrated over the energy region $[-0.47,-0.46]$ eV (i.e. outside of
the gap as marked with a green dashed line in Fig. 1(a)), is finite at any
point in the unit cell which marks the delocalized character of the
constituting states.
The precise localization of the gap states in the domain that we observe
irrespective of the domain wall width allows us to directly associate them
with the emergence of the band crossing in the electronic structure of the
model ferromagnetic system for the in-plane magnetization. Recently it was
shown that this band crossing acquires a non-trivial topological character
once the magnetization direction $\hat{\mathbf{m}}$ is included into the
topological analysis. In this sense the ferromagnetic 2D model that we study
is an example of what is called a mixed Weyl semimetal Hanke _et al._
(2017a); Niu _et al._ (2019) $-$ a term motivated by necessity of including
the so-called mixed Berry curvature into the analysis of the band topology.
Since the gap states that we observe arise as a result of the non-trivial
mixed topology of our model, we refer to them as mixed topology ring states
(MTRS). It has been shown, that, depending on the symmetry of the model, the
mixed Weyl points can be associated with a transition between the Chern
insulating phases with different Chern number arising for different directions
of $\hat{\mathbf{m}}$ Hanke _et al._ (2017a); Niu _et al._ (2019). This is
exactly the case for our model: as the direction of $\hat{\mathbf{m}}$ is
changed from along the $z$-axis to the opposite to it, the Chern number at
half-filling changes sign from $+2$ to $-2$. The skyrmion lattice that we
consider thus presents a lattice of domains with an opposite Chern number
separated by the domain walls, and the MTRS can be naturally interpreted as
the topological edge states localized at the boundary separating the two
domains.
### IV.2 Hall conductance and orbital magnetization in skyrmions of a mixed
Weyl semimetal
In this section we investigate the possible influence of MTRS on the Hall
transport properties and orbital magnetization exhibited by the skyrmion
lattices of mixed Weyl semimetals. In order to do that, we parametrize the
magnetization distribution within the skyrmion of radius $R$ in a way that
$\alpha(r)=\pi\left(1-\frac{\delta(r)-\delta(0)}{\delta(R_{\text{max}})-\delta(0)}\right)$
with $\delta(x)=-\operatorname{atan}\left(a\cdot(r-R)\right)+\frac{\pi}{2}$.
Unlike the previous parameterization used in section IV.A, which exhibits a
pronounced flat region in the middle of the wall, the parametrization we use
from here on is designed to model most closely the skyrmions in systems which
favor an out-of-plane magnetization direction, which leads to domain walls of
roughly constant magnetization gradient across the wall. When compared to the
approach used in the preceding subsection, the parametrization employed from
now on is very close to that routinely used to model skyrmion profiles in
micromagnetic studies Romming _et al._ (2015); foo . It is important to note
that the MTRS emerge for both types of the profile parametrization.
We first consider the case of $R=R_{max}/2$, and show the skyrmion profile
along the path between the skyrmion centers for various values of $a$ in Fig.
3(a). We present the results of our calculations of the Hall conductance and
orbital magnetization in the system in Fig. 3(b,c) as a function of band
filling and parameter $a$, of which the latter controls the domain wall width
and correspondingly the MTRS spread in real space. Concerning the overall
energy dependence for all values of $a$, while HC exhibits a symmetric
structure with respect to the middle of the bulk gap at $E_{F}=0$ eV, the OM
is antisymmetric, which originates in the symmetry of the ferromagnetic band
structure and the properties of the OM around the mixed Weyl points Niu _et
al._ (2019). Overall $-$ excluding the region of the bulk gap, marked with a
shaded area in Fig. 3(b,c), which will be considered in detail below $-$ the
computed HC appears to be extremely sensitive to the domain wall width. This
manifestly “non-topological” behavior of the HC emerges in contrast to the
expectations of the topological Hall effect in this system, insensitive to the
details of the spin distribution but determined by the overall topological
charge, with the latter remaining constant in our calculations.
The observed sensitivity of the HC to the domain wall width, i.e. to the
magnitude of the magnetization gradient in it, can be best understood by
referring to the novel phenomenon of the chiral Hall effect Lux _et al._
(2020), which has been recently shown to be prominent in chiral skyrmions with
interfacial spin-orbit coupling, and which is originated in the change of the
magnetization within the walls as given by their chirality along the line
which passes through the center of the skyrmion. Within the theory of the
chiral Hall effect, emerging already for spin-spiral solutions Lux _et al._
(2020), the chiral signal is proportional to the sense of the spin chirality
among the neighboring spins $\mathbf{S}_{i}\times\mathbf{S}_{j}$, and thus the
change in sign of the Hall conductivity upon changing the sign of the spin
chirality serves as one of the trademarks of the chiral Hall effect Kipp _et
al._ (2020). In Fig. 3(d) we present the results of the HC calculations as a
function of the domain wall width, but assuming that the domain wall has an
opposite sense of spin rotation from that shown in Fig. 3(a). While the
topological charge of the skyrmion does not depend on the chosen chirality,
the sign of the HC is opposite for two opposite chiralities for almost all
values of $a$ and majority of energies. This underpins the origin of the
observed HC in the chiral Hall effect originated in the domain walls of
skyrmions. A perfect reversal of the HC at a given energy with chirality (i.e.
reversal in sign but not in magnitude) is not expected given very strong spin-
orbit interaction strength which makes the electronic structure of two flavors
of skyrmions different.
On the other hand, the overall value of the orbital magnetization is
remarkably insensitive to the domain wall width. While the emergence of the
chiral orbital magnetization, arising in analogy to the chiral Hall effect,
would be also expected for interfacial skyrmions Lux _et al._ (2020), for our
particular choice of the model and symmetric parametrization of the skyrmion
profile the chiral part of the OM vanishes from symmetry arguments, thus
reducing to the well-known effect of topological orbital magnetization
Hoffmann _et al._ (2015); Hanke _et al._ (2016); dos Santos Dias _et al._
(2016); Hanke _et al._ (2017b); Lux _et al._ (2018). The topological OM
remains extremely robust with respect to the changes in the details of spin
distribution, as visible in Fig. 3(c), and its behavior is manifestly not very
sensitive to the change in the spin chirality of the skyrmion (not shown).
Figure 3: Transport properties of a skyrmion lattice of a mixed Weyl semimetal
with $R=R_{max}/2$. (a) The real-space cut through the skyrmion profile along
the path connecting two skyrmion centers for different values of parameter $a$
(see text for more details). (b,c) The dependence of the Hall conductance
$\sigma_{xy}$ (b), and orbital magnetization $M$ (c), on the band filling and
parameter $a$. The inset in (b) zooms into the region of quantization where
MTRS reside. (d) Displays the HC of the skyrmions with profile shown in (a)
but with reversed sense of the spin rotation in the walls (for impression see
the sketch shown as an inset). In all plots the color indicates the value of
$a$ in accordance to the color scale shown in (a).
In the following, we focus on the region in energy which corresponds to the
position of the bulk gap of the model for the out-of-plane magnetization,
marked with a shaded area in Fig. 3(b-c), and zoomed into in the inset of Fig.
3(b). The behavior of the HC, shown in the inset, displays isolated spikes on
the background of a plateau value of $-2$ $\frac{e^{2}}{h}$. The latter value
can be traced back to the quantized value of the HC for the out-of-plane
ferromagnetic model, see Fig. 1(e), as for the given value of $R$ the overall
out-of-plane magnetization in the unit cell is positive. On the other hand,
the spikes at isolated positions correspond to the contributions of ring
states to the HC. These contributions become somewhat more pronounced as the
width of the domain wall increases, owing to the increased spread of the ring
states and increased probability of electron hopping among the ring states
positioned in different unit cells. In order to investigate this effect in
more detail, we study the dependence of the HC and OM on the radius of the
skyrmion $R$, while assuming a very large value of parameter $a$, thus
localizing the ring states in a narrow region of the domain wall, see sketches
in the right column of Fig. 5. The overall qualitative behavior of the HC and
OM with $R$, discussed below, does not qualitatively depend on the value of
parameter $a$ chosen.
Figure 4: Transport properties of a skyrmion lattice of a mixed Weyl semimetal
in the region of the bulk gap as a function of relative radius $R/R_{max}$ and
a value of parameter $a$ of 105. (a) The energy dependence of the orbital
magnetization as a function of $R/R_{max}$. The color of the line corresponds
to the magnitude of the relative radius according to the color bar. (b) The
energy dependence of the Hall conductance as a function of $R/R_{max}$. The HC
of the systems with $R/R_{max}\leq 0.7$ are plotted with a solid line “$-$”,
with $R/R_{max}=0.8$ is plotted with a dashed line “$---$”, with
$R/R_{max}=0.9$ is plotted with a dashed-dotted line “$-\cdot-$”, and with
$R/R_{max}=1$ is plotted with a dotted line “$\cdot\cdot\cdot$”. Figure 5:
The evolution of the bandstructure around the Fermi energy in the region of
MTRS (in green, scale on the left) and of the orbital magnetization $M$ (in
blue, scale on top, as a function of band filling) with respect to
$R/R_{max}$. The positions of the jumps in the saw-tooth pattern of the
orbital magnetization coincide with the positions of the MTRSs. The sketches
on the left mark the division of the unit cell into regions with roughly
magnetization up (tainted in blue), and roughly magnetization down (tainted in
red), with $R/R_{max}=0.3$ in (d), in $R/R_{max}=0.8$ in (e), and
$R/R_{max}=1$ in (f).
We first discuss the evolution of the HC with the relative radius $R/R_{max}$,
presented in Fig. 4(b). Over a very large range of radii $R<0.8R_{max}$ the
conductance is ideally quantized to $-2$ $\frac{e^{2}}{h}$ in the gap, despite
a large number of MTRS, see Fig. 5(a-b). In this case the Hall effect is
dominated by the ferromagnetic background in between the skyrmions with
$\hat{\mathbf{m}}$ along $+z$ which provides a Chern number of $-2$. The ring
states are strongly localized in the wall region (which explains the flat
dispersion of the ring states visible in Fig. 5(a)), do not hybridize with
each other across the unit cell boundaries, and do not contribute to the HC.
In the other limit of $R=R_{max}$, see Fig. 5(f) for a sketch, the film
comprises mainly areas of the texture with $\hat{\mathbf{m}}$ along $-z$,
which explains mostly quantized value of the HC of $+2$ $\frac{e^{2}}{h}$.
The crossover point of the HC from $\sigma_{xy}=-2\,\frac{e^{2}}{h}$ to
$\sigma_{xy}=2\,\frac{e^{2}}{h}$ positioned between $R/R_{\text{max}}=0.7$ and
$R/R_{\text{max}}=0.8$ coincides with the point where the integrated over the
unit cell out-of-plane component of the magnetization changes sign as a result
of the competition between domains with $\hat{\mathbf{m}}$ along $+z$ and
domains with $\hat{\mathbf{m}}$ along $-z$. The region of $0.8R_{max}\leq
R\leq R_{max}$ is the region of where the MTRS interact with each other
strongly across the unit cell boundaries, exhibit a strong dispersion (see
Fig. 5(b)) and give rise to a drastic variation in the HC. In the limit of
$R=R_{max}$ the MTRS are localized in the unit cell corners, thus interacting
noticeably along the unit cell boundaries, which gives rise to a small but
finite dispersion of the bands (see Fig. 5(c)) and peak-like contributions to
the HC.
We come now to the discussion of the OM of the skyrmion lattice exhibited in
response to the variation of the radius. This behavior, presented in Fig.
4(a), is distinctly different from that exhibited by the Hall conductance.
Clearly, in the limiting cases of $R=0$ and $R=R_{max}$ the slope of the OM as
a function of energy in the gap has to be opposite, as follows from Eq. (6),
however, while in the case of the HC the transition between the two Chern
insulator cases is relatively abrupt, the same transition in terms of the OM
seems to be quite smooth, with the overall slope of the OM changing
continuously as a function of $R$, and the OM curves being quite smooth as a
function of energy, when compared to the case of the HC. This can be
understood by looking in detail at the contributions of the MTRS to the OM
variation in the gap, Fig. 5(a-c). Here, we observe that although the slope of
the OM in between the groups of MTRS is strictly constant and consistent with
the sign of the quantized value of the HC in the same region, Fig. 5(b), the
overall curve of the OM for all values of $R$ displays a characteristic saw-
tooth shape, where the overall slope is “renormalized” by the contributions
from MTRS. Interestingly, these contributions are consistent in sign among all
MTRS for a given $R$: they are positive in the region of $R<0.8R_{max}$, and
negative for $R\geq 0.8R_{max}$. This “coherence” of MTRS is the reason that
the slope of OM is getting consistenly and gradually increased as $R$ goes
from zero to $R_{max}$.
According to our calculations, the OM of MTRS is predominantly originated in
the local circulation part, which is consistent with the strong localization
of the ring states. In a sense, the ring states can be envisaged as a group of
localized electronic states which have a common unique sense of bound orbital
current circulation around the skyrmion center $-$ which we can naturally
refer to as having a specific sense of orbital chirality. And while in the
case of the Hall effect it is the Hall conductance which changes abruptly at
the point of phase transition between two areas of out-of-plane magnetized
domains, the corresponding marker in terms of orbital magnetism is the orbital
chirality of the ring states which exhibits a sharp transition. Manifestly,
as follows from our calculations of the OM with reversed spin chirality of
skyrmions, while the number and exact energetic positions of the states depend
on the width of the domain wall for both spin chiralities, the orbital
chirality of MTRS is remains a robust quantity even under the change in the
skyrmion chirality.
## V Discussion
In our work, we have considered the intrinsic Hall effect and orbital
magnetism exhibited by skyrmion lattices of mixed Weyl semimetals by referring
to the Berry phase framework. Our main finding is the discovery of the special
type of ring states which form as a result of complex mixed topology in real-
space textures of these materials. These electronic states reside at the
domain wall region, which serves as the boundary between the two Chern
insulator phases and separates the skyrmion cores from the ferromagnetic
background. We have shown that the degree of the localization of the ring
states and the strength of their inter-cell interaction can be controlled by
the parameters which determine the texture details. In turn, the ring states
mediate the transition between the two Chern insulating phases occurring upon
switching of the magnetization direction from out-of-plane to the opposite,
when e.g. an external magnetic field changes its strength and sign in
experiment. This concerns the Hall conductance, but mostly the orbital
magnetization of the samples, as the ring states carry a specific orbital
chirality and orbital momentum.
The scaling of the Hall conductance as a function of the skyrmions domain wall
width is a strong indicator of a prominent chiral Hall effect, where such a
behavior has recently been predicted and which could have possible
ramifications for the unambiguous electrical detection of magnetic skyrmions
Lux _et al._ (2020). In simplified model systems, the emergence of this
effect can be understood from the interplay of spin-orbit interaction and
exchange coupling which gives rise to an effective magnetic field Nakabayashi
and Tatara (2014). This phenomenology is similar to the topological Hall
effect (THE) Bruno _et al._ (2004), but in contrast, it materializes already
at the leading order perturbation theory whereas the THE is subleading Lux
_et al._ (2020). From the viewpoint of Weyl semimetals, the coaction of chiral
magnetism and spin-orbit coupling (which is responsible for the chiral Hall
effect) is directly linked to an emergent chiral anomaly Ilan _et al._
(2020). Our model elevates this phenomenological interpretation to a more
realistic setting. To which degree this explanation can be uphold will be the
subject of future investigations.
Ultimately, the ring states arise as a result of a strong variation in the
electronic structure around a specific magnetization direction, and thus it is
rewarding to explore in the future what impact can the ring states have on the
local real-space variation of the spin-orbit torques and the Dzyaloshinskii-
Moriya interaction, as well as current-induced skyrmion dynamics Hanke _et
al._ (2017a); Niu _et al._ (2019); Araki (2020); Hanke _et al._ (2020); Liu
_et al._ (2013); Kurebayashi and Nomura (2019); Kurebayashi and Nagaosa
(2019), where the ring states can play a role of local “hooks” strongly
coupling skyrmions to applied currents. On the other hand, the unique orbital
properties of the ring states lend themselves as a possible platform for
obtaining a detailed information on the skyrmion parameters with e.g. XMCD
type of techniques. Additionally, peculiar topological nature of these one-
dimensional edge states emerging in skyrmions of Weyl semimetals hints at
exceptional possibilities that the ring states can play in mediating the
physics of Majorana states emerging upon deposition of skyrmions on
superconductors Pershoguba _et al._ (2016); Güngördü _et al._ (2018);
Palacio-Morales _et al._ (2019); Yang _et al._ (2016); Rex _et al._ (2019);
Palacio-Morales _et al._ (2019), which presents an exciting future direction
to pursue.
## VI Acknowledgements
We gratefully acknowledge computing time on the supercomputers JUQUEEN and
JURECA at Jülich Supercomputing Center, and at the JARA-HPC cluster of RWTH
Aachen. We acknowledge funding under SPP 2137 “Skyrmionics” of Deutsche
Forschungsgemeinschaft (DFG), DARPA TEE program through grant MIPR#
HR0011831554 from DOI. We gratefully acknowledge financial support from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant No. 856538, project ”3D MAGiC”).
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|
2024-09-04T02:54:57.924517 | 2020-02-18T01:54:36 | 2003.03259 | {
"authors": "Mingxiao Li, Jingwei Ling, Yang He, Usman A. Javid, Shixin Xue and\n Qiang Lin",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26086",
"submitter": "Mingxiao Li",
"url": "https://arxiv.org/abs/2003.03259"
} | arxiv-papers | # Lithium niobate photonic-crystal electro-optic modulator
Mingxiao Li Department of Electrical and Computer Engineering, University of
Rochester, Rochester, NY 14627 Jingwei Ling Department of Electrical and
Computer Engineering, University of Rochester, Rochester, NY 14627 Yang He
Department of Electrical and Computer Engineering, University of Rochester,
Rochester, NY 14627 Usman A. Javid Institute of Optics, University of
Rochester, Rochester, NY 14627 Shixin Xue Institute of Optics, University of
Rochester, Rochester, NY 14627 Qiang Lin<EMAIL_ADDRESS>Department
of Electrical and Computer Engineering, University of Rochester, Rochester, NY
14627 Institute of Optics, University of Rochester, Rochester, NY 14627
###### Abstract
Modern advanced photonic integrated circuits require dense integration of
high-speed electro-optic functional elements on a compact chip that consumes
only moderate power. Energy efficiency, operation speed, and device dimension
are thus crucial metrics underlying almost all current developments of
photonic signal processing units. Recently, thin-film lithium niobate (LN)
emerges as a promising platform for photonic integrated circuits. Here we make
an important step towards miniaturizing functional components on this
platform, reporting probably the smallest high-speed LN electro-optic
modulators, based upon photonic crystal nanobeam resonators. The devices
exhibit a significant tuning efficiency up to 1.98 GHz/V, a broad modulation
bandwidth of 17.5 GHz, while with a tiny electro-optic modal volume of only
0.58 $\mu{\rm m}^{3}$. The modulators enable efficient electro-optic driving
of high-Q photonic cavity modes in both adiabatic and non-adiabatic regimes,
and allow us to achieve electro-optic switching at 11 Gb/s with a bit-
switching energy as low as 22 fJ. The demonstration of energy efficient and
high-speed electro-optic modulation at the wavelength scale paves a crucial
foundation for realizing large-scale LN photonic integrated circuits that are
of immense importance for broad applications in data communication, microwave
photonics, and quantum photonics.
## Introduction
High-speed electro-optic modulation underlies many important applications
ranging from optical communication Wooten00 , microwave photonics Capmany19 ,
computing Vladimir15 , frequency metrology Diddams18 , to quantum photonics
Smith17 . A variety of approaches have been employed for electro-optic
modulation, such as carrier plasma dispersion Reed10 ; Keeler18 , electro-
absorption Nakano18 ; Michel08 , and Pockels effect Wooten00 ; Koos15 , the
latter of which is particularly interesting since the Pockels effect offers an
ultrafast and pure refractive-index modulation over an extremely broad optical
spectrum while without introducing extra loss. The best-known electro-optic
Pockels material is probably lithium niobate (LN), which has been widely used
in telecommunication Wooten00 . Recently, thin-film monolithic LN Gunter12 ;
Bowers18 emerges as a promising platform, where low-loss and high-quality
photonic integration together with the strong Pockels effect enables superior
modulation performance Gunter07 ; Reano13 ; Reano14 ; Fathpour15 ; Fathpour16
; Amir17 ; Loncar18 ; Prather18 ; Loncar18_2 ; Shayan18 ; Fathpour18 ; Cai19 ;
Cai19_2 ; Loncar19 , showing great potential as an excellent medium for
photonic integrated circuits and future photonic interconnect.
Figure 1: Design of LN photonic crystal EOM. (a) Schematic of the LN photonic
crystal EOM. (b) The structure of the unit cell (top: top view; bottom: cross-
section view). The LN photonic crystal nanobeam has a width of $w=1200$ nm,
layer thickness of $t=300$ nm, and a partially etched wing layer with a
thickness of 150 nm. The elliptical hole has dimensions of $h_{x}=270$ nm and
$h_{y}=490$ nm, and a fully etched depth of 300 nm. The full cross section is
shown in (d). (c) Dispersion property of the partially etched LN photonic
crystal nanobeam, simulated by the finite element method (FEM). The blue open
circles show the dielectric and air bands. The red solid and open circles
denote the fundamental and second-order TE-like cavity modes shown in (f) and
(g). Our simulations show that there exhibits another mode with eigenfrequency
within the band gap (gray open circles). This mode, however, has only
negligible perturbation to the dielectric mode due to distinctive spatial
symmetry, thus not affecting the quality of the defect cavity mode. (d) Cross-
section schematic of the EOM structure, where the arrow profile shows the RF
electric field distribution and the color profile shows the optical cavity
mode field distribution, both simulated by the FEM method. (e) Lattice
constant as a function of position, which is optimized for low insertion loss
together with high radiation-limited optical Q. (f) Top view of the FEM-
simulated optical mode field profile of the fundamental TE-like cavity mode
$TE^{0}_{01}$. The left inset shows the orientation of the LN crystal where
the optical axis is along the $z$ direction. (e) Simulated optical mode field
profile of the second-order TE-like cavity mode $TE^{1}_{01}$.
Power efficiency is crucial for the application of electro-optic modulator
(EOM), which depends sensitively on the physical size of the device Miller17 .
Scaling an EOM down to a small footprint would reduce the device capacitance
and thus decrease the switching energy Miller17 ; Sorger15 , which is
indispensable for all practical applications. Among various device geometries,
photonic crystal nanoresonators are particularly beneficial in this regard,
given their exceptional capability of controlling light confinement and light-
matter interactions on the sub-wavelength scale. In the past decade, photonic-
crystal EOMs have been developed on various material platforms such as silicon
Notomi09 ; Baba12 ; Notomi14 , GaAs Jelena11 , InP Notomi19 , polymers
Krauss09 ; Chen16 , ITO AlanX19 , etc. For lithium niobate, however, the EOMs
developed so far Wooten00 ; Gunter07 ; Reano13 ; Reano14 ; Fathpour15 ;
Fathpour16 ; Amir17 ; Loncar18 ; Prather18 ; Loncar18_2 ; Shayan18 ;
Fathpour18 ; Cai19 ; Cai19_2 ; Loncar19 generally exhibit significant
dimensions, leading to significant power required to drive the EOMs. Although
attempts have been made to explore the electro-optic effect in LN photonic
crystals Bernal12 ; Bernal12_3 ; Bernal14 , the low device quality and poor
optoelectronic integration unfortunately limit seriously the operation speed.
To date, it remains an open challenge in realizing a high-speed and energy-
efficient modulator at the wavelength scale on the monolithic LN platform.
Figure 2: Scanning electron microscopic (SEM) image of a fabricated EOM
device. (a) Full SEM image of the whole device structure. The region
highlighted in red is the electrode used to drive the photonic crystal
nanoresonator. That highlighted in blue indicates the large metal pad used for
contacting the RF probe. The green region indicates the electrode that can be
shrunk to in the future design. (b) Zoom-in image of the photonic crystal
resonator and electrodes, corresponding to the dashed rectangular region in
(a). (c) Further zoom-in image showing the detailed structure of the photonic
crystal defects cavity, corresponding to the dashed rectangular region in (b).
Here we report high-speed and energy-efficient LN photonic crystal EOMs, which
exhibits a tiny electro-optic modal volume of only $\sim$ 0.58 $\mu{\rm
m}^{3}$, the smallest among all high-speed LN EOMs ever reported Wooten00 ;
Gunter07 ; Reano13 ; Reano14 ; Fathpour15 ; Fathpour16 ; Amir17 ; Loncar18 ;
Prather18 ; Loncar18_2 ; Shayan18 ; Fathpour18 ; Cai19 ; Cai19_2 ; Loncar19 .
The sub-wavelength-scale EOM cavity enables compact optoelectronic integration
to achieve not only a high electro-optic tuning efficiency up to 16 pm/V
(corresponding to 1.98 GHz/V) that is significantly beyond other LN EOM
resonators Gunter07 ; Reano13 ; Reano14 ; Fathpour15 ; Amir17 ; Loncar18 ;
Fathpour18 ; Loncar19 , but also a large modulation bandwidth up to 17.5 GHz
that reaches the photon-lifetime limit of the EOM cavity. The fully on-chip
design achieves a full-swing extinction ratio of 11.5 dB. With these devices,
we are able to realize efficient driving of the optical mode in both adiabatic
sideband-unresolved and non-adiabatic sideband-resolved regimes, and to
observe the transition in between. As an example application, we demonstrate
electro-optic switching of non-return-to-zero signal at a rate of 11 Gb/s,
with a switching energy as low as 22 fJ/bit that is more than one order of
magnitude smaller than other LN EOMs Wooten00 ; Gunter07 ; Reano13 ; Reano14 ;
Fathpour15 ; Fathpour16 ; Amir17 ; Loncar18 ; Prather18 ; Loncar18_2 ;
Shayan18 ; Fathpour18 ; Cai19 ; Cai19_2 ; Loncar19 . To the best of our
knowledge, this is the smallest LN EOM ever demonstrated with combined high
speed and energy efficient operation.
## Device design and fabrication
Recently, there have been significant advance in high-Q LN photonic-crystal
nanoresonators Liang17 ; Li19 ; Amir19 ; Mingxiao192 , which led to the
demonstration of intriguing phenomena and functionalities such as
photorefraction quenching Liang17 , harmonic generation Li19 , piezo-
optomechanics Amir19 , and all-optical resonance tuning Mingxiao192 .
For EOM, We adopt one-dimensional photonic-crystal nanobeam as the basic
underlying structure (Fig. 1(a)) since it supports compact optical and
electrical integration to enhance the electro-optic response. Due to the high
permittivity of LN at radio frequency, the commonly used full surrounding air
cladding Liang17 ; Mingxiao192 ; Amir19 is not suitable for EOM since it
would significantly reduce the coupling between the optical and electric
fields. To maximize the electro-optic interaction, we utilize a partially
etched structure with a rib-waveguide-like cross section, leaving a 150-nm-
thick wing layer for the electrodes to sit on (Fig. 1(a) and (d)). Although
the breaking of the mirror symmetry along the normal direction of the device
plane considerably alters the band gap of the photonic crystal (Fig. 1(c)),
optimization of the photonic potential via an appropriate pattern of lattice
constant (Fig. 1(e)) is still able to produce a well-confined point-defect
cavity, with a simulated optical Q of $\sim 10^{5}$ for the fundamental
transverse-electric-like (TE-like) cavity mode, $TE^{0}_{01}$, shown in Fig.
1(f). The cavity mode exhibits an extremely small electro-optic modal volume
of $1.52(\lambda/n)^{3}\sim 0.58~{}\mu{\rm m}^{3}$ (where $n$ is the
refractive index of LN), which is the smallest among all LN EOMs ever reported
Wooten00 ; Gunter07 ; Reano13 ; Reano14 ; Fathpour15 ; Fathpour16 ; Amir17 ;
Loncar18 ; Prather18 ; Loncar18_2 ; Shayan18 ; Fathpour18 ; Cai19 ; Cai19_2 ;
Loncar19 , to the best of our knowledge.
The photonic crystal cavity is oriented such that the dominant optical field
is in parallel with the optical axis of underlying LN medium (Fig. 1(f)), so
as to take advantage of the largest electro-optic component $r_{33}$ of
lithium niobate. The electrodes are designed to be placed close to the
photonic-crystal resonator (Fig. 1(d)) to maximize the in-plane electric field
${E_{z}}$, while preventing potential loss induced by metal absorption. The
compact device structure design results in a significant electro-optic tuning
efficiency of 1.81 GHz/V, simulated by the finite element method. The
electrodes are designed to have a length of 30 $\mu$m to ensure a full
coverage of the applied electric field over the entire photonic crystal
structure. Numerical simulations show that the device exhibits a small
capacitance C of $C$ = $\sim$22 fF, which is more than one order of magnitude
smaller than other LN EOMs Wooten00 ; Gunter07 ; Reano13 ; Reano14 ;
Fathpour15 ; Fathpour16 ; Amir17 ; Loncar18 ; Prather18 ; Loncar18_2 ;
Shayan18 ; Fathpour18 ; Cai19 ; Cai19_2 ; Loncar19 . Therefore, we expect our
devices to have much higher energy efficiency, as will be shown in the
following sections.
Figure 3: Experimental testing setup. Light is coupled into and out of the
EOM chip via one lensed fiber. The inset shows an optical microscopic image of
an EOM with the RF probe in contact. The equipment in the highlighted dashed
box is used for characterizing the performance of electro-optic modulation.
VOA: variable optical attenuator; MZI: Mach-Zehnder interferometer; EDFA:
erbium doped fiber amplifier; BPF: band pass filter; MNA: microwave network
analyzer; PRBS: pseudo-random binary sequence source
For simplicity of testing, the EOM is designed such that light is coupled into
and out of the EOM via only one side of the cavity (Fig. 1(a)). As such, the
photonic-crystal mirror on the right side of the defect cavity is designed to
be of 100% reflection, while that on the left side has decreased number of
holes (Fig. 1(e)) to enable a partial reflection/transmission, with the hole
number optimized for a critical coupling to the cavity. To support on-chip
integration, light is coupled to the EOM cavity via an on-chip waveguide (Fig.
1(a)), where an injector section (Fig. 1(e)), with the lattice constant
varying from 450 to 550 nm, is designed and placed in front of the left mirror
to reduce the coupling loss.
The devices were fabricated on a $300$-nm-thick x-cut single-crystalline LN
thin film bonded on a 3-${\mu}$m silicon dioxide layer sitting on a silicon
substrate (from NanoLN). The photonic crystal hole structure was patterned
with ZEP-$520$A positive resist via electron-beam lithography, which was then
transferred to the LN layer with an Ar+ plasma milling process to etch down
the full 300-nm depth. The resist residue was removed by a further O+ plasma
etching. A second exposure is then performed to define the waveguide
structure, which is partially etched by 150 nm with the same process. After
the residue removal, we used diluted hydrofluoric acid to undercut the buried
oxide layer to form a suspended photonic crystal membrane structure [Fig.
1(d)]. The metal electrode layer (10 nm Ti/500 nm Au) was deposited by an
electron-beam evaporator and the electrode structure was formed by a lift-off
process via ZEP-$520$A.
Figure 2 shows a fabricated device. The large metal pads (highlighted in blue
box) are used simply as the contacts for the air-coplanar probe (Formfactor
Acp65-A-GSG-100) for applying the RF driving signal (see also the inset of
Fig. 3). The impedance of the metallic structure is optimized to minimize the
coupling loss of the RF signal from the pads to the device. The high quality
of device fabrication as indicated by the device images implies high
performance of the EOM, as we will show below.
## Device characterization and electro-optic properties
Figure 4: Linear optical property of a fabricated LN photonic crystal EOM.
(a) Laser-scanned transmission spectrum in the telecom band. (b) Detailed
transmission spectrum of the fundamental TE-like cavity mode $TE^{0}_{01}$ at
wavelength of 1554.47 nm, with the experimental data shown in blue and the
theoretical fitting shown in red. Figure 5: Electro-optic tuning property of
an LN photonic crystal EOM. (a) Recorded transmission spectrum of the EOM
cavity as a function of applied DC voltage from 0 to 4.5 V, with a voltage
step of 0.5 V. (b) Recorded resonance shift as a function of applied DC
voltage, where the experimental data are shown in black dots and the blue line
is a linear fitting to the data.
To characterize the optical and electro-optic properties of the devices, a
continuous-wave tunable laser (Santec TSL-510) was launched onto the chip via
a lensed fiber. The light reflected from the EOM was collected by the same
lensed fiber, routed by a circulator, and then delivered to a photodiode for
detection. Figure 3 illustrates the schematic of the experimental testing
setup, where the inset shows an optical image of the device with the RF probe
in contact. The insertion loss from the on-chip coupling waveguide to the
photonic-crystal cavity is measured to be around 2.2 dB, calibrated by
subtracting the coupling loss from the facet and circulator transmission loss.
To characterize the performance of high-speed modulation, the majority of the
modulated light output was amplified by an erbium-doped fiber amplifier to
boost the power, passed through a bandpass filter to remove the amplifier
noise, and was then detected by a high-speed detector (New Focus 1024). The
detector output was recorded either by a microwave network analyzer (Keysight
N5235B) for characterizing the modulation bandwidth or by a sampling
oscilloscope module (Keysight 54754A) to record the eye diagram of the
switching signal.
Figure 4(a) shows the transmission spectrum of an EOM when the laser is
scanned in the telecom band. The device exhibits a resonance at 1554.47 nm,
which corresponds to the fundamental TE-like cavity mode $TE^{0}_{01}$ (Fig.
1(f)). As shown in Fig. 4(b), the $TE^{0}_{01}$ mode exhibits a high loaded
optical Q of $1.34\times 10^{5}$, which is very close to our numerical
simulation, indicating the negligible impact of the electrodes on the optical
quality. The cavity resonance exhibits a coupling depth of 93%, corresponding
to a full-swing extinction ratio of 11.5 dB. This value can be improved in the
future by further optimizing the partially reflective photonic-crystal mirror
(Fig. 1(e)). The device also exhibits a second-order TE-like cavity mode
$TE^{1}_{01}$ (Fig. 1(g)) at 1604.13 nm (not shown) with a loaded optical Q of
$3.03\times 10^{4}$.
To show the electro-optic tuning property, we applied a DC voltage to the chip
and monitored the cavity transmission spectrum of the $TE^{0}_{01}$ mode. As
shown in Fig. 5(a), the cavity resonance tunes smoothly with the applied
voltage, without any degradation to the lineshape or coupling depth, clearly
showing the pure dispersive electro-optic tuning as expected from the Pockels
effect. We have applied a voltage of 25 Volts to the device (not shown in the
figure) and did not observe any degradation. Figure 5(b) shows a clear linear
dependence of the induced resonance wavelength shift on the applied voltage,
from which we obtained a tuning slope of 16 pm/V (corresponding to a frequency
tuning slope of 1.98 GHz/V), close to our design. This value is significantly
larger than those in other LN EOM resonators Gunter07 ; Reano13 ; Reano14 ;
Fathpour15 ; Amir17 ; Loncar18 ; Fathpour18 ; Loncar19 , which is primarily
benefited from the strong optical field confinement, large optical and
electric field overlap, and the resulting compact optical and electric
integration offered by our devices.
## Electro-optic modulation
Figure 6: Electro-optic modulation of a high-Q optical cavity resonance. (a)
Recorded transmission spectra of the $TE^{0}_{01}$ cavity mode with RF driving
signal at 7 different powers from 0 to 12 mW, with a power step of 2 mW,
modulated at 0.6 GHz. (b) Same as (a) but with a modulation frequency of 2.0
GHz. (c) Detailed spectrum with RF driving signal at 2.0 GHz with power of 16
mW. The gray lines are in Lorentzian shapes and the sum of all gray lines are
showed in red lines fitted by the theory. The dashed lines note the spacing of
the sidebands. (d) Recorded transmission spectra at different RF modulation
frequency from 0.4 to 3.0 GHz, with a frequency step of 0.2 GHz. The RF
driving power is 16 mW.
The high efficiency of electro-optic tuning together with the high optical
quality of the EOM resonator enables efficient electrical driving of the
optical mode into different dynamic regimes. To show this phenomenon, we
applied a sinusoidal RF signal at a certain frequency to the EOM and monitored
the transmission spectrum of the device by scanning laser back and forth
across the cavity resonance. The laser wavelength is scanned at a repetition
rate of $\sim$15 Hz, so we primarily monitored the time-averaged cavity
transmission.
When the EOM is driven at a modulation frequency of 600 MHz much smaller than
the cavity linewidth of 1.4 GHz, increasing the driving power simply broadens
the transmission spectrum into one with two shallow side lobes, as shown in
Fig. 6(a), with a broadened spectral linewidth dependent on the driving power.
This is a typical signature of resonance modulation in the sideband-unresolved
regime, where the cavity resonance follows adiabatically the electric driving
signal in a sinusoidal fashion, resulting in a broadened average transmission
spectrum as shown in Fig. 6(a).
When the modulation frequency is increased to 2.0 GHz greater than the cavity
linewidth, the cavity is too slow to follow the electro-optic modulation,
which results in the frequency conversion of photons into sidebands with
frequency separation equal to the modulation frequency. Consequently, the
transmission spectrum transforms into a multi-resonance spectrum, as shown in
Fig. 6(b). Increasing the electrical driving power now does not perturb the
positions of the resonance dips, but rather changes their relative magnitudes
since the magnitudes of the created sidebands depends on the driving amplitude
Usman19 . This phenomenon is shown more clearly in Fig. 6(c), where a driving
power of 16 mW (corresponding peak-to-peak driving voltage, ${\rm V_{pp}}$, of
${\rm V_{pp}}=2.5$ V) splits the cavity resonance into five with notable
magnitudes (black curve), resulting in a cavity transmission with five side
lobes (blue curve).
Electro-optic modulation enables arbitrary modulation of cavity resonance
within the bandwidth allowed by the driving circuit. This is in strong
contrast to piezoelectric acoustic modulation which is confined to the
vicinity of mechanical resonance frequency Amir19 ; Loncar196 ; Piazza19 .
Such flexibility allows us to observe direct transition between the adiabatic
driving regime and the non-adiabatic regime simply by continuously sweeping
the modulation frequency to across the cavity linewidth. Figure 6(d) shows an
example. When the modulation frequency is below 1.0 GHz, The transmission
spectrum remains fairly similar regardless of modulation frequency, as
expected from the adiabatic driving discussed above. However, when the
modulation frequency is tuned above 1.0 GHz towards the cavity linewidth, the
two side lobes moves towards each other and the spectral shape is considerably
distorted, until around 1.8 GHz where the transmission spectrum splits into
three lobes, with the two side lobes located about 1.8 GHz from the center.
Further increase of the modulation frequency shifts apart the two side lobes
accordingly, with amplitude decreased, while the position of the center lobe
remains unchanged, as expected from the non-adiabatic driving. The flexible
electro-optic modulation shown here may offer a convenient method for
controlling the spectrotemporal properties of photons inside the cavity and
for creating exotic quantum states Usman19 that are crucial for quantum
photonic applications.
Figure 7: High-speed electro-optic switching. (a) Measured scattering
parameter $S_{21}$ for one device with Q around 14000 in blue and another with
Q around 20000 in orange. The gray dashed line notes the 3 dB cutoff frequency
with the gray region represents the bandwidth limit for two device
respectively. The inset shows the $S_{11}$ reflection scattering parameter for
both devices. (b) and (c) Eye diagrams of the photonic crystal EOM, measured
with $2^{7}$-1 NRZ PRBS with a driving voltage of Vpp=2V.
## Electro-optic switching
The electro-optic modulation demonstrated in the previous section indicates
the potential high-speed operation of the EOMs. To show this feature, we
selected another similar device on the same chip, which has a lower loaded
optical Q of 14000. Figure 7(a) shows the electro-optic modulation response of
the device (blue curve), which exhibits a 3-dB modulation bandwidth up to
around 17.5 GHz. This value primarily reaches the photon-lifetime limit of the
EOM cavity ($\sim$11 ps), as the electrode circuit has much broader spectral
response as indicated by the flat S11 reflection spectrum shown in the inset
of Fig. 7(a). As the modulation bandwidth is primarily related to the optical
Q of the device, it can be engineered flexibly for different application
purposes, simply by choosing device with appropriate optical Q. The red curve
in Fig. 7(a) shows another example of a device with optical Q of 20000, which
exhibits a 3-dB bandwidth of about 12.5 GHz.
The broad modulation bandwidth of these devices would thus enable high-speed
electro-optic switching. As an example application, we applied non-return-to-
zero (NRZ) signal with a ($2^{7}$-1)-bit pseudo-random binary sequence (PRBS)
to an EOM with a ${\rm V_{pp}}$ of 2.0 Volts. Figure 7(b) and (c) show the
recorded eye diagrams at two different bit rates of 9 and 11 Gb/s,
respectively, which show clear open eyes. The demonstrated bit rate is
currently limited by our PRBS generator (Agilent 70843B) which has a maximum
bit rate of 12 Gb/s. However, negligible degradation observed between Fig.
7(b) and (c) implies that the EOM could operate at higher bit rates, which
will left for future demonstration. The bit switching energy for NRZ signal is
given by $\frac{1}{4}CV_{\rm pp}^{2}$ Miller17 , which is about 22 fJ/bit in
our EOM. This value is the smallest switching energy ever reported for LN EOMs
Wooten00 ; Gunter07 ; Reano13 ; Reano14 ; Fathpour15 ; Fathpour16 ; Amir17 ;
Loncar18 ; Prather18 ; Loncar18_2 ; Shayan18 ; Fathpour18 ; Cai19 ; Cai19_2 ;
Loncar19 , clearly showing the high energy efficiency of our devices.
## Discussion and conclusion
The energy efficiency of the LN photonic crystal EOM can be further improved
since our current devices are not optimized. For example, the capacitance of
our device can be significantly decreased since the majority of the metallic
parts in the current devices are used for coupling the RF driving signal,
which can be removed in a future on-chip integration design. The 50-$\mu$m
width of the electrode (Fig. 2, red box) is used primarily for impedance
matching to the large metal pad for probe contact, which can be decreased to 3
$\mu$m for a fully on-chip operation Notomi19 . On the other hand, the
30-$\mu$m length of the electrode is overly conservative since it covers the
full length of photonic crystal structure including the injector, mirrors, and
the cavity (Fig. 1(e) and Fig. 2). Essentially, only the 10-$\mu$m long point-
defect cavity requires electric driving to achieve electro-optic modulation.
Therefore, the electrodes can be shrunk to $10\times 3~{}{\mu{\rm m}}^{2}$,
which would reduce the capacitance considerably to $\sim$0.27 fF ($\sim$1.0 fF
if including the integrated wires Notomi19 ), according to our FEM
simulations. On the other hand, the electrodes are currently placed far from
the photonic crystal cavity so as to leave the optical mode intact to achieve
a high optical Q. For the application of high-speed electro-optic switching,
our simulations show that the electrode-waveguide spacing can be decreased to
1.5 $\mu$m for an optical Q of $\sim$5000 (corresponding to a modulation
bandwidth of $\sim$ 45 GHz), which will improve the modulation efficiency to
2.38 GHz/V. We expect that these optimization would significantly improve the
energy efficiency of the LN photonic crystal EOM, further decreasing the
switching energy down to sub-femtoJoule level.
In the current EOMs shown above, light is coupled into and out of the EOMs via
a same side of the cavity, which is not convenient in practice since a
circulator is required to separate the modulated light for the laser input.
This can be changed simply by engineering the photonic crystal mirror on the
other side to function as the output port.
In summary, we have demonstrated high-speed LN EOMs with a broad modulation
bandwidth of 17.5 GHz, a significant tuning efficiency up to 1.98 GHz/V, and
an electro-optic modal volume as small as 0.58 $\mu{\rm m}^{3}$. We believe
this is the first LN EOM ever reported with such combined device
characteristics and modulation performance. With these devices, we are able to
demonstrate efficient electrical driving of high-Q cavity mode in both
adiabatic and non-adiabatic regimes and to observe transition in between. We
are also able to achieve high-speed electro-optic switching of at 11 Gb/s,
with switching energy as low as 22 fJ/bit. The demonstration of energy
efficient and high-speed EOM at the wavelength scale paves an important step
for device miniaturization and high density photonic integration on the
monolithic LN platform, which is expected to find broad applications in
communication, computing, microwave signal processing, and quantum photonic
information processing.
## Funding Information
National Science Foundation (NSF) (EFMA-1641099, ECCS-1810169, and
ECCS-1842691); the Defense Threat Reduction Agency-Joint Science and
Technology Office for Chemical and Biological Defense (grant No.
HDTRA11810047).
## Acknowledgment
The authors thank Professor Hui Wu and Professor Wayne Knox for the use of
their equipment. They also thank Wuxiucheng Wang, Lejie Lu, and Ming Gong for
valuable discussions and help on testing. This work was performed in part at
the Cornell NanoScale Facility, a member of the National Nanotechnology
Coordinated Infrastructure (National Science Foundation, ECCS-1542081).
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|
2024-09-04T02:54:57.935208 | 2020-03-06T16:23:42 | 2003.03298 | {
"authors": "Shubham Gupta",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26087",
"submitter": "Shubham Gupta",
"url": "https://arxiv.org/abs/2003.03298"
} | arxiv-papers | # Certain Diophantine tuples in imaginary quadratic fields
Shubham Gupta Shubham Gupta @Harish-Chandra Research Institute, HBNI,
Chhatnag Road, Jhunsi, Allahabad 211 019, India<EMAIL_ADDRESS>
###### Abstract.
Let $K$ be an imaginary quadratic field and $\mathcal{O}_{K}$ be its ring of
integers. A set
$\\{a_{1},a_{2},\cdots,a_{m}\\}\subset\mathcal{O}_{K}\setminus\\{0\\}$ is
called a Diophantine $m$-tuple in $\mathcal{O}_{K}$ with $D(-1)$ if
$a_{i}a_{j}-1=x_{ij}^{2}$, where $x_{ij}\in\mathcal{O}_{K}$ for all $i,j$ such
that $1\leq i<j\leq m$. Here we prove the non-existence of Diophantine
$m$-tuples in $\mathcal{O}_{K}$ with $D(-1)$ for $m>36$.
###### Key words and phrases:
Diophantine tuples, Imaginary quadratic fields, Pell equation, Simultaneous
approximation.
###### 2010 Mathematics Subject Classification:
Primary: 11D09, 11R11, Secondary: 11J68.
## 1\. Introduction
A set $\\{a_{1},a_{2},\cdots,a_{m}\\}$ of $m$ positive integers is called a
Diophantine $m$-tuple with $D(n)$ if $a_{i}a_{j}+n=x_{ij}^{2}$, where
$x_{ij}\in\mathbb{Z}$ and $n\in\mathbb{Z}$, for all $1\leq i<j\leq m$.
Diophantus found a set of four positive rationals {1/16, 33/16, 17/4, 105/16}
with the above property for $n=1$. The first Diophantine $4$-tuple with
$D(1)$, namely, $\\{1,3,8,120\\}$ was found by Fermat. Baker and Davenport [2]
proved that this particular quadruple cannot be extended to a Diophantine
$5$-tuple with $D(1)$. Now on whenever we say a $m$-tuple, it would mean a
Diophantine $m$-tuple as above.
Let $\\{a,b,c\\}$ be a $3$-tuple with $D(1)$. If there exists a
$d\in\mathbb{N}$ such that $\\{a,b,c,d\\}$ is a $4$-tuple with $D(1)$, then
there exist $x,y,z$ $\in$ $\mathbb{Z}$ such that
$ad+1=x^{2},\hskip 5.69046ptbd+1=y^{2},\hskip 5.69046pt\text{and }cd+1=z^{2}.$
Hence we get an elliptic curve $E$ over $\mathbb{Q}$
$E:(xyz)^{2}=(ad+1)(bd+1)(cd+1).$
As the number of integral points on an elliptic curve over $\mathbb{Q}$ is
finite([13, page 176]) so the number of possible choices of $d$ is finite.
Over the years due to the findings of many researchers there exist many
examples of $3$\- and $4$-tuples. In 2001, Dujella [5] proved that there are
atmost finitely many Diophantine 8-tuple with $D(1)$ and there does not exist
Diophantine $9$-tuple with $D(1)$. In 2004, he improved this result and proved
that there does not exist Diophantine 6-tuple with $D(1)$ and there exist
atmost finitely many Diophantine 5-tuple with $D(1)$ (see [6]). There was a
‘folklore’ conjecture that there does not exist Diophantine $5$-tuples with
$D(1)$. This is recently (in 2019) been settled by B. He et. al. [9] in a
pioneering work. Let
$S(n)=\max\\{|A|:A\text{ is a Diophantine}~{}~{}m-\text{tuple
with}~{}D(n)\\}.$
Thus from the work of He et.al. $S(1)\leq 4$. Dujella and Fuchs [7] showed
that there do not exist Diophnatine 5-tuples with $D(-1)$. Dujella, Fuchs and
Filipin [8] also proved that there exist atmost finitely many Diophnatine
4-tuple with $D(-1)$. Furthermore they showed that, any such Diophantine
4-tuple with $D(-1)$ $\\{a_{1},\cdots,a_{4}\\}$ should satisfy
$a_{4}<10^{903}$. This bound was further reduced to $3.01\times 10^{60}$ by
Trudgian [14].
###### Definition 1.1.
A set $\\{a_{1},a_{2},\cdots,a_{m}\\}\subset\mathcal{O}_{K}\setminus\\{0\\}$
is called Diophantine $m$-tuples in $\mathcal{O}_{K}$ with $D(n)$ if
$a_{i}a_{j}+n=x_{ij}^{2}$, $x_{ij}\in\mathcal{O}_{K}$ for all $1\leq i<j\leq
m$.
For the remainder of the article, $m$ and $n$ carry the same meaning as in
definition 1.1 above.
In 1997, Dujella proved that there does not exist Diophantine 4-tuple in
$\mathbb{Z}[i]$ with $D(a+bi)$ , where $b$ is odd or $a\equiv b\equiv
2\pmod{4}$ (see [4]). For $n=1$, Azadaga [1] proved that $m\leq 42$. For
$n=-1$, Soldo studied the extension of certain triples to quadruples (see
[11], [12]). In this paper, we studied the existence of $m$-tuple with $D(-1)$
and obtained the following:
###### Theorem 1.1.
Let $K$ be an imaginary quadratic field and $\mathcal{O}_{K}$ be its ring of
integers. Then there does not exist Diophantine $m$-tuple with $D(-1)$ for
$m>36$ in $\mathcal{O}_{K}$.
Here is a brief of how we proceed to prove the above result. We employ similar
techniques as that of Azadaga [1]. Let $\\{a,b,c\\}$ be a triple in
$\mathcal{O}_{K}$ with $D(-1)$. If $d\in\mathcal{O}_{K}$ such that
$\\{a,b,c,d\\}$ be a quadruple with $D(-1)$, then we get a system of Pellian
equations. Using the solution of these Pellian equations and a result of
Jedrizević-Zeigler [10], we will get an upper bound on $d$ in term of $c$, if
$\\{a,b,c,d\\}$ satisfies some conditions. Further using the regularity
condition (refer section 4 below) on $\\{a,b,c,d\\}$ one gets a lower bound,
i.e., $d\geq g(a)$ for some function $g$ in terms of $a$. We use SAGE for the
computations and prove Theorem 1.1 by contradiction. The lower and upper
bounds on $d$ will give the desired contradiction.
## 2\. System of Pellian equations
Let $K=\mathbb{Q}(\sqrt{-D})$ with $D$ a square free positive integer. We know
that $\mathcal{O}_{K}=\mathbb{Z}[\omega]=\\{a+b\omega:a,b\in\mathbb{Z}\\}$,
where
$\omega=\begin{cases}\sqrt{-D}&\text{if $-D\equiv 2,3\pmod{4}$},\\\
\dfrac{1+\sqrt{-D}}{2}&\text{if $-D\equiv 1\pmod{4}$}.\end{cases}$
If
$\alpha=\left(a+\dfrac{b}{2}\right)+\dfrac{b}{2}\sqrt{-D}\in\mathcal{O}_{K}$
then the norm of $\alpha$:
$||\alpha||=\left(a+\dfrac{b}{2}\right)^{2}+\dfrac{Db^{2}}{4},$
and in particular if $\alpha=a+b\sqrt{-D}$, then
$||\alpha||=a^{2}+Db^{2}.$
Then the absolute value of $\alpha\in\mathcal{O}_{K}$ (denoted as $|\alpha|$)
is defined as $|\alpha|=\sqrt{||\alpha||}$. When $D=1$ the units in
$\mathbb{Z}[i]$ are $\\{\pm 1,\pm i\\}$, when $D=3$ the units are $\left\\{\pm
1,\dfrac{\pm 1\pm\sqrt{-3}}{2}\right\\}$ and else the units are $\\{\pm 1\\}$.
Notations- Throughout, a triple $\\{a,b,c\\}$ will denote a Diophantine
$3$-tuple in $\mathcal{O}_{K}$ such that $0<|a|\leq|b|\leq|c|$ with property
$D(-1)$ and similarly other tuples. Let $r,s,t\in\mathcal{O}_{K}$ such that
$r=\sqrt{ab-1},\ s=\sqrt{ac-1}~{}~{}\text{and}~{}~{}t=\sqrt{bc-1},$
where $a,b,c,d$ form a quadruple.
###### Lemma 2.1.
Let $\mathcal{A}=\\{a_{1},a_{2},a_{3},\cdots,a_{m}\\}$ be a $m$-tuple in
$\mathcal{O}_{K}$ with $D(-1)$. Then, for $m\geq 4$, $a_{i}a_{j}$ is not a
square in $\mathcal{O}_{K}$ for all $1\leq i<j\leq m$. Also, for $m\geq 4$,
$a_{i}a_{j}$ is not a square in $K$.
###### Proof.
If $\\{a,b\\}$ be a pair in $\mathcal{A}$ such that $ab=x^{2}$ where
$x\in\mathcal{O}_{K}\setminus\\{0\\}$, then
$ab-1=r^{2}=x^{2}-1\Rightarrow 1=x^{2}-r^{2}=(x-r)(x+r)\Rightarrow x=0\hskip
5.69046pt\text{or}\hskip 5.69046ptr=0,$
so $r=0$ and hence $ab=1$. If $D=1$ then $a,b\in\\{i,-i\\}$ and it implies
that if $\\{a,b,c\\}$ be a triple then $c$ has to be one of $\\{\pm i\\}$. One
can easily check that $\\{a,b,c\\}$ is not triple in $\mathcal{O}_{K}$ with
$D(-1)$. On the other hand when $D=3$ then $a,b\in\Big{\\{}\dfrac{\pm
1\pm\sqrt{-3}}{2}\Big{\\}}$. It implies that if $\\{a,b,c\\}$ be a triple
then, $c$ is one of $\\{\pm 1\\}$. Thus only two pairs
$\Big{\\{}\dfrac{1+\sqrt{-3}}{2},\dfrac{1-\sqrt{-3}}{2}\Big{\\}}$ and
$\Big{\\{}\dfrac{-1+\sqrt{-3}}{2},\dfrac{-1-\sqrt{-3}}{2}\Big{\\}}$ survive.
The corresponding triples are
$\Big{\\{}\dfrac{1+\sqrt{-3}}{2},\dfrac{1-\sqrt{-3}}{2},1\Big{\\}}~{}~{}\text{and}~{}~{}\Big{\\{}\dfrac{-1+\sqrt{-3}}{2},\dfrac{-1-\sqrt{-3}}{2},-1\Big{\\}}.$
Note also that these pairs
$\Big{\\{}\dfrac{1+\sqrt{-3}}{2},\dfrac{1-\sqrt{-3}}{2}\Big{\\}}$ and
$\Big{\\{}\dfrac{-1+\sqrt{-3}}{2},\dfrac{-1-\sqrt{-3}}{2}\Big{\\}}$ cannot be
extended to quadruple. Now if $D\neq 1,3$ then the units are $\pm 1$ so either
$a=b=1$ or $a=b=-1$. Hence $ab$ is not a square in $\mathcal{O}_{K}$.
Now if $ab$ is a square in $K$, then it is a root of monic polynomial
$x^{2}-ab$. Since $\mathcal{O}_{K}$ is integrally closed, $ab$ is not a square
in $K$. Hence $ab$ is not a square in $K$. ∎
Let us suppose $\\{a,b,c\\}$ extends to a quadruple $\\{a,b,c,d\\}$. Thus
there exist $x,y,z\in\mathcal{O}_{K}$ such that
$ad-1=x^{2},\hskip 5.69046ptbd-1=y^{2},\hskip 5.69046ptcd-1=z^{2}.$
Thus there is a system of Pell’s equations:
$az^{2}-cx^{2}=c-a$ (2.1) $bz^{2}-cy^{2}=c-b$ (2.2)
with $d=\dfrac{z^{2}+1}{c}$.
## 3\. Upper bound of $d$ in term of $c$
Let $\\{a,b,c,d\\}$ be a quadruple. We will see that if $c$ is bounded by some
power of $b$ then $d$ is bounded by some power of $c$. In 1998, Bennett [3]
proved a theorem which is related to simultaneous approximations of rationals,
where these rationals have square roots close to one. Jadrijevic̀-Zeigler
proved the following theorem which is an analog to Bennett’s theorem.
###### Lemma 3.1.
$($Jadrijević -Zeigler [10, Theorem 7.3, 7.4]$)$ Let
$\theta_{i}=\sqrt{1+\dfrac{a_{i}}{T}}$, $i=1,2$ with $a_{1},a_{2}$ distinct
algebraic integers in $K$, and $T$ be any algebraic integer of $K$. Further,
let $M=\max\\{|a_{1}|,|a_{2}|$}, $|T|>M$, $a_{0}=0$ and
$L=\dfrac{27}{16|a_{1}|^{2}|a_{2}|^{2}|a_{1}-a_{2}|^{2}}(|T|-M)^{2}>1$
Then
$\max\\{|\theta_{1}-p_{1}/q|,|\theta_{2}-p_{2}/q|\\}>c_{1}|q|^{-\lambda}$
(3.1)
for all algebraic integers $p_{1},p_{2},q\in K$ where
$\displaystyle\lambda$ $\displaystyle=1+\dfrac{\log P}{\log L},\hskip
5.69046ptc_{1}^{-1}=4pP(\max\\{1,2l\\})^{\lambda-1},$ $\displaystyle l$
$\displaystyle=\dfrac{27|T|}{64(|T|-M)},\hskip
5.69046ptp=\sqrt{\dfrac{2|T|+3M}{2|T|-2M}},$ $\displaystyle P$
$\displaystyle=16\dfrac{|a_{1}|^{2}|a_{2}|^{2}|a_{1}-a_{2}|^{2}}{\min\\{|a_{1}|,|a_{2}|,|a_{1}-a_{2}|\\}^{3}}(2|T|+3M).$
###### Lemma 3.2.
Let $(x,y,z)$ be a solution of the system of equations (2.1) and (2.2). Assume
$|c|>4|b|$, $|a|\geq 2$. If
$\theta_{1}^{(1)}=\pm\dfrac{s}{a}\sqrt{\dfrac{a}{c}},\hskip
5.69046pt\theta_{1}^{(2)}=-\theta_{1}^{(1)}$ and
$\theta_{2}^{(1)}=\pm\dfrac{t}{b}\sqrt{\dfrac{b}{c}},\hskip
5.69046pt\theta_{2}^{(2)}=-\theta_{2}^{(1)}$ with ‘sign’ chosen so that
$\Big{|}\theta_{1}^{(1)}-\dfrac{sx}{az}\Big{|}\leq\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}$
and
$\Big{|}\theta_{2}^{(1)}-\dfrac{ty}{bz}\Big{|}\leq\Big{|}\theta_{2}^{(2)}-\dfrac{ty}{bz}\Big{|}$,
then
$\Big{|}\theta_{1}^{(1)}-\dfrac{sbx}{abz}\Big{|}\leq\dfrac{|s||a-c|}{|a|\sqrt{|ac|}}\times\dfrac{1}{|z|^{2}}<\dfrac{21|c|}{16|a|}\times\dfrac{1}{|z|^{2}}$
(3.2)
and
$\Big{|}\theta_{2}^{(1)}-\dfrac{tay}{abz}\Big{|}\leq\dfrac{|s||a-c|}{|b|\sqrt{|bc|}}\times\dfrac{1}{|z|^{2}}<\dfrac{21|c|}{16|a|}\times\dfrac{1}{|z|^{2}}.$
(3.3)
###### Proof.
We prove inequality (3.2) and similarly (3.3) can be proven. Consider
$\Big{|}\theta_{1}^{(1)}-\dfrac{sx}{az}\Big{|}=\dfrac{\Big{|}\theta_{1}^{(1)}-\dfrac{sx}{az}\Big{|}\times\Big{|}\theta_{1}^{(1)}+\dfrac{sx}{az}\Big{|}}{\Big{|}\theta_{1}^{(1)}+\dfrac{sx}{az}\Big{|}}=\dfrac{\Big{|}\Big{(}\theta_{1}^{(1)}\Big{)}^{2}-\dfrac{s^{2}x^{2}}{a^{2}z^{2}}\Big{|}}{\Big{|}\theta_{1}^{(1)}+\dfrac{sx}{az}\Big{|}}.$
We substitute $\theta_{1}^{(2)}=-\theta_{1}^{(1)}$ in above and get
$\displaystyle\dfrac{\Big{|}\Big{(}\theta_{1}^{(1)}\Big{)}^{2}-\dfrac{s^{2}x^{2}}{a^{2}z^{2}}\Big{|}}{\Big{|}\theta_{1}^{(1)}+\dfrac{sx}{az}\Big{|}}$
$\displaystyle=\Big{|}\dfrac{s^{2}}{a^{2}}\Big{|}\times\Big{|}\dfrac{a^{2}}{s^{2}}\times\Big{(}\theta_{1}^{(1)}\Big{)}^{2}-\dfrac{x^{2}}{z^{2}}\Big{|}\times\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}^{-1}$
$\displaystyle=\Big{|}\dfrac{s^{2}}{a^{2}}\Big{|}\times\Big{|}\dfrac{a}{c}-\dfrac{x^{2}}{z^{2}}\Big{|}\times\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}^{-1}$
$\displaystyle=\Big{|}\dfrac{s^{2}}{a^{2}}\Big{|}\times\Big{|}\dfrac{az^{2}-cx^{2}}{|cz^{2}|}\Big{|}\times\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}^{-1}$
$\displaystyle=\Big{|}\dfrac{s^{2}}{a^{2}}\Big{|}\times\dfrac{|c-a|}{|cz^{2}|}\times\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}^{-1}.$
This is because
$\displaystyle 2\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}$
$\displaystyle\geq\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}+\Big{|}\theta_{1}^{(1)}-\dfrac{sx}{az}\Big{|}$
$\displaystyle\geq\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}-\Big{(}\theta_{1}^{(1)}-\dfrac{sx}{az}\Big{)}\Big{|}$
$\displaystyle=\Big{|}\theta_{1}^{(2)}-\theta_{1}^{(1)}\Big{|}=2\Big{|}\dfrac{s}{a}\sqrt{\dfrac{a}{c}}\Big{|}.$
Thus
$\Big{|}\theta_{1}^{(2)}-\dfrac{sx}{az}\Big{|}\geq\Big{|}\dfrac{s}{a}\sqrt{\dfrac{a}{c}}\Big{|}.$
This implies that
$\Big{|}\theta_{1}^{(1)}-\dfrac{sbx}{abz}\Big{|}\leq\dfrac{|s||c-a|}{|a|\sqrt{|ac|}}\times\dfrac{1}{|z|^{2}}.$
For proving other part of the inequality (3.2), we want to show that
$|\sqrt{ac-1}|\times|c-a|<(21/16)\times|c|\times\sqrt{|ac|}$
and this holds if and only if
$\Big{|}\sqrt{1-\dfrac{1}{ac}}\Big{|}<\dfrac{21}{16}\times\dfrac{|c|}{|c-a|}.$
Now $|c|>4|a|$ implies that
$\dfrac{21}{16}\times\dfrac{|c|}{|c-a|}\geq\dfrac{21}{20}$
and then
$\displaystyle\Big{|}\sqrt{1-\dfrac{1}{ac}}\Big{|}$ $\displaystyle=$
$\displaystyle\sqrt{\Big{|}1-\dfrac{1}{ac}\Big{|}}$ $\displaystyle\leq$
$\displaystyle\sqrt{1+\dfrac{1}{|ac|}}<\dfrac{\sqrt{17}}{4}$ $\displaystyle<$
$\displaystyle\dfrac{21}{20}$ $\displaystyle\leq$
$\displaystyle\dfrac{21}{16}\times\dfrac{|c|}{|c-a|}.$
∎
Thus from Lemma 3.2 we conclude that
$\displaystyle\Big{|}\theta_{1}^{(2)}+\dfrac{sbx}{abz}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\theta_{1}^{(1)}-\dfrac{sbx}{abz}\Big{|}$
$\displaystyle\leq$
$\displaystyle\dfrac{|s||a-c|}{|a|\sqrt{|ac|}}\times\dfrac{1}{|z|^{2}}$
$\displaystyle<$ $\displaystyle\dfrac{21|c|}{16|a|}\times\dfrac{1}{|z|^{2}},$
and
$\displaystyle\Big{|}\theta_{2}^{(2)}+\dfrac{tay}{abz}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\theta_{2}^{(1)}-\dfrac{tay}{abz}\Big{|}$
$\displaystyle\leq$
$\displaystyle\dfrac{|s||a-c|}{|b|\sqrt{|bc|}}\times\dfrac{1}{|z|^{2}}$
$\displaystyle<$ $\displaystyle\dfrac{21|c|}{16|a|}\times\dfrac{1}{|z|^{2}}.$
###### Lemma 3.3.
Let {a, b, c, d} be a quadruple such that $|b|\geq(3/2)|a|$, $|b|\geq 22$,
$|a|\geq 2$ and $|c|>|b|^{16}$. Then
$|d|<(3956)^{10}|c|^{24}.$
###### Proof.
Let $\theta_{1}=\dfrac{s}{a}\sqrt{\dfrac{a}{c}}$ and
$\theta_{2}=\dfrac{t}{b}\sqrt{\dfrac{b}{c}}$. Then
$\displaystyle\theta_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{s^{2}a}{a^{2}c}}=\sqrt{1+\dfrac{(-b)}{abc}},~{}~{}\text{and}$
$\displaystyle\theta_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{t^{2}b}{b^{2}c}}=\sqrt{1+\dfrac{(-a)}{abc}}.$
If we write $a_{1}=-b$, $a_{2}=-a,$ $T=abc$ and $M=|b|$ then the claim is
that:
$l=\dfrac{27|abc|}{64(|abc|-|b|)}<\dfrac{1}{2}.$
Proving the above claim is equivalent to show that $27|abc|<32(|abc|-|b|)$ and
this holds if and only if $|ac|>(32/5)$. By hypothesis $|ac|\geq|b|\geq
22>(32/5)$ and thus the claim holds.
Now
$p=\sqrt{\dfrac{2|abc|+3|b|}{2|abc|-2|b|}}=\sqrt{1+\dfrac{5}{2(|ac|-1)}}\leq\sqrt{\dfrac{47}{42}}.$
Also $l<\dfrac{1}{2}$, one has $c_{1}^{-1}=4pP\times 1$ would give
$c_{1}\geq\dfrac{1}{4\times
P\times(\sqrt{47/42})}=\dfrac{\sqrt{42}}{\sqrt{47}(4P)}.$
Consider now
$P=16\times\dfrac{|-b|^{2}|-a|^{2}|-b+a|^{2}}{\min\\{|-a|,|-b|,|-a+b|\\}^{3}}\times\Big{(}2|abc|+3|b|\Big{)}.$
Since
$|-b+a|\geq|b|-|a|\geq\Big{(}\dfrac{3}{2}\times|a|-|a|\Big{)}=\dfrac{|a|}{2},$
so, $\min\\{|a|,|b|,|a-b|\\}\geq\dfrac{|a|}{2}$. Thus
$P\leq 128\cdot\dfrac{|b|^{2}|a|^{2}|b-a|^{2}|b|(2|ac|+3)}{|a|^{3}}.$
Hence
$P\leq\dfrac{128|b|^{3}|b-a|^{2}(2|ac|+3)}{|a|}.$ (3.4)
Let us now look at
$L=\dfrac{27}{16|-b|^{2}|-a|^{2}|-b+a|^{2}}\times\Big{(}|abc|-|b|\Big{)}^{2}=\dfrac{27(|ac|-1)^{2}}{16|a|^{2}|b-a|^{2}}.$
We claim that $L>1$. Which is equivalent to show
$27(|ac|-1)^{2}>16|a|^{2}|b-a|^{2}$. This holds if and only if
$3\sqrt{3}(|ac-1|)>4|a||b-a|$ which is equivalent to
$\dfrac{3\sqrt{3}}{4}\times(|ac|-1)>|a||b-a|.$
Since
$|ac|-1>|a||b|^{3}-1>2|a|^{2}|b|-1>|a||b|+|a|^{2}\geq|ab-b^{2}|=|a||b-a|$
the claim is validated.
Clearly $P>1$ and so $\lambda>1$. In fact $\lambda<1.8$.
Indeed, observe that $\lambda=1+\dfrac{\log P}{\log L}<1.8$ holds if and only
if $P<L^{0.8}$ , which is equivalent to
$P<\Big{(}\dfrac{27}{16}\Big{)}^{0.8}\times\Bigg{(}\dfrac{|ac|-1}{|a|(|b-a|)}\Bigg{)}^{1.6}.$
Appealing to inequality (3.4), we need to show
$\dfrac{128|b|^{3}|b-a|^{2}(2|ac|+3)}{|a|}<\Big{(}\dfrac{27}{16}\Big{)}^{0.8}\cdot\Big{(}\dfrac{|ac|-1}{|a||b-a|}\Big{)}^{1.6}.$
After rearranging the above inequality,
$128|b|^{3}|b-a|^{3.6}|a|^{0.6}(2|ac|+3)<\Big{(}\dfrac{27}{16}\Big{)}^{0.8}(|ac|-1)^{1.6}.$
We see that it suffices to show
$128|b|^{3}|b-a|^{3.6}(9/4)|a|^{0.6}<\Big{(}\dfrac{27}{16}\Big{)}^{0.8}(|ac|-1)^{0.6},$
(3.5)
as $|ac|-1>\dfrac{4}{9}(2|ac|+3)$. Since the function
$f(t)=(t-1)^{0.6}-t^{0.6}+1$ vanishes at $t=1$ and is increasing,
$|ac|^{0.6}-1<(|ac|-1)^{0.6}$. Thus (using $|c|>|b|^{16}$)
$|a|^{0.6}|b|^{9.6}-1=|a|^{0.6}|b|^{(16)\cdot(0.6)}-1<|ac|^{0.6}-1<(|ac|-1)^{0.6}.$
For proving inequality (3.5), it suffices to show
$128\times(9/4)|b|^{3}|b-a|^{3.6}|a|^{0.6}<\Big{(}\dfrac{27}{16}\Big{)}^{0.8}(|b|^{9.6}-1).$
(3.6)
Since we have $|a|\leq\dfrac{2}{3}(|b|)$,
$\displaystyle\Big{(}\dfrac{16}{27}\Big{)}^{0.8}\times
128\times(9/4)|b|^{3}|b-a|^{3.6}|a|^{0.6}$ $\displaystyle<$
$\displaystyle\Big{(}\dfrac{16}{27}\Big{)}^{0.8}\times
128\times(9/4)|b|^{3}(5/3)^{3.6}\cdot|b|^{3.6}\cdot\Big{|}\dfrac{2b}{3}\Big{|}^{0.6}$
$\displaystyle<$ $\displaystyle 936|b|^{7.2}.$
Thus inequality (3.6) holds if $936|b|^{7.2}<|b|^{9.6}-1$. This is obvious
since the function $f(t)=t^{9.6}-936t^{7.2}-1$ is increasing function for
$t\geq 15.5$ and $f(18)>0$. Hence our claim is proved.
Proceeding further, with $\theta_{1},\theta_{2}$ as above, take $p_{1}=\pm
sbx,p_{2}=\pm tay,q=abz$ (‘sign’ is chosen suitably) and upon applying Lemmas
3.1 and 3.2, we get
$\dfrac{21}{16}\cdot\dfrac{|c|}{|a|}\cdot\dfrac{1}{|z|^{2}}>\dfrac{\sqrt{42}}{\sqrt{47}(4P)}|abz|^{-\lambda}.$
From inequality (3.4), we get
$\dfrac{21}{16}\cdot\dfrac{|c|}{|a|}\cdot\dfrac{1}{|z|^{2}}>\dfrac{\sqrt{42}|a||abz|^{-\lambda}}{\sqrt{47}(4\cdot
128)\cdot|b|^{3}|b-a|^{2}(2|ac|+3)}.$
It implies that
$\dfrac{21}{16}\dfrac{4\sqrt{47}\times
128}{\sqrt{42}}\dfrac{|c|}{|a|^{2}}|b|^{3}|b-a|^{2}(2|ac|+3)\cdot|ab|^{\lambda}>|z|^{2-\lambda}>|z|^{0.2}.$
Hence
$|z|^{0.2}<712|c|\cdot
3\cdot|ac||b-a|^{2}|b|^{3+\lambda}|a|^{\lambda-2}<712\times
3|c|^{2}\cdot(2/3)|b|(5/3)^{2}|b|^{2}|b|^{4.8}.$
Using $|c|<|b|^{16}$, one further gets,
$|z|^{0.2}<3956\cdot|c|^{2}|b|^{7.8}<3956|c|^{2.49}.$
Hence
$|z|<(3956)^{5}|c|^{12.45}$
and finally
$|d|=\dfrac{|z^{2}-1|}{|c|}\leq\dfrac{|z|^{2}+1}{|c|}\leq\dfrac{(3956)^{10}|c|^{24.9}+1}{|c|}<3956^{10}|c|^{24}.$
∎
## 4\. Lower bound on $d$
A triple $\\{a,b,c\\}$ is said to be regular if $c=a+b\pm 2r$ (refer notation
above). If $\\{a,b,c,d\\}$ is a quadruple, then the use of this regularity
criterion gives us a lower bound on $d$ in terms of $a$. The following lemma
states this.
###### Lemma 4.1.
Let $\\{a,b,c,d\\}$ be a quadruple with $5<|a|\leq|b|\leq|c|\leq|d|$. Then
atleast one of $\\{a,b,c\\}$ and $\\{a,b,d\\}$ is not regular.
###### Proof.
If possible let both $\\{a,b,c\\}$ and $\\{a,b,d\\}$ are regular, i.e.,
$c=a+b+2r$ and $d=a+b-2r$. Substituting the value of $r$ gives
$cd-1=(a-b)^{2}+3$. As $\\{c,d\\}$ is a pair in $\mathcal{O}_{K}$ with
$D(-1)$, there exists a $z\in\mathcal{O}_{K}$ such that $cd-1=z^{2}$. Thus
$z^{2}=(a-b)^{2}+3$ and therefore $3=(z-(a-b))(z+(a-b))$. We take
$X=(z-(a-b))$ and $Y=(z+(a-b))$. Then
$XY=3$ (4.1)
and
$X+Y=2z.$ (4.2)
Taking norm on both sides in (4.1), we get $||X||\times||Y||=||3||=9$.
Case (i): $||X||=1$ or $||Y||=1$.
Assume that $||X||=1$, then $X$ is a unit.
If $D=1$, by equation (4.1), $(X,Y)\in\\{(1,3),(-1,-3),(i,-3i),(-i,3i)$}. This
implies that $X+Y=\pm 4,\pm 2i$ and therefore $z=\pm 2,\pm i$ (from the
equation (4.2)). Since $cd-1=z^{2}$, so either $cd=5$ or $cd=0$. Thus we get
$|d|\leq 5,$ which is a contradiction to our hypothesis.
If $D=3,$ by again using equation (4.1), we get
$\begin{split}(X,Y)\in\Bigg{\\{}(1,3),(-1,-3),\Big{(}\dfrac{1+\sqrt{-3}}{2},\dfrac{3(1-\sqrt{-3})}{2}\Big{)},\Big{(}\dfrac{1-\sqrt{-3}}{2},\dfrac{3(1+\sqrt{-3})}{2}\Big{)},\\\
\Big{(}\dfrac{-1+\sqrt{-3}}{2},\dfrac{3(-1-\sqrt{-3})}{2}\Big{)},\Big{(}\dfrac{-1-\sqrt{-3}}{2},\dfrac{3(-1+\sqrt{-3})}{2}\Big{)}\Bigg{\\}}.\end{split}$
From equation (4.2), it follows that $2z=\pm 4,\pm 2\pm\sqrt{-3}$. Since
$z\in\mathcal{O}_{K}$, therefore $z=\pm 2$. Thus $cd=5$. This implies that
$|d|\leq 5$, a contradiction.
If $D\neq 1,3,$ then $(X,Y)\in\\{(1,3),(-1,-3)\\}$ (from equation (4.1)).
Again using equation (4.2), we get $2z=\pm 4$ and hence $cd=5$. Again this
will give $|d|\leq 5$, contradiction.
Case (ii): $||X||=||Y||=3$.
If $D=1,$ then $||X||=3=a_{1}^{2}+b_{1}^{2}$ where $a_{1},b_{1}\in\mathbb{Z}$,
which is not possible.
If $D=2$, then $||X||=3=a_{1}^{2}+2b_{1}^{2}$ where
$a_{1},b_{1}\in\mathbb{Z}$. This implies that
$\displaystyle(X,Y)\in$
$\displaystyle\Big{\\{}\Big{(}1+\sqrt{-2},1-\sqrt{-2}\Big{)},\Big{(}1-\sqrt{-2},1+\sqrt{-2}\Big{)},$
$\displaystyle\Big{(}-1+\sqrt{-2},-1-\sqrt{-2}\Big{)},\Big{(}-1-\sqrt{-2},-1+\sqrt{-2}\Big{)}\Big{\\}}.$
Then $z=\pm 1$ and therefore $cd=2$. We conclude that $|d|\leq 2$.
If $D>3$ and $D\equiv 1,2\pmod{4},$ then $||X||=a_{1}^{2}+Db_{1}^{2}=3$ where
$a_{1},b_{1}\in\mathbb{Z}$ which is again not possible.
If $D=3$, then $||X||=\Big{(}a+\dfrac{b}{2}\Big{)}^{2}+\dfrac{3\cdot
b^{2}}{4}=3$. From equation (4.1), we get
$\begin{split}(X,Y)\in\Bigg{\\{}\Big{(}\dfrac{3}{2}+\dfrac{\sqrt{-3}}{2},\dfrac{3}{2}-\dfrac{\sqrt{-3}}{2}\Big{)},\Big{(}\dfrac{-3}{2}+\dfrac{\sqrt{-3}}{2},\dfrac{-3}{2}-\dfrac{\sqrt{-3}}{2}\Big{)},\Big{(}\dfrac{3}{2}-\dfrac{\sqrt{-3}}{2},\dfrac{3}{2}+\dfrac{\sqrt{-3}}{2}\Big{)},\\\
\Big{(}\dfrac{-3}{2}-\dfrac{\sqrt{-3}}{2},\dfrac{-3}{2}+\dfrac{\sqrt{-3}}{2}\Big{)},\Big{(}\sqrt{-3},-\sqrt{-3}\Big{)},\Big{(}-\sqrt{-3},\sqrt{-3}\Big{)}\Bigg{\\}}.\end{split}$
Using equation (4.2), $2z=0,\pm 3$. Since $z\in\mathcal{O}_{K}$, we get $z=0$
and therefore $cd=1$. This implies that $|d|\leq 1$, which is a contradiction.
Same way we can prove our lemma for $D\geq 7$ with $D\equiv 3\pmod{4}$. ∎
###### Lemma 4.2.
Let $\\{a,b,c,d\\}$ be a quadruple with $10\leq|a|\leq|b|\leq|c|\leq|d|$, then
$|d|\geq\dfrac{|ab|}{(330/65)}\geq\dfrac{|a|^{2}}{(330/65)}$.
###### Proof.
We assume that $\\{a,b,d\\}$ is not regular(from Lemma (4.1)). Define
$c_{\pm}=a+b+d-2abd\pm 2rxy,$
where $x,y\in\mathcal{O}_{K}$ such that, $ad-1=x^{2}$ and $bd-1=y^{2}$.
Claim: $c_{\pm}\neq 0$.
Suppose $c_{\pm}=0$. This implies that $a+b+d(1-2ab)=\mp 2rxy$. Squaring and
rearranging this equation we get, $d^{2}-2d(a+b)+(a-b)^{2}+4=0$. Therefore
$d=a+b+2r$ or $a+b-2r$. Since $\\{a,b,d\\}$ is not regular, this is a
contradiction.
Consider
$c_{+}c_{-}=(a+b+d-2abd)^{2}-4(rxy)^{2}=a^{2}+b^{2}+d^{2}-2ab-2ad-2bd+4$.
Therefore
$|c_{+}c_{-}|\leq|d^{2}|+|d^{2}|+|d^{2}|+2|d|^{2}+2|d|^{2}+2|d|^{2}+|d|^{2}\leq
10|d|^{2}$, also $|c_{+}+c_{-}|=2|a+b+d-2abd|$. We may assume that
$|c_{+}|\geq|c_{-}|$. Since
$2c_{+}=|c_{+}|+|c_{+}|\geq|c_{+}+c_{-}|=2|a+b+d-2abd|$, this implies that,
$|c_{+}|\geq|a+b+d-2abd|$
We have $10\leq|a|\leq|b|\leq|c|\leq|d|$, which follows that $|a+b+d|\leq
3|d|\leq\dfrac{3}{99}\cdot|abd|$. Thus
$|c_{+}|\geq|a+b+d-2abd|\geq 2|abd|-|a+b+d|\geq
2|abd|-(3/99)|abd|=\dfrac{65}{33}\cdot|abd|.$
We have proved that $|c_{+}c_{-}|\leq 10|d|^{2}$ which gives that
$|c_{-}|\leq\dfrac{10|d|^{2}}{|c_{+}|}\leq\dfrac{10|d|^{2}}{(65/33)|abd|}=\dfrac{(330)|d|}{(65)|ab|}$.
Since $c_{-}\neq 0$, $|c_{-}|\geq 1$ and this implies that
$\dfrac{330|d|}{65|ab|}\geq 1$. Hence
$|d|\geq\dfrac{|ab|}{(330/65)}\geq\dfrac{|a|^{2}}{(330/65)}.$ ∎
## 5\. Proof of the main theorem
Let $\\{a,b,c,d,e\\}$ be a quintuple with $|e|<15$. For $D<226$, we can check
that, by computer, there does not exist such type of quintuples, and for
$D\geq 226$, we can easily seen that $a,b,c,d,e\in\mathbb{Z}$. Therefore, if
$ab-1=(x+y\sqrt{-D})^{2}$, then $2xy=0$. This gives that either $x=0$ or
$y=0$. Now if $x=0$ then $ab-1=-Dy^{2}$. This implies that
$|ab-1|\leq|ab|+1<226$, and hence $x=0$ is not possible. Thus $y=0$. We
conclude that if $\\{a,b,c,d,e\\}$ is a quintuple, then $|e|\geq 15$.
Similarly, one can check that if $\\{a,b,c,d\\}$ is a quadruple, then $|d|\geq
12$.
Let $\mathcal{A}=\\{a_{1},a_{2},\cdots,a_{m}\\}$ be a Diophantine $m$-tuple in
$\mathcal{O}_{K}$ with $D(-1)$ such that $m\geq 37$. Thus
$\\{a_{4},a_{5},a_{6},a_{7}\\}$ is a quadruple. From Lemma (4.2), we get
$|a_{7}|\geq\dfrac{|a_{4}a_{5}|}{(330/65)}\geq\dfrac{12\cdot
15}{(330/65)}>35$.
By applying lemma (4.2) to quadruples
$\\{a_{7},a_{8},a_{9},a_{10}\\},\\{a_{10},a_{11},a_{12},a_{13}\\}$,$\cdots$,
$\\{a_{19},a_{20},\\\ a_{21},a_{22}\\}$ respectively, we get the following
inequalities
$\displaystyle|a_{10}|\geq\dfrac{|a_{7}|^{2}}{(330/65)},$
$\displaystyle|a_{13}|\geq\dfrac{|a_{10}|^{2}}{(330/65)}=\dfrac{|a_{7}|^{4}}{(330/65)^{3}},$
$\displaystyle|a_{22}|\geq\dfrac{|a_{7}|^{32}}{(330/65)^{31}}.$
Consider quadruples $\\{a_{4},a_{7},a_{22},a_{22+k}\\}$ for $k>0$. Since
$\\{a_{1},a_{2},a_{3},a_{4}\\}$ is a quadruple, $|a_{4}|\geq 12$. Quadruple
$\\{a_{4},a_{5},a_{6},a_{7}\\}$ implies that $|a_{7}|\geq|a_{5}|\geq 15$ and
from Lemma (4.2),
$|a_{7}|\geq\dfrac{|a_{4}a_{5}|}{(330/65)}\geq\dfrac{15|a_{4}|}{(330/65)}>\dfrac{3|a_{4}|}{2}$.
Inequality $|a_{22}|>|a_{7}|^{16}$ holds if
$\dfrac{|a_{7}|^{32}}{(330/65)^{31}}>|a_{7}|^{16}$, and this holds if
$|a_{7}|>24$. By Lemma(3.3),
$|a_{22+k}|<3956^{10}|a_{22}|^{24},\hskip 5.69046pt\text{ $k>$ 0}.$ (5.1)
Again we apply lemma (4.2) to quadruples
$\\{a_{22},a_{23},a_{24},a_{25}\\},\\{a_{25},a_{26},a_{27},a_{28}\\}$,
$\cdots$, $\\{a_{34},a_{35},a_{36},a_{37}\\}$ respectively, and get the
following inequalities
$\displaystyle|a_{25}|\geq\dfrac{|a_{22}|^{2}}{(330/65)},$
$\displaystyle|a_{28}|\geq\dfrac{|a_{25}|^{2}}{(330/65)}\geq\dfrac{|a_{22}|^{4}}{(330/65)^{3}},$
$\displaystyle|a_{37}|\geq\dfrac{|a_{22}|^{32}}{(330/65)^{31}}.$
From inequality (5.1), $3956^{10}|a_{22}|^{24}>|a_{37}|$.
Claim: $\dfrac{|a_{22}|^{32}}{(330/65)^{31}}>3956^{10}|a_{22}|^{24}$.
It is equivalent to showing $|a_{22}|^{8}\geq(330/65)^{31}\cdot 3956^{10}$,
and this inequality holds, if $|a_{22}|>1.8\times 10^{7}$. Since
$|a_{22}|\geq\dfrac{|a_{7}|^{32}}{(330/65)^{31}}\geq\dfrac{35^{32}}{(330/65)^{31}}>10^{27}$,
our claim is proved. Finally we get
$3956^{10}|a_{22}|^{24}>|a_{37}|\geq\dfrac{|a_{22}|^{32}}{(330/65)^{31}}>3956^{10}|a_{22}|^{24},$
which is a contradiction. Hence $m\leq 36$. This completes the proof.
We have an example of quadruple in $\mathbb{Z}[i]$ with $D(-1)$ which is
$\\{1,2,5,-24\\}$. Unfortunately, we do not know about the existence of
Diophantine $m$-tuple in $\mathcal{O}_{K}$ with $D(-1)$, for $m\geq 5$.
## 6\. Acknowledgement
The author is indebted to Prof. Kalyan Chakraborty for his suggestions and for
carefully going through the manuscript; The author is also thankful to Dr. A.
Hoque for introducing him to this area and for his encouragement throughout.
It is also a pleasure to acknowledge Mr. Mohit Mishra and Mr. Rishabh
Agnihotri their support throughout the preparation of this manuscript and for
providing all required assistance.
## References
* [1] N. Adzaga, On the size of Diophantine m-tuples in imaginary quadratic number rings, Bull. Math. Sci. 9(3) (2019) 1950020.
* [2] A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137.
* [3] M. A. Bennett, On the number of solutions of simultaneous Pell equations, J. ReineAngew. Math. 498 (1998) 173-199.
* [4] A. Dujella, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32 (1997), 1-10.
* [5] A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126-150.
* [6] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.
* [7] A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71 (2005), 33-52.
* [8] A. Dujella, A. Filipin and C. Fuchs, Effective solution of the D(-1)-quadruple conjecture, Acta Arith. 128 (2007), 319-338.
* [9] B. He, A. Togbé, V. Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc. 371 (2019), 6665-6709.
* [10] B. Jadrijević and V. Ziegler, A system of relative Pellian equations and a related family of relative Thue equations, Int. J. Number Theory 2(4) (2006) 569-590.
* [11] I. Soldo, On the extensibility of $D(-1)$-triples {1, b, c} in the ring $\mathbb{Z}[\sqrt{-t}],t>0$, Studia Sci. Math. Hungar. 50 (2013), 296-330.
* [12] I. Soldo, $D(-1)$-triples of the form {1, b, c} in the ring $\mathbb{Z}[\sqrt{-t}],t>0$, Bull. Malays. Math. Sci. Soc. 39 (2016), 1201-1224.
* [13] J. H. Silverman and J. Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
* [14] T. Trudgian, Bounds on the number of Diophantine quintuples, J.Number Theory 157 (2015), 233-249.
|
2024-09-04T02:54:57.945316 | 2020-03-06T17:45:31 | 2003.03323 | {
"authors": "Louisa Seelbach Benkner and Stephan Wagner",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26088",
"submitter": "Louisa Seelbach Benkner",
"url": "https://arxiv.org/abs/2003.03323"
} | arxiv-papers | 11institutetext: Department für Elektrotechnik und Informatik, Universität
Siegen, Hölderlinstrasse 3, 57076 Siegen, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of
Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland
7602, South Africa
22email<EMAIL_ADDRESS>33institutetext: Department of Mathematics, Uppsala
Universitet, Box 480, 751 06 Uppsala, Sweden
33email<EMAIL_ADDRESS>
# On the Collection of Fringe Subtrees in Random Binary Trees
Louisa Seelbach Benkner This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 731143 and the DFG research project LO
748/10-1 (QUANT-KOMP).11 Stephan Wagner 2233
###### Abstract
A fringe subtree of a rooted tree is a subtree consisting of one of the nodes
and all its descendants. In this paper, we are specifically interested in the
number of non-isomorphic trees that appear in the collection of all fringe
subtrees of a binary tree. This number is analysed under two different random
models: uniformly random binary trees and random binary search trees.
In the case of uniformly random binary trees, we show that the number of non-
isomorphic fringe subtrees lies between $c_{1}n/\sqrt{\ln n}(1+o(1))$ and
$c_{2}n/\sqrt{\ln n}(1+o(1))$ for two constants $c_{1}\approx 1.0591261434$
and $c_{2}\approx 1.0761505454$, both in expectation and with high
probability, where $n$ denotes the size (number of leaves) of the uniformly
random binary tree. A similar result is proven for random binary search trees,
but the order of magnitude is $n/\ln n$ in this case.
Our proof technique can also be used to strengthen known results on the number
of distinct fringe subtrees (distinct in the sense of ordered trees). This
quantity is of the same order of magnitude in both cases, but with slightly
different constants in the upper and lower bounds.
###### Keywords:
Uniformly Random Binary Trees Random Binary Search Trees Fringe Subtrees Tree
Compression
## 1 Introduction
A subtree of a rooted tree that consists of a node and all its descendants is
called a _fringe subtree_. Fringe subtrees are a natural object of study in
the context of random trees, and there are numerous results for various random
tree models, see e.g. [3, 9, 11, 13].
Fringe subtrees are of particular interest in computer science: One of the
most important and widely used lossless compression methods for rooted trees
is to represent a tree as a directed acyclic graph, which is obtained by
merging nodes that are roots of identical fringe subtrees. This compressed
representation of the tree is often shortly referred to as _minimal DAG_ and
its size (number of nodes) is the number of distinct fringe subtrees occurring
in the tree. Compression by minimal DAGs has found numerous applications in
various areas of computer science, as for example in compiler construction [2,
Chapter 6.1 and 8.5], unification [25], symbolic model checking (binary
decision diagrams) [7], information theory [21, 28] and XML compression and
querying [8, 20].
In this work, we investigate the number of fringe subtrees in random binary
trees, i.e. random trees such that each node has either exactly two or no
children. So far, this problem has mainly been studied with respect to ordered
fringe subtrees in random ordered binary trees: A _uniformly random ordered
binary tree_ of size $n$ (with $n$ leaves) is a random tree whose probability
distribution is the uniform probability distribution on the set of ordered
binary trees of size $n$. In [19], Flajolet, Sipala and Steyaert proved that
the expected number of distinct ordered fringe subtrees in a uniformly random
ordered binary tree of size $n$ is asymptotically equal to $c\cdot n/\sqrt{\ln
n}$, where $c$ is the constant $2\sqrt{\ln 4/\pi}$. This result of Flajolet et
al. was extended to unranked labelled trees in [6] (for a different constant
$c$). Moreover, an alternative proof to the result of Flajolet et al. was
presented in [26] in the context of simply-generated families of trees.
Another important type of random trees are so-called _random binary search
trees_ : A random binary search tree of size $n$ is a binary search tree built
by inserting the keys $\\{1,\dots,n\\}$ according to a uniformly chosen random
permutation on $\\{1,\dots,n\\}$. Random binary search trees naturally arise
in theoretical computer science, see e.g. [12]. In [17], Flajolet, Gourdon and
Martinez proved that the expected number of distinct ordered fringe subtrees
in a random binary search tree of size $n$ is $O(n/\ln n)$. This result was
improved in [10] by Devroye, who showed that the asymptotics $\Theta(n/\ln n)$
holds. Moreover, the result of Devroye was generalized from random binary
search trees to a broader class of random ordered binary trees in [27], where
the problem of estimating the expected number of distinct ordered fringe
subtrees in random binary trees was considered in the context of so-called
leaf-centric binary tree sources, which were introduced in [23, 28] as a
general framework for modeling probability distributions on the set of ordered
binary trees of size $n$.
In this work, we focus on estimating the number of _non-isomorphic_ fringe
subtrees in random ordered binary trees, where we call two binary trees non-
isomorphic if they are distinct as unordered binary trees. This question
arises quite naturally for example in the context of XML compression: Here,
one distinguishes between so-called document-centric XML, for which the
corresponding XML document trees are ordered, and data-centric XML, for which
the corresponding XML document trees are unordered. Understanding the
interplay between ordered and unordered structures has thus received
considerable attention in the context of XML (see, for example, [1, 5, 29]).
In particular, in [24], it was investigated whether tree compression can
benefit from unorderedness. For this reason, so-called _unordered minimal
DAGs_ were considered. An unordered minimal DAG of a binary tree is a directed
acyclic graph obtained by merging nodes that are roots of isomorphic fringe
subtrees, i.e. of fringe subtrees which are identical as unordered trees. From
such an unordered minimal DAG, an unordered representation of the original
tree can be uniquely retrieved. The size of this compressed representation is
the number of non-isomorphic fringe subtrees occurring in the tree. So far,
only some worst-case estimates comparing the size of a minimal DAG to the size
of its corresponding unordered minimal DAG are known: Among other things, it
was shown in [24] that the size of an unordered minimal DAG of a binary tree
can be exponentially smaller than the size of the corresponding (ordered)
minimal DAG.
However, no average-case estimates comparing the size of the minimal DAG of a
binary tree to the size of the corresponding unordered minimal DAG are known
so far. In particular, in [24] it is stated as an open problem to estimate the
expected number of non-isomorphic fringe subtrees in a uniformly random
ordered binary tree of size $n$ and conjectured that this number
asymptotically grows as $\Theta(n/\sqrt{\ln n})$.
In this work, as one of our main theorems, we settle this open conjecture by
proving upper and lower bounds of order $n/\sqrt{\ln n}$ for the number of
non-isomorphic fringe subtrees which hold both in expectation and with high
probability (i.e., with probability tending to $1$ as $n\to\infty$). Our
approach can also be used to obtain an analogous result for random binary
search trees, though the order of magnitude changes to $\Theta(n/\ln n)$.
Again, we have upper and lower bounds in expectation and with high
probability. Our two main theorems read as follows.
###### Theorem 1
Let $F_{n}$ be the total number of non-isomorphic fringe subtrees in a
uniformly random ordered binary tree with $n$ leaves. For two constants
$c_{1}\approx 1.0591261434$ and $c_{2}\approx 1.0761505454$, the following
holds:
1. (i)
$\displaystyle c_{1}\frac{n}{\sqrt{\ln n}}(1+o(1))\leq\mathbb{E}(F_{n})\leq
c_{2}\frac{n}{\sqrt{\ln n}}(1+o(1))$,
2. (ii)
$\displaystyle c_{1}\frac{n}{\sqrt{\ln n}}(1+o(1))\leq F_{n}\leq
c_{2}\frac{n}{\sqrt{\ln n}}(1+o(1))$ with high probability.
###### Theorem 2
Let $G_{n}$ be the total number of non-isomorphic fringe subtrees in a random
binary search tree with $n$ leaves. For two constants $c_{3}\approx
1.5470025923$ and $c_{4}\approx 1.8191392203$, the following holds:
1. (i)
$\displaystyle c_{3}\frac{n}{\ln n}(1+o(1))\leq\mathbb{E}(G_{n})\leq
c_{4}\frac{n}{\ln n}(1+o(1))$,
2. (ii)
$\displaystyle c_{3}\frac{n}{\ln n}(1+o(1))\leq G_{n}\leq c_{4}\frac{n}{\ln
n}(1+o(1))$ with high probability.
To prove the above Theorems 1 and 2, we refine techniques from [26]. Our proof
technique also applies to the problem of estimating the number of distinct
ordered fringe subtrees in uniformly random binary trees or in random binary
search trees. In this case, upper and lower bounds for the expected value have
already been proven by other authors. Our new contribution is to show that
they also hold with high probability.
###### Theorem 3
Let $H_{n}$ denote the total number of distinct fringe subtrees in a uniformly
random ordered binary tree with $n$ leaves. Then, for the constant
$c=2\sqrt{\ln 4/\pi}\approx 1.3285649405$, the following holds:
1. (i)
$\displaystyle\mathbb{E}(H_{n})=c\frac{n}{\sqrt{\ln n}}(1+o(1))$,
2. (ii)
$\displaystyle H_{n}=c\frac{n}{\sqrt{\ln n}}(1+o(1))$ with high probability.
Here, the first part (i) was already shown in [19] and [26], part (ii) is new.
Similarly, we are able to strengthen the results of [10] and [27]:
###### Theorem 4
Let $J_{n}$ be the total number of distinct fringe subtrees in a random binary
search tree with $n$ leaves. For two constants $c_{5}\approx 2.4071298335$ and
$c_{6}\approx 2.7725887222$, the following holds:
1. (i)
$\displaystyle c_{5}\frac{n}{\ln n}(1+o(1))\leq\mathbb{E}(J_{n})\leq
c_{6}\frac{n}{\ln n}(1+o(1))$,
2. (ii)
$\displaystyle c_{5}\frac{n}{\ln n}(1+o(1))\leq J_{n}\leq c_{6}\frac{n}{\ln
n}(1+o(1))$ with high probability.
The upper bound in part (i) can already be found in [17] and [10]. Moreover, a
lower bound of the form $\mathbb{E}(J_{n})\geq\frac{\alpha n}{\ln n}(1+o(1))$
was already shown in [10] for the constant $\alpha=(\ln 3)/2\approx
0.5493061443$ and in [27] for the constant $\alpha\approx 0.6017824584$. So
our new contributions are part (ii) and the improvement of the lower bound on
$\mathbb{E}(J_{n})$.
## 2 Preliminaries
Let $\mathcal{T}$ denote the set of ordered binary trees, i.e. of ordered
rooted trees such that each node has either exactly two or no children. We
define the _size_ $|t|$ of a binary tree $t\in\mathcal{T}$ as the number of
leaves of $t$ and by $\mathcal{T}_{k}$ we denote the set of binary trees of
size $k$ for every integer $k\geq 1$. It is well known that
$|\mathcal{T}_{k}|=C_{k-1}$, where $C_{k}$ denotes the $k$-th _Catalan number_
[18]: We have
$\displaystyle
C_{k}=\frac{1}{k+1}\binom{2k}{k}\sim\frac{4^{k}}{\sqrt{\pi}k^{3/2}}(1+O(1/k)),$
(1)
where the asymptotic growth of the Catalan numbers follows from Stirling’s
Formula [18]. Analogously, let $\mathcal{U}$ denote the set of unordered
binary trees, i.e. of unordered rooted trees such that each node has either
exactly two or no children. The _size_ $|u|$ of an unordered tree
$u\in\mathcal{U}$ is again the number of leaves of $u$ and by
$\mathcal{U}_{k}$ we denote the set of unordered binary trees of size $k$. We
have $|\mathcal{U}_{k}|=W_{k}$, where $W_{k}$ denotes the $k$-th _Wedderburn-
Etherington number_. Their asymptotic growth is
$\displaystyle W_{k}\sim A\cdot k^{-3/2}\cdot b^{k},$ (2)
for certain positive constants $A,b$ [4, 16]. In particular, we have $b\approx
2.4832535362$.
A _fringe subtree_ of a binary tree is a subtree consisting of a node and all
its descendants. For a binary tree $t$ and a given node $v\in t$, let $t(v)$
denote the fringe subtree of $t$ rooted at $v$. Two fringe subtrees are called
_distinct_ if they are distinct as ordered binary trees.
Every tree $t\in\mathcal{T}$ can be considered as an element of $\mathcal{U}$
by simply forgetting the ordering on $t$’s nodes. If two binary trees
$t_{1},t_{2}$ correspond to the same unordered tree $u\in\mathcal{U}$, we call
them _isomorphic_ : Thus, we obtain a partition of $\mathcal{T}$ into
isomorphism classes. If two binary trees $t_{1},t_{2}\in\mathcal{T}$ belong to
the same isomorphism class, we can obtain $t_{1}$ from $t_{2}$ and vice versa
by reordering the children of some of $t_{1}$’s (respectively, $t_{2}$’s)
inner nodes. An inner node $v$ of an ordered or unordered binary tree $t$ is
called a _symmetrical node_ if the fringe subtrees rooted at $v$’s children
are isomorphic. Let $\operatorname{sym}(t)$ denote the number of symmetrical
nodes of $t$. The cardinality of the automorphism group of $t$ is given by
$\lvert\operatorname{\mathrm{Aut}}(t)\rvert=2^{\operatorname{sym}(t)}$. Thus,
by the orbit-stabilizer theorem, there are $2^{k-1-\operatorname{sym}(t)}$
many ordered binary trees in the isomorphism class of $t\in\mathcal{T}_{k}$,
and likewise $2^{k-1-\operatorname{sym}(t)}$ many ordered representations of
$t\in\mathcal{U}_{k}$.
We consider two types of probability distributions on the set of ordered
binary trees of size $n$:
* (i)
The _uniform probability distribution_ on $\mathcal{T}_{n}$, that is, every
binary tree of size $n$ is assigned the same probability $\frac{1}{C_{n-1}}$.
A random variable taking values in $\mathcal{T}_{n}$ according to the uniform
probability distribution is called a _uniformly random (ordered) binary tree_
of size $n$.
* (ii)
The probability distribution induced by the so-called _Binary Search Tree
Model_ (see e.g. [12, 17]): The corresponding probability mass function
$P_{\operatorname{bst}}:\mathcal{T}_{n}\rightarrow[0,1]$ is given by
$\displaystyle P_{\operatorname{bst}}(t)=\prod_{v\in
t\atop|t(v)|>1}\frac{1}{|t(v)|-1},$ (3)
for every $n\geq 1$. A random variable taking values in $\mathcal{T}_{n}$
according to this probability mass function is called a _random binary search
tree_ of size $n$.
Before we start with proving our main results, we need two preliminary lemmas
on the number of fringe subtrees in uniformly random ordered binary trees and
in random binary search trees:
###### Lemma 1
Let $a,\varepsilon$ be positive real numbers with $\varepsilon<\frac{1}{3}$.
For every positive integer $k$ with $a\ln n\leq k\leq n^{\varepsilon}$, let
$\mathcal{S}_{k}\subset\mathcal{T}_{k}$ be a set of ordered binary trees with
$k$ leaves. We denote the cardinality of $\mathcal{S}_{k}$ by $s_{k}$. Let
$X_{n,k}$ denote the (random) number of fringe subtrees with $k$ leaves in a
uniformly random ordered binary tree with $n$ leaves that belong to
$\mathcal{S}_{k}$. Moreover, let $Y_{n,\varepsilon}$ denote the (random)
number of arbitrary fringe subtrees with more than $n^{\varepsilon}$ leaves in
a uniformly random ordered binary tree with $n$ leaves. We have
1. (1)
$\mathbb{E}(X_{n,k})=s_{k}4^{1-k}n\big{(}1+O(k/n)\big{)}$ for all $k$ with
$a\ln n\leq k\leq n^{\varepsilon}$, the $O$-constant being independent of $k$,
2. (2)
$\mathbb{V}(X_{n,k})=s_{k}4^{1-k}n(1+O(k^{-1/2}))$ for all $k$ with $a\ln
n\leq k\leq n^{\varepsilon}$, again with an $O$-constant that is independent
of $k$,
3. (3)
$\mathbb{E}(Y_{n,\varepsilon})=O(n^{1-\varepsilon/2})$ and
4. (4)
with high probability, the following statements hold simultaneously:
* (i)
$|X_{n,k}-\mathbb{E}(X_{n,k})|\leq s_{k}^{1/2}2^{-k}n^{1/2+\varepsilon}$ for
all $k$ with $a\ln n\leq k\leq n^{\varepsilon}$,
* (ii)
$Y_{n,\varepsilon}\leq n^{1-\varepsilon/3}$.
We emphasize (since it will be important later) that the inequality in part
(4), item (i), does not only hold with high probability for each individual
$k$, but that it is satisfied with high probability for all $k$ in the given
range simultaneously.
###### Proof
(1) Recall first that the number of ordered binary trees with $n$ leaves is
the Catalan number $C_{n-1}=\frac{1}{n}\binom{2n-2}{n-1}$. We observe that
every occurrence of a fringe subtree in $\mathcal{S}_{k}$ in a tree with $n$
leaves can be obtained by choosing an ordered tree with $n-k+1$ leaves,
picking one of the leaves and replacing it by a tree in $\mathcal{S}_{k}$.
Thus the total number of occurrences is
$\frac{1}{n-k+1}\binom{2n-2k}{n-k}\cdot(n-k+1)\cdot
s_{k}=\binom{2n-2k}{n-k}s_{k}.$
Consequently, the average number is
$\mathbb{E}(X_{n,k})=\frac{\binom{2n-2k}{n-k}s_{k}}{\frac{1}{n}\binom{2n-2}{n-1}}=s_{k}4^{1-k}n\big{(}1+O(k/n)\big{)},$
by Stirling’s formula (the $O$-constant being independent of $k$ in the
indicated range).
(2) The variance is determined in a similar fashion: we first count the total
number of pairs of fringe subtrees in $\mathcal{S}_{k}$ that appear in the
same ordered tree with $n$ leaves. Each such pair can be obtained as follows:
take an ordered tree with $n-2k+2$ leaves, pick two leaves, and replace them
by fringe subtrees in $\mathcal{S}_{k}$. The total number is thus
$\frac{1}{n-2k+2}\binom{2n-4k+2}{n-2k+1}\cdot\binom{n-2k+2}{2}\cdot
s_{k}^{2}=\frac{n-2k+1}{2}\binom{2n-4k+2}{n-2k+1}s_{k}^{2},$
giving us
$\mathbb{E}\Big{(}\binom{X_{n,k}}{2}\Big{)}=\frac{\frac{n-2k+1}{2}\binom{2n-4k+2}{n-2k+1}s_{k}^{2}}{\frac{1}{n}\binom{2n-2}{n-1}}=s_{k}^{2}4^{2-2k}\frac{n^{2}}{2}\big{(}1+O(k/n)\big{)},$
again by Stirling’s formula. The second moment and the variance are now
derived from this formula in a straightforward fashion: We find
$\displaystyle\mathbb{E}(X_{n,k}^{2})=2\mathbb{E}\Big{(}\binom{X_{n,k}}{2}\Big{)}+\mathbb{E}(X_{n,k})=\left(s_{k}^{2}4^{2-2k}n^{2}+s_{k}4^{1-k}n\right)\big{(}1+O(k/n)\big{)},$
and thus, as $s_{k}/C_{k-1}\leq 1$,
$\displaystyle\mathbb{V}(X_{n,k})$
$\displaystyle=\mathbb{E}(X_{n,k}^{2})-\mathbb{E}(X_{n,k})^{2}=O(s_{k}^{2}4^{2-2k}nk)+s_{k}4^{1-k}n(1+O(k/n))$
$\displaystyle=O\left(\frac{s_{k}^{2}}{C_{k-1}^{2}}\frac{n}{k^{2}}\right)+\frac{s_{k}}{C_{k-1}}\frac{n}{\sqrt{\pi}k^{3/2}}(1+O(1/k))=s_{k}4^{1-k}n(1+O(1/k^{1/2})).$
(3) To obtain the estimate for $\mathbb{E}(Y_{n,\varepsilon})$, we observe
that the average total number of fringe subtrees with $k$ leaves is
$\frac{\binom{2n-2k}{n-k}\cdot\frac{1}{k}\binom{2k-2}{k-1}}{\frac{1}{n}\binom{2n-2}{n-1}}=O\Big{(}\frac{n^{3/2}}{k^{3/2}(n-k+1)^{1/2}}\Big{)},$
where the estimate follows from Stirling’s formula again for
$k>n^{\varepsilon}$. Summing over all $k$, we get
$\mathbb{E}(Y_{n,\varepsilon})=O\Big{(}n^{3/2}\sum_{n^{\varepsilon}<k\leq
n}\frac{1}{k^{3/2}(n-k+1)^{1/2}}\Big{)}=O(n^{1-\varepsilon/2}).$
(4) For the second part, we apply Chebyshev’s inequality to obtain
concentration of $X_{n,k}$:
$\mathbb{P}\Big{(}\big{|}X_{n,k}-\mathbb{E}(X_{n,k})\big{|}\geq
s_{k}^{1/2}2^{-k}n^{1/2+\varepsilon}\Big{)}\leq\frac{\mathbb{V}(X_{n,k})}{s_{k}4^{-k}n^{1+2\varepsilon}}=O\big{(}n^{-2\varepsilon}\big{)}.$
Hence, by the union bound, the probability that the stated inequality fails
for any $k$ in the given range is only $O(n^{-\varepsilon})$, proving that the
first statement holds with high probability. Finally, Markov’s inequality
implies that
$\mathbb{P}\big{(}Y_{n,\varepsilon}>n^{1-\varepsilon/3}\big{)}\leq\frac{\mathbb{E}(Y_{n,\varepsilon})}{n^{1-\varepsilon/3}}=O(n^{-\varepsilon/6}),$
showing that the second inequality holds with high probability as well.
$\scriptstyle\blacksquare$
For the number of fringe subtrees in random binary search trees, a very
similar lemma holds:
###### Lemma 2
Let $a,\varepsilon$ be positive real numbers with $\varepsilon<\frac{1}{3}$
and let $n$ and $k$ denote positive integers. Moreover, for every $k$, let
$\mathcal{S}_{k}\subset\mathcal{T}_{k}$ be a set of ordered binary trees with
$k$ leaves and let $p_{k}$ denote the probability that a random binary search
tree is contained in $\mathcal{S}_{k}$, that is, $p_{k}=\sum
P_{\operatorname{bst}}(t)$, where the sum is taken over all binary trees in
$\mathcal{S}_{k}$. Let $X_{n,k}$ denote the (random) number of fringe subtrees
with $k$ leaves in a random binary search tree with $n$ leaves that belong to
$\mathcal{S}_{k}$. Moreover, let $Y_{n,\varepsilon}$ denote the (random)
number of arbitrary fringe subtrees with more than $n^{\varepsilon}$ leaves in
a random binary search tree with $n$ leaves. We have
* (1)
$\mathbb{E}(X_{n,k})=\frac{2p_{k}n}{k(k+1)}$ for $1\leq k<n$,
* (2)
$\mathbb{V}(X_{n,k})=O(p_{k}n/k^{2})$ for all $k$ with $a\ln n\leq k\leq
n^{\varepsilon}$, where the $O$-constant is independent of $k$,
* (3)
$\mathbb{E}(Y_{n,\varepsilon})=2n/\lceil
n^{\varepsilon}\rceil-1=O(n^{1-\varepsilon})$ and
* (4)
with high probability, the following statements hold simultaneously:
* (i)
$|X_{n,k}-\mathbb{E}(X_{n,k})|\leq p_{k}^{1/2}k^{-1}n^{1/2+\varepsilon}$ for
all $k$ with $a\ln n\leq k\leq n^{\varepsilon}$,
* (ii)
$Y_{n,\varepsilon}\leq n^{1-\varepsilon/2}$.
###### Proof
(1) In order to estimate $\mathbb{E}(X_{n,k})$, we define $Z_{n,k}$ as the
(random) number of arbitrary fringe subtrees with $k$ leaves in a random
binary search tree with $n$ leaves. That is, $Z_{n,k}=X_{n,k}$ for
$\mathcal{S}_{k}=\mathcal{T}_{k}$. Applying the law of total expectation, we
find
$\displaystyle\mathbb{E}(X_{n,k})=\sum_{m=0}^{n}\mathbb{E}(X_{n,k}\mid
Z_{n,k}=m)\mathbb{P}(Z_{n,k}=m).$
As $X_{n,k}$ conditioned on $Z_{n,k}=m$ for some integer $m$ is binomially
distributed with parameters $m$ and $p_{k}$, we find $\mathbb{E}(X_{n,k}\mid
Z_{n,k}=m)=mp_{k}$ and hence
$\displaystyle\mathbb{E}(X_{n,k})=p_{k}\sum_{m=0}^{n}m\mathbb{P}(Z_{n,k}=m)=p_{k}\mathbb{E}(Z_{n,k}).$
With $\mathbb{E}(Z_{n,k})=\frac{2n}{k(k+1)}$ (see for example [14]), the
statement follows.
(2) In order to estimate $\mathbb{V}(X_{n,k})$, we apply the law of total
variance:
$\displaystyle\mathbb{V}(X_{n,k})=\mathbb{V}(\mathbb{E}(X_{n,k}\mid
Z_{n,k}))+\mathbb{E}(\mathbb{V}(X_{n,k}\mid Z_{n,k})).$
Again as $X_{n,k}$ conditioned on $Z_{n,k}=m$ for some integer $m$ is
binomially distributed with parameters $m$ and $p_{k}$, we find
$\mathbb{E}(X_{n,k}\mid Z_{n,k})=p_{k}Z_{n,k}$ and $\mathbb{V}(X_{n,k}\mid
Z_{n,k})=Z_{n,k}p_{k}(1-p_{k})$. Thus, we have
$\displaystyle\mathbb{V}(X_{n,k})=\mathbb{V}(p_{k}Z_{n,k})+\mathbb{E}(Z_{n,k}p_{k}(1-p_{k}))=\mathbb{V}(Z_{n,k})p_{k}^{2}+\mathbb{E}(Z_{n,k})p_{k}(1-p_{k}).$
With $\mathbb{E}(Z_{n,k})=\frac{2n}{k(k+1)}$ and
$\mathbb{V}(Z_{n,k})=\frac{2(k-1)(4k^{2}-3k-4)n}{(k+1)^{2}k(2k-1)(2k+1)},$
(see for example [14]), this yields
$\displaystyle\mathbb{V}(X_{n,k})=\frac{2(k-1)(4k^{2}-3k-4)np_{k}^{2}}{(k+1)^{2}k(2k-1)(2k+1)}+\frac{2np_{k}(1-p_{k})}{k(k+1)}=O\left(\frac{np_{k}}{k^{2}}\right).$
(3) In order to estimate $\mathbb{E}(Y_{n,\varepsilon})$, first observe that
$\displaystyle\mathbb{E}(Y_{n,\varepsilon})=\sum_{k>n^{\varepsilon}}\mathbb{E}(Z_{n,k}).$
With $\mathbb{E}(Z_{n,k})=\frac{2n}{k(k+1)}$ for $n^{\varepsilon}<k<n$ and
$\mathbb{E}(Z_{n,n})=1$, this yields
$\displaystyle\mathbb{E}(Y_{n,\varepsilon})=\sum_{n^{\varepsilon}<k\leq
n-1}\frac{2n}{k(k+1)}+1=\frac{2n}{\lceil
n^{\varepsilon}\rceil}-1=O(n^{1-\varepsilon}).$
(4) For the second part of the statement, we apply Chebyshev’s inequality to
obtain:
$\displaystyle\mathbb{P}\left(\left|X_{n,k}-\frac{2np_{k}}{k(k+1)}\right|\geq
p_{k}^{1/2}n^{1/2+\varepsilon}k^{-1}\right)\leq\frac{\mathbb{V}(X_{n,k})k^{2}}{p_{k}n^{1+2\varepsilon}}=O(n^{-2\varepsilon}).$
Hence, by the union bound, the probability that the stated inequality fails
for any $k$ in the given range is $O(n^{-\varepsilon})$, proving that the
given statement holds with high probability. Furthermore, with Markov’s
inequality, we find
$\displaystyle\mathbb{P}(Y_{n,\varepsilon}>n^{1-\varepsilon/2})\leq\frac{\mathbb{E}(Y_{n,\varepsilon})}{n^{1-\varepsilon/2}}=O(n^{-\varepsilon/2}).$
Thus, the second inequality holds with high probability as well.
$\scriptstyle\blacksquare$
## 3 Fringe Subtrees in Uniformly Random Binary Trees
### 3.1 Ordered Fringe Subtrees
We provide the proof of Theorem 3 first, since it is simplest and provides us
with a template for the other proofs. Basically, it is a refinement of the
proof for the corresponding special case of Theorem 3.1 in [26]. In the
following sections, we refine the argument further to prove Theorems 1, 2 and
4.
###### Proof (Proof of Theorem 3)
We prove the statement in two steps: In the first step, we show that the upper
bound $H_{n}\leq cn/\sqrt{\ln n}(1+o(1))$ holds for $c=2\sqrt{\ln 4/\pi}$ both
in expectation and with high probability. In the second step, we prove the
corresponding lower bound.
_The upper bound:_ Let $k_{0}=\log_{4}n$. The number $H_{n}$ of distinct
fringe subtrees in a uniformly random ordered binary tree with $n$ leaves
equals (i) the number of such distinct fringe subtrees of size at most $k_{0}$
plus (ii) the number of such distinct fringe subtrees of size greater than
$k_{0}$. We upper-bound (i) by the number of all ordered binary trees of size
at most $k_{0}$ (irrespective of their occurrence as fringe subtrees), which
is
$\displaystyle\sum_{k=0}^{k_{0}-1}C_{k}=O\left(\frac{4^{k_{0}}}{k_{0}^{3/2}}\right)=O\left(\frac{n}{(\ln
n)^{3/2}}\right).$
This upper bound holds deterministically. Furthermore, we upper-bound (ii) by
the total number of fringe subtrees of size greater than $k_{0}$ occurring in
the tree: We apply Lemma 1 with $a=1/\ln 4$ and $\varepsilon=1/6$ and let
$\mathcal{S}_{k}$ denote the set $\mathcal{T}_{k}$, such that $s_{k}=C_{k-1}$,
to obtain:
$\displaystyle\left(\sum_{k_{0}<k\leq
n^{\varepsilon}}X_{n,k}\right)+Y_{n,\varepsilon}$
$\displaystyle=\frac{n}{\sqrt{\pi}}\sum_{k_{0}<k\leq
n^{\varepsilon}}k^{-3/2}\left(1+O(k^{-1})\right)+O(n^{1-\varepsilon/3})$
$\displaystyle=\frac{2\sqrt{\ln 4}}{\sqrt{\pi}}\cdot\frac{n}{\sqrt{\ln
n}}+O(n/(\ln n)^{3/2}),$
in expectation and with high probability as well, as the estimate from Lemma 1
(part (4)) holds with high probability simultaneously for all $k$ in the given
range. As we have
$\displaystyle H_{n}\leq\sum_{k\leq k_{0}}C_{k-1}+\Big{(}\sum_{k_{0}<k\leq
n^{\varepsilon}}X_{n,k}\Big{)}+Y_{n,\varepsilon},$
we can combine the two bounds to obtain the upper bound on $H_{n}$ stated in
Theorem 3, both in expectation and with high probability.
_The lower bound:_ Again, let $k_{0}=\log_{4}n$ and $\varepsilon=\frac{1}{6}$.
From the first part of the proof, we find that the main contribution to the
total number of fringe subtrees in a uniformly random binary tree of size $n$
comes from fringe subtrees of sizes $k$ with $k_{0}<k\leq n^{\varepsilon}$.
Hence, in order to lower-bound the number $H_{n}$ of distinct fringe subtrees
in a uniformly random binary tree with $n$ leaves, we only count distinct
fringe subtrees of sizes $k$ with $k_{0}<k\leq n^{\varepsilon}$ and show that
we did not overcount too much in the first part of the proof by upper-bounding
this number by the total number of fringe subtrees of sizes $k$. To this end,
let $X_{n,k}^{(2)}$ denote the number of pairs of identical fringe subtrees of
size $k$ in a uniformly random ordered binary tree of size $n$. Each such pair
can be obtained as follows: Take an ordered tree with $n-2k+2$ leaves, pick
two leaves, and replace them by the same ordered binary tree of size $k$. The
total number of such pairs of identical fringe subtrees of size $k$ is thus
$\displaystyle C_{n-2k+1}\cdot\binom{n-2k+2}{2}\cdot
C_{k-1}=\frac{4^{n-k}}{2\pi k^{3/2}}(n-2k+1)^{1/2}(1+O(1/k)).$
By dividing by $C_{n-1}$, i.e. the total number of binary trees of size $n$,
we thus obtain the expected value:
$\displaystyle\mathbb{E}(X_{n,k}^{(2)})=\frac{1}{C_{n-1}}\frac{4^{n-k}}{2\pi
k^{3/2}}(n-2k+1)^{1/2}(1+O(1/k))=O(4^{-k}n^{2}k^{-3/2}).$
Thus, we find
$\displaystyle\sum_{k_{0}<k\leq
n^{\varepsilon}}\mathbb{E}(X_{n,k}^{(2)})=O\left(n^{2}\frac{4^{-k_{0}}}{k_{0}^{3/2}}\right)=O\left(\frac{n}{(\ln
n)^{3/2}}\right).$
If a binary tree of size $k$ occurs $m$ times as a fringe subtree in a
uniformly random binary tree of size $n$, it contributes $m-\binom{m}{2}$ to
the random variable $X_{n,k}-X_{n,k}^{(2)}$. Since $m-\binom{m}{2}\leq 1$ for
all non-negative integers $m$, we find that $X_{n,k}-X_{n,k}^{(2)}$ is a lower
bound on the number of distinct fringe subtrees with $k$ leaves. Hence, we
have
$\displaystyle H_{n}\geq\sum_{k_{0}<k\leq
n^{\varepsilon}}X_{n,k}-\sum_{k_{0}<k\leq n^{\varepsilon}}X_{n,k}^{(2)}.$
The second sum is $O(n/(\ln n)^{3/2})$ in expectation and thus with high
probability as well by the Markov inequality. As the first sum is
$\frac{2\sqrt{\ln 4}}{\sqrt{\pi}}\cdot\frac{n}{\sqrt{\ln n}}(1+o(1)),$ both in
expectation and with high probability by our estimate from the first part of
the proof, the statement of Theorem 3 follows. $\scriptstyle\blacksquare$
As the main idea of the proof is to split the number of distinct fringe
subtrees into the number of distinct fringe subtrees of size at most $k_{0}$
plus the number of distinct fringe subtrees of size greater than $k_{0}$ for
some suitably chosen integer $k_{0}$, this type of argument is called a _cut-
point argument_ and the integer $k_{0}$ is called the _cut-point_ (see [17]).
This basic technique is applied in several previous papers to similar problems
(see for instance [10], [17], [26], [27]). Moreover, we remark that the
statement of Theorem 3 can be easily generalized to simply generated families
of trees.
### 3.2 Unordered Fringe Subtrees
In this subsection, we prove Theorem 1. For this, we refine the cut-point
argument we applied in the proof of Theorem 3: In particular, for the lower
bound on $F_{n}$, we need a result due to Bóna and Flajolet [4] on the number
of automorphisms of a uniformly random ordered binary tree. It is stated for
random phylogenetic trees in [4], but the two probabilistic models are
equivalent.
###### Theorem 5 ([4], Theorem 2)
Consider a uniformly random ordered binary tree $T_{k}$ with $k$ leaves, and
let $A_{k}=\lvert\operatorname{\mathrm{Aut}}(T_{k})\rvert$ be the cardinality
of its automorphism group. The logarithm of this random variable satisfies a
central limit theorem: For certain positive constants $\gamma$ and
$\sigma_{1}$, we have
$\mathbb{P}(A_{k}\leq 2^{\gamma
k+\sigma_{1}\sqrt{k}x})\overset{k\to\infty}{\to}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\,dt$
for every real number $x$. The numerical value of the constant $\gamma$ is
$0.2710416936$.
With Theorem 5, we are able to upper-bound the probability that two fringe
subtrees of the same size are isomorphic in our proof of Theorem 1:
###### Proof (Proof of Theorem 1)
We prove the statement in two steps: First, we show that the upper bound on
$F_{n}$ stated in Theorem 1 holds both in expectation and with high
probability, then we prove the respective lower bound.
_The upper bound:_ The proof for the upper bound in Theorem 1 exactly matches
the first part of the proof of Theorem 3, except that we choose a different
cut-point: Let $k_{0}=\log_{b}n$, where $b\approx 2.4832535362$ is the
constant in the asymptotic formula (2) for the Wedderburn-Etherington numbers.
We then find
$\displaystyle F_{n}\leq\sum_{k<k_{0}}W_{k}+\Big{(}\sum_{k_{0}\leq k\leq
n^{\epsilon}}X_{n,k}\Big{)}+Y_{n,\epsilon}=\frac{2\sqrt{\ln
b}}{\sqrt{\pi}}\cdot\frac{n}{\sqrt{\ln n}}+O(n(\ln n)^{-3/2}),$
both in expectation and with high probability, where the estimates for
$X_{n,k}$ and $Y_{n,\varepsilon}$ follow again from Lemma 1. We have
$2\sqrt{\ln b}/\sqrt{\pi}\approx 1.0761505454$.
_The lower bound:_ As a consequence of Theorem 5, the probability that the
cardinality of the automorphism group of a uniformly random binary tree
$T_{k}$ of size $k$ satisfies
$\lvert\operatorname{\mathrm{Aut}}(T_{k})\rvert\leq 2^{\gamma k-k^{3/4}}$
tends to $0$ as $k\to\infty$. We define $\mathcal{S}_{k}$ as the set of
ordered trees with $k$ leaves that do not satisfy this inequality, so that
$s_{k}=|\mathcal{S}_{k}|=C_{k-1}(1+o(1))$. Our lower bound is based on
counting only fringe subtrees in $\mathcal{S}_{k}$ for suitable $k$. The
reason for this choice is that we have an upper bound on the number of ordered
binary trees in the same isomorphism class for every tree in
$\mathcal{S}_{k}$. Recall that the number of possible ordered representations
of an unordered binary tree $t$ with $k$ leaves is given by
$2^{k-1}/\lvert\operatorname{\mathrm{Aut}}(t)\rvert$ by the orbit-stabiliser
theorem. Hence, the number of ordered binary trees in the same isomorphism
class as a tree $t\in\mathcal{S}_{k}$ is bounded above by $2^{k-1-\gamma
k+k^{3/4}}$.
Now set $k_{1}=\frac{1+\delta}{1+\gamma}\log_{2}n$ for some positive constant
$\delta<\frac{2}{3}$, and consider only fringe subtrees that belong to
$\mathcal{S}_{k}$, where $k_{1}\leq k\leq n^{\delta/2}$. By Lemma 1, the
number of such fringe subtrees in a random ordered binary tree with $n$ leaves
is
$s_{k}4^{1-k}n(1+O(k/n+s_{k}^{-1/2}2^{k}n^{(\delta-1)/2}))$
both in expectation and with high probability. Since $s_{k}=C_{k-1}(1+o(1))$,
the number of fringe subtrees that belong to $\mathcal{S}_{k}$ in a random
ordered binary tree of size $n$ becomes $\frac{n}{\sqrt{\pi k^{3}}}(1+o(1))$.
We show that most of these trees are the only representatives of their
isomorphism classes as fringe subtrees. To this end, we consider all fringe
subtrees in $\mathcal{S}_{k}$ for some $k$ that satisfies $k_{1}\leq k\leq
n^{\delta/2}$. Let the sizes of the isomorphism classes of trees in
$\mathcal{S}_{k}$ be $r_{1},r_{2},\ldots,r_{\ell}$, so that
$r_{1}+r_{2}+\cdots+r_{\ell}=s_{k}$. By definition of $\mathcal{S}_{k}$, we
have $r_{i}\leq 2^{k-1-\gamma k+k^{3/4}}$ for every $i$. Let us condition on
the event that their number $X_{n,k}$ is equal to $N$ for some $N\leq n$. Each
of these $N$ fringe subtrees $S_{1},S_{2},\ldots,S_{N}$ follows a uniform
distribution among the elements of $\mathcal{S}_{k}$, so the probability of
being in an isomorphism class with $r_{i}$ elements is $r_{i}/s_{k}$.
Moreover, the $N$ fringe subtrees are also all independent. Let
$X_{n,k}^{(2)}$ be the number of pairs of isomorphic trees among the fringe
subtrees with $k$ leaves. We have
$\displaystyle\mathbb{E}\big{(}X_{n,k}^{(2)}|X_{n,k}=N\big{)}$
$\displaystyle=\binom{N}{2}\sum_{i}\Big{(}\frac{r_{i}}{s_{k}}\Big{)}^{2}\leq\frac{n^{2}}{2s_{k}^{2}}\sum_{i}r_{i}^{2}\leq\frac{n^{2}}{s_{k}}2^{k-2-\gamma
k+k^{3/4}}.$
Since this holds for all $N$, the law of total expectation yields
$\mathbb{E}\big{(}X_{n,k}^{(2)}\big{)}\leq\frac{n^{2}}{s_{k}}2^{k-2-\gamma
k+k^{3/4}}=\sqrt{\pi}n^{2}k^{3/2}2^{-k-\gamma k+k^{3/4}}(1+o(1)).$
Since $k\geq k_{1}=\frac{1+\delta}{1+\gamma}\log_{2}n$, we find that
$\mathbb{E}\big{(}X_{n,k}^{(2)}\big{)}\leq
n^{2}2^{-(1+\gamma)k+O(k^{3/4})}\leq n^{1-\delta}\exp\big{(}O((\ln
n)^{3/4})\big{)}.$
Thus
$\sum_{k_{1}\leq k\leq n^{\delta/2}}\mathbb{E}\big{(}X_{n,k}^{(2)}\big{)}\leq
n^{1-\delta/2}\exp\big{(}O((\ln n)^{3/4})\big{)}=o(n/\sqrt{\ln n}).$
As in the previous proof, we see that $X_{n,k}-X_{n,k}^{(2)}$ is a lower bound
on the number of non-isomorphic fringe subtrees with $k$ leaves. This gives us
$F_{n}\geq\sum_{k_{1}\leq k\leq n^{\delta/2}}X_{n,k}-\sum_{k_{1}\leq k\leq
n^{\delta/2}}X_{n,k}^{(2)}.$
The second sum is negligible since it is $o(n/\sqrt{\ln n})$ in expectation
and thus also with high probability by the Markov inequality. For the first
sum, a calculation similar to that for the upper bound shows that it is
$\frac{2\sqrt{(1+\gamma)\ln 2}}{\sqrt{\pi(1+\delta)}}\cdot\frac{n}{\sqrt{\ln
n}}(1+o(1)),$
both in expectation and with high probability. Since $\delta$ is arbitrary, we
can choose any constant smaller than $\frac{2\sqrt{(1+\gamma)\ln
2}}{\sqrt{\pi}}\approx 1.0591261434$ for $c_{1}$. $\scriptstyle\blacksquare$
## 4 Fringe Subtrees in Random Binary Search Trees
In this section, we prove our results presented in Theorem 2 and Theorem 4 on
the number of distinct, respectively, non-isomorphic fringe subtrees in a
random binary search tree. In order to show the respective lower bounds of
Theorem 2 and Theorem 4, we need two theorems similar to Theorem 5: The first
one shows that the logarithm of the random variable
$B_{k}=P_{\operatorname{bst}}(T_{k})^{-1}$, where $T_{k}$ denotes a random
binary search tree of size $k$, satisfies a central limit theorem and is
needed to estimate the probability that two fringe subtrees in a random binary
search tree are identical. The second one transfers the statement of Theorem 5
from uniformly random binary trees to random binary search trees and is needed
in order to estimate the probability that two fringe subtrees in a random
binary search tree are isomorphic. The first of these two central limit
theorems is shown in [15]:
###### Theorem 6 ([15], Theorem 4.1)
Consider a random binary search tree $T_{k}$ with $k$ leaves, and let
$B_{k}=P_{\operatorname{bst}}(T_{k})^{-1}$. The logarithm of this random
variable satisfies a central limit theorem: For certain positive constants
$\mu$ and $\sigma_{2}$, we have
$\displaystyle\mathbb{P}\left(B_{k}\leq 2^{\mu
k+\sigma_{2}\sqrt{k}x}\right)\overset{k\to\infty}{\to}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\,dt$
for every real number $x$. The numerical value of the constant $\mu$ is
$\displaystyle\mu=\sum_{k=1}^{\infty}\frac{2\log_{2}k}{(k+1)(k+2)}\approx
1.7363771368.$
The second of these two central limit theorems follows from a general theorem
devised by Holmgren and Janson [22]: Let $f:\mathcal{T}\rightarrow\mathbb{R}$
denote a function mapping an ordered binary tree to a real number. Moreover,
given such a mapping $f$, define
$\mathcal{F}:\mathcal{T}\rightarrow\mathbb{R}$ by
$\displaystyle\mathcal{F}(t)=\sum_{v\in t}f(t(v)).$
The theorem by Holmgren and Janson states:
###### Theorem 7 ([22], Theorem 1.14)
Let $T_{k}$ be a random binary search tree of size $k$. If
$\displaystyle\sum_{k=1}^{\infty}\frac{\mathbb{V}(f(T_{k}))^{1/2}}{k^{3/2}}<\infty,\quad\lim_{k\rightarrow\infty}\frac{\mathbb{V}(f(T_{k}))}{k}=0\quad\text{
and }\quad$
$\displaystyle\sum_{k=1}^{\infty}\frac{\mathbb{E}(f(T_{k}))^{2}}{k^{2}}<\infty,$
then for certain constants $\nu$ and $\sigma\geq 0$, we have
$\mathbb{E}(\mathcal{F}(T_{k}))\sim\nu k\text{ and
}\mathbb{V}(\mathcal{F}(T_{k}))\sim\sigma^{2}k.$
Moreover, if $\sigma\neq 0$, then
$\displaystyle\mathbb{P}(\mathcal{F}(T_{k})\leq\nu
k+\sigma\sqrt{k}x)\overset{k\to\infty}{\to}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\,dt$
for every real number $x$. In particular, we have
$\displaystyle\nu=\sum_{k=1}^{\infty}\frac{2}{k(k+1)}\mathbb{E}(f(T_{k})).$
Note that in [22], the equivalent binary search model is considered that
allows binary trees to have unary nodes, so that the index of summation has to
be shifted in the sum defining $\nu$. Moreover, note that if we set
$f(t)=\log_{2}(|t|-1)$ for $|t|>1$ and $f(t)=0$ otherwise, we have
$\displaystyle\mathcal{F}(t)=\sum_{v\in t}f(t(v))=\sum_{v\in
t\atop|t(v)|>1}\log_{2}(|t(v)|-1)=\log_{2}(P_{\operatorname{bst}}(t)^{-1}),$
by definition of $P_{\operatorname{bst}}$ in (3), and thus Theorem 6 follows
as a special case of Theorem 7. This special case is also considered in
Example 8.13 of [22].
As our main application of Theorem 7, we transfer the statement of Theorem 5
from uniformly random binary trees to random binary search trees, that is, we
show that if the random number
$A_{k}=\lvert\operatorname{\mathrm{Aut}}(T_{k})\rvert$ denotes the size of the
automorphism group of a random binary search tree $T_{k}$ with $k$ leaves,
then the logarithm of this random variable satisfies a central limit theorem
as well. For this, we define the function $f:\mathcal{T}\rightarrow\mathbb{R}$
in Theorem 7 by
$\displaystyle f(t)=\begin{cases}1\quad&\text{if the root of }t\text{ is a
symmetrical node,}\\\ 0\quad&\text{otherwise.}\end{cases}$
We thus have
$\displaystyle\mathcal{F}(t)=\sum_{v\in t}f(t(v))=\operatorname{sym}(t),$
that is, $\mathcal{F}(t)$ evaluates to the number of symmetrical nodes in $t$.
Recall that $2^{\operatorname{sym}(t)}$ equals the size of the automorphism
group $\operatorname{\mathrm{Aut}}(t)$ of $t$. It is not difficult to check
that $f$ satisfies the conditions of Theorem 7: As $f(t)\in\\{0,1\\}$ for
every $t\in\mathcal{T}$, we have
$\mathbb{E}(f(T_{k})^{2})=\mathbb{E}(f(T_{k}))\in[0,1]$ and thus
$\mathbb{V}(f(T_{k}))\in[0,1]$ as well, so that the assumptions of Theorem 7
are satisfied. In order to determine the corresponding value $\nu$, we start
with estimating the expectation $\mathbb{E}(f(T_{k}))$:
If $k$ is odd, then $\mathbb{E}(f(T_{k}))=0$, as the fringe subtrees rooted at
the root’s children cannot be of the same size in this case and thus cannot be
isomorphic. If $k$ is even, then $\mathbb{E}(f(T_{k}))$ equals the probability
that these two subtrees are of the same size $\frac{k}{2}$ (which is
$\frac{1}{k-1}$) times the probability that these two subtrees of size
$\frac{k}{2}$ are isomorphic.
In order to estimate the latter probability, let $P_{k}^{r}$ for positive
integers $k$ and $r$ denote the probability that $2^{r}$ random binary search
trees of size $k$ are isomorphic, and let $\delta(k)=1$ if $k$ is even and
$\delta(k)=0$ otherwise. We find that $P_{k}^{r}$ satisfies the following
recurrence relation:
$\displaystyle
P_{k}^{r}=\smashoperator[]{\sum_{i=1}^{\lfloor\frac{k-1}{2}\rfloor}}\Big{(}\frac{2}{k-1}\Big{)}^{\\!2^{r}}\\!P_{i}^{r}P_{k-i}^{r}+\delta(k)\Big{(}\frac{1}{k-1}\Big{)}^{\\!2^{r}}\\!\Big{(}2^{2^{r}-1}\big{(}P_{k/2}^{r}\big{)}^{2}\\!-\\!\Big{(}2^{2^{r}-1}-1\Big{)}P_{k/2}^{r+1}\Big{)},$
with $P_{k}^{r}=1$ for $k\in\\{1,2,3\\}$ and every positive integer $r$. To
see that this recurrence relation holds, first consider the case that $k$ is
odd: If all the $2^{r}$ trees are isomorphic, then the respective sizes of the
fringe subtrees rooted at the root nodes’ children must coincide, that is,
there are integers $i$ and $k-i$ with $1\leq
i\leq\lfloor\frac{k-1}{2}\rfloor$, such that for each of the $2^{r}$ trees,
one of those subtrees is of size $i$ while the other is of size $k-i$. This
holds with probability $(2/(k-1))^{2^{r}}$. Moreover, all of the $2^{r}$
subtrees of size $i$ (respectively, $k-i$) have to be isomorphic, which holds
with probability $P_{i}^{r}$ (respectively, $P_{k-i}^{r}$).
If $k$ is even, we furthermore have to consider the case that $i=\frac{k}{2}$,
which holds with probability $(1/(k-1))^{2^{r}}$: In this case pick the first
of the $2^{r}$ trees and let $t_{1}$ (respectively, $t_{2}$) denote the fringe
subtree of size $\frac{k}{2}$ rooted at the root node’s left (respectively,
right) child. For all of the other $2^{r}-1$ trees, one of the fringe subtrees
rooted at the root node’s children has to be isomorphic to $t_{1}$, while the
other has to be isomorphic to $t_{2}$: This holds with probability
$(P_{k/2}^{r})^{2}$. Moreover, for each of those $2^{r}-1$ many trees, we can
choose whether the subtree rooted at the root’s left child or right child is
isomorphic to $t_{1}$, which gives us $2^{2^{r}-1}$ many possibilities.
However, in the case that $t_{1}$ is isomorphic to $t_{2}$ as well (which
means that all the $2^{r+1}$ subtrees are isomorphic, which holds with
probability $P_{k/2}^{r+1}$), this means some overcounting, which is taken
into account by the final term. Thus, the recursion for $P_{k}^{r}$ follows.
We find for a random binary search tree $T_{k}$ of size $k$:
$\displaystyle\mathbb{E}(f(T_{k}))=\begin{cases}\frac{1}{k-1}P^{1}_{k/2}\quad&\text{if
}k\text{ is even,}\\\ 0\quad&\text{otherwise.}\end{cases}$
Thus, we have
$\displaystyle\nu=\sum_{k=1}^{\infty}\frac{2}{k(k+1)}\mathbb{E}(f(T_{k}))=\sum_{k=1}^{\infty}\frac{P_{k}^{1}}{k(2k+1)(2k-1)}\approx
0.3795493473,$
where the numerical value for $\nu$ can be determined using the recurrence
relation for $P_{k}^{r}$. We remark that the constant $\sigma^{2}$ can also be
evaluated numerically. It is approximately $0.115$, thus in particular not
$0$, but we do not need its precise value. The following theorem now follows
from Theorem 7:
###### Theorem 8
Consider a random binary search tree $T_{k}$ with $k$ leaves, and let
$A_{k}=\lvert\operatorname{\mathrm{Aut}}(T_{k})\rvert$ be the cardinality of
its automorphism group. The logarithm of this random variable satisfies a
central limit theorem: for certain positive constants $\nu$ and $\sigma_{3}$,
we have
$\displaystyle\mathbb{P}(A_{k}\leq 2^{\nu
k+\sigma_{3}\sqrt{k}x})\overset{k\to\infty}{\to}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\,dt$
for every real number $x$. The numerical value of $\nu$ is $\nu\approx
0.3795493473$.
### 4.1 Ordered Fringe Subtrees in Random Binary Search Trees
We are now able to prove Theorem 4:
###### Proof (Proof of Theorem 4)
_The upper bound:_ Let $k_{0}=\log_{4}n$. We upper-bound the number $J_{n}$ of
distinct fringe subtrees in a random binary search tree with $n$ leaves as
follows: The number of distinct fringe subtrees with fewer than $k_{0}$ leaves
is trivially bounded from above by the number of all binary trees of size at
most $k_{0}$ (irrespective of their occurrence as fringe subtrees), which is
$\displaystyle\sum_{k<k_{0}}C_{k-1}=O\left(\frac{4^{k_{0}}}{k_{0}^{3/2}}\right)=O\left(\frac{n}{(\ln
n)^{3/2}}\right).$
This upper bound holds deterministically. The number of distinct fringe
subtrees with at least $k_{0}$ leaves is upper-bounded by the total number of
fringe subtrees with at least $k_{0}$ leaves: For this, we apply Lemma 2 with
$a=\frac{1}{\ln 4}$, $\varepsilon<\frac{1}{3}$ and
$\mathcal{S}_{k}=\mathcal{T}_{k}$, so that $p_{k}=1$ for $k_{0}\leq k\leq
n^{\varepsilon}$. Thus, both in expectation and with high probability, as the
estimate from Lemma 2 (part (4)) holds with high probability simultaneously
for all $k$ in the given range, we obtain:
$\displaystyle\left(\sum_{k_{0}\leq k\leq
n^{\varepsilon}}X_{n,k}\right)+Y_{n,\varepsilon}$
$\displaystyle=\sum_{k_{0}\leq k\leq
n^{\varepsilon}}\frac{2n}{k(k+1)}(1+O(kn^{-1/2+\varepsilon}))+O(n^{1-\frac{\varepsilon}{2}})$
$\displaystyle=2\ln 4\cdot\frac{n}{\ln n}(1+o(1)).$
Hence, we find that $J_{n}$ is both in expectation and with high probability
bounded from above by
$\displaystyle J_{n}\leq\sum_{k<k_{0}}C_{k}+\left(\sum_{k_{0}\leq k\leq
n^{\varepsilon}}X_{n,k}\right)+Y_{n,\varepsilon}=2\ln 4\cdot\frac{n}{\ln
n}(1+o(1)).$
The numerical value of the constant is $c_{6}=2\ln(4)\approx 2.7725887222$.
_The lower bound:_ As a consequence of Theorem 6, the probability that
$P_{\operatorname{bst}}(T_{k})^{-1}\leq 2^{\mu k-k^{3/4}}$ for a random binary
search tree $T_{k}$ of size $k$ tends to $0$ for $k\rightarrow\infty$. Let
$\mathcal{S}_{k}$ denote the set of binary trees of size $k$ that do not
satisfy this inequality: Thus, every binary tree $t\in\mathcal{S}_{k}$
satisfies $P_{\operatorname{bst}}(t)\leq 2^{-\mu k+k^{3/4}}$ and we have
$p_{k}=1+o(1)$. In order to prove the lower bound, we only consider fringe
subtrees in $\mathcal{S}_{k}$ for suitable $k$: Thus, we can suitably upper-
bound the probability that two fringe subtrees of size $k$ in a random binary
search tree are identical. Let $\delta$ denote a positive constant with
$\delta<\frac{2}{3}$, let $k_{1}=(1+\delta)\mu^{-1}\log_{2}n$ and let
$k_{1}\leq k\leq n^{\delta/2}$. By Lemma 2, the number of such fringe subtrees
in a random binary search tree with $n$ leaves is $2np_{k}/(k(k+1))$ in
expectation and
$\displaystyle\frac{2np_{k}}{k(k+1)}(1+O(p_{k}^{-1/2}n^{(\delta-1)/2}k))$
with high probability. Furthermore, let $X^{(2)}_{n,k}$ denote the (random)
number of pairs of identical fringe subtrees among the fringe subtrees with
$k$ leaves that belong to $\mathcal{S}_{k}$, for $k_{1}\leq k\leq
n^{\delta/2}$. Let us condition on the event that $X_{n,k}=N$ for some
nonnegative integer $N\leq n$. Those $N$ fringe subtrees are all independent
random binary search trees, and the probability that such a fringe subtree
equals a given binary tree $t\in\mathcal{S}_{k}$ is
$P_{\operatorname{bst}}(t)/p_{k}$. Thus, we have
$\displaystyle\mathbb{E}(X_{n,k}^{(2)}\mid
X_{n,k}=N)=\binom{N}{2}\sum_{t\in\mathcal{S}_{k}}\left(\frac{P_{\operatorname{bst}}(t)}{p_{k}}\right)^{2}\leq\frac{n^{2}}{2}\frac{1}{p_{k}^{2}}\sum_{t\in\mathcal{S}_{k}}P_{\operatorname{bst}}(t)^{2}.$
As by assumption, $P_{\operatorname{bst}}(t)\leq 2^{-\mu k+k^{3/4}}$ for every
$t\in\mathcal{S}_{k}$, the expected value is upper-bounded by
$\displaystyle\mathbb{E}(X_{n,k}^{(2)}\mid
X_{n,k}=N)\leq\frac{n^{2}}{2}\frac{2^{-\mu
k+k^{3/4}}}{p_{k}^{2}}\sum_{t\in\mathcal{S}_{k}}P_{\operatorname{bst}}(t)=\frac{n^{2}}{p_{k}}2^{-\mu
k-1+k^{3/4}}.$
Since this upper bound for $\mathbb{E}(X_{n,k}^{(2)}\mid X_{n,k}=N)$ holds
independently of $N$, the law of total expectation yields
$\displaystyle\mathbb{E}(X_{n,k}^{(2)})=\sum_{N=0}^{n}\mathbb{E}(X_{n,k}^{(2)}\mid
X_{n,k}=N)\mathbb{P}(X_{n,k}=N)\leq\frac{n^{2}}{p_{k}}2^{-\mu k-1+k^{3/4}}.$
With $p_{k}=1+o(1)$, we obtain
$\displaystyle\mathbb{E}(X_{n,k}^{(2)})\leq n^{2}2^{-\mu
k-1+k^{3/4}}(1+o(1))=n^{2}2^{-\mu k+O(k^{3/4})}.$
As $k\geq k_{1}=\frac{1+\delta}{\mu}\log_{2}n$, we find
$\displaystyle\mathbb{E}(X_{n,k}^{(2)})\leq
n^{2}2^{-(1+\delta)\log_{2}n+O(k^{3/4})}\leq
n^{1-\delta}2^{O((\log_{2}n)^{3/4})}.$
Thus,
$\displaystyle\sum_{k_{1}\leq k\leq n^{\delta/2}}\mathbb{E}(X_{n,k}^{(2)})\leq
n^{1-\delta/2}2^{O((\log_{2}n)^{3/4})}=o(n/\ln n).$
The (random) number $J_{n}$ of distinct fringe subtrees in a random binary
search tree of size $n$ is lower-bounded by the number of distinct fringe
subtrees of sizes $k$ for $k_{1}\leq k\leq n^{\delta/2}$ that belong to
$\mathcal{S}_{k}$, and this number is again lower-bounded by the sum over
$X_{n,k}-X_{n,k}^{(2)}$ for $k_{1}\leq k\leq n^{\delta/2}$. We thus have
$\displaystyle J_{n}\geq\sum_{k_{1}\leq k\leq
n^{\delta/2}}X_{n,k}-\sum_{k_{1}\leq k\leq n^{\delta/2}}X_{n,k}^{(2)}.$
The second sum is $o(n/\ln n)$ in expectation and hence by Markov’s inequality
with high probability as well. The first sum can be estimated using Lemma 2,
as in the proof of the upper bound, which yields
$\displaystyle\sum_{k_{1}\leq k\leq n^{\delta/2}}X_{n,k}=\frac{2\mu\ln
2}{1+\delta}\cdot\frac{n}{\ln n}(1+o(1)),$
in expectation and with high probability. Since $\delta$ can be chosen
arbitrarily, the desired statement holds for any constant
$\displaystyle c_{5}<2\mu\ln 2\approx 2.4071298335.$
### 4.2 Unordered Fringe Subtrees in Random Binary Search Trees
It remains to prove Theorem 2:
###### Proof (Proof of Theorem 2)
_The upper bound_ : The proof for the upper bound exactly matches the first
part of the proof of Theorem 4, except that we choose the cut-point
$k_{0}=\log_{b}(n)$, where $b\approx 2.4832535362$ is the constant determining
the asymptotic growth of the Wedderburn-Etherington numbers.
_The lower bound_ : Let $T_{k}$ denote a random binary search tree with $k$
leaves. As a consequence of Theorem 8, the probability that
$\lvert\operatorname{\mathrm{Aut}}(T_{k})\rvert=A_{k}\leq 2^{\nu k-k^{3/4}}$
tends to $0$ as $k\rightarrow\infty$. Moreover, by Theorem 6 the probability
that $B_{k}=P_{\operatorname{bst}}(T_{k})^{-1}\leq 2^{\mu k-k^{3/4}}$ tends to
$0$ for $k\rightarrow\infty$. Let $\mathcal{S}_{k}$ denote the set of ordered
binary trees with $k$ leaves for which neither of the two inequalities is
satisfied: Thus, every binary tree $t\in\mathcal{S}_{k}$ satisfies
$P_{\operatorname{bst}}(t)\leq 2^{-\mu k+k^{3/4}}$ and
$\lvert\operatorname{\mathrm{Aut}}(t)\rvert\geq 2^{\nu k-k^{3/4}}$, and we
have $p_{k}=1+o(1)$. By the orbit-stabilizer theorem, we find that the number
of ordered binary trees in the same isomorphism class as a tree
$t\in\mathcal{S}_{k}$ is bounded from above by
$\displaystyle\frac{2^{k-1}}{\lvert\operatorname{\mathrm{Aut}}(t)\rvert}\leq\frac{2^{k-1}}{2^{\nu
k-k^{3/4}}}=2^{(1-\nu)k-1+k^{3/4}}.$
In order to prove the lower bound, we only consider fringe subtrees in
$\mathcal{S}_{k}$ for suitable $k$: Thus, we are able to suitably upper-bound
the probability that two fringe subtrees of size $k$ in a random binary search
tree are identical as unordered binary trees. Let $\delta<\frac{2}{3}$, let
$k_{1}=(1+\delta)\log_{2}n/(\mu+\nu-1)$ and let $k_{1}\leq k\leq
n^{\delta/2}$. By Lemma 2, the number of fringe subtrees of size $k$ with
$k_{1}\leq k\leq n^{\delta/2}$ is $2np_{k}/(k(k+1))$ in expectation and
$\displaystyle\frac{2np_{k}}{k(k+1)}(1+O(p_{k}^{-1/2}n^{(\delta-1)/2}k))$
with high probability. For $k_{1}\leq k\leq n^{\delta/2}$, let $X_{n,k}^{(2)}$
denote the (random) number of pairs of isomorphic binary trees among the
fringe subtrees of size $k$ that belong to $\mathcal{S}_{k}$. Moreover, let
$l$ denote the number of isomorphism classes of binary trees in
$\mathcal{S}_{k}$ and for each isomorphism class, pick one representative: Let
$t_{1},t_{2},\dots,t_{l}$ denote those representatives. If a binary tree $t$
is in the same isomorphism class as tree $t_{i}$, then
$P_{\operatorname{bst}}(t)=P_{\operatorname{bst}}(t_{i})$ and
$\lvert\operatorname{\mathrm{Aut}}(t)\rvert=\lvert\operatorname{\mathrm{Aut}}(t_{i})\rvert$.
In particular, if $t\in\mathcal{T}_{k}$ is isomorphic to a representative
$t_{i}$, then $t\in\mathcal{S}_{k}$ as well, that is, all binary trees that
are isomorphic to a binary tree in $\mathcal{S}_{k}$ are automatically
contained in $\mathcal{S}_{k}$ as well. As there are
$2^{k-1}/\lvert\operatorname{\mathrm{Aut}}t_{i}\rvert$ many trees in the same
isomorphism class as the binary tree $t_{i}$, we find
$\displaystyle\sum_{i=1}^{l}P_{\operatorname{bst}}(t_{i})\frac{2^{k-1}}{\lvert\operatorname{\mathrm{Aut}}t_{i}\rvert}=\sum_{t\in\mathcal{S}_{k}}P_{\operatorname{bst}}(t)=p_{k}.$
Let us condition on the event that $X_{n,k}=N$ for some integer $0\leq N\leq
n$. Those $N$ fringe subtrees are all independent random binary search trees,
and the probability that such a fringe subtree is isomorphic to a given binary
tree $t_{i}\in\mathcal{S}_{k}$ is
$(P_{\operatorname{bst}}(t_{i})/p_{k})\cdot(2^{k-1}/\lvert\operatorname{\mathrm{Aut}}(t_{i})\rvert)$.
Thus, we find
$\displaystyle\mathbb{E}(X_{n,k}^{(2)}\mid
X_{n,k}=N)\\!=\\!\binom{N}{2}\sum_{i=1}^{l}\left(\frac{2^{k-1}P_{\operatorname{bst}}(t_{i})}{\lvert\operatorname{\mathrm{Aut}}t_{i}\rvert
p_{k}}\right)^{2}\\!\leq\frac{n^{2}}{2}\frac{1}{p_{k}^{2}}\sum_{i=1}^{l}\left(\frac{2^{k-1}P_{\operatorname{bst}}(t_{i})}{\lvert\operatorname{\mathrm{Aut}}t_{i}\rvert}\right)^{2}\\!.$
As $t_{1},\dots,t_{l}\in\mathcal{S}_{k}$, we have
$P_{\operatorname{bst}}(t_{i})\leq 2^{-\mu k+k^{3/4}}$ and thus
$2^{k-1}/\lvert\operatorname{\mathrm{Aut}}t_{i}\rvert\leq
2^{(1-\nu)k-1+k^{3/4}}$
for every $i\in\\{1,2,\dots,l\\}$. Hence
$\displaystyle\mathbb{E}(X_{n,k}^{(2)}\mid X_{n,k}=N)$
$\displaystyle\leq\frac{n^{2}2^{(1-\nu-\mu)k+2k^{3/4}-2}}{p_{k}^{2}}\sum_{i=1}^{l}\left(\frac{2^{k-1}P_{\operatorname{bst}}(t_{i})}{\lvert\operatorname{\mathrm{Aut}}t_{i}\rvert}\right)$
$\displaystyle=p_{k}^{-1}n^{2}2^{(1-\nu-\mu)k+2k^{3/4}-2}.$
As this upper bound on the expectation is independent of $N$, we find by the
law of total expectation:
$\displaystyle\mathbb{E}(X_{n,k}^{(2)})\leq
p_{k}^{-1}n^{2}2^{(1-\nu-\mu)k+2k^{3/4}-2}=n^{2}2^{-(\nu+\mu-1)k+2k^{3/4}-2}(1+o(1)).$
With $k\geq k_{1}=(1+\delta)/(\mu+\nu-1)\log_{2}n$, we obtain
$\displaystyle\mathbb{E}(X_{n,k}^{(2)})\leq
n^{2}2^{-(1+\delta)\log_{2}n+O((\log_{2}n)^{3/4})}=n^{1-\delta}2^{O((\log_{2}n)^{3/4})}.$
Thus,
$\displaystyle\sum_{k_{1}\leq k\leq n^{\delta/2}}\mathbb{E}(X_{n,k}^{(2)})\leq
n^{1-\frac{\delta}{2}}2^{O((\log_{2}n)^{3/4})}=o(n/\ln n).$
Analogously as in the previous proofs, we lower-bound the random number
$G_{n}$ of non-isomorphic fringe subtrees in a random binary search tree of
size $n$ by the number of such fringe subtrees of sizes $k$ for $k_{1}\leq
k\leq n^{\delta/2}$ that belong to $\mathcal{S}_{k}$, and this number is again
lower-bounded by the sum over $X_{n,k}-X_{n,k}^{(2)}$ for $k_{1}\leq k\leq
n^{\delta/2}$ by the inclusion-exclusion principle. We thus have
$\displaystyle G_{n}\geq\sum_{k_{1}\leq k\leq
n^{\delta/2}}X_{n,k}-\sum_{k_{1}\leq k\leq n^{\delta/2}}X_{n,k}^{(2)}.$
The second sum is $o(n/\ln n)$ in expectation and hence by Markov’s inequality
with high probability as well. The first sum is bounded similarly as in the
estimate for the upper bound, which yields
$\displaystyle\sum_{k_{1}\leq k\leq n^{\delta/2}}X_{n,k}=\frac{2(\mu+\nu-1)\ln
2}{1+\delta}\cdot\frac{n}{\ln n}(1+o(1)).$
Since $\delta$ can again be chosen arbitrarily, the desired statement holds
for any constant $c_{3}<2(\mu+\nu-1)\ln 2\approx 1.5470025923.$
$\scriptstyle\blacksquare$
## 5 Open Problems
The following natural question arises from our results: Is it possible to
determine constants $\alpha_{1},\alpha_{2},\alpha_{3}$ with
$c_{1}\leq\alpha_{1}\leq c_{2}$, $c_{3}\leq\alpha_{2}\leq c_{4}$ and
$c_{5}\leq\alpha_{3}\leq c_{6}$, such that
$\displaystyle\mathbb{E}(F_{n})=\frac{\alpha_{1}n}{\sqrt{\log n}}(1+o(1)),\
\mathbb{E}(G_{n})=\frac{\alpha_{2}n}{\log n}(1+o(1)),\
\mathbb{E}(J_{n})=\frac{\alpha_{3}n}{\log n}(1+o(1)),$
respectively, and
$\displaystyle\frac{F_{n}}{n/\sqrt{\log n}}\overset{P}{\to}\alpha_{1},\
\frac{G_{n}}{n/\log n}\overset{P}{\to}\alpha_{2},\ \text{and}\
\frac{J_{n}}{n/\log n}\overset{P}{\to}\alpha_{3}\ ?$
In order to prove such estimates, it seems essential to gain a better
understanding of the random variables $P_{\operatorname{bst}}(T_{k})^{-1}$ and
$\lvert\operatorname{\mathrm{Aut}}(T_{k})\rvert$, in particular their
distributions further away from the mean values, for random binary search
trees or uniformly random ordered binary trees $T_{k}$ of size $k$.
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|
2024-09-04T02:54:57.957653 | 2020-03-06T12:28:04 | 2003.03375 | {
"authors": "Eric Guizzo, Tillman Weyde, Jack Barnett Leveson",
"full_text_license": null,
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26089",
"submitter": "Eric Guizzo",
"url": "https://arxiv.org/abs/2003.03375"
} | arxiv-papers | # Multi-Time-Scale Convolution for Emotion Recognition
from Speech Audio Signals
###### Abstract
Robustness against temporal variations is important for emotion recognition
from speech audio, since emotion is expressed through complex spectral
patterns that can exhibit significant local dilation and compression on the
time axis depending on speaker and context. To address this and potentially
other tasks, we introduce the multi-time-scale (MTS) method to create
flexibility towards temporal variations when analyzing time-frequency
representations of audio data. MTS extends convolutional neural networks with
convolution kernels that are scaled and re-sampled along the time axis, to
increase temporal flexibility without increasing the number of trainable
parameters compared to standard convolutional layers. We evaluate MTS and
standard convolutional layers in different architectures for emotion
recognition from speech audio, using 4 datasets of different sizes. The
results show that the use of MTS layers consistently improves the
generalization of networks of different capacity and depth, compared to
standard convolution, especially on smaller datasets.
Index Terms— Convolutional Neural Network, Scale Invariance, Speech Emotion
Recognition
## 1 Introduction
Convolutional Neural Networks (CNNs) have been extremely successful in recent
years in a number of audio processing tasks, such as source separation, audio
denoising, speech enhancement, speech and music transcription [1, 2, 3, 4].
CNNs have also been extensively adopted for speech emotion recognition (SER)
[5, 6, 7].
Convolutional networks benefit from translation invariance of the processing
on the time and frequency axis of a spectrogram or other time-frequency
representations. However, in speech there are also variations in the speed of
articulation between speakers and even of the same speaker in different
situations. Therefore, allowing for matching the same kernel in multiple
versions that are scaled differently on the time axis is the main idea in this
work. We implement this in a self-contained layer architecture, the multi-
time-scale (MTS) convolution layer, which does not increase the number of
parameters and increases the temporal flexibility in our networks compared to
standard CNNs. Separate treatment of dimensions is useful for speech
processing with time-frequency representations, as opposed to image
processing, where scaling is normally applied to both dimensions.
The contributions of our work are specifically:
* •
a convolution layer design for audio emotion recognition that learns locally-
scale-invariant features in the time dimension
* •
an evaluation of our approach to 4 emotion-labelled speech datasets with 4
different network architectures
* •
an analysis of the experimental results, confirming the effectiveness of the
MTS approach.
The remainder of the paper is organized as follows: Section 2 contains a brief
review of relevant background literature, Section 3 introduces the
architecture of multi-time-scale convolution layer, Section 4 presents the
experimental results we obtained and Section 5 provides the conclusion of this
paper.
## 2 Related Work
Scale-invariance in convolutional neural networks has been addressed in a
number of ways. The most common approach for audio by far is data augmentation
[8, 9], which is frequently done by generating time-stretched variants of the
training data. This procedure is usually part of a pipeline of different
transformations, as in [10], which has proven effective in various tasks.
However, in this approach the different scales in the data need to be learned
by different filters in the network. Therefore, greater network capacity is
required and there is no guarantee that scale-invariance is consistently
achieved .
Another strategy for scale-invariance in neural networks is to design it into
the training and inference methods, so that it is applied consistently and
without the need for additional training examples. There are many existing
approaches to achieve this. The majority of them use a pyramidal structure, in
which the scale is progressively narrowed along the network. [11] use parallel
models trained with images at descending resolutions and then combine the
obtained predictions as an ensemble model. [12] achieve scale invariance with
multiple loss functions, separately computed in layers with different
resolutions within the network. Inception networks [12] use parallel
convolution layers with different filter sizes, matching features at different
scales, but also increasing the number of variables in the network. [13]
propose a convolutional architecture, in which a scaling factor is learned by
the network for every layer.
The majority of studies of scale-invariance in neural networks is focused on
computer vision tasks. In the acoustic domain, in addition to data
augmentation techniques, scale-invariance can also be addressed through
specific hard-coded transforms [14] that are robust to some extent to scale
variations. Nevertheless, since they are hard-coded, these methods need manual
intervention and are usually highly task-specific, while embedding scale-
invariance in the models provides a more generic solution that can be applied
to multiple domains. The work of [15] is an exception to this trend. They show
that a network with $n$ identically-sized filters performs worse than a
network with the same number of filters, but split in $3$ different sizes.
Nevertheless, their models learn independent filters at different scales,
increasing the number of free parameters.
Locally scale-invariant convolutional neural networks, as introduced for image
recognition in [16], are similar to our approach. This method consists of
performing feature-extraction through multiple parallel convolution layers,
whose outputs are locally merged through max-pooling. This produces a self-
contained structure that can substitute a canonical convolution layer. The key
feature of their approach is the possibility of matching a feature at multiple
scales without increasing the number of free variables in the network. It
permits introducing several re-scaled parallel branches at different points in
the network, providing higher flexibility then pyramidal architectures.
## 3 Method
Our approach is similar to [16], but specifically adapted to the audio domain,
where we analyse 2D magnitude spectrograms of speech audio. Since the time and
frequency dimensions are of different nature in this representation, we treat
them independently. Here, we focus on SER and address only time-scaling, while
image processing techniques apply re-scaling to both dimensions with the same
factor.
The core of our architecture is the multi-time-scale convolution layer (MTS),
a custom 2D-convolution layer that can replace a standard convolution layer in
a CNN design. The main feature of MTS is that it uses multiple versions of the
learned kernel that are re-sampled on the time axis and performs parallel
convolutions with them. This method enables the network to detect patterns at
multiple time scales.
Fig. 1: Example architecture of a Multi-Time-Scale convolution layer with 3
scale factors.
Figure 1 shows the architecture of one MTS layer with 3 parallel branches. In
this example, the 2D spectrogram input, is convolved in parallel with the
original kernel (in the center) and 2 time-stretched versions of the kernel
(on both sides). The latter are generated by re-sampling the original kernel,
applying linear interpolation. It is possible to independently apply different
scaling factors for the 2 dimensions. These parallel convolutions produce 3
different feature maps, matching the feature of the original kernel at 3
different time scales. After this stage, the scaled feature maps are re-
sampled again (applying linear interpolation) to match the shape of the
original feature map. Then, a 3D max-pooling function is applied to merge the
feature maps, selecting the scale with the maximal result in every time-
frequency point. Therefore, the pooled feature map maintains the same
dimension of the feature map generated by the original kernel. During the
training we average the weights of the original kernel and its scaled versions
after each update. There is no constraint by design on the number of parallel
branches that can be added to a MTS layer and MTS layers with different
numbers of branches can be placed at various positions in the network. It is
possible to fine-tune the scaling factors layer-by-layer. This approach
provides a high degree of flexibility in the network design and enables scale
invariance without increasing the number of free parameters. We have
implemented this method in PyTorch as open
source111https://github.com/ericguizzo/multi_time_scale.
Our method is different from [16] in that it re-scales only one dimension and
that we re-sample the kernels. Although re-sampling the data or kernel is
equivalent up to numerical variations, our method is somewhat more efficient.
Moreover, [16] augment test data by re-scaling. At least for SER tasks, we
believe that this practice would not give a good estimate of the
generalization capabilities of the models and thus we test without
augmentation.
## 4 Evaluation
We have evaluated the performance of MTS on 4 benchmark datasets for speech
emotion recognition:
1. 1.
EMODB, a database of German emotional speech [17]. 10 speakers, German
language, 535 utterances, 25 min of audio, 7 emotion labels: angry, bored,
disgusted, anxious/fearful, happy, sad. Actors pronounce 10 different
sentences which could be used in everyday communication.
2. 2.
RAVDESS, the Ryerson Audio Visual Database of Emotional Speech and Song [18].
24 speakers, English language, 2542 utterances, 2:47 hours of audio, 8 emotion
labels: happy, sad, angry, fearful, surprised, disgusted, calm, neutral.
Actors pronounce 2 sentences: “Kids are talking by the door” and “Dogs are
sitting by the door”.
3. 3.
TESS, the Toronto Emotional Speech Set [19]. 2 speakers, English language,
2800 utterances, 1:36 hours of audio, 7 emotion labels: happy, sad, angry,
disgusted, neutral, pleasant surprise, fearful. Actors say “Say the word …”
followed by 200 different words.
4. 4.
IEMOCAP, the Interactive Emotional Dyadic Motion Capture Database [20]. 5
speakers, English language, 7529 utterances, 9:32 hours of audio, 10 emotion
labels: neutral, angry, happy, excited, sad, frustrated, fearful, surprised,
disgusted, other. Actors perform improvisations or scripted scenarios on
defined topics.
For each dataset we keep only the audio information and the emotion labels,
discarding any other types of data. We also discard the “song” data from
RAVDESS. IEMOCAP is the only highly inbalanced dataset, therefore we removed
the rarest labels from it, keeping only neutral, angry, happy and sad samples.
Every sound file is pre-processed in 3 consecutive stages: re-sampling to 16
kHz, Short-Time Fourier Transform and normalization. For EMODB, RAVDESS and
TESS datasets every file is zero-padded to obtain equally-sized data. Since
the IEMOCAP dataset contains longer recordings we segmented them into 4-second
frames with 2-second overlap. The STFT is computed using 20 ms sliding windows
with 10 ms overlap. Then, we normalize the magnitude spectra to zero mean and
unit standard deviation.
Table 1: Accuracy results for all datasets. N ist the number of audio recordings per dataset. A1-4 are the network architectures. The usage factors relate to scaling factors in the same row. The best results per dataset are highlighted in bold font. Dataset | N | Type | A1 | A2 | A3 | A4 | Best scale factors | Use of parallel branches
---|---|---|---|---|---|---|---|---
EMODB | 535 | Standard | 64.3 | 66.26 | 66.91 | 62.75 | n/a | n/a
| 535 | MTS | 66.5 | 70.97 | 70.68 | 66.28 | 0.7, 1, 1.428 | 0.47, 0.05, 0.48
RAVDESS | 1440 | Standard | 42.09 | 39,84 | 42.56 | 47.41 | n/a | n/a
| 1440 | MTS | 47.85 | 44.95 | 51.32 | 55.85 | 0.5, 1, 2 | 0.45, 0.06, 0.49
TESS | 2800 | Standard | 47.45 | 49.6 | 50.61 | 40.78 | n/a | n/a
| 2800 | MTS | 51.76 | 48.75 | 53.05 | 51.71 | 0.5, 0.7, 1, 1.428, 2. | 0.41, 0.04, 0.05, 0.07, 0.43
IEMOCAP | 5531 | Standard | 48.93 | 50.48 | 49.0 | 54.96 | n/a | n/a
| 5531 | MTS | 49.0 | 50.84 | 49.86 | 55.01 | 0.5, 0.7, 1, 1.428, 2 | 0.39, 0.04, 0.04, 0.05, 0.48
We divide every dataset using approximately 70% of the data as training, 20%
for validation and 10% as test set. Furthermore, we perform every experiment
with 4-fold cross-validation. We make sure that samples from the same speaker
appear only in the same set, in order to get a meaningful measure of the
models’ capability to generalize to new speakers, because new speakers are
likely to produce patterns at different speeds. For this and other reasons,
our results are not directly comparable to most published results. Many
results are computed with randomly-split training, validation and test sets,
without separating speakers, as in [21]. Many rely on different preprocessing
[22, 23], on different architectures [22] or use multi-modal features rather
than only audio [23]. Rather than aiming at a state-of-art classification
accuracy for these datasets, we focus on evaluating the performance of MTS
layers compared to standard convolution with the same number of channels, i.e.
without increasing the number of trainable variables. Therefore, we arranged
our experiments in order to obtain consistent results within our set-up, with
the same conditions for all datasets. We perform this comparison for 4
different CNN architectures with different capacity:
1. A1:
Convolution (1 channel, [10,5] kernel) - fully connected (200 neurons) - fully
connected output layer.
2. A2:
Convolution (10 channels, [10,5] kernel) - fully connected (200 neurons) -
fully connected output layer.
3. A3:
Convolution (10 channels, [10,5] kernel) - max pooling ([2,2] kernel) -
convolution (10 channels, [10,5] kernel - fully connected (200 neurons) -
fully connected output layer.
4. A4:
AlexNet: 5 convolutions and max pooling, 2 fully connected layers. See [24]
for a detailed description.
The kernel dimensions above are in the form [time,frequency]. The activation
function is ReLU for hidden and softmax for output units. In all experiments
we use the ADAM optimizer with L2 regularization and Cross Entropy loss. We
perform a grid search to find the best regularization parameter. We train for
a maximum of 500 epochs, applying early stopping with 10 epochs patience for
validation loss improvement. In architectures A1, A2 and A3, MTS is applied to
all convolutional layers, while in A4 only the first 2 layers are augmented
with MTS. We tested MTS with 3, 5 and 7 parallel branches, using
logarithmically spaced scale factors in these combinations: (0.25, 1, 4),
(0.5, 1, 2), (0.7, 1, 1.428), (0.8, 1, 1.25), (0.9, 1, 1.111), (0.95, 1,
1.053), (0.25, 0.5, 1, 2, 4), (0.5, 0.7, 1, 1.428, 2), (0.8, 0.9, 1, 1.111,
1.25), (0.25, 0.5, 0.7, 1, 1.428, 2, 4), (0.7, 0.8, 0.9, 1, 1.111, 1.25,
1.428). In each experiment, we apply the same combination of stretch factors
to all MTS-enabled layers.
Table 1 shows the results we obtained for all datasets and all architectures.
The first 3 columns show the dataset, total number of data points and the type
of convolution layer(s). Columns A1-A4 show the mean test accuracy across
folds obtained with each architecture (as listed above). The last 2 columns
refer to the MTS model with the best accuracy in a row and show the scaling
factors applied to each parallel branch of MTS and their average percentage of
use.
The results clearly show that MTS consistently improves the generalization for
all datasets. We reach a maximum improvement of 8.04 percentage points
(RAVDESS) and with an average of 3.78 with a standard deviation of 3.45 across
all datasets and architectures. For all model/architecture combinations except
one (A2 with TESS), MTS outperforms standard convolution. We performed a two-
sided Wilcoxon signed-rank test comparing the standard and MTS results, which
shows statistical significance with $p<0.001$. The mean improvement is higher
for the smaller datasets, which confirms that enabling pattern recognition at
different time scales with MTS improves generalisation. Considering the
general scarcity of emotion-labelled speech data, this is a desirable feature
for SER applications.
The best performing models on different datasets used different combinations
of scaling factors. In particular, for the smaller datasets applying only 3
factors gives the best results. Architectures with 5 parallel branches perform
better for the larger datasets. MTS models tend to use mostly 2 scale factors
(see last column of table 1). In every case, at least 2 parallel branches give
a high contribution, confirming that MTS is actually matching patterns at
multiple time-scales.
We found that MTS is more effective at larger kernel sizes. In an experiment
with an MTS version of ResNet18, where most kernels are very small (3x3), we
achieved no improvement with MTS.
Training a MTS-enabled network generally takes longer than a standard CNN. In
a test with architecture A2, it took on average 1.3 times longer per epoch to
train MTS models with 3 branches and 1.52 times longer for MTS models 5
branches. Moreover, MTS networks need on average more epochs to converge
(27.85 vs 32.26 epochs for CNN vs MTS average overall).
We also tested modified variants of MTS:
* •
Applying a penalty to the re-sampled feature maps, to give the model a
preference for the unscaled kernel.
* •
Performing the training using standard convolution layers and substitute them
with MTS layers with shared weights only at inference time.
* •
Concatenating the used scaling factor for each time-frequency point to the
output feature map of an MTS layer.
Each of these modifications reduced the performance of MTS models. Therefore,
we kept the simplest variant described above.
## 5 Conclusions
In this paper, we propose a multi-time-scale convolution layer (MTS) for CNNs
applied to audio analysis, specifically emotion recognition from speech. The
MTS performs parallel 2D-convolutions using a standard kernel and its re-
sampled versions to match patterns at different time scales. This method
enables the network to learn to some extent time-invariant features without
increasing its number of trainable parameters or the number of training
examples. We evaluated our approach on speech emotion recognition with unknown
speakers, using 4 different datasets and applying it to networks of different
size and structure. We found a consistent and statistically significant
improvement in test accuracy across all datasets and models, up to 8.04
percentage points for RAVDESS and on average 3.78 across all datasets and
architectures. MTS is particularly effective on smaller datasets, which makes
MTS well suited for Speech Emotion Recognition where labelled data is scarce.
As future developments we intend to test more extensively the effectiveness of
MTS with larger datasets and explore more architectures and different
resampling techniques. Furthermore, we are going to apply the concept of MTS
in the context of convolution-based generative models, extending our multi-
branch approach also to transposed convolutions.
## References
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* [15] Zhenyao Zhu, Jesse H Engel, and Awni Hannun, “Learning multiscale features directly from waveforms,” arXiv:1603.09509, 2016.
* [16] Angjoo Kanazawa, Abhishek Sharma, and David Jacobs, “Locally scale-invariant convolutional neural networks,” in NIPS Deep Learning and Representation Learning Workshop, 12 2014\.
* [17] Felix Burkhardt, Astrid Paeschke, Miriam Rolfes, Walter F Sendlmeier, and Benjamin Weiss, “A database of german emotional speech,” in Eurospeech, 2005.
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* [19] Kate Dupuis and M Kathleen Pichora-Fuller, Toronto Emotional Speech Set (TESS), University of Toronto, Psychology Department, 2010.
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|
2024-09-04T02:54:57.967746 | 2020-03-06T19:02:55 | 2003.03394 | {
"authors": "Ikram Ullah, Umar Hayat and Miguel D. Bustamante",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26090",
"submitter": "Miguel D. Bustamante",
"url": "https://arxiv.org/abs/2003.03394"
} | arxiv-papers | ###### Abstract
We propose an image encryption scheme based on quasi-resonant Rossby/drift
wave triads (related to elliptic surfaces) and Mordell elliptic curves (MECs).
By defining a total order on quasi-resonant triads, at a first stage we
construct quasi-resonant triads using auxiliary parameters of elliptic
surfaces in order to generate pseudo-random numbers. At a second stage, we
employ an MEC to construct a dynamic substitution box (S-box) for the plain
image. The generated pseudo-random numbers and S-box are used to provide
diffusion and confusion, respectively, in the tested image. We test the
proposed scheme against well-known attacks by encrypting all gray images taken
from the USC-SIPI image database. Our experimental results indicate the high
security of the newly developed scheme. Finally, via extensive comparisons we
show that the new scheme outperforms other popular schemes.
###### keywords:
quasi-resonant Rossby/drift wave triads; Mordell elliptic curve; pseudo-random
numbers; substitution box
xx 1 5 Entropy 2020, 22, 454; doi.org/10.3390/e22040454; Received: 04 March
2020; Accepted: 14 April 2020 Image Encryption Using Elliptic Curves and
Rossby/Drift Wave Triads Ikram Ullah 1, Umar Hayat 1,* and Miguel D.
Bustamante 2,* Ikram Ullah, Umar Hayat and Miguel D. Bustamante
Correspondence<EMAIL_ADDRESS>(M.D.B<EMAIL_ADDRESS>(U.H.)
## 1 Introduction
The exchange of confidential images via the internet is usual in today’s life,
even though the internet is an open source that is unsafe and unauthorized
persons can steal useful or sensitive information. Therefore it is essential
to be able to share images in a secure way. This goal is achieved by using
cryptography. Traditional cryptographic techniques such as data encryption
standard (DES) and advanced encryption standard (AES) are not suitable for
image transmission because image pixels are usually highly correlated Mahmud
(2020); Zhang (2014). By contrast, DES and AES are ideal techniques for text
encryption El-Latif (2013), so researchers are trying to develop such
techniques to meet the demand for reliable image delivery.
A number of image encryption schemes have been developed using different
approaches Yang (2015); Zhong (2019); Li (2017); Hua (2018); Xie (2017); Azam
(2017); Luo (2018); Li (2015); Hua (2019); Wu (2019); Yousaf (2020). Hua et
al. Hua (2019) developed a highly secure image encryption algorithm, where
pixels are shuffled via the principle of the Josephus problem and diffusion is
obtained by a filtering technology. Wu et al. Wu (2019) proposed a novel image
encryption scheme by combining a random fractional discrete cosine transform
(RFrDCT) and the chaos-based Game of Life (GoL). In their scheme, the desired
level of confusion and diffusion is achieved by GoL and an XOR operation,
respectively. “Confusion” entails hiding the relation between input image,
secret keys and the corresponding cipher image, and “diffusion” is an
alteration of the value of each pixel in an input image Mahmud (2020).
One of the dominant trends in encryption techniques is chaos-based encryption
Ismaila (2020); Tang (2019); Abdelfatah (2019); Yu (2020); Zhu (2018);
ElKamchouchi (2020). The reason for this dominance is that the chaos-based
encryption schemes are highly sensitive to the initial parameters. However,
there are certain chaotic cryptosystems that exhibit a lower security level
due to the usage of chaotic maps with less complex behavior (see Zhou (2013)).
This problem is addressed in Hu (2019) by introducing a cosine-transform-based
chaotic system (CTBCS) for encrypting images with higher security. Xu et al.
Xu (2014) suggested an image encryption technique based on fractional chaotic
systems and verified experimentally the higher security of the underlying
cryptosystem. Ahmad et al. Ahmad (2015) highlighted certain defects of the
above-mentioned cryptosytem by recovering the plain image without the secret
key. Moreover, they proposed an enhanced scheme to thwart all kinds of
attacks.
The chaos-based algorithms also use pseudo-random numbers and substitution
boxes (S-boxes) to create confusion and diffusion Cheng (2015); Belazi (2016).
Cheng et al. Cheng (2015) proposed an image encryption algorithm based on
pseudo-random numbers and AES S-box. The pseudo-random numbers are generated
using AES S-box and chaotic tent maps. The scheme is optimized by combining
the permutation and diffusion phases, but the image is encrypted in rounds,
which is time consuming. Belazi et al. Belazi (2016) suggested an image
encryption algorithm using a new chaotic map and logistic map. The new chaotic
map is used to generate a sequence of pseudo-random numbers for masking phase.
Then eight dynamic S-boxes are generated. The masked image is substituted in
blocks via aforementioned S-boxes. The substituted image is again masked by
another pseudo-random sequence generated by the logistic map. Finally, the
encrypted image is obtained by permuting the masked image. The permutation is
done by a sequence generated by the map function. This algorithm fulfills the
security analysis but performs slowly due to the four cryptographic phases. In
Rehman (2016), an image encryption method based on chaotic maps and dynamic
S-boxes is proposed. The chaotic maps are used to generate the pseudo-random
sequences and S-boxes. To break the correlation, pixels of an input image are
permuted by the pseudo-random sequences. In a second phase the permuted image
is decomposed into blocks. Then blocks are encrypted by the generated S-boxes
to get the cipher image. From histogram analysis it follows that the suggested
technique generates cipher images with a nonuniform distribution.
Similar to the chaotic maps, elliptic curves (ECs) are sensitive to input
parameters, but EC-based cryptosystems are more secure than those of chaos Jia
(2016). Toughi et al. Toughi (2017) developed a hybrid encryption algorithm
using elliptic curve cryptography (ECC) and AES. The points of an EC are used
to generate pseudo-random numbers and keys for encryption are acquired by
applying AES to the pseudo-random numbers. The proposed algorithm gets the
promising security but pseudo-random numbers are generated via the group law,
which is time consuming. In El-Latif (2013), a cyclic EC and a chaotic map are
combined to design an encryption algorithm. The developed scheme overcomes the
drawbacks of small key space but is unsafe to the known-plaintext/chosen-
plaintext attack Liu (2014). Similarly, Hayat et al. Hayat (2019) proposed an
EC-based encryption technique. The stated scheme generates pseudo-random
numbers and dynamic S-boxes in two phases, where the construction of S-box is
not guaranteed for each input EC. Therefore, changing of ECs to generate an
S-box is a time-consuming work. Furthermore, the generation of ECs for each
input image makes it insufficient.
Based on the above discussion, we propose an improved image encryption
algorithm, based on quasi-resonant Rossby/drift wave triads Bustamante (2013);
Hayat (2016) (triads, for short) and Mordell elliptic curves (MECs). The
triads are utilized in the generation of pseudo-random numbers and MECs are
employed to create dynamic S-boxes. The proposed scheme is novel in that it
introduces the technique of pseudo-random numbers generation using triads,
which is faster than generating pseudo-random numbers by ECs. Moreover, the
scheme does not require to separately generate triads for each input image of
the same size. In the present scheme, MECs are used opposite to Hayat (2019),
in the sense that now, for each input image, the generation of a dynamic S-box
is guaranteed Azam (2018). Finally, extensive performance analyses and
comparisons reveal the efficiency of the proposed scheme.
This paper is organized as follows. Preliminaries are described in Section 2.
In Section 3, the proposed encryption algorithm is explained in detail.
Section 4 provides the experimental results as well as a comparison between
the proposed method and other existing popular schemes. Lastly, conclusions
are presented in Section 5.
## 2 Preliminaries
Barotropic vorticity equation: The barotropic vorticity equation (in the so-
called $\beta$-plane approximation) is one of the simplest two-dimensional
models of the large-scale dynamics of a shallow layer of fluid on the surface
of a rotating sphere. It is described in mathematical terms by the partial
differential equation
$\frac{\partial}{\partial
t}(\nabla^{2}\psi-F\psi)+\Big{(}\frac{\partial\psi}{\partial
x}\frac{\partial\nabla^{2}\psi}{\partial y}-\frac{\partial\psi}{\partial
y}\frac{\partial\nabla^{2}\psi}{\partial
x}\Big{)}+\gamma\frac{\partial\psi}{\partial x}=0,$ (1)
where $\psi(x,y,t)\in\mathbb{R}$ represents the geopotential height, $\gamma$
is the Coriolis parameter, a real constant measuring the variation of the
Coriolis force with latitude ($x$ represents longitude and $y$ represents
latitude) and $F$ is a non-negative real constant representing the inverse of
the square of the deformation radius. We assume periodic boundary conditions:
$\psi(x+2\pi,y,t)=\psi(x,y+2\pi,t)=\psi(x,y,t)$ for all $x,y,t\in\mathbb{R}$.
In the literature Equation (1) is also known as the Charney–Hasegawa–Mima
equation (CHM) Charney (1948); Hasegawa (1978); Connaughton (2010); Harris
(2013); Galperin (2019). This equation accepts harmonic solutions, known as
Rossby waves, which are solutions of both the linearized form and the whole
(nonlinear) form of Equation (1). A Rossby wave solution is given explicitly
by the parameterized function
$\psi_{(k,l)}(x,y,t)=\Re\\{A\,{\mathrm{e}}^{i(kx+ly-\omega(k,l)t)}\\}$, where
$A\in\mathbb{C}$ is an arbitrary constant,
$\omega(k,l)=-\frac{\gamma\,k}{k^{2}+l^{2}+F}$ is the so-called dispersion
relation, and $(k,l)\in\mathbb{Z}^{2}$ is called the wave vector. For
simplicity, we take $\gamma=-1$ and $F=0$ in what follows Bustamante (2013);
Hayat (2016).
Resonant triads: As Equation (1) is nonlinear, modes with different wave
vectors tend to couple and exchange energy. If the nonlinearity is weak, this
exchange happens to be quite slow and is more efficient amongst groups of
modes that are in _resonance_. To the lowest order of nonlinearity in Equation
(1), approximate solutions known as resonant triad solutions can be
constructed via linear combinations of the form
$\psi(x,y,t)=\Re\\{A_{1}\,{\mathrm{e}}^{i(k_{1}x+l_{1}y-\omega(k_{1},l_{1})t)}+A_{2}\,{\mathrm{e}}^{i(k_{2}x+l_{2}y-\omega(k_{2},l_{2})t)}+A_{3}\,{\mathrm{e}}^{i(k_{3}x+l_{3}y-\omega(k_{3},l_{3})t)}\\}\,,$
where $A_{1},A_{2},A_{3}$ are slow functions of time (they satisfy a closed
system of ODEs, not shown here), and the wave vectors
$(k_{1},l_{1}),(k_{2},l_{2})$ and $(k_{3},l_{3})$ satisfy the Diophantine
system of equations:
$k_{1}+k_{2}=k_{3},\,\,\,\,\,\,l_{1}+l_{2}=l_{3}\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\omega_{1}+\omega_{2}=\omega_{3},$ (2)
for $\omega_{i}=\omega(k_{i},l_{i}),i=1,2,3$. A set of three wavevectors
satisfying Equations (2) is called a resonant triad. Solutions can be found
analytically via a rational transformation to elliptic surfaces (see below).
Quasi-resonant triads and detuning level: If, in (2), the equation
$\omega_{1}+\omega_{2}=\omega_{3}$ is replaced by the inequality
$|\omega_{1}+\omega_{2}-\omega_{3}|\leq\delta^{-1}$, for a large positive
number $\delta$, then the triad becomes a quasi-resonant triad and
$\delta^{-1}$ is known as the detuning level of the quasi-resonant triad. It
is possible to construct quasi-resonant triads via downscaling of resonant
triads that have very large wave vectors Bustamante (2013). For simplicity, in
what follows we simply call a quasi-resonant triad a triad and denote it by
$\Delta$. Finally, to avoid over-counting of triads we will impose the
condition $k_{3}>0$.
Rational transformation: In Bustamante (2013), wave vectors are explicitly
expressed in terms of rational variables $X,Y$ and $D$ as follows:
$\frac{k_{1}}{k_{3}}=\frac{X}{Y^{2}+D^{2}},\,\,\,\,\,\,\frac{l_{1}}{k_{3}}=\Big{(}\frac{X}{Y}\Big{)}\Big{(}1-\frac{D}{Y^{2}+D^{2}}\Big{)},\,\,\,\,\,\,\frac{l_{3}}{k_{3}}=\frac{D-1}{Y}.$
(3)
In the case $F=0$, the rational variables $X,Y,D$ lie on an elliptic surface.
The transformation is bijective and its inverse mapping is given by:
$X=\frac{k_{3}(k_{1}^{2}+l_{1}^{2})}{k_{1}(k_{3}^{2}+l_{3}^{2})},\,\,\,\,\,\,Y=\frac{k_{3}(k_{3}l_{1}-k_{1}l_{3})}{k_{1}(k_{3}^{2}+l_{3}^{2})},\,\,\,\,\,\,D=\frac{k_{3}(k_{3}k_{1}-l_{1}l_{3})}{k_{1}(k_{3}^{2}+l_{3}^{2})}.$
(4)
New parameterization: In Kopp (2017), Kopp parameterized the resonant triads
and in terms of parameters $u$ and $t$ it follows by Kopp (2017) (Equation
(1.22)) that:
$\displaystyle\frac{k_{1}}{k_{3}}=$
$\displaystyle(t^{2}+u^{2})(t^{2}-2u+u^{2})/(1-2u),$ (5)
$\displaystyle\frac{l_{3}}{k_{3}}=$
$\displaystyle\big{(}u(2u-1)+(t^{2}+u^{2})(t^{2}-2u+u^{2})\big{)}/\big{(}t(1-2u)\big{)},$
(6) $\displaystyle\frac{l_{1}}{k_{3}}=$
$\displaystyle(t^{2}+u^{2})\big{(}(2u-1)+u(t^{2}-2u+u^{2})\big{)}/\big{(}t(1-2u)\big{)}.$
(7)
In $2019$, Hayat et al. Hayat (2016) found a new parameterization of $X,Y$ and
$D$ in terms of auxiliary parameters $a,b$ and hence
$\frac{k_{1}}{k_{3}},\frac{l_{3}}{k_{3}}$ and $\frac{l_{1}}{k_{3}}$ are given
by:
$\displaystyle\frac{k_{1}}{k_{3}}=$
$\displaystyle\frac{\big{(}a^{2}+b(2-3b)+1\big{)}^{3}}{(a^{2}-3b^{2}-2b+1)\big{(}2(11-3a^{2})b^{2}+(a^{2}+1)^{2}-16ab+9b^{4}\big{)}},$
(8) $\displaystyle\frac{l_{3}}{k_{3}}=$
$\displaystyle\frac{6(a^{2}+a-1)b^{2}-(a+1)^{2}(a^{2}+1)+4ab-9b^{4}}{(a^{2}-3b^{2}-1)(a^{2}-3b^{2}-2b+1)},$
(9) $\displaystyle\frac{l_{1}}{k_{3}}=$
$\displaystyle\frac{\big{(}a^{2}+b(2-3b)+1\big{)}}{\footnotesize\begin{matrix}(a^{2}-3b^{2}-1)(a^{2}-3b^{2}-2b+1)\big{(}2(11-2a^{2})b^{2}+(a^{2}+1)^{2}-16ab+9b^{4}\big{)}&\\\
\times[a^{6}+2a^{5}+a^{4}(-9b^{2}-6b+3)-4a^{3}(3b^{2}+2b-1)+3a^{2}(3b^{2}+2b-1)^{2}&\\\
+2a(9b^{4}+12b^{3}+14b^{2}-4b+1)-(3b^{2}+1)^{2}(3b^{2}+6b-1)]\end{matrix}}.$
(10)
Elliptic curve (EC): Let $\mathbb{F}_{p}$ be a finite field for any prime
$p$, then an EC $E_{p}$ over $\mathbb{F}_{p}$ is defined by
$y^{2}\equiv x^{3}+bx+c\pmod{p},$ (11)
where $b,c\in\mathbb{F}_{p}$. The integers $b,c$ and $p$ are called parameters
of an EC. The number of all $(x,y)\in\mathbb{F}_{p}^{2}$ satisfying the
congruence (11) is denoted by $\\#E_{p}$.
Mordell elliptic curve (MEC): In the special but important case $b=0$, the
above EC is known as an MEC and is represented by
$y^{2}\equiv x^{3}+c\pmod{p}.$ (12)
For $p\equiv 2\pmod{3}$, there are exactly $p+1$ points
$(x,y)\in\mathbb{F}_{p}^{2}$ satisfying the congruence (12), see Wash for
further details.
If points on $E_{p}$ are ordered according to some total order $\prec$ then
$E_{p}$ is said to be an ordered EC. Recall that total order is a binary
relation which possesses the reflexive, antisymmetric and transitive
properties. Azam et al. Azam1 (2019) introduced a total order known as a
natural ordering on MECs given by
$(x_{1},y_{1})\prec(x_{2},y_{2})\Leftrightarrow\begin{cases}{\rm either\
}x_{1}<x_{2},{\rm\ or}\\\\[5.0pt] x_{1}=x_{2}{\rm\ and\
}y_{1}<y_{2},\end{cases}$
and generated efficient S-boxes using the aforesaid ordering. We will use
natural ordering to generate S-boxes. Thus from here on $E_{p}$ stands for a
naturally ordered MEC unless it is specified otherwise.
## 3 The Proposed Encryption Scheme
The proposed encryption scheme is based on pseudo-random numbers and S-boxes.
The pseudo-random numbers are generated using quasi-resonant triads. To get an
appropriate level of diffusion we need to properly order the $\Delta$s. For
this purpose we define a binary relation $\lesssim$ as follows.
### 3.1 Ordering on Quasi-Resonant Triads
Let $\Delta,\Delta^{\prime}$ represent the triads
$(k_{i},l_{i}),(k_{i}^{\prime},l_{i}^{\prime}),i=1,2,3$, respectively, then
$\Delta\lesssim\Delta^{\prime}\Leftrightarrow\begin{cases}{\rm either\
}a<a^{\prime},{\rm\ or}\\\\[5.0pt] a=a^{\prime}{\rm\ and\ }b<b^{\prime},{\rm\
or}\\\\[5.0pt] a=a^{\prime},b=b^{\prime}{\rm\ and\ }k_{3}\leq
k_{3}^{\prime},\end{cases}$
where $a,b$ and $a^{\prime},b^{\prime}$ are the corresponding auxiliary
parameters of $\Delta$ and $\Delta^{\prime}$, respectively.
###### Lemma .
If $T$ denotes the set of $\Delta$s in a box of size $L$, then $\lesssim$ is a
total order on $T$.
###### Proof.
The reflexivity of $\lesssim$ follows from $a=a,b=b$ and $k_{3}=k_{3}$ and
hence $\Delta\lesssim\Delta.$ As for antisymmetry we suppose
$\Delta\lesssim\Delta^{\prime}$ and $\Delta^{\prime}\lesssim\Delta$. Then, by
definition $a\leq a^{\prime}$ and $a^{\prime}\leq a$, which imply
$a=a^{\prime}$. Thus we are left with two results: $b\leq b^{\prime}$ and
$b^{\prime}\leq b$, which imply $b=b^{\prime}$. Thus, we obtain the results
$k_{3}\leq k_{3}^{\prime}$ and $k_{3}^{\prime}\leq k_{3}$, which ultimately
give $k_{3}=k_{3}^{\prime}$. Solving Equations (8)–(10) for the obtained
values, we get $k_{1}=k_{1}^{\prime},l_{3}=l_{3}^{\prime}$ and from Equation
(2) it follows that $l_{2}=l_{2}^{\prime}$. Consequently
$\Delta=\Delta^{\prime}$ and $\lesssim$ is antisymmetric. As for transitivity,
let us assume $\Delta\lesssim\Delta^{\prime}$ and
$\Delta^{\prime}\lesssim\Delta^{\prime\prime}$. Then $a\leq a^{\prime}$ and
$a^{\prime}\leq a^{\prime\prime}$, implying $a\leq a^{\prime\prime}$. If
$a<a^{\prime\prime}$, then transitivity follows. If $a=a^{\prime\prime}$, then
$a^{\prime}=a^{\prime\prime}$ too. Thus, $b\leq b^{\prime}$ and
$b^{\prime}\leq b^{\prime\prime}$, so $b\leq b^{\prime\prime}$. If
$b<b^{\prime\prime}$, then transitivity follows. If $b=b^{\prime\prime}$, then
$b^{\prime}=b^{\prime\prime}$ too. Thus, $k_{3}\leq k_{3}^{\prime}$ and
$k_{3}^{\prime}\leq k_{3}^{\prime\prime}$, implying $k_{3}\leq
k_{3}^{\prime\prime}$ and hence transitivity follows:
$\Delta\lesssim\Delta^{\prime\prime}$. ∎
Let $\stackrel{{\scriptstyle*}}{{T}}$ stand for the set of $\Delta$s ordered
with respect to the order $\lesssim$. The main steps of the proposed scheme
are explained as follows.
### 3.2 Encryption
A. Public parameters: In order to exchange the useful information the sender
and receiver should agree on the public parameters described as below:
1. (1)
Three sets: choose three sets $\mathcal{A}_{i}=[A_{i},B_{i}],i=1,2,3$ of
consecutive numbers with unknown step sizes, where the end points
$A_{i},B_{i},i=1,2,3$ are rational numbers.
2. (2)
A total order: select a total order $\prec$ so that the triads generated by
the above-mentioned sets may be arranged with respect to that order.
Suppose that $P$ represents an image of size $m\times n$ to be encrypted, and
the pixels of $P$ are arranged in column-wise linear ordering. Thus, for
positive integer $i\leq mn$, $P(i)$ represents the $i$-th pixel value in
linear ordering. Define $S_{\rm P}$ as the sum of all pixel values of the
image $P$. Then the proposed scheme chooses the secret keys in the following
ways.
B. Secret keys: To generate confusion and diffusion in an image, the sender
chooses the secret keys as follows.
1. (1)
Step size: select positive integers $a_{i},b_{i}$ to construct the step sizes
$\alpha_{i}=\frac{a_{i}}{b_{i}}$ of $\mathcal{A}_{i},i=1,2$. Additionally,
choose a non-negative integer $a_{3}$ as a step size of $\mathcal{A}_{3}$ in
such a way that $\prod_{i=1}^{3}n_{i}\geq mn$, where
$\\#\mathcal{A}_{i}=n_{i}$ represents the number of elements in
$\mathcal{A}_{i}$.
2. (2)
Detuning level: fix some posive integer $\delta$ to find the detuning level
$\delta^{-1}$ allowed for the triads.
3. (3)
Bound: select a positive integer $L$ such that $|k_{i}|,|l_{i}|\leq L$ for
$i=1,2,3.$ This condition is imposed in order to bound the components of the
triad wave vectors. Furthermore, choose an integer $t$ to find $r=\lfloor
S_{\rm P}/t\rceil$, where $\lfloor\cdot\rceil$ gives the nearest integer when
$S_{\rm P}$ is divided by $t$. The reason for choosing such a $t$ is to
generate key-dependent S-boxes and the integer $r$ is used to diffuse the
components of triads.
4. (4)
A prime: select a prime $p\geq 257$ such that $p\equiv 2\pmod{3}$ as a secret
key for computing nonzero $c\equiv S_{\rm P}+t\pmod{p}$ to generate an S-box
$\zeta_{E_{p}}(p,t,S_{\rm P})$ on the $E_{p}$. The S-box construction
technique is made clear in Algorithm 1, and the S-box generated for
$p=1607,t=182$ and $S=0$ by Algorithm 1 is shown in Table 1. Furthermore, the
cryptographic properties of the said S-box are evaluated in Sections 4.1 and
4.2.
/* $B$ is a set of points $(x,y)$ satisfying $E_{p}$, $B(i)$ is $i$-th point
of $B$ and $y_{i}$ stands for $y$-component of point $B(i)$. */
Input : A prime $p\equiv 2\pmod{3}$ and two integers $t$ and $S$ such that
$c=S+t$ and $S+t\not\equiv 0\pmod{p}$.
Output : An S-box $\zeta_{E_{p}}(p,t,S)$.
1 $B:=\emptyset$;
2 $Y:=[0,(p-1)/2]$;
3 $i\leftarrow 0$;
4 for _$x\in[0,p-1]$_ do
5 for _$y\in Y$_ do
6 if _$y^{2}\equiv x^{3}+c\pmod{p}$_ then
7 $i\leftarrow i+1$; $B(i):=(x,y)$;
8 if _$y\not=0$_ then
9 $i\leftarrow i+1$; $B(i):=(x,p-y)$;
10 break;
11
12
13
14 $Y=Y-\\{y\\}$;
15$\zeta_{E_{p}}(p,t,S)=\\{y_{i}\in B(i):0\leq y_{i}<256\\}.$
Algorithm 1 Construction of $8\times 8$ S-box. Table 1: The obtained S-box
$\zeta_{E_{1607}}(1607,182,0)$.
220 | 118 | 17 | 158 | 25 | 138 | 33 | 196 | 247 | 252 | 15 | 226 | 135 | 177 | 232 | 83
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
161 | 70 | 107 | 186 | 137 | 236 | 21 | 142 | 131 | 103 | 54 | 58 | 217 | 181 | 201 | 172
91 | 84 | 223 | 89 | 29 | 156 | 136 | 14 | 69 | 99 | 164 | 171 | 35 | 188 | 76 | 139
153 | 16 | 198 | 227 | 32 | 10 | 115 | 122 | 184 | 61 | 208 | 225 | 213 | 106 | 94 | 56
165 | 40 | 245 | 189 | 163 | 239 | 193 | 194 | 129 | 175 | 241 | 141 | 130 | 231 | 215 | 127
151 | 199 | 105 | 22 | 148 | 39 | 179 | 173 | 78 | 248 | 81 | 23 | 75 | 55 | 146 | 109
195 | 251 | 178 | 170 | 162 | 206 | 228 | 169 | 147 | 28 | 210 | 221 | 80 | 121 | 202 | 77
9 | 74 | 197 | 31 | 26 | 154 | 145 | 44 | 47 | 82 | 43 | 60 | 117 | 250 | 88 | 191
67 | 8 | 174 | 93 | 1 | 20 | 128 | 53 | 218 | 237 | 96 | 72 | 3 | 65 | 6 | 253
150 | 101 | 119 | 87 | 160 | 133 | 108 | 57 | 41 | 64 | 51 | 49 | 185 | 243 | 2 | 249
167 | 50 | 205 | 183 | 97 | 114 | 48 | 27 | 246 | 254 | 124 | 92 | 19 | 134 | 159 | 95
24 | 224 | 111 | 62 | 116 | 168 | 200 | 86 | 79 | 143 | 126 | 112 | 45 | 71 | 125 | 13
5 | 216 | 187 | 222 | 7 | 113 | 238 | 36 | 204 | 52 | 140 | 46 | 240 | 85 | 207 | 4
152 | 104 | 235 | 190 | 242 | 68 | 63 | 203 | 230 | 176 | 180 | 59 | 157 | 244 | 66 | 212
34 | 90 | 120 | 0 | 30 | 166 | 37 | 255 | 38 | 110 | 211 | 233 | 11 | 155 | 209 | 219
192 | 12 | 144 | 73 | 182 | 132 | 98 | 214 | 42 | 102 | 18 | 149 | 123 | 229 | 100 | 234
The positive integers $a_{1},b_{1},a_{2},b_{2},a_{3},\delta,L,S_{\rm P},t$ and
$p$ are secret keys. Here it is mentioned that the parameters
$a_{1},b_{1},a_{2},b_{2},a_{3},\delta$ and $L$ are used to generate $mn$
triads in a box of size $L$. The generation of triads is explained step by
step in Algorithm 2. These triads along with keys $S_{\rm P}$ and $t$ are used
to generate the sequence $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm
P})$ of pseudo-random numbers.
/* T is a set containing the Quasi-resonant triads, while $m$ and $n$ are the
dimensions of an input image. */
Input : Three sets $\mathcal{A}_{i},i=1,2,3$, inverse detuning level $\delta$,
bound $L$, two positive integers $m$ and $n$.
Output : Quasi-resonant triads
1 $T:=\emptyset$;
2 $c_{1}\leftarrow 0,c_{2}\leftarrow 1$ ;
3 for _$a\in\mathcal{A}_{1}$_ do
4 for _$b\in\mathcal{A}_{2}$_ do
5 $c_{1}\leftarrow c_{1}+1$;
6 Calculate and store the values of
$k_{1}^{\prime}(c_{1}),l_{3}^{\prime}(c_{1})$, and $l_{1}^{\prime}(c_{1})$ for
each pair $(a,b)$ using Equations (8)–(10).
7
8for _$c_{2}\in[1,c_{1}]$_ do
9 for _$k_{3}\in\mathcal{A}_{3}$_ do
10
$k_{1}=\lfloor(k_{1}^{\prime}(c_{2})*k_{3})\rceil,l_{3}=\lfloor(l_{3}^{\prime}(c_{2})*k_{3})\rceil$
and $l_{1}=\lfloor(l_{1}^{\prime}(c_{2})*k_{3})\rceil$;
11 $k_{2}=k_{3}-k_{1},l_{2}=l_{3}-l_{1}$ and
$\omega_{i}=k_{i}/(k_{i}^{2}+l_{i}^{2}),i=1,2,3;$
12 $\omega_{4}=\omega_{3}-\omega_{2}-\omega_{1};$
13 if _$|\omega_{4}| <\delta^{-1}$ and $0<|k_{i}|,|l_{i}|<L,i=1,2,3$_ then
14 $T:=T\cup\\{\Delta\\}$;
15
16 if _#T=mn_ then
17 break;
18
19 break;
20
Sort $T$ with respect to the ordering $\lesssim$ to get
$\stackrel{{\scriptstyle*}}{{T}}$.
Algorithm 2 Generating quasi-resonant triads.
Thus $\Delta_{\rm j}$ represents the $j$-th triad in ordered set
$\stackrel{{\scriptstyle*}}{{T}}$. Moreover, $(k_{ji},l_{ji}),i=1,2,3$ are the
components of $\Delta_{\rm j}$ . In Algorithm 3, the generation of
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})$ is interpreted.
Input : An ordered set $\stackrel{{\scriptstyle*}}{{T}}$, an integer $t$ and a
plain image $P$.
Output : Random numbers sequence
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})$.
1 $Tr(j):=|{rk_{j1}}|+|{l_{j1}}|+|{k_{j2}}|$;
2 $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})(j)=(Tr(j)+S_{\rm
P})\pmod{256}$;
3
Algorithm 3 Generating the proposed pseudo-random sequence.
The proposed sequence $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})$
is cryptographically a good source of pseudo-randomness because triads are
highly sensitive to the auxiliary parameters $(a,b)$ Hayat (2016) and inverse
detuning level $\delta$. It is shown in Bustamante (2013) that the intricate
structure of clusters formed by triads depends on the chosen $\delta$, and the
size of the clusters increases as the inverse detuning level increases.
Moreover, the generation of triads is rapid due to the absence of modular
operation.
C. Performing diffusion. To change the statistical properties of an input
image, a diffusion process is performed. While performing the diffusion, the
pixel values are changed using the sequence
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})$. Let $M_{\rm P}$ denote
the diffused image for a plain image $P$. The proposed scheme alters the
pixels of $P$ according to:
$M_{\rm P}(i)=\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm
P})(i)+P(i)\pmod{256}.$ (13)
D. Performing confusion. A nonlinear function causes confusion in a
cryptosystem, and nonlinear components are necessary for a secure data
encryption scheme. The current scheme uses the dynamic S-boxes to produce the
confusion in an encrypted image. If $C_{\rm P}$ stands for the encrypted image
of $P$, then confusion is performed as follows:
$C_{\rm P}(i)=\zeta_{E_{p}}(p,t,S_{\rm P})(M_{\rm P}(i)).$ (14)
###### Lemma .
If $\\#\mathcal{A}_{i}=n_{i},i=1,2,3$ and $p$ is a prime chosen for the
generation of an S-box, then the time complexity of the proposed encryption
scheme is max$\\{\mathcal{O}(n_{1}n_{2}n_{3}),p^{2}\\}$.
###### Proof.
The computation of all possible values of $k_{1}^{\prime},l_{3}^{\prime}$ and
$l_{1}^{\prime}$ in Algorithm 2 takes $\mathcal{O}(n_{1}n_{2})$ time.
Similarly the time complexity for generating $\stackrel{{\scriptstyle*}}{{T}}$
is $\mathcal{O}(c_{1}n_{3})$ but $c_{1}$ executes $n_{1}n_{2}$ times. Thus the
time required by $\stackrel{{\scriptstyle*}}{{T}}$ and hence by
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})$ is
$\mathcal{O}(n_{1}n_{2}n_{3})$. Additionally, Algorithm 1 shows that the
proposed S-box can be constructed in $\mathcal{O}(p^{2})$ time. Thus the time
complexity of the proposed scheme is
max$\\{\mathcal{O}(n_{1}n_{2}n_{3}),p^{2}\\}$.
∎
Example 1. _In order to have a clear picture of the proposed cryptosystem, we
explain the whole procedure using the following hypothetical $4\times 4$
image. For example, let $I$ represent the plain image of Lena256×256, and let
$P$ be the subimage of $I$ consisting of the intersection of the first four
rows and the first four columns of $I$ as shown in Table 2, whereas the
column-wise linearly ordered image $P$ is shown in Table 3._
Table 2: Plain image $P$. 162 | 162 | 162 | 163
---|---|---|---
162 | 162 | 162 | 163
162 | 162 | 162 | 163
160 | 163 | 160 | 159
Table 3: Linear ordering of image $P$. $P(1)$ | $P(5)$ | $P(9)$ | $P(13)$
---|---|---|---
$P(2)$ | $P(6)$ | $P(10)$ | $P(14)$
$P(3)$ | $P(7)$ | $P(11)$ | $P(15)$
$P(4)$ | $P(8)$ | $P(12)$ | $P(16)$
We have $S_{\rm P}=2589$ and $c=247$ and the values of other parameters are
described in Section 4.3. The corresponding $16$ triads are obtained by
Algorithm 2 as shown in Table 4.
Table 4: The corresponding set $\stackrel{{\scriptstyle*}}{{T}}$ for image $P$. $\Delta_{j}$ | $k_{1}$ | $l_{1}$ | $k_{2}$ | $l_{2}$ | $k_{3}$ | $l_{3}$ | $\Delta_{j}$ | $k_{1}$ | $l_{1}$ | $k_{2}$ | $l_{2}$ | $k_{3}$ | $l_{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\Delta_{1}$ | $-$1128 | 1152 | 1529 | 668 | 401 | 1820 | $\Delta_{9}$ | $-$1240 | 1267 | 1681 | 735 | 441 | 2002
$\Delta_{2}$ | $-$1142 | 1167 | 1548 | 676 | 406 | 1843 | $\Delta_{10}$ | $-$1254 | 1282 | 1700 | 743 | 446 | 2025
$\Delta_{3}$ | $-$1156 | 1181 | 1567 | 685 | 411 | 1866 | $\Delta_{11}$ | $-$1268 | 1296 | 1719 | 751 | 451 | 2047
$\Delta_{4}$ | $-$1170 | 1195 | 1586 | 694 | 416 | 1889 | $\Delta_{12}$ | $-$1282 | 1310 | 1738 | 760 | 456 | 2070
$\Delta_{5}$ | $-$1184 | 1210 | 1605 | 701 | 421 | 1911 | $\Delta_{13}$ | $-$1296 | 1325 | 1757 | 768 | 461 | 2093
$\Delta_{6}$ | $-$1198 | 1224 | 1624 | 710 | 426 | 1934 | $\Delta_{14}$ | $-$1310 | 1339 | 1776 | 776 | 466 | 2115
$\Delta_{7}$ | $-$1212 | 1238 | 1643 | 719 | 431 | 1957 | $\Delta_{15}$ | $-$1325 | 1353 | 1796 | 785 | 471 | 2138
$\Delta_{8}$ | $-$1226 | 1253 | 1662 | 726 | 436 | 1979 | $\Delta_{16}$ | $-$1339 | 1368 | 1815 | 793 | 476 | 2161
From $S_{\rm P}=2589$ and $t=2$, it follows that $r=1295$ and hence by
application of Algorithm 3 the terms of
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)$ are listed in Table 5.
Moreover, the S-box $\zeta_{E_{293}}(293,2,2589)$ is constructed by Algorithm
1, giving the mapping
$\zeta_{E_{293}}(293,2,2589):\\{0,1,\ldots,255\\}\rightarrow\\{0,1,\ldots,255\\}$,
which maps the list $(0,\ldots,255)$ to the list
$(80,213,29,113,180,2,119,174,10,103,190,120,173,99,194,126,167,42,251,78,215,84,209,93,200,130,$
$163,32,17,117,176,62,231,110,183,56,237,75,218,127,166,73,220,13,91,202,28,129,164,118,175,69,$
$224,50,243,100,193,137,156,89,204,12,63,230,74,219,4,131,162,134,159,123,170,90,203,70,223,87,$
$206,59,234,145,148,58,235,57,236,65,228,15,112,181,52,241,76,217,60,233,121,172,68,225,51,242,$
$135,158,41,252,21,142,151,26,25,40,253,96,197,136,157,9,116,177,122,171,45,248,115,178,102,191,$
$67,226,95,198,143,150,133,160,98,195,3,94,199,30,104,189,132,161,8,64,229,144,149,140,153,14,$
$85,208,20,6,109,184,125,168,92,201,19,53,240,31,66,227,35,82,211,108,185,139,154,33,16,86,207,$
$128,165,5,71,222,38,255,23,0,81,212,1,141,152,111,182,138,155,49,244,22,106,187,105,188,36,54,$
$239,46,247,43,250,97,196,27,11,24,44,249,83,210,61,232,39,254,7,72,221,77,216,47,246,107,186,$
$48,245,55,238,124169,34,79,214,88,205,114,179,37,18,146,147,101,192)$.
Table 5: Pseudo-random sequence for plain image $P.$ $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(1)=188$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(5)=126$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(9)=65$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(13)=3$
---|---|---|---
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(2)=108$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(6)=47$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(10)=241$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(14)=180$
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(3)=29$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(7)=224$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(11)=162$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(15)=115$
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(4)=206$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(8)=144$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(12)=83$ | $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,2589)(16)=35$
Hence by the respective application of Equation (13) and the S-box
$\zeta_{E_{293}}(293,2,2589)$, the pixel values of diffused image $M_{\rm P}$
and encrypted image $C_{\rm P}$ are shown in Tables 6 and 7, respectively.
Table 6: Diffused image $M_{\rm P}.$ 94 | 32 | 227 | 166
---|---|---|---
14 | 209 | 147 | 87
191 | 130 | 68 | 22
110 | 51 | 243 | 194
Table 7: Encrypted image $C_{\rm P}.$ 76 | 231 | 254 | 19
---|---|---|---
194 | 54 | 161 | 65
0 | 67 | 162 | 209
151 | 69 | 34 | 1
### 3.3 Decryption
In our scheme the decryption process can take place by reversing the
operations of the encryption process. One should know the inverse S-box
$\zeta^{-1}_{E_{p}}(n,t,S_{\rm P})$ and the pseudo-random numbers
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})$. Assume the situation
when the secret keys $a_{1},b_{1},a_{2},b_{2},a_{3},\delta$, $L,S_{\rm P},t$
and $p$ are transmitted by a secure channel, so that the set
$\stackrel{{\scriptstyle*}}{{T}}$ is obtained using keys
$a_{1},b_{1},a_{2},b_{2},a_{3},\delta$ and $L$, and hence the S-box
$\zeta^{-1}_{E_{p}}(p,t,S_{\rm P})$ and the pseudo-random numbers
$\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm P})$ can be computed by
$S_{\rm P},t$ and $p$. Finally, the receiver gets the original image $P$ by
applying the following equations:
$\displaystyle M_{\rm P}(i)=\zeta_{E_{p}}^{-1}(p,t,S_{\rm P})(C_{\rm P}(i)),$
(15) $\displaystyle\begin{split}P(i)=M_{\rm
P}(i)-\beta_{\stackrel{{\scriptstyle*}}{{T}}}(t,S_{\rm
P})(i)\pmod{256}.\end{split}$ (16)
## 4 Security Analysis
In this section the cryptographic strength of both the S-box construction
technique and encryption scheme are analyzed in detail.
### 4.1 Evaluation of the Designed S-Box
An S-box with good cryptographic properties ensures the quality of an
encryption technique. Generally, some standard tests such as nonlinearity
(NL), linear approximation probability (LAP), strict avalanche criterion
(SAC), bit independence criterion (BIC) and differential approximation
probability (DAP) are used to evaluate the cryptographic strength of an S-box.
The NL Adams (1990) and the LAP Matsui (1994) are outstanding features of an
S-box, used to measure the resistance against linear attacks. The NL measures
the level of nonlinearity and the LAP finds the maximum imbalance value of an
S-box. The optimal value of the nonlinearity is $112$. A low value of LAP
corresponds to a high resistance. The minimum NL and the LAP values for the
displayed S-box are $106$ and $0.1484$, respectively. This ensures that the
proposed S-box is immune to linear attacks. Webster and Tavares Webster (1985)
developed the concepts of the SAC and the BIC, which are used to find the
confusion and diffusion creation potential of an S-box. In other words, the
SAC criterion measures the change in output bits when an input bit is altered.
Similarly, the BIC criterion explores the correlation in output bits when
change in a single input bit occurs. The average values of the SAC and the BIC
for the constructed S-box are $0.4951$ and $0.4988$, respectively, which are
close to the optimal value $0.5$. Thus, both tests are satisfied by the
suggested S-box. The DAP Biham (1991) is another important feature used to
analyze the capability of an S-box against differential attacks. The lowest
value of DAP for an S-box implies the highest security to the differential
attacks. Our DAP result is $0.0234$, which is good enough to resist
differential cryptanalysts.
### 4.2 Performance Comparison of the S-Box Generation Algorithm
After performing the rigorous analyses, the S-box constructed by the current
algorithm is compared with some cryptographically strong S-boxes developed by
recent schemes, as shown in Table 8.
Table 8: Comparison table of the proposed S-box
$\zeta_{E_{1607}}(1607,182,0)$.
S-Boxes | NL | LAP | SAC | BIC | DAP
---|---|---|---|---|---
| | | (min) | (avg) | (max) | (min) | (avg) | (max) |
Ours | 106 | 0.1484375 | 0.390625 | 0.49511719 | 0.609375 | 0.47265625 | 0.49888393 | 0.52539063 | 0.0234375
Ref. Hayat (2019) | 104 | 0.1484375 | 0.421900 | - | 0.6094 | 0.4629 | - | 0.5430 | 0.0469
Ref. Ye (2018) | 104 | 0.1328125 | 0.40625 | 0.49755859 | 0.625 | 0.46679688 | 0.50223214 | 0.5234375 | 0.0234375
Ref. Ozkaynak (2017) | 101 | 0.140625 | 0.421875 | 0.49633789 | 0.578125 | 0.46679688 | 0.49379185 | 0.51953125 | 0.03125
Ref. Çavuşoğlu (2017) | 104 | 0.140625 | 0.421875 | 0.50390625 | 0.59375 | 0.4765625 | 0.50585938 | 0.5390625 | 0.0234375
Ref. Belazi (2017) | 100 | 0.140625 | 0.40625 | 0.50097656 | 0.609375 | 0.44726563 | 0.50634766 | 0.53320313 | 0.03125
Ref. Özkaynak (2019) | 106 | 0.140625 | 0.390625 | 0.49414063 | 0.609375 | 0.47070313 | 0.50132533 | 0.53320313 | 0.0234375
Ref. Liu (2018) | 102 | 0.140625 | 0.421875 | 0.49804688 | 0.640625 | 0.4765625 | 0.50746373 | 0.53320313 | 0.0234375
Ref. Hayat (2018) | 104 | 0.0391 | 0.3906 | - | 0.6250 | 0.4707 | - | 0.53125 | 0.0391
Ref. Wang (2019) | 104 | 0.0547000 | 0.4018 | 0.4946 | 0.5781 | 0.4667969 | 0.4988839 | 0.5332031 | 0.0391
Ref. Alzaidi (2018) | 108 | 0.1328 | 0.40625 | 0.4985352 | 0.59375 | 0.46484375 | 0.5020229 | 0.52734375 | 0.0234375
From Table 8 it follows that the NL of $\zeta_{E_{1607}}(1607,182,0)$ is
greater than the S-boxes in Hayat (2019); Ye (2018); Ozkaynak (2017);
Çavuşoğlu (2017); Belazi (2017); Liu (2018); Hayat (2018); Wang (2019), equal
to that of Özkaynak (2019) and less than the S-box developed in Alzaidi
(2018), which indicates that $\zeta_{E_{1607}}(1607,182,0)$ is highly
nonlinear in comparison to the S-boxes in Hayat (2019); Ye (2018); Ozkaynak
(2017); Çavuşoğlu (2017); Belazi (2017); Liu (2018); Hayat (2018); Wang
(2019). Additionally, the LAP of $\zeta_{E_{1607}}(1607,182,0)$ is comparable
to all the S-boxes in Table 8. The SAC (average) value of
$\zeta_{E_{1607}}(1607,182,0)$ is greater than the S-boxes in Özkaynak (2019);
Wang (2019), and the SAC (max) value is less than or equal to the S-boxes in
Hayat (2019); Ye (2018); Belazi (2017); Özkaynak (2019); Liu (2018); Hayat
(2018). Similarly the BIC (min) value of $\zeta_{E_{1607}}(1607,182,0)$ is
closer to the optimal value $0.5$ than that of Hayat (2019); Ye (2018);
Ozkaynak (2017); Belazi (2017); Özkaynak (2019); Hayat (2018); Wang (2019);
Alzaidi (2018), and the BIC (max) value of the new S-box is better than that
of the S-boxes in Hayat (2019); Çavuşoğlu (2017); Belazi (2017); Özkaynak
(2019); Liu (2018); Hayat (2018); Wang (2019); Alzaidi (2018). Thus the
confusion/diffusion creation capability of $\zeta_{E_{1607}}(1607,182,0)$ is
better than Hayat (2019); Belazi (2017); Özkaynak (2019); Liu (2018); Hayat
(2018); Alzaidi (2018). The DAP value of our suggested S-box
$\zeta_{E_{1607}}(1607,182,0)$ is lower than the DAP of the S-boxes presented
in Hayat (2019); Ozkaynak (2017); Belazi (2017); Hayat (2018); Wang (2019) and
equal to that of Ye (2018); Çavuşoğlu (2017); Özkaynak (2019); Liu (2018);
Alzaidi (2018). Thus from the above discussion it follows that the newly
designed S-box shows high resistance to linear as well as differential
attacks.
### 4.3 Evaluation of the Proposed Encryption Technique
In this section the current scheme is implemented on all gray images of the
USC-SIPI Image Database Dbase . The USC-SIPI database contains images of size
$m\times m$, $m$ = 256,512,1024. Furthermore, some security analyses that are
explained one by one in the associated subsections are presented. To validate
the quality of the proposed scheme, the experimental results are compared with
some other encryption schemes. The parameters used for the experiments are
$A_{1}=A_{2}=-1.0541,A_{3}=401,B_{1}=B_{2}=-0.8514$ and $B_{3}=691,3036,5071$
for $m$ = 256,512,1024, respectively;
$a_{1}=2,b_{1}=1000,a_{2}=19,b_{2}=1000,a_{3}=5,\delta=1000,t=2,p=293,L$ =
90,000 and $S_{\rm P}$ varies for each $P$. The experiments were performed
using Matlab R$2016$a on a personal computer with a $1.8$ GHz Processor and
$6$ GB RAM. All encrypted images of the database along with histograms are
available at github . Some plain images, House256×256, Stream512×512,
Boat512×512 and Male1024×1024 and their cipher images are displayed in Figure
1.
a
b
c
d
e
f
g
h
Figure 1: (a)–(d) Plain images House, Stream, Boat and Male; (e)–(h) cipher
images of the plain images (a)–(d), respectively.
#### 4.3.1 Statistical Attack
A cryptosystem is said to be secure if it has high resistance against
statistical attacks. The strength of resistance against statistical attacks is
measured by entropy, correlation and histogram tests. All of these tests are
applied to evaluate the performance of the discussed scheme.
1. (1)
Histogram. A histogram is a graphical way to display the frequency
distribution of pixel values of an image. A secure cryptosystem generates
cipher images with uniform histograms. The histograms of the encrypted images
using the proposed method are available at github . However, the respective
histograms for the images in Figure 1 are shown in Figure 2. The histograms of
the encrypted images are almost uniform. Moreover, the histogram of an
encrypted image is totally different from that of the respective plain image,
so that it does not allow useful information to the adversaries, and the
proposed algorithm can resist any statistical attack.
a
b
c
d
e
f
g
h
Figure 2: (a)–(d) Histograms of Figure 1(a)–(d); (e)–(h) histograms of Figure
1(e)–(h).
2. (2)
Entropy. Entropy is a standout feature to measure the disorder. Let $I$ be a
source of information over a set of symbols $N$. Then the entropy of $I$ is
defined by:
${\rm H}(I)=\sum_{i=1}^{\\#N}p(I_{i}){\rm log}_{2}\frac{1}{p(I_{i})},$ (17)
where $p(I_{i})$ is the probability of occurrence of symbol $i.$ The ideal
value of ${\rm H}(I)$ is ${\rm log}_{2}{(\\#N)}$, if all symbols of $N$ occur
in $I$ with the same probability. Thus, an image $I$ emanating $256$ gray
levels is highly random if ${\rm H}(I)$ is close to $8$ (notice, however, that
this definition of entropy does not take into account pixel correlations). The
entropy results for all images encrypted by the suggested technique are shown
in Figure 3, where the minimum, average and maximum values are $7.9966,7.9986$
and $7.9999$, respectively. These results are close to $8$, and hence the
developed mechanism is secure against entropy attacks.
3. (3)
Pixel correlation. A meaningful image has strong correlation among the
adjacent pixels. In fact, a good cryptosystem has the ability to break the
pixel correlation and bring it close to zero. For any two gray values $x$ and
$y$, the pixel correlation can be computed as:
$C_{xy}=\frac{E\big{[}(x-E[x])(y-E[y])\big{]}}{\sqrt{K[x]K[y]}},$ (18)
where $E[x]$ and $K[x]$ denote expectation and variance of $x$, respectively.
The range of $C_{xy}$ is $-1$ to $1$. The gray values $x$ and $y$ are in low
correlation if $C_{xy}$ is close to zero. As the pixels may be adjacent in
horizontal, diagonal and vertical directions, the correlation coefficients of
all encrypted images along all three directions are shown in Figure 3, where
the respective ranges of $C_{xy}$ are [$-0.0078$, $0.0131$],
[$-0.0092$,$0.0080$] and [$-0.0100$,$0.0513$]. These results show that the
presented method is capable of reducing the pixel correlation near to zero.
a
b
c
d
Figure 3: (a)–(c) The horizontal, diagonal and vertical correlations among
pixels of each image in USC-SIPI database; (d) the entropy of each image in
USC-SIPI database.
In addition, $2000$ pairs of adjacent pixels of the plain image and cipher
image of Lena512×512 are randomly selected. Then correlation distributions of
the adjacent pixels in all three directions are shown in Figure 4, which
reveals the strong pixel correlation in the plain image but a weak pixel
correlation in the cipher image generated by the current scheme.
a
b
c
d
e
f
g
h
Figure 4: (b)–(d) The distribution of pixels of the plane image in the
horizontal, diagonal and vertical directions; (f)–(h) the distribution of
pixels of the cipher image in the horizontal, diagonal and vertical
directions.
#### 4.3.2 Differential Attack
In differential attacks the opponents try to get the secret keys by studying
the relation between the plain image and cipher image. Normally attackers
encrypt two images by applying a small change to these images, then compare
the properties of the corresponding cipher images. If a minor change in the
original image can cause a significant change in the encrypted image, then the
cryptosystem has a high security level. The two tests NPCR (number of pixels
change rate) and UACI (unified average changing intensity) are usually used to
describe the security level against differential attacks. For two plain images
$P$ and $P^{{}^{\prime}}$ different at only one pixel value, let $C_{\rm P}$
and $C_{\rm P^{{}^{\prime}}}$ be the cipher images of $P$ and
$P^{{}^{\prime}}$, respectively, then NPCR and UACI are calculated as:
$\displaystyle{\rm NPCR}$
$\displaystyle=\sum_{u=1}^{m}\sum_{v=1}^{n}\frac{\tau(u,v)}{m\times n},$ (19)
$\displaystyle{\rm UACI}$
$\displaystyle=\sum_{u=1}^{m}\sum_{v=1}^{n}\frac{|C_{\rm P}(u,v)-C_{\rm
P^{{}^{\prime}}}(u,v)|}{255\times m\times n},$ (20)
where $\tau(u,v)=0$ if $C_{\rm P}(u,v)=C_{\rm P^{{}^{\prime}}}(u,v)$ and
$\tau(u,v)=1,$ otherwise. The expected values of NPCR and UACI for 8-bit
images are $0.996094$ and $0.334635$, respectively Wu (2019). We applied the
above two tests to each image of the database by randomly changing the pixel
value of each image. The experimental results are shown in Figure 5, giving
average values of NPCR and UACI of $0.9961$ and $0.3334$, respectively. It
follows from the obtained results that our scheme is capable of resisting a
differential attack.
a
b
c
d
Figure 5: (a–b) The NPCR and UACI results for each image in the USC-SIPI
database; (c) First $256$ pseudo-random numbers and (d) two S-boxes generated
for Lena512×512 with a small change in an input key $t$.
#### 4.3.3 Key Analysis
For a secure cryptosystem it is essential to perform well against key attacks.
A cryptosystem is highly secure against key attacks if it has key sensitivity
and large key space and strongly opposes the known-plaintext/chosen-plaintext
attack. The proposed scheme is analyzed against key attacks as follows.
1. (1)
Key sensitivity. Attackers usually use slightly different keys to encrypt a
plain image and then compare the obtained cipher image with the original
cipher image to get the actual keys. Thus, high key sensitivity is essential
for higher security. That is, cipher images of a plain image generated by two
slightly different keys should be entirely different. The difference of the
cipher images is quantified by Equations (19) and (20). In experiments we
encrypted the whole database by changing only one key, while other keys remain
unchanged. The key sensitivity results are shown in Table 9, where the average
values of NPCR and UACI are $0.9960$ and $0.3341$, respectively, which specify
the remarkable difference in the cipher images. Moreover, our cryptosytem is
based on the pseudo-random numbers and S-boxes. The sensitivity of pseudo-
random numbers sequences $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(2,S_{\rm
P})$ and $\beta_{\stackrel{{\scriptstyle*}}{{T}}}(1,S_{\rm P})$ and S-boxes
$\zeta_{E_{p}}(p,2,S_{\rm P})$ and $\zeta_{E_{p}}(p,1,S_{\rm P})$ for
Lena512×512 is shown in Figure 5.
Table 9: Difference between two encrypted images when key $t=2$ is changed to
$t=1$. NPCR: number of pixels change rate; UACI: unified average changing
intensity.
Image | NPCR(%) | UACI(%) | Image | NPCR(%) | UACI(%) | Image | NPCR(%) | UACI(%)
---|---|---|---|---|---|---|---|---
Female | 99.62 | 33.39 | House | 99.62 | 33.23 | Couple | 99.56 | 33.30
Tree | 99.59 | 33.35 | Beans | 99.64 | 33.23 | Splash | 99.60 | 33.97
2. (2)
Key space. In order to resist a brute force attack, key space should be
sufficiently large. For any cryptosystem, key space represents the set of all
possible keys required for the encryption process. Generally, the size of the
key space should be greater than $2^{128}.$ In the present scheme the
parameters $a_{1},b_{1},a_{2},b_{2},a_{3},\delta,L,S_{\rm P},t$ and $p$ are
used as secret keys, and we store each of them in $28$ bits. Thus the key
space of the proposed cryptosystem is $2^{280}$ which is larger than $2^{128}$
and hence capable to resist a brute force attack.
3. (3)
Known-plaintext/chosen-plaintext attack. In a known-plaintext attack, the
attacker has partial knowledge about the plain image and cipher image, and
tries to break the cryptosystem, while in a chosen-plaintext attack the
attacker encrypts an arbitrary image to get the encryption keys. An all-
white/black image is usually encrypted to test the performance of a scheme
against these powerful attacks Wang (2019); Toughi (2017). We analyzed our
scheme by encrypting an all-white/black image of size $256\times 256$. The
results are shown in Figure 6 and Table 10, revealing that the encrypted
images are significantly randomized. Thus the proposed system is capable of
preventing the above mentioned attacks.
a
b
c
d
e
f
Figure 6: (a) All-white; (b) all-black; (c)–(d) cipher images of (a)–(b);
(e)–(f) histograms of (c)–(d).
Table 10: Security analysis of all-white/black encrypted images by the
proposed encryption technique.
Plain Image | Entropy | Correlation of Plain Image | NPCR (%) | UACI (%)
---|---|---|---|---
Hori. | Diag. | Ver.
All-white | 7.9969 | 0.0027 | 0.0020 | $-$0.0090 | 99.60 | 33.45
All-black | 7.9969 | $-$0.0080 | 0.0035 | 0.0057 | 99.62 | 33.41
### 4.4 Comparison and Discussion
Apart from security analyses, the proposed scheme is compared with some well-
known image encryption techniques. The gray scale images of Lena256×256 and
Lena512×512 are encrypted using the presented method, and experimental results
are listed in Table 11.
Table 11: Comparison of the proposed encryption scheme with several existing
cryptosystems for image Lenam×m, $m$ = 256,512.
Size $m$ | Algorithm | Entropy | Correlation | NPCR (%) | UACI(%) | $\\#$ | Dynamic
---|---|---|---|---|---|---|---
Hori. | Diag. | Ver. | S-Boxes | S-Boxes
$256$ | Ours | 7.9974 | 0.0001 | $-$0.0007 | $-$0.0001 | 99.91 | 33.27 | 1 | Yes
Ref. Hayat (2019) | 7.9993 | 0.0012 | 0.0003 | 0.0010 | 99.60 | 33.50 | 1 | Yes
Ref. El-Latif (2013) | 7.9973 | - | - | - | 99.50 | 33.30 | 0 | -
Ref. Rehman (2016) | 7.9046 | 0.0164 | $-$0.0098 | 0.0324 | 98.92 | 32.79 | >1<50 | Yes
Ref. Belazi (2016) | 7.9963 | $-$0.0048 | $-$0.0045 | $-$0.0112 | 99.62 | 33.70 | 8 | Yes
Ref. Wu (2017) | 7.9912 | $-$0.0001 | 0.0091 | 0.0089 | 100 | 33.47 | 0 | -
Ref. Wan (2020) | 7.9974 | 0.0020 | 0.0020 | 0.0105 | 99.59 | 33.52 | 0 | -
$512$ | Ours | 7.9993 | 0.0001 | 0.0042 | 0.0021 | 99.61 | 33.36 | 1 | Yes
Ref. Cheng (2015) | 7.9992 | 0.0075 | 0.0016 | 0.0057 | 99.61 | 33.38 | 1 | No
Ref. Toughi (2017) | 7.9993 | $-$0.0004 | 0.0001 | $-$0.0018 | 99.60 | 33.48 | 1 | No
- | Ref. Tong (2016) | 7.9970 | $-$0.0029 | 0.0135 | 0.0126 | 99.60 | 33.48 | 0 | -
Ref. Zhang (2014) | 7.9994 | 0.0018 | $-$0.0012 | 0.0011 | 99.62 | 33.44 | >1 | Yes
Ref. Zhang (2014) | 7.9993 | 0.0032 | 0.0011 | $-$0.0002 | 99.60 | 33.47 | >1 | Yes
It is deduced that our scheme generates cipher images with comparable
security. Furthermore, we remark that the scheme in Toughi (2017) generates
pseudo-random numbers using group law on EC, while the proposed method
generates pseudo-random numbers by constructing triads using auxiliary
parameters of elliptic surfaces. Group law consists of many operations, which
makes the pseudo-random number generation process slower than the one we
present here. The scheme in Belazi (2016) decomposes an image to eight blocks
and uses dynamic S-boxes for encryption purposes. The computation of multiple
S-boxes takes more time than computing only one S-box. Similarly the
techniques in Zhang (2014); Rehman (2016) use a set of S-boxes and encrypt an
image in blocks, while our newly developed scheme encrypts the whole image
using only one dynamic S-box. Thus, our scheme is faster than the schemes in
Zhang (2014); Rehman (2016). The security system in Tong (2016) uses a chaotic
system to encrypt blocks of an image. The results in Table 11 reveal that our
proposed system is cryptographically stronger than the scheme in Tong (2016).
The algorithms in Wu (2017); El-Latif (2013) combine chaotic systems and
different ECs to encrypt images. It follows from Table 11 that the security
level of our scheme is comparable to that of the schemes in Wu (2017); El-
Latif (2013). The technique in Wan (2020) uses double chaos along with DNA
coding to get good results, as shown in Table 11, but the results obtained by
the new scheme are better than that of Wan (2020). Similarly the technique in
Hayat (2019) encrypts images using ECs but does not guarantee an S-box for
each set of input parameters, thus making our scheme faster and more robust
than the scheme developed in Hayat (2019).
Furthermore, the following facts put our scheme in a favorable position:
* (i)
Our scheme uses a dynamic S-box for each input image while the S-box used in
Toughi (2017) is a static one, which is vulnerable Rosenthal (2003) and less
secure than a dynamic one Kazlauskas (2009).
* (ii)
The presented scheme guarantees an S-box for each image, which is not the case
in Hayat (2019).
* (iii)
To get random numbers, the described scheme generates triads for all images of
the same size, while in Hayat (2019) the computation of an EC for each input
image is necessary, which is time consuming.
* (iv)
The scheme in Belazi (2016) uses eight dynamic S-boxes for a plain image,
while the current scheme uses only one dynamic S-box for each image to get the
desired cryptographic security.
## 5 Conclusions
An image encryption scheme based on quasi-resonant triads and MECs was
introduced. The proposed technique constructs triads to generate pseudo-random
numbers and computes an MEC to construct an S-box for each input image. The
pseudo-random numbers and S-box are then used for altering and scrambling the
pixels of the plain image, respectively. As for the advantages of our proposed
method, firstly triads are based on auxiliary parameters of elliptic surfaces,
and thus pseudo-random numbers and S-boxes generated by our method are highly
sensitive to the plain image, which prevents adversaries from initiating any
successful attack. Secondly, generation of triads using auxiliary parameters
of elliptic surfaces consumes less time than computing points on ECs (we find
a 4x speed increase for a range of image resolutions $m\in[128,512]$), which
makes the new encryption system relatively faster. Thirdly, our algorithm
generates the cipher images with an appropriate security level.
In summary, all of the above analyses imply that the presented scheme is able
to resist all attacks. It has high encryption efficiency and less time
complexity than some of the existing techniques. In the future, the current
scheme will be further optimized by means of new ideas to construct the
S-boxes using the constructed triads, so that we will not need to compute an
MEC for each input image.
All authors contributed equally to this work.
This research is funded through the HEC project NRPU-7433.
###### Acknowledgements.
We thank Gene Kopp for useful comments and suggestions. The authors declare no
conflict of interest. The funding sponsors had no role in the design of the
study; in the collection, analyses, or interpretation of data; in the writing
of the manuscript, and in the decision to publish the results. The following
abbreviations are used in this manuscript:
MEC | Mordell elliptic curve
---|---
S-box | Substitution box
EC | Elliptic curves
no References
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|
2024-09-04T02:54:57.980511 | 2020-03-06T21:19:44 | 2003.03444 | {
"authors": "Yuval Pinter and Cassandra L. Jacobs and Max Bittker",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26091",
"submitter": "Yuval Pinter",
"url": "https://arxiv.org/abs/2003.03444"
} | arxiv-papers | # NYTWIT: A Dataset of Novel Words in the New York Times
Yuval Pinter
Georgia Institute of Technology
Atlanta, GA, USA
<EMAIL_ADDRESS>
&Cassandra L. Jacobs
University of Wisconsin
Madison, WA, USA
<EMAIL_ADDRESS>
&Max Bittker
School for Poetic Computation
New York, NY, USA
<EMAIL_ADDRESS>
###### Abstract
We present the New York Times Word Innovation Types dataset, or NYTWIT, a
collection of over 2,500 novel English words published in the New York Times
between November 2017 and March 2019, manually annotated for their class of
novelty (such as lexical derivation, dialectal variation, blending, or
compounding). We present baseline results for both uncontextual and contextual
prediction of novelty class, showing that there is room for improvement even
for state-of-the-art NLP systems. We hope this resource will prove useful for
linguists and NLP practitioners by providing a real-world environment of novel
word appearance.
## 1 Introduction
Novel words, or Out-Of-Vocabulary words (OOVs), are a pervasive problem in
modern natural language processing [Brill, 1995, Young et al., 2018]. A common
scenario in which this problem appears is that of a pre-trained model
containing a word representation component such as an embedding table,
encountering a previously-unseen word in a downstream task such as question
answering or natural language inference. Multiple lines of work attempt to
alleviate the downstream effect of OOV words [Müller and Schütze, 2011, Pinter
et al., 2017], but most tend to focus on individual categories of OOVs:
typographical errors [Sakaguchi et al., 2017], domain-specific terminology [Du
et al., 2016], stylistic variability [Eisenstein, 2013, van der Goot, 2019],
morphological productivity [Bhatia et al., 2016], or novel named entities
[Hoffart et al., 2014]. In reality, unseen texts contain all these classes of
novelty, and more. OOVs are a typically presented as a significant challenge
for generalization or understanding in noisy user-generated text (e.g.
Twitter) and/or domain-specific content. Nevertheless, even large corpora that
are narrow in domain (edited news stories) contain linguistic innovations,
including but not limited to novel morphological processes, typographical
errors, and loan words.
In this paper, we present a dataset of novel words in English relative to the
corpus of articles published by the New York Times (NYT), as collected
automatically in real time by a Twitter bot. We name it the New York Times
Word Innovation Types corpus, or NYTWIT for short. We annotated each word for
one of eighteen linguistically-informed categories of novelty within the
context of the NYT corpus, as well as for its date of publication and a
retrieval document identifier to enable context extraction.111Context article
excerpts are not freely available without copyright licensing from the New
York Times, who have ignored all contact attempts to date. To our knowledge,
this is the first resource to include novel words along with their contextual
information in addition to linguistically-informed annotation, a method that
enables expansion beyond dictionary-based methods [Cook and Stevenson, 2010,
Dhuliawala et al., 2016, Ahmad, 2000] and decontextualized neologisms
[Kulkarni and Wang, 2018]. In contrast with resources which provide examples
and attestations to lexical forms, NYTWIT was constructed in a corpus-
comprehensive manner where novelty guides curation and not vice versa.
In addition, we provide results for the task of classifying words into their
categories based on word form and contextual information, a task which can
both provide data for linguistic analysis of lexical enrichment and serve as a
processing step for NLP systems which may work better if different modules are
applied to different classes of novel words. We show that both character-level
models and large pre-trained sentence encoders struggle on this task,
illustrating the challenges of modeling language innovation.
We release the data at https://github.com/yuvalpinter/nytwit under the GNU
General Public License v3.0. The project is ongoing, and this document
pertains to version 1.1.
## 2 The New York Times Word Innovation Types Dataset
Our dataset is built upon two bots developed by the third author. The first
stage of data collection relies on tweets from the NYT_First_Said
bot222https://twitter.com/NYT_First_Said, which operates by scraping new
articles as they post on the NYT site and tweeting out novel words following a
filtering process which we will describe at a high level.333The code for the
bot is available at https://github.com/MaxBittker/NYT-first-said. After
tokenizing on white space and punctuation, the precision-oriented script
rejects capitalized words in order to avoid proper nouns (at the cost of
missing sentence-initial true OOVs). langid [Lui and Baldwin, 2012] is used to
reject non-English sentences, while still allowing loanwords in English
sentences. Words are queried against the historical NYT search API to detect
unpublished words.444We note that the search index relies on imperfect,
although extensive, digitization artifacts. At the time of writing, in a
sample of 450 terms from our dataset, [Cassie: 4] were entries in the Oxford
English Dictionary, nearly all of which belong to the domain or foreign
categories. For the time range of our collected corpus, November 7, 2017 to
March 28, 2019, a bandwidth limit of five words per 30 minutes was imposed,
but we confirmed that this did not have a substantial effect on OOV coverage,
leaving our artifacts distributionally representative for the news domain.
An associated context bot replies to the tweets with links to the original
articles.555https://twitter.com/NYT_Said_Where We used the URLs from this
bot’s posts as the main reference for the words’ contexts. For 17 words, the
article URL was retrieved manually by searching for the target article
directly.666One term lacks a context because neither the NYT search engine nor
the API support the letter é. As the articles are subject to edits long after
publication, there is an increasing but small portion of articles which no
longer contain the context, although at time of publication these mostly
include the removal of typographical errors from the stories and which are
ultimately filtered by our annotation process (see below).
### 2.1 Annotation
The extracted data was independently annotated and filtered by the first two
authors. Initially, all 2,587 words were assigned one of 20+ tags inspired by
the word formation literature [Kiparsky, 1982, Klymenko, 2019]. Certain
categories were filter categories intended to capture and exclude false
positives from the final dataset: Duplicate for inflections of words already
appearing in the dataset in a morphologically simpler form, e.g. batchcode and
batchcodes; Foreign and PRP for foreign words and proper names (mostly all-
lowercase Twitter usernames) which were not caught by the automatic filtering;
Spaces and Typo for unintended cases of space deletion and typographical
errors which were not caught by NYT editors.777The overwhelming share of these
words have indeed since been deleted from the NYT website. The filtered items
are provided in the dataset under the label Filtered.
Agreement between the annotators at the preliminary phase was 68% over all
labels, and 0.65 Cohen’s Kappa. Following category filtering, amounting to 40%
of the original dataset, agreement over the remaining 1,550 words was
calculated to be 65% at 0.61 Kappa. At the coarse-grained level, agreement on
the four themes (lexical / morphological / syntactic / sociopragmatic) was 89%
at 0.75 Kappa.888A reviewer noted that these are low agreement rates, and
compared the task to part-of-speech annotation. We dispute the comparison,
both on grounds of the novelty of the forms involved and of the mechanical
syntactic nature of the majority of POS tagging decisions.
The annotators then examined each other’s annotations and agreed on some
consolidation of rarely-occurring original labels, as well as introduction of
new labels deemed useful post-hoc.
### 2.2 Novel Word Taxonomy
We describe the eighteen categories in the finalized dataset, organized by a
thematic grouping not explicitly annotated. Counts for each category are
provided [in brackets].
##### Lexical OOVs.
We deem certain categories to arise from the fact that the NYT, while being
interested in many aspects of life, has not had the chance to delve into each
and every one at depth over its 168 years of existence. These are the Domain
label for technical terms from uncommon domains (e.g. glossopoeia) [285]; the
innovation label for terms coined with no discernable prevailing linguistic
process (e.g. swanicles, a term from a work of fiction) [11]; and the
Onomatopeia label for sound-based sequences (e.g. ktktk) [23], which includes
cases of verbatim vocalization such as trololo.
##### Morphological OOVs.
In this group we include categories of words composed of meaning-carrying
units present in existing English words which have appeared in the NYT before,
manifested in a new form. In increasing order of syntactic and semantic
novelty, they are: Infl, unseen inflections of existing wordforms: same part-
of-speech, different syntactic attributes (e.g. pennyloafers) [53];999We
include the negating prefixes in- and un-, which despite change a word’s
meaning, but retain its part-of-speech. Deriv, unseen derivations of existing
words into new parts-of-speech which carry no semantic distancing beyond that
implicit in the new part-of-speech itself (e.g. foamability) [215]; Affix,
affixation of very distinct base words which are typically derivational in
nature but include a semantic charge (e.g. extraphotographic, pizzaless)
[483]; Affix_Libfix, affixation of distinct base words with particles
extracted from another word in a process known as libfixation [Zwicky, 2010]
or splintering [Berman, 1961], where the liberated affix still elicits the
originating word but can be freely attached to a growing selection of words
(e.g. dripware) [18]; Compound_Comp, a concatenation of two complete words
each contributing essential semantics to the final form in a way we deem
(subjectively, with help of context) to be compositional (e.g. smellwalks,
strolls focusing on olfactory input) [121];101010One compound in our dataset,
dramatotherapy, adds characters for cadence; another, laysoccerperson, is
nonlinear. Compound_New, a concatenation of base words resulting in a new
semantic concept deemed remote from the bases (e.g. nothingbuffet, a play on
nothingburger) [49]; and Blend, a fusion of two or more base forms together
where original characters are lost or shared, or new ones are added (e.g.
chipster, a chicano hipster) [142].111111A single blend, pregret, has just one
base fused with a prefix.
##### Syntactic OOVs.
This group consists solely of the Synth category of tokens which synthesize
multiple syntactic words into one form, a rare formation process in English
limited typically to auxiliary contractions (e.g. this’ll) [6].
##### Sociopragmatic OOVs.
Words in this group exhibit an orthographic diversion from standard English
usually intended as a statement of register or status, or as a faithful
representation of a certain linguistic style or sentiment. Archaic, a register
of older variants of English or an ironic semblance of such (e.g. shooketh, a
mock-archaic form of shake using Middle English morphology) [14]; Dialect, a
geographically- or demographically-specific form of a word typically spelled
differently in the NYT (e.g. skwarsh, an r-full squash) [46]; Infix, a
morphological tool reserved in English for expletive emphasis [McCawley, 1978]
(e.g. unfreakingbelievable) [2]; Phonaestheme, a phonological duplication
phenomenon used in contemporary English nearly only as derisive echo
reduplication borrowed from Yiddish [Wales and Ramsaran, 1990] (e.g.
schmarket) [6]; Lengthening, a written manifestation of the expressive
elongation of phonetic segments (e.g. greaaaaat) [53]; Variant, spelling
alternations or intentional typos which are not intended to be read
differently from the standard form of the word, used for branding and jest
(e.g. kyllyng) [18]; and Spaces_Sic, the removal of whitespace to simulate
breathlessness (e.g. lineafterlineafterline) [5].
#### 2.2.1 Difficult Distinctions
Naturally, some annotation cases are not clear-cut, as evidenced by the
imperfect inter-annotator agreement. We found the most challenging cases to be
among the morphological categories, where an affix is either semantically null
(Deriv / Infl) or not (Affix) (14% and 15% of disagreements, respectively);
where a sense of the nearest in-vocabulary word can signal the difference
between Infl and Deriv (3.4%); where an Affix_Libfix has been liberated enough
from the underlying word such that it is now simply an Affix (does cyber-
still envoke the full word cybernetics? Does crypto- envoke cryptography?); if
it has not been liberated yet, it should be a Blend or a Compound. In
addition, the pre-processing phase required a demarcation between Domain and
Foreign which was not easy to make given the heavy foreign-word influence in
certain knowledge domains such as cuisine (e.g. dinkelbrot). Words adapted
into English morphology would usually lead to a Domain label (Domain vs.
Compound: 4%). In many cases, we found the contexts in which the words were
introduced to give sufficient disambiguation (so, e.g., cybercoach is an
affix, but cyberinvasion is a compound).
We invite readers to email errata to either of the first two authors, or
submit a pull request on Github.
## 3 OOV Classification Task
The task of classifying OOVs, i.e. assigning a novel word with a label from
the taxonomy we defined above, can be beneficial from both an analytical
linguistic standpoint, and from an NLP standpoint concerned with model
performance on downstream language understanding tasks. To get a sense of the
predictability of the various OOV classes in the dataset, we present several
baselines for this straightforward task. The uniqueness of our dataset allows
us to apply both type-level and context-dependent systems, the latter
operating in the real-world scenario of encountering a word for the first time
in the actual context of its introduction to the corpus.
First, our Majority class baseline assumes all OOVs are the result of
affixation.
For all following models we trained a ridge classifier with default
regularization parameters in scikit-learn. Scores for all supervised models
are reported via 10-fold cross-validation using the same folds for all
systems. Due to the class imbalance, we chose to implement training in such a
way that upsampled rare classes with replacement at each iteration to equal
frequency as the most common class. We report accuracy (Acc) and macro F1
scores.
##### Contextless features.
We compare and contrast several input features to our classifier that only
have access to the form of the OOV, without consideration of the context:
* •
Character n-grams. We extract bag-of-character features ranging from one to
three characters for each OOV. The feature vocabulary is estimated on the
training set and applied to the test set.
* •
FastText. We infer fasttext vectors [Bojanowski et al., 2017], applying its
3–6 character-ngram representations, from the subword model trained on English
Wikipedia.121212wiki.en.bin file obtained May 25, 2020.
* •
ELMo embeddings. We use the word-level embeddings from ELMo [Peters et al.,
2018], obtained via a pre-trained character-level convolutional net for each
OOV presented in isolation, with no surrounding sentence context.
* •
BERT no-context. We apply BERT-Base [Devlin et al., 2019] to the OOVs in a
null [CLS] ___ [SEP] . context. The averaged top-layer vectors from all the
OOV’s word pieces are passed to the classifier.131313Using just the embedding
of the final word piece produced similar results.
##### Context-aware features.
* •
Character RNN. We train a 2-layer forward- (backward-) character-level GRU
language model on 100,000 Wikipedia documents and run it through the beginning
(end) of the sentence until the OOV, then use the concatenated final hidden
states from each direction as features.
* •
ELMo. We obtain contextualized embeddings for all words in our sentences and
select the top layer representation associated with each OOV.
* •
BERT. We apply BERT-Base to the entire sentence in which the OOV appears, and
use the averaged top-layer embeddings at the indices of each OOV.
### 3.1 Results
The results, presented in Table 1, show that pretrained contextual models not
only trail behind a contextless, un-pretrained character n-gram baseline, they
even fail to improve over their own uncontextualized variants. An analysis of
class-specific F1 scores across the different models exposed two general
patterns in classifier performance: in all models, performance on the Affix
class was in the top four, and the same for Lengthening except for Character
RNN. We also observed that models that encode contextual, sentence-level
properties are typically better at encoding genre phenomena (e.g. Domain was a
top-four category for BERT, Character RNN, fastText, and ELMo). However, for
some classes of models, there was a clear benefit to memorizing word forms.
All count-based feature representations (e.g. bag-of-character ngrams, bag-of-
wordpieces) led to better performance on orthographic properties, namely
Phonaestheme, Synth, and Onomatopoiea. These results demonstrate the power
that simple surface-form signals from character sequences still possess in
meaningful NLP tasks. In future work, we will attempt to supplement the
contextual models with auxiliary mechanisms and perform fine-tuning.
Contextless | Acc | | F1 | Contextual | Acc | | F1
---|---|---|---|---|---|---|---
Majority class | .312 | | .026 | | | |
Character n-grams | .484 | | .323 | | | |
FastText | .433 | | .241 | Character RNN | .128 | | .054
ELMo embeddings | .365 | | .203 | ELMo | .324 | | .135
BERT no-context | .442 | | .288 | BERT | .469 | | .269
Table 1: Baseline results for OOV classification ($N=1550$, $|C|=18$).
## 4 Conclusion
We presented a novel dataset of OOVs along with their contexts and linguistic
novelty class annotations. We showed that contextual information in the form
of other parts of the sentence provides some signal, but simple models relying
on character n-gram information alone achieve high performance.
The availability of broader document contexts in which these neologisms occur
enables many linguistic and technical applications. From the perspective of
the study of language growth and formation, the dataset may be used to assess
the morphological productivity of different affixes and roots, or the
prevalence of the different word formation processes in a realistic setting;
or perform in-depth analysis on any of the specific types of innovations we
identified. In addition, the in-vivo nature of the dataset provides a
reference for neologisms which may or may not be later adopted into everyday
use, allowing diachronic studies anchored in the time of word introduction.
Analysis of the phonological, morphological, and discourse-level properties of
these words may provide insight into lexical adoption dynamics.
For NLP researchers, an important component of text applications is proper
normalization and segmentation of word forms. Our experiment shows that
popular word form encoders, such as ELMo or BERT’s WordPiece, still have a
long way to go in terms of recognizing the origins of a novel form. Errors at
this stage might lead to inability to handle morphologically complex OOVs in
downstream semantic applications [Pinter et al., 2020], although further study
of such effects and of the utility of OOV classification in alleviating them
is still necessary. Properly leveraging context for morphological
decomposition of complex forms also remains an open problem.
The resource is an ongoing project; the repository includes plans for the next
versions, including increasing the dataset size by including newer words from
the bot, and annotating additional information such as part-of-speech tags.
## Acknowledgments
We thank Jacob Eisenstein, Kyle Gorman, Arya McCarthy, Sandeep Soni, and the
anonymous reviewers for their valuable notes. Yuval Pinter is a Bloomberg Data
Science PhD Fellow. Cassandra Jacobs is supported on NSF BCS Grant 1849236
awarded to Maryellen MacDonald.
## References
* [Ahmad, 2000] Khurshid Ahmad. 2000\. Neologisms, nonces and word formation. In Proceedings of the Ninth EURALEX International Congress, pages 711–730.
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|
2024-09-04T02:54:58.004399 | 2020-03-06T22:54:46 | 2003.03461 | {
"authors": "Weili Nie, Tero Karras, Animesh Garg, Shoubhik Debnath, Anjul Patney,\n Ankit B. Patel, Anima Anandkumar",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26092",
"submitter": "Weili Nie",
"url": "https://arxiv.org/abs/2003.03461"
} | arxiv-papers | # Semi-Supervised StyleGAN for Disentanglement Learning
Weili Nie∗ Tero Karras Animesh Garg Shoubhik Debnath Anjul Patney Ankit B.
Patel Anima Anandkumar
###### Abstract
Disentanglement learning is crucial for obtaining disentangled representations
and controllable generation. Current disentanglement methods face several
inherent limitations: difficulty with high-resolution images, primarily
focusing on learning disentangled representations, and non-identifiability due
to the unsupervised setting. To alleviate these limitations, we design new
architectures and loss functions based on StyleGAN (Karras et al., 2019), for
semi-supervised high-resolution disentanglement learning. We create two
complex high-resolution synthetic datasets for systematic testing. We
investigate the impact of limited supervision and find that using only
0.25%$\sim$2.5% of labeled data is sufficient for good disentanglement on both
synthetic and real datasets. We propose new metrics to quantify generator
controllability, and observe there may exist a crucial trade-off between
disentangled representation learning and controllable generation. We also
consider semantic fine-grained image editing to achieve better generalization
to unseen images.
Machine Learning, ICML
## 1 Introduction
Disentanglement learning with deep generative models has attracted much
attention recently (Chen et al., 2016; Higgins et al., 2017; Locatello et al.,
2019). This is crucial for controllable generation, where the style codes
specified to the generator need to separately control various factors of
variation for faithful generation. Another goal is learning disentangled
representations where the input samples can be encoded to latent factors that
are disentangled. This has been argued as a key to success of deep learning
(Achille & Soatto, 2018). Previous works have primarily focused on only one of
the above two objectives. Ideally, the ultimate goal of disentanglement
learning is to achieve both the objectives at the same time, especially on
more complex high-resolution images, and we pursue this goal in this paper. We
first list the three main limitations of the current disentanglement methods.
First, much effort has focused on unsupervised disentanglement methods (Chen
et al., 2016; Higgins et al., 2017; Nguyen-Phuoc et al., 2019). This is
because a large number of fully annotated samples is expensive to obtain.
These methods suffer from non-identifiability, which means multiple repeated
runs will not reliably observe the same latent representations (Hyvärinen &
Pajunen, 1999; Locatello et al., 2019). In addition, human feedback is needed
to discern (i) what factors of variation the model has learnt (e.g., object
shape and color), and (ii) what semantic meaning different values of the
discovered factor code represent (e.g., red and blue in a color factor code).
To reliably control generation for practical use, adding a small amount of
labeled data may resolve the non-identifiability issue and lead to
interpretable factors. Hence, we investigate the impact of limited supervision
on disentanglement learning.
Second, current disentanglement methods (Locatello et al., 2019, 2020) are
mainly developed and evaluated on relatively simple low-resolution images,
such as dSprites (Matthey et al., 2017) and 3DShapes (Kim & Mnih, 2018), which
raises concerns about their ability to scale up to more diverse, higher-
resolution images. For example, the use of 3D representations to disentangle
the 3D pose may not easily apply to high-resolution images due to the
computational cost. The difficulty of some deep generative models at
generating realistic images also limits their application in more complex
domains. Furthermore, although there exist many real image datasets of high
resolution, the latent factors are typically only partially observed or
unbalanced, which makes it hard to scientifically study disentanglement. To
gain practically useful insights, it is critical to first test disentanglement
methods on complex, high-resolution, synthetic images wherein ground-truth
factors are easy to obtain.
Third, most previous works (Kim & Mnih, 2018; Chen et al., 2018) have
primarily focused on learning disentangled representations by quantifying
encoder disentanglement quality, in the hopes that a better disentangled
encoder might also lead to a better disentangled generator. However, to the
best of our knowledge, there is no clear evidence supporting that a
proportional relationship between encoder and generator disentanglement
quality always exists. A good analogy is that an art critic can disentangle
various painting styles and skills, but may not be able to create a good
painting by combining these styles and skills. Thus, results based solely on
evaluating encoder disentanglement may be misleading, especially in tasks
where the requirement for controllable generation is more critical. This
highlights the importance of measuring the generator disentanglement quality
in order to properly evaluate and compare different methods.
#### Main contributions.
In this work, we investigate semi-supervised disentanglement learning based on
StyleGAN (Karras et al., 2019), one of the state-of-the-art generative
adversarial networks (GANs), for complex high-resolution images. In summary,
our main contributions are as follows,
* •
We first justify the advantages of the StyleGAN architecture in
disentanglement learning, by showing that StyleGAN augmented with a mutual
information loss (called Info-StyleGAN) ourperforms most state-of-the-art
unsupervised disentanglement methods.
* •
We propose Semi-StyleGAN that achieves near fully-supervised disentanglement
quality with limited supervision (0.25%$\sim$2.5%) on synthetic and real data.
* •
We propose new metrics (termed as MIG-gen and L2-gen) to evaluate the
generator controllability, and reveal a crucial trade-off between learning
disentangled representations and controllable generation.
* •
We then extend Semi-StyleGAN to an image-to-image model, enabling semantic
fine-grained image editing with better generalization to unseen images.
* •
We create two high-quality datasets with much higher resolution, better
photorealism, and richer factors of variation than existing disentanglement
datasets.
## 2 Background and Related Work
#### StyleGAN.
GANs are a family of generative models that have shown great success. Among
various GANs, StyleGAN (Karras et al., 2019) is a state-of-the-art GAN
architecture for unsupervised image generation, particularly for high-fidelity
human faces of resolution up to 1024x1024. StyleGAN comprises a mapping
network whose role is to map a latent vector $z$ to an intermediate space,
which then controls the styles at each convolutional layer in the synthesis
network with adaptive instance normalization (AdaIN) (Ulyanov et al., 2016;
Huang & Belongie, 2017). StyleGAN also enables the separation of fine-grained
and coarse-grained features. For example, modifying the styles of low-
resolution blocks affects only coarse-grained features (e.g. overall pose and
presence of eyeglasses), while modifying the styles of high-resolution blocks
affects only fine-grained features (e.g. color scheme and microstructure).
These nice properties make it a potentially good candidate for disentanglement
learning of high-resolution images.
#### Disentanglement learning.
In terms of unsupervised disentanglement learning, there exists much prior
work based on either Variational Autoencoders (VAEs) (Higgins et al., 2017;
Kim & Mnih, 2018; Chen et al., 2018) or GANs (Chen et al., 2016; Lin et al.,
2019; Nguyen-Phuoc et al., 2019). The basic idea in disentangled VAEs is to
encourage a factorization of the latent code by regularizing the total
correlation (Chen et al., 2018). They have shown the state-of-the-art
performance on many standard disentanglement benchmarks (Matthey et al., 2017;
Kim & Mnih, 2018). Many GAN-based models reply on maximizing the mutual
information between the observation and factor code, such as InfoGAN (Chen et
al., 2016) and its variants (Lin et al., 2019). Other GANs also learn
disentanglement by designing a domain-specific generator architecture to add
model-inductive bias, represented by HoloGAN (Nguyen-Phuoc et al., 2019).
However, the use of 3D representations in HoloGAN may not scale up to higher-
resolution images.
Another line of work in disentanglement learning is to use explicit
supervision. (Kulkarni et al., 2015) applies a supervised training procedure
to encourage each group of the graphics code to distinctly represent a
specific factor of variation. (Bouchacourt et al., 2018) proposes the Multi-
Level VAE (ML-VAE) to learn disentanglement from the supervision of group
information. (Xiao et al., 2017) develops a supervised disentanglement
algorithm called DNA-GAN using a swapping policy. (Narayanaswamy et al., 2017)
proposes a semi-supervised VAE by employing a general graphical model
structure in the encoder and decoder. However, it still remains unclear how
the use of supervision impacts the disentanglement learning. (Spurr et al.,
2017) proposes ss-InfoGAN by adding few labels to InfoGAN to learn
semantically meaningful data representations. More recently, (Locatello et
al., 2020) shows the benefits of adding limited supervision into learning
disentangled presentations in VAEs. We extend these results to GANs on more
complex and higher-resolution images, and also quantify the impact of limited
supervision on the generator controllability.
#### Conditional GANs.
A class of conditional GANs conditions on the class labels for better image
generation quality, such as cGAN (Mirza & Osindero, 2014), AC-GAN (Odena et
al., 2017), Projection Discriminator (Miyato & Koyama, 2018). The
architectural component of semi-StyleGAN makes it a special case of semi-
supervised conditional GANs, as it conditions on partially available factor
codes. However, the tasks and loss functions of Semi-StyleGAN differ greatly
from those of these conditional GANs. First, these conditional GANs mainly
focus on generating more realistic images, whereas semi-StyleGAN focuses on
two joint tasks: (i) disentangled representation learning and (ii)
controllable generation. Second, to this end, both the mutual information loss
and a new smoothness regularization are introduced in our loss functions for a
better trade-off in disentanglement learning.
Another class of conditional GANs conditions on the given images for image-to-
image translation, where many works focus on multi-attribute editing, and two
representatives are StarGAN (Choi et al., 2018) and AttGAN (He et al., 2019).
Although the proposed Semi-StyleGAN-fine in Section 5 share the similar
objectives, there are many differences: StarGAN needs to condition on images
of a specific domain, namely a set of images sharing the same attribute, which
limits itself to only performing discrete/binary attribute control. However,
our work is capable of doing continuous style manipulation. AttGAN requires an
encoder-decoder structure in the generator while we do not need to. Instead,
our work adds controllable fine-grained factors along with a super-resolution
process, which is especially suitable for editing fine styles of high-
resolution images with a good generalization ability.
Figure 1: An illustration of disentanglement learning based on StyleGAN, where
the mapping network in the generator conditions on the factor code and the
encoder (which shares all layers in the discriminator except for the last
layer) predicts its value.
## 3 Why StyleGAN for Disentanglement?
In unsupervised disentanglement learning, there exist many state-of-the-art
VAE-based models, such as $\beta$-VAE (Higgins et al., 2017), FactorVAE (Kim &
Mnih, 2018) and $\beta$-TCVAE (Chen et al., 2018). More recently, GAN-based
models, such as InfoGAN-CR (Lin et al., 2019), have also achieved competitive
performance by adding more tuning heuristics and regularization. Here, we
consider Info-StyleGAN, enabling StyleGAN with a mutual information loss, and
show that the structural advance of StyleGAN provides a stronger prior for
disentanglement learning compared to regularization used previously in VAEs or
GANs.
As shown in Figure 1, the mapping network in the generator of Info-StyleGAN
now conditions on a factor code, a vector representing each factor of
variation in each dimension, by simply concatenating it with the latent code
$z$. The output of the mapping network, called conditional styles will
modulate each block in the synthesis network using AdaIN. Similar to InfoGAN,
the encoder in Info-StyleGAN shares all the network layers except the last one
with the discriminator and predicts the value of factor code. Thus, we use
$G/D/E$ to represent the generator, discriminator and encoder, respectively.
The mutual information loss of InfoGAN can be approximated by an unsupervised
code reconstruction loss (Chen et al., 2016), which is
$\displaystyle\mathcal{L}_{\text{unsup}}=\sum\nolimits_{c\sim\mathcal{C},z\sim
p_{z}}\left\|E(G(c,z))-c\right\|_{2}$ (1)
where $\mathcal{C}$ denotes the set of all factor codes, and $p_{z}$ denote
the prior distribution of latent code $z$. The respective loss functions for
$G$ and $(D,E)$ are given by
$\displaystyle\begin{split}\mathcal{L}^{(G)}=&\mathcal{L}_{\text{GAN}}+\gamma\mathcal{L}_{\text{unsup}}\\\
\mathcal{L}^{(D,E)}=&-\mathcal{L}_{\text{GAN}}+\gamma\mathcal{L}_{\text{unsup}}\end{split}$
(2)
where we keep the GAN loss function $\mathcal{L}_{\text{GAN}}$ the same as in
(Karras et al., 2019). The hyperparameter $\gamma$ controls a trade-off
between image realism and disentanglement quality.
### 3.1 Experimental Setup
#### Datasets and evaluation metrics.
We consider two datasets to compare Info-StyleGAN and state-of-the-art
disentanglement models: dSprites (Matthey et al., 2017) and our proposed
Isaac3D (See details in Section 4.1). dSprites is a commonly used dataset in
disentanglement learning, with 737,280 images and each of resolution 64x64.
For experiments on dSprites, we use both Factor score (Kim & Mnih, 2018) and
Mutual Information Gap (MIG) (Chen et al., 2018) to evaluate disentanglement.
For experiments on Isaac3D, we first downscale the resolution of each image to
128x128 because VAEs have difficulties in generating higher-resolution images.
We use MIG to evaluate disentanglement quality and Frechet Inception Distance
(FID) (Heusel et al., 2017) to evaluate image quality.
#### Experimental protocol.
We consider $\beta$-VAE, FactorVAE, $\beta$-TCVAE and InfoGAN-CR for
comparison, where all the models are trained based on the implementation in
(Locatello et al., 2019). For VAEs, we set $\beta=6$ for $\beta$-VAE,
$\gamma=30$ for FactorVAE and $\beta=8$ for $\beta$-TCVAE after a grid search
over different hyperparameters. For InfoGAN-CR, we use default hyperparameters
on dSprites from the original paper, and perform a grid search over different
hyperparameters on Isaac3D to report the best results. For Info-StyleGAN, we
keep $\gamma=10$ on dSprites and $\gamma=1$ on Isaac3D. We also keep the
progressive training (Karras et al., 2017), as we show that it helps improve
disentanglement in Appendix B.1. Because Info-StyleGAN and state-of-the-art
disentanglement models have largely different network architectures, for a
fairer comparison, we also try to keep their network sizes to be the same. See
Appendix B.2 for how we decrease the network size of Info-StyleGAN (called
Info-StyleGAN∗) to match those of previous models.
Methods | # Params | Factor Score $\uparrow$ | MIG $\uparrow$
---|---|---|---
$\beta$-VAE | 0.69M | 0.713 $\pm$ 0.095 | 0.132 $\pm$ 0.031
FactorVAE | 5.70M | 0.764 $\pm$ 0.098 | 0.175 $\pm$ 0.057
$\beta$-TCVAE | 0.69M | 0.731 $\pm$ 0.097 | 0.174 $\pm$ 0.046
InfoGAN-CR | 0.76M | 0.853 $\pm$ 0.046 | 0.270 $\pm$ 0.034
Info-StyleGAN* | 0.74M | 0.769 $\pm$ 0.144 | 0.274 $\pm$ 0.096
Info-StyleGAN | 47.89M | 0.840 $\pm$ 0.090 | 0.290 $\pm$ 0.098
(a) dSprites with resolution 64x64
Methods | # Params | FID $\downarrow$ | MIG $\uparrow$
---|---|---|---
$\beta$-VAE | 1.91M | 122.6 $\pm$ 2.0 | 0.231 $\pm$ 0.068
FactorVAE | 6.93M | 305.8 $\pm$ 142.1 | 0.245 $\pm$ 0.034
$\beta$-TCVAE | 1.91M | 155.4 $\pm$ 13.6 | 0.216 $\pm$ 0.074
InfoGAN-CR | 3.29M | 80.72 $\pm$ 30.79 | 0.342 $\pm$ 0.139
Info-StyleGAN* | 3.44M | 8.10 $\pm$ 2.25 | 0.404 $\pm$ 0.085
Info-StyleGAN | 49.05M | 2.19 $\pm$ 0.48 | 0.328 $\pm$ 0.057
(b) Isaac3D with resolution 128x128
Table 1: Comparison of Info-StyleGAN and state-of-the-art disentanglement
models on dSprites and (downscaled) Isaac3D. Note that the scores of VAEs are
obtained based on the implementation in (Locatello et al., 2019). Info-
StyleGAN* represents the smaller version of Info-StyleGAN, in which its number
of parameters (i.e., # Params) is similar to that of previous models.
### 3.2 Key Results
Table 1 shows that Info-StyleGAN and its variant with smaller network size,
termed as Info-StyleGAN∗, consistently outperform state-of-the-art VAE-based
methods by a large margin on both dSprites and Isaac3D. Meanwhile, Info-
StyleGAN achieves competitive or even better disentanglement performance than
the strong GAN baseline. Although unsupervised disentanglement learning is
impossible without supervision or inductive bias (Locatello et al., 2019),
this result reveals that the network structural improvement of StyleGAN
provides a stronger prior for disentanglement learning compared to different
explicit loss regularizations in disentangled VAEs or InfoGAN-CR. Besides, we
observe that previous methods have much higher FID scores on (downscaled)
Isaac3D, along with their poor generated samples in Appendix B.3. We have also
increased the capacity of VAEs but the improvement of image quality still
cannot close the gap with Info-StyleGAN, as shown in Appendix B.4. These
results show that previous disentanglement methods have difficulties on more
diverse and complex data, such as Isaac3D, while StyleGAN does not.
## 4 Semi-StyleGAN
As pointed out by (Locatello et al., 2019) that unsupervised disentangled
methods are formally non-identifiable (Hyvärinen & Pajunen, 1999), the impact
of limited supervision on both learning disentangled representations and
controllable generation, which has been rarely explored, becomes crucial. In
this section, we propose to add semi-supervision into Info-StyleGAN to get a
semi-supervised disentanglement model – Semi-StyleGAN. Based on Semi-StyleGAN,
we systematically analyze the role of limited supervision on both synthetic
and real data.
A naive way of applying (semi-)supervision is to add a supervised code
reconstruction term for the small amount of labeled data into Eq (2), similar
to (Kingma et al., 2014; Odena et al., 2017; Locatello et al., 2020). That is,
$\displaystyle\mathcal{L}_{\text{sup}}=\sum\nolimits_{(x,c)\sim\mathcal{J}}\|E(x)-c\|_{2}$
(3)
where $\mathcal{J}$ represents the set of labeled pairs of real image and
factor code. When considering the limited supervision, we assume the
cardinality of the labeled set $\mathcal{J}$ satisfies that
$|\mathcal{J}|\ll|\mathcal{X}|$, with $\mathcal{X}$ being the set of all real
images. Thus, the semi-supervised loss functions become
$\displaystyle\begin{split}\mathcal{L}^{(G)}=&\mathcal{L}_{\text{GAN}}+\gamma_{G}\mathcal{L}_{\text{unsup}}\\\
\mathcal{L}^{(D,E)}=&-\mathcal{L}_{\text{GAN}}+\gamma_{E}\mathcal{L}_{\text{unsup}}+\beta\mathcal{L}_{\text{sup}}\end{split}$
(4)
where $\beta$ is the weight of the supervised term $\mathcal{L}_{\text{sup}}$,
and we use different $\gamma$’s (denoted by $\gamma_{G}$ and $\gamma_{E}$) to
separately represent the weight of the unsupervised term in
$\mathcal{L}^{(G)}$ and $\mathcal{L}^{(D,E)}$. As we show later, $\gamma_{G}$
and $\gamma_{E}$ play an important role in controlling the trade-off between
encoder and generator disentanglement. Note that the supervised term
$\mathcal{L}_{\text{sup}}$ does not update $G$ directly as shown in Eq. (4).
While semi-supervised learning for image recognition is an active research
area, many algorithms may not be directly applied to disentanglement learning.
Take the consistency regularization (Sajjadi et al., 2016) as an example.
Commonly used data perturbations, such as image rotation and color
randomization will inevitably cause inconsistency if the considered factors of
variation include object rotation or color. In contrast, encouraging
smoothness in the latent space of GANs may help improve disentanglement
(Karras et al., 2019). Thus, we propose to explicitly add a smoothness
regularization by using the idea of MixUp (Zhang et al., 2018; Berthelot et
al., 2019).
Formally, given a labeled observation-code pair $(x,c)\sim\mathcal{J}$ and a
generated pair $(x^{\prime},c^{\prime})$ where $x^{\prime}=G(z,c^{\prime})$,
we get a set of mixed observation-code pairs
$\mathcal{M}=\left\\{(\tilde{x},\tilde{c})\right\\}$ by
$\displaystyle\begin{split}\lambda\sim&\text{Beta}(\xi,\xi),\;\;\lambda^{\prime}=\max(\lambda,1-\lambda)\\\
\tilde{x}&=\lambda^{\prime}x+(1-\lambda^{\prime}){x}^{\prime}\\\
\tilde{c}&=\lambda^{\prime}c+(1-\lambda^{\prime})c^{\prime}\end{split}$ (5)
where $\xi$ is a hyperparameter. Thus, the smoothness regularization term is
$\displaystyle\mathcal{L}_{\text{sr}}=\sum\nolimits_{(x,c)\sim\mathcal{M}}\|E(x)-c\|_{2}$
(6)
and the new semi-supervised loss functions with smoothness regularization
become
$\displaystyle\begin{split}\mathcal{L}^{(G)}=&\mathcal{L}_{\text{GAN}}+\gamma_{G}\mathcal{L}_{\text{unsup}}+\alpha\mathcal{L}_{\text{sr}}\\\
\mathcal{L}^{(D,E)}=&-\mathcal{L}_{\text{GAN}}+\gamma_{E}\mathcal{L}_{\text{unsup}}+\beta\mathcal{L}_{\text{sup}}+\alpha\mathcal{L}_{\text{sr}}\end{split}$
(7)
where $\alpha$ is the weight of the smoothness term $\mathcal{L}_{\text{sr}}$.
Different from (Zhang et al., 2018; Berthelot et al., 2019) that combines
labeled and unlabeled real data, the MixUp in (5) is performed between real
labeled data and generated data. This way, it not only encourages smooth
behaviors of both the generator and encoder, but also takes good advantages of
enormous fake data for disentanglement.
### 4.1 New Datasets
Current disentanglement datasets, such as dSprites (Matthey et al., 2017),
3DShapes (Kim & Mnih, 2018) and MPI3D (Gondal et al., 2019), are of low
resolution and mostly lack photorealism. We create two new datasets – Falcor3D
and Isaac3D, with much higher resolution, better photorealism and richer
factors of variations, as shown in Table 2.
#### Falcor3D.
It contains 233,280 images and each has a resolution of 1024x1024. This
dataset is based on the 3D scene of a living room, where we move the camera
positions and change the lighting conditions. Each image is paired with a
ground-truth factor code, consisting of 7 factors of variation: lighting
intensity (5), lighting $x$-dir (6), lighting $y$-dir (6), lighting $z$-dir
(6), camera $x$-pos (6), camera $y$-pos (6), and camera $z$-pos (6). The
number $m$ behind each factor represents that the factor has $m$ possible
values, uniformly sampled from $[0,1]$. For example, “lighting $x$-dir (6)”
represents the lighting direction moving along the $x$-axis and “camera
$z$-pos (6)” denotes the camera position moving along the $z$-axis. Both
factors have 6 possible values.
#### Isaac3D.
It contains 737,280 images and each has a resolution of 512x512. This dataset
is based on the 3D scene of a kitchen, where we move the camera positions and
vary the lighting conditions. There is a robotic arm inside, grasping an
object. The robotic arm has two degrees of freedom: $x$-movement (horizontal)
and $y$-movement (vertical). The attached object could change its shape, scale
or color. All objects in the 3D scene are properly textured for better
photorealism. Similarly, each image is paired with a ground-truth factor code,
consisting of 9 factors of variation: lighting intensity (4), lighting $y$-dir
(6), object color (4), wall color (4), object shape (3), object scale (4),
camera height (4), robot $x$-movement (8), and robot $y$-movement (5). The
number $m$ behind each factor represents that it has $m$ possible values,
uniformly sampled from $[0,1]$.
Datasets | # Images | # Factors | Resolution | 3D
---|---|---|---|---
dSprites | 737,280 | 5 | 64x64 | ✗
Noisy dSprites | 737,280 | 7 | 64x64 | ✗
Scream dSprites | 737,280 | 7 | 64x64 | ✗
SmallNORB | 48,600 | 5 | 128x128 | ✓
Cars3D | 17,568 | 3 | 64x64 | ✓
3DShapes | 480,000 | 7 | 64x64 | ✓
MPI3D | 640,800 | 7 | 64x64 | ✓
Falcor3D | 233,280 | 7 | 1024x1024 | ✓
Isaac3D | 737,280 | 9 | 512x512 | ✓
Table 2: Summary of the proposed two datasets, compared with currently
commonly used datasets (Gondal et al., 2019). We can see that the proposed two
datasets – Faclor3D and Isaac3D have much larger resolutions than previous
datasets, along with the largest number of factors. More importantly, the
proposed datasets are of much higher photorealism, as shown in Appendix A.
(a) Isaac3D
(b) Falcor3D
Figure 2: Semi-StyleGAN with the default setting $\gamma_{G}=\beta=\gamma$,
$\gamma_{E}=0$, $\alpha=1$ where $\gamma\in\\{1,10\\}$ on (a) Isaac3D and (b)
Falcor3D. We vary the portion of labeled data $\eta$ to show the impact of
semi-supervision by comparing with Info-StyleGAN (i.e. the unsupervised
baseline), and the fully-supervised one ($\eta=1$). Only using 0.25$\sim$2.5%
of labeled data achieves near fully-supervised disentanglement.
Methods | MIG $\uparrow$ | L2 $\downarrow$ | MIG-gen $\uparrow$ | L2-gen $\downarrow$
---|---|---|---|---
Encoder-only | 0.731 $\pm$ 0.009 | 0.379 $\pm$ 0.002 | - | -
Encoder-only w/ MixUp | 0.834 $\pm$ 0.004 | 0.279 $\pm$ 0.005 | - | -
Semi-StyleGAN | 0.812 $\pm$ 0.020 | 0.301 $\pm$ 0.012 | 0.965 $\pm$ 0.014 | 0.052 $\pm$ 0.016
\+ Remove smoothness consistency | 0.765 $\pm$ 0.042 | 0.347 $\pm$ 0.019 | 0.945 $\pm$ 0.011 | 0.072 $\pm$ 0.008
\+ Increase the $\mathcal{L}_{\text{unsup}}$ term in $E$ ($\gamma_{E}=10$) | 0.880 $\pm$ 0.120 | 0.225 $\pm$ 0.222 | 0.888 $\pm$ 0.087 | 0.283 $\pm$ 0.247
\+ Remove the $\mathcal{L}_{\text{unsup}}$ term in $G$ | 0.719 $\pm$ 0.014 | 0.490 $\pm$ 0.024 | 0.130 $\pm$ 0.054 | 1.514 $\pm$ 0.003
(a) Isaac3D ($\eta$ = 0.1%)
Methods | MIG $\uparrow$ | L2 $\downarrow$ | MIG-gen $\uparrow$ | L2-gen $\downarrow$
---|---|---|---|---
Encoder-only | 0.690 $\pm$ 0.007 | 0.271 $\pm$ 0.002 | - | -
Encoder-only w/ MixUp | 0.701 $\pm$ 0.005 | 0.265 $\pm$ 0.003 | - | -
Semi-StyleGAN | 0.704 $\pm$ 0.007 | 0.285 $\pm$ 0.002 | 0.754 $\pm$ 0.017 | 0.205 $\pm$ 0.022
\+ Remove smoothness consistency | 0.674 $\pm$ 0.011 | 0.296 $\pm$ 0.017 | 0.632 $\pm$ 0.058 | 0.303 $\pm$ 0.088
\+ Increase the $\mathcal{L}_{\text{unsup}}$ term in $E$ ($\gamma_{E}=10$) | 0.643 $\pm$ 0.035 | 0.343 $\pm$ 0.016 | 0.636 $\pm$ 0.065 | 0.346 $\pm$ 0.070
\+ Remove the $\mathcal{L}_{\text{unsup}}$ term in $G$ | 0.680 $\pm$ 0.016 | 0.300 $\pm$ 0.010 | 0.034 $\pm$ 0.028 | 1.096 $\pm$ 0.086
(b) Falcor3D ($\eta$ = 0.5%)
Table 3: Ablation studies of Semi-StyleGAN on (a) Isaac3D and (b) Falcor3D,
where the default setting is $\gamma_{G}=\beta=10$, $\gamma_{E}=0$,
$\alpha=1$. “Encoder-only” means we train the encoder by minimizing the L2
score with the labeled data only, a supervised baseline for the encoder
disentanglement. “Encoder-only w/ MixUp” means we train the encoder by using
MixUp (Zhang et al., 2018), a semi-supervised baseline for the encoder
disentanglement. We set $\eta=0.1\%$ on Isaac3D and $\eta=0.5\%$ on Falcor3D,
respectively.
### 4.2 New Metrics
Many metrics have been proposed for evaluating disentanglement, such as Factor
score (Kim & Mnih, 2018), MIG (Chen et al., 2018), DCI score (Ridgeway &
Mozer, 2018), and SAP score (Kumar et al., 2017). See the prior work
(Locatello et al., 2019) for their more implementation details. However, they
all have some inherent limitations in quantifying semi-supervised
disentanglement methods.
First, these metrics are designed for unsupervised disentanglement methods,
which are non-identifiable (Locatello et al., 2019). But with supervision, the
model become identifiable and thus we need to evaluate the semantic meaning of
learned representations as well. A simple solution here is to measure the
average $L2$ distance between the ground-truth factor code and the prediction
of its paired observation using the considered encoder, termed as $L2$ score.
Second, these metrics only evaluate the the encoder disentanglement while
ignoring the generator controllability, another important characteristic of
disentanglement learning. However, there may exist a trade-off between the
encoder and generator disentanglement. That is, a high MIG score does not mean
a good model in terms of the controllable generation ability. Therefore, we
propose new metrics to quantify the generator controllability.
Specifically, given a generator $G$ to be evaluated and an oracle encoder
$E_{\text{oracle}}$ that can perfectly predict the factor code, we first
sample $N$ generated observation-code pairs
$({x^{\prime}}^{(n)},{c^{\prime}}^{(n)})$ where
${x^{\prime}}^{(n)}=G(z,{c^{\prime}}^{(n)})$. We then pass the generated
sample ${x^{\prime}}^{(n)}$ into $E_{\text{oracle}}$ to get its factor code
prediction $\hat{c}^{\prime(n)}=E_{\text{oracle}}(x^{\prime(n)})$.
Accordingly, we measure the correlation between $\hat{c}^{\prime(n)}$ and
${c^{\prime}}^{(n)}$ in the same way with prior disentanglement metrics. In
particular, we define an MIG-like metric, called MIG-gen, to evaluate the
generator,
$\displaystyle\text{MIG-gen}=\frac{1}{NK}$
$\displaystyle\sum\limits_{n=0}^{N-1}\sum\limits_{k=0}^{K-1}\frac{1}{H({\hat{c}^{(n)}_{k}})}\cdot$
$\displaystyle\left(I(\hat{c}^{\prime(n)}_{j_{k}};{c^{\prime}}^{(n)}_{k})-\max\limits_{j\neq
j_{k}}I(\hat{c}^{\prime(n)}_{j};{c^{\prime}}^{(n)}_{k})\right)$
where $K$ is the length of factor code, $H(\cdot)$ and $I(\cdot;\cdot)$ denote
the entropy and mutual information, respectively, and
$j_{k}=\arg\max\nolimits_{j}I(\hat{c}^{\prime(n)}_{j},{c^{\prime}}^{(n)}_{k})$.
Similarly, we also introduce L2-gen to measure the semantic correctness of the
generator,
$\text{L2-gen}=\frac{1}{N}\sum\limits_{n=0}^{N-1}\|E_{\text{oracle}}({x}^{\prime(n)})-{c^{\prime}}^{(n)}\|_{2}$
Intuitively, if the oracle encoder is perfect for every ground-truth
observation-code pair, any mismatch between its prediction and the
corresponding factor code should be contributed to the generator instead.
Thus, both MIG-gen and L2-gen can effectively measure the generator
controllability. To obtain an oracle encoder for each dataset, such as the
proposed Falcor3D and Isaac3D, we pre-train a separate encoder network by
minimizing the $L2$ score with all the ground-truth observation-code pairs.
### 4.3 Experimental Setup
#### Datasets and evaluation metrics.
To test the proposed Semi-StyleGAN on complex high-resolution images that many
prior works have difficulty with, we focus on three datasets: Isaac3D with
resolution 512x512, Falcor3D with resolution 512x512 and CelebA with
resolution 256x256. For the proposed Isaac3D and Falcor3D, we use MIG and L2
to measure the encoder disentanglement, and MIG-gen and L2-gen to measure the
generator controllability. For experiments on CelebA, we focus on the latent
traversals to qualitatively measure the disentanglement quality.
#### Experimental protocol.
Before training, we first get the labeled set $\mathcal{J}$, by randomly
sampling observation-code pairs from each dataset with a probability $\eta$.
All the remaining observations form as the unlabeled set. The value of
$\eta\in[0,1]$ controls the portion of labeled data during training.
Particularly Semi-StyleGAN becomes a fully-supervised method if $\eta=1$, and
reduces to Info-StyleGAN if $\eta=0$. In experiments, we set $\xi=0.75$ in Eq.
(5) to be the same with (Berthelot et al., 2019). For the hyperparameters
$\\{\gamma_{G},\gamma_{E},\beta,\alpha\\}$, we find that setting
$\gamma_{G}=\beta=\gamma$, $\gamma_{E}=0$, $\alpha=1$ works well across
different datasets, where we vary $\gamma\in\\{1,10\\}$. Thus, without stated
otherwise, we use the above setting by default in Semi-StyleGAN.
Our experiments mainly include four aspects. (i) We first vary the supervision
rate $\eta$ to show the impact of limited supervision. (ii) Given the
supervision rate $\eta$, we train the encoder alone with the labeled data only
(called Encoder-only) and with the MixUp (called Encoder-only w/ MixUp),
respectively, as supervised and semi-supervised baselines for the encoder
disentanglement. (iii) For ablation studies, we vary $\gamma_{G},\gamma_{E}$
to reveal the trade-off between the encoder and generator disentanglement, and
vary $\alpha$ to show the impact of smoothness regularization. (iv) We show
latent traversal results on both synthetic and real datasets.
### 4.4 Key Results
#### Impact of limited supervision.
Figure 2 shows the quantitative results of varying $\eta$ in Semi-StyleGAN on
Isaac3D and Falcor3D, where we consider two cases of $\gamma\in\\{1,10\\}$.
For Isaac3D, only using 0.25% of the labeled data can achieve a very close
performance to the fully-supervised one ($\eta=1$) in terms of both the
encoder and generator disentanglement. Similarly for Falcor3D, only using 2.5%
of the labeled data can also achieve near the fully-supervised
disentanglement. It means adding a very small amount of labeled data
(0.25%$\sim$2.5%) into the training dataset could benefit significantly the
disentanglement learning with Semi-StyleGAN. Besides, we can see that the
generator disentanglement is more sensitive to the choice of $\gamma$ than the
encoder disentanglement, particularly on Falcor3D.
(a) Semi-StyleGAN on Isaac3D with 0.5% of labeled data
(b) Semi-StyleGAN on Falcor3D with 1% of labeled data
Figure 3: Latent traversal of Semi-StyleGAN on Isaac3D and Falcor3D where
$\gamma=10$. Images in the first column (marked by red box) are randomly
sampled real images and the rest images in each row are their interpolations,
by uniformly varying the given factor from 0 to 1. See Appendix C.1 and C.2
for more results.
#### Ablation studies and comparison with baselines.
First, Table 3 shows there may exist a crucial trade-off between the encoder
and generator disentanglement, governed by the interplay between the
unsupervised and supervised loss terms. For example, we can see that Semi-
StyleGAN gets the best generator disentanglement by slightly sacrificing the
encoder disentanglement, in particular on Isaac3D. Second, by removing the
smoothness consistency, we can see a large performance drop in terms of all
metrics on both datasets, which demonstrates the effectiveness of the
smoothness regularization in Semi-StyleGAN. Besides, there exist a trade-off
between the generator controllability and encoder disentanglement. For
example, increasing the $\mathcal{L}_{\text{unsup}}$ term in $E$
($\gamma_{E}=10$) worsens the generator controllability while improving the
encoder disentanglement on Isaac3D. Removing the $\mathcal{L}_{\text{unsup}}$
term in $G$ ($\gamma_{G}=0$), we can still get a decent encoder
disentanglement despite the generator controllability totally fails. This
trade-off also depends on the datasets, evidenced by different behaviors on
Isaac3D and Falcor3D. Finally, Table 3 shows the results of comparing Semi-
StyleGAN with the supervised and semi-supervised baselines. We can see that
Semi-StyleGAN significantly outperforms the supervised baseline (i.e.,
Encoder-only) on Isaac3D, and also gets a better MIG score on Falcor3D. With
good hyperparameters which potentially weigh more on the encoder side, we can
also achieve on par or better encoder disentanglement than the semi-supervised
baseline (i.e., Encoder-only w/ MixUp).
#### Latent traversal on synthetic and real data.
Qualitatively, we show the latent traversal results on both synthetic and real
data in Figure 3 and 4. When only 0.5% or 1% of the labeled data is available,
each factor in the interpolated images changes smoothly without affecting
other factors. For Isaac3D and Falcor3D, all the interpolated images visually
look the same with their reference real image except for the considered
factor, verifying the semantic correctness of Semi-StyleGAN with very limited
supervision. For CelebA, we use a higher resolution than the prior work (Chen
et al., 2016; Higgins et al., 2017; Nguyen-Phuoc et al., 2019), and achieve
visually better disentanglement quality together with a higher image quality.
It means that the insights gained in the synthetic datasets also applies to
the real domain. With very limited supervision, Semi-StyleGAN can achieve good
disentanglement on real data.
Figure 4: Latent traversal of Semi-StyleGAN on CelebA with resolution 256x256
by using 0.5% of the labeled data, where we use $\gamma=1$ and disentangle all
40 binary attributes. See Appendix C.3 for the results of other attributes.
## 5 An Extension for Better Generalization
Although Semi-StyleGAN performs well in both synthetic and real data, it
cannot generalize to unseen data whose high-level content does not match the
training data but whose fine-grained styles might. For instance, Semi-StyleGAN
trained on Isaac3D cannot generate an image in which the robot arm stands on
the right (instead of in the middle as in the training data). In this section,
we design a new GAN architecture that extends Semi-StyleGAN to an image-to-
image model, that we call Semi-StyleGAN-fine. This model achieves better
generalization to unseen data.
Inspired by the observations that lower resolution blocks in the StyleGAN
generator learn coarse-grained features while its high-resolution blocks
account for fine-grained styles, we change the StyleGAN generator to not
contain lower-resolution blocks. As shown in Figure 5, the generator instead
takes the real image as one of its inputs by downscaling it to a lower
resolution $\phi$ (e.g., $\phi=32$ in Figure 5). Accordingly, it generates the
high-resolution image by only modulating the (fine-grained) factor code into
higher resolution blocks. Also, the encoder predicts the value of factor code
from the block with resolution $\phi$, instead of the last output block. The
intuition is that lower-resolution blocks in the encoder also have less
relationship with fine-grained styles, and thus the code prediction better use
its higher resolution blocks only. This way, the generator in Semi-StyleGAN-
fine does a semi-supervised controllable fine-grained image editing while its
encoder infers the fine-grained factor code that the generator has used.
Finally, the loss functions remain the same as in Eq. (7).
Figure 5: An illustration of Semi-StyleGAN-fine, where we downsample the real
image into 32x32 resolution and replace the lower resolution blocks (4x4 -
32x32) in the generator by a new input block. Also, the encoder predicts the
value of (fine-grained) factor code from the 32x32 block instead.
### 5.1 Experimental Setup
We mainly focus on the latent traversals of Semi-StyleGAN-fine on Isaac3D and
CelebA to qualitatively test its generalization ability. In the training time,
we train Semi-StyleGAN-fine on Isaac3D and CelebA, respectively. In the test
time, we apply novel test images (with the different high-level content) as
the input to evaluate the proposed method. For Isaac3D, the novel test images
are given by: (i) shifting the robot position to the right side (instead of
standing right in the middle for all the training data), and (ii) attaching
the robot arm with an unseen object. For CelebA, we simply download some new
face images from online, following by aligning and cropping them into 256x256.
Figure 6: Generalized latent traversal results of Semi-StyleGAN-fine trained
on Isaac3D with 1% of labeled data where we set $\phi=64$ and interpolate the
shown fine styles. In the test images, we shift the position of the robot arm
to the right side, and also attach it with an unseen object (i.e., a
tetrahedron). Figure 7: Generalized latent traversal results of Semi-
StyleGAN-fine trained on CelebA with 1% of labeled data where we set $\phi=64$
and control the shown fine styles.
### 5.2 Key Results
Figure 6 and 7 show the results of interpolating fine-grained factors in the
Isaac3D dataset and the CelebA dataset, respectively, with different test
images where we set $\eta=0.01$ and $\phi=64$. We can see that each considered
fine-grained factor in both datasets keeps changing smoothly during its
interpolations without affecting other factors, implying good generalized
disentanglement. Particularly, the interpolations of the Isaac3D test images
all maintain the new robot position and new object shape (i.e., a tetrahedron)
with relatively high image quality. See Appendix D for test images with
another novel object shape. The interpolations of the CelebA test images also
keep the same identities and other coarse-grained features with the given
input images. It is worthy to note that the good generalized disentanglement
of Semi-StyleGAN-fine has been achieved by using only 1% of labeled data, and
further increasing the supervision rate $\eta$ does not visually improve
performance. Therefore, these results demonstrate the ability of Semi-
StyleGAN-fine in semantic fine-grained image editing with limited supervision
that generalizes well to unseen novel images.
## 6 Conclusions
In this paper, we designed new loss functions and architectures based on
StyleGAN for semi-supervised high-resolution disentanglement learning. We
first showed that Info-StyleGAN largely outperforms most state-of-the-art
unsupervised disentanglement methods, which justified the advantages of the
StyleGAN architecture in disentanglement learning. We then proposed Semi-
StyleGAN that achieved near fully-supervised disentanglement with limited
supervision (0.25%$\sim$2.5%) on complex high-resolution synthetic and real
data. We also proposed new metrics to quantify the generator controllability.
To the best of our knowledge, we were the first to reveal that there exists a
trade-off between learning disentangled representations and controllable
generation. Besides, we extended Semi-StyleGAN to do semantic fine-grained
image editing with better generalization to unseen images. Finally, we created
two high-quality synthetic datasets to serve as new disentanglement
benchmarks.
In the future, we want to apply Semi-StyleGAN to even larger-scale high-
resolution real datasets. We are aware of the gender and racial biases in the
CelebA dataset (Kärkkäinen & Joo, 2019), and thus hope to create better
datasets and find other ways to address the algorithmic bias. Besides, It
would be interesting to extend Semi-StyleGAN to the weakly-supervised, semi-
supervised scenario, where the factors of variation are only partially
observed.
## Acknowledgement
Thanks to the anonymous reviewers for useful comments. We also thank Zhiding
Yu, Anuj Pahuja, Yaosheng Fu, Tan Minh Nguyen and many others at Nvidia for
helpful discussions on this work. WN and ABP were supported by IARPA via
DoI/IBC contract D16PC00003.
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## Appendix
## Appendix A More details of Two New Datasets
### A.1 Examples of the Isaac3D Dataset
Figure 8: Examples of the Isaac3D dataset where we vary each factor of
variation individually to see how each factor of variation changes with its
corresponding ground-truth factor code.
### A.2 Examples of the Falcor3D Dataset
Figure 9: Examples in the Falcor3D dataset where we vary each factor of
variation individually to see how each factor of variation changes with its
corresponding ground-truth factor code.
## Appendix B More results of Info-StyleGAN
### B.1 Progressive Training for Disentanglement Learning
Progressive growing has been shown to improve the image quality of GANs
(Karras et al., 2017, 2019), however, its impact on disentanglement learning
remains unknown so far. Thus, we compare the MIG scores of Info-StyleGAN with
progressive and non-progressive growing, respectively, on both dSprites and
Isaac3D, and the results are shown in Figure 10. We can see that with
progressive growing, the disentanglement quality tends to be better on both
two datasets. Besides, the gap of average MIG scores with different values of
the hyperparameter $\gamma$ is much smaller if the progressive growing is
applied, which implies Info-StyleGAN with progressive growing seems to be less
sensitive to hyperparameters. Therefore, unless stated otherwise, we use
progressive growing for the GAN training in all the experiments.
Figure 10: The impact of progressive training on the disentanglement learning
with Info-StyleGAN. We can see that with progressive growing, the
disentanglement quality tends to be better on both two datasets. Besides, the
gap of average MIG scores with different values of the hyperparameter $\gamma$
is much smaller if the progressive growing is applied, which implies Info-
StyleGAN with progressive growing seems to be less sensitive to
hyperparameters.
### B.2 How to Get Info-StyleGAN* with Smaller Network Size
Mapping Network
---
(FC $\times$ $n_{\text{mp}}$) $f_{\text{mp}}$ $\times$ $f_{\text{mp}}$
Synthesis Network
(4$\times$4 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(4$\times$4 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(8$\times$8 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(8$\times$8 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(16$\times$16 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(16$\times$16 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(32$\times$32 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(32$\times$32 Conv) 3$\times$3$\times$ $f_{\text{0}}$ $\times$ $f_{\text{0}}$
(64$\times$64 Conv) 3$\times$3$\times$ $\frac{f_{\text{0}}}{2}$ $\times$
$\frac{f_{\text{0}}}{2}$
(64$\times$64 Conv) 3$\times$3$\times$ $\frac{f_{\text{0}}}{2}$ $\times$
$\frac{f_{\text{0}}}{2}$
(a) Generator
(64$\times$64 Conv) 3$\times$3$\times$ $\frac{f_{\text{0}}}{2}$ $\times$
$\frac{f_{\text{0}}}{2}$
---
(64$\times$64 Conv) 3$\times$3$\times$ $\frac{f_{\text{0}}}{2}$ $\times$
$\frac{f_{\text{0}}}{2}$
(32$\times$32 Conv) 3$\times$3$\times$${f_{\text{0}}}$$\times$${f_{\text{0}}}$
(32$\times$32 Conv) 3$\times$3$\times$${f_{\text{0}}}$$\times$${f_{\text{0}}}$
(16$\times$16 Conv) 3$\times$3$\times$${f_{\text{0}}}$$\times$${f_{\text{0}}}$
(16$\times$16 Conv) 3$\times$3$\times$${f_{\text{0}}}$$\times$${f_{\text{0}}}$
(8$\times$8 Conv) 3$\times$3$\times$${f_{\text{0}}}$$\times$${f_{\text{0}}}$
(8$\times$8 Conv) 3$\times$3$\times$${f_{\text{0}}}$$\times$${f_{\text{0}}}$
(4$\times$4 Conv) 3$\times$3$\times$${f_{\text{0}}}$$\times$${f_{\text{0}}}$
(4$\times$4 FC) (16$f_{\text{0}}$) $\times$ 64
(4$\times$4 FC) 64$\times$(1+ code_length)
(b) Discriminator / Encoder
Table 4: Generator and discriminator (or encoder) architectures in the
implementation of Info-StylGAN for generating the image of resolution
$128\times 128$, where we use “FC $\times n_{\text{mp}}$” to denote that there
are $n_{\text{mp}}$ dense layers in the given block, and use “8$\times$8 Conv”
to denote the convolutional layer in the 8$\times$8 resolution block. Note
that there is not the last block (i.e., 64$\times$64 Conv) in the generator
and not the first block (i.e., 64$\times$64 Conv) in the encoder if we want to
generate the image of resolution $64\times 64$. For the original Info-
StyleGAN, we have $n_{\text{mp}}=8$, $f_{\text{0}}=512$, $f_{\text{0}}=512$.
For Info-StyleGAN* on dSprites, we set $n_{\text{mp}}=3$, $f_{\text{mp}}=64$,
$f_{\text{0}}=64$ with 0.74M parameters in total. For Info-StyleGAN* on
Downscaled Isaac3D, we set $n_{\text{mp}}=3$, $f_{\text{mp}}=256$,
$f_{\text{0}}=128$ with 3.44M parameters in total.
### B.3 Randomly Generated Samples of Baseline Models and Info-StyleGAN*
(with Smaller Network Size)
(a) $\beta$-VAE (FID=120.3)
(b) FactorVAE (FID=358.2)
(c) $\beta$-TCVAE (FID=143.8)
(d) InfoGAN-CR (FID=73.43)
(e) Info-StyleGAN* (FID=8.38)
Figure 11: Randomly sampled images of baseline models and Info-StyleGAN* on
(downscaled) Isaac3D of resolution 128x128. Note that for VAE-based models, we
use the similar network architectures as in (Locatello et al., 2019), and
Info-StyleGAN* denotes the smaller version of Info-StyleGAN, in which its
number of parameters is similar to baseline models. We can see that compared
with Info-StyleGAN* (of the same network size), the generated images of VAE-
based models (i) tend to be quite blurry and of low quality, and (ii) fail to
cover all the variations in the dataset. As a strong GAN baseline, InfoGAN-CR
is also significantly worse than Info-StyleGAN in terms of image quality. The
results demonstrate that our proposed dataset can serve as a new challenging
benchmark for disentanglement learning, in particular regarding the much
higher resolution, and larger variation of factors.
### B.4 Randomly Generated Samples of baseline models† (with Larger Network
Size) and Info-StyleGAN
(a) $\beta$-VAE† (FID=60.71)
(b) FactorVAE† (FID=60.67)
(c) $\beta$-TCVAE† (FID=77.48)
(d) InfoGAN-CR (FID=30.41)
(e) Info-StyleGAN (FID=2.50)
Figure 12: Randomly sampled images of baseline models† and Info-StyleGAN on
(downscaled) Isaac3D of resolution 128x128. Note that for VAE-based models, we
increase the number of featuremaps ($\times$8) in each layer of the network
architectures in (Locatello et al., 2019), so that their number of parameters
is similar to Info-StyleGAN. We also apply the same operations for InfoGAN-CR
to match the network size of In-StyleGAN. We can see that (i) the image
quality gets better after increasing the network size, (ii) compared with
Info-StyleGAN (of the same network size), the generated images of VAE-based
models still have issues with blurriness and failure in capturing all
variations, (iii) the generated images of InfoGAN-CR are better than VAEs but
still worse than Info-StyleGAN.
### B.5 Other Experimental Settings
Our experiments are based on the StyleGAN implementation (Karras et al.,
2019), where the GAN loss $L_{\text{GAN}}$, batch sizes, learning rates for
both generator and discriminator, and other hyperparameters in the Adam
optimizer and weights in each resolution black are all kept the same with
(Karras et al., 2019), unless stated otherwise. Different from the original
StyleGAN implementation, we do not apply truncation tricks. We also do not add
noise inputs to introduce another randomness, as we consider the case where
the factor code and latent $z$ will capture all the factors in the data. For
all the quantitative results in the paper, we report the error bars by taking
the mean and standard deviation of four runs with random seeds. For the
implementation of evaluation metrics, we use 50K random sampled real images
and fake images to calculate the FID score. We use 5K ground-truth
observation-code pairs as training samples and 2K ground-truth observation-
code pairs as test samples to evaluate the Factor score. We use 10K ground-
truth observation-code pairs and 10K generated observation-code pairs to
calculate the MIG and MIG-gen scores, respectively. We also use 1K ground-
truth observation-code pairs and 1K generated observation-code pairs to
calculate the L2 and L2-gen scores, respectively.
## Appendix C More Results of Semi-StyleGAN
### C.1 Semi-StyleGAN with 0.5% of Labeled Data on Isaac3D with Resolution
512x512
Figure 13: Latent traversal of Semi-StyleGAN on Isaac3D by using 0.5% of the
labeled data. Images in the first column (marked by red box) are randomly
sampled real images of resolution 512x512 and the rest images in each row are
their interpolations, respectively, by uniformly varying the given factor from
0 to 1. We can see that each factor changes smoothly during its interpolation
without affecting other factors, and the interpolated images in each row
visually look almost the same with their input image except the considered
varying factor. Also, the image quality does not get worse during the
interpolations.
### C.2 Semi-StyleGAN with 1% of Labeled Data on Falcor3D with Resolution
512x512
Figure 14: Latent traversal of Semi-StyleGAN on Falcor3D by using 0.5% of the
labeled data. Images in the first column (marked by red box) are randomly
sampled real images of resolution 512x512 and the rest images in each row are
their interpolations, respectively, by uniformly varying the given factor from
0 to 1. We can see that each factor changes smoothly during its interpolation
without affecting other factors, and the interpolated images in each row
visually look almost the same with their input image except the considered
varying factor. Also, the image quality does not get worse during the
interpolations.
### C.3 Semi-StyleGAN with 0.5% of Labeled Data on CelebA with Resolution
256x256
Figure 15: More latent traversal results of Semi-StyleGAN on CelebA with
resolution 256x256 by using 0.5% of the labeled data, where we control all 40
binary attributes at the same time. We can see that Semi-StyleGAN with only
0.5% of the labeled data is capable of controlling the considered attributes.
We note that the image background may also change slightly over interpolations
of some attributions. We argue that it is because the other nuisance factors
(i.e., those not in the set of considered 40 attributes) including background
strongly confound the observed factors, which has been a common and difficult
problem in high-dimensional partially observed latent variables models. We
leave the investigation into how to further alleviate this confounding issue
as the future work.
Figure 16: Latent traversal of Semi-StyleGAN on CelebA with resolution 256x256
by using 0.5% of the labeled data, where we control all 40 binary attributes
at the same time. We can see that Semi-StyleGAN with only 0.5% of the labeled
data is capable of controlling the considered attributes. We note that the
image background may also change slightly over interpolations of some
attributions. We argue that it is because the other nuisance factors (i.e.,
those not in the set of considered 40 attributes) including background
strongly confound the observed factors, which has been a common and difficult
problem in high-dimensional partially observed latent variables models. We
leave the investigation into how to further alleviate this confounding issue
as the future work.
## Appendix D More results on Semi-StyleGAN-fine
### D.1 Semi-StyleGAN-fine with 1% of Labeled Data on Isaac3D Novel Images
with Resolution 512x512
Figure 17: Generalization of Semi-StyleGAN-fine with 1% of the labeled data
where we set $\phi=64$ and interpolate three fine styles: (lighting intensity,
object color, wall color). In the test image, we shift the position of the
robot arm to the right side, and also attach it with an unseen object (i.e.,
octahedron). Images in the first column (marked by red box) are real novel
images of resolution $512\times 512$ and the rest images in each row are their
interpolations, respectively, by uniformly varying the given factor from 0 to
1. We can see that Semi-StyleGAN-fine with only 1% of the labeled data is
capable of controlling the considered fine-grained attributes without
affecting the coarse-grained factors.
### D.2 Semi-StyleGAN-fine with 1% of Labeled Data on CelebA Novel Images
with Resolution 256x256
Figure 18: Generalized latent traversal results of Semi-StyleGAN-fine trained
on CelebA with 1% of labeled data where we set $\phi=64$ and control the shown
fine styles. Images in the first column (marked by red box) are real novel
images of resolution $256\times 256$ and the rest images in each row are their
interpolations, respectively, by uniformly varying the given factor from 0 to
1. We can see that Semi-StyleGAN-fine with only 1% of the labeled data is
capable of controlling the considered fine-grained attributes without
affecting the coarse-grained factors, in particular the personal identity.
|
2024-09-04T02:54:58.016641 | 2020-03-07T00:17:58 | 2003.03475 | {
"authors": "Rajan Puri",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26093",
"submitter": "Rajan Puri",
"url": "https://arxiv.org/abs/2003.03475"
} | arxiv-papers | # Beta Critical for the Schrödinger Operator with Delta Potential
Rajan Puri
###### Abstract.
For the one dimensional Schrödinger operator in the case of Dirichlet boundary
condition, we show that $\beta_{cr}$ is positive and zero for the case of
Neumann and Robin boundary condition considering the potential energy of the
form $V(x)=-\beta\delta(x-a)$ where, $\beta\geq 0,\ a>0.$ We prove that the
$\beta_{cr}$ goes to infinity when the delta potential moves towards the
boundary in dimension one with Dirichlet boundary condition. We also show that
the $\beta_{cr}>0$ and $\beta\in(0,\frac{1}{2})$ considering Dirichlet problem
with delta potential on the circle in dimension two.
###### Key words and phrases:
Schrodinger Operator, Delta Potential, Critical Value, Beta Critical, Negative
Eigenvalues.
###### 2010 Mathematics Subject Classification:
35J25, 35P15, 47A10, 47D07
Department of Mathematics and Statistics, Wake Forest University, Winston
Salem, NC 27109, USA, ([email protected]).
## 1\. Introduction
This paper is a continuation of our work [4] where we studied the beta
critical of the coupling constant of exterior elliptic problems and proved
that $\beta_{cr}>0$ in the case of Dirichlet boundary condition and
$\beta_{cr}=0$ in the case of the Neumann boundary condition in the dimension
d=1 and 2 by studying the truncated resolvent operator. It was shown that the
choice $\beta_{cr}>0$ or $\beta_{cr}=0$ depends on whether the truncated
resolvent is bounded or goes to infinity when $\lambda\to 0^{-}$. In fact,
$\beta_{cr}$ was expressed through truncated resolvent operator. There has
been a considerable interest in the study of coupling constant and the problem
was investigated by several researchers like Barry Simon in[5], Martin Klaus
in [3], Cranston, Koralov, Molchanov and Vainberg in [1], and Yuriy Golovaty
in [2]. It is known that the spectrum of $-\Delta-\beta V(x)$ consists of the
absolutely continuous part $[0,\infty)$ and at most a finite number of of
negative eigenvalues.
$\sigma(-\Delta-\beta V(x))=\\{\lambda_{j}\\}\cup[0,\infty),\ 0\leq j\leq
N,\quad\lambda_{j}\leq 0.$
We proved the dependence of beta critical with the boundary condition and
dimension in our earlier paper [4]. Namely,
###### Theorem 1.1 ([4]).
Consider the following elliptic problems in $\Omega$.
$H_{0}u-\beta V(x)u-\lambda u=f,\ \ x\in\Omega,$ (1)
where $H_{0}=-\text{div}(a(x)\nabla)$, the potential $V(x)\geq 0$ is compactly
supported and continuous, $\beta\geq 0$ , $a(x)>0\ ,\ a(x)\in C^{1}(\Omega)$,
and $a=1\ \text{when}\ |x|>>1$. If $d=1$ or $2$ then $\beta_{cr}>0$ in the
case of the Dirichlet boundary condition, and $\beta_{cr}=0$ in the cases of
the Neumann boundary condition.
In [4], we also studied the dependence of $\beta_{cr}$ on the distance between
the support of the potential and the boundary of the domain. In fact, it was
proven that, in the case of the Dirichlet boundary condition in the dimension
one, $\beta_{cr}$ tends to infinity as potential moves towards the boundary.
In dimension two with the Dirichlet boundary condition, the behavior of
$\beta_{cr}$ was interesting and depends on the relation between the rates of
the shrinking of the support of the potential and the speed of its motion
towards the boundary. We did not consider the Neumann boundary condition when
$d=1$ or $2$ since $\beta_{cr}$ is always zero. In particular, we proved the
following theorem in [4].
###### Theorem 1.2 ([4]).
If $d=1$, then $\beta_{cr}$ for the Schrödinger operator $-\Delta-\beta V(x)$
goes to infinity as $n\to\infty$ for the Dirichlet boundary condition. The
same is true if $d=2$ and $|x(n)-x_{0}|<C/n,~{}n\to\infty$. If $d=2$ and
$|x(n)-x_{0}|\to 0,~{}|x(n)-x_{0}|>C/n^{\delta},~{}n\to\infty,$ with some
$\delta\in(0,1)$, then $\beta_{cr}$ remains bounded as $n\to\infty$. If $d\geq
3$, then $\beta_{cr}$ remains bounded as $n\to\infty$ for both the Dirichlet
and Neumann boundary conditions.
## 2\. Schrödinger operator with Delta Potential
In this paper, we would present results on the beta critical in the case of
one and two dimensional Schrödinger equation with delta potential given by
$-y^{{}^{\prime\prime}}-\beta\delta(x-a)y(x)=\lambda y(x),$ (2)
where $\beta\geq 0,\ a>0$. The delta function is a infinitely high,
infinitesimally narrow spike at $x=a$. This allows solutions for both the
bound states $\lambda<0$ and scattering states $\lambda>0.$ The classification
of the spectrum into discrete and continuous parts usually corresponds to a
classification of the dynamics into localized (bound) states and locally
decaying states when time increases (scattering), respectively. The lower
bound, $0$, of the absolutely continuous spectrum is called the ionization
threshold. This follows from the fact that the particle is no longer
localized, but moves freely when $\lambda>0$. This classification is related
to the space-time behaviour of solutions of the corresponding Schrödinger
equation.
We are interested to study the beta critical, the critical value the coupling
constant denoted by $\beta_{cr},$ the value of $\beta$ such that equation (2)
does not have negative eigenvalues for $\beta<\beta_{cr}$ and has them if
$\beta>\beta_{cr}$. We can find the solution to Schrodinger equation (2) in
the region I and II as shown in the figure 1.
Figure 1. Delta Potential at x=a.
In both regions: region I or $0\leq x<a$ and region 2 or $x>a$, the potential
is $V(x)=0.$
###### Theorem 2.1.
If we consider the following one dimensional Schrödinger equation in the half
axis
${}-y^{{}^{\prime\prime}}-\beta\delta(x-a)y(x)=\lambda y(x),\
\lambda=-k^{2}<0\ x\in[0,\infty).$ (3)
then the $\beta_{cr}>0$ in the case of Dirichlet Boundary condition and
$\beta_{cr}=0$ in the case of Neumann and Robin boundary condition.
###### Proof.
The Dirichlet problem is given by
${}-y^{{}^{\prime\prime}}-\beta\delta(x-a)y(x)=\lambda y(x),\ y(0)=0,\
y(a)=1,\ \lambda=-k^{2}<0.$ (4)
The solution of the problem (4) is given by
$y(x)=Pe^{-kx}+Qe^{kx}.$
We can determine the value of constant P and Q by using the given condition
$y(0)=0,y(a)=1$ with the continuity of the solution. Actually we can divide
the solution of the problem (4) in to two different regions.
$\begin{cases}y_{1}(x)=\frac{\sinh kx}{\sinh ka},&\text{if }0\leq x\leq a\\\
y_{2}(x)=e^{k(a-x)},&\text{if }a\leq x.\end{cases}$
We integrated the Schrödinger equation with respect to $x$ over a small
interval $\Delta\epsilon$.
$\int_{a-\epsilon}^{a+\epsilon}(-y^{{}^{\prime\prime}}-\beta\delta(x-a)y\
)dx=\int_{a-\epsilon}^{a+\epsilon}(\lambda y)dx.$
The integral of the second derivative is just the first derivative function
and the integral over the function in the right side goes to zero, since it is
a continuous, single valued function. We get,
$-y^{{}^{\prime}}|_{a-\epsilon}^{a+\epsilon}-\beta y(a)=0.$
When $\epsilon\xrightarrow{}0,$ $k+k\coth ka=\beta.$ which gives,
$\frac{k}{\beta}=\frac{1}{1+\coth ka}$ and $\frac{ka}{\beta
a}=\frac{1}{1+\coth ka}.$ Let $ka=A,\beta a=B,$ then $e^{-2A}=1-\frac{2A}{B}.$
Again let $2A=z,$ then we have,
${}e^{-z}=1-\frac{z}{B}.$ (5)
From equation (5) $1-\frac{z}{B}\geq 0.$ Hence, we get $\frac{\beta}{2}\geq
k,$ and it follows $\frac{\beta^{2}}{4}\geq k^{2}.$ Since
$\lambda=-k^{2}\geq\frac{-\beta^{2}}{4}.$ Hence, if $\beta=0$ then there is no
possibility of having negative eigenvalues so $\beta_{cr}$ must be greater
than zero to produce negative eigenvalues.
If we consider the Neumann boundary condition then the equation (3) becomes
${}-y^{{}^{\prime\prime}}-\beta\delta(x-a)y(x)=\lambda y(x),\
y^{{}^{\prime}}(0)=0,\ y(a)=1,\ \lambda=-k^{2}<0,k>0.$ (6)
As above, we can divide the solution of this problem in to two different
regions, i.e region (I) with $0\leq x<a$ and region (II) with $a<x$.
$\begin{cases}y_{1}(x)=\frac{\cosh kx}{\cosh ka},&\text{if }0\leq x\leq a\\\
y_{2}(x)=e^{k(a-x)},&\text{if }a\leq x.\end{cases}$
We will again integrate the Schrödinger equation with respect to $x$ over a
small interval and take $\epsilon\xrightarrow{}0.$ Then, $k+k\tanh ka=\beta.$
Which gives, $\frac{k}{\beta}=\frac{1}{1+\tanh ka}$ and $\frac{ka}{\beta
a}=\frac{1}{1+\tanh ka}.$ Let $ka=A,\ \beta a=B$ then
$e^{-2A}=\frac{2A}{B}-1.$ Again let $2A=z,$ then $e^{-z}=\frac{z}{B}-1.$ It
tells us that $\frac{z}{B}-1\geq 0.$ From here, we get the following relation
$\frac{\beta}{2}\leq k$ which follows $\frac{\beta^{2}}{4}\leq k^{2}.$ Since
$\lambda=-k^{2}\leq\frac{-\beta^{2}}{4}.$ Hence, if $\beta=0$ then there is
still a possibility of having a negative eigenvalues so $\beta_{cr}=0.$
If we consider the Robin boundary condition then the equation (3) becomes
${}-y^{{}^{\prime\prime}}-\beta\delta(x-a)y(x)=\lambda y(x),\
\frac{dy}{dx}+y\bigg{|}_{x=0}=0,\ y(a)=1,\ \lambda=-k^{2}<0.$ (7)
We can divide the solution of this problem in to two different region as
described below.
$\begin{cases}y_{1}(x)=\frac{k\cosh kx-\sinh kx}{k\cosh ka-\sinh ka},&\text{if
}0\leq x\leq a\\\ y_{2}(x)=e^{k(a-x)},&\text{if }a\leq x.\end{cases}$
As above, we will integrate the Schrödinger equation with respect to $x$ over
a small interval $\Delta\epsilon$.
$\int_{a-\epsilon}^{a+\epsilon}(-y^{{}^{\prime\prime}}-\beta\delta(x-a)y\
)dx=\int_{a-\epsilon}^{a+\epsilon}(\lambda y)dx.$
After solving this integral problem, we come up with the following equation
when $\epsilon\xrightarrow{}0,$
$k+k\bigg{(}\frac{k-\coth ka}{k\coth ka-1}\bigg{)}=\beta.$
which gives,
$\frac{k}{\beta}=\frac{1}{1+\bigg{(}\frac{k-\coth ka}{k\coth ka-1}\bigg{)}},$
and
$\frac{ka}{\beta a}=\frac{1}{1+\bigg{(}\frac{k-\coth ka}{k\coth
ka-1}\bigg{)}}.$
Let $ka=A,\beta a=B$ then
$e^{-2A}=\frac{2A^{2}-2Aa-AB+Ba}{AB+Ba}.$
It tells us that $\frac{2A^{2}-2Aa-AB+Ba}{AB+Ba}$ must be $\geq 0.$ After
solving this inequality, we get the following relation
$\frac{\beta}{2}\leq k,$
which tells us that $\frac{\beta^{2}}{4}\leq k^{2}.$ Since
$\lambda=-k^{2}\leq\frac{-\beta^{2}}{4}.$ Hence if $\beta=0$ then there is a
possibility of having negative eigenvalues so $\beta_{cr}$ must be zero. ∎
Now, It will be shown that the beta critical goes to infinity in the case of
Dirichlet boundary condition in the dimension one considering delta potential.
We will not consider the Neumann and Robin boundary condition since
$\beta_{cr}=0.$
###### Theorem 2.2.
Consider the Dirichlet boundary condition where the delta potential is located
at $x=a_{n}$.
${}-y^{{}^{\prime\prime}}-\beta\delta(x-a_{n})y(x)=\lambda y(x),\ y(0)=0,\
y(a_{n})=1,\ \lambda=-k^{2}<0,k>0,a_{n}>0.$ (8)
Then, the $\beta_{cr}\xrightarrow{}\infty$ as $a_{n}\xrightarrow{}0.$
###### Proof.
We will not have a solution of the equation (5) if $\frac{1}{B}\geq 1.$ That
means there is no negative eigenvalues when $\frac{1}{a}\geq\beta.$ However,
if we have the case $\frac{1}{B}\leq 1$ then we would have a solution of the
equation (5). It tells us that when $\frac{1}{a}\leq\beta,$ we will have the
existence of negative eigenvalues. As we defined $\beta_{cr},$ the value of
$\beta$ such that equation (4) does not have negative eigenvalues for
$\beta<\beta_{cr}$ and has them if $\beta>\beta_{cr}$, we conclude that
$\beta_{cr}=\frac{1}{a}$ for the equation (5).
Similarly, if we consider the case where the delta potential is located at
$x=a_{n}$ where $a_{n}\xrightarrow{}0$ as $n\xrightarrow{}\infty,$ given by
(8) then
$\beta_{cr}=\frac{1}{a_{n}}.$
If we approach the potential towards to the boundary which is
$a_{n}\xrightarrow{}0.$ We get, $\beta_{cr}\xrightarrow{}\infty$ as
$a_{n}\xrightarrow{}0.$ ∎
## 3\. Dirichlet Boundary condition for $d=2$ with delta potential on the
circle
###### Theorem 3.1.
If we consider the case of Dirichlet Boundary condition for $d=2$ with delta
potential on the circle is given by
${}-\Delta y(x)-\beta\delta_{1+a}y(x)=\lambda y(x),\ y(1)=0,\ y(1+a)=1,\
\lambda=-k^{2}<0,k>0.$ (9)
then the $\beta_{cr}>0$ and $\beta\in(0,\frac{1}{2}).$
Figure 2. Delta potential in dimension 2.
The rotational invariance suggests that the two dimensional Laplacian should
take a particularly simple form in polar coordinates. We use polar coordinates
$(r,\theta)$ and look for solutions depending only on r. Thus equation (9)
becomes
${}y^{{}^{\prime\prime}}+\frac{y^{\prime}}{r}-\beta\delta_{1+a}\ y(r)=\lambda
y(r),\ y(1)=0,\ y(1+a)=1,\\\ \lambda=-k^{2}<0,k>0.$ (10)
###### Proof.
We will divide the solution of this problem (10) into two different regions,
i.e region (I) with $1\leq r<1+a$ and region (II) with $1+a<r$.
$\begin{cases}y_{1}(r)=\frac{Y_{0}(k)J_{0}(kr)-J_{0}(k)Y_{0}(kr)}{Y_{0}(k)J_{0}(k(1+a))-J_{0}(k)Y_{0}(k(1+a))},&\text{if
}1\leq r\leq 1+a\\\ y_{2}(r)=\frac{K_{0}(kr)}{K_{0}(k(1+a))},&\text{if
}1+a\leq r.\end{cases}$
As above,
$-y^{{}^{\prime}}\bigg{|}_{1+a-\epsilon}^{1+a+\epsilon}-\beta y(1+a)=0.$
Which gives,
$-\big{(}y_{2}^{{}^{\prime}}|_{1+a+\epsilon}-y_{1}^{{}^{\prime}}|_{1+a-\epsilon}\big{)}=\beta.$
When $\epsilon\xrightarrow{}0,$
${}\frac{-Y_{0}(k)J_{1}(k(1+a)+J_{0}(k)Y_{1}(k(1+a))}{Y_{0}(k)J_{0}(k(1+a))-J_{0}(k)Y_{0}(k(1+a))}-\frac{-K_{1}(k(1+a))}{K_{0}(k(1+a)}=\frac{\beta}{k}.$
(11)
Where $Y_{0}$ and $Y_{1}$ are Bessel function of second kind and and $J_{0}$
and $J_{1}$ are Bessel function of first kind. Similarly, $K_{0}$ and $K_{1}$
are a modified Bessel function of second kind. Define
$g(k,a)=\frac{g_{1}(k,a)}{g_{2}(k,a)}=\frac{-Y_{0}(k)J_{1}(k(1+a)+J_{0}(k)Y_{1}(k(1+a))}{Y_{0}(k)J_{0}(k(1+a))-J_{0}(k)Y_{0}(k(1+a))}.$
It seems that $g(k,a)=-1$ for all the values of $a$ as shown in the figure
below.
Figure 3. Graph of the function g(k,.5).
Figure 4. Graph of the function g(k,2).
Figure 5. Graph of the function g(k,4).
Figure 6. Graph of the function g(k,10).
From equation (11), we get
${}-1+\frac{K_{1}(k(1+a))}{K_{0}(k(1+a)}=\frac{\beta}{k}.$ (12)
We will use the following fact from [6] to prove that $\beta_{cr}>0.$
###### Lemma 3.2 ([6]).
Let $p,q\geq 0.$ Then the double inqualities
${}1+\frac{1}{2(x+p)}<\frac{K_{1}(x)}{K_{0}(x)}<1+\frac{1}{2(x+q)}$ (13)
hold for all $x>0$ if only if $p\geq 1/4$ and $q=0.$
Now, from equation (12) and (13) we get,
$\frac{1}{2(x+p)}<\frac{K_{1}(x)}{K_{0}(x)}-1<\frac{1}{2(x+q)}.$
When $x=k(a+1)>0,$
$\frac{1}{2(k(a+1)+p)}<\frac{K_{1}(k(a+1))}{K_{0}(k(a+1))}-1<\frac{1}{2(k(a+1)+q)}.$
$\frac{1}{2(k(a+1)+p)}<\frac{\beta}{k}<\frac{1}{2(k(a+1)+q)}.$
$\frac{1}{2((a+1)+\frac{p}{k})}<\beta<\frac{1}{2((a+1)+\frac{q}{k})}.$
Since $a>0,p\geq\frac{1}{4},k>0$, which tells us that $\beta>0$ and hence
$\beta_{cr}>0$ and $\beta\in(0,\frac{1}{2}).$ ∎
## References
* [1] M Cranston, L Koralov, S Molchanov, and B Vainberg. Continuous model for homopolymers. Journal of Functional Analysis, 256(8):2656–2696, 2009.
* [2] Yuriy Golovaty. On coupling constant thresholds in one dimension. arXiv preprint arXiv:1905.10766, 2019.
* [3] M Klaus. On the bound state of schrödinger operators in one dimension. Annals of Physics, 108(2):288–300, 1977.
* [4] Rajan Puri and Boris Vainberg. On the critical value of the coupling constant in exterior elliptic problems. Applicable Analysis, pages 1–10, 2020.
* [5] Barry Simon. The bound state of weakly coupled schrödinger operators in one and two dimensions. Annals of Physics, 97(2):279–288, 1976.
* [6] Zhen-Hang Yang and Yu-Ming Chu. On approximating the modified bessel function of the second kind. Journal of Inequalities and Applications, 2017(1):1–8, 2017.
|
2024-09-04T02:54:58.025364 | 2020-03-07T03:04:56 | 2003.03495 | {
"authors": "Ryota Nakano and Masatoshi Hirabayashi",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26094",
"submitter": "Ryota Nakano",
"url": "https://arxiv.org/abs/2003.03495"
} | arxiv-papers | # Mass shedding activities of Asteroid (3200) Phaethon enhanced by its
rotation
Ryota Nakano Department of Aerospace Engineering
Auburn University, 211 Davis Hall
Auburn, AL 36849, USA Masatoshi Hirabayashi Department of Aerospace
Engineering
Auburn University, 211 Davis Hall
Auburn, AL 36849, USA
###### Abstract
Asteroid (3200) Phaethon, a B-type asteroid, has been active during its
perihelion passages. This asteroid is considered to be a source of the Geminid
meteor stream. It is reported that this asteroid is spinning at a rotation
period of $3.60\ hr$ and has a top shape (an oblate body with an equatorial
ridge) with a mean equatorial diameter of $6.25\ km$. Here, we report that
Phaethon’s rotation state may be close to or above its critical rotation
period when the bulk density is $0.5\ -\ 1.5\ {g/cm^{3}}$ (a typical bulk
density of a B-type asteroid). We found that in this condition, the structure
of Phaethon is sensitive to failure unless the cohesive strength is ${\sim}50\
Pa\ -\ {\sim}260\ Pa$. This result implies that if there are some surface
processes driven by, for example, thermal waves, large-scaled deformation may
happen and cause mass shedding. From this interpretation, we propose the
processes that produced the Geminid meteor stream in the past and dust tails
recently. Phaethon initially rotated at a spin period shorter than the current
period. The magnitude of structural deformation at this stage was higher than
the present spin condition, and a large mass shedding event, i.e., the Geminid
meteor stream, occurred. After this deformation process, the body became more
oblate, and its spin slowed down. At this point, while the spin was high
enough for the body to have mass shedding events, the magnitude of these
events became small.
comets: general — meteorites, meteors, meteoroids — minor planets, asteroids:
general — minor planets, asteroids: individual (3200 Phaethon)
††journal: ApJL
## 1 Introduction
Asteroid (3200) Phaethon may be a source of the Geminid meteor stream
(Whipple, 1983; Gustafson, 1989; Williams & Wu, 1993) and has been observed
for a long time (Cochran & Barker, 1984; Chamberlin et al., 1996; Hsieh &
Jewitt, 2005; Wiegert et al., 2008). In 2009 (Jewitt & Li, 2010) and 2012 (Li
& Jewitt, 2013), dust tails from Phaethon were observed during the perihelion
passage, revealing that this asteroid is indeed an active asteroid. However,
the activities detected near the perihelion left a mystery of the production
of the Geminid meteor stream. The dust mass inferred from the observations is
${\sim}3\times 10^{5}\ kg$ (Li & Jewitt, 2013), which is much smaller than the
estimated mass of the Geminid meteor stream, ${10}^{12}-{10}^{13}\ kg$ (Hughes
& McBride, 1989; Jenniskens, 1994). Also, the estimated average mass-loss rate
of $3\ {kg}/s$ is too small to replenish the Geminid meteor stream mass
(Jewitt et al., 2013), if the dynamical lifetime of the Geminid meteor stream
is ${\sim}10^{3}\ yr$ (Gustafson, 1989; Ryabova, 2007). While most of the
proposed mechanisms were found incapable of producing dust tails, thermal
fracture and cracking due to dehydration in surface materials might be
reasonable processes to generate dust tails (Jewitt & Li, 2010; Jewitt, 2012;
Li & Jewitt, 2013).
Radar observations during the 2017 apparition revealed Phaethon’s shape.
Taylor et al. (2019) reported that Phaethon might be an oblate shape with an
equatorial ridge, or the so-called top shape. The equivalent diameter of this
asteroid was estimated to be $6\ km$ in diameter. The examples of top-shaped
asteroids are OSIRIS-REx’s target (101955) Bennu (Lauretta et al., 2019) and
Hayabusa2’s target (162173) Ryugu (Watanabe et al., 2019; Sugita et al., 2019;
Kitazato et al., 2019). Phaethon is currently spinning at a rotation period of
$3.60\ hr$ (Taylor et al., 2019). The radar albedo is reported to be the
lowest among the cataloged near-Earth asteroids (Taylor et al., 2019),
implying that its spectral type is consistent with a B-type (Taylor et al.,
2019), and thus the bulk density may be as low as ${\sim}1.0\ g/{cm}^{3}$
(Scheeres et al., 2019; Watanabe et al., 2019).
We hypothesize that the mass shedding activities of Phaethon may have been
enhanced by fast rotation, given recent work arguing that the equatorial
ridges of top-shaped asteroids were evolved by rotationally driven reshaping
(Walsh et al., 2008, 2012). We propose that structural failure on the surface
and/or inside the body triggered by fast rotation plays an important role in
mass shedding mechanism. This study provides better insights into the physical
activities of Phaethon to support DESTINY+, a planned flyby mission concept
led by the Japan Aerospace Exploration Agency (Arai et al., 2018).
## 2 Semi-analytical model for structural failure in a top-shaped body
Phaethon was reported to be a top-shaped body. Recent work has shown that the
global failure condition in uniformly rotating top shapes can be roughly
determined by assuming that they are triaxial ellipsoids (Hirabayashi, 2015).
For Phaethon, the shape is not well known at present while an effort in
analyzing it from radar observation data is being made (Taylor, personal
communication). Therefore, a simplified model that uses a triaxial ellipsoid
can still reasonably provide structural failure in this asteroid. We note that
the heterogeneity in structural failure may be critical once the detailed
shape of this asteroid is considered (Hirabayashi & Scheeres, 2019).
We consider that Phaethon has a mean equatorial diameter of $6.25\ km$ and an
oblateness (the ratio of semi-minor axis to semi-major axis) of $0.889$, which
is the same as Bennu’s (Barnouin et al., 2019). We note that Taylor et al.
(2019) did not specify the oblateness of Phaethon but implied that it would be
similar to that of Bennu. For the oblateness, we do not account for the semi-
intermediate axis to simplify the discussion. We assume that Phaethon’s
structure is uniform because the internal condition is unknown. We consider
three types of the bulk density, $0.5\ g/{cm}^{3}$, $1.0\ g/{cm}^{3}$, and
$1.5\ g/{cm}^{3}$, the averaged of which is consistent with that of a B-type
asteroid (Scheeres et al., 2019). Later, we denote the oblateness, the bulk
density, and the gravitational constant as $\epsilon$, $\rho$, and $G$,
respectively. Phaethon is assumed to be rotating along the maximum principal
axis. We define a three-dimensional Cartesian coordinate system such that the
$z$ axis is lined up along the rotation axis, and the $x$ and $y$ axes are
along the maximum and intermediate moment of inertia axes, respectively, on
the equatorial plane. Using these definitions, we compute how Phaethon
experiences structural failure at a give rotation period, $P$.
To determine the failure condition of Phaethon, we extend the technique by
Hirabayashi (2015), who only considered a sphere. We analyze when the stress
field in a given element reaches its yield condition. In this model, the
material behavior is assumed to be elastic-perfectly plastic, where a plastic
flow begins once the stress reaches its yield condition without material
hardening and softening. To describe such a material behavior, we use the
following material properties: Poisson’s ratio, $\nu$; Young’s modulus, $E$;
the angle of friction, $\phi$; and the cohesive strength, $Y$. It is worth
noting that the evolution of plastic deformation is a function of loading
paths. However, our purpose is not to track plastic deformation but to
determine what element would first have its plastic state in a quasi-static
condition where deformation is small enough that the time-variation is
negligible (Hirabayashi, 2015).
Although we follow the terminologies of the cohesive strength by Hirabayashi
(2015), we reintroduce them here to facilitate the following discussion. We
compute the minimum cohesive strength that can prevent structural failure of a
given element in an asteroid rotating at $P$. We call this strength ‘critical
cohesive strength’ and denote it as $Y^{*}$. On the other hand, we use ‘actual
cohesive strength’ as an assumed strength that an asteroid may have. We denote
this as $Y$. Also, the critical rotation period $P_{c}$ is the rotation period
at which a small particle on the equatorial surface of Phaethon gains the
centrifugal acceleration larger than the gravitational acceleration and is
lifted off from the surface.
### 2.1 Stress field computation
Similar to Hirabayashi et al. (2015) and Hirabayashi (2015), we apply a
technique by Dobrovolskis (1982) and Holsapple (2001) to provide an elastic
stress in a triaxial ellipsoid that is uniformly spinning at rotation period
of $P$. Here, while noting that the details are found in Dobrovolskis (1982),
we briefly introduce the formulation process. The displacement $u$ in
Cartesian coordinates can be expressed in terms of twelve unknown constants
$A$ through $L$:
$u_{x}=x\left[A+B\frac{x^{2}}{a^{2}}+C\frac{y^{2}}{b^{2}}+D\frac{z^{2}}{c^{2}}\right],$
(1)
$u_{y}=y\left[E+F\frac{x^{2}}{a^{2}}+G\frac{y^{2}}{b^{2}}+H\frac{z^{2}}{c^{2}}\right],$
(2)
$u_{z}=z\left[I+J\frac{x^{2}}{a^{2}}+K\frac{y^{2}}{b^{2}}+L\frac{z^{2}}{c^{2}}\right].$
(3)
where $a$, $b$, and $c$ $(a\geq b\geq c)$ are the principle semi-axes of the
triaxial ellipsoid. The strain is obtained by:
$\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right).$ (4)
Applying Hooke’s law, the stress tensor is also expressed in terms of $A$
through $L$:
$\sigma_{ij}=\lambda\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij},$ (5)
where $\epsilon_{kk}=\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}$, and
$\delta_{ij}$ is the Kronecker delta. $\lambda$ and $\mu$ are the Lame’s
constants obtained from
$\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)},$ (6) $\mu=\frac{E}{2(1+\nu)}.$ (7)
To determine the twelve unknown constants, we must impose twelve linearly
independent relations. The stresses $\sigma_{ij}$ in a body in equilibrium
under body forces must satisfy the stress equilibrium equations:
$\frac{\partial}{\partial x_{j}}\sigma_{ji}=-\rho b_{i},$ (8)
where $b_{i}$ is the body force, which is driven by gravitational and
rotational effects in our problem. Equation (8) provides three constraints out
of the twelve required relations. Remaining nine relations are then imposed
from the traction free boundary condition:
$\sigma_{ij}n_{j}=0,$ (9)
where $n_{j}$ is the unit normal to the surface and given by:
$n_{x}=\frac{x}{a^{2}w},$ (10)
$n_{y}=\frac{y}{b^{2}w},$ (11)
$n_{z}=\frac{z}{c^{2}w},$ (12)
where
$w=\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}+\frac{z^{2}}{c^{4}}\right)^{1/2}$
(13)
The twelve unknown constants are, hence, provided by twelve linearly
independent relations.
### 2.2 Structural failure condition
Once the stress field is obtained from the previous section, we use it to
determine the structural failure condition of a given element. Here, we apply
the Drucker-Prager yield criterion, a smooth approximation of the Mohr-Coulomb
yield criterion (Chen & Han, 1988):
$f=\alpha I_{1}+\sqrt{J_{2}}-s\leq 0.$ (14)
$I_{1}$ and $J_{2}$ are the stress invariants:
$I_{1}=\sigma_{1}+\sigma_{2}+\sigma_{3},$ (15)
$J_{2}=\frac{1}{6}\\{(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\\},$
(16)
where $\sigma_{i}$ ($i=1,2,3$) is the principal stress component, which can be
obtained from the derived stress field above. $\alpha$ and $s$ are material
constants and given by (Chen & Han, 1988):
$\alpha=\frac{2\sin{\phi}}{\sqrt{3}(3-\sin{\phi)}},$ (17)
$s=\frac{6Y\cos{\phi}}{\sqrt{3}(3-\sin{\phi)}}.$ (18)
From the equal condition of Equation (14), we obtain the following expression
for $Y^{*}$:
$Y^{*}=\frac{\sqrt{3}(3-\sin{\phi})}{6\cos{\phi}}\left(\alpha
I_{1}+\sqrt{J_{2}}\right).$ (19)
If Equation (19) becomes negative, we consider $Y^{*}$ to be $0\ Pa$, which
means that no strength is necessary for an element to keep the original shape.
## 3 Results
We investigate the critical cohesive strength based on the following assumed
parameters, $\nu=0.25$, $E=10^{7}\ Pa$, and $\phi=35\degree$, to make our
discussion consistent with earlier work (e.g., Hirabayashi & Scheeres 2015).
We note that Young’s modulus does not influence our stress field calculation
(Love, 2013), and the variations in Poisson’s ratio and the angle of friction
do not affect our results for geological materials significantly (Lambe &
Whitman, 1969; Hirabayashi & Scheeres, 2019). The current rotation period is
set to be $3.60\ hr$ (Taylor et al., 2019). We consider the bulk density to be
$0.5$, $1.0$, and $1.5\ g/{cm}^{3}$. Table 1 lists the parameters considered
in the current study.
We find that the failure mode varies with the bulk density. We plot the
distribution of $Y^{*}$ at the $x$-$z$ plane for different bulk densities in
Figure 1. Panels a, b, and c describe the bulk densities of $0.5$, $1.0$, and
$1.5\ g/{cm}^{3}$, respectively. In this range of the bulk density, the body
should have a cohesive strength to keep the original shape. For the case of
$\rho=0.5\ g/{cm}^{3}$, $P_{c}$ is found to be $4.83\ hr$. Therefore, the
current rotation period is shorter than $P_{c}$, indicating that materials
should be shed and highly sensitive to structural failure. Figure 1a shows
$Y^{*}>0$ everywhere except the pole region. $Y^{*}$ is higher in the interior
than on the surface, and its maximum value is $259\ Pa$ at the center. This
indicates that the interior is more sensitive to structural failure than the
surface. If the actual cohesive strength $Y$ is smaller than $Y^{*}$, the
central region structurally fails first. For the case of $\rho=1.0\
g/{cm}^{3}$, $P_{c}$ is found to be $3.42\ hr$ and is shorter than the current
period. However, the interior still exhibits high $Y^{*}$ in the major regions
(Panel b). The maximum value of $Y^{*}$ is $179\ Pa$ and is located at the
center. For the case of $\rho=1.5\ g/{cm}^{3}$, $P_{c}$ is found to be $2.79\
hr$. Unlike the other two cases, the interior has $Y^{*}=0$ in the most areas;
however, high $Y^{*}$ is still distributed beneath the surface (${\sim}50\
Pa$)(Panel c).
All these three cases show the sensitivity of Phaethon to structural failure.
This body needs a cohesive strength to keep the original condition without
shape deformation. However, if there is a trigger of reshaping, it is likely
that the deformation process would be enhanced by rotation, as seen from the
derived sensitivity. Because of the observed activities, the cohesive strength
of Phaethon is ${\sim}50\ Pa\ -\ {\sim}260\ Pa$, depending on the bulk
density. This range is consistent with that of observed small bodies (Scheeres
& Sánchez, 2018).
Table 1: Parameter settings Parameter | Symbol | Value | Units
---|---|---|---
Gravitational constant | G | 6.6738 ×10^-11 | m^3 ⋅kg^-1 ⋅s^-2
Semi-major axis | a | 3200 | m
Semi-minor axis | c | 2847 | m
Oblateness | ϵ | 0.889 | -
Current rotation period | P | 3.60 | hr
Bulk density | ρ | 0.5, 1.0, 1.5 | g ⋅cm^-3
Poisson’s ratio | ν | 0.25 | -
Elastic modulus | E | 10^7 | Pa
Friction angle | ϕ | 35 | deg
Figure 1: Distribution of $Y^{*}$ at the $x$-$z$ plane. The rotation period is
$3.60\ hr$. Panels a, b, and c describe bulk densities of $0.5$, $1.0$, and
$1.5\ g/{cm}^{3}$, respectively.
## 4 Discussions
Generation of dust tails at present
During perihelion passage in 2009 and 2012, the observations of Phaethon’s
dust tails revealed that this asteroid was an active asteroid (Jewitt & Li,
2010; Li & Jewitt, 2013). While the detailed mechanisms are not well known,
thermal waves may be one of plausible explanations for the reported mass
shedding activities (Li & Jewitt, 2013).
At the current rotation period of $3.60\ hr$, the body is sensitive to
structural failure regardless of the bulk density and thus needs cohesive
strength to maintain its shape. The derived cohesive strength of Phaethon is
less than ${\sim}50\ Pa\ -\ {\sim}260\ Pa$, which is consistent with that of
small bodies ranging up to a few hundred pascals (Scheeres & Sánchez, 2018).
We interpret this sensitivity as a potential enhancement of mass shedding. If
there is a trigger of reshaping even at small scales, the structure of Phaeton
would be perturbed, leading to rotationally driven reshaping at larger scales.
Such a trigger can be thermal waves in thin surface layers (Jewitt & Li, 2010;
Jewitt, 2012; Li & Jewitt, 2013). Micrometeoroid impacts or other processes
may also be possible, as seen on Bennu (Lauretta et al., 2019), although
thermal waves, again, are more consistent to explain the activities of
Phaethon around its perihelion passage (Li & Jewitt, 2013).
Possible source of the Geminid meteor stream
We expect that rotationally induced structural failure makes Phaethon more
oblate, i.e., $\epsilon$ becomes lower (Hirabayashi, 2015). Because the
angular momentum is constant during deformation, Phaethon may have been less
oblate and rotated faster at an earlier stage before it had large deformation.
Figure 2 shows the dependence of $P$ on $\epsilon$ and $P_{c}$ for different
bulk densities. We find that if Phaethon is less oblate, the rotation period
is shorter, and thus $Y^{*}$ should be higher at a shorter rotation period. If
Phaethon is a sphere ($\epsilon=1.0$; $a=b=c$), which is the shape condition
that is the least affected by rotation, the rotation period should become
$P=3.38\ hr$. Figure 3 shows the distribution of $Y^{*}$ for this case.
Similar features observed in Figure 1 are evident in this case; however,
$Y^{*}$ is higher, implying that Phaethon should have been more sensitive to
structural failure at a shorter rotation period. Therefore, the failure mode
may become severer, and more materials should be shed significantly at a
shorter rotation period.
Figure 2: Rotation period as a function of the oblateness (the blue solid
line). $P=3.60\ hr$ when $\epsilon=0.889$ and $P=3.38\ hr$ when
$\epsilon=1.0$. The dotted lines indicate $P_{c}$ for different bulk
densities: the red, green, and magenta show bulk densities of $0.5$, $1.0$,
and $1.5\ g/{cm}^{3}$, respectively.
Figure 3: Distribution of $Y^{*}$ at the $x$-$z$ plane. The rotation period is
$3.38\ hr$. Panels a, b, and c describe bulk densities of $0.5$, $1.0$, and
$1.5\ g/{cm}^{3}$, respectively.
From these results, we propose a possible evolution scenario of Phaethon
(Figure 4). Phaethon was originally less oblate and spinning at a shorter
rotation period than the current period. This stage is before the Geminid
meteor stream was generated. Possible initiation processes such as thermal
waves during Phaethon’s apparition passages triggered reshaping, and
rotational deformation enhanced this reshaping process significantly. Because
the rotation period at this stage was closer to or above $P_{c}$, the
reshaping process caused mass shedding at large scale, which became a source
of the Geminid meteor stream. Thus, the current oblate shape maybe a remnant
of this large deformation event. When the oblateness evolved, the rotation of
Phaethon slowed down. However, the structure was still sensitive to failure.
When there is similar perturbation such as thermal waves at present,
rotationally driven failure can be triggered; however, because the centrifugal
effect is less significant, the magnitude of mass shedding is less intense
than that in the past.
While we cannot strongly constrain whether the generation of the Geminid
meteor stream was a single event or episodic, our study gives some hints of
large-scaled mass shedding processes as a source of the Geminid meteor stream.
Because the generation of the Geminid meteor stream may have occurred within
the last $1\ ka$ (Gustafson, 1989; Williams & Wu, 1993; Ryabova, 2007), the
large-scaled reshaping and mass shedding processes may have occurred in this
timescale. These processes should be more rapid and intense to be completed
(within the last $1\ ka$) than the YORP driven evolution, which may take
${\sim}1\ Ma$ based on earlier work (Čapek & Vokrouhlický, 2004; Bottke et
al., 2006). Given this short timescale, a possible explanation of the
deformation mode may be internal failure, which can provide large-scaled
deformation and thus mass shedding at fast rotation (Hirabayashi, 2015).
We note that the total mass of the Geminid meteor stream is about $2.5\%$ of
that of Phaethon. If a mass shedding even that can produce the same magnitude
of the Geminid meteor stream occurs every $1ka$, the lifetime of Phaethon
would only be ${\sim}40\ ka$, which may be much shorter than that predicted by
the orbital evolution, ${\sim}26\ Ma$ (De León et al., 2010). While there is
no decisive evidence, the YORP effect may give some insights into this
discrepancy. The YORP evolution timescale of Phaethon may become longer by
many different factors such as the mass, shape, surface composition, and
stochastic evolution (Bottke et al., 2015). Furthermore, its highly eccentric
orbit, $e=0.890$, with small perihelion distance of 0.14 AU (JPL Small-Body
Database) may give strong variations in solar radiation acting on Phaethon.
Thus, it may be possible that Phaethon has stochastically spun up by the YORP
effect for the entire orbital age, ${\sim}26Ma$ (De León et al., 2010), and it
recently experienced large-scaled mass shedding that formed the Geminid meteor
stream. Then, a large mass shedding event that produced the Geminid meteor
stream may have decelerated Phaethon’s spin, but the spin state after this
event may have been still high enough to have some mass shedding events at
small level, similar to what we observed in 2009 and 2012 (Jewitt & Li, 2010;
Jewitt, 2012; Li & Jewitt, 2013). To fully address the detailed timescale of
the rotationally driven reshaping process is beyond the scope of this study;
we leave this problem as future work.
Finally, We assumed that Phaethon currently has a top shape having Bennu’s
oblateness, $\epsilon=0.889$, by following Taylor et al. (2019). To check if
this setting is consistent with other top-shaped asteroids, we consider six
top-shaped near-earth asteroids: Ryugu (Watanabe et al., 2019), 1994 KW4
(Ostro et al., 2006), 2008 EV5 (Busch et al., 2011), 1994 CC (Brozović et al.,
2011), 2001 SN263 (Becker et al., 2015), and 2000 DP107 (Naidu et al., 2015).
We find the range of $\epsilon$ to be $0.873$ – $0.968$, and thus Bennu’s
$\epsilon$ is within this range. We conduct the same analyses for these
objects and found trends similar to Bennu’s. The variation in the oblateness
at this magnitude does not affect the results of $Y^{*}$ significantly, which
is less than 12%. Therefore, we conclude that our oblateness setting is
meaningful to capture a possible scenario of Phaethon’s activities.
Figure 4: Possible evolution scenario of Phaethon. Phaethon has a highly
eccentric orbit ($e=0.890$) with small perihelion distance of 0.14 AU. At an
earlier stage, Phaethon was less oblate and rotating faster, leading to mass
shedding at larger scale. For both stages, perturbation such as thermal waves
may be a possible trigger for mass shedding.
Potential issues
We finally address issues of our analysis approach. This study explored
Phaethon’s rotationally induced structural failure by modeling Phaethon as a
triaxial ellipsoid. We did not account for local topographic features;
therefore, our semi-analytical model does not capture local deformations which
may differ from global deformations (Hirabayashi & Scheeres, 2019; Hirabayashi
et al., 2019). However, Hirabayashi (2015), who compared an analytical
solution and a FEM solution, found that there was no significant variation
between the two. Therefore, we conclude that our semi-analytical model can
provide solutions with reasonable accuracy. It is our future work to perform
FEM analysis using a shape model to investigate the local deformations.
Another issue to be addressed is that our model does not assert the detailed
failure mode. While numerous research studies have been undertaken, it is
still not well determined how top shape asteroids deform. Hirabayashi (2015)
showed that depending on the internal structure and the bulk density, the
failure mode may become different – either surface processing or internal
deformation. However, because our semi-analytical model can only describe a
homogeneous structure, we cannot infer how Phaethon’s deformation is
controlled by the heterogeneity. Therefore, the detailed failure mode cannot
be specified. If Phaethon’s heterogeneous structure is revealed, we need to
employ different approach (i.e. Hirabayashi et al. 2015).
There still remain many unknowns regarding Phaethon’s physical properties. We
will further elaborate our approach to give constraints on the activity of
Phaethon. Concurrently, further constraints are vitally important to support
DESTINY+.
RN and MH acknowledge support from NASA/Solar System Workings
(NNH17ZDA001N/80NSSC19K0548) and Auburn University/Intramural Grant Program.
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|
2024-09-04T02:54:58.036023 | 2020-03-07T03:44:21 | 2003.03508 | {
"authors": "Marnus Stoltz, Gene Stoltz, Kazushige Obara, Ting Wang, David Bryant",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26095",
"submitter": "Stephanus Marnus Stoltz",
"url": "https://arxiv.org/abs/2003.03508"
} | arxiv-papers | # A 1000-fold Acceleration of Hidden Markov Model Fitting using Graphical
Processing Units, with application to Nonvolcanic Tremor Classification.
Marnus Stoltz1 Gene Stoltz2,3 Kazushige Obara4 Ting Wang1 David Bryant1
$1.$ Department of Mathematics and Statistics, University of Otago, New
Zealand
$2.$ Council for Scientific and Industrial Research of South Africa, Pretoria,
South Africa
$3.$ Department of Electronic Engineering, University of Johannesburg, South
Africa
$4.$ Earthquake Research Institute, University of Tokyo, Japan
$*$ Corresponding author<EMAIL_ADDRESS>
###### Abstract
Hidden Markov models (HMMs) are general purpose models for time-series data
widely used across the sciences because of their flexibility and elegance.
However fitting HMMs can often be computationally demanding and time
consuming, particularly when the the number of hidden states is large or the
Markov chain itself is long. Here we introduce a new Graphical Processing Unit
(GPU) based algorithm designed to fit long chain HMMs, applying our approach
to an HMM for nonvolcanic tremor events developed by Wang et al. (2018). Even
on a modest GPU, our implementation resulted in a 1000-fold increase in speed
over the standard single processor algorithm, allowing a full Bayesian
inference of uncertainty related to model parameters. Similar improvements
would be expected for HMM models given large number of observations and
moderate state spaces (<80 states with current hardware). We discuss the
model, general GPU architecture and algorithms and report performance of the
method on a tremor dataset from the Shikoku region, Japan.
Keywords— Bayesian inference, Computational hardware, Seismology, Algorithm
design.
## missingum@section Introduction
Slow slip events (SSEs), a type of slow earthquakes, play an important role in
releasing strain energy in subduction zones, the region where one tectonic
plate moves underneath another tectonic plate and sinks. It is currently
understood that SSEs occur as shear slips on the bottom tip of subduction
zones that transition between a fixed region above and slipping region below
(Beroza and Ide, 2011). Recent evidence suggest that nonvolcanic tremors are
observed in close association with SSEs, however the causal relationship
between the two phenomena is not yet well understood. Classifying nonvolcanic
tremors helps to better understand this link but can be time consuming when
typically done by hand.
Recently, an automated procedure was developed by Wang et al. (2018) to
classsify spatio-temporal migration patterns of nonvolcanic tremors. The
procedure classifies tremor source regions into distinct segments in 2-D space
using a Hidden Markov Model. The model is fitted using the Expectation
Maximisation algorithm. Here we implement a Bayesian approach. However,
fitting the model in either a frequentest or Bayesian framework is extremely
demanding computationally, often taking days or weeks for large dataset with
moderate state space. Fortunately, technological advances in hardware have the
potential to solve this issue. Specifically, we make use of fast and
affordable graphic processing units (GPUs).
In recent years HMM algorithms on GPUs have been implemented in various
fields. A non-exhaustive list includes implementations in bioinformatics (Yao
et al., 2010), speech recognition (Yu et al., 2015), a registered patent in
speech matching (Chong et al., 2014) and workload classification (Cuzzocrea et
al., 2016), as well as HMMer (Horn et al., 2005) an open-source project for
use with protein databases. The HMM implementations are application specific
often with large number of states and mostly focused on increasing throughput
of the Verterbi and Baum-Welch algorithms (Zhang et al., 2009; Li et al.,
2009; Liu, 2009). This leads to a range of concurrent approaches. Here we
focus on the efficient implementation of the forward algorithm of an HMM model
given a large number of observations and a moderate number of states.
The outline of the paper is as follows: In Section 2 we describe the HMM for
classifying nonvolcanic tremors and discuss the likelihood algorithm in a
serial and parallel context. Thereafter we give details on the OpenCL
implementation of the parallel likelihood algorithm. In Section 3 we discuss
performance of the OpenCL implementation and compare it to the standard
Forward algorithm. In Section 4 we report our analysis on a large tremor
dataset from the Shikoku region, Japan.
## missingum@section An HMM for classifying nonvolcanic tremors
Nonvolcanic tremor activity is clustered spatially and each spatial cluster
seems to recur episodically. To represent this phenomenon using an HMM, Wang
et al. (2018) introduce one hidden state for each spatial cluster. The tremors
themselves (including the absence of a tremor) are the observations. The
frequency and spatial distribution of tremors changes according to the hidden
state.
More formally, we suppose that the observations of nonvolcanic tremors are a
sample path of a stochastic process
$\\{X_{i}\\}_{i=0,\dots,N}$
with observations represented in the state space
$I=\\{\emptyset,\mathbb{R}^{2}\\}$
generated under an HMM with $K$ numbered hidden states. For each hidden state
$k=1,\dots,K$ we introduce parameters $p_{k}$, $\bm{\mu}^{(k)}$ and
$\mathbf{\Sigma}^{(k)}$, where $p_{k}$ is the probability of observing a
tremor and $\bm{\mu}^{(k)}$, $\mathbf{\Sigma}^{(k)}$ are the mean and variance
of a bivariate normal distribution modelling where a tremor is likely to
occur, if it does occur.
To simplify notation we introduce for each observation $\mathbf{x}$ a $K\times
K$ diagonal matrix $\mathbf{P}(\mathbf{x})$, also called the emission matrix,
with the $k$th diagonal element corresponding to the probability of observing
$\mathbf{x}$ given state $k$
$\bm{P}(\bm{x})_{kk}=\begin{cases}p_{k}\phi(\bm{x}|\bm{\mu}^{(k)},\mathbf{\Sigma}^{(k)})\\\
1-p_{k}.\end{cases}$ (1)
Here $\phi(.)$ is the density function of bivariate normal distribution. Let
$\mathbf{\Gamma}=(\Gamma_{ij})$ denote the $K\times K$ transition matrix of
the HMM, where $\Gamma_{ij}$ indicate the the transition probability from
hidden state $K=i$ to $K=j$. Also, let
$\bm{\delta}=\delta_{1},\dots,\delta_{K}$ denote the vector of probabilities
for the initial state.
Now the likelihood function for the parameters given the observed data can be
written as
$\mathrm{L}\big{(}\mathbf{\Gamma},\bm{\delta},\\{p_{k},\bm{\mu}^{(k)},\mathbf{\Sigma}^{(k)}\\}_{k=1,\ldots,K}|\mathbf{x}_{0},\ldots,\mathbf{x}_{N}\big{)}=\bm{\delta}^{T}\mathbf{\Gamma}\mathbf{P}(\mathbf{x}_{0})\dots\mathbf{\Gamma}\mathbf{P}(\mathbf{x}_{N})\mathbf{1}.$
(2)
## missingum@section GPU computing framework
GPUs have had a large impact across statistical and computing sciences due to
cost-effect parallelism (Kindratenko, 2014). However in order to translate an
algorithm from CPU to GPU some careful consideration is needed in terms of
1. 1.
Reducing latency (how to concurrently execute instructions on GPU in order to
optimise data throughput.)
2. 2.
Managing memory (how to effectively distribute and utilise memory across
processors to avoid bandwidth bottlenecks).
3. 3.
Designing robust algorithms with respect to varying GPU architecture between
models and vendors as well as the rapidly changing landscape of computational
hardware.
Frameworks like OpenCL and CUDA, allow programmers to implement GPU algorithms
with some level of generality. The implementation we describe here was carried
out in the OpenCL framework. OpenCL is an open standard maintained by the non-
profit technology consortium Khronos Group, see https://www.khronos.org for
more details on the non-profit organisation.
The OpenCL framework consists of a host (CPU; terms in brackets relate to
computation on GPU architecture) controlling one or more compute devices (We
just used one GPU). Each compute device (GPU) is divided into compute units
(streaming multiprocessors). Compute units are further divided into compute
elements (microprocessors or cores). Each compute unit has access to global
memory of the compute device. This access though is slow. Each compute unit
also has a shared memory to allow efficient data exchange between compute
elements. Each compute element has exclusive access to private memory
(registers) for computation.
## missingum@section The likelihood algorithm
### 4.1 Overview
Our implementation will work well on a range of GPU models. For our studies we
used a NVIDIA GeForce GTX 1080 Ti GPU with 28 compute units (streaming
multiprocessors) each with 48KB of shared memory, 128 compute elements (cores)
and a register file that can contain up to 32,768 32-bit elements distributed
across the compute elements (cores). For the host we used an Intel Core
i7-7700K CPU at 4.20GHz. There are two main limitation for the OpenCL
algorithm in terms of hardware specifications
1. 1.
The number of registers per compute element.
2. 2.
The size of shared memory on a compute unit.
For example given the hardware described above we have (32,768/128)=256
registers per compute element. This implies that we can store up to roughly
200 32-bit matrix elements on a compute element (we also need some registers
left to store counters and other meta variables). Our implementation assumes
that at least two matrix rows can fit into the registers of a compute element.
This gives an upper limit for the number of hidden states of $K<100$. In order
to efficiently distribute rows of a matrix and update matrix elements we need
space for two matrices in the shared memory of the compute unit. Our
configuration has 48KB of shared memory per compute unit. Implying that we can
fit a total of $(48\cdot 2^{10})/4=12288$ 32-bit matrix elements per compute
unit. This gives a second upperlimit for number of hidden states of $K<80$. To
handle a large number of states, alternative parallel computing strategies
should be used (Horn et al., 2005; Yu et al., 2015).
First we consider how the algorithm for the likelihood (2) would be
implemented on a single processor unit. To avoid matrix-matrix multiplications
we would start with the stationary vector $\bm{\delta}$ on the left, and then
sequentially multiply that by transition matrices and emission matrices:
Algorithm 1 The Forward algorithm on a CPU
1:procedure Compute-
likelihood($\\{p_{k},\bm{\mu}^{(k)},\mathbf{\Sigma}^{(k)}\\}_{k=1,\ldots,K},\\{\mathbf{x}_{0},\dots,\mathbf{x}_{N}\\}$)
2: $\bm{v}\leftarrow\bm{\delta}^{T}$
3: for $k$ from $0$ to $N$ do
4: Compute $\bm{P}(x_{k})$
5: $\bm{v}\leftarrow\bm{v}\bm{\Gamma}$
6: $\bm{v}\leftarrow\bm{v}\bm{P}(x_{k})$
7: return $\bm{v}\bm{1}$
Running time will be dominated by the matrix-vector multiplication in steps 5
and 6, taking $\mathcal{O}(K^{2})$ time per iteration. Hence the running time,
or work, for this implementation is $\mathcal{O}(NK^{2})$. Next we compare it
with the parallel implementation.
The overview of our implementation is as follows:
1. 1.
We compute all of the emission matrices $\bm{P}(x_{0}),\ldots\bm{P}(x_{N})$ in
parallel
2. 2.
We then multiply the emission matrices by the transition matrices, all in
parallel storing N matrices
$\bm{\Gamma}\bm{P}(x_{0}),\dots,\bm{\Gamma}\bm{P}(X_{N})$.
3. 3.
Instead of computing $\bm{\Gamma}\bm{P}(x_{0}),\dots,\bm{\Gamma}\bm{P}(X_{N})$
as a single sequence of vector-matrix multiplications, we multiply the
matrices $(\bm{\Gamma}\bm{P}(x_{0})),\dots,(\bm{\Gamma}\bm{P}(X_{N}))$
together.
This increases the work done: we are carrying out matrix-matrix
multiplications instead of matrix-vector multiplications, but it allows us to
spread the computation over multiple processors. We now discuss steps (1) to
(3) in greater detail.
### 4.2 Step 1: The emission probability evaluation on GPU
The goal in this step is to compute the emission matrices $\mathbf{P}(x_{i})$
for each observation $x_{i}$. The emission probability is defined by (1) and
makes use of the parameters $p_{k},\bm{\Sigma}_{k},\bm{\mu}_{k}$ for each
hidden state $k$. These parameters are initially copied to the registers of
each core and remains there until all the datapoints have been evaluated. The
compute elements work in parallel. Each is allocated a data point $x_{i}$,
uses the stored values to compute $\bm{P}(x_{i})$ and copies the diagonal
matrix computed to global memory. Note that a compute element can request and
copy the next data point at the same time as it processes the current data
point.
For this step there is no data sharing between compute elements, allowing for
data-level parallelism. Therefore it is more efficient to allow compute device
compiler to optimise the work-load scheduling and data transfer between
compute units in order to fully utilise SIMD (Single instruction multiple
data) instructions. Output from compute elements are collected and copied to
global memory to form the list of new inputs
$\\{\mathbf{\Gamma},\mathbf{P}(x_{0}),\dots,\mathbf{P}(x_{N})\\}$ for the next
kernel.
### 4.3 Step 2: The transmission-emission matrix multiplication on GPU
During the next step we compute $\bm{\Gamma}\bm{P}(x_{i})$ for all data points
$x_{i}$, again in parallel. At this point we run into limitations with memory.
While the register of a single compute element is large enough to store the
diagonal matrix $\bm{P}(x_{i})$, it is not large enough to store the full
transition matrix $\bm{\Gamma}$ nor the product matrix
$\bm{\Gamma}\bm{P}(x_{i})$. The solution is to break down the multiplication
of $\bm{\Gamma}$ and $\bm{P}(x_{i})$ by computing only a few rows at once.
We query the register size for each compute element to determine how many rows
of $\mathbf{\Gamma}$ can be copied. The rows remain in the register until all
data points have been evaluated. Thereafter the next set of rows is copied
into the registers and the data points is evaluated again until all the rows
of $\mathbf{\Gamma}\mathbf{P_{r}}$ for $r=0,\dots,N$ have been computed. As
$\bm{P}(x_{i})$ is diagonal, the product of rows of $\bm{\Gamma}$ with
$\bm{P}(x_{i})$ is computed by simply rescaling the corresponding columns.
The next diagonal matrix subset is requested while scaling subset for current
data point. Again, there is no data sharing between compute elements, allowing
for optimal data-level parallelism. Output from compute elements are collected
and a new list of inputs, namely $\\{(\mathbf{\Gamma
P_{0}}),\dots,(\mathbf{\Gamma P_{N}})\\}$ is compiled for the final GPU
kernel.
### Step 3: The Square Matrix-Chain Multiplication on GPU
The third step is the most time-consuming, and also the most involved. The
general idea is to avoid the long sequence of matrix vector calculations
$\bm{\delta}^{T}\mathbf{\Gamma}\mathbf{P}(x_{0})\dots\mathbf{\Gamma}\mathbf{P}(x_{N})\bm{1}$
which cannot be readily parallelized, by instead multiplying the matrices
together in parallel. Our algorithm here roughly follows (Masliah et al.,
2016).
Recall the general hierarchical structure of a GPU calculations, as described
above. The CPU controls the GPU. Each compute GPU is divided into compute
units (streamline multiprocessors). Compute units are further divided into
compute elements (microprocessors or cores). The CPU is actually faster than
the compute units for individual computations, the speed of GPUs being due to
parallelism. Our algorithm takes advantage of all three levels: The sequence
of matrices (known in computer science as a matrix chain) is divided into
multiple segments, one for each compute unit. The compute units then carry out
matrix multiplication directly, making use of multiple compute elements to
share out the rows in each matrix-matrix computation. We then use the CPU to
carry out the final sequence of matrix-vector computations, using the matrices
returned by the computational units of the GPU.
Note that in practise we compute $\log$L rather than L and shift the registers
either up or down using the scale coefficients from compute units to avoid
underflow.
## missingum@section Performance assessment of OpenCL implementation
### 5.1 OpenCL algorithm vs Forward algorithm
One of the factors that influence the use of an algorithm on GPUs is whether
it is actually faster than a Forward algorithm. To check this we compare
computational times of the GPU algorithm with the Forward algorithm from the
software library Tensor flow. First we fixed the number of HMM states to
$K=25$ while increasing the number of datapoints over a range of magnitude
orders $N=10^{2},\dots,10^{5}$. Thereafter we fixed the number of datapoints
to $N=100,000$ and increased the number of HMM states for $K=5,10,\dots,50$.
In each case model parameters were drawn from the prior disribution (discussed
in the next section) and thereafter data was simulated using the R software
package in Wang et al. (2018). The results are shown in Figure 1 and Figure 2.
We see that the GPU algorithm executes orders of magnitude faster then a
Forward algorithm.
$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$Number
of DatapointsExecution Time [ms]Execution Time for 25 StatesForwardOpenCL-GPU
Figure 1: We compare computational time of OpenCL algorithm on GPU with a
Forward algorithm on CPU. Computational time is indicated on the y-axis and
number of datapoints are indicated by the x-axis. We see that with $10^{5}$
datapoints, the GPU algorithm runs $\sim 10^{3}$ times faster.
$10$$20$$30$$40$$50$$60$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$Matrix
SizeExecution Time [ms]Execution Time for 100,000 Data pointsForwardOpenCL-GPU
Figure 2: We compare computational time of OpenCL algorithm on GPU with a
Forward algorithm on CPU. Computational time is indicated on the y-axis and
number of HMM states are indicated by the x-axis. We see that the GPU
algorithm slows down as the register capacity of compute elements is reached.
However it still outperforms the Forward algorithm by orders of magnitude.
### 5.2 Comparing execution time of matrix-chain multiplication
Here we specifically compare computation time of step 3 in the OpenCL
algorithm with matrix-chain multiplication using popular GPU BLAS (Basic
Linear Algebra Subprograms) libraries. We use subroutines from the CLBlast
library as well as the MAGMA BLAS libary to do the matrix-chain
multiplication. CLBLast is a general BLAS library in OpenCL that automatically
tunes subroutines for specific hardware based on compile time. MAGMA BLAS is a
CUDA library exclusively available for NVIDIA GPUs. We followed the same
procedure as in the previous two experiments except that we fixed the number
of HMM states to $K=50$. We show results in Figure 3 and Figure 4. Using the
MAGMA library gives roughly the same performance as the OpenCL algorithm for
small matrices. We note that using these libraries in the OpenCL algorithm is
not straightforward due to small tweaks and scaling coefficients that we keep
track of in addition to performing the matrix-chain multiplication. The
algorithm became very slow when the HMM had more than 100 states due to memory
limitations previously discussed.
$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$Data
PointsExecution Time [ms]Speed Comparison for 50 StatesCLBlastMAGMAOWN Figure
3: For this computational comparison (in miliseconds) with the BLAS libraries
we fix the number of HMM states to $K=50$ and increase the number of
datapoints over a range of magnitude orders.
$10$$20$$30$$40$$50$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$Matrix
SizeExecution Time [ms]Speed Comparison for 100000 Data PointsCLBlastMAGMAOWN
Figure 4: For this computational comparison (in miliseconds) with the BLAS
libraries we fix the number of datapoints to $N=100,000$ and increase the
number of HMM states for $K=5,10,\dots,50$.
## missingum@section Bayesian analysis of nonvolcanic tremor data
### 6.1 Monte Carlo Markov Chains
Bayesian techniques have become a popular method of statistical inference
across a broad range of sciences (Jóhannesson et al., 2016; Kruschke, 2010;
Moore and Zuev, 2005; Stoltz et al., 2019; Turner et al., 2016; Woolrich et
al., 2009). This is due to advances in numerical techniques and the
affordability of powerful computers (Andrieu et al., 2004). In a Bayesian
analysis the aim is to compute the joint posterior distribution of model
parameters, simply referred to as the posterior distribution. The posterior
distribution summarizes the uncertainty related to model parameters.
Typically due to model complexity the posterior distribution is an
analytically intractable function. However methods such as Monte Carlo Markov
Chains (MCMCs) use random walks to estimate the posterior distribution. The
basic concept behind MCMCs is that a Markov chain can be constructed with a
stationary distribution, where the stationary distribution is in fact the
posterior distribution.
An MCMC is initialized by choosing a random state, typically by drawing a
sample from the prior distribution (we discuss prior distributions below). The
MCMC is then simulated by accepting or rejecting proposed MCMC states based on
a ratio of the likelihood function and prior distribution of both the current
and proposed MCMC state. The MCMC is simulated until after the stationary
distribution is reached. Stationarity of an MCMC is assessed by looking at
trace plots of parameters as well as computing the number of effective
independent samples. Samples of the stationary MCMC is then used to
approximate the posterior distribution.
### 6.2 Model priors
Using ratios of the likelihood and prior distributions is an elegant way of
sampling from the posterior distribution. It sidesteps some nasty calculations
if we were to compute the posterior distribution directly instead. Roughly
speaking prior distributions is a way to incorporate knowledge about model
parameters before looking at the data. However prior distributions can easily
be neglected but they are in fact an important part of the model. Therefore
choosing prior distributions needs to be carefully considered and requires
some justification. For instance, it is known that tremors occur in sequence
bursts that cluster around the same area (Wang et al., 2018). This observation
we translate into the model by specifying a model prior centred around sparse
transition matrices. More formally, we specify a symmetric Dirichlet prior
with concentration parameter 0.01 on $\bm{\Gamma}$ (formulas for prior
densities are given in Appendix A). Furthermore we expect that for some hidden
states we are more likely to observe tremors than others. Therefore we specify
independent gamma distributions on state probabilities
$\\{p_{k}\\}_{k=1,\ldots,K}$ , half of the state probabilities with mean 0.1
and variance 0.001 and the other half with mean 0.9 and variance 0.001. Also,
we specify a uniform prior on hidden state means
$\\{\bm{\mu}^{(k)}\\}_{k=1,\ldots,K}$ restricted to a rectangular domain that
contains all observations. We have no prior information on the shape of the
hidden states therefore we specify an uninformative Inverse-Wishart prior on
the covariance matrices $\\{\mathbf{\Sigma}^{(k)}\\}_{k=1,\ldots,K}$ with
degrees of freedom equal to the number of states $K$ and scale matrix set to a
$K\times K$ identity matrix.
### 6.3 GPUeR-hmmer
In order to simulate MCMCs for the model, we incorporated the GPU likelihood
algorithm along with the prior distributions into a general purpose MCMC
sampler (Christen et al., 2010). This R package bundle is freely available at
https://github.com/genetica/HMMTremorRecurrencePatterns. Note that OpenCL 1.2
and Python 3.6 (or later versions) needs to be separately installed on a
system in order to support the back-end of the R package. The R package also
contains a simple example using simulated data from the HMM described in
Section 2. Additionally we provide instructions on how to modify the OpenCL
code if an HMM with a different emission function is required. In order to
assess convergence of the MCMC chains we used Tracer. For a brief tutorial on
how to use Tracer to assess convergence, see
https://beast.community/analysing_beast_output. Furthermore if some problems
are encountered with convergence of the simulated MCMCs see
https://beast.community/tracer_convergence for some recommendations.
### 6.4 Tremor dataset of the Shikoku region
We use a large tremor dataset from the Shikoku region, Japan to demonstrate
the sort of Bayesian analysis that can be done with GPUeR-hmmer. The Shikoku
region is one the three major regions in Japan (the other two being the Tokai
region and Kii region) in which nonvolcanic tremor occurrences have been
repeatedly detected. Tremor activity spans along the strike of the Phillipines
Sea plate for about 600km and the depth ranges from 30 to 45 km on the plate
interface. The original waveform data is supplied by the High Sensivity
Seismograph Network of the National institute for Earth Sciences and Disaster
prevention in Japan. The dataset analysed by Wang et al. (2018) was extracted
from the waveform data. It consists of $105,000$ data point measurements
between $2001$ and $2012$. It is hourly control measurements determined using
clustering and correlation methods described in Obara et al. (2010).
### 6.5 Model fitting
A full Bayesian analysis of the model will sample the number of hidden states
along with the rest of the model parameters. However sampling from different
parameter spaces is quite challenging and is an active and ongoing area of
research (Lunn et al., 2009). Instead we incorporate the choice of number of
hidden states $K$ into the model fitting process.
We start with a small number of hidden states and incrementally increase the
number of hidden states, while doing so we assess the posterior distribution
for each case. The posterior distribution of each model is estimated by
running the MCMC sampler for 1,000,000 iterations. Running each chain took
approximately $\sim 4$ hours.
In Figure 7 we summarize the posterior distributions for model fitted with
number of hidden states $K=5,10,\dots,30$. Typically, the background states
(i.e states that cover large areas) have the highest variance in posterior
distribution. Whereas states covering smaller areas have considerably less
variance in posterior distribution of parameters. We also see in Figure 7(d)
that parameters used in Wang et al. (2018) are recovered by the posterior
distribution. Typically as we increase the number of states some states are
divided into two, with rare new clusters. Furthermore we see for $K=30$ that
some additional hidden states ($k=4,8,26$) doesn’t fit over one particular
cluster of points, covers a large area, has a low probability of observing
tremors and a low stationary probability (i.e time spent in state). Thereafter
we also fitted models with hidden states for $K=26,27$ (see Appendix B) and we
find that additional hidden states have the same undesirable properties and
therefore use $K=25$ as our choice for number of hidden states for the model
(see MCMC summary statistics in Appendix C).
(a) $K=5$
(b) $K=10$
(a) $K=15$
(b) $K=20$
(a) $K=25$
(b) $K=30$
Figure 7: Posterior distributions of fitted models with number of hidden
states $K=5,10,\dots,30$ for tremor occurrences in Shikoku region. Ellipses
each map represent the 2D normal density of one hidden state for one sample
from the posterior distribution. States are numbered in red. Colour of an
ellipse indicate how likely a tremor will occur given the process is in the
hidden state. In the bottom right corner of each map we give the mean
transition matrix of the posterior distribution. Transition probabilities
(array entries) and state probabilities (colour of ellipse) both use same
colormap given in bottom right corner. Furthermore grey dots represent the
Shikoku tremor data points. Black ellipses and -dots represent mean
parameters.
### 6.6 Forecasting
We carry out a Bayesian forecast from the model for a 5 day period (from
December 11, 2012 to December 16, 2012). Note that the data for this period
was excluded in the model fitting process. In order to forecast tremors we
simulated 120 hourly datapoints (i.e for 5 days) from the model (with fixed
number of hidden states $K=25$) for every 1000th MCMC sample (total of 500
simulations) of the approximate posterior distribution. Note that we used the
same realization of the MCMC that was generated in the model fitting process
(see previous section). We used the HMM simulator in the R package HMMextra0s
(freely available at
https://rdrr.io/cran/HMMextra0s/man/HMMextra0s-package.html).
We summarize the 500 forecast simulations as a density in a longitude plot
over time and a latitude plot over time (Figure 8). Furthermore we plot the
actual data as a scatterplot with red datapoints. We also include the last day
(December 10, 2012) of the data used for model fitting (as a scatterplot with
black datapoints).
We see that the model works well for the first two days. It captures nicely in
which area the tremors occur. We also see that see that we get coverage from
the forecast (density plot) for all the data points except for one outlier.
Furthermore we see that the variance in the model predictions increase with
time. This is not unexpected since the further away our forecasts are from the
present the less information our data contains about the future states of the
process. It would be very unlikely to make an accurate forecast of more than a
week.
(a)
(b)
Figure 8: We summarize the forecast simulations as two density plots. (a)
Latitude predictions and data plotted against time (in hours). (b) Longitude
predictions and data plotted against time (in hours). The red dots in both
figures are the hourly Shikoku data for the time period from December 11, 2012
to December 30, 2012 (not included in data used for model fitting).
Furthermore black dots in both figures are the hourly Shikoku data for the
time period December 10, 2012 (included in data used for model fitting).
## missingum@section Discussion
In this paper we present an algorithm for evaluating HMM likelihoods that can
run several orders of magnitude faster than the traditional Forward algorithm.
Our algorithm requires more work, but the high level of parallization of the
likelihood calculation translates into high data throughput.
We have implemented the algorithm for an HMM model that categorises
nonvolcanic tremor data. Furthermore we have integrated the algorithm as part
of an R package for Bayesian analysis using the OpenCL framework with Python
under the hood. It is however expected that a CUDA implementation for NVIDIA
GPUs will achieve higher data throughput but this limits the algorithm to a
single vendor. OpenCL on the other hand allows execution of the algorithm on
any OpenCL compliant device such as Intel CPUs, AMD CPUs and GPUs, Qualcomm
processors, Xilinx FPGAs (Field-programmable gate array) and even NVIDIA GPUs.
We have reported some runtime comparisons with implementations of the Forward
algorithm. The efficieny gains in computation of the likelihood allowed us to
conduct a detailed Bayesian analysis for tremor data of Shikoku region of
Japan.
Lastly, the OpenCL algorithm can be easily modified for other HMM models. In
some cases only the evaluation function of the emission matrix needs to be
updated.
## missingum@section Acknowledgement
Marnus Stolz received a doctoral scholarship from the NZ Marsden Fund (PIs
David Bryant and Steven Higgins).
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## APPENDIX A: Formulas for prior distributions
### Symmetric Dirichlet distributions
$f(\gamma_{1},\dots,\gamma_{K^{2}},\alpha)=\frac{\Gamma(\alpha
K^{2})}{\alpha^{K^{2}}}\prod_{i=1}^{K^{2}}\gamma_{i}^{\alpha-1}\qquad\text{,
for }\alpha<1,$
we note that probability mass is sparsely distributed among
$\gamma_{1},\dots,\gamma_{K^{2}}$ if $\alpha<1.$
### Inverse-wishart distributions
Suppose $\Psi$ is the scale matrix and $\nu$ the degrees of freedom then
$f(\mathbf{x},\Psi,\nu)=\frac{|\Psi|^{\nu/2}}{2^{\nu
p/2}\Gamma_{K}(\frac{\nu}{2})}=|\mathbf{x}|^{-(\nu+K+1)/2}e^{-\frac{1}{2}tr(\Psi
x^{-1})},$
where $\Gamma_{K}$ is a multivariate gamma function
$\Gamma_{K}\left(\frac{\nu}{2}\right)=\pi^{(\frac{\nu}{2})(\frac{\nu}{2}-1)/4}\prod^{K}_{j=1}\Gamma\left(\left(\frac{\nu}{2}\right)+(1-j)/2\right).$
### Gamma distribution
$f(x,\alpha,\beta)=\frac{\beta^{\alpha}x^{\alpha-1})e^{-\beta
x}}{\Gamma(\alpha)},\qquad\text{for $x>0$}\qquad\alpha,\beta>0.$
## APPENDIX B: Additionals model fitted for nonvolcanic tremor data
(a) $K=26$
(b) $K=27$
Figure 9: Posterior distributions of fitted models with number of hidden
states $K=26,27$ for tremor occurrences in Shikoku region. Ellipses each map
represent the 2D normal density of one hidden state for one sample from the
posterior distribution. States are numbered in red. Colour of an ellipse
indicate how likely a tremor will occur given the process is in the hidden
state. In the bottom right corner of each map we give the mean transition
matrix of the posterior distribution. Transition probabilities (array entries)
and state probabilities (colour of ellipse) both use same colormap given in
bottom right corner. Furthermore grey dots represent the Shikoku tremor data
points. Black ellipses and -dots represent mean parameters.
## APPENDIX C: Tabulated posterior statistics for number of hidden states K=25
Table 1: Combined GPU-hmmer parameter summary after 5,000,000 MCMC iterations for soybean dataset hmmer | mean | variance | HPD | ACT | ESS |
---|---|---|---|---|---|---
posterior | -1.0987E+03 | 7.6624E+04 | -1.6110E+03 | -4.0624E+02 | 35830.8997 | 239.5444
Gamma1 | 8.6710E-01 | 2.4186E-04 | 8.3770E-01 | 8.9248E-01 | 29571.9564 | 169.0791
Gamma2 | 1.6289E-02 | 4.7111E-05 | 3.8358E-03 | 2.8083E-02 | 14455.9226 | 345.8790
Gamma3 | 1.8235E-03 | 8.1549E-06 | 5.3391E-06 | 5.9633E-03 | 31080.9139 | 160.8704
Gamma4 | 1.8604E-03 | 9.1692E-06 | 8.0933E-06 | 6.5181E-03 | 23964.1982 | 208.6446
Gamma5 | 1.4205E-03 | 1.2756E-06 | 4.4740E-05 | 3.3164E-03 | 14213.5729 | 351.7764
Gamma6 | 3.4741E-03 | 7.6753E-06 | 2.8037E-05 | 8.2833E-03 | 21277.4832 | 234.9902
Gamma7 | 2.9676E-03 | 3.5263E-06 | 4.7983E-05 | 5.7603E-03 | 15272.0127 | 327.3963
Gamma8 | 1.9523E-03 | 5.1619E-06 | 1.8840E-05 | 4.8152E-03 | 27378.2848 | 182.6265
Gamma9 | 8.8053E-04 | 1.4839E-06 | 6.1831E-07 | 2.6187E-03 | 15509.9601 | 322.3735
Gamma10 | 4.0934E-03 | 1.6603E-05 | 8.1382E-15 | 1.2107E-02 | 57380.8895 | 87.1370
Gamma11 | 9.3274E-03 | 1.4699E-05 | 1.9398E-03 | 1.5410E-02 | 45068.2804 | 110.9428
Gamma12 | 6.1730E-04 | 3.1284E-07 | 4.1541E-07 | 1.7926E-03 | 14440.9950 | 346.2365
Gamma13 | 3.5715E-03 | 5.4052E-05 | 1.2686E-04 | 8.2704E-03 | 24168.9799 | 206.8767
Gamma14 | 9.6052E-04 | 7.1797E-07 | 8.2637E-06 | 2.5498E-03 | 18857.9341 | 265.1404
Gamma15 | 1.3255E-03 | 1.4142E-06 | 4.1034E-06 | 4.0367E-03 | 31384.2830 | 159.3154
Gamma16 | 1.5935E-03 | 1.1240E-05 | 7.3288E-06 | 4.9660E-03 | 20798.4511 | 240.4025
Gamma17 | 3.8764E-02 | 3.7291E-05 | 2.6857E-02 | 4.8477E-02 | 48946.9321 | 102.1514
Gamma18 | 1.0956E-03 | 1.9142E-06 | 7.6476E-06 | 3.7071E-03 | 45031.8581 | 111.0325
Gamma19 | 1.0625E-03 | 4.7366E-06 | 5.9749E-06 | 2.6002E-03 | 17168.5493 | 291.2302
Gamma20 | 2.6584E-03 | 5.6930E-06 | 2.0232E-05 | 6.8404E-03 | 35644.4221 | 140.2744
Gamma21 | 2.9296E-02 | 2.9234E-05 | 1.9465E-02 | 4.0877E-02 | 42258.6322 | 118.3190
Gamma22 | 1.9319E-03 | 3.1337E-06 | 8.6148E-05 | 6.0423E-03 | 16100.7869 | 310.5438
Gamma23 | 2.3521E-03 | 3.8152E-06 | 3.0464E-05 | 6.1173E-03 | 22249.4145 | 224.7250
Gamma24 | 2.8243E-03 | 7.0229E-06 | 1.4940E-04 | 7.1177E-03 | 15713.5026 | 318.1977
Gamma25 | 7.5430E-04 | 1.2261E-06 | 0.0000E+00 | 2.8100E-03 | 33773.0435 | 148.0471
Gamma26 | 7.5800E-04 | 9.2460E-07 | 2.7447E-06 | 2.1539E-03 | 19423.7468 | 257.4169
Gamma27 | 9.7455E-01 | 1.8772E-05 | 9.6572E-01 | 9.8144E-01 | 16385.9590 | 305.1393
Gamma28 | 1.1656E-03 | 5.5348E-07 | 3.8569E-06 | 2.4886E-03 | 433749.2599 | 11.5274
Gamma29 | 2.6091E-04 | 1.4724E-07 | 6.0330E-07 | 5.3802E-04 | 34952.5560 | 143.0511
Gamma30 | 1.1413E-03 | 2.5184E-07 | 3.8769E-04 | 1.6722E-03 | 12690.4915 | 393.9958
Gamma31 | 2.5300E-03 | 2.2597E-07 | 1.7208E-03 | 3.7603E-03 | 62747.9452 | 79.6839
Gamma32 | 7.2782E-04 | 6.3201E-07 | 8.2677E-05 | 1.5357E-03 | 22661.4152 | 220.6394
Gamma33 | 4.7610E-03 | 2.8677E-06 | 2.1785E-03 | 7.5344E-03 | 128377.9942 | 38.9475
Gamma34 | 1.4483E-04 | 3.3611E-07 | 0.0000E+00 | 4.4469E-04 | 18494.6559 | 270.3484
Gamma35 | 8.8254E-04 | 9.1352E-06 | 1.1430E-179 | 2.6530E-03 | 17104.6382 | 292.3184
Gamma36 | 2.2049E-04 | 5.6664E-07 | 3.2433E-07 | 8.8779E-04 | 13504.7449 | 370.2402
Gamma37 | 2.8423E-04 | 1.8785E-06 | 1.3566E-07 | 5.3904E-04 | 16705.1430 | 299.3090
Gamma38 | 1.8184E-03 | 8.9467E-07 | 1.0682E-03 | 2.9130E-03 | 12987.5478 | 384.9841
Gamma39 | 1.5827E-04 | 3.8431E-07 | 8.2793E-240 | 5.5617E-04 | 10408.8774 | 480.3592
Gamma40 | 1.5324E-04 | 5.9919E-08 | 1.7997E-06 | 3.9441E-04 | 35391.8500 | 141.2755
Gamma41 | 1.1195E-04 | 6.7680E-07 | 4.5519E-07 | 1.0529E-04 | 6175.3438 | 809.6715
Gamma42 | 1.6970E-04 | 5.8092E-08 | 2.3201E-68 | 4.8800E-04 | 19975.0180 | 250.3127
Gamma43 | 5.4145E-04 | 1.7014E-06 | 8.0357E-07 | 1.2248E-03 | 12214.7293 | 409.3419
Gamma44 | 9.4007E-05 | 1.2845E-07 | 0.0000E+00 | 3.0288E-04 | 15527.8979 | 322.0011
Gamma45 | 5.8817E-04 | 2.5773E-06 | 2.6848E-07 | 1.7851E-03 | 16773.6991 | 298.0857
Gamma46 | 2.7130E-04 | 1.6124E-06 | 0.0000E+00 | 9.7746E-04 | 12013.6336 | 416.1938
Gamma47 | 8.3608E-04 | 8.7118E-07 | 8.1343E-06 | 1.6621E-03 | 25000.7219 | 199.9942
Gamma48 | 6.1769E-03 | 1.6042E-06 | 3.7912E-03 | 8.2029E-03 | 72454.3676 | 69.0090
Gamma49 | 1.3814E-03 | 1.2691E-07 | 8.2454E-04 | 1.8834E-03 | 16414.5669 | 304.6075
Gamma50 | 2.7088E-04 | 1.8704E-07 | 1.9386E-06 | 8.3363E-04 | 20676.1070 | 241.8250
Gamma51 | 3.8265E-03 | 1.4101E-05 | 1.5164E-06 | 1.1256E-02 | 54832.1289 | 91.1874
Gamma52 | 4.1702E-02 | 2.7147E-04 | 1.4143E-02 | 7.0675E-02 | 177277.4811 | 28.2044
Gamma53 | 6.5895E-01 | 1.1852E-02 | 5.3762E-01 | 8.6165E-01 | 179266.1419 | 27.8915
Gamma54 | 5.2686E-02 | 1.3514E-04 | 3.0989E-02 | 7.5725E-02 | 20727.5986 | 241.2243
Gamma55 | 3.7342E-03 | 8.7625E-06 | 4.2153E-06 | 8.8395E-03 | 35075.5349 | 142.5495
Gamma56 | 3.7250E-03 | 1.4367E-05 | 2.1957E-05 | 1.0399E-02 | 13703.4629 | 364.8713
Gamma57 | 3.4035E-03 | 7.6310E-06 | 1.1752E-06 | 9.5530E-03 | 60398.7204 | 82.7832
Gamma58 | 1.1569E-02 | 5.7239E-05 | 1.4684E-03 | 2.6915E-02 | 126050.2525 | 39.6667
Gamma59 | 8.3384E-03 | 5.0755E-05 | 2.4430E-05 | 1.9246E-02 | 108972.2649 | 45.8832
Gamma60 | 7.5111E-03 | 1.8539E-05 | 1.3374E-03 | 1.5506E-02 | 35526.5778 | 140.7397
Gamma61 | 2.4433E-03 | 5.7828E-06 | 3.7794E-05 | 7.1885E-03 | 23869.7025 | 209.4706
Gamma62 | 2.9865E-03 | 1.1750E-05 | 0.0000E+00 | 9.4605E-03 | 24301.5009 | 205.7486
Gamma63 | 4.6511E-03 | 1.1309E-05 | 4.0228E-04 | 1.1406E-02 | 17927.0292 | 278.9085
Gamma64 | 3.4551E-03 | 1.0811E-05 | 8.0421E-06 | 9.2846E-03 | 60480.2163 | 82.6717
Gamma65 | 1.7704E-03 | 2.9336E-06 | 2.0011E-05 | 5.5824E-03 | 31567.4252 | 158.3911
Gamma66 | 1.8257E-03 | 5.5040E-06 | 4.8305E-06 | 6.7129E-03 | 21359.3236 | 234.0898
Gamma67 | 4.4132E-03 | 1.0302E-05 | 2.3097E-04 | 1.0175E-02 | 30529.5466 | 163.7758
Gamma68 | 3.3537E-03 | 1.3710E-05 | 4.6203E-06 | 1.1097E-02 | 47189.7634 | 105.9552
Gamma69 | 2.5590E-03 | 1.2021E-05 | 1.6972E-05 | 8.6447E-03 | 17168.6412 | 291.2286
Gamma70 | 3.6075E-03 | 2.1892E-05 | 1.3317E-05 | 1.4296E-02 | 51779.6476 | 96.5630
Gamma71 | 2.7341E-03 | 1.0429E-05 | 6.6654E-05 | 9.0064E-03 | 50454.1729 | 99.0998
Gamma72 | 1.6084E-01 | 8.9549E-03 | 1.7265E-04 | 2.6376E-01 | 52563.5955 | 95.1229
Gamma73 | 4.7693E-03 | 1.3051E-05 | 9.0943E-04 | 1.3995E-02 | 31721.7813 | 157.6204
Gamma74 | 3.2346E-03 | 1.1070E-05 | 2.3274E-05 | 8.6432E-03 | 32056.6747 | 155.9738
Gamma75 | 1.9117E-03 | 3.3446E-06 | 0.0000E+00 | 5.6489E-03 | 15101.7754 | 331.0869
Gamma76 | 4.0374E-03 | 1.3862E-05 | 5.6963E-06 | 1.1135E-02 | 17448.4236 | 286.5588
Gamma77 | 1.8735E-02 | 1.3793E-04 | 1.4735E-03 | 3.7747E-02 | 153773.7616 | 32.5153
Gamma78 | 5.2840E-02 | 1.4351E-03 | 8.7997E-03 | 1.2569E-01 | 109036.5946 | 45.8562
Gamma79 | 7.7194E-01 | 4.7660E-04 | 7.3193E-01 | 8.1135E-01 | 97858.7582 | 51.0940
Gamma80 | 2.3710E-03 | 6.3079E-06 | 8.1711E-209 | 6.0531E-03 | 40191.5185 | 124.4044
Gamma81 | 9.8015E-03 | 2.1754E-05 | 2.7665E-03 | 1.9849E-02 | 14021.4458 | 356.5966
Gamma82 | 8.0723E-03 | 1.8660E-05 | 1.5913E-03 | 1.6722E-02 | 13912.5818 | 359.3869
Gamma83 | 4.2189E-03 | 1.5787E-05 | 5.4825E-05 | 1.2599E-02 | 17447.4497 | 286.5748
Gamma84 | 1.5088E-02 | 5.6666E-05 | 2.5035E-03 | 3.0542E-02 | 25580.4554 | 195.4617
Gamma85 | 9.6710E-03 | 1.6331E-05 | 2.9544E-03 | 1.8682E-02 | 31452.3565 | 158.9706
Gamma86 | 1.6917E-03 | 2.8601E-06 | 8.1084E-06 | 5.7956E-03 | 8240.6163 | 606.7507
Gamma87 | 5.7498E-03 | 2.7984E-05 | 2.2783E-04 | 1.5747E-02 | 21734.0305 | 230.0540
Gamma88 | 6.3461E-03 | 1.9530E-05 | 4.5535E-04 | 1.4662E-02 | 26108.6270 | 191.5076
Gamma89 | 5.2471E-03 | 1.3105E-05 | 2.0134E-04 | 1.0240E-02 | 13017.1322 | 384.1092
Gamma90 | 2.0111E-03 | 4.2089E-06 | 1.1327E-05 | 5.9145E-03 | 25705.1264 | 194.5137
Gamma91 | 1.0309E-03 | 1.6301E-06 | 0.0000E+00 | 3.8256E-03 | 28956.6968 | 172.6716
Gamma92 | 2.2777E-03 | 3.1269E-06 | 1.0209E-05 | 5.9816E-03 | 14923.2556 | 335.0475
Gamma93 | 2.1677E-03 | 9.3605E-06 | 0.0000E+00 | 5.6929E-03 | 23761.2254 | 210.4269
Gamma94 | 1.0397E-03 | 3.5146E-06 | 4.8823E-06 | 4.4321E-03 | 23742.1096 | 210.5963
Gamma95 | 3.3163E-03 | 5.2410E-06 | 1.4686E-04 | 7.9886E-03 | 30114.5464 | 166.0327
Gamma96 | 2.5090E-03 | 1.6163E-05 | 2.7992E-05 | 7.1467E-03 | 41015.2414 | 121.9059
Gamma97 | 5.8867E-02 | 1.3932E-03 | 2.0425E-05 | 1.0662E-01 | 31943.0005 | 156.5288
Gamma98 | 5.7913E-03 | 2.4696E-05 | 4.1050E-50 | 1.4469E-02 | 35391.9791 | 141.2749
Gamma99 | 1.4880E-03 | 2.3845E-06 | 0.0000E+00 | 4.5140E-03 | 25023.0598 | 199.8157
Gamma100 | 3.6941E-03 | 6.4817E-06 | 3.0497E-04 | 7.7939E-03 | 20250.6950 | 246.9051
Gamma101 | 1.0241E-02 | 7.2072E-05 | 1.0568E-04 | 2.7836E-02 | 20759.9054 | 240.8489
Gamma102 | 1.0804E-01 | 4.6584E-04 | 6.1602E-02 | 1.4489E-01 | 14058.9871 | 355.6444
Gamma103 | 5.6760E-03 | 2.0145E-05 | 5.4058E-06 | 1.4865E-02 | 82129.7028 | 60.8793
Gamma104 | 2.8431E-03 | 6.8754E-06 | 1.2424E-04 | 8.9155E-03 | 34299.2175 | 145.7759
Gamma105 | 7.1477E-01 | 7.0890E-04 | 6.6167E-01 | 7.6297E-01 | 11826.7772 | 422.7694
Gamma106 | 6.5209E-03 | 1.7024E-05 | 1.5508E-04 | 1.4459E-02 | 18234.4629 | 274.2060
Gamma107 | 2.8461E-02 | 2.2138E-04 | 7.8455E-03 | 5.4417E-02 | 17921.7664 | 278.9904
Gamma108 | 3.3291E-03 | 8.5009E-06 | 1.2591E-05 | 9.5390E-03 | 26847.8163 | 186.2349
Gamma109 | 1.0648E-02 | 4.2232E-05 | 3.1849E-04 | 2.3514E-02 | 37656.6887 | 132.7785
Gamma110 | 1.4558E-02 | 9.6667E-05 | 1.4384E-03 | 3.0609E-02 | 19327.2137 | 258.7026
Gamma111 | 2.6155E-03 | 1.2609E-05 | 0.0000E+00 | 1.0664E-02 | 35246.2265 | 141.8592
Gamma112 | 3.1530E-02 | 9.3212E-05 | 1.5939E-02 | 5.2513E-02 | 21814.3469 | 229.2070
Gamma113 | 3.0332E-03 | 9.1017E-06 | 2.1723E-05 | 8.8238E-03 | 21228.0396 | 235.5375
Gamma114 | 2.8459E-03 | 5.0074E-06 | 8.5689E-05 | 8.0017E-03 | 20882.2793 | 239.4375
Gamma115 | 3.2308E-03 | 1.2224E-05 | 5.1372E-05 | 9.8159E-03 | 26005.5177 | 192.2669
Gamma116 | 1.7967E-03 | 3.4193E-06 | 5.5578E-06 | 5.4687E-03 | 14293.7529 | 349.8032
Gamma117 | 5.1239E-03 | 2.2861E-05 | 7.6876E-06 | 1.4478E-02 | 14718.0883 | 339.7180
Gamma118 | 3.9297E-03 | 1.7284E-05 | 0.0000E+00 | 1.0973E-02 | 30107.3335 | 166.0725
Gamma119 | 2.4660E-03 | 6.7519E-06 | 3.6767E-05 | 8.8143E-03 | 39887.9865 | 125.3510
Gamma120 | 5.9192E-03 | 2.0708E-05 | 2.1575E-05 | 1.4863E-02 | 41440.4025 | 120.6552
Gamma121 | 3.3092E-03 | 7.1388E-06 | 1.2865E-04 | 9.1760E-03 | 19966.6857 | 250.4171
Gamma122 | 1.4188E-02 | 1.4186E-04 | 1.0583E-04 | 3.8582E-02 | 29696.9340 | 168.3675
Gamma123 | 6.1469E-03 | 3.7944E-05 | 3.5337E-103 | 2.0600E-02 | 14266.5070 | 350.4712
Gamma124 | 5.3542E-03 | 6.3464E-05 | 2.1460E-05 | 1.3922E-02 | 16095.5744 | 310.6444
Gamma125 | 3.4203E-03 | 2.1335E-05 | 0.0000E+00 | 1.0130E-02 | 45807.5273 | 109.1524
Gamma126 | 1.4618E-02 | 1.1325E-04 | 2.8701E-04 | 3.6125E-02 | 13691.7749 | 365.1827
Gamma127 | 3.2206E-01 | 1.0197E-03 | 2.6087E-01 | 3.7722E-01 | 49086.4278 | 101.8612
Gamma128 | 7.2857E-03 | 3.1361E-05 | 1.3948E-04 | 1.8385E-02 | 42005.7489 | 119.0313
Gamma129 | 1.6891E-02 | 9.2670E-05 | 6.3445E-04 | 3.3494E-02 | 7378.6105 | 677.6344
Gamma130 | 4.6442E-03 | 1.3438E-05 | 1.1728E-04 | 1.1089E-02 | 13249.6606 | 377.3682
Gamma131 | 4.9890E-01 | 1.3868E-03 | 4.2268E-01 | 5.6160E-01 | 22054.7456 | 226.7086
Gamma132 | 8.1591E-03 | 2.6096E-05 | 2.4168E-04 | 1.6574E-02 | 11093.0986 | 450.7307
Gamma133 | 5.9883E-03 | 2.4691E-05 | 0.0000E+00 | 1.6412E-02 | 26398.9966 | 189.4011
Gamma134 | 7.2804E-03 | 2.6287E-05 | 8.8405E-05 | 1.8001E-02 | 18358.7256 | 272.3501
Gamma135 | 8.7256E-03 | 2.1526E-05 | 2.2706E-04 | 1.7155E-02 | 23918.9752 | 209.0391
Gamma136 | 2.9880E-03 | 7.3198E-06 | 2.0184E-06 | 7.9991E-03 | 19008.6479 | 263.0382
Gamma137 | 4.1896E-03 | 1.5155E-05 | 0.0000E+00 | 1.0915E-02 | 25792.6040 | 193.8540
Gamma138 | 4.0759E-03 | 1.7561E-05 | 0.0000E+00 | 1.2611E-02 | 26961.9908 | 185.4462
Gamma139 | 1.9013E-03 | 2.7532E-06 | 3.6100E-06 | 5.6579E-03 | 24439.0003 | 204.5910
Gamma140 | 3.0420E-03 | 9.0646E-06 | 0.0000E+00 | 9.7842E-03 | 13533.4384 | 369.4553
Gamma141 | 5.9292E-03 | 9.9349E-05 | 0.0000E+00 | 1.8168E-02 | 26232.0800 | 190.6063
Gamma142 | 3.5377E-03 | 9.6379E-06 | 0.0000E+00 | 1.0215E-02 | 23203.7127 | 215.4828
Gamma143 | 1.5168E-02 | 6.2250E-05 | 1.9025E-03 | 3.0656E-02 | 15722.1103 | 318.0235
Gamma144 | 5.7933E-03 | 4.4980E-05 | 1.7462E-289 | 1.6168E-02 | 22946.4046 | 217.8991
Gamma145 | 5.7463E-03 | 3.2700E-05 | 2.3855E-05 | 1.7080E-02 | 21825.2432 | 229.0925
Gamma146 | 2.6456E-03 | 1.2255E-05 | 2.5832E-06 | 9.3531E-03 | 24569.0253 | 203.5083
Gamma147 | 1.4579E-02 | 1.2048E-04 | 0.0000E+00 | 3.9936E-02 | 16737.9921 | 298.7216
Gamma148 | 8.6204E-03 | 5.7072E-05 | 2.7282E-06 | 2.0417E-02 | 38093.7914 | 131.2550
Gamma149 | 1.6937E-02 | 6.7864E-05 | 4.1349E-03 | 3.4704E-02 | 17925.6478 | 278.9299
Gamma150 | 1.0294E-02 | 4.6247E-05 | 1.4572E-03 | 2.1769E-02 | 18136.6081 | 275.6855
Gamma151 | 1.3152E-02 | 7.3651E-05 | 4.6341E-05 | 3.0154E-02 | 15556.8030 | 321.4028
Gamma152 | 3.7244E-02 | 2.8304E-04 | 2.8950E-03 | 6.5610E-02 | 33478.3899 | 149.3501
Gamma153 | 1.3758E-02 | 6.6246E-05 | 4.0890E-04 | 3.0621E-02 | 29346.9173 | 170.3756
Gamma154 | 5.9445E-03 | 2.1308E-05 | 1.4043E-04 | 1.5699E-02 | 32066.2635 | 155.9271
Gamma155 | 2.8742E-02 | 1.1982E-04 | 7.3737E-03 | 5.2091E-02 | 33898.0279 | 147.5012
Gamma156 | 2.5899E-03 | 7.0253E-06 | 7.4161E-06 | 7.8707E-03 | 18689.6816 | 267.5273
Gamma157 | 7.4292E-01 | 1.0465E-03 | 6.7896E-01 | 7.9747E-01 | 17523.3619 | 285.3334
Gamma158 | 3.7707E-03 | 1.0295E-05 | 0.0000E+00 | 8.9664E-03 | 39356.2073 | 127.0448
Gamma159 | 4.6426E-02 | 1.6495E-04 | 1.9526E-02 | 6.8351E-02 | 20248.3966 | 246.9331
Gamma160 | 9.4584E-03 | 2.2591E-05 | 2.0420E-03 | 1.9575E-02 | 15479.8442 | 323.0007
Gamma161 | 2.4250E-03 | 6.0102E-06 | 2.2032E-05 | 6.5940E-03 | 24814.9610 | 201.4914
Gamma162 | 1.1943E-02 | 5.1395E-05 | 1.7040E-03 | 2.6546E-02 | 19386.5635 | 257.9106
Gamma163 | 2.5954E-03 | 5.8674E-06 | 1.6756E-05 | 7.9368E-03 | 48028.9088 | 104.1040
Gamma164 | 2.1189E-03 | 4.6098E-06 | 6.1450E-06 | 5.7563E-03 | 21749.6357 | 229.8889
Gamma165 | 2.2330E-03 | 1.8865E-05 | 8.4136E-06 | 6.9105E-03 | 32478.0637 | 153.9501
Gamma166 | 3.6582E-03 | 8.4473E-06 | 9.5463E-05 | 9.4273E-03 | 22048.8286 | 226.7694
Gamma167 | 3.6427E-03 | 1.1416E-05 | 1.0440E-05 | 9.4321E-03 | 28140.4471 | 177.6802
Gamma168 | 1.1421E-02 | 6.9643E-05 | 4.6150E-04 | 2.6784E-02 | 25694.6466 | 194.5931
Gamma169 | 5.1223E-03 | 4.4866E-05 | 1.0712E-04 | 1.6117E-02 | 24383.3685 | 205.0578
Gamma170 | 4.9605E-03 | 2.9550E-05 | 1.9398E-05 | 1.4858E-02 | 19684.0267 | 254.0131
Gamma171 | 2.9722E-03 | 1.0856E-05 | 0.0000E+00 | 7.2370E-03 | 19550.7361 | 255.7448
Gamma172 | 1.9379E-02 | 2.5865E-04 | 2.0458E-04 | 5.2602E-02 | 42800.9521 | 116.8198
Gamma173 | 1.3881E-02 | 7.6848E-05 | 6.7001E-04 | 3.1088E-02 | 23168.4207 | 215.8110
Gamma174 | 6.8947E-03 | 1.6571E-05 | 1.2776E-03 | 1.5726E-02 | 18513.8606 | 270.0679
Gamma175 | 2.7526E-03 | 4.9136E-06 | 1.9243E-63 | 7.1387E-03 | 36917.1373 | 135.4385
Gamma176 | 2.6324E-02 | 2.7210E-04 | 8.3009E-04 | 5.7787E-02 | 34809.0990 | 143.6406
Gamma177 | 5.5414E-01 | 2.2137E-03 | 4.7383E-01 | 6.4615E-01 | 47621.5790 | 104.9944
Gamma178 | 2.5632E-02 | 2.7222E-04 | 3.2199E-03 | 6.4055E-02 | 63701.0484 | 78.4916
Gamma179 | 1.3130E-02 | 9.7449E-05 | 8.0479E-04 | 3.2826E-02 | 19797.6125 | 252.5557
Gamma180 | 5.8898E-03 | 2.2534E-05 | 1.5316E-04 | 1.6794E-02 | 45691.9265 | 109.4285
Gamma181 | 7.4855E-03 | 2.9769E-05 | 7.2334E-04 | 1.9642E-02 | 46600.4791 | 107.2950
Gamma182 | 4.8547E-03 | 1.6843E-05 | 0.0000E+00 | 1.3644E-02 | 15670.4387 | 319.0721
Gamma183 | 8.8900E-02 | 1.2533E-03 | 2.7782E-02 | 1.5979E-01 | 55484.9668 | 90.1145
Gamma184 | 6.7210E-03 | 6.9651E-05 | 3.3967E-06 | 2.5447E-02 | 22950.5821 | 217.8594
Gamma185 | 6.8905E-03 | 2.4856E-05 | 2.6100E-05 | 1.6220E-02 | 23021.4924 | 217.1884
Gamma186 | 1.0912E-02 | 1.1490E-04 | 1.9169E-05 | 2.9730E-02 | 24510.3336 | 203.9956
Gamma187 | 7.3192E-03 | 7.6226E-05 | 8.4817E-07 | 2.4485E-02 | 59524.0225 | 83.9997
Gamma188 | 1.2526E-02 | 5.8547E-05 | 7.1780E-04 | 2.5975E-02 | 18057.2891 | 276.8965
Gamma189 | 7.0914E-03 | 4.2373E-05 | 0.0000E+00 | 1.9989E-02 | 26258.6102 | 190.4137
Gamma190 | 5.6499E-03 | 2.0119E-05 | 3.7507E-04 | 1.5291E-02 | 18594.2158 | 268.9008
Gamma191 | 5.3477E-03 | 5.5532E-05 | 1.4105E-214 | 2.0262E-02 | 27848.3877 | 179.5436
Gamma192 | 1.1914E-02 | 1.2016E-04 | 4.8203E-04 | 4.0740E-02 | 42037.8420 | 118.9405
Gamma193 | 8.6947E-02 | 1.2275E-03 | 2.4052E-02 | 1.4001E-01 | 65323.2487 | 76.5424
Gamma194 | 4.3828E-03 | 1.4208E-05 | 5.8816E-05 | 1.2065E-02 | 24807.7751 | 201.5497
Gamma195 | 6.2229E-03 | 4.6592E-05 | 1.5081E-126 | 2.0269E-02 | 19968.8234 | 250.3903
Gamma196 | 3.9935E-03 | 1.6995E-05 | 5.8948E-06 | 1.1468E-02 | 22964.2384 | 217.7298
Gamma197 | 4.8131E-02 | 1.0506E-03 | 1.8951E-03 | 1.1265E-01 | 61968.1768 | 80.6866
Gamma198 | 3.8953E-02 | 4.6221E-04 | 4.3143E-03 | 7.6449E-02 | 103496.0248 | 48.3110
Gamma199 | 6.8304E-03 | 3.0269E-05 | 0.0000E+00 | 1.6100E-02 | 23752.2087 | 210.5067
Gamma200 | 3.8137E-03 | 1.2918E-05 | 0.0000E+00 | 1.1751E-02 | 29431.9507 | 169.8834
Gamma201 | 2.2061E-03 | 2.8632E-05 | 5.1399E-06 | 4.8039E-03 | 19079.3054 | 262.0640
Gamma202 | 3.9100E-03 | 1.0463E-05 | 1.0212E-05 | 1.0611E-02 | 42334.9988 | 118.1056
Gamma203 | 1.3941E-02 | 5.6850E-05 | 4.6010E-03 | 3.2667E-02 | 73870.7415 | 67.6858
Gamma204 | 8.8065E-03 | 4.3442E-05 | 2.1139E-04 | 1.6809E-02 | 25373.7124 | 197.0543
Gamma205 | 3.5903E-03 | 6.3083E-06 | 1.0064E-04 | 8.4176E-03 | 26858.5310 | 186.1606
Gamma206 | 1.8215E-03 | 3.9859E-06 | 0.0000E+00 | 5.5891E-03 | 21877.8619 | 228.5415
Gamma207 | 2.7125E-02 | 5.0313E-05 | 1.2766E-02 | 3.8572E-02 | 18578.7242 | 269.1250
Gamma208 | 4.5804E-03 | 1.6699E-05 | 1.4838E-05 | 1.1127E-02 | 36375.1408 | 137.4565
Gamma209 | 8.0613E-01 | 1.7607E-03 | 7.3269E-01 | 8.7467E-01 | 124530.5793 | 40.1508
Gamma210 | 8.4669E-03 | 1.9079E-05 | 2.2761E-03 | 1.5011E-02 | 20560.2395 | 243.1878
Gamma211 | 2.4497E-03 | 1.0519E-05 | 0.0000E+00 | 9.1696E-03 | 13154.7921 | 380.0896
Gamma212 | 2.7867E-02 | 8.6381E-05 | 1.1267E-02 | 4.2506E-02 | 27908.3662 | 179.1577
Gamma213 | 1.4054E-03 | 2.2605E-06 | 1.4660E-24 | 4.9417E-03 | 18466.9677 | 270.7537
Gamma214 | 1.2470E-02 | 4.7618E-05 | 1.3578E-03 | 2.4040E-02 | 57251.1230 | 87.3345
Gamma215 | 1.9740E-03 | 3.5207E-06 | 2.6479E-05 | 5.2427E-03 | 13247.3981 | 377.4326
Gamma216 | 2.5117E-03 | 2.6384E-05 | 2.9458E-05 | 6.2596E-03 | 19544.4539 | 255.8271
Gamma217 | 2.2047E-03 | 3.0671E-06 | 2.7441E-05 | 5.7185E-03 | 21443.2562 | 233.1735
Gamma218 | 2.1137E-02 | 1.0274E-04 | 3.6511E-03 | 3.9757E-02 | 32932.5392 | 151.8255
Gamma219 | 2.2999E-03 | 1.0651E-05 | 4.5831E-06 | 9.0745E-03 | 42058.9521 | 118.8808
Gamma220 | 2.8658E-03 | 2.5555E-05 | 1.5123E-05 | 9.6544E-03 | 22096.7836 | 226.2773
Gamma221 | 1.2031E-03 | 1.5336E-06 | 1.4189E-05 | 4.1287E-03 | 19430.3636 | 257.3292
Gamma222 | 3.4537E-02 | 5.9361E-04 | 1.6772E-04 | 7.8791E-02 | 189963.0833 | 26.3209
Gamma223 | 3.0790E-03 | 7.7608E-06 | 1.2559E-05 | 7.7606E-03 | 20839.1635 | 239.9329
Gamma224 | 1.6715E-03 | 1.8892E-06 | 4.5916E-05 | 4.3769E-03 | 15313.2087 | 326.5155
Gamma225 | 1.7481E-03 | 7.3093E-06 | 2.5438E-159 | 5.0969E-03 | 12502.8650 | 399.9083
Gamma226 | 3.6707E-02 | 9.9152E-04 | 3.0241E-04 | 1.1157E-01 | 47271.4019 | 105.7722
Gamma227 | 6.2283E-02 | 3.8712E-03 | 2.9392E-05 | 2.0028E-01 | 94429.1491 | 52.9498
Gamma228 | 2.7096E-02 | 2.2985E-04 | 4.3516E-03 | 5.4892E-02 | 26164.9348 | 191.0955
Gamma229 | 3.2028E-02 | 1.4356E-04 | 1.2579E-02 | 5.5498E-02 | 29452.0847 | 169.7673
Gamma230 | 1.0139E-02 | 5.1313E-05 | 4.9129E-04 | 2.3844E-02 | 65162.2063 | 76.7316
Gamma231 | 2.1222E-02 | 2.0195E-04 | 3.6462E-03 | 4.7430E-02 | 22587.0898 | 221.3654
Gamma232 | 3.0705E-02 | 1.8114E-04 | 8.2274E-03 | 5.9118E-02 | 36407.2808 | 137.3352
Gamma233 | 1.5053E-02 | 3.4100E-04 | 0.0000E+00 | 4.8016E-02 | 80339.2098 | 62.2361
Gamma234 | 3.7322E-02 | 1.6725E-04 | 1.0257E-02 | 5.9128E-02 | 12156.2454 | 411.3112
Gamma235 | 2.1579E-01 | 3.2940E-03 | 1.0663E-01 | 3.2862E-01 | 73331.2251 | 68.1838
Gamma236 | 1.2151E-02 | 9.8877E-05 | 8.8663E-05 | 3.5255E-02 | 23871.2828 | 209.4567
Gamma237 | 1.8690E-02 | 7.5997E-05 | 3.7458E-03 | 3.5588E-02 | 13149.2648 | 380.2494
Gamma238 | 6.3248E-03 | 4.0885E-05 | 2.4533E-07 | 2.1520E-02 | 31778.5257 | 157.3390
Gamma239 | 4.9375E-02 | 6.0725E-04 | 1.1591E-02 | 9.9927E-02 | 45728.0585 | 109.3421
Gamma240 | 6.7224E-03 | 4.2430E-05 | 0.0000E+00 | 1.8560E-02 | 36849.3569 | 135.6876
Gamma241 | 9.4317E-02 | 6.8467E-04 | 5.8860E-02 | 1.4398E-01 | 31663.8131 | 157.9090
Gamma242 | 1.7808E-02 | 2.4746E-04 | 3.8456E-06 | 5.3202E-02 | 45864.0363 | 109.0179
Gamma243 | 1.5343E-02 | 1.2346E-04 | 1.0887E-03 | 3.9775E-02 | 25997.7812 | 192.3241
Gamma244 | 5.1805E-02 | 2.3883E-04 | 2.8589E-02 | 8.9097E-02 | 14533.6772 | 344.0286
Gamma245 | 2.0058E-02 | 3.1468E-04 | 1.1706E-04 | 5.0474E-02 | 42293.9325 | 118.2203
Gamma246 | 7.4123E-02 | 5.8961E-04 | 3.8454E-02 | 1.3592E-01 | 35161.7526 | 142.2000
Gamma247 | 1.9739E-02 | 2.0427E-04 | 1.2378E-04 | 4.3046E-02 | 18502.4938 | 270.2338
Gamma248 | 4.6480E-02 | 7.2386E-04 | 1.1234E-02 | 1.0265E-01 | 31479.8738 | 158.8316
Gamma249 | 1.9907E-02 | 1.7395E-04 | 1.2233E-03 | 4.3735E-02 | 38284.4223 | 130.6014
Gamma250 | 5.8817E-02 | 3.0505E-04 | 2.9232E-02 | 8.8807E-02 | 48014.8455 | 104.1345
Gamma251 | 4.8917E-02 | 9.7887E-04 | 1.9972E-03 | 1.0607E-01 | 36523.3956 | 136.8986
Gamma252 | 6.4240E-03 | 5.2194E-05 | 8.1154E-06 | 2.3883E-02 | 33883.8243 | 147.5630
Gamma253 | 8.1575E-03 | 2.8191E-05 | 2.8640E-04 | 1.7617E-02 | 15709.6296 | 318.2761
Gamma254 | 5.3155E-03 | 2.9752E-05 | 0.0000E+00 | 1.2059E-02 | 28570.1134 | 175.0081
Gamma255 | 4.4334E-03 | 1.9147E-05 | 0.0000E+00 | 1.4631E-02 | 34892.3956 | 143.2977
Gamma256 | 5.4825E-03 | 2.9670E-05 | 1.3524E-04 | 1.4669E-02 | 19045.1706 | 262.5337
Gamma257 | 6.8300E-03 | 4.7635E-05 | 0.0000E+00 | 1.8941E-02 | 29033.1484 | 172.2169
Gamma258 | 1.3275E-02 | 9.1295E-05 | 7.4201E-04 | 3.0903E-02 | 28043.2345 | 178.2961
Gamma259 | 3.8815E-03 | 1.7372E-05 | 0.0000E+00 | 1.2537E-02 | 24454.0425 | 204.4652
Gamma260 | 1.0610E-02 | 5.7112E-05 | 3.7106E-04 | 2.4722E-02 | 31783.3337 | 157.3152
Gamma261 | 6.3711E-01 | 2.6489E-03 | 5.4235E-01 | 7.2877E-01 | 65788.9660 | 76.0006
Gamma262 | 8.8268E-03 | 9.8065E-05 | 3.1832E-05 | 2.7775E-02 | 24850.7622 | 201.2011
Gamma263 | 5.9210E-03 | 1.9170E-05 | 8.2042E-05 | 1.4713E-02 | 20750.6234 | 240.9566
Gamma264 | 7.3070E-02 | 3.7953E-04 | 4.4710E-02 | 1.1973E-01 | 40442.9323 | 123.6310
Gamma265 | 4.1779E-03 | 2.8985E-05 | 2.9634E-06 | 1.1795E-02 | 27245.2398 | 183.5183
Gamma266 | 3.6660E-03 | 1.0923E-05 | 0.0000E+00 | 1.1055E-02 | 28282.6364 | 176.7869
Gamma267 | 8.0333E-02 | 5.5329E-04 | 3.7376E-02 | 1.2613E-01 | 24671.3012 | 202.6646
Gamma268 | 3.5894E-02 | 5.3182E-04 | 9.1184E-04 | 7.6760E-02 | 22406.7080 | 223.1475
Gamma269 | 5.0000E-03 | 1.6646E-05 | 8.0503E-06 | 1.2885E-02 | 20377.3673 | 245.3703
Gamma270 | 6.0987E-03 | 5.0104E-05 | 0.0000E+00 | 1.9194E-02 | 20392.5245 | 245.1879
Gamma271 | 5.9221E-03 | 4.8925E-05 | 2.2163E-06 | 1.9486E-02 | 26498.4931 | 188.6900
Gamma272 | 5.4337E-03 | 3.5837E-05 | 3.2224E-05 | 1.7387E-02 | 24207.3908 | 206.5485
Gamma273 | 5.5041E-03 | 2.6215E-05 | 0.0000E+00 | 1.5638E-02 | 25953.9721 | 192.6487
Gamma274 | 5.7329E-03 | 1.9426E-05 | 6.6190E-06 | 1.4090E-02 | 13558.9442 | 368.7603
Gamma275 | 3.9834E-03 | 1.6668E-05 | 8.5724E-05 | 1.2867E-02 | 33148.8933 | 150.8346
Gamma276 | 2.9681E-03 | 7.5712E-06 | 3.9389E-05 | 7.4899E-03 | 24184.5647 | 206.7434
Gamma277 | 3.7934E-03 | 2.5369E-05 | 7.5299E-06 | 1.0718E-02 | 21527.6928 | 232.2590
Gamma278 | 4.1185E-03 | 1.0647E-05 | 0.0000E+00 | 1.0465E-02 | 33540.4810 | 149.0736
Gamma279 | 4.2983E-03 | 1.5452E-05 | 3.3467E-06 | 1.0948E-02 | 25237.1937 | 198.1203
Gamma280 | 1.0858E-02 | 2.4034E-05 | 9.7111E-04 | 2.0302E-02 | 13321.8237 | 375.3240
Gamma281 | 5.0261E-03 | 8.7573E-06 | 1.0469E-03 | 1.1525E-02 | 13832.8982 | 361.4572
Gamma282 | 6.5803E-03 | 1.8902E-05 | 2.3867E-05 | 1.5060E-02 | 19289.4499 | 259.2091
Gamma283 | 5.0972E-03 | 2.5601E-05 | 3.0547E-05 | 1.6494E-02 | 20443.3423 | 244.5784
Gamma284 | 4.7241E-02 | 9.0826E-05 | 2.9230E-02 | 6.4696E-02 | 12236.9987 | 408.5969
Gamma285 | 5.9855E-03 | 1.7391E-05 | 1.7215E-05 | 1.3667E-02 | 16319.6658 | 306.3788
Gamma286 | 6.7989E-03 | 4.6773E-05 | 4.4576E-06 | 1.2852E-02 | 37887.2385 | 131.9706
Gamma287 | 7.0529E-01 | 6.5625E-04 | 6.5503E-01 | 7.5514E-01 | 69707.2689 | 71.7285
Gamma288 | 4.1422E-03 | 1.0170E-05 | 1.0113E-04 | 1.0852E-02 | 19079.0956 | 262.0669
Gamma289 | 4.8746E-02 | 1.5836E-04 | 3.0056E-02 | 7.2797E-02 | 26323.2367 | 189.9462
Gamma290 | 2.5543E-03 | 2.4406E-05 | 2.8525E-05 | 9.1834E-03 | 17013.3777 | 293.8864
Gamma291 | 1.1412E-03 | 3.3345E-06 | 0.0000E+00 | 2.4611E-03 | 17679.4058 | 282.8149
Gamma292 | 2.8599E-03 | 6.6291E-06 | 1.2848E-04 | 8.7528E-03 | 22401.1303 | 223.2030
Gamma293 | 1.1679E-01 | 2.8930E-04 | 8.4514E-02 | 1.4702E-01 | 67755.4724 | 73.7948
Gamma294 | 1.9097E-03 | 3.0563E-06 | 0.0000E+00 | 4.9579E-03 | 24372.6881 | 205.1477
Gamma295 | 2.5202E-03 | 6.4498E-06 | 0.0000E+00 | 6.4033E-03 | 38714.7146 | 129.1499
Gamma296 | 1.5994E-03 | 7.4314E-06 | 0.0000E+00 | 4.0390E-03 | 23306.1716 | 214.5354
Gamma297 | 3.7095E-03 | 6.1932E-05 | 7.2249E-05 | 1.0261E-02 | 17278.2065 | 289.3819
Gamma298 | 2.3703E-03 | 6.8958E-06 | 0.0000E+00 | 6.7324E-03 | 21231.4268 | 235.5000
Gamma299 | 1.8840E-03 | 6.6609E-06 | 1.8302E-06 | 6.1451E-03 | 19788.1064 | 252.6770
Gamma300 | 1.7226E-03 | 3.0834E-06 | 7.0403E-06 | 5.5546E-03 | 44347.4723 | 112.7460
Gamma301 | 2.9327E-02 | 4.2531E-04 | 3.2592E-03 | 6.3432E-02 | 30510.9965 | 163.8753
Gamma302 | 1.2290E-01 | 6.3671E-04 | 7.8302E-02 | 1.7072E-01 | 24623.5504 | 203.0576
Gamma303 | 7.1168E-03 | 2.6278E-05 | 2.5334E-04 | 1.8564E-02 | 15769.2543 | 317.0727
Gamma304 | 5.7247E-03 | 3.1242E-05 | 5.5859E-05 | 1.8575E-02 | 27674.4285 | 180.6722
Gamma305 | 4.6415E-03 | 6.6660E-05 | 1.1610E-04 | 1.1787E-02 | 18508.2806 | 270.1494
Gamma306 | 3.0414E-03 | 7.5922E-06 | 1.2860E-05 | 8.4326E-03 | 18594.0679 | 268.9030
Gamma307 | 3.2661E-03 | 1.1904E-05 | 2.5661E-06 | 1.0671E-02 | 61553.7933 | 81.2298
Gamma308 | 3.6162E-03 | 9.8520E-06 | 5.5995E-06 | 9.4542E-03 | 29464.3761 | 169.6964
Gamma309 | 1.8790E-03 | 3.4469E-06 | 0.0000E+00 | 5.8284E-03 | 28845.1062 | 173.3396
Gamma310 | 4.3399E-03 | 2.5936E-05 | 3.4381E-06 | 1.1586E-02 | 17978.2400 | 278.1140
Gamma311 | 4.5431E-03 | 9.5163E-06 | 2.4009E-04 | 9.5250E-03 | 28937.6912 | 172.7850
Gamma312 | 1.5384E-03 | 2.1657E-06 | 1.7842E-06 | 4.4478E-03 | 25970.4808 | 192.5263
Gamma313 | 7.3209E-01 | 1.1326E-03 | 6.6362E-01 | 7.8946E-01 | 16866.8588 | 296.4393
Gamma314 | 1.8622E-03 | 3.4278E-06 | 6.9320E-07 | 5.5181E-03 | 36575.9053 | 136.7020
Gamma315 | 3.1327E-03 | 4.8260E-06 | 1.2577E-06 | 7.2462E-03 | 16861.5038 | 296.5335
Gamma316 | 2.5437E-03 | 2.2604E-05 | 3.6101E-05 | 7.5467E-03 | 15749.6754 | 317.4669
Gamma317 | 9.1956E-03 | 3.5181E-05 | 4.3308E-05 | 2.1015E-02 | 27536.5460 | 181.5769
Gamma318 | 5.6658E-03 | 2.1964E-05 | 5.5923E-05 | 1.5150E-02 | 16059.1005 | 311.3499
Gamma319 | 4.1368E-03 | 2.9159E-05 | 5.8963E-05 | 1.1556E-02 | 21005.5369 | 238.0325
Gamma320 | 5.5329E-03 | 1.8947E-05 | 1.9608E-04 | 1.5499E-02 | 16956.3464 | 294.8748
Gamma321 | 2.4158E-02 | 6.8231E-05 | 1.0261E-02 | 3.7861E-02 | 6766.1119 | 738.9768
Gamma322 | 3.6947E-03 | 1.1449E-05 | 6.7122E-138 | 9.9310E-03 | 20442.0729 | 244.5936
Gamma323 | 9.9583E-03 | 5.0129E-05 | 5.4610E-04 | 2.2615E-02 | 24769.9109 | 201.8578
Gamma324 | 2.0962E-03 | 5.0383E-06 | 0.0000E+00 | 6.3166E-03 | 18788.8331 | 266.1155
Gamma325 | 4.0082E-03 | 1.7494E-05 | 1.9199E-05 | 1.4044E-02 | 35960.8150 | 139.0402
Gamma326 | 2.4669E-03 | 6.8498E-06 | 2.6688E-06 | 7.8518E-03 | 30845.1923 | 162.0998
Gamma327 | 2.7149E-03 | 1.0769E-05 | 2.8700E-246 | 7.3696E-03 | 20326.7589 | 245.9812
Gamma328 | 4.2796E-03 | 1.2370E-05 | 9.5174E-05 | 1.0872E-02 | 32612.1957 | 153.3169
Gamma329 | 2.5964E-03 | 7.0825E-06 | 5.3002E-06 | 7.6621E-03 | 20614.7673 | 242.5446
Gamma330 | 1.7047E-03 | 1.7693E-05 | 9.7357E-06 | 5.5125E-03 | 23923.4003 | 209.0004
Gamma331 | 1.0921E-03 | 2.7578E-06 | 5.4262E-06 | 3.8869E-03 | 23087.1659 | 216.5705
Gamma332 | 1.5254E-03 | 4.0615E-06 | 2.2535E-05 | 3.9127E-03 | 8463.8927 | 590.7447
Gamma333 | 4.5659E-03 | 9.2828E-06 | 1.0744E-04 | 1.0630E-02 | 43707.7232 | 114.3963
Gamma334 | 8.1321E-03 | 2.3897E-05 | 1.5494E-03 | 1.8872E-02 | 22710.8803 | 220.1588
Gamma335 | 1.6005E-02 | 4.9995E-05 | 5.6207E-03 | 2.8615E-02 | 64528.7687 | 77.4848
Gamma336 | 2.3665E-02 | 6.6610E-05 | 1.0211E-02 | 4.3433E-02 | 21460.8618 | 232.9823
Gamma337 | 4.5288E-02 | 9.5400E-05 | 2.6571E-02 | 6.3111E-02 | 35760.0416 | 139.8209
Gamma338 | 1.9542E-03 | 2.6936E-06 | 1.1975E-105 | 5.3739E-03 | 16936.6509 | 295.2178
Gamma339 | 7.3149E-01 | 6.6108E-04 | 6.8932E-01 | 7.8483E-01 | 41576.3816 | 120.2606
Gamma340 | 1.4309E-03 | 1.5812E-05 | 8.7102E-06 | 3.1581E-03 | 17139.3611 | 291.7262
Gamma341 | 1.5732E-03 | 4.4854E-06 | 0.0000E+00 | 4.3259E-03 | 24026.0963 | 208.1070
Gamma342 | 5.0500E-03 | 1.5580E-05 | 2.3646E-04 | 1.1701E-02 | 33677.8372 | 148.4656
Gamma343 | 1.2851E-01 | 3.7326E-04 | 9.3754E-02 | 1.6243E-01 | 50482.0442 | 99.0451
Gamma344 | 1.1096E-03 | 1.0636E-05 | 1.0640E-103 | 3.2372E-03 | 20442.1127 | 244.5931
Gamma345 | 1.1524E-03 | 1.5336E-06 | 2.3512E-05 | 3.4638E-03 | 15780.5214 | 316.8463
Gamma346 | 2.6502E-03 | 7.4835E-06 | 1.1649E-209 | 6.1499E-03 | 23170.2820 | 215.7937
Gamma347 | 2.4587E-03 | 1.0587E-05 | 2.4929E-66 | 7.9508E-03 | 26962.0248 | 185.4460
Gamma348 | 2.5180E-03 | 4.8702E-06 | 0.0000E+00 | 7.5496E-03 | 18700.3745 | 267.3743
Gamma349 | 2.8176E-03 | 9.8475E-06 | 5.2767E-05 | 8.0141E-03 | 20869.7338 | 239.5814
Gamma350 | 3.2411E-03 | 2.9135E-05 | 2.8385E-06 | 1.0693E-02 | 25938.5472 | 192.7633
Gamma351 | 2.4793E-02 | 3.3018E-04 | 9.0062E-04 | 5.9350E-02 | 33327.8158 | 150.0248
Gamma352 | 5.1001E-02 | 6.1874E-04 | 2.6070E-03 | 9.2298E-02 | 37286.2379 | 134.0977
Gamma353 | 6.1075E-03 | 6.7308E-05 | 0.0000E+00 | 2.5894E-02 | 38588.0048 | 129.5739
Gamma354 | 3.7852E-03 | 1.4613E-05 | 0.0000E+00 | 1.2001E-02 | 8916.7918 | 560.7398
Gamma355 | 7.7790E-03 | 6.9994E-05 | 2.7015E-05 | 2.5643E-02 | 31182.7687 | 160.3450
Gamma356 | 4.9178E-03 | 3.2680E-05 | 0.0000E+00 | 1.7044E-02 | 34944.3649 | 143.0846
Gamma357 | 6.1894E-03 | 5.0146E-05 | 0.0000E+00 | 1.8185E-02 | 29584.5974 | 169.0069
Gamma358 | 7.1705E-03 | 6.8995E-05 | 0.0000E+00 | 2.7270E-02 | 24876.9587 | 200.9892
Gamma359 | 4.6590E-03 | 2.1364E-05 | 6.4461E-06 | 1.3523E-02 | 28535.3499 | 175.2213
Gamma360 | 9.1733E-03 | 5.5219E-05 | 3.8856E-05 | 2.3279E-02 | 37390.2793 | 133.7246
Gamma361 | 6.2608E-03 | 7.4079E-05 | 0.0000E+00 | 2.5291E-02 | 40482.6649 | 123.5097
Gamma362 | 3.8615E-03 | 1.0624E-05 | 7.8225E-06 | 9.8337E-03 | 14735.0059 | 339.3280
Gamma363 | 7.0250E-03 | 5.3764E-05 | 0.0000E+00 | 2.0566E-02 | 24005.2184 | 208.2880
Gamma364 | 4.3183E-03 | 2.7370E-05 | 1.8785E-06 | 1.2305E-02 | 29506.0729 | 169.4566
Gamma365 | 6.0414E-01 | 3.5649E-03 | 4.6988E-01 | 7.0603E-01 | 45696.0153 | 109.4187
Gamma366 | 1.6645E-02 | 1.3086E-04 | 4.1311E-04 | 3.2876E-02 | 12823.9967 | 389.8940
Gamma367 | 3.3010E-02 | 3.7879E-04 | 6.9911E-03 | 6.6756E-02 | 28993.9751 | 172.4496
Gamma368 | 1.8315E-02 | 1.6982E-04 | 3.3724E-04 | 4.5120E-02 | 12876.7647 | 388.2963
Gamma369 | 6.8801E-03 | 4.4235E-05 | 1.1847E-04 | 2.0388E-02 | 28752.5357 | 173.8977
Gamma370 | 1.0269E-01 | 1.5614E-03 | 2.4741E-02 | 1.7153E-01 | 36482.7202 | 137.0512
Gamma371 | 1.8484E-02 | 1.1383E-04 | 1.8662E-04 | 3.6468E-02 | 22382.4058 | 223.3897
Gamma372 | 1.0651E-02 | 1.2899E-04 | 0.0000E+00 | 3.1775E-02 | 20097.0589 | 248.7926
Gamma373 | 1.7435E-02 | 1.5667E-04 | 3.3653E-04 | 3.8187E-02 | 20262.4381 | 246.7620
Gamma374 | 8.3769E-03 | 9.3021E-05 | 0.0000E+00 | 3.0907E-02 | 36945.3840 | 135.3349
Gamma375 | 1.6333E-02 | 1.4734E-04 | 8.4483E-04 | 4.1077E-02 | 27353.5296 | 182.7918
Gamma376 | 2.5206E-03 | 5.0930E-06 | 2.9205E-05 | 7.2401E-03 | 20979.2645 | 238.3306
Gamma377 | 3.1381E-03 | 9.6389E-06 | 0.0000E+00 | 9.6390E-03 | 23175.4021 | 215.7460
Gamma378 | 3.2783E-03 | 4.6262E-06 | 1.4914E-04 | 7.0207E-03 | 18371.5781 | 272.1595
Gamma379 | 1.9466E-03 | 4.3353E-06 | 1.9932E-05 | 6.0063E-03 | 37457.2433 | 133.4855
Gamma380 | 1.5848E-03 | 3.9237E-06 | 0.0000E+00 | 4.5166E-03 | 20140.3158 | 248.2583
Gamma381 | 3.8456E-03 | 2.2601E-05 | 2.3276E-05 | 1.4362E-02 | 28454.7058 | 175.7179
Gamma382 | 2.3889E-03 | 1.7898E-05 | 5.8433E-06 | 7.2489E-03 | 20698.5296 | 241.5631
Gamma383 | 1.9750E-03 | 4.4970E-06 | 5.0407E-06 | 5.5307E-03 | 20741.2644 | 241.0653
Gamma384 | 3.1523E-03 | 6.6443E-06 | 1.1953E-04 | 7.5092E-03 | 20241.7043 | 247.0148
Gamma385 | 3.1986E-02 | 6.3682E-05 | 1.9275E-02 | 4.8261E-02 | 20170.9672 | 247.8810
Gamma386 | 1.9295E-03 | 5.7482E-06 | 9.4874E-06 | 6.3732E-03 | 21484.8529 | 232.7221
Gamma387 | 1.3705E-03 | 2.1416E-06 | 9.1667E-06 | 4.3620E-03 | 30328.2155 | 164.8630
Gamma388 | 1.6270E-03 | 2.5758E-06 | 1.9328E-07 | 4.6204E-03 | 27632.8669 | 180.9439
Gamma389 | 1.7055E-03 | 2.6181E-06 | 0.0000E+00 | 5.0687E-03 | 27168.1551 | 184.0390
Gamma390 | 1.0854E-02 | 2.1510E-05 | 3.3281E-03 | 2.0403E-02 | 29059.2892 | 172.0620
Gamma391 | 7.1509E-01 | 5.9987E-04 | 6.6839E-01 | 7.6232E-01 | 16853.6152 | 296.6723
Gamma392 | 2.8702E-03 | 1.3946E-05 | 0.0000E+00 | 1.1007E-02 | 33926.4352 | 147.3777
Gamma393 | 3.2691E-03 | 1.4818E-05 | 5.2630E-06 | 1.2622E-02 | 12896.3590 | 387.7063
Gamma394 | 3.4814E-02 | 9.0375E-05 | 1.6997E-02 | 5.2017E-02 | 22320.0818 | 224.0135
Gamma395 | 1.3375E-01 | 2.6785E-04 | 9.8756E-02 | 1.6443E-01 | 31435.8188 | 159.0542
Gamma396 | 3.6185E-03 | 1.2274E-05 | 2.1532E-04 | 1.0387E-02 | 17364.7334 | 287.9399
Gamma397 | 3.5225E-03 | 1.0651E-05 | 0.0000E+00 | 9.8298E-03 | 23028.6780 | 217.1206
Gamma398 | 1.2023E-02 | 4.8850E-05 | 5.7406E-04 | 2.4273E-02 | 43936.2306 | 113.8013
Gamma399 | 7.1335E-03 | 2.2365E-05 | 2.8489E-04 | 1.5523E-02 | 19134.3241 | 261.3105
Gamma400 | 1.0605E-02 | 3.5265E-05 | 2.2637E-03 | 1.9447E-02 | 18513.1736 | 270.0780
Gamma401 | 1.4145E-01 | 5.6709E-04 | 9.5771E-02 | 1.8805E-01 | 46561.6260 | 107.3846
Gamma402 | 1.1253E-02 | 6.0266E-05 | 8.2796E-06 | 2.6947E-02 | 31745.4089 | 157.5031
Gamma403 | 1.0800E-02 | 2.5388E-05 | 1.8994E-03 | 2.0844E-02 | 14008.5219 | 356.9256
Gamma404 | 1.4723E-03 | 2.4668E-06 | 8.5575E-07 | 4.0427E-03 | 27293.1690 | 183.1960
Gamma405 | 5.1690E-03 | 6.1864E-06 | 8.0014E-04 | 1.0367E-02 | 17573.1706 | 284.5246
Gamma406 | 2.4912E-03 | 4.0882E-06 | 1.5439E-04 | 6.6381E-03 | 17474.4905 | 286.1314
Gamma407 | 2.6837E-03 | 5.2157E-06 | 3.7603E-05 | 7.0031E-03 | 25231.3303 | 198.1663
Gamma408 | 2.7472E-03 | 7.4211E-06 | 4.1349E-05 | 8.4109E-03 | 83638.5474 | 59.7810
Gamma409 | 2.4838E-03 | 8.3758E-06 | 3.0316E-05 | 8.3449E-03 | 14372.5505 | 347.8854
Gamma410 | 9.7625E-03 | 3.1184E-05 | 8.8820E-04 | 2.0287E-02 | 23276.3846 | 214.8100
Gamma411 | 2.5801E-02 | 7.9244E-05 | 6.7707E-03 | 4.3522E-02 | 31942.0713 | 156.5334
Gamma412 | 3.0284E-03 | 6.9727E-06 | 1.9368E-05 | 7.8618E-03 | 24849.1321 | 201.2143
Gamma413 | 4.9011E-03 | 1.0844E-05 | 9.8124E-05 | 1.0737E-02 | 12502.6231 | 399.9161
Gamma414 | 6.3975E-03 | 1.7203E-05 | 1.3161E-03 | 1.6126E-02 | 13817.0982 | 361.8705
Gamma415 | 5.2195E-03 | 1.0166E-05 | 1.4682E-04 | 1.1186E-02 | 17327.3056 | 288.5619
Gamma416 | 2.4188E-03 | 3.8366E-06 | 0.0000E+00 | 5.6751E-03 | 13496.7349 | 370.4600
Gamma417 | 7.0950E-01 | 6.3763E-04 | 6.5662E-01 | 7.5761E-01 | 46169.2866 | 108.2971
Gamma418 | 1.0090E-02 | 4.1201E-05 | 2.1459E-04 | 1.9922E-02 | 49123.0598 | 101.7852
Gamma419 | 2.8005E-03 | 7.3404E-06 | 4.7947E-05 | 9.7403E-03 | 25931.2775 | 192.8173
Gamma420 | 4.8523E-03 | 2.1230E-05 | 1.3610E-05 | 1.5738E-02 | 16063.6154 | 311.2624
Gamma421 | 2.1635E-02 | 6.2529E-05 | 6.5801E-03 | 3.7325E-02 | 16900.7761 | 295.8444
Gamma422 | 2.8975E-03 | 6.7063E-06 | 0.0000E+00 | 8.1080E-03 | 27776.0669 | 180.0111
Gamma423 | 5.6688E-03 | 2.0444E-05 | 6.2900E-05 | 1.4027E-02 | 15309.3126 | 326.5986
Gamma424 | 2.8593E-03 | 8.1163E-06 | 8.5713E-06 | 8.6236E-03 | 64005.0544 | 78.1188
Gamma425 | 1.6124E-03 | 2.2344E-06 | 3.4202E-06 | 4.5395E-03 | 25226.5574 | 198.2038
Gamma426 | 2.0740E-03 | 4.0899E-06 | 2.0857E-05 | 6.1500E-03 | 31370.0390 | 159.3878
Gamma427 | 9.9544E-03 | 2.0338E-05 | 6.3560E-04 | 1.6414E-02 | 43002.3537 | 116.2727
Gamma428 | 2.9525E-03 | 7.1782E-06 | 2.1226E-04 | 7.0641E-03 | 25471.4357 | 196.2983
Gamma429 | 1.6369E-03 | 2.7502E-06 | 9.2684E-06 | 4.9787E-03 | 24336.0265 | 205.4567
Gamma430 | 1.3922E-03 | 3.1853E-06 | 0.0000E+00 | 5.0678E-03 | 32406.8350 | 154.2884
Gamma431 | 1.4245E-03 | 6.8981E-06 | 5.7031E-06 | 4.4679E-03 | 26461.6035 | 188.9530
Gamma432 | 1.4795E-03 | 2.3451E-06 | 6.7581E-06 | 3.8377E-03 | 28039.3865 | 178.3206
Gamma433 | 1.5035E-02 | 6.3199E-05 | 3.4628E-03 | 2.6602E-02 | 43256.3488 | 115.5900
Gamma434 | 5.0448E-03 | 7.4097E-06 | 3.6836E-04 | 1.0538E-02 | 33605.9976 | 148.7830
Gamma435 | 1.7197E-03 | 9.7988E-06 | 6.9005E-06 | 4.5959E-03 | 14081.1043 | 355.0858
Gamma436 | 3.7395E-03 | 1.7747E-05 | 9.4233E-05 | 1.1876E-02 | 17872.4177 | 279.7607
Gamma437 | 4.4831E-02 | 8.2069E-05 | 2.4317E-02 | 5.9053E-02 | 55420.5429 | 90.2193
Gamma438 | 2.0099E-03 | 2.9398E-06 | 4.0242E-06 | 5.3215E-03 | 23310.4666 | 214.4959
Gamma439 | 5.7917E-02 | 5.4776E-05 | 4.4385E-02 | 7.1556E-02 | 18974.4135 | 263.5128
Gamma440 | 1.2071E-03 | 8.7368E-07 | 2.0884E-05 | 3.0730E-03 | 21556.4801 | 231.9488
Gamma441 | 7.7877E-04 | 1.2320E-06 | 3.9643E-06 | 2.7132E-03 | 34442.7165 | 145.1686
Gamma442 | 2.8372E-03 | 6.8045E-06 | 3.3803E-06 | 5.9055E-03 | 14566.5388 | 343.2524
Gamma443 | 8.3304E-01 | 1.6314E-04 | 8.0840E-01 | 8.5844E-01 | 32451.6730 | 154.0753
Gamma444 | 7.0302E-04 | 5.6680E-07 | 2.1415E-235 | 1.9467E-03 | 22644.6403 | 220.8028
Gamma445 | 1.2805E-03 | 1.7479E-06 | 0.0000E+00 | 3.6397E-03 | 22109.1297 | 226.1509
Gamma446 | 7.8547E-04 | 8.4742E-07 | 1.3043E-05 | 2.7585E-03 | 31651.8098 | 157.9689
Gamma447 | 2.0870E-03 | 4.3782E-06 | 1.0134E-05 | 6.9000E-03 | 20642.3162 | 242.2209
Gamma448 | 2.9245E-03 | 1.2113E-05 | 2.6652E-06 | 6.9217E-03 | 13529.3141 | 369.5679
Gamma449 | 1.1445E-03 | 1.0144E-06 | 1.9544E-05 | 3.1386E-03 | 20352.2298 | 245.6733
Gamma450 | 2.0047E-03 | 3.2108E-06 | 3.2630E-06 | 5.7242E-03 | 17725.7266 | 282.0759
Gamma451 | 7.1105E-03 | 4.8820E-05 | 5.8517E-06 | 2.3692E-02 | 26338.7144 | 189.8346
Gamma452 | 1.2013E-02 | 1.0252E-04 | 1.1897E-05 | 3.1603E-02 | 23952.7096 | 208.7447
Gamma453 | 4.1419E-03 | 3.2262E-05 | 8.8946E-05 | 1.6086E-02 | 48457.2786 | 103.1837
Gamma454 | 6.0563E-03 | 1.1015E-04 | 0.0000E+00 | 1.7585E-02 | 22774.3455 | 219.5453
Gamma455 | 5.1958E-03 | 1.9394E-05 | 1.4232E-04 | 1.3593E-02 | 29107.8441 | 171.7750
Gamma456 | 9.3003E-03 | 3.1841E-05 | 3.9921E-04 | 2.1125E-02 | 24715.7050 | 202.3005
Gamma457 | 5.6489E-03 | 2.3863E-05 | 0.0000E+00 | 1.6950E-02 | 25998.2037 | 192.3210
Gamma458 | 2.2142E-03 | 5.1309E-06 | 0.0000E+00 | 7.4352E-03 | 33010.7600 | 151.4658
Gamma459 | 3.2802E-03 | 1.5509E-05 | 0.0000E+00 | 8.8765E-03 | 11562.4319 | 432.4350
Gamma460 | 2.5709E-02 | 1.0707E-04 | 7.3453E-03 | 4.4883E-02 | 19696.9727 | 253.8461
Gamma461 | 3.2744E-03 | 8.1703E-06 | 2.0800E-05 | 8.9429E-03 | 50279.2180 | 99.4447
Gamma462 | 3.7221E-03 | 1.0454E-05 | 2.1471E-06 | 1.1410E-02 | 21433.3031 | 233.2818
Gamma463 | 3.6508E-03 | 1.4389E-05 | 1.4977E-05 | 1.0892E-02 | 28069.2042 | 178.1312
Gamma464 | 4.3139E-03 | 1.7651E-05 | 5.4271E-05 | 1.2524E-02 | 21386.6295 | 233.7909
Gamma465 | 6.8082E-03 | 2.5981E-05 | 3.0086E-04 | 1.5746E-02 | 15262.2073 | 327.6066
Gamma466 | 6.6788E-02 | 2.5808E-04 | 3.8270E-02 | 9.6085E-02 | 33162.6412 | 150.7721
Gamma467 | 4.7363E-03 | 2.5146E-05 | 2.1756E-05 | 1.3580E-02 | 24894.7832 | 200.8453
Gamma468 | 2.5679E-03 | 1.2170E-05 | 0.0000E+00 | 9.7807E-03 | 45902.5130 | 108.9265
Gamma469 | 5.5808E-01 | 1.4357E-03 | 4.7133E-01 | 6.2205E-01 | 28278.1542 | 176.8149
Gamma470 | 1.6805E-01 | 8.0602E-04 | 1.2388E-01 | 2.3427E-01 | 35056.1824 | 142.6282
Gamma471 | 4.1599E-03 | 2.0401E-05 | 1.7421E-06 | 1.2528E-02 | 17364.7550 | 287.9396
Gamma472 | 7.6984E-03 | 5.7707E-05 | 1.7294E-66 | 2.1739E-02 | 33902.9657 | 147.4797
Gamma473 | 5.9583E-02 | 5.5918E-04 | 1.4012E-02 | 9.9320E-02 | 102335.1947 | 48.8590
Gamma474 | 5.1308E-03 | 3.7137E-05 | 2.9308E-05 | 1.9383E-02 | 18410.6335 | 271.5822
Gamma475 | 2.0768E-02 | 6.8060E-05 | 6.6840E-03 | 3.7132E-02 | 18835.5876 | 265.4550
Gamma476 | 4.5246E-03 | 2.9527E-05 | 1.2246E-05 | 1.5367E-02 | 28418.4006 | 175.9423
Gamma477 | 7.9723E-03 | 3.4245E-05 | 3.6387E-04 | 1.6831E-02 | 28131.9693 | 177.7337
Gamma478 | 1.8001E-03 | 3.8621E-06 | 3.6605E-06 | 5.4703E-03 | 34500.3679 | 144.9260
Gamma479 | 2.1294E-03 | 2.6664E-06 | 2.6298E-06 | 5.2907E-03 | 10274.7777 | 486.6285
Gamma480 | 1.8446E-03 | 3.9387E-06 | 2.2398E-05 | 5.2873E-03 | 45227.5387 | 110.5521
Gamma481 | 1.7509E-03 | 3.8855E-06 | 3.6611E-97 | 4.4719E-03 | 21426.9780 | 233.3507
Gamma482 | 2.7513E-03 | 1.2285E-05 | 1.4434E-05 | 6.9678E-03 | 25084.5453 | 199.3259
Gamma483 | 3.2090E-03 | 9.4208E-06 | 6.0894E-05 | 8.6159E-03 | 35241.6479 | 141.8776
Gamma484 | 1.2637E-03 | 1.5646E-06 | 0.0000E+00 | 3.2678E-03 | 26065.9621 | 191.8210
Gamma485 | 6.6660E-03 | 1.2467E-05 | 1.1263E-03 | 1.3715E-02 | 21821.2805 | 229.1341
Gamma486 | 9.9720E-04 | 1.2262E-06 | 0.0000E+00 | 3.8547E-03 | 40447.7451 | 123.6163
Gamma487 | 1.6581E-03 | 5.8962E-06 | 8.4528E-06 | 5.3982E-03 | 11862.0810 | 421.5112
Gamma488 | 3.1126E-03 | 6.7946E-06 | 6.7868E-05 | 7.3754E-03 | 23350.4316 | 214.1288
Gamma489 | 1.2443E-03 | 2.4282E-06 | 6.0437E-06 | 3.9047E-03 | 26562.3243 | 188.2365
Gamma490 | 1.1049E-02 | 2.9569E-05 | 8.7339E-04 | 1.9780E-02 | 23895.0789 | 209.2481
Gamma491 | 4.8344E-02 | 1.0015E-04 | 2.9564E-02 | 6.9302E-02 | 25991.7975 | 192.3684
Gamma492 | 2.3050E-03 | 3.3382E-06 | 9.9957E-05 | 5.4752E-03 | 17442.8134 | 286.6510
Gamma493 | 1.8940E-03 | 1.0811E-05 | 0.0000E+00 | 4.7140E-03 | 30163.3613 | 165.7640
Gamma494 | 3.2402E-02 | 5.7514E-05 | 2.1006E-02 | 4.7081E-02 | 24356.8705 | 205.2809
Gamma495 | 8.2957E-01 | 3.3422E-04 | 7.9682E-01 | 8.6667E-01 | 20813.6358 | 240.2271
Gamma496 | 1.7734E-03 | 2.7448E-06 | 1.5778E-06 | 5.3464E-03 | 27772.7873 | 180.0323
Gamma497 | 2.8291E-03 | 7.4174E-06 | 2.0672E-238 | 7.7762E-03 | 66433.4908 | 75.2632
Gamma498 | 1.9807E-02 | 1.1193E-04 | 4.7007E-03 | 3.9870E-02 | 55844.1523 | 89.5349
Gamma499 | 3.7384E-03 | 1.0750E-05 | 6.1443E-06 | 9.4147E-03 | 27539.7138 | 181.5560
Gamma500 | 5.3652E-03 | 9.2246E-06 | 8.6062E-04 | 1.1890E-02 | 72948.9489 | 68.5411
Gamma501 | 1.3157E-01 | 2.7214E-04 | 1.0350E-01 | 1.6442E-01 | 16829.9330 | 297.0897
Gamma502 | 3.6686E-03 | 1.1225E-05 | 1.7460E-06 | 1.0535E-02 | 37864.7786 | 132.0488
Gamma503 | 8.1465E-03 | 1.5319E-05 | 2.2713E-03 | 1.5033E-02 | 16359.8070 | 305.6271
Gamma504 | 3.0005E-03 | 7.1040E-06 | 2.8331E-07 | 8.7553E-03 | 22353.0670 | 223.6830
Gamma505 | 2.3504E-03 | 4.2734E-06 | 6.3767E-06 | 6.3886E-03 | 19613.4301 | 254.9274
Gamma506 | 3.4265E-03 | 6.4921E-06 | 1.4338E-04 | 7.8648E-03 | 28198.0034 | 177.3175
Gamma507 | 2.2086E-03 | 4.2489E-06 | 6.6654E-06 | 6.9498E-03 | 28960.2604 | 172.6504
Gamma508 | 1.8329E-03 | 3.4288E-06 | 2.5296E-05 | 4.4587E-03 | 20578.9871 | 242.9663
Gamma509 | 1.7202E-03 | 3.1004E-06 | 0.0000E+00 | 5.2366E-03 | 22627.6876 | 220.9682
Gamma510 | 2.4846E-02 | 5.0564E-05 | 8.7965E-03 | 3.6573E-02 | 19652.0538 | 254.4263
Gamma511 | 3.0427E-03 | 7.0590E-06 | 4.7212E-07 | 7.9829E-03 | 24954.0345 | 200.3684
Gamma512 | 1.6581E-03 | 2.4683E-06 | 5.2152E-07 | 4.7452E-03 | 42106.3515 | 118.7469
Gamma513 | 2.7028E-03 | 7.0896E-06 | 3.5590E-205 | 8.0023E-03 | 33245.8452 | 150.3947
Gamma514 | 2.5833E-03 | 8.3656E-06 | 1.1496E-06 | 6.5102E-03 | 21024.7301 | 237.8152
Gamma515 | 1.4949E-02 | 2.1112E-05 | 7.6734E-03 | 2.3874E-02 | 14787.2077 | 338.1301
Gamma516 | 4.5785E-03 | 9.8908E-06 | 1.4013E-04 | 9.7391E-03 | 28731.1410 | 174.0272
Gamma517 | 2.1947E-02 | 1.2392E-04 | 7.0573E-03 | 4.2178E-02 | 11922.9571 | 419.3591
Gamma518 | 2.0214E-03 | 3.1687E-06 | 1.8265E-05 | 5.6039E-03 | 16504.4170 | 302.9492
Gamma519 | 1.9063E-03 | 1.5384E-05 | 0.0000E+00 | 5.5721E-03 | 17525.8421 | 285.2930
Gamma520 | 2.5680E-03 | 1.0041E-05 | 1.2352E-253 | 7.7990E-03 | 17917.6273 | 279.0548
Gamma521 | 7.4956E-01 | 6.0872E-04 | 7.0241E-01 | 7.8861E-01 | 30806.1306 | 162.3054
Gamma522 | 2.2303E-03 | 4.3630E-06 | 1.5289E-05 | 5.6639E-03 | 9786.4206 | 510.9120
Gamma523 | 3.3383E-03 | 2.8680E-05 | 1.7586E-06 | 1.4263E-02 | 31699.7803 | 157.7298
Gamma524 | 1.2529E-03 | 1.4225E-06 | 0.0000E+00 | 3.6943E-03 | 12450.4991 | 401.5903
Gamma525 | 2.8892E-03 | 3.2576E-05 | 2.7673E-06 | 1.0769E-02 | 24567.9091 | 203.5175
Gamma526 | 1.0670E-03 | 1.3093E-06 | 3.7588E-305 | 3.7336E-03 | 25525.8323 | 195.8800
Gamma527 | 6.8261E-03 | 2.3397E-05 | 2.8033E-04 | 1.6208E-02 | 51743.5942 | 96.6303
Gamma528 | 3.3637E-02 | 2.6144E-04 | 2.8814E-03 | 5.1314E-02 | 56874.1537 | 87.9134
Gamma529 | 1.2611E-02 | 3.3020E-05 | 1.6657E-03 | 2.2728E-02 | 169947.5985 | 29.4208
Gamma530 | 1.6999E-03 | 5.2823E-06 | 1.3537E-05 | 4.3540E-03 | 25726.4129 | 194.3528
Gamma531 | 3.5967E-03 | 3.5060E-06 | 8.3529E-06 | 7.0724E-03 | 17345.1166 | 288.2656
Gamma532 | 3.1536E-03 | 5.6114E-06 | 7.2577E-113 | 7.7095E-03 | 33188.7328 | 150.6535
Gamma533 | 8.3641E-03 | 1.9661E-05 | 1.6485E-03 | 1.7416E-02 | 26295.3614 | 190.1476
Gamma534 | 8.4514E-03 | 2.7209E-05 | 2.8089E-04 | 1.7232E-02 | 124216.1730 | 40.2524
Gamma535 | 2.6716E-03 | 1.3608E-05 | 4.0078E-06 | 7.4991E-03 | 81553.5622 | 61.3094
Gamma536 | 9.3193E-04 | 2.9252E-06 | 7.9932E-06 | 3.8403E-03 | 18179.8773 | 275.0294
Gamma537 | 1.1114E-03 | 1.3498E-06 | 6.6574E-06 | 3.2932E-03 | 19534.9622 | 255.9514
Gamma538 | 1.5258E-03 | 3.8594E-06 | 2.1416E-06 | 4.9796E-03 | 15821.6917 | 316.0218
Gamma539 | 1.3697E-03 | 6.3098E-06 | 2.7123E-06 | 5.3163E-03 | 18226.7989 | 274.3213
Gamma540 | 1.8537E-03 | 8.1875E-06 | 9.3136E-07 | 8.5293E-03 | 103611.6168 | 48.2571
Gamma541 | 1.5786E-03 | 8.1487E-06 | 8.1707E-06 | 4.0892E-03 | 21485.0467 | 232.7200
Gamma542 | 1.4865E-03 | 6.0516E-06 | 0.0000E+00 | 5.9249E-03 | 27535.5807 | 181.5832
Gamma543 | 1.1454E-03 | 2.7441E-06 | 9.3162E-06 | 4.0491E-03 | 28161.6175 | 177.5466
Gamma544 | 1.0501E-03 | 3.8784E-06 | 9.7212E-238 | 4.1070E-03 | 17075.4013 | 292.8189
Gamma545 | 1.1498E-03 | 1.1439E-06 | 2.2421E-06 | 3.1272E-03 | 12510.7885 | 399.6551
Gamma546 | 8.7337E-04 | 2.8368E-06 | 1.0350E-05 | 2.3897E-03 | 12978.9781 | 385.2383
Gamma547 | 8.8294E-01 | 2.7952E-04 | 8.5074E-01 | 9.1435E-01 | 18172.1093 | 275.1469
Gamma548 | 5.6532E-03 | 2.3565E-05 | 6.5690E-05 | 1.6203E-02 | 123254.6503 | 40.5664
Gamma549 | 1.4508E-03 | 1.4327E-05 | 6.4857E-07 | 4.0106E-03 | 10878.6212 | 459.6171
Gamma550 | 1.3803E-02 | 1.8331E-04 | 2.2917E-03 | 4.4794E-02 | 34666.3023 | 144.2323
Gamma551 | 6.7310E-03 | 2.7680E-05 | 1.1266E-04 | 1.7192E-02 | 35142.0623 | 142.2796
Gamma552 | 2.4209E-01 | 8.7090E-03 | 3.8008E-02 | 3.7133E-01 | 162869.1094 | 30.6995
Gamma553 | 3.1470E-03 | 6.9356E-06 | 2.8290E-05 | 8.0054E-03 | 33272.2618 | 150.2753
Gamma554 | 3.9453E-03 | 1.3509E-05 | 1.0103E-04 | 1.2127E-02 | 24520.4763 | 203.9112
Gamma555 | 3.6845E-03 | 9.3521E-06 | 5.2495E-05 | 9.5950E-03 | 24818.6934 | 201.4610
Gamma556 | 7.3746E-03 | 2.0730E-05 | 1.2310E-03 | 1.6435E-02 | 29287.3525 | 170.7222
Gamma557 | 4.5495E-03 | 1.0316E-05 | 3.0097E-04 | 1.0594E-02 | 21781.4116 | 229.5535
Gamma558 | 9.9648E-03 | 5.5931E-05 | 9.4161E-04 | 2.7924E-02 | 32907.5859 | 151.9407
Gamma559 | 1.9224E-03 | 3.7264E-06 | 1.4318E-05 | 5.3916E-03 | 25555.6939 | 195.6511
Gamma560 | 8.1419E-03 | 2.9276E-05 | 1.1023E-03 | 1.9852E-02 | 81626.1404 | 61.2549
Gamma561 | 1.9963E-03 | 7.0339E-06 | 1.4485E-286 | 5.6362E-03 | 39974.1572 | 125.0808
Gamma562 | 1.4446E-03 | 2.2048E-06 | 1.4879E-05 | 4.5418E-03 | 44515.1380 | 112.3213
Gamma563 | 6.0201E-03 | 5.3759E-05 | 9.1406E-05 | 1.8311E-02 | 43368.0244 | 115.2923
Gamma564 | 2.6932E-03 | 6.5481E-06 | 8.6165E-05 | 7.5424E-03 | 19725.0931 | 253.4842
Gamma565 | 4.9005E-03 | 8.1615E-06 | 3.5200E-04 | 1.0600E-02 | 45810.4813 | 109.1453
Gamma566 | 6.7075E-03 | 6.6712E-05 | 3.4673E-06 | 1.4944E-02 | 50579.6758 | 98.8539
Gamma567 | 3.6168E-03 | 1.1999E-05 | 1.0541E-04 | 1.0273E-02 | 22432.4382 | 222.8915
Gamma568 | 5.4354E-03 | 2.0577E-05 | 4.0865E-07 | 1.4374E-02 | 34782.8385 | 143.7491
Gamma569 | 1.3744E-02 | 4.3454E-05 | 4.8660E-03 | 2.7866E-02 | 89800.7279 | 55.6788
Gamma570 | 2.0294E-02 | 2.9998E-04 | 2.1426E-03 | 6.0205E-02 | 108491.3389 | 46.0866
Gamma571 | 1.2219E-03 | 2.3025E-06 | 0.0000E+00 | 3.5279E-03 | 26854.9602 | 186.1853
Gamma572 | 1.0003E-02 | 3.9214E-05 | 6.9248E-04 | 2.2608E-02 | 70626.9782 | 70.7945
Gamma573 | 6.1290E-01 | 1.9880E-02 | 3.9848E-01 | 9.0596E-01 | 63459.6414 | 78.7902
Gamma574 | 7.6587E-03 | 2.6901E-05 | 1.7922E-04 | 1.6135E-02 | 37523.9088 | 133.2484
Gamma575 | 9.8089E-03 | 4.2591E-05 | 1.0069E-03 | 2.3786E-02 | 63298.2454 | 78.9911
Gamma576 | 8.3441E-03 | 5.9303E-05 | 1.0641E-04 | 2.4359E-02 | 48248.9802 | 103.6291
Gamma577 | 1.1359E-01 | 5.5179E-04 | 7.9416E-02 | 1.6391E-01 | 29470.7144 | 169.6600
Gamma578 | 2.0572E-03 | 4.9978E-06 | 1.9914E-06 | 5.1430E-03 | 19695.0448 | 253.8710
Gamma579 | 5.0992E-03 | 1.8647E-05 | 1.0823E-05 | 1.2779E-02 | 45835.5124 | 109.0857
Gamma580 | 3.0145E-03 | 1.6992E-05 | 9.0696E-06 | 9.2287E-03 | 18390.8363 | 271.8745
Gamma581 | 6.3175E-03 | 2.1863E-05 | 2.0929E-04 | 1.5309E-02 | 37372.7549 | 133.7873
Gamma582 | 5.5664E-03 | 1.5330E-05 | 2.5135E-04 | 1.2761E-02 | 27636.1311 | 180.9226
Gamma583 | 7.1495E-03 | 2.5749E-05 | 2.6355E-05 | 1.7411E-02 | 16700.0796 | 299.3998
Gamma584 | 2.3959E-03 | 9.8418E-06 | 0.0000E+00 | 8.6192E-03 | 31609.9241 | 158.1782
Gamma585 | 1.3901E-02 | 5.7898E-05 | 1.1593E-04 | 2.6953E-02 | 28774.4422 | 173.7653
Gamma586 | 4.5053E-03 | 1.5922E-05 | 1.3999E-05 | 1.1896E-02 | 24670.6474 | 202.6700
Gamma587 | 2.3749E-03 | 4.3713E-06 | 4.6294E-06 | 6.3873E-03 | 18980.6039 | 263.4268
Gamma588 | 6.2443E-03 | 1.9280E-05 | 6.4128E-04 | 1.6022E-02 | 19087.7909 | 261.9475
Gamma589 | 2.6017E-03 | 7.3585E-06 | 0.0000E+00 | 7.6996E-03 | 29942.2713 | 166.9880
Gamma590 | 3.9186E-03 | 3.8209E-05 | 8.5562E-07 | 8.2811E-03 | 38640.8552 | 129.3967
Gamma591 | 3.3120E-02 | 1.3032E-04 | 1.4226E-02 | 5.1841E-02 | 14839.2761 | 336.9437
Gamma592 | 1.1356E-02 | 6.4249E-05 | 3.9847E-04 | 2.5152E-02 | 36637.9856 | 136.4704
Gamma593 | 2.5612E-03 | 7.2270E-06 | 1.8899E-05 | 8.2262E-03 | 43793.8830 | 114.1712
Gamma594 | 4.0829E-03 | 9.9005E-06 | 1.9282E-139 | 9.1253E-03 | 26159.3926 | 191.1359
Gamma595 | 1.6455E-02 | 1.4091E-04 | 9.8861E-04 | 4.0410E-02 | 25487.0538 | 196.1780
Gamma596 | 4.6105E-03 | 7.1124E-05 | 1.6447E-05 | 2.3031E-02 | 12820.6463 | 389.9959
Gamma597 | 1.2425E-02 | 1.0413E-04 | 2.9849E-04 | 3.5192E-02 | 29425.0675 | 169.9231
Gamma598 | 5.9868E-03 | 2.9668E-05 | 9.7568E-307 | 1.4641E-02 | 19891.5709 | 251.3628
Gamma599 | 7.1850E-01 | 1.4458E-03 | 6.4775E-01 | 7.9540E-01 | 43147.8720 | 115.8806
Gamma600 | 3.8137E-03 | 1.5407E-05 | 3.7176E-107 | 1.2743E-02 | 36807.9025 | 135.8404
Gamma601 | 4.5700E-03 | 2.5936E-05 | 1.2255E-05 | 1.4629E-02 | 61878.8181 | 80.8031
Gamma602 | 5.9703E-02 | 5.4771E-04 | 8.3209E-03 | 9.8580E-02 | 97691.4403 | 51.1816
Gamma603 | 3.7547E-03 | 1.2219E-05 | 1.3979E-06 | 1.0969E-02 | 18740.0983 | 266.8076
Gamma604 | 3.3626E-03 | 3.0958E-05 | 7.2764E-06 | 9.5617E-03 | 23786.6937 | 210.2016
Gamma605 | 2.8119E-03 | 1.2861E-05 | 1.2029E-05 | 1.0376E-02 | 39130.1276 | 127.7788
Gamma606 | 8.8095E-03 | 3.7352E-05 | 7.7500E-04 | 2.0562E-02 | 16802.9664 | 297.5665
Gamma607 | 5.9677E-03 | 2.1292E-05 | 1.5958E-05 | 1.4337E-02 | 26793.4490 | 186.6128
Gamma608 | 6.4906E-03 | 3.8496E-05 | 8.2154E-05 | 1.8456E-02 | 25738.6885 | 194.2601
Gamma609 | 3.4075E-03 | 1.1365E-05 | 1.9948E-06 | 9.8028E-03 | 19701.1370 | 253.7925
Gamma610 | 3.7020E-02 | 1.5706E-04 | 1.8155E-02 | 6.0903E-02 | 30342.1325 | 164.7874
Gamma611 | 2.7671E-03 | 7.2091E-06 | 1.7110E-06 | 8.7172E-03 | 13700.4153 | 364.9524
Gamma612 | 2.5592E-03 | 7.2477E-06 | 0.0000E+00 | 7.1332E-03 | 42692.5417 | 117.1165
Gamma613 | 5.1943E-03 | 1.5530E-05 | 1.3503E-06 | 1.2906E-02 | 51838.7161 | 96.4530
Gamma614 | 3.7065E-03 | 1.0699E-05 | 2.1408E-06 | 1.0606E-02 | 24862.7780 | 201.1038
Gamma615 | 4.0503E-03 | 9.8144E-06 | 4.5949E-05 | 1.0229E-02 | 42991.9561 | 116.3008
Gamma616 | 1.5509E-02 | 6.6455E-05 | 5.7150E-03 | 3.5507E-02 | 34656.8095 | 144.2718
Gamma617 | 5.9456E-03 | 2.8665E-05 | 1.7264E-04 | 1.7430E-02 | 31807.6331 | 157.1950
Gamma618 | 1.0786E-02 | 5.9151E-05 | 5.5395E-05 | 2.5488E-02 | 16244.7560 | 307.7916
Gamma619 | 8.3767E-03 | 2.0980E-05 | 3.4127E-04 | 1.7033E-02 | 16412.7446 | 304.6413
Gamma620 | 1.8622E-02 | 2.2345E-04 | 2.2046E-04 | 5.1229E-02 | 134100.5341 | 37.2855
Gamma621 | 3.7281E-03 | 1.7661E-05 | 0.0000E+00 | 1.0159E-02 | 30196.9492 | 165.5796
Gamma622 | 9.0645E-02 | 2.0393E-03 | 2.7301E-02 | 1.7825E-01 | 13895.2991 | 359.8339
Gamma623 | 1.4109E-02 | 1.2263E-04 | 8.9471E-05 | 3.6890E-02 | 39533.6827 | 126.4744
Gamma624 | 5.2883E-03 | 2.2810E-05 | 0.0000E+00 | 1.3935E-02 | 33696.7899 | 148.3821
Gamma625 | 6.7281E-01 | 1.8216E-03 | 5.8844E-01 | 7.4477E-01 | 102972.9902 | 48.5564
mean00 | 3.3153E+01 | 6.2187E-06 | 3.3148E+01 | 3.3157E+01 | 92575.8051 | 54.0098
mean01 | 1.3207E+02 | 2.1123E-05 | 1.3206E+02 | 1.3207E+02 | 40296.2464 | 124.0810
mean02 | 3.3530E+01 | 1.1900E-05 | 3.3524E+01 | 3.3535E+01 | 23306.5483 | 214.5320
mean03 | 1.3223E+02 | 1.5145E-05 | 1.3223E+02 | 1.3224E+02 | 18904.7864 | 264.4833
mean04 | 3.3143E+01 | 3.1642E-05 | 3.3134E+01 | 3.3154E+01 | 312025.9022 | 16.0243
mean05 | 1.3222E+02 | 9.8978E-05 | 1.3221E+02 | 1.3224E+02 | 185982.0670 | 26.8843
mean06 | 3.3227E+01 | 4.6647E-05 | 3.3215E+01 | 3.3240E+01 | 142592.6571 | 35.0649
mean07 | 1.3218E+02 | 3.5168E-05 | 1.3217E+02 | 1.3219E+02 | 45558.0184 | 109.7502
mean08 | 3.3646E+01 | 4.3518E-05 | 3.3640E+01 | 3.3660E+01 | 36341.5325 | 137.5836
mean09 | 1.3244E+02 | 1.0831E-05 | 1.3244E+02 | 1.3245E+02 | 14098.1236 | 354.6571
mean10 | 3.3349E+01 | 2.8868E-05 | 3.3337E+01 | 3.3358E+01 | 42864.7199 | 116.6460
mean11 | 1.3212E+02 | 3.0243E-05 | 1.3211E+02 | 1.3213E+02 | 19599.8797 | 255.1036
mean12 | 3.3524E+01 | 5.5641E-05 | 3.3512E+01 | 3.3540E+01 | 55988.4161 | 89.3042
mean13 | 1.3238E+02 | 4.4698E-05 | 1.3237E+02 | 1.3239E+02 | 40444.0229 | 123.6277
mean14 | 3.3468E+01 | 4.4562E-05 | 3.3455E+01 | 3.3476E+01 | 136234.5080 | 36.7014
mean15 | 1.3274E+02 | 4.3066E-05 | 1.3274E+02 | 1.3276E+02 | 142630.1781 | 35.0557
mean16 | 3.3375E+01 | 9.4142E-05 | 3.3356E+01 | 3.3389E+01 | 113550.4454 | 44.0333
mean17 | 1.3251E+02 | 7.9729E-05 | 1.3249E+02 | 1.3252E+02 | 42178.8880 | 118.5427
mean18 | 3.3736E+01 | 6.2646E-04 | 3.3699E+01 | 3.3785E+01 | 214245.2839 | 23.3377
mean19 | 1.3315E+02 | 1.3344E-03 | 1.3309E+02 | 1.3322E+02 | 104416.6575 | 47.8851
mean20 | 3.3720E+01 | 7.8763E-05 | 3.3704E+01 | 3.3737E+01 | 69508.4940 | 71.9337
mean21 | 1.3294E+02 | 9.5506E-05 | 1.3292E+02 | 1.3296E+02 | 108031.1650 | 46.2829
mean22 | 3.3536E+01 | 5.8907E-06 | 3.3531E+01 | 3.3541E+01 | 18076.7866 | 276.5978
mean23 | 1.3269E+02 | 4.6310E-05 | 1.3268E+02 | 1.3271E+02 | 66099.5260 | 75.6435
mean24 | 3.3957E+01 | 2.6005E-05 | 3.3950E+01 | 3.3965E+01 | 4633.3471 | 1079.1335
mean25 | 1.3324E+02 | 1.9592E-05 | 1.3323E+02 | 1.3324E+02 | 13775.2710 | 362.9693
mean26 | 3.3621E+01 | 1.9563E-05 | 3.3615E+01 | 3.3627E+01 | 75168.1397 | 66.5175
mean27 | 1.3286E+02 | 2.7768E-05 | 1.3285E+02 | 1.3287E+02 | 72290.7471 | 69.1651
mean28 | 3.3967E+01 | 1.9263E-06 | 3.3964E+01 | 3.3969E+01 | 7635.2819 | 654.8547
mean29 | 1.3344E+02 | 9.1469E-06 | 1.3344E+02 | 1.3345E+02 | 21692.2676 | 230.4969
mean30 | 3.3965E+01 | 1.7468E-06 | 3.3962E+01 | 3.3968E+01 | 30107.9168 | 166.0693
mean31 | 1.3372E+02 | 2.5383E-05 | 1.3371E+02 | 1.3373E+02 | 29180.7359 | 171.3459
mean32 | 3.3807E+01 | 9.2728E-06 | 3.3800E+01 | 3.3811E+01 | 51407.7307 | 97.2616
mean33 | 1.3317E+02 | 5.2719E-05 | 1.3316E+02 | 1.3318E+02 | 64555.7808 | 77.4524
mean34 | 3.3589E+01 | 1.6618E-05 | 3.3583E+01 | 3.3596E+01 | 24340.1354 | 205.4220
mean35 | 1.3289E+02 | 1.0972E-05 | 1.3288E+02 | 1.3289E+02 | 25239.1536 | 198.1049
mean36 | 3.3957E+01 | 5.4964E-06 | 3.3953E+01 | 3.3961E+01 | 33494.4226 | 149.2786
mean37 | 1.3388E+02 | 1.4084E-05 | 1.3387E+02 | 1.3389E+02 | 19086.3787 | 261.9669
mean38 | 3.3869E+01 | 1.1937E-04 | 3.3852E+01 | 3.3888E+01 | 146107.2442 | 34.2214
mean39 | 1.3381E+02 | 2.1818E-04 | 1.3379E+02 | 1.3383E+02 | 126098.6469 | 39.6515
mean40 | 3.3899E+01 | 6.3637E-06 | 3.3896E+01 | 3.3903E+01 | 29684.1219 | 168.4402
mean41 | 1.3334E+02 | 1.2572E-05 | 1.3334E+02 | 1.3335E+02 | 28284.2642 | 176.7767
mean42 | 3.4005E+01 | 8.7598E-05 | 3.3990E+01 | 3.4022E+01 | 34455.4313 | 145.1150
mean43 | 1.3439E+02 | 7.7585E-05 | 1.3437E+02 | 1.3441E+02 | 44015.9496 | 113.5952
mean44 | 3.4089E+01 | 1.0985E-05 | 3.4087E+01 | 3.4092E+01 | 10783.1209 | 463.6876
mean45 | 1.3389E+02 | 2.1354E-06 | 1.3389E+02 | 1.3390E+02 | 16814.1550 | 297.3685
mean46 | 3.4030E+01 | 1.4371E-05 | 3.4023E+01 | 3.4037E+01 | 24344.4476 | 205.3856
mean47 | 1.3368E+02 | 6.3650E-06 | 1.3368E+02 | 1.3369E+02 | 26869.3262 | 186.0858
mean48 | 3.3974E+01 | 1.8305E-05 | 3.3969E+01 | 3.3982E+01 | 16220.0577 | 308.2603
mean49 | 1.3418E+02 | 4.3357E-05 | 1.3417E+02 | 1.3419E+02 | 33984.2140 | 147.1271
sig0 | 1.0397E-02 | 5.1172E-06 | 7.3910E-03 | 1.4208E-02 | 101753.1033 | 49.1386
sig1 | 1.1494E-02 | 2.2762E-05 | 6.9227E-03 | 1.6940E-02 | 89742.0891 | 55.7152
sig2 | 2.2147E-02 | 7.5291E-06 | 1.7002E-02 | 2.6493E-02 | 18702.3423 | 267.3462
sig3 | 2.5312E-02 | 1.3278E-05 | 2.0509E-02 | 3.3648E-02 | 16758.0436 | 298.3642
sig4 | 6.7377E-02 | 3.4771E-05 | 5.8035E-02 | 7.7321E-02 | 90886.8280 | 55.0135
sig5 | 1.2037E-01 | 8.2671E-05 | 1.0577E-01 | 1.3751E-01 | 79241.5240 | 63.0982
sig6 | 7.6083E-02 | 1.2216E-05 | 6.9972E-02 | 8.3122E-02 | 55230.0747 | 90.5304
sig7 | 8.9054E-02 | 1.8748E-05 | 8.0247E-02 | 9.5237E-02 | 33024.4230 | 151.4031
sig8 | 2.3056E-02 | 3.7189E-06 | 1.8870E-02 | 2.6342E-02 | 23756.2145 | 210.4712
sig9 | 2.7127E-02 | 3.8963E-06 | 2.4081E-02 | 3.1347E-02 | 28984.7066 | 172.5048
sig10 | 4.4864E-02 | 1.5913E-05 | 3.8071E-02 | 5.1483E-02 | 21330.6113 | 234.4049
sig11 | 2.8569E-02 | 1.6763E-05 | 2.2633E-02 | 3.5289E-02 | 79364.6376 | 63.0004
sig12 | 7.3945E-02 | 1.4626E-05 | 6.6090E-02 | 8.0964E-02 | 16359.3461 | 305.6357
sig13 | 6.0247E-02 | 4.9425E-05 | 5.0572E-02 | 7.5667E-02 | 45307.7279 | 110.3564
sig14 | 4.7560E-02 | 6.7658E-05 | 3.7400E-02 | 5.9961E-02 | 71546.1485 | 69.8850
sig15 | 4.5253E-02 | 5.1485E-05 | 3.2712E-02 | 5.8367E-02 | 122438.7684 | 40.8367
sig16 | 8.1149E-02 | 6.4461E-05 | 7.3386E-02 | 9.7219E-02 | 35379.4699 | 141.3249
sig17 | 9.8157E-02 | 1.0560E-04 | 8.1357E-02 | 1.1856E-01 | 65447.2037 | 76.3975
sig18 | 1.7481E-01 | 4.9086E-04 | 1.4254E-01 | 2.1546E-01 | 118957.1502 | 42.0319
sig19 | 3.7826E-01 | 2.1703E-03 | 2.9221E-01 | 4.5453E-01 | 98079.9965 | 50.9788
sig20 | 6.3216E-02 | 3.0365E-05 | 5.2831E-02 | 7.1833E-02 | 63138.9261 | 79.1905
sig21 | 8.3011E-02 | 3.0218E-05 | 7.2707E-02 | 9.3286E-02 | 67121.1605 | 74.4922
sig22 | 4.2591E-02 | 5.3225E-06 | 3.8782E-02 | 4.7331E-02 | 11863.5828 | 421.4578
sig23 | 7.9679E-02 | 1.6019E-05 | 7.4067E-02 | 8.7572E-02 | 30533.1187 | 163.7566
sig24 | 4.0758E-02 | 1.3113E-05 | 3.5724E-02 | 4.5320E-02 | 18413.7036 | 271.5369
sig25 | 3.6311E-02 | 5.4108E-05 | 2.5687E-02 | 4.5761E-02 | 319437.7767 | 15.6525
sig26 | 3.3717E-02 | 1.8502E-05 | 2.7022E-02 | 4.1190E-02 | 38797.8225 | 128.8732
sig27 | 3.2957E-02 | 1.9160E-05 | 2.7969E-02 | 4.3278E-02 | 85318.1770 | 58.6042
sig28 | 1.3705E-02 | 1.3658E-06 | 1.1576E-02 | 1.5649E-02 | 11658.0919 | 428.8867
sig29 | 1.7861E-02 | 5.2670E-05 | 1.3820E-02 | 2.7240E-02 | 8839.8629 | 565.6196
sig30 | 2.3851E-02 | 4.9658E-06 | 2.1458E-02 | 2.6093E-02 | 20196.0798 | 247.5728
sig31 | 6.3047E-02 | 1.3221E-05 | 5.5082E-02 | 6.8559E-02 | 20046.4462 | 249.4208
sig32 | 3.3400E-02 | 3.7970E-06 | 2.9305E-02 | 3.6581E-02 | 50200.1253 | 99.6013
sig33 | 8.9384E-02 | 1.3365E-05 | 8.3359E-02 | 9.6302E-02 | 40222.3843 | 124.3089
sig34 | 1.6552E-02 | 7.3247E-05 | 1.2323E-02 | 2.3667E-02 | 20320.9017 | 246.0521
sig35 | 1.9078E-02 | 9.0345E-06 | 1.5246E-02 | 2.6690E-02 | 18222.0431 | 274.3929
sig36 | 2.9753E-02 | 3.2640E-06 | 2.6535E-02 | 3.3308E-02 | 12711.3278 | 393.3499
sig37 | 3.9156E-02 | 9.5006E-06 | 3.3603E-02 | 4.3555E-02 | 15530.8432 | 321.9400
sig38 | 4.3736E-02 | 8.1981E-05 | 3.3489E-02 | 5.9352E-02 | 87473.4563 | 57.1602
sig39 | 5.6483E-02 | 7.0326E-05 | 4.4777E-02 | 7.5767E-02 | 44498.5518 | 112.3632
sig40 | 3.2181E-02 | 1.0031E-05 | 2.9040E-02 | 3.4087E-02 | 18767.5229 | 266.4177
sig41 | 5.8665E-02 | 7.4578E-06 | 5.3512E-02 | 6.2982E-02 | 8008.1789 | 624.3617
sig42 | 4.3597E-02 | 2.8268E-05 | 3.5905E-02 | 5.3344E-02 | 35259.9516 | 141.8039
sig43 | 4.2660E-02 | 1.8198E-05 | 3.4373E-02 | 5.0306E-02 | 31057.7121 | 160.9906
sig44 | 2.0933E-02 | 2.7451E-05 | 1.7862E-02 | 2.2511E-02 | 15465.7156 | 323.2957
sig45 | 2.1647E-02 | 1.2810E-06 | 1.9749E-02 | 2.4108E-02 | 18882.7570 | 264.7918
sig46 | 3.4843E-02 | 1.1288E-05 | 3.0400E-02 | 4.1280E-02 | 12880.6637 | 388.1788
sig47 | 2.9820E-02 | 4.6206E-06 | 2.5811E-02 | 3.3832E-02 | 14872.4660 | 336.1917
sig48 | 3.8268E-02 | 1.7943E-05 | 3.2641E-02 | 4.4989E-02 | 37424.5456 | 133.6022
sig49 | 7.3317E-02 | 4.4131E-05 | 6.2792E-02 | 8.3872E-02 | 37615.4550 | 132.9241
rho0 | -1.6297E-02 | 5.2185E-02 | -4.6878E-01 | 3.7495E-01 | 26512.5997 | 188.5896
rho1 | -6.2773E-01 | 9.5917E-03 | -7.8494E-01 | -4.5777E-01 | 26590.6405 | 188.0361
rho2 | 9.3617E-01 | 1.7745E-04 | 9.1451E-01 | 9.5750E-01 | 230515.6332 | 21.6905
rho3 | 3.8187E-01 | 2.3841E-03 | 2.6518E-01 | 4.5914E-01 | 44177.0986 | 113.1808
rho4 | -4.7966E-01 | 4.1922E-03 | -5.9941E-01 | -3.5422E-01 | 19050.5638 | 262.4594
rho5 | 1.1060E-01 | 1.5303E-02 | -1.2180E-01 | 2.9967E-01 | 57679.1470 | 86.6864
rho6 | -4.9259E-01 | 2.0951E-02 | -6.5592E-01 | -8.0627E-02 | 116877.5009 | 42.7798
rho7 | -9.3090E-01 | 3.7978E-04 | -9.5990E-01 | -8.9145E-01 | 105535.7505 | 47.3773
rho8 | 7.6369E-01 | 8.3119E-04 | 6.9430E-01 | 8.1311E-01 | 46400.7472 | 107.7569
rho9 | 7.9118E-01 | 1.3539E-02 | 5.9939E-01 | 9.1923E-01 | 145895.9604 | 34.2710
rho10 | 2.3060E-02 | 3.1655E-02 | -2.6114E-01 | 3.6095E-01 | 144155.6079 | 34.6847
rho11 | -6.2912E-02 | 7.3775E-03 | -1.9036E-01 | 1.2205E-01 | 27496.0159 | 181.8445
rho12 | -2.5315E-01 | 7.2828E-03 | -3.9030E-01 | -6.6908E-02 | 38868.7962 | 128.6379
rho13 | 5.4412E-01 | 5.7701E-03 | 4.2760E-01 | 7.1766E-01 | 86363.9160 | 57.8945
rho14 | 3.1846E-02 | 6.8279E-03 | -1.2925E-01 | 1.6057E-01 | 39234.2156 | 127.4398
rho15 | -4.9963E-03 | 3.9516E-03 | -1.1434E-01 | 1.0178E-01 | 42726.6201 | 117.0231
rho16 | 3.6540E-01 | 3.1031E-03 | 2.8682E-01 | 4.8095E-01 | 126902.3632 | 39.4004
rho17 | -7.3498E-01 | 5.5361E-03 | -8.5863E-01 | -6.2345E-01 | 23100.7809 | 216.4429
rho18 | -5.7640E-01 | 3.0451E-03 | -6.7629E-01 | -4.8114E-01 | 29505.4889 | 169.4600
rho19 | -9.6126E-01 | 1.9381E-04 | -9.8534E-01 | -9.3612E-01 | 42478.2570 | 117.7073
rho20 | -4.5866E-01 | 1.6254E-03 | -5.2164E-01 | -3.5626E-01 | 36729.7248 | 136.1295
rho21 | 6.6679E-01 | 6.4360E-03 | 5.3659E-01 | 8.0395E-01 | 25260.9133 | 197.9343
rho22 | -5.5560E-01 | 1.3427E-03 | -6.1193E-01 | -4.7605E-01 | 25955.2914 | 192.6389
rho23 | 2.9981E-01 | 4.8525E-03 | 1.7641E-01 | 4.3718E-01 | 14442.0845 | 346.2104
rho24 | -8.1558E-01 | 1.4636E-03 | -8.8326E-01 | -7.4729E-01 | 51720.3490 | 96.6737
p0 | 9.4841E-03 | 8.8157E-06 | 3.7773E-03 | 1.4609E-02 | 19695.7896 | 253.8614
p1 | 1.3091E-03 | 5.0283E-07 | 6.7313E-04 | 1.9354E-03 | 19060.7841 | 262.3187
p2 | 5.2091E-01 | 3.1162E-02 | 2.0918E-01 | 7.0672E-01 | 39399.0310 | 126.9067
p3 | 6.8907E-01 | 1.1123E-03 | 6.3295E-01 | 7.5842E-01 | 122348.5804 | 40.8668
p4 | 4.8204E-01 | 1.1005E-03 | 4.0973E-01 | 5.4387E-01 | 25151.2564 | 198.7972
p5 | 6.1286E-01 | 3.2653E-03 | 5.2855E-01 | 7.4532E-01 | 45865.3068 | 109.0149
p6 | 4.5864E-01 | 1.9018E-03 | 3.7606E-01 | 5.4930E-01 | 33871.6865 | 147.6159
p7 | 6.4258E-01 | 3.2882E-02 | 3.4405E-01 | 1.0000E+00 | 257471.7527 | 19.4196
p8 | 4.9533E-01 | 2.1998E-03 | 4.2800E-01 | 5.9699E-01 | 82995.0090 | 60.2446
p9 | 9.6980E-01 | 1.4932E-03 | 8.9655E-01 | 9.9985E-01 | 38433.8119 | 130.0938
p10 | 7.1425E-01 | 3.0174E-03 | 6.2227E-01 | 7.9271E-01 | 15718.6406 | 318.0937
p11 | 7.7223E-01 | 9.7882E-04 | 7.0868E-01 | 8.3627E-01 | 45482.2677 | 109.9330
p12 | 4.1091E-01 | 2.1401E-03 | 3.2501E-01 | 4.9506E-01 | 29664.4736 | 168.5518
p13 | 7.0933E-01 | 8.8617E-04 | 6.5631E-01 | 7.6445E-01 | 47446.1868 | 105.3825
p14 | 5.4156E-01 | 5.2855E-03 | 4.2643E-01 | 6.7479E-01 | 24435.7993 | 204.6178
p15 | 8.4811E-01 | 6.0109E-04 | 8.0251E-01 | 8.9474E-01 | 20822.5178 | 240.1247
p16 | 6.0290E-01 | 8.1748E-04 | 5.4996E-01 | 6.6087E-01 | 22349.5548 | 223.7181
p17 | 5.6851E-02 | 7.7910E-05 | 3.9254E-02 | 7.0971E-02 | 14496.7123 | 344.9058
p18 | 8.3788E-01 | 2.4751E-03 | 7.5064E-01 | 9.5049E-01 | 31895.1454 | 156.7637
p19 | 3.6076E-02 | 7.1633E-05 | 1.8826E-02 | 5.1437E-02 | 48798.1506 | 102.4629
p20 | 7.9248E-01 | 1.0195E-03 | 7.4217E-01 | 8.4501E-01 | 19238.1132 | 259.9007
p21 | 2.4458E-02 | 3.2593E-04 | 1.0902E-02 | 6.8324E-02 | 37652.1742 | 132.7945
p22 | 2.1440E-01 | 7.4085E-03 | 7.4024E-02 | 3.8601E-01 | 108925.6785 | 45.9029
p23 | 4.4622E-01 | 2.4041E-03 | 3.5150E-01 | 5.2919E-01 | 20916.1565 | 239.0497
p24 | 4.5593E-01 | 1.6771E-03 | 3.7982E-01 | 5.3879E-01 | 129941.7537 | 38.4788
|
2024-09-04T02:54:58.066099 | 2020-03-07T04:21:02 | 2003.03513 | {
"authors": "Thoan Pham Duc, Tuyen Nguyen Dang and Vangty Noulorvang",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26096",
"submitter": "Duc Thoan Pham",
"url": "https://arxiv.org/abs/2003.03513"
} | arxiv-papers | # Finiteness of meromorphic mappings from
Kähler manifold into projective space
Pham Duc Thoan Department of Mathematics, National University of Civil
Engineering
55 Giai Phong street, Hai Ba Trung, Hanoi, Vietnam<EMAIL_ADDRESS>,
Nguyen Dang Tuyen Department of Mathematics, National University of Civil
Engineering
55 Giai Phong street, Hai Ba Trung, Hanoi, Vietnam<EMAIL_ADDRESS>and
Noulorvang Vangty Department of Mathmatics, National University of Education
136-Xuan Thuy str., Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.
The purpose of this paper is to prove the finiteness theorems for meromorphic
mappings of a complete connected Kähler manifold into projective space sharing
few hyperplanes in subgeneral position without counting multiplicity, where
all zeros with multiplicities more than a certain number are omitted. Our
results are extensions and generalizations of some recent ones.
††footnotetext: 2010 Mathematics Subject Classification: Primary 32H30, 32A22;
Secondary 30D35.
Key words and phrases: finiteness theorems, meromorphic mapping, complete
Kähler manifold.
## 1\. Introduction
Let $f$ be a non-constant meromorphic mapping of $\mathbb{C}^{m}$ into
$\mathbb{P}^{n}(\mathbb{C})$ and let $H$ be a hyperplane in
$\mathbb{P}^{n}(\mathbb{C})$. Denote by $\nu_{(f,H_{j})}(z)$ the intersecting
multiplicity of the mapping $f$ with the hyperplane $H_{j}$ at the point
$f(z)$.
For a divisor $\nu$ on $\mathbb{C}^{m}$ and for a positive integer $k$ or
$k=+\infty$, we set
$\nu_{\leqslant k}(z)=\begin{cases}0&{\text{ if }}\nu(z)>k,\\\ \nu(z)&{\text{
if }}\nu(z)\leqslant k.\end{cases}$
Similarly, we define $\nu_{>k}(z).$ If $\varphi$ is a meromorphic function,
the zero divisor of $\varphi$ is denoted by $\nu_{\varphi}.$
Let $H_{1},H_{2},\ldots,H_{q}$ be hyperplanes of $\mathbb{P}^{n}(\mathbb{C})$
(in subgeneral position or in general position) and let $k_{1},\ldots,k_{q}$
be positive integers or $+\infty$. Assume that $f$ is a meromorphic mapping
satisfying
$\dim\\{z:\nu_{(f,H_{i}),\leqslant k_{i}}(z)\cdot\nu_{(f,H_{j}),\leqslant
k_{j}}(z)\\}\leqslant m-2\ \ (1\leqslant i<j\leqslant q).$
Let $d$ be an integer number. We denote by
$\mathcal{F}(f,\\{H_{j},k_{j}\\}_{j=1}^{q},d)$ the set of all meromorphic
mappings $g:\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ satisfying the
following two conditions:
* (a)
$\min(\nu_{(f,H_{j}),\leqslant k_{j}},d)=\min(\nu_{(g,H_{j}),\leqslant
k_{j}},d)$ ($1\leqslant j\leqslant q$).
* (b)
$f(z)=g(z)$ on $\bigcup_{j=1}^{q}\\{z:\nu_{(f,H_{j}),\leqslant
k_{j}}(z)>0\\}$.
If $k_{1}=\cdots=k_{q}=+\infty$, we will simply use notation
$\mathcal{F}(f,\\{H_{j}\\}_{j=1}^{q},d)$ instead of
$\mathcal{F}(f,\\{H_{j},\infty\\}_{j=1}^{q},d).$
In 1926, Nevanlinna [8] showed that two distinct nonconstant meromorphic
functions $f$ and $g$ on the complex plane cannot have the same inverse images
for five distinct values, and that $g$ is a linear fractional transformation
of $f$ if they have the same inverse images counted with multiplicities for
four distinct values. After that, many authors have extended and improved
Nevanlinna’s results to the case of meromorphic mappings into complex
projective spaces such as Fujimoto [3, 5, 6], Smiley [15], Ru-Sogome [14],
Chen-Yan [1], Dethloff-Tan [2], Quang [16, 17, 18, 19], Nhung-Quynh [9]….
These theorems are called uniqueness theorems or finiteness theorems. The
first finiteness theorem for the case of meromorphic mappings from
$\mathbb{C}^{m}$ into complex projective space $\mathbb{P}^{n}(\mathbb{C})$
sharing $2n+2$ hyperplanes is given by Quang [17] in 2012 and its correction
[20] in 2015. Recently, he [18] extended his results and obtained the
following finiteness theorem, in which he did not need to count all zeros with
multiplicities more than certain values. Theorem A (see [18, Theorem 1.1]) Let
$f$ be a linearly nondegenerate meromorphic mapping of $\mathbb{C}^{m}$ into
$\mathbb{P}^{n}(\mathbb{C})$. Let $H_{1},\ldots,H_{2n+2}$ be $2n+2$
hyperplanes of $\mathbb{P}^{n}(\mathbb{C})$ in general position and let
$k_{1},\ldots,k_{2n+2}$ be positive integers or $+\infty$. Assume that
$\sum_{i=1}^{2n+2}\frac{1}{k_{i}+1}<\min\left\\{\frac{n+1}{3n^{2}+n},\frac{5n-9}{24n+12},\frac{n^{2}-1}{10n^{2}+8n}\right\\}.$
Then $\sharp\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)\leqslant 2.$
Note that the condition
$\displaystyle\sum_{i=1}^{2n+2}\frac{1}{k_{i}+1}<\min\left\\{\frac{n+1}{3n^{2}+n},\frac{5n-9}{24n+12},\frac{n^{2}-1}{10n^{2}+8n}\right\\}$
in Theorem A becomes
$\displaystyle\sum_{i=1}^{2n+2}\frac{1}{k_{i}+1}<\frac{n+1}{3n^{2}+n}$ when
$n\geq 5.$
We now consider the general case, where $f:M\to\mathbb{P}^{n}(\mathbb{C})$ is
a meromorphic mapping of an $m$-dimensional complete connected Kähler manifold
$M$, whose universal covering is biholomorphic to a ball
$B(R_{0})=\\{z\in{\mathbf{C}}^{m}\ :\ ||z||<R_{0}\\}$
$(0<R_{0}\leqslant\infty)$, into $\mathbb{P}^{n}(\mathbb{C})$.
Let $H_{1},\ldots,H_{q}$ be hyperplanes of $\mathbb{P}^{n}(\mathbb{C})$ and
let $k_{1},\ldots,k_{q}$ be integers or $+\infty$. Then, the family
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},d)$ are defined similarly as above,
where $d$ is an integer number.
For $\rho\geqslant 0,$ we say that $f$ satisfies the condition $(C_{\rho})$ if
there exists a nonzero bounded continuous real-valued function $h$ on $M$ such
that
$\rho\Omega_{f}+dd^{c}\log h^{2}\geq\text{Ric}\omega,$
where $\Omega_{f}$ is the full-back of the Fubini-Study form $\Omega$ on
$\mathbb{P}^{n}(\mathbb{C})$,
$\omega=\dfrac{\sqrt{-1}}{2}\sum_{i,j}h_{i\bar{j}}dz_{i}\wedge
d\overline{z}_{j}$ is Kähler form on $M$,
$\text{Ric}\omega=dd^{c}\log(det(h_{i\overline{j}}))$,
$d=\partial+\overline{\partial}$ and
$d^{c}=\dfrac{\sqrt{-1}}{4\pi}(\overline{\partial}-\partial)$.
Very recently, Quang [19] obtained a finiteness theorem for meromorphic
mappings from such Kähler manifold $M$ into $\mathbb{P}^{n}(\mathbb{C})$
sharing hyperplanes regardless of multiplicities by giving new definitions of
”functions of small intergration” and ”functions of bounded intergration” as
well as proposing a new method to deal with the difficulties when he met on
the Kähler manifold. We would like to emphasize that Quang’s result is also
the first finiteness theorem for meromorphic mappings on the Kähler manifold,
although the uniqueness theorems were discovered early by Fujimoto [5] and
later by many authors such as Ru-Sogome [14] or Nhung-Quynh [9] and others.
Here is his result. Theorem B (see [19, Theorem 1.1]). Let $M$ be an
$m$-dimensional connected Kähler manifold whose universal covering is
biholomorphic to $\mathbb{C}^{m}$ or the unit ball $B(1)$ of $\mathbb{C}^{m}$,
and let $f$ be a linearly nondegenerate meromorphic mapping of $M$ into
$\mathbb{P}^{n}(\mathbb{C})\ (n\geqslant 2)$. Let $H_{1},\ldots,H_{q}$ be $q$
hyperplanes of $\mathbb{P}^{n}(\mathbb{C})$ in general position. Assume that
$f$ satisfies the condition $(C_{\rho})$. If
$\displaystyle
q>n+1+\frac{3nq}{6n+1}+\rho\frac{(n^{2}+4q-3n)(6n+1)}{6n^{2}+2}$
then $\sharp\mathcal{F}(f,\\{H_{i}\\}_{i=1}^{q},1)\leqslant 2.$
Unfortunately, in this result, all zeros with multiplicities must need to be
counted and hence Theorem B can not be an extension or a generalization of
Theorem A.
Our purpose in this article is to prove a similar result to Theorems A and B
for the case of a meromorphic mapping from a complete connected Kähler
manifold into projective space, in which all zeros with multiplicities more
than a certain number are omitted. However, the key used in the proof of
Theorem A is technique “rearranging counting functions” to compare counting
functions with characteristic functions, which is not valid on the Kähler
manifold. In addition, the proof of Theorem B cannot work on the case of
$k_{i}<\infty$. To overcome these difficulties, we use the technique in [22]
and the methods in [19], as well as considering new auxiliary functions to
obtain a new finiteness theorem which will generalize and extend the theorems
cited above. Namely, we will prove the following theorem.
###### Theorem 1.1.
Let $M$ be an $m$-dimensional connected Kähler manifold whose universal
covering is biholomorphic to $\mathbb{C}^{m}$ or the unit ball $B(1)$ of
$\mathbb{C}^{m}$, and let $f$ be a linearly nondegenerate meromorphic mapping
of $M$ into $\mathbb{P}^{n}(\mathbb{C})\ (n\geqslant 2)$. Let
$H_{1},\ldots,H_{q}$ be $q$ hyperplanes of $\mathbb{P}^{n}(\mathbb{C})$ in
$N$-subgeneral position and let $k_{1},\ldots,k_{q}$ be integers or $+\infty$.
Assume that $f$ satisfies the condition $(C_{\rho})$. Let $k$ be the largest
integer number not exceeding $\dfrac{q-2N-2}{2}$ and let $l$ be the smallest
integer number not less than $\dfrac{2N-2}{k+2}+2$ if $k>0$ or let $l=2N+1$ if
$k=0.$ Then $\sharp\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)\leqslant 2$ if
$\displaystyle q$
$\displaystyle>2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{4(q-n)n}{n-1}\big{)}$
$\displaystyle+\max\left\\{\frac{3nq}{2\big{(}3n+1+\frac{n-1}{l}\big{)}},\frac{4q+3nq-14}{4q+3n-14},\frac{3nq^{2}}{6nq+(n-2)(q-2)+4q-6n-8}\right\\}.$
Remark 1. It is easy to see that
$\dfrac{3nq}{2\big{(}3n+1+\frac{n-1}{l}\big{)}}<\dfrac{3nq}{6n+2}<\dfrac{3nq}{6n+1},$
and
$\dfrac{3nq^{2}}{6nq+(n-2)(q-2)+4q-6n-8}<\dfrac{3nq^{2}}{6nq+q}=\dfrac{3nq}{6n+1},\forall
n\geq 2.$
We now show that
$\frac{4q+3nq-14}{4q+3n-14}<\dfrac{3nq}{6n+1},\forall n\geq 3.$
Indeed, it suffices to prove that $12nq^{2}-9n^{2}q-69nq-4q+84n+14>0$ for all
$n\geq 3.$ Since $q\geq 2n+2$, we have $12nq^{2}-9n^{2}q-69nq-4q\geq
q(15n^{2}-45n-4)>0$ for all $n\geq 4.$ For $n=3,$ we have
$12nq^{2}-9n^{2}q-69nq-4q+84n+14=36q^{2}-292q+266>0$ since $q\geq 8.$
Hence, when $k_{1}=\cdots=k_{q}=+\infty$ and $N=n$, Theorem 1.1 is an
extension of Theorem B.
When $q=2n+2$, $M=\mathbb{C}^{n}$ and $H_{1},\ldots,H_{q}$ are in general
position, by $\rho=0$, $N=n$, $k=0$ and $l=2n+1,$ we obtain the following
corollary from Theorem 1.1.
###### Corollary 1.2.
Let $f$ be a linearly nondegenerate meromorphic mapping of $\mathbb{C}^{m}$
into $\mathbb{P}^{n}(\mathbb{C})$. Let $H_{1},\ldots,H_{2n+2}$ be $2n+2$
hyperplanes of $\mathbb{P}^{n}(\mathbb{C})$ in general position and let
$k_{1},\ldots,k_{n+2}$ be positive integers or $+\infty$. Then
$\sharp\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)\leqslant 2$ provided
$\sum_{i=1}^{2n+2}\frac{1}{k_{i}+1}<\min\left\\{\frac{1}{2n},\frac{n^{3}+2n+3}{n(7n^{2}+5n+3)}\right\\}.$
In particular, if $n\geq 4$ then
$\sharp\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{2n+2},1)\leqslant 2$ provided
$\sum_{i=1}^{2n+2}\frac{1}{k_{i}+1}<\frac{1}{2n}.$
Remark 2. Consider the quantities
$A=\min\left\\{\frac{n+1}{3n^{2}+n},\frac{5n-9}{24n+12},\frac{n^{2}-1}{10n^{2}+8n}\right\\}$
in Theorem A and
$B=\min\left\\{\frac{1}{2n},\frac{n^{3}+2n+3}{n(7n^{2}+5n+3)}\right\\}$ in
Corollary 1.2. We have the following estimates.
$\bullet$ For $n\geq 5$, $A=\frac{n+1}{3n^{2}+n}<\frac{1}{2n}=B.$
$\bullet$ For $n=4$, $A=\frac{n^{2}-1}{10n^{2}+8n}<\frac{1}{2n}=B.$
$\bullet$ For $n=3$,
$A=\frac{n^{2}-1}{10n^{2}+8n}<\frac{n^{3}+2n+3}{n(7n^{2}+5n+3)}=B.$
$\bullet$ For $n=2$,
$A=\frac{5n-9}{24n+12}<\frac{n^{3}+2n+3}{n(7n^{2}+5n+3)}=B.$
In all the cases, always $A<B$. Therefore, Corollary 1.2 is a nice improvement
of Theorem A.
In order to prove our results, we first give an new estimate of the counting
function of the Cartan’s auxiliary function (see Lemma 2.8). We second improve
the algebraically dependent theorem of three meromorphic mappings (see Lemma
3.3). After that we use arguments similar to those used by Quang [19] to
finish the proofs.
## 2\. Basic notions and auxiliary results from Nevanlinna theory
We will recall some basic notions in Nevanlinna theory due to [13, 21].
2.1. Counting function. We set
$||z||=\big{(}|z_{1}|^{2}+\dots+|z_{n}|^{2}\big{)}^{1/2}$ for
$z=(z_{1},\dots,z_{n})\in\mathbb{C}^{m}$ and define
$\displaystyle B(r):=\\{z\in\mathbb{C}^{m}:||z||<r\\},\quad
S(r):=\\{z\in\mathbb{C}^{m}:||z||=r\\}\ (0<r\leqslant\infty),$
where $B(\infty)=\mathbb{C}^{m}$ and $S(\infty)=\emptyset$.
Define
$v_{m-1}(z):=\big{(}dd^{c}||z||^{2}\big{)}^{m-1}\quad\quad\text{and}$
$\sigma_{m}(z):=d^{c}\text{log}||z||^{2}\land\big{(}dd^{c}\text{log}||z||^{2}\big{)}^{m-1}\text{on}\quad\mathbb{C}^{m}\setminus\\{0\\}.$
A divisor $E$ on a ball $B(R_{0})$ is given by a formal sum
$E=\sum\mu_{\nu}X_{\nu}$, where $\\{X_{\nu}\\}$ is a locally family of
distinct irreducible analytic hypersurfaces in $B(R_{0})$ and
$\mu_{\nu}\in\mathbb{Z}$. We define the support of the divisor $E$ by setting
$\mathrm{Supp}\,(E)=\cup_{\mu_{\nu}\neq 0}X_{\nu}$. Sometimes, we identify the
divisor $E$ with a function $E(z)$ from $B(R_{0})$ into $\mathbb{Z}$ defined
by $E(z):=\sum_{X_{\nu}\ni z}\mu_{\nu}$.
Let $M,k$ be positive integers or $+\infty$. We define the truncated divisor
$E^{[M]}$ by
$\displaystyle E^{[M]}:=\sum_{\nu}\min\\{\mu_{\nu},M\\}X_{\nu},$
and the truncated counting function to level $M$ of $E$ by
$\displaystyle
N^{[M]}(r,r_{0};E):=\int\limits_{r_{0}}^{r}\frac{n^{[M]}(t,E)}{t^{2m-1}}dt\quad(r_{0}<r<R_{0}),$
where
$\displaystyle n^{[M]}(t,E):=\begin{cases}\int\limits_{\mathrm{Supp}\,(E)\cap
B(t)}E^{[M]}v_{m-1}&\text{ if }m\geqslant 2,\\\ \sum_{|z|\leqslant
t}E^{[M]}(z)&\text{ if }m=1.\end{cases}$
We omit the character [M] if $M=+\infty$.
Let $\varphi$ be a non-zero meromorphic function on $B(R_{0})$. We denote by
$\nu^{0}_{\varphi}$ (resp. $\nu^{\infty}_{\varphi}$) the divisor of zeros
(resp. divisor of poles ) of $\varphi$. The divisor of $\varphi$ is defined by
$\nu_{\varphi}=\nu^{0}_{\varphi}-\nu^{\infty}_{\varphi}.$
For a positive integer $M$ or $M=\infty$, we define the truncated divisors of
$\nu_{\varphi}$ by
$\nu^{[M]}_{\varphi}(z)=\min\
\\{M,\nu_{\varphi}(z)\\},\quad\nu^{[M]}_{\varphi,\leqslant
k}(z):=\begin{cases}\nu^{[M]}_{\varphi}(z)&\text{ if
}\nu^{[M]}_{\varphi}(z)\leqslant k,\\\ 0&\text{ if
}\nu^{[M]}_{\varphi}(z)>k.\end{cases}$
For convenience, we will write $N_{\varphi}(r,r_{0})$ and
$N^{[M]}_{\varphi,\leqslant k}(r,r_{0})$ for $N(r,r_{0};\nu^{0}_{\varphi})$
and $N^{[M]}(r,r_{0};\nu^{0}_{\varphi,\leqslant k})$ respectively.
2.2. Characteristic function. Let
$f:B(R_{0})\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a meromorphic
mapping. Fix a homogeneous coordinates system $(w_{0}:\cdots:w_{n})$ on
$\mathbb{P}^{n}(\mathbb{C})$. We take a reduced representation
$f=(f_{0}:\cdots:f_{n})$, which means $f_{i}\ (0\leqslant i\leqslant n)$ are
holomorphic functions and $f(z)=\big{(}f_{0}(z):\dots:f_{n}(z)\big{)}$ outside
the analytic subset $\\{f_{0}=\dots=f_{n}=0\\}$ of codimension at least two.
Set $\|f\|=\big{(}|f_{0}|^{2}+\dots+|f_{n}|^{2}\big{)}^{1/2}$. Let $H$ be a
hyperplane in $\mathbb{P}^{n}(\mathbb{C})$ defined by
$H=\\{(\omega_{0},\ldots,\omega_{n}):a_{0}\omega_{0}+\cdots+a_{n}\omega_{n}=0\\}$.
We set $H(f)=a_{0}f_{0}+\cdots+a_{n}f_{n}$ and
$\|H\|=\big{(}|a_{0}|^{2}+\dots+|a_{n}|^{2}\big{)}^{1/2}.$
The characteristic function of $f$ (with respect to Fubini Study form
$\Omega$) is defined by
$\displaystyle
T_{f}(r,r_{0}):=\int_{t=r_{0}}^{r}\dfrac{dt}{t^{2m-1}}\int_{B(t)}f^{*}\Omega\wedge
v_{m-1},\quad\quad 0<r_{0}<r<R_{0}.$
By Jensen’s formula we have
$\displaystyle
T_{f}(r,r_{0})=\int_{S(r)}\log||f||\sigma_{m}-\int_{S(r_{0})}\log||f||\sigma_{m},\quad\quad
0<r_{0}<r<R_{0}.$
Through this paper, we assume that the numbers $r_{0}$ and $R_{0}$ are fixed
with $0<r_{0}<R_{0}$. By notation “$||\ P$”, we mean that the asseartion $P$
hold for all $r\in[r_{0},R_{0}]$ outside a set $E$ such that
$\int_{E}dr<\infty$ in case $R_{0}=\infty$ and
$\int_{E}\dfrac{1}{R_{0}-r}dr<\infty$ in case $R_{0}<\infty$.
2.3. Functions of small intergration. We recall some definitions due to Quang
[19].
Let $f^{1},\ldots,f^{k}$ be $k$ meromorphic mappings from the complete Kähler
manifold $B(1)$ into $\mathbb{P}^{m}(\mathbb{C})$, which satisfies the
condition $(C_{\rho})$ for a non-negative number $\rho$. For each $1\leqslant
u\leqslant k$, we fix a reduced representation
$f^{u}=(f_{0}^{u}:\cdots:f_{n}^{u})$ of $f^{u}$.
A non-negative plurisubharmonic function $g$ on $B(1)$ is said to be of small
intergration with respective to $f^{1},\ldots,f^{k}$ at level $l_{0}$ if there
exists an element $\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{N}^{m}$
with $|\alpha|\leqslant l_{0}$, a positive number $K$, such that for every
$0\leqslant tl_{0}<p<1$ then
$\int_{S(r)}|z^{\alpha}g|^{t}\sigma_{m}\leqslant
K\left(\frac{R^{2m-1}{R-r}}{\sum}_{u=1}^{m}T_{f^{u}}(r,r_{0})\right)^{p}$
for all $r$ with $0<r_{0}<r<R<1,$ where $z^{\alpha}=z_{1}^{\alpha_{1}}\cdots
z_{m}^{\alpha_{m}}.$
We denote by $S(l_{0};f^{1},\ldots,f^{k})$ the set of all non-negative
plurisubharmonic functions on $B(1)$ which are of small intergration with
respective to $f^{1},\ldots,f^{k}$ at level $l_{0}.$ We see that, if $g\in
S(l_{0};f^{1},\ldots,f^{k})$ then $g\in S(l;f^{1},\ldots,f^{k})$ for all
$l>l_{0}.$ Moreover, if $g$ is a constant function then $g\in
S(0;f^{1},\ldots,f^{k})$.
By [19, Proposition 3.2], if $g_{i}\in S(l_{i};f^{1},\ldots,f^{k})$, then
$g_{1}\cdots g_{s}\in S(\sum_{i=1}^{s}l_{i};f^{1},\ldots,f^{k})$.
A meromorphic function $h$ on $B(1)$ is said to be of bounded intergration
with bi-degree $(p,l_{0})$ for the family $\\{f^{1},\ldots,f^{k}\\}$ if there
exists $g\in S(l_{0};f^{1},\ldots,f^{k})$ satisfying
$|h|\leqslant||f^{1}||^{p}\cdots||f^{u}||^{p}\cdot g,$
outside a proper analytic subset of $B(1).$
We denote by $B(p,l_{0};f^{1},\ldots,f^{k})$ the set of all meromorphic
functions on $B(1)$ which are of bounded intergration of bi-degree $p,l_{0}$
for $\\{l_{0};f^{1},\ldots,f^{k}\\}$. We have the following assertions:
$\bullet$ For a meromorphic mapping $h$, $|h|\in S(l_{0};f^{1},\ldots,f^{k})$
iff $h\in B(0,l_{0};f^{1},\ldots,f^{k})$.
$\bullet$ $B(p,l_{0};f^{1},\ldots,f^{k})\subset B(p,l;f^{1},\ldots,f^{k})$ for
all $0\leqslant l_{0}<l.$
$\bullet$ If $h_{i}\in B(p_{i},l_{i};f^{1},\ldots,f^{k})$ then $h_{1}\cdots
h_{s}\in B(\sum_{i=1}^{s}p_{i},\sum_{i=1}^{s}l_{i};f^{1},\ldots,f^{k})$.
2.4. Some Lemmas and Propositions.
###### Lemma 2.1.
[6, Lemma 3.4] If $\Phi^{\alpha}(F,G,H)=0$ and
$\Phi^{\alpha}\left(\frac{1}{F},\frac{1}{G},\frac{1}{H}\right)=0$ for all
$\alpha$ with $|\alpha|\leqslant 1$, then one of the following assertions
holds:
(i) $F=G,G=H$ or $H=F$.
(ii) $\frac{F}{G},\frac{G}{H}$ and $\frac{H}{F}$ are all constants.
###### Proposition 2.2 (see [11, 12]).
_Let $H_{1},\ldots,H_{q}$ $(q>2N-n+1)$ be hyperplanes in
$\mathbb{P}^{n}(\mathbb{C})$ located in $N$-subgeneral position. Then there
exists a function $\omega:\\{1,\ldots,q\\}\to(0,1]$ called a Nochka weight and
a real number $\tilde{\omega}\geqslant 1$ called a Nochka constant satisfying
the following conditions:
(i) If $j\in\\{1,\ldots,q\\}$, then $0<\omega_{j}\tilde{\omega}\leqslant 1.$
(ii) $q-2N+n-1=\tilde{\omega}(\sum^{q}_{j=1}\omega_{j}-n-1).$
(iii) For $R\subset\\{1,\ldots,q\\}$ with $|R|=N+1$, then $\sum_{i\in
R}\omega_{i}\leqslant n+1.$
(iv) $\frac{N}{n}\leqslant\tilde{\omega}\leqslant\frac{2N-n+1}{n+1}.$
(v) Given real numbers $\lambda_{1},\ldots,\lambda_{q}$ with
$\lambda_{j}\geqslant 1$ for $1\leqslant j\leqslant q$ and given any
$R\subset\\{1,\ldots,q\\}$ and $|R|=N+1,$ there exists a subset $R^{1}\subset
R$ such that $|R^{1}|=\text{rank}\\{H_{i}\\}_{i\in R^{1}}=n+1$ and_
$\prod_{j\in R}\lambda_{j}^{\omega_{j}}\leqslant\prod_{i\in
R^{1}}\lambda_{i}.$
###### Proposition 2.3 (see [21], Lemma 3.2).
Let $\\{H_{i}\\}_{i=1}^{q}\ (q\geqslant n+1)$ be a set of hyperplanes of
$\mathbb{P}^{n}(\mathbb{C})$ satisfying $\cap_{i=1}^{q}H_{i}=\emptyset$ and
let $f:B(R_{0})\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a meromorphic
mapping. Then there exist positive constants $\alpha$ and $\beta$ such that
$\alpha\|f\|\leqslant\max\limits_{i\in\\{1,\ldots,q\\}}|H_{i}(f)|\leqslant\beta\|f\|.$
###### Proposition 2.4 (see [4], Proposition 4.5).
Let $F_{1},\ldots,F_{n+1}$ be meromorphic functions on
$B(R_{0})\subset\mathbb{C}^{m}$ such that they are linearly independent over
$\mathbb{C}$. Then there exists an admissible set
$\\{\alpha_{i}=(\alpha_{i1},\ldots,\alpha_{im})\\}_{i=1}^{n+1}$ with
$\alpha_{ij}\geq 0$ being integers,
$|\alpha_{i}|=\sum_{j=1}^{m}|\alpha_{ij}|\leqslant i$ for $1\leqslant
i\leqslant n+1$ such that the generalized Wronskian
$W_{\alpha_{1},\ldots,\alpha_{n+1}}(F_{1},\ldots,F_{n+1})\not\equiv 0$ where
$W_{\alpha_{1},\ldots,\alpha_{n+1}}(F_{1},\ldots,F_{n+1})=det\left(\mathcal{D}^{\alpha_{i}}F_{j}\right)_{1\leqslant
i,j\leqslant n+1}.$
Let $L_{1},\ldots,L_{n+1}$ be linear forms of $n+1$ variables and assume that
they are linearly independent. Let
$F=(F_{1}:\cdots:F_{n+1}):B(R_{0})\to\mathbb{P}^{n}(\mathbb{C})$ be a
meromorphic mapping and $(\alpha_{1},\ldots,\alpha_{n+1})$ be an admissible
set of $F$. Then we have following proposition.
###### Proposition 2.5 (see [13], Proposition 3.3).
In the above situation, set $l_{0}=|\alpha_{1}|+\cdots+|\alpha_{n+1}|$ and
take $t,p$ with $0<tl_{0}<p<1.$ Then, for $0<r_{0}<R_{0}$ there exists a
positive constant $K$ such that for $r_{0}<r<R<R_{0},$
$\int\limits_{S(r)}\left|z^{\alpha_{1}+\cdots+\alpha_{n+1}}\dfrac{W_{\alpha_{1},\ldots,\alpha_{n+1}}(F_{1},\ldots,F_{n+1})}{L_{1}(F)\cdots
L_{n+1}(F)}\right|^{t}\sigma_{m}\leqslant
K\left(\dfrac{R^{2m-1}}{R-r}T_{F}(R,r_{0})\right)^{p},$
where $z^{\alpha}=z_{1}^{\alpha_{1}}\cdots z_{m}^{\alpha_{m}}$ for
$z=(z_{1},\ldots,z_{m})$ and $\alpha=(\alpha_{1},\ldots,\alpha_{m})$.
For convenience of presentation, for meromorphic mappings
$f^{u}:B(R)\to\mathbb{P}^{n}(\mathbb{C})$ and hyperplanes
$\\{H_{i}\\}_{i=1}^{q}$ of $\mathbb{P}^{n}(\mathbb{C})$, we denote by
$\mathcal{S}$ the closure of
$\cup_{1\leqslant u\leqslant 3}I(f^{u})\cup\cup_{1\leqslant i<j\leqslant
q}\\{z:\nu_{(f,H_{i}),\leqslant k_{i}}(z)\cdot\nu_{(f,H_{j}),\leqslant
k_{j}}(z)>0\\}.$
We see that $\mathcal{S}$ is an analysis subset of codimension two of $B(R)$.
###### Lemma 2.6.
[22, Lemma 2.6] Let $f^{1},f^{2},f^{3}$ be three mappings in
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$. Suppose that there exist
$s,t,l\in\\{1,\ldots,q\\}$ such that
$P:=Det\left(\begin{array}[]{ccc}(f^{1},H_{s})&(f^{1},H_{t})&(f^{1},H_{l})\\\
(f^{2},H_{s})&(f^{2},H_{t})&(f^{2},H_{l})\\\
(f^{3},H_{s})&(f^{3},H_{t})&(f^{3},H_{l})\end{array}\right)\not\equiv 0.$
Then we have
$\displaystyle\nu_{P}(z)\geq\sum_{i=s,t,l}(\min_{1\leqslant u\leqslant
3}\\{\nu_{(f^{u},H_{i}),\leqslant
k_{i}}(z)\\}-\nu^{[1]}_{(f^{1},H_{i}),\leqslant
k_{i}}(z))+2\sum_{i=1}^{q}\nu^{[1]}_{(f^{1},H_{i}),\leqslant k_{i}}(z),\forall
z\not\in\mathcal{S}.$
###### Lemma 2.7.
[22, Lemma 2.7] Let $f$ be a linearly nondegenerate meromorphic mapping from
$B(R_{0})$ into $\mathbb{P}^{n}(\mathbb{C})$ and let
$H_{1},H_{2},\ldots,H_{q}$ be $q$ hyperplanes of $\mathbb{P}^{n}(\mathbb{C})$
in $N$-subgeneral position. Set $l_{0}=|\alpha_{0}|+\cdots+|\alpha_{n}|$ and
take $t,p$ with $0<tl_{0}<p<1.$ Let $\omega(j)$ be Nochka weights with respect
to $H_{j}$, $1\leqslant j\leqslant q$ and let $k_{j}\ (j=1,\ldots,q)$ be
positive integers not less than $n$. For each $j$, we put
$\hat{\omega}(j):=\omega{(j)}\big{(}1-\frac{n}{k_{j}+1}).$ Then, for
$0<r_{0}<R_{0}$ there exists a positive constant $K$ such that for
$r_{0}<r<R<R_{0},$
$\int\limits_{S(r)}\left|z^{\alpha_{0}+\cdots+\alpha_{n}}\frac{W_{\alpha_{0}\ldots\alpha_{n}}(f)}{(f,H_{1})^{\hat{\omega}(1)}\cdots(f,H_{q})^{\hat{\omega}(q)}}\right|^{t}\bigl{(}\|f\|^{\sum_{j=1}^{q}\hat{\omega}(j)-n-1}\bigr{)}^{t}\sigma_{m}\leqslant
K\bigl{(}\frac{R^{2m-1}}{R-r}T_{f}(R,r_{0})\bigl{)}^{p},$
In fact, Lemma 2.7 is another version of Lemma 8 in [10], in which
$\omega{(j)}$ is replaced by $\hat{\omega}(j)$.
###### Lemma 2.8.
Let $M$, $f$ and $H_{1},H_{2},\ldots,H_{q}$ be as in Theorem 1.1. Let $P$ be a
holomorphic function on $M$ and $\beta$ be a positive real number such that
$P^{\beta}\in B(\alpha,l_{0};f^{1},f^{2},f^{3})$ and
$\displaystyle\sum_{u=1}^{3}\sum_{i=1}^{q}\nu^{[n]}_{H_{i}(f^{u}),\leqslant
k_{i}}\leqslant\beta\nu_{P},$
where $f^{1},f^{2},f^{3}\in\mathcal{F}(f,\\{H_{j},k_{j}\\}_{j=1}^{q},1)$. Then
$q\leqslant
2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{2}{3}l_{0}\big{)}+{\alpha}.$
###### Proof.
Let $F_{u}=(f^{u}_{0}:\cdots:f^{u}_{n})$ be a reduced representation of
$f^{u}\ (1\leqslant u\leqslant 3)$. By routine arguments in the Nevanlinna
theory and using Proposition 2.2 (i), we have
$\displaystyle\sum\limits_{i=1}^{q}\omega_{i}\nu_{H_{i}(f^{u})}(z)$
$\displaystyle-\nu_{W_{\alpha_{u,0}\cdots\alpha_{u,n}}(F_{u})}(z)$
$\displaystyle\leqslant\sum\limits_{i=1}^{q}\omega_{i}\min\\{n,\nu_{H_{i}(f^{u})}(z)\\}$
$\displaystyle=\sum\limits_{i=1}^{q}\omega_{i}\min\\{n,\nu_{H_{i}(f^{u}),\leqslant
k_{i}}(z)\\}+\sum\limits_{i=1}^{q}\omega_{i}\min\\{n,\nu_{H_{i}(f^{u}),>k_{i}}(z)\\}$
$\displaystyle\leqslant\sum\limits_{i=1}^{q}\frac{1}{\tilde{\omega}}\nu^{[n]}_{H_{i}(f^{u}),\leqslant
k_{i}}(z)+\sum\limits_{i=1}^{q}\omega_{i}\dfrac{n}{k_{i}+1}\nu_{H_{i}(f^{u})}(z).$
Hence, it is easy to see from the assumption that
(2.9)
$\displaystyle\sum_{i=1}^{q}{\hat{\omega}_{i}}(\nu_{H_{i}(f^{1})}+\nu_{H_{i}(f^{2})}+\nu_{H_{i}(f^{3})})-(\nu_{W_{\alpha_{1}}(F_{1})}+\nu_{W_{\alpha_{2}}(F_{2})}+\nu_{W_{\alpha_{3}}(F_{3})})\leqslant\frac{\beta}{\tilde{\omega}}\nu_{P},$
where $\hat{\omega}_{i}:=\omega_{i}\big{(}1-\dfrac{n}{k_{i}+1}\big{)}$ for all
$1\leqslant i\leqslant q$.
Since the universal covering of $M$ is biholomorphic to
$B(R_{0}),0<R_{0}\leqslant\infty$, by using the universal covering if
necessary, we may assume that $M=B(R_{0})\subset{\mathbf{C}}^{m}$. We consider
the following cases.
$\bullet$ First case: $R_{0}=\infty$ or $\lim\sup_{r\to
R_{0}}\dfrac{T_{f^{1}}(r,r_{0})+T_{f^{2}}(r,r_{0})+T_{f^{3}}(r,r_{0})}{\log(1/(R_{0}-r))}=\infty$.
Integrating both sides of inequality (2.9), we get
(2.10) $\displaystyle\beta N_{P}(r)$
$\displaystyle\geqslant{\tilde{\omega}}\sum_{u=1}^{3}(\sum_{i=1}^{q}{\omega_{i}}N_{H_{i}(f^{u})}(r,r_{0})-N_{W_{\alpha}(F_{u})}(r,r_{0}))-\sum_{u=1}^{3}\sum_{i=1}^{q}\frac{\tilde{\omega}\omega_{i}n}{k_{i}+1}(T_{f^{u}}(r,r_{0})+O(1).$
Applying Lemma 2.7 to $\omega_{i}\ (1\leqslant i\leqslant q),$ we have
$\int\limits_{S(r)}\left|z^{\alpha_{0}+\cdots+\alpha_{n}}\frac{W_{\alpha_{0}\ldots\alpha_{n}}(F_{u})}{H_{1}^{{\omega}_{1}}(f^{u})(z)\cdots
H_{q}^{{\omega}_{q}}(f^{u})(z)}\right|^{t_{u}}\left(\|f^{u}\|^{\sum_{i=1}^{q}{\omega}_{i}-n-1}\right)^{t_{u}}\sigma_{m}\leqslant
K\bigl{(}\frac{R^{2m-1}}{R-r}T_{f^{u}}(R,r_{0})\bigl{)}^{p_{u}}.$
By the concativity of the logarithmic function, we obtain
$\displaystyle\int\limits_{S(r)}\log|z^{\alpha_{0}+\cdots+\alpha_{n}}|\sigma_{m}$
$\displaystyle+(\sum_{i=1}^{q}{\omega}_{i}-n-1)\int\limits_{S(r)}\log||f^{u}||\sigma_{m}+\int\limits_{S(r)}\log|W_{\alpha_{0}\ldots\alpha_{n}}(F_{u})|\sigma_{m}$
$\displaystyle-\sum_{i=1}^{q}\omega_{i}\int\limits_{S(r)}\log|H_{i}(f^{u})|\sigma_{m}\leqslant\frac{p_{u}K}{t_{u}}\big{(}\log^{+}\frac{1}{R_{0}-r}+\log^{+}T_{f^{u}}(r,r_{0})\big{)}.$
By the definition of the characteristic function and the counting function, we
get the following estimate
$\displaystyle||\ (\sum_{i=1}^{q}{\omega}_{i}-n-1)T_{f^{u}}(r,r_{0})$
$\displaystyle\leqslant\sum_{i=1}^{q}\omega_{i}N_{H_{i}(f^{u})}(r,r_{0})-N_{W_{\alpha_{1}\ldots\alpha_{n}}F_{u})}(r)$
$\displaystyle+K_{1}\big{(}\log^{+}\frac{1}{R_{0}-r}+\log^{+}T_{f^{u}}(r,r_{0})\big{)}.$
Using Proposition 2.2 (ii), we get
$\displaystyle||\ (q-2N+n-1)T_{f^{u}}(r,r_{0})$
$\displaystyle\leqslant{\tilde{\omega}}\left(\sum_{i=1}^{q}{\omega_{i}}N_{H_{i}(f^{u})}(r,r_{0})-N_{W_{\alpha_{0}\ldots\alpha_{n}}(F_{u})}(r,r_{0})\right)$
$\displaystyle+{\tilde{\omega}}{K_{1}}\big{(}\log^{+}\frac{1}{R_{0}-r}+\log^{+}T_{f^{u}}(r,r_{0})\big{.}$
Combining these inequalities with (2.10) and noticing that
$\tilde{\omega}\omega_{i}\leqslant 1$, we get
(2.11) $\displaystyle||\ \beta N_{P}(r)$
$\displaystyle\geqslant(q-2N+n-1)T(r,r_{0})-\sum_{i=1}^{q}\frac{n}{k_{i}+1}T(r,r_{0})+O(1),$
where $T(r,r_{0}):=T_{f}(r,r_{0})+T_{g}(r,r_{0}).$
Since the assumption $P^{\beta}\in B(\alpha,l_{0};f^{1},f^{2},f^{3})$, there
exists $g\in S(l_{0};f^{1},f^{2},f^{3})$ satisfying
$|P|^{\beta}\leqslant||f^{1}||^{\alpha}\cdot||f^{2}||^{\alpha}\cdot||f^{3}||^{\alpha}\cdot
g,$
outside a proper analytic subset of $B(1).$ Hence, by Jensen’s formula and the
definition of the characteristic function, we have the following estimate
(2.12) $\displaystyle||\ \beta N_{P}(r)=$
$\displaystyle\int_{S(r)}\log|P|^{\beta}\sigma_{n}+O(1)$
$\displaystyle\leqslant$
$\displaystyle\int_{S(r)}({\alpha}\sum_{u=1}^{3}\log||f^{u}||+\log||g||)\sigma_{n}+O(1)$
$\displaystyle=$ $\displaystyle{\alpha}T_{f}(r,r_{0})+o(T(r,r_{0})).$
Together (2.11) with (2.12), we obtain
$\displaystyle(q-2N+n-1)T(r,r_{0})-\sum_{i=1}^{q}\frac{n}{k_{i}+1}T(r,r_{0})\leqslant{\alpha}T(r,r_{0})+o(T(r,r_{0}))$
for every $r$ outside a Borel finite measure set. Letting
$r\rightarrow\infty$, we deduce that
$q-2N+n-1-\sum_{i=1}^{q}\frac{n}{k_{i}+1}\leqslant\rho\big{(}n(2N-n+1)+\frac{2}{3}l_{0}\big{)}+{\alpha}$
with $\rho=0.$
$\bullet$ Second Case: $R_{0}<\infty$ and $\lim\sup_{r\to
R_{0}}\dfrac{T_{f^{1}}(r,r_{0})+T_{f^{2}}(r,r_{0})+T_{f^{3}}(r,r_{0})}{\log(1/(R_{0}-r))}<\infty$.
It suffices to prove the lemma in the case where $B(R_{0})=B(1)$.
Suppose that
$q>2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{2}{3}l_{0}\big{)}+{\alpha}.$
Then, we have
$q>2N-n+1+\sum_{i=1}^{q}{\tilde{\omega}}{\omega_{i}}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{2}{3}l_{0}\big{)}+\alpha.$
It follows from Proposition 2.2 ii), iv) that
$\displaystyle\sum_{i=1}^{q}{{\omega_{i}}}\big{(}1-\frac{n}{k_{i}+1}\big{)}-(n+1)-\dfrac{\alpha}{\tilde{\omega}}$
$\displaystyle>\rho\big{(}\frac{n(2N-n+1)}{\tilde{\omega}}+\frac{2}{3}\frac{l_{0}}{\tilde{\omega}}\big{)}$
$\displaystyle\geqslant\rho\big{(}n(n+1)+\frac{2}{3}\frac{l_{0}}{\tilde{\omega}}\big{)}.$
Put
$t=\dfrac{\frac{2\rho}{3}}{\displaystyle\sum_{i=1}^{q}{\hat{\omega}_{i}}-(n+1)-\dfrac{\alpha}{\tilde{\omega}}}.$
It implies that
(2.13)
$\displaystyle\big{(}\frac{3n(n+1)}{2}+\frac{l_{0}}{\tilde{\omega}}\big{)}t<1.$
Put
$\psi_{u}=z^{\alpha_{u,0}+\cdots+\alpha_{u,n}}\dfrac{W_{\alpha_{u,0}\cdots\alpha_{u,n}}(F_{u})}{H_{1}^{\hat{\omega}_{1}}(f^{u})\cdots
H_{q}^{\hat{\omega}_{q}}(f^{u})}\ \ (1\leqslant u\leqslant 3)$. It follows
from (2.9) that
$\psi_{1}^{t}\psi_{2}^{t}\psi_{3}^{t}P^{\frac{t\beta}{\tilde{\omega}}}$ is
holomorphic. Hence
$a=\log|\psi_{1}^{t}\psi_{2}^{t}\psi_{3}^{t}P^{\frac{t\beta}{\tilde{\omega}}}|$
is plurisubharmonic on $B(1)$.
We now write the given Kähler metric form as
${\omega}=\frac{\sqrt{-1}}{2\pi}\sum\limits_{i,j}h_{i\bar{j}}dz_{i}\wedge
d\bar{z}_{j}.$
From the assumption that $f^{1}$, $f^{2}$ and $f^{3}$ satisfy condition
$(C_{\rho})$, there are continuous plurisubharmonic functions $a^{\prime}_{u}$
on $B(1)$ such that
$e^{a^{\prime}_{u}}\text{det}(h_{i\bar{j}})^{\frac{1}{2}}\leqslant\|f^{u}\|^{\rho},u=1,2,3.$
Put $a_{u}=\frac{2}{3}a^{\prime}_{u}$, $u=1,2,3$ and we get
$e^{a_{u}}\text{det}(h_{i\bar{j}})^{\frac{1}{3}}\leqslant\|f^{u}\|^{\frac{2\rho}{3}}.$
Therefore, by the definition of $t$, we get
$\displaystyle e^{a+a_{1}+a_{2}+a_{3}}\text{det}(h_{i\bar{j}})$
$\displaystyle\leqslant
e^{a}\|f^{1}\|^{\frac{2\rho}{3}}\|f^{2}\|^{\frac{2\rho}{3}}\|f^{3}\|^{\frac{2\rho}{3}}$
$\displaystyle=|\psi_{1}|^{t}|\psi_{2}|^{t}|\psi_{3}|^{t}|P|^{\frac{t\beta}{\tilde{\omega}}}\|f^{1}\|^{\frac{2\rho}{3}}\|f^{2}\|^{\frac{2\rho}{3}}\|f^{3}\|^{\frac{2\rho}{3}}$
$\displaystyle\leqslant|\psi_{1}|^{t}|\psi_{2}|^{t}|\psi_{3}|^{t}\big{(}\|f^{1}\|\|f^{2}\|\|f^{3}|\big{)}^{\frac{t\alpha}{\tilde{\omega}}}\|f^{1}\|^{\frac{2\rho}{3}}\|f^{2}\|^{\frac{2\rho}{3}}\|f^{3}\|^{\frac{2\rho}{3}}\cdot|g|^{\frac{t}{\tilde{\omega}}}$
$\displaystyle=|\psi_{1}|^{t}|\psi_{2}|^{t}|\psi_{3}|^{t}\big{(}\|f^{1}\|\|f^{2}\|\|f^{3}\|\big{)}^{t(\frac{\alpha}{\tilde{\omega}}+\frac{2\rho}{3t})}\cdot|g|^{\frac{t}{\tilde{\omega}}}$
$\displaystyle=|\psi_{1}|^{t}|\psi_{2}|^{t}|\psi_{3}|^{t}\big{(}\|f^{1}\|\|f^{2}\|\|f^{3}\|\big{)}^{t(\sum_{i=1}^{q}\hat{\omega}_{i}-n-1)}\cdot|g|^{\frac{t}{\tilde{\omega}}}.$
Note that the volume form on $B(1)$ is given by
$dV:=c_{m}\text{det}(h_{i\bar{j}})v_{m};$
therefore,
$\int\limits_{B(1)}e^{a+a_{1}+a_{2}+a_{3}}dV\leqslant
C\int\limits_{B(1)}\prod_{u=1}^{3}\big{(}|\psi_{u}|\|f^{u}\|^{\sum_{i=1}^{q}\hat{\omega}_{i}-n-1}\big{)}^{t}\cdot|g|^{\frac{t}{\tilde{\omega}}}v_{m},$
with some positive constant $C.$
Setting $x=\dfrac{l_{0}/\tilde{\omega}}{3n(n+1)/2+l_{0}/\tilde{\omega}},\
y=\dfrac{n(n+1)/2}{3n(n+1)/2+l_{0}/\tilde{\omega}}$, then $x+3y=1$. Thus, by
the Hölder inequality and by noticing that
$v_{m}=(dd^{c}\|z\|^{2})^{m}=2m\|z\|^{2m-1}\sigma_{m}\wedge d\|z\|,$
we obtain
$\displaystyle\int\limits_{B(1)}e^{a+a_{1}+a_{2}+a_{3}}dV$
$\displaystyle\leqslant
C\prod_{u=1}^{3}\left(\int\limits_{B(1)}\big{(}|\psi_{u}|\|f^{u}\|^{\sum_{i=1}^{q}\hat{\omega}_{i}-n-1}\big{)}^{\frac{t}{y}}v_{m}\right)^{y}\left(\int\limits_{B(1)}|z^{\beta}g|^{\frac{t}{x\tilde{\omega}}}v_{m}\right)^{x}$
$\displaystyle\leqslant
C\prod_{u=1}^{3}\bigl{(}2m\int\limits_{0}\limits^{1}r^{2m-1}\bigl{(}\int\limits_{S(r)}\big{(}|\psi_{u}|\|f^{u}\|^{\sum_{i=1}^{q}\hat{\omega}_{i}-n-1}\big{)}^{\frac{t}{y}}\sigma_{m}\bigl{)}dr\bigl{)}^{y}$
$\displaystyle\times\bigl{(}2m\int\limits_{0}\limits^{1}r^{2m-1}\bigl{(}\int\limits_{S(r)}|z^{\beta}g|^{\frac{t}{x\tilde{\omega}}}\sigma_{m}\bigl{)}dr\bigl{)}^{x}.$
We see from (2.13) that
$\dfrac{l_{0}t}{\tilde{\omega}x}=\big{(}\dfrac{3n(n+1)}{2}+\dfrac{l_{0}}{\tilde{\omega}}\big{)}t<1$
and
$\sum\limits_{s=0}^{n}|\alpha_{u,s}|\dfrac{t}{y}\leqslant\dfrac{n(n+1)}{2}\dfrac{t}{y}=\big{(}\dfrac{3n(n+1)}{2}+\dfrac{l_{0}}{\tilde{\omega}}\big{)}t<1.$
Then, we can choose a positive number $p$ such that
$\dfrac{l_{0}t}{\tilde{\omega}x}<p<1$ and
$\sum\limits_{s=0}^{n}|\alpha_{u,s}|\dfrac{t}{y}<p<1.$ Applying Lemma 2.7 to
$\hat{\omega}_{i}$, and from the property of $g$, we get
$\int\limits_{S(r)}\big{(}|\psi_{u}|\|f^{u}\|^{\sum_{i=1}^{q}\hat{\omega}_{i}-n-1}\big{)}^{\frac{t}{y}}\sigma_{m}\leqslant
K_{1}\left(\frac{R^{2m-1}}{R-r}T_{f}^{u}(R,r_{0})\right)^{p}$
and
$\int\limits_{S(r)}|z^{\beta}g|^{\frac{t}{\tilde{\omega}x}}\sigma_{m}\leqslant
K\left(\frac{R^{2m-1}}{R-r}T_{g}(R,r_{0})\right)^{p}$
outside a subset $E\subset[0,1]$ such that
$\displaystyle\int\limits_{E}\dfrac{1}{1-r}dr\leqslant+\infty.$ Choosing
$R=r+\dfrac{1-r}{eT_{f^{u}}(r,r_{0})},$ we have
$T_{f^{u}}(R,r_{0})\leqslant 2T_{f^{u}}(r,r_{0}),$
Hence, the above inequality implies that
$\int\limits_{S(r)}\big{(}|\psi_{u}|\|f^{u}\|^{\sum_{i=1}^{q}\hat{\omega}_{i}-n-1}\big{)}^{\frac{t}{y}}\sigma_{m}\leqslant\frac{K_{2}}{(1-r)^{p}}(T_{f^{u}}(r,r_{0}))^{2p}\leqslant\frac{K_{2}}{(1-r)^{p}}(\log\frac{1}{1-r})^{2p},$
since $\lim\limits_{r\to
R_{0}}\sup\dfrac{T_{f^{1}}(r,r_{0})+T_{f^{2}}(r,r_{0})+T_{f^{2}}(r,r_{0})}{\log(1/(R_{0}-r))}<\infty.$
It implies that
$\int\limits_{0}\limits^{1}r^{2m-1}\left(\int\limits_{S(r)}\big{(}|\psi_{u}|\|f^{u}\|^{\sum_{i=1}^{q}\hat{\omega}_{i}-n-1}\big{)}\sigma_{m}\right)dr\leqslant\int\limits_{0}\limits^{1}r^{2m-1}\frac{K_{2}}{(1-r)^{p}}\left(\log\frac{1}{1-r}\right)^{2p}dr<\infty.$
Similarly,
$\int\limits_{0}\limits^{1}r^{2m-1}\left(\int\limits_{S(r)}|z^{\beta}g|^{\frac{t}{\tilde{\omega}x}}\sigma_{m}\right)dr\leqslant\int\limits_{0}\limits^{1}r^{2m-1}\frac{K_{2}}{(1-r)^{p}}\left(\log\frac{1}{1-r}\right)^{2p}dr<\infty.$
Hence, we conclude that $\int\limits_{B(1)}e^{a+a_{1}+a_{2}+a_{3}}dV<\infty,$
which contradicts Yau’s result [23] and Karp’s result [7]. The proof of Lemma
2.8 is complete. ∎
## 3\. Proof of Theorem 1.1
###### Lemma 3.1 (see [22], Lemma 3.1).
If $q>2N+1+\sum_{v=1}^{q}\frac{n}{k_{v}+1}+\rho n(2N-n+1)$, then every
$g\in\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$ is linearly nondegenerate.
###### Lemma 3.2 (see [10], Lemma 12).
Let $q,N$ be two integers satisfying $q\geq 2N+2$, $N\geq 2$ and $q$ be even.
Let $\\{a_{1},a_{2},\ldots,a_{q}\\}$ be a family of vectors in a 3-dimensional
vector space such that $\text{rank}\\{a_{j}\\}_{j\in R}=2$ for any subset
${R}\subset Q=\\{1,\ldots,q\\}$ with cardinality $|R|=N+1$. Then there exists
a partition $\bigcup_{j=1}^{q/2}I_{j}$ of $\\{1,\ldots,q\\}$ satisfying
$|I_{j}|=2$ and $\text{rank}\\{a_{i}\\}_{i\in I_{j}}=2$ for all
$j=1,\ldots,q/2.$
We need the following result which slightly improves [22, Theorem 1.3].
###### Lemma 3.3.
Let $k$ be the largest integer number not exceeding $\dfrac{q-2N-2}{2}$. If
$n\geqslant 2$ then $f^{1}\wedge f^{2}\wedge f^{3}\equiv 0$ for every
$f^{1},f^{2},f^{3}\in\mathcal{(}f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$ provided
$q>2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho
n(2N-n+1)+\frac{3nq}{2\big{(}q+(n-1)\frac{l+1}{l}\big{)}},$
where $l$ is the smallest integer number not less than $\dfrac{2N+2+2k}{k+2}$
if $k>0$ or $l=2N+1$ if $k=0.$
###### Proof.
We consider $\mathcal{M}^{3}$ as a vector space over the field $\mathcal{M}$
and denote $Q=\\{1,\ldots,q\\}$. For each $i\in Q$, we set
$V_{i}=\left((f^{1},H_{i}),(f^{2},H_{i}),(f^{3},H_{i})\right)\in\mathcal{M}^{3}.$
By Lemma 3.1, $f^{1},f^{2},f^{3}$ are linearly nondegenerate. Suppose that
$f^{1}\wedge f^{2}\wedge f^{3}\not\equiv 0$. Since the family of hyperplanes
$\\{H_{1},H_{2},\ldots,H_{q}\\}$ are in $N$-subgeneral position, for each
subset $R\subset Q$ with cardinality $|R|=N+1$, there exist three indices
$l,t,s\in R$ such that the vectors $V_{l},V_{t}$ and $V_{s}$ are linearly
independent. This means that
$P_{I}:=\det\left(\begin{array}[]{ccc}(f^{1},H_{l})&(f^{1},H_{t})&(f^{1},H_{s})\\\
(f^{2},H_{l})&(f^{2},H_{t})&(f^{2},H_{s})\\\
(f^{3},H_{l})&(f^{3},H_{t})&(f^{3},H_{s})\end{array}\right)\not\equiv 0,$
where $I:=\\{l,t,s\\}.$ We separate into the following cases.
$\bullet$ Case 1: $q\mod 2=0$
By the assumption, we have $q=2N+2+2k$ $(k\geq 0)$. Applying Lemma 3.2, we can
find a partition $\\{J_{1},\ldots,J_{q/2}\\}$ of $Q$ satisfying $|J_{j}|=2$
and $\text{rank}\\{V_{v}\\}_{v\in J_{j}}=2$ for all $j=1,2,\ldots,q/2.$ Take a
fixed subset $S_{j}=\\{j_{1},\ldots,j_{k+2}\\}\subset\\{1,\ldots,q\\}$. We
claim that:
There exists a partition $J^{j}_{1},\ldots,J^{j}_{N+1+k}$ with $k+2$ indices
${r^{j}_{1},\ldots,r^{j}_{k+2}}\in\\{1,\ldots,N+1+k\\}$ satisfying
$\text{rank}\\{V_{v},V_{j_{i}}\\}_{v\in J^{j}_{r^{j}_{i}}}=3$ for all
$1\leqslant i\leqslant k+2$.
Indeed, consider $N$ sets $J_{1},\ldots,J_{N}$ and $j_{1}$. Assume that
$\text{rank}\\{V_{j_{1}},V_{t_{2}}\ldots,V_{t_{u}}\\}=1$ where $u$ is maximal.
By the assumption, we have $1\leqslant u\leqslant N-1.$ It follows that there
exist $N-u$ pairs, for instance $\\{V_{v}\\}_{v\in
J_{1}},\ldots,\\{V_{v}\\}_{v\in J_{N-u}}$ which do not contain $V_{j_{1}}$ or
$V_{t_{i}}$ with $2\leqslant i\leqslant u$. Obviously, $N-u\geq 1$. Without
loss of generality, we can assume that $V_{j_{1}}\in\\{V_{v}\\}_{v\in J_{N}}$.
If $u=N-1$ then obviously, $\text{rank}\\{V_{v},V_{j_{1}}\\}_{v\in J_{1}}=3$
since
$\sharp(\\{V_{j_{1}},V_{t_{2}},\ldots,V_{t_{N-1}}\\}\cup\\{V_{v}\\}_{v\in
J_{1}})=N+1.$
If $u\leqslant N-2$, there are at least two pairs vectors, which do not
contain $V_{j_{1}}$ or $V_{t_{i}}$ with $2\leqslant i\leqslant u$. Assume that
$V_{j_{1}}\in$ span$\\{V_{v}\\}_{v\in J_{r_{1}}}$ with some
$r_{1}\in\\{1,\ldots,N-u\\}$, there exists at least one pair, for instance
$\\{V_{v}\\}_{v\in J_{j_{0}}}$ with $j_{0}\in\\{1,\ldots,N-u\\}$ such that
$\text{rank}\\{V_{v}\\}_{v\in(J_{r_{1}}\cup J_{j_{0}})}=3$. Indeed, otherwise
$\text{rank}\\{V_{v}\\}_{v\in(\cup_{i=1}^{N-u}J_{i})\cup\\{j_{1},t_{2}\ldots,t_{u}\\}}=\text{rank}\\{V_{v}\\}_{v\in
J_{r_{1}}}=2$. This is impossible since
$\\{V_{v}\\}_{v\in(\cup_{i=1}^{N-u}J_{i})\cup\\{j_{1},t_{2}\ldots,t_{u}\\}}$
has at least $N+2$ vectors. From sets $\\{V_{v}\\}_{v\in J_{r_{1}}}$ and
$\\{V_{v}\\}_{v\in J_{j_{0}}}$, we can rebuild two linearly independent pairs
$\\{V_{i_{1}},V_{i_{2}}\\}$ and $\\{V_{i_{3}},V_{i_{4}}\\}$ such that
$\text{rank}\\{V_{i_{1}},V_{i_{2}},V_{j_{1}}\\}=3$, where
$\\{i_{1},i_{2},i_{3},i_{4}\\}=J_{r_{1}}\cup J_{j_{0}}.$ We redenote by
$J_{r_{1}}=\\{i_{1},i_{2}\\}$ and $J_{j_{0}}=\\{i_{3},i_{4}\\}$.
Therefore, we obtain a partition still denoted by $J_{1},\ldots,J_{N+1+k}$
such that there exists an index ${r^{j}_{1}}\in\\{1,\ldots,N\\}$ satisfying
$\text{rank}\\{V_{v},V_{j_{1}}\\}_{v\in J_{r^{j}_{1}}}=3$.
Next, we consider $N$ sets
$J_{1},\ldots,J_{r^{j}_{1}-1},J_{r^{j}_{1}+1},\ldots,J_{N+1}$ and $j_{2}$.
Repeating the above argument, we get a partition still denoted by
$J_{1},\ldots,J_{q/2}$ such that there exists an index
${r^{j}_{2}}\in\\{1,\ldots,{r^{j}_{1}-1},{r^{j}_{1}+1},\ldots,N+1\\}$
satisfying $\text{rank}\\{V_{v},V_{j_{2}}\\}_{v\in J_{r^{j}_{2}}}=3$. Of
course, this partition still satisfies $\text{rank}\\{V_{v},V_{j_{1}}\\}_{v\in
J_{r^{j}_{1}}}=3$.
Continue to the process, after $k+2$ times, we will obtain a new partition
denoted by $J^{j}_{1},\ldots,J^{j}_{N+1+k}$ such that there exists $k+2$
indices ${r^{j}_{1},\ldots,r^{j}_{k+2}}\in\\{1,\ldots,N+1+k\\}$ satisfying
$\text{rank}\\{V_{v},V_{j_{i}}\\}_{v\in J^{j}_{r^{j}_{i}}}=3$ for all
$1\leqslant i\leqslant k+2$. The claim is proved.
Put $I^{j}_{r^{j}_{i}}=J^{j}_{r^{j}_{i}}\cup\\{j_{i}\\}$, then
$P_{{I^{j}_{r^{j}_{i}}}}\not\equiv 0$ for all $1\leqslant i\leqslant k+2$.
For each remained index
$i\in\\{1,\ldots,N+1+k\\}\setminus\\{r^{j}_{1},\ldots,r^{j}_{k+2}\\}$, we
choose a vector $V_{s_{i}}$ such that $\text{rank}\\{V_{v}\\}_{v\in
J^{j}_{i}\cup\\{s_{i}\\}}=3.$ Put $I^{j}_{i}=J^{j}_{i}\cup\\{s_{i}\\}$, then
$P_{{I^{j}_{i}}}\not\equiv 0$ for all $i.$
$\bullet$ If $k=0$ then $l=2N+1$ and $q=2N+2$. Put
$S_{1}=\\{1\\},S_{2}=\\{2\\},\ldots,S_{l-1}=\\{2N\\},S_{l}=\\{2N+1,2N+2\\}.$
$\bullet$ If $k>0$ then $q=(k+2)(l-1)+t$ with $0<t\leqslant k+2.$ Put
$S_{1}=\\{1,\ldots,k+2\\},S_{2}=\\{(k+2)+1,\ldots,2(k+2)\\},\ldots,S_{l-1}=\\{(k+2)(l-2)+1,\ldots,(k+2)(l-1)\\},S_{l}=\\{(k+2)(l-1)+1,\ldots,2N+2+2k\\}.$
Applying the claim to each set $S_{j}$ $(1\leqslant j\leqslant l)$, we get a
partition $J^{j}_{1},\ldots,J^{j}_{N+1+k}$ with $s_{j}=\sharp S_{j}$ indices
${r^{j}_{1},\ldots,r^{j}_{s_{j}}}\in\\{1,\ldots,N+1+k\\}$ satisfying
$\text{rank}\\{V_{v},V_{u}\\}_{v\in J^{j}_{r^{j}_{i}},u\in S_{j}}=3$ for all
$1\leqslant i\leqslant s_{j}$.
We put
$P_{Q}=\prod_{j=1}^{l}\prod_{i=1}^{N+1+k}P_{I^{j}_{i}},$
where $I^{j}_{i}$ is defined as in the above.
Since $(\min\\{a,b,c\\}-1)\geq\min\\{a,n\\}+\min\\{b,n\\}+\min\\{c,n\\}-2n-1$
for any positive integers $a,b,c$, we have
$\displaystyle\min_{1\leqslant u\leqslant 3}\\{\nu_{(f^{u},H_{v}),\leqslant
k_{v}}(z)\\}-\nu^{[1]}_{(f^{k},H_{v}),\leqslant k_{v}}(z)$
$\displaystyle\geq\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z)-(2n+1)\nu^{[1]}_{(f^{k},H_{v}),\leqslant k_{v}}(z),$
for all $z\in\mathrm{Supp}\,\nu_{(f^{k},H_{v}),\leqslant k_{v}}$.
Putting $\nu_{v}(z)=\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z)-(2n+1)\nu^{[1]}_{(f^{k},H_{v}),\leqslant k_{v}}(z)\ (1\leqslant
k\leqslant 3,\ v\in Q),$ from Lemma 2.6, we have
$\displaystyle\nu_{P_{{I}^{j}_{i}}}(z)\geq\sum_{v\in{I}^{j}_{i}}\nu_{v}(z)+2\sum_{v=1}^{q}\nu^{[1]}_{(f^{k},H_{v}),\leqslant
k_{v}}(z)$
and
$\displaystyle\nu_{P_{{I}^{j}_{i}}}(z)\geq\sum_{v\in{J}^{j}_{i}}\nu_{v}(z)+2\sum_{v=1}^{q}\nu^{[1]}_{(f^{k},H_{v}),\leqslant
k_{v}}(z).$
Note that for $k=0$ then $l(q-2N-1)-(2N+1)=0$. For $k>0$ then
$2N+1\leqslant\frac{q}{k+2}(2k+1)\leqslant l(2k+1)=l(q-2N-1)$. Therefore, we
always have $l(q-2N-1)-(2N+1)\geqslant 0.$ It implies that
$l(q-2n-1)-(2n+1)\geqslant 0$ since $N\geq n.$ Then, for all
$z\not\in\mathcal{S}$, we obtain
$\displaystyle\nu_{P_{Q}}(z)$ $\displaystyle\geq
l\sum_{v=1}^{q}\nu_{v}(z)+\sum_{v=1}^{q}\nu_{v}(z)+lq\sum_{v=1}^{q}\nu^{[1]}_{(f^{k},H_{v}),\leqslant
k_{v}}(z)$
$\displaystyle=(l+1)\sum_{v=1}^{q}(\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z)-(2n+1)\nu^{[1]}_{(f^{k},H_{v}),\leqslant
k_{v}}(z))+lq\sum_{v=1}^{q}\nu^{[1]}_{(f^{k},H_{v}),\leqslant k_{v}}(z)$
$\displaystyle=(l+1)\sum_{v=1}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z)+\big{(}l(q-2n-1)-(2n+1)\big{)}\sum_{v=1}^{q}\nu^{[1]}_{(f^{k},H_{v}),\leqslant
k_{v}}(z)$
$\displaystyle\geq\left(l+1+\frac{l(q-2n-1)-(2n+1)}{3n}\right)\sum_{v=1}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z)$
$\displaystyle\geq\frac{l(q+n-1)+n-1}{3n}\sum_{v=1}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z).$
We put $P:=P_{Q}$. The above inequality implies that
$\displaystyle\sum_{v=1}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z)\leqslant\frac{3n}{l(q+n-1)+n-1}\nu_{P}(z),\forall
z\not\in\mathcal{S}.$
Define $\beta:=\dfrac{3n}{l(q+n-1)+n-1}$ and $\gamma:=\dfrac{lq}{2}$.
$\bullet$ Case 2: $q\mod 2=1$.
By the assumption, we have $q-1=2N+2+2k.$ We consider any subset
${R}=\\{j_{1},\ldots,j_{q-1}\\}$ of $\\{1,\ldots,q\\}$. By the same argument
as in Case 1 for $R$, we get
$\displaystyle\nu_{P_{R}}(z)$
$\displaystyle\geq(l+1)\sum_{v=1}^{q-1}\nu_{j_{v}}(z)+l(q-1)\sum_{v=1}^{q}\nu^{[1]}_{(f^{k},H_{v}),\leqslant
k_{v}}(z),\forall z\not\in\mathcal{S}.$
We now define $P:=\prod_{|R|=q-1}P_{R},$ so we obtain
$\displaystyle\nu_{P}(z)$ $\displaystyle=\sum_{|R|=q-1}\nu_{P_{R}}$
$\displaystyle\geq(q-1)(l+1)\sum_{v=1}^{q}\nu_{v}(z)+ql(q-1)\sum_{v=1}^{q}\nu^{[1]}_{(f^{k},H_{v}),\leqslant
k_{v}}(z)$
$\displaystyle\geq(q-1)\frac{l(q+n-1)+n-1}{3n}\sum_{v=1}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z).$
Hence, we have
$\displaystyle\sum_{v=1}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{v}),\leqslant
k_{v}}(z)\leqslant\frac{3n}{(l(q+n-1)+n-1)(q-1)}\nu_{P}(z),\forall
z\not\in\mathcal{S}.$
Define $\beta:=\dfrac{3n}{\big{(}l(q+n-1)+n-1\big{)}(q-1)}$ and
$\gamma:=\dfrac{(q-1)lq}{2}.$ Then, from all the above cases, we always get
$\alpha:=\beta\gamma=\dfrac{3nlq}{2(l(q+n-1)+n-1)}=\frac{3nq}{2\big{(}q+(n-1)\frac{l+1}{l}\big{)}},$
and
$\displaystyle\sum_{u=1}^{3}\sum_{v=1}^{q}\nu_{(f^{u},H_{v}),\leqslant
k_{v}}^{[n]}(z)\leqslant\beta\nu_{P}(z),\forall z\not\in\mathcal{S}.$
It is easy to see that $|P|^{\beta}\leqslant
C(\|f^{1}\|\|f^{2}\|\|f^{3}\|)^{\beta\gamma}=C(\|f^{1}\|\|f^{2}\|\|f^{3}\|)^{\alpha}$,
where $C$ is some positive constant. This means that $P^{\beta}\in
B(\alpha,0;f^{1},f^{2},f^{3})$. Applying Lemma 2.8, we obtain
$\displaystyle q$ $\displaystyle\leqslant
2N-n+1+\sum_{j=1}^{q}\frac{n}{k_{j}+1}+\rho n(2N-n+1)+\alpha$
$\displaystyle=2N-n+1+\sum_{j=1}^{q}\frac{n}{k_{j}+1}+\rho
n(2N-n+1)+\frac{3nq}{2\big{(}q+(n-1)\frac{l+1}{l}\big{)}},$
which contradicts the assumption. Therefore, $f^{1}\wedge f^{2}\wedge
f^{3}\equiv 0$ on $M$. The proof of Lemma 3.3 is complete. ∎
By basing on the proofs of Quang [18, Lemma 3.3, 3.4, 3.5, 3.6] or [19, Lemma
4.4, 4.5, 4.6, 4.8], we obtain the following Lemmas which are necessary for
the proof of our theorem.
The first, for three mappings
$f^{1},f^{2},f^{3}\in\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$, we define
$\bullet F^{ij}_{k}=\frac{(f^{k},H_{i})}{(f^{k},H_{i})},\ \ 0\leqslant
k\leqslant 2,\ 1\leqslant i,j\leqslant q,$
$\bullet
V_{i}=((f^{1},H_{i}),(f^{2},H_{i}),(f^{3},H_{i}))\in\mathcal{M}^{3}_{m},$
$\bullet\nu_{i}:\text{ the divisor whose support is the closure of the set }$
$\\{z:\nu_{(f^{u},H_{i}),\leqslant
k_{i}}(z)\geqslant\nu_{(f^{v},H_{i}),\leqslant
k_{i}}(z)=\nu_{(f^{t},H_{i}),\leqslant k_{i}}(z)\text{ for a permutation
}(u,v,t)\text{ of }(1,2,3)\\}.$
We write $V_{i}\cong V_{j}$ if $V_{i}\wedge V_{j}\equiv 0$, otherwise we write
$V_{i}\not\cong V_{j}$. For $V_{i}\not\cong V_{j}$, we write $V_{i}\sim V_{j}$
if there exist $1\leqslant u<v\leqslant 3$ such that $F_{u}^{ij}=F_{v}^{ij}$,
otherwise we write $V_{i}\not\sim V_{j}.$
###### Lemma 3.4.
[18, Lemma 3.3] or [19, Lemma 4.4] With the assumption of Theorem 1.1, let $h$
and $g$ be two elements of the family
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$. If there exists a constant
$\lambda$ and two indices $i,j$ such that
$\frac{(h,H_{i})}{(h,H_{j})}=\lambda\frac{(g,H_{i})}{(g,H_{j})},$ then
$\lambda=1.$
###### Lemma 3.5.
[18, Lemma 3.4] or [19, Lemma 4.5] Let $f^{1},f^{2},f^{3}$ be three elements
of $\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$. Suppose that $f^{1}\wedge
f^{2}\wedge f^{3}\equiv 0$ and $V_{i}\sim V_{j}$ for some distinct indices $i$
and $j$. Then $f^{1},f^{2},f^{3}$ are not distinct.
###### Lemma 3.6.
[18, Lemma 3.5] or [19, Lemma 4.6] With the assumption of Theorem 1.1, let
$f^{1},f^{2},f^{3}$ be three maps in
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$. Suppose that
$f^{1},f^{2},f^{3}$ are distinct and there are two indices
$i,j\in\\{1,2,\ldots,q\\}\ (i\not=j)$ such that $V_{i}\not\cong V_{j}$ and
$\Phi^{\alpha}_{ij}:=\Phi^{\alpha}(F_{1}^{ij},F_{2}^{ij},F_{3}^{ij})\equiv 0$
for every $\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{Z}^{m}_{+}$ with
$|\alpha|=1.$ Then for every $t\in\\{1,\ldots,q\\}\setminus\\{i\\}$, the
following assertions hold:
(i) $\Phi^{\alpha}_{it}\equiv 0$ for all $|\alpha|\leqslant 1,$
(ii) if $V_{i}\not\cong V_{t}$, then $F^{ti}_{1},F^{ti}_{2},F^{ti}_{3}$ are
distinct and there exists a meromorphic function $h_{it}\in
B(0,1;f^{1},f^{2},f^{3})$ such that
$\displaystyle\nu_{h_{ti}}\geq-\nu^{[1]}_{(f,H_{i}),\leqslant
k_{i}}-\nu^{[1]}_{(f,H_{t}),\leqslant
k_{t}}+\sum_{j\not=i,t}\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}.$
###### Lemma 3.7.
[18, Lemma 3.6] or [19, Lemma 4.8] With the assumption of Theorem 1.1, let
$f^{1},f^{2},f^{3}$ be three maps in
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$. Assume that there exist
$i,j\in\\{1,2,\ldots,q\\}\ (i\not=j)$ and $\alpha\in\mathbb{Z}^{m}_{+}$ with
$|\alpha|=1$ such that $\Phi^{\alpha}_{ij}\not\equiv 0$. Then there exists a
holomorphic function $g_{ij}\in B(1,1;f^{1},f^{2},f^{3})$ such that
$\displaystyle\nu_{g_{ij}}$
$\displaystyle\geq\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}+\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{j}),\leqslant
k_{j}}+2\sum_{t=1,t\not=i,j}\nu^{[1]}_{(f,H_{t}),\leqslant
k_{t}}-(2n+1)\nu^{[1]}_{(f,H_{i}),\leqslant k_{i}}$
$\displaystyle-(n+1)\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}+\nu_{j}.$
We now prove Theorem 1.1.
Suppose that there exist three distinct meromorphic mappings
$f^{1},f^{2},f^{3}$ belonging to
$\mathcal{F}(f,\\{H_{i},k_{i}\\}_{i=1}^{q},1)$. By Lemma 3.3, we get
$f^{1}\wedge f^{2}\wedge f^{3}\equiv 0.$ We may assume that
${\mathrel{\mathop{{\underbrace{{V_{1}\cong\cdots\cong
V_{l_{1}}}}}}\limits_{{group\
1}}}}\not\cong{\mathrel{\mathop{{\underbrace{V_{l_{1}+1}\cong\cdots\cong
V_{l_{2}}}}}\limits_{{group\
2}}}}\not\cong{\mathrel{\mathop{{\underbrace{V_{l_{2}+1}\cong\cdots\cong
V_{l_{3}}}}}\limits_{{group\
3}}}}\not\cong\cdots\not\cong{\mathrel{\mathop{{\underbrace{V_{l_{s-1}+1}\cong\cdots\cong
V_{l_{s}}}}}\limits_{{group\ s}}}},$
where $l_{s}=q.$
Denote by $P$ the set of all $i\in\\{1,\ldots,q\\}$ satisfying that there
exists $j\in\\{1,\ldots,q\\}\setminus\\{i\\}$ such that $V_{i}\not\cong V_{j}$
and $\Phi^{\alpha}_{ij}\equiv 0$ for all $\alpha\in\mathbb{Z}^{m}_{+}$ with
$|\alpha|\leqslant 1.$ We separate into three cases.
$\bullet$ Case 1: $\sharp P\geq 2.$ It follows that $P$ contains two elements
$i,j.$ We get $\Phi^{\alpha}_{ij}=\Phi^{\alpha}_{ji}=0$ for all
$\alpha\in\mathbb{Z}^{m}_{+}$ with $|\alpha|\leqslant 1.$ By Lemma 2.1, there
exist two functions, for instance $F_{1}^{ij}$ and $F_{2}^{ij}$, and a
constant $\lambda$ such that $F_{1}^{ij}=\lambda F_{2}^{ij}.$ Applying Lemma
3.4, we have $F_{1}^{ij}=F_{2}^{ij}$. Hence, since Lemma 3.6 (ii), we can see
that $V_{i}\cong V_{j}$, i.e., $V_{i}$ and $V_{j}$ belong to the same group in
the partition. We may assume that $i=1$ and $j=2.$ Since our assumption
$f^{1},f^{2},f^{3}$ are distinct, the number of each group in the partition is
less than $N+1.$ Thus, we get $V_{1}\cong V_{2}\not\cong V_{t}$ for all
$t\in\\{N+1,\ldots,q\\}.$ By Lemma 3.6 (ii), we obtain
$\displaystyle\nu_{h_{1t}}\geq-\nu^{[1]}_{(f,H_{1}),\leqslant
k_{1}}-\nu^{[1]}_{(f,H_{t}),\leqslant
k_{t}}+\sum_{s\not=1,t}\nu^{[1]}_{(f,H_{s}),\leqslant k_{s}},$
and
$\displaystyle\nu_{h_{2t}}\geq-\nu^{[1]}_{(f,H_{2}),\leqslant
k_{2}}-\nu^{[1]}_{(f,H_{t}),\leqslant
k_{t}}+\sum_{s\not=2,t}\nu^{[1]}_{(f,H_{s}),\leqslant k_{s}}.$
By summing up both sides of the above two inequalities, we have
$\displaystyle\nu_{h_{1t}}+\nu_{h_{2t}}\geq-2\nu^{[1]}_{(f,H_{t})\leqslant
k_{t}}+\sum_{s\not=1,2,t}\nu^{[1]}_{(f,H_{s}),\leqslant k_{s}}.$
Summing up both sides of the above inequalities over all
$t\in\\{N+1,\ldots,q\\},$ we obtain
$\displaystyle\sum_{t=N+1}^{q}(\nu_{h_{1t}}+\nu_{h_{2t}})$
$\displaystyle\geq(q-N)\sum_{t=3}^{N}\nu^{[1]}_{(f,H_{t})\leqslant
k_{t}}+(q-N-3)\sum_{t=N+1}^{q}\nu^{[1]}_{(f,H_{t})\leqslant k_{t}}$
$\displaystyle\geq(q-N-3)\sum_{t=3}^{q}\nu^{[1]}_{(f,H_{t})\leqslant
k_{t}}\geq\frac{q-N-3}{3n}\sum_{u=1}^{3}\sum_{t=3}^{q}\nu^{[n]}_{(f,H_{t})\leqslant
k_{t}}.$
Hence, we get
$\displaystyle\sum_{u=1}^{3}\sum_{t=3}^{q}\nu^{[n]}_{(f,H_{t})\leqslant
k_{t}}\leqslant\frac{3n}{q-N-3}\nu_{\prod_{t=N+1}^{q}}(h_{1t}h_{2t}).$
Since $({\prod_{t=N+1}^{q}}(h_{1t}h_{2t}))^{\frac{3n}{q-N-3}}\in
B(0,2(q-N)\frac{3n}{q-N-3};f^{1},f^{2},f^{3})$, applying Lemma 2.8, we obtain
$q-2\leqslant
2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+4(q-N)\frac{n}{q-N-3}\big{)}.$
From the definition of $l$ and the condition of $q$, it is easy to see that
$l\geq 3.$ It is easy to see that
$2\leqslant\frac{3nq}{2\big{(}q+n-1+\frac{n-1}{3}\big{)}}\leqslant\frac{3nq}{2\big{(}q+n-1+\frac{n-1}{l}\big{)}},$
and
$4(q-N)\frac{n}{q-N-3}\leqslant\frac{4(q-n)n}{n-1}.$
These inequalities imply that
$q\leqslant
2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{4(q-n)n}{n-1}\big{)}+\frac{3nq}{2\big{(}q+n-1+\frac{n-1}{l}\big{)}},$
which is a contradiction.
$\bullet$ Case 2: $\sharp P=1$. We assume that $P=\\{1\\}.$ It is easy to see
that $V_{1}\not\cong V_{i}$ for all $i=2,\ldots,q$. By Lemma 3.6 (ii), we
obtain
$\displaystyle\nu_{h_{1i}}\geq-\nu^{[1]}_{(f,H_{1})\leqslant
k_{1}}-\nu^{[1]}_{(f,H_{i})\leqslant
k_{i}}+\sum_{s\not=1,t}\nu^{[1]}_{(f,H_{s})\leqslant k_{s}}.$
Summing up both sides of the above inequalities over all $i=2,\ldots,q,$ we
have
(3.8)
$\displaystyle\sum_{i=2}^{q}\nu_{h_{1i}}\geq(q-3)\sum_{i=2}^{q}\nu^{[1]}_{(f,H_{i})\leqslant
k_{i}}-(q-1)\nu^{[1]}_{(f,H_{1})\leqslant k_{1}}.$
Obviously, $i\not\in P$ for all $i=2,\ldots,q.$ Now put
$\sigma(i)=\begin{cases}i+N,&\text{ if }i+N\leqslant q\\\ i-N-q+1,&\text{ if
}i+N>q,\end{cases}$
then $i$ and $\sigma(i)$ belong to distinct groups, i.e., $V_{i}\not\cong
V_{\sigma(i)}$ for all $i=2,\ldots,q$ and hence
$\Phi^{\alpha}_{i\sigma(i)}\not\equiv 0$ for some
$\alpha\in\mathbb{Z}^{m}_{+}$ with $|\alpha|\leqslant 1.$ By Lemma 3.7, we get
$\displaystyle\nu_{g_{i\sigma(i)}}$
$\displaystyle\geq\sum_{u=1}^{3}\sum_{t=i,\sigma(i)}\nu^{[n]}_{(f^{u},H_{t})\leqslant
k_{t}}-(2n+1)\nu^{[1]}_{(f,H_{i})\leqslant
k_{i}}-(n+1)\nu^{[1]}_{(f,H_{\sigma(i)})\leqslant k_{\sigma(i)}}$
$\displaystyle+2\sum_{t=1,t\not=i,\sigma(i)}\nu^{[1]}_{(f,H_{t})\leqslant
k_{t}}.$
Summing up both sides of this inequality over all $i\in\\{2,\ldots,q\\}$ and
using (3.8), we obtain
$\displaystyle\sum_{i=2}^{q}\nu_{g_{i\sigma(i)}}$ $\displaystyle\geq
2\sum_{i=2}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}+(2q-3n-8)\sum_{i=2}^{q}\nu^{[1]}_{(f,H_{i}),\leqslant
k_{i}})+2(q-1)\nu^{[1]}_{(f,H_{1}),\leqslant k_{1}}$ $\displaystyle\geq
2\sum_{i=2}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}+\frac{4q-3n-14}{3}\sum_{u=1}^{3}\sum_{i=2}^{q}\nu^{[1]}_{(f^{u},H_{i}),\leqslant
k_{i}})-2\sum_{i=2}^{q}\nu_{h_{1i}}$
$\displaystyle\geq\frac{4q+3n-14}{3n}\sum_{i=2}^{q}\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}-2\sum_{i=2}^{q}\nu_{h_{1t}}.$
It implies that
$\sum_{u=1}^{3}\sum_{i=2}^{q}\nu^{[n]}_{(f^{u},H_{i})}\leqslant\frac{3n}{4q+3n-14}\nu_{\prod_{i=2}^{q}(g_{i\sigma(i)}h^{2}_{1i})}.$
Obviously, $\prod_{i=2}^{q}(g_{i\sigma(i)}h^{2}_{1i})\in
B(q-1,3(q-1);f^{1},f^{2},f^{3})$. Applying Lemma 2.8, we obtain
$q-1\leqslant
2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{6n(q-1)}{4q+3n-14}\big{)}+\frac{3n(q-1)}{4q+3n-14}.$
Since $q\geq 2n+2$ and by the simple calculation, we have
$\frac{6n(q-1)}{4q+3n-14}\leqslant\frac{6n(q-1)}{11n-6}<\frac{4(q-n)n}{n-1}.$
It implies that
$q\leqslant
2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{4(q-n)n}{n-1}\big{)}+\frac{4q+3nq-14}{4q+3n-14},$
which is a contradiction.
$\bullet$ Case 3: $\sharp P=0$. By Lemma 3.7, for all $i\not=j$, we get
$\displaystyle\nu_{g_{ij}}$
$\displaystyle\geq\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}+\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{j}),\leqslant
k_{j}}+2\sum_{t=1,t\not=i,j}\nu^{[1]}_{(f,H_{t}),\leqslant
k_{t}}-(2n+1)\nu^{[1]}_{(f,H_{i}),\leqslant k_{i}}$
$\displaystyle-(n+1)\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}+\nu_{j}.$
Put
$\gamma(i)=\begin{cases}i+N&\text{ if }i\leqslant q-N\\\ i+N-q&\text{ if
}i>q-N.\end{cases}$
By summing up both sides of the above inequality over all pairs
$(i,\gamma(i)),$ we obtain
(3.9) $\displaystyle\sum_{i=1}^{q}\nu_{g_{i\gamma(i)}}\geq
2\sum_{u=1}^{3}\sum_{i=1}^{q}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}+(2q-3n-6)\sum_{t=1}^{q}\nu^{[1]}_{(f,H_{t}),\leqslant
k_{t}}+\sum_{t=1}^{q}\nu_{t}.$
By Lemma 3.5, we can see that $V_{j}\not\sim V_{l}$ for all $j\not=l.$ Thus,
we have
$P^{i\gamma(i)}_{st}:=(f^{s},H_{i})(f^{t},H_{\gamma(i)})-(f^{t},H_{\gamma(i)})(f^{s},H_{i})\not\equiv
0,\ s\not=t,1\leqslant i\leqslant q.$
We claim that: With $i\not=j\not=\gamma(i)$, for every $z\in f^{-1}(H_{j})$,
we have
$\sum_{1\leqslant s<t\leqslant 3}\nu_{P^{i\gamma(i)}_{st}}(z)\geq
4\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}-\nu_{j}(z).$
Indeed, for $z\in f^{-1}(H_{j})\cap\mathrm{Supp}\,{\nu_{j}},$ we have
$4\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}(z)-\nu_{j}(z)\leqslant
4-1=3\leqslant\sum_{1\leqslant s<t\leqslant 3}\nu_{P^{i\gamma(i)}_{st}}.$
For $z\in f^{-1}(H_{j})\setminus\mathrm{Supp}\,\nu_{j}$, we assume that
$\nu_{(f^{1},H_{j}),\leqslant k_{j}}(z)<\nu_{(f^{2},H_{j}),\leqslant
k_{j}}(z)\leqslant\nu_{(f^{3},H_{j}),\leqslant k_{j}}(z).$ Since $f^{1}\wedge
f^{2}\wedge f^{3}\equiv 0,$ we have $\det(V_{i},V_{\gamma(i)},V_{j})\equiv 0,$
and hence
$(f^{1},H_{j})P^{i\gamma(i)}_{23}=(f^{2},H_{j})P^{i\gamma(i)}_{13}-(f^{3},H_{j})P^{i\gamma(i)}_{12}.$
It implies that $\nu_{P^{i\gamma(i)}_{23}}\geq 2$ and so
$\sum_{1\leqslant s<t\leqslant 3}\nu_{P^{i\gamma(i)}_{st}}(z)\geq
4=4\nu^{[1]}_{(f,H_{i}),\leqslant k_{i}}(z)-\nu_{j}(z).$
The claim is proved.
On the other hand, with $j=i$ or $j=\sigma(i)$, for every $z\in
f^{-1}(H_{j})$, we see that
$\displaystyle\nu_{P^{i\gamma(i)}_{st}}(z)$
$\displaystyle\geq\min\\{\nu_{(f^{s},H_{j}),\leqslant
k_{j}}(z),\nu_{(f^{t},H_{j}),\leqslant k_{j}}(z)\\}$
$\displaystyle\geq\nu^{[n]}_{(f^{s},H_{j}),\leqslant
k_{j}}(z)+\nu^{[n]}_{(f^{t},H_{j}),\leqslant
k_{j}}(z)-n\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}(z).$
Hence, $\sum_{1\leqslant s<t\leqslant 3}\nu_{P^{i\gamma(i)}_{st}}(z)\geq
2\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{j}),\leqslant
k_{j}}(z)-3n\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}(z).$ Together this
inequality with the above claim, we obtain
$\displaystyle\sum_{1\leqslant s<t\leqslant 3}\nu_{P^{i\gamma(i)}_{st}}(z)$
$\displaystyle\geq\sum_{j=i,\gamma(i)}\big{(}2\sum_{u=1}^{3}\nu^{[n]}_{(f^{u},H_{j}),\leqslant
k_{j}}(z)-3n\nu^{[1]}_{(f,H_{j}),\leqslant k_{j}}(z)\big{)}$
$\displaystyle+\sum_{j=1,j\not=i,\gamma(i)}(4\nu^{[1]}_{(f,H_{j}),\leqslant
k_{j}}(z)-\nu_{j}(z)).$
On the other hand, it is easy to see that $\prod_{1\leqslant s<t\leqslant
3}P^{i\gamma(i)}_{st}\in B(2,0;f^{1},f^{2},f^{3}).$ Summing up both sides of
the above inequality over all $i,$ we obtain
$\displaystyle\sum_{i=1}^{q}\sum_{1\leqslant s<t\leqslant
3}\nu_{P^{i\gamma(i)}_{st}}\geq
4\sum_{u=1}^{3}\sum_{i=1}^{q}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}+(4q-6n-8)\sum_{i=1}^{q}\nu^{[1]}_{(f,H_{i}),\leqslant
k_{i}}-(q-2)\sum_{i=1}^{q}\nu_{i}.$
Thus,
$\sum_{i=1}^{q}\nu_{i}+\frac{1}{q-2}\sum_{i=1}^{q}\sum_{1\leqslant
s<t\leqslant
3}\nu_{P^{i\gamma(i)}_{st}}\geq\frac{4}{q-2}\sum_{u=1}^{3}\sum_{i=1}^{q}\nu^{[n]}_{(f^{u},H_{i}),\leqslant
k_{i}}+\frac{4q-6n-8}{q-2}\sum_{i=1}^{q}\nu^{[1]}_{(f,H_{i}),\leqslant
k_{i}}.$
Using this inequality and (3.9), we have
$\displaystyle\sum_{i=1}^{q}\nu_{g_{i\gamma(i)}}$
$\displaystyle+\frac{1}{q-2}\sum_{i=1}^{q}\sum_{1\leqslant s<t\leqslant
3}\nu_{P^{i\gamma(i)}_{st}}$
$\displaystyle\geq\big{(}2+\frac{4}{q-2}\big{)}\sum_{u=1}^{q}\sum_{t=1}^{q}\nu^{[n]}_{(f^{u},H_{t}),\leqslant
k_{t}}+\big{(}n-2+\frac{4q-6n-8}{q-2}\big{)}\sum_{i=1}^{q}\nu^{[1]}_{(f^{u},H_{i}),\leqslant
k_{i}}$
$\displaystyle\geq\big{(}2+\frac{4}{q-2}+\frac{n-2}{3n}+\frac{4q-6n-8}{3n(q-2)}\big{)}\sum_{u=1}^{q}\sum_{t=1}^{q}\nu^{[n]}_{(f^{u},H_{t}),\leqslant
k_{t}}.$
It implies that
$\sum_{u=1}^{q}\sum_{t=1}^{q}\nu^{[n]}_{(f^{u},H_{t}),\leqslant
k_{t}}\leqslant\frac{3n}{6nq+(n-2)(q-2)+4q-6n-8}\nu_{\prod_{i=1}^{q}(g^{q-2}_{i\gamma(i)}P^{i\gamma(i)}_{12}P^{i\gamma(i)}_{13}P^{i\gamma(i)}_{23})}.$
Observe that
$\prod_{i=1}^{q}g^{q-2}_{i\gamma(i)}P^{i\gamma(i)}_{12}P^{i\gamma(i)}_{13}P^{i\gamma(i)}_{23}\in
B(q^{2},q(q-2);f^{1},f^{2},f^{3})$, hence applying Lemma 2.8, we obtain
$\displaystyle q$ $\displaystyle\leqslant
2N-n+1+\sum_{i=1}^{q}\frac{n}{k_{i}+1}+\rho\big{(}n(2N-n+1)+\frac{2nq(q-2)}{6nq+(n-2)(q-2)+4q-6n-8}\big{)}$
$\displaystyle+\frac{3nq^{2}}{6nq+(n-2)(q-2)+4q-6n-8},$
which is impossiple since
$\frac{2nq(q-2)}{6nq+(n-2)(q-2)+4q-6n-8}<\frac{2nq(q-2)}{6nq+q-2}=\frac{2n(q-2)}{6n+1}\leqslant\frac{4(q-n)n}{n-1}.$
The proof of Theorem 1.1 is complete. $\square$
Acknowledgement: This work was done while the first author was staying at the
Vietnam Institude for Advanced Study in Mathematics (VIASM). He would like to
thank VIASM for the support.
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|
2024-09-04T02:54:58.083935 | 2020-03-07T15:25:34 | 2003.03591 | {
"authors": "Ira B Schwartz, Victoria Edwards, Sayomi Kamimoto, Klimka Kasraie,\n Ioana Triandaf, M. Ani Hsieh and Jason Hindes",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26097",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/2003.03591"
} | arxiv-papers | # Torus bifurcations of large-scale swarms having range dependent
communication delay
Ira B. Schwartz<EMAIL_ADDRESS>U.S. Naval Research Laboratory,
Code 6792, Plasma Physics Division, Washington, DC 20375, USA Victoria
Edwards U.S. Naval Research Laboratory, Code 5514, Navy Center for Applied
Research in Artificial Intelligence, Washington, DC 20375, USA Sayomi
Kamimoto Department of Mathematics, George Mason University, Fairfax
Virginia, 22030, USA Klimka Kasraie Aerospace, Transportation and Advanced
Systems Laboratory of the Georgia Tech Research Institute, Atlanta, GA 30332
M. Ani Hsieh Mechanical Engineering and Applied Mechanics University of
Pennsylvania, Philadelphia, PA 19104 USA Ioana Triandaf1 Jason Hindes1
###### Abstract
Dynamical emergent patterns of swarms are now fairly well established in
nature, and include flocking and rotational states. Recently, there has been
great interest in engineering and physics to create artificial self-propelled
agents that communicate over a network and operate with simple rules, with the
goal of creating emergent self-organizing swarm patterns. In this paper, we
show that when communicating networks have range dependent delays, rotational
states which are typically periodic, undergo a bifurcation and create swarm
dynamics on a torus. The observed bifurcation yields additional frequencies
into the dynamics, which may lead to quasi-periodic behavior of the swarm.
Swarming behavior occurs when a large number of self-propelled agents interact
using simple rules. Natural swarms of biological systems have been observed at
a range of length scales forming complex emergent patterns. Engineers have
drawn inspiration from these natural systems, resulting in the translation of
swarm theory to communicating robotic systems. Example applications of
artificial swarms include: exploration and mapping, search and rescue, and
distributed sensing and estimation. Through continued development, an
additional parameter of delay in communication between artificial agents has
become important to consider. Specifically, it was previously discovered, that
communication delay will create new rotational patterns which are not observed
without delay, both theoretically and experimentally. Here we extend the
understanding of communication delays to reveal the effects of range dependent
delay, where the communication between agents depends on the distance between
agents. The results of the research show that by including range dependent
delay, new rotational states are introduced. We show how these new states
emerge, discuss their stability, and discuss how they may be realized in large
scale robotic systems. In improving our theoretical understanding of predicted
swarm behavior modeled in simulation we can better anticipate what will happen
experimentally. Additionally, it is possible to leverage the predicted
autonomous behaviors to try and force different swarm behavior.
## I Introduction
Swarming behavior, which we define as the emergence of spatio-temporal group
behaviors from simple local interactions between pairs of agents, is
widespread and observed over a range of application domains. Examples can be
found in biological systems over a range of length scales, from aggregates of
bacterial cells and dynamics of skin cells in wound healing Budrene and Berg
(1995); Polezhaev _et al._ (2006); Lee _et al._ (2013) to dynamic patterns
of fish, birds, bats, and even humans Tunstrø m _et al._ (2013); Helbing and
Molnar (1995); Giuggioli, McKetterick, and Holderied (2015a); Lee (2006).
These systems are particularly interesting because they allow simple
individual agents to achieve complex tasks in ways that are scalable,
extensible, and robust to failures of individual agents. In addition, these
swarming behaviors are able to form and persist in spite of complicating
factors such as delayed actuation, latent communication, localized number of
neighbors each agent is able to interact with, heterogeneity in agent
dynamics, and environmental noise. These factors have been the focus of
previous theoretical research in describing the bifurcating spatial-temporal
patterns in swarms, as seen for example in Refs. Topaz and Bertozzi (2004);
Szwaykowska, Romero, and Schwartz (2015); Mier-y-Teran Romero, Forgoston, and
Schwartz (2011); Hindes, Szwaykowska, and Schwartz (2016). Likewise, the
application of swarms have been experimentally realized in areas, such as
mappingRamachandran, Elamvazhuthi, and Berman (2018), leader-followingMorgan
and Schwartz (2005); Wiech, Eremeyev, and Giorgio (2018), and density
controlLi _et al._ (2017). To guarantee swarming behavior experimentally,
control is typically employed Tanner, Jadbabaie, and Pappas (2007); Gazi
(2005); Jadbabaie, Jie Lin, and Morse (2003); Viragh _et al._ (2014); Desai,
Ostrowski, and Kumar (2001) to prove convergence to a given state by relying
on strict assumptions to guarantee the desired behavior. However, by relaxing
certain assumptions, a number of studies show that even with simple
interaction protocols, swarms of agents are able to converge to organized,
coherent behaviors in a self-emergent manner; i.e. autonomously without
control. Different mathematical approaches yielded a wide selection of both
agent-based Helbing and Molnar (1995); Lee (2006); Vicsek _et al._ (2006);
Tunstrø m _et al._ (2013) and continuum models that predict swarming
dynamics. Edelstein-Keshet, Grunbaum, and Watmough (1998); Topaz and Bertozzi
(2004); Polezhaev _et al._ (2006). In almost all models, since the agents
have just a few simple rules, there exists only a relatively small number of
controllable parameters. The parameter set usually consists of a self-
propulsion force, a potential function governing attracting and repelling
forces between agents, and a communicating radius governing the local
neighborhood at which the agents can sense and interact with each other.
In both robotic and biological swarms, an additional parameter appears as a
delay between the time information is perceived and the actuation (reaction)
time of an agent. Such delays have now been measured in swarms of bats, birds,
fish, and crowds of peopleGiuggioli, McKetterick, and Holderied (2015b); Nagy
_et al._ (2010); Fehrenbach _et al._ (2014). The measured delays are longer
than the typical relaxation times of the agents, and may be space and time
dependent. Robotic swarms experience communication delays which provide
similar effects to the delay experienced in natural swarms. Incorporating
stationary delays along with a minimal set of parameters in swarm models
results in multi-stability of rotational patterns in space Mier-y-Teran
Romero, Forgoston, and Schwartz (2012); Szwaykowska _et al._ (2016); Edwards
_et al._ (2020); Hindes and Schwartz (2018); Szwaykowska, Schwartz, and Carr
(2018). In particular, for delays that equal and fixed, one observes three
basic swarming states or modes: Flocking, which is a translating center of
mass, Ring state, where the agents are splayed out on a ring in in phase about
a stationary center of mass, and a Rotating state, where the center of mass
itself rotates.
Synthetic robotic swarms have communication delays that naturally occur over
wireless networks, as a result of low bandwidthKomareji, Shang, and Bouffanais
(2018) resulting in delayed communication and multi-hop communicationOliveira,
Almeida, and Lima (2015). In cases where the delays are fixed and equal, and
the communication occurs on a homogeneous network, it is known that delays
create new rotational patterns , as has been verified both theoretically and
experimentally Szwaykowska _et al._ (2016); Edwards _et al._ (2020).
However, in situations with robots, even simple communication models are based
on the distance between agents ying Ani Hsieh _et al._ (2004); Hsieh _et
al._ (2007). Following from these models, if one assumes that the delays are
range dependent, the problem becomes one of studying state dependent delays
where delays depend implicitly on the relative positions between agents.
When placing swarms in realistic complex environments, delays are not
necessarily a continuous function of range, but rather it is the increasing
probability of delays increasing stochastically when agents move further away
from one another beyond a certain radius Fink, Ribeiro, and Kumar (2012,
2013). That is, the rate of communication becomes spatially dependent, whereby
near agents see a signal with a fast rate of communication, but due to shading
and fading of signals, communication rates are slowed and complex outside a
given radius. Underwater communication is an excellent swarm example where
delays outside a significant radius impart rates of communication of one to
two orders of magnitude greater than local communication rates Arrichiello
_et al._ (2009).
The swarm model that follows takes a globally coupled swarm, and explicitly
relaxes the fixed delay assumption, by including range dependent delay based
on a fixed communication radius. We show that when range dependent delays are
included, new frequencies are introduced and generate bifurcations to a torus.
The result is a milling type of swarm that depends on just a few parameters.
The results here are important for robotic swarming where one of the goals is
to produce desired patterns autonomously, without external controls. The
pattern formations predicted here show how delayed information, whether coming
from communication, actuations, or both, impacts the stability of swarm
states, such as ring and/or rotating states. By revealing those parameter
regions where patterns are destabilized, we provide a comprehensive
characterization of the autonomously accessible swarm states in the presence
of range-dependent delay.
## II The swarm model
Consider a swarm of delay-coupled agents in $\bm{{R}}^{2}$. Each agent is
indexed by $i\in\\{1,\ldots,N\\}$. We use a simple but general model for
swarming motion. Each agent has a self-propulsion force that strives to
maintain motion at a preferred speed and a coupling force that governs its
interaction with other agents in the swarm. The interaction force is defined
as the negative gradient of a pairwise interaction potential $U(\cdot,\cdot)$.
All agents follow the same rules of motion; however, mechanical differences
between agents may lead to heterogeneous dynamics; this effect is captured by
assigning different acceleration factors (denoted $\kappa_{i}$) to the agents.
In this paper, we assume $\kappa_{i}=1$ for all $i$. For the effect of
heterogeneity on the swarm bifurcations, see Szwaykowska, Romero, and Schwartz
(2015).
Agent-to-agent interactions occur along a graph
$\mathcal{{G}}=\\{\mathcal{{V}},\mathcal{{E}}\\}$, where $\mathcal{{V}}$ is
the set of vertexes $v_{i}$ in the graph and $\mathcal{{E}}$ is the set of
edges $e_{ij}$. The vertices correspond to individual swarm agents, and edges
represent communication links; that is, agents $i$ and $j$ communicate with
each other if and only if $e_{ij}\in\mathcal{{E}}$. All communications links
are assumed to be bi-directional, and all communications occur with a time
delay $\tau$. That is, range dependence is not included. Let
${\bm{{r}}}_{i}\in\mathbb{R}^{2}$ denote the position of the agent $i$ and let
$\mathcal{{N}}_{i}=\\{v_{j}\in\mathcal{{V}}\mathrel{\mathop{\mathchar
58\relax}}e_{ij}\in\mathcal{{E}}\\}$ denote its set of neighbors of agent $i$.
The motion of agent $i$ is governed by the following equation:
$\ddot{\mathbf{r}}_{i}=\kappa_{i}(1-\mathinner{\\!\left\lVert\dot{\mathbf{r}}_{i}\right\rVert}^{2})\dot{\mathbf{r}}_{i}-\kappa_{i}\sum_{j\in\mathcal{{N}}_{i}}\nabla_{x}U({\mathbf{r}}_{i}(t),{\mathbf{r}}_{j}^{\tau}(t)),$
(1)
where superscript $\tau$ is used to denote time delay, so that
${\mathbf{r}}_{j}^{\tau}(t)={\mathbf{r}}_{j}(t-\tau)$,
$\mathinner{\\!\left\lVert\cdot\right\rVert}$ denotes the Euclidean norm, and
$\nabla_{x}$ denotes the gradient with respect to the first argument of $U$.
The first term in Eq. 1 governs self-propulsion, where the speed has been
normalized to unity. That is, without coupling the agents always asymptote to
unit speed.
To analyze the dynamics of a large scale swarm, we use a harmonic interaction
potential with short-range repulsion.
$U(\mathbf{r}_{i},\mathbf{r}_{j}^{\tau})=c_{r}e^{-\frac{\mathinner{\\!\left\lVert\mathbf{r}_{i}-\mathbf{r}_{j}\right\rVert}}{l_{r}}}+\frac{a}{2N}\mathinner{\\!\left\lVert\mathbf{r}_{i}-\mathbf{r}_{j}^{\tau}\right\rVert}^{2}.$
(2)
In Eq. 1, it is assumed that the communication delay, $\tau$, is independent
of the distance, or range, between any pair of agents. (Notice that the
exponent of the repulsion term is independent of the delay since the repulsion
force is local.) With the addition of delays in the network, it was shown in
homogeneous communication networks that in addition to the usual dynamical
translating and milling (or ring) states, for sufficiently large $\tau$, new
rotational states emerge Szwaykowska _et al._ (2016). In particular, for a a
given attractive coupling strength, there is a delay that destabilizes the
periodic ring state into a rotating state, in which the agents coalesce to a
small group and move around a fixed center of rotation; this behavior is quite
different from the ring state where agents are spread out in a splay state
phase. The rotating state is only observed with delay introduced in the
communication network, and it appears through a Hopf bifurcation.
However, in real-world robotic swarms, communication delays are not uniform
between all pairs of agents; delays may be stochastic or even state-dependent.
For example, if agents are communicating over a multi-hop network, the delay
will increase with the number of hops required to send a message from one
agent to the other, and in general will scale with the separation between
them. In order to handle range dependent delays, we will make an approximation
that depends on a communication range radius.
### II.1 Approximating range dependent delayed coupling
For the coupling term, we are interested in introducing an approximation to
range based coupling delay. Since all communicating agents send signals with
some delay, we compute relative distances defined as
$\displaystyle D^{\tau}_{i,j}$
$\displaystyle\equiv||{\mathbf{r}}_{i}-{\mathbf{r}}_{j}^{\tau}||.$ (3)
We define a Heaviside function, $H(x)$, that is zero when $x\leq 0$ and 1
otherwise, and we employ global coupling based on a spring potential. For our
range dependent metric, we let $\epsilon\geq 0$ denote the range radius.
Suppose that when the separation between two agents is small, that is less
than $\epsilon$, then sensing between two agents is almost immediate. In
practice, the time needed for sensing depends on several factors, such as
actuation times, and so distances in practice are computed with delay.
Therefore, we model the coupling term for the $i^{th}$ agent as
$\displaystyle
C_{i}({\mathbf{r}}_{i},{\mathbf{r}}_{j},{\mathbf{r}}_{j}^{\tau},\epsilon)=-\frac{a}{N}(\nabla_{x}U({\mathbf{r}}_{i}(t),{\mathbf{r}}_{j}^{\tau}(t)))H(D^{\tau}_{i,j}-\epsilon)$
$\displaystyle-\frac{a}{N}(\nabla_{x}U({\mathbf{r}}_{i}(t),{\mathbf{r}}_{j}(t)))(1-H(D^{\tau}_{i,j}-\epsilon)),$
(4)
where the first coupling term has delay turned on since the distance is
outside a ball of radius $\epsilon$, while the second term has no delay since
the distance is within the $\epsilon$ ball. The resulting swarm model with
range dependence from Eq. II.1 is now
$\ddot{\mathbf{r}}_{i}=\kappa_{i}(1-\mathinner{\\!\left\lVert\dot{\mathbf{r}}_{i}\right\rVert}^{2})\dot{\mathbf{r}}_{i}-\kappa_{i}\sum_{j\in\mathcal{{N}}_{i}}C_{i}({\mathbf{r}}_{i},{\mathbf{r}}_{j},{\mathbf{r}}_{j}^{\tau},\epsilon).$
(5)
If the delayed distance is within an $\epsilon$ ball, then we evaluate the
coupling without delay. Otherwise the coupling is delayed. Thus the coupling
function takes into account when delay is active or not between pairs of
communicating agents, and depends on the range radius, $\epsilon$.
The Heaviside function of the right hand side of Eq. 9 renders the
differential delay equation derivatives discontinuous, and as such poses a
numerical integration problem. To mollify the lack of smoothness, we
approximate $H(x)$ by letting
$H(x)\approx\frac{1}{\pi}\arctan(kx)+\frac{1}{2}$, where $k\gg 1$ and
constant, and limits on the Heaviside function as $k\rightarrow\infty$.
Using only the delayed distance to compute a range dependent coupling assumes
that any measurement is not instantaneous. If one were to be able to compute
the ideal situation where delay would not be a sensing factor, then certain
issues would need to be resolved, which we do not consider here.
### II.2 Numerical simulations of full swarms
Examples of simulations using the swarm model with the range dependent
coupling are shown below. Here the number of agents, $N=150$, and the coupling
strength, $a=2.0$. For the remainder of the analysis, we set $c_{r}=0$, and
note that the attractors persist when the repulsive amplitude is sufficiently
small Szwaykowska _et al._ (2016). (See supplementary material for a video of
the dynamics with small repulsion.)
Figure 1: Three snapshots of swarm state in space for
$\epsilon=0.01,a=2.0,\tau=1.75$. Sample times
$t_{0},t_{1}=t_{0}+20,t_{2}=t_{0}+40$ .
Note that even when $\epsilon$ is very small, as shown in Fig. 1, we observe a
mix of clustered states which are a combination of pure ring and rotation
states. The agents tend to cluster into local groups, and the clusters move in
clockwise and counter-clockwise directions as in the ring state. Here,
however, the phase differences between agents are non-uniform. When examining
a single random agent, as shown in Fig 2, it is periodic with a sharp
frequency of rotation, and the relative positions of all individual agents are
phase locked. When considering the center of mass of the positions over all
agents, $\bm{{R}}\equiv\frac{1}{N}\sum_{i}\bm{{r}}_{i}$, the center of mass
does small amplitude oscillations about a fixed point (not shown).
Figure 2: Swarm ring state for $\epsilon=0.01,a=2.0,\tau=1.75$. (a)Time series
of the x-component of a single agent. (b)The power spectrum showing a sharp
frequency. (c)A phase portrait of the orbit of a single agent. The red point
denotes the center of mass.
As the radius $\epsilon$ increases, instability of the periodic mixed state
occurs, giving rise to more complicated behavior, as seen in Fig. 3. New
frequencies are introduced, causing the ring state to appear as a quasi-
periodic attractor. Moreover, the dynamics of the center of mass has its own
non-trivial dynamics which includes the effects of new frequencies. By
examining the Poincare map of the attractors, the instability gives rise to
dynamics which we conjecture is motion on a torus. Letting $(M_{x},M_{y})$
denote the time averaged center of mass over all agents, we compute the
sequence ${x(t_{i}),i=1..M}$ when $y(t_{i})=0$ and $x(t_{i})>M_{x}$. The
result is shown in the two panels in Fig. 4. Panel (a) shows a complicated
toroidal motion after transients are removed of the center of mass in Fig. 3c.
For a single frequency, the dynamics of the center of mass would be a single
fixed point. The addition of new frequencies is revealed in the Poincare map
as complicated motion on a torus. For larger values of $\epsilon$, the motion
on the torus converges to a periodic attractor in panel (b).
Figure 3: Swam instability $\epsilon=0.25,a=2.0,\tau=1.75$. (a)Time series of
the x-component of a single agent. (b)The Power spectrum showing a slight
broadening and birth of a new frequency. (c)A phase portrait of the orbit of a
single agent.
Figure 4: Poincare map of Eqs. 1-II.1 for (a) $\epsilon=0.25$, (b)
$\epsilon=0.5$. Other parameters are fixed: $a=2.0,\tau=1.75$. See text for
details.
## III Mean-Field Equation of Range Dependent Delay Coupled Swarm
In order to shed some light on the origin of the bifurcation to dynamics on a
torus, we examine the full swarm model from a mean-field perspective. The mean
field is much lower dimensional, and a full bifurcation analysis may be done.
We consider the case of all-to-all communication. Let
$\bm{R}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}$
and
$\mathbf{r}_{i}=\bm{R}+\delta\mathbf{r}_{i},$
where $\delta\mathbf{r}_{i}$ is a fluctuation term with the identity, and
$\sum_{i=1}^{N}\delta\mathbf{r}_{i}=0.$ (6)
Then we can write Eq. 5 as
$\displaystyle\ddot{\bm{R}}+\delta\ddot{\mathbf{r}}_{i}$
$\displaystyle=(1-|\dot{\bm{R}}+\delta\dot{\mathbf{r}}_{i}|^{2})(\dot{\bm{R}}+\delta\dot{\mathbf{r}}_{i})$
$\displaystyle-\frac{a}{N}\sum_{j=1,j\neq
i}^{N}((\bm{R}+\delta\mathbf{r}_{i})-(\bm{R}^{\tau}+\delta\mathbf{r}_{j}^{\tau}))C_{1,i}$
$\displaystyle-\frac{a}{N}\sum_{j=1,j\neq
i}^{N}((\bm{R}+\delta\mathbf{r}_{i})-(\bm{R}+\delta\mathbf{r}_{j}))C_{2,i},$
(7)
where
$\displaystyle C_{1,i}$
$\displaystyle=H(\lVert{\mathbf{r}_{i}-\mathbf{r}_{j}^{\tau}\rVert}-\epsilon)$
$\displaystyle=H(\lVert{(\bm{R}+\delta\mathbf{r}_{i})-(\bm{R}^{\tau}+\delta\mathbf{r}_{j}^{\tau})\rVert}-\epsilon)$
$\displaystyle=H(\lVert{\bm{R}-\bm{R}^{\tau}+\delta\mathbf{r}_{i}-\delta\mathbf{r}_{j}^{\tau}\rVert}-\epsilon)$
and
$C_{2,i}=1-C_{1,i}.$
We use the following to reduce the equations of motion to the mean field: From
Eq. 6, we note
$\displaystyle\sum_{i=1}^{N}\delta\mathbf{r}_{i}^{\tau}$
$\displaystyle=\sum_{j=1,j\neq
i}^{N}\delta\mathbf{r}_{j}^{\tau}+\delta\mathbf{r}_{i}^{\tau}=0\iff$
$\displaystyle-\sum_{j=1,j\neq
i}^{N}\delta\mathbf{r}_{j}^{\tau}=\delta\mathbf{r}_{i}^{\tau}.$ (8)
We further assume that all perturbations from the mean,
$\delta\mathbf{r_{i}}$, are all negligible. (This is always true if the
coupling amplitude is sufficiently large.) In addition, we use the fact that
$\displaystyle{\frac{a(N-1)}{N}}$ limits to $a$, as $N\to\infty$. Therefore,
we obtain mean field approximation for the center of mass of range dependent
coupled delay case:
$\ddot{\bm{R}}=(1-\lvert\dot{\bm{R}}\rvert^{2})\cdot\dot{\bm{R}}-a(\bm{R}-\bm{R}^{\tau})\cdot
H(\lVert{\bm{R}-\bm{R}^{\tau}\rVert}-\epsilon)$ (9)
## IV Numerical Analysis of the mean field equation
### IV.1 Examples of rotational attractors
As in the case for the full multi-agent system, we see the existence of
periodic behavior for $\tau$ sufficiently below an instability threshold, as
shown in the time series of Fig. 5. As we increase $\tau$, we expect the
periodic orbit to lose stability, resulting in a new attractor. In particular,
one notices the emergence of a new frequency in addition to the existing
dominant one, as shown in Fig. 6 The additional frequency usually implies a
bifurcation to dynamics on a torus, or a higher dimensional torus.
Figure 5: Periodic motion of the mean field Eq. 9 for
$\epsilon=0.01,a=0.64,\tau=1.6$. (a) Time series of the x-component of the
mean field. (b) Power spectra of the time series.
We now investigate this transition by tracking the stability via monitoring
the Floquet exponents corresponding to the periodic orbit. For a general
differential delay equation given by
$\dot{\bm{x}}(t)=\bm{F}(\bm{x}(t),\bm{x}(t-\tau))$, if
$\bm{\phi}(t)=\bm{\phi}(t+T)$ for all $t\geq 0$, then stability is determined
by examining the linearized equation along $\bm{\phi}(t)$:
$\displaystyle\dot{\bm{X}}(t)$
$\displaystyle=\frac{\partial\bm{F}}{\partial\bm{x}(t)}(\bm{\phi}(t),\bm{\phi}(t-\tau))\bm{X}(t)$
$\displaystyle+\frac{\partial\bm{F}}{\partial\bm{x}(t-\tau)}(\bm{\phi}(t),\bm{\phi}(t-\tau))\bm{X}(t-\tau).$
(10)
The stability of the periodic solution is determined by the spectrum of the
time integration operator $U(T,0)$ which integrates Eq. IV.1 around $\phi(t)$
from time t = 0 to t = T. This operator is called the monodromy operator and
its (infinite number of) eigenvalues, which are independent of the initial
state, are called the Floquet multipliers Hale (1977). For autonomous systems,
it is necessary and sufficient there exists a trivial Floquet multiplier at 1,
corresponding to a perturbation along the periodic solution Hartung _et al._
(2006); Hale and Lunel (1993). The periodic solution is stable provided all
multipliers (except the trivial one) have modulus smaller than 1; it is
unstable if there exists a multiplier with modulus larger than 1. Bifurcations
occur whenever Floquet multipliers move into or out of the unit circle.
Generically three types of bifurcations occur in a one parameter continuation
of periodic solutions: a turning point, a period doubling, and a torus
bifurcation where a branch of quasi-periodic solutions originates and where a
complex pair of multipliers crosses the unit circle Hale (1977).
Figure 6: Quasi-periodic motion of the mean field Eq. 9. (a) Time series of
the x-component of the mean field. Solid (red) line denotes period length of
dominant spectral peak. Dashed line denotes period length of secondary peak.
(b) Power spectra of the time series.
We have tracked a set of stable periodic orbits for various radii of
$\epsilon$, and located the change in stability by computing the Floquet
multipliers. The results plotted in Fig. 7 show that for a range of radii
$\epsilon$, there exists a bifurcation to a torus at some delay. Notice that
as $\epsilon$ increases, there results an increase in the size of the orbits,
which qualitatively agrees with our full agent based simulations.
Figure 7: Bifurcation plot showing the norm of the periodic orbits as a
function of delay $\tau$. Parameter a=0.68. Red (blue) markers denote unstable
(stable) orbits.Cyan symbols denote the change in stability where a pair of
complex eigenvalues cross the imaginary axis.
Since there exists a range of delays which destabilize periodic swarm dynamics
for each $\epsilon$, we summarize the onset of torus bifurcations by plotting
the locus of points at which stability changes as a a function of coupling
amplitude and delay. The results are plotted in Fig. 8.
Figure 8 is revealing in that it shows a functional relationship of the
bifurcation onset that is similar over a range of $\epsilon$. For larger
values of $\epsilon$, it is clear that lower values of delay and coupling are
required to generate bifurcations. This holds true over two orders of
$\epsilon$. For a fixed value of $\epsilon$, we also see monotonic
relationship between delay and coupling strength, so that it is easier for
smaller delays to destabilize periodic motion for larger coupling strengths.
Figure 8: Plotted is the locus of points at which torus bifurcations emerge as
a function of coupling amplitude $a$, delay $\tau$ for various range radii
$\epsilon$ for the mean field Eq. 9.
## V Conclusions
We considered a new model of a swarm with delay coupled communication network,
where the delay is considered to be range dependent. That is, given a range
radius, delay is on if two agents are outside the radius, and zero otherwise.
The implication is that small delays do not matter if the agents are close to
each other.
The additional range dependence creates a new set of bifurcations not
previously seen. For general swarms without delay, the usual states consist of
flocking (translation) or ring / rotational state (milling), with agents
spread in phase. With the addition of a fixed delay, a rotational state
bifurcates that has all agents in phase and rotate together Hindes _et al._
(2020). Range dependence introduces a new rotational bifurcating state that
exhibits behavior observed as a new mixed state combining dynamics of both
ring and rotating states.
The radius parameter $\epsilon$, was used to quantify the bifurcation of the
rotational mixed state. For small $\epsilon$, we see dynamics for the full
swarm shows clustered counter-rotational behavior that is periodic. This
agrees for small radius values in the mean field description as well. As the
radius increases, the mixed periodic state generates new frequencies in the
full model, which are manifested as torus bifurcations in the mean field. Mean
field analysis was done by tracking Floquet multipliers that cross the
imaginary axis as complex pairs. Frequency analysis explicitly shows the
additional frequencies in the mean field.
Finally, we tracked the locus of coupling amplitudes and delay for various
values of $\epsilon$ locating the parameters at which torus bifurcation occur.
The results reveal that as $\epsilon$ increases, torus bifurcations onset at
lower values of coupling amplitude and delay. The implications are that more
complicated behavior than periodic motion has a greater probability of being
observed in both theory and experiment if range dependence of delay is
included.
## VI Supplementary Material
The videos show the attractor of a swarm consisting of N=300 agents. Fixed
parameters for the three videos are $a=2.0,\tau=1.75$ The parameters for zero
radius (delay is on all the time) are $\epsilon=0.0,c_{r}=0.05$, and
$l_{r}=0.05$ for a baseline, are shown in Video1_eps_0p0.mp4.
The parameters corresponding to Fig. 2 are $\epsilon=0.01,c_{r}=0.01$, and
$l_{r}=0.05$ are shown in Video2_eps_0p01.mp4. The video shows that the
attractor persists when repulsive forces are local and weak. Similar behavior
is observed when N=150, which is used in Fig. 1 without repulsion; i.e.,
$c_{r}=0$. The parameters for corresponding to Fig. 3 are
$\epsilon=0.25,c_{r}=0.05$, and $l_{r}=0.05$, shown in Video3_eps_0p25.mp4.
###### Acknowledgements.
IBS, JH, IT and KK gratefully acknowledge ONR for their support under
N0001412WX20083, N0001420WX00034, and the NRL Base Research Program
N0001420WX00410. VE is supported under the NRL Karles Fellowship Program, JON
55-N2Q4-09. SK was supported through the GMU Provost PhD award as part of the
Industrial Immersion Program. MAH is supported by ONR No. N00014-18-1-2580 and
ARL DCIST CRA W911NF-17-2-0181.
Data sharing is not applicable to this article as no new data were created in
this study.
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|
2024-09-04T02:54:58.098869 | 2020-03-07T17:28:54 | 2003.03614 | {
"authors": "Batuhan Kaplan, \\.Ibrahim Kahraman, Ali G\\\"or\\c{c}in, Hakan Ali\n \\c{C}{\\i}rpan, Ali R{\\i}za Ekti",
"full_text_license": null,
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26098",
"submitter": "Batuhan Kaplan",
"url": "https://arxiv.org/abs/2003.03614"
} | arxiv-papers | # Measurement based FHSS–type Drone Controller Detection at 2.4GHz: An STFT
Approach ††thanks: This paper has been accepted for the presentation in the
2020 IEEE 91st Vehicular Technology Conference (VTC2020-Spring).
Batuhan Kaplan12, İbrahim Kahraman13, Ali Görçin14, Hakan Ali Çırpan2, Ali
Rıza Ekti15 1 Informatics and Information Security Research Center (BİLGEM),
TÜBİTAK, Kocaeli, Turkey 2 Department of Electronics and Communication
Engineering, İstanbul Technical University, İstanbul, Turkey 3 Department of
Electrical and Electronics Engineering, Boğaziçi University, İstanbul, Turkey
4 Faculty of Electronics and Communications Engineering, Yıldız Technical
University, İstanbul, Turkey 5 Department of Electrical and Electronics
Engineering, Balıkesir University, Balıkesir, Turkey
Emails: batuhan.kaplan<EMAIL_ADDRESS>
[email protected],hakan.cirpan<EMAIL_ADDRESS>
###### Abstract
The applications of the unmanned aerial vehicles increase rapidly in everyday
life, thus detecting the UAVs and/or its pilot is a crucial task. Many UAVs
adopt frequency hopping spread spectrum (FHSS) technology to efficiently and
securely communicate with their radio controllers where the signal follows a
hopping pattern to prevent harmful interference. In order to realistically
distinguish the frequency hopping (FH) RC signals, one should consider the
real–world radio propagation environment since many UAVs communicate with RCs
from a far distance in which signal faces both slow and fast fading
phenomenons. Therefore, in this study different from the literature, we
consider a system that works under real–conditions by capturing over–the–air
signals at hilly terrain suburban environments in the presence of foliages. We
adopt the short–time Fourier transform (STFT) approach to capture the hopping
sequence of each signal. Furthermore, time guards associated with each hopping
sequence are calculated using the autocorrelation function (ACF) of the STFT
which results in differentiating the each UAV RC signal accurately. In order
to validate the performance of the proposed method, the results of normalized
mean square error (MSE) respect to different signal–to–noise ratio (SNR),
window size and Tx–Rx separation values are given.
###### Index Terms:
short–time Fourier transform, frequency hopping, UAV remote controller
detection
## I Introduction
Unmanned aerial vehicles have become a prevalent part of the daily life with
their applications to many fields such as mapping and surveying,
transportation, surveillance, law enforcement, aerial imaging and agriculture
[1]. Besides the aforementioned use of UAVs in many areas, one should keep in
mind that UAVs can also be used dangerously to create unwanted incidents
especially when they are diverted to the sensitive airspace near airports and
their presence may cause accidents which can result in fatal crashes [2].
Moreover, UAVs can be utilized for collecting information about people,
organisations, and companies without their consent. Therefore, identification
of UAV systems and their communication are great importance, especially to
prevent unwanted situations. In this context, it is known that most of the
communication between the UAVs and wireless RC utilize the spread spectrum
technology of FHSS on industrial, scientific, and medical (ISM) band at 2.4GHz
[3]. Therefore a method to detect and classify these kinds of signals in this
band would lead to the identification of the communication between the UAV and
the controller.
In literature, most well known FHSS signal detection methods adopt the
time–frequency analysis and wavelet analysis [4, 5, 6, 7]. It is shown that
Wigner Ville distribution, Wavelet transform method, and array signal
processing methods can be used to identify FHSS signals with the burden of
heavy computational complexity which make them hard to satisfy the real–time
implementation. Since STFT based detection method does not need any prior
information about received signal compared to computationally complex
algorithms, STFT becomes the designated optimum detector in the absence of
information regarding the received signal [8]. However, one should note that
in order to achieve good results with STFT, SNR value needs to be high. Thus,
in this study, we propose STFT based blind signal detection method for FHSS
UAV RC signals by using over–the–air signal measurements which accounts for
the signal imperfections present in the nature of real–world conditions (e.g.
fading, multipath, and so on) by considering the hilly terrain suburban
environments in the presence of foliage. Furthermore, the literature utilizes
the simulated data instead of over–the–air signals in general and these
simulations assume that there is no time guards between hops. This assumption
makes differentiation of FH signals easier, however, many hopping signals use
time guards and also these time guards are different for different signal
sources. The proposed approach is also able to detect the time guards between
each hopping sequence using the ACF [9] of the STFT which results in
differentiating the each drone RC signal accurately.
The paper is organized as follows, Section II details the system model.
Section III presents the proposed method. The measurement setup is explained
in Section IV. In Section V, measurement results are given and discussed.
Finally, Section VI concludes the paper.
## II Signal Model
Drone controller signals might be FH signals that have temporal statistical
characteristics and they can be written as [6],
$x(t)=s(t)\sum_{m=0}^{M-1}e^{j2\pi f_{c_{m}}t_{m}+\theta_{m}}$ (1)
where $\theta_{m}$ and $f_{c_{m}}$ are the carrier phase and carrier frequency
of $m_{th}$ hop, respectively. Also, $s(t)$ denotes the complex baseband
equivalent of the information bearer for $t\in[0,T]$, $M$ stands for the total
number of hop of a signal, $t_{m}$ is the duration time of $m_{th}$ hop that
might be a uniform distribution or not.
The received controller signal which is complex baseband equivalent of the
received passband signal and can be expressed as,
$r(t)=\sum_{n=0}^{N-1}y_{n}(t)+n(t)+I(t)$ (2)
where $y_{n}(t)$ is an $n_{th}$ FH signal source, $n(t)$ denotes the complex
Additive White Gaussian Noise (AWGN) in which I and Q components are i.i.d
with $\mathcal{N}(0,\,\sigma^{2})$, $I(t)$ stands for the interference signal.
### II-A Short–Time Fourier Transform
STFT approach is utilized to analyze the FH signals as a method to observe the
frequency content of this type of non–stationary signals over time.
Mathematical expression of STFT of the time-domain signal $z(t)$ can be
written as [8],
$STFT\Big{\\{}z(t)\Big{\\}}=\int_{-\infty}^{\infty}[z(t)w(t-\tau)]e^{-j2\pi
f\tau}d\tau$ (3)
where $w(t)$ is the window function. The STFT matrix
$S=[s_{1}[f],s_{2}[f],...,s_{K}[f]]$ such that $i_{th}$ element of this matrix
is a column vector determined by discrete Fourier transform of $r[n]w[n-iR]$,
$s_{i}[f]=\sum_{n=0}^{N-1}r[n]w[n-iR]e^{-j2\pi fn}$ (4)
where $r[n]$ is sampled version of $r(t)$ by considering the anti–aliasing
property and $R$ denotes the shifting length.
One should keep in mind that adjusting the time and frequency resolution is
crucial point for STFT analysis [10] due to the trade off between them. The
length of the time point can be calculated as [11]
$m=\left\lfloor\frac{N_{x}-L}{M-L}\right\rfloor$ (5)
where $N_{x}$ is length of the signal, $L$ denotes the number of overlap in
the Fourier transform, $M$ represents the window size, $\lfloor\cdot\rfloor$
stands for the floor operator.
## III Proposed Method
The received controller signal, $r(t)$, analyzed by STFT method. As depicted
in Fig. 1, the flow graph explains how the system works in brief. After the
signal is received in the first stage, optimal window time length is decided
to get the optimum resolution at (5) based on maximizing the number of
elements on the matrix $S$ in the second stage of the flowchart. STFT is
calculated in a dBm unit according to power spectral density (PSD) in the same
step.
Figure 1: The flowchart of the proposed detection method.
A binarization process is conducted at the following steps of the flowchart;
the STFT matrix is converted to a binarized matrix, $Z(k,l)$, with the
utilization of a threshold $\mu$. Also, in the flowchart PSD refers to each
point in the STFT matrix. Based on the dynamically calculated threshold value,
whether the signal presents or not is decided. Therefore, the problem
statement can be indicated as the identification of the presence of the
unknown FH signal. When the signal is present, $Z(k,l)$ is evaluated as 1 and
the new binarized matrix, $Z(k,l)$, is given as,
$Z(k,l)=\begin{cases}1,&S(k,l)\geq\mu\\\ 0,&S(k,l)<\mu\end{cases}$ (6)
To determine the threshold value, each element of STFT matrix are concatenated
and a sorting algorithm implemented to list the power levels of each point of
the STFT matrix in a ascending order. Then, we assume that the majority of the
received signal comprise of noise, therefore, we take the mean value of the
top $20\%$ of the sorted values of STFT matrix to determine a lower bound for
the computation of the threshold. Thus, the threshold is calculated as
$\mu=\frac{S_{max}+\sigma_{20\%}}{2}$ (7)
where $S_{max}$ is the maximum value of the STFT matrix, $\sigma_{20\%}$
denotes the mean of the top $20\%$ samples. Fig. 2 shows $S_{max}$,
$\sigma_{20\%}$, and the threshold value. Please note that even for the very
low SNR regimes, the measurement results indicate the feasibility of this
threshold selection process.
Figure 2: Estimated threshold representation with $S_{max}$ and
$\sigma_{20\%}$ values over real recorded signal.
Due to the wireless impairments on the received signal, some portions of the
$Z$ matrix might be missing as shown in Fig. 3(a). In order to represent the
signal in a more plausible way, we adopt a widely used morphological dilation
and erosion processes from the domain of image processing [12, 13] to recover
the received signal properly. A signal with impairments and the output of
dilation and erosion processes are shown in Fig. 3.
(a) Recorded FH drone controller signal with channel impairments.
(b) FH drone controller signal after correction.
Figure 3: Result of dilation process.
After the recovery process, it becomes possible to extract the signal
parameters such as start time, stop time, center frequency and difference
between start time and stop time (dwell time) accurately. The parameter
estimation algorithm is given in Algorithm 1. This process is implemented to
the $Z$ matrix and the parameters of each signal in the spectrum are
extracted. The algorithm simply detects the signals from $Z$ and extracts the
duration of each independent transmission which can be part of same or
different signal sources.
Input: $Z(k,l)$
Output: start time, stop time, center frequency, dwell time, bandwidth
1
2for _$i\leftarrow 1$ to row_ do
3 for _$j\leftarrow 1$ to column_ do
4 $count\leftarrow 0$
5 $bandwidth\leftarrow 0$
6 if _$Z(i,j)$ == to $1$_ then
7 start time $\leftarrow j$
8 $f_{start}\leftarrow i$
9 while _$Z(i,j)$ $==$ $1$ _ do
10 $count\leftarrow count+1$
11 $j\leftarrow j+1$
12 while _$Z(i,j-1)$ $==$ $1$ _ do
13 $bandwidth\leftarrow bandwidth+1$ $i\leftarrow i+1$
14 stop time $\leftarrow j-1$
15 $f_{stop}\leftarrow i-1$
16 center frequency $\leftarrow\frac{f_{start}+f_{stop}}{2}$
17 dwell time $\leftarrow count$
18
19 Assign $0$ to rectangle that founded above.
20
21
Algorithm 1 Parameter Extraction Algorithm
In the last step of the flowchart, controlling for each hop is done to decide
whether it belongs to FH signal or not. Due to the signal structure which will
be explained in Section IV, time guards are inserted during hopping and this
makes classification complicated. Time guards may lead to miss matches between
hops because there can be some other signals between them and these miss
matches should be corrected. In this paper ACF is utilized for this purpose
since the highest correlation peaks occur as the signal matches itself
perfectly and this is the case for drone controller FH signals based on the
fact that time and frequency characteristics are fixed. ACF is used on $Z$
matrix and result of ACF gives the highest peak at $T_{1}$, which is the
fundamental period (6.8ms) for the particular drone controller FH signal. The
rest of the peaks give information in regards to hops which are generated from
the same source. Thus, we define a set T = $\\{T_{1},T_{2},...T_{n}\\}$ which
represents the locations of all the peak values where $T_{1}=\sup\hskip
2.84526pt\text{T}$. In other words, we put all the local extremum for interval
$(0,T_{1})$ to the set T as shown in Fig. 4 meaning that we obtained all the
required T values (guard and dwell times) to track and classify hops. Finally,
the signal classification block in the flowchart is executed by the procedure:
$\exists T_{i}\in\text{T}$ and any two hops, $hop\\_a$ and $hop\\_b$, are
emitted from the same signal source if
$\text{start time($hop\\_a$)}-\text{start time($hop\\_b$)}\equiv T_{i}\hskip
2.84526pt(\text{mod }T_{1})$ (8)
where start time is the estimated parameter from Algorithm 1. Thus, hop
separation is also possible with the proposed method. Please note that even
though the method is applied to a particular set of FH signals, it can be
utilized to detect any kinds of FH signal with observable time and frequency
domain hopping pattern.
Figure 4: Time difference estimation between hops utilizing auto correlation
function.
## IV Measurement Setup
Experimental setup for FH signal parameter estimation is performed in
Scientific and Technological Research Council of Turkey (TÜBİTAK) Informatics
and Information Security Research Center (BİLGEM). Data collection of FH
signal is conducted with different distances from 5m to 135m with 10m
intervals between each step. Fig. 5 shows the fixed location of our receiver
and locations of the drone controller (transmitter).
Figure 5: The aerial map of data collection locations.
### IV-A Hardware Setup
The testbed used in the data acquisition procedure consists of signal source
(drone controller) and spectrum analyzer to record the signals.
Figure 6: The hopping pattern of Drone controller signal: Futaba T8J RC as
signal source.
Futaba T8J RC is used during the experiment as a FH signal source. Futaba T8J
RC operates in the 2.4 gigahertz (GHz) ISM band over the half of the spectrum
band (up to 2.45 GHz). When analyzing the signal, it is discovered that the RC
transmitter is behaved differently compared to the standard FH communication
systems (e.g. Bluetooth). An illustration of the periodic hopping sequence for
the Futaba T8J RC is shown in Fig. 6. Also, in the figure for $\tau_{dwell}$
represents dwell time and fundamental period becomes
$\Delta t_{1}<\Delta t_{2}<\Delta t_{3}$ (9) $3\tau_{dwell}+\Delta
t_{1}+\Delta t_{2}+\Delta t_{3}=T_{1}=0.0068sec$ (10)
TABLE I: FH signal characteristics FH Signal |
---|---
Dwell Time | 1.44ms
Center Frequency Set | | 2.4GHz-2.45GHz interval
---
Hopping Sequence | f1 f1 f2 f3 f3 f4 …
The parameters of Futaba T8J RC FH signal source are listed in Table I. In the
receiver side, Rohde&Schwarz FSW 26 signal and spectrum analyzer (SSA) is
utilized to record FH signals. SSA can support the frequency range from 2
hertz (Hz) to 26.5GHz. The device provides real–time spectral analysis up to
160 megahertz (MHz) bandwidth. The signals are recorded over the 2.4GHz ISM
spectrum band with an omnidirectional antenna.
### IV-B Experimental Procedures
In data collection from real–world, it is assumed that transmitter operates
between 2.4GHz–2.48GHz ISM spectrum band. The center frequency of SSA is set
to 2.44GHz and bandwidth of interest is adjusted to 80MHz for the purpose of
full coverage. Also, SSA is connected to external computer via an Ethernet
cable in favor of achieving data storage with ease. The sampling rate depends
on the analysis bandwidth of real–time spectrum which is selected as 80MS/s.
Each measurement is captured as an I/Q samples and collected 20M I/Q samples
during 250ms. However, considering the processing limit, the collected data is
divided into pieces with each has 4M samples. Each divided data was considered
when calculating the performance of the proposed method. Finally, captured I/Q
data is fed into the computer which runs MATLAB R2015b software.
## V Measurement Results
Over–the–air data collection is realized and the performance of time–frequency
analysis method is evaluated. Please note that all the captured data includes
real–world propagation effects such as multipath fading, interference, carrier
frequency offset (CFO). Fig. 7 shows how over–the–air recorded FH signal
source behaves during the observation time. Please also note that some of the
estimated parameters of the real signal which measured at 25m distance with
5dB SNR can be found in Table II.
Figure 7: Recorded 2.4GHz spectrum that comprised of drone controller FH
signals.
In order to validate the performance of the system, normalized MSE was
considered. Normalized MSE can be calculated as[14]
$\text{NMSE}=\frac{1}{N}\sum_{i=1}^{N}\Big{(}\frac{\hat{t}_{i}-t}{t}\Big{)}^{2}$
(11)
where $\hat{t}_{i}$ is the estimated hopping time of $i_{th}$ hop and $t$
denotes the true value of hopping time. $N$ represents the total number of hop
that must be found. When calculating the error, it is assumed that $N$ equals
to $22$ in the observation time. Also, $0$ values have been added to extracted
parameters if there is some undetected signal.
TABLE II: Estimated parameters of FH signal | Start Time
---
(ms)
| Stop Time
---
(ms)
| Dwell Time
---
(ms)
| Center Frequency
---
(GHz)
1.7930 | 3.2403 | 1.4472 | 2.4271
5.1742 | 6.6086 | 1.4344 | 2.4414
6.7623 | 8.1967 | 1.4344 | 2.4414
8.6066 | 10.0410 | 1.4344 | 2.4211
11.9749 | 13.4221 | 1.4472 | 2.4039
Figure 8: Estimated hopping time errors vs. distance in terms of normalized
MSE.
The error curve of an estimated hopping time from measured data is plotted in
Fig. 8. In here, error values are determined for three different SNR values
and distances. It is clearly seen that as the distance increases, the error of
estimation increases. Also poor SNR condition adversely affects the
performance. Moreover, even with the same SNR condition, there may be a
missing hop of an FH signal in the signal received from the farther point.
As discussed before, another important issue is selection of window size
($M$). In this regard, precision of window size is also studied. It can be
seen that window size directly effects the accuracy of the estimation. While
window size decreases, we get high resolution in time but low resolution in
frequency. In contrast, increasing window size implies high resolution in
frequency but low resolution in time. Because both the frequency and time
information are main features to classify the hopping signals, there should be
some optimum window size providing better performance. It is empirically shown
in Fig. 9 that the optimum window size for STFT is $M=2048$.
Since the previous works utilize STFT only for deciding whether a signal is
hopping between different frequencies, ACF is applied only in time–domain for
the signals with no guard times, and the simulations were considered instead
of measurements, comparison of the performance of the proposed method with
these simulations are avoided in this work.
Figure 9: Estimated hopping time errors vs. distance for different window
sizes and for SNR $=$ 0 dB.
## VI Concluding Remarks and Future Directions
In this work, an FH drone controller signal detection algorithm is proposed
and the performance of the algorithm is evaluated using measurements of UAV RC
in real–world wireless environment. The algorithm can estimate the guards of
the each hopping sequence successfully by using the ACF of STFT, which also
leads to the accurate identification whether a FH signal is present or not.
The performance of the proposed method is also quantified by normalized MSE
for over–the–air signal which are recorded at different distances and SNRs.
Measurement results show that reasonable input parameters will improve the
performance of frequency hopping signal parameter estimation. In future
studies, we will consider adopting the recently emerging deep learning
algorithms to distinguish multiple standard based wireless hopping signals.
## VII Acknowledgement
This publication was made possible by NPRP12S-0225-190152 from the Qatar
National Research Fund (a member of The Qatar Foundation). The statements made
herein are solely the responsibility of the author[s].
## References
* [1] DHL, “DHL launches first commercial drone ’parcelcopter’ delivery service,” Available: https://www.theguardian.com/technology/2014/sep/25/german-dhl-launches-first-commercial-drone-delivery-service, Accessed: Oct. 29, 2019.
* [2] FAA, “Federal Aviation Administration UAS Sightings Report,” Available: https://www.faa.gov/uas/resources/public_records/uas_sightings_report/, Accessed: Oct. 29, 2019.
* [3] P. Popovski, H. Yomo, and R. Prasad, “Strategies for adaptive frequency hopping in the unlicensed bands,” _IEEE Wireless Communications_ , vol. 13, no. 6, pp. 60–67, Dec 2006.
* [4] A. Kanaa and A. Z. Sha’ameri, “A robust parameter estimation of fhss signals using time–frequency analysis in a non-cooperative environment,” _Physical Communication_ , vol. 26, pp. 9–20, 2018.
* [5] B. Boashash, _Time-frequency signal analysis and processing: a comprehensive reference_. Academic Press, 2015.
* [6] S. Wei, M. Zhang, G. Wang, X. Sun, L. Zhang, and D. Chen, “Robust multi-frame joint frequency hopping radar waveform parameters estimation under low signal-noise-ratio,” _IEEE Access_ , 2019.
* [7] X. Zhang, X. Wang, and X.-m. Du, “Blind parameter estimation of frequency-hopping signals based on wavelet transform [j],” _Journal of Circuits and Systems_ , vol. 4, 2009.
* [8] X. Ouyang and M. G. Amin, “Short-time fourier transform receiver for nonstationary interference excision in direct sequence spread spectrum communications,” _IEEE Trans. Signal Process._ , vol. 49, no. 4, pp. 851–863, 2001.
* [9] C.-D. Chung and A. Polydoros, “Parameter estimation of random fh signals using autocorrelation techniques,” _IEEE Trans. Commun._ , vol. 43, no. 2/3/4, pp. 1097–1106, 1995.
* [10] N.-K. Kim and S.-J. Oh, “Comparison of methods for parameter estimation of frequency hopping signals,” in _International Conference on Information and Communication Technology Convergence (ICTC)_ , 2017, pp. 567–569.
* [11] J. Smith, S. U. C. for Computer Research in Music, Acoustics, and S. U. D. of Music, _Spectral Audio Signal Processing_. W3K, 2011, online book, 2011 edition. [Online]. Available: https://ccrma.stanford.edu/~jos/sasp/
* [12] R. M. Haralick, S. R. Sternberg, and X. Zhuang, “Image analysis using mathematical morphology,” _IEEE Trans. Pattern Anal. Mach. Intell._ , no. 4, pp. 532–550, 1987.
* [13] L. Luo _et al._ , “Detection of an unknown frequency hopping signal based on image features,” in _2nd International Congress on Image and Signal Processing_ , 2009, pp. 1–4.
* [14] Y. Ma and Y. Yan, “Blind detection and parameter estimation of single frequency-hopping signal in complex electromagnetic environment,” in _2016 Sixth International Conference on Instrumentation & Measurement, Computer, Communication and Control (IMCCC)_, 2016, pp. 370–374.
*[UAVs]: Unmanned aerial vehicle
*[FHSS]: frequency hopping spread spectrum
*[FH]: frequency hopping
*[RC]: radio controller
*[RCs]: radio controller
*[STFT]: short–time Fourier transform
*[ACF]: autocorrelation function
*[UAV]: Unmanned aerial vehicle
*[MSE]: mean square error
*[SNR]: signal–to–noise ratio
*[ISM]: industrial, scientific, and medical
*[AWGN]: Additive White Gaussian Noise
*[PSD]: power spectral density
*[TÜBİTAK]: Scientific and Technological Research Council of Turkey
*[BİLGEM]: Informatics and Information Security Research Center
*[GHz]: gigahertz
*[SSA]: signal and spectrum analyzer
*[Hz]: hertz
*[MHz]: megahertz
*[CFO]: carrier frequency offset
|
2024-09-04T02:54:58.109592 | 2020-03-07T19:07:31 | 2003.03638 | {
"authors": "James D. Brunner and Nicholas Chia",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26099",
"submitter": "James Brunner",
"url": "https://arxiv.org/abs/2003.03638"
} | arxiv-papers |
Minimizing the number of optimizations for efficient community dynamic flux
balance analysis.
James D. Brunner1†*,Nicholas Chia1†
1Department of Surgery, Center for Individualized Medicine Microbiome Program,
Mayo Clinic, Rochester, MN, USA
†Current Address: Mayo Clinic, 200 First St. SW, Rochester, MN, USA
*<EMAIL_ADDRESS>
## Abstract
Dynamic flux balance analysis uses a quasi-steady state assumption to
calculate an organism’s metabolic activity at each time-step of a dynamic
simulation, using the well-known technique of flux balance analysis. For
microbial communities, this calculation is especially costly and involves
solving a linear constrained optimization problem for each member of the
community at each time step. However, this is unnecessary and inefficient, as
prior solutions can be used to inform future time steps. Here, we show that a
basis for the space of internal fluxes can be chosen for each microbe in a
community and this basis can be used to simulate forward by solving a
relatively inexpensive system of linear equations at most time steps. We can
use this solution as long as the resulting metabolic activity remains within
the optimization problem’s constraints (i.e. the solution to the linear system
of equations remains a feasible to the linear program). As the solution
becomes infeasible, it first becomes a feasible but degenerate solution to the
optimization problem, and we can solve a different but related optimization
problem to choose an appropriate basis to continue forward simulation. We
demonstrate the efficiency and robustness of our method by comparing with
currently used methods on a four species community, and show that our method
requires at least $91\%$ fewer optimizations to be solved. For
reproducibility, we prototyped the method using Python. Source code is
available at `https://github.com/jdbrunner/surfin_fba`.
## Author summary.
The standard methods in the field for dynamic flux balance analysis (FBA)
carries a prohibitively high computational cost because it requires solving a
linear optimization problem at each time-step. We have developed a novel
method for producing solutions to this dynamical system which greatly reduces
the number of optimization problems that must be solved. We prove
mathematically that we can solve the optimization problem once and simulate
the system forward as an ordinary differential equation (ODE) for some time
interval, and solutions to this ODE provide solutions to the optimization
problem. Eventually, the system reaches an easily check-able condition which
implies that another optimization problem must be solved. We compare our
method against typically used methods for dynamic FBA to validate that it
provides equivalent solutions while requiring fewer linear-program solutions.
## Introduction.
### Microbial communities and human health.
The makeup of microbial communities is often complex, dynamic, and hard to
predict. However, microbial community structure has a profound effect on human
health and disease [1, 2, 3, 4, 5, 6, 7]. These two facts have lead to
significant interest in mathematical models which can predict relative
abundances among microbes in a community. Various dynamical models have been
proposed to explain and predict microbial community population dynamics [8, 9,
10, 11, 12]. Among these are models which propose that interactions between
species are mediated by the metabolites that each species produces and
consumes [13, 14], and there is significant evidence that these models perform
better than models which depend on direct interaction between species [15,
16].
Recently, advances in genetic sequencing have allowed the creation of genome-
scale models (GEMs) that reflect the internal network of cellular metabolism,
and can therefore be used to predict metabolite use and production [17, 18,
19]. This technique can be extended to microbial community modeling by
combining GEMs of different species. There has been significant interest in
using GEMs to predict relative populations of stable microbial communities
[20, 21, 22, 23, 24, 25, 26]. Community metabolic modeling can not only
predict relative populations, but also holds the potential to predict and
explain the community metabolite yield, which can have a profound effect on
health [4]. Furthermore, model repositories such as the online bacterial
bioinformatics resource _PATRIC_ [27] or the _BiGG model database_ [28] make
it possible to build community models using information from individual
species investigations.
GEMs can be used to predict microbial growth rates as well as metabolite
consumption and production rates using a process called _flux balance
analysis_ (FBA). Because these predictions appear in the form of rates of
change, they can be used to define a metabolite mediated dynamical model,
simply by taking as a vector field the rates of change predicted by FBA. We
can therefore combine the techniques of metabolite mediated dynamic modeling
and community metabolic modeling to produce dynamic predictions of microbial
community population size and metabolite yield. This strategy is called
_dynamic FBA_ [29, 30, 31], and has recently been used to model microbial
communities [32, 33, 34].
Dynamic FBA, when implemented naïvely, requires a linear optimization problem
to be repeatedly solved, and carries a high computational cost for even small
communities. Furthermore, _in silico_ experiments may need to be repeated many
times over various environmental conditions or using various parameter choices
in order to make robust conclusions or to accurately fit model parameters. As
a result, implementations of dynamic FBA which depend on optimization at every
time-step carry a prohibitively high computational cost when used to simulate
larger microbial communities. The implementation of dynamic FBA in the popular
COBRA toolbox software package [17] is done in this way, and essentially all
more efficient available tools for simulating dynamic FBA fundamentally use an
ODE solver approach with optimization at each time-step [31, 35, 36, 24, 37,
38]. Dynamic FBA can be improved by taking advantage of the linear structure
of the optimization problem which provides a choice of basis for an optimal
solution that may be reused at future time-steps [39, 40]. However, the
optimizations that are required by this strategy involve solutions with non-
unique bases. This means that a basis chosen at random may not provide an
optimal solution to the linear program at future time-steps because it
provides a solution that is non-optimal or infeasible.
In order to implement dynamic FBA without optimizing at each time step, we use
an optimal basic set for the FBA linear optimization problem to create a
system of linear equations whose solutions at future time-steps coincide with
the solutions to the FBA optimization problem. To solve the problem of non-
uniqueness among bases, we prove that there exists a choice of basis that
allows forward simulation for a given optimal flux solution and provide a
method to choose this basis. Note that this method does not choose among a set
of non-unique optimal flux solutions, but instead chooses a basis for a single
given optimum. To choose among multiple optimal flux solutions, biological,
rather than mathematical, considerations should be used.
In this manuscript, we detail how dynamic FBA can be simulated forward without
re-optimization for some time interval, and give a method for doing so. We
propose conditions on an optimal basic set for the FBA linear optimization
problem which allows for forward simulation, and we prove that such a choice
exists. We then detail how to choose this basis set, and finally give examples
of simulations which demonstrate the power of our method. For reproducibility,
we make a prototype implementation of our method in the Python language
available at `https://github.com/jdbrunner/surfin_fba`.
## Background
### Flux balance analysis.
With the advent of genetic sequencing and the resulting genome scale
reconstruction of metabolic pathways, methods have been developed to analyze
and draw insight from such large scale models [18]. To enable computation of
relevant model outcomes, constraint based reconstruction and analysis (COBRA)
is used to model steady state fluxes $v_{i}$ through a microorganism’s
internal metabolic reactions under physically relevant constraints [18]. One
of the most basic COBRA methods, called _flux balance analysis_ (FBA)
optimizes some combination of reaction fluxes $\sum\gamma_{i}v_{i}$ which
correspond to increased cellular biomass, subject to the constraint that the
cell’s internal metabolism is at equilibrium:
$\Gamma\bm{v}=0$ (1)
where $\Gamma$ is the _stoichiometric matrix_ , a matrix describing the
stoichiometry of the metabolic model.
This optimization is chosen because it reflects the optimization carried out
by nature through evolution [18]. The vector
$\bm{\gamma}=(\gamma_{1},\gamma_{2},...,\gamma_{d})$ is an encoding of
cellular objectives, reflecting the belief that the cell will be optimized to
carry out these objectives. The constraint Eq. 1 means that any optimal set of
fluxes found by FBA corresponds to a steady state of the classical model of
chemical reaction networks [41]. This reflects the assumption that the cell
will approach an internal chemical equilibrium.
The optimization is done over a polytope of feasible solutions defined by the
inequalities $v_{i,min}\leq v_{i}\leq v_{i,max}$, or possibly more complicated
linear constraints. See Fig. 1 for a geometric representation of an example of
the type of linear optimization problem that is carried out. By convention,
forward and reverse reactions are not separated and so negative flux is
allowed. Linear optimization problems like FBA often give rise to an infinite
set of optimal flux vectors $\bm{v}=(v_{1},v_{2},...,v_{d})$. Geometrically,
this set will correspond to some face of the polytope of feasible solutions.
To draw conclusions despite this limitation, many methods have been developed
to either characterize the set of optimal solutions, as with flux variability
analysis (FVA), or enforce more constraints on the network to reduce the size
of this set, as with loopless FVA [18].
### Dynamic FBA.
FBA provides a rate of increase of biomass which can be interpreted as a
growth rate for a cell. Furthermore, a subset of the reactions of a GEM
represent metabolite exchange between the cell and its environment. By
interpreting constraints on nutrient exchange reactions within the metabolic
network as functions of the available external metabolites and fluxes of
exchange reactions as metabolite exchange rates between the cell and its
environment, the coupled system can be modeled. The simplest way to do this is
to use an Euler method, as in [30].
In addition to Euler’s method, more sophisticated ODE solvers may be used in
the so-called “direct” method of simply recomputing the FBA optimization at
every time-step. This can provide better solution accuracy and potentially
larger time-steps, but may also require more than one FBA optimization at each
time-step. For instance, the Runge-Kutta fourth order method [42] requires
four FBA solutions at each time step. Direct methods are implemented in the
COBRA toolbox [17] and are the central algorithm in many modern tools,
including those of Zhuang et al. [31, 35], Harcombe et al. [36], Zomorrodi et
al. [24], Louca and Doebeli [37], and Popp and Centler [38]. Notably, any
direct method requires at least one complete recalculation of the network
fluxes _at each time-step_.
However, resolving the system at each time step is not necessary, as the
solution the optimization problem at some initial time can actually be used to
compute future optimal solutions. Höffner et al., [40], used this observation
to introduce a variable step-size method for dynamic FBA. In that method a
basic index set is chosen by adding biological constraints to the optimization
problem hierarchically until a unique optimal flux vector is found. The
challenge of such an approach is in choosing the basis for the optimal
solution, as the optimal basis is not guaranteed to be unique even for a
unique optimal flux solution. In fact, due to the nature of the method of
Höffner et al. and of our method, any optimization past the initial solution
that must be carried out is guaranteed to have a solution with a non-unique
basis. Furthermore, many choices of optimal basis will not provide a solution
for future time-steps, so that choosing among these bases must be done
intelligently. Unfortunately, Höffner et al. [40] do not provide a method for
choosing among non-unique bases for a single linear program solution.
Our method seeks to solve this problem by choosing a basis from among the
possibilities provided from an FBA solution which is most likely to remain
optimal as simulation proceeds forward. We therefore prioritize reducing the
number of times the linear program must be solved, choosing our basis based on
the mathematical properties of the system which gives the best chance of
providing a solution at future time-steps.
Additionally, a method described as the “dynamic optimization approach” was
introduced in Mahadevan et al., [29], however this method is computationally
expensive. In particular, the method given in [29] involves optimizing over
the entire time-course simulated, and so is formulated as a non-linear program
which only needs to be solved once. While this method requires only one
optimization, this optimization is itself prohibitively difficult due to the
dimensionality of the problem growing with the fineness of time-
discretization.
### The dynamic FBA model for communities.
We can write a metabolite mediated model for the population dynamics of a
community of organisms $\bm{x}=(x_{1},...,x_{p})$ on a medium composed of
nutrients $\bm{y}=(y_{1},...,y_{m})$:
$\displaystyle\dot{x}_{i}$ $\displaystyle=g_{i}(\bm{\psi}_{i}(\bm{y}))x_{i}$
(2) $\displaystyle\dot{y}_{j}$
$\displaystyle=-\sum_{i=1}^{p}\psi_{ij}(\bm{y})x_{i}$ (3)
where $\bm{\psi}_{i}$ is a vector of the fluxes of nutrient exchange reactions
for organism $x_{i}$ as determined by FBA. Using FBA to determine
$\bm{\psi}_{i}$ is therefore a quasi-steady state assumption on the internal
metabolism of the organisms $x_{i}$[43, 44, 45].
Recall that the basic assumption of flux balance analysis is that, given a
matrix $\Gamma_{i}$ that gives the stoichiometry of the network of reactions
in a cell of organism $x_{i}$ that growth $g_{i}(\bm{y})$ is the maximum
determined by solving the following linear program [18]:
$\left\\{\begin{array}[]{r}\max(\bm{v}_{i}\cdot\bm{\gamma}_{i})\\\
\Gamma_{i}\bm{v}_{i}=0\\\
\bm{c}^{1}_{i}\leq\bm{v}\leq\bm{c}^{2}_{i}(\bm{y})\end{array}\right\\}$ (4)
where $\bm{c}^{1}_{i}$ is some vector of lower flux bounds while
$\bm{c}^{2}_{i}(\bm{y})$ is some vector-valued function of the available
metabolites which represents upper flux bounds. The key observation allowing
dynamic FBA is that the optimal solution to this problem also determines
$\bm{\psi}_{i}$ simply by taking $\psi_{ij}$ to be the value of the flux
$v_{ij}$ of the appropriate metabolite exchange reaction. For clarity, we will
relabel the elements of $\bm{v}_{i}$ so that $\psi_{ik}=v_{ij}$ if $v_{ij}$ is
the $k^{th}$ exchange flux, and $\phi_{ik}=v_{ij}$ if $v_{ij}$ is the $k^{th}$
internal flux. The objective vector $\bm{\gamma}_{i}$ indicates which
reactions within the cell contribute directly to cellular biomass, and so is
non-zero only in elements corresponding to internal fluxes. We can therefore
rewrite this vector to include only elements corresponding to internal fluxes,
so that the objective of the optimization is to maximize
$\bm{\gamma}_{i}\cdot\bm{\phi}_{i}$.
The stoichiometry of metabolite exchange reactions is represented by standard
basis vectors [18]. Therefore, we can partition $\Gamma_{i}$ as
$\Gamma_{i}=\begin{bmatrix}I&-\Gamma_{i}^{*}\\\
0&\Gamma_{i}^{\dagger}\end{bmatrix}$ (5)
where $I$ is the identity matrix of appropriate size, and $\Gamma_{i}^{*}$ and
$\Gamma_{i}^{\dagger}$ contain the stoichiometry of the internal reactions
[18, 46, 47]. Making this change in notation allows us to see that the
optimization problem of flux balance analysis is essentially internal to the
cell, with external reactions providing constraints.
We can see from Eq. 5 that $\ker(\Gamma_{i})$ is isomorphic to
$\ker(\Gamma^{\dagger}_{i})$, and so we can maximize over this kernel. Then,
the exchange reaction fluxes are determined by the internal fluxes according
to the linear mapping $\bm{\psi}_{i}=\Gamma^{*}_{i}\bm{\phi}_{i}$ . The
maximization of FBA becomes a maximization problem over the internal
fluxes111In fact, we can project onto the kernel of the matrix
$\Gamma^{\dagger}_{i}$ and so reduce the dimensionality of the problem.
However, in practice this projection is not numerically stable.. We rewrite
Eq. 4 using Eq. 5 and combine with Eqs. 2 and 3 to form the differential
algebraic system
$\displaystyle\frac{dx_{i}}{dt}=x_{i}(\bm{\gamma}_{i}\cdot\bm{\phi}_{i})$ (6)
$\displaystyle\frac{d\bm{y}}{dt}=-\sum_{i}x_{i}\Gamma^{*}_{i}\bm{\phi}_{i}$
(7)
$\displaystyle\left\\{\begin{array}[]{r}\max(\bm{\phi}_{i}\cdot\bm{\gamma}_{i})\\\
\Gamma^{\dagger}_{i}\bm{\phi}_{i}=0\\\
\bm{c}^{1}_{i}\leq\begin{bmatrix}\Gamma^{*}_{i}\\\
I\end{bmatrix}\bm{\phi}_{i}\leq\bm{c}^{2}_{i}(\bm{y})\end{array}\right\\}$
(11)
where each $\bm{\phi}_{i}$ is determined by the optimization Eq. 11, all
carried out separately. Note that this is a metabolite mediated model of
community growth as defined in [15]. That is, the coupling of the growth of
the separate microbes is due to the shared pool of metabolites $\bm{y}$. Each
separate optimization which determines $\bm{\phi}_{i}$ at a single time-step
depends on $\bm{y}$, and each $\bm{\phi}_{i}$ determines some change in
$\bm{y}$. Furthermore, each optimization is carried out in a manner that
depends only the status of the metabolite pool and is independent from the
optimizations of other organisms. There is therefore no shared “community
objective”. Instead, each organism optimizes according to only its own
internal objective.
We write, for full generality, upper and lower dynamic bounds on internal and
exchange reactions, and assume that each function $c_{ij}(\bm{y})\in
C^{\infty}$. We let
$A_{i}=\begin{bmatrix}(\Gamma_{i}^{*})^{T},-(\Gamma_{i}^{*})^{T},I,-I,\end{bmatrix}^{T}$
(12)
so that we can rewrite the optimization problem Eq. 11 as
$\left\\{\begin{array}[]{r}\max(\bm{\phi}_{i}\cdot\bm{\gamma}_{i})\\\
A_{i}\bm{\phi}_{i}\leq\bm{c}_{i}(\bm{y},t)\\\
\Gamma^{\dagger}_{i}\bm{\phi}_{i}=\bm{0}\end{array}\right\\}$ (13)
for ease of notation.
We now hope to select a basic index set $\mathcal{I}_{i}$ for Eq. 13 for each
organism $x_{i}$ so that each $\bm{\phi}_{i}(t)$ is a solution to the
resulting linear system of equations.
## Methods.
### Linear optimization preliminaries.
In this manuscript, we will rewrite the FBA optimization problem in the form
$\left\\{\begin{array}[]{c}\max(\bm{\phi}\cdot\bm{\gamma})\\\
A\bm{\phi}\leq\bm{c}\\\ \Gamma^{\dagger}\bm{\phi}=0\end{array}\right\\}$ (14)
where the matrices $A$ and $\Gamma^{\dagger}$ are derived from the
stoichiometric matrix and flux constraints. Such a problem is often referred
to as a _linear program_ (LP). We now recall some well known results from the
study of linear programming (see, for example [48, 40]).
First, we note that Eq. 14 can be rewritten in the so-called _standard form_
with the addition of _slack variables_ $\bm{s}=(s_{1},...,s_{n})$ which
represent the distance each of the $n$ constraints is from its bound as
follows:
$\left\\{\begin{array}[]{c}\max(\bm{\tilde{\phi}}\cdot\bm{\tilde{\gamma}})\\\
\begin{bmatrix}\tilde{A}&I\end{bmatrix}\begin{bmatrix}\bm{\tilde{\phi}}\\\
\bm{s}\end{bmatrix}=\bm{c}\\\ \tilde{\phi}_{i}\geq 0,s_{i}\geq
0\end{array}\right\\}.$ (15)
Standard form requires that we rewrite $\phi_{i}=\phi_{i}^{+}-\phi_{i}^{-}$
and then define
$\bm{\tilde{\phi}}=(\phi_{1}^{+},\phi_{2}^{+},...,\phi_{d}^{+},\phi_{1}^{-},\phi_{2}^{-},...,\phi_{d}^{-})$
so that we require non-negativity of each variable, and the matrix
$\tilde{A}=\left[A\;B\right]$, $B=-A$. We rewrite the problem in this form to
make use of established results, and for ease of notation will write
$\bm{\phi}$ instead of $\bm{\tilde{\phi}}$ when it is clear which form of the
problem we are discussing.
We will make use of the well-known result that there exists an _optimal basis_
or _basic set_ for a bounded linear program [49]. To state this result, we
first define the notation $B_{\mathcal{J}}$ to be the matrix with columns of
$[\tilde{A}\;I]$ corresponding to some index set
$\\{k_{1},k_{2},...,k_{n}\\}=\mathcal{J}$, and if $B_{\mathcal{J}}$ is
invertible we define the notation $\bm{w}_{\mathcal{J}}(\bm{a})$ so that
$(\bm{w}_{\mathcal{J}}(\bm{a}))_{l}=\left\\{\begin{array}[]{cc}(B^{-1}_{\mathcal{I}}\bm{a})_{j}&l=k_{j}\in\mathcal{J}\\\
0&l\not\in\mathcal{J}\end{array}\right.$ (16)
for any $\bm{a}\in\mathbb{R}^{n}$. We may now define an _optimal basis_ and
_optimal basic set_.
###### Definition 1.
A _basic optimal solution_ to a linear program is an optimal solution along
with some index set $\\{k_{1},k_{2},...,k_{n}\\}=\mathcal{I}$ such that
$\bm{w}=\bm{w}_{\mathcal{I}}(\bm{c})$, where $\bm{c}$ is the vector of
constraints as in Eq. 15. The variables $\\{\bm{w}_{i}|i\in\mathcal{I}\\}$ are
referred to as _basic variables_ , and the index set $\mathcal{I}$ is referred
to as the _basic index set_.
Finally, if there exists a bounded, optimal solution to Eq. 15, then there
exists a basic optimal solution and corresponding basic index set.
For a given basic optimal solution vector $\bm{w}$, there may be more than one
basic index set $\mathcal{I}$ such that $\bm{w}=\bm{w}_{\mathcal{I}}(\bm{b})$.
Such a solution is called _degenerate_. Clearly a necessary condition for such
non-uniqueness is that there exists some $k\in\mathcal{I}$ such that
$w_{k}=0$. This is also a sufficient condition as long as there is some column
of $[\tilde{A}\,I]$ which is not in the column space of
$B_{\mathcal{I}\setminus\\{k\\}}$.
### Forward simulation without re-solving.
Consider again Eq. 13, the linear program that must be solved at each time
point of the dynamical system for each microbial population. Information from
prior solutions can inform future time-steps as long as the region of feasible
solutions has not qualitatively changed. Thus, we may only need to solve the
optimization problem a few times over the course of a simulation. The key
observation making this possible is that the simplex method of solving a
linear program provides an optimal basis for the solution. We may often re-use
this basis within some time interval, and therefore find optimal solutions
without re-solving the linear program.
In order to do this, we need to find a form of the solution which may be
evolved in time. Thus, we turn the system of linear inequalities given in the
linear program into a system of linear equations. Then, if this system has a
unique solution we have reduced the task to solving a system of equations
rather than optimizing over a system of inequalities. We can find such a
system of equations by solving the linear program once, and using this
solution to create a system of equations whose solution provides the optimal
flux $\bm{\phi}_{i}$, as described above. We then use this same system to
simulate forward without the need to re-solve the solution to the system of
equations until there is no longer a feasible solution to the linear program.
First, the linear program Eq. 13 is transformed into standard form (Eq. 15).
Then, a basic optimal solution is found with corresponding basic index set
$\mathcal{I}_{i}$. The dynamical system Eqs. 6, 7 and 13 can then be evolved
in time using Eq. 16. This evolution is accurate until some $w_{ij}$ becomes
negative (meaning that the solution is no longer a feasible solution to the
linear program). At this point, a new basis must be chosen. That is, until
$\bm{w}_{\mathcal{I}_{i}}(\bm{c}(t))$ becomes infeasible, we let
$(\phi_{j_{1}}(\bm{c}_{i}(t)),...,\phi_{j_{m}}(\bm{c}_{i}(t)),s_{1}(\bm{c}_{i}(t)),...,s_{n}(\bm{c}_{i}(t)))=\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$
and replace Eqs. 6, 7 and 13 with
$\displaystyle\frac{dx_{i}}{dt}$
$\displaystyle=x_{i}(\bm{\gamma}_{i}\cdot\bm{\phi}_{i}(\bm{c}_{i}(t)))$ (17)
$\displaystyle\frac{d\bm{y}}{dt}$
$\displaystyle=-\sum_{i}x_{i}\Gamma^{*}_{i}\bm{\phi}_{i}(\bm{c}_{i}(t))$ (18)
One major difficulty in this technique is that a unique $\bm{w}_{i}$ does not
guarantee a unique basis set $\mathcal{I}_{i}$. If we have some
$(w_{\mathcal{I}_{i}})_{j}=0$ for $j\in\mathcal{I}_{i}$, then there exists
some alternate set $\hat{\mathcal{I}}_{i}$ such that
$\bm{{w}}_{\hat{\mathcal{I}}_{i}}=\bm{{w}}_{\mathcal{I}_{i}}$. Such a solution
$\bm{{w}}_{\mathcal{I}_{i}}$ is called _degenerate_. In a static
implementation of a linear program, the choice of basis of a degenerate
solution is not important, as one is interested in the optimal vector and
optimal value. However, as we will demonstrate with Example 1, the choice of
basis of a degenerate solution is important in a dynamic problem. In fact, if
the system given in Eqs. 17 and 18 is evolved forward until
$\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$ becomes infeasible, the time at
which the system becomes infeasible is the time at which we have some
$(w_{\mathcal{I}_{i}})_{j}=0$ for $j\in\mathcal{I}_{i}$. Thus, we need to
resolve Eq. 13 whenever $\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$ becomes
degenerate, which will be the final time-point at which the
$\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$ is feasible.
###### Example 1.
Consider the dynamic linear program
$\left\\{\begin{array}[]{c}\max((1,1)\cdot\bm{v})\\\ \begin{bmatrix}1&0\\\
0&1\\\ 1&2\end{bmatrix}\bm{v}\leq\begin{bmatrix}10\\\ 10\\\
30-t\end{bmatrix}\\\ v_{i}\geq 0\end{array}\right\\}$ (19)
In standard form at $t=0$, this linear program becomes
$\left\\{\begin{array}[]{c}\max((1,1)\cdot\bm{v})\\\
\begin{bmatrix}1&0&1&0&0\\\ 0&1&0&1&0\\\
1&2&0&0&1\end{bmatrix}\begin{bmatrix}\bm{v}\\\
\bm{s}\end{bmatrix}=\begin{bmatrix}10\\\ 10\\\ 30\end{bmatrix}\\\
v_{i},s_{i}\geq 0\end{array}\right\\}$ (20)
which has the unique solution $\bm{w}=(10,10,0,0,0)$. There are three choices
of basic index sets: $\mathcal{I}_{1}=\\{1,2,3\\}$,
$\mathcal{I}_{2}=\\{1,2,4\\}$, and $\mathcal{I}_{3}=\\{1,2,5\\}$. The
resulting bases are
$\ B_{\mathcal{I}_{1}}=\begin{bmatrix}1&0&1\\\ 0&1&0\\\
1&2&0\end{bmatrix}\quad B_{\mathcal{I}_{2}}=\begin{bmatrix}1&0&0\\\ 0&1&1\\\
1&2&0\end{bmatrix}\quad B_{\mathcal{I}_{3}}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\
1&2&1\end{bmatrix}$
Computing Eq. 16 at $t>0$ for each, we have that $B_{\mathcal{I}_{1}}$ yields
$\bm{w}_{\mathcal{I}_{1}}(\bm{c}(t))=(10-t,10,t,0,0)$, $B_{\mathcal{I}_{2}}$
yields
$\bm{w}_{\mathcal{I}_{2}}(\bm{c}(t))=(10,10-\nicefrac{{t}}{{2}},0,\nicefrac{{t}}{{2}},0)$,
and $B_{\mathcal{I}_{3}}$ yields
$\bm{w}_{\mathcal{I}_{3}}(\bm{c}(t))=(10,10,0,0,-t)$, shown in Fig. 1 for
$t>0$. Thus, only $\bm{w}_{\mathcal{I}_{2}}(\bm{c}(t))$ solves the dynamic
problem because $\bm{w}_{\mathcal{I}_{1}}(\bm{c}(t))$ is not optimal and
$\bm{w}_{\mathcal{I}_{3}}(\bm{c}(t))$ is not feasible for $t>0$. We may follow
$\bm{w}_{\mathcal{I}_{2}}$ and be insured of remaining at an optimal solution
to the linear program until $t=20+\varepsilon$, at which point
$\bm{w}_{\mathcal{I}_{2}}=(10,-\varepsilon/2,0,10,0)$, which is not a feasible
solution to the linear program. At time $t=20$, a re-optimization is required
to choose a new basis.
Notice that the correct choice of basis fundamentally depends on the time-
varying bound function $\bm{c}(t)=(10,10,30-t)$. To see this, consider other
possible time-varying bounds $\bm{c}(t)$ which have $\bm{c}(0)=(10,10,30)$.
For example, if $\bm{c}(t)=(10-t,10-t,30)$, then only $B_{\mathcal{I}_{3}}$
would give the correct $\bm{w}(\bm{c}(t))$ for $t>0$.
Fig 1: Geometric representation of Example 1 for $t_{3}>t_{2}>t_{1}>0$,
showing the three options for bases which are equivalent at $t=0$. Note that
the best choice depends on the function $\bm{c}(t)=(10,10,30-t)$ and cannot be
chosen using the static problem alone. The feasible region of the optimization
problem is shown in gray.
### A basis for the flux vector.
We now provide a method to choose a basis $\mathcal{I}_{i}$ for each organism
$x_{i}$ in the case of a degenerate solution. Consider an optimal solution
$\bm{w}_{i}$ to the linear program Eq. 15. To simulate forward according to
Eqs. 17 and 18, we need for each organism $x_{i}$ a basic index set
$\mathcal{I}_{i}$ such that
$\left\\{\begin{array}[]{c}\bm{\dot{w}_{i}}=\bm{w}_{\mathcal{I}_{i}}\left(\frac{d}{dt}\bm{c}_{i}\right)\\\
\begin{bmatrix}\tilde{A}&I\end{bmatrix}\bm{\dot{w}}=\frac{d}{dt}\bm{c}_{i}\\\
(\bm{w}_{\mathcal{I}_{i}})_{j}=0\Rightarrow\dot{w}_{ij}\geq
0\end{array}\right\\}$ (21)
so that the solution remains feasible, and furthermore that $\bm{\dot{w}}_{i}$
is optimal over the possible choice of basic index sets for $\bm{w}_{i}$. This
is obviously a necessary condition for forward simulation within some non-
empty time interval, and can be made sufficient (although no longer necessary)
by making the inequality
$(\bm{w}_{\mathcal{I}_{i}})_{j}=0\Rightarrow\dot{w}_{ij}\geq 0$ strict. We use
the relaxed condition for more practical applicability.
In order to develop a method based on the above observation (i.e., Eq. 21), we
must know that Eq. 15 has such a solution. We therefore require the following
lemma, which is proved in Appendix A:
###### Lemma 1.
For a linear program with the form given in Eq. 15 with a basic optimal
solution $\bm{w}$, there exists a basic index set $\mathcal{I}$ such that Eq.
21 holds and $\bm{\dot{w}}$ is optimal over the possible choice of basic index
sets for $\bm{w}$.
If Eq. 15 has only a non-degenerate solution, the unique basis will satisfy
this requirement. The challenge remains to choose from among the possible
bases of a degenerate solution.
To do this, we form a second linear program analogous to Eq. 21 in the
following way. We first find all constraints $\bm{a}_{j}$ (i.e. rows of
$A_{i}$ or $\Gamma^{\dagger}_{i}$) such that
$\bm{a}_{ij}\cdot\bm{\phi}_{i}=c_{ij}(t)$, calling this set $\mathcal{S}_{i}$.
Note that this set contains all the rows of $\Gamma^{\dagger}_{i}$, for which
we regard $c_{ij}(t)=0$ for all $t>0$. Note that if the solution given is a
basic optimal solution, the rank of the matrix whose rows are $\bm{a}_{ij}$
for $\bm{a}_{ij}\in\mathcal{S}_{i}$ is $d$, where again $d$ is the number of
internal fluxes. This is true because we include constraints of the type
$a<\phi_{ij}<b$ as rows of $A_{i}$.
Then, we solve the linear program
$\left\\{\begin{array}[]{c}\max(\bm{\dot{w}}_{i}\cdot\bm{\gamma}_{i})\\\
\bm{a}_{j}\cdot\bm{\dot{\phi}}_{i}\leq\frac{dc_{ij}}{dt},\;\;\bm{a}_{j}\in\mathcal{S}_{i}\end{array}\right\\}$
(22)
We may then use any basis $B_{\mathcal{I}}^{i}$ which solves Eq. 22 as long as
it has exactly $d$ non-basic slack variables. Lemma 1 tells us that such a
choice exists, although it may be necessary to manually pivot non-slack
variables into the basis set given by the numerical solver222In testing the
algorithm, this was necessary when using IBM ILOG CPLEX Optimization Studio to
solve, but not when using The Gurobi Optimizer.. Note that we do not need the
entire basis $B_{\mathcal{I}}^{i}$, but instead only need the $d\times d$
submatrix formed by rows of $A_{i}$ or $\Gamma_{i}^{\dagger}$ which correspond
to non-basic slack variables in the solution to Eq. 22. These appear as rows
$(\bm{a}_{i},\bm{0})$ in $B_{\mathcal{I}}^{i}$, and so this sub-matrix
uniquely determines $\bm{\phi}_{i}$. We call this smaller matrix $B_{i}$, and
label the set of row indices as $\mathcal{J}$.
The chosen basis $\mathcal{J}$ and corresponding constraints is used to
simulate forward until that particular solution becomes infeasible. At that
time, we have an optimal solution to Eq. 13 simply by continuity. We therefore
do not need to resolve Eq. 13 but instead re-form and solve Eq. 22.
### Pseudo-Code of the method.
Below, we present as pseudo-code an outline of the method. A practical
implication may need to adaptively adjust the time-step $\Delta t$ to insure
that no resource is artificially over-depleted past $0$.
Input: Final time $T$, initial microbial biomasses $x_{i}(0)$, initial
nutrient concentrations $y_{j}(0)$, maximum inflow rates of nutrients
$\alpha_{i}$, stoichiometric matrices $\Gamma_{i}$
Output: Timecourse simulation of biomass and nutrient concentrations
1 for _each microbial population $i$_ do
2 Set $\bm{w}_{i}(0)$ to be solution to eq. (13) which lies on a vertex of the
feasible polytope.;
3 Solve eq. (21) to find initial basis $B_{i}$
4 end for
5
6while _$t <T$_ do
7 Integrate eqs. (14) and (15) from $t$ to $t+\Delta t$ with
$\bm{\phi}_{i}=B_{i}^{-1}\bm{c}_{\mathcal{J}}(\bm{y}(t),t)$;
8 if _$B_{i}^{-1}\bm{c}_{\mathcal{J}}(\bm{y}(t+\Delta t),t+\Delta t)$ is not
a feasible solution_ then
9 reset $x_{i}=x_{i}(t)$, $y_{j}=y_{j}(t)$;
10 Solve eq. (21) to find new basis $B_{i}$, with additional constraints
representing bounds violated by $B_{i}^{-1}\bm{c}_{\mathcal{J}}(\bm{y}(t),t)$.
11 end if
12
13 end while
Algorithm 1 Dynamic FBA algorithm following Lemma 1. Note that for numerical
stability and speed, we may store the matrices $Q_{i},R_{i}$ such that
$Q_{i}R_{i}=B_{i}$ is the QR-factorization of $B_{i}$ rather than either
storing $B^{-1}_{i}$ or solving completely during each time step of numerical
integration.
## Results.
### Number of optimizations.
We can compare the efficiency of Algorithm 1 with modern dynamic FBA methods
by counting the number of times a large linear program must be carried out
over the course of a simulation. At their core, state-of-art dynamic FBA tools
such as _d-OptCom_ [24] and _COMETS_ [36] employ the direct method of calling
an ODE-solving method with the linear program set as the right-hand-side. In
the case of Euler’s method, the resulting ODE can be integrated by hand
between time-steps. This last strategy is often referred to as the “static
optimization approach” [40].
We compared simulation of various combinations of the organisms _Escherichia
coli str. K-12 substr. MG1655_ (model iJR904), _Saccharomyces cerevisiae
S288C_ (model iND705), _Pseudomonas putida KT2440_ (model iJN746) and
_Mycobacterium tuberculosis H37Rv_ (model iEK1008), using models from the BiGG
database [28] (see S2 File. for details). We counted the optimizations
required for our model, as well as for direct methods using the numerical ODE
solvers _vode_ , _zvode_ , _lsoda_ , _dopri5_ , and _dop853_ from the SciPy
library. All of these numerical ODE solvers use adaptive step sizes for
accuracy and stability, and so represent optimized choices of time-steps.
Additionally, we compared the method of Höffner et al. as implemented in the
MatLab package _DFBAlab_ [39].
For our method and the direct method, we allowed exchange of every metabolite
detailed in S1 File. with initial metabolite concentrations given by that same
file, and with initial biomass of $0.3$ for each species. The file
`sim_comm.py` in the supplementary repository S3 Software. contains complete
simulation set-up.
To compare with the method of Höffner et al. [40], we use the newly available
Python package from the research group of Dr. David Tourigny titled _dynamic-
fba_ [50] for single organisms. This package allows simulation without
secondary optimizations, as our does, and so is more similar to our prototype
tool for comparison. Unfortunately, this package is currently only able to
simulate single organisms at the time of publishing. For microbial
communities, we can compare with the MatLab package DFBAlab [39] which
requires all dynamics variables to be optimized in a secondary optimization.
For simulations with DFBAlab, we use only the low-concentration metabolites
D-glucose, oxygen, and cob(I)alamin from the M9 medium detailed in S1 File. as
dynamically varying metabolites. It is worth noting that these are the most
favorable conditions we could find for the method of H”offner [40, 39] et al.
which are still biologically equivalent to our other simulations.
| Solution Method
---|---
Model Combination | Algorithm 1 | Höffner | vode | zvode | lsoda | dopri5 | dop853
iJR904 | 7 | 1 | 62 | 62 | 116 | 3313 | 6228
iND750 | 4 | 1 | 91 | 91 | 85 | 3508 | 6514
iJN746 | 4 | 13 | 166 | 167 | 376 | 1176 | 2249
iEK1008 | 4 | 4 | 120 | 120 | 208 | 2768 | 5148
iJR904 + iND750 | 4 | 24 | 240 | 211 | 346 | 5586 | 10469
iJR904 + iJN746 | 30 | 479 | 420 | 420 | 744 | 2695 | 5579
iJR904 + iEK1008 | 20 | 136 | 216 | 216 | 454 | 3385 | 6411
iND750 + iEK1008 | 8 | 32 | 311 | 311 | 509 | 5284 | 9888
iJR904 + iND750 + iEK1008 | 18 | 32* | 451 | 451 | 1282 | 6225 | 11961
iJR904 + iND750 + iJN746 + iEK1008 | 56 | 672 | 1122 | 1122 | 2242 | 6837 | 13529
Table 1: Number of realizations required to simulate to time $t=5$ with no
cell death or metabolite flow, using M9 minimal medium. *Simulation failed at
$t=3.034277$. Fig 2: Time-points of re-optimizations required in simulations
using the proposed method, the method of Höffner et al. [40] and various
direct methods, shown in blue. Shown in orange are times at which the direct
method solver encountered an infeasible linear program due to numerical error.
### Error estimation.
Our method provides much less theoretical error in dynamic FBA solutions than
traditional methods. In fact, Algorithm 1 implies that a simulation of a
microbial community can be divided into time intervals on which the algorithm
is exact. Of course, this assumes that the linear ODE solved in these
intervals is solved exactly rather than numerically.
Precisely, there exits some sequence $t_{0}=0<t_{1}<\cdots<t_{n-1}<t_{n}=T$
such that if we know the optimal flux vectors $\bm{w}_{i}(t_{l})$ at time
$t_{l}$, then Lemma 1 implies the existence of a set of invertible matrices
$B_{i}^{l}$ such that solutions to Eqs. 17 and 18 are solutions to Eqs. 6, 7
and 13 for $t\in[t_{l},t_{l+1}]$. Therefore, if we are able to identify the
$t_{l}$ exactly, then Algorithm 1 provides exact solutions to the dynamic FBA
problem Eqs. 6, 7 and 13. Of course, numerical limitations imply that we will
not re-optimize precisely at each $t_{l}$, and so we must investigate the
impact of this error. However, once re-optimization is done, the method is
again exact. The result is that we have no local truncation error for any time
step taken between re-optimization after $t_{l}$ and the interval endpoint
$t_{l+1}$, except for error due to numerical integration. In comparison,
direct methods from some integration error at every time step. This error
depends on the integration strategy used, and so for example the Euler’s
method based static optimization approach carries first order local truncation
error at each time step. This can easily lead to ODE overshoot and infeasible
linear programs at future time-step.
Assume that $t_{l-1}$ is known exactly, and $N$ is such that
$t^{1}=t_{l-1}+(N-1)\Delta t\leq t_{l}<t_{l-1}+N\Delta t=t^{2}$, so that there
is some possible error in the interval $[t^{1},t^{2}]$. We can estimate the
accumulated error in this time interval using a power series expansion. Let
$\bm{x}(t),\bm{y}(t)$ be solutions to Eqs. 6, 7 and 13 and
$\bm{\tilde{x}},\bm{\tilde{y}}$ be solutions given by Algorithm 1 for
$t\in[t^{1},t^{2})$. Furthermore, let $B_{i}^{l-1}$ be the invertible matrices
derived by solving Eq. 13 at $t_{l-1}$ and $B_{i}^{l}$ those derived by
solving at $t_{l}$. Then, $\bm{x}(t^{1})=\bm{\tilde{x}}(t^{1})$ and
$\bm{y}(t^{1})=\bm{\tilde{y}}(t^{1})$. For each $x_{i}$ we expand, assuming
some regularity of the functions $\bm{c}(\bm{y})$,
$x_{i}(t^{2})-\tilde{x}_{i}(t^{2})=(\Delta
t)x_{i}(t_{1})(\bm{\gamma}_{i}\cdot\left((B_{i}^{l-1})^{-1}-(B_{i}^{l-1})^{-1}\right)\bm{\hat{c}}_{i}(\bm{y}(t^{1}))+o(\Delta
t)$ (23)
and see that this method gives first order local error in time steps that
require a re-optimization.
The local error, while first order, only appears at time steps in which a re-
optimization occurred, and so global error will scale with the number of
necessary re-optimizations. This is in contrast with the classical use of
Euler’s method, which gives first order local error at every time-step, or any
other direct ODE method, whose error is dependent on the solver used.
We may compare the solutions provided by direct methods with those provided by
the method presented in Algorithm 1 and by the method of Höffner et al. [40].
The root-sum-square ($l_{2}$) difference in results are shown in Table 2. As
we argue above, direct methods are less accurate in theory that the algorithm
presented in Algorithm 1. Furthermore, direct simulations routinely failed to
simulate to time $t=5$ without encountering an infeasible linear program. This
infeasibility is the result of numerical error accumulating throughout the
simulation. The comparisons in Table 2 can be summarized by three distinct
characteristics. First, in the case of _S.cerevisiae_ , the direct methods
agree well with the newly presented method. Secondly, in the case of _E.coli_
and _M.tuberculosis_ , error seems to begin accumulating immediately. Finally,
in the case of _P.putida_ , the simulations agree well up to some time-point
at which the direct method fails and either quits entirely (as in the case of
the _dopri5_ solver which returns small error) or continues at a constant
value.
We note that discrepancies in dynamic FBA simulation may not always be due to
numerical error, but instead due to non-uniqueness in optimal flux solutions.
Our method provides a strategy for choosing between non-unique representations
(in the form of a basis) of a single optimal flux solution. The method of
Höffner et al. [40] provides a lexicographic strategy for choosing between
non-unique optimal flux solutions based on biological, rather than
mathematical, considerations. We note that for complete reproducibility, our
method should be integrated with some biologically based strategy for choosing
between non-unique optima.
| vode | zvode | lsoda | dopri5 | dop853 | Hoffner et al.
---|---|---|---|---|---|---
E.coli | 5.09933 | 5.09933 | 4.61467 | 5.09928 | 5.09928 | 4.68578
M.tuberculosis | 1.45401 | 1.45401 | 1.45417 | 1.45415 | 1.45415 | 2.48691
S.cerevisiae | 0.00426 | 0.00426 | 0.00430 | 0.00429 | 0.00429 | 3.06105
P.putida | 15.29177 | 15.29177 | 0.07080 | 15.23826 | 15.26221 | 4.78751
Table 2: $l_{2}$ difference in solutions to single-organism simulations
between direct methods and the method presented in Algorithm 1. Fig 3:
Simulations of _E.coli_ , _S.cerevisae_ , _M.tuberculosis_ and _P.putida_
using Algorithm 1, direct solvers, and the method of Höffner et al. In
simulations of _E.coli_ _M.tuberculosis_ , there is discrepancy early in the
simulation. In contrast, simulations of _P.putida_ agree up to the point that
an ODE solver fails.
## Examples & applications.
There has been a recent surge in interest in modeling microbial communities
using genome-scale metabolic models, much of which has focused on equilibrium
methods [22, 21, 4, 51, 26]. In order to capture transient behavior and
dynamic responses to stimuli, dynamic FBA has also been applied to microbial
communities [24, 52, 34]. However, community dynamic FBA invariable leads to a
large dynamical system with a high-dimensional parameter space, often with
little to know knowledge of parameter values. Any parameter fitting therefore
requires repeated numerical simulation of the system. Existing tools to do
this are built around a direct simulation approach, requiring many linear
program solutions. By drastically reducing the number of optimizations
required for numerical simulation, our approach offers the promise of
efficient numerical simulation of dynamic FBA which will make parameter
fitting more tractable, and may even allow conclusions without well-fit
parameters.
Below, we demonstrate that the problem of parameter fitting is an important
one by show that experimental outcome in even small communities is sensitive
to changes in kinetic parameters. Precisely, the kinetic parameters governing
the uptake rate of nutrients (i.e., the parameters of the functions
$\bm{c}^{2}_{i}$ in Eq. 4) have a profound effect on species competition.
Next, we show how repeated simulation with randomly sampled parameters can
provide some insight into community structure even without a well-fit set of
nutrient uptake parameters. These examples demonstrate the importance of
efficient dynamic FBA to microbial community modeling.
### Prediction dependence on nutrient uptake.
The set of unknown functions $\bm{c}_{i}^{2}(\bm{y})$ in Eq. 4 present a
profound problem for dynamic FBA simulation. If the behavior of the system is
sensitive to the functions chosen and parameters of those functions, a single
simulation will be of little use in drawing biological conclusion. In order to
demonstrate that such a sensitivity exists, we repeatedly simulated the same
simple community with different randomly drawn parameters. While a more
realistic choice of function may be saturating or sigmoidal (as with Hill or
Michaelis-Menten kinetics), for the following experiment we take these
functions to be linear:
$c_{ij}^{2}(\bm{y})=\kappa_{ij}y_{j},$ (24)
meaning that the maximum uptake rate of nutrient $y_{j}$ by organism $x_{i}$
is proportional to the concentration of $y_{j}$. This choice minimizes the
number of parameters that must be chosen for our analysis of parameter
sensitivity, and is in line with an assumption of simple mass action
kinetics[53, 54].
The choice of $\kappa_{ij}$ may have a profound effect on the outcome of a
community simulation, as it represents how well an organism can sequester a
resource when this will optimize the organism’s growth. In order study this
effect in a small community, we sampled a three-species community model with
$\kappa_{ij}\in(0,1)$ chosen uniformly at random. We used models for _E.coli_
, _S.cerevisiae_ and _M.tuberculosis_ downloaded from the BiGG model
database[28].
We simulated with no dilution of metabolites or microbes, and no replenishment
of nutrients. In every simulation, some critical metabolite was eventually
depleted and the organisms stopped growing. We recorded the simulated final
biomass of each organism from each simulation, and the results are shown in
Fig. 4.
Fig 4: (Top) Histogram of the final simulated biomass of each of _E.coli_ ,
_S.cerevisiae_ and _M.tuberculosis_ from 95 simulations, each with different
metabolite uptake rates $\kappa_{ij}$. (Bottom) Pair-wise comparison of the
final simulated biomass densities using a kernel density estimation. In red is
the result of uniform uptake rates $\kappa_{ij}=1$ for all $i,j$.
### Community growth effects.
As we saw in previous section, community growth outcomes depend on the choice
of nutrient uptake rates $\kappa_{ij}$. Using Algorithm 1, we can perform
Monte-Carlo sampling in order to understand the possible effects on some
microorganism growing in some community. To do this, we randomly sample the
set of uptake rates $\kappa_{ij}$ and run simulations of various communities
for the chosen uptake rates. Then, the correlation between communities of
final simulated biomass of some organism can be interpreted as the effect of
the community on the growth of that organism. A correlation less than $1$
between growth of an organism in different communities indicates that the
community is having some effect. To see the direction of this effect, we can
fit a simple linear regression model (best fit line) to the final simulated
biomasses. Then, the slope of this line tells us if the organism benefits or
is harmed by being in one community over another.
We again simulated _E.coli_ , _S.cerevisiae_ and _M.tuberculosis_ downloaded
from the BiGG model database [28]. Simulations were run with the M9 medium
described in S1 File., with no replenishment of resources.
Each organism grew to a larger final simulated biomass when alone compared to
when in a trio with the other two, which is unsurprising given the finite
resources. This difference was the least pronounced for _S.cerevisiae_ ,
suggesting that this organism is the least negatively effected by the
competition. However, this can be seen as only a preliminary observation
without better estimates of uptake parameters. Best-fit lines are shown in
Fig. 5. Efficient dynamic FBA allows repeated simulation with randomly sampled
parameters, which gives an indication of likely behavior even without accurate
parameter fitting.
Fig 5: Final simulated biomass of _E.coli_ , _S.cerevisiae_ and
_M.tuberculosis_ when grown alone or in pairs, for randomly sampled modeled
parameters. Best fit lines indicate the average effect of the community on an
organism’s growth.
## Conclusion
Understanding, predicting, and manipulating the make-up of microbial
communities requires understanding a complex dynamic process. Genome-scale
metabolic models provide an approximation to this process through the quasi-
steady state assumption which leads to dynamic flux balance analysis. However,
this system is large and hard to simulate numerically, let alone analyze for
qualitative behaviors. As a first step towards a thorough analysis of
community of organisms modeled with dynamic FBA, an efficient method of
numerical simulation would provide an essential tool. However, modern tools
for simulating dynamic FBA rely on repeatedly solving an optimization problem
at every time step [31, 35, 36, 24, 37, 38].
Dynamic FBA simulation can be improved by considering the structure of these
linear programs so that many fewer optimizations are required. As of now, the
algorithm of Höffner et al. [40] is the only published method which takes
advantage of this observation. However, that method does not account for the
degeneracy of solutions to the relevant linear programs, meaning that it can
choose a solution that cannot be carried forward in time. We present a method
that chooses a basis to for forward simulation. In contrast to the method of
Höffner et al., we choose this basis in such a way that increases the
likelihood that this forward simulation is actually possible.
Efficient dynamic FBA will allow better parameter fitting to time-longitudinal
data. Furthermore, it allows for a search of parameter space which can help
predict likely model outcomes or learn maps from parameter values to model
outcomes.
## Supporting information.
#### S1 File.
M9 medium File. `m9med.csv` defines an M9 minimal medium as adapted from Monk
et al. [55].
#### S2 File.
List of Models Used. `modelsUsed.csv` provides name, ID, and URL for the four
models used in analysis of the method.
#### S3 Software.
`https://github.com/jdbrunner/surfin_fba`. Available code for the algorithm
described in the Python language. This code requires the popular COBRAPy
package for metabolic models.
## Acknowledgments
This work was supported by funding from the DeWitt and Curtiss Family
Foundation, National Cancer Institute grant R01 CA179243, and the Center for
Individualized Medicine, Mayo Clinic.
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## Appendix A Existence of desired optimal basis.
###### Lemma 1.
For a linear program with the form given in Eq. 15 with a basic optimal
solution $\bm{w}$, there exists a basic index set $\mathcal{I}$ such that Eq.
21 holds and $\bm{\dot{w}}$ is optimal over the possible choice of basic index
sets for $\bm{w}$.
###### Proof.
For convenience, we now restate Eq. 15:
$\left\\{\begin{array}[]{c}\max(\bm{\tilde{\phi}}\cdot\bm{\tilde{\gamma}})\\\
\begin{bmatrix}\tilde{A}&I\end{bmatrix}\begin{bmatrix}\bm{\tilde{\phi}}\\\
\bm{s}\end{bmatrix}=\bm{c}\\\ \tilde{\phi}_{i}\geq 0,s_{i}\geq
0\end{array}\right\\}$
where we write $(\bm{\tilde{\phi}},\bm{s})=\bm{w}$.
We note that there is a finite number of basic index sets for $\bm{w}$, and so
we need only show that there exists $\mathcal{I}$ such that Eq. 21 holds.
Then, the existence of an optimal such $\mathcal{I}$ follows trivially.
If $\bm{w}$ is not degenerate, then the unique choice of basic index set
$\mathcal{I}$ satisfies Eq. 21. To see this, simply note that if $\bm{w}$ is
non-degenerate, then for every $i\in\mathcal{I}$, $w_{i}>0$. Thus, Eq. 21 only
includes non-negativity constraints on $\dot{w}_{i}$ if $i\not\in\mathcal{I}$,
and for any $i\not\in\mathcal{I}$, $\dot{w}_{i}=0$. Thus, the non-negativity
constraints are enforced. The equality constraints are enforced by the
definition of $\bm{w}_{\mathcal{I}}(\bm{a})$ given in Eq. 16, which implies
that $[\tilde{A}\;I]\bm{w}_{\mathcal{I}}(\bm{a})=\bm{a}$ for any vector
$\bm{a}\in\mathbb{R}^{n}$.
In the case of a degenerate solution $\bm{w}$, we use the following procedure
to choose a set of basic variables. Let $\mathcal{J}\subset\\{1,...,n\\}$ be
the indices of the $n_{1}$ slack variables such that $s_{j}=0$ if
$j\in\mathcal{J}$ (recalling that each $s_{i}$ is a component of the vector
$\bm{w}$). Then, let $\tilde{A}_{\mathcal{J}}$ be the matrix with rows $m_{j}$
of $\tilde{A}$ for $j\in\mathcal{J}$. Next, let $\mathcal{J}^{*}$ be the
indices of the $n_{2}$ non-slack variables such that $\phi_{j}=0$ and
$I_{\mathcal{J}^{*}}$ the corresponding rows of the identity matrix $I$.
Notice that we now have that
$M\bm{\tilde{\phi}}=\begin{bmatrix}\tilde{A}_{\mathcal{J}}\\\
-I_{\mathcal{J}^{*}}\end{bmatrix}\bm{\tilde{\phi}}=\begin{bmatrix}\bm{c}_{\mathcal{J}}\\\
\bm{0}\end{bmatrix}.$ (25)
and that if $w_{j}=0$ then either $j\in\mathcal{J}^{*}$ or $w_{j}=s_{k}$ where
$k\in\mathcal{J}$ so that $\bm{m}_{k}\cdot\bm{\tilde{\phi}}=c_{k}$ (i.e.
$s_{k}$ is a slack variable and $s_{k}=0$). Notice that because Eq. 15 has a
bounded solution, then we can assume without loss of generality that if
$M\in\mathbb{R}^{q\times r}$, then $\mathit{rank}(M)=r$ (i.e. $M$ is full
rank) because $\bm{w}$ must satisfy at least $r$ linearly independent
constraints. If this is not the case, then the problem can be projected onto a
lower dimensional subspace.
Consider the linear program
$\left\\{\begin{array}[]{c}\max(\bm{y}\cdot\bm{\gamma})\\\
\begin{bmatrix}M&I\end{bmatrix}\begin{bmatrix}\bm{y}_{\bm{\tilde{\phi}}}\\\
\bm{y}_{\bm{s}}\end{bmatrix}=\begin{bmatrix}\frac{d}{dt}\bm{c}_{\mathcal{J}}\\\
\bm{0}\end{bmatrix}\\\ y_{j}\geq 0\end{array}\right\\}.$ (26)
Assume that there is some basic optimal solution to Eq. 26 with a basic index
set $\hat{\mathcal{I}}$ such that exactly $r$ slack variables are non-basic,
where again $r=|\bm{\phi}|$ is the rank of the matrix $M$. This implies that
there are $r$ linearly independent rows of $M$ (which we index by
$\mathcal{J}^{{\dagger}}$) which form an invertible matrix $\tilde{M}$ such
that
$\tilde{M}\bm{y}_{\bm{\tilde{\phi}}}=\begin{bmatrix}\frac{d}{dt}\bm{c}_{\mathcal{J}^{{\dagger}}}\\\
\bm{0}\end{bmatrix}$ (27)
and we can then determine $\bm{y}_{\bm{s}}$ by
$\bm{y}_{\bm{s}}=\begin{bmatrix}\frac{d}{dt}\bm{c}_{\mathcal{J}}\\\
\bm{0}\end{bmatrix}-M\bm{y}_{\bm{\tilde{\phi}}}$ (28)
and note that each $(\bm{y}_{\bm{s}})_{i}\geq 0$. We now rewrite
$\bm{\dot{w}}=(\bm{\dot{w}}_{\bm{\tilde{\phi}}},\bm{\dot{w}}_{\bm{s}})$ from
Eq. 21 and define
$\bm{\dot{w}}_{\bm{\tilde{\phi}}}=\bm{y}_{\bm{\tilde{\phi}}}$ and
$\bm{\dot{w}}_{\bm{s}}=\frac{d}{dt}\bm{c}-M\bm{\dot{w}}_{\bm{\tilde{\phi}}}$
(29)
and conclude that this satisfies the constraints of Eq. 21. Next, we take
$\bm{\tilde{\phi}}$ to be the unique solution to
$\tilde{M}\bm{\tilde{\phi}}=\begin{bmatrix}\bm{c}_{\mathcal{J}^{{\dagger}}}\\\
\bm{0}\end{bmatrix}$ (30)
and $\bm{s}=\bm{c}-\tilde{A}\bm{\tilde{\phi}}$.
Finally, we take
$\mathcal{I}=(\hat{\mathcal{I}}\setminus\mathcal{J}^{*})\cup\mathcal{J}^{c}$
and note that this basis set enforces exactly the same $r$ linearly
independent constraints as $\tilde{M}$333In practice, we may simply use
$\tilde{M}$ to find $\tilde{\bm{\phi}}$.
We now prove that there is some basic optimal solution to Eq. 26 with a basic
index set $\hat{\mathcal{I}}$ such that exactly $r$ slack variables are non-
basic, where $r$ is the rank of the matrix $M$.
First we note that for any basic optimal solution, if there are $r^{*}>r$
slack variables which are non-basic, then there are $r^{*}$ rows of
$B_{\hat{\mathcal{I}}}$ which are non-zero only in the columns of ${M}$.
Therefore, $B_{\hat{\mathcal{I}}}$ is not invertible. We can conclude that the
number of non-basic slack variables is at most $r$.
Next, suppose $\bm{\dot{w}}^{*}$ is a basic optimal solution with basis
$\mathcal{I}^{*}$ such that there are $r^{*}<r$ slack variables which are non-
basic.
We would like to assume that there are at least $r$ slack variables
$s_{k}^{*}$ corresponding to $r$ linearly independent constraints such that
$s_{k}^{*}=0$. Recall that $\tilde{A}$ was formed with repeated (negated)
columns in order write the problem in standard form (the non-negativity bounds
of Eq. 15 are artificial). Therefore, we can find some vector $\bm{x}$ in the
kernel of the matrix formed by the rows of $\tilde{A}$ corresponding to zero
slacks which also has $\bm{x}\cdot\bm{\gamma}=0$. We can therefore find a
vector $\bm{y}$ in the kernel of
$\begin{bmatrix}\tilde{A}_{\mathcal{J}}&I&0\\\
-I_{\mathcal{J}^{*}}&0&I\end{bmatrix}$
which has $y_{k}=0$ if $s_{k}=0$ and $y_{j}\neq 0$ if $s_{j}\neq 0$ and
$s_{j}$ corresponds to a constraint that is not a linear combination of the
constraints corresponding to the $s_{k}=0$. There is at least one such
constraint as long as the $0$ slack variables correspond to constraints with
span less than dimension $r$, and so we can take $\bm{\dot{w}}+\lambda\bm{y}$
for some $\lambda$ and so increase the number of non-zero slack variables. We
can therefore assume without loss of generality that there are at least $r$
slack variables $s_{k}^{*}$ corresponding to $r$ linearly independent
constraints such that $s_{k}^{*}=0$, as was desired.
We can finally choose some linearly independent set of $r$ constraints which
correspond to $0$ slack variables, and call the matrix whose rows are these
constraint vectors $M^{*}$. Now, because there are $r^{*}<r$ non-slack basic
variables, there is some non-slack, non-basic variable $v_{j}$ such that the
column $m_{j}^{*}$ of $M^{*}$ (and ${m}_{j}$ of ${M}$) is linearly independent
from the columns corresponding to the $r^{*}$ non-slack basic variables. We
can conclude that if
$B_{\mathcal{I}^{*}}\bm{\lambda}={m}_{j}$ (31)
then there is some $\lambda_{k}\neq 0$ where $k$ corresponds to the index of a
slack variable with $s_{k}=0$. We can remove $k$ from the basic index set and
add $j$ without changing $\bm{\dot{w}}^{*}$, and therefore preserving
optimality and feasibility. We have then increased the number of non-basic
slack variables, and we can repeat if necessary to form $\hat{\mathcal{I}}$
with exactly $r$ non-basic slack variables.
∎
|
2024-09-04T02:54:58.132395 | 2020-03-07T23:33:34 | 2003.03685 | {
"authors": "Jakob Runge",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26100",
"submitter": "Jakob Runge",
"url": "https://arxiv.org/abs/2003.03685"
} | arxiv-papers | # Discovering contemporaneous and lagged causal relations in autocorrelated
nonlinear time series datasets
Jakob Runge
German Aerospace Center
Institute of Data Science
07745 Jena, Germany
and
Technische Universität Berlin
10623 Berlin, Germany
###### Abstract
The paper introduces a novel conditional independence (CI) based method for
linear and nonlinear, lagged and contemporaneous causal discovery from
observational time series in the causally sufficient case. Existing CI-based
methods such as the PC algorithm and also common methods from other frameworks
suffer from low recall and partially inflated false positives for strong
autocorrelation which is an ubiquitous challenge in time series. The novel
method, PCMCI+, extends PCMCI [Runge et al., 2019b] to include discovery of
contemporaneous links. PCMCI+ improves the reliability of CI tests by
optimizing the choice of conditioning sets and even benefits from
autocorrelation. The method is order-independent and consistent in the oracle
case. A broad range of numerical experiments demonstrates that PCMCI+ has
higher adjacency detection power and especially more contemporaneous
orientation recall compared to other methods while better controlling false
positives. Optimized conditioning sets also lead to much shorter runtimes than
the PC algorithm. PCMCI+ can be of considerable use in many real world
application scenarios where often time resolutions are too coarse to resolve
time delays and strong autocorrelation is present.
## 1 INTRODUCTION
A number of frameworks address the problem of causal discovery from
observational data utilizing different assumptions. Next to Bayesian score-
based methods [Chickering, 2002], classical Granger causality (GC) [Granger,
1969], and the more recent restricted structural causal models (SCM) framework
[Peters et al., 2017, Spirtes and Zhang, 2016], conditional independence (CI)
based network learning algorithms [Spirtes et al., 2000] form a main pillar. A
main representative of the CI framework in the causally sufficient case (no
unobserved common drivers) is the PC algorithm [Spirtes and Glymour, 1991].
Its advantages lie, firstly, in the flexibility of utilizing a wide and
growing class of CI tests, from linear partial correlation (ParCorr) and non-
parametric residual-based approaches [Ramsey, 2014, Runge et al., 2019b] to
Kernel measures [Zhang et al., 2011], tests based on conditional mutual
information [Runge, 2018b], and neural networks [Sen et al., 2017]. Secondly,
the PC algorithm utilizes sparsity making it applicable also to large numbers
of variables while score- and SCM-based methods are more difficult to adapt to
nonlinear high-dimensional causal discovery.
Causal discovery in the time series case is partially less and partially more
challenging [Runge et al., 2019a]. Obviously, time-order greatly helps in
identifying causal directions for lagged links (causes precede effects). This
forms the basis of GC which, however, cannot deal with contemporaneous links
and suffers from the curse of dimensionality [Runge et al., 2019b]. SCM-based
methods such as LiNGAM [Hyvärinen et al., 2010] and also CI-based methods
[Runge et al., 2019b, Entner and Hoyer, 2010, Malinsky and Spirtes, 2018] have
been adapted to the time series case. In [Moneta et al., 2011] GC is augmented
by the PC algorithm. However, properties such as non-stationarity and
especially autocorrelation can make causal discovery much less reliable.
Here I show that autocorrelation, an ubiquitous property of time series (e.g.,
temperature data), is especially detrimental and propose a novel CI-based
method, PCMCI+, that extends the PCMCI method from [Runge et al., 2019b] to
also include discovery of contemporaneous links, which requires substantial
changes. PCMCI+ is based on two central ideas that deviate from the PC
algorithm and the time-series adaptations of FCI in [Entner and Hoyer, 2010,
Malinsky and Spirtes, 2018]: First, an edge removal phase is conducted
separately for lagged and contemporaneous conditioning sets and the lagged
phase uses much fewer CI tests. Secondly, and more importantly, PCMCI+
optimizes the choice of conditioning sets for the individual CI tests to make
them better calibrated under autocorrelation and increase detection power by
utilizing the momentary conditional independence idea [Runge et al., 2019b].
The paper is structured as follows. Section 2 briefly introduces the problem
and Sect. 3 describes the method and states theoretical results. Numerical
experiments in Sect. 4 show that PCMCI+ benefits from strong autocorrelation
and yields much more adjacency detection power and especially more orientation
recall for contemporaneous links while better controlling false positives at
much shorter runtimes than the PC algorithm. A Supplementary Material (SM)
contains proofs and further numerical experiments.
## 2 TIME SERIES CAUSAL DISCOVERY
### 2.1 PRELIMINARIES
We are interested in discovering time series graphs (e.g., [Runge, 2018a])
that can represent the temporal dependency structure underlying complex
dynamical systems. Consider an underlying discrete-time structural causal
process $\mathbf{X}_{t}=(X^{1}_{t},\ldots,X^{N}_{t})$ with
$\displaystyle X^{j}_{t}$
$\displaystyle=f_{j}\left(\mathcal{P}(X^{j}_{t}),\,\eta^{j}_{t}\right)$ (1)
where $f_{j}$ are arbitrary measurable functions with non-trivial dependencies
on their arguments and $\eta^{j}_{t}$ represents mutually ($i\neq j$) and
serially ($t^{\prime}\neq t$) independent dynamical noise. The nodes in a time
series graph $\mathcal{G}$ (example in Fig. 1) represent the variables
$X^{j}_{t}$ at different lag-times and the set of variables that $X^{j}_{t}$
depends on defines the causal parents
$\mathcal{P}(X^{j}_{t})\subset\mathbf{X}^{-}_{t+1}=(\mathbf{X}_{t},\mathbf{X}_{t-1},\ldots){\setminus}\\{X^{j}_{t}\\}$.
We denote _lagged parents_ by
$\mathcal{P}^{-}_{t}(X^{j}_{t})=\mathcal{P}(X^{j}_{t})\cap\mathbf{X}^{-}_{t}$.
A lagged ($\tau>0$) or contemporaneous ($\tau=0$) causal link
$X^{i}_{t-\tau}\to X^{j}_{t}$ exists if
$X^{i}_{t-\tau}\in\mathcal{P}(X^{j}_{t})$. Throughout this work the graph
$\mathcal{G}$ is assumed _acyclic_ and the causal links _stationary_ meaning
that if $X^{i}_{t-\tau}\to X^{j}_{t}$ for some $t$, then
$X^{i}_{t^{\prime}-\tau}\to X^{j}_{t^{\prime}}$ for all $t^{\prime}\neq t$.
Then we can always fix one variable at $t$ and take $\tau\geq 0$. Note that
the stationarity assumption may be relaxed. The graph is actually infinite in
time, but in practice only considered up to some maximum time lag
$\tau_{\max}$. We define the set of adjacencies $\mathcal{A}(X^{j}_{t})$ of a
variable $X^{j}_{t}$ to include all $X^{i}_{t-\tau}$ for $\tau\geq 0$ that
have a (lagged or contemporaneous) link with $X^{j}_{t}$ in $\mathcal{G}$. We
define contemporaneous adjacencies as
$\mathcal{A}_{t}(X^{j}_{t})=\mathcal{A}(X^{j}_{t})\cap\mathbf{X}_{t}$. A
sequence of $m$ contemporaneous links is called a _directed contemporaneous
path_ if for all $k\in\\{1,\ldots,m\\}$ the link $X^{i+k-1}_{t}\to
X^{i+k}_{t}$ occurs. We call $X^{i}_{t}$ a _contemporaneous ancestor_ of
$X^{j}_{t}$ if there is a directed contemporaneous path from $X^{i}_{t}$ to
$X^{j}_{t}$ and we denote the set of all contemporaneous ancestors as
$\mathcal{C}_{t}(X^{j}_{t})$ (which excludes $X^{j}_{t}$ itself). We denote
separation in the graph by $\bowtie$, see [Runge, 2018a] for further notation
details.
### 2.2 PC ALGORITHM
The PC algorithm is the most wide-spread CI-based causal discovery algorithm
for the causally sufficient case and utilizes the Markov and Faithfulness
assumptions as formally defined in Sect. S1. Adapted to time series
(analogously to the methods for the latent case in [Entner and Hoyer, 2010,
Malinsky and Spirtes, 2018]), it consists of three phases: First, a skeleton
of adjacencies is learned based on iteratively testing which pairs of
variables (at different time lags) are conditionally independent at some
significance level $\alpha_{\rm PC}$ (Alg. 2 with the PC option). For lagged
links, time-order automatically provides orientations, while for
contemporaneous links a collider phase (Alg. S2) and rule phase (Alg. S3)
determine the orientation of links. CI-based discovery algorithms can identify
the contemporaneous graph structure only up to a Markov equivalence class
represented as a completed partially directed acyclic graph (CPDAG). We denote
links for which more than one orientation occurs in the Markov equivalence
class by $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. Here we consider a
modification of PC that removes an undesired dependence on the order of
variables, called PC-stable [Colombo and Maathuis, 2014]. These modifications
also include either the _majority_ or _conservative_ [Ramsey et al., 2006]
rule for handling ambiguous triples where separating sets are inconsistent,
and conflicting links where different triples in the collider or orientation
phase lead to conflicting link orientations. With the _conservative_ rule the
PC algorithm is consistent already under the weaker Adjacency Faithfulness
condition [Ramsey et al., 2006]. Another approach for the time series case
(considered in the numerical experiments) is to combine vector-autoregressive
modeling to identify lagged links with the PC algorithm for the
contemporaneous causal structure [Moneta et al., 2011].
### 2.3 AUTOCORRELATION
Figure 1: The curse and blessing of autocorrelation. Linear example of model
(3) with ground truth links shown for the PCMCI+ case (right panel). All
autodependency coefficients are $0.95$ (except $0.475$ for $X^{5,6}$) and all
cross-coupling coefficients are $0.4$ ($\pm$ indicated by red/blue links). The
graphs show true and false link detection rates as the link width (if $>$
0.06) for true (color indicating ParCorr) and incorrect links (grey) for the
PC algorithm, Alg. 1, and the variants PCMCI+ and PCMCI${}^{+}_{0}$ as
explained in the text (detection rates based on $500$ realizations run at
$\alpha_{\rm PC}=0.01$ for $T=500$).
To illustrate the challenge of autocorrelation, in Fig. 1 we consider a linear
example with lagged and contemporaneous ground truth links shown for the
PCMCI+ case (right panel). The PC algorithm (Alg. 2 with ParCorr CI test)
starts by testing all unconditional independencies ($p=0$). Here the coupled
pairs $(X^{5},X^{6})$ as well as $(X^{7},X^{8})$ are independent of the other
variables and removed from each others adjacency sets, which shows how PC
exploits sparsity and reduces the estimation dimension compared to fitting a
full model on the whole past as in the GC framework. Due to the strong
autocorrelation the remaining variables, on the other hand, are almost all
adjacent to each other at multiple time lags in this iteration. In the next
iteration, CI for all remaining links is tested conditional on all one-
dimensional ($p=1$) conditioning sets. Here the PC algorithm removes the true
lagged link $X^{1}_{t-1}\to X^{0}_{t}$ (black dots) due to the incorrect CI
result $X^{1}_{t-1}\perp\\!\\!\\!\perp X^{0}_{t}|X^{1}_{t-2}$ (condition
marked by blue box). Later this then leads to the false positive
$X^{1}_{t-2}\to X^{0}_{t}$ (grey link) since $X^{1}_{t-1}$ is not conditioned
on. In a similar way the true link $X^{1}_{t-2}\to X^{3}_{t}$ is missed
leading to the false positive $X^{0}_{t-1}\to X^{3}_{t}$. Further, the true
contemporaneous link $X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ (similarly
$X^{3}_{t}{\circ\\!{\\--}\\!\circ}X^{4}_{t}$) is removed when conditioning on
$\mathcal{S}=(X^{4}_{t-1},X^{3}_{t-1})$ (blue boxes), which leads to the false
positive autodependencies at lag $2$ for $X^{2}_{t},X^{4}_{t}$, while the
false autodependency $X^{3}_{t-2}\to X^{3}_{t}$ is due to missing
$X^{1}_{t-2}\to X^{3}_{t}$. This illustrates the pattern of a cascade of false
negative errors (missing links) leading to false positives in later stages of
the PC algorithm.
What determines the removal of a true link in the finite sample case?
Detection power depends on sample size, the significance level $\alpha_{\rm
PC}$, the CI test dimension ($p+2$), and effect size, e.g., the absolute
ParCorr (population) value, here denoted
$I(X^{i}_{t-\tau};X^{j}_{t}|\mathcal{S})$ for some conditioning set
$\mathcal{S}$. Within each $p$-iteration the sample size, $\alpha_{\rm PC}$,
and the dimension are the same and a link will be removed if
$I(X^{i}_{t-\tau};X^{j}_{t}|\mathcal{S})$ falls below the $\alpha_{\rm
PC}$-threshold for _any_ considered $\mathcal{S}$. Hence, the overall minimum
effect size $\min_{\mathcal{S}}[I(X^{i}_{t-\tau};X^{j}_{t}|\mathcal{S})]$
determines whether a link is removed. The PC algorithm will iterate through
_all_ subsets of adjacencies such that this minimum can become very small. Low
effect size can be understood as a low (causal) signal-to-noise ratio: Here
$I(X^{1}_{t-1};X^{0}_{t}|X^{1}_{t-2})$ is small since the signal $X^{1}_{t-1}$
is reduced by conditioning on its autodependency $X^{1}_{t-2}$ and the ‘noise’
in $X^{0}_{t}$ is large due to its strong autocorrelation.
But autocorrelation can also be a blessing. The contemporaneously coupled pair
$(X^{7},X^{8})$ illustrates a case where autocorrelation helps to identify the
orientation of the link. Without autocorrelation the output of PC would be an
unoriented link to indicate the Markov equivalence class. On the other hand,
the detection rate here is rather weak since, as above, the signal (link from
$X^{8}_{t}$) is small compared to the noise (autocorrelation in $X^{7}$).
This illustrates the curse and blessing of autocorrelation. In summary, the PC
algorithm often results in false negatives (low recall) and these then lead to
false positives. Another reason for false positives are ill-calibrated tests:
To correctly model the null distribution, each individual CI test would need
to account for autocorrelation, which is difficult in a complex multivariate
and potentially nonlinear setting [Runge, 2018a]. In the experiments we will
see that the PC algorithm features inflated false positives.
As a side comment, the pair $(X^{5},X^{6})$ depicts a feedback cycle. These
often occur in real data and the example shows that time series graphs allow
to resolve time-delayed feedbacks while an aggregated _summary graph_ would
contain a cyclic dependency and summary graph-based methods assuming acyclic
graphs would not work. The orientation of the contemporaneous link
$X^{6}_{t}\to X^{5}_{t}$ is achieved via rule R1 in the orientation phase of
PC (Alg. S3).
## 3 PCMCI+
Figure 2: Schematic of PCMCI+. Note that for ease of visualization repeating
edges due to stationarity are only partially shown.
### 3.1 ALGORITHM
The goal of PCMCI+ is to optimize the choice of conditioning sets in CI tests
in order to increase detection power and at the same time maintain well-
calibrated tests. The approach is based on two central ideas, (1) separating
the skeleton edge removal phase into a lagged and contemporaneous conditioning
phase with much fewer CI tests and (2) utilizing the momentary conditional
independence (MCI) test [Runge et al., 2019b] idea in the contemporaneous
conditioning phase. Below, I explain the reasoning behind. Figure 2
illustrates the steps.
First, the goal of PC’s skeleton phase is to remove all those adjacencies that
are due to indirect paths and common causes by conditioning on subsets
$\mathcal{S}$ of the variables’ neighboring adjacencies in each iteration.
Consider a variable $X^{j}_{t}$. If we test lagged adjacencies from nodes
$X^{i}_{t-\tau}\in\mathbf{X}^{-}_{t}$ conditional on the whole past, i.e.,
$\mathcal{S}=\mathbf{X}^{-}_{t}\setminus\\{X^{i}_{t-\tau}\\}$, the only
indirect adjacencies remaining are due to paths through contemporaneous
parents of $X^{j}_{t}$. This is in contrast to conditioning sets on
contemporaneous adjacencies which can also open up paths $X^{j}_{t}\to
X^{k}_{t}\leftarrow X^{i}_{t-\tau}$ if $X^{k}_{t}$ is conditioned on. One
reason why the PC algorithm tests _all_ combinations of subsets $\mathcal{S}$
is to avoid opening up such collider paths. Therefore, one approach would be
to start by $\mathcal{S}=\mathbf{X}^{-}_{t}\setminus\\{X^{i}_{t-\tau}\\}$ and
then iterate through contemporaneous conditions. A similar idea lies behind
the combination of GC and the PC algorithm in [Moneta et al., 2011]. However,
conditioning on large-dimensional conditioning sets strongly affects detection
power [Runge et al., 2019b]. To avoid this, the lagged conditioning phase of
PCMCI+ (Alg. 1, see Fig. 2 left panels) tests all pairs
$(X^{i}_{t-\tau},X^{j}_{t})$ for $\tau>0$ conditional on only the _strongest_
$p$ adjacencies of $X^{j}_{t}$ in each $p$-iteration without going through all
$p$-dimensional subsets of adjacencies. This choice $(i)$ improves the causal
signal-to-noise ratio and recall since for a given test
$X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}$ the ‘noise’ in
$X^{j}_{t}$ due to other lagged adjacencies is conditioned out, $(ii)$ leads
to fewer CI tests further improving recall, and $(iii)$ speeds up the skeleton
phase. We denote the lagged adjacency set resulting from Alg. 1 as
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$. Lemma 1 in Sect. 3.2 states that
the only remaining indirect adjacencies in
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ are then due to paths passing
through contemporaneous parents of $X^{j}_{t}$. In the schematic in Fig. 2
this is the link $Y_{t-1}\to X_{t}$.
Secondly, in Alg. 2 (Fig. 2 center panels) the graph $\mathcal{G}$ is
initialized with all contemporaneous adjacencies plus all lagged adjacencies
from $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$. Algorithm
2 tests all (unordered lagged and ordered contemporaneous) adjacent pairs
$(X^{i}_{t-\tau},X^{j}_{t})$ and iterates through contemporaneous conditions
$\mathcal{S}\subseteq\mathcal{A}_{t}(X^{j}_{t})$ with the MCI test
$\displaystyle X^{i}_{t-\tau}$
$\displaystyle{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau}).$
(2)
In the schematic in Fig. 2 the condition on $\mathcal{S}=Y_{t}$, as part of
the full conditioning set
$\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$,
removes the link $X_{t}{\circ\\!{\\--}\\!\circ}Z_{t}$. The condition on
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ blocks paths through lagged parents
and the advantage of the additional conditioning on
$\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ is discussed in the
following. We denote the variant without the condition on
$\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ as PCMCI${}^{+}_{0}$.
Both versions are followed by the collider orientation phase (Alg. S2) and
rule orientation phase (Alg. S3) which are deferred to the SM since they are
equivalent to the PC algorithm with the modification that the additional CI
tests in the collider phase for the conservative or majority rule are also
based on the test (2) (Fig. 2 right panel).
We now discuss PCMCI${}^{+}_{0}$ and PCMCI+ on the example in Fig. 1.
Algorithm 1 tests $X^{1}_{t-1}\to X^{0}_{t}$ conditional on
$\mathcal{S}=\\{X^{0}_{t-1}\\}$ for $p=1$ and
$\mathcal{S}=\\{X^{0}_{t-1},X^{1}_{t-2}\\}$ for $p=2$ as the two strongest
adjacencies (as determined by the test statistic value, see pseudo-code). In
both of these tests the effect size $I$ (causal signal-to-noise ratio) is much
larger than for the condition on $\mathcal{S}=\\{X^{1}_{t-2}\\}$ which lead to
the removal of $X^{1}_{t-1}\to X^{0}_{t}$ in the PC algorithm. In Sect. 3.2 we
elaborate more rigorously on effect size. In the example
$\widehat{\mathcal{B}}^{-}_{t}(X^{2}_{t})$ is indicated as blue boxes in the
second panel and contains lagged parents as well as adjacencies due to paths
passing through contemporaneous parents of $X^{2}_{t}$. One false positive,
likely due to an ill-calibrated test caused by autocorrelation, is marked by a
star.
Based on these lagged adjacencies, Alg. 2 with the PCMCI${}^{+}_{0}$ option
then recovers all lagged links (3rd panel), but it still the misses
contemporaneous adjacencies $X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ and
$X^{3}_{t}{\circ\\!{\\--}\\!\circ}X^{4}_{t}$ and we also see strong lagged
false positives from $X^{3}$ to $X^{2}$ and $X^{4}$. What happened here? The
problem are now tests on contemporaneous links: The CI test for
PCMCI${}^{+}_{0}$ in the $p=0$ loop, like the original PC algorithm, will test
_ordered_ contemporaneous pairs. Hence, first
$X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ conditional on
$\widehat{\mathcal{B}}^{-}_{t}(X^{3}_{t})$ and, if the link is not removed,
$X^{3}_{t}{\circ\\!{\\--}\\!\circ}X^{2}_{t}$ conditional on
$\widehat{\mathcal{B}}^{-}_{t}(X^{2}_{t})$. Here
$X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ is removed conditional on
$\widehat{\mathcal{B}}^{-}_{t}(X^{3}_{t})$ (indicated by blue boxes in the
panel) because
$I(X^{2}_{t};X^{3}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{3}_{t}))$ falls below
the significance threshold.
The second central idea of PCMCI+ is to improve the effect size of CI tests
for contemporaneous links by conditioning on _both_ lagged adjacencies
$\widehat{\mathcal{B}}^{-}_{t}$ in the CI test (2) (see blue and red boxes in
Fig. 1 right panel). At least for the initial phase $p=0$ one can prove that
for non-empty $\widehat{\mathcal{B}}^{-}_{t}$ the effect size of the PCMCI+ CI
test is always strictly larger than that of the PCMCI${}^{+}_{0}$ test (Thm.
4). I conjecture that this similarly holds for PCMCI+ vs. the PC algorithm.
Higher effect size leads to higher recall and PCMCI+ now recovers all lagged
as well as contemporaneous links and also correctly removes the lagged false
positives that PCMCI${}^{+}_{0}$ obtains. Also the contemporaneous coupled
pair $(X^{7},X^{8})$ is now much better detected since the MCI effect size
$I(X^{7}_{t};X^{8}_{t}|X^{7}_{t-1})$ is larger than $I(X^{7}_{t};X^{8}_{t})$,
one of the two PCMCI${}^{+}_{0}$ and PC algorithm effect sizes tested here.
Another advantage, discussed in [Runge et al., 2019b] is that PCMCI+ CI tests
are better calibrated, in contrast to PCMCI${}^{+}_{0}$ and PC algorithm
tests, since the condition on both parents removes autocorrelation effects.
Note that for lagged links the effect size of PCMCI+ is generally smaller than
that of PCMCI${}^{+}_{0}$ since the extra condition on
$\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ can only reduce
effect size (see [Runge et al., 2012]). This is the cost of avoiding inflated
false positives.
In summary, the central PCMCI+ idea is to increase effect size in individual
CI tests to achieve higher detection power and at the same time maintain well-
controlled false positives also for high autocorrelation. Correct adjacency
information then leads to better orientation recall in Alg. S2, S3. The other
advantage of PCMCI+ compared to the PC algorithm is a much faster and, as
numerical examples show, also much less variable runtime.
The full algorithm is detailed in pseudo-code Algorithms 1,2,S2,S3 with
differences to PC and PCMCI${}^{+}_{0}$ indicated. Note that pairs
$(X^{i}_{t-\tau},X^{j}_{t})$ in lines 5 and 6 of Alg. 2 are ordered for
$\tau=0$ and unordered for $\tau>0$. One can construct (rather conservative)
$p$-values for the skeleton adjacencies $(X^{i}_{t-\tau},X^{j}_{t})$ by taking
the maximum $p$-value over all CI tests conducted in Alg. 2. A link strength
can be defined corresponding to the test statistic value of the maximum
$p$-value. Based on the PC stable variant, PCMCI+ is fully order-independent.
Here shown is the majority-rule implementation of the collider phase, the
version without handling ambiguous triples and for the conservative rule are
detailed in Alg. S2. Note that the tests in the collider phase also use the CI
tests (2).
Like other CI-based methods, PCMCI+ has the free parameters $\alpha_{\rm PC}$,
$\tau_{\max}$, and the choice of the CI test. $\alpha_{\rm PC}$ can be chosen
based on cross-validation or an information criterion (implemented in
tigramite). $\tau_{\max}$ should be larger or equal to the maximum true time
lag of any parent and can in practice also be chosen based on model selection.
However, the numerical experiments indicate that, in contrast to GC, a too
large $\tau_{\max}$ does not degrade performance much and $\tau_{\max}$ can
also be chosen based on the lagged dependence functions, see [Runge et al.,
2019b]. PCMCI+ can flexibly be combined with different CI tests for nonlinear
causal discovery, and for different variable types (discrete or continuous,
univariate or multivariate).
The computational complexity of PCMCI+ strongly depends on the network
structure. The sparser the causal dependencies, the faster the convergence.
Compared to the original PC algorithm with worst-case exponential complexity,
the complexity is much reduced since Alg. 1 only has polynomial complexity
[Runge et al., 2019b] and Alg. 2 only iterates through contemporaneous
conditioning sets, hence the worst-case exponential complexity only applies to
$N$ and not to $N\tau_{\max}$.
Algorithm 1 (PCMCI+ / PCMCI${}^{+}_{0}$ lagged skeleton phase)
1:Time series dataset $\mathbf{X}=(X^{1},\,\ldots,X^{N})$, max. time lag
$\tau_{\max}$, significance threshold $\alpha_{\rm PC}$, CI test ${\rm
CI}(X,\,Y,\,\mathbf{Z})$ returning $p$-value and test statistic value $I$
2:for all $X^{j}_{t}$ in $\mathbf{X}_{t}$ do
3: Initialize
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){=}\mathbf{X}^{-}_{t}{=}(\mathbf{X}_{t-1},\dots,\mathbf{X}_{t-\tau_{\max}})$
and
$I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\infty~{}~{}\forall~{}X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$
4: Let $p=0$
5: while any $X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$
satisfies
$|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq
p$ do
6: for all $X^{i}_{t-\tau}$ in $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$
satisfying
$|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq
p$ do
7: $\mathcal{S}=$ first $p$ variables in
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\}$
8: $(\text{$p$-value},\,I)\leftarrow$
CI($X^{i}_{t-\tau},\,X^{j}_{t},\,\mathcal{S}$)
9:
$I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\min(|I|,I^{\min}(X^{i}_{t-\tau},X^{j}_{t}))$
10: if $p$-value $>\alpha_{\rm PC}$ then mark $X^{i}_{t-\tau}$ for removal
11: Remove non-significant entries and sort
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ by
$I^{\min}(X^{i}_{t-\tau},X^{j}_{t})$ from largest to smallest
12: Let $p=p+1$
13:return $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in
$\mathbf{X}_{t}$
Algorithm 2 (PCMCI+ / PCMCI${}^{+}_{0}$ contemporaneous skeleton phase / PC
full skeleton phase)
1:Time series dataset $\mathbf{X}=(X^{1},\,\ldots,X^{N})$, max. time lag
$\tau_{\max}$, significance threshold $\alpha_{\rm PC}$, ${\rm
CI}(X,\,Y,\,\mathbf{Z})$, PCMCI+ / PCMCI${}^{+}_{0}$:
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in
$\mathbf{X}_{t}$
2:PCMCI+ / PCMCI${}^{+}_{0}$: Form time series graph $\mathcal{G}$ with lagged
links from $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in
$\mathbf{X}_{t}$ and fully connect all contemporaneous variables, i.e., add
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ for all $X^{i}_{t}\neq
X^{j}_{t}\in\mathbf{X}_{t}$
3:PC: Form fully connected time series graph $\mathcal{G}$ with lagged and
contemporaneous links
4:PCMCI+ / PCMCI${}^{+}_{0}$: Initialize contemporaneous adjacencies
$\widehat{\mathcal{A}}(X^{j}_{t}):=\widehat{\mathcal{A}}_{t}(X^{j}_{t})=\\{X^{i}_{t}{\neq}X^{j}_{t}\in\mathbf{X}_{t}:X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}~{}\text{in
$\mathcal{G}$}\\}$
5:PC: Initialize full adjacencies $\widehat{\mathcal{A}}(X^{j}_{t})$ for all
(lagged and contemporaneous) links in $\mathcal{G}$
6:Initialize $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\infty$ for all links in
$\mathcal{G}$
7:Let $p=0$
8:while any adjacent pairs $(X^{i}_{t-\tau},X^{j}_{t})$ for $\tau\geq 0$ in
$\mathcal{G}$ satisfy
$|\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq p$ do
9: Select new adjacent pair $(X^{i}_{t-\tau},X^{j}_{t})$ for $\tau\geq 0$
satisfying
$|\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq p$
10: while $(X^{i}_{t-\tau},X^{j}_{t})$ are adjacent in $\mathcal{G}$ and not
all
$\mathcal{S}\subseteq\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$
with $|\mathcal{S}|=p$ have been considered do
11: Choose new
$\mathcal{S}{\subseteq}\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$
with $|\mathcal{S}|{=}p$
12: PCMCI+: Set
$\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau}))$
13: PCMCI${}^{+}_{0}$: Set
$\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\})$
14: PC: Set $\mathbf{Z}{=}\mathcal{S}$
15: $(\text{$p$-value},\,I)\leftarrow$
CI($X^{i}_{t{-}\tau},X^{j}_{t},\mathbf{Z})$
16:
$I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\min(|I|,I^{\min}(X^{i}_{t-\tau},X^{j}_{t}))$
17: if $p$-value $>\alpha_{\rm PC}$ then
18: Delete link $X^{i}_{t-\tau}\to X^{j}_{t}$ for $\tau>0$ (or
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ for $\tau=0$) from $\mathcal{G}$
19: Store (unordered) ${\rm sepset}(X^{i}_{t-\tau},X^{j}_{t})=\mathcal{S}$
20: Let $p=p+1$
21: Re-compute $\widehat{\mathcal{A}}(X^{j}_{t})$ from $\mathcal{G}$ and sort
by $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})$ from largest to smallest
22:return $\mathcal{G}$, sepset
### 3.2 THEORETICAL RESULTS
This section states asymptotic consistency, finite sample order-independence,
and further results regarding effect size and false positive control. The
consistency of network learning algorithms is separated into _soundness_ ,
i.e., the returned graph has correct adjacencies, and _completeness_ , i.e.,
the returned graph is also maximally informative (links are oriented as much
as possible). We start with the following assumptions.
###### Assumptions 1 (Asymptotic case).
Throughout this paper we assume Causal Sufficiency, the Causal Markov
Condition, the Adjacency Faithfulness Conditions, and consistent CI tests
(oracle). In the present time series context we also assume stationarity and
time-order and that the maximum time lag
$\tau_{\max}\geq\tau^{\mathcal{P}}_{\max}$, where $\tau^{\mathcal{P}}_{\max}$
is the maximum time lag of any parent in the SCM (1). Furthermore, we rule out
_selection variables_ and _measurement error_.
Definitions of these assumptions, adapted from [Spirtes et al., 2000] to the
time series context, are in Sect. S1 and all proofs are in Sect. S2. We start
with the following lemma.
###### Lemma 1.
Under Assumptions 1 Alg. 1 returns a set that always contains the parents of
$X^{j}_{t}$ and, _at most_ , the lagged parents of all contemporaneous
ancestors of $X^{j}_{t}$, i.e.,
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})=\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$.
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ contains _all_ lagged parents of
all contemporaneous ancestors if the weaker Adjacency Faithfulness assumption
is replaced by standard Faithfulness.
This establishes that the conditions
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ estimated in the first phase of
PCMCI+ will suffice to block all lagged confounding paths that do not go
through contemporaneous links. This enables to prove the soundness of Alg. 2,
even though Alg. 2 is a variant of the PC algorithm that only iterates through
contemporaneous conditioning sets.
###### Theorem 1 (Soundness of PCMCI+).
Algorithm 2 returns the correct adjacencies under Assumptions 1, i.e.,
$\widehat{\mathcal{G}^{*}}=\mathcal{G}^{*}$, where the $\mathcal{G}^{*}$
denotes the skeleton of the time series graph.
To prove the completeness of PCMCI+, we start with the following observation.
###### Lemma 2.
Due to time-order and the stationarity assumption, the considered triples in
the collider phase (Alg. S2) and rule orientation phase (Alg. S3) can be
restricted as follows: In the collider orientation phase only unshielded
triples $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for
$\tau>0$) or
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$
(for $\tau=0$) in $\mathcal{G}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not
adjacent are relevant. For orientation rule R1 triples $X^{i}_{t-\tau}\to
X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ where $(X^{i}_{t-\tau},X^{j}_{t})$
are not adjacent, for orientation rule R2 triples $X^{i}_{t}\to X^{k}_{t}\to
X^{j}_{t}$ with $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$, and for
orientation rule R3 pairs of triples
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}\to X^{j}_{t}$ and
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{l}_{t}\to X^{j}_{t}$ where
$(X^{k}_{t},X^{l}_{t})$ are not adjacent and
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ are relevant. These restrictions
imply that only contemporaneous parts of separating sets are relevant for the
collider phase.
###### Theorem 2 (PCMCI+ is complete).
PCMCI+ (Algorithms 1,2,S2,S3) when used with the conservative rule for
orienting colliders in Alg. S2 returns the correct CPDAG under Assumptions 1.
Under standard Faithfulness also PCMCI+ when used with the majority rule or
the standard orientation rule is complete.
Also the proof of order-independence follows straightforwardly from the proof
in [Colombo and Maathuis, 2014]. Of course, order independence does not apply
to time-order.
###### Theorem 3 (Order independence).
Under Assumptions 1 PCMCI+ with the conservative or majority rule in Alg. S2
is independent of the order of variables $(X^{1},\ldots,X^{N})$.
Next, we consider effect size. The toy example showed that a major problem of
PCMCI${}^{+}_{0}$ (and also PC) is lack of detection power for contemporaneous
links. A main factor of statistical detection power is effect size, i.e., the
population value of the test statistic considered (e.g., absolute partial
correlation). In the following, I will base my argument in an information-
theoretic framework and consider the conditional mutual information as a
general test statistic, denoted $I$. In Alg. 2 PCMCI${}^{+}_{0}$ will test a
contemporaneous dependency $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ first
with the test statistic
$I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}))$, and, if
that test was positive, secondly with
$I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})))$. If either
of these tests finds (conditional) independence, the adjacency is removed.
Therefore, the minimum test statistic value determines the relevant effect
size. On the other hand, PCMCI+ treats both cases symmetrically since the test
statistic is always
$I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t}))$.
###### Theorem 4 (Effect size of MCI tests for $p=0$).
Under Assumptions 1 the PCMCI+ oracle case CI tests in Alg. 2 for $p=0$ for
contemporaneous true links $X^{i}_{t}\to X^{j}_{t}\in\mathcal{G}$ have an
effect size that is always greater than that of the PCMCI${}^{+}_{0}$ CI
tests, i.e.,
$I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t}))>\min(I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})),\,I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})))$
if both $X^{i}_{t}$ and $X^{j}_{t}$ have parents that are not shared with the
other.
I conjecture that this result holds similarly for $p>0$ and also that PCMCI+
has greater effect sizes than the PC algorithm since the latter iterates over
_all_ subsets of adjacencies and, hence, the minimum is taken generally over
an even larger set leading to even smaller effect sizes. For lagged links the
effect size of the PCMCI+ tests is always smaller (or equal) than that of the
PCMCI${}^{+}_{0}$ tests (see [Runge et al., 2012]).
Last, we discuss false positive control. While the effect size result regards
detection power, in the following I give a mathematical intuition why the MCI
tests are better calibrated than the PC algorithm CI tests and control false
positives below the expected significance level. Lemma 1 implies that even
though Alg. 1 does not aim to estimate the contemporaneous parents, it still
yields a set of conditions that shields $X^{j}_{t}$ from the ‘infinite’ past
$\mathbf{X}^{-}_{t}$, either by blocking the parents of $X^{j}_{t}$ or by
blocking indirect contemporaneous paths through contemporaneous ancestors of
$X^{j}_{t}$. Blocking paths from the infinite past, I conjecture, is key to
achieve well-calibrated CI tests in Alg. 2. The authors in [Runge et al.,
2019b] showed that under certain model assumptions the MCI tests reduce to CI
tests among the noise terms $\eta$ from model (1) which are assumed to be
i.i.d. and help to achieve well-calibrated CI tests. In the numerical
experiments below we can see that the PC algorithm has inflated false positive
for high autocorrelation, while PCMCI+ well controls false positives, but a
formal proof of correct false positive control for this challenging nonlinear,
high-dimensional setting is beyond the scope of this paper.
## 4 NUMERICAL EXPERIMENTS
We consider a number of typical challenges [Runge et al., 2019a],
contemporaneous and time lagged causal dependencies, strong autocorrelation,
large number of variables and considered time lags, different noise
distributions and nonlinearity, in the following additive variant of model
(1):
$\displaystyle X_{t}^{j}$
$\displaystyle=a_{j}X^{j}_{t-1}+\textstyle{\sum_{i}}c_{i}f_{i}(X^{i}_{t-\tau_{i}})+\eta^{j}_{t}$
(3)
for $j\in\\{1,\ldots,N\\}$. Autocorrelations $a_{j}$ are uniformly drawn from
$[\max(0,a-0.3),\,a]$ for $a$ as indicated in Fig. 3 and $\eta^{j}$ is
_i.i.d._ and follows a zero-mean Gaussian $\mathcal{N}$ or Weibull
$\mathcal{W}$ (scale parameter $2$) distribution (depending on setup) with
standard deviation drawn from $[0.5,\,2]$. In addition to autodependency
links, for each model $L=\lfloor 1.5\cdot N\rfloor$ (except for $N=2$ with
$L=1$) cross-links are chosen whose functional dependencies are linear or
$f_{i}(x)=f^{(2)}(x)=(1+5xe^{-x^{2}/20})x$ (depending on setup), with
$f^{(2)}$ designed to yield more stationary dynamics. Coefficients $c_{i}$ are
drawn uniformly from $\pm[0.1,0.5]$. 30% of the links are contemporaneous
($\tau_{i}=0$) and the remaining $\tau_{i}$ are drawn from $[1,\,5]$. Only
stationary models are considered. We have an average cross-in-degree of
$d=1.5$ for all network sizes (plus an auto-dependency) implying that models
become sparser for larger $N$. We consider several model setups: linear
Gaussian, linear mixed noise (among the $N$ variables: 50% Gaussian, 50%
Weibull), and nonlinear mixed noise (50% linear, 50% $f^{(2)}(x)$; 66%
Gaussian, 34% Weibull).
For the linear model setups we consider the PC algorithm and PCMCI+ in the
majority-rule variant with ParCorr and compare these with GCresPC [Moneta et
al., 2011], a combination of GC with PC applied to residuals, and a
autoregressive model version of LiNGAM [Hyvärinen et al., 2010], a
representative of the SCM framework (implementation details in Sect. S4). For
the LiNGAM implementation I could not find a way to set a significance level
and used the LASSO option which prunes ‘non-active’ links to zero. Both
GCresPC and LiNGAM assume linear dependencies and LiNGAM also non-Gaussianity.
For the nonlinear setup the PC algorithm and PCMCI+ are implemented with the
GPDC test [Runge et al., 2019b] that is based on Gaussian process regression
and a distance correlation test on the residuals, which is suitable for a
large class of nonlinear dependencies with additive noise.
Performance is evaluated as follows: True (TPR) and false positive rates (FPR,
shown to evaluate false positive control, not applicable to LiNGAM) for
adjacencies are distinguished between lagged cross-links ($i\neq j$),
contemporaneous, and autodependency links. Due to time order, lagged links
(and autodependencies) are automatically oriented. Contemporaneous orientation
precision is measured as the fraction of correctly oriented links
(${\circ\\!{\\--}\\!\circ}$ or $\to$) among all estimated adjacencies, and
recall as the fraction of correct orientations among all true contemporaneous
links. Further shown is the fraction of conflicting links among all detected
contemporaneous adjacencies (not applicable to LiNGAM). All metrics (and their
std. errors) are computed across all estimated graphs from $500$ realizations
of model (3) at time series length $T$. The average runtimes were evaluated on
Intel Xeon Platinum 8260. In Fig. 3 results for the linear Gaussian setup with
default model parameters $N=5,\,T=500,\,a=0.95$ and method parameters
$\tau_{\max}=5$ and $\alpha=0.01$ (not applicable to LiNGAM) are shown. Each
of the four panels shows results for varying one of $a,\,N,\,T,\,\tau_{\max}$.
The insets show ANOVA statistics $r\pm\bar{\Delta}r$ [per unit], where $r$ is
the performance metric at the leftmost parameter on the $x$-axis
($a,\,N,\,T,\,\tau_{\max}$, respectively) and $\bar{\Delta}r$ denotes the
average change per parameter unit. In the adjacency subplots the statistics
refer to lagged links.
Figure 3: Numerical experiments with linear Gaussian setup for varying (A)
autocorrelation strength $a$ (B) number of variables $N$ (C) sample size $T$
and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters
indicated in the top right. Errorbars show std. errors or the 90% range (for
runtime). The insets show ANOVA statistics.
Figure 3A demonstrates that the TPR of PCMCI+ and GCresPC for contemporaneous
links is stable even under high autocorrelation while PC and LiNGAM show
strong declines. Since LiNGAM has no $\alpha_{\rm PC}$ for FPR-control we
focus on its relative changes rather than absolute performance. Lagged TPR
decreases strongly for PC while the other methods are more robust. FPR is
well-controlled for PCMCI+ while PC and slightly also GCresPC show inflated
lagged FPR for high autocorrelation. LiNGAM features a strong increase of
lagged FPR. These adjacency results translate into higher contemporaneous
orientation recall for PCMCI+ which increases with autocorrelation, while it
decreases for all other methods. GCresPC has steady low recall since it does
not use lagged links in the orientation phase. Except for GCresPC, all methods
have increasing precision with PCMCI+ and PC outperforming LiNGAM. PCMCI+
shows almost no conflicts while PC’s conflicts increase with autocorrelation
until low power reduces them again. Finally, runtimes are almost constant for
GCresPC and LiNGAM, while they increase for PCMCI+ and much stronger for PC.
Figure 3B shows that PCMCI+ and GCresPC have the highest TPR for increasing
number of variables $N$, especially for contemporaneous links. FPR is well
controlled only for PCMCI+ while PC has false positives for small $N$ where
model connectivity is denser and false negatives are more likely leading to
false positives. For high $N$ PC has false positives only regarding
autodependencies while inflated FPR appears for GCresPC. PCMCI+ has more than
twice as much contemporaneous recall compared to the other methods and is
almost not affected by higher $N$. Orientation precision is decreasing for all
methods (except PC) with a higher decrease for PCMCI+. Runtime is increasing
at a much smaller rate for PCMCI+ compared to PC, which also has a very high
runtime variability across the different model realizations. LiNGAM and
especially GCresPC are fastest.
PCMCI+, GCresPC, and LiNGAM benefit similarly and PC less so for increasing
sample size regarding TPR (Fig. 3C). FPR is still not controlled for PC for
large sample sizes, lagged FPR increases for GCresPC. PCMCI+ shows the highest
increases in contemporaneous recall and precision. Runtime increases are
moderate compared to PC, conflicts decrease.
Last, Fig. 3D shows that all methods are relatively robust to large maximum
time lags $\tau_{\max}$ (beyond the true max. time lag $5$) for the considered
sample size $T=500$. Contemporaneous FPR and runtime increase for PC.
In the SM further results are shown. For too large $N\tau_{\max}$ (relative to
$T$) GCresPC and LiNGAM (despite LASSO-regularization) sharply drop in
performance. For the linear mixed noise setup (Fig. S2) results are almost
unchanged for all methods except for LiNGAM for which recall and precision
rise, as expected. Recall is then higher than PCMCI+ for low autocorrelation,
but still much lower for high autocorrelation and large $N$ or $\tau_{\max}$,
at similar precision.
In the nonlinear mixed noise setup (Fig. S3), the difference between PC and
PCMCI+ is similar. We observe slight FPR inflation for high autocorrelation.
GPDC seems to not work well in high-dimensional, highly autocorrelated
settings. Runtime for GPDC compared to ParCorr is orders of magnitude longer,
especially for PC. Further figures in the SM show many combinations of
$a,\,N,\,T,\,\tau_{\max}$ and $\alpha_{\rm PC}$ for the model setups and
demonstrate that the above findings are robust.
## 5 CONCLUSIONS
PCMCI+ improves the reliability of CI tests by optimizing the choice of
conditioning sets and yields much higher recall, well-controlled false
positives, and faster runtime than the original PC algorithm for highly
autocorrelated time series, while maintaining similar performance for low
autocorrelation. The algorithm well exploits sparsity in high-dimensional
settings and can flexibly be combined with different CI tests for nonlinear
causal discovery, and for different variable types (discrete or continuous,
univariate or multivariate). Autocorrelation is actually key to increase
contemporaneous orientation recall since it creates triples $X^{i}_{t-1}\to
X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ that can often be oriented while
an isolated link $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ stays undirected
in the Markov equivalence class, a drawback of CI-based methods. If the data
is at least non-Gaussian, a SCM method like LiNGAM can exploit this property
and recover directionality in such cases. Still, we saw that LiNGAM suffers
from large autocorrelation. PCMCI+ is available as part of the _tigramite_
Python package at https://github.com/jakobrunge/tigramite. A next step will be
to extend the present ideas to an algorithm accounting for latent confounders
and to explore combinations between SCM-based methods and PCMCI+. The
numerical results will be contributed to the causality benchmark platform
`www.causeme.net` [Runge et al., 2019a] to facilitate a further expanded
method evaluation.
#### Acknowledgments
DKRZ provided computational resources (grant no. 1083). I thank Andreas
Gerhardus for helpful comments.
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## Appendix S1 Definitions
The following definitions are adaptations of the standard assumptions of
causal discovery to the time series case. Here we consider the causally
sufficient case and assume that all variables
$\mathbf{X}=(X^{1},\ldots,X^{N})$ of the underlying SCM (1) are observed.
Additionally, we assume that the maximum PCMCI+ time lag
$\tau_{\max}\geq\tau^{\mathcal{P}}_{\max}$, where $\tau^{\mathcal{P}}_{\max}$
is the maximum time lag of any parent in the SCM (1).
###### Definition S1 (Causal Markov Condition).
The joint distribution of a process $\mathbf{X}$ whose causal structure can be
represented in a time series graph $\mathcal{G}$ fulfills the Causal Markov
Condition iff for all $X^{j}_{t}\in\mathbf{X}_{t}$ every non-descendent of
$X^{j}_{t}$ in $\mathcal{G}$ is independent of $X^{j}_{t}$ given the parents
$\mathcal{P}(X^{j}_{t})$. In particular,
$\mathbf{X}_{t}^{-}{\setminus}\mathcal{P}(X^{j}_{t})\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t})$ since all variables in
$\mathbf{X}_{t}^{-}$ are non-descendants of $X^{j}_{t}$ by time-order.
Note that for the SCM (1) with independent noise terms the Causal Markov
Condition is automatically fulfilled.
###### Definition S2 (Adjacency and standard faithfulness Condition).
The joint distribution of a process $\mathbf{X}$ whose causal structure can be
represented in a time series graph $\mathcal{G}$ fulfills the Adjacency
Faithfulness Condition iff for all disjoint
$X^{i}_{t-\tau},X^{j}_{t},\mathcal{S}\in\mathbf{X}^{-}_{t+1}$ with $\tau>0$
$\displaystyle X^{i}_{t-\tau}$ $\displaystyle\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t-\tau}\to
X^{j}_{t}\notin\mathcal{G}$ $\displaystyle X^{i}_{t-\tau}$ $\displaystyle\to
X^{j}_{t}\in\mathcal{G}~{}\Rightarrow~{}X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}~{}~{}\text{(contrapositive)}$
and with $\tau=0$
$\displaystyle X^{i}_{t}$ $\displaystyle\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}\notin\mathcal{G}$
$\displaystyle X^{i}_{t}$
$\displaystyle{\circ\\!{\\--}\\!\circ}X^{j}_{t}\in\mathcal{G}~{}\Rightarrow~{}X^{i}_{t}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}~{}~{}\text{(contrapositive)}\,.$
Furthermore, the variables fulfill the (standard) Faithfulness Condition iff
for $\tau\geq 0$
$\displaystyle X^{i}_{t-\tau}$ $\displaystyle\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t-\tau}\bowtie
X^{j}_{t}~{}|~{}\mathcal{S}$ $\displaystyle X^{i}_{t-\tau}$
$\displaystyle\cancel{\bowtie}X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}~{}~{}\text{(contrapositive)}\,.$
## Appendix S2 Proofs
### S2.1 Proof of Lemma 1
We first consider the following Lemma:
###### Lemma S1.
Algorithm 1 returns a superset of lagged parents under Assumptions 1, i.e.,
$\mathcal{P}^{-}_{t}(X^{j}_{t})\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$
for all $X^{j}_{t}$ in $\mathbf{X}_{t}$.
###### Proof.
We need to show that for arbitrary $(X^{i}_{t-\tau},X^{j}_{t})$ with $\tau>0$
we have
$X^{i}_{t-\tau}\notin\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})~{}\Rightarrow~{}X^{i}_{t-\tau}\notin\mathcal{P}^{-}_{t}(X^{j}_{t})$.
Algorithm 1 removes $X^{i}_{t-\tau}$ from
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ iff
$X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}$ for some
$\mathcal{S}\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\}$
in the iterative CI tests. Then Adjacency Faithfulness directly implies that
$X^{i}_{t-\tau}$ is not adjacent to $X^{j}_{t}$ and in particular
$X^{i}_{t-\tau}\notin\mathcal{P}^{-}_{t}(X^{j}_{t})$. ∎
With this step we can prove Lemma 1.
###### Proof.
The lemma states that under Assumptions 1 with Adjacency Faithfulness replaced
by standard Faithfulness Alg. 1 for all $X^{j}_{t}\in\mathbf{X}_{t}$ returns
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})=\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$
where $\mathcal{C}_{t}(X^{j}_{t})$ denotes the contemporaneous ancestors of
$X^{j}_{t}$. We need to show that for arbitrary
$X^{i}_{t-\tau},X^{j}_{t}\in\mathbf{X}^{-}_{t+1}$ with $\tau>0$: (1)
$X^{i}_{t-\tau}\notin\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})~{}\Rightarrow~{}X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$
and (2)
$X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})~{}\Rightarrow~{}X^{i}_{t-\tau}\in\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$.
Ad 1) Algorithm 1 removes $X^{i}_{t-\tau}$ from
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ iff
$X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}$ for some
$\mathcal{S}\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$
in the iterative CI tests. Then standard Faithfulness implies that
$X^{i}_{t-\tau}\bowtie X^{j}_{t}~{}|~{}\mathcal{S}$ and in particular
$X^{i}_{t-\tau}\notin\mathcal{P}^{-}_{t}(X^{j}_{t})$, as proven already in
Lemma S1 under the weaker Adjacency Faithfulness Condition. To show that
$X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$
we note that
$\mathcal{S}\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$
does not include any contemporaneous conditions and, hence, all
contemporaneous directed paths from contemporaneous ancestors of $X^{j}_{t}$
are open and also paths from parents of those ancestors are open. If
$X^{i}_{t-\tau}\in\bigcup_{X^{i}_{t}\in\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$,
by the contraposition of standard Faithfulness we should observe
$X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}$. Then
the fact that on the contrary we observe $X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S}$ implies that
$X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$.
Ad 2) Now we have $X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$
which implies that
$X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$
in the last iteration step of Alg. 1. By (1),
$\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ is a superset of
$\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$.
Define the lagged extra conditions as
$W^{-}_{t}=\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t}),X^{i}_{t-\tau}\\}$.
Since $W^{-}_{t}$ is lagged, it is a non-descendant of $X^{j}_{t}$ or any
$X^{k}_{t}\in\mathcal{C}_{t}(X^{j}_{t})$. We now proceed by a proof by
contradiction. Suppose to the contrary that
$X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$.
The Causal Markov Condition applies to both $X^{i}_{t-\tau}$ and $W^{-}_{t}$
and implies that $(X^{i}_{t-\tau},W^{-}_{t})\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$.
From the weak union property of conditional independence we get
$X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t}),W^{-}_{t}$
which is equivalent to $X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\}$,
contrary to the assumption, hence
$X^{i}_{t-\tau}\in\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$.
∎
### S2.2 Proof of Theorem 1
###### Proof.
The theorem states that under Assumptions 1
$\widehat{\mathcal{G}^{*}}=\mathcal{G}^{*}$, where the $\mathcal{G}^{*}$
denotes the skeleton of the time series graph. We denote the two types of
skeleton links $\to$ and ${\circ\\!{\\--}\\!\circ}$ here generically as
${\star\\!{\\--}\\!\star}$ and can assume $\tau_{\max}\geq\tau\geq 0$. We need
to show that for arbitrary $X^{i}_{t-\tau},X^{j}_{t}\in\mathbf{X}^{-}_{t+1}$:
(1)
$X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\widehat{\mathcal{G}^{*}}~{}\Rightarrow~{}X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\mathcal{G}^{*}$
and (2)
$X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\mathcal{G}^{*}~{}\Rightarrow~{}X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\widehat{\mathcal{G}^{*}}$.
Ad (1): Algorithm 2 deletes a link
$X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}$ from
$\widehat{\mathcal{G}^{*}}$ iff $X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$
for some $\mathcal{S}\subseteq\widehat{\mathcal{A}}_{t}(X^{j}_{t})$ in the
iterative CI tests with $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ estimated
in Alg. 1. $\widehat{\mathcal{A}}_{t}(X^{j}_{t})$ denotes the contemporaneous
adjacencies. Then Adjacency Faithfulness directly implies that
$X^{i}_{t-\tau}$ is not adjacent to $X^{j}_{t}$:
$X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\mathcal{G}^{*}$.
Ad (2): By Lemma 1 we know that $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ is
a superset of the lagged parents of $X^{j}_{t}$. Denote the lagged, extra
conditions occurring in the CI tests of Alg. 2 as
$W^{-}_{t}=(\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau}))\setminus\mathcal{P}(X^{j}_{t})$.
$W^{-}_{t}$ does not contain parents of $X^{j}_{t}$ and by the assumption also
$X^{i}_{t-\tau}$ is not a parent of $X^{j}_{t}$. We further assume that for
$\tau=0$ $X^{i}_{t}$ is also not a descendant of $X^{j}_{t}$ since that case
is covered if we exchange $X^{i}_{t}$ and $X^{j}_{t}$. Then the Causal Markov
Condition implies $(X^{i}_{t-\tau},W^{-}_{t})\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t})$. By the weak union property of
conditional independence this leads to $X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t}),W^{-}_{t}$ which is equivalent to
$X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau})$.
Now Alg. 2 iteratively tests $X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau})$
for all $\mathcal{S}\subseteq\widehat{\mathcal{A}_{t}}(X^{j}_{t})$. By the
first part of this proof, the estimated contemporaneous adjacencies are always
a superset of the true contemporaneous adjacencies, i.e.,
$\mathcal{A}_{t}(X^{j}_{t})\subseteq\widehat{\mathcal{A}_{t}}(X^{j}_{t})$, and
by Lemma 1 $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ is a superset of the
lagged parents. Hence, at some iteration step
$\mathcal{S}=\mathcal{P}_{t}(X^{j}_{t})$ and Alg. 2 will find
$X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau})$
and remove $X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}$ from
$\widehat{\mathcal{G}^{*}}$. ∎
For empty conditioning sets $\mathcal{S}$ ($p=0$), Alg. 2 is equivalent to the
MCI algorithm [Runge et al., 2019b] with the slight change that the latter is
initialized with a fully connected (lagged) graph, which has no effect
asymptotically. In [Runge et al., 2019b] the authors prove the consistency of
PCMCI assuming no contemporaneous causal links under the standard Faithfulness
Condition. The proof above implies that PCMCI is already consistent under the
weaker Adjacency Faithfulness Condition.
### S2.3 Proof of Lemma 2
###### Proof.
Time order and stationarity can be used to constrain the four cases as
follows. Let us first consider a generic triple
$X^{i}_{t_{i}}{\star\\!{\\--}\\!\star}X^{k}_{t_{k}}{\star\\!{\\--}\\!\star}X^{j}_{t_{j}}$.
By stationarity we can fix $t=t_{j}$. We only need to consider cases with
$t_{i},t_{k}\leq t$. If $t_{k}>t_{j}$, the triple is oriented already by time
order and the case $t_{i}>t_{j}$ is symmetric.
The possible triples in the collider phase of the original PC algorithm are
$X^{i}_{t_{i}}{\star\\!{\\--}\\!\star}X^{k}_{t_{k}}{\star\\!{\\--}\\!\star}X^{j}_{t}$
where $(X^{i}_{t_{i}},X^{j}_{t})$ are not adjacent. For $t_{k}<t$ the time-
order constraint automatically orients $X^{k}_{t_{k}}\to X^{j}_{t}$ and hence
$X^{k}_{t_{k}}$ is a parent of $X^{j}_{t}$ and must always be in the
separating set that makes $X^{i}_{t_{i}}$ and $X^{j}_{t}$ independent. Hence
we only need to consider $t_{k}=t$ and can set $\tau=t-t_{i}$
($\tau_{\max}\geq\tau\geq 0$), leaving the two cases of unshielded triples
$X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau>0$)
or
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$
(for $\tau=0$) in $\mathcal{G}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not
adjacent. Since $X^{k}_{t}$ is contemporaneous to $X^{j}_{t}$, this
restriction implies that only contemporaneous parts of separating sets are
relevant for the collider orientation phase.
For rule R1 in the orientation phase the original PC algorithm considers the
remaining triples with $X^{i}_{t-\tau}\to X^{k}_{t}$ that were not oriented by
the collider phase (or by time order). This leaves $X^{i}_{t-\tau}\to
X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ where $\tau_{\max}\geq\tau\geq 0$.
For rule R2 the original PC algorithm considers $X^{i}_{t_{i}}\to
X^{k}_{t_{k}}\to X^{j}_{t}$ with
$X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. The latter type of link
leads to $t_{i}=t$ and time order restricts the triples to $X^{i}_{t}\to
X^{k}_{t}\to X^{j}_{t}$ with $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$.
For rule R3 the original PC algorithm considers
$X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{k}_{t_{k}}\to X^{j}_{t}$ and
$X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{l}_{t_{l}}\to X^{j}_{t}$ where
$(X^{k}_{t_{k}},X^{l}_{t_{l}})$ are not adjacent and
$X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. The latter constraint leads
to $t_{i}=t$ and $X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{k}_{t_{k}}$ and
$X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{k}_{t_{l}}$ imply $t_{k}=t_{l}=t$.
Hence we only need to check triples
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}\to X^{j}_{t}$ and
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{l}_{t}\to X^{j}_{t}$ where
$(X^{k}_{t},X^{l}_{t})$ are not adjacent and
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. ∎
### S2.4 Proof of Theorem 2
###### Proof.
We first consider the case under Assumptions 1 with Adjacency Faithfulness and
PCMCI+ in conjunction with the conservative collider orientation rule in Alg.
S2. We need to show that all separating sets estimated in Alg. S2 during the
conservative orientation rule are correct. From the soundness (Theorem 1) and
correctness of the separating sets follows the correctness of the collider
orientation phase and the rule orientation phase which implies the
completeness.
By Lemma 2 we only need to prove that in Alg. S2 for unshielded triples
$X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau>0$)
or
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$
(for $\tau=0$) the separating sets among subsets of contemporaneous neighbors
of $X^{j}_{t}$ and, if $\tau=0$, of $X^{i}_{t}$, are correct. Algorithm S2
tests $X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$
for all
$\mathcal{S}{\subseteq}\widehat{\mathcal{A}}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$
and for all
$\mathcal{S}{\subseteq}\widehat{\mathcal{A}}_{t}(X^{i}_{t}){\setminus}\\{X^{j}_{t}\\}$
(if $\tau=0$). Since PCMCI+ is sound, all adjacency information is correct and
since all CI tests are assumed correct, all information on separating sets is
correct. Furthermore, with the conservative rule those triples where only
Adjacency Faithfulness, but not standard Faithfulness, holds will be correctly
marked as ambiguous triples.
Under standard Faithfulness the completeness requires to prove that PCMCI+
without the conservative orientation rule yields correct separating set
information. By Lemma 2 also here we need to consider only separating sets
among subsets of contemporaneous neighbors of $X^{j}_{t}$. Algorithm 2 tests
$X^{i}_{t-\tau}\perp\\!\\!\\!\perp
X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$
for all
$\mathcal{S}{\subseteq}\widehat{\mathcal{A}}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$.
And again, since PCMCI+ is sound, all adjacency information is correct and
since all CI tests are assumed correct, all information on separating sets is
correct, from which the completeness for this case follows. ∎
### S2.5 Proof of Theorem 3
###### Proof.
Order-independence follows straightforwardly from sticking to the PC algorithm
version in [Colombo and Maathuis, 2014]. In particular, Alg. 1 and Alg. 2 are
order-independent since they are based on PC stable where adjacencies are
removed only after each loop over conditions of cardinality $p$. Furthermore,
the collider phase (Alg. S2) and rule orientation phase (Alg. S3) are order-
independent by marking triples with inconsistent separating sets as ambiguous
and consistently marking conflicting link orientations by ${x\\!{\\--}\\!x}$.
∎
### S2.6 Proof of Theorem 4
###### Proof.
The theorem states that under Assumptions 1 the effect size for the PCMCI+
oracle case CI tests in Alg. 2 for $p=0$ for contemporaneous true links
$X^{i}_{t}\to X^{j}_{t}\in\mathcal{G}$ is greater than that of
PCMCI${}^{+}_{0}$:
$I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t}))>\min(I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})),\,I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})))$
if both $X^{i}_{t}$ and $X^{j}_{t}$ have parents that are not shared with the
other. We will use an information-theoretic framework here and consider the
conditional mutual information.
To prove this statement, we denote by
$\mathcal{B}_{i}=\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})\setminus\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$
the lagged conditions of $X^{i}_{t}$ that are not already contained in those
of $X^{j}_{t}$ and, correspondingly,
$\mathcal{B}_{j}=\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})$.
Since both $X^{i}_{t}$ and $X^{j}_{t}$ have parents that are not shared with
the other and we assume the oracle case, both these sets are non-empty.
Further, we denote the common lagged conditions as
$\mathcal{B}_{ij}=\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\cap\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})$
and make use of the following conditional independencies, which hold by the
Markov assumption: (1) $\mathcal{B}_{i}\perp\\!\\!\\!\perp
X^{j}_{t}|\mathcal{B}_{j},\mathcal{B}_{ij},X^{i}_{t}$ and (2)
$\mathcal{B}_{j}\perp\\!\\!\\!\perp
X^{i}_{t}|\mathcal{B}_{i},\mathcal{B}_{ij}$. We first prove that, given a
contemporaneous true link $X^{i}_{t}\to X^{j}_{t}\in\mathcal{G}$,
$I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})>I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})$
by using the following two ways to apply the chain rule of conditional mutual
information:
$\displaystyle
I(X^{i}_{t},\mathcal{B}_{i};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij})=$
$\displaystyle=I(X^{i}_{t},\mathcal{B}_{i};\mathcal{B}_{j}|\mathcal{B}_{ij})+I(X^{i}_{t},\mathcal{B}_{i};X^{j}_{t}|\mathcal{B}_{ij}\mathcal{B}_{j})$
$\displaystyle=I(\mathcal{B}_{i};\mathcal{B}_{j}|\mathcal{B}_{ij})+\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i})}_{=0~{}~{}\text{(Markov)}}$
$\displaystyle\phantom{=}+I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})+\underbrace{I(\mathcal{B}_{i};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j},X^{i}_{t})}_{=0~{}~{}\text{(Markov)}}$
(S1)
and
$\displaystyle
I(X^{i}_{t},\mathcal{B}_{i};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij})=$
$\displaystyle=I(\mathcal{B}_{i};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij})+I(X^{i}_{t};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij}\mathcal{B}_{i})$
$\displaystyle=I(\mathcal{B}_{i};\mathcal{B}_{j}|\mathcal{B}_{ij})+\underbrace{I(\mathcal{B}_{i};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})}_{>0~{}~{}\text{since
$X^{i}_{t}\to X^{j}_{t}$}}$
$\displaystyle\phantom{=}+I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})+\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i},X^{j}_{t})}_{>0~{}~{}\text{since
$X^{i}_{t}\to X^{j}_{t}$}}$ (S2)
where (S1) and (S2) denote two different applications of the chain rule. From
this is follows that
$I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})>I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})$.
Hence, it remains to prove that
$I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j},\mathcal{B}_{i})>I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})$,
which we also do by the chain rule:
$\displaystyle
I(X^{i}_{t};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i})=$
$\displaystyle=I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})+\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i},X^{j}_{t})}_{>0~{}~{}\text{since
$X^{i}_{t}\to X^{j}_{t}$}}$ (S3)
$\displaystyle=\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i})}_{=0~{}~{}\text{(Markov)}}+I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i},\mathcal{B}_{j})$
(S4)
∎
## Appendix S3 Further pseudo code
Algorithms S2 and S3 detail the pseudo-code for the PCMCI+ / PCMCI${}^{+}_{0}$
/ PC collider phase with different collider rules and the orientation phase.
Algorithm S2 (Detailed PCMCI+ / PCMCI${}^{+}_{0}$ / PC collider phase with
different collider rules)
1:$\mathcal{G}$ and sepset from Alg. 2, rule $=\\{$’none’, ’conservative’,
’majority’$\\}$, time series dataset $\mathbf{X}=(X^{1},\,\ldots,X^{N})$,
significance threshold $\alpha_{\rm PC}$, ${\rm CI}(X,\,Y,\,\mathbf{Z})$,
PCMCI+ / PCMCI${}^{+}_{0}$: $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all
$X^{j}_{t}$ in $\mathbf{X}_{t}$
2:for all unshielded triples $X^{i}_{t-\tau}\to
X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ ($\tau>0$) or
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$
($\tau=0$) in $\mathcal{G}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not
adjacent do
3: if rule $=$ ’none’ then
4: if $X^{k}_{t}$ is not in sepset$(X^{i}_{t-\tau},X^{j}_{t})$ then
5: Orient $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$
($\tau>0$) or
$X^{i}_{t-\tau}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$
($\tau=0$) as $X^{i}_{t-\tau}\to X^{k}_{t}\leftarrow X^{j}_{t}$
6: else
7: PCMCI+ / PCMCI${}^{+}_{0}$: Define contemporaneous adjacencies
$\widehat{\mathcal{A}}(X^{j}_{t})=\widehat{\mathcal{A}}_{t}(X^{j}_{t})=\\{X^{i}_{t}{\neq}X^{j}_{t}\in\mathbf{X}_{t}:X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}~{}\text{in
$\mathcal{G}$}\\}$
8: PC: Define full adjacencies $\widehat{\mathcal{A}}(X^{j}_{t})$ for all
(lagged and contemporaneous) links in $\mathcal{G}$
9: for all for all
$\mathcal{S}{\subseteq}\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$
and for all
$\mathcal{S}{\subseteq}\widehat{\mathcal{A}}(X^{i}_{t}){\setminus}\\{X^{j}_{t}\\}$
(if $\tau=0$) do
10: Evaluate CI($X^{i}_{t{-}\tau},X^{j}_{t},\mathbf{Z})$ with
11: PCMCI+:
$\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau}))$
12: PCMCI${}^{+}_{0}$:
$\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\})$
13: PC: $\mathbf{Z}{=}\mathcal{S}$
14: Store all subsets $\mathcal{S}$ with $p$-value $>\alpha_{\rm PC}$ as
separating subsets
15: if no separating subsets are found then
16: Mark triple as ambiguous
17: else
18: Compute fraction $n_{k}$ of separating subsets that contain $X^{k}_{t}$
19: if rule $=$ ’conservative’ then
20: Orient triple as collider if $n_{k}{=}0$, leave unoriented if $n_{k}{=}1$,
and mark as ambiguous if $0{<}n_{k}{<}1$
21: else if rule $=$ ’majority’ then
22: Orient triple as collider if $n_{k}{<}0.5$, leave unoriented if
$n_{k}{>}0.5$, and mark as ambiguous if $n_{k}{=}0.5$
23: Mark links in $\mathcal{G}$ with conflicting orientations as
${x\\!{\\--}\\!x}$
24:return $\mathcal{G}$, sepset, ambiguous triples, conflicting links
Algorithm S3 (Detailed PCMCI+ / PCMCI${}^{+}_{0}$ / PC rule orientation phase)
1:$\mathcal{G}$, ambiguous triples, conflicting links
2:while any unambiguous triples suitable for rules R1-R3 are remaining do
3: Apply rule R1 (orient unshielded triples that are not colliders):
4: for all unambiguous triples $X^{i}_{t-\tau}\to
X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ where $(X^{i}_{t-\tau},X^{j}_{t})$
are not adjacent do
5: Orient as $X^{i}_{t-\tau}\to X^{k}_{t}\to X^{j}_{t}$
6: Mark links with conflicting orientations as ${x\\!{\\--}\\!x}$
7: Apply rule R2 (avoid cycles):
8: for all unambiguous triples $X^{i}_{t}\to X^{k}_{t}\to X^{j}_{t}$ with
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ do
9: Orient as $X^{i}_{t}\to X^{j}_{t}$
10: Mark links with conflicting orientations as ${x\\!{\\--}\\!x}$
11: Apply rule R3 (orient unshielded triples that are not colliders and avoid
cycles):
12: for all pairs of unambiguous triples
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}\to X^{j}_{t}$ and
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{l}_{t}\to X^{j}_{t}$ where
$(X^{k}_{t},X^{l}_{t})$ are not adjacent and
$X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ do
13: Orient as $X^{i}_{t}\to X^{j}_{t}$
14: Mark links with conflicting orientations as ${x\\!{\\--}\\!x}$
15:return $\mathcal{G}$, conflicting links
## Appendix S4 Implementation details
In the linear and nonlinear numerical experiments PCMCI+ is compared with the
PC algorithm, both implemented with the appropriate CI test (ParCorr for the
linear case, GPDC for the nonlinear case). For the linear numerical
experiments we additionally consider representatives from two further
frameworks: GCresPC, a combination of GC with PC applied to residuals, and an
autoregressive model version of LiNGAM [Hyvärinen et al., 2010], a
representative of the SCM framework. Their implementations are as follows.
### S4.1 LiNGAM
For LiNGAM the code was taken from https://github.com/cdt15/lingam which
provides a class VARLiNGAM. The method was called follows:
Input: data, tau_max
model = lingam.VARLiNGAM(lags=tau_max, criterion=None, prune=True)
model.fit(data)
val_matrix = model.adjacency_matrices_.transpose(2,1,0)
graph = (val_matrix != 0.).astype(’int’)
Output: graph
The causal graph `graph` encodes the causal relations in an array of shape
`(N, N, tau_max + 1)`. The option `criterion=None` just ignores the optional
automatic selection of `lags`, which is here set to the same `tau_max` for all
methods. I could not find a way to obtain p-values in the VARLiNGAM
implementation, but with the parameter setting `prune=True` the resulting
adjacency matrices are regularized with an adaptive LASSO approach using the
BIC criterion to find the optimal regularization hyper-parameter
(`sklearn.LassoLarsIC(criterion=’bic’)`). Non-zero adjacencies were then
evaluated as causal links. Note that all other methods can be intercompared at
different $\alpha_{\rm PC}$ levels while for comparison against LiNGAM we
focus on its relative changes rather than absolute performance.
### S4.2 GCresPC
There was no code available for the method proposed in [Moneta et al., 2011].
The present implementation first fits a VAR model up to $\tau_{\max}$ and
applies the PC algorithm on the residuals. To remove spurious lagged links
(due to contemporaneous paths), the PC algorithm was additionally run on
significant lagged and contemporaneous links, but the orientation phase was
restricted to contemporaneous links, as proposed in [Moneta et al., 2011]. The
following Python pseudo-code utilizes functionality from the tigramite
package, numpy, and statsmodels:
Input: data, tau_max, alpha
import functions/classes ParCorr, PCMCI, DataFrame from tigramite
graph = np.zeros((N, N, tau_max + 1))
# 1. Estimate lagged adjacencies (to be updated in step 3.)
tsamodel = tsa.var.var_model.VAR(data)
results = tsamodel.fit(maxlags=tau_max, trend=’nc’)
pvalues = results.pvalues
values = results.coefs
residuals = results.resid
lagged_parents = significant lagged links at alpha
# 2. Run PC algorithm on residuals (with tau_max=0)
pcmci = PCMCI(dataframe=DataFrame(residuals), cond_ind_test=ParCorr())
pcmcires = pcmci.run_pcalg(pc_alpha=alpha,
tau_min=0,
tau_max=0)
# Update contemporaneous graph
graph[:,:,0] = pcmcires[’graph’][:,:,0]
# 3. Run PC algorithm on significant lagged and contemporaneous adjacencies
# to remove spurious lagged links due to contemporaneous parents
selected_links = lagged_parents + significant contemporaneous adjacencies
pcmci = PCMCI(dataframe=DataFrame(data), cond_ind_test=ParCorr())
pcmcires = pcmci.run_pcalg(selected_links=selected_links,
pc_alpha=alpha,
tau_min=0,
tau_max=tau_max)
# Update lagged part of graph
graph[:,:,1:] = pcmcires[’graph’][:,:,1:]
Output: graph
Note that the contemporaneous graph structure in `graph` comes only from
applying the PC algorithm to the residuals and, hence, does not utilize
triples containing lagged adjacencies. Step 3 is necessary to remove spurious
lagged links due to contemporaneous parents. The output of GCresPC depends on
$\alpha_{\rm PC}$ as for PCMCI+ and the PC algorithm.
## Appendix S5 Further numerical experiments
Next to repeating the overview figure for the linear Gaussian model setup from
the main text in Fig. S1, in Fig. S2 we show the linear mixed noise setup, and
in Fig. S3 the nonlinear mixed noise setup. The remaining pages contain
results of further numerical experiments that evaluate different
$a,\,N,\,T,\,\tau_{\max}$ and $\alpha_{\rm PC}$ for the linear model setups.
All results and more will be contributed to the causality benchmark platform
`www.causeme.net` [Runge et al., 2019a] to facilitate a further expanded
method evaluation.
Figure S1: Numerical experiments with linear Gaussian setup for varying (A)
autocorrelation strength $a$ (B) number of variables $N$ (C) sample size $T$
and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters
indicated in the top right. Errorbars show std. errors or the 90% range (for
runtime). The insets show ANOVA statistics.
Figure S2: Numerical experiments with linear mixed noise setup for varying
(A) autocorrelation strength $a$ (B) number of variables $N$ (C) sample size
$T$ and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters
indicated in the top right. Errorbars show std. errors or the 90% range (for
runtime). The insets show ANOVA statistics.
Figure S3: Numerical experiments with nonlinear mixed noise setup for varying
(A) autocorrelation strength $a$ (B) number of variables $N$ (C) sample size
$T$ and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters
indicated in the top right. Errorbars show std. errors or the 90% range (for
runtime). The insets show ANOVA statistics.
Figure S4: Numerical experiments with linear Gaussian setup for varying
autocorrelation $a$ and $T=200$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
$N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are
indicated in the upper right of each panel.
Figure S5: Numerical experiments with linear Gaussian setup for varying
autocorrelation $a$ and $T=500$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
$N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are
indicated in the upper right of each panel.
Figure S6: Numerical experiments with linear Gaussian setup for varying
autocorrelation $a$ and $T=1000$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
$N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are
indicated in the upper right of each panel.
Figure S7: Numerical experiments with linear Gaussian setup for varying
number of variables $N$ and $T=200$ . The left (right) column shows results
for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results
for increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S8: Numerical experiments with linear Gaussian setup for varying
number of variables $N$ and $T=500$ . The left (right) column shows results
for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results
for increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S9: Numerical experiments with linear Gaussian setup for varying
number of variables $N$ and $T=1000$ . The left (right) column shows results
for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results
for increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S10: Numerical experiments with linear Gaussian setup for varying
sample size $T$ for $N=5$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S11: Numerical experiments with linear Gaussian setup for varying
sample size $T$ for $N=10$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S12: Numerical experiments with linear Gaussian setup for varying
sample size $T$ for $N=20$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S13: Numerical experiments with linear Gaussian setup for varying
maximum time lag $\tau_{\max}$ and $T=200$ . The left (right) column shows
results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict
results for increasing autocorrelations $a$ (top to bottom). All model and
method parameters are indicated in the upper right of each panel.
Figure S14: Numerical experiments with linear Gaussian setup for varying
maximum time lag $\tau_{\max}$ and $T=500$ . The left (right) column shows
results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict
results for increasing autocorrelations $a$ (top to bottom). All model and
method parameters are indicated in the upper right of each panel.
Figure S15: Numerical experiments with linear Gaussian setup for varying
maximum time lag $\tau_{\max}$ and $T=1000$ . The left (right) column shows
results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict
results for increasing autocorrelations $a$ (top to bottom). All model and
method parameters are indicated in the upper right of each panel.
Figure S16: Numerical experiments with linear mixed noise setup for varying
autocorrelation $a$ and $T=200$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
$N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are
indicated in the upper right of each panel.
Figure S17: Numerical experiments with linear mixed noise setup for varying
autocorrelation $a$ and $T=500$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
$N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are
indicated in the upper right of each panel.
Figure S18: Numerical experiments with linear mixed noise setup for varying
autocorrelation $a$ and $T=1000$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
$N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are
indicated in the upper right of each panel.
Figure S19: Numerical experiments with linear mixed noise setup for varying
number of variables $N$ and $T=200$ . The left (right) column shows results
for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results
for increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S20: Numerical experiments with linear mixed noise setup for varying
number of variables $N$ and $T=500$ . The left (right) column shows results
for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results
for increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S21: Numerical experiments with linear mixed noise setup for varying
number of variables $N$ and $T=1000$ . The left (right) column shows results
for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results
for increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S22: Numerical experiments with linear mixed noise setup for varying
sample size $T$ for $N=5$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S23: Numerical experiments with linear mixed noise setup for varying
sample size $T$ for $N=10$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S24: Numerical experiments with linear mixed noise setup for varying
sample size $T$ for $N=20$ . The left (right) column shows results for
significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for
increasing autocorrelations $a$ (top to bottom). All model and method
parameters are indicated in the upper right of each panel.
Figure S25: Numerical experiments with linear mixed noise setup for varying
maximum time lag $\tau_{\max}$ and $T=200$ . The left (right) column shows
results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict
results for increasing autocorrelations $a$ (top to bottom). All model and
method parameters are indicated in the upper right of each panel.
Figure S26: Numerical experiments with linear mixed noise setup for varying
maximum time lag $\tau_{\max}$ and $T=500$ . The left (right) column shows
results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict
results for increasing autocorrelations $a$ (top to bottom). All model and
method parameters are indicated in the upper right of each panel.
Figure S27: Numerical experiments with linear mixed noise setup for varying
maximum time lag $\tau_{\max}$ and $T=1000$ . The left (right) column shows
results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict
results for increasing autocorrelations $a$ (top to bottom). All model and
method parameters are indicated in the upper right of each panel.
|
2024-09-04T02:54:58.146594 | 2020-03-08T01:10:11 | 2003.03694 | {
"authors": "Yu-Hui Chen, Ronnie R. Tamming, Kai Chen, Zhepeng Zhang, Yanfeng\n Zhang, Justin M. Hodgkiss, Richard J. Blaikie, Boyang Ding, Min Qiu",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26101",
"submitter": "Boyang Ding",
"url": "https://arxiv.org/abs/2003.03694"
} | arxiv-papers | ††thanks: These authors contributed equally††thanks: These authors contributed
equally
# Bandgap Control in Two-Dimensional Semiconductors via Coherent Doping of
Plasmonic Hot Electrons
Yu-Hui Chen School of Physics, Beijing Institute of Technology, Beijing
10081, China Ronnie R. Tamming MacDiarmid Institute for Advanced Materials
and Nanotechnology, Dodd-Walls Centre for Photonic and Quantum Technologies,
School of Chemical and Physical Sciences, Victoria University of Wellington,
Wellington 6012, New Zealand Kai Chen MacDiarmid Institute for Advanced
Materials and Nanotechnology, Dodd-Walls Centre for Photonic and Quantum
Technologies, School of Chemical and Physical Sciences, Victoria University of
Wellington, Wellington 6012, New Zealand Zhepeng Zhang Department of
Materials Science and Engineering, College of Engineering, Center for
Nanochemistry (CNC), College of Chemistry and Molecular Engineering, Academy
for Advanced Interdisciplinary Studies, Peking University, Beijing 100871,
China Yanfeng Zhang Department of Materials Science and Engineering, College
of Engineering, Center for Nanochemistry (CNC), College of Chemistry and
Molecular Engineering, Academy for Advanced Interdisciplinary Studies, Peking
University, Beijing 100871, China Justin M. Hodgkiss MacDiarmid Institute for
Advanced Materials and Nanotechnology, Dodd-Walls Centre for Photonic and
Quantum Technologies, School of Chemical and Physical Sciences, Victoria
University of Wellington, Wellington 6012, New Zealand Richard J. Blaikie
MacDiarmid Institute for Advanced Materials and Nanotechnology, Dodd-Walls
Centre for Photonic and Quantum Technologies, Department of Physics,
University of Otago, PO Box 56, Dunedin 9016, New Zealand Boyang Ding
<EMAIL_ADDRESS>MacDiarmid Institute for Advanced Materials and
Nanotechnology, Dodd-Walls Centre for Photonic and Quantum Technologies,
Department of Physics, University of Otago, PO Box 56, Dunedin 9016, New
Zealand Min Qiu<EMAIL_ADDRESS>Key Laboratory of 3D Micro/Nano
Fabrication and Characterization of Zhejiang Province, School of Engineering,
Westlake University, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province,
China Institute of Advanced Technology, Westlake Institute for Advanced Study,
18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China
###### Abstract
Bandgap control is of central importance for semiconductor technologies. The
traditional means of control is to dope the lattice chemically, electrically
or optically with charge carriers. Here, we demonstrate for the first time a
widely tunable bandgap (renormalisation up to 650 meV at room-temperature) in
two-dimensional (2D) semiconductors by coherently doping the lattice with
plasmonic hot electrons. In particular, we integrate tungsten-disulfide (WS2)
monolayers into a self-assembled plasmonic crystal, which enables coherent
coupling between semiconductor excitons and plasmon resonances. Accompanying
this process, the plasmon-induced hot electrons can repeatedly fill the WS2
conduction band, leading to population inversion and a significant
reconstruction in band structures and exciton relaxations. Our findings
provide an innovative and effective measure to engineer optical responses of
2D semiconductors, allowing a great flexiblity in design and optimisation of
photonic and optoelectronic devices.
Two-dimensional (2D) semiconductors, such as transition metal dichalcogenides
(TMDCs)Mak _et al._ (2010); Splendiani _et al._ (2010), have direct bandgap
at their monolayer limit, exhibiting tremendous potential in development of
next-generation nanoscale devices. Like in their bulk counterparts, bandgap
control plays a vital role in 2D semiconductor technoglogies, since it enables
the creation of desirable optoelectronic properties that are required in
numerous applications, ranging from lasersYe _et al._ (2015) to modulatorsMak
and Shan (2016), photodetectorsLopez-Sanchez _et al._ (2013) and
photocatalysisVoiry _et al._ (2013). The traditional means of control is to
dope the lattice chemicallyKim _et al._ (2015), electricallyChernikov _et
al._ (2015a) or opticallyChernikov _et al._ (2015b) with charge carriers, the
practicality of which is, however, limited by many factors, e.g. the
irreversible bandgap modification, contact-type control and requirement of
ultrastrong pump.
Here we report that one can flexibly and effectively modify the electronic
band structures of 2D semiconductors by establishing coherent strong coupling
between the semiconductor excitons and a plasmonic resonatorEbbesen _et al._
(1998); Liu and Lalanne (2008). In particular, plasmonic resonators are
metallic nanostructures that support collective oscillation of electrons,
known as plasmons. The excitation of plasmons can produce hot electrons, i.e.
highly energetic electrons with non-equilibrium thermal distributionsClavero
(2014); Brongersma _et al._ (2015), which, in the strong coupling regime, can
repeatedly dope the lattice along with the coherent plasmon-exciton energy
exchange. As a result, the bandgap of 2D semiconductors is significantly
renormalised and the renormalisation can be easily altered through changing
the detuning between plasmons and excitons.
The schematic of our sample in Fig.1a demonstrates a WS2 monolayer (ML)
deposited onto a plasmonic crystal (PC)Ding _et al._ (2013, 2019), which
comprises of a periodic array of silver capped silica nanospheres that are
coated with an ultrathin Al2O3 spacer. This metal-insulator-semiconductor
configuration constitutes PC-WS2 hybrid systems, supporting plasmon lattice
modes propagating on the PC-WS2 interface. Here the top WS2 MLs belong to the
family of atomically thin TMDCs, having been extensively studiedYe _et al._
(2014); Sie _et al._ (2017); Ruppert _et al._ (2017); Cunningham _et al._
(2017); Steinhoff _et al._ (2017) for their unusual exciton-dominated optical
responses, such as high absorption and emission efficiency. These properties
make the PC-WS2 systems a suitable platform to study plasmon-exciton
interactionsDing _et al._ (2019).
The PC geometries were chosen to excite plasmon lattice modesEbbesen _et al._
(1998); Liu and Lalanne (2008); Ding _et al._ (2013, 2019) that can match the
frequency of exciton A in WS${}_{\text{2}}$ MLs at certain incident angles
$\theta$. The plasmon modes show red-shift dispersion at higher $\theta$
(yellow curve in Fig.1b), matching the frequency of exciton A ($E=2.061$ eV)
at $\theta=22^{\circ}$. In this case, plasmon modes can coherently couple with
excitons, leading to the formation of plasmon-exciton polaritons, i.e. half-
light half-matter quasiparticles that inherit properties from both the
plasmonic and excitonic components. As a result, the transmission maxima
exhibit pronounced splitting features that follow the dispersions of upper
polariton (UP) and lower polariton (LP), indicating the establishment of
strong coupling between plasmons and excitons. When the frequency of the
plasmon mode is tuned in resonance with exciton A ($\theta=22^{\circ}$), the
hybrid system is characterised by a vacuum Rabi splitting of
$\hbar\cdot\Omega_{\text{R}}\approx 136$ meV. More detailed analysis of strong
plasmon-exciton coupling in equilibrium states can be found in a previous
workDing _et al._ (2019) and Fig.S1 in the Supplementary Information (SI).
Upon photoexcitation, the transient optical responses of PC-WS2 samples can be
characterised using femtosecond transient absorption (TA) spectroscopy (Fig.2a
and Methods), which enables incident angle-resolved probes of the optical
properties and dynamics of WS2 MLs that are strongly coupled with plasmon
resonancesDarby _et al._ (2016). Fig.2b shows the transient transmission
spectra ($\Delta\text{T}/\text{T}$) with a pump fluence of
$12\mu\text{J/cm}^{2}$ as a function of time delay and energy at the tuned
state ($\theta=22^{\circ}$), which displays two split relaxation traces
flanking the spectral position of exciton A ($E=2.061$ eV), corresponding to
UP and LP. This sharply contrasts with the single-trace relaxations of exciton
B ($E=2.471$ eV, Fig.2b) and uncoupled exciton A in bare WS2 MLs (Fig.S2 in
SI). When the PC is detuned from exciton A, e.g. at $\theta=30^{\circ}$
(Fig.2c), a single relaxation trace appears, highly resembling the trace of
bare exciton A. These ultrashort timescale results confirm again the strong
coupling nature of our PC-WS2 systems.
It is worth noting that the photoinduced absorption minimum associated with
tuned polaritons appears at the 1 to 10 ps range (blue area centred at
$E=1.946$ eV in Fig. 2b and the corresponding $\Delta\text{T}/\text{T}$
transient with negative magnitudes in Fig.2f), obviously delayed compared to
the minimum near exciton B (Fig.2b) and its counterpart in the detuned
polaritons (Fig. 2c), which all emerge simultaneously after the arrival of the
pump pulse. Similar postponed minima have been found in transient spectra of
bare TMDC MLs, which typically arise from enhanced exciton-exciton and/or
exciton-electron interactions under high-power pump that can populate high-
density carriers in the latticeCeballos _et al._ (2016); Ruppert _et al._
(2017); Cunningham _et al._ (2017); Sie _et al._ (2017) (see Section 2 in SI
for detailed discussions). What is different is that, in our hybrid systems,
the delayed minima appear under much lower pump intensity than that in the
reference experiments for bare WS2 MLs and are only associated with tuned
polaritons.
More importantly, it is noted that a $\Delta\text{T}/\text{T}$ maximum lasting
for $\sim 1$ ps in $E=1.6$ to $1.8$ eV arise in the tuned polariton spectra
(Fig.2b), which, in contrast, is remarkably weaker in the detuned state
(Fig.2c) and is completely absent in bare WS2 MLs (Fig.S2 in SI). The
integrated $\Delta\text{T}/\text{T}$ spectrum near zero probe delay (Fig.2d)
shows that the broad maximum has positive magnitudes, which indicates negative
optical absorption or positive gain, being a clear evidence of bandgap
renormalisation accompanied by population inversion.Chernikov _et al._
(2015b) Such phenomena are typically induced by the population of high-density
carriers in 2D semiconductor latticeMeckbach _et al._ (2018), which leads to
the non-equilibrium occupation of electron and/or hole states that can induce
the formation of new quasiparticle bandgaps. This process can be decribed
byPeyghambarian _et al._ (1993):
$\Delta E_{\text{g}}=-\underset{q\neq
0}{\sum}V_{\text{s}}(q)\,[f_{\text{e}}(q)+f_{\text{h}}(q)]-\underset{q\neq
0}{\sum}[V_{\text{s}}(q)-V(q)]$ (1)
where $V_{\text{s}}(q)$ and $V(q)$ represent fourier transforms of screened
and unscreened Coulomb potentials, while $f_{\text{e}}(q)$ and
$f_{\text{h}}(q)$ are occupation probabilities of electron and hole with
momentum $q$. The onset of the new bandgap can be extracted from the low-
energy end of the broad maximum. It means that in our experiments, the
renormalised bandgap starts at $E_{\text{g}}\approx 1.60$ eV, lying $\sim 400$
meV below LP and $\sim 650$ meV below the initial bandgap of WS2 MLs (given
that the binding energy of exciton A is $\sim 200$ meVCunningham _et al._
(2017)). This is, to the best of our knowledge, the largest bandgap
renormalisation in 2D semiconductors under such a low pump intensity
(12$\mu$J$/$cm2) to date, which, in the meanwhile, results in the inversion of
carrier population near the newly formed band edgeChernikov _et al._ (2015b);
Meckbach _et al._ (2018), presenting as optical gains, i.e. the broad maximum
in Fig.2b and 2d.
These unusual spectral and transient features are broadly understood as the
presence of high-density carriers, which, in our tuned PC-WS2 systems,
surprisingly have been achieved under room temperature and extremely low pump
intensity ($12\mu$J$/$cm2). This sharply contrasts with similar
observationsChernikov _et al._ (2015b) in bare WS2 single/bi-layers with
ultrastrong photoexcitation ($840\mu$J$/$cm2 at 70 K or $3400\mu$J$/$cm2 at
room temperature). In their study, the population of high-density carriers are
a result of Mott-transition, which are induced by enhanced exciton-exciton
interactions under high-power pump, reducing exciton binding energy, finally
breaking excitons into unbound electron-hole plasmaChernikov _et al._
(2015b); Steinhoff _et al._ (2017). In our experiments, the pump power is too
low to develop a Mott-transition, suggesting that there must be other sources
that can provide large numbers of additional carriers.
To understand the origin of these carriers, we turn to discuss one unique
property of plasmon-exciton polaritons, i.e. the generation of hot electrons
that are inherited from the polaritons’ plasmon root. In particular, hot
electrons are electrons with non-equilibrium thermal distributions, generated
by plasmon dephasing from wave-like states through non-radiative
decayBrongersma _et al._ (2015), which can electrically dope adjacent
semiconductorsFang _et al._ (2012), modifying their photovoltaic and
photocatalytic performanceClavero (2014). When plasmons are coupled to
exciton-like resonances in semiconductors, the hot electron density can be
highly enhanced in the lattice through direct electron tunnelingGarcía De
Arquer _et al._ (2013) or dipole-dipole interactionCushing _et al._ (2012).
Therefore it is very likely that the high-density carriers in tuned PC-WS2
systems are the hot electrons introduced during strong coupling process. (See
Section 3 in SI for detailed discussions)
The analyses of relaxation dynamics of tuned and detuned polaritons support
the hot electron model. We note that both the UP and LP in Fig.2f demonstrate
slower decays than that of detuned states in Fig.2g (Table S2 in SI for
fitting parameters). This observation coincides with a previous studyBoulesbaa
_et al._ (2016), clearly indicating the involvement of plasmonic hot electrons
in the strong plasmon-exciton coupling process. Specifically, as the system
sits in the strong coupilng regime, after photoexcitation, excitons and
plasmons coherently exchange energy at the Rabi frequency ($\sim 136$ meV)Vasa
_et al._ (2013), while the plasmon-to-exciton process is accompanied by hot
electron population in the lattice. Such a charge population runs at an
ultrashort period of $\sim 30$ fs ($T_{\text{R}}=2\pi/\Omega_{\text{R}}$),
which is too short to be caught by our equipment, also greatly shorter than
the exciton formation ($<1$ ps)Ceballos _et al._ (2016), the non-radiative
decay (at scales of $10$ ps) and the radiative decay process (up to few-
hundred ps) in WS2 MLsRuppert _et al._ (2017); Sie _et al._ (2017). This
means that during exciton relaxation, there is frequent tunneling/generation
of hot electrons that can repeatedly fill the unoccupied states in conduction
band of WS2 monolayers, which slow down the exciton bleaching via Pauli
blocking and lead to the extended lifetimes (Section 4 in SI for more
details).
Given that there is little evidence for other possible carrier sources, e.g.
polariton condensates Byrnes _et al._ (2014), we conclude that coherent
doping of plasmonic hot electrons is the origin of the spectral and transient
features that require high-density population. In particular, the hot electron
population repeatedly takes place throughout the whole relaxation process,
while the Al2O3 spacer can form a Schottky-like barrier that prevents charges
from returning back to the metalsCushing _et al._ (2012); García De Arquer
_et al._ (2013). As a result, hot electrons can be accumulated in the lattice
before they decay (within 1 psBrongersma _et al._ (2015)), which
simultaneously competes with rapid exciton relaxations, transiently converting
the intrinsic WS2 monolayers to ”n-doped” ones. This leads to the giant
bandgap renormalisation with population inversion that peak at few-hundred
femtoseconds (Fig.S10 in SI), and also induces the delayed absorption minima
in Fig.2b and 2f (Section 5 in SI).
To confirm our observations, we have performed meaurements under $\sim 10$
times higher pump fluence ($100\mu$J$/$cm2) (Fig. 3a). Apart from the
pronounced broad maxima at low energies, we can see large spectral shift as
well as remarkably delayed occurance of UP and LP maxima, revealing that the
accumulation of hot electrons competes with the relaxation dynamics, which
significantly enhances the systems’ nonlinear responses on ultrashort
timescales. (Detailed discussions in Section 6 of SI). Similar to the low-
power case, the transient variation of the broad maximum (Fig. 3c) under
intense photoexcitation takes $\sim 1.5$ ps from initial excitation to fading.
Fig. 3d shows the evolution of population inversion, where the magnitude and
width of the maximum is highly dependent on pump intensity. Under
$100\mu$J$/$cm2 pump fluence, the full-width at half-maximum can reach at
$\sim 200$ meV with highly enhanced magnitudes as compared to the maximum
under $5\mu$J$/$cm2 pump, also contrasting the unchanged flat spectral
features in bare WS2 MLs. But, even in this case, the pump fluence is still
sigfinicantly lower than that in Ref.Chernikov _et al._ (2015b).
As discussed above, the strong plasmon-exction coupling dramatically modifies
the electronic band structures of WS2 monolayers, which are induced, to a
large degree, by plasmonic hot electron doping via strong coupling. This
effect is extremely hard to observe in traditional exciton-polaritonsByrnes
_et al._ (2014), being a non-trivial factor that has to be considered when
studying light-matter interactions using plasmonic resonators, which, on the
other hand, provides new and effective measures to engineer bandgap of 2D
semiconductors.
## Acknowledgments
The authors acknowledge the New Idea Research Funding 2018 (Dodd-Walls Centre
for photonic and quantum technologies), the Marsden Fast-start Fund by Royal
Society of New Zealand through contract MFP-UOO1827 and the Smart Ideas Fund
by Ministry of Business, Innovation and Employment, New Zealand through
contract UOOX1802. In addition, this work was supported in part by the
National Key Research and Development Program of China (no. 2017YFA0205700)
and the National Natural Science Foundation of China (nos. 61425023, 61235007,
61575177 and 51861135201). The authors also acknowledge the visiting
Fellowship awarded by New Zealand Centre at Peking University. We thank Dr. M.
Yan and Dr. F. Hong for their help with thin-film deposition, AFM, and SEM
measurements.
## Author Contributions
B.D. and Y.-H.C. conceived the project; Z.Z, and B.D. prepared the samples;
R.T., K.C., Y.-H.C. and B.D. carried out the optical and other
characterization; Y.-H.C. and B.D. performed the simulation; Y.Z., M.Q.,
R.J.B., and B.D. supervised the projects; Y.-H.C. and B.D. prepared the
manuscript; all authors discussed and analyzed the results.
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Figure 1: Structures of a PC-WS${}_{\text{2}}$ sample and steady-state optical
properties. a, schematic of polariton formation in a WS${}_{\text{2}}$ ML that
is supported on a self-assembled plasmonic crystal. The Al2O3 spacer is not
depicted for similicity. right insets: side and top-view scanning electron
microscope (SEM) images; b, angle-resolved transmission spectra under
p-polarised illumination and their projection (top x-y plane), in which the
spectral positions of exciton A (X${}_{\text{A}}$) and B (X${}_{\text{B}}$),
calculated dispersions of plasmon lattice modes (yellow curve), and upper and
lower branches of polaritons (orange curves) are indicated. The tuned angle
($\theta=22^{\circ}$) is marked with a blacked dahsed line. Refer to Section 1
in the SI for detailed discussion of the strong plasmon-exciton coupling and
its dispersion. Figure 2: Transient optical responses. a, schematic of angle-
resolved ultrafast pump-probe spectroscopy; b, d and f refer to normalised
differential transmission spectra ($\Delta\text{T}/\text{T}$) at the tuned
angle ($\theta=22^{\circ}$), while c, e and g refer to
$\Delta\text{T}/\text{T}$ at the detuned angle ($\theta=30^{\circ}$); b and c
are intensity plots of $\Delta\text{T}/\text{T}$ as function of time delay and
probe photon energy, using the same colour bar (which is also used by Fig.3a);
d and e are $\Delta\text{T}/\text{T}$ spectra averaged within the time span
from 0.1 to 0.7 ps after pump; f and g are $\Delta\text{T}/\text{T}$ transient
at specific energies (labelled with different colours), in which scatter
symbols and solid curves represent measured and fitted data, respectively.
Dashed frames in panel b, d and e mark the spectral region of the broad maxima
(see main text). All measurements were carried out using 400 nm ($E=3.1$ eV)
pump pulses that have 100 fs duration and pump fluence of 12 $\mu$J/cm2 at
room temperature. The instrument-response-function is shown as the grey area
in panel g Figure 3: Bandgap renormalisation and evloution of population
inversion. a, intensity plot of $\Delta\text{T}/\text{T}$ spectra of PC-WS2
under $100\mu$J$/$cm2 pump fluence at $\theta=22^{\circ}$, where orange (blue)
colour represents the maximum (minimum) value. b, delay time dependent spectra
($\Delta\text{T}/\text{T}$) at energies of UP, LP and exciton B extracted from
panel a. Solid curves are plotted only for visual guidance c,
$\Delta\text{T}/\text{T}$ spectra at different delay times, extracted from the
white dashed frame in panel a; red dashed vertical line indicates the onset of
renormalised bandgap. d, comparison of $\Delta\text{T}/\text{T}$ spectra at
delay of 0.96 ps between PC-WS2 (left) and WS2 MLs (right) under gradually
increasing pump fluence.
|
2024-09-04T02:54:58.158122 | 2020-03-08T07:12:20 | 2003.03734 | {
"authors": "Qingxia Liu, Gong Cheng, Kalpa Gunaratna, Yuzhong Qu",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26102",
"submitter": "Gong Cheng",
"url": "https://arxiv.org/abs/2003.03734"
} | arxiv-papers | 11institutetext: National Key Laboratory for Novel Software Technology,
Nanjing University, China
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext:
Samsung Research America, Mountain View CA, USA
22email<EMAIL_ADDRESS>
# ESBM: An Entity Summarization BenchMark
Qingxia Liu 11 Gong Cheng 11 Kalpa Gunaratna 22 Yuzhong Qu 11
###### Abstract
Entity summarization is the problem of computing an optimal compact summary
for an entity by selecting a size-constrained subset of triples from RDF data.
Entity summarization supports a multiplicity of applications and has led to
fruitful research. However, there is a lack of evaluation efforts that cover
the broad spectrum of existing systems. One reason is a lack of benchmarks for
evaluation. Some benchmarks are no longer available, while others are small
and have limitations. In this paper, we create an Entity Summarization
BenchMark (ESBM) which overcomes the limitations of existing benchmarks and
meets standard desiderata for a benchmark. Using this largest available
benchmark for evaluating general-purpose entity summarizers, we perform the
most extensive experiment to date where 9 existing systems are compared.
Considering that all of these systems are unsupervised, we also implement and
evaluate a supervised learning based system for reference.
###### Keywords:
Entity summarization Triple ranking Benchmarking.
## 1 Introduction
RDF data describes entities with triples representing property values. In an
RDF dataset, the description of an entity comprises all the RDF triples where
the entity appears as the subject or the object. An example entity description
is shown in Fig. 1. Entity descriptions can be large. An entity may be
described in dozens or hundreds of triples, exceeding the capacity of a
typical user interface. A user served with all of those triples may suffer
information overload and find it difficult to quickly identify the small set
of triples that are truly needed. To solve the problem, an established
research topic is _entity summarization_ [15], which aims to compute an
optimal compact summary for the entity by selecting a size-constrained subset
of triples. An example entity summary under the size constraint of 5 triples
is shown in the bottom right corner of Fig. 1.
Figure 1: Description of entity Tim Berners-Lee and a summary thereof.
Entity summarization supports a multiplicity of applications [6, 21]. Entity
summaries constitute entity cards displayed in search engines [9], provide
background knowledge for enriching documents [26], and facilitate research
activities with humans in the loop [3, 4]. This far-reaching application has
led to fruitful research as reviewed in our recent survey paper [15]. Many
entity summarizers have been developed, most of which generate summaries for
general purposes.
Research Challenges. However, two challenges face the research community.
First, there is a _lack of benchmarks_ for evaluating entity summarizers. As
shown in Table 1, some benchmarks are no longer available. Others are
available [22, 7, 8] but they are small and have limitations. Specifically,
[22] has a task-specific nature, and [7, 8] exclude classes and/or literals.
These benchmarks could not support a comprehensive evaluation of general-
purpose entity summarizers. Second, there is a _lack of evaluation efforts_
that cover the broad spectrum of existing systems to compare their performance
and assist practitioners in choosing solutions appropriate to their
applications.
Contributions. We address the challenges with two contributions. First, we
create an Entity Summarization BenchMark (ESBM) which overcomes the
limitations of existing benchmarks and meets the desiderata for a successful
benchmark [18]. ESBM has been published on GitHub with extended documentation
and a permanent identifier on w3id.org111https://w3id.org/esbm under the ODC-
By license. As the largest available benchmark for evaluating general-purpose
entity summarizers, ESBM contains 175 heterogeneous entities sampled from two
datasets, for which 30 human experts create 2,100 general-purpose ground-truth
summaries under two size constraints. Second, using ESBM, we evaluate 9
existing general-purpose entity summarizers. It represents the most extensive
evaluation effort to date. Considering that existing systems are unsupervised,
we also implement and evaluate a supervised learning based entity summarizer
for reference.
In this paper, for the first time we comprehensively describe the creation and
use of ESBM. We report ESBM v1.2—the latest version, while early versions have
successfully supported the entity summarization shared task at the EYRE 2018
workshop222https://sites.google.com/view/eyre18/sharedtasks and the EYRE 2019
workshop.333https://sites.google.com/view/eyre19/sharedtasks We will also
educate on the use of ESBM at an ESWC 2020 tutorial on entity
summarization444https://sites.google.com/view/entity-summarization-
tutorials/eswc2020.
The remainder of the paper is organized as follows. Section 2 reviews related
work and limitations of existing benchmarks. Section 3 describes the creation
of ESBM, which is analyzed in Section 4. Section 5 presents our evaluation. In
Section 6 we discuss limitations of our study and perspectives for future
work.
Table 1: Existing benchmarks for evaluating entity summarization. | Dataset | Number of entities | Availability
---|---|---|---
WhoKnows?Movies! [22] | Freebase | 60 | Available111http://yovisto.com/labs/iswc2012
Langer et al. [13] | DBpedia | 14 | Unavailable
FRanCo [1] | DBpedia | 265 | Unavailable
Benchmark for evaluating RELIN [2] | DBpedia | 149 | Unavailable
Benchmark for evaluating DIVERSUM [20] | IMDb | 20 | Unavailable
Benchmark for evaluating FACES [7] | DBpedia | 50 | Available222http://wiki.knoesis.org/index.php/FACES
Benchmark for evaluating FACES-E [8] | DBpedia | 80 | Available222http://wiki.knoesis.org/index.php/FACES
## 2 Related Work
We review methods and evaluation efforts for entity summarization.
Methods for Entity Summarization. In a recent survey [15] we have categorized
the broad spectrum of research on entity summarization. Below we briefly
review _general-purpose_ entity summarizers which mainly rely on generic
technical features that can apply to a wide range of domains and applications.
We will not address methods that are domain-specific (e.g., for movies [25] or
timelines [5]), task-specific (e.g., for facilitating entity resolution [3] or
entity linking [4]), or context-aware (e.g., contextualized by a document [26]
or a query [9]).
RELIN [2] uses a weighted PageRank model to rank triples according to their
statistical informativeness and relatedness. DIVERSUM [20] ranks triples by
property frequency and generates a summary with a strong constraint that
avoids selecting triples having the same property. SUMMARUM [24] and LinkSUM
[23] mainly rank triples by the PageRank scores of property values that are
entities. LinkSUM also considers backlinks from values. FACES [7], and its
extension FACES-E [8] which adds support for literals, cluster triples by
their bag-of-words based similarity and choose top-ranked triples from as many
different clusters as possible. Triples are ranked by statistical
informativeness and property value frequency. CD [28] models entity
summarization as a quadratic knapsack problem that maximizes the statistical
informativeness of the selected triples and in the meantime minimizes the
string, numerical, and logical similarity between them. In ES-LDA [17], ES-
LDAext [16], and MPSUM [27], a Latent Dirichlet Allocation (LDA) model is
learned where properties are treated as topics, and each property is a
distribution over all the property values. Triples are ranked by the
probabilities of properties and values. MPSUM further avoids selecting triples
having the same property. BAFREC [12] categorizes triples into meta-level and
data-level. It ranks meta-level triples by their depths in an ontology and
ranks data-level triples by property and value frequency. Triples having
textually similar properties are penalized to improve diversity. KAFCA [11]
ranks triples by the depths of properties and values in a hierarchy
constructed by performing the Formal Concept Analysis (FCA). It tends to
select triples containing infrequent properties but frequent values, where
frequency is computed at the word level.
Limitations of Existing Benchmarks. For evaluating entity summarization,
compared with task completion based _extrinsic evaluation_ , ground truth
based _intrinsic evaluation_ is more popular because it is easy to perform and
the results are reproducible. Its idea is to create a benchmark consisting of
human-made ground-truth summaries, and then compute how much a machine-
generated summary is close to a ground-truth summary.
Table 1 lists known benchmarks, including dedicated benchmarks [22, 13, 1] and
those created for evaluating a particular entity summarizer [2, 20, 7, 8]. It
is not surprising that these benchmarks are not very large since it is
expensive to manually create high-quality summaries for a large set of
entities. Unfortunately, some of these benchmarks are not publicly available
at this moment. Three are available [22, 7, 8] but they are relatively small
and have limitations. Specifically, WhoKnows?Movies! [22] is not a set of
ground-truth summaries but annotates each triple with the ratio of movie
questions that were correctly answered based on that triple, as an indicator
of its importance. This kind of task-specific ground truth may not be suitable
for evaluating general-purpose entity summarizers. The other two available
benchmarks were created for evaluating FACES/-E [7, 8]. Classes and/or
literals are not included because they could not be processed by FACES/-E and
hence were filtered out. Such benchmarks could not comprehensively evaluate
most of the existing entity summarizers [2, 20, 28, 27, 12, 11] that can
handle classes and literals. These limitations of available benchmarks
motivated us to create a new ground truth consisting of _general-purpose
summaries_ for a _larger set of entities_ involving _more comprehensive
triples_ where property values can be entities, classes, or literals.
## 3 Creating ESBM
To overcome the above-mentioned limitations of existing benchmarks, we created
a new benchmark called ESBM. To date, it is the largest available benchmark
for evaluating general-purpose entity summarizers. In this section, we will
first specify our design goals. Then we describe the selection of entity
descriptions and the creation of ground-truth summaries. We partition the data
to support cross-validation for parameter fitting. Finally we summarize how
our design goals are achieved and how ESBM meets standard desiderata for a
benchmark.
### 3.1 Design Goals
The creation of ESBM has two main design goals. First, a successful benchmark
should meet seven desiderata [18]: accessibility, affordability, clarity,
relevance, solvability, portability, and scalability, which we will detail in
Section 3.5. Our design of ESBM aims to satisfy these basic requirements.
Second, in Section 2 we discussed the limitations of available benchmarks,
including task specificness, small size, and triple incomprehensiveness.
Besides, all the existing benchmarks use a single dataset and hence may weaken
the generalizability of evaluation results. We aim to overcome these
limitations when creating ESBM. In Section 3.5 we will summarize how our
design goals are achieved.
### 3.2 Entity Descriptions
To choose entity descriptions to summarize, we sample entities from selected
datasets and filter their triples. The process is detailed below.
Datasets. We sample entities from two datasets of different kinds: an
encyclopedic dataset and a domain-specific dataset. For the encyclopedic
dataset we choose DBpedia [14], which has been used in other benchmarks [13,
1, 2, 7, 8]. We use the English version of DBpedia
2015-10555http://wiki.dbpedia.org/dbpedia-dataset-version-2015-10—the latest
version when we started to create ESBM. For the domain-specific dataset we
choose LinkedMDB [10], which is a popular movie database. The movie domain is
also the focus of some existing benchmarks [22, 20] possibly because this
domain is familiar to the lay audience so that it would be easy to find
qualified human experts to create ground-truth summaries. We use the latest
available version of
LinkedMDB.666http://www.cs.toronto.edu/~oktie/linkedmdb/linkedmdb-latest-
dump.zip
Entities. For DBpedia we sample entities from five large classes: Agent,
Event, Location, Species, and Work. They collectively contain 3,501,366
entities (60%) in the dataset. For LinkedMDB we sample from Film and Person,
which contain 159,957 entities (24%) in the dataset. Entities from different
classes are described by very different properties as we will see in Section
4.3, and hence help to assess the generalizability of an entity summarizer.
According to the human efforts we could afford, from each class we randomly
sample 25 entities. The total number of selected entities is 175. Each
selected entity should be described in at least 20 triples so that
summarization would not be a trivial task. This requirement follows common
practice in the literature [1, 2, 20, 7] where a minimum constraint in the
range of 10–20 was posed.
(a) Average number of triples describing an entity.
(b) Average number of distinct properties describing an entity.
Figure 2: Composition of entity descriptions (the left bar in each group),
top-5 ground-truth summaries (the middle bar), and top-10 ground-truth
summaries (the right bar), grouped by class in DBpedia (D) and LinkedMDB (L).
Triples. For DBpedia, entity descriptions comprise triples in the following
dump files: _instance types_ , _instance types transitive_ , _YAGO types_ ,
_mappingbased literals_ , _mappingbased objects_ , _labels_ , _images_ ,
_homepages_ , _persondata_ , _geo coordinates mappingbased_ , and _article
categories_. We do not import dump files that provide metadata about Wikipedia
articles such as _page links_ and _page length_. We do not import _short
abstracts_ and _long abstracts_ as they provide handcrafted textual entity
summaries; it would be inappropriate to include them in a benchmark for
evaluating entity summarization. For LinkedMDB we import all the triples in
the dump file except sameAs links which do not express facts about entities
but are of more technical nature. Finally, as shown in Fig. 2a (the left bar
in each group), the mean number of triples in an entity description is in the
range of 25.88–52.44 depending on the class, and the overall mean value is
37.62.
### 3.3 Ground-Truth Summaries
We invite 30 researchers and students to create ground-truth summaries for
entity descriptions. All the participants are familiar with RDF.
Task Assignment. Each participant is assigned 35 entities consisting of 5
entities randomly selected from each of the 7 classes in ESBM. The assignment
is controlled to ensure that each entity in ESBM is processed by 6
participants. A participant creates two summaries for each entity description
by selecting different numbers of triples: a _top-5 summary_ containing 5
triples, and a _top-10 summary_ containing 10 triples. Therefore, we will be
able to evaluate entity summarizers under different size constraints. The
choice of these two numbers follows previous work [2, 7, 8]. Participants work
independently and they may create different summaries for an entity. It is not
feasible to ask participants to reach an agreement. It is also not reasonable
to merge different summaries into a single version. So we keep different
summaries and will use all of them in the evaluation. The total number of
ground-truth summaries is $175\cdot 6\cdot 2=2,100$.
Figure 3: User interface for creating ground-truth entity summaries.
Procedure. Participants are instructed to create _general-purpose summaries_
that are not specifically created for any particular task. They read and
select triples using a Web-based user interface shown in Fig. 3. All the
triples in an entity description are listed in random order but those having a
common property are placed together for convenient reading and comparison. For
IRIs, their human-readable labels (rdfs:label) are shown if available. To help
participants understand a property value that is an unfamiliar entity, a click
on it will open a pop-up showing a short textual description extracted from
the first paragraph of its Wikipedia/IMDb page. Any triple can be selected
into the top-5 summary, the top-10 summary, or both. The top-5 summary is not
required to be a subset of the top-10 summary.
### 3.4 Training, Validation, and Test Sets
Some entity summarizers need to tune hyperparameters or fit models. To make
their evaluation results comparable with each other, we specify a split of our
data into training, validation, and test sets. We provide a partition of the
175 entities in ESBM into 5 equally sized subsets $P_{0},\ldots,P_{4}$ to
support 5-fold cross-validation. Entities of each class are partitioned evenly
among the subsets. For $0\leq i\leq 4$, the $i$-th fold uses
$P_{i},P_{i+1\text{ mod }5},P_{i+2\text{ mod }5}$ as the training set (e.g.,
for model fitting), uses $P_{i+3\text{ mod }5}$ for validation (e.g., tuning
hyperparameters), and retains $P_{i+4\text{ mod }5}$ as the test set.
Evaluation results are averaged over the 5 folds.
### 3.5 Conclusion
ESBM overcomes the limitations of available benchmarks discussed in Section 2.
It contains 175 entities which is 2–3 times as large as available benchmarks
[22, 7, 8]. In ESBM, property values are not filtered as in [7, 8] but can be
any entity, class, or literal. Different from the task-specific nature of
[22], ESBM provides general-purpose ground-truth summaries for evaluating
general-purpose entity summarizers.
Besides, ESBM meets the seven desiderata proposed in [18] as follows.
* •
Accessibility. ESBM is publicly available and has a permanent identifier on
w3id.org.
* •
Affordability. ESBM is with an open-source program and example code for
evaluation. The cost of using ESBM is minimized.
* •
Clarity. ESBM is documented clearly and concisely.
* •
Relevance. ESBM samples entities from two real datasets that have been widely
used. The summarization tasks are natural and representative.
* •
Solvability. An entity description in ESBM has at least 20 triples and a mean
number of 37.62 triples, from which 5 or 10 triples are to be selected. The
summarization tasks are not trivial and not too difficult.
* •
Portability. ESBM can be used to evaluate any general-purpose entity
summarizer that can process RDF data.
* •
Scalability. ESBM samples 175 entities from 7 classes. It is reasonably large
and diverse to evaluate mature entity summarizers but is not too large to
evaluate research prototypes.
However, ESBM has its own limitations, which we will discuss in Section 6.
## 4 Analyzing ESBM
In this section, we will first characterize ESBM by providing some basic
statistics and analyzing the triple composition and heterogeneity of entity
descriptions. Then we compute inter-rater agreement to show how much consensus
exists in the ground-truth summaries given by different participants.
### 4.1 Basic Statistics
The 175 entity descriptions in ESBM collectively contain 6,584 triples, of
which 37.44% are selected into at least one top-5 summary and 58.15% appear in
at least one top-10 summary, showing a wide selection by the participants.
However, many of them are selected only by a single participant; 20.46% and
40.23% are selected by different participants into top-5 and top-10 summaries,
respectively. We will further analyze inter-rater agreement in Section 4.4.
We calculate the overlap between the top-5 and the top-10 summaries created by
the same participant for the same entity. The mean overlap is in the range of
4.80–4.99 triples depending on the class, and the overall mean value is 4.91,
showing that the top-5 summary is usually a subset of the top-10 summary.
### 4.2 Triple Composition
In Fig. 2 we present the composition of entity descriptions (the left bar in
each group) and their ground-truth summaries (the middle bar for top-5 and the
right bar for top-10) in ESBM, in terms of the average number of triples
describing an entity (Fig. 2a) and in terms of the average number of distinct
properties describing an entity (Fig. 2b). Properties are divided into
literal-valued, class-valued, and entity-valued. Triples are divided
accordingly.
In Fig. 2a, both class-valued and entity-valued triples occupy a considerable
proportion of the entity descriptions in DBpedia. Entity-valued triples
predominate in LinkedMDB. Literal-valued triples account for a small
proportion in both datasets. However, they constitute 30% in top-5 ground-
truth summaries and 25% in top-10 summaries. Entity summarizers that cannot
process literals [24, 23, 7, 17] have to ignore these notable proportions,
thereby significantly influencing their performance.
In Fig. 2b, in terms of distinct properties, entity-valued and literal-valued
triples have comparable numbers in entity descriptions since many entity-
valued properties are multi-valued. Specifically, an entity is described by
13.24 distinct properties, including 5.31 literal-valued (40%) and 6.93
entity-valued (52%). Multi-valued properties appear in every entity
description and they constitute 35% of the triples. However, in top-5 ground-
truth summaries, the average number of distinct properties is 4.70 and is very
close to 5, indicating that the participants are not inclined to select
multiple values of a property. Entity summarizers that prefer diverse
properties [20, 7, 8, 28, 27, 12] may exhibit good performance.
Figure 4: Jaccard similarity between property sets describing different
classes. Table 2: Popular properties in ground-truth summaries.
In top-5 summaries | In top-10 summaries
---|---
Agent | Event | Location | Species | Work | Film | Person | Agent | Event | Location | Species | Work | Film | Person
type | type | type | type | type | director | type | type | type | type | family | type | director | type
birthDate | date | country | family | | type | actor | subject | subject | country | type | subject | actor | actor
| | | | | | | birthDate | date | subject | order | genre | type | label
| | | | | | | | label | | class | | writer | page
| | | | | | | | | | genus | | producer |
| | | | | | | | | | subject | | date |
| | | | | | | | | | kingdom | | language |
### 4.3 Entity Heterogeneity
Entities from different classes are described by different sets of properties.
For each class we identify the set of properties describing at least one
entity from the class. The Jaccard similarity between properties sets for each
pair of classes is very low, as shown in Fig. 4. Such heterogeneous entity
descriptions help to assess the generalizability of an entity summarizer.
Table 2 shows popular properties that appear in at least 50% of the ground-
truth summaries for each class. Some universal properties like rdf:type and
dct:subject are popular for most classes. We also see class-specific
properties, e.g., dbo:birthDate for Agent, dbo:family for Species. However,
the results suggest that it would be unrealistic to generate good summaries by
manually selecting properties for each class. For example, among 13.24
distinct properties describing an entity, only 1–2 are popular in top-5
ground-truth summaries. The importance of properties is generally
contextualized by concrete entities.
### 4.4 Inter-Rater Agreement
Recall that each entity in ESBM has six top-5 ground-truth summaries and six
top-10 summaries created by different participants. We calculate the average
overlap between these summaries in terms of the number of common triples they
contain. As shown in Table 3, the results are generally comparable with those
reported for other benchmarks in the literature. There is a moderate degree of
agreement between the participants.
Table 3: Inter-rater agreement. | ESBM | [2] | [7] | [8]
---|---|---|---|---
Overlap between top-5 summaries | 1.99 (39.8$\%$) | 2.91 (58.2$\%$) | 1.92 (38.4$\%$) | 2.12 (42.4$\%$)
Overlap between top-10 summaries | 5.42 (54.2$\%$) | 7.86 (78.6$\%$) | 4.64 (46.4$\%$) | 5.44 (54.4$\%$)
Ground-truth summaries per entity | 6 | 4.43 | $\geq$ 7 | $\geq$ 4
## 5 Evaluating with ESBM
We used ESBM to perform the most extensive evaluation of general-purpose
entity summarizers to date. In this section, we will first describe evaluation
criteria. Then we introduce the entity summarizers that we evaluate. Finally
we present evaluation results.
### 5.1 Evaluation Criteria
Let $S_{m}$ be a machine-generated entity summary. Let $S_{h}$ be a human-made
ground-truth summary. To compare $S_{m}$ with $S_{h}$ and assess the quality
of $S_{m}$ based on how much $S_{m}$ is close to $S_{h}$, it is natural to
compute precision (P), recall (R), and F1. The results are in the range of
0–1:
$\text{P}=\frac{|S_{m}\cap S_{h}|}{|S_{m}|}\,,\quad\text{R}=\frac{|S_{m}\cap
S_{h}|}{|S_{h}|}\,,\quad\text{F1}=\frac{2\cdot\text{P}\cdot\text{R}}{\text{P}+\text{R}}\,.$
(1)
In the experiments we configure entity summarizers to output at most $k$
triples and we set $k=|S_{h}|$, i.e., $k=5$ and $k=10$ are our two settings
corresponding to the sizes of ground-truth summaries. We will trivially have
P$=$R$=$F1 if $|S_{m}|=|S_{h}|$. However, some entity summarizers may output
less than $k$ triples. For example, DIVERSUM [20] disallows an entity summary
to contain triples having the same property. It is possible that an entity
description contains less than $k$ distinct properties and hence DIVERSUM has
to output less than $k$ triples. In this case, P$\neq$R and one should rely on
F1.
In the evaluation, for each entity in ESBM, we compare a machine-generated
summary with each of the 6 ground-truth summaries by calculating F1, and take
their aggregation value. Finally we report the mean F1 over all the entities.
For aggregation function, we report the results of average, to show an overall
match with all the different ground truths; on the website we also give the
results of maximum, to show the best match with each individual ground truth.
### 5.2 Participating Entity Summarizers
We not only evaluate existing entity summarizers but also compare them with
two special entity summarizers we create: an oracle entity summarizer which is
used to show the best possible performance on ESBM, and a new supervised
learning based entity summarizer.
Existing Entity Summarizers. We evaluate 9 out of the 12 general-purpose
entity summarizers reviewed in Section 2. We re-implement RELIN [2], DIVERSUM
[20], LinkSUM [23], FACES [7], FACES-E [8], and CD [28], while MPSUM [27],
BAFREC [12], and KAFCA [11] are open source. We exclude SUMMARUM [24], ES-LDA
[17], and ES-LDAext [16] because LinkSUM represents an extension of SUMMARUM,
and MPSUM represents an extension of ES-LDA and ES-LDAext.
We follow the original implementation and suggested configuration of existing
entity summarizers as far as possible. However, for RELIN, we replace its
Google-based relatedness measure with a string metric [19] because Google’s
search API is no longer free. We also use this metric to replace the
unavailable UMBC’s SimService used in FACES-E. For DIVERSUM, we ignore its
witness count measure since it does not apply to ESBM. For LinkSUM, we obtain
backlinks between entities in LinkedMDB via their corresponding entities in
DBpedia.
RELIN, CD, and LinkSUM compute a weighted combination of two scoring
components. We tune these hyperparameters in the range of 0–1 in 0.01
increments. Since these summarizers are unsupervised, we use both the training
set and the validation set described in Section 3.4 for tuning
hyperparameters.
Oracle Entity Summarizer. We implement an entity summarizer denoted by ORACLE
to approximate the best possible performance on ESBM and form a reference
point used for comparisons. ORACLE simply outputs $k$ triples that are
selected by the most participants into ground-truth summaries.
Supervised Learning Based Entity Summarizer. Existing general-purpose entity
summarizers are unsupervised. We implement a supervised learning based entity
summarizer with features that are used by existing entity summarizers. A
triple with property $p$ and value $v$ describing entity $e$ is represented by
the following features:
* •
$\mathtt{gf}_{\mathbb{T}}$: the number of triples in the dataset where $p$
appears [23, 12],
* •
$\mathtt{lf}$: the number of triples in the description of $e$ where $p$
appears [20, 23],
* •
$\mathtt{vf}_{\mathbb{T}}$: the number of triples in the dataset where $v$
appears [7, 8, 12], and
* •
$\mathtt{si}$: the self-information of the triple [2, 7, 8, 28].
We also add three binary features:
* •
$\mathtt{isC}$: whether $v$ is a class,
* •
$\mathtt{isE}$: whether $v$ is an entity, and
* •
$\mathtt{isL}$: whether $v$ is a literal.
Based on the training and validation sets described in Section 3.4, we
implement and tune 6 pointwise learning to rank models provided by Weka:
SMOreg, LinearRegression, MultilayerPerceptron, AdditiveRegression, REPTree,
and RandomForest. Each model outputs $k$ top-ranked triples as a summary.
### 5.3 Evaluation Results
We first report the overall evaluation results to show which entity summarizer
generally performs better. Then we break down the results into different
entity types (i.e., classes) for detailed comparison. Finally we present and
analyze the performance of our supervised learning based entity summarizer.
Table 4: Average F1 over all the entities in a dataset. For the nine existing
entity summarizers, significant improvements and losses over each other are
indicated by $\blacktriangle$ and $\blacktriangledown$ ($p<0.05$),
respectively. Insignificant differences are indicated by $\circ$.
| DBpedia | LinkedMDB
---|---|---
| $k=5$ | $k=10$ | $k=5$ | $k=10$
RELIN | 0.242 ${}^{\text{-}\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.455 ${}^{\text{-}\blacktriangledown\circ\circ\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.203 ${}^{\text{-}\circ\circ\blacktriangledown\circ\blacktriangle\blacktriangledown\circ\blacktriangledown}$ | 0.258 ${}^{\text{-}\blacktriangledown\circ\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$
DIVERSUM | 0.249 ${}^{\circ\text{-}\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.507 ${}^{\blacktriangle\text{-}\blacktriangle\circ\circ\circ\circ\circ\circ}$ | 0.207 ${}^{\circ\text{-}\circ\blacktriangledown\circ\blacktriangle\blacktriangledown\circ\blacktriangledown}$ | 0.358 ${}^{\blacktriangle\text{-}\blacktriangle\circ\circ\blacktriangle\blacktriangledown\circ\blacktriangledown}$
FACES | 0.270 ${}^{\circ\circ\text{-}\circ\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.428 ${}^{\circ\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.169 ${}^{\circ\circ\text{-}\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.263 ${}^{\circ\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$
FACES-E | 0.280 ${}^{\blacktriangle\circ\circ\text{-}\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.488 ${}^{\circ\circ\blacktriangle\text{-}\circ\circ\circ\circ\circ}$ | 0.313 ${}^{\blacktriangle\blacktriangle\blacktriangle\text{-}\blacktriangle\blacktriangle\blacktriangledown\blacktriangle\circ}$ | 0.393 ${}^{\blacktriangle\circ\blacktriangle\text{-}\blacktriangle\blacktriangle\circ\circ\circ}$
CD | 0.283 ${}^{\blacktriangle\blacktriangle\circ\circ\text{-}\circ\blacktriangledown\circ\circ}$ | 0.513 ${}^{\blacktriangle\circ\blacktriangle\circ\text{-}\circ\circ\circ\circ}$ | 0.217 ${}^{\circ\circ\blacktriangle\blacktriangledown\text{-}\blacktriangle\blacktriangledown\circ\blacktriangledown}$ | 0.331 ${}^{\blacktriangle\circ\blacktriangle\blacktriangledown\text{-}\blacktriangle\blacktriangledown\blacktriangledown\blacktriangledown}$
LinkSUM | 0.287 ${}^{\blacktriangle\blacktriangle\circ\circ\circ\text{-}\blacktriangledown\circ\circ}$ | 0.486 ${}^{\circ\circ\blacktriangle\circ\circ\text{-}\circ\circ\circ}$ | 0.140 ${}^{\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.279 ${}^{\circ\blacktriangledown\circ\blacktriangledown\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\blacktriangledown}$
BAFREC | 0.335 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\text{-}\circ\circ}$ | 0.503 ${}^{\blacktriangle\circ\blacktriangle\circ\circ\circ\text{-}\circ\circ}$ | 0.360 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\text{-}\blacktriangle\blacktriangle}$ | 0.402 ${}^{\blacktriangle\blacktriangle\blacktriangle\circ\blacktriangle\blacktriangle\text{-}\circ\circ}$
KAFCA | 0.314 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\circ\circ\circ\text{-}\circ}$ | 0.509 ${}^{\blacktriangle\circ\blacktriangle\circ\circ\circ\circ\text{-}\circ}$ | 0.244 ${}^{\circ\circ\blacktriangle\blacktriangledown\circ\blacktriangle\blacktriangledown\text{-}\circ}$ | 0.397 ${}^{\blacktriangle\circ\blacktriangle\circ\blacktriangle\blacktriangle\circ\text{-}\circ}$
MPSUM | 0.314 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\circ\circ\circ\circ\text{-}}$ | 0.512 ${}^{\blacktriangle\circ\blacktriangle\circ\circ\circ\circ\circ\text{-}}$ | 0.272 ${}^{\blacktriangle\blacktriangle\blacktriangle\circ\blacktriangle\blacktriangle\blacktriangledown\circ\text{-}}$ | 0.423 ${}^{\blacktriangle\blacktriangle\blacktriangle\circ\blacktriangle\blacktriangle\circ\circ\text{-}}$
ORACLE | 0.595 | 0.713 | 0.619 | 0.678
SMOreg | 0.279 | 0.543 | 0.403 | 0.472
LinearRegression | 0.319 | 0.556 | 0.401 | 0.471
MultilayerPerceptron | 0.340 | 0.560 | 0.390 | 0.477
AdditiveRegression | 0.345 | 0.558 | 0.415 | 0.510
REPTree | 0.392 | 0.570 | 0.455 | 0.538
RandomForest | 0.399 | 0.576 | 0.449 | 0.506
Overall Results of Existing Entity Summarizers. Table 4 presents the results
of all the participating entity summarizers on two datasets under two size
constraints. We compare nine existing summarizers using one-way ANOVA post-hoc
LSD and we show whether the difference between each pair of them is
statistical significant at the 0.05 level. Among existing summarizers, BAFREC
achieves the highest F1 under $k=5$. It significantly outperforms six existing
summarizers on DBpedia and outperforms all the eight ones on LinkedMDB. It is
also among the best under $k=10$. MPSUM follows BAFREC under $k=5$ but
performs slightly better under $k=10$. Other top-tier results belong to KAFCA
on DBpedia and FACES-E on LinkedMDB.
The F1 scores of ORACLE are in the range of 0.595–0.713. It is impossible for
ORACLE or any other summarizer to reach $\text{F1}=1$, because for each entity
in ESBM there are six ground-truth summaries which are often different and
hence cannot simultaneously match a machine-generated summary. However, the
gap between the results of ORACLE and the best results of existing summarizers
is still as large as 0.20–0.26, suggesting that there is much room for
improvement.
Results on Different Entity Types. We break down the results of existing
entity summarizers into 7 entity types (i.e., classes). When $k=5$ in Fig. 5,
there is no single winner on every class, but BAFREC and MPSUM are among top
three on 6 classes, showing relatively good generalizability over different
entity types. Some entity summarizers have limited generalizability and they
perform not well on certain classes. For example, RELIN and CD mainly rely on
the self-information of a triple, while for Location entities their latitudes
and longitudes are often unique in DBpedia but such triples with large self-
information rarely appear in ground-truth summaries. Besides, most summarizers
generate low-quality summaries for Agent, Film, and Person entities. This is
not surprising since these entities are described in more triples and/or by
more properties according to Fig. 2. Their summarization is inherently more
difficult. When $k=10$ in Fig. 6, MPSUM is still among top three on 6 classes.
KAFCA also shows relatively good generalizability—among top three on 5
classes.
Figure 5: Average F1 over all the entities in each class under $k=5$. Figure
6: Average F1 over all the entities in each class under $k=10$.
Results of Supervised Learning. As shown in Table 4, among the six supervised
learning based methods, RandomForest and REPTree achieve the highest F1 on
DBpedia and LinkedMDB, respectively. Four methods (MultilayerPerceptron,
AdditiveRegression, REPTree, and RandomForest) outperform all the existing
entity summarizers on both datasets under both size constraints, and two
methods (SMOreg and LinearRegression) only fail to outperform in one setting.
The results demonstrate the powerfulness of supervised learning for entity
summarization. Further, recall that these methods only use standard models and
rely on features that are used by existing entity summarizers. It would be
reasonable to predict that better results can be achieved with specialized
models and more advanced features. However, creating a large number of ground-
truth summaries for training is expensive, and the generalizability of
supervised methods for entity summarization still needs further exploration.
Moreover, we are interested in how much the seven features contribute to the
good performance of supervised learning. Table 5 shows the results of
RandomForest after removing each individual feature. Considering statistical
significance at the 0.05 level, two features $\mathtt{gf}_{\mathbb{T}}$ and
$\mathtt{lf}$ show effectiveness on both datasets under both size constraints,
and two features $\mathtt{vf}_{\mathbb{T}}$ and $\mathtt{si}$ are only
effective on LinkedMDB. The usefulness of the three binary features
$\mathtt{isC}$, $\mathtt{isE}$, and $\mathtt{isL}$ is not statistically
significant.
Table 5: F1 of RandomForest after removing each individual feature, its
difference from using all features ($\Delta\%$), and the significance level
for the difference ($p$).
DBpedia | LinkedMDB
---|---
$k=5$ | $k=10$ | $k=5$ | $k=10$
| F1 | $\Delta\%$ | $p$ | | F1 | $\Delta\%$ | $p$ | | F1 | $\Delta\%$ | $p$ | | F1 | $\Delta\%$ | $p$
All | 0.399 | — | — | All | 0.576 | — | — | All | 0.449 | — | — | All | 0.506 | — | —
-$\mathtt{gf}_{\mathbb{T}}$ | 0.346 | $-$5.360 | 0.000 | -$\mathtt{lf}$ | 0.546 | $-$0.030 | 0.000 | -$\mathtt{gf}_{\mathbb{T}}$ | 0.383 | $-$0.066 | 0.000 | -$\mathtt{lf}$ | 0.473 | $-$0.033 | 0.008
-$\mathtt{lf}$ | 0.366 | $-$3.307 | 0.000 | -$\mathtt{gf}_{\mathbb{T}}$ | 0.551 | $-$0.025 | 0.000 | -$\mathtt{lf}$ | 0.413 | $-$0.036 | 0.025 | -$\mathtt{vf}_{\mathbb{T}}$ | 0.477 | $-$0.029 | 0.010
-$\mathtt{isC}$ | 0.392 | $-$0.720 | 0.261 | -$\mathtt{vf}_{\mathbb{T}}$ | 0.569 | $-$0.007 | 0.198 | -$\mathtt{vf}_{\mathbb{T}}$ | 0.414 | $-$0.035 | 0.022 | -$\mathtt{gf}_{\mathbb{T}}$ | 0.479 | $-$0.027 | 0.007
-$\mathtt{isE}$ | 0.397 | $-$0.267 | 0.720 | -$\mathtt{isE}$ | 0.570 | $-$0.006 | 0.262 | -$\mathtt{si}$ | 0.442 | $-$0.007 | 0.574 | -$\mathtt{si}$ | 0.486 | $-$0.020 | 0.009
-$\mathtt{si}$ | 0.400 | $+$0.027 | 0.973 | -$\mathtt{isC}$ | 0.571 | $-$0.005 | 0.303 | -$\mathtt{isE}$ | 0.455 | $+$0.005 | 0.651 | -$\mathtt{isL}$ | 0.491 | $-$0.015 | 0.079
-$\mathtt{isL}$ | 0.401 | $+$0.160 | 0.816 | -$\mathtt{si}$ | 0.572 | $-$0.004 | 0.402 | -$\mathtt{isL}$ | 0.456 | $+$0.007 | 0.504 | -$\mathtt{isE}$ | 0.492 | $-$0.014 | 0.148
-$\mathtt{vf}_{\mathbb{T}}$ | 0.407 | $+$0.720 | 0.346 | -$\mathtt{isL}$ | 0.578 | $+$0.002 | 0.683 | -$\mathtt{isC}$ | 0.463 | $+$0.013 | 0.281 | -$\mathtt{isC}$ | 0.514 | $+$0.008 | 0.396
Conclusion. Among existing entity summarizers, BAFREC generally shows the best
performance on ESBM while MPSUM seems more robust. However, none of them are
comparable with our straightforward implementation of supervised learning,
which in turn is still far away from the best possible performance represented
by ORACLE. Therefore, entity summarization on ESBM is a non-trivial task. We
invite researchers to experiment with new ideas on ESBM.
## 6 Discussion and Future work
We identify the following limitations of our work to be addressed in future
work.
Evaluation Criteria. We compute F1 score in the evaluation, which is based on
common triples but ignores semantic overlap between triples. A triple $t$ in a
machine-generated summary $S$ may partially cover the information provided by
some triple $t^{\prime}$ in the ground-truth summary. It may be reasonable to
not completely penalize $S$ for missing $t^{\prime}$ but give some reward for
the presence of $t$. However, it is difficult to quantify the extent of
penalization for all possible cases, particularly when multiple triples
semantically overlap with each other. In future work, we will explore more
proper evaluation criteria.
Representativeness of Ground Truth. The ground-truth summaries in ESBM are not
supposed to represent the view of the entire user population. They are
intrinsically biased towards their creators. Besides, these ground-truth
summaries are created for general purposes. Accordingly, we use them to
evaluate general-purpose entity summarizers. However, for a specific task,
these summaries may not show optimality, and the participating systems may not
represent the state of the art. Still, we believe it is valuable to evaluate
general-purpose systems not only because of their wide range of applications
but also because their original technical features have been reused by task-
specific systems. In future work, we will extend ESBM to a larger scale, and
will consider benchmarking task-specific entity summarization.
Form of Ground Truth. ESBM provides ground-truth summaries, whereas some other
benchmarks offer ground-truth scores of triples [22, 13, 1]. Scoring-based
ground truth may more comprehensively evaluate an entity summarizer than our
set-based ground truth because it not only considers the triples in a machine-
generated summary but also assesses the rest of the triples. However, on the
other hand, a set of top-scored triples may not equal an optimal summary
because they may cover limited aspects of an entity and show redundancy.
Therefore, both methods have their advantages and disadvantages. In future
work, we will conduct scoring-based evaluation to compare with the current
results.
## Acknowledgments
This work was supported in part by the NSFC under Grant 61772264 and in part
by the Qing Lan Program of Jiangsu Province.
## References
* [1] Bobic, T., Waitelonis, J., Sack, H.: FRanCo - A ground truth corpus for fact ranking evaluation. In: SumPre 2015 & HSWI 2015 (2015)
* [2] Cheng, G., Tran, T., Qu, Y.: RELIN: relatedness and informativeness-based centrality for entity summarization. In: ISWC 2011, Part I. pp. 114–129 (2011). https://doi.org/10.1007/978-3-642-25073-6_8
* [3] Cheng, G., Xu, D., Qu, Y.: C3D+P: A summarization method for interactive entity resolution. J. Web Sem. 35, 203–213 (2015). https://doi.org/10.1016/j.websem.2015.05.004
* [4] Cheng, G., Xu, D., Qu, Y.: Summarizing entity descriptions for effective and efficient human-centered entity linking. In: WWW 2015. pp. 184–194 (2015). https://doi.org/10.1145/2736277.2741094
* [5] Gottschalk, S., Demidova, E.: EventKG - the hub of event knowledge on the web \- and biographical timeline generation. Semantic Web 10(6), 1039–1070 (2019). https://doi.org/10.3233/SW-190355
* [6] Gunaratna, K.: Semantics-based Summarization of Entities in Knowledge Graphs. Ph.D. thesis, Wright State University (2017)
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* [9] Hasibi, F., Balog, K., Bratsberg, S.E.: Dynamic factual summaries for entity cards. In: SIGIR 2017. pp. 773–782 (2017). https://doi.org/10.1145/3077136.3080810
* [10] Hassanzadeh, O., Consens, M.P.: Linked movie data base. In: LDOW 2009 (2009)
* [11] Kim, E.K., Choi, K.S.: Entity summarization based on formal concept analysis. In: EYRE 2018 (2018)
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|
2024-09-04T02:54:58.169588 | 2020-03-08T07:15:48 | 2003.03736 | {
"authors": "Qingxia Liu, Gong Cheng, Yuzhong Qu",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26103",
"submitter": "Gong Cheng",
"url": "https://arxiv.org/abs/2003.03736"
} | arxiv-papers | 11institutetext: National Key Laboratory for Novel Software Technology,
Nanjing University, China
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# DeepLENS: Deep Learning for Entity Summarization
Qingxia Liu Gong Cheng Yuzhong Qu
###### Abstract
Entity summarization has been a prominent task over knowledge graphs. While
existing methods are mainly unsupervised, we present DeepLENS, a simple yet
effective deep learning model where we exploit textual semantics for encoding
triples and we score each candidate triple based on its interdependence on
other triples. DeepLENS significantly outperformed existing methods on a
public benchmark.
## 1 Introduction
Entity summarization is the task of computing a compact summary for an entity
by selecting an optimal size-constrained subset of entity-property-value
triples from a knowledge graph such as an RDF graph [7]. It has found a wide
variety of applications, for example, to generate a compact entity card from
Google’s Knowledge Graph where an entity may be described in dozens or
hundreds of triples. Generating entity summaries for general purposes has
attracted much research attention, but existing methods are mainly
unsupervised [2, 9, 3, 4, 13, 10, 6, 5, 11]. One research question that
naturally arises is _whether deep learning can much better solve this task_.
To the best of our knowledge, ESA [12] is the only supervised method in the
literature for this task. ESA encodes triples using graph embedding (TransE),
and employs BiLSTM with supervised attention mechanism. Although it
outperformed unsupervised methods, the improvement reported in [12] was rather
marginal, around $+7\%$ compared with unsupervised FACES-E [4] on the ESBM
benchmark [8]. It inspired us to explore more effective deep learning models
for the task of general-purpose entity summarization.
In this short paper, we present DeepLENS,111https://github.com/nju-
websoft/DeepLENS a novel Deep Learning based approach to ENtity Summarization.
DeepLENS uses a simple yet effective model which addresses the following two
limitations of ESA, and thus achieved significantly better results in the
experiments.
1. 1.
Different from ESA which encodes a triple using graph embedding, we use word
embedding because we consider textual semantics more useful than graph
structure for the entity summarization task.
2. 2.
Whereas ESA encodes a set of triples as a sequence and its performance is
sensitive to the chosen order, our aggregation-based representation satisfies
permutation invariance and hence more suitable for entity summarization.
In the remainder of the paper, Section 2 details DeepLENS, Section 3 presents
experiment results, and Section 4 concludes the paper.
## 2 Approach
#### 2.0.1 Problem Statement
An RDF graph $T$ is a set of triples. The _description_ of entity $e$ in $T$,
denoted by $\mathtt{Desc}(e)\subseteq T$, comprises triples where $e$ is the
subject or object. Each triple $t\in\mathtt{Desc}(e)$ describes a property
$\mathtt{prop}(t)$ which is the predicate of $t$, and gives a value
$\mathtt{val}(t)$ which is the object or subject of $t$ other than $e$. For a
size constraint $k$, a _summary_ of $e$ is a subset of triples
$S\subseteq\mathtt{Desc}(e)$ with $|S|\leq k$. We aim to generate an optimal
summary for general purposes.
#### 2.0.2 Overview of DeepLENS
Our approach DeepLENS generates an optimal summary by selecting $k$ most
salient triples. As a supervised approach, it learns salience from labeled
entity summaries. However, two issues remain unsolved. First, knowledge graph
like RDF graph is a mixture of graph structure and textual content. The
effectiveness of a learning-based approach to entity summarization relies on a
_proper representation of entity descriptions of such mixed nature_. Second,
the salience of a triple is not absolute but dependent on the context, i.e.,
the set of other triples in the entity description. It is essential to
_represent their independence_. DeepLENS addresses these issues with the
scoring model presented in Fig. 1. It has three modules which we will detail
below: triple encoding, entity description encoding, and triple scoring.
Finally, the model scores each candidate triple $t\in\mathtt{Desc}(e)$ in the
context of $\mathtt{Desc}(e)$.
Figure 1: Model of DeepLENS.
#### 2.0.3 Triple Encoding
For entity $e$, a triple $t\in\mathtt{Desc}(e)$ provides a property-value pair
$\langle\mathtt{prop}(t),\mathtt{val}(t)\rangle$ of $e$. Previous research
[12] leverages graph embedding to encode the structural features of
$\mathtt{prop}(t)$ and $\mathtt{val}(t)$. By contrast, for the task of entity
summarization we consider textual semantics more important than graph
structure, and we _solely exploit textual semantics_ for encoding $t$.
Specifically, for RDF resource $r$, we obtain its _textual form_ as follows.
For an IRI or a blank node, we retrieve its rdfs:label if it is available,
otherwise we have to use its local name; for a literal, we take its lexical
form. We represent each word in the textual form by a pre-trained word
embedding vector, and we average these vectors over all the words to represent
$r$, denoted by $\text{Embedding}(r)$. For triple $t\in\mathtt{Desc}(e)$, we
generate and concatenate such vector representations for $\mathtt{prop}(t)$
and $\mathtt{val}(t)$ to form $\boldsymbol{t}$, the _initial representation_
of $t$. Then $\boldsymbol{t}$ is fed into a multi-layer perceptron (MLP) to
generate $\boldsymbol{h}$, the _final representation_ of $t$:
$\boldsymbol{t}=\left[\text{Embedding}(\mathtt{prop}(t));~{}\text{Embedding}(\mathtt{val}(t))\right]\,,\quad\boldsymbol{h}=\text{MLP}_{\text{C}}(\boldsymbol{t})\,.\\\
$ (1)
#### 2.0.4 Entity Description Encoding
To score a candidate triple in the context of other triples in the entity
description, previous research [12] captures the independence between triples
in $\mathtt{Desc}(e)$ using BiLSTM to pass information. Triples are fed into
BiLSTM as a sequence. However, $\mathtt{Desc}(e)$ is a set and the triples
lack a natural order. The performance of this model is unfavourably sensitive
to the order of input triples. Indeed, as we will show in the experiments,
different orders could lead to considerably different performance.
To generate a representation for $\mathtt{Desc}(e)$ that is _permutation
invariant_ , we perform aggregation. Specifically, let
$\boldsymbol{t_{1}},\ldots,\boldsymbol{t_{n}}$ be the initial representations
of triples in $\mathtt{Desc}(e)$ computed by Eq. (1). We feed a MLP with each
$\boldsymbol{t_{i}}$ for $1\leq i\leq n$ and generate their final
representations $\boldsymbol{g_{1}},\ldots,\boldsymbol{g_{n}}$, which in turn
are weighted using attention mechanism from $\boldsymbol{h}$ computed by Eq.
(1), the final representation of the candidate triple $t$ to be scored. We
calculate the sum of these weighted representations of triples to represent
$\mathtt{Desc}(e)$, denoted by $\boldsymbol{d}$:
$\boldsymbol{g_{i}}=\text{MLP}_{\text{D}}(\boldsymbol{t_{i}})\,,\quad
a_{i}=\frac{\exp(\cos(\boldsymbol{h},\boldsymbol{g_{i}}))}{\sum_{j}\exp(\cos(\boldsymbol{h},\boldsymbol{g_{j}}))}\,,\quad\boldsymbol{d}=\sum_{i=1}^{n}{a_{i}\boldsymbol{g_{i}}}\,.\\\
$ (2)
The result of summation is not sensitive to the order of triples in
$\mathtt{Desc}(e)$.
#### 2.0.5 Triple Scoring
For each candidate triple $t\in\mathtt{Desc}(e)$ to be scored, we concatenate
its final representation $\boldsymbol{h}$ and the representation
$\boldsymbol{d}$ for $\mathtt{Desc}(e)$. We feed the result into a MLP to
compute the context-based salience score of $t$:
$\mathtt{score}(t|\mathtt{Desc}(e))=\text{MLP}_{\text{S}}(\left[\boldsymbol{h};~{}\boldsymbol{d}\right])\,.$
(3)
Parameters of the entire model are jointly trained based on the mean squared
error loss, supervised by labeled entity summaries.
## 3 Experiments
### 3.1 Datasets
We used ESBM v1.2, the largest available benchmark for evaluating general-
purpose entity summarization.222https://w3id.org/esbm For each of 125 entities
in DBpedia and 50 entities in LinkedMDB, this benchmark provided 6 ground-
truth summaries created by different human experts under $k=5$, and another 6
ground-truth summaries under $k=10$. We used the train-valid-test split
specified in the benchmark to perform five-fold cross-validation.
### 3.2 Participating Methods
We compared DeepLENS with 10 baseline methods.
Unsupervised Methods. We compared with 9 unsupervised methods that had been
tested on ESBM: RELIN [2], DIVERSUM [9], FACES [3], FACES-E [4], CD [13],
LinkSUM [10], BAFREC [6], KAFCA [5], and MPSUM [11]. We directly presented
their results reported on the ESBM website.
Supervised Methods. We compared with ESA [12], the only supervised method in
the literature to our knowledge. We reused its open-source implementation and
configuration.333https://github.com/WeiDongjunGabriel/ESA We fed it with
triples sorted in alphabetical order.
For our approach DeepLENS, we used 300-dimensional fastText [1] word embedding
vectors trained on Wikipedia to generate initial representations of triples.
The numbers of hidden units in $\text{MLP}_{\text{C}}$,
$\text{MLP}_{\text{D}}$, and $\text{MLP}_{\text{S}}$ were [64, 64], [64, 64],
and [64, 64, 64], respectively. All hidden layers used ReLU as activation
function. The final output layer of $\text{MLP}_{\text{S}}$ consisted of one
linear unit. We trained the model using Adam optimizer with learning rate
0.01.
For both ESA and DeepLENS, we performed early stopping on the validation set
to choose the number of training epochs from 1–50.
Oracle Method. ORACLE approximated the best possible performance on ESBM and
formed a reference point used for comparisons. It outputted $k$ triples that
most frequently appeared in ground-truth summaries.
### 3.3 Results
Following ESBM, we compared machine-generated summaries with ground-truth
summaries by calculating F1 score, and reported the mean F1 achieved by each
method over all the test entities in a dataset.
Table 1: Average F1 over all the test entities. Significant and insignificant
differences ($p<0.01$) between DeepLENS and each baseline are indicated by
$\blacktriangle$ and $\circ$, respectively.
| DBpedia | LinkedMDB
---|---|---
| $k=5$ | $k=10$ | $k=5$ | $k=10$
RELIN [2] | 0.242 | 0.455 | 0.203 | 0.258
DIVERSUM [9] | 0.249 | 0.507 | 0.207 | 0.358
FACES [3] | 0.270 | 0.428 | 0.169 | 0.263
FACES-E [4] | 0.280 | 0.488 | 0.313 | 0.393
CD [13] | 0.283 | 0.513 | 0.217 | 0.331
LinkSUM [10] | 0.287 | 0.486 | 0.140 | 0.279
BAFREC [6] | 0.335 | 0.503 | 0.360 | 0.402
KAFCA [5] | 0.314 | 0.509 | 0.244 | 0.397
MPSUM [11] | 0.314 | 0.512 | 0.272 | 0.423
ESA [12] | 0.331 | 0.532 | 0.350 | 0.416
DeepLENS | 0.402 ▲▲▲▲▲▲▲▲▲▲ | 0.574 ▲▲▲▲▲▲▲▲▲▲ | 0.474 ▲▲▲▲▲▲▲▲▲▲ | 0.493 ▲▲▲▲▲▲▲▲▲▲
ORACLE | 0.595 | 0.713 | 0.619 | 0.678
Comparison with Baselines. As shown in Table 1, supervised methods were
generally better than unsupervised methods. Our DeepLENS outperformed all the
baselines including ESA. Moreover, two-tailed t-test showed that all the
differences were statistically significant ($p<0.01$) in all the settings.
DeepLENS achieved new state-of-the-art results on the ESBM benchmark. However,
the notable gaps between DeepLENS and ORACLE suggested room for improvement
and were to be closed by future research.
Table 2: Average F1 over all the test entities achieved by different variants of ESA. | DBpedia | LinkedMDB
---|---|---
| $k=5$ | $k=10$ | $k=5$ | $k=10$
ESA | 0.331 | 0.532 | 0.350 | 0.416
ESA-text | 0.379 | 0.558 | 0.390 | 0.418
ESA-rnd | 0.116$\pm$0.008 | 0.222$\pm$0.007 | 0.113$\pm$0.015 | 0.219$\pm$0.011
Ablation Study. Compared with ESA, we attributed the better performance of
DeepLENS to two improvements in our implementation: the exploitation of
textual semantics, and the permutation invariant representation of triple set.
They were demonstrated by the following ablation study of ESA.
First, we compared two variants of ESA by encoding triples in different ways.
For triple $t$, the original version of ESA encoded the structural features of
$\mathtt{prop}(t)$ and $\mathtt{val}(t)$ using TransE. We implemented ESA-
text, a variant that encoded both $\mathtt{prop}(t)$ and $\mathtt{val}(t)$
using fastText as in our approach. As shown in Table 2, ESA-text slightly
outperformed ESA, showing the usefulness of textual semantics compared with
graph structure used by ESA.
Second, we compared two variants of ESA by feeding with triples in different
orders. The default version of ESA was fed with triples sorted in alphabetical
order for both training and testing. We implemented ESA-rnd, a variant that
was fed with triples in alphabetical order for training but in random order
for testing. We tested ESA-rnd 20 times and reported its mean F1 with standard
deviation. In Table 2, the notable falls from ESA to ESA-rnd showed the
unfavourable sensitivity of BiLSTM used by ESA to the order of input triples.
## 4 Conclusion
We presented DeepLENS, a simple yet effective deep learning model for general-
purpose entity summarization. It has achieved new state-of-the-art results on
the ESBM benchmark, significantly outperforming existing methods. Thus, entity
summarization becomes another research field where a combination of deep
learning and knowledge graph is likely to shine. However, in DeepLENS we only
exploit textual semantics. In future work, we will incorporate ontological
semantics into our model. We will also revisit the usefulness of structural
semantics.
## Acknowledgments
This work was supported by the National Key R&D Program of China under Grant
2018YFB1005100 and by the Qing Lan Program of Jiangsu Province.
## References
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|
2024-09-04T02:54:58.183852 | 2020-03-08T09:26:26 | 2003.03751 | {
"authors": "Nathan Bowler, Ting Su",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26104",
"submitter": "Ting Su",
"url": "https://arxiv.org/abs/2003.03751"
} | arxiv-papers | # Classification of Doubly Distributive skew Hyperfields and Stringent
hypergroups
Nathan Bowler and Ting Su<EMAIL_ADDRESS>Department of
Mathematics, Universität Hamburg, Germany<EMAIL_ADDRESS>Department
of Mathematics, Universität Hamburg, Germany
###### Abstract.
A hypergroup is stringent if $a\boxplus b$ is a singleton whenever $a\neq-b$.
A hyperfield is stringent if the underlying additive hypergroup is. Every
doubly distributive skew hyperfield is stringent, but not vice versa. We
present a classification of stringent hypergroups, from which a classification
of doubly distributive skew hyperfields follows. It follows from our
classification that every such hyperfield is a quotient of a skew field.
###### Key words and phrases:
hypergroup, hyperring, hyperfield, double distributivity
## 1\. Introduction
The notion of hyperfield was first introduced by Krasner in [Kra57, Kra83]. It
is an algebraic structure similar to a field except that its addition
$\boxplus$ is multivalued. In [Vir10], Viro provided an excellent introduction
to and motivation for hyperfields and introduced several good examples of
hyperfields, including the tropical hyperfield $\mathbb{T}_{+}$, the tropical
real hyperfield $\mathbb{TR}$ and the ultratriangle hyperfield
$\mathbb{T}\triangle$. Viro has also illustrated the utility of
$\mathbb{T}_{+}$ for the foundations of tropical geometry in several
interesting papers (cf. [Vir10, Vir11]).
In [BB16], Baker and Bowler presented an algebraic framework which
simultaneously generalizes the notion of linear subspaces, matroids, oriented
matroids, and valuated matroids, and called the resulting objects matroids
over hyperfields. A matroid over a field $F$ corresponds to a subspace of some
$F^{n}$. A $\mathbb{K}$-matroid is just a matroid. An $\mathbb{S}$-matroid is
an oriented matroid. And a $\mathbb{T}\triangle$-matroid is a valuated
matroid, as defined in [DW92].
Baker and Bowler also provided two natural notions of matroids over a
hyperfield $F$, weak $F$-matroids and strong $F$-matroids, and showed that the
two notions coincide when $F$ has a property called double distributivity. A
hyperfield $F$ is doubly distributive if $(a\boxplus b)(c\boxplus
d)=ac\boxplus ad\boxplus bc\boxplus bd$ for any $a,b,c,d\in F$. Fields,
$\mathbb{K}$, $\mathbb{S}$ and $\mathbb{T}\triangle$ are all doubly
distributive. So too are the other two hyperfields mentioned above,
$\mathbb{T}_{+}$ and $\mathbb{TR}$.
It is these the results in tropical geometry and matroid theory which motivate
our interest in doubly distributive hyperfields. More generally, we are also
interested in doubly distributive hyperrings, which were also analysed by
Baker and Bowler. In fact, rather than just hyperfields, they worked with a
more general kind of algebraic object known as tracts (cf. [BB19]). The other
important example of tracts other than hyperfields is given by partial fields,
which have also been the subject of much fruitful study. Baker and Bowler
defined a special class of tracts called partial hyperfields, objects based on
hyperrings which generalize both hyperfields and partial fields in a natural
way. The property of double distributivity also extends to hyperrings and thus
to partial hyperfields.
We will classify the doubly distributive skew hyperfields in Section 5. The
classification itself will be described in Section 4, but has the following
important consequence:
###### Definition 1.1.
A valuation $\nu$ of a skew hyperfield $F$ is a map from $F$ to
$G\cup\\{-\infty\\}$, where $(G,<)$ is a linearly ordered group, satisfying
1. (1)
$\nu(x)=-\infty$ if and only if $x=0$.
2. (2)
$\nu(xy)=\nu(x)\cdot\nu(y)$.
3. (3)
$\nu(x)>\nu(y)$ implies $x\boxplus y=\\{x\\}$.
###### Theorem 1.2.
For every doubly distributive skew hyperfield $F$, there is always a valuation
$\nu$ of $F$ such that $\nu^{-1}(1_{G})$ is either the Krasner hyperfield, or
the sign hyperfield, or a skew field.
This compact description is from the paper [BP19].
In particular, since any nontrivial ordered group is infinite, it follows from
our results that the only finite doubly distributive hyperfields are the
Krasner hyperfield, the sign hyperfield and the finite fields.
This classification has a number of applications. For example, we use it in
Section 7 to show that any doubly distributive skew hyperfield is a quotient
of a skew field. Bowler and Pendavingh used it in [BP19] to show that any
doubly distributive skew hyperfield is perfect and to provide vector axioms
for matroids over such skew hyperfields.
Our classification uses a property of the underlying hypergroup which we call
stringency. A hyperfield $F$ is stringent, if $a\boxplus b$ is a singleton
whenever $a\neq-b$.
###### Proposition 1.3.
Every doubly distributive skew hyperfield is stringent.
###### Proof.
Let $F$ be a doubly distributive skew hyperfield. Let $a,b\in F^{\times}$ be
such that $a\neq-b$. Let $x,y\in F^{\times}$ be such that $x,y\in a\boxplus
b$. By double distributivity, we have
$(a\boxplus b)(x^{-1}\boxplus-y^{-1})=(a\boxplus b)\cdot
x^{-1}\boxplus(a\boxplus b)\cdot(-y^{-1})\supseteq x\cdot x^{-1}\boxplus
y\cdot(-y^{-1})=1\boxplus-1\ni 0.$
As $a\neq-b$, then $x^{-1}=y^{-1}$, and so $x=y$. So $a\boxplus b$ is a
singleton if $a\neq-b$. ∎
However, not every stringent skew hyperfield is doubly distributive. The
following is a counterexample.
###### Example 1.4.
Let $F:=\mathbb{Z}\cup\\{-\infty\\}$ be the stringent hyperfield with
multiplication given by $a\odot b=a+b$ and multiplicative identity $0$.
Hyperaddition is given by
$a\boxplus b=\begin{cases}\\{\max(a,b)\\}&\text{ if $a\neq b$,}\\\
\\{c\,|\,c<a\\}&\text{ if $a=b$,}\end{cases}$
so that the additive identity is $-\infty$. Here we use the standard total
order on $\mathbb{Z}$ and set $-\infty<x$ for all $x\in\mathbb{Z}$.
$F$ is not doubly distributive because
$\displaystyle(0\boxplus 0)\odot(0\boxplus 0)$
$\displaystyle=\\{z\,|\,z<0\\}\odot\\{z\,|\,z<0\\}=\\{z\,|\,z<-1\\},$
$\displaystyle 0\boxplus 0\boxplus 0\boxplus 0$
$\displaystyle=\\{z\,|\,z<0\\}\boxplus\\{z\,|\,z<0\\}=\\{z\,|\,z<0\\}.$
We use our classification of stringent skew hyperfields to derive a
classification of stringent skew hyperrings in Section 6. However, this does
not give a classification of doubly distributive skew hyperrings, since not
every doubly distributive skew hyperring is stringent (see Example 6.2).
In fact, we classify all stringent hypergroups, and our classification of
doubly distributive skew hyperfields follows from this.
###### Definition 1.5.
Let $(G,<)$ be a totally ordered set, let $(F_{g}\,|\,g\in G)$ be a family of
hypergroups with a common identity element $0$ in each $F_{g}$ but otherwise
disjoint, and let $\psi$ be the surjective function from $\bigcup_{g\in
G}F_{g}^{\times}$ to $G$ sending $f$ in $F_{g}^{\times}$ to $g$. We denote the
hyperaddition of $F_{g}$ by $\boxplus_{g}$. For any $g\in G$ we denote by
$g\downarrow$ the set of $h\in G$ with $h<g$.
Then the wedge sum $F=\bigvee_{g\in G}{F_{g}}$ is the hypergroup with ground
set $\bigcup_{g\in G}F_{g}$ and hyperaddition given by
$\displaystyle x\boxplus 0$ $\displaystyle=0\boxplus x=\\{x\\},$
$\displaystyle x\boxplus y$ $\displaystyle=\begin{cases}\\{x\\}&\text{if
$\psi(x)>\psi(y)$,}\\\ \\{y\\}&\text{if $\psi(x)<\psi(y)$,}\\\
x\boxplus_{\psi(x)}y&\text{if $\psi(x)=\psi(y)$ and $0\not\in
x\boxplus_{\psi(x)}y$,}\\\
(x\boxplus_{\psi(x)}y)\cup\psi^{-1}(\psi(x)\downarrow)&\text{if
$\psi(x)=\psi(y)$ and $0\in x\boxplus_{\psi(x)}y$. }\end{cases}$
We can also define $\bigcup_{g\in G}F_{g}$ up to isomorphism if the
$F_{g}^{\prime}s$ don’t have the same identity or aren’t otherwise disjoint,
by replacing the $F_{g}^{\prime}s$ with suitably chosen isomorphic copies.
We will show in Section 3 that this construction always yields a hypergroup,
and we classify the stringent hypergroups as follows:
###### Theorem 1.6.
Every stringent hypergroup is a wedge sum $\bigvee_{g\in G}{F_{g}}$ where each
$F_{g}$ is either a copy of the Krasner hypergroup, or a copy of the sign
hypergroup, or a group.
This classification of hypergroups is used to derive the classification of
doubly distributive skew hyperfields discussed above.
### 1.1. Structure of the paper
After the classification of stringent hypergroups in Section 3, we show in
Section 4 that every stringent skew hyperfield arises from a short exact
sequence of groups, where the first group in the sequence is the
multiplicative group of either the Krasner hyperfield or the sign hyperfield
or a skew field, and the last group in the sequence is a totally ordered
group. The underlying additive hypergroup is a wedge sum of isomorphic copies
of hypergroups. Then we present the classification of doubly distributive skew
hyperfields in Section 5 following from the classification of stringent skew
hyperfields. We show the surprising result that every stringent skew hyperring
is either a skew ring or a stringent skew hyperfield in Section 6. We use our
classification to show that every stringent skew hyperfield is a quotient of a
skew field by some normal subgroup in Section 7. In Appendix A we present a
proof that a construction really gives a skew field and in Appendix B we talk
about the semirings associated to doubly distributive hyperfields.
### Acknowledgements
We thank Matthew Baker and Laura Anderson (second author’s PhD advisor) for
introducing the two authors to each other. We thank Laura Anderson and Tom
Zaslavsky, who gave us important comments on early versions of the work.
Thanks also to Pascal Gollin for asking whether our classification might hold
for all stringent hypergroups.
## 2\. Background
###### Notation 2.1.
Throughout $G$ and $H$ denotes groups.
For a hypergroup (or skew hyperring) $S$, $S^{\times}$ denotes $S-\\{0\\}$.
For a function $f$ from a hypergroup (or skew hyperring) $A$ to a hypergroup
(or skew hyperring) $B$, $\operatorname{supp}(f)$ denotes the set of support
of $f$ (the elements of $A$ where the function value is not zero).
### 2.1. Hypergroups, hyperrings and hyperfields
###### Definition 2.2.
A hyperoperation on a set $S$ is a map $\boxplus$ from $S\times S$ to the
collection of non-empty subsets of $S$.
If $A$, $B$ are non-empty subsets of $S$, we define
$A\boxplus B:=\bigcup_{a\in A,b\in B}a\boxplus b$
and we say that $\boxplus$ is associative if $a\boxplus(b\boxplus
c)=(a\boxplus b)\boxplus c$ for all $a,b,c\in S$.
All hyperoperations in this paper will be associative.
###### Definition 2.3.
[Vir10] A hypergroup is a tuple $(G,\boxplus,0)$ where $\boxplus$ is an
associative hyperoperation on $G$ such that:
1. (1)
$0\boxplus x=x\boxplus 0=\\{x\\}$ for all $x\in G$.
2. (2)
For every $x\in G$ there is a unique element $x^{\prime}$ of $G$ such that
$0\in x\boxplus x^{\prime}$ and there is a unique element $x^{\prime\prime}$
of $G$ such that $0\in x^{\prime\prime}\boxplus x$. Furthermore,
$x^{\prime}=x^{\prime\prime}$. This element is denoted by $-x$ and called the
hyperinverse of $x$.
3. (3)
(Invertibility of sums) $x\in y\boxplus z$ if and only if $-x\in-z\boxplus-y$.
A hypergroup is said to be commutative if
4. (4)
$x\in y\boxplus z$ if and only if $x\in z\boxplus y$.
###### Theorem 2.4.
[Vir10] In Definition 2.3, the axiom (3) can be replaced by
(Reversibility property) $x\in y\boxplus z$ implies $y\in x\boxplus-z$ and
$z\in-y\boxplus x$.
The Reversibility property was introduced by Marshall in [Mar06].
###### Definition 2.5.
A skew hyperring is a tuple $(R,\odot,\boxplus,1,0)$ such that:
1. (1)
$(R,\odot,1)$ is a monoid.
2. (2)
$(R,\boxplus,0)$ is a commutative hypergroup.
3. (3)
(Absorption rule) $x\odot 0=0\odot x=0$ for all $x\in R$.
4. (4)
(Distributive Law) $a\odot(x\boxplus y)=(a\odot x)\boxplus(a\odot y)$ and
$(x\boxplus y)\odot a=(x\odot a)\boxplus(y\odot a)$ for all $a,x,y\in R$.
A hyperring is a skew hyperring with commutative multiplication.
A skew hyperring $F$ is called a skew hyperfield if $0\neq 1$ and every non-
zero element of $F$ has a multiplicative inverse.
A hyperfield is then a skew hyperfield with commutative multiplication.
###### Definition 2.6.
Let $F$ and $G$ be skew hyperrings. We may define a skew hyperring $F\times G$
with $(x_{1},y_{1})\boxplus(x_{2},y_{2})$ defined as
$(x_{1}\boxplus_{F}x_{2})\times(y_{1}\boxplus_{G}y_{2})$ and multiplication
defined pointwise. Its additive identity is $(0_{F},0_{G})$ and its
multiplicative identity is $(1_{F},1_{G})$. We call $F\times G$ the product of
$F$ and $G$.
Let $x,y\in F$, we will sometimes write $xy$ instead of $x\odot y$ if there is
no risk of confusion.
###### Example 2.7.
In [Vir10], Viro provided a good introduction to hyperfields. Several of the
following hyperfields were first introduced there.
1. (1)
If $F$ is a field, then $F$ is a hyperfield with $a\odot b=a\cdot b$ and
$a\boxplus b=\\{a+b\\}$, for any $a,b\in F$.
2. (2)
The Krasner hyperfield $\mathbb{K}:=\\{0,1\\}$ has the usual multiplication
rule and hyperaddition is defined by $0\boxplus x=\\{x\\}$ for
$x\in\mathbb{K}$ and $1\boxplus 1=\\{0,1\\}$.
3. (3)
The sign hyperfield $\mathbb{S}:=\\{0,1,-1\\}$ has the usual multiplication
rule and hyperaddition is defined by $0\boxplus x=\\{x\\},x\boxplus x=\\{x\\}$
for $x\in\mathbb{S}$, and $1\boxplus-1=\\{0,1,-1\\}$.
4. (4)
The triangle hyperfield $\triangle:=\mathbb{R}_{\geq 0}$ has the usual
multiplication rule and hyperaddition is defined by $x\boxplus
y=\\{z\,|\,|x-y|\leq z\leq x+y\\}$.
5. (5)
The tropical hyperfield $\mathbb{T}_{+}:=\mathbb{R}\cup\\{-\infty\\}$ has
multiplication defined by $x\odot y=x+y$ (with $-\infty$ as an absorbing
element), for $x,y\in\mathbb{T}_{+}$. Hyperaddition is defined by
$x\boxplus y=\begin{cases}\\{\max(x,y)\\}&\text{ if $x\neq y$,}\\\
\\{z\,|\,z\leq x\\}&\text{ if $x=y$.}\end{cases}$
Here we use the standard total order on $\mathbb{R}$ and set $-\infty<x$ for
all $x\in\mathbb{R}$. The additive identity is $-\infty$ and the
multiplicative identity is $0$.
6. (6)
The tropical phase hyperfield $\Phi:=S^{1}\cup\\{0\\}$ has the usual
multiplication rule and hyperaddition is defined by $0\boxplus x=\\{x\\}$,
$x\boxplus-x=S^{1}\cup\\{0\\}$ and $x\boxplus
y=\\{\frac{ax+by}{|ax+by|}\,|\,a,b\in\mathbb{R}_{\geq 0},a+b\neq 0\\}$ for
$x,y\in S^{1}$ with $y\neq-x$.111This is called phase hyperfield in Viro’s
paper, but more recent papers have often worked with the phase hyperfield (7)
described next. The confusion on this point is exacerbated by the fact that
Viro incorrectly claims that his phase hyperfield is the same as the quotient
hyperfield of the complex numbers by the positive real numbers, but this
construction actually gives the hyperfield (7).
7. (7)
The phase hyperfield $\mathbb{P}:=S^{1}\cup\\{0\\}$ has the usual
multiplication rule and hyperaddition is defined by $0\boxplus x=\\{x\\}$,
$x\boxplus-x=\\{x,-x,0\\}$ and $x\boxplus
y=\\{\frac{ax+by}{|ax+by|}\,|\,a,b\in\mathbb{R}_{>0}\\}$ for $x,y\in S^{1}$
with $y\neq-x$.
8. (8)
The tropical real hyperfield $\mathbb{TR}:=\mathbb{R}$ has the usual
multiplication rule and hyperaddition is defined by
$x\boxplus y=\begin{cases}\\{x\\}&\text{ if $|x|>|y|$,}\\\ \\{y\\}&\text{ if
$|x|<|y|$,}\\\ \\{x\\}&\text{ if $x=y$,}\\\ \\{z\,|\,|z|\leq|x|\\}&\text{ if
$x=-y$.}\end{cases}$
9. (9)
The tropical complex hyperfield $\mathbb{TC}:=\mathbb{C}$ has the usual
multiplication rule and hyperaddition is defined by
$x\boxplus y=\begin{cases}\\{x\\}&\text{if $|x|>|y|$,}\\\ \\{y\\}&\text{if
$|x|<|y|$,}\\\ \\{|x|\dfrac{ax+by}{|ax+by|}\,|\,a,b\in\mathbb{R}_{\geq
0},a+b\neq 0\\}&\text{if $|x|=|y|$ and $x\neq-y$,}\\\
\\{z\,|\,|z|\leq|x|\\}&\text{if $x=-y$.}\end{cases}$
10. (10)
The ultratriangle hyperfield $\mathbb{T}\triangle:=\mathbb{R}_{\geq 0}$
(denoted by $\mathbb{Y}_{\times}$ in [Vir10] and $\mathbb{T}$ in [BB19]) has
the usual multiplication rule and hyperaddition is defined by
$x\boxplus y=\begin{cases}\\{\max(x,y)\\}&\text{ if $x\neq y$,}\\\
\\{z\,|\,z\leq x\\}&\text{ if $x=y$.}\end{cases}$
###### Definition 2.8.
[Vir10, BB19] A skew hyperring $R$ is said to be doubly distributive if for
any $a$, $b$, $c$ and $d$ in $R$, we have $(a\boxplus b)(c\boxplus
d)=ac\boxplus ad\boxplus bc\boxplus bd.$
###### Example 2.9.
Fields, $\mathbb{K}$, $\mathbb{S}$, $\mathbb{T}_{+}$, $\mathbb{TR}$,
$\mathbb{T}\triangle$ are all doubly distributive, but $\triangle$,
$\mathbb{P}$, $\Phi$ and $\mathbb{TC}$ are not doubly distributive.
###### Definition 2.10.
A hypergroup $G$ is said to be stringent if for any $a,b\in G$ the set
$a\boxplus b$ is a singleton whenever $a\neq-b$.
A skew hyperring is said to be stringent if the underlying hypergroup $F$ is
stringent.
### 2.2. Homomorphism
###### Definition 2.11.
[BB16, Pen18] A hypergroup homomorphism is a map $f:G\rightarrow H$ such that
$f(0)=0$ and $f(x\boxplus y)\subseteq f(x)\boxplus f(y)$ for all $x,y\in G$.
A skew hyperring homomorphism is a map $f:R\rightarrow S$ which is a
homomorphism of additive commutative hypergroups as well as a homomorphism of
multiplicative monoids (i.e., $f(1)=1$ and $f(x\odot y)=f(x)\odot f(y)$ for
$x,y\in R$).
A skew hyperfield homomorphism is a homomorphism of the underlying skew
hyperrings.
A hypergroup (resp. skew hyperring, skew hyperfield) isomorphism is a
bijection $f:G\rightarrow H$ which is a hypergroup (resp. skew hyperring, skew
hyperfield) homomorphism and whose inverse is also a hypergroup (resp. skew
hyperring, skew hyperfield) homomorphism.
###### Example 2.12.
The map $\exp:\mathbb{T}_{+}\rightarrow\mathbb{T}\triangle$ is a hyperfield
isomorphism.
## 3\. Classification of stringent hypergroups
Our aim in this section is to prove Theorem 1.6, the Classification Theorem
for stringent hypergroups. We will work with the definition of wedge sums
given as Definition 1.5. First we will show that $F:=\bigvee_{g\in G}F_{g}$ is
indeed a hypergroup.
###### Lemma 3.1.
$F$ is again a hypergroup. If every hypergroup in $(F_{g}\,|\,g\in G)$ is
stringent, then so is $F$. If every hypergroup in $(F_{g}\,|\,g\in G)$ is
commutative, then so is $F$.
###### Proof.
For associativity, suppose we have $x_{1},x_{2},x_{3}\in F$. If any of them is
0, then associativity is clear, so suppose that each $x_{i}$ is in $H$. If one
of the elements $\psi(x_{i})$ of $G$, say $\psi(x_{i_{0}})$, is bigger than
the others, then $x_{1}\boxplus(x_{2}\boxplus
x_{3})=\\{x_{i_{0}}\\}=(x_{1}\boxplus x_{2})\boxplus x_{3}$. If one of the
$\psi(x_{i})$ is smaller than the others, then both
$x_{1}\boxplus(x_{2}\boxplus x_{3})$ and $(x_{1}\boxplus x_{2})\boxplus x_{3}$
evaluate to the sum of the other two $x_{j}$. So we may suppose that all
$\psi(x_{i})$ are equal, taking the common value $g$. If $0\not\in
x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3}$, then both
$x_{1}\boxplus(x_{2}\boxplus x_{3})$ and $(x_{1}\boxplus x_{2})\boxplus x_{3}$
evaluate to $x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3}$, whereas if $0\in
x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3}$, then both evaluate to
$(x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3})\cup\psi^{-1}(g\downarrow)$. The
hyperinverse of 0 is 0 and the hyperinverse of any other $x$ is its
hyperinverse in $F_{\psi(x)}$, and $0$ is the additive identity.
For invertibility of sums, suppose we have $x,y,z\in F$. We would like to show
that $x\in y\boxplus z$ if and only if $-x\in-z\boxplus-y$. It suffices to
prove one direction, say if $x\in y\boxplus z$, then $-x\in-z\boxplus-y$. If
$\psi(y)<\psi(z)$, then $x\in y\boxplus z=\\{z\\}$ and $\psi(-y)<\psi(-z)$. So
$-z\boxplus-y=\\{-z\\}=\\{-x\\}$. Similarly, we have if $\psi(y)>\psi(z)$,
then $-z\boxplus-y=\\{-x\\}$. If $\psi(x)=\psi(y)=\psi(z)$, then the statement
holds by the reversibility of the hypergroup
$(F_{\psi(y)},\boxplus_{\psi(y)},0)$. Otherwise we have
$\psi(x)<\psi(y)=\psi(z)$, and so $y=-z$. Then $\psi(-x)<\psi(-y)=\psi(-z)$
and so $-x\in-z\boxplus-y$.
Then, we would like to show that $F$ is stringent if every hypergroup $F_{g}$
in $(F_{g}\,|\,g\in G)$ is. By definition of $F$, we just need to show that
for any $x,y\in F$ with $\psi(x)=\psi(y)$ and $0\not\in x\boxplus_{\psi(x)}y$,
$x\boxplus y$ is a singleton. As $F_{\psi(x)}$ is stringent and $0\not\in
x\boxplus_{\psi(x)}y$, then $x\boxplus_{\psi(x)}y$ is a singleton. So
$x\boxplus y=x\boxplus_{\psi(x)}y$ is also a singleton.
Finally, it is clear that $\boxplus$ is commutative if each $\boxplus_{g}$ is
commutative. ∎
Now we begin the proof of the Classification Theorem. We first introduce a
useful lemma. Note that this lemma automatically holds for stringent
commutative hypergroups, so readers only interested in that case may skip the
proof.
###### Lemma 3.2.
Let $F$ be a stringent hypergroup. If $y\in x\boxplus y$, then $y\in y\boxplus
x$.
###### Proof.
We will divide the proof into four cases.
_Case 1:_ If $x=y$, this is immediate.
_Case 2:_ If $x=-y$, then by reversibility we get
$y\in x\boxplus y\Rightarrow y\in-y\boxplus y\Rightarrow y\in y\boxplus
y\Rightarrow y\in y\boxplus-y\Rightarrow y\in y\boxplus x.$
_Case 3:_ If $y=-y$, then by reversibility and case 2 we get
$y\in x\boxplus y\Rightarrow x\in y\boxplus-y\Rightarrow x\in-y\boxplus
y\Rightarrow y\in y\boxplus x.$
_Case 4:_ Now we suppose $x\notin\\{y,-y\\}$ and $y\neq-y$. Let $z\in
F^{\times}$ be such that $y\boxplus x=\\{z\\}$ and let $t\in F^{\times}$ be
such that $-y\boxplus-y=\\{t\\}$. Then by associativity we get
$z\boxplus y\boxplus t=(y\boxplus x)\boxplus
y\boxplus(-y\boxplus-y)=y\boxplus(x\boxplus y)\boxplus-y\boxplus-y=y\boxplus
y\boxplus-y\boxplus-y\ni 0.$
So we get $0\in z\boxplus y\boxplus t$, thus $-z\in y\boxplus t$. As
$t\in-y\boxplus-y$, we have $-y\in y\boxplus t$. So $-z,-y\in y\boxplus t$.
Then by stringency we get either $z=y$ or $t=-y$. If $z=y$, then we are done.
Now assume $t=-y$. Thus $-z\in y\boxplus t=y\boxplus-y$, and so
$-y\in-y\boxplus z$. As $y\boxplus x=\\{z\\}$, we have $x\in-y\boxplus z$. So
$-y,x\in-y\boxplus z$. Then by stringency we get either $x=-y$ or $z=y$. By
case 2, the statement holds. ∎
Now we define a relation on $F^{\times}$ which roughly corresponds to the
ordering of $G$.
###### Definition 3.3.
We define a relation $<_{F}$ on $F^{\times}$ by $x<_{F}y$ if $x\boxplus
y=y\boxplus x=\\{y\\}$ but $x\neq y$.
###### Lemma 3.4.
$<_{F}$ is a strict partial order on $F^{\times}$.
###### Proof.
Irreflexivity is built into the definition, so it remains to check
transitivity. Suppose that $x<_{F}y<_{F}z$. Then $x\boxplus z=x\boxplus
y\boxplus z=y\boxplus z=\\{z\\}$. Similarly, $z\boxplus x=\\{z\\}$. We cannot
have $x=z$, since then $\\{y\\}=y\boxplus x=y\boxplus z=\\{z\\}$, so $y=z$,
which is a contradiction. ∎
###### Lemma 3.5.
If $x<_{F}y$, then
1. (1)
$\pm x<_{F}\pm y$.
2. (2)
for any $z\in F^{\times}$ we have either $x<_{F}z$ or $z<_{F}y$.
###### Proof.
1. (1)
It suffices to prove that $-x<_{F}y$ by invertibility of sums. As $x<_{F}y$,
then $x\neq-y$ since $0\in-y\boxplus y$. As $x\boxplus y=\\{y\\}$, then
$y\in-x\boxplus y$. By stringency, $-x\boxplus y=\\{y\\}$. Similarly,
$y\boxplus-x=\\{y\\}$. So $-x<_{F}y$.
2. (2)
By (1), we have $\pm x<_{F}\pm y$. Suppose that $z\not<_{F}y$. If
$z\in\\{y,-y\\}$, then we have $x<_{F}z$. Otherwise, $y\not\in z\boxplus y$
and $y\notin y\boxplus z$ by Lemma 3.2. Then $0\not\in z\boxplus y\boxplus-y$
and $0\not\in-y\boxplus y\boxplus z$. So by stringency, we have $z\boxplus
y\boxplus-y=\\{z\\}$ and $-y\boxplus y\boxplus z=\\{z\\}$. However, $x\in
y\boxplus-y$ and $x\in-y\boxplus y$, since $x<_{F}y$. So $z\boxplus x=\\{z\\}$
and $x\boxplus z=\\{z\\}$. Now if $z\neq x$ this implies that $x<_{F}z$, but
if $z=x$ then we have $z<_{F}y$.
∎
Now we define a relation $\sim_{F}$ on $F^{\times}$ by $x\sim_{F}y$ if and
only if both $x\not<_{F}y$ and $y\not<_{F}x$.
###### Lemma 3.6.
$\sim_{F}$ is an equivalence relation.
###### Proof.
$\sim_{F}$ is clearly reflexive and symmetric. For transitivity, suppose that
$x\sim_{F}y$ and $y\sim_{F}z$. If $x<_{F}z$ then either $x<_{F}y$,
contradicting $x\sim_{F}y$, or else $y<_{F}z$, contradicting $y\sim_{F}z$, so
this is impossible. Similarly we have $z\not<_{F}x$. So $x\sim_{F}z$. ∎
The following results are obvious and we will put them together.
###### Lemma 3.7.
1. (1)
If $x\sim_{F}y<_{F}z$ or $x<_{F}y\sim_{F}z$, then $x<_{F}z$.
2. (2)
The relation $<_{F}$ could lift to a relation (denoted by $<_{F}^{\prime}$) on
the set $G$ of $\sim_{F}$-equivalence classes and $(G,<_{F}^{\prime})$ is a
totally ordered set.
3. (3)
For every $x\in F^{\times}$, $-x\sim_{F}x$.
4. (4)
Let $x,y,z\in F^{\times}$ with $x\neq-y$, $y\neq-z$ and $z\neq-x$. If $0\in
x\boxplus y\boxplus z$, then $x\sim_{F}y\sim_{F}z$.
###### Proof.
For (1), the proof is trivial. (1) implies (2).
(3) As $0\in x\boxplus-x$, we have $x\not<_{F}-x$ and $-x\not<_{F}x$. So
$-x\sim_{F}x$.
(4) If not, then without loss of generality we have $x<_{F}y$, and so $-z\in
x\boxplus y=\\{y\\}$, giving $y=-z$, contradicting our assumptions. ∎
###### Lemma 3.8.
Let $(x_{i}\,|\,i\in I)$ be a finite family of elements of $F$, and $z\in F$
with $x_{i}<_{F}z$ for all $i\in I$. Then for any $y\in\boxplus_{i\in I}x_{i}$
we have $y<_{F}z$.
###### Proof.
It suffices to prove this when $I$ has just two elements, say $x_{1}$ and
$x_{2}$, since the general result then follows by induction. Suppose
$x_{1},x_{2}<_{F}z$ and $y\in x_{1}\boxplus x_{2}$, then we have
$y\boxplus z\subseteq x_{1}\boxplus x_{2}\boxplus z=x_{1}\boxplus z=\\{z\\}$
and
$z\boxplus y\subseteq z\boxplus x_{1}\boxplus x_{2}=z\boxplus x_{2}=\\{z\\}.$
So $y\boxplus z=\\{z\\}$ and $z\boxplus y=\\{z\\}$. If $z\in x_{1}\boxplus
x_{2}$ then $-x_{1}\in x_{2}\boxplus-z=\\{-z\\}$, contradicting $x_{1}<_{F}z$.
So $z\not\in x_{1}\boxplus x_{2}$, and so $z\neq y$. So $y<_{F}z$. ∎
It follows from the above results that the sum $x\boxplus y$ is given by
$\\{x\\}$ if $x>_{F}y$, by $\\{y\\}$ if $x<_{F}y$, by $\\{z\\}$ for some $z$
in the $\sim_{F}$-equivalence class of $x$ and $y$ if $x\sim_{F}y$ but
$x\neq-y$, and by some subset of that class together with
$\\{t\,|\,t<_{F}x\\}\cup\\{0\\}$ if $x=-y$. This looks very similar to the
hyperaddition given in Definition 1.5.
We now want to consider the structure of the equivalence classes. Let $g$ be
an equivalence class in $G$ and let $F_{g}$ be the set $g\cup\\{0\\}$. We can
define a multivalued binary operation $\boxplus_{g}$ on $F_{g}$ by
$x\boxplus_{g}y=(x\boxplus y)\cap F_{g}$.
###### Lemma 3.9.
For any element $g$ in $G$, $F_{g}$ is again a hypergroup, with hyperaddition
given by $\boxplus_{g}$.
###### Proof.
For every $x\in F_{g}$, we have $0\boxplus_{g}x=\\{x\\}\cap F_{g}=\\{x\\}$.
Suppose $0\in x\boxplus_{g}y$, then $0\in x\boxplus y$, and so $y=-x$.
Similarly, if $0\in y\boxplus_{g}x$, then $y=-x$.
For invertibility of sums, let $x,y,z\in F_{g}$ with $x\in y\boxplus_{g}z$.
Then we have $x\in y\boxplus z$. By invertibility of sums of $F$,
$-x\in-z\boxplus-y$. So $-x\in-z\boxplus_{g}-y$.
For associativity, suppose we have $x,y,z\in F_{g}$. We would like to show
that
$(x\boxplus_{g}y)\boxplus_{g}z=x\boxplus_{g}(y\boxplus_{g}z).$
Let $t\in F_{g}$. Let us first show that $t\in x\boxplus_{g}(y\boxplus_{g}z)$
if and only if $t\in x\boxplus(y\boxplus z)$. It is clear that
$x\boxplus_{g}(y\boxplus_{g}z)\subseteq x\boxplus(y\boxplus z)$. So it
suffices to prove the other direction. We suppose that $t\in
x\boxplus(y\boxplus z)$. Then there exists $k\in F$ such that $k\in y\boxplus
z$ and $t\in x\boxplus k$. If $k\in F_{g}$, then we are done. If not, we have
$y=-z$ and $k<_{F}y$. So we also have $k<_{F}x$, and so $t=x\in
x\boxplus_{g}0\subseteq x\boxplus_{g}(y\boxplus_{g}z)$. Similarly, we can also
get $t\in(x\boxplus_{g}y)\boxplus_{g}z$ if and only if $t\in(x\boxplus
y)\boxplus z$. By associativity of $F$, $(x\boxplus y)\boxplus
z=x\boxplus(y\boxplus z)$. So
$(x\boxplus_{g}y)\boxplus_{g}z=x\boxplus_{g}(y\boxplus_{g}z).$ ∎
###### Lemma 3.10.
For any element $g$ in $G$, $F_{g}$ is either isomorphic to $\mathbb{K}$ or
isomorphic to $\mathbb{S}$ or is a group.
###### Proof.
For any $y$ and any $x$ with $x\in y\boxplus-y$, we have $y\in x\boxplus y$
and so $x<_{F}y$ unless $x\in\\{-y,0,y\\}$. So for any $y\in F_{g}$ we have
$y\boxplus_{g}-y\subseteq\\{-y,0,y\\}$. Now suppose that there is some $y\in
F_{g}$ with $y\boxplus_{g}-y\neq\\{0\\}$. Then $y$ is nonzero and
$y,-y\in-y\boxplus_{g}y$. Suppose for a contradiction that there is some $z\in
F_{g}\setminus\\{-y,0,y\\}$, and let $t$ be the unique element of $-y\boxplus
z$. Then by Lemma 3.5, $t\notin\\{y,-y\\}$, since $z\not<_{F}-y$. So
$y\boxplus t=\\{z\\}$. Thus
$y\in
y\boxplus_{g}0\subseteq(-y\boxplus_{g}y)\boxplus_{g}(t\boxplus_{g}-t)=-y\boxplus_{g}(y\boxplus_{g}t)\boxplus_{g}-t=-y\boxplus_{g}z\boxplus_{g}-t=t\boxplus_{g}-t,$
and so $y\in\\{-t,0,t\\}$, which is the desired contradiction.
So if there is any $y$ with $y\boxplus_{g}-y\neq\\{0\\}$, then
$F_{g}=\\{-y,0,y\\}$. It is now not hard to check that in this case if $y=-y$
then $F_{g}\cong\mathbb{K}$, and if $y\neq-y$ then $F_{g}\cong\mathbb{S}$. On
the other hand, if there is no such $y$ then the hyperaddition on $F_{g}$ is
single-valued, and so $F_{g}$ is a group. ∎
We can finally prove the Classification Theorem.
###### Proof of Theorem 1.6.
Let $H$ be $F^{\times}$, let $G$ be given as above and let $\psi$ be the map
sending an element $h$ of $H$ to its equivalence class in $G$. For any $x$ and
$y$ in $H$, if $\psi(x)>_{F}^{\prime}\psi(y)$ then $x>_{F}y$ and so $x\boxplus
y=\\{x\\}$. Similarly if $\psi(x)<_{F}^{\prime}\psi(x)$ then $x\boxplus
y=\\{y\\}$. If $\psi(x)=\psi(y)$ then $x\boxplus_{\psi(x)}y=(x\boxplus y)\cap
F_{\psi(x)}$. So by the remarks following Lemma 3.8 we have that the
hyperaddition of $F$ agrees with that of $\bigvee_{g\in G}F_{g}$ in this case
as well. ∎
## 4\. Classification of stringent skew hyperfields
In this section, we will present the classification of stringent skew
hyperfields. We will first introduce a construction of skew hyperfields
arising from short exact sequences.
###### Definition 4.1.
Let $F$ be a skew hyperfield and let $G$ be a totally ordered group. Suppose
that we have a short exact sequence of groups
$1\to F^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,.$
Since $\varphi$ is injective, by replacing $H$ with an isomorphic copy if
necessary we may (and shall) suppose that $\varphi$ is the identity. As usual,
we define $x^{h}$ to be $h^{-1}\cdot x\cdot h$ for $x,h\in H$. We extend this
operation by setting $0_{F}^{h}:=0_{F}$. We say that the short exact sequence
has stable sums if for each $h\in H$ the operation $x\mapsto x^{h}$ is an
automorphism of $F$ (as a skew hyperfield). Since this operation clearly
preserves the multiplicative structure, this is equivalent to the condition
that it is always an automorphism of the underlying additive hypergroup.
Furthermore, any short exact sequence as above with $H$ abelian automatically
has stable sums.
Suppose now that we have a short exact sequence with stable sums as above.
Then we may define a hyperfield with multiplicative group $H$ as follows. We
begin by choosing some object not in $H$ to serve as the additive identity,
and we denote this object by $0$. For each $g$ in $G$, let $A_{g}$ be
$\psi^{-1}(g)\cup\\{0\\}$. For any $h$ in $\psi^{-1}(g)$ there is a bijection
$\lambda_{h}$ from $F$ to $A_{g}$ sending $0_{F}$ to $0$ and $x$ to $h\cdot x$
for $x\in F^{\times}$, and so there is a unique hypergroup structure on
$A_{g}$ making $\lambda_{h}$ an isomorphism of hypergroups. Furthermore, this
structure is independent of the choice of $h$ since for
$h_{1},h_{2}\in\psi^{-1}(g)$ the map
$\lambda_{h_{1}}^{-1}\cdot\lambda_{h_{2}}$ is just left multiplication by
$h_{1}^{-1}\cdot h_{2}$, which is an automorphism of the additive hypergroup
of $F$. In this way we obtain a well defined hypergroup structure on $A_{g}$,
whose hyperaddition we denote by $\boxplus_{g}$.
Then the $G$-layering $F\rtimes_{H,\psi}G$ of $F$ along this short exact
sequence has as ground set $H\cup\\{0\\}$. Multiplication is given by $x\cdot
y=0$ if $x$ or $y$ is $0$ and by the multiplication of $H$ otherwise.
$H\cup\\{0\\}$ is the underlying set of the hypergroup $\bigvee_{g\in
G}A_{g}$, and we take the hyperaddition of $F\rtimes_{H,\psi}G$ to be given by
that of this hypergroup. Explicitly; the hyperaddition is given by taking 0 to
be the additive identity and setting
$x\boxplus y=\begin{cases}\\{x\\}&\text{if }\psi(x)>\psi(y),\\\
\\{y\\}&\text{if }\psi(x)<\psi(y),\\\ x\boxplus_{\psi(x)}y&\text{if
}\psi(x)=\psi(y)\text{ and }0\not\in x\boxplus_{\psi(x)}y,\\\
(x\boxplus_{\psi(x)}y)\cup\psi^{-1}(\psi(x)\downarrow)&\text{if
}\psi(x)=\psi(y)\text{ and }0\in x\boxplus_{\psi(x)}y.\end{cases}$
###### Lemma 4.2.
$F\rtimes_{H,\psi}G$ is again a skew hyperfield. If $F$ is stringent, then so
is $F\rtimes_{H,\psi}G$.
###### Proof.
As shown in Lemma 3.1, $\bigvee_{g\in G}A_{g}$ is a commutative hypergroup. So
it suffices to show that $\cdot$ distributes over $\boxplus$. For left
distributivity, we must prove an equation of the form
$x_{1}\cdot(x_{2}\boxplus x_{3})=x_{1}\cdot x_{2}\boxplus x_{1}\cdot x_{3}$.
As usual, if any of the $x_{i}$ is 0, then this is trivial, so we suppose that
each $x_{i}$ is in $H$. If $\psi(x_{2})>\psi(x_{3})$, then both sides are
equal to $x_{1}\cdot x_{2}$. If $\psi(x_{2})<\psi(x_{3})$, then both sides are
equal to $x_{1}\cdot x_{3}$. So we may assume that $\psi(x_{2})=\psi(x_{3})$
and we call their common value $g$. Then
$x_{2}\boxplus_{g}x_{3}=\lambda_{x_{2}}(1\boxplus_{F}x_{2}^{-1}\cdot x_{3})$
and $(x_{1}\cdot x_{2})\boxplus_{\psi(x_{1})\cdot g}(x_{1}\cdot
x_{3})=\lambda_{x_{1}\cdot x_{2}}(1\boxplus_{F}x_{2}^{-1}\cdot x_{3})$. So if
$0\not\in x_{2}\boxplus_{g}x_{3}$, then also $0\not\in(x_{1}\cdot
x_{2})\boxplus_{\psi(x_{1})\cdot g}(x_{1}\cdot x_{3})$, and so both sides of
the equation are equal to $x_{1}\cdot(x_{2}\boxplus_{g}x_{3})$. If $0\in
x_{2}\boxplus_{g}x_{3}$, then also $0\in(x_{1}\cdot
x_{2})\boxplus_{\psi(x_{1})\cdot g}(x_{1}\cdot x_{3})$, and so both sides of
the equation are equal to $x_{1}\cdot(x_{2}\boxplus_{g}x_{3})\cup
x_{1}\cdot\psi^{-1}(g\downarrow)$.
For the right distributivity, we need to consider bijections
$\lambda_{h}^{\prime}\colon F\to A_{\psi(h)}$ similar to the $\lambda_{h}$. We
take $\lambda_{h}^{\prime}(x)$ to be $x\cdot h$ for $x\in F^{\times}$ and to
be $0$ for $x=0_{F}$. Then since $\lambda_{h}^{\prime}(x)=\lambda_{h}(x)^{h}$
for any $x$ and the short exact sequence has stable sums, the
$\lambda_{h}^{\prime}$ are also hyperfield isomorphisms. So we may argue as
above but with the $\lambda_{h}^{\prime}$ in place of the $\lambda_{h}$.
Finally, we must show that $F\rtimes_{H,\psi}G$ is stringent if $F$ is. By
definition of $F\rtimes_{H,\psi}G$, we just need to show that for $x,y\in
F\rtimes_{H,\psi}G$ with $\psi(x)=\psi(y)$ and $0\not\in
x\boxplus_{\psi(x)}y$, $x\boxplus y$ is a singleton. As $F$ is stringent and
$0\not\in x\boxplus_{\psi(x)}y$, then $x\boxplus_{\psi(x)}y$ is a singleton.
So $x\boxplus y=x\boxplus_{\psi(x)}y$ is also a singleton. ∎
Now let us see some interesting examples of hyperfields constructed in this
way.
###### Example 4.3.
If $F$ is the Krasner hyperfield, $G$ and $H$ are both the additive group of
real numbers, and $\psi$ is the identity, then $F\rtimes_{H,\psi}G$ is the
tropical hyperfield.
###### Example 4.4.
$F:=\mathbb{Z}\cup\\{-\infty\\}$ in Example 1.4 can arise from the short exact
sequence of groups
$0\to
GF(2)^{\times}\xrightarrow{\varphi}\mathbb{Z}\xrightarrow{\psi}\mathbb{Z}\to
0.$
###### Example 4.5.
In [AD19], Anderson and Davis drew a diagram encoding many popular and
important hyperfields and homomorphisms, see as follows.
$\mathbb{TR}$$\mathbb{TC}$$\mathbb{T}\triangle$$\mathbb{S}$$\mathbb{K}$$\Phi$$\mathbb{P}$$\mathbb{R}$$\mathbb{C}$$\triangle$$|\,\,|$$|\,\,|$$|\,\,|$$|\,\,|$phphphphphph
The diagram with the solid arrows commutes. The four dashed arrows are
inclusions giving sections (one-sided inverses). Here ph is the _phase map_
ph$(x)=x/|x|$ if $x=0$ and ph$(0)=0$. In each of the ten hyperfields, the
underlying set is a subset of the complex numbers closed under multiplication.
And in each hyperfield, multiplication, the additive identity, and the
multiplicative identity coincides with that of the complex numbers.
Our classification gives a good relationship between the hyperfields in each
column and we can construct each hyperfield in the bottom row from the
corresponding element of the row just above the bottom and the ordered group
$\mathbb{R}_{>0}$.
1. (1)
From the short exact sequence of groups
$1\to\mathbb{S}^{\times}\rightarrow\mathbb{R}^{\times}\rightarrow\mathbb{R}_{>0}\to
1,$
we can get the tropical real hyperfield
$\mathbb{TR}=\mathbb{S}\rtimes\mathbb{R}_{>0}$.
2. (2)
From the short exact sequence of groups
$1\to\Phi^{\times}\rightarrow\mathbb{C}^{\times}\rightarrow\mathbb{R}_{>0}\to
1,$
we can get the tropical complex hyperfield
$\mathbb{TC}=\Phi\rtimes\mathbb{R}_{>0}$.
3. (3)
From the short exact sequence of groups
$1\to\mathbb{K}^{\times}\rightarrow\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}\to
1,$
we can get the ultratriangle hyperfield
$\mathbb{T}\triangle=\mathbb{K}\rtimes\mathbb{R}_{>0}$.
Since in each column the second element is obtained as a quotient of the first
by a $\mathbb{R}_{>0}-$subgroup, this operation of putting back the factor of
$\mathbb{R}_{>0}$ yields a hyperfield on the same ground set as the top
element of the column.
Our aim is to show that every stringent skew hyperfield is of the form
$F\rtimes_{H,\psi}G$ with $F$ either the Krasner hyperfield or the sign
hyperfield or a skew field. Let’s start with a stringent skew hyperring.
Let $R$ be a stringent skew hyperring. By Theorem 1.6, we can classify $R$ to
be the wedge sum $\bigvee_{g\in G}R_{g}$ with a surjective mapping $\psi$ from
$R^{\times}$ to the set $G$ defined in last section and an ordering
$<_{R}^{\prime}$ on $G$ by $\psi(x)<_{R}^{\prime}\psi(y)$ if and only if
$x\boxplus y=\\{y\\}$ but $x\neq y$, where the hypergroup $R_{g}$ is either
isomorphic to $\mathbb{K}$ or isomorphic to $\mathbb{S}$ or is a group. Thus
by distributivity of $R$, $\psi(x)<_{R}^{\prime}\psi(y)$ if and only if
$\psi(ax)<_{R}^{\prime}\psi(ay)$ if and only if
$\psi(xa)<_{R}^{\prime}\psi(ya)$ for $a\in R^{\times}$. So the multiplication
of $R$ lifts to a multiplication on $G$ respecting the ordering, with identity
$\psi(1):=1_{G}$. By Lemma 3.7(2), we can easily get the following lemma.
###### Lemma 4.6.
$(G,\cdot,<_{R}^{\prime})$ is a totally ordered monoid. If $R$ is a skew
hyperfield, then $G$ is a totally ordered group.
Now we want to consider the structure of $R_{g}$.
###### Lemma 4.7.
$R_{1_{G}}$ is again a skew hyperring, with hyperaddition given by
$\boxplus_{1_{G}}$ and multiplication by that of $R$.
###### Proof.
By Lemma 3.9, it suffices to check the distributivity. To prove left
distributivity we must show that any element $t\in R_{1_{G}}$ of
$x\cdot(y\boxplus z)$ is also an element of the same expression evaluated in
$R_{1_{G}}$. So let $w$ be an element of $y\boxplus z$ with $x\cdot w=t$. This
second equation implies that the equivalence class of $w$ is $1_{G}$, as
desired. The right distributivity is similar. ∎
###### Lemma 4.8.
If $R$ is a skew hyperfield, $R_{1_{G}}$ is either the Krasner hyperfield or
the sign hyperfield or a skew field.
###### Proof.
By Lemma 3.10 and Lemma 4.7, we can get $R_{1_{G}}$ is either the Krasner
hyperfield or the sign hyperfield or a skew ring.
Since $\sim_{R}$ respects the multiplication, the multiplicative inverse of
anything equivalent to $1_{R}$ is again equivalent to $1_{R}$, so that
$R_{1_{G}}$ is a skew field if it is a skew ring. ∎
###### Lemma 4.9.
For every $g\in G$, the hypergroup of $R_{g}$ is isomorphic to the hypergroup
of $R_{1_{G}}$.
###### Proof.
Let $a\in R_{g}^{\times}$. Define $f:R_{g}\rightarrow R_{1_{G}}$ by sending
$0$ to $0$ and $x$ in $R_{g}^{\times}$ to $a^{-1}\cdot x$. Since $f$ has an
inverse operation, namely left multiplication by $a$, this is a bijection. Now
we would like to show $f(x\boxplus_{g}y)=f(x)\boxplus_{1_{G}}f(y)$.
$\displaystyle f(x\boxplus_{g}y)$ $\displaystyle=a^{-1}\cdot(x\boxplus_{g}y)$
$\displaystyle=a^{-1}\cdot\big{(}(x\boxplus y)\cap R_{g}\big{)}$
$\displaystyle=\big{(}a^{-1}\cdot(x\boxplus y)\big{)}\cap(a^{-1}\cdot R_{g})$
$\displaystyle=\big{(}(a^{-1}\cdot x)\boxplus(a^{-1}\cdot y)\big{)}\cap
R_{1_{G}}$ $\displaystyle=(a^{-1}\cdot x)\boxplus_{1_{G}}(a^{-1}\cdot y)$
$\displaystyle=f(x)\boxplus_{1_{G}}f(y)$
∎
Now using the above results, we can classify the stringent skew hyperfields as
follows.
###### Theorem 4.10.
Any stringent skew hyperfield $R$ has the form $F\rtimes_{H,\psi}G$, where $F$
is either the Krasner hyperfield or the sign hyperfield or a skew field.
###### Proof.
Let $F$ be $R_{1_{G}}$, let $H$ be $R^{\times}$ and let $G$ be given as above.
Let $\varphi$ be the injection of $F^{\times}$ as a subgroup of $H$ and let
$\psi$ be the map sending an element $h$ of $H$ to its equivalence class in
$G$. Then
$1\to F^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1$
is a short exact sequence. For any $x$ and $y$ in $H$, if
$\psi(x)>_{R}^{\prime}\psi(y)$, then $x>_{R}y$ and so $x\boxplus y=\\{x\\}$.
Similarly if $\psi(x)<_{R}^{\prime}\psi(x)$, then $x\boxplus y=\\{y\\}$. If
$\psi(x)=\psi(y)$, then $x\boxplus_{\psi(x)}y=x\cdot((1\boxplus x^{-1}\cdot
y)\cap R_{1_{G}})=(x\boxplus y)\cap R_{\psi(x)}$. So by the remarks following
Lemma 3.8 we have that the hyperaddition of $R$ agrees with that of
$F\rtimes_{H,\psi}G$ in this case as well. ∎
Using results of Marshall’s paper [Mar06], we can show that the structure is
even more constrained if the multiplication of $R$ is commutative (so that $R$
is a stringent hyperfield) and $R_{1_{G}}$ is the Krasner or the sign
hyperfield.
###### Proposition 4.11.
Let $R$ be a stringent skew hyperfield with $R_{1_{G}}=\mathbb{S}$ and let
$a\in R^{\times}-\\{1,-1\\}$. Then $a^{2}\notin\\{1,-1\\}$.
###### Proof.
As $a\notin\\{1,-1\\}$, then $a\not\sim_{R}1$. So $\psi(a)\neq 1$. Then
$\psi(a^{2})=(\psi(a))^{2}\neq 1$, since $G$ is a totally ordered group. So
$a^{2}\not\sim_{R}1$. That is $a^{2}\notin\\{1,-1\\}$. ∎
Following are some useful Lemmas in Marshall’s paper (cf. section 3,[Mar06]).
###### Definition 4.12.
[Mar06] Let $R$ be a hyperfield. A subset $P$ of $R$ is called an ordering if
$P\boxplus P\subseteq P,P\odot P\subseteq P,P\cup-P=R\text{ and
}P\cap-P=\\{0\\}.$
###### Definition 4.13.
[Mar06] A hyperfield $R$ is said to be real if $-1\notin R^{2}\boxplus R^{2}$
where $R^{2}:=\\{a^{2}\,|\,a\in R\\}$.
###### Lemma 4.14.
[Mar06, Lemma 3.3] Let $R$ be a hyperfield. $R$ has an ordering if and only if
$R$ is real.
###### Lemma 4.15.
[Mar06, Lemma 3.2, 3.3] Let $R$ be a hyperfield with $1\neq-1$. If $R$ has an
ordering $P$, then $-1\notin P$.
Based on above lemmas, we get the following.
###### Proposition 4.16.
If $R$ is a stringent hyperfield with $R_{1_{G}}=\mathbb{S}$, then $R$ has an
ordering.
###### Proof.
By Lemma 4.14, we just need to show that $R$ is real.
Suppose that $-1\in R^{2}\boxplus R^{2}$. Then there exist $a,b\in R$ such
that $-1\in a^{2}\boxplus b^{2}$. By Proposition 4.11, $a^{2}\neq-1$ and
$b^{2}\neq-1$. Thus $a\neq 0$ and $b\neq 0$. And by reversibility, $-b^{2}\in
1\boxplus a^{2}$. As $a^{2}\neq-1$, then $1\boxplus
a^{2}\subseteq\\{1,a^{2}\\}$. Thus $-b^{2}=a^{2}$. Then
$-1=a^{2}b^{-2}=(ab^{-1})^{2}$, a contradiction to Proposition 4.11.
So $R$ is real, and therefore has an ordering. ∎
###### Theorem 4.17.
If $R$ is a stringent hyperfield with
$R_{1_{G}}\in\\{\mathbb{K},\mathbb{S}\\}$, then $R$ arises from a short exact
sequence
$1\to R_{1_{G}}^{\times}\xrightarrow{\varphi}R_{1_{G}}^{\times}\times
G\xrightarrow{\psi}G\to 1.$
###### Proof.
If $R_{1_{G}}=\mathbb{K}$, this is trivial.
If $R_{1_{G}}=\mathbb{S}$, by Theorem 4.10, we may suppose
$R=\mathbb{S}\rtimes_{H,\psi}G=H\cup\\{0\\}$ with a short exact sequence of
groups
$1\to\mathbb{S}^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1.$
By Proposition 4.16, we know that $R$ has an ordering $P$. Let $O=P-\\{0\\}$.
As $P\cup-P=R$ and $P\cap-P=\\{0\\}$, then $R=O\mbox{\hskip
1.49994pt$\cup$$\cdot$\hskip 3.99994pt}-O\cup\\{0\\}$. By Lemma 4.15,
$-1\notin P$. Then $-1\notin O$, thus $1\in O$. And as $P\odot P\subseteq P$,
then $O\odot O\subseteq O$. For any $a\in O$, $a^{-1}\in O$. Otherwise
$-a^{-1}\in O$. Then $a\odot-a^{-1}=-1\in O$, which is a contradiction. So $O$
is a multiplicative group with $1\in O$ and $R=O\mbox{\hskip
1.49994pt$\cup$$\cdot$\hskip 3.99994pt}-O\cup\\{0\\}$, and $\psi\restriction
O$ is an isomorphism from $O$ to $G$.
Now we can identify $x\in H$ with $(1,\psi(x))$ if $x\in O$, and with
$(-1,\psi(x))$ if $x\notin O$, giving a bijection from $H$ to
$\mathbb{S}^{\times}\times G$.
So $R\cong(\mathbb{S}^{\times}\times G)\cup\\{0\\}$.
∎
It is not clear whether this result extends to stringent skew hyperfields.
## 5\. Classification of Doubly Distributive Skew Hyperfields
In this section, we will present the classification of doubly distributive
skew hyperfields.
###### Proposition 5.1.
The doubly distributive skew hyperfields are precisely those of the form
$F\rtimes_{H,\psi}G$ of exactly one of the following types:
1. (1)
$F$ is the Krasner hyperfield,
2. (2)
$F$ is the sign hyperfield,
3. (3)
$F$ is a skew field and $G$ satisfies
$\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}.$
The following example is a doubly distributive hyperfield of type (3) in
Proposition 5.1.
###### Example 5.2.
Let $F:=\mathbb{R}$ be the hyperfield with usual multiplication and
hyperaddition given by
$x\boxplus y=\begin{cases}x&\text{ if }|x|>|y|,\\\ y&\text{ if }|x|<|y|,\\\
\\{z\,|\,|z|<|x|\\}&\text{ if }|x|=|y|.\end{cases}$
This hyperfield is stringent and arises from the short exact sequence of
groups
$1\to
GF(3)^{\times}\xrightarrow{\varphi}\mathbb{R}^{\times}\xrightarrow{\psi}\mathbb{R}_{>0}\to
1\,.$
Another natural example of doubly distributive skew hyperfields of type (3) in
Proposition 5.1 can be found in [Pen18], which is built around a
(noncommutative) such hyperfield (the one called $L^{\sigma}$).
Before showing the proof of Proposition 5.1, We’ll first introduce a useful
lemma.
###### Lemma 5.3.
Let $R$ be a stringent skew hyperfield. $R$ is doubly distributive if and only
if
$(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$
###### Proof.
By Definition 2.8, $R$ is doubly distributive if and only if $(a\boxplus
b)(c\boxplus d)=ac\boxplus ad\boxplus bc\boxplus bd$, for any $a,b,c,d\in R$.
As $R$ is stringent, we have $u\boxplus v$ is a singleton if $u\neq-v$. So if
either $a\neq-b$ or $c\neq-d$, then the equation above is just about
distributivity. It already holds.
If both $a=-b$ and $c=-d$, then
$(a\boxplus b)(c\boxplus
d)=(a\boxplus-a)(c\boxplus-c)=a(1\boxplus-1)(1\boxplus-1)c$
and
$ac\boxplus ad\boxplus bc\boxplus bd=ac\boxplus-ac\boxplus-ac\boxplus
ac=a(1\boxplus-1\boxplus 1\boxplus-1)c.$
So $R$ is doubly distributive if and only if
$(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$
∎
Now we will present the proof of Proposition 5.1.
###### Proof of Proposition 5.1.
By Proposition 1.3 and Theorem 4.10, we know that a doubly distributive skew
hyperfield $R$ also has the form $F\rtimes_{H,\psi}G$, where $F$ is either the
Krasner hyperfield or the sign hyperfield or a skew field, with a short exact
sequence of groups
$1\to F^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,.$
So it suffices to show that the hyperfields of type (1) and (2) are doubly
distributive and the hyperfields with $F$ a skew field are doubly distributive
if and only if they are of type (3).
_Case 1:_ When $F=\mathbb{K}=\\{1,0\\}$, hyperaddition is defined by
$\displaystyle x\boxplus 0$ $\displaystyle=\\{x\\},$ $\displaystyle x\boxplus
y$ $\displaystyle=\begin{cases}\\{x\\}&\text{ if $\psi(x)>\psi(y)$,}\\\
\\{y\\}&\text{ if $\psi(x)<\psi(y)$,}\\\
\\{z\,|\,\psi(z)\leq\psi(x)\\}\cup\\{0\\}&\text{ if $\psi(x)=\psi(y)$, that is
$x=y$.}\end{cases}$
By Lemma 5.3, $R$ is doubly distributive if and only if
$(1\boxplus 1)(1\boxplus 1)=1\boxplus 1\boxplus 1\boxplus 1.$
$(1\boxplus 1)(1\boxplus 1)=(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})\cdot(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\},$
and
$1\boxplus 1\boxplus 1\boxplus 1=(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})\boxplus(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\}.$
So $R$ is doubly distributive when $F=\mathbb{K}$.
_Case 2:_ When $F=\mathbb{S}=\\{1,-1,0\\}$, hyperaddition is defined by
$\displaystyle x\boxplus 0$ $\displaystyle=\\{x\\},$ $\displaystyle x\boxplus
y$ $\displaystyle=\begin{cases}\\{x\\}&\text{ if $\psi(x)>\psi(y)$,}\\\
\\{y\\}&\text{ if $\psi(x)<\psi(y)$,}\\\ \\{x\\}&\text{ if $x=y$,}\\\
\\{z\,|\,\psi(z)\leq\psi(x)\\}\cup\\{0\\}&\text{ if $x=-y$.}\end{cases}$
By Lemma 5.3, $R$ is doubly distributive if and only if
$(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$
$(1\boxplus-1)(1\boxplus-1)=(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})\cdot(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\},$
and
$1\boxplus-1\boxplus 1\boxplus-1=(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})\boxplus(\\{z\,|\,\psi(z)\leq
1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\}.$
So $R$ is doubly distributive when $F=\mathbb{S}$.
_Case 3:_ When $F$ is a skew field, hyperaddition is defined by
$\displaystyle x\boxplus 0$ $\displaystyle=\\{x\\},$ $\displaystyle x\boxplus
y$ $\displaystyle=\begin{cases}\\{x\\}&\text{ if $\psi(x)>\psi(y)$,}\\\
\\{y\\}&\text{ if $\psi(x)<\psi(y)$,}\\\ x\boxplus_{\psi(x)}y&\text{ if
$\psi(x)=\psi(y)$ and $0\notin x\boxplus_{\psi(x)}y$,}\\\
\\{z\,|\,\psi(z)<\psi(x)\\}\cup\\{0\\}&\text{ if $\psi(x)=\psi(y)$ and $0\in
x\boxplus_{\psi(x)}y$.}\\\ \end{cases}$
By Lemma 5.3, $R$ is doubly distributive if and only if
$(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$
$(1\boxplus-1)(1\boxplus-1)=(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})\cdot(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})=\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\},$
and
$1\boxplus-1\boxplus
1\boxplus-1=(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})\boxplus(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\}.$
So $R$ is doubly distributive if and only if
$\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\}=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\}.$
We claim that
$\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\}=\psi^{-1}(\psi(1)\downarrow)\cup\\{0\\}=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\},$
if and only if
$\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}.$
$(\Rightarrow):$ If
$\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\}=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\}$,
the direction $\subseteq$ is clear. We just need to consider the other
direction. Let $c\in G$ be such that $c<1_{G}$ and let $z\in\psi^{-1}(c)$.
Then there exist $x,y\in H$ such that $z=xy$ and $\psi(x),\psi(y)<1$ by our
assumption. So $c=\psi(z)=\psi(xy)=\psi(x)\psi(y)$. We have
$c\in\\{ab\,|\,a,b<1_{G}\\}$.
$(\Leftarrow):$ If $\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}$, the direction
$\subseteq$ is also clear. We just need to consider the other direction. Let
$z\in H$ be such that $\psi(z)<1$ and let $c=\psi(z)$. Then there exist
$a,b\in G$ such that $c=ab$ and $a,b<1_{G}$ by our assumption. Let $x\in H$ be
such that $\psi(x)=a<1_{G}$ and let $y=x^{-1}z$. We have
$\psi(y)=\psi(x^{-1}z)=a^{-1}c=b<1_{G}$ and $z=xy$. So
$z\in\\{xy\,|\,\psi(x),\psi(y)<1\\}$.
So $R$ is doubly distributive if and only if
$\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}.$
∎
## 6\. Reduction of stringent skew hyperrings to hyperfields
In this section, we will show that stringent skew hyperrings are very
restricted.
###### Theorem 6.1.
Every stringent skew hyperring is either a skew ring or a stringent skew
hyperfield.
###### Proof.
If $G$ is trivial, then $R=R_{1_{G}}$. So $R$ is either $\mathbb{K}$, or
$\mathbb{S}$, or a skew ring.
If $G$ is nontrivial, we would like to show that every element $x$ in
$R^{\times}$ is a unit. Now let $s$ and $t$ in $R^{\times}$ be such that
$x\cdot s>_{R}1$ and $t\cdot x>_{R}1$. Then by the remarks after Lemma 3.8, we
have
$1\in x\cdot s\boxplus-x\cdot s=x\cdot(s\boxplus-s),$ $1\in t\cdot
x\boxplus-t\cdot x=(t\boxplus-t)\cdot x.$
So there exists $y\in s\boxplus-s$ and $z\in t\boxplus-t$ such that $1=x\cdot
y=z\cdot x$. Thus $y=(z\cdot x)\cdot y=z\cdot(x\cdot y)=z$. So $x$ has a
multiplicative inverse $y$ in $R$. Then $x$ is a unit of $R$.
So every stringent skew hyperring is either a skew ring or a stringent skew
hyperfield. ∎
We cannot classify doubly distributive hyperring using our classification
because not every doubly distributive hyperring is stringent. The following is
a counterexample.
###### Example 6.2.
The hyperring $\mathbb{K}\times\mathbb{K}$ that is the square of the Krasner
hyperfield is doubly distributive but not stringent.
## 7\. Every stringent skew hyperfield is a quotient of a skew field
In this section, we would like to show that every stringent skew hyperfield is
a quotient of a skew field by some normal subgroup. In particular, every
stringent hyperfield is a quotient of a field by some special kind of
subgroups, called ‘hüllenbildend’. This kind of subgroups was studied by
Diller and Grenzdörffer in [DG73] when they tried to unify the treatment of
various notions of convexity in projective spaces over a field $K$ by
introducing for any subgroup $U\leq K^{\times}$ the notion of $U$-convexity.
They showed that this notion is reasonably well behaved if and only if $U$ is
as follows.
###### Definition 7.1.
[DG73] Let $K$ be a field and let $U\leq K^{\times}$. $U$ is called
$U$-‘hüllenbildend’ (hull producing) if $U$ satisfies
$x,y\in K,x+y-xy\in U\rightarrow x\in U\text{ or }y\in U.$ (1)
In [Dre77], Dress presented a simple complete classification of such
‘hüllenbildend’ subgroups $U$. We will combine our classification of stringent
(skew) hyperfields into three types with Dress’s classification of such
subgroups.
###### Theorem 7.2.
[Dre77, Theorem 1] Let $U\leq K^{\times}$ satisfy (1) and let $S_{U}=\\{x\in
K\,|\,x\notin U\text{ and }x+U\subseteq U\\}$. Then $S_{U}$ is the maximal
ideal of a valuation ring $R=R_{U}(=\\{x\in K\,|\,x\cdot S_{U}\subseteq
S_{U}\\})$ in $K$, $U$ is contained in $R$,
$\overline{U}=\\{\overline{x}\in\overline{K}_{U}=R_{U}/S_{U}\,|\,x\in U\\}$ is
either a domain of positivity in $\overline{K}_{U}$ (if $-1\notin U$, $2\in
U$) or $\overline{U}=\\{\overline{1}\\}$ or
$\overline{U}=\overline{K}_{U}^{\times}$ and, in any case, $U=\\{x\in
R_{U}\,|\,\overline{x}\in\overline{U}\\}$.
We will first explain how to choose the suitable subgroup $U$ in the case of
stringent hyperfield and then give the proof in full generality for stringent
skew hyperfields.
From our classification in Theorem 4.10, we know that every stringent
hyperfield $F$ has the form $M\rtimes_{H,\psi}G$, where $M$ is either
$\mathbb{K}$, or $\mathbb{S}$, or a field. To show that $F$ is a quotient by
some subgroup $U$, we will choose $U$ with $\overline{U}=\\{\overline{1}\\}$
if $M$ is $\mathbb{K}$, choose $U$ with $\overline{U}$ a domain of positivity
if $M$ is $\mathbb{S}$, and choose $U$ with
$\overline{U}=\overline{K}_{U}^{\times}$ if $M$ is a field.
Now we begin the proof that every stringent skew hyperfield is a quotient of a
skew field $K$ by some normal subgroup $U$. First, we recall the definition of
quotient hyperfield. The quotient hyperfield $K/U=\\{[g]=gU\,|\,g\in K\\}$ was
introduced by Krasner in [Kra83] with multiplication given by
$[g]\cdot[h]=[gh]$, for $[g],[h]\in K/U$. Hyperaddition is given by
$[g]\boxplus[0]=[g]$ and $[g]\boxplus[h]=\\{[f]\subseteq K/U\,|\,f\in
gU+hU\\}$, for $[g],[h]\in(K/U)^{\times}$. As the subgroup $U$ we are choosing
would be normal, so this quotient works in the skew case.
We may suppose that a stringent skew hyperfield $F=M\rtimes_{H,\psi}G$ arises
from a short exact sequence of groups
$1\to M^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,,$
where $G$ is a totally ordered group equipped with a total order $\leq$ and
$M$ is either $\mathbb{K}$, or $\mathbb{S}$, or a skew field. We define an
order $\leq^{\prime}$ on $G$ such that $x\leq^{\prime}y$ if and only if $y\leq
x$. So $\leq^{\prime}$ is also a total order on $G$. Similarly as in the non-
skew case, we will also choose $U$ with $\overline{U}=\\{\overline{1}\\}$ if
$M$ is $\mathbb{K}$, choose $U$ with $\overline{U}$ a domain of positivity if
$M$ is $\mathbb{S}$, and choose $U$ with
$\overline{U}=\overline{K}_{U}^{\times}$ if $M$ is a skew field.
Our difficulty now is to choose a suitable skew field $K$ for the quotient
hyperfield corresponding to each $U$. We will introduce two different
constructions of skew fields, as follows.
###### Example 7.3.
[FS01] (Construction 1) Let $k$ be an arbitrary field. Define $K=k((G))$ to be
the ring of formal power series whose powers come from $G$, that is, the
elements of $K$ are functions from $G$ to $k$ such that the support of each
function is a well-ordered subset of $(G,\leq^{\prime})$. Addition is
pointwise, and multiplication is the Cauchy product or convolution, that is
the natural operation when viewing the functions as power series
$\sum_{{a\in G}}p(a)x^{a}$
It is well known (and easy to check) that $K$ is a skew field.
We will construct a skew field $K=k((G))$ as in Example 7.3 by choosing $k$ to
be an arbitrary field when $M$ is $\mathbb{K}$ and choosing $k$ to be the
field $\mathbb{R}$ of real numbers (or any other ordered field) when $M$ is
$\mathbb{S}$.
The second construction is for a stringent skew hyperfield
$F=M\rtimes_{H,\psi}G$ when $M$ is a skew field.
###### Example 7.4.
(Construction 2) We define $K=M[[G]]$ to be the set of formal sums of elements
of $H$ all from different layers such that the support is well-ordered, that
is, an element of $K$ is a function $p$ from $G$ to $H$ such that for any $g$
in $G$, $p(g)\in\psi^{-1}(g)\cup\\{0\\}=A_{g}$ and the support of each
function is a well-ordered subset of $(G,\leq^{\prime})$. As $M$ is a skew
field and each $\lambda_{h}$ with $h\in H$ is an isomorphism of hypergroups,
then $(A_{g},\boxplus_{g},0)$ is always an abelian group. We claim that $K$ is
a skew field, viewing functions as power series
$\sum_{{a\in G}}p(a)x^{a}\,,$
with addition $+_{K}$ given by
$\sum_{{a\in G}}p(a)x^{a}+_{K}\sum_{{a\in G}}q(a)x^{a}=\sum_{{a\in
G}}(p(a)\boxplus_{a}q(a))x^{a},$
and the additive identity is $\sum_{{a\in G}}0x^{a}$. Multiplication
$\cdot_{K}$ is given by
$\Big{(}\sum_{{a\in G}}p(a)x^{a}\Big{)}\cdot_{K}\big{(}\sum_{{a\in
G}}q(a)x^{a}\Big{)}=\sum_{{s\in
G}}\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q),}{\underset{g\in\operatorname{supp}(p),}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}x^{s},$
and the multiplicative identity is $1x^{1_{G}}$. Since the proof that this
really gives a skew field is a long calculation and is very similar to that
for $k((G))$, we do not give it here but in Appendix A.
Now we divide the proof into three cases and show that $F\cong K/U$ in each
case. For simplicity, we denote $\min(\operatorname{supp}(p))$ by $m_{p}$ for
$p\in K^{\times}$.
1. _Case_ 1:
If $M$ is $\mathbb{K}$, then let $U=\\{p\in K^{\times}\,|\,m_{p}=1_{G}\\}$.
It’s easy to check that $U$ is normal. Then the quotient hyperfield
$K/U=\\{[q]=qU\,|\,q\in K\\}$ has
$\displaystyle[q]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\\},$
$\displaystyle[0]$ $\displaystyle=\\{0_{K}\\}.$
So we can identify $[q]$ in $(K/U)^{\times}$ with $m_{q}$ in $G$ and identify
$[0]$ in $K/U$ with $0$. So we have $K/U\cong(G\cup\\{0\\},\boxplus,\cdot)$
with multiplication given by
$\displaystyle 0\cdot g$ $\displaystyle=0,$ $\displaystyle g\cdot h$
$\displaystyle=g\cdot_{G}h,$
where $g,h\in G$. And hyperaddition is given by
$\displaystyle g\boxplus 0$ $\displaystyle=\\{g\\},$ $\displaystyle g\boxplus
h$ $\displaystyle=\begin{cases}\\{g\\}&\text{ if $g<^{\prime}h$, that is
$g>h$, }\\\ \\{h\\}&\text{ if $g>^{\prime}h$, that is $g<h$, }\\\ \\{f\in
G\,|\,f\geq^{\prime}g\\}\cup\\{0\\}=\\{f\in G\,|\,f\leq g\\}\cup\\{0\\}&\text{
if $g=h$,}\end{cases}$
where $g,h\in G$.
Now it is clear to see that
$K/U\cong(G\cup\\{0\\},\boxplus,\cdot)\cong\mathbb{K}\rtimes_{H,\psi}G=F$.
2. _Case_ 2:
If $M$ is $\mathbb{S}$, $k=\mathbb{R}$ (or any other ordered field) and
$K=k((G))$, then let $U=\\{p\in K^{\times}\,|\,m_{p}=1_{G}\text{ and
}p(1_{G})>0\\}$. It’s easy to check that $U$ is normal. Then the quotient
hyperfield $K/U=\\{[q]=qU\,|\,q\in K\\}$ has
$\displaystyle[q]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and
}p(m_{p})>0\\}\text{ if }q(m_{q})>0,$ $\displaystyle[q]$
$\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and
}p(m_{p})<0\\}\text{ if }q(m_{q})<0,$ $\displaystyle[0]$
$\displaystyle=\\{0_{K}\\}.$
We can identify $[q]$ in $(K/U)^{\times}$ with $(1,m_{q})$ if $q(m_{q})>0$,
identify $[q]$ in $(K/U)^{\times}$ with $(-1,m_{q})$ if $q(m_{q})<0$, and
identify $[0]$ with $0$. So we have $K/U\cong((\mathbb{S}^{\times}\times
G)\cup\\{0\\},\boxplus,\cdot)$ with multiplication given by
$\displaystyle(r,g)\cdot 0$ $\displaystyle=0,$
$\displaystyle(r_{1},g_{1})\cdot(r_{2},g_{2})$
$\displaystyle=(r_{1}\cdot_{\mathbb{S}}r_{2},g_{1}\cdot_{G}g_{2}),$
where $r,r_{1},r_{2}\in\mathbb{S}^{\times}$ and $g,g_{1},g_{2}\in G$. And
hyperaddition is given by
$\displaystyle(r,g)\boxplus 0$ $\displaystyle=\\{(r,g)\\},$
$\displaystyle(r_{1},g_{1})\boxplus(r_{2},g_{2})$
$\displaystyle=\begin{cases}\\{(r_{1},g_{1})\\}&\text{if
$g_{1}<^{\prime}g_{2}$, that is $g_{1}>g_{2}$, }\\\
\\{(r_{2},g_{2})\\}&\text{if $g_{1}>^{\prime}g_{2}$, that is $g_{1}<g_{2}$,
}\\\ \\{(r_{1},g_{1})\\}&\text{if $g_{1}=g_{2}$ and $r_{1}=r_{2}$ }\\\
\\{(r,f)\,|\,f\geq^{\prime}g\\}\cup\\{0\\}=\\{(r,f)\,|\,f\leq
g\\}\cup\\{0\\}&\text{if $g_{1}=g_{2}$ and $r_{1}=-r_{2}$,}\end{cases}$
where $r,r_{1},r_{2}\in\mathbb{S}^{\times}$ and $g,g_{1},g_{2}\in G$.
So by Theorem 4.17, $K/U\cong((\mathbb{S}^{\times}\times
G)\cup\\{0\\},\boxplus,\cdot)\cong\mathbb{S}\rtimes_{H,\psi}G=F$.
3. _Case_ 3:
If $M$ is a skew field and $K=M[[G]]$, then let $U=\\{p\in
K^{\times}\,|\,m_{p}=1_{G}\text{ and }p(1_{G})=1\\}$. It’s easy to check that
$U$ is normal. Then the quotient hyperfield $K/U=\\{[q]=qU\,|\,q\in K\\}$ has
$\displaystyle[q]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and
}p(m_{p})=q(m_{q})\\},$ $\displaystyle[0]$ $\displaystyle=0_{K}.$
We can identify $[q]$ in $(K/U)^{\times}$ with $q(m_{q})$ in $H$ (clearly
$\psi(q(m_{q}))=m_{q}$) and identify $[0]$ with $0_{F}$. So we have $K/U\cong
F$ with multiplication given by
$\displaystyle[q]\cdot 0$ $\displaystyle=0,$ $\displaystyle[q]\cdot[h]$
$\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\cdot_{G}m_{h}\text{ and
}p(m_{p})=q(m_{q})\cdot_{H}h(m_{h})\\}=[p]$
Hyperaddition is given by
$[q]\boxplus 0=[q],$ $\displaystyle[q]\boxplus[h]$ $\displaystyle=$
$\displaystyle\begin{cases}\\{[q]\\}&\text{if $m_{q}<^{\prime}m_{h}$, that is
$m_{q}>m_{h}$, }\\\ \\{[h]\\}&\text{if $m_{q}>^{\prime}m_{h}$, that is
$m_{q}<m_{h}$, }\\\ \\{[p]=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and
}p(m_{p})=q(m_{q})\boxplus_{m_{q}}h(m_{h})\\}\\}&\text{if $m_{q}=m_{h}$ and
$0\notin q(m_{q})\boxplus_{m_{q}}h(m_{h})$,}\\\
\\{[p]\in(K/U)^{\times}\,|\,m_{p}>^{\prime}m_{q}\\}\cup\\{0\\}=\\{[p]\in(K/U)^{\times}\,|\,m_{p}<m_{q}\\}\cup\\{0\\}&\text{if
$m_{q}=m_{h}$ and $0\in q(m_{q})\boxplus_{m_{q}}h(m_{h}).$}\end{cases}$
where $[q],[h]\in(K/U)^{\times}$.
So $K/U\cong M\rtimes_{H,\psi}G=F.$
###### Theorem 7.5.
Every stringent skew hyperfield is a quotient of a skew field.
###### Corollary 7.6.
Every doubly distributive skew hyperfield is a quotient of a skew field.
It follows from the construction that the same statements with all instances
of the word ‘skew’ removed also hold.
## Appendix A The construction 2 in Example 7.4 gives a skew field
###### Lemma A.1.
Let $F=M\rtimes_{H,\psi}G$ be a stringent skew hyperfield arising from a short
exact sequence of groups
$1\to M^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,,$
where $G$ is a totally ordered group and $M$ is a skew field. Define
$K=k[[G]]$ as we did in section 7. Then $K$ is a skew field.
###### Proof.
The commutativity and associativity of $(K,+_{K},\sum_{{a\in G}}0x^{a})$
follow from those of $(H\cup\\{0\\},\boxplus,0)$. So we only need to show the
associativity of $(K,\cdot_{K},1x^{1_{G}})$, the existence of a multiplicative
inverse for every element and the distributivity.
An important principle which we will need again and again as we go along is a
kind of distributivity of the composition of $H$ over the various additions
$\boxplus_{g}$. To express it cleanly, we begin by extending $\cdot_{H}$ to
$H\cup\\{0\\}$ by setting $x\cdot 0=0\cdot x=0$ for all $x\in H\cup\\{0\\}$.
Suppose that we have elements $x$ and $y_{1},y_{2}\ldots y_{n}$ of $H$ with
$\psi(y_{i})=u\in G$ for all $i$, so that $\boxplus_{i=1}^{n}y_{i}$ is
defined. Let $v\in G$ be such that $v=\psi(x)$. Then $z\mapsto x\cdot_{H}z$ is
a bijection from $A_{u}$ to $A_{v\cdot u}$ whose composition with
$\lambda_{y_{1}}$ is $\lambda_{x\cdot_{H}y_{1}}$, so it must also be an
isomorphism of hypergroups. Thus
$x\cdot_{H}\big{(}\underset{1\leq i\leq
n}{\boxplus_{u}}y_{i}\big{)}=\underset{1\leq i\leq n}{\boxplus_{v\cdot
u}}x\cdot_{H}y_{i}\,.$
A similar argument using the $\lambda^{\prime}_{h}$ defined in the proof of
Lemma 4.2 shows
$\big{(}\underset{1\leq i\leq
n}{\boxplus_{u}}y_{i}\big{)}\cdot_{H}x=\underset{1\leq i\leq
n}{\boxplus_{u\cdot v}}y_{i}\cdot_{H}x\,.$
To show the associativity of $(K,\cdot_{K},1x^{1_{G}})$, we let $p,q,w\in K$.
Then for $s\in G$,
$\displaystyle\big{(}(p\cdot_{K}q)\cdot_{K}w\big{)}(s)$
$\displaystyle=\underset{g\cdot_{G}c=s}{\underset{c\in\operatorname{supp}(w)}{\boxplus_{s}}}\Big{(}\big{(}\underset{a\cdot_{G}b=g}{\underset{b\in\operatorname{supp}(q)}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{g}}}}p(a)\cdot_{H}q(b)\big{)}\cdot_{H}w(c)\Big{)}$
$\displaystyle=\underset{g\cdot_{G}c=s}{\underset{c\in\operatorname{supp}(w)}{\boxplus_{s}}}\Big{(}\underset{a\cdot_{G}b=g}{\underset{b\in\operatorname{supp}(q)}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}}p(a)\cdot_{H}q(b)\cdot_{H}w(c)\Big{)}$
$\displaystyle=\underset{a\cdot_{G}b\cdot_{G}c=s}{\underset{c\in\operatorname{supp}(w)}{{\underset{b\in\operatorname{supp}(q)}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}}}}p(a)\cdot_{H}q(b)\cdot_{H}w(c),$
$\displaystyle=\underset{a\cdot_{G}h=s}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}\Big{(}\underset{b\cdot_{G}c=h}{\underset{c\in\operatorname{supp}(w)}{\underset{b\in\operatorname{supp}(q)}{\boxplus_{s}}}}p(a)\cdot_{H}q(b)\cdot_{H}w(c)\Big{)}$
$\displaystyle=\underset{a\cdot_{G}h=s}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}\Big{(}p(a)\cdot_{H}\big{(}\underset{b\cdot_{G}c=h}{\underset{c\in\operatorname{supp}(w)}{\underset{b\in\operatorname{supp}(q)}{\boxplus_{h}}}}q(b)\cdot_{H}w(c)\big{)}\Big{)}$
$\displaystyle=\big{(}p\cdot_{K}(q\cdot_{K}w)\big{)}(s)\,.$
So $(p\cdot_{K}q)\cdot_{K}w=p\cdot_{K}(q\cdot_{K}w)$.
Next we will show that each element of $K$ has a multiplicative inverse. We do
this first for those $p\in K=k[[G]]$ such that $m_{p}=1_{G}$ and $p(m_{p})=1$.
Let $S$ be the set of finite sums of elements of $\operatorname{supp}(p)$. $S$
is well founded.
Define $q\in K=k[[G]]$ such that $q(1_{G}):=1$, $q(s):=0$ for $s\notin S$ and,
for $s\in S$, define $q(s)$ recursively by
$q(s):=-\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in
S-\\{s\\}}{\underset{g\in\operatorname{supp}(p)-\\{1_{G}\\}}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}.$
So
$\displaystyle p\cdot_{K}q(1_{G})$ $\displaystyle=1,$ $\displaystyle
p\cdot_{K}q(s)$ $\displaystyle=0$ if $s\notin S$, $\displaystyle
p\cdot_{K}q(s)$
$\displaystyle=\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q)}{\underset{g\in\operatorname{supp}(p)}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)$
$\displaystyle=\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q)-\\{s\\}}{\underset{g\in\operatorname{supp}(p)-\\{1_{G}\\}}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}\boxplus_{s}p(1)\cdot_{H}q(s)$
$\displaystyle=\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q)-\\{s\\}}{\underset{g\in\operatorname{supp}(p)-\\{1_{G}\\}}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}\boxplus_{s}q(s)$
$\displaystyle=0$ if $s\in S-\\{1_{G}\\}$.
So $p\cdot_{K}q$ is the identity. Therefore, $q$ is the multiplicative inverse
of $p$.
Next we consider elements of $K$ with only a single summand, that is, those of
the form $ax^{g}$. It is clear that each such element also has a
multiplicative inverse, namely $a^{-1}x^{g^{-1}}$.
Now every element of $K$ can be expressed as a product $p_{1}\cdot p_{2}$,
with $m_{p_{1}}=1_{G}$ and $p_{1}(m_{p_{1}})=1$ and such that $p_{2}$ has only
a single summand. As seen above, each of $p_{1}$ and $p_{2}$ has a
multiplicative inverse, and hence $p_{1}\cdot p_{2}$ also has one, namely
$p_{2}^{-1}\cdot p_{1}^{-1}$.
For distributivity, we first would like to show that
$p\cdot_{K}(q+_{K}w)=p\cdot_{K}q+_{K}p\cdot_{K}w$. For $s\in G$,
$\displaystyle(p\cdot_{K}(q+_{K}w))(s)$
$\displaystyle=\underset{g\cdot_{G}h=s}{\boxplus_{s}}p(g)\cdot_{H}(q(h)\boxplus_{h}w(h))$
$\displaystyle=\underset{g\cdot_{G}h=s}{\boxplus_{s}}\big{(}p(g)\cdot_{H}q(h)\boxplus_{s}p(g)\cdot_{H}w(h)\big{)}$
$\displaystyle=\big{(}\underset{g\cdot_{G}h=s}{\boxplus_{s}}p(g)\cdot_{H}q(h)\big{)}\boxplus_{s}\big{(}\underset{g\cdot_{G}h=s}{\boxplus_{s}}p(g)\cdot_{H}w(h)\big{)},$
$\displaystyle=(p\cdot_{K}q+_{K}p\cdot_{K}w)(s)\,.$
So $p\cdot_{K}(q+_{K}w)=p\cdot_{K}q+_{K}p\cdot_{K}w$. A similar calculation
shows that $(p+_{K}q)\cdot_{K}w=p\cdot_{K}w+_{K}q\cdot_{K}w$.
So $K=k[[G]]$ is a skew field. ∎
## Appendix B The semirings associated to doubly distributive hyperfields
In [GJL17], Lemma 6.2(2) provides a way to build a semiring out of a doubly
distributive hyperfield.222In [Row16, Theorem 2.5], Rowen extended the theory
of constructing the semiring to every hyperfield. In this section, we would
like to talk about these semirings.
For any doubly distributive hyperfield $H$ we can define binary operations
$\oplus$ and $\odot$ on $\mathcal{P}H$ by setting $A\oplus B:=\bigcup_{a\in
A,b\in B}a\boxplus b$ (this is just the extension of $\boxplus$ to subsets of
$H$ from Definition 2.2) and $A\odot B:=\\{ab\colon a\in A,b\in B\\}$. Let
$\langle H\rangle$ be the substructure of $(\mathcal{P}H,\oplus,\odot)$
generated from the singletons of elements of $H$. So $\langle H\rangle$ is a
semiring. We will refer $\langle H\rangle$ as the associated semiring to $H$.
Using our classification, we can easily determine all such associated
semirings. Surprisingly, some of the basic examples have already been
intensively studied and play an important role in the foundations of tropical
geometry. In each case, we find that $\langle H\rangle$ contains only few
elements in addition to the singletons of elements of $H$.
We have seen that any doubly distributive hyperfield has the form
$F\rtimes_{H,\psi}G$, where $F$ is the Krasner hyperfield, the sign hyperfield
or a field. We divide into cases according the value of $F$.
### B.1. Supertropical semirings
If $F$ is the Krasner hyperfield then $\psi\colon H^{\times}\to G$ is an
isomorphism, and we can take it to be the identity. Then the elements of
$\langle H\rangle$ are the singletons of elements of $H$ and the sets
$g^{\nu}:=\\{h\in G\colon h\leq g\\}\cup\\{0\\}$. To simplify the definition
of the addition we define an operation $\nu$ on $\langle
H\rangle\setminus\\{\\{0\\}\\}$ by $\nu(\\{g\\})=\nu(g^{\nu})=g^{\nu}$ and we
transfer the total order of $G$ to the $g^{\nu}$ in the obvious way. Then
addition is given by $x\oplus\\{0\\}=x$ for any $x$ and otherwise by
$x\oplus y=\begin{cases}x&\text{if $\nu(x)>\nu(y)$,}\\\ y&\text{if
$\nu(x)<\nu(y)$,}\\\ \nu(x)&\text{if $\nu(x)=\nu(y)$.}\\\ \end{cases}$
Multiplication is given by $x\odot\\{0\\}=\\{0\\}$, by
$\\{g\\}\odot\\{h\\}=\\{g\cdot h\\}$, by $\\{g\\}\odot h^{\nu}=(g\cdot
h)^{\nu}$ and by $g^{\nu}\odot h^{\nu}=(g\cdot h)^{\nu}$. In the case that $G$
is the ordered group of real numbers, this is simply the supertropical
semiring introduced by Izhakian in [Izh09]. This associated semiring has also
been studied by Rowen in [Row16]. It would be reasonable to call such
semirings in general supertropical semirings.
### B.2. Symmetrised $(\max,+)$-semirings
If $F$ is the sign hyperfield then by Theorem 4.17 without loss of generality
it arises from a short exact sequence
$1\to\mathbb{S}^{\times}\to\mathbb{S}^{\times}\times G\to G\to 1\,.$
The elements of $\langle H\rangle$ then have the form $0:=\\{0_{H}\\}$,
$\oplus g:=\\{(1,g)\\}$, $\ominus g:=\\{(-1,g)\\}$, or
$g^{\circ}:=\\{(i,h)\colon i\in\mathbb{S}^{\times},h\leq g\\}\cup\\{0_{H}\\}$.
There is an obvious projection map $\pi$ from $\langle
H\rangle\setminus\\{0\\}$ to $G$. Then addition is given by $x\oplus 0=x$ for
any $x$, by $x\oplus y=x$ if $\pi(x)>\pi(y)$, by $x\oplus g^{\circ}=g^{\circ}$
if $\pi(x)=g$, by $(\oplus g)\oplus(\oplus g)=\oplus g$, by $(\ominus
g)\oplus(\ominus g)=\ominus g$ and by $(\oplus g)\oplus(\ominus g)=g^{\circ}$.
Multiplication is given by $x\odot 0=0$ for any $x$, by $x\odot
g^{\circ}=(\pi(x)\cdot g)^{\circ}$, by $(\oplus g)\odot(\oplus
h)=\oplus(g\cdot h)$, by $(\ominus g)\odot(\ominus h)=\oplus(g\cdot h)$ and by
$(\oplus g)\odot(\ominus h)=\ominus(g\cdot h)$.
In the case that $G$ is the ordered group of real numbers, this is simply the
symmetrised $(\max,+)$-semiring introduced by Akian et al in [ACG+91]. So it
would be reasonable to call such semirings in general symmetrised
$(max,+)$-semirings.
### B.3. Linearised $(\max,+)$-semirings
If $F$ is a field, then the elements of $\langle H\rangle$ are the singletons
of elements of $H$ (which are in canonical bijection with $H$) and the sets
$\psi^{-1}(g\downarrow)\cup\\{0\\}$ (which are in canonical bijection with
$G$). So $\langle H\rangle$ is isomorphic to the semiring on $H\cup G$ with
$x\oplus y$ for $x,y\in H$ given by the unique element of $x\boxplus y$ if
this set is a singleton and by $\psi(x)$ otherwise, with $x\oplus g$ for $x\in
H$ and $g\in G$ given by $x$ if $\psi(x)\geq g$ and by $g$ otherwise, and with
$g\oplus h$ for $g,h\in G$ given by $\max(g,h)$. For multiplication, $x\odot
y=x\cdot y$ for $x,y\in H$ and $x\odot g=\psi(x)\cdot g$ for $x\in H$ and
$y\in G$ and finally $g\odot h=g\cdot h$ for $g,h\in G$.
By analogy to the previous construction, we could refer to such semirings as
linearised $(\max,+)$-semirings. So far as we know, such semirings have not
yet been seriously investigated.
## References
* [ACG+91] Marianne Akian, G. Cohen, S. Gaubert, Ramine Nikoukhah, and J.P. Quadrat. Linear systems in (max, +) algebra. In Proceedings of the 29th IEEE Conference on Decision and Control, pages 151 – 156 vol.1, 01 1991.
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* [BB19] Matthew Baker and Nathan Bowler. Matroids over partial hyperstructures. Adv. Math., 343:821–863, 2019.
* [BP19] Nathan Bowler and Rudi Pendavingh. Perfect matroids over hyperfields. arXiv:1908.03420, 2019.
* [DG73] Justus Diller and Jochen Grenzdörffer. $G$-Hüllen metrischer Teilräume. Math. Ann., 200:151–164, 1973.
* [Dre77] A. Dress. On orderings and valuations of fields. Geometriae Dedicata, 6(3):259–266, 1977.
* [DW92] Andreas W. M. Dress and Walter Wenzel. Valuated matroids. Adv. Math., 93(2):214–250, 1992.
* [FS01] László Fuchs and Luigi Salce. Modules over non-Noetherian domains, volume 84 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.
* [GJL17] Jeffrey Giansiracusa, Jaiung Jun, and Oliver Lorscheid. On the relation between hyperrings and fuzzy rings. Beitr. Algebra Geom., 58(4):735–764, 2017.
* [Izh09] Zur Izhakian. Tropical arithmetic and matrix algebra. Communications in Algebra, 37(4):1445–1468, 2009.
* [Kra57] Marc Krasner. Approximation des corps valués complets de caractéristique $p\not=0$ par ceux de caractéristique $0$. In Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, pages 129–206. Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1957.
* [Kra83] Marc Krasner. A class of hyperrings and hyperfields. Internat. J. Math. Math. Sci., 6(2):307–311, 1983.
* [Mar06] M. Marshall. Real reduced multirings and multifields. J. Pure Appl. Algebra, 205(2):452–468, 2006.
* [Pen18] Rudi Pendavingh. Field extensions, derivations, and matroids over skew hyperfields. arXiv:1802.02447, 2018.
* [Row16] Louis Halle Rowen. Algebras with a negation map. arXiv:1602.00353, 2016.
* [Vir10] Oleg Viro. Hyperfields for tropical geometry i. hyperfields and dequantization. arXiv:1006.3034v2, 2010.
* [Vir11] O. Ya. Viro. On basic concepts of tropical geometry. Tr. Mat. Inst. Steklova, 273(Sovremennye Problemy Matematiki):271–303, 2011.
|
2024-09-04T02:54:58.201400 | 2020-03-08T19:28:53 | 2003.03837 | {
"authors": "Lucileide M. D. da Silva, Maria G. F. Coutinho, Carlos E. B. Santos,\n Mailson R. Santos, Luiz Affonso Guedes, M. Dolores Ruiz, Marcelo A. C.\n Fernandes",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26105",
"submitter": "Marcelo Fernandes",
"url": "https://arxiv.org/abs/2003.03837"
} | arxiv-papers | # Hardware Architecture Proposal for TEDA algorithm to Data Streaming Anomaly
Detection
Lucileide M. D. da Silva<EMAIL_ADDRESS>Maria G. F. Coutinho
<EMAIL_ADDRESS>Carlos E. B. Santos<EMAIL_ADDRESS>Mailson
R. Santos<EMAIL_ADDRESS>Luiz Affonso Guedes<EMAIL_ADDRESS>M. Dolores Ruiz<EMAIL_ADDRESS>Marcelo A. C. Fernandes111Present address:
John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA<EMAIL_ADDRESS>Laboratory of
Machine Learning and Intelligent Instrumentation, Federal University of Rio
Grande do Norte, Natal 59078-970, Brazil. Federal Institute of Education,
Science and Technology of Rio Grande do Norte, Paraiso, Santa Cruz, RN,
59200-000, Brazil. Department of Statistics and Operations Research,
University of Granada, Spain. Department of Computer Engineering and
Automation, Federal University of Rio Grande do Norte, Natal, RN, 59078-970,
Brazil.
###### Abstract
The amount of data in real-time, such as time series and streaming data,
available today continues to grow. Being able to analyze this data the moment
it arrives can bring an immense added value. However, it also requires a lot
of computational effort and new acceleration techniques. As a possible
solution to this problem, this paper proposes a hardware architecture for
Typicality and Eccentricity Data Analytic (TEDA) algorithm implemented on
Field Programmable Gate Arrays (FPGA) for use in data streaming anomaly
detection. TEDA is based on a new approach to outlier detection in the data
stream context. In order to validate the proposals, results of the occupation
and throughput of the proposed hardware are presented. Besides, the bit
accurate simulation results are also presented. The project aims to Xilinx
Virtex-6 xc6vlx240t-1ff1156 as the target FPGA.
###### keywords:
FPGA , TEDA , data streaming , reconfigurable computing
††journal: arXiv.org
## 1 Introduction
Outlier detection or anomaly detection consists in detect rare events in a
data set. It is a central problem in many application areas such as time
series forecasting, data mining and industrial process monitoring. Due the
increasing number of sensors in the most diverse areas and applications, there
is a huge raise in the availability of data from time series. Thus, outlier
detection for temporal data has become a central problem [1], especially when
data are captured and processed continuously in online way. In this case, the
data are considered as data streams [2].
Some important aspects need to be considered when choosing an anomaly
detection method, such as the computational effort to handle large streaming
data. Since the received information need to be stored and analyzed without
compromising memory and run-time. Many of the solutions presented in the
literature require prior knowledge of the process and system, such as
mathematical models, data distribution, and predefined parameters [3]. Anomaly
detection is traditionally done from statistical analysis, using probability
and making a series of initial assumptions that in most cases are not in
practice applied.
A disadvantage of the traditional statistical method is comparing a single
point with the average of all points rather than comparing with sample or data
pairs. This way, the information is no longer punctual and local. Moreover,
probability theory was developed from examples where processes and variables
are purely random. However, real processes are not purely random and shows
dependency between samples. Thus, real problems are addressed from offline
processes, where the entire data set needs to be known. Being a potential
problem of the traditional method. Another problem with traditional approaches
is that they often use an offline dataset. Thus, all samples must be
previously available from the beginning of the algorithm execution [4], making
it impossible to use in real-time and data stream applications. This type of
data presents new technical challenges and opportunities in new fields of
work. Detecting real-time anomalies can provide valuable information in
critical scenarios, but it is a high computational demand problem that still
lacks reliable solutions capable of providing high processing capabilities.
Typicality and Eccentricity Data Analytic (TEDA) is based on new approach to
outlier detection in data stream context [5] and it can applied with an
algorithm to detect autonomous behavior in industrial process operation, for
example. TEDA analyzes the density of each sample of data read, calculated
according to the distance from the sample to the other samples previously
read. It is an online algorithm that learns autonomously without the need for
prior knowledge about the process or parameters. Therefore, the computational
effort required is smaller, allowing the use in real time applications [3].
TEDA can be used as an alternative statistical framework for analyzing most
data, except for purely random processes. It is based on new metrics, all
based on similarity/proximity of data in the data space, not in density or
entropy, as in traditional methods. The metrics used with TEDA are typicality,
defined in [5] as the extent to which objects are ?good examples? of a
concept, and eccentricity, defined as how distinct the object is from the rest
of the group. A high eccentricity data has a low typicality and is usually an
outlier [3].
Eccentricity can be very useful for anomaly detection, image processing, fault
detection, particle physics, etc. Allows analysis for data samples (which can
also be done in real time for data stream) [6]. It is also relevant in
clustering processes, since elements of a cluster are naturally opposed to the
atypical [5].
Another area where anomaly detection has been increasingly used is in industry
4.0 projects. One of the challenges of the Industry 4.0 is the detection of
production failures and defects [7]. New technologies aim to add value and
increase process productivity, but face difficulties in performing complex and
massive-scale computing due to the large amount of data generated [8]. The
huge accumulation of real time data to flow in a network, for example, can
quickly overload traditional computing systems due to the large amount of data
that originates from the sensors and the requirement for intensive processing
and high performance. The development of specialized hardware presents itself
as a possible solution to overcome the bottlenecks, making it possible to
create solutions for mass data processing and, at the same time, consider
ultra-low-latency, low-power, high-throughput, security and ultra-reliable
conditions, important requirements for increasing productivity and quality in
industry 4.0 processes.
Thinking about the challenges presented, this work proposes a specialized
hardware architecture of TEDA for anomaly detection. The development of the
hardware technique allows systems to be made even faster than their software
counterparts, extending the possibilities of use for situations where time
constraints are even more severe. In addition allowing its use in applications
with large data processing. The works [9, 10, 11, 12, 13] were developed in
hardware, specifically on FPGA, for the acceleration of complex algorithms.
The development of machine learning algorithms in hardware has grown
significantly. This is justified from performance data with respect to system
sampling times compared to software equivalents. One of the motivations for
this work is the possibility of accelerating the TEDA algorithm and handling
large data streams, such as streaming and real-time.
In this work, all validation and synthesis results was made using a FPGA
Virtex 6 xc6vlx240t1ff1156. The FPGA choice was because it has high
performance. Modern FPGAs can deliver performance and density comparable to
Application Specific Integrated Circuits (ASICs), without the disadvantages of
high development time and enabling reprogramming, as FPGAs have a flexible
architecture.
The rest of this paper is organized as follows: This first section presented a
introduction about the work explaining the motivation behind it and major
contributions. Section 2 discusses some related works and the state of the
art. In Section 3 will be presented a theoretical foundation regarding the
TEDA technique. Section 4 presents the implementation description details for
the architecture proposed. Section 5 will present the validation and synthesis
results of the proposed hardware, as well as comparisons with software
implementations. Finally, Section 6 will present the conclusions regarding the
obtained results.
## 2 Related work
Real-time anomaly detection in data stream has potential applications in many
areas. Such as: preventive maintenance, fault detection, fraud detection,
signals monitoring, among others. Concepts that can be used in many different
ranges of industry, such as information technology, finance, medicine,
security, energy, e-commerce, agriculture, social media, among others. In the
literature there are some uses of the TEDA technique for anomaly detection and
even for classification.
The article presented in [6] shows a proposal for a new TEDA-based anomaly
detection algorithm. The proposed method, called by the author $\sigma$ gap,
combines the accumulated proximity information for all samples with the
comparison of specific point pairs suspected of being anomalies. Using local
spatial distribution information about the vicinity of the suspect point. In
the journal, TEDA is compared to an approach using traditional statistical
methods, emphasizing that the set of initial assumptions is different. TEDA
has been shown to be a generalization of traditional statistics compared to a
known analysis, n $\sigma$, which is a widely used principle for threshold
anomaly detection. The same result was obtained for both approaches, although
TEDA does not need the initial assumptions. In addition, for various types of
proximity measurements (such as Euclidean, Cosine, Mahalanobis), it has been
shown that due to the recursion feature, TEDA is computationally more
efficient and suitable for online and real-time applications.
In the work [14] a study is presented about the use of TEDA for fault
detection in industrial processes. The work is pioneering the use of this
approach for real industry data. For the experiments, TEDA was applied online
to the dataset provided by the DAMADICS (Development and Application of
Methods for Actuator Diagnosis in Industrial Control Systems) database, one of
the most widely used benchmarks in fault detection and diagnosis applications.
The experiments showed good results both in accuracy and execution time, which
shows the suitability of this approach for real-time industrial applications.
Finally, it was found that the TEDA algorithm is capable of dealing with the
limitations and particularities of the industrial environment.
The paper of [15] is intended to enable the use of TEDAClass, which consists
of the TEDA algorithm for classification, in big data processing. The main
feature of the proposed algorithm, called TEDAClassBDp, is the processing of
block data, where each block uses the TEDAClass so that all blocks operate in
parallel. As with TEDAClass, the proposed algorithm does not require
information from previous data, and its operation occurs recursively, online
and in real-time. The results indicated a reduction in time and computational
complexity, without significantly compromising its accuracy, which indicates
the strong possibility of using the proposal in problems where it is necessary
to process large volumes of data quickly.
The work presented in [16] proposes a new non-frequency and density-based data
analysis tool. Classified by the author as a further development of TEDA and
an effective alternative to the probability distribution function (pdf).
Typicality Distribution Function (TDF) can provide valuable information for
extreme process analysis, fault detection and identification, where the number
of extreme event or fault observations is often disproportionately small. The
proposed offers a closed non-parametric analytical (quadratic) description,
extracted from the actual realizations of the data exactly in contrast to the
usual practice in which these distributions are being assumed or approximated.
In addition, for various types of proximity and similarity measures (such as
Euclidean, Mahalonobis, and cosine distances), it can be recursively
calculated, thus computationally efficient and suitable for online and real-
time algorithms. As a fundamental theoretical innovation, TDF and TEDA
application areas can range from anomaly detection, grouping, classification,
prediction, control, filter regression (similar to Kalman). Practical
applications may be even broader, so it is difficult to list them all.
The paper presented in [3] proposes the application of TEDA for fault
detection in industrial processes. The effectiveness of the proposal has been
demonstrated with two real industrial plants, using data streaming, and
compared with traditional failure detection methods. This paper presents a
practical application of the TEDA algorithm for two fault detection problems
of real industrial plants. The first application uses a well-known database,
DAMADICS, a database that provides actual data on the water evaporation
process of an operating plant of a Polish sugar manufacturing plant. The
second application was made from data analysis of a pilot plant of the
authors’ university laboratory. A plant equipped with real industrial
instruments used for process control.
The work of [4] presents a new proposal for unsupervised fuzzy classifier,
capable of aggregating the main characteristics of evolving classifiers, as
well as making fuzzy classifications of real time data streams completely
online. The proposed algorithm uses TEDA concepts, replacing traditional
clusters with data clouds, granular structures without shape or predefined
boundaries. For data classification, the proposed approach uses the concepts
of soft-labeling rather than mutually exclusive classes. Experiments performed
using data obtained from different operational failures of a real industrial
plant, showed very significant results regarding unsupervised as well as semi-
supervised learning, requiring minimal human interaction.
The manuscript presented in [2] brings a new algorithm for detecting anomalies
based on an online memory sequence algorithm called Hierarchical Temporal
Memory (HTM). The performance of the proposed algorithm was evaluated and
compared with a set of real time anomaly detection algorithms. Comparative
analysis was performed as a way to evaluate anomaly detection algorithms for
data streaming. All analyzes were performed from the Numenta Anomaly Benchmark
(NAB) [17], which is a benchmark of actual streaming data.
The paper published by [18] brings a study for anomaly detection in TCP / IP
networks. The purpose of the paper is to detect computer network anomalies in
the process of virtual machine (VM) live migration from local to cloud, by
comparing this approach between TEDA, clustering K-Means, and static analysis.
They used the tuple - Source IP, Destination IP, Source Port, and Destination
Port \- to create a signature process and validate errors, including those of
traffic flow hidden in the legitimate network. Testing was done using the
SECCRIT (SEcure Cloud Computing for CRitical Infrastructure IT -
http://www.seccrit.eu) project dataset, which allows anomalies or
environmental attacks to be analyzed with Live Migration and other background
traffic conditions. The results demonstrate that the proposed method makes it
possible to automatically and successfully detect anomalies in attacks,
network port scan (NPS) and network scan (NS). A major difficulty is
distinguishing a high-volume attack from a denial of service (DoS) attack, for
example. Accuracy and false negative rate calculations were made for
comparison with K-Means and the proposed solution, with TEDA having better
rates in almost all measurements performed.
As the amount of data that needs to be processed grows exponentially and
autonomous systems become increasingly important and necessary. Implementation
of machine learning and streaming algorithms have been studying in literature.
The work presented in [19] describes how to use run-time reconfiguration on
FPGAs to improve the efficiency of streaming data transmission in shared
communication channel with real-time applications. The reconfigurable
architecture proposed consists of two subsystems: the reconfiguration
subsystem, which running the modules, and the scheduling subsystem, that
controls which modules are loaded to the reconfiguration subsystem.
Besides, many works in the literature have been studied fault and anomaly
detection in hardware. In work [20], an implementation of target and anomaly
detection algorithms for real-time hyper-spectral imaging was proposed on
FPGA. The algorithms were implemented in streaming fashion, similar to this
work. The results, obtained from a Kintex-7 FPGA using fixed point structure,
were very satisfactory and demonstrated that the implementation can be used in
different detection circumstances. The work [21] presented a study of the
impact of Neural Network architectures compared to statistical methods in the
implementation of an Electrocardiogram (ECG) anomaly detection algorithm on
FPGA. The fixed point implementation contributes to reduce the amount of
needed resources. However, the design was made with High Level Sinthesys
(HLS), witch could not optimize the FPGA resources consumption. In relation to
the TEDA algorithm, no studies in the literature aimed at exploring its
hardware implementation on FPGA were identified to date this paper had been
write, which this work proposes to accomplish in a pioneering manner.
## 3 TEDA
TEDA was introduced by [22] as a statistical framework, influenced by
recursive density estimation algorithms. However, unlike algorithms that uses
data density as a measure of similarity, TEDA uses concepts of typicity and
eccentricity to infer whether a given sample is normal or abnormal to the
dataset. The methodology used in TEDA does not require the use of a previous
data information, and can be applied to problems involving fault detection,
clustering, classification, among others [22].
TEDA is a data structure-based anomaly detection algorithm that aims to
generalize and avoid the need for well-known, but very restrictive, initial
conditions inherent in traditional statistics and probability theory [23]. The
approach presented in the TEDA has some advantages over traditional
statistical anomaly detection methods. Its recursive feature allows it to
handle large volumes of data, such as data streams, with low computational
cost and online, enabling faster processing.
TEDA main features include [6]:
* 1.
It is entirely based on data and its distribution in data spaces;
* 2.
No previous assumptions are made;
* 3.
Limits and parameters does not need to be pre-specified;
* 4.
No sample independence required;
* 5.
An infinite number of observations are not required.
The typicality of TEDA is the similarity of a given data sample to the rest of
the dataset samples to which it belongs. Eccentricity, on the other hand, is
the opposite of typicality, which indicates how much a sample is dissociated
from the other samples in its set. Thus, an outlier can be defined as a sample
with high eccentricity and low typicality, considering a threshold established
for comparison. It is important to note that for eccentricity and typicality
calculations no parameter or threshold is required.
To calculate the eccentricity of each sample, TEDA uses the sum of the
geometric distances between the analyzed sample $\bm{x}_{k}$ and the other
samples in the set. Thus, the higher this value, the greater the eccentricity
of the sample, and consequently, the lower its typicality. [6] proposed
recursively calculating eccentricity. Thus, the eccentricity, $\xi$ can be
expressed as
$\xi_{k}(x)=\frac{1}{k}+\frac{(\bm{\mu}_{k}^{x}-\bm{x}_{k})^{T}(\bm{\mu}^{x}_{k}-\bm{x}_{k})}{k[\sigma^{2}]^{x}_{k}},[\sigma^{2}]^{x}_{k}>0$
(1)
where $k$ is discreization instant; $\bm{x}_{k}$ is a input set of N elements
in the k-th iteration, $\bm{x}_{k}=[x_{k}^{1}\ x_{k}^{2}\ ...\ x_{k}^{N}]$;
$\bm{\mu}_{x}^{k}$ is also a N elements vector, equal to the average of
$\bm{x}_{k}$ at the $k$-th iteration and $[\sigma^{2}]^{x}_{k}$ is the
variance of $\bm{x}_{k}$ at the $k$-th iteration. The calculation of
$\bm{\mu}_{x}^{k}$ and $[\sigma^{2}]^{x}_{k}$ is also recursively done, using
the following equation
$\bm{\mu}^{x}_{k}=\frac{(k-1)}{k}\bm{\mu}^{x}_{k-1}+\frac{1}{k}\bm{x}_{k},\
k\geq 1,\ \bm{\mu}^{x}_{0}=0$ (2)
and
$[\sigma^{2}]^{x}_{k}=\frac{(k-1)}{k}[\sigma^{2}]^{x}_{k-1}+\frac{1}{k}\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2},\
k\geq 1,\ [\sigma^{2}]^{x}_{0}=0.$ (3)
The typicality of a given sample $\bm{x}_{k}$, at the $k$-th iteration, can be
expressed as a complement to eccentricity [6], as follows
$\tau_{k}(x)=1-\xi_{k}(x).$ (4)
In addition, [6] also defined that normalized eccentricity can be calculated
as
$\zeta_{k}(x)=\frac{\xi_{k}(x)}{2},\sum^{k}_{i=1}\xi_{k}(x)=1,\ k\geq 2.$ (5)
In order to separate normal state data from abnormal state data, it is
necessary to define a comparison threshold. For anomaly detection, the use of
the $m\sigma$ [24] threshold is widespread. However, this principle must first
assume the distribution of the analyzed data, such as the Gaussian
distribution [6]. Chebyshev inequality can be used for any data distribution,
assuming that the probability that the data samples are more than $m\sigma$
from the average is less than or equal to $1/m^{2}$, where $\sigma$ is the
standard deviation of the data [25].
The condition that produces the same results as Chebyshev’s inequality,
discarding any assumptions about data and its independence, can be expressed
as [6]
$\zeta_{k}>\frac{m^{2}+1}{2k},\ m>0$ (6)
where $m$ corresponds to the comparison threshold.
For a better understanding of the hardware implemented technique in this work,
the Algorithm 1 details the operation of TEDA, based on the equations
presented above.
Input: $\mathbf{x}_{k}$: $k$-th sample; $m$: threshold
Output: outlier: sample classification as abnormal or normal
1 begin
2 while _receive $\mathbf{x}$_ do
3 if _k=1_ then
4 $\bm{\mu}^{x}_{k}\leftarrow\mathbf{x}_{k}$;
5 $[\sigma^{2}]^{x}_{k}\leftarrow 0$;
6
7 else
8 update $\bm{\mu}^{x}_{k}$ using equation 2;
9 update $[\sigma^{2}]_{k}^{x}$ using equation 3;
10 update $\xi_{k}(x)$ using equation 1;
11 update $\zeta_{k}(x)$ using equation 5;
12 if _$\zeta_{k}(x) >\frac{m^{2}+1}{2k}$_ then
13 $outlier\leftarrow true$;
14
15 else
16 $outlier\leftarrow false$;
17
18
19 $k\leftarrow k+1$;
20
21
22
Algorithm 1 TEDA
As presented in the Algorithm 1, only input data samples, $\bm{x}_{k}$, and a
comparison threshold, $m$, are used as input to the algorithm. The output for
each entry, $\bm{x}_{k}$, is the indication of the sample’s classification as
abnormal (outlier = true) or normal (outlier = false).
## 4 Implementation description
In this work, a TEDA FPGA proposal was implemented using Register Transfer
Level (RTL) such as works presented in [9, 10, 11, 12, 13]. In the following
section characteristics of the proposal will be presented, as well as details
regarding processing time. A design overview can be seen in Figure 1.
### 4.1 Architecture proposal overview
As illustrated in the Figure 1, the proposed implementation of TEDA has four
different block structures: The MEAN module, which implements the average
described in Equation 2; The VARIANCE module, responsible for calculate the
variance as presented at the equation 3; The ECCENTRICITY module, which
calculates the eccentricity, as presented in the equation 1; and the OUTLIER
module, a block used to normalize the eccentricity as in equation 5 and
compare with the threshold, as showed in equation 6. The architecture was
developed in an attempt to pipeline the operations presented in Algorithm 1 in
order to decrease the TEDA processing time. So, the output of the ECCENTRICITY
and OUTLIER modules are one clock cycle delayed in relation to VARIANCE module
and two in relation to MEAN module. As well as VARIANCE module is one clock
cycle delayed in relation to MEAN module. Each of the modules are detailed
later in the following sections.
The implementation has the Algorithm 1 as reference. The system receives the
FPGA clock and the $k$-th sample vector $\mathbf{x}_{k}$ as inputs. The $k$-th
iteration number is updated from the increment of a counter and the $m$
threshold is used as a constant, stored at OUTLIER module. As in the algorithm
line 1, the MEAN module do each single element average of $\mathbf{x}_{k}$
vector. It is possible to observe that there are $N$ MEAN blocks, where $N$ is
the vector size. This block will be detailed in section 4.2. After this step,
moving to the next line (1), the calculation of variance is done in VARIANCE
Module, this block is detailed in the section 4.3. ECCENTRICITY block has as
inputs the signals that left the block VARIANCE and $k$, as referred at line 1
and detailed in subsection 4.4. OUTLIER block is detailed in subsection 4.5.
It receives the ECCENTRICITY, $\xi_{k}(x)$, and calculate the normalized
eccentricity to compare with the threshold as presented in lines 1, 1 and 1.
Figure 1: General architecture overview.
### 4.2 Module I - MEAN
Each n-th MEAN module computes the average of each one of n-th elements vector
$\bm{x}_{k}$ acquired at run time. The implementations is based on Equation 2
and it is detailed in Figure 2. In addition to receiving the n-th element of
vector $\bm{x}_{k}$ as an input, the MEAN block uses a counter to define the
number of sample interaction, k. The implementation uses a comparator block
identified at the Figure 2 as MCOMPn witch is used to verify if the system is
in the first iteration as Line 1 of Algorithm 1. The MMUXn is a multiplexer
that acts as a conditional evaluation, using as selecting value the output of
MCOMPn comparator. The register MREGn is storing the n-th $\bm{\mu}^{x}_{k}$
element ($\mu^{n}_{k}$). The $\mu^{n}_{k}$ value stored in MREGn is multiplied
with $\frac{k-1}{k}$ in MMULT1n and added in MSUMn with the output of MMULT2n
that has as input $x^{n}_{k}$ and the inverse value of $k$. Each n-th element
of vector $\bm{x}_{k}$, $x^{n}_{k}$, requires a MEAN block.
Figure 2: MEAN module.
### 4.3 Module II - VARIANCE
The VARIANCE module is illustrated in Figure 3. It computes the variance of
$\bm{x}_{k}$ vector samples by receiving the $\bm{x}_{k}$ vector itself and
its average, $\bm{\mu}^{x}_{k}$, calculated in the previous MEAN blocks.
The VARIANCE module, as the MEAN module, uses a comparator identified at the
Figure 3 as VCOMP1 also to verify if the system is in the first iteration
(Line 1 of Algorithm 1). The VMUX1 is a multiplexer that also implements a
conditional evaluation to release the value $0$ in the register output VREG1
in the first iteration. The register VREG1 stores the variance value,
$[\sigma^{2}]^{x}_{k}$, from the second iteration. The other registers in the
block, VREG2 register and the N VREG$n$ registers, are used to delay by one
clock cycle the iteration number $k$ and the elements of $\bm{x}_{k}$
respectively.
Figure 3: VARIANCE module.
As demonstrated in Equation 3, the variance calculation is done recursively.
It is necessary to calculate $\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ and
to do that, N subtractors (VSUB$n$) and N multipliers (VMULT1_$n$) are used,
as well a adder (VSUM1) with N inputs. Each element of vector
$\bm{\mu}^{x}_{k}$ is subtracted from its respective element in vector
$\bm{x}_{k}$ and the result of this operation is multiplied by itself
(squared) and then added to the other results. The
$\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ value is the multiplied (at
VMULT2) by $1/k$. It is then added at VSUM2 adder with the variance calculated
in the previous iteration, $[\sigma^{2}]^{x}_{k}$, multiplied (VMULT3) by
$(k-1)/k$. From the second iteration on, this value passes through the VMUX1
multiplexer to the VREG1 register, delivering the calculation of the variance
value at the VARIANCE block output. The values of
$\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ and $1/k$ are also delivered at
the output of the VARIANCE block to avoid redundant operations as they will be
used in the next block, the ECCENTRICITY block.
### 4.4 Module III - ECCENTRICITY
The ECCENTRICITY module is a simpler block than those previously presented.
This is because it uses operations already performed in the VARIANCE block to
calculate eccentricity. The geometric distance
$\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ (equivalent to
$(\bm{\mu}_{k}^{x}-\bm{x}_{k})^{T}(\bm{\mu}^{x}_{k}-\bm{x}_{k})$) is stored in
register EREG3 and $1/k$ is stored in EREG4 register. As the ECCENTRICITY
module is the architecture design of Equation 1 (Algorithm 1 line 1), the
variance value $[\sigma^{2}]^{x}_{k}$ is multiplied by $k$ (EMULT1) and used
to divise (EDIV1) the geometric distance
$(\bm{\mu}_{k}^{x}-\bm{x}_{k})^{T}(\bm{\mu}^{x}_{k}-\bm{x}_{k})$. This
operation output is added to $1/k$ in the ESUM1 adder, calculating the
eccentricity of the samples ($\xi_{k}(x)$) and delivering to the ECCENTRICITY
block output.
Figure 4: ECCENTRICITY module.
### 4.5 Module IV - OUTLIER
Finally, in the OUTLIER block, the samples are classified into abnormal
(outlier = true) or normal (outlier = false). The design module can be seen in
Figure 5. To classify the samples, the OUTLIER block normalizes eccentricity
by dividing (ODIV1) by a constant, as shown in Equation 5, and compares
(OCOMP1 this normalized eccentricity with a threshold as shown in the Lines 1,
1, 1 and 1 of the Algorithm 1. The register OREG1 and OREG2 are used to
synchronize the iteration number $k$, since as the modules act in pipeline,
the operations carried out in the OUTLIER block (as well as in ECCENTRICITY
module) are delayed by two clock cycles in relation to the system input.
Figure 5: OUTLIER module.
### 4.6 Processing time
The proposed architecture has an initial delay, $d$, that can be expressed as
$d=3\times t_{c}$ (7)
where $t_{c}$ is the system critical path time.
The execution time of the circuit implemented for TEDA algorithm is determined
by the system critical path time, $t_{c}$. So, after the initial delay, the
execution time of the proposed TEDA, $t_{TEDA}$, can be expressed as
$t_{TEDA}=t_{c}$ (8)
thus, in every $t_{TEDA}$ it is possible to obtain the output of a sample
inserted, that is, the sample classification as abnormal or normal.
The throughput of the implementation, $th_{TEDA}$, in samples per second (SPS)
can be expressed as
$th_{TEDA}=\frac{1}{t_{TEDA}}.$ (9)
## 5 Results
In this section will be presented the hardware validation and synthesis
results for the architecture proposed in this work. All cases were validated
and synthesized on floating point. Validation results were used to verify the
hardware functionality, while synthesis results allow the system to be
analyzed for important parameters for the design of hardware architectures
such as hardware occupancy and processing time, considering factors such as
throughput and speedup.
### 5.1 Validation results
To validate the hardware architecture of the TEDA algorithm, we used the
DAMADICS (Development and Application of Methods of the Actuator Diagnosis in
Industrial Control Systems) benchmark dataset [26]. The benchmark provides a
real data set of the water evaporation process in a Polish sugar factory. It
is a plant with three actuators; a control valve, which controls the flow of
water in the pipes; a pneumatic motor, which controls variable valve openings
and a positioner. This dataset has faults at different times of the day on
specific days. There are four different fault types, as shown in Table 1.
Table 1: Fault types [26]. Fault | Description
---|---
f16 | Positioner supply pressure drop
f17 | Unexpected pressure change across the valve
f18 | Fully or partly opened bypass valves
f19 | Flow rate sensor fault
Artificial failures were introduced on specific days to plant operation data.
The dataset has a set of $19$ faults in these $3$ actuators. As a way to
validate the architecture, actuator $1$ failures were simulated. Table 2 shows
a detailed description of some introduced faults for actuator $1$.
Table 2: List of artificial failures introduced to actuator 1 [26]. Item | Fault | Sample | Date | Description
---|---|---|---|---
1 | f18 | 58800-59800 | Oct 30, 2001 | Partly opened bypass valve
2 | f16 | 57275-57550 | Nov 9, 2001 | Positioner supply pressure drop
3 | f18 | 58830-58930 | Nov 9, 2001 | Partly opened bypass valve
4 | f18 | 58520-58625 | Nov 9, 2001 | Partly opened bypass valve
5 | f18 | 54600-54700 | Nov 17, 2001 | Partly opened bypass valve
6 | f16 | 56670-56770 | Nov 17, 2001 | Positioner supply pressure drop
7 | f17 | 37780-38400 | Nov 20, 2001 | Unexpected pressure drop across the valve
Figure 6 shows the results obtained for the item 1 signal of Table 2. Figure
6(a) illustrates the behavior of two simulated input variables in hardware
architecture ($\bm{x}_{k}=[x_{k}^{1}\ x_{k}^{2}]$). It is possible to observe
that a failure happens between the moments $k$=58900 and $k$=59800. In Figure
6(b) it is possible to observe that there is a sudden change in the behavior
of the eccentricity (black curve), surpassing the value of the comparison
threshold with $m=3$ (red curve).
(a) Fault item 1 - input vector $\bm{x}_{k}$.
(b) Fault item 1 - normalized eccentricity $\zeta_{k}(x)$ with $5/k$ $(m=3)$
threshold.
Figure 6: Detection of outliers in the dataset: Behavior of fault item 1.
In Figure 7 it is possible to observe the results obtained for the item 7
signal, from Table 2. As within Figure 6, Figures 7(a) illustrates the
behavior of two elements of input $\bm{x}_{k}=[x^{1}_{k}\ x^{2}_{k}]$ in
hardware architecture and in Figure 7(b) it is possible to observe that there
is a change of eccentricity (black curve), surpassing the value of the
comparison threshold (red curve) also to $m=3$. The failure happens between
moments $k=37700$ and $k=38400$.
(a) Fault item 7 - input vector $\bm{x}_{k}$.
(b) Fault item 7 - normalized eccentricity $\zeta_{k}(x)$ with $5/k$ $(m=3)$
threshold.
Figure 7: Detection of outliers in the dataset: Behavior of fault item 7.
Validation results in hardware architecture were compared with results
obtained in a python software implementation of the algorithm TEDA. The
hardware architecture was designed with floating point number format.
### 5.2 Synthesis results
After performing to validate the implemented circuit, the hardware synthesis
was performed to obtain the FPGA resource occupation report, as well as the
critical time information used to calculate the proposed implementation
processing time. The floating point synthesis results were obtained for a
Xilinx Virtex 6 xc6vlx240t-1ff1156 FPGA.
#### 5.2.1 Hardware occupation
Table 3 presents data related to the hardware occupation of the circuit
implemented in the target FPGA. The first column shows the number of
multipliers used, the second column displays the number of registers, and the
third column shows the number of logical cells used as LUT ($n_{LUT}$)
throughout the circuit.
Table 3: Hardware occupation. Multipliers | | Registers
---
$n_{LUT}$
$27$ ($3$%) | $414$ ($<1$%) | $11.567$ ($7$%)
Analyzing the data presented in Table 3 it can be seen that even using a
floating point resolution, which demands a greater amount of hardware
resources than a fixed point implementation, only a small portion of the
resources were occupied from the target FPGA, with a total of only about $3\%$
from multipliers, less than $1\%$ from registers, and about $7\%$ from logical
cells used as LUT. With this, we found that the proposed circuit could also be
applied in low cost FPGAs, where the amount of available hardware resources is
even smaller. In addition, multiple TEDA modules could be applied in parallel
for anomaly detection in the same dataset, in order to further reduce
processing time.
#### 5.2.2 Processing time
Table 4 presents information about the processing time (from Line 1 to Line 1
in Algorithm 1) of the architecture implemented for the TEDA technique. The
first column indicates the circuit critical time, $t_{c}$, the second column
shows the initial delay, expressed by Equation 7, the third column the TEDA
run-time, expressed by Equation 8, and the last column the implementation
throughput in samples per second (SPS), expressed by Equation 9, which
consists of the amount of samples processed and classified (as normal or
outlier) by TEDA every second.
Table 4: Processing time. Critical time | | Delay
---
TEDA time | Throughput
$138\,\text{ns}$ | $414\,\text{ns}$ | $138\,\text{ns}$ | $7.2$ MSPS
The data presented in Table 4 are quite expressive. The circuit critical time,
which also corresponds in the TEDA run-time, was only $t_{c}=138\,\text{ns}$.
Thus, after the $414\,\text{ns}$ delay, it is possible to get output for a
processed sample sorted every $138\,\text{ns}$, which guarantees a throughput
of $7.2$ million sorted samples per second. These results indicate the
feasibility of using the proposal presented in this work to manipulate large
data flows in real time.
### 5.3 Platforms comparison
To date, no previous literature has been found to explore TEDA hardware
implementations. Thus, this paper presents, for the first time, a proposal to
implement the TEDA technique on FPGA. To verify the advantages of the hardware
application proposed here over implementations on other software platforms,
some comparisons of the FPGA processing time with the processing time of other
software implementations were made. Table 5 presents the results of the
comparisons made. The first column indicates the hardware used, the second
presents the processing time required to obtain the classification of each
sample, and the third column, the speedup achieved by the proposal presented
in this paper.
Table 5: Software implementations comparison. Platform | Time | Speedup
---|---|---
This work proposal on FPGA | $138\,\text{ns}$ | $-$
Python (Colab without GPU) | $435\,\text{ms}$ | $3\text{,}000\text{,}000\times$
Python (Colab with Tesla K80 GPU) | $39.2\,\text{ms}$ | $280\text{,}000\times$
Python (Local execution with 940 MX GPU) | $23.1\,\text{ms}$ | $167\text{,}000\times$
The data presented in Table 5 reaffirm the importance of this work. The
hardware implementation on FPGA proposed here has been able to achieve
speedups of up to $3$ million times compared to a Pyhton TEDA implementation
using the Colab tool (without GPU processing). For the same Pyhton
implementation using the Tesla K80 GPU processing Colab tool, a speedup of
$280$ thousand times was obtained. In addition, when compared to a Python
implementation on Intel(R) Core(TM) i7-7500U<EMAIL_ADDRESS>with 16 GB of RAM and
GeForce 940 MX GPU, the hardware implementation on FPGA still had a $167$
thousand times advantage. Results that prove the advantages of using the
proposal presented in this work to accelerate the TEDA technique, through the
implementation on FPGA.
## 6 Conclusion
This work presented a proposal for hardware implementation of the TEDA data
streaming anomaly detection technique. The hardware was implemented in RTL
using floating point format. Synthesis results were obtained for a Xilinx
Virtex 6 xc6vlx240t-1ff1156 FPGA. The proposed implementation used a small
portion of the target FPGA resources, besides allowing the results to be
obtained in a short processing time. The high speedups obtained in comparison
with other software platforms reaffirmed the importance of this work, which is
pioneering the hardware implementation of the TEDA technique on FPGA. The
proposed architecture is feasible to be used in practical fault detection
applications in real industrial processes with severe time constraints, as
well as to handle large data volumes, such as data streaming, using low
processing time.
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|
2024-09-04T02:54:58.219827 | 2020-03-08T23:23:31 | 2003.03867 | {
"authors": "Wojciech Jamroga, Wojciech Penczek, and Teofil Sidoruk",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26106",
"submitter": "Wojciech Jamroga",
"url": "https://arxiv.org/abs/2003.03867"
} | arxiv-papers |
# Strategic Abilities of Asynchronous Agents:
Semantic Side Effects and How to Tame Them
Wojciech Jamroga1,2 Wojciech Penczek1 Teofil Sidoruk1,3
1Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland
2Interdisciplinary Centre on Security, Reliability and Trust, SnT, University
of Luxembourg
3Faculty of Mathematics and Information Science, Warsaw University of
Technology
{jamroga, penczek<EMAIL_ADDRESS>
###### Abstract
Recently, we have proposed a framework for verification of agents’ abilities
in asynchronous multi-agent systems (MAS), together with an algorithm for
automated reduction of models (?). The semantics was built on the modeling
tradition of distributed systems. As we show here, this can sometimes lead to
counterintuitive interpretation of formulas when reasoning about the outcome
of strategies. First, the semantics disregards finite paths, and yields
unnatural evaluation of strategies with deadlocks. Secondly, the semantic
representations do not allow to capture the asymmetry between proactive agents
and the recipients of their choices. We propose how to avoid the problems by a
suitable extension of the representations and change of the execution
semantics for asynchronous MAS. We also prove that the model reduction scheme
still works in the modified framework.
## 1 Introduction
Modal logics of strategic ability. _Alternating-time temporal logic_
$\mathbf{ATL_{\mathrm{}}^{*}}$ (?; ?; ?) is probably the most popular logic to
describe interaction of agents in multi-agent systems. Formulas of
$\mathbf{ATL_{\mathrm{}}^{*}}$ allow to express statements about what agents
(or groups of agents) can achieve. For example,
$\langle\\!\langle{taxi}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{fatality}}$
says that the autonomous cab can drive in such a way that nobody is ever
killed, and
$\langle\\!\langle{taxi,passg}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{destination}}$
expresses that the cab and the passenger have a joint strategy to arrive at
the destination, no matter what any other agents do. Such statements allow to
express important functionality and safety requirements in a simple and
intuitive way. Moreover, the provide input to algorithms and tools for
verification of strategic abilities, that have been in constant development
for over 20 years (?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?). Still,
there are two caveats.
First, all the realistic scenarios of agent interaction, that one may want to
specify and verify, involve imperfect information. That is, the agents in the
system do not always know exactly the global state of the system, and thus
they have to make their decisions based on their local view of the situation.
Unfortunately, verification of agents with imperfect information is hard to
very hard – more precisely, $\mathbf{{\Delta_{{2}}^{\mathbf{{P}}}}}$-complete
to undecidable, depending on the syntactic and semantic variant of the logic
(?; ?; ?). Also, the imperfect information semantics of
$\mathbf{ATL_{\mathrm{}}^{*}}$ does not admit alternation-free fixpoint
characterizations (?; ?; ?), which makes incremental synthesis of strategies
impossible, or at least difficult to achieve (?; ?; ?; ?; ?).
Secondly, the semantics of strategic logics is traditionally based on
synchronous concurrent game models. In other words, one implicitly assumes the
existence of a global clock that triggers subsequent global events in the
system; at each tick of the clock, all the agents choose their actions, and
the system proceeds accordingly with a global transition. However, many real-
life systems are inherently asynchronous, and do not operate on a global clock
that perfectly synchronizes the atomic steps of all the components. Moreover,
many systems that are synchronous at the implementation level can be more
conveniently modeled as asynchronous on a more abstract level. In many
scenarios, both aspects combine. For example, when modeling an anti-poaching
operation (?), one may take into account the truly asynchronous nature of
events happening in different national parks, but also the best level of
granularity for modeling the events happening within a single nature reserve.
Asynchronous semantics and partial-order reduction. We have recently proposed
how to adapt the semantics of $\mathbf{ATL_{\mathrm{}}^{*}}$ to asynchronous
MAS (?). We also showed that the technique of _partial order reduction (POR)_
(?; ?; ?; ?; ?; ?; ?) can be adapted to verification of strategic abilities in
asynchronous MAS. In fact, the (almost 30 years old) POR for linear time logic
$\mathbf{LTL}$ can be taken off the shelf and applied to a significant part of
$\mathbf{ATL^{*}_{\mathrm{ir}}}$, the variant of
$\mathbf{ATL_{\mathrm{}}^{*}}$ based on strategies with imperfect information
and imperfect recall. This is very important, as the practical verification of
asynchronous systems is often impossible due to the state- and transition-
space explosion resulting from interleaving of local transitions. POR allows
for a significant, sometimes even exponential, reduction of the models.
Semantic side effects. While the result is appealing, there is a sting in its
tail: the $\mathbf{ATL_{\mathrm{}}^{*}}$ semantics in (?) leads to
counterintuitive interpretation of strategic properties. First, it disregards
finite paths, and evaluates some intuitively losing strategies as winning (and
vice versa). Secondly, it provides a flawed interpretation of the _concurrency
fairness_ assumption. Thirdly, the representations and their execution
semantics do not allow to capture the asymmetry between the agents that
control which synchronization branch will be taken, and those influenced by
their choices. We tentatively indicated some of the problems in the extended
abstract (?). In this paper, we demonstrate them carefully, and propose how
they can be avoided.
Contribution. Our contribution is threefold. First, we discuss in detail the
semantic side effects of adding strategic reasoning on top of classical models
of concurrent systems (?). We identify the reasons, and demonstrate the
problematic phenomena on simple examples. Secondly, we show how to avoid these
pitfalls by extending the class of representations and slightly changing the
execution semantics of strategies. Specifically, we add “silent”
$\epsilon$-transitions in the models and on outcome paths of strategies, and
allow for nondeterministic choices in the agents’ repertoires. We also
identify a family of fairness-style conditions, suitable for the interaction
of proactive and reactive agents. No less importantly, we prove that partial
order reduction is still correct in the modified framework.
Motivation. The variant of $\mathbf{ATL_{\mathrm{}}^{*}}$ for asynchronous
systems in (?) was proposed mainly as a framework for formal verification.
This was backed by the results showing that it submits to partial order
reduction. However, a verification framework is only useful if it allows to
specify requirements in an intuitive way, so that the property we _think_ we
are verifying is indeed _the one being verified_. In this paper, we show that
this was not the case. We also propose how to overcome the problems without
spoiling the efficient reduction scheme. The solutions are not merely
technical. In fact, they lead to a better understanding of how strategic
activity influences the overall behavior of the system, and how it should be
integrated with the traditional models of asynchronous interaction.
## 2 Models of Multi-agent Systems
We first recall the models of asynchronous interaction in MAS, proposed in (?)
and inspired by (?; ?; ?).
### 2.1 Asynchronous Multi-agent Systems
In logical approaches to MAS, one usually assumes synchronous actions of all
the agents (?; ?). However, many agent systems are inherently asynchronous, or
it is useful to model them without assuming precise timing relationships
between the actions of different agents. As an example, consider a team of
logistic robots running in a factory (?). Often no global clock is available
to all the robots, and even if there is one, the precise relative timing for
robots operating in different places is usually irrelevant.
Such a system can be conveniently represented with a set of automata that
execute asynchronously by interleaving local transitions, and synchronize
their moves whenever a shared event occurs. The idea is to represent the
behavior of each agent by a finite automaton where the nodes and transitions
correspond, respectively, to the agent’s local states and the events in which
it can take part. Then, the global behavior of the system is obtained by the
interleaving of local transitions, assuming that, in order for a shared event
to occur, all the corresponding agents must execute it in their automata. This
motivates the following definition.
###### Definition 2.1 (Asynchronous MAS).
An _asynchronous multi-agent system (AMAS)_ S consists of $n$ agents
${\mathbb{A}\mathrm{gt}}=\\{{1,\dots,n}\\}$,111 We do not consider the
environment component, which may be added with no technical difficulty. each
associated with a tuple
$A_{i}=(L_{i},\iota_{i},\mathit{Evt}_{i},R_{i},T_{i}{,\mathcal{PV}_{i},V_{i}})$
including a set of _possible local states_
$L_{i}=\\{l_{i}^{1},l_{i}^{2},\dots,l_{i}^{n_{i}}\\}$, an _initial state_
$\iota_{i}\in L_{i}$, and a set of _events_
$\mathit{Evt}_{i}=\\{\alpha_{i}^{1},\alpha_{i}^{2},\ldots,\alpha_{i}^{m_{i}}\\}$.
An agent’s _repertoire of choices_ 222 In interpreted systems, this function
is usually referred to as a _protocol_. Here, we opt for a different name to
avoid possible confusion, e.g., with security protocols. $R_{i}:L_{i}\to
2^{\mathit{Evt}_{i}}\setminus\\{\emptyset\\}$ selects the events available at
each local state. $T_{i}:L_{i}\times\mathit{Evt}_{i}\rightharpoonup L_{i}$ is
a (partial) _local transition function_ such that $T_{i}(l_{i},\alpha)$ is
defined iff $\alpha\in R_{i}(l_{i})$. That is, $T_{i}(l,\alpha)$ indicates the
result of executing event $\alpha$ in local state $l$ from the perspective of
agent $i$.
Let $\mathit{Evt}=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}\mathit{Evt}_{i}$ be
the set of all events, and $Loc=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}L_{i}$ be
the set of all local states in the system. For each event
$\alpha\in\mathit{Evt}$,
$Agent(\alpha)=\\{{i\in{\mathbb{A}\mathrm{gt}}\mid\alpha\in\mathit{Evt}_{i}}\\}$
is the set of agents which have $\alpha$ in their repertoires; events shared
by multiple agents are jointly executed by all of them. We assume that each
agent $i$ in the AMAS is endowed with a disjoint set of its _local
propositions $\mathcal{PV}_{i}$_, and their valuation $V_{i}:L_{i}\rightarrow
2^{\mathcal{PV}_{i}}$. The overall set of propositions
$\mathcal{PV}=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}\mathcal{PV}_{i}$ collects
all the local propositions.
As our working example, we use the following scenario.
###### Example 2.2 (Conference in times of epidemic).
Consider the AMAS in Figure 1, consisting of the Steering Committee Chair
($sc$), the General Chair ($gc$), and the Organizing Committee Chair ($oc$).
Faced with the Covid-19 epidemics, $sc$ can decide to give up the conference,
or send a signal to $gc$ to proceed and open the meeting. Then, $gc$ and $oc$
jointly decide whether the conference will be run on site or online. In the
former case, the epidemiologic risk is obviously much higher, indicated by the
atomic proposition $\mathsf{{epid}}$.
The set of events, the agents’ repertoires of choices, and the valuation of
atomic propositions can be easily read from the graph. For easier reading, all
the private events are shown in grey. Note that event $proceed$ is shared by
agents $sc$ and $gc$, and can only be executed jointly. Similarly, $onsite$
and $online$ are shared by $gc$ and $oc$. All the other events are private,
and do not require synchronization.
gc | oc | sc
---|---|---
|
$0$$1$$\mathsf{{open}}$$2$$3$$\boldsymbol{proceed}$$\boldsymbol{onsite}$${online}$$\boldsymbol{rest}$$\boldsymbol{rest}$
---
|
$0$$1$$\mathsf{{epid}}\quad{}$$2$$3$$\mathsf{{closed}}\quad{}$$onsite$$\boldsymbol{online}$$\boldsymbol{handle}$$\boldsymbol{handle}$$\boldsymbol{idle}$
---
| $0$$1$$2$${proceed}$$giveup$$proceed$$giveup$
---
Figure 1: Simple asynchronous MAS: agents $gc$, $oc$, and $sc$. A joint
strategy of agents $\\{{gc,oc}\\}$ is highlighted.
### 2.2 Interleaved Interpreted Systems
To understand the interaction between asynchronous agents, we use the standard
execution semantics from concurrency models, i.e., interleaving with
synchronization on shared events. To this end, we compose the network of local
automata (i.e., AMAS) to a single automaton based on the notions of _global
states_ and _global transitions_ , see below.
###### Definition 2.3 (Model).
Let $S$ be an AMAS with $n$ agents. Its _model_ $IIS(S)$ extends $S$ with: (i)
the set of global states $St\subseteq L_{1}\times\ldots\times L_{n}$,
including the _initial state_ $\iota=(\iota_{1},\dots,\iota_{n})$ and all the
states reachable from $\iota$ by $T$ (see below); (ii) the _global transition
function_ $T:St\times\mathit{Evt}\rightharpoonup St$, defined by
$T(g_{1},\alpha)=g_{2}$ iff $T_{i}(g_{1}^{i},\alpha)=g^{i}_{2}$ for all $i\in
Agent(\alpha)$ and $g_{1}^{i}=g^{i}_{2}$ for all
$i\in{\mathbb{A}\mathrm{gt}}\setminus Agent(\alpha)$; (iii) the _global
valuation_ of propositions $V:St\rightarrow 2^{\mathcal{PV}}$, defined as
$V(l_{1},\dots,l_{n})=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}V_{i}(l_{i})$.
Models, sometimes called _interleaved interpreted systems_ (IIS), are used to
provide an execution semantics to AMAS, and consequently provide us with
semantic structures to reason about AMAS. Intuitively, the global states in
$IIS(S)$ can be seen as the possible configurations of local states of all the
agents. Moreover, the transitions are labeled by events that are
simultaneously selected (in the current configuration) by all the agents that
have the event in their repertoire. Clearly, private events (i.e., events such
that $Agent(\alpha)$ is a singleton) require no synchronization.
###### Example 2.4 (Conference).
The model for the asynchronous MAS of Example 2.2 is shown in Figure 1.
We say that event $\alpha\in\mathit{Evt}$ is _enabled_ at $g\in St$ if
$T(g,\alpha)=g^{\prime}$ for some $g^{\prime}\in St$. The set of events
enabled at $g$ is denoted by $enabled(g)$. The global transition function is
assumed to be serial, i.e., at each $g\in St$ there exists at least one
enabled event.
Discussion. This modeling approach is standard in theory of concurrent
systems, where it dates back to the early 1980s and the idea of APA Nets
(asynchronous, parallel automata nets) (?). Note that APA Nets and their
models were _not_ proposed with causal interpretation in mind. In particular,
they were _not_ meant to capture the interaction of purposeful agents that
freely choose their strategies, but rather a set of reactive components
converging to a joint behavior. Despite superficial differences, the same
applies to process-algebraic approaches to concurrency, such as CSP (?), CCS
(?), ACP (?), and $\pi$-calculus (?).
Definition 2.1 extends that with the repertoire functions from synchronous
models of MAS (?; ?). Agent $i$’s repertoire lists the events available to
$i$, and is supposed to define the space of $i$’s strategies. As we show
further, this is not enough in case of asynchronous MAS.
## 3 Reasoning About Abilities: ATL*
_Alternating-time temporal logic_ $\mathbf{ATL_{\mathrm{}}^{*}}$ (?; ?; ?)
generalizes the branching-time temporal logic $\mathbf{CTL^{*}}$ (?) by
replacing the path quantifiers $\mathsf{E},\mathsf{A}$ with _strategic
modalities_ $\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$,
expressing that agents $A$ can enforce the temporal property $\gamma$. While
the semantics of $\mathbf{ATL_{\mathrm{}}^{*}}$ is typically defined for
models of synchronous systems, a variant for asynchronous MAS was proposed
recently (?). We summarize the main points in this section.
### 3.1 Syntax
Let $\mathcal{PV}$ be a set of propositional variables and
${\mathbb{A}\mathrm{gt}}$ the set of all agents. The language of
$\mathbf{ATL_{\mathrm{}}^{*}}$ is defined as below.
$\varphi::=\mathsf{{p}}\mid\neg\varphi\mid\varphi\wedge\varphi\mid\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$,
$\gamma::=\varphi\mid\neg\gamma\mid\gamma\land\gamma\mid\mathrm{X}\,\gamma\mid\gamma\,\mathrm{U}\,\gamma$,
where $\mathsf{p}\in\mathcal{PV}$, $A\subseteq{\mathbb{A}\mathrm{gt}}$,
$\mathrm{X}\,$ stands for “next”, and $\,\mathrm{U}\,$ for “strong until”
($\gamma_{1}\,\mathrm{U}\,\gamma_{2}$ denotes that $\gamma_{1}$ holds until
$\gamma_{2}$ becomes true). The other Boolean operators and constants are
defined as usual. “Release” can be defined as
$\gamma_{1}\,\mathrm{R}\,\gamma_{2}\equiv\neg((\neg\gamma_{1})\,\mathrm{U}\,(\neg\gamma_{2}))$.
“Eventually” and “always” can be defined as
$\mathrm{F}\,\gamma\equiv\mathit{true}\,\mathrm{U}\,\gamma$ and
$\mathrm{G}\,\gamma\equiv\mathit{false}\,\mathrm{R}\,\gamma$. Moreover, the
$\mathbf{CTL^{*}}$ operator “for all paths” can be defined as
$\mathsf{A}\gamma\equiv\langle\\!\langle{\emptyset}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$.
###### Example 3.1 (Conference).
Formula
$\langle\\!\langle{sc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{open}}$
expresses that the Steering Chair can enforce that the conference is
eventually opened. Moreover, formula
$\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{epid}}$
says that the General Chair and the Organizing Chair have a joint strategy to
avoid high epidemiological risk.
### 3.2 Strategies and Outcomes
We adopt Schobbens’ taxonomy and notation for strategy types (?):
$\mathrm{ir}$, $\mathrm{Ir}$, $\mathrm{iR}$, and $\mathrm{IR}$, where _I_
(resp. _i_) denotes perfect (resp. imperfect) _information_ , and _R_ (resp.
_r_) denotes perfect (resp. imperfect) _recall_. In particular, an _imperfect
information/imperfect recall strategy ($\mathrm{ir}$-strategy) for $i$_ is a
function $\sigma_{i}\colon L_{i}\to\mathit{Evt}_{i}$ s.t. $\sigma_{i}(l)\in
R_{i}(l)$ for each $l\in L_{i}$. We denote the set of such strategies by
$\Sigma_{i}^{\mathrm{ir}}$. A _collective strategy_ $\sigma_{A}$ for a
coalition $A=(1,\dots,m)\subseteq{\mathbb{A}\mathrm{gt}}$ is a tuple of
strategies, one per agent $i\in A$. The set of $A$’s collective $\mathrm{ir}$
strategies is denoted by $\Sigma_{A}^{\mathrm{ir}}$. We will sometimes use
$\sigma_{A}(g)=(\sigma_{a_{1}}(g),\dots,\sigma_{a_{m}}(g))$ to denote the
tuple of $A$’s selections at state $g$.
###### Example 3.2 (Conference).
A collective strategy for the General Chair and the OC Chair in the conference
scenario is shown in Figure 1.
An infinite sequence of global states and events
$\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots$ is called an (interleaved)
_path_ if $g_{i}\stackrel{{\scriptstyle\alpha_{i}}}{{\longrightarrow}}g_{i+1}$
for every $i\geq 0$. $\mathit{Evt}(\pi)=\alpha_{0}\alpha_{1}\alpha_{2}\ldots$
is the sequence of events in $\pi$, and $\pi[i]=g_{i}$ is the $i$-th global
state of $\pi$. $\Pi_{M}(g)$ denotes the set of all paths in model $M$
starting at $g$. Intuitively, the outcome of $\sigma_{A}$ in $g$ is the set of
all the paths that can occur when the agents in $A$ follow $\sigma_{A}$ and
the agents in ${\mathbb{A}\mathrm{gt}}\setminus A$ freely choose events from
their repertoires. To define it formally, we first refine the concept of an
enabled event, taking into account the choices of $A$ in strategy
$\sigma_{A}$.
###### Definition 3.3 (Enabled events).
Let $A=(1,\dots,m)$, $g\in St$, and let
$\overrightarrow{\alpha}_{A}=(\alpha_{1},\dots,\alpha_{m})$ be a tuple of
events such that every $\alpha_{i}\in R_{i}(g^{i})$. That is, every
$\alpha_{i}$ can be selected by its respective agent $i$ at state $g$. We say
that event $\beta\in\mathit{Evt}$ is _enabled by $\overrightarrow{\alpha}_{A}$
at $g\in St$_ iff
* •
for every $i\in Agent(\beta)\cap A$, we have $\beta=\alpha_{i}$, and
* •
for every $i\in Agent(\beta)\setminus A$, it holds that $\beta\in
R_{i}(g^{i})$.
Thus, $\beta$ is enabled by $\overrightarrow{\alpha}_{A}$ if all the agents
that “own” $\beta$ can choose $\beta$ for execution, even when
$\overrightarrow{\alpha}_{A}$ has been selected by the coalition $A$. We
denote the set of such events by $enabled(g,\overrightarrow{\alpha}_{A})$.
Clearly, $enabled(g,\overrightarrow{\alpha}_{A})\subseteq enabled(g)$.
###### Example 3.4 (Conference).
Consider state $g=000$ and the choices of agents $A=\\{{gc,oc}\\}$ shown in
Figure 1, i.e., $\overrightarrow{\alpha}_{A}=(proceed,online)$. The only
events enabled by $\overrightarrow{\alpha}_{A}$ are $proceed$ and $giveup$.
Event $onsite$ is not enabled because $A$ chose different events for
execution; $online$ is not enabled because it requires synchronization which
is impossible at $000$.
###### Definition 3.5 (Outcome paths).
The _outcome_ of strategy $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ in state
$g\in St$ is the set $\mathit{out}_{M}(g,\sigma_{A})\subseteq\Pi_{M}(g)$ such
that
$\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots\in\mathit{out}_{M}(g,\sigma_{A})$
iff $g_{0}=g$, and $\forall i\geq 0\quad\alpha_{i}\in
enabled(\pi[i],\sigma_{A}(\pi[i]))$.
One often wants to look only at paths that do not consistently ignore agents
whose choice is always enabled. Formally, a path $\pi$ satisfies _concurrency-
fairness_ (CF) if there is no event $\alpha$ enabled in all states of $\pi$
from $\pi[n]$ on and such that for every $\alpha_{i}$ actually executed in
$\pi[i]$, $i=n,n+1,\dots$, we have $Agent(\alpha)\cap
Agent(\alpha_{i})=\emptyset$. We denote the set of all such paths starting at
$g$ by $\Pi_{M}^{\textbf{CF}}(g)$.
###### Definition 3.6 (CF-outcome).
The _CF -outcome_ of $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ is defined as
$\mathit{out}^{\textbf{CF}}_{M}(g,\sigma_{A})=\mathit{out}_{M}(g,\sigma_{A})\cap\Pi_{M}^{\textbf{CF}}(g)$.
### 3.3 Strategic Ability for Asynchronous Systems
The semantics of $\mathbf{ATL_{\mathrm{ir}}^{*}}$ in AMAS is defined by the
following clause for strategic modalities (?):
$M,g\models_{{}_{\mathrm{ir}}}\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$
iff there is a strategy $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ s.t.
$\mathit{out}_{M}(g,\sigma_{A})\neq\emptyset$ and, for each path
$\pi\in\mathit{out}_{M}(g,\sigma_{A})$, we have
$M,\pi\models_{{}_{\mathrm{ir}}}\gamma$.
The clauses for Boolean and temporal operators are standard. Moreover, the
_concurrency-fair semantics_ $\models_{{}_{\mathrm{ir}}}^{\textbf{CF}}$ of
$\mathbf{ATL_{\mathrm{}}}$ and $\mathbf{ATL_{\mathrm{}}^{*}}$ is obtained by
replacing $\mathit{out}_{M}(g,\sigma_{A})$ with
$\mathit{out}_{M}^{\textbf{CF}}(g,\sigma_{A})$ in the above clause.
###### Example 3.7 (Conference).
Clearly, formula
$\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{epid}}$
holds in $(M_{\mathit{conf}},000)$, in both $\models_{{}_{\mathrm{ir}}}$ and
$\models_{{}_{\mathrm{ir}}}^{\textbf{CF}}$ semantics. To see that, fix
$\sigma_{gc}(0)=proceed$ and $\sigma_{gc}(1)=\sigma_{oc}(0)=online$ in the
collective strategy of $\\{{gc,oc}\\}$. Note also that
$M_{\mathit{conf}},000\models_{{}_{\mathrm{ir}}}\neg\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{closed}}$
because, after executing $proceed$ and $online$ (or $onsite$), event $rest$
may be selected forever. On the other hand, such paths are not concurrency-
fair, and thus
$M_{\mathit{conf}},000\models_{{}_{\mathrm{ir}}}^{\textbf{CF}}\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{closed}}$.
Discussion. Strategic play assumes proactive attitude: the agents in
$\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}$ are free to choose
_any_ available strategy $\sigma_{A}$. This is conceptually consistent with
the notion of agency (?). At the same time, it is somewhat at odds with the
standard semantics of concurrent processes, where the components cannot
stubbornly refuse to synchronize if that is the only way to proceed with a
transition. This seems a minor problem, but it is worrying that a strategy can
have the empty set of outcomes, and equally worrying that such strategies are
treated differently from the other ones. Indeed, as we will show in the
subsequent sections, the semantics proposed in (?) leads to a counterintuitive
interpretation of strategic formulas.
## 4 Semantic Problems and How to Avoid Them
$000$$101$$\mathsf{{open}}$$002$$211$$\mathsf{{epid}}$$321$$231$$\mathsf{{closed}}$$331$$\mathsf{{closed}}$$proceed$$giveup$$onsite$$online$$giveup$$rest$$handle$$rest$$handle$$rest$$idle$$rest$$idle$
---
Figure 2: Model $M_{\mathit{conf}}$ for the conference scenario. We highlight
the transitions enabled by the strategy in Figure 1, and the resulting
reachable states.
Starting with this section, we describe some problematic phenomena that follow
from the straightforward combination of strategic ability with models of
concurrent systems, proposed in (?). We also show how to extend the
representations and modify their execution semantics to avoid the
counterintuitive interpretation of formulas.
### 4.1 Deadlock Strategies and Finite Paths
An automata network is typically required to produce no deadlock states, i.e.,
every global state in its composition must have at least one outgoing
transition. Then, all the maximal paths are infinite, and it is natural to
refer to only infinite paths in the semantics of temporal operators. In case
of AMAS, the situation is more delicate. Even if the AMAS as a whole produces
no deadlocks, some strategies might, which makes the interpretation of
strategic modalities cumbersome. We illustrate this on the following example.
###### Example 4.1 (Conference).
Recall the 3-agent AMAS of Figure 1, together with its model
$M_{\mathit{conf}}$ (Figure 2). Clearly, $M_{\mathit{conf}}$ has no deadlock
states. Let us now look at the collective strategies of coalition
$\\{{gc,oc}\\}$, with agent $sc$ serving as the opponent. It is easy to see
that the coalition has no way to prevent the opening of the conference, i.e.,
it cannot prevent the system from reaching state $101$. However, the strategy
depicted in Figure 1 produces only one _infinite_ path:
$(000\,giveup\,002\,giveup\,\dots)$. Since the $\mathbf{ATL_{\mathrm{}}^{*}}$
semantics in Section 3 disregards finite paths, we get that
$M_{\mathit{conf}},000\models\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{open}}$,
which is counterintuitive.
Things can get even trickier. In particular, the outcome of a strategy can be
empty – in fact, it may even happen that a coalition has only strategies with
empty outcomes.
|
$0$$1$$2$$3$$\mathsf{{voted_{a}}}$$4$$\mathsf{{voted_{b}}}$$vote_{a}$$vote_{b}$$send$$send$$idle_{v}$$idle_{v}$
---
| $0$$1$$vote_{a}$$vote_{b}$$send$$idle_{ebm}$
---
Figure 3: Casting a ballot: voter $v$ (left) and EBM $ebm$ (right)
###### Example 4.2 (Voting).
Consider the AMAS in Figure 3 that depicts a simple voting scenario. A voter
$v$ can fill in an electronic ballot with a vote for candidate $\mathsf{{a}}$
or $\mathsf{{b}}$, and then push the $send$ button. The Electronic Ballot
Machine $ebm$ duly registers the choices of the voter. Note that all the
_joint_ strategies of $\\{{v,ebm}\\}$ produce only finite sequences of
transitions. This is because $ebm$ must choose a single event at location $0$
in a memoryless strategy, and thus $v$ and $ebm$ are bound to “miscoordinate”
either at the first or at the second step. Since finite paths are not included
in the outcome sets, and the semantics in Section 3.3 rules out strategies
with empty outcomes, we get that
$IIS(S_{vote}),00\models\neg\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\top$,
which is quite strange.
Notice that removing the non-emptiness requirement from the semantic clause in
Section 3.3 does not help. In that case, any joint strategy of $\\{{v,ebm}\\}$
could be used to demonstrate that
$\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\bot$.
### 4.2 Solution: Adding Silent Transitions
To deal with the problem, we augment the model of the system with special
“silent” transitions, labeled by $\epsilon$, that are fired whenever no “real”
transition can occur. In our case, the $\epsilon$-transitions account for the
possibility that some agents miscoordinate and thus block the system.
Moreover, we redefine the outcome set of a strategy so that an
$\epsilon$-transition is taken whenever such miscoordination occurs.
###### Definition 4.3 (Undeadlocked IIS).
Let $S$ be an AMAS, and assume that no agent in S has $\epsilon$ in its
alphabet of events. The _undeadlocked model of S_ , denoted
$M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$, extends the model
$M=IIS(S)$ as follows:
* •
$\mathit{Evt}_{M^{\text{$\epsilon$}}}=\mathit{Evt}_{M}\cup\\{{\epsilon}\\}$,
where $Agent(\epsilon)=\emptyset$;
* •
For each $g\in St$, we add the transition
$g\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}g$ iff there is a
selection of agents’ choices
$\overrightarrow{\alpha}_{A}=(\alpha_{1},\dots,\alpha_{k})$, $\alpha_{i}\in
R_{i}(g)$, such that $enabled_{M}(g,\overrightarrow{\alpha}_{A})=\emptyset$.
In that case, we also fix
$enabled_{M^{\text{$\epsilon$}}}(g,\overrightarrow{\alpha}_{A})=\\{{\epsilon}\\}$.
In other words, “silent” loops are added in the states where a combination of
the agents’ actions can block the system.
Paths are defined as in Section 2.2. The following is trivial.
###### Proposition 4.4.
For any AMAS $S$, any state $g\in IIS^{\text{$\epsilon$}}(S)$, and any
strategy $\sigma_{A}$, we have that
$enabled_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A}(state))\neq\emptyset$.
###### Example 4.5 (Conference).
The undeadlocked model $M_{\mathit{conf}}^{\text{$\epsilon$}}$ of the
conference scenario (Example 2.2) extends the model in Figure 2 with one
$\epsilon$-loop at state $101$. The loop models the situation when the agents
choose $(onsite,online,proceed)$ or $(online,onsite,proceed)$. We leave it for
the reader to check that, at the other states, all the combinations of choices
enable at least one transition.
For the strategy in Example 4.1, notice that its outcome in
$M_{\mathit{conf}}^{\text{$\epsilon$}}$ contains _two_ infinite paths: not
only $(000\,giveup\,002\,giveup\,002\dots)$, but also
$(000\,proceed\,101\,\epsilon\,101\dots)$. Since the latter path invalidates
the temporal formula $\mathrm{G}\,\neg\mathsf{{open}}$, we get that
$M_{\mathit{conf}},000\not\models\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{open}}$,
as expected.
$00$$10$$20$$31$$\mathsf{{voted_{a}}}$$41$$\mathsf{{voted_{b}}}$$\epsilon$$vote_{a}$$vote_{b}$$\epsilon$$send$$\epsilon$$send$$\genfrac{}{}{0.0pt}{}{idle_{v}}{idle_{ebm}}$$\genfrac{}{}{0.0pt}{}{idle_{v}}{idle_{ebm}}$
Figure 4: Undeadlocked IIS for the voting scenario
###### Example 4.6 (Voting).
The undeadlocked model for the voting scenario is presented in Figure 4. Note
that formula
$\neg\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\top$
does not hold anymore, because the joint strategies of $\\{{v,ebm}\\}$ have
nonempty outcomes in $IIS^{\text{$\epsilon$}}(S_{vote})$. On the other hand,
the formula
$\langle\\!\langle{v}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$
(and even
$\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$)
does not hold, which is contrary to the intuition behind the modeling. We will
come back to this issue in Section 7.
Discussion. Adding “silent” transitions to account for the control flow when
no observable event occurs is pretty standard. The crucial issue is _where_ to
add them. Here, we add the $\epsilon$-transitions whenever a subset of agents
might choose to miscoordinate (and stick to their choices). Again, this is in
line with the notion of agency and strategic play in MAS (?; ?). In the next
section, we will discuss a concept of “agent fairness” where the addition of
$\epsilon$-transitions is constrained by the assumption that only a given
subset of agents is fully proactive.
The examples used in this section expose an important feature of agent
systems. The execution semantics of concurrent processes is often defined by a
state-transition graph (or, alternatively, by the tree of paths generated by
the graph, i.e., the tree unfolding of the graph). For systems that involve
proactive agents, this is not enough. Rather, the execution semantics should
map from the possible coalitions and their available strategies to the outcome
sets of those strategies. In this sense, the possible behaviors of an agent
system should be understood via the _set of possible execution trees_ , rather
than a single tree. This is consistent with the theoretical model of MAS in
(?), based on path effectivity functions.
An alternative way out of the problem is to include finite maximal paths in
the outcomes of strategies. However, the interpretation of strategic
modalities over finite paths is rather nonstandard (?) and may pose new
problems in the asynchronous setting. Moreover, our approach allows to reuse
the existing techniques and tools, which are typically built for infinite path
semantics, including the verification and partial order reduction
functionalities of tools like SPIN (?) and STV (?). In general, this is a
design dilemma between changing the logical semantics of the formulas vs.
updating the execution semantics of the representations. Here, we choose the
latter approach.
## 5 Playing Against Reactive Opponents
The solution proposed in Section 4.2 is based on the assumption that an agent
is free to choose any event in its repertoire – even one that prevents the
system from executing anything. The downside is that, for most systems, only
safety goals can be achieved (i.e., properties specified by
$\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\varphi$).
For reachability, there is often a combination of the opponents’ choices that
blocks the execution early on, and prevents the coalition from reaching their
goal. In this section, we define a fairness-style condition that constrains
the choices of more “reactive” opponents. We also show a construction to
verify the abilities of the coalition over the resulting paths in a
technically simpler way.
### 5.1 Opponent-Reactiveness
Given a strategy $\sigma_{A}$, the agents in $A$ are by definition assumed to
be proactive. Below, we propose an execution semantics for $\sigma_{A}$ which
assumes that $A$ cannot be stalled forever by miscoordination on the part of
the opponents.
###### Definition 5.1 (Opponent-reactiveness).
A path $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots$ in
$IIS^{\text{$\epsilon$}}(S)$ is _opponent-reactive for strategy $\sigma_{A}$_
iff we have that $\alpha_{n}=\epsilon$ implies
$enabled(g_{n},\sigma_{A}(g_{n}))=\\{{\epsilon}\\}$. In other words, whenever
the agents outside $A$ have a way to proceed, they must proceed. The _reactive
outcome_ (or _React-outcome_) of $\sigma_{A}$ in $g$, denoted
$\mathit{out}^{\textup{React}}(g,\sigma_{A})$, is the restriction of
$\mathit{out}(g,\sigma_{A})$ to its opponent-reactive paths.
###### Example 5.2 (Conference).
Consider the undeadlocked model $M_{\mathit{conf}}^{\text{$\epsilon$}}$ of
Example 4.5. Path $(000\,proceed\,101\,\epsilon\,101\dots)$ is opponent-
reactive for the strategy of agents $\\{{gc,oc}\\}$ shown in Figure 1.
On the other hand, consider coalition $\\{{gc,sc}\\}$, and the following
strategy of theirs:
$\sigma_{gc}(0)=proceed,\sigma_{gc}(1)=onsite,\sigma_{sc}(0)=proceed$. The
same path is _not_ opponent-reactive for the strategy because the only
opponent ($oc$) has a response at state $101$ that enables a “real” transition
($onsite$).
###### Proposition 5.3.
In $\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$,
the only possible occurrence of $\epsilon$ is as an infinite sequence of
$\epsilon$-transitions following a finite prefix of “real” transitions.
###### Proof.
Take any
$\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots\in\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$
such that $\epsilon$ occurs on $\pi$, and let $i$ be the first position on
$\pi$ st. $\alpha_{i}=\epsilon$. By Definition 5.1, we get that
$enabled(g_{i},\sigma_{A}(g_{i}))=\\{{\epsilon}\\}$. Moreover,
$state_{i+1}=g_{i}$, so also
$enabled(g_{i+1},\sigma_{A}(g_{i+1}))=\\{{\epsilon}\\}$. Thus,
$\alpha_{i+1}=\epsilon$. It follows by simple induction that
$\alpha_{j}=\epsilon$ for every $j\geq i$. ∎
The _opponent-reactive semantics_
$\models_{{}_{\mathrm{ir}}}^{\textup{React}}$ of
$\mathbf{ATL_{\mathrm{}}^{*}}$ is obtained by replacing
$\mathit{out}_{M}(g,\sigma_{A})$ with
$\mathit{out}_{M}^{\textup{React}}(g,\sigma_{A})$ in the semantic clause
presented in Section 3.3.
### 5.2 Encoding Strategic Deadlock-Freeness Under Opponent-Reactiveness in
AMAS
If we adopt the assumption of opponent-reactiveness for coalition $A$, there
is an alternative, technically simpler way to obtain the same semantics of
strategic ability as in Section 4.2. The idea is to introduce the “silent”
transitions already at the level of the AMAS.
###### Definition 5.4 (Undeadlocked AMAS).
The _undeadlocked variant of $S$_ is constructed from $S$ by adding an
auxiliary agent $A_{\epsilon}$ with $L_{\epsilon}=\\{{q_{0}^{\epsilon}}\\}$,
$\iota_{\epsilon}=q_{0}^{\epsilon}$,
$\mathit{Evt}_{\epsilon}=\\{{\epsilon}\\}$,
$R_{\epsilon}(q_{0}^{\epsilon})=\\{{\epsilon}\\}$,
$T_{i}(q_{0}^{\epsilon},\epsilon)=q_{0}^{\epsilon}$, and
$\mathcal{PV}_{\epsilon}=\emptyset$. In other words, we add a module with a
single local state and a “silent” loop labeled by $\epsilon$, as in Figure 5.
We will denote the undeadlocked variant of $S$ by $S^{\text{$\epsilon$}}$.
Note that $S^{\text{$\epsilon$}}$ can be seen as a special case of AMAS. Thus,
the outcome sets and reactive outcomes of strategies in
$IIS(S^{\text{$\epsilon$}})$ are defined exactly as before.
$q_{0}^{\epsilon}$$\epsilon$ Figure 5: The auxiliary agent added in
$S^{\text{$\epsilon$}}$
###### Example 5.5 (Voting).
The undeadlocked AMAS $\mathrm{ASV}_{1,2}^{\epsilon}$ is obtained by
augmenting $\mathrm{ASV}_{1,2}$ with the auxiliary agent in Figure 5.
Obviously, the extra agent adds $\epsilon$-loops to the model of $S$, i.e., to
$IIS(S)$. We show now that, under the assumption of opponent-reactiveness, the
view of $A$’s strategic ability in the undeadlocked AMAS
$S^{\text{$\epsilon$}}$ corresponds precisely to $A$’s abilities in the
undeadlocked model of the original AMAS $S$, i.e.,
$IIS^{\text{$\epsilon$}}(S)$. This allows to deal with deadlocks and finite
paths without redefining the execution semantics for AMAS, set in Definition
2.3, and thus use the existing tools such as SPIN (?) in a straightforward
way.
###### Proposition 5.6.
Let $A\subseteq{\mathbb{A}\mathrm{gt}}$. In
$\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})$,
the only possible occurrence of $\epsilon$ is as an infinite suffix of
$\epsilon$-transitions.
###### Proof.
Analogous to Proposition 5.3. ∎
###### Theorem 5.7.
For every strategy $\sigma_{A}$ in $S$, we have that
$\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})=\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A}).$
###### Proof.
$\boldsymbol{\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})}$:
Consider any
$\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots\in\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$.
If there are no $\epsilon$-transitions on $\pi$, we have that
$\pi\in\mathit{out}^{\textup{React}}_{IIS(S)}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})$,
QED. Suppose that $\pi$ includes $\epsilon$-transitions, with $\alpha_{i}$
being the first one. Then, we have that $\alpha_{j}\neq\epsilon$ and
$\alpha_{j}\in enabled_{IIS^{\text{$\epsilon$}}(S)}(g_{j},\sigma_{A}(g_{j}))$
for every $j<i$, hence also $\alpha_{j}\in
enabled_{IIS(S)}(g_{j},\sigma_{A}(g_{j}))\subseteq
enabled_{IIS(S^{\text{$\epsilon$}})}(g_{j},\sigma_{A}(g_{j}))$. (*)
By Proposition 5.3, $g_{j}=g_{i}$ and $\alpha_{j}=\epsilon$ for every $j\geq
i$. By Definition 5.1,
$enabled_{IIS^{\text{$\epsilon$}}(S)}(g_{j},\sigma_{A}(g_{j}))=\\{{\epsilon}\\}$.
Hence, $enabled_{IIS(S)}(g_{j},\sigma_{A}(g_{j}))=\emptyset$ and
$enabled_{IIS(S^{\text{$\epsilon$}})}(g_{j},\sigma_{A}(g_{j}))=\\{{\epsilon}\\}$.
(**)
Thus, by (*) and (**),
$\pi\in\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})$,
QED.
$\boldsymbol{\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})}$:
Analogous, with Proposition 5.6 used instead of Proposition 5.3. ∎
Discussion. Opponent-reactiveness is to strategic properties what fairness
conditions are to temporal properties of asynchronous systems. If an important
property cannot be satisfied in all possible executions, it may at least hold
under some reasonable assumptions about which events can be selected by whom
in response to what. Clearly, the condition can be considered intuitive by
some and problematic by others. The main point is, unlike in the previous
semantics, now it is made explicit, and can be adopted or rejected depending
on the intuition. Note that the semantic extensions proposed in this paper
(silent transitions and nondeterministic choices for strategies) make sense
both with and without opponent-reactiveness.
Note that, under the reactiveness assumption, we have that
$M_{\mathit{conf}}^{\text{$\epsilon$}},000\models_{{}_{\mathrm{ir}}}^{\textup{React}}\langle\\!\langle{gc,sc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{epid}}$
and
$M_{\mathit{conf}}^{\text{$\epsilon$}},000\models_{{}_{\mathrm{ir}}}^{\textup{React}}\langle\\!\langle{oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{epid}}$.
This seems to contradict the commonly accepted requirement of _regularity_ in
games (?). However, the contradiction is only superficial, as the two formulas
are evaluated _under different execution assumptions_ : for the former, we
assume agent $oc$ to be reactive, whereas the latter assumes $gc$ and $sc$ to
react to the strategy of $oc$.
## 6 Concurrency-Fairness Revisited
In Def. 3.6, we recalled the notion of concurrency-fair outcome of (?). The
idea was to remove from $out(g,\sigma_{A})$ the paths that consistently ignore
agents whose events are enabled _at the level of the whole model_.
Unfortunately, the definition has unwelcome side effects, too.
### 6.1 Problems with Concurrency-Fairness
We first show that, contrary to intuition, Definition 3.6 automatically
disregards _deadlock paths_ , i.e., paths with finitely many “real”
transitions.
###### Proposition 6.1.
Consider an AMAS $S$ and a path $\pi$ in $IIS^{\text{$\epsilon$}}(S)$ such
that, from some point $i$ on, $\pi$ includes only $\epsilon$-transitions.
Then, for every strategy $\sigma_{A}$ in $S$, we have that
$\pi\notin\mathit{out}^{\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$.
###### Proof.
Take $\pi$ as above, i.e., $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}\dots
g_{i}\epsilon g_{i}\epsilon g_{i}\dots$. Since the transition function in
$IIS^{\text{$\epsilon$}}(S)$ is serial, there must be some event
$\beta\neq\epsilon$ enabled in $g_{i}$. In consequence, $\beta$ is always
enabled from $i$ on, but none of its “owners” in $Agent(\beta)$ executes an
event on $\pi$ after $i$. Hence, $\pi$ does not satisfy CF, and does not
belong to
$\mathit{out}^{\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$ for
any strategy $\sigma_{A}$. ∎
Thus, the CF condition eliminates all the deadlock paths from the outcome of a
strategy (for instance, the path $(000\,proceed\,101\,\epsilon\,101\dots)$ in
Example 4.5). In consequence, reasoning about concurrency-fair paths suffers
from the problems that we identified in Section 4.1, even for undeadlocked
models. Moreover, combining the temporal and strategic fairness (i.e., CF and
React) collapses the undeadlocked execution semantics altogether, see below.
###### Proposition 6.2.
Reasoning about reactive _and_ fair outcomes in an undeadlocked model reduces
to reasoning about the fair executions in the original model without
$\epsilon$-transitions. Formally, let
$\mathit{out}^{\textup{React},\textbf{CF}}_{M}(g,\sigma_{A})=\mathit{out}^{\textup{React}}_{M}(g,\sigma_{A})\cap\mathit{out}^{\textbf{CF}}_{M}(g,\sigma_{A})$.
For any AMAS $S$ and any strategy $\sigma_{A}$ in $S$, we have:
$\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})=\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})$.
###### Proof.
Clearly, we have
$\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$,
since
$\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$
can only add to $\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})$ new paths
that include $\epsilon$-transitions.
For the other direction, take any
$\pi\in\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$,
and suppose that it contains an $\epsilon$-transition. By Proposition 5.3, it
must have an infinite suffix consisting only of $\epsilon$-transitions. Then,
by Proposition 6.1,
$\pi\notin\mathit{out}^{\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$,
which leads to a contradiction. Thus, $\pi$ contains only transitions from
$IIS(S)$, and hence $\pi\in\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})$,
QED. ∎
### 6.2 Strategic Concurrency-Fairness
So, how should fair paths be properly defined for strategic reasoning? The
answer is simple: in relation to the outcome of the strategy being executed.
###### Definition 6.3 (Strategic CF).
$\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots$ is a _concurrency-fair path for
strategy $\sigma_{A}$ and state $g$_ iff $g_{0}=g$, and there is no event
$\alpha$ s.t., for some $n$ and all $i\geq n$, we have $\alpha\in
enabled(\pi[i],\sigma_{A}(\pi[i]))$ and $Agent(\alpha)\cap
Agent(\alpha_{i})=\emptyset$. That is, agents with an event always enabled _by
$\sigma_{A}$_ cannot be ignored forever.
The _SCF -outcome_ of $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ is defined as
$\mathit{out}^{\textbf{SCF}}_{M}(g,\sigma_{A})=\\{\pi\in\mathit{out}_{M}(g,\sigma_{A})\mid\pi\text{
is concurrency-fair for }\sigma_{A},g\\}$.
The following formal results show that SCF does not suffer from the problems
demonstrated in Section 6.1.
###### Proposition 6.4.
There is an AMAS $S$, a strategy $\sigma_{A}$ in $S$, and a deadlock path
$\pi$ in $IIS^{\text{$\epsilon$}}(S)$ such that $\pi$ is concurrency-fair for
$\sigma_{A}$.
###### Proof.
To demonstrate the property, it suffices to take the AMAS and the strategy of
$\\{{gc,oc}\\}$ depicted in Figure 1, and the path
$\pi=(000\,proceed\,101\,\epsilon\,101\dots)$. ∎
###### Theorem 6.5.
Opponent-reactiveness and strategic concurrency-fairness are incomparable.
Formally, there exists an AMAS $S$, a state $g$ in
$IIS^{\text{$\epsilon$}}(S)$, and a strategy $\sigma_{A}$ such that
$\mathit{out}^{\textbf{SCF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})\not\subseteq\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$,
and vice versa.
###### Proof.
Consider the undeadlocked model $M_{\mathit{conf}}^{\text{$\epsilon$}}$ in
Example 4.5, and the strategy discussed in Example 5.2:
$\sigma_{gc}(0)=proceed$, $\sigma_{gc}(1)=onsite$, $\sigma_{sc}(0)=proceed$.
Let
$\pi_{1}=(000\,proceed\,101\,\epsilon\,101\,onsite\,211\,rest\,211\linebreak
handle\,211\,rest\,211\dots)$. We have
$\pi_{1}\in\mathit{out}^{\textbf{SCF}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$,
but
$\pi_{1}\notin\mathit{out}^{\textup{React}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$.
On the other hand, for path
$\pi_{2}=(000\,proceed\,101\,onsite\,211\,rest\,211\,rest\,\dots)$, we have
that
$\pi_{2}\notin\mathit{out}^{\textbf{SCF}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$,
but
$\pi_{2}\in\mathit{out}^{\textup{React}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$.
∎
Discussion. Theorem 6.5 suggests that reactiveness and fairness conditions
arise from orthogonal concerns. The two concepts refer to different factors
that influence which sequences of events can occur. Opponent-reactiveness
constrains the choices that (a subset of) the agents can select. Concurrency-
fairness and its strategic variant restrict the way in which the “scheduler”
(Nature, Chance, God…) can choose from the events selected by the agents.
## 7 Strategies in Asymmetric Interaction
Now, we point out that AMAS are too restricted to model the strategic aspects
of asymmetric synchronization in a natural way (e.g., a sender sending a
message to a receiver).
### 7.1 Simple Choices are Not Enough
We demonstrate the problem on an example.
###### Example 7.1 (Voting).
As already pointed out, we have
$IIS^{\text{$\epsilon$}}(S_{vote}),00\not\models\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$
in the model of Example 4.2. This is because receiving a vote for
$\mathsf{{a}}$, a vote for $\mathsf{{b}}$, and the signal to send the vote,
belong to _different choices_ in the repertoire of the EBM, and the agent can
only select one of them in a memoryless strategy. Moreover, formula
$\langle\\!\langle{ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$
holds under the condition of opponent-reactiveness, i.e., the EBM can force a
reactive voter to vote for a selected candidate. Clearly, it was not the
intention behind the AMAS: the EBM is supposed to _listen_ to the choice of
the voter. No matter whose strategies are considered, and who reacts to whose
actions, the EBM should have no influence on what the voter votes for.
The problem arises because the repertoire functions in AMAS are based on the
assumption that the agent can choose any single event in $R_{i}(l_{i})$. This
does not allow for natural specification of situations when the exact
transition is determined by another agent. For the AMAS in Example 4.2, the
decision to vote for candidate $\mathsf{{a}}$ or $\mathsf{{b}}$ (or to press
$send$) should belong solely to the voter. Thus, setting the EBM repertoire as
$R_{ebm}(0)=\\{{vote_{a},vote_{b},send}\\}$ does not produce a good model of
strategic play in the scenario.
### 7.2 AMAS with Explicit Control
As a remedy, we extend the representations so that one can indicate which
agent(s) control the choice between events.
###### Definition 7.2 (AMAS with explicit control).
Everything is exactly as in Definition 2.1, except for the repertoires of
choices, which are now functions $R_{i}:L_{i}\to
2^{2^{\mathit{Evt}_{i}}\setminus\\{\emptyset\\}}\setminus\\{\emptyset\\}$.
That is, $R_{i}(l)$ lists nonempty subsets of events
$X_{1},X_{2},\dots\subseteq\mathit{Evt}_{i}$, each capturing an available
choice of $i$ at the local state $l$. If the agent chooses
$X_{j}=\\{{\alpha_{1},\alpha_{2},\dots}\\}$, then only an event in that set
can be executed within the agent’s module; however, the agent has no firmer
control over which one will be fired. Accordingly, we assume that
$T_{i}(l,\alpha)$ is defined iff $\alpha\in\bigcup R_{i}(l)$.333 For a set of
sets $X$, we use $\bigcup X$ to denote its “flattening” $\bigcup_{x\in X}x$.
Notice that the AMAS of Definition 2.1 can be seen as a special case where
$R_{i}(l)$ is always a list of singletons. The definitions of IIS and
undeadlocked IIS stay the same, as agents’ repertoires of choices are not
actually used to generate the state-transition structure for the model of $S$.
Moreover, undeadlocked AMAS with explicit control can be obtained analogously
to Definition 5.4 by adding the auxiliary “epsilon”-agent with
$R_{\epsilon}(q_{0}^{\epsilon})=\\{{\\{{\epsilon}\\}}\\}$ in its sole local
state.
Strategies still assign choices to local states; hence, the type of agent
$i$’s strategies is now $\sigma_{i}\colon L_{i}\to
2^{\mathit{Evt}_{i}}\setminus\\{\emptyset\\}$ s.t. $\sigma_{i}(l)\in
R_{i}(l)$. The definition of the outcome set is updated accordingly, see
below.
###### Definition 7.3 (Outcome sets for AMAS with explicit control).
First, we lift the set of events enabled by
$\overrightarrow{\alpha}_{A}=(\alpha_{1},\dots,\alpha_{m})$ at $g$ to match
the new type of repertoires and strategies. Formally, $\beta\in
enabled(g,\overrightarrow{\alpha}_{A})$ iff: (1) for every $i\in
Agent(\beta)\cap A$, we have $\beta\in\alpha_{i}$, and (2) for every $i\in
Agent(\beta)\setminus A$, it holds that $\beta\in\bigcup R_{i}(g^{i})$.
The outcome, React-outcome, and SCF-outcome of $\sigma_{A}$ in $M,g$ are given
as in Definitions 3.5, 5.1, and 6.3.
###### Example 7.4 (Voting).
We improve our voting model by assuming repertoires of choices for the voter
and the EBM as follows: $R_{ebm}(0)=\\{{\\{{vote_{a},vote_{b},send}\\}}\\}$,
$R_{v}(0)=\\{{\\{{vote_{a}}\\},\\{{vote_{b}}\\}}\\}$,
$R_{v}(1)=R_{v}(2)=\\{{\\{{send}\\}}\\}$, etc. That is, the voter’s choices
are as before, but the EBM only listens to what the voter selects.
Clearly,
$\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$
holds in the new AMAS. Moreover,
$\langle\\!\langle{ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$
does not hold anymore, even assuming opponent-reactiveness.
It is easy to see that Propositions 4.4, 5.3, 5.6, and 6.4, as well as
Theorems 5.7 and 6.5 still hold in AMAS with explicit control.
Discussion. When reasoning about strategic play of asynchronous agents, two
kinds of asymmetry come into the picture. On the one hand, the processes
(agents) being modeled often synchronize in an asymmetric way. For example,
the sender chooses which message to send to the receiver. On the other hand,
the agents $A$ in formula
$\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\varphi$ choose the
strategy and thus push the other agents to respond accordingly. The variant of
AMAS introduced in (?) does not allow to capture the former kind of asymmetry.
In consequence, the choice between the available synchronization branches
belongs solely to the agents indicated by the formula. Unfortunately, there is
no natural way to model the converse situation, i.e., when the agents in
$\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}$ are forced by the
choices of their opponents. With the new variant of AMAS, we extend the
representations so that the modeler can explicitly specify the degree of
autonomy of each participating agent. Without that, the degree of autonomy is
implicit and comes from the formula being evaluated.
Related modeling approaches. Various forms of asymmetric synchronization are
present in most process algebras. For example, $\pi$-calculus distinguishes
between the action $\overline{c}\langle a\rangle$ of sending the value $a$ on
channel $c$, and action $c(x)$ of listening on channel $c$ and storing
whatever comes in variable $x$. CSP goes further, and allows for a similar
degree of flexibility to ours through suitable combinations of deterministic
choice, nondeterministic choice, and interface parallel operators. Other
synchronization primitives are also possible, see e.g. (?) for an overview.
Instead of allowing for multiple synchronization primitives, we come up with a
single general primitive that can be instantiated to cover different kinds of
interaction.
We note in passing the similarity of our new repertoire functions in
Definition 7.2 to state effectivity functions (?; ?) and especially
alternating transition systems (?).
## 8 Partial Order Reduction Still Works
_Partial order reduction (POR)_ has been defined for temporal and temporal-
epistemic logics without “next” (?; ?; ?; ?), and recently extended to
strategic specifications (?). The idea is to take a network of automata (AMAS
in our case), and use depth-first search through the space of global states to
generate a reduced model that satisfies exactly the same formulas as the full
model. Essentially, POR removes paths that change only the interleaving order
of an “irrelevant” event with another event. Importantly, the method generates
the reduced model directly from the representation, without generating the
full model at all.
### 8.1 Correctness of POR in the New Semantics
POR is a powerful technique to contain state-space explosion and facilitate
verification, cf. e.g. the experimental results in (?). In this paper, we
extend the class of models, and modify their execution semantics. We need to
show that the reduction algorithm in (?), defined for the flawed semantics of
ability, is still correct after the modifications. Our main technical result
in this respect is Theorem A.11, presented below. The detailed definitions,
algorithms and proofs are technical (and rather tedious) adaptations of those
in (?). We omit them here for lack of space, and refer the inquisitive reader
to Appendix A.
Theorem A.11. Let $M=\mathit{IIS}(S^{\text{$\epsilon$}})$,
$M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$ and let
$A\subseteq{\mathbb{A}\mathrm{gt}}$ be a subset of agents. Moreover, let
${M{{}^{\prime}}}\subseteq M$ and $M^{\text{$\epsilon$}}{{}^{\prime}}\subseteq
M^{\text{$\epsilon$}}$ be the reduced models generated by DFS with the choice
of enabled events $E(g^{\prime})$ given by conditions C1, C2, C3 and the
independence relation $I_{A,\mathit{PV}}$. For each
$\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ formula $\varphi$ over
$\mathit{PV}$, that refers only to coalitions $\hat{A}\subseteq A$, we have:
1. 1.
$M,\iota\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$ iff
${M{{}^{\prime}}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$,
and
2. 2.
$M^{\text{$\epsilon$}},\iota\models_{{}_{\mathrm{ir}}}\varphi$ iff
$M^{\text{$\epsilon$}}{{}^{\prime}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}\varphi$.
Thus, the reduced models can be used to model-check the
$\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ properties of the full models.
Proof idea. We aim at showing that the full model $M$ and the reduced one
$M^{\prime}$ satisfy the same formulas of
$\mathbf{ATL_{\mathrm{\mathrm{ir}}}^{*}}$ referring only to coalitions
$\hat{A}\subseteq A$ and containing no nested strategic operators. Thanks to
the restriction on the formulas, the proof can be reduced to showing that
${M{{}^{\prime}}}$ satisfies the condition $\textbf{AE}_{A}$, which states
that for each strategy and for each path of the outcome of this strategy in
$M$ there is an equivalent path in the outcome of the same strategy in
$M^{\prime}$. In order to show that $\textbf{AE}_{A}$ holds, we use the
conditions on the selection of events $E(g^{\prime})$ to be enabled at state
$g^{\prime}$ in $M^{\prime}$. The conditions include the requirement that
$\epsilon$ is always selected, together with the three conditions ${\bf
C1,C2,C3}$ adapted from (?; ?; ?).
Intuitively, ${\bf C1}$ states that, along each path $\pi$ in $M$ which starts
at $g^{\prime}$, each event that is dependent on an event in $E(g^{\prime})$
cannot be executed in $M$ unless an event in $E(g^{\prime})$ is executed first
in $M$. ${\bf C2}$ says that $E(g^{\prime})$ either contains all the events,
or only events that do not change the values of relevant propositions. ${\bf
C3}$ guarantees that for every cycle in $M^{\prime}$ containing no
$\epsilon$-transitions, there is at least one node $g^{\prime}$ in the cycle
for which all the enabled events of $g^{\prime}$ are selected.
First, we show that $M$ and $M^{\prime}$ are stuttering-equivalent, i.e., they
have the same sets of paths modulo stuttering (that is, finite repetition of
states on a path). The crucial observation here is that the reduction of $M$
under the conditions C1, C2, C3 is equivalent to the reduction of $M$ without
the $\epsilon$-loops under the conditions C1, C2, C3 of (?), and then adding
the $\epsilon$-loops to all the states of the reduced model. Therefore, for
the paths without $\epsilon$-loops the stuttering equivalence can be shown
similarly to (?, Theorem 12) while for the paths with $\epsilon$-loops we need
more involved arguments in the proof. It turns out that in addition to the
fact that $M$ and $M^{\prime}$ are stuttering equivalent, we can show that
stuttering equivalent paths of $M$ and $M^{\prime}$ have the same maximal
sequence of visible events. From that, we can prove that $\textbf{AE}_{A}$
holds.
## 9 Conclusions
In this paper, we reconsider the asynchronous semantics of strategic ability
for multi-agent systems, proposed in (?). We have already hinted at certain
problems with the semantics in the extended abstract (?). Here, we demonstrate
in detail how the straightforward combination of strategic reasoning and
models of distributed systems leads to counterintuitive interpretation of
formulas. We identify three main sources of problems. First, the execution
semantics does not handle reasoning about deadlock-inducing strategies well.
Secondly, fairness conditions need to be redefined for strategic play.
Thirdly, the class of representations lacks constructions to resolve the
tension between the asymmetry imposed by strategic operators on the one hand,
and the asymmetry of interaction, e.g., between communicating parties.
We deal with the problems as follows. First, we change the execution semantics
of strategies in asynchronous MAS by adding “silent” $\epsilon$-transitions in
states where no “real” event can be executed. We also propose and study the
condition of _opponent-reactiveness_ that assumes the agents outside the
coalition to not obstruct the execution of the strategy forever. Note that,
while the assumption may produce similar interpretation of formulas as in (?),
it is now explicit – as opposed to (?), where it was “hardwired” in the
semantics. The designer or verifier is free to adopt it or reject it,
depending on their view of how the agents in the system behave and choose
their actions.
Secondly, we propose a new notion of _strategic concurrency-fairness_ that
selects the fair executions of a strategy. Thirdly, we allow for
nondeterministic choices in agents’ repertoires. This way, we allow to
explicitly specify that one agent has more control over the outcome of an
event than the other participants of the event.
The main technical result consists in proving that partial order reduction for
strategic abilities (?) is still correct after the semantic modifications.
Thus, the new, more intuitive semantics admits efficient verification.
Beyond $\mathbf{ATL_{\mathrm{ir}}}$. In this study, we have concentrated on
the logic $\mathbf{ATL^{*}_{\mathrm{ir}}}$, i.e., the variant of
$\mathbf{ATL_{\mathrm{}}^{*}}$ based on memoryless imperfect information
strategies. Clearly, the concerns raised here are not entirely (and not even
not primarily) logical. $\mathbf{ATL^{*}_{\mathrm{ir}}}$ can be seen as a
convenient way to specify the players and the winning conditions in a certain
class of games (roughly speaking, $1.5$-player games with imperfect
information, positional strategies, and $\mathbf{LTL}$ objectives). The
semantic problems, and our solutions, apply to all such games interpreted over
arenas given by asynchronous MAS.
Moreover, most of the claims presented here are not specific to
$\mathrm{ir}$-strategies. In fact, we conjecture that our examples of semantic
side effects carry over to the other types of strategies (except for the
existence of coalitions whose all strategies have empty outcomes, which can
happen for neither perfect information nor perfect recall). Similarly, our
technical results should carry over to the other strategy types (except for
the correctness of POR, which does not hold for agents with perfect
information). We leave the formal analysis of those cases for future work.
Other issues. An interesting question concerns the relationship between
asynchronous and synchronous models. We conjecture that AMAS with explicit
control can be simulated by concurrent game structures and alternating
transition systems. Similarly, it should be possible to simulate CGS and ATS
by AMAS with explicit control, at the expense of using a huge space of fully
synchronized actions. For the model checking complexity in AMAS with explicit
control, we expect the same results as in (?).
## Acknowledgements
We thank the anonymous reviewers for their insightful comments. The authors
acknowledge the support of the National Centre for Research and Development,
Poland (NCBR), and the Luxembourg National Research Fund (FNR), under the
PolLux/FNR-CORE project STV (POLLUX-VII/1/2019). W. Penczek and T. Sidoruk
acknowledge support from CNRS/PAS project PARTIES.
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## Appendix A Partial Order Reduction: Details
All the results in this appendix are formulated and proved for the semantics
of $\mathbf{ATL_{\mathrm{ir}}}$ over undeadlocked AMAS with explicit control.
Also, we restrict the formulas to $\mathbf{ATL_{\mathrm{}}^{*}}$ without
nested strategic modalities and the next step operator $\mathrm{X}\,$ (“simple
$\mathbf{ATL_{\mathrm{}}^{*}}$”, or $\mathbf{sATL_{\mathrm{}}^{*}}$). As noted
in (?), $\mathbf{sATL_{\mathrm{}}^{*}}$ is sufficient for most practical
specifications and much more expressive than $\mathbf{LTL}$. Yet, as we prove
below, it enjoys the same efficiency of partial order reduction.
We begin by introducing the relevant notions of equivalence. Then, we propose
conditions on reduced models that preserve the stuttering equivalence with and
without the assumption of _opponent-reactiveness_ (React). We point out
algorithms that generate such models, and prove their correctness.
It should be stressed that the reduction scheme proposed here is general, in
the sense that it preserves equivalent representatives of both fair and unfair
paths in the model. In particular, we do _not_ propose a variant of POR,
optimized for strategic concurrency-fair paths, analogous to reductions of (?)
for CF. A variant of POR for $\mathbf{sATL_{\mathrm{\mathrm{ir}}}}$ under the
SCF assumption is planned for future work.
### A.1 Properties of Submodels
Given an undeadlocked AMAS $S^{\text{$\epsilon$}}$, partial order reduction
attempts to generate only a subset of states and transitions that is
sufficient for verification of $S^{\text{$\epsilon$}}$, i.e., a relevant
_submodel_ of $\mathit{IIS}(S^{\text{$\epsilon$}})$.
###### Definition A.1 (Submodel).
Let models $M,{M{{}^{\prime}}}$ extend the same AMAS $S^{\text{$\epsilon$}}$,
so that $St^{\prime}\subseteq St$, $\iota\in St^{\prime}$, $T$ is an extension
of $T^{\prime}$, and $V^{\prime}=V|_{St^{\prime}}$. Then, we write
${M{{}^{\prime}}}\subseteq M$ and call ${M{{}^{\prime}}}$ a _submodel_ of $M$.
Note that, for each $g\in St^{\prime}$, we have
$\Pi_{{M{{}^{\prime}}}}(g)\subseteq\Pi_{M}(g)$.
###### Lemma A.2.
Let ${M{{}^{\prime}}}\subseteq M$, $A\in{\mathbb{A}\mathrm{gt}}$,
$\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$. Then, we have
$\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{A})=\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})\cap\Pi_{{M{{}^{\prime}}}}(\iota)$.
_Proof._ Note that each joint $\mathrm{ir}$-strategy in $M$ is also a well
defined $\mathrm{ir}$-joint strategy in ${M{{}^{\prime}}}$ as it is defined on
the local states of each agent of an AMAS which is extended by both $M$ and
${M{{}^{\prime}}}$. The lemma follows directly from the definition of React-
outcome (Def. 5.1 and 7.3), plus the fact that
$\Pi_{{M{{}^{\prime}}}}(\iota)\subseteq\Pi_{M}(\iota)$. $\blacksquare$
###### Lemma A.3.
Let $M$ be a model, $\pi,\pi^{\prime}\in\Pi_{M}(\iota)$, and for some
$i\in{\mathbb{A}\mathrm{gt}}:$
$\mathit{Evt}(\pi)\mid_{\mathit{Evt}_{i}}=\mathit{Evt}(\pi^{\prime})\mid_{\mathit{Evt}_{i}}$.
Then, for each $\mathrm{ir}$-strategy $\sigma_{i}$, we have
$\pi\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$ iff
$\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$.
_Proof._ Let $\mathit{Evt}(\pi)\mid_{\mathit{Evt}_{i}}=b_{0}b_{1}\ldots$ be
the sequence of the events of agent $i$ in $\pi$. For each $b_{j}$ let
$\pi[b_{j}]$ denote the global state from which $b_{j}$ is executed in $\pi$.
By induction we can show that for each $j\geq 0$, we have
$\pi[b_{j}]^{i}=\pi^{\prime}[b_{j}]^{i}$. For $j=0$ it is easy to see that
$\pi[b_{0}]^{i}=\pi[b_{0}]^{i}=\iota^{i}$. Assume that the thesis holds for
$j=k$. The induction step follows from the fact the local evolution $T_{i}$ is
a function, so if $\pi[b_{k}]^{i}=\pi^{\prime}[b_{k}]^{i}=l$ for some $l\in
L_{i}$, then $\pi[b_{k+1}]^{i}=\pi^{\prime}[b_{k+1}]^{i}=T_{i}(l,b_{k})$.
Thus, by Def. 5.1 and 7.3, for each $\mathrm{ir}$-strategy $\sigma_{i}$ we
have $\pi\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$ iff
$\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$, which
concludes the proof. $\blacksquare$
Lemma A.3 can be easily generalized to joint strategies
$\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$.
### A.2 Stuttering Equivalence
Let $M$ be a model, ${M{{}^{\prime}}}\subseteq M$, and
$\mathit{PV}\subseteq\mathcal{PV}$ a subset of propositions. Stuttering
equivalence says that two paths can be divided into corresponding finite
segments, each satisfying exactly the same propositions. Stuttering path
equivalence444 The property is usually called _stuttering trace equivalence_
(?). We use a slightly different name to avoid confusion with Mazurkiewicz
traces, also used in this paper. requires two models to always have
corresponding, stuttering-equivalent paths.
###### Definition A.4 (Stuttering equivalence).
Two paths $\pi\in\Pi_{M}(\iota)$ and
$\pi^{\prime}\in\Pi_{{M{{}^{\prime}}}}(\iota)$ are stuttering equivalent,
denoted $\pi\equiv_{s}\pi^{\prime}$, if there exists a partition
$B_{0}=(\pi[0],\dots,\pi[i_{1}-1]),\ B_{1}=(\pi[i_{1}],\dots,\pi[i_{2}-1]),\
\ldots$ of the states of $\pi$, and an analogous partition
$B^{\prime}_{0},B^{\prime}_{1},\ldots$ of the states of $\pi^{\prime}$, s.t.
for each $j\geq 0:$ $B_{j}$ and $B^{\prime}_{j}$ are nonempty and finite, and
$V(g)\cap\mathit{PV}=V^{\prime}(g^{\prime})\cap\mathit{PV}$ for every $g\in
B_{j}$ and $g^{\prime}\in B^{\prime}_{j}$.
Models $M$ and ${M{{}^{\prime}}}$ are stuttering path equivalent, denoted
$M\equiv_{s}{M{{}^{\prime}}}$ if for each path $\pi\in\Pi_{M}(\iota)$, there
is a path $\pi^{\prime}\in\Pi_{{M{{}^{\prime}}}}(\iota)$ such that
$\pi\equiv_{s}\pi^{\prime}$.555Typically, the definition also contains the
symmetric condition which in our case always holds for $M$ and its submodel
${M{{}^{\prime}}}$, as $\Pi_{{M{{}^{\prime}}}}(\iota)\subseteq\Pi_{M}(\iota)$.
###### Theorem A.5 ((?)).
If $M\equiv_{s}{M{{}^{\prime}}}$, then we have $M,\iota\models\varphi$ iff
${M{{}^{\prime}}},\iota^{\prime}\models\varphi$, for any $\mathbf{LTL_{-X}}$
formula $\varphi$ over $\mathit{PV}$.
### A.3 Independence of Events
Intuitively, an event is invisible iff it does not change the valuations of
the propositions.666 This concept of invisibility is technical, and is not
connected to the view of any agent in the sense of (?). Additionally, we can
designate a subset of agents $A$ whose events are visible by definition.
Furthermore, two events are independent iff they are not events of the same
agent and at least one of them is invisible.
###### Definition A.6 (Invisible events).
Consider a model $M$, a subset of agents $A\subseteq{\mathbb{A}\mathrm{gt}}$,
and a subset of propositions $\mathit{PV}\subseteq\mathcal{PV}$. An event
$\alpha\in\mathit{Evt}$ is invisible wrt. $A$ and $\mathit{PV}$ if
$Agent(\alpha)\cap A=\emptyset$ and for each two global states
$g,g^{\prime}\in St$ we have that
$g\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}g^{\prime}$ implies
$V(g)\cap\mathit{PV}=V(g^{\prime})\cap\mathit{PV}$. The set of all invisible
events for $A,\mathit{PV}$ is denoted by $Invis_{A,\mathit{PV}}$, and its
closure – of visible events – by $Vis_{A,\mathit{PV}}=\mathit{Evt}\setminus
Invis_{A,\mathit{PV}}$.
###### Definition A.7 (Independent events).
The notion of _independence_
$I_{A,\mathit{PV}}\subseteq\mathit{Evt}\times\mathit{Evt}$ is defined as:
$I_{A,\mathit{PV}}=\\{(\alpha,\alpha^{\prime})\in\mathit{Evt}\times\mathit{Evt}\mid
Agent(\alpha)\cap Agent(\alpha^{\prime})=\emptyset\\}\ \setminus\
(Vis_{A,\mathit{PV}}\times Vis_{A,\mathit{PV}})$. Events
$\alpha,\alpha^{\prime}\in\mathit{Evt}$ are called dependent if
$(\alpha,\alpha^{\prime})\not\in I_{A,\mathit{PV}}$. If it is clear from the
context, we omit the subscript $\mathit{PV}$.
### A.4 Preserving Stuttering Equivalence
Rather than generating the full model $M=\mathit{IIS}(S^{\text{$\epsilon$}})$,
one can generate a reduced model ${M{{}^{\prime}}}$ satisfying the following
property:
$\textbf{AE}_{A}:\>\forall\sigma_{A}\\!\in\\!\Sigma_{A}^{\mathrm{ir}}\quad\forall\pi\\!\in\\!\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})$
$\qquad\exists\pi^{\prime}\\!\in\\!\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{A})\quad\pi\\!\equiv_{s}\\!\pi^{\prime}$.
We define a class of algorithms that generate reduced models satisfying
$\textbf{AE}_{A}$ (Section A.4), and then prove that these models preserve
$\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ (Section A.4).
Algorithms for partial order reduction. POR is used to reduce the size of
models while preserving satisfaction for a class of formulas. The standard DFS
(?) or DDFS (?) is modified in such a way that from each visited state $g$ an
event $\alpha$ to compute the successor state $g_{1}$ such that
$g\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}g_{1}$, is selected from
$E(g)\cup\\{{\epsilon}\\}$ such that $E(g)\subseteq
enabled(g)\setminus\\{{\epsilon}\\}$. That is, the algorithm always selects
$\epsilon$, plus a subset of the enabled events at $g$. Let
$A\subseteq{\mathbb{A}\mathrm{gt}}$. The conditions on the heuristic selection
of $E(g)$ given below are inspired by (?; ?; ?).
C1
Along each path $\pi$ in $M$ that starts at $g$, each event that is dependent
on an event in $E(g)$ cannot be executed in $\pi$ without an event in $E(g)$
being executed first in $\pi$. Formally, $\forall\pi\in\Pi_{M}(g)$ such that
$\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}\ldots$ with $g_{0}=g$, and $\forall
b\in\mathit{Evt}$ such that $(b,c)\notin I_{A}$ for some $c\in E(g)$, if
$\alpha_{i}=b$ for some $i\geq 0$, then $\alpha_{j}\in E(g)$ for some $j<i$.
C2
If $E(g)\neq enabled(g)\setminus\\{{\epsilon}\\}$, then $E(g)\subseteq
Invis_{A}$.
C3
For every cycle in ${M{{}^{\prime}}}$ containing no $\epsilon$-transitions,
there is at least one node $g$ in the cycle for which
$E(g)=enabled(g)\setminus\\{{\epsilon}\\}$, i.e., for which all the successors
of $g$ are expanded.
###### Theorem A.8.
Let $A\subseteq{\mathbb{A}\mathrm{gt}}$,
$M=\mathit{IIS}(S^{\text{$\epsilon$}})$, and ${M{{}^{\prime}}}\subseteq M$ be
the reduced model generated by DFS with the choice of $E(g^{\prime})$ for
$g^{\prime}\in St^{\prime}$ given by conditions C1, C2, C3 and the
independence relation $I_{A}$. Then, ${M{{}^{\prime}}}$ satisfies
$\textbf{AE}_{A}$.
_Proof._ Let ${M{{}^{\prime}}}\subseteq M=\mathit{IIS}(S^{\text{$\epsilon$}})$
be the reduced model generated as specified. Notice that the reduction of $M$
under the conditions C1, C2, C3 above is equivalent to the reduction of $M$
without the $\epsilon$-loops under the conditions C1, C2, C3 of (?), and then
adding the $\epsilon$-loops to all the states of the reduced model. Although
the setting is slightly different, it can be shown similarly to (?, Theorem
12) that the conditions C1, C2, C3 guarantee that the models: (i) $M$ without
$\epsilon$-loops and (ii) ${M{{}^{\prime}}}$ without $\epsilon$-loops are
stuttering path equivalent. More precisely, for each path
$\pi=g_{0}a_{0}g_{1}a_{1}\cdots$ with $g_{0}=\iota$ (without
$\epsilon$-transitions) in $M$ there is a stuttering equivalent path
$\pi^{\prime}=g^{\prime}_{0}a^{\prime}_{0}g^{\prime}_{1}a^{\prime}_{1}\cdots$
with $g^{\prime}_{0}=\iota$ (without $\epsilon$-transitions) in $M^{\prime}$
such that
$\mathit{Evt}(\pi)|_{Vis_{A}}=\mathit{Evt}(\pi^{\prime})|_{Vis_{A}}$, i.e.,
$\pi$ and $\pi^{\prime}$ have the same maximal sequence of visible events for
$A$. (*)
We will now prove that this implies $M\equiv_{s}{M{{}^{\prime}}}$. Removing
the $\epsilon$-loops from $M$ eliminates two kinds of paths: (a) paths with
infinitely many “proper” events, and (b) paths ending with an infinite
sequence of $\epsilon$-transitions. Consider a path $\pi$ of type (a) from
$M$. Notice that the path $\pi_{1}$, obtained by removing the
$\epsilon$-transitions from $\pi$, is stuttering-equivalent to $\pi$.
Moreover, by (*), there exists a path $\pi_{2}$ in ${M{{}^{\prime}}}$ without
$\epsilon$-transitions, which is stuttering-equivalent to $\pi_{1}$. By
transitivity of the stuttering equivalence, we have that $\pi_{2}$ is
stuttering equivalent to $\pi$. Since $\pi_{2}$ must also be a path in
${M{{}^{\prime}}}$, this concludes this part of the proof.
Consider a path $\pi$ of type (b) from $M$, i.e., $\pi$ ends with an infinite
sequence of $\epsilon$-transitions. Let $\pi_{1}$ be the sequence obtained
from $\pi$ after removing $\epsilon$-transitions, and $\pi_{2}$ be any
infinite path without $\epsilon$-transitions such that $\pi_{1}$ is its
prefix. Then, it follows from (*) that there is a stuttering equivalent path
$\pi_{2}^{\prime}=g^{\prime}_{0}a^{\prime}_{0}g^{\prime}_{1}a^{\prime}_{1}\cdots$
with $g^{\prime}_{0}=\iota$ in $M^{\prime}$ such that
$\mathit{Evt}(\pi_{2})|_{Vis_{A}}=\mathit{Evt}(\pi_{2}^{\prime})|_{Vis_{A}}$.
Consider the minimal finite prefix $\pi_{1}^{\prime}$ of $\pi_{2}^{\prime}$
such that
$\mathit{Evt}(\pi_{1}^{\prime})|_{Vis_{A}}=\mathit{Evt}(\pi_{1})|_{Vis_{A}}$.
Clearly, $\pi_{1}^{\prime}$ is a sequence in $M^{\prime}$ and can be extended
with an infinite number of $\epsilon$-transitions to the path $\pi^{\prime}$
in $M^{\prime}$. It is easy to see that $\pi$ and $\pi^{\prime}$ are
stuttering equivalent.
So far, we have shown that our reduction under the conditions C1, C2, C3
guarantees that the models $M$ and ${M{{}^{\prime}}}$ are stuttering path
equivalent, and more precisely that for each path
$\pi=g_{0}a_{0}g_{1}a_{1}\cdots$ with $g_{0}=\iota$ in $M$ there is a
stuttering equivalent path
$\pi^{\prime}=g^{\prime}_{0}a^{\prime}_{0}g^{\prime}_{1}a^{\prime}_{1}\cdots$
with $g^{\prime}_{0}=\iota$ in $M^{\prime}$ such that
$\mathit{Evt}(\pi)|_{Vis_{A}}=\mathit{Evt}(\pi^{\prime})|_{Vis_{A}}$, i.e.,
$\pi$ and $\pi^{\prime}$ have the same maximal sequence of visible events for
$A$. To show that ${M{{}^{\prime}}}$ satisfies $\textbf{AE}_{A}$, consider an
$\mathrm{ir}$-joint strategy $\sigma_{A}$ and
$\pi\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})$. As demonstrated
above, there is $\pi^{\prime}\in\Pi_{{M{{}^{\prime}}}}(\iota)$ such that
$\pi\equiv_{s}\pi^{\prime}$ and
$\mathit{Evt}(\pi)|_{Vis_{A}}=\mathit{Evt}(\pi^{\prime})|_{Vis_{A}}$. Since
$\mathit{Evt}_{i}\subseteq Vis_{A}$ for each $i\in A$, the same sequence of
events of each $\mathit{Evt}_{i}$ is executed in $\pi$ and $\pi^{\prime}$.
Thus, by the generalization of Lemma A.3 to $\mathrm{ir}$-joint strategies we
get $\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})$. So,
by Lemma A.2 we have
$\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M{{}^{\prime}}}(\iota,\sigma_{A})$.
$\blacksquare$
Algorithms generating reduced models, in which the choice of $E(g)$ is given
by similar conditions, can be found for instance in (?; ?; ?; ?; ?; ?).
POR for proactive opponents. The same reduction still works without the
assumption of opponent-reactiveness (React).
###### Theorem A.9.
Let $M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$ be an undeadlocked IIS.
Then, its reduced model $M^{\text{$\epsilon$}}{{}^{\prime}}$, generated as in
Theorem A.8, satisfies $\textbf{AE}_{A}$.
_Proof (Sketch)._ In this setting, there is no auxiliary agent in the AMAS,
and $\epsilon$-transitions are added directly to the IIS in accordance with
Definition 4.3. Hence, not every global state of $M^{\text{$\epsilon$}}$
necessarily has an $\epsilon$ loop, but only those where a miscoordinating
combination of events exists. However, this does not impact the reduction
itself.
First, note that Lemma A.2 still holds, directly from the definition of
outcome (Definition 3.5). Furthermore, because in the undeadlocked IIS
$M^{\text{$\epsilon$}}$ the $\epsilon$-transitions do not belong to any agent,
Lemma A.3, where sequences of some agent $i$’s events are considered, also
holds. Note that the React condition only restricts the outcome sets, and not
the model itself: both $M=IIS(S^{\epsilon})$ and $M^{\text{$\epsilon$}}$
contain the same two types (a) and (b) of paths with $\epsilon$-transitions as
discussed in Theorem A.8. Hence, following its reasoning, it can first be
shown that models $M^{\text{$\epsilon$}}$ and
$M^{\text{$\epsilon$}}{{}^{\prime}}$ without their $\epsilon$-transitions are
stuttering path equivalent, and that it remains the case also when both types
of paths including $\epsilon$ loops are included.
Note that the remark about ${M{{}^{\prime}}}$ being equivalent to reducing $M$
without $\epsilon$ loops and adding them to each global state obviously does
not apply to $M^{\text{$\epsilon$}}$ (not every global state of
$M^{\text{$\epsilon$}}$ has them in the first place). However, this
observation has no bearing on the proof. As before, $\epsilon$ is explicitly
stated to be selected for the subset $E(g)$, ensuring preservation of
representative paths with $\epsilon$ in $M^{\text{$\epsilon$}}{{}^{\prime}}$.
$\blacksquare$
Correctness of reductions satisfying $\textbf{AE}_{A}$. We show that the
reduced models satisfying $\textbf{AE}_{A}$ preserve
$\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$.
###### Theorem A.10.
Let $A\subseteq{\mathbb{A}\mathrm{gt}}$, and let models
${M{{}^{\prime}}}\subseteq M$, $M^{\text{$\epsilon$}}{{}^{\prime}}\subseteq
M^{\text{$\epsilon$}}$ satisfy $\textbf{AE}_{A}$. For each
$\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ formula $\varphi$ over
$\mathit{PV}$, that refers only to coalitions $\hat{A}\subseteq A$, we have
that:
1. 1.
$M,\iota\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$ iff
${M{{}^{\prime}}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$,
and
2. 2.
$M^{\text{$\epsilon$}},\iota\models_{{}_{\mathrm{ir}}}\varphi$ iff
$M^{\text{$\epsilon$}}{{}^{\prime}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}\varphi$.
_Proof._ Proof by induction on the structure of $\varphi$. We show the case
$\varphi=\langle\\!\langle{\hat{A}}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$.
The cases for $\neg,\land$ are straightforward.
Notice that
$\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{\hat{A}})\subseteq\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{\hat{A}})$,
which together with the condition $\textbf{AE}_{A}$ implies that the sets
$\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{\hat{A}})$ and
$\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{\hat{A}})$ are
stuttering path equivalent. Analogously, this is the case for
$\mathit{out}_{{M{{}^{\prime}}}}(\iota,\sigma_{\hat{A}})\subseteq\mathit{out}_{M}(\iota,\sigma_{\hat{A}})$,
i.e. without the React assumption. Hence, (1) and (2) follow from Theorem A.5.
$\blacksquare$
Together with Theorems A.8 and A.9, we obtain the following.
###### Theorem A.11.
Let $M=\mathit{IIS}(S^{\text{$\epsilon$}})$,
$M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$ and let
${M{{}^{\prime}}}\subseteq M$ and $M^{\text{$\epsilon$}}{{}^{\prime}}\subseteq
M^{\text{$\epsilon$}}$ be the reduced models generated by DFS with the choice
of $E(g^{\prime})$ for $g^{\prime}\in St^{\prime}$ given by conditions C1, C2,
C3 and the independence relation $I_{A,\mathit{PV}}$. For each
$\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ formula $\varphi$ over
$\mathit{PV}$, that refers only to coalitions $\hat{A}\subseteq A$, we have:
1. 1.
$M,\iota\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$ iff
${M{{}^{\prime}}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$,
and
2. 2.
$M^{\text{$\epsilon$}},\iota\models_{{}_{\mathrm{ir}}}\varphi$ iff
$M^{\text{$\epsilon$}}{{}^{\prime}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}\varphi$.
This concludes the proof that the adaptation of POR for $\mathbf{LTL_{-X}}$ to
$\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$, originally presented in (?),
remains sound in the updated semantics proposed in Sections 4 and 7. That is,
the structural condition $\textbf{AE}_{A}$ is sufficient to obtain correct
reductions for $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ with and without the
new opponent-reactiveness assumption (Theorem A.11). Thanks to that, one can
potentially reuse or adapt the existing POR algorithms and tools for
$\mathbf{LTL_{-X}}$, and the actual reductions are likely to be substantial.
|
2024-09-04T02:54:58.237523 | 2020-03-09T02:21:30 | 2003.03891 | {
"authors": "Nursefa Zengin, Baris Fidan",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26107",
"submitter": "Nursefa Zengin",
"url": "https://arxiv.org/abs/2003.03891"
} | arxiv-papers | # Adaptive Extremum Seeking Using Recursive Least Squares
Nursefa Zengin, Baris Fidan School of Engineering, University of Waterloo, ON,
Canada<EMAIL_ADDRESS>
###### Abstract
Extremum seeking (ES) optimization approach has been very popular due to its
non-model based analysis and implementation. This approach has been mostly
used with gradient based search algorithms. Since least squares (LS)
algorithms are typically observed to be superior, in terms of convergence
speed and robustness to measurement noises, over gradient algorithms, it is
expected that LS based ES schemes will also provide faster convergence and
robustness to sensor noises. In this paper, with this motivation, a recursive
least squares (RLS) estimation based ES scheme is designed and analysed for
application to scalar parameter and vector parameter static map and dynamic
systems. Asymptotic convergence to the extremum is established for all the
cases. Simulation studies are provided to validate the performance of proposed
scheme.
## I Introduction
Extremum seeking (ES) is a popular technique for adaptive optimization of the
performance of dynamic systems by tuning certain system parameters based on
measurements. The main advantage of this technique is that limited or no
knowledge of the plant model is required. ES is suitable for optimization of
the performance of systems with complex dynamics, unavailable suitable
measurements to validate the model, and time-varying disturbances that are
difficult to model accurately ([1]).
The most common ES algorithm used in the literature is the classical band-pass
filtering based one, in which the gradient of the output with respect to the
input will determine the direction of adjusting the input variables. This
method was successfully applied to different application areas including
biochemical reactors [[2, 3]], ABS control in automotive brakes ([4, 1, 5, 6,
7]), mobile robots ([8, 9, 10]), mobile sensor networks ([11, 12, 13]).
Among other types of ES algorithms, perturbation based ES relies on added
perturbation signals to estimate the gradient of the output by correlating the
perturbations. To overcome the implementation drawbacks of introducing
perturbation signals, some methods that are free of perturbation signals have
been developed by [14, 15, 16].
Convergence rate of conventional ES algorithms is a limiting factor in many
applications. Recursive Least Squares (RLS) based estimation has significant
potential in relaxing this limitation and improving robustness to measurement
noises. [17, 15, 18] used certain LS based techniques in their ES algorithms
to obtain better convergence results. [17] estimated the gradient of the
output with respect to the input using a LS based adaptive law for a class of
nonlinear dynamic systems together with a sinusoidal perturbation signal. [15]
used past data of a performance map to estimate the gradient of this
performance map by a first order LS fit. The proposed method used no dither
signal, but utilized a time window of history data of the performance map.
[18] provided general results and a framework for the design of ES schemes
applied to systems with parametric uncertainties and used LS algorithm to
estimate unknown parameters of the known system.
In absence of the parameter knowledge, a series of control/optimization
schemes have been proposed in the literature utilizing certain ES tools such
as switching methods ([19]), signal perturbation for persistence excitation,
and band pass filtering ([19],[20],[21],[18]. [22] and [23] used a discrete
time ES scheme to estimate the gradient as a time-varying parameter using LS
like update laws. They removed the need for averaging system in order to
achieve the convergence of ES. The designs are simulated for static unknown
maps, systems with unknown discrete-time dynamics and sampled-data systems.
In this paper, a continuous time RLS parameter estimation based ES scheme is
designed and analysed for scalar parameter and vector parameter static map and
dynamic systems. Asymptotic convergence to the extremum is established for
each case. Numerical simulation examples are provided to validate the
performance of proposed scheme comparing the results with gradient parameter
estimation based one. A specific simulation example, antilock braking systems
(ABS), in [1] is studied to compare the performance of RLS estimation based ES
with classical gradient based ES.
Contents of this paper are as follows. Section II is dedicated to the problem
statement. In Section III, existing classical perturbation based ES is
reviewed. Proposed RLS estimation based adaptive ES is developed for scalar
parameter systems in Section IV, and for vector parameter systems in Section
V. Comparative simulation examples are presented in Section VI. Finally,
conclusions of the paper are given in Section VII.
## II Problem Statement
The ES problem of interest is defined for static map systems and dynamic
systems separately in the following subsections.
### II-A Static Maps
Consider a concave static map system
$\displaystyle
y=h_{s}(u)=\bar{h}_{s}(\theta^{*},u),\quad\theta^{*}=\begin{bmatrix}\theta^{*}_{1}&\cdots&\theta^{*}_{N}\end{bmatrix}^{T},$
(1)
where $\theta^{*}\in\mathbb{R}^{N}$ is a fixed unknown parameter vector,
$u\in\mathbb{R}^{m}$ is the input and $y\in\mathbb{R}$ is the output of the
system. Assume that the control input signal $u$ is generated by a smooth
control law
$u=\alpha(\theta)$ (2)
parametrized by a control parameter vector $\theta\in\mathbb{R}^{N}$.
###### Assumption 1
The static map $\bar{h}_{s}(\theta^{*},u)$ is smoothly differentiable.
###### Assumption 2
$h_{s}(u)=\bar{h}_{s}(\theta^{*},u)$ has a single extremum (maximum) $y^{*}$
at $u=\alpha(\theta^{*}).$
The control objective is to maximize the steady-state value of $y$ but without
requiring the knowledge of $\theta^{*}$ or the system function $h_{s}$.
### II-B Dynamic Systems
Consider a general multi-input-single-output (MISO) nonlinear system
$\dot{x}=f(x,u)=\bar{f}(\theta^{*},x,u),$ (3)
$y=h_{d}(x)=\bar{h}_{d}(\theta^{*},\theta)=h(\theta),$ (4) $\theta=\pi(x)$ (5)
where $x\in\mathbb{R}^{n}$ is the state, $u\in\mathbb{R}^{m}$ is the input,
$y\in\mathbb{R}$ is the output, all measurable, and
$f:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ and
$h_{d}=h\circ\pi$ are smooth functions. Assume that the control input signal
$u$ is in the form (2), the control parameter $\theta\in\mathbb{R}^{N}$ is
dependant on $x$ through a map
$\pi(.):\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$.
The closed loop system can be written as follows:
$\dot{x}=f(x,\alpha(\theta))=f(x,\alpha(\pi(x)).$ (6)
The equilibria of (6) can be parameterized by $\theta$. The following
assumptions about the closed loop system (3) are made, similarly to [21].
###### Assumption 3
There exists a smooth function $l:\mathbb{R}^{N}\rightarrow\mathbb{R}^{m}$
such that
$f(x,\alpha(x,\theta))=0\quad\text{if and only if}\quad x=l(\theta),$ (7)
for any $(x,\theta)\in\mathbb{R}^{m}\times\mathbb{R}^{N}.$ For each
$\theta\in\mathbb{R}^{N}$, the equilibrium $x_{e}=l(\theta)$ of the system (6)
is locally exponentially stable with decay and overshoot constants uniformly
dependent on $\theta$.
###### Assumption 4
There exists $\theta^{*}\in\mathbb{R}^{N}$ such that for all admissible $x$
values, $h_{d}(x)$ has its unique maximum at $x=x^{*}=l(\theta^{*}),$
$y^{\prime}(x^{*})={\frac{\partial h}{\partial
x}}\Bigr{|}_{\begin{subarray}{c}x=x^{*}\end{subarray}}=0,$ (8)
and the $m\times m$ Hessian matrix
$y^{\prime\prime}(x^{*})={\frac{\partial^{2}h}{\partial
x^{2}}}\Bigr{|}_{\begin{subarray}{c}x=x^{*}\end{subarray}}$ is negative
definite.
The control objective is to maximize the steady-state value of $y$ but without
requiring the knowledge of $\theta^{*}$ or the system functions $h_{d},f$.
This objective could be perfectly performed if $\theta^{*}$ was known and
substituted in (2).
The control parameter vector estimation can be done in different ways, leading
to different ES schemes, even for the fixed control structure (2). The
assumption that $h$ has a maximum is without loss of generality, considering a
maximum seeking task. Minimum seeking case would be treated identically,
replacing $y$ with $-y$ in the subsequent feedback design.
In the next section, existing classical perturbation based ES approach will be
reviewed to give an idea about our proposed design and to later use in
simulation comparisons.
## III Classical Perturbation Based Extremum Seeking for Dynamic Systems
In the classical ES approach shown in Fig.1, a high pass filter, a multiplier,
and a lowpass filter are used to find the extremum. A general single input
nonlinear system is considered in the design of [21]. A multi input ES
approach is examined in [24].
Figure 1: Classic perturbation based ES scheme for multi input dynamic systems
given by [24].
In the approach in [1, 21], the control law (2) feeding the plant (3) is tuned
via the time-varying parameter
$\theta=\begin{bmatrix}\theta_{1},\theta_{2},\cdots,\theta_{N}\end{bmatrix}^{T}$
that is produced by
$\displaystyle\theta(t)=\hat{\theta}(t)+S(t),$ (9)
where
$\displaystyle S(t)$
$\displaystyle=\begin{bmatrix}a_{1}sin(\omega_{1}t)&a_{2}sin(\omega_{2}t)&\cdots&a_{N}sin(\omega_{N}t)\end{bmatrix}^{T},$
(10)
and $\hat{\theta}(t)$ is generated by
$\displaystyle\dot{\hat{\theta}}(t)$ $\displaystyle=k\hat{G}(t),$ (11)
$\displaystyle\dot{\hat{G}}(t)$
$\displaystyle=\omega_{l}M(t)(y(t)-\eta(t))-\omega_{l}\hat{G}(t),$
$\displaystyle\dot{\eta}(t)$
$\displaystyle=\omega_{h}\left(y(t)-\eta(t)\right).$
Perturbation signal is selected as
$M(t)=\begin{bmatrix}\frac{2}{a_{1}}sin(\omega_{1}t)&\frac{2}{a_{2}}sin(\omega_{2}t)&...&\frac{2}{a_{N}}sin(\omega_{N}t)\end{bmatrix}^{T}$.
In the next two sections, we develop RLS estimation based ES scheme with
forgetting factor instead of the approach of Section 3. Our proposed RLS
estimation based adaptive ES scheme will be separately developed for two
cases: for scalar parameter $(N=1)$ systems and for vector parameter $(N>1)$
systems, in Sections 4 and 5, respectively.
## IV RLS based ES Design for Scalar Parameter Systems
### IV-A Static Maps
Consider the static map (1) and the control law (2) for scalar case, $N=1$,
under Assumptions 1 and 2 about the closed-loop system. The proposed scheme is
depicted in Fig. 2.
RLS estimation based ES block shown in Fig. 2 consists of two parts: an RLS
based adaptive parameter identifier estimating the gradient
$h_{\theta}=\frac{\partial y}{\partial\theta}$ and a control law to be fed by
this estimate.
Figure 2: RLS based ES scheme for scalar parameter static maps.
Consider the static map equation (1). In this equation, the time derivative of
the output $y$ is given by
$\dot{y}=h_{\theta}\dot{\theta}.$ (12)
Design of the RLS based estimator to generate $\hat{h}_{\theta}$ considers the
relation (12) that is in the linear parametric model form.
$z=h_{\theta}\phi.$ (13)
where
$z=\dot{y},\quad\phi=\dot{\theta}.$ (14)
If $\dot{y}$ is not available for measurement, then the regressor signals can
be generated as
$\displaystyle
z=\frac{s}{s+\omega_{l}}[y],\quad\phi=\frac{1}{s+\omega_{l}}[\dot{\theta}],$
(15)
i.e.,
$\displaystyle\dot{z}=-\omega_{l}z+\dot{y},\quad\dot{\phi}=-\phi\omega_{l}+\dot{\theta},$
(16)
where $\omega_{l}>0$ is a constant design parameter. The control law
generating $\theta$ is proposed to be
$\dot{\theta}=k\hat{h}_{\theta},\quad k>0.$ (17)
Assuming that the time variation of $h_{\theta}$ is sufficiently slow, we
design an RLS estimator for the parametric model (13) as follows:
$\dot{\hat{h}}_{\theta}=p\epsilon\phi,$ (18) $\dot{p}=\beta p-p^{2}\phi^{2},$
(19) $\epsilon=z-\hat{h}_{\theta}\phi,$ (20)
where $\beta>0$ is forgetting factor and $p$ is the covariance term. The
overall ES scheme producing $\theta(t)$ can be summarized by (17), (18), (19),
and (20).
### IV-B Dynamic Systems
The RLS estimation based ES control scheme (17)-(20) applies to the dynamic
system (3)-(5) for $N=1$ with the control law (2) under Assumptions 3 and 4.
The proposed ES scheme is depicted in Fig. 3.
Figure 3: RLS based ES scheme for scalar parameter dynamic systems.
### IV-C Stability Analysis
In this section, stability proof of the proposed schemes in Sections IV-A and
IV-B will be presented. We know that $\theta^{*}$ is the equilibrium point and
the estimated gradient will be $h_{\theta}=0$ at the equilibrium point
$\theta=\theta^{*}$. We can write our stability result as follows:
###### Theorem IV.1
Consider the RLS estimation based ES scheme given in Figs. 2, 3 and defined in
(17) - (20) with $z$ and $\phi$ as given in (14) or (15), and Assumptions 1 \-
4. For any initial condition $\hat{\theta}(0)\in\mathbb{R}^{N}$ and adaptation
gain $k$, $\theta(t)$ asymptotically converges to small neighborhood of
extremum parameter $\theta^{*}$.
###### Proof:
We consider the Lyapunov function as
$V(\theta(t))=\frac{1}{2}\left(\theta(t)-\theta^{*}\right)^{2}=\frac{1}{2}\tilde{\theta}^{2}.$
(21)
We write the time derivative of $V$ along the solutions of (17) as
$\dot{V}=\dot{\theta}\left(\theta(t)-\theta^{*}\right)=\dot{\theta}\tilde{\theta}.$
(22)
Substituting (17) into (22), we obtain
$\dot{V}=k\hat{h}_{\theta}\tilde{\theta}.$ (23)
For the maximum case, $k>0$. Negative definiteness of (23) depends on the
initial condition $\theta_{0}$ that determines the signs of $\hat{h}_{\theta}$
and $\tilde{\theta}$. If $\theta(0)<\theta^{*}$, then $\hat{h}_{\theta}>0$ and
$\tilde{\theta}<0$. On the other hand, if $\theta(0)>\theta^{*}$, then
$\hat{h}_{\theta}<0$ and $\tilde{\theta}>0$. Hence, for both cases
$\dot{V}<0$. We also need to examine the forgetting factor $\beta$ and the
persistent excitation (PE) of $\phi$. If $\phi$ is PE, then (17) guarantees
that $p\in\mathcal{L}_{\infty}$ and $\theta(t)\to\theta^{*}$ as $t\to\infty$.
When $\beta>0$, the convergence of $\theta(t)\to\theta^{*}$ is exponential
([25]).
∎
## V RLS based ES Design for Vector Parameter Systems
In this section, the proposed RLS estimation based ES scheme is extended to
the systems with vector parameters $(N>1)$. Similar to the classical gradient
based analysis, small sinusoidal perturbation signals with different
frequencies ($\omega_{1},\cdots,\omega_{N}$) are added to the control signals
to provide sufficiently rich excitation.
### V-A Static Maps
Figure 4: RLS based ES scheme for vector parameter static maps.
Consider the block diagram in Fig. 4 for the static map in (1). The time
derivative of (1) is given by
$\dot{y}=h_{\theta}^{T}\dot{\theta},$ (24)
which, similarly to (13), can be written in the linear parametric form
$z=h_{\theta}^{T}\phi,$ (25)
where $z$ and $\phi$ are again defined by either (14) or (15). The control law
(17) is used for updating $\theta$ in the vector case as well. The design of
the RLS estimator to produce $\hat{h}_{\theta}$ is based on the parametric
model (25) and is given as follows ([25]):
$\dot{\hat{h}}_{\theta}=P\epsilon\phi,$ (26) $\dot{P}=\beta P-P\phi\phi^{T}P,$
(27) $\epsilon=z-\hat{h}_{\theta}^{T}\phi,$ (28)
where $\beta$ is the forgetting factor and $P$ is the covariance matrix of the
RLS algorithm. The control law generating $\theta$ is proposed to be
$\dot{\hat{\theta}}=k\hat{h}_{\theta},\quad k>0.$ (29)
$\theta(t)=\hat{\theta}(t)+S(t),$ (30)
where $S(t)$ is defined as in (10). Different from scalar parameter systems,
we use perturbation signals, $S(t)$. The need to use of dither signals in
vector parameter systems is that dither signals with different frequencies can
be implemented on each input signal to achieve overall PE.
### V-B Dynamic Systems
Figure 5: RLS based ES scheme for vector parameter dynamic systems.
The RLS estimation based ES scheme (26) - (29) applies to the dynamic system
(3)-(5) with control law (2) under Assumptions 3 and 4 for vector parameter
systems. Block diagram of the proposed ES scheme is given in Fig.5.
### V-C Stability Analysis
The intuition in (30) is to satisfy persistence of excitation for
$N$-dimensional $\phi$ by introducing at least one distinct dither frequency
for each input, following the standard perturbation based ES control
approaches mentioned in Section III. Similar to the analysis in Section IV-C,
consider the Lyapunov function as
$V(\tilde{\theta}(t))=\frac{1}{2}\tilde{\theta}^{T}\tilde{\theta}.$ (31)
We write the time derivative of $V$ along the solutions of (29) as
$\dot{V}=\tilde{\theta}^{T}\dot{\tilde{\theta}}=\tilde{\theta}^{T}\dot{\theta}.$
(32)
Substituting (30) into (32), we obtain
$\dot{V}=\tilde{\theta}^{T}(k\hat{h}_{\theta}+\dot{S}).$ (33)
The relationship between $\tilde{\theta}$ and $\hat{h}_{\theta}$ in Section
4.3 applies to vector parameter case. The stability again depends on $k$,
initial condition $\theta(0)$, forgetting factor $\beta$, and PE of $\phi$,
that is guaranteed by addition of dither signals in (30). Hence,
$P\in\mathcal{L}_{\infty}$ and $\theta(t)\to\theta^{*}$ as $t\to\infty$.
## VI Simulations
In this section, we present simulation results to show the validity of the
proposed schemes. We will present two examples for scalar parameter and vector
parameter cases with their comparison results with classical ES method in
Section III.
### VI-A Scalar Parameter Simulation Example
Consider the following model
$\displaystyle y$ $\displaystyle=10m(u),$ (34) $\displaystyle m(u)$
$\displaystyle=k_{1}\left(1-e^{-k_{2}u}\right)-k_{3}u$ $\displaystyle u$
$\displaystyle=\theta,$
where $\theta^{*}=0.3$. $\theta_{0}=0.01$ is chosen as initial value for both
schemes. $k_{1}=1.05,k_{2}=23,k_{3}=0.52$ are given. For RLS estimation based
ES scheme, the following parameters are used: $k_{ls}=0.01$, $p_{0}=10^{3}$,
and $\beta=0.98$ are given. For classical ES scheme, the following parameters
are given: $k=0.08$, $\omega_{h}=0.6$, $\omega_{l}=0.8$, $S(t)=0.01\sin 3t$,
and $M(t)=sin3t$. We apply the Gaussian measurement noise as ($\sigma=0.05$)
for both gradient and RLS algorithms. We apply RLS estimation based ES scheme
in Fig.3. The results for this example is given in Fig.6. It is obvious that
proposed scheme can reach a neighborhood of the extremum point
$\theta^{*}=0.3$ at $y^{*}=8.85$ less than 2 second while classical ES finds
the extremum point very late and cannot maintain that extremum point under
measurement noise.
Figure 6: Single parameter RLS estimation based ES results.
### VI-B Vector Parameter Simulation Example
Consider the following model
$\displaystyle y$ $\displaystyle=y_{1}+y_{2},$ (35) $\displaystyle y_{1}$
$\displaystyle=am(u_{1}),\leavevmode\nobreak\
m(u_{1})=(2m^{*}_{1}u^{*}_{1}u_{1})/(u^{*2}_{1}+u^{2}_{1}),$ $\displaystyle
y_{2}$ $\displaystyle=am(u_{2}),\leavevmode\nobreak\
m(u_{2})=(2m^{*}_{2}u^{*}_{2}u_{2})/(u^{*2}_{2}+u^{2}_{2}),$ $\displaystyle u$
$\displaystyle=[u_{1},\leavevmode\nobreak\
u_{2}]=[\theta_{1},\leavevmode\nobreak\ \theta_{2}].$
where $[\theta^{*}_{1},\leavevmode\nobreak\
\theta^{*}_{2}]=[0.2,\leavevmode\nobreak\ 0.3]$. For both schemes, initial
values are given as $u_{0}=[0.1,\leavevmode\nobreak\ 0.1].$ We aim to reach
$y^{*}_{1}(\theta^{*}_{1})=5$ and $y^{*}_{2}(\theta^{*}_{2})=9$. For RLS
estimation based ES scheme, the following parameters are used:
$k=[0.01,\leavevmode\nobreak\ 0.01]$, $P_{0}=10^{4}$, $\beta=0.98$, and
$S(t)=[0.01\sin 7t,\leavevmode\nobreak\ 0.01\sin 10t]$ are given. For
classical ES scheme, the following parameters are given:
$k=[0.02,\leavevmode\nobreak\ 0.01]$, $\omega_{h}=[0.6,\leavevmode\nobreak\
0.6]$, $\omega_{l}=[0.8,\leavevmode\nobreak\ 0.8]$, $S(t)=[0.01\sin
t,\leavevmode\nobreak\ 0.01\sin 2t]$, and $M(t)=[4.5\sin
5t,\leavevmode\nobreak\ 11\sin 5t]$. We apply the Gaussian measurement noise
as ($\sigma=0.05$) for both gradient and RLS algorithms. Simulation results
are given in Fig.7 for both RLS estimation based and classical ES schemes. It
is clear that the results taken with RLS can converge the extremum point and
find the maximized output $y^{*}$ while classical ES scheme has difficulty to
reach the extremum point. One reason for this difficulty is that in classical
ES scheme has many tuning parameters that must be tuned accordingly. For
vector case, we also emphasize the need to apply perturbation terms to the
scheme in order to observe multiple input channels separately. When there is
no perturbation signal applied, the inputs cannot be distinguished and
converge to an average value that caused to reach a value near the maximum.
Similar to scalar case, RLS estimation based ES scheme outweighs classical ES
scheme in terms of reaching extremum under measurement noises.
### VI-C ABS Simulation Example
In this section, we also tested our ES scheme in ABS using MATLAB/Simulink.
Then, we compared its performance with gradient based ES scheme developed by
[1]. The wheel characteristics are given by the following set of equations
Figure 7: Vector parameter RLS estimation based ES results.
$\displaystyle m\dot{\nu}$ $\displaystyle=N\mu(\lambda),$ (36) $\displaystyle
I\dot{\omega}$ $\displaystyle=-B\omega-NR\mu(\lambda)+\tau,$
where $v,\omega,m,N,R,I$ are linear velocity, angular velocity, the mass, the
weight, radius, and the moment of inertia of the wheel, respectively.
$B\omega$ is the bearing friction torque, $\tau$ is braking torque,
$\mu(\lambda)$ is the friction force coefficient. $\lambda$ is the wheel slip
which is defined as
$\lambda(v,\omega)=\frac{R\omega-\nu}{\nu}.$ (37)
Controller design procedure are identical to the design in [1]. The parameters
that are identical in both schemes are given as follows: $m=400kg$, $R=0.3m$,
$I=1.7kgm^{2}$, $B=0.01kg/s$. Perturbation signal amplitude and frequency is
selected as $a=0.01$, $\omega=3$, high pass, low pass and regulation gain are
selected as $\omega_{h}=0.6$, $\omega_{l}=0.8$, $k=6$ in gradient based scheme
equations (9), (10), and (11). $k=-0.01$ is used in ABS case and $\beta=0.95$
is selected for RLS based scheme. The simulation for both gradient and RLS
schemes is performed under the Gaussian noise ($\sigma=0.1$) in longitudinal
acceleration measurement, $\dot{v}$. Initial conditions are selected the same
in both schemes for a fair comparison. We use the approximation model (38) in
simulations to see the effect of the proposed schemes.
$\mu(\lambda)=2\mu_{max}\frac{\lambda^{*}\lambda}{{{\lambda}^{*2}}+\lambda^{2}},$
(38)
where (38) has a maximum at $\lambda=\lambda^{*}$ with
$\mu(\lambda^{*})=\mu_{m}$. For simulation, we choose wet road since it is one
of the safety critical conditions. Simulation results of ABS for gradient/RLS
based scheme comparison are given in Fig.8. Results show that vehicle stopping
time of RLS parameter estimation based ES in an emergency situation is less
than that of gradient one. Slip ratio estimation is almost 2 sec quicker with
RLS parameter estimation, can be seen in Fig. 8(a). RLS based ES scheme gives
better results under measurement noise and can reach the maximum deceleration
in less time.
(a) Friction force coefficient and estimated slip results for ABS.
(b) Braking torque, velocity and deceleration results for ABS.
Figure 8: Wet road comparison results for ABS.
## VII Conclusion
This paper focuses on designing an RLS parameter estimation based ES scheme
for scalar parameter and vector parameter static map and dynamic systems.
Their stability conditions are stated for each case. The proposed ES scheme
does not need perturbation signals for scalar parameter systems; however, the
proposed ES scheme needs perturbation signals with different frequencies for
vector parameter systems. Proposed scheme is applied to different simulation
scenarios and compared to classical gradient estimation based ES under
measurement noise. The results show the validity and effectiveness of RLS
parameter estimation based ES scheme over gradient one.
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|
2024-09-04T02:54:58.256546 | 2020-03-09T07:49:06 | 2003.03956 | {
"authors": "Jihyun Bhom, Marcin Chrzaszcz",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26108",
"submitter": "Jihyun Bhom",
"url": "https://arxiv.org/abs/2003.03956"
} | arxiv-papers | # HEPLike: an open source framework for experimental likelihood evaluation
Jihyun Bhom Marcin Chrzaszcz Henryk Niewodniczanski Institute of Nuclear
Physics Polish Academy of Sciences, Krak´ow, Poland
###### Abstract
We present a computer framework to store and evaluate likelihoods coming from
High Energy Physics experiments. Due to its flexibility it can be interfaced
with existing fitting codes and allows to uniform the interpretation of the
experimental results among users. The code is provided with large open
database, which contains the experimental measurements. The code is of use for
users who perform phenomenological studies, global fits or experimental
averages.
###### keywords:
experimental high energy physics , likelihoods
††journal: Computer Physics Communications
PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY
Program Title: HEPLike
Licensing provisions(please choose one): GPLv3
Programming language: C++
Supplementary material:
Journal reference of previous version: FIRST VERSION OF PROGRAM
Nature of problem(approx. 50-250 words): Provide a uniform way of store, share
and evaluate experimental likelihoods in a proper statistical manner. The code
can be easily interfaced with existing global fitting codes. In addition a
large database with the measurements is published. The program targets users
who perform in their scientific work: phenomenological studies, global fits or
measurements averages. The HEPLike has been created for FlavBit project[1],
which was used to perform several analysis[2,3] and here we present an updated
version, which can be used in standalone mode.
Solution method(approx. 50-250 words): C++ code that evaluates the statistical
properties of the measurements without user intervention. The large open
database is provided as well. The measurements are stored in YAML files
allowing for easy readability and extensions.
## References
* [1] arXiv: 1705.07933
* [2] arXiv: 1705.07935
* [3] arXiv: 1705.07917
## 1 Introduction
In the High Energy Physics (HEP) the experimental measurements are performed
by several collaborations, which measure various different observables. The
experimental results are presented in various ways; some being as simple as a
measurement with an Gaussian error, some more complicated as multiple
correlated measurements with asymmetric errors or in some places even a full
likelihood function is being published. To make things more complicated in
some cases multiple representations of the same measurement are being
published. All of this makes it hard to directly use and compare various
different results. It also leaves a room for misinterpreting the results by
theorists, which use these inputs to their studies. It happens that the
asymmetric errors are being symmetrized, instead of using the full likelihood
only central value with approximated asymmetric error is being used.
The High Energy Physics Likelihoods (HEPLike) is a computer program that
allows to store and share the likelihoods of various measured quantities. The
published code can be useful for users performing phenomenological studies
using experimental results, global fitting collaborations or experimental
averages. Thanks to its structure it is easy to be interface with existing
codes. It simplifies the work of people as instead of looking up the
appropriate measurement and coding up their own likelihood they can download
the database of measurements and choose the one they need. Furthermore, it
shifts the burden of constructing the proper likelihood functions back to the
experimentalists, which performed the measurement at the first place and are
clearly the most appropriate people to handle this task.
The computer code described in this paper is written in C++, making it useful
for majority of fitting programs available on the market [1, 2, 3, 4, 5, 6].
The library can be used in both the $\chi^{2}$ and likelihood fits. Moreover,
it contains a statistical module with useful functions that can be used in the
statistical analysis. Besides the computer code a database with the
likelihoods is being published. The measurements are stored in the YAML files
making them easy to read by both the machine and human. This database can be
easily extended by adding new YAML files if new measurement becomes available.
With the software we provide useful utilities, which allows to perform
searches inside the database, create BiBtex containing publications, which
have been in the fit, etc.
The paper is organized as follows: in Sec. 2 construction of the likelihood
functions is presented. Sec. 3 explains the detailed code implementations and
data storage, while Sec. 4 describes how to install and use HEPLike software.
## 2 Likelihood constructions
In this section we will present how likelihoods in HEPLike are stored and
constructed. Each measurement is stored in separate YAML file. There are
several ways collaborations published their results depending on the
measurements itself:
* 1.
Upper limits,
* 2.
Single measurement with symmetric uncertainty,
* 3.
Single measurement with asymmetric uncertainty,
* 4.
Multiple measurements with symmetric uncertainty,
* 5.
Multiple measurements with asymmetric uncertainty,
* 6.
One dimensional likelihood function,
* 7.
n-dimensional likelihood function.
In addition, there is growing interest from the community that the
experimental collaborations instead of only the results of the analysis
publish also the dataset that has been used to obtain the result. For this
future occasion we have also implement a way that this data can be directly
used in the fits.
Each of these cases has a different module of HEPLike that is designed to
evaluate the likelihood functions. In this section we will present the
statistical treatment of the above cases and the modules that are responsible
for their evaluation. Each of the YAML files is required to have the following
information (here for example we use the
$R_{\mathup{{{K}}^{\scriptstyle{\ast}}}}$ measurement [7]):
⬇
1BibCite: Aaij:2017vbb
2BibEntry: ’@article{Aaij:2017vbb,
3 author = ”Aaij, R. and others”,
4 title = ”{Test of lepton universality
5 with $B^{0} \rightarrow
6 K^{*0}\ell^{+}\ell^{-}$ decays}”,
7 collaboration = ”LHCb”,
8 journal = ”JHEP”,
9 volume = ”08”,
10 year = ”2017”,
11 pages = ”055”,
12 doi = ”10.1007/JHEP08(2017)055”,
13 eprint = ”1705.05802”,
14 archivePrefix = ”arXiv”,
15 primaryClass = ”hep-ex”,
16 reportNumber = ”LHCB-PAPER-2017-013,
17 CERN-EP-2017-100”,
18 SLACcitation = ”%%CITATION = ARXIV:1705.05802;%%”
19 }
20 ’
21DOI: 10.1007/JHEP08(2017)055
22Process: R_{Kstar^{*}}
23FileName: RKstar.yaml
24Name: RKstar
25Source: HEPDATA
26SubmissionYear: 2017
27PublicationYear: 2018
28Arxiv: 1705.05802
29Collaborations: LHCb
30Kinematics: q2>1.1 && q2<6\.
31HLAuthor: Gal Anonim
32HLEmail<EMAIL_ADDRESS>
33HLType: HL_ProfLikelihood
The above informations are used to store the information relevant for the
bookkeeping. For instance the entries BibCite and BibEntry correspond to the
information that are used to generate a BiBtex citation file with the
measurements that have been used in the studies. The DOI corresponds to the
digital object identifier of the publication. The Decay defines the process
that has been studied. It can also be replaced by the Process entry. The Name
is a unique name of this measurement type. If the measurement gets updated
with more data or by other collaboration the Name entry in the new YAML file
should be the same as in the old one. Source entry corresponds to the source
of the measurement. This can be either a HEPData or the collaboration itself.
The SubmissionYear (PublicationYear) refers to the year of appearance
(publication) of the result. The Arxiv codes the Arxiv number, while the
Collaborations stores the information which experimental collaboration has
performed the measurement. Finally, the Kinematics stores additional
information about the kinematic region that has been measured. The HLAuthor
and HLEmail encode the information about the YAML file author and his email in
case user needs further information about the encoded measurement. Last but
not least the entry HLType contains the information about which HEPLike object
should be used to read the file.
Reading of this content in the YAML is implemented in the HL_Data class. All
other classes that construct the likelihood functions inherit from this class
its capabilities. Please note that if the information is missing in the YAML
file the program will omit reading this entry. The only exception is the
FileName, which is mandatory. If a user wants to be notified by the program
that some informations are missing the HL_debug_yaml variable has to be set to
true (default value is false).
### 2.1 Upper limits
In case where the measurement did not observe a significant access of signal
candidates the collaborations usually report an upper limit on the measured
quantity. Commonly $90\%$ or $95\%$ upper limits are quoted. Experiments use
various statistical approaches to compute this limits. It can be the $\rm
CL_{s}$ method [8], Feldman–Cousins [9] or some variation of Bayesian methods
[10]. Publication of only an upper limits does not provide enough information
to use the result in global fits. However, nowadays experiments besides the
aforementioned upper limits publish a full p-value scans. Examples of such
scans are shown in Fig. 1. The plots are usually available in digital format,
which allows the information to be extracted and used in computer program.
Figure 1: Example of p-value scans for the
$\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mathup{{{\tau}}^{\scriptstyle{-}}}\mathup{{{\tau}}^{\scriptstyle{+}}}$
[11] (left) and $\mathup{{{D}}}\to\mathup{{{e}}}\mathup{{{\mu}}}$ [8] (right).
Please note that the $\rm CL_{s}$ value can be interpreted as p-value as
explained in [12]. The black line corresponds to the observed $\rm
CL_{s}$/p-value.
In HEPLike a class HL_Limit is responsible for handling this type of
measurements. It reads the YAML file that contains the standard information
about the measurement (see Sec. 2 for details). The additional information of
the observed $\rm CL_{s}$/p-value is stored in the YAML file in the following
way111Please note that the besides this information the previous information
from Sec. 2 should be included.:
⬇
1Cls:
2- [0.0, 1.0]
3- [1.0e-10, 0.977091694706]
4- [2.0e-10, 0.954375824297]
5- [3.0e-10, 0.93200355343]
6- [4.0e-10, 0.910630700546]
7- [5.0e-10, 0.889382721809]
The Cls can be replaced in the YAML file by p-value as they correspond to the
same information. The first number in each array is the value of tested
hypothesis (for example branching fraction), while the second is the
corresponding $\rm CL_{s}$/p-value. These values are then interpreted using a
$\chi^{2}$ distribution with one degree of freedom:
$pdf(x)=\frac{1}{2^{1/2}\Gamma(1/2)}x^{1/2-1}e^{-x/2},$ (1)
which had the cumulative distribution function defined as:
$cdf(x)=\frac{1}{\Gamma(1/2)}\gamma(1/2,x/2).$ (2)
In the above equations the $\Gamma(x)$ and $\gamma(k,x)$ correspond to Gamma
and incomplete gamma functions. By revering the $cdf(x)$ one can obtain the
$\chi^{2}$ value:
$\chi^{2}=cdf^{-1}(1-p),$ (3)
where p corresponds to the p-value of a given x hypothesis. This $\chi^{2}$
can be then translated to the log-likelihood via Wilks theorem [13]:
$-\log(\mathcal{L})=\frac{1}{2}\chi^{2},$ (4)
where the $\mathcal{L}$ is the likelihood. The user can choose if he wants to
obtain the $\chi^{2}$, likelihood or a log-likelihood value of a given
hypothesis.
### 2.2 Single measurement with symmetric uncertainties
The simplest case of a published experimental result is a single value with a
symmetric uncertainty. This is for example a typical result of an PDG of HFLAV
average [14, 15]. The measurement is coded in the YAML file as:
⬇
1Observables:
2- [ ”Br_A2BCZ”, 0.1, 0.05, 0.01 ]
The first argument in the array ”Br_A2BCZ” corresponds to the observable name.
Then the first number corresponds to the measured central value. The 2nd and
the 3rd number are the statistical and systematic uncertainties. In case where
only one uncertainty is available the 3rd number should be omitted and it will
be automatically set to 0 in the software. We have decided to keep the plural
Observables to be more uniform in case where more observables are measured.
The module responsible for reading this YAML file is called HL_Gaussian, it
calculates the $\chi^{2}$ for an $x$ hypothesis:
$\chi^{2}=\frac{(x_{obs}-x)^{2}}{\sigma_{stat}^{2}+\sigma_{syst}^{2}},$ (5)
where the $x_{obs}$ correspond to the measured central value in the YAML file
and the $\sigma_{stat}$ and $\sigma_{syst}$ are the statistical and systematic
uncertainties respectively. This can be the again translated to the likelihood
and log-likelihood value using Eq. 4.
### 2.3 Single measurement with asymmetric uncertainties
A simple extension of the Gaussian uncertainty is when an asymmetric
uncertainty is reported. This type of measurements although less frequent
appear in the literature. The publication in this case reports the central
value and two uncertainties: $\sigma_{+}$ and $\sigma_{-}$, which correspond
to the right (for values larger than the measured central value) and left (for
values smaller than the measured central value) uncertainty. In HEPLike we
have created a HL_BifurGaussian class, which reads the following entry in the
YAML file:
⬇
1Observables:
2- [ ”Br_A2BCZ”, 0.1, 0.05, -0.06, 0.01, -0.02 ]
The first argument is again the name of the observable and the second one is
its central value. The third and fourth arguments correspond to the
statistical $\sigma_{+}$ and $\sigma_{-}$ uncertainties, while the fifth and
sixth to the systematical $\sigma_{+}$ and $\sigma_{-}$ uncertainties. It is
important to keep the minus sign before the left side uncertainties. The code
will indicate the error in case of missing sign. In some cases the
systematical uncertainty is reported to be symmetric. In such case the last
number can be omitted in the YAML entry.
In the literature there exist number of ways to interpret asymmetric
uncertainties [16]. We have chosen the most commonly used one which is the so-
called bifurcated Gaussian:
$\displaystyle\chi^{2}=\begin{cases}\frac{(x_{obs}-x)^{2}}{\sigma_{+}^{2}},&\text{if
}x\geq x_{obs}\\\ \frac{(x_{obs}-x)^{2}}{\sigma_{-}^{2}},&\text{if
}x<x_{obs},\\\ \end{cases}$ (6)
where the $\sigma_{\pm}^{2}$ is the sum of squared statistical and systematic
uncertainties for right/left case. Once $\chi^{2}$ is calculated it can be
translated to the log-likelihood using Eq. 4.
### 2.4 Multiple measurements with symmetric uncertainties
Nowadays the most common are simultaneous measurements of various quantities,
which are correlated between each other. For instance cross section
measurements in different kinematic bins, or measurements of angular
coefficients in heavy mesons decays. In HEPLike the class responsible for
handling these cases is called HL_nDimGaussian. It reads the following
information from the YAML file:
⬇
1Observables:
2- [ ”BR1”, 0.1, 0.02]
3- [ ”BR2”, 0.2, 0.01, 0.01]
4- [ ”BR3”, 0.4, 0.04]
5Correlation:
6- [ ”BR1”, ”BR2”, ”BR3”]
7- [ 1. , 0.2 , 0 ]
8- [ 0.2, 1., 0. ]
9- [ 0 , 0., 1. ]
The information in the “Observables” entry is exactly the same as in the
HL_Gaussian class. Please note that similarly to the previous class the
systematic uncertainty is not mandatory and in case if it is not provided the
code will treat it as 0. The next entry in the YAML file is the “Correlation”,
which encodes the correlation matrix. The first ”row” is the names of the
variables it is important to keep the same order of variables as in the
“Observables” entry. The HL_nDimGaussian evaluates the $\chi^{2}$ in the
following way:
$\displaystyle\chi^{2}=V^{T}{\rm Cov}^{-1}V,$ (7)
where V is a column matrix, which is the difference between the measured and
the tested i-th observable value. The $\rm Cov$ is a square matrix,
constructed from the correlation matrix (${\rm Corr}$): ${\rm Cov}_{i,j}={\rm
Corr}_{i,j}\sigma_{i}\sigma_{j}$.
Often a user does not want to use the full set of measured quantities but just
their subset. In this case a function Restrict(vector<string>) can be used. By
passing in a form of vector the list of observables to be used, the program
will create new smaller covariance matrix, which will be used to evaluate the
$\chi^{2}$. In a similar manner the $\chi^{2}$ can be translated to the
likelihood and log-likelihood value by Eq. 4.
### 2.5 Multiple measurements with asymmetric uncertainties
More complicated case is when multiple correlated measurements are reported
with asymmetric uncertainties. The case is similar to the one discussed in
Sec. 2.3 and same statistic comments apply in this case. The YAML file
encoding such a measurement will contain the following entries:
⬇
1Observables:
2- [ ”BR1”, 0.1, +0.02, -0.01, 0.02]
3- [ ”BR2”, 0.2, +0.01, -0.05, +0.03, -0.02]
4- [ ”BR3”, 0.3, +0.04, -0.03, 0.05]
5Correlation:
6- [ ”BR1”, ”BR2”, ”BR3”]
7- [ 1. , 0.1 , 0.2 ]
8- [ 0.1, 1., 0.1 ]
9- [ 0.2 , 0.1, 1. ]
The meaning of the “Observables” entry is the same as in the previous class
(cf. Sec. 2.3) and the “Correlation” encodes the same information as in the
HL_nDimGaussian class (cf.. Sec. 2.4). The rules about the minus sign and
symmetric systematic uncertainty are the same as in case of the
HL_BifurGaussian (cf. Sec. 2.3). The difference arises when one evaluates the
$\chi^{2}$, namely the $\rm cov$ matrix is constructed depending if
$\sigma_{+}$ and $\sigma_{-}$ uncertainty is relevant:
$\displaystyle{\rm Cov}_{i,j}=\begin{cases}{\rm
Corr}_{i,j}~{}\sigma^{i}_{+}\sigma^{j}_{+},&\text{if }x^{i}\geq
x^{i}_{obs}\text{ and }x^{j}\geq x^{j}_{obs}\\\ {\rm
Corr}_{i,j}~{}\sigma^{i}_{+}\sigma^{j}_{-},&\text{if }x^{i}\geq
x^{i}_{obs}\text{ and }x^{j}<x^{j}_{obs}\\\ {\rm
Corr}_{i,j}~{}\sigma^{i}_{-}\sigma^{j}_{+},&\text{if }x^{i}<x^{i}_{obs}\text{
and }x^{j}\geq x^{j}_{obs}\\\ {\rm
Corr}_{i,j}~{}\sigma^{i}_{-}\sigma^{j}_{-},&\text{if }x^{i}<x^{i}_{obs}\text{
and }x^{j}<x^{j}_{obs}\\\ \end{cases}$ (8)
The obtained $\rm Cov$ matrix is then used to calculate the $\chi^{2}$ using
Eq. 7. The rest follows the same procedure as described in Sec. 2.4.
### 2.6 One dimensional likelihood function
The best way a result can be published is by providing the (log-)likelihood
function. This type of results are more and more common in the literature. The
most easy is the one-dimensional likelihood scans as can be presented in form
of a figure, which examples are shown in Fig. 2.
Figure 2: Examples of published one-dimensional likelihoods in the Lepton
Universality Violation of the
$\mathup{{{B}}}\to\mathup{{{K}}^{\scriptstyle{\ast}}}\ell\ell$ [7] (left) and
$\mathup{{{B}}}\to\mathup{{{K}}}\ell\ell$ [17] (right).
The biggest advantage of publishing the results in this form is its
completeness. The (log-)likelihood curve contains all the information about
all the non-Gaussian effects and incorporates the systematic uncertainties.
The technical problem is how to publish such information. Usually plots are
published in the pdf or png formats which makes them hard to be used. Since
experiments are mostly using ROOT [18] framework the plots are saved also in
the C format, which contains the points in the form of arrays. This of course
makes the points accessible however it is not easy to automate retrieving this
data from the C file. The best solution is provided by the HEPData portal
[19]. It allows to download the data in a user preferred format. In HEPLike we
have chosen to use the ROOT format by default, in which the data points are
saved in the form of a TGraph object, which is also the way experimentalists
like to store this information. In the YAML file we specify the path of the
ROOT in the following way:
⬇
1ROOTData: data/HEPData-ins1599846-v1-Table_1.root
2TGraphPath: ”Table 1/Graph1D_y1”
The ROOTData encodes the location of the ROOT file, while the TGraphPath
encodes the location of the TGraph object in that ROOT file. In HEPLike the
class HL_ProfLikelihood is responsible for reading and encoding this
likelihood. The value of the log-likelihood can be ten translated again into
the $\chi^{2}$ with Eq. 4.
### 2.7 n-dimensional likelihood function
The natural extension of one dimensional likelihood is an n-dim likelihood,
where $n\geq 2$. Currently experimental collaborations publish only 2-dim
likelihood functions (cf. Fig. 3).
Figure 3: Examples of published two-dimensional likelihoods. The
$\mathcal{B}(\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mu\mu)$
vs
$\mathcal{B}(\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mu\mu)$
likelihood [20] (left) and
$\sigma(\mathup{{{t}}}\mathup{{\overline{{t}}}}\mathup{{{Z}}})$ vs
$\sigma(\mathup{{{t}}}\mathup{{\overline{{t}}}}\mathup{{{W}}})$ likelihood
[21] (right).
The natural way of encoding such information is a histogram: TH2D or TH3D and
we have chosen this way to store this information. The corresponding entry in
the YAML file looks as following:
⬇
1ROOTData: data/LHCb/RD/Bs2mumu_5fb/histB2mumu.root
2TH2Path: ”h_2DScan”
Similar to the one dimensional likelihood (Sec. 2.6) the ROOTData encodes the
location of the ROOT file, while the TH2Path(TH3Path) encodes the location of
the TH2D(TH3D) object. In the long run the community will have to address the
question how to publish higher dimensional likelihoods and this module
(HL_nDimLikelihood) will have to be extended for such use cases.
### 2.8 Fits to experimental data
It is possible that in the future experimental collaborations besides the
results will made the datasets public. The procedure and the form in which the
data should be published is not decided and there is an ongoing debate if the
published data should correspond to the raw detector data, the final selected
points used in the analysis or something between? Clearly publishing a raw
data is problematic, as people outside the collaboration do not have the
necessary knowledge about the calibration and efficiency correction procedures
or data taking conditions. The most useful way to publish the dataset is to
allow the experimentalists to perform all the selection, all the necessary
efficiency corrections and publish the final dataset that has been used for
analysis. This would allow the theory community to use the dataset directly in
their fits without knowing the technicalities about the experimental data
analysis. For this case in HEPLike we have implemented such a class
HL_ExpPoints.
The data are stored in the TTree structure located in the ROOT file. The YAML
file encodes this information in form:
⬇
1ROOTData: data/toy/data.root
2TTreePath: t
3Observables:
4- [ x ]
5- [ y ]
6- [ z ]
7Weight: w
where the ROOTData points to the ROOT file and the TTreePath stores the
information of the TTree location inside the ROOT file. It is assumed that the
experiments will provide all the corrections in form of event-by-event
weights. The name of the weight inside the TTree is encoded in the Weight
entry. In general the data points are elements of $\mathcal{R}^{n}$ vector
space, which coordinates are stored in the Observables entry.
The only thing that user needs to provide to the HL_ExpPoints object is a
pointer to the function to be fitted. The function should have a form: double
(*fun)(vector<double> par , vector<double> point), where the par vector
encodes the parameters that want to be fitted and the point corresponds to a
data point. The HL_ExpPoints will then evaluate the likelihood:
$\mathcal{L}(\omega)=f(\textbf{x}|\omega)^{w(\textbf{x})}$ (9)
for the whole dataset. In the above the x correspondents to the n-dimensional
point, $\omega$ denotes the parameters that want to be fitted par, and $f$
denotes the fitting function (fun). The HEPLike does not provide a minimalizer
or a scanner tool as it is not purpose of this type of software. It has to be
interfaced with proper scanner tool for example [1]. Again the user can decide
if he/she prefers to perform a $\chi^{2}$ or log-likelihood fit.
The biggest advantage of such format is the compatibility with the
experimental analysis. Experimentalist can in principle publish as well the
function that they have used to fit this data and therefore a theorists
reproduce the experimental result and start where the experimentalists
finished.
## 3 Code implementation
In this section we will discuss the implementation of the code used to create
likelihoods discussed in Sec. 2. The code is build in several classes:
* 1.
HL_Data: base class from which other classes inherit their base functionality.
* 2.
HL_Limit: class that handles the upper limit measurements.
* 3.
HL_Gaussian: class that handles measurements with Gaussian uncertainty.
* 4.
HL_BifurGaussian: class that handles measurements with asymmetric uncertainty.
* 5.
HL_nDimGaussian: class that handles measurements with n-dimensional Gaussian
uncertainties.
* 6.
HL_nDimBifurGaussian: class that handles measurements with n-dimensional
asymmetric uncertainties.
* 7.
HL_ProfLikelihood: class that handles measurements with one-dimensional
likelihood function.
* 8.
HL_nDimLikelihood: class that handles measurements with 2(3)-dimensional
likelihood function.
* 9.
HL_ExpPoints: class that allows to perform the fits to experimental datasets.
In Tab. LABEL:tab:functions we present the functionality of these classes. In
addition we present the hierarchy of the structure of the class inheritance in
Fig. 4.
Table 1: Functions available in the HEPLike software. Function | Description
---|---
HL_Data() | Constructor of the HL_Data class.
HL_Data(string) | Constructor of the HL_Data class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_Limit() | Constructor of the HL_Limit class.
HL_Limit(string) | Constructor of the HL_Limit class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_Gaussian() | Constructor of the HL_Gaussian class.
HL_Gaussian(string) | Constructor of the HL_Gaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_BifurGaussian() | Constructor of the HL_BifurGaussian class.
HL_BifurGaussian(string) | Constructor of the HL_Gaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_nDimGaussian() | Constructor of the HL_nDimGaussian class.
HL_nDimGaussian(string ) | Constructor of the HL_nDimGaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_nDimBifurGaussian() | Constructor of the HL_nDimBifurGaussian class.
HL_nDimBifurGaussian(string) | Constructor of the HL_nDimBifurGaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_ProfLikelihood() | Constructor of the HL_ProfLikelihood class.
HL_ProfLikelihood(string) | Constructor of the HL_ProfLikelihood class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_nDimLikelihood() | Constructor of the HL_nDimLikelihood class.
HL_ProfLikelihood(string) | Constructor of the HL_nDimLikelihood class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
HL_ExpPoints() | Constructor of the HL_ExpPoints class.
HL_ExpPoints(string) | Constructor of the HL_ExpPoints class. The argument that is taken by constructor is the path for the YAML file encoding the measurement.
read_standard() | Function that reads the general information about the measurement from the YAML file.
set_debug_yaml(bool) | Function that enables debugging the YAML file. By default the debugging is switched off and can be switched on by passing a true bool argument to this function. Debugging will print a message that for a given information in the YAML file is missing.
Read() | Function reading the YAML file. The function
GetChi2(double) | Function that returns the $\chi^{2}$ value for a given point (passed to the function as double). Function is available for all classes besides HL_Data.
GetLogLikelihood(double) | Function that returns the log-likelihood value for a given point (passed to the function as double). Function is available for all classes besides HL_Data.
GetLikelihood(double) | Function that returns the likelihood value for a given point (passed to the function as double). Function is available for all classes besides HL_Data.
GetCLs(double) | Function that returns $\rm CL_{s}$ or p-value for a given point (passed to the function as double). The function is a member of the HL_Limit class.
Restrict(vector<string>) | Function that restricts number of observables from the YAML file. Function is a member of the HL_nDimGaussian, HL_nDimBifurGaussian and HL_nDimLikelihood classes.
InitData() | Function of HL_ExpPoints class that reads to the memory the data from the TTree object.
Profile() | Function of HL_nDimLikelihood class that creates the profile log-likelihood projections.
SetFun() | Function of HL_ExpPoints class, that sets the pointer to the function to be fitted.
Figure 4: Diagram of class inheritance of the HEPLike package.
## 4 Installation and usage
In this chapter we will present the requirements and installation for the
HEPLike package. The software is distributed via the github site:
https://github.com/mchrzasz/HEPLike.
In order to compile HEPLike the following packages (and the minimal version)
needs to be installed:
* 1.
git
* 2.
cmake, 2.8
* 3.
yaml-cpp, 1.58.0
* 4.
gsl, 2.1
* 5.
Boost, 1.58.0
* 6.
ROOT, 6.08
The compilation is done in the following way:
⬇
1cd <instalation dir>
2git clone https://github.com/mchrzasz/HEPLike.git
3cd HEPLike
4mkdir build
5cd build
6cmake ..
7make
In the above the make can be replaced with make -jN, where N is the number of
threads that user wants to be used for compilation. Please note that in case
of non standard installation of some packages one might have to provide cmake
with a proper path to the library. After successful compilation a libHEPLike.a
and libHEPLike.solibraries will be created in the build directory.
The HEPLike is provided with seven examples:
* 1.
Br_example.cc: example program showing the usage of the HL_Gaussian class.
* 2.
BrBifurGaussian_example.cc: example program showing the usage of the
HL_BifurGaussian class.
* 3.
Data_Fit_example.cc: example program showing the usage of the HL_ExpPoints
class.
* 4.
Limit_example.cc: example program showing the usage of the HL_Limit class.
* 5.
Ndim_BifurGaussian_example.cc: example program showing the usage of the
HL_nDimBifurGaussian class.
* 6.
Ndim_Gaussian.cc: example program showing the usage of the HL_nDimGaussian
class.
* 7.
Ndim_Likelihood_example.cc: example program showing the usage of the
HL_nDimLikelihood class.
* 8.
ProfLikelihood_example.cc: example program showing the usage of the
HL_ProfLikelihood class.
To compile them a proper variable has to be set during the cmake stage:
⬇
1 cd build
2 cmake -DEXECUTABLE=TRUE ..
3 make
After the compilation in the build directory will contain executables from the
following examples. The HEPLike package comes also with test procedures for
each of the classes. To perform the tests user has to perform the command:
⬇
ctest
or an equivalent:
⬇
make test
If the HEPLike was successfully installed the output will look as following:
⬇
Test project /storage/github/HEPLike/build
Start 1: HL_Test_YAML
1/7 Test #1: HL_Test_YAML ………………… Passed 0.01 sec
Start 2: HL_Limit
2/7 Test #2: HL_Limit ……………………. Passed 0.27 sec
Start 3: HL_Br_example
3/7 Test #3: HL_Br_example ……………….. Passed 0.02 sec
Start 4: HL_BrBifurGaussian_example
4/7 Test #4: HL_BrBifurGaussian_example ……. Passed 0.01 sec
Start 5: HL_Ndim_Gaussian
5/7 Test #5: HL_Ndim_Gaussian …………….. Passed 0.01 sec
Start 6: HL_ProfLikelihood_example
6/7 Test #6: HL_ProfLikelihood_example …….. Passed 0.25 sec
Start 7: HL_Ndim_BifurGaussian_example
7/7 Test #7: HL_Ndim_BifurGaussian_example …. Passed 0.01 sec
100% tests passed, 0 tests failed out of 7
Total Test time (real) = 0.57 sec
### 4.1 Available measurement
The YAML files that contain the stored measurements are located in a second
independent repository. The reason for this separation is that the YAML files
are expected to be updated more frequently then the code itself. It is
expected that users and experiments will contribute to this repository. By
implementing such model it is ensured that the repository will contain the
most up to date measurements.
The repository can be found at: https://github.com/mchrzasz/HEPLikeData. The
repository should be downloaded or cloned:
⬇
1cd <some new dir>
2git clone https://github.com/mchrzasz/HEPLikeData.git
Since the repository contains only YAML files there is no need for any
compilation. The repository contains a directory data, where all the YAML
files are kept. It should be linked by a symbolic link to the HEPLike package.
Inside the data the measurements are grouped by experiments (ex. LHCb, ATLAS,
CMS, etc.). Inside the experiment directory the measurements are grouped
according to type of measurement in the collaborations, for example: RD,
Semileptonic, Charmless, Exotica, etc. The names of the YAML files should be
named accordingly to publication report number. For example: CERN-
EP-2018-331.yaml. If a single publication produced more independent
measurements, user might code them in the independent files and give further
information at the end of the file, for example:CERN-PH-
EP-2015-314_q2_01_0.98.yaml.
Currently we are publishing the measurements that have been used by us in
other projects [22, 23, 24]. The list of YAML files with the context is
presented in Tab. LABEL:tab:yaml.
Table 2: Functions available in the HEPLike software. |
---|---
File | Description
CERN-EP-2017-100.yaml | YAML file encoding the measurement of branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mu\mu$ and $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mu\mu$ decays [20].
PH-EP-2015-314_q2_0.1_0.98.yaml PH-EP-2015-314_q2_11.0_12.5.yaml PH-EP-2015-314_q2_1.1_2.5.yaml PH-EP-2015-314_q2_15.0_19.yaml PH-EP-2015-314_q2_2.5_4.0.yaml PH-EP-2015-314_q2_4.0_6.0.yaml PH-EP-2015-314_q2_6.0_8.0.yaml | YAML files encoding the measurements of the angular coefficients of $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mathup{{{K}}^{\scriptstyle{\ast}}}\mu\mu$ decay in different $q^{2}$ regions [25].
CERN-EP-2016-141_q2_0.1_0.98.yaml CERN-EP-2016-141_q2_11.0_12.5.yaml CERN-EP-2016-141_q2_1.1_2.5.yaml CERN-EP-2016-141_q2_15.0_19.yaml CERN-EP-2016-141_q2_2.5_4.0.yaml CERN-EP-2016-141_q2_4.0_6.0.yaml CERN-EP-2016-141_q2_6.0_8.0.yaml | YAML files encoding the measurements of the branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mathup{{{K}}^{\scriptstyle{\ast}}}\mu\mu$ decay in different $q^{2}$ regions [26].
CERN-EP-2016-215_q2_0.1_0.98.yaml CERN-EP-2016-215_q2_1.1_2.5.yaml CERN-EP-2016-215_q2_2.5_4.yaml CERN-EP-2016-215_q2_4_6.yaml CERN-EP-2016-215_q2_6_8.yaml | YAML files encoding the measurements of the branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mathup{{{K}}}\mathup{{{\pi}}}\mu\mu$ decay in different $q^{2}$ regions [27].
CERN-PH-EP-2015-145_0.1_2.yaml CERN-PH-EP-2015-145_11_12.5.yaml CERN-PH-EP-2015-145_15_19.yaml CERN-PH-EP-2015-145_1_6.yaml CERN-PH-EP-2015-145_2_5.yaml CERN-PH-EP-2015-145_5_8.yaml | YAML files encoding the measurements of the branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mathup{{{\phi}}}\mu\mu$ decay in different $q^{2}$ regions [27].
CERN-EP-2019-043.yaml | YAML file encoding the measurement of the $R_{K}$ [28].
CERN-EP-2017-100_q2_0.045_1.1.yaml CERN-EP-2017-100_q2_1.1_6.yaml | YAML file encoding the measurement of the $R_{\mathup{{{K}}^{\scriptstyle{\ast}}}}$ [7].
b2sgamma.yaml | YAML file encoding the HFLAV average of the $\mathup{{{b}}}\to\mathup{{{s}}}\mathup{{{\gamma}}}$ [15].
RD_RDstar.yaml | YAML file encoding the HFLAV average of the $R(\mathup{{{D}}})$ and $R(\mathup{{{D}}^{\scriptstyle{\ast}}})$ [15].
HFLAV_2016_157.yaml HFLAV_2016_160.yaml HFLAV_2016_161.yaml HFLAV_2016_162.yaml HFLAV_2016_164.yaml HFLAV_2016_165.yaml HFLAV_2016_166.yaml HFLAV_2016_167.yaml HFLAV_2016_168.yaml HFLAV_2016_169.yaml HFLAV_2016_170.yaml HFLAV_2016_171.yaml HFLAV_2016_176.yaml HFLAV_2016_177.yaml HFLAV_2016_178.yaml HFLAV_2016_179.yaml HFLAV_2016_180.yaml HFLAV_2016_181.yaml HFLAV_2016_182.yaml HFLAV_2016_183.yaml HFLAV_2016_211.yaml HFLAV_2016_212.yaml | YAML files encoding the upper limits of $\tau$ Lepton Flavour Violation decays [27].
As already mentioned the measurements are constantly growing and there is
expected that the community will contribute to develop this repository. When a
new YAML file is wrote before merging it with the repository it should be
checked if it contains all the necessary information. It can be checked with
the Test_YAML.cc program. It can be used in the following way:
⬇
1cd HEPLike
2./build/Test_YAML <PATH_TO_YAML>
If an entry is missing the user will be notified by a printout. The
HEPLikeData repository contains also a template YAML (data/template.yaml) file
which can be used to create new measurements YAML files.
As already mentioned we provide useful utilities for the encoded measurements.
The first is the ability to create BiBtex file for the measurements that have
been used. The user should store the BiBtex items or YAML file names:
⬇
1Aaij:2017vbb
2b2mumu.yaml
To prepare the BiBtex file user should run the make_citations.py script
located in the utils directory:
⬇
1cd utils
2python make_citations.py list.txt
after this command a new file references.bib, will be created, which will
contain the full BiBtex entries. This can be directly used in preparing the
publication.
Another useful feature of HEPLike is the ability to search the measurement
database for relevant measurements. The script allowing for that utility is
also located in the utils. Currently the database can be searched for using
the year of publication, Arxiv number, author of the YAML file or the unique
name of the measurements. The syntax for running a search is the following:
⬇
1python lookup.py –Arxiv 1705.05802
2Found files:
3../data/examples/RKstar_lowq2.yaml
To see all available search options in the following script user can run it
with help option: python lookup.py -h.
## 5 Summary
We have presented a computer program HEPLike that enables to construct and
evaluate experimental likelihoods. The software is designed to handle the
interpretation of wide range of published results. It also allows to perform
direct fits to data once it is provided by the experimental collaborations.
The program can be easily interfaced with other computer programs and is aimed
to help users, who perform fits to experimental results in their scientific
work. It is especially useful for large fitting collaborations, which till now
had to implement the experimental measurements on their own. The measurement
themselves are stored in YAML files in separate repository. This allows for
easy extensions of the database without the need of compilation. Furthermore,
users and experimental collaborations can share their encoded measurements
with the community.
## Acknowledgments
This work is partly supported by the CERN FCC Design Study Program. The
research of M. Chrzaszcz is funded by the Polish National Agency for Academic
Exchange under the Bekker program. M. Chrzaszcz is also grateful to Foundation
for Polish Science (FNP) for its support.
We would like thank Mike Williams, Patrick Koppenburg, Pat Scott, Danny van
Dyk and Maria Moreno Llacer for invaluable comments about our manuscript.
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|
2024-09-04T02:54:58.268761 | 2020-03-09T09:01:53 | 2003.03977 | {
"authors": "Nikhil Iyer, V Thejas, Nipun Kwatra, Ramachandran Ramjee, Muthian\n Sivathanu",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26109",
"submitter": "Nikhil Iyer",
"url": "https://arxiv.org/abs/2003.03977"
} | arxiv-papers | # Wide-minima Density Hypothesis and the
Explore-Exploit Learning Rate Schedule
Nikhil Iyer
Microsoft Research India<EMAIL_ADDRESS>V Thejas 11footnotemark: 1
Atlassian India<EMAIL_ADDRESS>Nipun Kwatra
Microsoft Research India<EMAIL_ADDRESS>Ramachandran Ramjee
Microsoft Research India<EMAIL_ADDRESS>Muthian Sivathanu
Microsoft Research India<EMAIL_ADDRESS>Work done during an internship
at Microsoft Research India
###### Abstract
Several papers argue that wide minima generalize better than narrow minima. In
this paper, through detailed experiments that not only corroborate the
generalization properties of wide minima, we also provide empirical evidence
for a new hypothesis that the density of wide minima is likely lower than the
density of narrow minima. Further, motivated by this hypothesis, we design a
novel explore-exploit learning rate schedule. On a variety of image and
natural language datasets, compared to their original hand-tuned learning rate
baselines, we show that our explore-exploit schedule can result in either up
to 0.84% higher absolute accuracy using the original training budget or up to
57% reduced training time while achieving the original reported accuracy. For
example, we achieve state-of-the-art (SOTA) accuracy for IWSLT’14 (DE-EN)
dataset by just modifying the learning rate schedule of a high performing
model.
Keywords: deep learning, generalization, learning rate schedule, optimization
## 1 Introduction
One of the fascinating properties of deep neural networks (DNNs) is their
ability to generalize well, i.e., deliver high accuracy on the unseen test
dataset. It is well-known that the learning rate (learning rate) schedules
play an important role in the generalization performance (Keskar et al., 2016;
Wu et al., 2018; Goyal et al., 2017). In this paper, we study the question,
what are the key properties of a learning rate schedule that help DNNs
generalize well during training?
We start with a series of experiments training Resnet18 on Cifar-10 over 200
epochs. We vary the number of epochs trained at a high learning rate of $0.1$,
called the explore epochs, from 0 to 100 and divide up the remaining epochs
equally for training with learning rates of $0.01$ and $0.001$. Note that the
training loss typically stagnates around 50 epochs with $0.1$ learning rate.
Despite that, we find that as the number of explore epochs increase to 100,
the average test accuracy also increases. We also find that the minima found
in higher test accuracy runs are wider than the minima from lower test
accuracy runs, corroborating past work on wide-minima and generalization
(Keskar et al., 2016; Hochreiter and Schmidhuber, 1997; Jastrzebski et al.,
2017; Wang et al., 2018). Moreover, what was particularly surprising was that,
even when using fewer explore epochs, a few runs out of many trials still
resulted in high test accuracies!
Thus, we not only find that an initial exploration phase with a high learning
rate is essential to the good generalization of DNNs, but that this
exploration phase needs to be run for sufficient time, even if the training
loss stagnates much earlier. Further, we find that, even when the exploration
phase is not given sufficient time, a few runs still see high test accuracy
values.
To explain these observations, we hypothesize that, in the DNN loss landscape,
the density of narrow minima is significantly higher than that of wide minima.
Intuitively, a large learning rate can escape narrow minima easily (as the
optimizer can jump out of them with large steps). However, once it reaches a
wide minima, it is likely to get stuck in it (if the ”width” of the wide
minima is large compared to the step size). With fewer explore epochs, a large
learning rate might still get lucky occasionally in finding a wide minima but
invariably finds only a narrower minima due to their higher density. As the
explore duration increases, the probability of eventually landing in a wide
minima also increases. Thus, a minimum duration of explore is necessary to
land in a wide minimum with high probability.
An observation on the rarity of wide minima has been hinted at by prior work
(Wu et al., 2018; Baldassi et al., 2020) based on theoretical analysis of
simple neural networks (see Section 2). In this paper, we add significant
empirical evidence to these theoretical observations. We believe that all
these results together constitute sufficient evidence for this observation to
now be classified as a hypothesis, that we term the wide-minima density
hypothesis.
The hypothesis helps explain not only our experiments but also the
generalization out-performance of prior heuristic-based learning rate decay
schemes such as cosine decay (Loshchilov and Hutter, 2016). Cosine decay
implicitly maintains a higher learning rate during the first half of training
compared to schemes like linear decay. Based on the hypothesis, the higher
learning rate allows cosine decay to find wider minima with higher
probability, resulting in cosine decay’s better generalization compared to
linear decay.
Apart from helping explain empirical observations, the hypothesis also enables
a principled learning rate schedule design that explicitly accounts for the
requisite explore duration. Motivated by the hypothesis, we design a novel
Explore-Exploit learning rate schedule, where the initial explore phase
optimizes at a high learning rate in order to arrive in the vicinity of a wide
minimum. This is followed by an exploit phase which descends to the bottom of
this wide minimum. We give explore phase enough time so that the probability
of landing in a wide minima is high. For the exploit phase, we experimented
with multiple schemes, and found a simple, parameter-less, linear decay to
zero to be effective. Thus, our proposed learning rate schedule optimizes at a
constant high learning rate for a given duration, followed by a linear decay
to zero. We call this learning rate schedule the Knee schedule.
We extensively evaluate the Knee schedule across a wide range of models and
datasets, ranging from NLP (BERT pre-training, Transformer on WMT’14(EN-DE)
and IWSLT’14 (DE-EN)) to CNNs (ImageNet on ResNet-50, Cifar-10 on ResNet18),
and spanning multiple optimizers: SGD Momentum, Adam, RAdam, and LAMB. In all
cases, Knee schedule improves the test accuracy of state-of-the-art hand-tuned
learning rate schedules, when trained using the original training budget. The
explore duration is a hyper-parameter in Knee schedule but even if we set the
explore duration to a fixed 50% fraction of total training budget, we find
that it still outperforms prior schemes.
We also experimented with reducing the training budget, and found that Knee
schedule can achieve the same accuracy as the baseline under significantly
reduced training budgets. For the BERTLARGE pretraining, WMT’14(EN-DE) and
ImageNet experiments, we are able to train in 33%, 57% and 44% less training
budget, respectively, for the same test accuracy. This corresponds to
significant savings in GPU compute, e.g. savings of over 1000 V100 GPU-hours
for BERTLARGE pretraining.
The main contributions of our work 111Our work is available at:
https://github.com/nikhil-iyer-97/wide-minima-density-hypothesis are:
1. nosep
A hypothesis of lower density of wide minima in the DNN loss landscape, backed
by extensive experiments, that explains why a high learning rate needs to be
maintained for sufficient duration to achieve good generalization.
2. nosep
The hypothesis explains the good performance of heuristic-based schemes such
as cosine decay, and promotes a principled design of learning rate decay
schemes.
3. nosep
Motivated by the hypothesis, we design an Explore-Exploit learning rate
schedule called Knee schedule that outperforms prior heuristic-based learning
rate schedules, including achieving state-of-the-art results on the IWSLT’14
(DE-EN) dataset.
## 2 Related Work
Generalization. There has been a lot of work on understanding the
generalization characteristics of DNNs. Kawaguchi (2016) found that DNNs have
many local minima, but all local minima were also the global minima. It has
been observed by several authors that wide minima generalize better than
narrow minima (Arora et al., 2018; Hochreiter and Schmidhuber, 1997; Keskar et
al., 2016; Jastrzebski et al., 2017; Wang et al., 2018) but there have been
other works questioning this hypothesis as well (Dinh et al., 2017; Golatkar
et al., 2019; Guiroy et al., 2019; Jastrzebski et al., 2019; Yoshida and
Miyato, 2017).
Keskar et al. (2016) found that small batch SGD generalizes better and lands
in wider minima than large batch SGD. However, recent work has been able to
generalize quite well even with very large batch sizes (Goyal et al., 2017;
McCandlish et al., 2018; Shallue et al., 2018), by scaling the learning rate
linearly as a function of the batch size. Jastrzebski et al. (2019) analyze
how batch size and learning rate influence the curvature of not only the SGD
endpoint but also the whole trajectory. They found that small batch or large
step SGD have similar characteristics, and yield smaller and earlier peak of
spectral norm as well as smaller largest eigenvalue. Chaudhari et al. (2019);
Baldassi et al. (2019) propose methods to drive the optimizer to wide minima.
Wang et al. (2018) analytically show that generalization of a model is related
to the Hessian and propose a new metric for the generalization capability of a
model that is unaffected by model reparameterization of Dinh et al. (2017).
Yoshida and Miyato (2017) argue that regularizing the spectral norm of the
weights of the neural network help them generalize better. On the other hand,
Arora et al. (2018) derive generalization bounds by showing that networks with
low stable rank (high spectral norm) generalize better. Guiroy et al. (2019)
looks at generalization in gradient-based meta-learning and they show
experimentally that generalization and wide minima are not always correlated.
Finally, Golatkar et al. (2019) show that regularization results in higher
test accuracy specifically when it is applied during initial phase of
training, similar to the importance of Knee schedule’s explore phase during
initial phase of training. In a similar vein, Li et al. (2019) explain the
regularization benefits of the initial higher learning rate by showing that
higher learning rate helps networks learn easier-to-fit general patterns.
Neural network loss landscapes. The landscape of loss in neural networks have
been extensively studied (Draxler et al., 2018; Freeman and Bruna, 2016;
Garipov et al., 2018; Sagun et al., 2017). These papers point out that the
loss landscape contains both wide and narrow minima, and there may even exist
a path from one minima to another without barriers. However, there are
multiple paths between these minima and some paths indeed face barriers (e.g.,
see Figure 1 in Draxler et al. (2018)). Since we don’t know which path SGD and
other optimizers might follow, even if wide and narrow minima are part of a
single basin, SGD and other optimizers might still require higher learning
rates to navigate from narrow to wide minima.
Lower density of wide minima. Wu et al. (2018) compares the sharpness of
minima obtained by full-batch gradient descent (GD) with different learning
rates for small neural networks on FashionMNIST and Cifar10 datasets. They
find that GD with a given learning rate finds the theoretically sharpest
feasible minima for that learning rate. Thus, in the presence of several
flatter minimas, GD with lower learning rates does not find them, leading to
the conjecture that density of sharper minima is perhaps larger than density
of wider minima. Baldassi et al. (2020) show analytically for simple, two-
layer non-convex networks that wide minima exists and are rare, compared to
narrow minima, local minima and saddle points. In this paper, we add
significant evidence to these theoretical observations based on empirical
results obtained on large-scale, state-of-the-art neural networks through
carefully designed experiments.
## 3 Wide-Minima Density Hypothesis
Many popular learning rate schedules, such as the step decay schedules for
image datasets, start the training with high learning rate, and then reduce
the learning rate periodically. For example, consider the case of Cifar-10 on
Resnet-18, trained using a typical step learning rate schedule of $0.1,0.01,$
and $0.001$ for 100, 50, 50 epochs each. In many such schedules, even though
training loss stagnates after several epochs of high learning rate, one still
needs to continue training at high learning rate in order to get good
generalization.
For example, Figure 2 shows the training loss for Cifar-10 on Resnet-18,
trained with a fixed learning rate of 0.1 (orange curve), compared to a model
trained via a step schedule with learning rate reduced at epoch 50 (blue
curve). As can be seen from the figure, the training loss stagnates after
$\approx$ 50 epochs for the orange curve, and locally it makes sense to reduce
the learning rate to decrease the loss. However, as shown in Table 2,
generalization is directly correlated with duration of training at high
learning rate, with the highest test accuracy achieved when the high learning
rate is used for 100 epochs, well past the point where training loss
stagnates. Note that the final training loss remains similar for all runs.
To understand the above phenomena, we perform another experiment. We train
Cifar-10 on Resnet-18 for 200 epochs, using a high learning rate of $0.1$ for
only 30 epochs and then use learning rate of $0.01$ and $0.001$ for 85 epochs
each. We repeat this training 50 times with different random weight
initializations. On an average, as expected, this training yields a low test
accuracy of $94.81$. However, in 1 of the 50 runs, we find that the test
accuracy reaches $95.24$, even higher than the average accuracy of $95.1$
obtained while training at high learning rate for 100 epochs!
Figure 1: Training loss for Cifar-10 on Resnet-18. Orange plot uses a fixed
learning rate of 0.1, while in blue plot, the learning rate is reduced from
0.1 to 0.01 at epoch 50.
Figure 2: Cifar-10 on Resnet-18 trained for 200 epochs with Momentum. A learning rate of 0.1 is used for the explore epochs. Half the remaining epochs are trained at 0.01 and the other half at 0.001. Reported results are average over 4 runs. Epochs at | Test Accuracy | Train Loss
---|---|---
0.1 LR | Avg. (Std. Dev) | Avg. (Std. Dev.)
0 | 94.34 (0.13) | 0.0017 (8e-5)
30 | 94.81 (0.15) | 0.0017 (8e-5)
40 | 94.91 (0.14) | 0.0018 (9e-5)
60 | 95.01 (0.14) | 0.0018 (1e-4)
80 | 95.05 (0.15) | 0.0019 (1e-4)
100 | 95.10 (0.14) | 0.0021 (1e-4)
### 3.1 Hypothesis
To explain the above observations, i.e., using a high learning rate for short
duration results in low average test accuracy with rare occurrences of high
test accuracy, while using the same high learning rate for long duration
achieves high average test accuracy and frequent occurrences of high test
accuracy, we introduce a new hypothesis. We hypothesize that, in the DNN loss
landscape, the density of narrow minima is significantly higher than that of
wide minima.
Intuitively, a large learning rate can escape narrow minima “valleys” easily
(as the optimizer can jump out of them with large steps). However, once it
reaches a wide minima “valley”, it is likely to get stuck in it (if the
“width” of the wide valley is large compared to the step size). This intuition
is backed by theoretical results from Xie et al. (2020) that show that the
time to escape a minimum using SGD is exponential in the inverse of learning
rate as well as inverse of the sharpness (measured by eigenvalue of the
Hessian at the minima). Thus, large learning rates escape narrow minima
exponentially faster than wide minima.
If wide and narrow minima were uniformly distributed, SGD with a large LR
would be able to quickly escape the narrow minima, land on a wide minima and
get stuck there. Yet, we see that we need to maintain large LR for significant
duration for landing in a wide minima with high probability. On the other
hand, if our hypothesis is true, i.e., wide minima are much fewer than narrow
minima, the probability of landing in a wide minima after escaping a narrow
minima is low, and the optimizer needs to take a lot of steps to have a high
probability of eventually landing in a wide minimum. Thus, the hypothesis is a
better explanation for the observation in Table 2, where the average accuracy
continues to improve as we increase the number of high learning rate training
steps. The hypothesis also explains why very few (just 1) of the 50 runs
trained at $0.1$ learning rate for just 30-epochs also manages to attain high
accuracy—these runs just got lucky in a probabilistic sense and landed in a
wide minimum even with a shorter duration of explore.
| | |
---|---|---|---
(a) Explore 0 | (b) Explore 30 | (c) Explore 60 | (d) Explore 100
Figure 3: Histogram of minima sharpness (Keskar et al., 2016) for 50 random
trials of Cifar-10 on Resnet-18. Each figure shows histograms for runs with
different number of explore epochs. The distribution moves toward lower
sharpness and tightens as the number of explore epochs increase.
| | |
---|---|---|---
(a) Explore 0 | (b) Explore 30 | (c) Explore 60 | (d) Explore 100
Figure 4: Histogram of test accuracy for 50 random trials of Cifar-10 on
Resnet-18. Each figure shows histograms for runs with different number of
explore epochs. The distribution moves toward higher test accuracy and
sharpens as the number of explore epochs increase.
To validate this hypothesis further, we run experiments similar to the one in
Table 2. Specifically, we train Cifar-10 on Resnet-18 model for 200 epochs
using a standard step schedule with learning rate of $0.1,0.01,0.001$. We vary
the number of epochs trained using the high learning rate of 0.1, called the
explore epochs, from 0 to 100 epochs, and divide up the rest of the training
equally between 0.01 and 0.001. For each experimental setting, we conduct 50
random trials and plot the distributions of final test accuracy and the minima
sharpness as defined by the metric in Keskar et al. (2016) (see section 3.2).
If our hypothesis is true, then the more you explore, the higher the
probability of landing (and getting stuck) in a wide minima region, which
should cause the distribution to tighten and move towards wider minima (lower
sharpness), as the number of explore steps increase. This is exactly what is
observed in Figure 4. Also since wide minima correlate with higher test
accuracy, we should see the test accuracy distribution move towards higher
accuracy and sharpen, as the number of explore steps increase. This is
confirmed as well in Figure 4.
Longer training with low learning rate is not sufficient. Finally, to verify
whether explore at high learning rate is essential, we train Cifar-10 for
10,000 epochs at a fixed lower learning rate of 0.001. The training loss
converged but the final test accuracy was only 93.9, compared to an accuracy
of over 95% in 200 epochs in Table 2. Thus, even training $50\times$ longer at
low learning rate is not sufficient to achieve good generalization. Again,
this observation ties in well with the theoretical results from Xie et al.
(2020) where the authors show that the time to escape a minimum using SGD is
exponential in the inverse of learning rate. Thus, this result adds further
evidence to our density hypothesis, since even training $50\times$ longer at a
low learning rate is not sufficient to land in a wide minima.
Multi-scale. Given the importance of explore at high learning rate, a natural
question that may arise is whether explore is necessary at smaller learning
rate as well. To answer this, we train the same network for a total of 200
epochs with an initial high learning rate of $0.1$ for 100 epochs, but now we
vary the number of epochs trained with the learning rate of $0.01$ (we call
this finer-scale explore), and train with learning rate of $0.001$ for the
remaining epochs. As can be seen from Table 1, although the final training
loss remains similar, we find that finer-scale explore also plays a role
similar to the initial explore in determining the final test accuracy. This
indicates that our hypothesis about density of wide/narrow regions indeed
holds at multiple scales.
Table 1: Cifar-10 on Resnet-18 trained for 200 epochs. A learning rate of 0.1 is used for the first 100 epochs. We then vary the number of epochs trained with learning rate of $0.01$ (called finer-scale explore), and train the remaining epochs with a learning rate of $0.001$. We report averages values over 3 runs. Explore Epochs (Finer-scale) | Test Accuracy | Training Loss | Sharpness
---|---|---|---
10 | 94.78 | 0.0031 | 5.48
20 | 94.91 | 0.0026 | 4.47
30 | 95.00 | 0.0023 | 4.02
40 | 95.02 | 0.0021 | 3.91
50 | 95.10 | 0.0021 | 3.54
### 3.2 Minima Sharpness
Our hypothesis predicts that higher explore helps the optimizer land in a
wider minimum, which in turn helps generalization. We demonstrated this
empirically in Figure 4, where we plotted the distribution of the minima
sharpness, as measured by the sharpness metric introduced by (Keskar et al.,
2016). In this section, we describe Keskar’s sharpness metric in detail. We
also introduce a simple projected gradient ascent scheme to compute this
metric efficiently, which scales well to large networks. Finally, we also
evaluate our hypothesis with a different metric for minima sharpness, the
Fisher Score, which is based on the Fisher information matrix.
#### 3.2.1 Keskar’s Sharpness Metric
Keskar’s sharpness metric is based on measuring the maximum jump in the
network’s output function $F$ in a small neighborhood around the minimum.
After a few simplifications, Keskar’s metric for sharpness around a point $x$
can be written as:
$S_{x,F}(\epsilon):=\frac{(max_{y\in
C_{\epsilon}(x)}F(x+y))-F(x)}{1+F(x)}\times 100,$ (1)
where $C_{\epsilon}(x)$ is an $\epsilon$ neighborhood around $x$. Keskar et
al. (2016) mentions that under certain conditions and for small values of
$\epsilon$, $S_{x,F}$ is proportional to the largest eigenvalue of the
Hessian. Please see Keskar et al. (2016) for more details. For our
measurements we choose an $\epsilon$ of $1e^{-4}$.
For solving the maximization problem in Equation 1, Keskar et al. (2016) uses
a second-order L-BFGS-B (Byrd et al., 2003) optimization scheme. However, in
our experiments we found the method to be very slow. To combat this, Keskar et
al. (2016) limited their runs to 10 iterations but we found that results were
suboptimal using few iterations. Instead, we employed a projected gradient
ascent scheme to solve Equation 1. In each optimization step, we took a small
step with a learning rate of 0.001 in the gradient direction and projected the
updated point to lie inside $C_{\epsilon}(x)$. Because of the first order
nature, this method is much faster. We found that even 1000 iterations were
fast to compute and the results were much better than the second order method
in all cases we evaluated.
Using Keskar’s sharpness metric, we had shown in Figure 4 that the
distribution of minima sharpness moves towards lower values as the number of
explore epochs increase. In Table 2, we also report the average sharpness of
the minima for varying explores. As predicted by our hypothesis, average
sharpness decreases as number of explore epochs increase.
Table 2: Keskar’s sharpness metric for Cifar-10 on Resnet-18 trained for 200
epochs with Momentum. A learning rate of 0.1 is used for the explore epochs.
Half the remaining epochs are trained at 0.01 and the other half at 0.001. We
report the average sharpness over 50 different trials.
Explore Epochs | Sharpness
---|---
0 | 10.56
30 | 5.43
60 | 3.86
100 | 3.54
#### 3.2.2 Fisher Score
The maximum Eigen value of the Fisher Information Matrix (FIM) estimates the
highest curvature at a point, and is used as another metric to measure minima
sharpness (Sokol and Park, 2018). We used an unbiased estimate of the true
Fisher matrix (see Kunstner et al. (2019)) using 10 unbiased samples per
training data. Table 3 shows the average Fisher scores for the Cifar-10
experiments at varying explores. Again, the sharpness measured by the Fisher
score decreases as the number of explore epochs increase.
Table 3: Fisher Score for Cifar-10 on Resnet-18 trained for 200 epochs with
Momentum. A learning rate of 0.1 is used for the explore epochs. Half the
remaining epochs are trained at 0.01 and the other half at 0.001. We report
the average Fisher score over 10 different trials.
Explore Epochs | FIM score
---|---
0 | 0.051
30 | 0.046
60 | 0.043
100 | 0.042
## 4 Explore-Exploit Learning Rate Schedule
Given that we need to explore at multiple scales for good generalization, how
do we go about designing a good learning rate schedule? The search space of
the varying learning rate steps and their respective explore duration is
enormous.
Fortunately, since the explore at the initial scale is searching over the
entire loss surface while explore at finer-scales is confined to exploring
only the wide-minima region identified by the initial explore, the former is
more crucial. In our experiments as well, we found that the initial portion of
training is much more sensitive to exploration and needs a substantial number
of explore steps, while after this initial phase, several decay schemes worked
equally well. This is similar to the observations in (Golatkar et al., 2019)
where the authors found that regularization such as weight-decay and data
augmentation mattered significantly only during the initial phase of training.
The above observations motivate our Explore-Exploit learning rate schedule,
where the explore phase first optimizes at a high learning rate for some
minimum time in order to land in the vicinity of a wide minima. We should give
the explore phase enough time (a hyper-parameter), so that the probability of
landing in a wide minima is high. After the explore phase, we know with a high
probability, that the optimizer is in the vicinity of a wide region. We now
start the exploit phase to descend to the bottom of this wide region while
progressively decreasing the learning rate. Any smoothly decaying learning
rate schedule can be thought of as doing micro explore-exploit at
progressively reduced scales. A steady descent would allow more explore
duration at all scales, while a fast descent would explore less at higher
learning rates. We experimented with multiple schedules for the exploit phase,
and found a simple linear decay to zero, that does not require any hyper-
parameter, to be effective in all the models/datasets we tried. We call our
proposed learning rate schedule which starts at a constant high learning rate
for some minimum time, followed by a linear decay to zero, the Knee schedule.
Note that any learning rate decay scheme incorporates an implicit explore
during the initial part, where the learning rate stays high enough. To
evaluate the benefit of an explicit explore phase, we compare Knee schedule
against several decay schemes such as linear and cosine. Interestingly, the
results depend on the length of training. For long budget experiments, simple
decay schemes perform comparable to Knee schedule in some experiments, since
the implicit explore duration is also large, helping these schemes achieve
good generalization. However for short budget experiments, these schemes
perform significantly worse than Knee schedule, since the implicit explore
duration is much shorter. See Table 4 , 5 and 6 for the comparison.
Warmup. Some optimizers such as Adam use an initial warmup phase to slowly
increase the learning rate. However, as shown in Liu et al. (2019), learning
rate warmup is needed mainly to reduce variance during initial training stages
and can be eliminated with an optimizer such as RAdam. Learning rate warmup is
also used for large-batch training (Goyal et al., 2017). Here, warmup is
necessary since the learning rate is scaled to a very large value to
compensate for the large batch size. This warmup is complementary and can be
incorporated into Knee schedule. For example, we do this for BERTLARGE
pretraining experiment where a large 16k batch size was used.
## 5 Evaluation
In this section we present extensive empirical evaluation of Knee schedule on
multiple models and datasets across various optimizers, and compare Knee
schedule against the original hand-tuned learning rate baselines. We first
provide an overview of our main results followed by detailed experimental
results. We then run further experiments to validate our wide-minima density
hypothesis, as well as run sensitivity analysis of seed learning rate on the
Knee schedule.
Note that, for completeness, we present a detailed comparison of Knee schedule
with many other learning rate schedules in literature such as linear decay,
cosine decay (Loshchilov and Hutter, 2016), one-cycle (Smith, 2018) in
Appendix A.
### 5.1 Experiments
We evaluate Knee schedule on multiple models and datasets spanning both vision
and NLP problems. The training of these models spanned various optimizers
including SGD Momentum, Adam (Kingma and Ba, 2014a), RAdam (Liu et al., 2019)
and LAMB (You et al., 2019). For all experiments, we used an out of the box
policy, where we only change the learning rate schedule, without modifying
anything else. We evaluate on multiple image datasets – Imagenet on Resnet-50,
Cifar-10 on Resnet-18; as well as various NLP datasets – pretraining BERTLARGE
on Wikipidea+BooksCorpus and fine-tuning it on SQuADv1.1; and WMT’14 (EN-DE),
IWSLT’14 (DE-EN) on Transformers.
### 5.2 Results Overview
In all our experiments, we find that Knee schedule shows an improvement in
test accuracy over the original hand-tuned learning rate baseline as well as
various other learning rate schedules in the literature. Further, we also find
that Knee schedule can achieve the same accuracy as the baseline with a much
reduced training budget.
Table 4: We report the top-1 accuracy for ImageNet and Cifar-10, BLEU score for IWSLT’14 and WMT’14 and F1 score for BERT on SQuAD. All values are averaged over multiple runs for each experiment. Experiment details are mentioned in the individual sections of the experiments. | | | Knee | | | |
---|---|---|---|---|---|---|---
Experiment | Training | Knee | Schedule | Baseline | One-Cycle | Cosine | Linear
| Budget | Schedule | (Fixed 50% | | Decay | Decay |
| (epochs) | | explore) | | | |
ImageNet | 90 | 76.71 | 76.58 | 75.87 | 75.39 | 76.41 | 76.54
Cifar-10 | 200 | 95.26 | 95.26 | 95.10 | 94.09 | 95.23 | 95.18
IWSLT | 50 | 35.53 | 35.23 | 34.97 | 34.77 | 35.21 | 34.97
WMT’14 | 70 | 27.53 | 27.41 | 27.29 | 27.19 | 27.35 | 27.29
BERTLARGE | 31250 (iters) | 91.51 | 91.51 | 91.34 | - | - | 91.34
Table 5: Shorter budget training: Test accuracy on all learning rate schedules tried in this paper, but trained with a shortened budget. We report same metrics as Table 4. Knee schedule achieves the same accuracy as baseline schedules using much lower budget, saving precious GPU-hours. | Shortened Training | Knee | | Cosine | Linear | Saving
---|---|---|---|---|---|---
Experiment | Budget | Schedule | One-Cycle | Decay | Decay | ( V100 GPU
| (epochs) | | | | | hours)
ImageNet | 50 | 75.92 | 75.36 | 75.71 | 75.82 | 27
Cifar-10 | 150 | 95.14 | 93.84 | 95.06 | 95.02 | 0.25
IWSLT | 35 | 35.08 | 34.43 | 34.46 | 34.16 | 0.75
WMT’14 | 30 | 27.28 | 26.80 | 26.95 | 26.77 | 80
BERTLARGE | 20854 (iters) | 91.29 | - | - | 90.64 | 1002
Table 6: Epochs required by different LR schedules to reach the target accuracy. The target accuracy is chosen based on Knee schedule’s results with a reduced budget. Experiment | Target BLEU Score | Knee schedule | Cosine Decay | Linear Decay
---|---|---|---|---
IWSLT | 35.08 | 35 | 45 | 60
WMT’14 | 27.28 | 30 | 60 | 70
Table 4 shows the test accuracies of the various experiments, when trained
with the original budget; while Table 5 shows the results when trained with a
reduced budget. As shown, for the original budget runs, Knee schedule improves
on the test accuracies in all experiments. Note that in Knee schedule, the
explore duration is a hyperparameter. To avoid tuning this hyperparameter, we
experimented with a fixed 50% explore duration for the full budget runs. Even
the fixed 50% explore Knee schedule outperforms all the other baselines. Also
noteworthy is that Knee schedule is able to achieve the same test accuracies
as the baseline’s full budget runs with a much lower training budget, saving
precious GPU cycles (Table 5).
While the difference in accuracy values between the various schedules might
appear deceptively small in absolute terms, achieving these gains require a
large amount of compute. For example, the number of epochs needed by each
scheme to reach the target BLEU score for IWSLT’14 DE-EN and WMT’14 EN-DE with
the Transformer network is shown in Table 6. One can see that Knee schedule is
significantly more efficient as compared to say Cosine Decay, which takes 100%
more training time to achieve the same accuracy for WMT‘14 EN-DE. Thus, the
accuracy and/or compute gains achieved by Knee schedule is significant.
A summary of our main experimental results is as follows:
1. 1.
Imagenet on Resnet-50: We show an absolute gain of 0.8% in top-1 accuracy
against the competitive step schedule baseline for this model. Also, Knee
schedule can achieve the same accuracy as baseline in $\sim$45% less training
epochs.
2. 2.
BERTLARGE pre-training on Wikipedia+BooksCorpus dataset: Compared to the
baseline of You et al. (2019), we improve the F1 score on SQuAD v1.1 fine-
tuning task by 0.2% (91.51 compared to 91.34). Also, we were able to achieve
similar accuracy as baseline in 33% less training steps (a saving of
$\sim$1002 V100 GPU-hours!).
3. 3.
WMT’14 and IWSLT machine translation on Transformers: Compared to competitive
baselines, we were able to improve the BLEU scores by 0.24 and 0.56 points for
the two tasks. Moreover, Knee schedule was able to achieve the same accuracy
as baselines in 57% and 30% less training times.
4. 4.
State of the Art (SOTA) Results: We also attain state of the art results on
the IWSLT’14(DE-EN) machine translation dataset by simply replacing the
learning rate schedule of the current SOTA model (Shen et al., 2020) with
Knee. We were able to improve the BLEU score by 0.18, reaching a new SOTA
score of 37.78. Moreover, Knee can achieve the current SOTA baseline value in
30% less training time.
### 5.3 Detailed Results
We now describe each of our main experimental results in detail.
#### 5.3.1 ImageNet Image Classification on Resnet-50
We train ImageNet dataset (Russakovsky et al., 2015) on Resnet-50 network (He
et al., 2016) which has 25 million parameters, with a batch size of 256 and a
seed learning rate of 0.1. Random cropping and random horizontal flipping
augmentations were applied to the training dataset. We use SGD optimizer with
momentum of 0.9 and weight decay of $1e^{-4}$. For baseline runs, we used the
standard hand-tuned step learning rate schedule of 0.1, 0.01 and 0.001 for 30
epochs each. For Knee schedule we used a seed learning rate of 0.1 (same as
baseline). We trained with the original budget of 90 epochs as well as with a
reduced budget of 50 epochs. We used 30 explore epochs for the two
experiments. 222We used the opensource implementation at:
https://github.com/cybertronai/imagenet18_old
Table 7 shows the training loss and test accuracies for our experiments. Knee
schedule comfortably beats the test accuracy of baseline in the full budget
run (with absolute gains of 0.8% and 0.4% in top-1 and top-5 accuracy,
respectively), while meeting the baseline accuracy even with a much shorter
budget. The fact that the baseline schedule takes almost $80\%$ more training
time than Knee schedule for the same test accuracy, shows the effectiveness of
our Explore-Exploit scheme. See Figure 6 in Appendix B for training curves.
Table 7: ImageNet on Resnet-50 results. We report mean (stddev) over 3 runs. LR Schedule | Test Top 1 Acc. | Test Top 5 Acc. | Training Loss | Training Epochs
---|---|---|---|---
Baseline | 75.87 (0.035) | 92.90 (0.015) | 0.74 (1e-3) | 90
Knee | 76.71 (0.097) | 93.32 (0.031) | 0.79 (1e-3) | 90
Knee (short budget) | 75.92 (0.11) | 92.90 (0.085) | 0.90 (3e-3) | 50
#### 5.3.2 Cifar-10 Image Classification on Resnet-18
We train Cifar-10 dataset (Krizhevsky et al., 2009) on Resnet-18 network (He
et al., 2016) which has around 11 million parameters. SGD optimizer is used
with momentum of 0.9 and weight decay of $5e^{-4}$. Random cropping and random
horizontal flipping augmentations were applied to the training dataset. 333We
used the open-source implementation at: https://github.com/kuangliu/pytorch-
cifar.
For baseline, we used the hand-tuned step learning rate schedule of 0.1, 0.01
and 0.001 for 100, 50, 50 epochs, respectively. With Knee schedule, we train
the network with the original budget of 200 epochs, as well as a reduced
budget of 150 epochs. We used 100 explore epochs for both runs, and a seed
learning rate of 0.1 (same as baseline). Table 8 shows the training loss and
test accuracies for the experiments. Knee schedule beats the test accuracy of
baseline in the full budget run, while meeting the baseline test accuracy in
$25\%$ less budget. Refer to figure 7 in Appendix B for detailed comparisons
of training loss, test accuracy, and learning rate.
Table 8: Training loss and Test accuracy for Cifar-10 on Resnet-18. We report mean (stddev) over 7 runs. LR Schedule | Test Accuracy | Training Loss | Training Epochs
---|---|---|---
Baseline | 95.10 (0.14) | 0.002 (1e-4) | 200 epochs
Knee | 95.26 (0.11) | 0.002 (1e-4) | 200 epochs
Knee (short budget) | 95.14 (0.18) | 0.004 (3e-4) | 150 epochs
#### 5.3.3 BERTLARGE Pre-training
We pretrain on BERTLARGE on Wikipedia+BooksCorpus dataset with LAMB optimizer
(You et al. (2019)). BERTLARGE has around 330 million parameters and the pre-
training is divided into two phases with different sequence lengths. The first
phase consists of 90% steps with sequence length of 128 and the second phase
consists of the remaining 10% steps with sequence length of 512 (Devlin et al.
(2018)). We used a batch size of 16384 in both phases of training 444We used
the open-source implementation at:
https://github.com/NVIDIA/DeepLearningExamples/tree/master/PyTorch/LanguageModeling/BERT.
We use the same training budget of 31250 steps mentioned in (You et al.
(2019)). We also train the model on a shortened training budget of 2/3rd the
original steps (20854 steps).
Since large batch training requires learning rate warmup (see Goyal et al.
(2017)), we incorporate it into the Knee schedule by first doing a warmup of
10% as suggested in (You et al., 2019) followed by the explore-exploit phases.
We used an explore of 50% of the total steps available for both phases of BERT
training. For baseline, we use the warmup (10%) + linear decay (90%) schedule
(You et al., 2019; Devlin et al., 2018). The pre-trained models are evaluated
on the SQuAD v1.1 (Rajpurkar et al., 2016) dataset by fine-tuning on the
dataset for 2 epochs. See Table 9 for the results. For the full budget run,
Knee schedule improves the baseline by 0.2%, while for the reduced budget we
achieved similar fine-tuning accuracy as baseline. The baseline schedule
achieves a much lower accuracy with shorter budget training, showing the
efficacy of Knee schedule. BERT pre-training is extremely compute expensive
and takes around 47 hours on 64 V100 GPUs (3008 V100 GPU-hrs) on cloud VMs.
The reduced budget amounts to a saving of approximately 1002 V100 GPU-hours!
Table 9: BERTLARGE results. We report the pre-training train loss, and the test F1 accuracy on SQuAD v1.1 after fine-tuning. See figure 9 in Appendix B for training curves. LR Schedule | F1 score on SQuAD v1.1 | Training loss | Total Training Steps
---|---|---|---
Knee | 91.51 | 1.248 | 31250
Baseline (You et al., 2019) | 91.34 | - | 31250
Baseline (short budget) | 90.64 | 1.336 | 20854
Knee (short budget) | 91.29 | 1.275 | 20854
#### 5.3.4 Machine Translation on Transformer Network with WMT’14 and IWSLT
In the second NLP task, we train the Transformer (base model) (Vaswani et al.,
2017) on the IWSLT’14 (De-En) (Cettolo et al., 2014) and WMT’14 (En-De) (Bojar
et al., 2014) datasets with the RAdam (Liu et al., 2019) optimizer.
##### WMT’14 (EN-DE):
We use the default implementation provided by the fairseq package (Ott et al.,
2019) 555https://github.com/pytorch/fairseq. We train WMT’14 (EN-DE) dataset
on the TransformerBASE (Vaswani et al., 2017) model which has around 86
million parameters and use the RAdam (Liu et al., 2019) optimizer with
$\beta_{1}$ of 0.9 and $\beta_{2}$ of 0.999. Label smoothed cross entropy was
used as the objective function with an uncertainty of 0.1. A dropout of 0.1,
clipping norm of 25 and weight decay of $1e^{-4}$ is used. Each training batch
contains approximately 30000 tokens.
The baseline schedule uses a linear decay for 70 epochs (Liu et al., 2019).
With Knee schedule, we trained with the original budget of 70 epochs, as well
as a reduced budget of 30 epochs. We used 50 and 25 explore epochs for the two
runs, respectively and a seed learning rate of $3e^{-4}$ for both Knee
schedule and baseline. In all cases we use the model checkpoint with least
loss on the validation set for computing BLEU scores on the test set. Table 10
shows the training loss and test accuracy averaged over 3 runs. Knee schedule
improves the test BLEU score of baseline in the full budget run by 0.24
points. In the shorter budget run, Knee schedule matches the test accuracy of
the baseline while taking $57\%$ less training time (a saving of 80 V100 GPU-
hours!). See Figure 10 in Appendix B for training curves.
##### IWSLT’14 (DE-EN):
For IWSLT’14 (DE-EN) we use the same configuration as WMT’14 (EN-DE), except
for a dropout of 0.3 following Fairseq’s out-of-box implementation. Each
training batch contains approximately 4000 tokens. For Knee schedule, we
choose explore as 30 epochs for short budget runs and 40 epochs for full
budget runs.
The baseline schedule uses a linear decay for 50 epochs (Liu et al., 2019).
With Knee schedule, we trained with the original budget of 50 epochs, as well
as a reduced budget of 35 epochs. We used 40 and 30 explore epochs for the two
runs, respectively and a seed learning rate of $3e^{-4}$ for both Knee
schedule and baseline. In all cases we use the model checkpoint with least
loss on the validation set for computing BLEU scores on the test set. Knee
schedule improves the baseline test BLEU score by 0.56 points in the full
budget run. In the shorter budget run, Knee schedule matches the test accuracy
of the baseline schedule while taking $30\%$ less training time. See Figure 11
in Appendix B for training curves.
Table 10: Results for WMT’14 (EN-DE) on Transformer networks. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report mean (stdev) over 3 runs. LR Schedule | Test BLEU | Train | Validation | Training
---|---|---|---|---
Score | Perplexity | Perplexity | Epochs
Baseline | 27.29 (0.06) | 3.87 (0.017) | 4.89 (0.02) | 70
Knee | 27.53 (0.12) | 3.89 (0.017) | 4.87 (0.006) | 70
Knee (short budget) | 27.28 (0.17) | 4.31 (0.02) | 4.92 (0.007) | 30
Table 11: Training, validation perplexity and test BLEU scores for IWSLT on Transformer networks. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU | Train | Validation | Training
---|---|---|---|---
Score | Perplexity | Perplexity | Epochs
Baseline | 34.97 (0.035) | 3.36 (0.001) | 4.91 (0.035) | 50
Knee | 35.53 (0.06) | 3.00 (0.044) | 4.86 (0.02) | 50
Knee (short budget) | 35.08 (0.12) | 3.58 (0.049) | 4.90 (0.063) | 35
#### 5.3.5 SQuAD-v1.1 fine-tuning on BERTBASE
We also evaluate Knee schedule on the task of fine-tuning BERTBASE model
Devlin et al. (2018) on SQuAD v1.1 Rajpurkar et al. (2016) with the Adam
Kingma and Ba (2014b) optimizer 666We used the implementation at:
https://github.com/huggingface/transformers. BERT fine-tuning is prone to
overfitting because of the huge model size compared to the small fine-tuning
dataset, and is typically run for only a few epochs. For baseline we use the
linear decay schedule mentioned in Devlin et al. (2018). We use a seed
learning rate of $3e^{-5}$ and train for 2 epochs. For Knee schedule, we train
the network with 1 explore epoch with the same seed learning rate of
$3e^{-5}$. Table 12 shows our results over 3 runs. We achieve a mean EM score
of 81.4, compared to baseline’s 80.9, a 0.5% absolute improvement. We don’t do
a short budget run for this example, as the full budget is just 2 epochs.
Please refer to Figure 14 in Appendix B for the training loss, test accuracy
and learning rate curves.
Table 12: SQuAD fine-tuning on BERTBASE. We report the average training loss, and average test EM, F1 scores over 3 runs. LR Schedule | EM | F1 | Train Loss | Training Epochs
---|---|---|---|---
Baseline | 80.89 (0.15) | 88.38 (0.032) | 1.0003 (0.004) | 2
Knee schedule | 81.38 (0.02) | 88.66 (0.045) | 1.003 (0.002) | 2
#### 5.3.6 State of the Art Result
To further demonstrate the effectiveness of Knee schedule, we took a recent
high performing model, Cutoff (Shen et al., 2020)777We used the code available
at https://github.com/dinghanshen/Cutoff, which had reported state-of-the-art
accuracy on the IWSLT’14 (DE-EN) dataset. They reported a BLEU score of 37.6
when trained with an inverse square root learning rate schedule for 100
epochs, with the first 6000 steps allocated for warmup. We simply retrained
the model with our Knee schedule, and achieved a new SOTA BLEU score of 37.78
(an absolute increase of 0.18). See Table 13 for the BLEU scores, training and
validation perplexities.
We also show that Knee schedule can train the model in 30% less training time
(70 epochs), while achieving slightly better accuracy of 37.66 BLUE score
compared to the 100 epoch baseline. The baseline schedule when run for 70
epochs achieves a much worse accuracy of 37.31.
For both the full budget (100 epochs) and the short budget (70 epochs) Knee
runs, we choose 50% of the total training epochs as explore epochs. We also
perform warmup for the same number of steps as baseline. For all runs (Knee
and baseline), we report the BLEU score obtained by averaging the last 5
checkpoints and computing on the test set. See Figure 12 and 13 in Appendix B
for training curves.
Table 13: Training, validation perplexity and test BLEU scores for IWSLT’14 DE-EN on Cutoff. The test BLEU scores are computed by averaging the last 5 checkpoints LR Schedule | Test BLEU | Train | Validation | Training
---|---|---|---|---
Score | Perplexity | Perplexity | Epochs
Inv. Sqrt | 37.60 | 3.46 | 4.24 | 100
Knee | 37.78 | 3.29 | 4.13 | 100
Inv. Sqrt (short budget) | 37.31 | 3.76 | 4.29 | 70
Knee (short budget) | 37.66 | 3.48 | 4.18 | 70
### 5.4 Hypothesis Validation with Knee schedule on Language Tasks
For validating our hypothesis on the density of wide minima vs narrow minima,
we did multiple experiments on vision tasks, most of which were discussed in
Section 3. To summarize, in Figures 4 and 4, we showed that for Cifar-10 on
Resnet-18, as the number of explore steps increase, the distribution of minima
width and test accuracy sharpens and shifts towards wider minima and better
accuracy, respectively.
Table 14: IWSLT’14 (DE-EN) on the Transformer network trained with the Knee schedule. The explore duration is varied, while keeping the total training budget fixed at 50 epochs. We report averages over 3 runs. Explore Epochs | Test BLEU score | Training Perplexity
---|---|---
5 | 34.93 | 3.29
10 | 35.02 | 3.22
15 | 35.08 | 3.11
20 | 35.10 | 3.08
25 | 35.23 | 3.02
30 | 35.28 | 2.99
40 | 35.53 | 3.00
We now perform similar experiments on the IWSLT’14 German to English dataset
(Cettolo et al., 2014) trained on Transformer networks (Vaswani et al., 2017)
to demonstrate that our hypothesis holds even on a completely different NLP
dataset and network architecture. We train with the Knee schedule for a total
budget of 50 epochs with explore lr as $3e^{-4}$, but keep varying the number
of explore epochs. As shown in Table 14, the test BLEU score increases as we
increase the number of explore epochs. Further, we found that among multiple
trials, a 20 epoch explore run had a high BLEU score of 35.29, suggesting that
the run got lucky. Thus, these results on the IWSLT’14 (DE-EN) dataset add
more evidence to the wide-minima density hypothesis.
### 5.5 Learning Rate Sensitivity for Knee schedule
We performed sensitivity analysis of the starting learning rate, referred to
as the seed learning rate, for Knee schedule. We trained the Cifar-10 dataset
on Resnet-18 with the Knee schedule for a shortened budget of 150 epochs,
starting at different seed learning rates. For each experiment, we do a simple
linear search to find the best explore duration. The test accuracies and
optimal explore duration for the different seed learning rate choices is shown
in Table 15. As shown, the seed learning rate can impact the final accuracy,
but Knee schedule is not highly sensitive to it. In fact, we can achieve the
target accuracy of 95.1 with multiple seed learning rates of 0.05, 0.075,
0.0875 and 0.115, as compared to the original seed learning rate of 0.1, by
tuning the number of explore epochs.
Another interesting observation is that the optimal explore duration varies
inversely with the seed learning rate. Since a bigger learning rate has higher
probability of escaping narrow minima compared to a lower learning rate, it
would, on an average, require fewer steps to land in a wide minima. Thus,
larger learning rates can explore faster, and spend more time in the exploit
phase to go deeper in the wide minimum. This observation is thus consistent
with our hypothesis and further corroborates it.
We also note that by tuning both seed learning rate and explore duration, we
can achieve the twin objectives of achieving a higher accuracy, as well as a
shorter training time – e.g. here we are able to achieve an accuracy of 95.34
in 150 epochs (seed learning rate 0.075), compared to 95.1 achieved by the
baseline schedule in 200 epochs.
Table 15: Seed learning rate sensitivity analysis. Cifar-10 on Resnet-18 trained for 150 epochs with Knee schedule. We vary the seed learning rate and explore epochs to get the best test accuracy for the particular setting. We report averages over 3 runs. Seed LR | Test Accuracy | Optimal Explore Epochs
---|---|---
0.03 | 95.07 | 120
0.05 | 95.12 | 120
0.0625 | 95.15 | 120
0.075 | 95.34 | 100
0.0875 | 95.22 | 100
0.1 | 95.14 | 100
0.115 | 95.20 | 60
0.125 | 95.06 | 60
0.15 | 95.04 | 30
## 6 Conclusions
In this paper, we make an observation that an initial explore phase with a
high learning rate is essential for good generalization of DNNs. Further, we
find that a minimum explore duration is required even if the training loss
stops improving much earlier. We explain this observation via our hypothesis
that in the DNN loss landscape, the density of wide minima is significantly
lower than that of narrow minima. Motivated by this hypothesis, we present an
Explore-Exploit based learning rate schedule, called the Knee schedule. We do
extensive evaluation of Knee schedule on multiple models and datasets. In all
experiments, the Knee schedule outperforms prior hand-tuned baselines,
including achieving SOTA test accuracies, when trained with the original
training budget, and achieves the same test accuracy as the baseline when
trained with a much shorter budget.
## 7 Acknowledgement
We would like to thank Sanjith Athlur for his help in setting up the VM
cluster for large training runs and Harshay Shah for helpful discussions on
minima width computation.
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## A Comparisons with Other Baseline Learning Rate Schedules
In this section we compare Knee schedule against several other learning rate
schedules – one-cycle, linear decay and cosine decay.
One-Cycle: The one-cycle learning rate schedule was proposed in Smith (2018)
(also see Smith (2017)). This schedule first chooses a maximum learning rate
based on an learning rate range test. The learning rate range test starts from
a small learning rate and keeps increasing the learning rate until the loss
starts exploding (see figure 5). Smith (2018) suggests that the maximum
learning rate should be chosen to be bit before the minima, in a region where
the loss is still decreasing. There is some subjectivity in making this
choice, although some blogs and libraries888See e.g.
https://towardsdatascience.com/finding-good-learning-rate-and-the-one-cycle-
policy-7159fe1db5d6 and https://sgugger.github.io/how-do-you-find-a-good-
learning-rate.html. Also see https://docs.fast.ai/callbacks.lr_finder.html and
https://docs.fast.ai/callbacks.one_cycle.html suggest using a learning rate
one order lower than the one at minima. We go with this choice for all our
runs.
Once the maximum learning rate is chosen, the one-cycle schedule proceeds as
follows. The learning rate starts at a specified fraction999See div_factor in
https://docs.fast.ai/callbacks.one_cycle.html. We chose the fraction to be 0.1
in our experiments. of the maximum learning rate and is increased linearly to
the maximum learning rate for 45 percent of the training budget and then
decreased linearly for the remaining 45. For the final 10 percent, the
learning rate is reduced by a large factor (we chose a factor of 10). We used
an opensource implementation
101010https://github.com/nachiket273/One_Cycle_Policy for our experiments.
(a) LR range test for CIFAR-10
(b) LR range test for IWSLT’14 DE-EN
(c) LR range test for WMT’14 EN-DE
(d) LR range test for ImageNet
Figure 5: learning rate range test for selecting the maximum learning rate. A
good choice is the learning rate is a bit before the minima in a region where
the loss is still decreasing.
Linear Decay: The linear decay learning rate schedule simply decays the
learning rate linearly to zero starting from a seed learning rate.
Cosine Decay: The cosine decay learning rate schedule decays the learning rate
to zero following a cosine curve, starting from a seed learning rate.
### A.1 Cifar-10
Figure 5(a) shows the learning rate range test for Cifar-10 with the Resnet-18
network. The minima occurs around learning rate of 0.09, and we choose
$9e^{-3}$ as the maximum learning rate for the One-Cycle runs. For linear,
cosine decay schedules we start with a seed learning rate of 0.1 as used in
the standard baselines. The training loss and test accuracy for the various
schedules are shown in Table 16 for the full budget runs (200 epochs), and in
Table 17 for the short budget runs (150 epochs).
Table 16: Cifar-10 on Resnet-18 full budget training (200 epochs): Training loss and Test accuracy for more learning rate schedules. We report the mean and standard deviation over 7 runs. LR Schedule | Test Accuracy | Train Loss
---|---|---
One-Cycle | 94.08 (0.07) | 0.0041 (6e-5)
Cosine Decay | 95.23 (0.11) | 0.0023 (9e-5)
Linear Decay | 95.18 (0.15) | 0.0018 (7e-5)
Knee schedule | 95.26 (0.11) | 0.0023 (1e-4)
Table 17: Cifar-10 on Resnet-18 short budget training (150 epochs): Training loss and Test accuracy for more learning rate schedules. We report the mean and standard deviation over 7 runs. LR Schedule | Test Accuracy | Train Loss
---|---|---
One-Cycle | 93.84 (0.082) | 0.0052 (7e-5)
Cosine Decay | 95.06 (0.16) | 0.0030 (2e-4)
Linear Decay | 95.02 (0.10) | 0.0021 (1e-4)
Knee schedule | 95.14 (0.18) | 0.0044 (3e-4)
### A.2 ImageNet
Figure 5(d) shows the learning rate range test for ImageNet with the Resnet-50
network. The minima occurs around learning rate of 2.16, and we choose $0.216$
as the maximum learning rate for One-Cycle runs. For linear, cosine decay
schedules we start with a seed learning rate of 0.1 as used in the standard
baselines. The training loss and test accuracy for the various schedules are
shown in Table 18 for the full budget runs (90 epochs), and in Table 19 for
the short budget runs (50 epochs).
Table 18: ImageNet with ResNet-50 full budget training (90 epochs): Training loss, Test Top-1 and Test Top-5 for more learning rate schedules. We report the mean and standard deviation over 3 runs. LR Schedule | Test Top-1 | Test Top-5 | Train Loss (av)
---|---|---|---
One Cycle | 75.39 (0.137) | 92.56 (0.040) | 0.96 (0.003)
Cosine Decay | 76.41 (0.212) | 93.28 (0.066) | 0.80 (0.002)
Linear decay | 76.54 (0.155) | 93.21 (0.051) | 0.75 (0.001)
Knee schedule | 76.71 (0.097) | 93.32 (0.031) | 0.79 (0.001)
Table 19: ImageNet with ResNet-50 short budget training (50 epochs): Training loss, Test Top-1 and Test Top-5 for more learning rate schedules. We report the mean and standard deviation over 3 runs. LR Schedule | Test Top-1 | Test Top-5 | Train Loss (av)
---|---|---|---
One Cycle | 75.36 (0.096) | 92.53 (0.079) | 1.033 (0.004)
Cosine Decay | 75.71 (0.116) | 92.81 (0.033) | 0.96 (0.002)
Linear decay | 75.82 (0.080) | 92.84 (0.036) | 0.91 (0.002)
Knee schedule | 75.92 (0.11) | 92.90 (0.085) | 0.90 (0.003)
### A.3 WMT’14 EN-DE
Figure 5(c) shows the learning rate range test for WMT’14 EN-DE on the
transformer networks. The minima occurs near $1.25e^{-3}$. For the maximum
learning rate, we choose $2.5e^{-4}$ for the default one-cycle policy. For
linear, cosine decay schedules we start with a seed learning rate of $3e^{-4}$
as used in the standard baselines The training, validation perplexity and BLEU
scores for the various schedules are shown in Table 20 for the full budget
runs (70 epochs), and in Table 21 for the short budget runs (30 epochs).
Table 20: WMT’14 (EN-DE) on Transformer networks full budget training (70 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl
---|---|---|---
One-Cycle | 27.19 (0.081) | 3.96 (0.014) | 4.95 (0.013)
Cosine Decay | 27.35 (0.09) | 3.87 (0.011) | 4.91 (0.008)
Linear Decay | 27.29 (0.06) | 3.87 (0.017) | 4.89 (0.02)
Knee schedule | 27.53 (0.12) | 3.89 (0.017) | 4.87 (0.006)
Table 21: WMT’14 (EN-DE) on Transformer networks short budget training (30 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl
---|---|---|---
One-Cycle | 26.80 (0.2) | 4.38 (0.017) | 5.02 (0.007)
Cosine Decay | 26.95 (0.23) | 4.32 (0.013) | 4.99 (0.011)
Linear Decay | 26.77 (0.12) | 4.36 (0.092) | 5.02 (0.01)
Knee schedule | 27.28 (0.17) | 4.31 (0.02) | 4.92 (0.007)
### A.4 IWSLT’14 DE-EN
Figure 5(b) shows the learning rate range test for IWSLT on the transformer
networks. The minima occurs near $2.5e^{-3}$. For the maximum learning rate,
we choose $2.5e^{-4}$ for the default one-cycle policy. For linear, cosine
decay schedules we start with a seed learning rate of $3e^{-4}$ as used in the
standard baselines The training, validation perplexity and BLEU scores for the
various schedules are shown in Table 22 for the full budget runs (50 epochs),
and in Table 23 for the short budget runs (35 epochs).
Table 22: IWSLT’14 (DE-EN) on Transformer networks full budget training (50 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl
---|---|---|---
One-Cycle | 34.77 (0.064) | 3.68 (0.009) | 4.97 (0.010)
Cosine Decay | 35.21 (0.063) | 3.08 (0.004) | 4.88 (0.014)
Linear Decay | 34.97 (0.035) | 3.36 (0.001) | 4.92 (0.035)
Knee schedule | 35.53 (0.06) | 3.00 (0.044) | 4.86 (0.02)
Table 23: IWSLT’14 (DE-EN) on Transformer networks short budget training (35 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl
---|---|---|---
One-Cycle | 34.43 (0.26) | 3.98 (0.028) | 5.09 (0.017)
Cosine Decay | 34.46 (0.33) | 3.86 (0.131) | 5.06 (0.106)
Linear Decay | 34.16 (0.28) | 4.11 (0.092) | 5.14 (0.066)
Knee schedule | 35.08 (0.12) | 3.58 (0.063) | 4.90 (0.049)
### A.5 SQuAD-v1.1 finetuning with BERTBASE
We choose $1e^{-5}$ as the maximum learning rate for One-Cycle runs as the
minima occurs close to $1e^{-4}$ . For linear, cosine decays we start with a
seed learning rate of $3e^{-5}$ as used in standard baselines. Table 24 show
the average training loss, average test EM and F1 scores for the various
schedules. We did not do a short budget training for this dataset, as the full
budget is just 2 epochs.
Table 24: SQuAD-v1.1 fine-tuning on BERTBASE for more learning rate schedules. We report the average training loss, average test EM, F1 scores over 3 runs. LR Schedule | EM (av) | F1 (av) | Train Loss (av)
---|---|---|---
One Cycle | 79.9 (0.17) | 87.8 (0.091) | 1.062 (0.003)
Cosine Decay | 81.31 (0.07) | 88.61 (0.040) | 0.999 (0.003)
Linear decay | 80.89 (0.15) | 88.38 (0.042) | 1.0003 (0.004)
Knee schedule | 81.38 (0.02) | 88.66 (0.045) | 1.003 (0.002)
## B Detailed Plots
Figure 6: ImageNet on Resnet-50 trained with Momentum. Shown are the training
loss, top-1/top-5 test accuracy and learning rate as a function of epochs, for
the baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is
split into 3 parts to permit higher fidelity in the y-axis range.
Figure 7: Cifar-10 on Resnet-18 trained with Momentum. Shown are the training
loss, test accuracy and learning rate as a function of epochs, for the
baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is split
into 3 parts to permit higher fidelity in the y-axis range.
Figure 8: BERTLARGE pretraining for batch size of 16k with LAMB optimizer for
the short budget runs. Shown are the training loss and learning rate as a
function of steps, for the baseline scheme short budget (orange) vs the Knee
schedule scheme short budget (blue). The plot is split into 2 parts to give a
clear picture of the two phases of training Devlin et al. (2018). Note that
even though the training loss curves look similar for the two runs, we see a
significant gap in F1 score obtained when we fine-tune the model checkpoints
on SQuAD-v1.1 Rajpurkar et al. (2016). See Table 9 for details.
LR Schedule | F1 - Trial 1 | F1 - Trial 2 | F1 - Trial 3 | F1 avg. | F1 max
---|---|---|---|---|---
Baseline (short budget) | 90.39 | 90.64 | 90.53 | 90.52 | 90.64
Knee schedule ( short budget ) | 91.22 | 91.29 | 91.18 | 91.23 | 91.29
Knee schedule ( full budget ) | 91.45 | 91.41 | 91.51 | 91.46 | 91.51
Figure 9: SQuAD fine-tuning on BERTLARGE. We report F1 scores for 3 different
trials as well as the maximum and average values.
Figure 10: WMT’14 (EN-DE) on TransformerBASE network trained with RAdam. Shown
are the training perplexity, validation perplexity and learning rate as a
function of epochs, for the baseline scheme (orange) vs the Knee schedule
scheme (blue). The plot is split into 3 parts to permit higher fidelity in the
y-axis range.
Figure 11: IWSLT’14 (DE-EN) on TransformerBASE network trained with RAdam.
Shown are the training perplexity, validation perplexity and learning rate as
a function of epochs, for the baseline scheme (orange) vs the Knee schedule
scheme (blue). The plot is split into 3 parts to permit higher fidelity in the
y-axis range.
Figure 12: IWSLT’14 (DE-EN) on the SOTA model Cutoff(Shen et al., 2020),
trained with Adam. Shown are the training perplexity, validation perplexity
and learning rate as a function of epochs, for the baseline scheme (orange) vs
the Knee schedule scheme (blue).
Figure 13: IWSLT’14 (DE-EN) on the SOTA model Cutoff(Shen et al., 2020),
trained with Adam with a reduced training budget of 70 epochs. Shown are the
training perplexity, validation perplexity and learning rate as a function of
epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue).
Figure 14: SQuAD-v1.1 fine-tuning on BERTBASE trained with Adam. Shown are the
training loss, test EM score, and learning rate as a function of epochs, for
the baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is
split into 2 parts to permit higher fidelity in the y-axis range. It is clear
that with Knee schedule the network starts to overfit after the 2nd epoch,
where the testing loss continues to go down, but generalization suffers. We
saw similar behavior with different seeds, and thus need to train with Knee
schedule for only 2 epochs.
|
2024-09-04T02:54:58.282887 | 2020-03-09T09:26:23 | 2003.03988 | {
"authors": "Nasir Ahmad, Luca Ambrogioni, Marcel A. J. van Gerven",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26110",
"submitter": "Nasir Ahmad",
"url": "https://arxiv.org/abs/2003.03988"
} | arxiv-papers | Original Article Journal Section STDWI, Spike-Timing-Dependent Weight
Inference; LIF, Leaky Integrate and Fire; RDD, Regression Discontinuity
Design. Nasir Ahmad<EMAIL_ADDRESS>
# Overcoming the Weight Transport Problem via Spike-Timing-Dependent Weight
Inference
Nasir Ahmad Department of Artificial Intelligence, Donders Institute for
Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands
Luca Ambrogioni Department of Artificial Intelligence, Donders Institute for
Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands
Marcel van Gerven Department of Artificial Intelligence, Donders Institute
for Brain, Cognition and Behaviour, Radboud University, Nijmegen, the
Netherlands
###### Abstract
We propose a solution to the weight transport problem, which questions the
biological plausibility of the backpropagation algorithm. We derive our method
based upon a theoretical analysis of the (approximate) dynamics of leaky
integrate-and-fire neurons. Our results demonstrate that the use of spike
timing alone outcompetes existing biologically plausible methods for synaptic
weight inference in spiking neural network models. Furthermore, our proposed
method is more flexible, being applicable to any spiking neuron model, is
conservative in how many parameters are required for implementation and can be
deployed in an online-fashion with minimal computational overhead. These
features, together with its biological plausibility, make it an attractive
mechanism for weight inference at single synapses.
###### keywords:
Weight Transport Problem, Biologically Plausible Learning, Spiking Neural
Network
## 1 Introduction
Backpropagation of error is a successful approach for training rate-based
neural network models [1, 2]. However, since its inception it has been
criticised for its lack of biological plausibility [3, 4]. In particular, in
order to update individual synaptic connections weights within a network,
information is required about distant error signals and the weights of other
synaptic connections of the network – information which is not available
locally to the synapse. However, backpropagation’s flexibility, unrelenting
success in application-based research, and most significantly its capacity for
modelling and reproducing neural response statistics has contributed to a
recent re-examination of its potential role and plausibility in neural systems
[5, 6, 7, 8, 9].
A number of attempts have been made to explain mechanisms by which
backpropagation’s implausibilities can be addressed. These can be divided into
methods which propose alternative implementations of backpropagation, namely
energy-based and dynamical systems methods which converge to backpropagation
of error [10, 11, 12], for an overview see [6], and methods which show that
components which are considered implausible can be approximated using
alternative and plausible computations [13, 14, 15, 16, 17]. We focus on the
latter approaches in this study.
Figure 1: The weight transport problem in backpropagation of error. A. The
computations involved in the forward-pass of an example feedforward neural
network model. B. The backpropagation of error method. Specifically, the
derivative of the loss function can be computed with respect to each weight
matrix in our example network. Observe that the derivative of the loss
function with respect to a weight matrix ($W_{1}$) deep in the network depends
upon the weight matrices in the higher layers ($W_{2}$). C. Backpropagation of
error requires a copy of the weights of the forward network.
One particularly difficult-to-reconcile component of backpropagation is the
need to propagate error signals backwards through a network (see Fig. 1). This
requires that the backward propagating error signals between layers of neurons
is weighted according to the forward synaptic connection weights, leading to a
situation in which feedback weight matrices are copies of the feedforward
matrices. This duplication of weights has been identified as particularly
troubling in terms of a plausible biological implementation and is known as
the weight transport problem [3].
Early attempts to address the weight transport problem included proposals that
the feedback weights can converge to the values of the feedforward weights by
applying the same weight changes to both matrices during training (see [13]).
This explanation was criticised for simply shifting the problem from
transporting weights to transporting weight changes in a network. More
recently, feedback-alignment was proposed as a method to completely sidestep
the need for weight symmetry [14]. It was empirically shown that by having
fixed random feedback weight matrices between the layers of a network, the
feedforward weight matrices are modified by backpropagation such they they
come into alignment with the feedback matrices. This approach can also be
implemented with a randomly weighted direct feedback error to every layer
(direct feedback alignment, [18]), a method which has also been applied in
spiking neural networks [19]. Though such an error distribution process is
biologically plausible, the effectiveness of the approach is limited to
shallow networks and the accuracy of deep networks appears to suffer severely
under such a protocol [20]. Beyond static feedback matrices, matrices with
arbitrary magnitudes but alignment of the signs of weights (i.e. positive
feedforward weights are mirrored with positive feedback weights and vice
versa) show greatly improved performance over feedback alignment [21, 22].
However, propagating the sign of feedback weights is itself a transport
problem and performance with this method is less than optimal.
Recently, methods have been proposed by which the symmetric feedback weight
matrices could be learned by biologically plausible methods (using local only
information). Specifically, methods have emerged which carry out a process of
synaptic weight inference [15, 16]. In essence the backwards synaptic
connections (which would propagate the error) attempt to infer the feedforward
weight between two neurons by observation of their activity alone. This is a
process in which, based upon the activity patterns of a pair of neurons, a
feedback synapse can infer (and thereby copy) the strength of the feedforward
synapse. Such a method was successfully applied in a rate-based neural network
by Akrout et al. [15] (hereafter referred to as the Akrout method). This
method makes use of inference phases during which neurons are randomly
stimulated and their activation is correlated in order to infer synaptic
weights. Alternative rate-based methods are available though we do not
consider them given their non-locality [17]. A more recent proposal [16]
considers a spiking neural network model and makes use of the spiking
threshold of neurons to implement a quasi-experimental, causal inference
method known as regression discontinuity design (we hereafter refer to this
method as RDD, also see [23]). This method similarly uses inference phases in
between training epochs in order to infer the backward synaptic weight
matrices.
These inference methods have proven successful in inferring the feedforward
synaptic weights for use in the feedback weight matrices but also suffer from
a number of drawbacks. First, the Akrout method operates on firing rates and
requires a demeaning process which is carried out in batches. This demeaning
and batching process is particularly troublesome when applied to spiking
networks where the learning must therefore be carried out offline and firing
rates measured by aggregating spikes at specific intervals. In the RDD method,
weight inference requires a piece-wise linear fitting process in order to
infer the synaptic weights. This procedure requires the storage of four times
more parameters per synapse (than just the synaptic weight), a second state
variable per neuron and a high computational complexity per update. Though
these components and the calculation protocols might be possible for a neuron
to compute, they incur a significant computational cost.
To overcome these issues, we propose a spike-timing-dependent weight inference
(STDWI) mechanism for solving the weight transport problem in spiking neural
networks. Our method is motivated by analysis of the time-to-spike of various
neuron models under the influence of an incident spikes. In order to estimate
this in a biologically plausible and computationally efficient manner, we make
use of local information for this computation, in particular just the spike
times of the pre- and post-synaptic neurons. We show that under a number of
conditions our method outperforms both the Akrout and RDD methods when applied
to weight estimation in spiking neural network models. We also compare our
method to an optimal Bayesian update rule for an integrate-and-fire neuron
with stochastic input. Our rule proves effective as an approximation of this
update rule. Furthermore, for networks in which the neurons emit action
potentials at random times (i.e. without a correlation structure), our
learning rule can analytically be shown to approximate a rate-based learning
rule similar to the Akrout method. Finally, the update rule we propose is
computationally cheap and can be applied in an online fashion.
## 2 Methods
To address the weight transport problem, it has been proposed that network
weights can be inferred from activity [15, 16]. We can formulate this problem
as follows: Consider two neurons, labelled “A” and “B”, embedded in a larger
network structure. Amongst other connections, there exists a ‘forward’
synaptic connection from neuron A to neuron B. Therefore, the activity of
neuron B is dependent upon some internal dynamics as well as the network
activity as a whole, including the activity of neuron A, via incoming synaptic
connections. Let us also now consider a pseudo synaptic connection from B to
A, a connection meant to carry error information backward through the network
(note that this work and prior work do not describe this synapse as having an
impact upon the network activity during inference). According to the
backpropagation of error algorithm, the optimal value of this synaptic
connection weight should be equivalent to the weight of the forward synapse
from A to B. How the forward synaptic weight can be copied to the backward
synaptic connection is the problem at hand.
Here we address how to infer the forward synaptic weight value at the
backwards synapse given knowledge of the spike times (indexed $k$) of the
neurons A and B ($t_{A}^{k}$ and $t_{B}^{k}$ respectively) and by accounting
for some average impact from all other synapses. We derive a computationally
simple and biologically plausible method which, by use of appropriate
approximations, achieves this aim and could be employed at the synaptic level
to learn feedback weights for error propagation.
### 2.1 Derivation of the weight inference method
In order to derive our proposed weight inference rule, we analyse a simplified
deterministic leaky integrate-and-fire (LIF) neuron with instantaneous
synaptic inputs from a single source and drift (where drift is a placeholder
for the unknown impact of all other incident synapses) and then consider the
impact of noise upon this model.
A deterministic LIF neuron with drift $\mu$ has voltage dynamics
$\tau_{m}\frac{dv(t)}{dt}=v_{r}-v(t)+\mu\,.$ (1)
In the absence of any input spikes, this equation can be solved, for an
arbitrary initial condition $v_{0}$, at time $t_{0}$, yielding
$v(t)=(v_{r}+\mu)\left(1-e^{-(t-t_{0})/\tau_{m}}\right)+v_{0}e^{-(t-t_{0})/\tau_{m}}\,.$
(2)
With this expression we can now consider two cases, one in which the neuron is
not stimulated by any incoming spikes from neuron $j$ and, beginning at
voltage $v_{0}$ at time $t_{0}$, it spikes with some time delay $\hat{T}$
(purely under the influence of drift). The other case is one in which the
neuron received an additional instantaneous voltage injection of magnitude $w$
at time $t_{0}$ (i.e. a spike arrives and stimulates the neuron) and it spikes
with a different time delay, $T$ (such that the second case involves
replacement of $v_{0}$ with $v_{0}+w$). These cases can be subtracted at
threshold in order to give an expression for $w$, the stimulation magnitude,
of the form
$w={e^{T/\tau_{m}}}(v_{r}+\mu-
v_{0})\left(e^{-T/\tau_{m}}-e^{-\hat{T}/\tau_{m}}\right).$ (3)
Equation (3) provides an exact solution for determining the amount of
instantaneous voltage ($w$) injected to a neuron at some time $t_{0}$ given
that its spike time was modified from an expected time $\hat{T}$ to the time
$T$. This is under the assumption that other than the instantaneous voltage
injection and a background drift, there are no other inputs to the neuron
during this time.
We wish to make use of this deterministic solution for application to noisy
conditions. In particular, when the background drift is considered as due to
input from many other neurons it would inherently be noisy (unlike our
derivation above). However, the current expression includes a number of terms
which are highly susceptible to noise. First, the exponential term,
$e^{T/\tau_{m}}$ is a strictly positive function which is exponential in the
time that the neuron took to spike. If we consider a case in which $T$ is
noisy, this term scales our noise exponentially but never changes sign.
Second, the expected time to spike, $\hat{T}$ is difficult to estimate in a
noisy neuron. However, this term is crucial for our ability to accurately
identify positive and negative weights and it must, therefore, be
approximated.
First we consider the exponential term $e^{T/\tau_{m}}$. Though this term
might (in the noiseless case) aid in producing a highly accurate weight
estimation, in the face of noise it introduces a significant error.
Furthermore, in the noiseless case (where a single estimate of the weight is
exact), its biggest function is to scale the estimated weight based upon the
time taken to spike. This, in essence, reweighs cases in which the neuron
dynamics take excess time to reach threshold – due to beginning far from
threshold (low $v_{0}$), having a small drift, and/or when the incident spike
arrives from a synapse with a small/negative weight. This is therefore a
mechanism to ensure that, for cases in which our system setup results in
dynamics which take some time to reach threshold, the weight scale is treated
sensibly. However, in the coming derivation we intend to sample over multiple
instances of such an estimation in a noisy system such that there is an
unreliable signal of ‘time to spike’. And given that this term is heavily
influenced by noise we wish to ignore it. Therefore, given its function, our
intention to sample, and its susceptibility to noise, we test in this work the
removal of this term from our weight estimation and instead propose weight
estimation without this scaling. We empirically find this approach successful.
Thus, our approach to (approximate) weight estimation can be described as
$\tilde{w}=C(v_{r}+\mu-
v_{0})\left(e^{-T/\tau_{m}}-e^{-\hat{T}/\tau_{m}}\right)$ (4)
where $\tilde{w}$ is an approximate estimation of the weight (ignoring a
rescaling based upon time to spike) and we have introduced a general constant
$C$ to allow linear rescaling of weight estimates.
Next, we wish to approximate $\hat{T}$ in the face of noisy samples. For this
purpose, let us average our estimate of the weight over $K$ observations. In
particular, let us consider a set of samples $T^{k}$, indexed by $k$, each of
which correspond to the time to spike given that the ‘output’ neuron started
from some initial voltage $v_{0}^{k}$ at the moment of an incident spike. For
each of these samples, there exists an “expected” time from incident spike to
neuron spike, $\hat{T}^{k}$, which corresponds to when the neuron would have
spiked if not for this incident spike. Taking an average of the weight
estimate over these $K$ samples yields an estimated weight
$\tilde{w}^{K}=\frac{C}{K}\sum_{k=0}^{K}\left(v_{r}+\mu-
v_{0}^{k}\right)\left(e^{-T^{k}/\tau_{m}}-e^{-\hat{T}^{k}/\tau_{m}}\right)$
(5)
with $K$ indicating the number of observations/samples taken. If we assume
that our $K$ samples are chosen independently of the incident activity (i.e.
the incident spikes are random), then the values of the initial voltage,
$v_{0}^{k}$, and expected times to spike, $\hat{T}^{k}$, are both independent
of the sampling process (and of $w^{k}$ and $T^{k}$). Therefore, these can be
independently averaged and, hence, replaced with $\langle v_{0}\rangle$ and
$\langle\hat{T}\rangle$. Thus, we arrive at an expression
$\tilde{w}^{K}=\frac{D}{K}\sum_{k=0}^{K}\left(e^{-T^{k}/\tau_{m}}-e^{-\langle\hat{T}\rangle/\tau_{m}}\right)\,,$
(6)
where $D=C(v_{r}+\mu-\langle v_{0}\rangle)$ combines the various constants and
scales our estimate of the weights.
If we now finally consider how we ought to update our estimate of $w$ when we
receive an additional $(K+1)$-th sample, we arrive at
$\Delta\tilde{w}=\tilde{w}^{K+1}-\tilde{w}^{K}=\frac{1}{K+1}\left(D\left(e^{-T^{K+1}/\tau_{m}}-e^{-\langle\hat{T}\rangle/\tau_{m}}\right)-w^{K}\right).$
(7)
Inspecting our derived update rule, the first exponential term in Eq. (7) is
exponential in the time since an incident spike arrived. Given this, it is
equivalent to sampling a trace which continuously measures the (fast
exponential) instantaneous firing rate of the neuron from which the incident
spike is arriving. The second exponential term is exponential in the average
time since incident spikes ‘should’ arrive if the weight had been zero,
$\langle\hat{T}\rangle$, an measure of the incident spike-rate. This term can
be approximated as a sampling of a slow exponential measure of the average
rate of the neuron from which incident spikes arrive. Finally, the constant
term $D=C(v_{r}+\mu-\langle v_{0}\rangle)$, has a factor of the drift term
$\mu$. In our model assumption, this drift is background input aside from the
synapse under inference and affects the baseline time to spike of our output
unit. This drift therefore scales up with the output neuron’s average firing
rate. With these observations, we can make appropriate replacements in order
to describe a local spike-timing-dependent weight inference rule.
### 2.2 Spike-timing-dependent weight inference
We propose a spike-timing-dependent rule for the purpose of weight inference
(STDWI) which can be deployed for parallel and online updates with minimal
computational complexity. Our method maintains multiple online estimates of
neuron firing rates through eligibility traces [24, 25] and makes use of these
for synaptic weight estimation. In particular, each neuron (indexed $j$)
maintains a fast trace $\epsilon_{j}^{f}(t)$ and a slow trace
$\epsilon_{j}^{s}(t)$. The dynamics of the fast and slow traces are calculated
for each neuron as
$\tau_{f}\frac{d\epsilon_{j}^{f}(t)}{dt}=-\epsilon_{j}^{f}(t)+S_{j}(t)\quad\text{and}\quad\tau_{s}\frac{d\epsilon_{j}^{s}(t)}{dt}=-\epsilon_{j}^{s}(t)+\frac{\tau_{f}}{\tau_{s}}S_{j}(t)\,,$
(8)
where $\tau_{f}$ and $\tau_{s}$ are the decay constants of the fast and slow
traces respectively, and $S_{j}(t)$ is the spike train of the $j$th neuron.
These traces are computed during simulation in a time-stepping method with the
analytic (exponential) solution to these traces computed at every timestep.
This spike train is computed from the set of $k$ spike times of the $j$th
neuron, $t^{k}_{j}$, such that $S_{j}(t)=\sum_{k}\delta(t-t^{k}_{j})$, where
$\delta(\cdot)$ is the Dirac delta function. Note that these two traces have
an equal area (across time) when they both start with an initial value of zero
due to the normalization of the slow trace spike-train, scaling factor
$\tau_{f}/\tau_{s}$. This property ensures that both eligibility traces act to
measure the firing rate of the neurons with the same scale. Having defined
these eligibility traces, we define our weight inference rule as
$\frac{dw_{ji}}{dt}=\alpha
S_{i}(t)\left(\epsilon_{i}^{s}(t)\left(\epsilon_{j}^{f}(t)-\epsilon_{j}^{s}(t)\right)-\eta
w_{ji}\right)\,,$ (9)
where this rule describes inference of the weight of the forward synapse at
the backward synapse (from neuron $i$ to neuron $j$), $w_{ji}$, with $\alpha$
as the learning rate and $\eta$ as the relative level of weight decay (both
constant hyper-parameters). This learning rule and the fast and slow measures
of the neuron’s firing rates are inspired by the synaptic inference rule
derived in Section 2.1. Note that though this rule is given as a differential
equation, since updates are gated by neuron $i$’s spike times, it is
implemented as updating the synaptic weights such that $w_{ji}\leftarrow
w_{ji}+dw_{ji}/dt$ at every timepoint where neuron $i$ spikes.
Figure 2: A) Illustration of the difference between our derived method for
weight inference by analysis of a deterministic LIF neuron (left) versus our
proposed STDWI method (right) which uses a fast trace to measure the
instantaneous firing rate (the first exponential term in Eq. (7)) and a slow
trace to measure the average firing rate (second exponential term in Eq. (7)).
B) Assuming regular neuron firing conditions, our method can be interpreted as
an STDP rule of the form shown inset, where $T$ is the post minus pre-synaptic
neuron spike time. Note pre and post-synaptic are termed relative to the
backward synaptic connection.
The formulation for the weight update given in Eq. (7) and our proposed STDWI
rule given in Eq. (9) have corresponding terms, see Figure 2A. Both of these
formulations include updates which occur only upon our pre-synaptic neuron
spikes. Note we use the terms post-synaptic/pre-synaptic relative to the
backward synaptic connection for application to the weight transport problem.
In our approximation, we replace the first exponential term of Eq. (7) (an
exponential measure of the time since the post-synaptic neuron’s last spike)
with a fast timescale measure of the post-synaptic neuron’s firing rate (the
fast trace) and we use a slow timescale measure of the post-synaptic neuron’s
firing rate (the slow trace) to approximate the second exponential term (which
computes a trace tracking the average post-synaptic neuron’s firing rate).
Finally, we include a slow measure of the pre-synaptic neuron’s firing rate as
a multiplicative factor, which is intended to capture the dependence of the
weight estimate upon the pre-synaptic neuron drift. Figure 2A depicts how
updates are calculated upon pre-synaptic neuron spikes for both the
deterministic LIF and STDWI update, highlighting both the similarities and key
differences between these implementations.
Note that the learning rule being proposed here relates in a curious form to
traditional Spike-Timing Dependent Plasticity (STDP) rules. In particular, the
sign of the weight update is determined by the spike-timings and firing-rates
of the pre and post-synaptic units. In general, if we assume some fixed
regular firing rate of the post-synaptic neuron, then depending upon the
spike-time of the pre-synaptic neuron relative to this regular firing, we
encounter positive or negative weight estimation. This rule therefore appears
in a mirrored-form to the commonly cited STDP observations [26], see Figure
2B.
### 2.3 Spiking neuron model
For simulations in this study, we consider neurons with membrane leakage and
conductance-based synaptic kernels whose membrane voltage dynamics can be
described by
$\tau_{m}\frac{dv_{i}(t)}{dt}=(v_{r}-v_{i}(t))+\frac{g_{D}}{g_{L}}\bigg{(}\sum_{j}w_{ij}\kappa_{j}(t)-v_{i}(t)\bigg{)}\,,$
(10)
where $\tau_{m}$ is the leakage time constant, $v_{r}$ is the rest voltage,
$g_{D}$ and $g_{L}$ are the dendritic and somatic leakage conductances,
respectively, $w_{ij}$ is the weighting of the forward synaptic connection
from the $j$th neuron to the $i$th neuron and $\kappa_{j}$ describes a
filtered form of the $j$th neuron’s spike train. The form of the synaptic
filtering kernel is taken as a double exponential with a fast rise and slow
decay, such that
$\kappa_{j}(t)=\frac{1}{\tau_{2}-\tau_{1}}\sum_{k}H(t-t^{k}_{j})\left(e^{-\frac{t-t^{k}_{j}}{\tau_{2}}}-e^{-\frac{t-t^{k}_{j}}{\tau_{1}}}\right)\,,$
(11)
where $\tau_{1}$ and $\tau_{2}$ are the timescales of the fast rise and slow
decay, taken to be 3ms and 10ms respectively, and $H(\cdot)$ is the Heaviside
step function.
When the membrane voltage, $v_{i}(t)$, reaches a threshold, $\theta$, an
action potential is recorded and propagated. The membrane voltage is
thereafter reset to a reset voltage $v_{\text{reset}}$. For the simulations in
this study, we do not implement a refractory period explicitly. This is not
expected to cause much deviation of the analysis in our low firing-rate
regime.
### 2.4 Comparison against alternative weight inference methods
In Section 3.2, we compare our method (STDWI) to alternative methods (RDD and
Akrout methods) proposed for the local inference of weights in a network. The
inference is carried out in networks composed of a group of spiking input
neurons connected via a single forward weight matrix to a group of spiking
output neuron. The spiking neuron dynamics are equivalent to those used in the
work which introduced RDD as a causal weight inference method [16], see
Section 2.3. The network is stimulated by selectively exciting input neurons,
collecting the responses of input and output neurons, and applying the set of
techniques to these neural data.
During simulation, some percent of the input neurons are randomly sampled
every 100ms and these are excited with background random Poisson distributed
spike trains (with a fixed positive synaptic connection weight from
stimulation nodes to the input neurons). Every 100ms the input neurons being
stimulated are re-sampled. During this stimulation, non-selected neurons are
left unstimulated with zero input. The STDWI and RDD methods are applied in a
continuous form, paying no attention to the 100ms stimulation periods. The
Akrout method was proposed for rate-based neural networks and makes use of a
batch-wise de-meaning of firing rates. Therefore, the 100ms stimulation
periods are considered as individual ’stimuli’ for the Akrout method, and the
firing rates computed for each of these ‘stimuli’. These individual stimuli
are then grouped into batches (batch-size chosen by grid search) and used to
update the inferred weight according to the Akrout method.
The spiking dynamics with stimulation, as described above, were simulated for
2500s. During weight inference, these 2500s of dynamics were then looped ten
times in order to produce a total 25,000s training time. This looping was
necessary due to memory and storage constraints and can be interpreted as ten
epochs of training.
All methods were trained with a learning rate of $5\times 10^{-5}$. This
learning rate was chosen by iteratively reducing the learning rate until
stable weight inference was clear for all methods. Conclusions are (and should
be) drawn based upon asymptotic performance and not speed given that this
hyperparameter was not tuned on a method-by-method basis. Free parameters were
optimized for by measurement of sign-accuracy and Pearson correlation (best
average performance) using a grid search carried out with a single seed of the
network simulation. Selected parameters were then applied for inference to
five other network seeds, and results collected. See Appendix B for the grid
search results.
## 3 Results
To validate our approach, we compare it against a Bayes-optimal method for a
simple neuron model that affords an analytical solution. Furthermore, we
compare it to two state-of-the-art synaptic weight inference methods for
estimation of the connectivity of simulated spiking neural networks (see
models described in Section 2.3). Code to reproduce results is available at
https://github.com/nasiryahm/STDWI.
### 3.1 Comparison of STDWI to a Bayesian optimal method
To verify the validity of our proposed STDWI rule and demonstrate its
flexibility, we compare it against a Bayes-optimal method for inferring
synaptic inputs to a neuron with internal state modelled by a Wiener process
(Figure 3). Unlike a stochastic LIF neuron model, this model has a tractable
hitting-time analysis and thereby we can form a Bayesian update rule for
estimating the size of a synaptic input given a subsequent output neuron spike
time. A detailed derivation of the Bayes-optimal method is provided in
Appendix A.
Figure 3: Weight inference accuracy of Bayesian and STDWI approaches applied
to a pure Wiener process with jumps. Panels A and B show scatter plots of the
true and inferred weights for the Bayesian and STDWI approach, respectively,
at the end of the training time ($t=50s$). Panels C and D show how Pearson
correlation and sign alignment between the true and inferred weights evolve
through the training process. The standard deviation of the measures across 10
random network seeds are shown as envelopes about the curves.
The Bayesian update rule occurs upon every pre-synaptic neuron spike and is
based upon knowledge of when the last post-synaptic spike occurred (rather
than knowledge of all past post-synaptic spikes), it would be an improper
comparison to test the optimal Bayesian method against our full STDWI rule
(which makes use of all previous spikes in its eligibility traces). Therefore,
to ensure a fair comparison we modify our STDWI rule (Eq. 9) to use only
single spikes. To do this we replaced the slow eligibility traces,
$\epsilon_{j}^{s}(t)$, with a constant (optimally set as the average of the
fast traces), and replaced the fast trace, $\epsilon_{j}^{f}(t)$, with a term
which is exponential in the time since the last spike alone (rather than a
decaying trace of all past post-synaptic spikes). This modification is
equivalent to Eq. 7 if we treat the second exponential terms as a constant and
use an arbitrary learning rate.
We repeatedly simulated stochastic neurons, each with a single forward
synaptic input connection but with varying synaptic connection strengths
across simulations. We simulated the systems for 50s and thereafter used the
network activity in this time period for synaptic weight inference. We
repeated this analysis for synaptic weight strengths over a wide range to
attempt inference of many different synaptic strengths. Figure 3 shows various
measures of the similarity between the true and inferred jump widths for this
simulation when using either the Bayesian or our derived method for weight
inference. Both the scatter plots and learning curves show that the STDWI
method closely matches the Bayes-optimal results, supporting the theoretical
soundness of our approach.
### 3.2 Comparison of STDWI to alternative weight inference methods
Figure 4: Weight inference accuracy comparison between the RDD, Akrout, and
STDWI approaches for a network of LIF neurons with conductance-based synapses.
Panels A, B and C show scatter plots of the true and inferred weights for each
method at the end of training for a single network. Panels D and E show
Pearson correlation and sign alignment between the inferred and true weights.
Solid lines show the mean of these measures across ten randomly seeded
networks and the shaded areas show the standard deviation across these
networks. Panel F shows the convergence of the inferred weights for each
method. The inferred weights for the 75% largest inferred weight (by
magnitude) were collected, individually normalized to their final value and
their average plot with standard deviation shown, as before, by the shaded
area.
We also compared our proposed STDWI approach to two existing methods for
synaptic weight inference. In particular, we compare against the RDD and
Akrout methods. Details of both methods are provided in Appendix B.
To simulate a neural network model which is amenable to all of these weight
inference methods, we use the same neural network models and setup as that
described in [16]. This network is composed of LIF neurons with kernel-
filtered, conductance-based synaptic inputs. We simulate two-layer network
models with an input layer of 100 LIF neurons fully connected to an output
layer of 10 LIF neurons. The synaptic weight matrix connecting these is drawn
from a normal distribution with a small but positive mean. It is this weight
matrix which must be inferred by the range of methods.
The network is stimulated by selectively exciting input neurons.
Some percentage of the input neurons are randomly sampled every 100ms and
these are excited with background Poisson distributed input spike trains (with
a fixed positive synaptic connection weight from stimulation nodes to the
neurons). Every 100ms the input neurons being stimulated are re-sampled.
During this stimulation process, non-selected neurons are left unstimulated
with zero input.
Figure 4 shows the result of weight inference with the range of methods
discussed above for networks in which 20% of input neurons are simultaneously
stimulated (sparse random stimulation). Scatter plots of the inferred vs true
weights (see Panels 4A-C) show the strength of the STDWI method, which
produces a tighter distribution of weights than competitors. Note that the
scale of the synaptic weights inferred differs from the true weights for all
methods, relating to the approximate nature of the methods. In practice, a
rescaling could be applied to improve the correspondence to the scale of the
true weights though none of our measures were sensitive to inferred weight
scale and therefore this was disregarded.
Panels 4D and E show the evolution of the Pearson correlation and sign
alignment between the true and inferred weights through training for the range
of methods. As can be seen, our proposed STDWI method outperforms both the RDD
and Akrout methods, though the difference in the Pearson correlation of all
three methods is small. Note that RDD outperforms Akrout in terms of sign
accuracy.
Finally, Panel 4F shows the successful convergence of inferred weights for all
three methods. This plot shows the normalized weights (normalised through a
division by the final converged weight value) of the top 75% largest magnitude
network weights. These weights had a relatively unambiguous sign, hence their
selection. This plot is provided to support the argument that the
hyperparameter selections made were sufficient for stable inference by these
methods.
### 3.3 The impact of stimulation protocol on weight inference
It is also instructive to investigate how different stimulation protocols
affect weight inference. To this end, and in contrast to the sparse
stimulation in the previous section, we assume that all input neurons are
stimulated (dense stimulation). Furthermore, we investigate how input timing
correlations affect weight inference. Since input neurons are stimulated by
random Poisson spike trains, we can create correlation between individual
Poisson spike trains by a thinning process (via a Single Process Interaction
Model, see [27]). Figure 5 shows results for this dense stimulation regime.
Figure 5: Weight inference accuracy comparison between the RDD, Akrout, and
STDWI approaches for a network of LIF neurons with conductance-based synapses
when all input neurons are stimulated. Panels A, B and C show scatter plots of
the true and inferred weights for each method at the end of training for a
single network. Panels D and E show Pearson correlation and sign alignment
between the inferred and true weights in the uncorrelated spiking case during
training. Panels F and G show the final results (post-training) under varying
input spike-time correlation. Solid lines (points) show the mean of these
measures across five randomly seeded networks and the shaded areas (error
bars) show the standard deviation across these networks.
Scatter plots of the true vs inferred weights (see Panels 5A-C) again show
that STDWI produces a tighter distribution of weights than its competitors.
This highlights the smaller impact of stimulation density upon the STDWI
inference method compared with the Akrout or RDD methods. These scatter plots
show inferred weights for the dense stimulation case with zero correlation in
timing between the various input stimulation spike-trains.
Panels 5D and E show that the STDWI method remains most successful (as
measured by the Pearson correlation and sign alignment mean) when compared
with RDD and Akrout methods under dense stimulation. However, the Akrout
method benefits significantly from dense stimulation (whereas the RDD method
appears to suffer somewhat). Thus, the RDD method does not systematically
outperform the Akrout method as previously reported (cf. Panels 4D and 5E).
Panels 5F and G demonstrate how weight inference is affected by input timing
correlations. STDWI remains largely successful, however as input spike timing
correlation increases, the RDD method performs favourably. This may be
expected as unlike the STDWI and Akrout methods, the RDD method compares only
events which are near-threshold to establish synaptic weights. This filtering
of events by which inference is done may be favourable in the regime of high
input spike timing correlation, though the benefit only exists for some
parameter range.
## 4 Discussion
Our results demonstrate the efficacy of STDWI for synaptic weight inference
across a range of network models and stimulation protocols. We have shown that
our approach successfully approximates Bayes-optimal results in a simple
neuron model and outperforms existing methods for weight inference. Our
results also highlight the attention that must be paid to the employed
stimulation protocols since the efficacy of different synaptic weight
inference method has been shown to crucially depend on these.
Existing methods cannot be so indiscriminately applied to arbitrary neuron
models. For example, the RDD method requires a neuron model which has a second
state variable mimicking the membrane voltage. This state variable should
relax to the same value as the membrane voltage when the neuron is not spiking
and otherwise should reflect how “driven” the neuron is when it spikes.
However, such a state variable is not necessarily constructable for an
arbitrary neuron model. In contrast, STDWI makes use of spike timing alone and
is therefore agnostic to the neuron dynamics being simulated.
In our analyses, we reported both on Pearson correlation and sign accuracy.
STDWI systematically outperformed the alternative approaches on both measures
for a range of parameters. One exception is in the investigation of timing-
based correlations (Figure 5F and G), in which RDD outperformed the other
methods for inference in the case of the medium correlation regime. This
suggests a particular regime might favour the RDD method, however its failure
in other regimes suggests that the current formulation of RDD for analysis
about a spike-threshold may not be most effective.
It is also important to realize that the number of variables stored per
synaptic connection is greater for RDD compared to either the STDWI or Akrout
methods. RDD requires a fitting process using data-points corresponding to
events within which the neuron’s membrane voltage was near the spiking
threshold. Aside from the selection of events close to threshold, the RDD
method also uses four variables per synaptic connection to characterise a
piece-wise linear function (with linear functions fit above and below the
spiking threshold). By comparison, STDWI uses two variables for the fast and
slow eligibility traces of each neuron and the Akrout method uses two
variables storing the firing rate and mean firing rate of each unit within a
batch.
To derive our learning rule, we made use of a deterministic analysis of a LIF
neuron and considered the spike times of a single input neuron. Our
deterministic analyses later required approximations in order to remove terms
which are highly affected by noise. Ideally we would instead have carried out
a stochastic process analysis for a LIF neuron. The particular stochastic
process to which our leaky neuron model corresponds is known as an Ornstein-
Uhlenbeck (OU) process. Unfortunately a general analysis of the OU process
that describes when we ought to expect such a neuron to spike (the hitting
time) is non-trivial [28]. Nonetheless, the efficacy of our assumptions is
validated by the quality of our results. Furthermore, under a rate-based
analysis of our proposed STDWI rule, we can show a correspondence to the
Akrout rule (see Appendix C).
A limitation of our approach is that the inference process considers the spike
times of a single input neuron. Instead, a multivariate approach which would
take into account the spike-times of all input neurons to infer synaptic
weights could prove even more powerful and accurate for weight inference.
Indeed multivariate analyses, often making use of such multi-neuron spiking
information along with cross-correlation measures and statistical significance
testing, have been applied previously in approaches which aim to infer neural
circuit connectivity from neural data [29, 30, 31, 32]. These approaches,
however, make use of globally available network information and are not
concerned with whether this information is locally available at the synapse.
Instead, we took a simplified but powerful approach which could plausibly be
implemented at the single synapse level, providing a candidate solution to the
weight transport problem.
Concluding, we have shown that STDWI outperforms existing approaches for
solving the weight transport problem. Moreover, it is more flexible, being
capable of application to any spiking network data, while requiring minimal
computational overhead. The benefits of data efficiency and online computation
along with its computational simplicity and accuracy make STDWI a promising
biologically plausible mechanism for gradient-based learning in spiking neural
networks.
## acknowledgements
We thank Blake Richards and Jordan Guerguiev for their correspondence and for
providing us with the code they used for the RDD method.
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## Appendix A Bayesian weight estimation for a stochastic neuron model
As a method of verification of our proposed STDWI rule and an exhibition of
its flexibility, we compare it against an optimal Bayesian method for
inferring a single synaptic input to a neuron with internal state modelled by
Brownian motion with drift and diffusion (a Wiener process). Unlike a
stochastic leaky integrate and fire neuron model, this model has a tractable
hitting-time analysis and thereby we can form an optimal Bayesian update rule
for estimating the size of a synaptic input given a subsequent output neuron
spike time. This synaptic weight inference analysis for this simple neuron
model and its similarity to our STDWI rule is described in the following
section.
### A.1 Bayesian estimation of synaptic weights
We wish to estimate the weight of synaptic connection given local-only
information. In particular, this involves estimating the weight of a synaptic
connection given the spike times of an input and output neuron (input and
output relative to the forward synaptic connection) as well as the output
neuron’s membrane voltages.
Constraining this further, let us estimate a synaptic connection weight, $w$,
between two neurons given a single input spike time, $t_{\text{in}}$, and the
first output spike time which follows this input spike, $t_{\text{out}}$ where
$t_{\text{out}}>t_{\text{in}}$. If we carry out all analysis relative to the
input spike time, $t_{\text{in}}$, we can define the key dependent factors.
First, the output neuron’s time to spike (the hitting time), following the
input neuron spike, is a key measure which we define as
$T=t_{\text{out}}-t_{\text{in}}$. The initial state of the output neuron is
also a determining factor in this analysis as it defines the distance to
threshold $\Delta$, which we elaborate on below. Given this setup and by
Bayes’ rule, we aim to compute
$p(w\mid T,\Delta)\propto p(T\mid w,\Delta)p(w)\,.$ (12)
The likelihood term $p(T\mid w,\Delta)$ can be computed through analysis of
the neural dynamics. To compute it, we must account for the impact of spikes
from all other input neurons. In general this is non-trivial. To simplify this
analysis, we consider the case of a non-leaky integrate-and-fire neuron driven
by random input.
### A.2 Stochastic neuron model
We consider a spiking neural network of neurons with membrane voltage under
the effect of Brownian motion. As such, changes in the membrane voltage,
$v(t)$, can be described by
$\frac{dv(t)}{dt}=I(t)\,,$ (13)
where $I(t)$ is the total input to the cell at time $t$. Notably, this change
in membrane voltage is agnostic to the current voltage $v(t)$ (meaning there
is no leakage effect). When this membrane voltage meets the threshold,
$\theta$, an action potential is emitted and the membrane voltage is directly
reset to the reset voltage $v_{\text{reset}}$.
Let us consider the input, $I(t)$, as composed of input from the single
synaptic connection and some background stochastic process. The synaptic
connection is modelled as a producing instantaneous voltage injections which
occur upon the spike times of the input neurons. The amplitudes of the
instantaneous voltage injections induced by input spikes are equal to the
weight of the synaptic connection from input to output neuron, $w$.
Aside from these synaptic inputs, we also consider some background input which
is a stochastic process. Assuming that there are a large number of randomly
spiking input neurons, we can approximate their impact as a random Gaussian
input with some mean and variance. This describes a stochastic process, known
as a Wiener process, with some drift (mean input) and a diffusion constant
(variance). This approximation for a neuron’s membrane voltage is valid in the
limit of a large number of synaptic connections with small synaptic weight
magnitudes.
The above details are all approximations but provide us with a simple
description of the neural dynamics such that
$dv(t)=w\delta(t-t_{\text{in}})dt+\sqrt{D}dX_{i}(t)\,,$ (14)
where $\delta(\cdot)$ is the Dirac-delta function and $X(t)$ is a Wiener
process with drift $\mu$ and variance scaled by $D$.
### A.3 The hitting time of a non-leaky neuron
We can now attempt to determine the “hitting time” of this system, i. e., the
time $T$ at which it makes contact with our neuron membrane voltage threshold.
The hitting-time density for a Wiener process with drift (by which we are
approximating our non-leaky neuron) can be calculated as;
$f(T\mid\Delta)=\frac{\Delta}{\sqrt{2D\pi T^{3}}}\exp\left(-\frac{(\Delta-\mu
T)^{2}}{2DT}\right)\,,$ (15)
where $\Delta=\theta-v_{0}$ is the membrane voltage distance to threshold (
where $v_{0}=v(t_{\text{in}})$), $T=t_{\text{out}}-t_{\text{in}}$ is defined
as above, $\mu$ is the drift of our Wiener process, and $D$ is the variance of
our Wiener process. In our neuron model, $\Delta$ corresponds to the
difference between some initial membrane voltage $v_{0}$ and the threshold
$\theta$, whereas $\mu$ corresponds to the average input to the output neuron
from all input synapses in volts.
The description assumes that the membrane voltage starts at some value $v_{0}$
and is under constant drift. However, instead we wish to assume that at the
initial time, $t_{0}=t_{\text{in}}$, our input neuron fired and added some
unknown voltage $w$ to the membrane voltage. Furthermore, rather than
computing a probability distribution over the possible times at which the
output neuron might spike, we instead know the next spike time of the output
neuron, $t_{\textrm{out}}$, and wish to use it to infer the weight, $w$.
We can therefore assume that for a given pair of input and output spikes, we
have a fixed hitting time, $T$, as described above. Furthermore, under our
synapse description for the non-leaky neuron (where synaptic inputs cause an
instantaneous change in output neuron membrane voltage of size proportional to
the synaptic weight) our initial membrane voltage, $v_{0}$, can be represented
as the membrane voltage just prior to the input spike, plus the synaptic
weight. That is, we take the limit of $v_{0}$ from below, i.e.,
$v_{0}=\lim_{t\to t_{0}^{-}}v(t)+w.$ This allows us to augment our first-
passage density in terms of $w$ such that
$f(T\mid w,\Delta)=\frac{(\Delta-w)}{\sqrt{2D\pi
T^{3}}}\exp\bigg{(}-\frac{(\Delta-w-\mu T)^{2}}{2DT}\bigg{)}\,,$ (16)
where we now define $\Delta=\theta-\lim_{t\to t_{0}^{-}}\>v(t)$. With this
formulation of the hitting-time density, we can compute an estimate of the
weight $w$ given a particular set of input and output neuron spike times.
Thereafter we can update our estimate of the synaptic weight of interest
through Eq. (12).
To make our inference of $w$ tractable, we first take a Laplace approximation
of Eq. (16). This produces a Gaussian with mean weight
$\hat{w}=\Delta-\frac{\mu T+\sqrt{(\mu T)^{2}+4DT}}{2}\,,$ (17)
calculated as the maximum of our likelihood $f(T\mid w,\Delta)$, and a
variance
$\hat{\sigma}=1/((\Delta-\hat{w})^{-2}+(DT)^{-1})\,.$ (18)
Since we have Gaussian distributions for our likelihood, we can take a
Gaussian conjugate prior with mean $\mu_{0}$ and variance $\sigma^{2}_{0}$ and
obtain a closed-form solution to our posterior weight when given a single
input-output spike pair as
${w}_{p}=\frac{1}{\sigma_{0}^{-2}+\hat{\sigma}^{-2}}\bigg{(}\frac{w_{0}}{\sigma_{0}^{2}}+\frac{\hat{w}}{\hat{\sigma}^{2}}\bigg{)}\,.$
(19)
Similarly, we can compute the posterior variance as
${\sigma}_{p}^{2}=\bigg{(}\sigma_{0}^{-2}+\sigma^{-2}\bigg{)}^{-1}\,.$ (20)
### A.4 Weight estimation under negligible drift
Let us assume that the diffusion term, $D$, is sufficiently small compared to
the drift $\mu$ (such that $\mu\gg D$). This allows us to ignore the diffusion
term in the numerator of Eq. (17). Having assumed this small diffusion scale,
we can then describe the maximum likelihood estimate of the weight as
$\hat{w}\approx\Delta-\mu T\,.$ (21)
Furthermore, recall that $\Delta$ is the distance to threshold when the input
neuron spikes, $\Delta=\theta-v(t_{\textrm{in}})$. By dividing this distance,
$\Delta$, by the drift, $\mu$, we can calculate the expected time of the
output spike under drift alone, $\hat{T}$, such that
$\frac{\Delta}{\mu}=\hat{T}\implies\Delta=\mu\hat{T}\,.$ (22)
Given these assumptions, we can approximate Eq. (17) as
$\hat{w}\approx\mu\hat{T}-\mu T=\mu(\hat{T}-T).$ (23)
This formulation can be understood well if we consider a non-leaky neuron
under the effect of drift alone (without any stochastic input) and a single
input neuron providing instantaneous voltage injections. In such a case, with
knowledge of the initial membrane voltage and drift of the output neuron, we
have a deterministic system which will spike at a specific time, $\hat{T}$. If
we perturb this system with a spike from an input neuron (which causes a jump
in the membrane voltage), we can decode the synaptic weight by simply
measuring the effect on the timing of the output neuron spike time. The
induced change in the output spike time is linearly proportional to the
synaptic weight.
## Appendix B Details on baseline methods
The STDWI method is compared to existing methods for synaptic weight
inference. We provide more details on these methods below.
### B.1 The Akrout method
In our simulations of LIF neurons, we compare against the Akrout method [15].
This rate-based method makes use of an inference phase in which neurons are
stimulated (with mean zero) and then the levels of activity of input and
output neurons are correlated to form a weight estimate. This approach was
shown to be highly successful for weight inference and thereby training of
rate-based neural network models. However, since we simulate spiking neurons,
which cannot have a negative firing rate, we instead demean the neuron firing
ratesand randomly stimulate the input neurons (post-synaptic from the
perspective of the backward synapse). In particular, we use an update rule of
the form
$\Delta w_{ji}=\eta(r_{i}-\langle r_{i}\rangle)(r_{j}-\langle
r_{j}\rangle)-\eta\lambda w_{ji}\,,$ (24)
where $\Delta w_{ji}$ is the update to backward synaptic weight, from a neuron
indexed $j$ to a neuron indexed $i$, which is attempting to estimate the
weight of the forward synaptic connection, $w_{ij}$. $r_{i}$ and $r_{j}$
denote the firing rates of the $i$th and $j$th neurons, and
$\langle\cdot\rangle$ indicates an average of these over time. Parameters
$\eta$ and $\lambda$ are the learning rate and the weight decay respectively.
The learning rate is fixed at with value $\eta=0.0001$ and the weight decay
determined by grid search, see below. The firing rates $r_{i}$ and $r_{j}$ are
calculated by computing the firing rates within the non-overlapping 100ms
stimulation periods of the network. These stimulation periods are then grouped
into batches (of size again determined by grid search) for calculation of the
mean firing rates for this batch ($\langle r_{j}\rangle$ and $\langle
r_{i}\rangle$ respectively) according to the weight-mirror gradient descent
method described in [15].
### B.2 Regression discontinuity design
We also compare against the regression discontinuity design (RDD) method,
which was proposed for application in spiking neural networks [16]. It makes
use of all times at which a neuron spiked or almost spiked (i.e. its membrane
voltage came within some margin of the spiking threshold but never reached
it). It thereafter separately fits the almost-spiked and spiked events
linearly against the membrane voltage. Notably, for the spiking events, a non-
reset version of the membrane voltage is used for the linear fitting.
Following a fitting process, the discontinuity of these linear fits at the
spiking threshold is used as a measure of the synaptic weight. For full
details of the RDD implementation, see [16].
### B.3 Grid-based optimization of free parameters
The methods compared have a number of free parameters that can be optimized
for. In case of STDWI these are the time constants of the fast ($\tau_{f}$)
and slow ($\tau_{s})$ traces. In case of RDD these are the distance to
threshold at which samples are initiated and the window duration of a sample.
For the Akrout method, the weight decay scaling and the batch-size are
hyperparameters.
These parameters are chosen from a grid-search using a single test network’s
spike trains. The parameters producing highest average sign accuracy and
Pearson correlation between the inferred and true weights are then chosen for
analysis of a further four networks (each with a different random seed for
input stimulation and the synaptic weight matrix).
Figure 6: Variation in the performance of the STDWI, RDD, and Akrout methods
with changes in the method parameters. The best parameter sets are highlighted
with a black box. These were the parameter used to analyse all other seeded
networks and produce the main results.
Figure 6 shows the parameters maps for the grid searches carried out to select
parameters for Figure 4. The same grid search parameter sweeps were repeated
in order to choose parameters for Figure 5.
## Appendix C Rate-based analysis of the STDWI rule
To appreciate the effect of STDWI rule, we can consider its approximate rate-
based form under the assumption of random Poisson process sampled neuron
spikes (for a review of the rationale of such methods see [33]). This produces
an update rule based upon the firing rates of the neurons. Note that below, as
in Section 2.2, we refer to pre/post-synaptic relative to a ‘backward’
synapse.
In our case, the dependence upon the post-synaptic firing rate has two forms
which correspond to a quickly-adapting exponential average,
${\lambda}_{\textrm{j}}^{\textrm{f}}$, and a slowly-adapting exponential
average, ${\lambda}_{\textrm{j}}^{\text{s}}$. Similarly there is a dependence
upon the pre-synaptic firing rate as a slowly-adapting exponential average,
${\lambda}_{\textrm{i}}^{\text{s}}$. Taking the assumption of Poisson random
spiking, we can describe our weight update in a rate-based form as
$\frac{d\hat{w}_{ji}}{dt}=\alpha
S_{i}(t)\left(\lambda_{\textrm{i}}^{\textrm{s}}({\lambda}_{\textrm{j}}^{\textrm{f}}-{\lambda}_{\textrm{j}}^{\textrm{s}})-\eta\hat{w}_{ji}\right).$
(25)
We can solve this equation for its fixed point ($\frac{d\hat{w_{ji}}}{dt}=0$),
producing an expression for the fixed-point weight as
$\hat{w}_{ji}^{*}=\frac{1}{\eta}{\lambda}_{\textrm{i}}^{\textrm{s}}\left({\lambda}_{\textrm{j}}^{\textrm{f}}-{\lambda}_{\textrm{j}}^{\textrm{s}}\right)$
(26)
when $S_{i}(t)$ is non-zero.
For networks with solely positive firing rates, Akrout et al. [15] proposed
correlating the demeaned firing rates of pre and post-synaptic neurons in
order to estimate synaptic weights. If we here interpret the slow firing rate
measure of the input neuron activity as an approximation of its average value,
then our method similarly correlates pre-synaptic firing rate with the
demeaned post-synaptic neuron firing rate. Though this rate-based analysis
shows similarities to the Akrout method, our spike timing implementation is
unique in that it makes use of asymmetric causal kernels and has a demeaning
process which slowly tracks the firing rates of neurons (rather than making
use of batches). We attribute our performance gains to these features.
Furthermore, given the spike-timing-dependent feature of the rule, weight
updates can be computed in an event-driven fashion and with minimal
communication between neurons (weight updates requiring communication only
upon spike times).
If we compare Eqs. (26) and (23) then we can also appreciate the
correspondence of the STDWI rule and the Bayesian estimate. The STDWI update,
instead of making use of an estimate of the a drift, $\mu$, makes use of the
pre-synaptic neuron firing rate as a proxy. This is appropriate given the
linear relationship between drift and firing rate for a non-leaky neuron.
Furthermore, rather than directly comparing the expected and true time to
spike, $\hat{T}$ and $T$ respectively, the STDWI rule keeps track of a slow
and fast estimate of the post-synaptic neuron firing rate, through
$\lambda_{\textrm{j}}^{\text{s}}$ and $\lambda_{\textrm{j}}^{\text{f}}$
respectively. The subtraction of these firing rate estimates in Eq. (26)
provides a measure with a similar form to the subtraction of expected and true
spike times ($\hat{T}-T$). Specifically, an earlier than average spike time
induces a positive weight estimate and a later than average spike time induces
a negative weight estimate.
|
2024-09-04T02:54:58.299400 | 2020-03-09T09:34:20 | 2003.03994 | {
"authors": "Divyam Aggarwal, Dhish Kumar Saxena, Thomas B\\\"ack, Michael Emmerich",
"full_text_license": null,
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26111",
"submitter": "Divyam Aggarwal",
"url": "https://arxiv.org/abs/2003.03994"
} | arxiv-papers | WGMwgm QEqe EPep PMSpms BECbec DEde
[orcid=0000-0003-0740-780X]
[orcid=0000-0001-7809-7744] [1]
[orcid=0000-0001-6768-1478]
[orcid=0000-0002-7342-2090]
[cor1]Corresponding author; Email Address<EMAIL_ADDRESS>Postal
Address: Room No.-231, East Block, MIED, IIT Roorkee, Roorkee,
Uttarakhand-247667, India; Phone: +91-8218612326
# Airline Crew Pairing Optimization Framework for Large Networks with Multiple
Crew Bases and Hub-and-Spoke Subnetworks
Divyam Aggarwal<EMAIL_ADDRESS>Department of Mechanical & Industrial
Engineering (MIED), Indian Institute of Technology Roorkee, Roorkee,
Uttarakhand-247667, India Dhish Kumar Saxena<EMAIL_ADDRESS>Thomas Bäck<EMAIL_ADDRESS>Leiden Institute of Advanced
Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333 CA Leiden,
the Netherlands Michael Emmerich<EMAIL_ADDRESS>
###### Abstract
Crew Pairing Optimization aims at generating a set of flight sequences (crew
pairings), covering all flights in an airlines’ flight schedule, at minimum
cost, while satisfying several legality constraints. CPO is critically
important for airlines’ business viability considering that the crew operating
cost is second only to the fuel cost. It poses an NP-hard combinatorial
optimization problem, to tackle which, the state-of-the-art relies on relaxing
the underlying Integer Programming Problem (IPP) into a Linear Programming
Problem (LPP), solving the latter through Column Generation (CG) technique,
and integerization of the resulting LPP solution. However, with the growing
scale and complexity of the airlines’ networks (those with large number of
flights, multiple crew bases and/or multiple hub-and-spoke subnetworks), the
efficacy of the conventionally used exact CG-implementations is severely
marred, and their utility has become questionable. This paper proposes an
Airline Crew Pairing Optimization Framework, $AirCROP$, whose constitutive
modules include the Legal Crew Pairing Generator, Initial Feasible Solution
Generator, and an Optimization Engine built on heuristic-based CG-
implementation. $AirCROP$’s novelty lies in not just the design of its
constitutive modules but also in how these modules interact. In that, insights
in to several important questions which the literature is otherwise silent on,
have been shared. These relate to sensitivity analysis of $AirCROP^{\prime}s$
performance in terms of final solutions’ cost quality and run-time, with
respect to - sources of variability over multiple runs for a given problem;
cost quality of the initial solution and the run-time spent to obtain it; and
termination parameters for LPP-solutioning and IPP-solutioning. In addition,
the efficacy of the $AirCROP$ has been: (a) demonstrated on real-world airline
flight networks with an unprecedented conjunct scale-and-complexity, marked by
over 4200 flights, 15 crew bases, and billion-plus pairings, and (b) validated
by the research consortium’s industrial sponsor. It is hoped that with the
emergent trend of conjunct scale and complexity of airline networks, this
paper shall serve as an important milestone for affiliated research and
applications.
###### keywords:
Airline Crew Scheduling Crew Pairing Combinatorial Optimization Column
Generation Mathematical Programming Heuristics
## 1 Introduction
Airline scheduling poses some of the most challenging optimization problems
encountered in the entire Operations Research (OR) domain. For a large-scale
airline, the crew operating cost constitutes the second-largest cost
component, next to the fuel cost, and even its marginal improvements may
translate to annual savings worth millions of dollars. Given the potential for
huge cost-savings, Airline Crew Scheduling is recognized as a critical
planning activity. It has received an unprecedented attention from the
researchers of the OR community over the last three decades. Conventionally,
it is tackled by solving two problems, namely, Crew Pairing Optimization
Problem (CPOP) and Crew Assignment Problem, in a sequential manner. The former
problem is aimed at generating a set of flight sequences (each called a crew
pairing) that covers all flights from an airlines’ flight schedule, at minimum
cost, while satisfying several legality constraints linked to federations’
rules, labor laws, airline-specific regulations, etc. These optimally-derived
crew pairings are then fed as input to the latter problem, which is aimed to
generate a set of pairing sequences (each sequence is a schedule for an
individual crew member), while satisfying the corresponding crew requirements.
Being the foremost step of the airline crew scheduling, CPOP is the main focus
of this paper, and interested readers are referred to Barnhart . (2003) for a
comprehensive review of the airline crew scheduling.
CPOP is an NP-hard111For NP-hard (NP-complete) problems, no polynomial time
algorithms on sequential computers are known up to now. However, verification
of a solution might be (can be) accomplished efficiently, i.e., in polynomial
time. combinatorial optimization problem (Garey Johnson, 1979). It is modeled
as either a set partitioning problem (SPP) in which each flight is allowed to
be covered by only one pairing, or a set covering problem (SCP) in which each
flight is allowed to be covered by more than one pairing. In CPOP, a crew
pairing has to satisfy hundreds of legality constraints (Section 2.2) to be
classified as legal, and it is imperative to generate legal pairings in a
time-efficient manner to assist optimization search. Several legal pairing
generation approaches, based on either a flight- or a duty-network, have been
proposed in the literature (Aggarwal ., 2018). Depending upon how the legal
pairing generation module is invoked, two CPOP solution-architectures are
possible. In the first architecture, all possible legal pairings are
enumerated a priori the CPOP-solutioning. However, this is computationally-
tractable only for small-scale CPOPs (with $\approx$¡1000 flights).
Alternatively, legal pairings are generated during each iteration of the CPOP-
solutioning, but only for a subset of flights, so the CPOP solution could be
partially improved before triggering the next iteration. Such an architecture
mostly suits medium- to large-scale CPOPs (with $\approx\geq$1000 flights)
involving millions/billions of legal pairings, whose complete-enumeration is
computationally-intractable.
In terms of solution-methodologies, heuristic-based optimization techniques
and mathematical programming techniques, are commonly employed (Section 2.3).
In the former category, Genetic Algorithms (GAs) which are population-based
randomized-search heuristics (Goldberg, 2006) are most commonly used. However,
they are found to be efficient only for tackling very small-scale CPOPs
(Ozdemir Mohan, 2001). Alternatively, several mathematical programming based
approaches do exist to solve CPOPs of varying-scales. CPOP is inherently an
Integer Programming Problem (IPP), and some approaches have used standard
Integer Programming (IP) techniques to find a best-cost pairing subset from a
pre-enumerated pairings’ set (Hoffman Padberg, 1993). However, these
approaches have proven effective only with small-scale CPOPs with up to a
million pairings. This perhaps explains the prevalence of an altogether
different strategy, in which the original CPOP/IPP is relaxed into a Linear
Programming Problem (LPP); the LPP is solved iteratively by invoking a LP
solver and relying on Column Generation (CG) technique to generate new
pairings as part of the pricing sub-problem; and finally, the resulting LPP
solution is integerized using IP techniques and/or some special connection-
fixing heuristics. The challenge associated with this strategy is that even
though the LPP solver may lead to a near-optimal LPP solution, the scope of
finding a good-cost IPP solution is limited to the pairings available in the
LPP solution. To counter this challenge, heuristic implementations of branch-
and-price framework (Barnhart . (1998)) in which CG is utilized during the
integerization phase too, have been employed to generate new legal pairings at
nodes of the IP-search tree. However, the efficacy of such heuristic
implementations depends on a significant number of algorithmic-design choices
(say, which branching scheme to adopt, or how many CG-iterations to perform at
the nodes). Furthermore, it is noteworthy that the scale and complexity of
flight networks have grown alarmingly over the past decades. As a result, an
inestimably large number of new pairings are possible under the pricing sub-
problem, given which most existing solution methodologies are rendered
computationally-inefficient. Recognition of such challenges have paved the way
towards domain-knowledge driven CG strategies to generate a manageable, yet
crucial part of the overall pairings’ space under the pricing sub-problem
(Zeren Özkol, 2016). Though rich in promise, the efficacy of this approach is
yet to be explored vis-$\grave{a}$-vis the emergent large-scale and complex
flight networks characterized by multiple crew bases and/or multiple hub-and-
spoke subnetworks where billions of legal pairings are possible.
In an endeavor to address airline networks with conjunct scale and complexity,
this paper proposes an Airline Crew Pairing Optimization Framework ($AirCROP$)
based on domain-knowledge driven CG strategies, and:
* •
presents not just the design of its constitutive modules (including Legal Crew
Pairing Generator, Initial Feasible Solution Generator, and Optimization
Engine powered by CG-driven LPP-solutioning and IPP-solutioning), but also how
these modules interact
* •
discusses how sensitive its performance is to - sources of variability over
multiple runs for a given problem; cost quality of the initial solution and
the run-time spent to obtain it; and termination parameters for LPP-
solutioning and IPP-solutioning. Such an investigation promises important
insights for researchers and practitioners on critical issues which are
otherwise not discussed in the existing literature.
* •
presents empirical results for real-world, large-scale (over 4200 flights),
complex flight network (over 15 crew bases and multiple hub-and-spoke
subnetworks) for a US-based airline, the data for which has been provided by
the research consortium’s industrial partner.
The outline of the remaining paper is as follows. Section 2 discusses the
underlying concepts, related work, and problem formulation; Section 3 entails
the details of the proposed $AirCROP$; Section 4 presents the results of the
computational experiments along with the corresponding observations; and
Section 5 concludes the paper as well as briefly describes the potential
future directions.
## 2 Crew Pairing Optimization: Preliminaries, Related Work and Problem
Formulation
This section first describes the preliminaries, including the associated
terminology, pairings’ legality constraints, and pairings’ costing criterion.
Subsequently, the related work is presented in which the existing CPOP
solution approaches are discussed. Lastly, the airline CPOP formulation is
presented.
### 2.1 Associated Terminology
In airline crew operations, each crew member is assigned a fixed (home)
airport, called a crew base. A crew pairing (or a pairing) is a flight
sequence operated by a crew, that begins and ends at the same crew base, and
satisfies the given pairing legality constraints (detailed in Section 2.2). An
example of a crew pairing with the Dallas (DAL) airport as the crew base is
illustrated in Figure 1. In a crew pairing, the legal sequence of flights
operated by a crew in a single working day (not necessarily equivalent to a
calendar day) is called a crew duty or a duty. A sit-time or a connection-time
is a small rest-period, provided between any two consecutive flights within a
duty for facilitating operational requirements such as aircraft changes by the
crew, turn-around operation for the aircraft, etc. An overnight-rest is a
longer rest-period, provided between any two consecutive duties within a
pairing. Moreover, two short-periods, provided in the beginning and ending of
any duty within a pairing, are called briefing and de-briefing time,
respectively. The total time elapsed in a crew pairing, i.e., the time for
which a crew is away from its crew base is called the time away from base
(TAFB). Sometimes, it is required for a crew to be transported at an airport
to fly their next flight. For this, the crew travels as passenger in another
flight, flown by another crew, to arrive at the required airport. Such a
flight is called a deadhead flight or a deadhead for the crew traveling as
passenger. It is desired by an airline to minimize the number of deadheads
(ideally zero), as it affects the airline’s profit in two-folds. Firstly, the
airline suffers a loss of the revenue on the passenger seat being occupied by
the deadhead-ing crew, and secondly, the airline has to pay the hourly wages
to the deadhead-ing crew even when it is not operating the flight.
Figure 1: An example of a crew pairing starting from Dallas (DAL) crew base
### 2.2 Crew Pairing: Legality Constraints and Costing Criterion
To govern the safety of crew members, airline federations such as European
Aviation Safety Agency, Federal Aviation Administration, and others, have laid
down several rules and regulations, which in addition to the airline-specific
regulations, labor laws, etc. are required to be satisfied by a pairing to be
“legal”. These legality constraints could be broadly categorized as follows:
* •
Connection-city constraint ($\mathcal{C}_{connect}$): this constraint requires
the arrival airport of a flight (or the last flight of a duty) within a
pairing to be same as the departure airport of its next flight (or the first
flight of its next duty).
* •
Sit-time ($C_{sit}$) and Overnight-rest ($\mathcal{C}_{night}$) constraints:
these constraints imposes the respective maximum and minimum limits on the
duration of sit-times and overnight-rests, where these limits are governed by
airlines and federations’ regulations.
* •
Duty constraints ($\mathcal{C}_{duty}$): these constraints govern the
regulations linked to the crew duties. For instance, they impose maximum
limits on the– number of flights allowed in a duty of a pairing; duty elapsed-
time and the corresponding flying-time; number of duties allowed in a pairing,
etc.
* •
Start- and end-city constraint ($\mathcal{C}_{base}$): this constraint
requires the beginning airport (departure airport of the first flight) and
ending airport (arrival airport of the last flight) of a pairing, to be the
same crew base.
* •
Other constraints ($\mathcal{C}_{other}$): Airlines formulate some specific
constraints, according to their operational requirements, so as to maximize
their crew utilization. For example, a pairing is refrained from involving
overnight-rests at the airports that belong to the same city as the crew base
from which the pairing started, etc.
Considering the multiplicity of the above constraints, it is critical to
develop a time-efficient legal crew pairing generation approach, enabling
their prompt availability, when their requirement arises during the
optimization.
In general, a pairing’s cost could be split into the flying cost and non-
flying (variable) cost. The flying cost is the cost incurred in actually
flying all the given flights, and is computed on hourly-basis. The variable
cost is the cost incurred during the non-flying hours of the pairing, and is
made up of two sub-components, namely, hard cost and soft cost. The hard cost
involves the pairing’s hotel cost, meal cost, and excess pay– the cost
associated with the difference between the guaranteed hours of pay and the
actual flying hours. Here, the pairing’s hotel cost is the lodging cost
incurred during its overnight-rests, and its meal cost is computed as a
fraction of its TAFB. The soft cost is the undesirable cost associated with
the number of aircraft changes (during flight-connections) in the pairing,
etc.
### 2.3 Related Work
As mentioned in Section 1, the existing CPOP solution approaches are based on
either heuristic or mathematical programming techniques. Among the heuristic-
based approaches, GA is the most widely adopted technique, and Beasley Chu
(1996) is the first instance to customize a GA (using guided GA-operators) for
solving a general class of SCPs. In that, the authors validated their proposed
approach on small-scale synthetic test cases (with over 1,000 rows and just
10,000 columns). The important details of the GA-based CPOP solution
approaches, available in the literature, are reported in Table 1.
Table 1: Key facts around the GA-based CPOP solution approaches, available in
the literature
Literature Studies | Modeling | Timetable | Airline Test Cases* | Airlines
---|---|---|---|---
Levine (1996) | Set Partitioning | - | 40R; 823F; 43,749P | -
Ozdemir Mohan (2001) | Set Covering | Daily | 28R; 380F; 21,308P | Multiple Airlines
Kornilakis Stamatopoulos (2002) | Set Covering | Monthly | 1R; 2,100F; 11,981P | Olympic Airways
Zeren Özkol (2012) | Set Covering | Monthly | 1R; 710F; 3,308P | Turkish Airlines
Deveci Demirel (20181) | Set Covering | - | 12R; 714F; 43,091P | Turkish Airlines
R represents the number of real-world test cases considered; F and P
represents the maximum number of flights and pairings covered, therein.
Notably, the utility of the studies reported in the table, have been
demonstrated on CPOPs with reasonably small number of flights, leading to
relatively smaller number of pairings. Though, CPOPs with 2,100 and 710
flights have been tackled by Kornilakis Stamatopoulos (2002) and Zeren Özkol
(2012) respectively, only a subset of all possible legal pairings has been
considered by them for finding the reported solutions. Zeren Özkol (2012)
proposed a GA with highly-customized operators, which efficiently solved
small-scale CPOPs but failed to solve large-scale CPOPs with the same search-
efficiency. Furthermore, Aggarwal, Saxena . (20202) tackled a small-scale CPOP
(with 839 flights and multiple hub-and-spoke sub-networks) using a customized-
GA (with guided operators) as well as mathematical programming techniques. The
authors concluded that customized-GAs are inefficient in solving complex
versions of even small-scale flight networks, compared to a mathematical
programming-based solution approach.
Several mathematical programming-based CPOP solution approaches have been
proposed in the literature over past few decades, and based on the size and
characteristics of the flight network being tackled, these approaches have
been categorized into either of the three general classes. In the first class
of approaches, all legal pairings or a subset of good pairings are enumerated
prior to the CPOP-solutioning, and the corresponding CPOP/IPP model is solved
using standard IP techniques (such as branch-and-bound algorithm (Land Doig,
1960)). Gershkoff (1989) proposed an iterative solution approach, which is
initialized using a set of artificial pairings (each covering a single flight
at a high pseudo-cost). In that, each iteration involves selection of very few
pairings (5 to 10); enumeration of all legal pairings using the flights
covered in the selected pairings; optimization of the resulting SPP to find
the optimal pairings; and lastly, replacement of the originally selected
pairings with the optimal pairings, only if the latter offers a better cost.
The search-efficiency of such an approach is highly dependent on the sub-
problem-size (handled up to 100 flights and 5,000 pairings), as the length and
breadth of the branching tree increases drastically with an increase in sub-
problem-size. Hoffman Padberg (1993) proposed an alternative approach to
tackle SPPs with up to 825 flights and 1.05 million pairings in which all
possible pairings are enumerated a priori, and the resulting SPP is solved to
optimality using a branch-and-cut algorithm222The branch-and-cut algorithm was
first proposed by Padberg Rinaldi (1991) to solve Mixed Integer Programs
(MIP), by integrating the standard branch-and-bound and cutting-plane
algorithms. For comprehensive details of the MIP solvers, interested readers
are referred to Lodi (2009); Linderoth Lodi (2011); Achterberg Wunderling
(2013).. Such approaches are efficient only in tackling small-scale CPOPs,
that too with up to a million pairings. However, even small-scale CPOPs may
involve large number of pairings (an instance reported in Vance . (1997) had
250 flights and over five million pairings), rendering it computationally-
intractable to use such approaches.
The second class of approaches relies on relaxing the integer constraints in
the original CPOP/IPP to form an LPP, which is then solved iteratively by–
invoking an LP solver and generating new pairings using CG; and integerizing
the resulting LPP solution. In any iteration of the LPP-solutioning (referred
to as an LPP iteration), an LP solver (based on either a simplex method or an
interior-point method333The class of interior-point methods was first
introduced by Karmarkar (1984). In that, a polynomial-time algorithm, called
Karmarkar’s algorithm, was proposed, which, in contrary to simplex method,
searches for the best solution by traversing the interior of the feasible
region of the search space.) is invoked on the input pairing set to find the
LPP solution and its corresponding dual information (shadow price
corresponding to each flight-coverage constraint), which are then utilized to
generate new pairings as part of the pricing sub-problem, promising the
corresponding cost-improvements. For the first LPP iteration, any set of
pairings covering all the flights becomes the input to the LP solver, and for
any subsequent LPP iteration, the current LPP solution and the set of new
pairings (from the pricing sub-problem) constitute the new input. For more
details on how new pairings are generated under the pricing sub-problem in the
CG technique, interested readers are referred to Vance . (1997); Lübbecke
Desrosiers (2005). As cited in Zeren Özkol (2016), the CG technique has
several limitations, out of which the prominent ones are– heading-in effect
(poor dual information in initial LPP iteration leads to generation of
irrelevant columns), bang-bang effect (dual variables oscillate from one
extreme point to another, leading to poor or slower convergence), and tailing-
off effect (the cost-improvements in the later LPP iterations taper-off).
While, different stabilization techniques are available for CG in the
literature Du Merle . (1999); Lübbecke (2010), the use of interior point
methods is gaining prominence. Anbil . (1991) presented the advancements at
the American Airlines, and enhanced the approach proposed by Gershkoff (1989)
(discussed above), by leveraging the knowledge of dual variables to screen-
out/price-out the pairings from the enumerated set at each iteration, enabling
it to solve larger sub-problems (up to 25 flights and 100,000 pairings). As an
outcome of a collaboration between IBM and American Airlines, Anbil . (1992)
proposed an iterative global solution approach (though falling short of global
optimization) in which an exhaustive set of pairings ($\approx$5.5 million) is
enumerated a priori. Several thousands of these pairings are used to
initialize the iterative procedure, and in each of its iterations, a sub-
problem is solved to obtain optimal dual variables, which are then used to
price-out all 5.5 million pairings to select a sufficiently-sized set of good
pairings ($\approx$5,000 pairings). For integerization of the LPP solution,
the literature points to two prominent strategies. The first strategy is based
on utilizing either a branch-and-bound, or a branch-and-cut algorithm. The
other strategy utilizes some special “connection-fixing” heuristics either
solely for integerization (Anbil ., 1992; Marsten, 1994), or during the
iterations of the LPP-solutioning (Zeren Özkol, 2016) to boost the performance
of the subsequent MIP solver (in some cases, may even get integer solution
without using the MIP solver). These heuristics eliminate some irrelevant
pairings by exploiting the knowledge of their linear variables and fixing some
specific flight-connections. The limitation in this class of approaches is
that even though, a good IPP solution to the original CPOP may exist and the
LPP-solutioning leads to a near-optimal LPP solution, the pairings available
in it may not fit well together to constitute a good-cost IPP solution.
The third class of approaches share a similar solution-architecture as of the
preceding class, however, differs in terms of the integerization of the LPP
solution. In that, a heuristic branch-and-price framework444The branch-and-
price algorithm was originally proposed by Barnhart . (1998) as an exact
algorithm to solve then-known large-scale IPPs, and has been utilized to solve
a variety of combinatorial optimization problems in transportation such as
Desrosiers . (1984); Desrochers Soumis (1989); Desrochers . (1992). is
adopted, wherein, CG is utilized during the integerization phase too, to
generate new legal pairings at nodes of the MIP-search tree. Desrosiers .
(1991) is the first instance that solved CPOP using a branch-and-price
framework. However, given the inestimable number of legal pairings possible
for even medium-scale CPOPs, numerous branch-and-price based heuristic-
approaches have been proposed over the last three decades (Desaulniers .,
1997; Vance ., 1997; Anbil ., 1998; Desaulniers Soumis, 2010). Notably, the
development of these approaches, being heuristic in nature, require a
significant number of algorithmic-design choices to be taken empirically,
which may vary with the characteristics of the flight networks being solved
for. To name a few such decisions, which branching scheme should be employed
(branching on linear variables, branching on flight-connections, or others),
should CG be performed on each node of the MIP-search tree, how many CG
iterations to be performed each time, etc. Furthermore, the commercial LP and
MIP solvers are not much open to modifications, making it difficult for the
new researchers to implement a computationally- and time-efficient branch-and-
price framework from scratch. For further details of the existing CPOP
solution approaches, interested readers are referred to recent survey
articles– Kasirzadeh . (2017); Deveci Demirel (20182).
In addition to the above classification of solution approaches, the literature
differs on the notion of how the pricing sub-problem is modeled and solved to
generate new legal pairings during the LPP iterations. However, the focus of
this paper is not on the solution to the pricing sub-problem step, but on the
interactions between different modules of a CG-based CPOP solution approach.
Hence, for details on the existing work related to the pricing sub-problem
step, interested readers are referred to Vance . (1997); Aggarwal, Saxena .
(20201).
### 2.4 Integer Programming Problem Formulation
As mentioned earlier, a CPOP is intrinsically an IPP, modeled either as a SCP
or a SPP. Notably, the SCP formulation provides higher flexibility during its
solutioning compared to the SPP formulation by accommodating deadhead flights
in the model, possibly resulting in faster convergence (Gustafsson, 1999). For
a given flight set $\mathcal{F}$ (including $F$ flights) that could be covered
in numerous ways by a set of legal pairings $\mathcal{P}$ (including $P$
pairings), the set covering problem is aimed to find a subset of pairings
($\in\mathcal{P}$), say $\mathcal{P}_{IP}^{*}$, which not only covers each
flight ($\in\mathcal{F}$) at least once, but does it at a cost lower than any
alternative subset of pairings in $\mathcal{P}$. In that, while finding
$\mathcal{P}_{IP}^{*}$ ($\subseteq\mathcal{P}$), each pairing
$p_{j}\in\mathcal{P}$ corresponds to a binary variable $x_{j}$, which
represents whether the pairing $p_{j}$ is included in $\mathcal{P}_{IP}^{*}$
(marked by $x_{j}=1$) or not ($x_{j}=0$). Here, $p_{j}$ is a $F$-dimensional
vector, whose each element, say $a_{ij}$, represents whether the flight
$f_{i}$ is covered by pairing $p_{j}$ (marked by $a_{ij}=1$) or not
($a_{ij}=0$). In this background, the IPP formulation, as used in this paper,
is as follows.
$\displaystyle\text{Minimize}~{}Z_{IP}=\sum_{j=1}^{P}c_{j}x_{j}+\psi_{D}\cdot\left(\sum_{i=1}^{F}\left(\sum_{j=1}^{P}a_{ij}x_{j}-1\right)\right),$
(1) $\displaystyle\text{subject to}\quad\sum_{j=1}^{P}a_{ij}x_{j}\geq
1,\quad~{}~{}~{}~{}~{}\forall i\in\\{1,2,...,F\\}$ (2)
$\displaystyle\qquad\qquad\quad
x_{j}\in\mathbb{Z}=\\{0,1\\},~{}~{}~{}~{}\forall j\in\\{1,2,...,P\\}$ (3)
$\displaystyle\text{where},c_{j}$ $\displaystyle:~{}\text{the cost of a legal
pairing }p_{j},$ $\displaystyle\psi_{D}$ $\displaystyle:~{}\text{an airline-
defined penalty cost against each deadhead in the solution},$
$\displaystyle\quad a_{ij}$ $\displaystyle=~{}1,~{}\text{if
flight}~{}f_{i}~{}\text{is covered in pairing}~{}p_{j};\text{ else }0$
$\displaystyle x_{j}$ $\displaystyle=~{}1,~{}\text{if
pairing}~{}p_{j}~{}\text{contributes to Minimum}~{}Z;\text{ else }0$
In the objective function (Equation 1), the first component gives the sum of
the individual costs of the pairings selected in the solution, while the other
component gives the penalty cost for the deadheads incurred in the solution
(note, $(\sum_{j=1}^{P}a_{ij}x_{j}-1)$ gives the number of deadheads,
corresponding to the flight $f_{i}$). Notably, in the above formulation, it is
assumed that the set of all possible legal pairings, namely, $\mathcal{P}$,
are available a priori, and the task is to determine $\mathcal{P}_{IP}^{*}$.
However, the generation of $\mathcal{P}$ a priori is computationally-
intractable for large-scale CPOPs, as mentioned in Section 2.3. Hence, the
solution to the CPOP/IPP is pursued in conjunction with the corresponding LPP
(formulation deferred till Section 3.3.1) assisted by the CG technique.
## 3 Proposed Airline Crew Pairing Optimization Framework (AirCROP)
This section presents the constitutive modules of the proposed optimization
framework - $AirCROP$, their working, and their interactions. As per the
schematic in Figure 2, $AirCROP$ accepts a set of given flights $\mathcal{F}$
along with the pairings’ legality constraints and costing criterion as input,
and outputs a minimal-cost set of legal pairings $\mathcal{P}_{IP}^{\star}$,
that covers all given flights. This transition from the input to output is
enabled by the constitutive modules, namely, the Legal Crew Pairing Generator,
the Initial Feasible Solution Generator, and an Optimization Engine in turn
enabled by CG-driven LPP-solutioning and IPP-solutioning submodules and their
intermittent interactions. While parts of these modules have been presented
elsewhere (Aggarwal ., 2018; Aggarwal, Saxena ., 20201) in isolation, these
are being detailed below towards a holistic view on the experimental results
presented later.
Figure 2: A schematic of $AirCROP$ illustrating the interactions between its
constitutive modules– Legal Crew Pairing Generator, Initial Feasible Solution
Generator, Optimization Engine (CG-driven LPP-solutioning interacting with
IPP-solutioning). The CG heuristic in LPP-solutioning generates a set of fresh
pairings $\mathcal{P}_{CG}^{t}$ at any LPP iteration $t$ using the following
CG strategies: Deadhead reduction ($CGD$, generating $\mathcal{P}_{CGD}^{t}$),
Crew Utilization enhancement ($CGU$, generating $\mathcal{P}_{CGU}^{t}$),
Archiving ($CGA$, generating $\mathcal{P}_{CGA}^{t}$), and Random exploration
($CGR$, generating $\mathcal{P}_{CGR}^{t}$). The interactions between LPP-
solutioning and IPP-solutioning are tracked by the counter $T$.
### 3.1 Legal Crew Pairing Generator
This module enables generation of the legal pairings in a time-efficient
manner, so they could feed real-time into the other modules - Initial Feasible
Solution Generator and the optimization engine. For time-efficiency, it
employs a parallel, duty-network based legal pairing generation approach,
whose distinctive contributions are two-folds. Firstly, a crew base centric
parallel architecture is adopted considering that several duty- and pairing-
constitutive constraints do vary with crew bases. In that, for an input flight
set, the legal pairing generation process is decomposed into independent sub-
processes (one for each crew base), running in parallel on idle-cores of the
central processing unit (CPU). This leads to a significant reduction in the
pairing generation time ($\approx$10 folds for a CPOP with 15 crew bases, as
demonstrated in Aggarwal . (2018)). Secondly, the set of all possible legal
duties and the corresponding duty overnight-connection graph with-respect-to
each crew base are enumerated and stored a priori the CPOP-solutioning. In a
duty overnight-connection graph, a node represents a legal duty, and an edge
between any two nodes represents a legal overnight-rest connection between the
respective duties. Such a preprocessing ensures that all the connection-city,
sit-time, duty, and overnight-rest constraints get naturally satisfied,
eliminating the need for their re-evaluation during the generation of legal
pairings, and leading to a significant reduction in the legal pairing
generation time.
The implementation of this module, formalized in Algorithms 1 & 2, is
elaborated below.
Input: $\mathcal{F}$; $\mathcal{B}$; and constraints:
$\mathcal{C}_{connect},~{}\mathcal{C}_{sit},~{}\mathcal{C}_{duty}~{}\&~{}\mathcal{C}_{night}$
Output: $\mathcal{D}_{b}$ & $\mathcal{G}^{d}_{b}~{}\forall b\in\mathcal{B}$
$\mathcal{G}^{f}\leftarrow$ Generate the flight-connection graph by evaluating
$\mathcal{C}_{connect}~{}\&~{}\mathcal{C}_{sit}$ between each pair of flights
$\in\mathcal{F}$
$\triangleright$
$\mathcal{G}^{f}\equiv\left(\mathcal{F},\mathcal{E}^{f}\right)$
1 for _each crew base $b\in\mathcal{B}$ in parallel_ do
2 for _each flight $f\in\mathcal{F}$_ do
3 Push $f$ into an empty $duty$
4 if _updated flight-sequence in $duty$ satisfies constraints in
$\mathcal{C}_{duty}$_ then
5 Add $duty$ to $\mathcal{D}_{b}$
6 if _$f$ has at least one flight-connection in $\mathcal{G}^{f}$_ then
7 $\texttt{DFS(}duty,f,\mathcal{G}^{f},\mathcal{C}_{duty}\texttt{)}$, and add
the enumerated duties to $\mathcal{D}_{b}$
8
9 end if
10
11 end if
12 Pop out $f$ from $duty$
13
14 end for
15 $\mathcal{G}^{d}_{b}\leftarrow$ Generate the duty overnight-connection
graph by evaluating $\mathcal{C}_{night}$ between each pair of duties
$\in\mathcal{D}_{b}$
16 end for
17return $\mathcal{D}_{b}$ & $\mathcal{G}^{d}_{b}~{}\forall b\in\mathcal{B}$
$\triangleright$ DFS($duty,parent,\mathcal{G}^{f},\mathcal{C}_{duty}$)
18 for _each $child$ of $parent$ in $\mathcal{G}^{f}$_ do
19 Push $child$ into $duty$
20 if _updated flight-sequence in $duty$ satisfies $\mathcal{C}_{duty}$_ then
21 yield $duty$ to $\mathcal{D}_{b}$
22 if _$child$ has at least one connection in $\mathcal{G}^{f}$_ then
23 DFS($duty,child,\mathcal{G}^{f},\mathcal{C}_{duty}$)
24
25 end if
26
27 end if
28 Pop out $child$ from $duty$
29
30 end for
Algorithm 1 Procedure for enumeration of legal duties and duty overnight-
connection graphs
Input: $\mathcal{F}_{*}\text{ or
}\mathcal{D}_{*};~{}\mathcal{B};~{}\mathcal{D}_{b}~{}\&~{}\mathcal{G}^{d}_{b}~{}\forall
b\in\mathcal{B}$; and constraints:
$\mathcal{C}_{base}~{}\&~{}\mathcal{C}_{other}$
Output: $\mathcal{P}_{*}$
1 for _each crew base $b\in\mathcal{B}$ in parallel_ do
2 Update $\mathcal{D}_{b}~{}\&~{}\mathcal{G}^{d}_{b}$ by removing duties
$\notin\mathcal{D}_{*}$ if $\mathcal{D}_{*}$ is input, or by removing those
duties which cover flights $\notin\mathcal{F}_{*}$ if $\mathcal{F}_{*}$ is
input
3 for _each $duty\in\mathcal{D}_{b}$_ do
4 if _departure airport of $duty$ is $b$_ then
5 Push $duty$ into an empty $pairing$
6 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{other}$_
then
7 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{base}$_ then
8 Add $pairing$ to $\mathcal{P}_{*}$
9
10 else if _$duty$ has at least one duty overnight-connection in
$\mathcal{G}^{d}_{b}$_ then
11
$\texttt{DFS(}pairing,duty,\mathcal{G}^{d}_{b},\mathcal{C}_{base}\cup\mathcal{C}_{other}\texttt{)}$,
and add enumerated pairings to $\mathcal{P}_{*}$
12
13 end if
14
15 end if
16 Pop out $duty$ from $pairing$
17
18 end if
19
20 end for
21
22 end for
23return $\mathcal{P}_{*}$
$\triangleright$
DFS($pairing,parent,\mathcal{G}^{d}_{b},\mathcal{C}_{base}\cup\mathcal{C}_{other}$)
24 for _each $child$ of $parent$ in $\mathcal{G}^{d}_{b}$_ do
25 Push $child$ into $pairing$
26 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{other}$_
then
27 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{base}$_
then
28 yield $pairing$ to $\mathcal{P}_{*}$
29
30 else if _$child$ has at least one duty overnight-connection in
$\mathcal{G}^{d}_{b}$_ then
31
$\texttt{DFS(}pairing,child,\mathcal{G}^{d}_{b},\mathcal{C}_{base}\cup\mathcal{C}_{other}\texttt{)}$
32
33 end if
34
35 end if
36 Pop out $child$ from $pairing$
37
38 end for
Algorithm 2 Procedure for enumeration of legal pairings from an input flight
set $\mathcal{F}_{*}\text{ or a duty set }\mathcal{D}_{*}$
For solving any CPOP, the foremost step of the $AirCROP$ is to preprocess the
entire duty-connection network– set of legal duties $\mathcal{D}_{b}$ and duty
overnight-connection graph
$\mathcal{G}^{d}_{b}\left(\equiv\left(\mathcal{D}_{b},~{}\mathcal{E}^{d}_{b}\right)\right)$
for each crew base $b$ in the given set of crew bases $\mathcal{B}$, where
$\mathcal{E}^{d}_{b}$ is the set of legal overnight-rest connections between
duty-pairs $\in\mathcal{D}_{b}$. The procedure for the above preprocessing is
presented in Algorithm 1. In that, the first step is the generation of a
flight-connection graph (denoted by $\mathcal{G}^{f}$) by evaluating the
legality of connection-city ($\mathcal{C}_{connect}$) and sit-time
($\mathcal{C}_{sit}$) constraints between every flight-pair in the given
flight schedule $\mathcal{F}$ (line 1). Here, in
$\mathcal{G}^{f}~{}\left(\equiv\left(\mathcal{F},\mathcal{E}^{f}\right)\right)$,
$\mathcal{F}$ is the set of nodes (flights) and $\mathcal{E}^{f}$ is the set
of edges (legal flight connections). Subsequently, $\mathcal{G}^{f}$ is used
for legal duty enumeration, by decomposing the process into independent sub-
processes, one for each crew base $b\in\mathcal{B}$, and executing them in
parallel (lines 2-12). In each of these sub-processes, enumeration of legal
duties, starting from each flight $f\in\mathcal{F}$, is explored. In that:
* •
flight $f$ is added to an empty candidate duty stack, given by $duty$ (line
4).
* •
the flight-sequence in $duty$ is checked for satisfaction of duty constraints
$\mathcal{C}_{duty}$, and if satisfied, $duty$ is added to the desired legal
duty set $\mathcal{D}_{b}$ (lines 5-6). Notably, if $f$ has at least one
connection with another flight in $\mathcal{G}^{f}$, and if the duty
constraints permit, then more flights could be accommodated in $duty$, leading
to enumeration of other legal duties (lines 7-9).
* •
a Depth-first Search (DFS) algorithm (Tarjan, 1972) is adapted, which is
called recursively to enumerate legal duties, starting from a parent flight
node ($parent$), by exploring its all successive paths in $\mathcal{G}^{f}$ in
a depth-first manner (lines 16-25). In each recursion, a child flight node
($child$) is pushed into $duty$, the updated flight-sequence is checked for
satisfaction of $\mathcal{C}_{duty}$, and if satisfied, $duty$ is yielded to
$\mathcal{D}_{b}$, followed by another recursion of DFS() with $child$ as the
new $parent$. In this way, all legal duties, starting from flight $f$, are
enumerated.
Subsequently, $f$ is popped out from $duty$, and duty enumeration using other
flights in $\mathcal{F}$ is explored (lines 3 & 11). The resulting set
$\mathcal{D}_{b}$ is then used to generate the duty overnight-connection graph
$\mathcal{G}^{d}_{b}$), by evaluating the legality of connection-city
($\mathcal{C}_{connect}$) and overnight-rest ($\mathcal{C}_{night}$)
constraints between every duty-pair $\in\mathcal{D}_{b}$ (line 13). Here, in
$\mathcal{G}^{d}_{b}~{}\left(\equiv\left(\mathcal{D}_{b},\mathcal{E}^{d}_{b}\right)\right)$,
$\mathcal{D}_{b}$ is the set of nodes (legal duties), and
$\mathcal{E}^{d}_{b}$ is the set of edges (legal overnight-rest connections).
The preprocessed sets of legal duties and the corresponding duty overnight-
connection graphs are utilized to enumerate legal pairings for any input
flight set (say $\mathcal{F}_{*}$) or a duty set (say $\mathcal{D}_{*}$), when
required in real-time in other modules of the $AirCROP$. Its procedure,
formalized in Algorithm 2, is elaborated below. For legal pairing enumeration,
the same crew base driven parallel architecture is utilized in which the
process is decomposed into independent sub-processes, one for each crew base
$b\in\mathcal{B}$, running in parallel on idle-cores of the CPU (line 1). In
each of these sub-processes, the first step is to update $\mathcal{D}_{b}$ and
$\mathcal{G}^{d}_{b}$, by removing duties $\notin\mathcal{D}_{*}$ if
$\mathcal{D}_{*}$ is input, or those duties that cover flights
$\notin\mathcal{F}_{*}$ if $\mathcal{F}_{*}$ is input (line 2). Subsequently,
the enumeration of legal pairings, starting from each duty ($duty$)
$\in\mathcal{D}_{b}$, is explored (line 3). In that:
* •
the $duty$ is pushed into an empty candidate pairing stack, given by
$pairing$, only if the departure airport of $duty$ is same as the crew base
$b$ (lines 4-5).
* •
the $pairing$ is checked for satisfaction of pairing constraints
$\mathcal{C}_{other}$, and if satisfied, $pairing$ is further checked for
satisfaction of end-city constraint $\mathcal{C}_{base}$, which ensures that
the arrival airport of the $pairing$’s last duty is same as the crew base $b$.
* –
If $pairing$ satisfies $\mathcal{C}_{base}$, it is classified as legal, and is
added to the desired pairing set $\mathcal{P}_{*}$ (lines 7-8).
* –
If $pairing$ does not satisfy $\mathcal{C}_{base}$, it is not complete, and
more duties are required to be covered in it to complete the legal duty-
sequence. This is only possible if $duty$ has at least one overnight-rest
connection in $\mathcal{G}^{d}_{b}$. And if it does, the DFS() sub-routine,
similar to the one used in legal duty enumeration, is called recursively to
enumerate legal pairings, starting from a parent duty node ($parent$), by
exploring its all successive paths in $\mathcal{G}^{d}_{b}$ in a depth-first
manner (lines 18-28). In each recursion:
* $\circ$
a child duty node ($child$) is pushed into the $pairing$ (line 19).
* $\circ$
the updated duty-sequence in $pairing$ is checked for satisfaction of first
$\mathcal{C}_{other}$ and then $\mathcal{C}_{base}$ (lines 20-21).
* $\circ$
if it satisfies both constraints, then $pairing$ is complete (legal), and is
yielded to the desired pairing set $\mathcal{P}_{*}$ (line 22).
* $\circ$
if it satisfies $\mathcal{C}_{other}$ but not $\mathcal{C}_{base}$, then
another recursion of DFS() with $child$ as new $parent$ is called, only if
$child$ has at least one duty overnight-rest connection in
$\mathcal{G}^{d}_{b}$ (lines 23-25).
In the above way, all legal pairings, starting from $duty$, are enumerated
using the DFS() sub-routine. Subsequently, $duty$ is popped out of $pairing$
(line 13), and the legal pairing enumeration using other duties
$\in\mathcal{D}_{b}$ is explored (line 3). Once, all the sub-processes are
complete, the desired pairing set $\mathcal{P}_{*}$ is returned (line 17).
### 3.2 Initial Feasible Solution Generator
An initial feasible solution (IFS) is any set of pairings, covering all
flights in the given flight schedule, which is used to initialize a CPOP
solution approach. For large-scale CPOPs, generation of an IFS standalone is a
computationally-challenging task. This module is designed to generate a
reasonably-sized IFS in a time-efficient manner for large and complex flight
networks, which is then used to initialize the Optimization Engine of
$AirCROP$. For this, it employs a novel Integer Programming based Divide-and-
cover Heuristic (IPDCH), which relies on: (a) a divide-and-cover strategy to
decompose the input flight schedule into sufficiently-small flight subsets,
and (b) integer programming to find a lowest-cost pairing set, covering the
maximum possible flights for each of the decomposed flight subsets.
The procedure of the proposed IPDCH, formalized in Algorithm 3, is elaborated
below.
Input: $\mathcal{F},~{}K,~{}\texttt{Pairing\\_Gen()}$
Output: $\mathcal{P}_{IFS}$
1 while _all flights $\in\mathcal{F}$ are not covered in $\mathcal{P}_{IFS}$_
do
$\mathcal{F}_{K}\leftarrow$ Select $K$ random flights from $\mathcal{F}$
without replacement
$\triangleright$ $K<F$
2
$\mathcal{P}_{K}\leftarrow~{}\texttt{Pairing\\_Gen(}\mathcal{F}_{K}\texttt{)}$
$\mathcal{F}_{K^{\prime}}\leftarrow$ Flights covered in $\mathcal{P}_{K}$
$\triangleright$ $K^{\prime}\leq K$
3 Add remaining flights
$\left(\mathcal{F}_{K}\backslash\mathcal{F}_{K^{\prime}}\right)$ back to
$\mathcal{F}_{K}$
4 Formulate the IPP using flights in $\mathcal{F}_{K^{\prime}}$ and pairings
in $\mathcal{P}_{K}$
5 $\mathcal{P}_{IP}\leftarrow$ Solve the IPP using an MIP solver, and select
pairings corresponding to non-zero variables
6 Add pairings from $\mathcal{P}_{IP}$ to $\mathcal{P}_{IFS}$
7 Replace flights in $\mathcal{F}$ if it becomes empty
8
9 end while
10return $\mathcal{P}_{IFS}$
Algorithm 3 Procedure for IFS generation using the proposed IPDCH
Being an iterative heuristic, IPDCH terminates when all flights in the input
set are covered by pairings in the desired IFS, notated as $\mathcal{P}_{IFS}$
(lines 1). The input to the heuristic involves the given flight schedule
$\mathcal{F}$ (with $F$ number of flights), the pairing generation sub-routine
Pairing_Gen() (presented in Section 3.1), and a pre-defined decomposition
parameter $K$, which regulates the number of flights to be selected from
$\mathcal{F}$ in each IPDCH-iteration. The setting of $K$ largely depends upon
the available computational resources, and the characteristics of the input
flight dataset (as highlighted in Section 4.3.3). In each IPDCH-iteration,
first a flight subset, say $\mathcal{F}_{K}$ $\left(K<F\right)$, is formed by
randomly selecting $K$ number of flights from $\mathcal{F}$ without
replacement (line 2). Subsequently, $\mathcal{F}_{K}$ is fed as input to the
Pairing_Gen() sub-routine to enumerate the set of all possible legal pairings,
say $\mathcal{P}_{K}$ (line 3). Notably, all flights in $\mathcal{F}_{K}$ may
not get covered by pairings in $\mathcal{P}_{K}$, as random selection of
flights does not guarantee legal connections for all selected flights. Let
$\mathcal{F}_{K^{\prime}}$ $\left(K^{\prime}\leq K\right)$ be the set of
flights covered in $\mathcal{P}_{K}$ (line 4). The remaining flights, given by
$\mathcal{F}_{K}\backslash\mathcal{F}_{K^{\prime}}$, are added back to
$\mathcal{F}$ (line 5). Subsequently, $\mathcal{F}_{K^{\prime}}$ and
$\mathcal{P}_{K}$ are used to formulate the corresponding IPP (line 6), which
is then solved using a commercial off-the-shelf MIP solver to find the optimal
IPP solution, say $\mathcal{P}_{IP}$, constituted by pairings corresponding to
only non-zero variables (line 7). The pairings in $\mathcal{P}_{IP}$ are then
added to the desired set $\mathcal{P}_{IFS}$ (line 8). Lastly, the flights in
$\mathcal{F}$ are replaced if it becomes empty (line 9). As soon as
$\mathcal{P}_{IFS}$ covers all the required flights, IPDCH is terminated, and
$\mathcal{P}_{IFS}$ is passed over to the Optimization Engine for its
initialization.
### 3.3 Optimization Engine: Interactions between CG-driven LPP-solutioning
and IPP-solutioning
The search for minimal cost, full flight-coverage CPOP solution is enabled by
an optimization engine. It tackles the underlying LPP and IPP through
intermittent interactions of two submodules, namely, CG-driven LPP-solutioning
and IPP-solutioning, tracked by a counter $T$. These submodules are presented
below.
#### 3.3.1 CG-driven LPP-solutioning
As illustrated in Figure 2, this submodule entails several iterations (each
referred to as an LPP iteration, and is tracked by $t$) in each of which: (a)
an LP solver is invoked on the input pairing set, leading to the current LPP
solution $\mathcal{P}_{LP}^{t}$, (b) the corresponding dual od the LPP is
formulated using $\mathcal{P}_{LP}^{t}$, which is then solved to fetch dual
variables (given by vector $Y^{t}$), and (c) a fresh set of pairings
$\mathcal{P}_{CG}^{t}$, that promises associated cost-improvement, is
generated using a domain-knowledge driven CG heuristic. For the first LPP
iteration ($t=1$), the input to the LP solver is either $\mathcal{P}_{IFS}$ if
$T=1$, or $\mathcal{P}_{IP}^{T-1}$ if $T>1$. For any subsequent LPP iteration
($t>1$), the input comprises of the current $\mathcal{P}_{CG}^{t}$ and
$\mathcal{P}_{LP}^{t}$. In this background, each of these LPP iterations are
implemented in the following three phases555For ease of reference, the
notations introduced in these phases are kept independent of the LPP iteration
counter $t$. However, these notations are super-scripted by $t$ in the
corresponding discussions and pseudocodes with reference to a particular LPP
iteration.:
* •
In the first phase, a primal of the LPP (Equations 4 to 6) is formulated from
the input pairing set, and is solved using an interior-point method based
commercial off-the-shelf LP solver (Gurobi Optimization, 2019). In the
resulting LPP solution, a primal variable $x_{j}$, varying from $0$ to $1$, is
assigned to each pairing $p_{j}$ in the input pairing set. These $x_{j}$s
together constitute the primal vector, notated as
$X~{}\left(=[x_{1}~{}x_{2}~{}x_{3}~{}...~{}x_{P}]^{\mathsf{T}}\right)$. The
set of $x_{j}$s with non-zero values ($x_{j}\neq 0$) and the set of
corresponding pairings are notated as $X_{LP}$ and $\mathcal{P}_{LP}$,
respectively.
$\displaystyle\text{Minimize}~{}Z_{LP}^{p}=\sum_{j=1}^{P}c_{j}x_{j}+\psi_{D}\cdot\left(\sum_{i=1}^{F}\left(\sum_{j=1}^{P}a_{ij}x_{j}-1\right)\right)=\sum_{j=1}^{P}\left(c_{j}+\psi_{D}\cdot\sum_{i=1}^{F}a_{ij}\right)x_{j}-F\cdot\psi_{D},$
(4) $\displaystyle\text{subject to}\quad\sum_{j=1}^{P}a_{ij}x_{j}\geq
1,\qquad\quad\forall i\in\\{1,2,...,F\\}$ (5) $\displaystyle\qquad\qquad\quad
x_{j}\in\mathbb{R}=[0,1],\qquad\forall j\in\\{1,2,...,P\\}$ (6)
It is to be noted that the minimization of $Z_{LP}^{p}$ will always lead to a
solution with all primal variables $x_{j}\leq 1$, even without explicitly
involving the corresponding constraint– Equation 6 (Vazirani, 2003). Hence,
the contribution of each pairing in the LPP solution, given by its $x_{j}$,
could be effectively treated as $x_{j}\in\mathbb{R}_{\geq 0}$ instead of
Equation 6.
* •
In the second phase, dual variables are extracted from the current LPP
solution. For this, the dual of the LPP (Equations 7 to 9) is formulated using
the pairing set $\mathcal{P}_{LP}$, and is solved using an interior-point
method (Andersen Andersen, 2000) based non-commercial LP solver (Virtanen .,
2020), to fetch the optimal dual solution. In that, a dual variable $y_{i}$
represents a shadow price corresponding to an $i^{th}$ flight-coverage
constraint in the primal. The optimal dual vector, constituted by all $y_{i}$s
in the optimal dual solution, is notated as
$Y~{}\left(=[y_{1}~{}y_{2}~{}y_{3}~{}...~{}y_{F}]^{\mathsf{T}}\right)$, whose
dimension is equal to $F$.
$\displaystyle\text{Maximize}~{}Z_{LP}^{d}=\sum_{i=1}^{F}y_{i}-F\cdot\psi_{D},$
(7) $\displaystyle\text{subject
to}\quad\sum_{i=1}^{F}a_{ij}y_{i}\leq\left(c_{j}+\psi_{D}\cdot\sum_{i=1}^{F}a_{ij}\right),~{}~{}~{}~{}\forall
j\in\\{1,2,...,P_{LP}\\}$ (8)
$\displaystyle\qquad\qquad\qquad\quad~{}~{}y_{i}\in\mathbb{R}\geq
0,\qquad\qquad\qquad~{}~{}~{}~{}\forall i\in\\{1,2,...,F\\}$ (9)
$\displaystyle\text{where},\qquad P_{LP}:~{}\text{is the number of pairings in
the set}~{}\mathcal{P}_{LP}$
$\displaystyle\qquad\qquad\quad~{}~{}~{}y_{i}:~{}\text{dual variable,
corresponding to an $i^{th}$ flight-coverage constraint},$
Notably, in a conventional approach, the optimal $Y$ is directly computed from
the optimal basis of the primal solution (obtained in the first phase), using
the principles of duality theory, particularly the theorem of complementary
slackness (Bertsimas Tsitsiklis, 1997), without explicitly solving the
corresponding dual. However, in the second phase, solving the dual explicitly
using the interior-point method (Andersen Andersen, 2000), in a sense, helps
in stabilizing the oscillating behavior of dual variables over the successive
LPP iterations (bang-bang effect, as discussed in Section 2.3). Moreover, this
interior-point method is available via only a non-commercial LP solver
(Virtanen ., 2020), and to ensure a time-efficient search, the above dual is
formulated using the pairings $\in\mathcal{P}_{LP}$, instead of pairings from
the large-sized input pairing set.
* •
In the last phase, the availability of dual variables from the second phase
paves the way for solution to the pricing sub-problem. It is aimed to generate
those legal pairings (non-basic), which if included as part of the input to
the next LPP iteration, promise a better-cost (at least a similar-cost) LPP
solution compared to the current solution. Such non-basic pairings are
identified using a reduced cost metric, given by $\mu_{j}$ (Equation 10),
which if negative (as CPOP is a minimization problem) indicates the potential
in the pairing to further reduce the cost of the current LPP solution
$Z_{LP}^{p}$, when included in the current basis (Bertsimas Tsitsiklis, 1997).
Moreover, the potential of such a pairing to further reduce the current
$Z_{LP}^{p}$, is in proportion to the magnitude of its $\mu_{j}$ value.
$\displaystyle\mu_{j}=c_{j}-\mu d_{j},~{}\text{where,}~{}\mu
d_{j}=\sum_{i=1}^{F}\left(a_{ij}\cdot y_{i}\right)~{}=\sum_{f_{i}\in
p_{j}}y_{i}~{}~{}(~{}\text{represents the dual cost component of}~{}\mu_{j})$
(10)
As mentioned in Section 2.3, the standard CG practices generate a complete
pricing network and solves it as a resource-constrained shortest-path
optimization problem, to identify only the pairing(s) with negative reduced
cost(s). However, generation of a complete pricing network for CPOPs with
large-scale and complex flight networks is computationally-intractable. To
overcome this challenge, a domain-knowledge driven CG heuristic (Aggarwal,
Saxena ., 20201) is employed here to generate a set of promising pairings (of
pre-defined size, criterion for which is discussed in Section 4.2). Notably,
the merit of this CG heuristic lies in the fact that from within the larger
pool of pairings with negative $\mu_{j}$, besides selecting pairings randomly,
it also selects pairings in a guided manner. In that, the selection of such
pairings is guided by optimal solution features at a set level and an
individual pairing level, and re-utilization of the past computational
efforts. These optimal solution features are related to the minimization of
deadheads and maximization of the crew utilization, respectively. In essence,
while the standard CG practices present equal opportunity for any pairing with
a negative $\mu_{j}$ to qualify as an input for the next LPP iteration, this
CG heuristic, besides ensuring that the pairings have negative $\mu_{j}$,
prioritizes some pairings over the others via its two-pronged strategy–
exploration of the new pairings’ space and re-utilization of pairings from the
past LPP iterations. In that:
* –
the exploration of the new pairings’ space is guided by three CG strategies,
which are elaborated below.
* $\circ$
Deadhead Reduction strategy ($CGD$): this strategy prioritizes a set of legal
pairings that is characterized by low deadheads, a feature which domain
knowledge recommends for optimality at a set level. To exploit this optimality
feature, $CGD$ generates a new paring set $\mathcal{P}_{CGD}$, which not only
provides an alternative way to cover the flights involved in a subset of the
current $\mathcal{P}_{LP}$, but also ensures that some of these flights get
covered with zero deadheads. It promises propagation of the zero deadhead
feature over successive LPP iterations, as: (a) $\mathcal{P}_{CGD}$ alongside
the current $\mathcal{P}_{LP}$ forms a part of the input for the next LPP
iteration; (b) $\mathcal{P}_{CGD}$ provides a scope for better coverage (zero
deadhead) of some flights, compared to the current $\mathcal{P}_{LP}$; and (c)
$\mathcal{P}_{CGD}$ may focus on zero deadhead coverage for different flights
in different LPP iterations.
* $\circ$
Crew Utilization enhancement strategy ($CGU$): this strategy prioritizes a set
of legal pairings each member of which is characterized by high crew
utilization, a feature which domain knowledge recommends for optimality at an
individual pairing level. To exploit this optimality feature, $CGU$: (a)
introduces a new measure, namely, crew utilization ratio, given by
$\gamma_{j}$ (Equation 11), to quantify the degree of crew utilization in a
pairing $p_{j}$ at any instant; (b) identifies pairings from the current
$\mathcal{P}_{LP}$, which are characterized by high dual cost component ($\mu
d_{j}$, Equation 10), reflecting in turn on those constitutive flights that
have high value of dual variables $y_{i}$, and hence, on the potential of
these flights to generate new pairings with more negative $\mu_{j}$; and (c)
utilizes these flights to generate promising pairings from which only the ones
with high $\gamma_{j}$ are picked to constitute the new pairing set
$\mathcal{P}_{CGU}$.
$\displaystyle\gamma_{j}=\frac{1}{\text{Number of duties in
}p_{j}}\cdot\sum_{d\in p_{j}}\frac{\text{Working hours in
duty}~{}d}{\text{Permissible hours of duty }d}$ (11)
In doing so, $CGD$ promises propagation of the higher crew utilization ratio
over successive LPP iterations, given that in each LPP iteration,
$\mathcal{P}_{CGU}$ alongside the current $\mathcal{P}_{LP}$ forms a part of
the input for the next LPP iteration.
* $\circ$
Random exploration strategy ($CGR$): this strategy, unlike $CGU$ and $CGD$
which are guided by optimal solution features, pursues random and unbiased
exploration of the new pairings’ space, independent of the current LPP
solution. It involves generation of new pairings for a random selected set of
legal duties from which only the pairings with negative reduced cost are
selected to constitute the new pairing set $\mathcal{P}_{CGR}$. Here, a random
set of legal duties is used instead of a random set of flights, as the former
has a higher probability of generating legal pairings, given that a majority
of pairing legality constraints get satisfied with the preprocessing of legal
duties.
* –
the re-utilization of pairings from the past LPP iterations is guided by an
Archiving strategy ($CGA$), that prioritizes a set of legal pairings
comprising of those flight-pairs, which as per the existing LPP solution, bear
better potential for improvement in the objective function. Such a pairing
set, originating from the flight-pair level information, is extracted from an
archive (denoted by $\mathcal{A}$) of the previously generated pairings. In
doing so, $CGA$ facilitates re-utilization of the past computational efforts,
by providing an opportunity for a previously generated pairing to be re-
inducted in the current pairing pool. For this, $CGA$:
* $\circ$
updates the archive $\mathcal{A}$ in each LPP iteration such that any pairing
is stored/retrieved with reference to a unique index $(f_{m},f_{n})$ reserved
for any legal flight-pair in that pairing.
* $\circ$
introduces a new measure, namely, reduced cost estimator, given by $\eta_{mn}$
(Equation 12), for a flight-pair $(f_{m},f_{n})$ in $\mathcal{A}$. In each LPP
iteration, this estimator is computed for all the flight-pairs present in
$\mathcal{A}$, by fetching $f_{m}$, $f_{n}$, $y_{m}$ and $y_{n}$.
$\displaystyle\eta_{mn}$
$\displaystyle=\texttt{flying\\_cost($f_{m}$)}+\texttt{flying\\_cost($f_{n}$)}-y_{m}-y_{n}=\sum_{i\in\\{m,n\\}}\left(\texttt{flying\\_cost($f_{i}$)}-y_{i}\right)$
(12)
Notably, this formulation is analogous to Equation 10, just that instead of
the complete cost of a pairing, only the flying costs corresponding to the
flights in a legal flight-pair are accounted for. Given this, $\eta_{mn}$ may
be seen as an indicator of $\mu_{j}$ at the flight-pair level.
* $\circ$
recognizes that towards further improvement in the current LPP solution, it
may be prudent to include as a part of the input for the next LPP iteration–
the new pairing set $\mathcal{P}_{CGA}$, constituted by preferentially picking
pairings from $\mathcal{A}$, that cover flight-pairs with lower $\eta_{mn}$
value.
In doing so, $CGA$ pursues the goal of continual improvement in the objective
function, while relying on the flight-pair level information embedded in the
LPP solution of current LPP iteration, and re-utilizing the computational
efforts spent till that LPP iteration.
For further details and associated nitty-gritty of the above domain-knowledge
driven CG heuristic, interested readers are referred to the authors’ previous
work– Aggarwal, Saxena . (20201). Once this CG heuristic generates a set of
promising pairings $\mathcal{P}_{CG}$ of pre-defined size, it is merged with
the current $\mathcal{P}_{LP}$, and fed as the input to the next LPP iteration
($t\mathrel{+}=1$).
These LPP iterations are repeated until the cost-improvements over a pre-
specified number of successive LPP iterations falls below a pre-specified
cost-threshold (settings given in Section 4.2). In this submodule, these LPP
iterations are repeated, until its termination criterion is not met. In that,
the cost-improvement over LPP iterations is observed, and if it falls below a
pre-specified cost-threshold, say $Th_{cost}$, over a pre-specified number of
successive LPP iterations, say $Th_{t}$, then it is terminated. The settings
of these pre-specified limits– $Th_{cost}$ and $Th_{t}$, are highlighted in
Section 4.2. After termination, the final LPP solution $\mathcal{P}_{LP}^{T}$
is then passed over to the IPP-solutioning submodule for its integerization.
#### 3.3.2 IPP-solutioning
This submodule receives as input, the LPP solution $\mathcal{P}^{T}_{LP}$, and
aims to find therein a full-coverage integer solution, notated as
$\mathcal{P}^{T}_{IP}$. Towards it, an IPP (Equations 1 to 3) is formulated
using $\mathcal{P}^{T}_{LP}$ and $\mathcal{F}$, and solved using a branch-and-
cut algorithm based off-the-shelf commercial MIP solver (Gurobi Optimization,
2019). At each node of the MIP-search tree, this solver maintains a valid
lower bound (cost of the LPP solution) and a best upper bound (cost of the IPP
solution), and it self-terminates if the gap between these two bounds becomes
zero, or all branches in the MIP-search tree have been explored. Considering
that the MIP-search for large-scale CPOPs is extremely time-consuming, a pre-
defined time limit, notated as $Th_{ipt}$ (setting highlighted in Section
4.2), is used to terminate this MIP solver, if it does not terminate by itself
a priori. Once the $\mathcal{P}^{T}_{IP}$ is obtained, it is passed back to
the previous submodule for the next LPP-IPP interaction ($T\mathrel{+}=1$),
only if the termination criterion of the Optimization Engine is not satisfied.
Overarching Optimization Engine
In the wake of the above, the procedure of the overarching Optimization
Engine, formalized in Algorithm 4, is elaborated below.
Input:
$\mathcal{F},~{}\mathcal{P}_{IFS},~{}Th_{cost},~{}Th_{t},~{}Th_{ipt},~{}\texttt{Pairing\\_Gen()},~{}\texttt{CGD()},~{}\texttt{CGU()},~{}\texttt{CGR()},~{}\texttt{CGA()}$
Output: $\mathcal{P}^{\star}_{IP}$
1 $T\leftarrow 1$ while _termination criterion of Optimization Engine is not
met_ do
2 $\triangleright$ CG-driven LPP-solutioning: $t\leftarrow 1$ while
_termiantion criterion of CG-driven LPP-solutioning is not met_ do
3 if _$t=1$ and $T=1$_ then
4 Formulate the primal of the LPP using $\mathcal{P}_{IFS}$ and $\mathcal{F}$
5 else if _$t=1$ and $T>1$_ then
6 Formulate the primal of the LPP using $\mathcal{P}^{T-1}_{IP}$ and
$\mathcal{F}$
7 else
8 Formulate the primal of the LPP using
$\mathcal{P}^{t-1}_{CG}\cup\mathcal{P}^{t-1}_{LP}$ and $\mathcal{F}$
9 end if
10 $\mathcal{P}^{t}_{LP},~{}X^{t}_{LP}\leftarrow$ Solve the primal using the
interior-point method based LP solver $\triangleright$ Termination of the CG-
driven LPP-solutioning: if _cost-improvements $\leq Th_{cost}$ over last
$Th_{t}$ number of successive LPP iterations_ then
11 $\mathcal{P}^{T}_{LP}\leftarrow\mathcal{P}^{t}_{LP}$ Break
12 end if
13 Formulate the dual of the LPP using $\mathcal{F}$ and
$\mathcal{P}^{t}_{LP}$ $Y^{t}\leftarrow$ Solve the dual using the interior-
point method based LP solver $\triangleright$ Solution to pricing sub-problem
using the CG heuristic:
$\mathcal{P}^{t}_{CGD}\leftarrow\texttt{CGD($\mathcal{P}^{t}_{LP},X^{t}_{LP},Y^{t},\ldots$)}$
$\mathcal{P}^{t}_{CGU}\leftarrow\texttt{CGU($\mathcal{P}^{t}_{LP},X^{t}_{LP},Y^{t},\ldots$)}$
$\mathcal{P}^{t}_{CGR}\leftarrow\texttt{CGR($Y^{t},\ldots$)}$
$\mathcal{P}^{t}_{CGA}\leftarrow\texttt{CGA($\mathcal{P}^{t}_{LP},X^{t}_{LP},Y^{t},\ldots$)}$
$\mathcal{P}^{t}_{CG}\leftarrow\mathcal{P}^{t}_{CGD}\cup\mathcal{P}^{t}_{CGU}\cup\mathcal{P}^{t}_{CGR}\cup\mathcal{P}^{t}_{CGA}$
$t\mathrel{+}=1$
14 end while
15 $\triangleright$ IPP-solutioning: Formulate the IPP using
$\mathcal{P}^{T}_{LP}$ and $\mathcal{F}$ $\mathcal{P}^{T}_{IP}\leftarrow$
Solve the IPP using a branch-and-cut algorithm based MIP solver until its run-
time becomes $\geq Th_{ipt}$ $\triangleright$ Termination of the Optimization
Engine: if _$Z^{T}_{IP}\left(\text{cost of
}\mathcal{P}^{T}_{IP}\right)=Z^{T}_{LP}\left(\text{cost of
}\mathcal{P}^{T}_{LP}\right)$_ then
16 $\mathcal{P}^{\star}_{IP}\leftarrow\mathcal{P}^{T}_{IP}$ Break
17 end if
18 $T\mathrel{+}=1$
19 end while
return $\mathcal{P}_{IP}^{\star}$
Algorithm 4 Procedure for the Optimization Engine
Its input involves the given flight set $\mathcal{F}$; the generated IFS
$\mathcal{P}_{IFS}$; the pre-defined termination parameters– $Th_{cost}$ &
$Th_{t}$ (for CG-driven LPP-solutioning) and $Th_{ipt}$ (for IPP-solutioning);
and the sub-routines for Legal Crew Pairing Generator (Pairing_Gen()) and the
four CG strategies ($\texttt{CGD()},~{}\texttt{CGU()},~{}\texttt{CGR()}~{}$and
CGA()) in the proposed CG heuristic. In each LPP-IPP interaction of the
Optimization Engine, first, the CG-driven LPP-solutioning is executed (lines
3-25). It entails several LPP iterations (tracked by $t$), in each of which
the first step is to formulate the primal using $\mathcal{F}$ and the
respective input pairing set. This input pairing set is:
* •
$\mathcal{P}_{IFS}$, if the first LPP iteration ($t=1$) of the first LPP-IPP
interaction ($T=1$) is being executed (lines 5-6).
* •
$\mathcal{P}^{T-1}_{IP}$, if the first LPP iteration ($t=1$) of any subsequent
LPP-IPP interaction ($T>1$) is being executed (lines 7-8).
* •
$\mathcal{P}^{t-1}_{CG}\cup\mathcal{P}^{t-1}_{LP}$, if any subsequent LPP
iteration ($t>1$) of any LPP-IPP interaction ($T\geq 1$) is being executed
(lines 9-11).
Once the primal is formulated, it is solved using the corresponding LP solver
to obtain the current optimal LPP solution, constituted by
$\mathcal{P}^{t}_{LP}~{}$and$~{}X^{t}_{LP}$ (line 12). Subsequently, the
termination criterion of CG-driven LPP-solutioning is checked (lines 13-16).
If it is terminated, then the current LPP solution $\mathcal{P}^{t}_{LP}$ is
fetched as the final LPP solution $\mathcal{P}^{T}_{LP}$ of this LPP-IPP
interaction. If not, then a dual is formulated using $\mathcal{P}^{t}_{LP}$
and $\mathcal{F}$ (line 17), which is then solved using the corresponding LP
solver to obtain the current optimal dual vector $Y^{t}$ (line 18). Using the
current $\mathcal{P}^{t}_{LP}$, $X^{t}_{LP}$ and $Y^{t}$, a fresh set of
pairings $\mathcal{P}^{t}_{CG}$ is obtained using the CG heuristic, which is
constituted by the new pairing sets from the four underlying CG strategies
(lines 19-23). At the end of the LPP iteration $t$, the fresh set of pairings
$\mathcal{P}^{t}_{CG}$ is combined with the current $\mathcal{P}^{t}_{LP}$ to
serve as input pairing set for the subsequent LPP iteration
($t\mathrel{+}=1$). Once this submodule is terminated, the resulting
$\mathcal{P}^{T}_{LP}$ is passed over to the IPP-solutioning for its
integerization, wherein, the MIP solver is used to obtain the IPP solution
$\mathcal{P}^{T}_{IP}$ (lines 26 and 27). In that, the pre-defined $Th_{ipt}$
time-limit is used to terminate the MIP-search, if it does not self-terminate
a priori. Subsequently, the resulting $\mathcal{P}^{T}_{IP}$ is passed back to
the CG-driven LPP-solutioning for the next LPP-IPP interaction
($T\mathrel{+}=1$), or returned as the final integer solution
$\mathcal{P}^{\star}_{IP}$, depending upon the termination condition of the
Optimization Engine (lines 28-32). In that, if the cost of
$\mathcal{P}^{T}_{IP}~{}\left(Z_{IP}^{T}\right)$, matches the cost of
$\mathcal{P}^{T}_{LP}~{}\left(Z_{LP}^{p,T}\right)$, then the Optimization
Engine is terminated.
## 4 Computational Experiments
This section first presents the test cases and the computational setup, used
to investigate the utility of $AirCROP$, its modules, and their interactions.
Subsequently, the settings of parameters involved in different modules of
$AirCROP$ are presented. Lastly, the experimental results are discussed.
### 4.1 Test Cases and Computational Setup
The real-world airline test cases, used for experimentation, are detailed in
Table 2. Each of these test cases involves a weekly flight schedule, and have
been provided by the research consortium’s industrial sponsor (from the
networks of US-based airlines).
Table 2: Real-world airline test cases used in this research work Test Cases | $\\#$Flights | $\\#$Crew Bases | $\\#$Airports | $\\#$Legal Duties
---|---|---|---|---
TC-1 | 3202 | 15 | 88 | 454205
TC-2 | 3228 | 15 | 88 | 464092
TC-3 | 3229 | 15 | 88 | 506272
TC-4 | 3265 | 15 | 90 | 446937
TC-5 | 4212 | 15 | 88 | 737184
(a)
(b)
Figure 3: (a) Geographical representation of TC-5 flight network, where the
red nodes, green edges and yellow nodes represent the airports, scheduled
flights and crew bases, respectively, and (b) legal flight-connections, each
represented by a point in the plot, where for a flight marked on the y-axis,
the connecting flight is marked on the x-axis.
The columns in Table 2, in order of their occurrence, highlight the notations
for the different test cases; the number of its constituent flights; the
number of constituent crew bases; and the total number of legal duties
involved, respectively. It is critical to recognize that the challenge
associated with solutioning of these test cases, depends not just on the
number of flights involved but also to the fact that these flights are part of
complex flight networks, characterized by a multiplicity of hubs as opposed to
a single hub, and multiplicity of crew bases as opposed to a single crew base.
In that, the number of legal pairings possible, grow exponentially with the
number of hubs and crew bases. As a sample instance, the geographical
representation of the flight network associated with TC-5, and the legal
flight connections involved in it, are portrayed in Figure 3. Notably, in
Figure 3(a), the presence of multiple hub-and-spoke subnetworks and multiple
crew bases (highlighted in yellow color) is evident. Furthermore, the pattern
visible in Figure 3(b) could be attributed to the (minimum and maximum) limits
on the sit-time and overnight-rest constraints. For instance, a flight, say
$f_{500}$, has legal connections only with those flights that depart from the
arrival airport of $f_{500}$, and whose departure-time gap (difference between
its departure-time and the arrival time of $f_{500}$) lies within the minimum
and maximum allowable limits, of the sit-time or the overnight-rest.
All the experiments in this research have been performed on an HP Z640
Workstation, which is powered by two Intel® Xeon® E5-2630v3 processors, each
with 16 cores at 2.40 GHz, and 96 GBs of RAM. All codes related to the
$AirCROP$ have been developed using the Python scripting language in alignment
with the Industrial sponsor’s larger vision and preference. Furthermore:
* •
the interior-point method from Gurobi Optimizer 8.1.1 (Gurobi Optimization,
2019) is used to solve the primal in the CG- driven LPP-solutioning submodule.
* •
the interior-point method (Andersen Andersen, 2000) from SciPy’s linprog
library (Virtanen ., 2020) is used to solve the dual in the CG-driven LPP-
solutioning submodule.
* •
the branch-and-cut algorithm based MIP solver from Gurobi Optimizer 8.1.1 is
used to solve the IPP in the Initial Feasible Solution Generator and the IPP-
solutioning submodule.
* •
an $AirCROP$-run, in principle, terminates when the cost of the IPP solution
matches the cost of its input LPP solution in a particular LPP-IPP
interaction. However, for practical considerations on the time-limit, an
$AirCROP$-run is allowed to terminate if the IPP and LPP costs do not conform
with each other even after 30 LPP-IPP interactions are over, or 30 hours of
total run-time is elapsed.
### 4.2 Parameter Settings
The settings of the parameters associated with different modules and
submodules of the $AirCROP$ are, as highlighted below.
* •
Initial Feasible Solution Generator: here, the proposed IPDCH involves the
decomposition parameter $K$, which regulates the size of flight subsets formed
in each of IPDCH-iteration. As mentioned before, the setting of $K$ is
dependent on the characteristics of input flight dataset and the configuration
of available computational resources. Here, the aim is to cover all given
flights in a time-efficient manner. Hence, it is important to understand the
effect of setting of $K$ on the time-performance of IPDCH, which is
highlighted below.
* –
For a relatively lower value of $K$, smaller flight subsets with lesser number
of legal flight-connections would be formed in each IPDCH-iteration, leading
to coverage of relatively lesser number of unique flights in each of them.
Though, this by itself is not a challenge, but this would necessitate a
significant number of additional IPDCH-iterations (and the respective run-
time), since the number of unique flights covered per IPDCH-iteration, which
by construct reduces with the iterations, would get further reduced with
relatively smaller flight subsets.
* –
On the flip side, for a relatively higher value of $K$, bigger flight subsets
would be formed that would lead to coverage of higher number of unique flights
per IPDCH-iteration. Though, this may reduce the total number of IPDCH-
iterations required to generate the desired IFS, the overall run-time of the
IPDCH may increase drastically. The rationale being that with bigger flight
subset in each IPDCH-iteration, the number of possible legal pairings would
increase drastically, leading to huge run-time for their generation as well as
for the subsequent MIP-search.
The above considerations suggest that $K$ should be reasonably-sized.
Considering the given computational resources and the results of initial
exploration around the possible number of pairings for differently-sized
flight sets, the value of $K$ in each IPDCH-iteration is guided by a random
integer between one-eighth and one-fourth of the size of the input flight set
$\mathcal{F}$. It may be noted that this setting of $K$ has been selected
considering the scale and complexity of the given test cases, and it needs to
be re-visited if the scale and complexity of the flight network changes
drastically.
* •
CG-driven LPP-solutioning: The parameters involved in the termination
criterion for this submodule– $Th_{cost}$ & $Th_{t}$, are set as 100 USD & 10
iterations respectively, to achieve an LPP solution with a sufficiently good
cost in a reasonably good time. Moreover, the sensitivity of these parameters
towards the $AirCROP$’s performance is discussed in Section 4.3.4. Moreover,
the effect of the parameter– size of $\mathcal{P}^{t}_{CG}$, on the
performance of this submodule (the final LPP solution’s cost and required run-
time), and the demand on the computational resources (dominantly, RAM) is
highlighted below.
* –
for a relatively small-sized $\mathcal{P}^{t}_{CG}$, the alternative pairings
available to foster further cost improvement shall be quite limited, amounting
to smaller cost benefits in each phase of the CG-driven LPP-solutioning. This
would necessitate far more LPP-IPP interactions, to reach the near-optimal
cost. This pre se is not a challenge, however, significant amount of
additional run-time may be required, since: (a) each call for CG-driven LPP-
solutioning demands a minimum of 10 LPP iterations, before it could be
terminated, (b) such calls when invoked repeatedly, may consume significant
run-time, yet, without reasonable cost benefit.
* –
On the other hand, for a very large-sized $\mathcal{P}_{CG}^{t}$, though the
potential for significant cost benefits may exist, the demand on the RAM may
become overwhelming for any CG-driven LPP-solutioning phase to proceed.
The above considerations suggest that the size of $\mathcal{P}_{CG}^{t}$ may
neither be too small nor too large. Factoring these, the experiments here aim
at $\mathcal{P}_{CG}^{t}$ sized approximately of a million pairings
(significant size, yet, not overwhelming for 96 GB RAM). Furthermore, for a
search that is not biased in favor of any particular CG strategy, the number
of pairings from each CG strategy towards the overall CG heuristic are kept
equable.
* •
IPP-solutioning: As mentioned before, the MIP-search on a large-scale IPP is
time-intensive. Hence, the termination parameter– $Th_{ipt}$, that restricts
the run-time of any IPP-solutioning phase if not self-terminated a priori, is
reasonably set as 20 minutes, and its sensitivity on the $AirCROP$’s
performance is discussed in Section 4.3.4.
### 4.3 Results & Observations
This section presents the experimental results and associated inferences, in
the order highlighted below.
1. 1.
The performance of the proposed $AirCROP$ on the given test cases with the
aforementioned parameter settings is discussed.
2. 2.
The phenomenon referred to as performance variability (Lodi Tramontani, 2013)
is discussed in the context of $AirCROP$. This aspect is pertinent since some
variability in performance (even for the same random seed) is inevitable owing
to $AirCROP$’s reliance on the mathematical programming solvers, which over
the different runs may pick different permutations of the rows (flight-
coverage) or columns (pairings).
3. 3.
The impact of the initialization methods: (a) the proposed IPDCH, (b) an
Enhanced-DFS heuristic, earlier proposed by the authors (Aggarwal ., 2018),
and (c) a commonly adopted Artificial Pairings method (Hoffman Padberg, 1993;
Vance ., 1997), on the final performance of $AirCROP$ is investigated.
4. 4.
The sensitivity of $AirCROP$’s performance to the termination parameters in
the Optimization Engine’s submodules (CG-driven LPP-solutioning and IPP-
solutioning) has been discussed.
#### 4.3.1 AirCROP’s Performance
The results of the $AirCROP$-runs on the given test cases (TC-1 to TC-5) with
the aforementioned parameter settings are reported in Table 3. In that, for
each test case:
* •
the first row marked by “$\mathcal{P}_{IFS}$” highlights the cost associated
with the IFS that initializes the $AirCROP$-run and the run-time consumed in
its generation.
* •
the subsequent rows present the results of the LPP-IPP interactions (marked by
the counter $T$). In that, for a particular $T$, the cost of the LP-solution
passed on for its integerization and the associated time are highlighted. Also
the cost of the IP-solution returned and the associated time are highlighted.
Here, the unit of cost is USD, and the time corresponds to the HH:MM format.
* •
the final crew pairing solution ($\mathcal{P}^{\star}_{IP}$) is highlighted in
the last row (emboldened) marked by “Final Solution”.
It may be noted that the experimental results in the subsequent sections are
presented in the same format, unless any digression is specifically
highlighted.
Table 3: $AirCROP$’s performance∗ on the given test cases
LPP-IPP | TC-1 | TC-2 | TC-3 | TC-4 | TC-5
---|---|---|---|---|---
Interactions
$T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time
$\mathcal{P}_{IFS}$ | 85893202 | 00:05 | 81950079 | 00:05 | 51552744 | 00:03 | 131716653 | 00:08 | 89690776 | 00:06
1 | $\mathcal{P}_{LP}^{1}$ | 3468349 | 03:56 | 3493986 | 03:56 | 3483057 | 05:18 | 3595565 | 03:27 | 4583484 | 07:48
$\mathcal{P}_{IP}^{1}$ | 3689420 | 00:20 | 3715798 | 00:20 | 3697204 | 00:20 | 3807233 | 00:20 | 4930789 | 00:20
2 | $\mathcal{P}_{LP}^{2}$ | 3467837 | 02:18 | 3494675 | 01:19 | 3484645 | 02:42 | 3600195 | 01:17 | 4588740 | 02:49
$\mathcal{P}_{IP}^{2}$ | 3557615 | 00:20 | 3587139 | 00:20 | 3590336 | 00:20 | 3679138 | 00:20 | 4734553 | 00:20
3 | $\mathcal{P}_{LP}^{3}$ | 3469591 | 00:47 | 3495254 | 01:22 | 3486614 | 01:59 | 3600813 | 01:16 | 4592143 | 01:46
$\mathcal{P}_{IP}^{3}$ | 3518161 | 00:02 | 3546777 | 00:02 | 3523538 | 00:02 | 3639313 | 00:01 | 4654258 | 00:20
4 | $\mathcal{P}_{LP}^{4}$ | 3471619 | 01:13 | 3496797 | 00:57 | 3491000 | 01:13 | 3601168 | 01:27 | 4593422 | 02:17
$\mathcal{P}_{IP}^{4}$ | 3489534 | 00:01 | 3505941 | 00:01 | 3496142 | 00:01 | 3621723 | 00:01 | 4634187 | 00:01
5 | $\mathcal{P}_{LP}^{5}$ | 3472403 | 00:31 | 3497106 | 00:23 | 3490420 | 00:56 | 3604082 | 00:37 | 4594282 | 02:14
$\mathcal{P}_{IP}^{5}$ | 3484783 | 00:01 | 3497106 | 00:01 | 3490420 | 00:01 | 3612845 | 00:01 | 4617838 | 00:01
6 | $\mathcal{P}_{LP}^{6}$ | 3473238 | 00:30 | | | | | 3604753 | 00:28 | 4595481 | 01:53
$\mathcal{P}_{IP}^{6}$ | 3473238 | 00:01 | | | | | 3604753 | 00:01 | 4615272 | 00:01
7 | $\mathcal{P}_{LP}^{7}$ | | | | | | | | | 4596466 | 01:12
$\mathcal{P}_{IP}^{7}$ | | | | | | | | | 4600428 | 00:01
8 | $\mathcal{P}_{LP}^{8}$ | | | | | | | | | 4595613 | 01:42
$\mathcal{P}_{IP}^{8}$ | | | | | | | | | 4595613 | 00:01
Final Solution | 3473238 | 10:05 | 3497106 | 08:46 | 3490420 | 12:55 | 3604753 | 09:24 | 4595613 | 22:52
∗All values in the “Cost” columns are in USD, and all corresponding real
values are rounded-off to the next integer values. All values in the “Time”
columns are in HH:MM format, and all corresponding seconds’ values are
rounded-off to the next minute values.
The above results have been tested by the research consortium’s industrial
sponsor, and verified to be highly-competitive compared to the best practice
solutions known, for different test cases. In general, the obtained solutions
have been found to be superior by about 1.5 to 3.0% in terms of the hard cost,
which reportedly is one of the most important solution quality indicator. For
reference, a comparison of the obtained solution vis-$\grave{a}$-vis the best
known solution has been drawn for TC-5, in Table 4, where a significant
difference in terms of the size of pairings can be observed. Notably, the key
features contributing to lower hard cost relate to presence of pairings with
relatively lower - TAFB, overnight rests and meal cost. However, the obtained
solution also entails more crew changes, some of which (involving aircraft
change) negatively impact the soft cost. Hence, there appears to be a trade-
off between the hard cost and the soft cost.
Table 4: Salient features of $\mathcal{P}^{\star}_{IP}$ for TC-5: $AirCROP$’s solution vis-$\grave{a}$-vis the best practice solution Features | $\bm{AirCROP}$’s solution | Best practice solution
---|---|---
$\\#$ pairings | 926 | 783
$\\#$ unique flights covered | 4,212 | 4,212
$\\#$ deadhead flights | 3 | 3
$\\#$ overnight-rests | 1,203 | 1,279
$\\#$ crew changes | 1,002 | 825
$\\#$ average crew changes per pairing | 1.082 | 1.054
Total TAFB (HH:MM) | 37444:54 | 38189:39
$\\#$ pairings covering 2 flights | 303 | 205
$\\#$ pairings covering 3 flights | 17 | 31
$\\#$ pairings covering 4 flights | 170 | 95
$\\#$ pairings covering 5 flights | 63 | 37
$\\#$ pairings covering 6 flights | 202 | 153
$\\#$ pairings covering 7 flights | 59 | 62
$\\#$ pairings covering 8 flights | 83 | 90
$\\#$ pairings covering 9 flights | 19 | 49
$\\#$ pairings covering 10 flights | 8 | 45
$\\#$ pairings covering 11 flights | 1 | 10
$\\#$ pairings covering 12 flights | 1 | 5
$\\#$ pairings covering 13 flights | 0 | 0
$\\#$ pairings covering 14 flights | 0 | 1
Hotel cost (USD) | 166,240 | 176,170
Meal cost (USD) | 157,269 | 160,397
Hard cost (USD) | 340,671 | 350,818
Soft cost (USD) | 51,600 | 42,750
Actual flying cost (USD) | 4,203,342 | 4,203,342
Total cost (USD) | 4,595,613 | 4,596,910
#### 4.3.2 Performance Variability in AirCROP
These section investigates the sensitivity of $AirCROP$ with respect to the
sources of variability over multiple runs, even for the same problem. This
study assumes importance, considering that performance variability is rather
inevitable when the mathematical programming based solution approaches are
employed (Koch ., 2011). As cited by Lodi Tramontani (2013), variability in
the performance of LP & MIP solvers may be observed on – changing the
computing platform (which may change the floating-point arithmetic), permuting
the constraints/variables of the respective mathematical models, or changing
the pseudo-random numbers’ seed. These changes/permutations may lead to an
entirely different outcome of the respective search algorithms (LP & MIP), as
highlighted below.
* •
The root source for the performance variability in MIP is the imperfect tie-
breaking. A majority of the decisions to be taken during an MIP-search are
dependent on– the ordering of the candidates according to an interim score as
well as the selection of the best candidate (one with the best score value). A
perfect score that could fully-distinguish between the candidates is not-known
mostly due to the lack of theoretical knowledge, and even if it is known, it
may be too expensive to compute666For instance, in a strong branching scheme,
the best variable to branch at each node is decided after simulating one-level
of branching for each fractional variable, however, it is performed
heuristically to make it a computationally-affordable task for MIP solvers
(Linderoth Lodi, 2011). Furthermore, additional ties or tiebreaks could be
induced by changing the floating-point operations, which inherently may change
when the computing platform is changed. Amidst such an imperfect tie-breaking,
the permutation of the variables/constraints changes the path within the MIP-
search tree, leading to a completely different evolution of the algorithm with
rather severe consequences.
* •
Depending upon the floating-point arithmetic or the sequence of variables
loaded in an LPP, the performance of the simplex and interior-point methods
may vary.
* •
The performance of the LP and MIP solvers is also affected by the choice of
pseudo-random numbers’ seed, wherever the decisions are made heuristically.
For instance, an interior-point method in the LP solvers performs a (random)
crossover to one of the vertices of the optimal face when the search reaches
its (unique) center.
Table 5: Performance variability assessment for $AirCROP$ on two test
instances∗ (TC-2 and TC-5)
Test | LPP-IPP | Runs with performance variability | Runs without performance variability
---|---|---|---
Case | Interactions | Run-1 | Run-2 | Run (Seed-$\alpha$) | Run (Seed-$\beta$) | Run (Seed-$\gamma$)
$T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time
TC-2 | $\mathcal{P}_{IFS}$ | 81950079 | 00:05 | 74533686 | 00:04 | 129221508 | 00:08 | 114054265 | 00:07 | 52515476 | 00:04
1 | $\mathcal{P}_{LP}^{1}$ | 3493986 | 03:56 | 3494580 | 03:57 | 3495054 | 03:51 | 3493757 | 03:52 | 3493909 | 03:52
$\mathcal{P}_{IP}^{1}$ | 3715798 | 00:20 | 3746847 | 00:20 | 3769811 | 00:20 | 3711267 | 00:20 | 3722248 | 00:20
2 | $\mathcal{P}_{LP}^{2}$ | 3494675 | 01:19 | 3494540 | 02:18 | 3495311 | 01:57 | 3496733 | 01:57 | 3494657 | 03:02
$\mathcal{P}_{IP}^{2}$ | 3587139 | 00:20 | 3621066 | 00:20 | 3628514 | 00:20 | 3581978 | 00:20 | 3620745 | 00:20
3 | $\mathcal{P}_{LP}^{3}$ | 3495254 | 01:22 | 3496475 | 01:41 | 3497558 | 00:52 | 3499651 | 00:54 | 3496398 | 01:23
$\mathcal{P}_{IP}^{3}$ | 3546777 | 00:02 | 3555152 | 00:06 | 3566092 | 00:11 | 3536050 | 00:01 | 3551149 | 00:03
4 | $\mathcal{P}_{LP}^{4}$ | 3496797 | 00:57 | 3497750 | 01:36 | 3499237 | 01:26 | 3500818 | 01:03 | 3496069 | 01:37
$\mathcal{P}_{IP}^{4}$ | 3505941 | 00:01 | 3525600 | 00:01 | 3516807 | 00:01 | 3520552 | 00:01 | 3543236 | 00:02
5 | $\mathcal{P}_{LP}^{5}$ | 3497106 | 00:23 | 3498588 | 01:40 | 3500169 | 00:42 | 3500504 | 01:02 | 3496706 | 01:01
$\mathcal{P}_{IP}^{5}$ | 3497106 | 00:01 | 3498588 | 00:01 | 3517585 | 00:01 | 3500504 | 00:01 | 3501210 | 00:01
6 | $\mathcal{P}_{LP}^{6}$ | | | | | 3501523 | 00:43 | | | 3499063 | 00:41
$\mathcal{P}_{IP}^{6}$ | | | | | 3504085 | 00:01 | | | 3499063 | 00:01
7 | $\mathcal{P}_{LP}^{7}$ | | | | | 3502118 | 00:31 | | | |
$\mathcal{P}_{IP}^{7}$ | | | | | 3502118 | 00:01 | | | |
Final Solution | 3497106 | 08:46 | 3498588 | 12:05 | 3502118 | 11:05 | 3500504 | 09:38 | 3499063 | 12:27
TC-5 | $\mathcal{P}_{IFS}$ | 89690776 | 00:06 | 92080420 | 00:05 | 131443284 | 00:09 | 847887053 | 00:56 | 470430395 | 00:29
1 | $\mathcal{P}_{LP}^{1}$ | 4583484 | 07:48 | 4583476 | 08:00 | 4584525 | 07:28 | 4581988 | 08:47 | 4580130 | 07:36
$\mathcal{P}_{IP}^{1}$ | 4930789 | 00:20 | 4973580 | 00:20 | 4974341 | 00:20 | 4925863 | 00:20 | 4949616 | 00:20
2 | $\mathcal{P}_{LP}^{2}$ | 4588740 | 02:49 | 4588938 | 05:59 | 4589091 | 02:25 | 4584956 | 04:51 | 4584273 | 03:22
$\mathcal{P}_{IP}^{2}$ | 4734553 | 00:20 | 4765453 | 00:20 | 4782657 | 00:20 | 4749664 | 00:20 | 4753133 | 00:20
3 | $\mathcal{P}_{LP}^{3}$ | 4592143 | 01:46 | 4591571 | 02:35 | 4589952 | 02:14 | 4587812 | 03:02 | 4585046 | 03:40
$\mathcal{P}_{IP}^{3}$ | 4654258 | 00:20 | 4661078 | 00:20 | 4736313 | 00:20 | 4653279 | 00:20 | 4666390 | 00:20
4 | $\mathcal{P}_{LP}^{4}$ | 4593422 | 02:17 | 4595741 | 01:49 | 4591145 | 02:36 | 4589247 | 02:00 | 4588952 | 02:56
$\mathcal{P}_{IP}^{4}$ | 4634187 | 00:01 | 4624039 | 00:01 | 4654627 | 00:20 | 4614651 | 00:01 | 4628239 | 00:01
5 | $\mathcal{P}_{LP}^{5}$ | 4594282 | 02:14 | 4599006 | 01:14 | 4592463 | 02:03 | 4590573 | 01:05 | 4589577 | 02:02
$\mathcal{P}_{IP}^{5}$ | 4617838 | 00:01 | 4613385 | 00:01 | 4632708 | 00:02 | 4603938 | 00:01 | 4618710 | 00:01
6 | $\mathcal{P}_{LP}^{6}$ | 4595481 | 01:53 | 4598727 | 01:11 | 4593094 | 02:00 | 4591176 | 01:15 | 4589874 | 01:48
$\mathcal{P}_{IP}^{6}$ | 4615272 | 00:01 | 4605126 | 00:01 | 4625993 | 00:01 | 4591176 | 00:01 | 4607590 | 00:01
7 | $\mathcal{P}_{LP}^{7}$ | 4596466 | 01:12 | 4598412 | 01:39 | 4593431 | 01:04 | | | 4590674 | 01:24
$\mathcal{P}_{IP}^{7}$ | 4600428 | 00:01 | 4598412 | 00:01 | 4619643 | 00:01 | | | 4605058 | 00:01
8 | $\mathcal{P}_{LP}^{8}$ | 4595613 | 01:42 | | | 4594146 | 01:03 | | | 4591065 | 02:10
$\mathcal{P}_{IP}^{8}$ | 4595613 | 00:01 | | | 4594146 | 00:01 | | | 4591065 | 00:01
Final Solution | 4595613 | 22:52 | 4598412 | 23:37 | 4594146 | 22:27 | 4591176 | 22:59 | 4591065 | 26:32
∗All values in the “Cost” columns are in USD, and all the corresponding real
values are rounded-off to the next integer values. All values in the “Time”
columns are in HH:MM, and all the corresponding seconds’ values are rounded-
off to the next minute values.
In the above background, the plausible reasons for variability in $AirCROP$’s
performance are elaborated below.
* •
Generation of new legal pairings using a parallel architecture: in any LPP
iteration $t$, new legal pairings are generated in parallel, by allocating the
sub-processes to the idle-cores of the CPU. These sub-processes return their
respective pairing sets as soon as they are terminated. This by itself is not
a challenge, however, when the $AirCROP$ is re-run, the order in which these
sub-processes terminate may not be same as before (as it depends on the state
of the CPU), permuting the pairings in the cumulative pairing set
$\mathcal{P}_{CG}^{t}$. This permuted pairing set, when fed as part of the
input to the LP solver in the next LPP iteration, may lead to a different LPP
solution, leading to a different outcome of the subsequent $AirCROP$’s search.
To curb this, the pairings in the set that trigger the LP solver are sorted in
lexicographical order of their representative strings. These strings are
constructed from the indices of the flights covered in the corresponding
pairings. For instance, the string corresponding to a pairing that covers
flights $f_{1}$, $f_{10}$, $f_{100}$ & $f_{200}$ is $1\\_10\\_100\\_200$.
Given that the pairings are distinct, the resulting strings are distinct too,
allowing for a crisp sorting criterion and ensuring a fixed pairing sequence
in each $AirCROP$-run.
* •
Numerical seed for generation of pseudo-random numbers: variability may also
be introduced if the numerical seed employed to generate pseudo-random numbers
for use in the proposed modules or the utilized LP & MIP solvers, varies. For
instance, use of the default seed method of Python (i.e., the current time of
the computing system) across different $AirCROP$ runs may lead to different
pseudo-random numbers, each time. This in turn would trigger variability in
the IFS generated by IPDCH (since the random selection of flights in each of
its iterations, is impacted), and the pairing set resulting from the CG
heuristic (since each of the underlying CG strategy is impacted). Such
variability could be negated by use of a fixed numerical seed, instead of a
time dependent one.
The intriguing questions for researchers could relate to the impact that
presence or absence of causes of variability may have on the quality of
$AirCROP$’s solutions, in terms of both cost and run-time. Table 5 attempts to
shed light on these questions through empirical evidence for two test cases
involving 3228 flights (TC-2) and 4212 flights (TC-5), respectively. In each
of these test cases, the effect of variability is revealed through:
* •
two independent runs (Run-1 and Run-2), in each of which the causes of
variability exist, that is: (a) the permutations of pairings generated using
the parallel architecture is possible, and (b) the default seed method of
Python, based on the time of the computing system applies.
* •
three independent runs, in each of which the causes of variability have been
eliminated, that is: (a) the lexicographical order of the pairings is imposed,
and (b) a fixed numerical seed has been fed for random number generation. For
these runs, the numerical seeds are given by $\alpha=0$, $\beta=1$, and
$\gamma=2$, respectively.
The key observations and inferences that could be drawn from each test case in
Table 5 are highlighted below.
* •
understandably, the Run-1 and Run-2 (corresponding to the same numerical
seed), yield different cost solutions over different run-time. Importantly,
the variation in cost (despite the presence of causes of variability) is not
alarming, though significantly different run-times may be required.
* •
each run (corresponding to Seed-$\alpha$, Seed-$\beta$, and Seed-$\gamma$,
respectively) where the causes of variability have been negated, if repeated,
yield the same cost solution in the same run-time though it has not been shown
in the table for paucity of space.
* •
the runs corresponding to the numerical seeds given by $\alpha$, $\beta$, and
$\gamma$, respectively, differ solely due to the difference in the
corresponding random numbers generated, and subsequently utilized. It can be
observed that the change in numerical seed does not significantly affect the
cost-quality of the final $AirCROP$ solution though the associated run-time
may vary significantly.
The fact that $AirCROP$ can offer final solutions with comparable cost
quality, regardless of the presence or absence of causes of variability,
endorses the robustness of the constitutive modules of the $AirCROP$. Also,
the variation in run-time could be attributed to different search trajectories
corresponding to different permutations of variables or different random
numbers. It may be noted that for the subsequent runs the lexicographical
order of the pairings and a fixed numerical seed (Seed-$\alpha=0$) have been
utilized.
#### 4.3.3 Impact of Initialization on AirCROP’s Performance
This section investigates the sensitivity of $AirCROP$ with respect to the
cost quality of the initial solution and the run-time spent to obtain it.
Towards it, the initial solution is obtained using three different methods
(offering three input alternatives with varying cost and run-time) and the
cost quality of $AirCROP^{\prime}s$ final solution alongside the necessary
run-time is noted.
Notably, in an initial attempt to generate IFS for large-scale CPOPs, the
authors proposed a DFS algorithm based heuristic, namely, Enhanced-DFS
heuristic (Aggarwal ., 2018). Its performance across the five test cases has
been highlighted in Table 6. In that, TC-1 emerges as an outlier owing to
alarmingly high run-time, when compared to all other test cases.
Table 6: Performance of Enhanced-DFS heuristic (Aggarwal ., 2018) for IFS generation. Here, the real valued “Cost” is rounded-off to the next integer value, and the seconds’ in the “Time” column are rounded-off to the next minute values. Test Cases | Time (HH:MM) | Cost (USD) | # Pairings
---|---|---|---
TC-1 | 01:48 | 3863669070 | 477617
TC-2 | 00:02 | 167405376 | 26678
TC-3 | 00:03 | 167967482 | 26871
TC-4 | 00:13 | 1072078483 | 135269
TC-5 | 00:04 | 325922318 | 51920
A plausible explanation behind this aberration is that TC-1 involves some
flights with very few legal flight connections, and a DFS based algorithm may
have to exhaustively explore several flight connections, to be able to
generate an IFS with full flight coverage. The need to do away with reliance
on DFS so as to have equable run-time across different data sets explains the
motivation for:
* •
proposition of IPDCH in this paper, which as highlighted in Section 3.2,
relies on: (a) a divide-and-cover strategy to decompose the input flight
schedule into sufficiently-small flight subsets, and (b) IP to find a lowest-
cost pairing set that covers the maximum-possible flights for each of the
decomposed flight subsets.
* •
consideration of a commonly adopted Artificial Pairings method (Vance .,
1997), that constructs a pairing set which covers all the flights, though
some/all the pairings may not be legal. Hence, for this method the initial
solution would be referred as $\mathcal{P}_{IS}$ instead of
$\mathcal{P}_{IFS}$.
Table 7: Performance assessment of $AirCROP$ on TC-1 and TC-5 when initialized
using the proposed IPDCH, the Artificial Pairings method, and the Enhanced-DFS
heuristic.
LPP-IPP | TC-1 | TC-5
---|---|---
Interactions | Enhanced-DFS | IPDCH | Artificial Pairings | Enhanced-DFS | IPDCH | Artificial Pairings
$T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time
$\mathcal{P}_{IFS}/\mathcal{P}_{IS}$ | 3863669070 | 01:48 | 74945982 | 00:05 | 14604919138 | $\approx$00:00 | 325922318 | 00:04 | 131443284 | 00:09 | 25409939785 | $\approx$00:00
1 | $\mathcal{P}_{LP}^{1}$ | 3463560 | 04:14 | 3465379 | 04:19 | 17589714 | 03:10 | 4583664 | 06:57 | 4584525 | 07:28 | 4585380 | 07:29
$\mathcal{P}_{IP}^{1}$ | 3650828 | 00:20 | 3689312 | 00:20 | 17833718 | 00:20 | 4943531 | 00:20 | 4974341 | 00:20 | 4960813 | 00:20
2 | $\mathcal{P}_{LP}^{2}$ | 3464678 | 01:51 | 3466567 | 01:32 | 17589851 | 01:29 | 4586675 | 03:41 | 4589091 | 02:25 | 4589470 | 03:34
$\mathcal{P}_{IP}^{2}$ | 17731125 | 00:20 | 3566415 | 00:20 | 3578030 | 00:20 | 4773342 | 00:20 | 4809348 | 00:20 | 4782657 | 00:20
3 | $\mathcal{P}_{LP}^{3}$ | 3466217 | 01:38 | 3467848 | 01:33 | 3466868 | 02:02 | 4586581 | 04:54 | 4589952 | 02:14 | 4593117 | 02:05
$\mathcal{P}_{IP}^{3}$ | 3531694 | 00:13 | 3556499 | 00:20 | 3553432 | 00:20 | 4701607 | 00:20 | 4736313 | 00:20 | 4672696 | 00:20
4 | $\mathcal{P}_{LP}^{4}$ | 3467672 | 01:19 | 3468777 | 01:26 | 3467935 | 00:47 | 4589568 | 01:51 | 4591145 | 02:36 | 4593938 | 02:18
$\mathcal{P}_{IP}^{4}$ | 3507987 | 00:01 | 3517901 | 00:01 | 3516376 | 00:02 | 4651824 | 00:20 | 4654627 | 00:20 | 4650449 | 00:06
5 | $\mathcal{P}_{LP}^{5}$ | 3469533 | 00:44 | 3468894 | 00:49 | 3468332 | 00:40 | 4591698 | 02:03 | 4592463 | 02:03 | 4596256 | 02:24
$\mathcal{P}_{IP}^{5}$ | 3483690 | 00:01 | 3499531 | 00:01 | 3496156 | 00:01 | 4616605 | 00:01 | 4632708 | 00:02 | 4620903 | 00:01
6 | $\mathcal{P}_{LP}^{6}$ | 3469276 | 00:52 | 3469352 | 00:48 | 3469095 | 00:47 | 4591969 | 01:05 | 4593094 | 02:00 | 4597203 | 00:49
$\mathcal{P}_{IP}^{6}$ | 3469276 | 00:01 | 3477354 | 00:01 | 3491947 | 00:01 | 4606253 | 00:01 | 4625993 | 00:01 | 4612164 | 00:01
7 | $\mathcal{P}_{LP}^{7}$ | | | 3469950 | 00:42 | 3469543 | 00:52 | 4592860 | 01:15 | 4593431 | 01:04 | 4597913 | 01:17
$\mathcal{P}_{IP}^{7}$ | | | 3469950 | 00:01 | 3487562 | 00:01 | 4592860 | 00:01 | 4619643 | 00:01 | 4606368 | 00:01
8 | $\mathcal{P}_{LP}^{8}$ | | | | | 3470100 | 00:38 | | | 4594146 | 01:03 | 4597730 | 02:00
$\mathcal{P}_{IP}^{8}$ | | | | | 3478057 | 00:01 | | | 4594146 | 00:01 | 4604551 | 00:01
9 | $\mathcal{P}_{LP}^{9}$ | | | | | 3470355 | 00:28 | | | | | 4597929 | 00:50
$\mathcal{P}_{IP}^{9}$ | | | | | 3470355 | 00:01 | | | | | 4597929 | 00:01
Final Solution | 3469276 | 13:22 | 3469950 | 12:18 | 3470355 | 12:00 | 4592860 | 23:13 | 4594146 | 22:27 | 4597929 | 23:57
∗All values in the “Cost” columns are in USD, where the real values are
rounded-off to the next integer values. All values in the “Time” columns are
in HH:MM, where the seconds’ values are rounded-off to the next minute values.
A comparison of the above three methods has been drawn in Table 7, for TC-1
(posing challenge to Enhanced-DFS) and TC-5 (largest flight set). In that,
besides the cost and run-time of the initial solution for each test case, the
results of all the iterations of $AirCROP$ leading up to the final solution
have been presented. The latter is done to shed light on whether
$AirCROP^{\prime}s$ final solution cost quality strongly depends on the cost
of the initial solution. The prominent observations from the Table 7 include:
* •
In terms of run-time: IPDCH could outperform the Enhanced-DFS, as its run-time
happened to be less than ten minutes in both the test cases. The Artificial
pairing method even out performs IPDCH, since its run-time happened to be in
milliseconds (formatted to 0 minutes in the table).
* •
In terms of initial cost: IPDCH could again outperform the Enhanced-DFS. This
could be attributed to the use of IP to find a lowest-cost pairing set that
covers the maximum-possible flights for each of the decomposed flight subsets.
In contrast, the cost associated with the Artificial pairing method, is the
worst. This is owing to a very high pseudo-cost attached to the pairings to
offset their non-legality.
Critically, regardless of the significantly varying run-time and the initial
cost associated with the three methods, the variation in the cost of the final
solution offered by $AirCROP$ is not significant. This endorses the robustness
of its constitutive modules.
#### 4.3.4 Impact of Termination Settings of Optimization Engine’s Submodules
on AirCROP’s Performance
This section investigates the sensitivity of $AirCROP$ to the termination
parameter settings of the Optimization Engine’s submodules, namely, LPP-
solutioning and IPP-solutioning. The parameters involved in LPP-solutioning
are $Th_{cost}$ and $Th_{t}$, while $Th_{ipt}$ is involved in IPP-solutioning.
To assess their impact on $AirCROP^{\prime}s$ performance, experiments are
performed with three different sets of parameter settings each, for both the
submodules.
Impact of Termination Settings of CG-driven LPP-solutioning:
As mentioned earlier, the CG-driven LPP-solutioning is terminated if the cost-
improvement per LPP iteration falls below the pre-specified threshold
$Th_{cost}$ (in USD) over $Th_{t}$ number of successive LPP iterations. To
achieve a reasonable balance between $AirCROP^{\prime}s$ run time on the one
hand and the cost reduction of the crew pairing solution on the other hand,
three different sets of parameter settings are chosen, and experimented with.
These settings of $\\{Th_{cost},Th_{t}\\}$ including $\\{500,5\\}$,
$\\{100,10\\}$, and $\\{50,15\\}$ symbolize relaxed, moderate and strict
settings, respectively, since the criterion for $AirCROP^{\prime}s$
termination gets more and more difficult as the settings change from
$\\{500,5\\}$ to $\\{50,15\\}$. The results of the $AirCROP$-runs
corresponding to these termination settings are reported in Table 8, and the
key observations are as highlighted below.
* •
As the termination settings transition through relaxed, moderate and strict
settings, the run-time to obtain the final solution increases, while the cost
of the final solution decreases. An apparent exception to this trend is
observed in TC-5 with the strict setting, but this could be explained by the
fact that the upper limit of 30 hours set for $AirCROP^{\prime}s$ run time
under practical considerations was exceeded during the fourth LPP-IPP
interaction ($T=4$). It implies that due to the enforced termination in this
particular case, $AirCROP$ could not fully utilize the potential for cost
reduction.
* •
Despite the variation in the termination settings, the cost quality of
$AirCROP^{\prime}s$ final solution does not vary as drastically, as its run
time. For instance, as the settings switched from relaxed to moderate: an
additional saving of 6384 USD could be achieved at the expense of additional
5:20 run time in the case of TC-2, while these indicators stand at 13388 USD
and 10:25, respectively, in the case of TC-5. It can also be inferred that
$\\{Th_{cost},Th_{t}\\}$ set as $\\{100,10\\}$ possibly offers a fair balance
between solution’s cost quality and run time, and this explains why these
settings have been used as the base settings for the experimental results
presented in this paper, beginning with Table 3 and ending with Table 9.
It is important to recognize that as the termination settings for LPP-
solutioning are made stricter, its run time is bound to increase. It is also
fair to expect that the cost quality of the final solution may be better,
though it cannot be guaranteed. Any such departures from the expected trend
may be due to the dependence of the quality of the final solution on the
quality of the IPP-solution for each $T$. In that, if an IPP-solution for a
particular $T$ may largely fail to approach the lower bound set by the
corresponding LPP-solution, it may negatively influence the cost quality
obtained in subsequent LPP- and IPP-solutioning phases. While such a
possibility remains, it did not surface in the experiments above.
Table 8: Performance assessment of $AirCROP$ on TC-2 and TC-5, against three
different termination settings (Relaxed, Moderate and Strict Settings) of the
CG-driven LPP-solutioning∗
LPP-IPP | TC-2 | TC-5
---|---|---
Interactions | Relaxed Setting | Moderate Setting | Strict Setting | Relaxed Setting | Moderate Setting | Strict Setting
| | $\bm{Th_{cost}=}$ 500, $\bm{Th_{t}=}$ 5 | $\bm{Th_{cost}=}$ 100, $\bm{Th_{t}=}$ 10 | $\bm{Th_{cost}=}$ 50, $\bm{Th_{t}=}$ 15 | $\bm{Th_{cost}=}$ 500, $\bm{Th_{t}=}$ 5 | $\bm{Th_{cost}=}$ 100, $\bm{Th_{t}=}$ 10 | $\bm{Th_{cost}=}$ 50, $\bm{Th_{t}=}$ 15
$T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time
$\mathcal{P}_{IFS}$ | 129221508 | 00:08 | 129221508 | 00:08 | 129221508 | 00:08 | 131443284 | 00:09 | 131443284 | 00:09 | 131443284 | 00:09
1 | $\mathcal{P}_{LP}^{1}$ | 3510231 | 01:12 | 3495054 | 03:51 | 3489337 | 08:21 | 4603984 | 02:43 | 4584525 | 07:28 | 4581711 | 10:05
$\mathcal{P}_{IP}^{1}$ | 3844119 | 00:20 | 3769811 | 00:20 | 3698316 | 00:20 | 5165821 | 00:20 | 4974341 | 00:20 | 4946510 | 00:20
2 | $\mathcal{P}_{LP}^{2}$ | 3510105 | 00:26 | 3495311 | 01:57 | 3491725 | 03:36 | 4605049 | 01:12 | 4589091 | 02:25 | 4582977 | 09:32
$\mathcal{P}_{IP}^{2}$ | 3729820 | 00:20 | 3628514 | 00:20 | 3607470 | 00:20 | 4962643 | 00:20 | 4782657 | 00:20 | 4780005 | 00:20
3 | $\mathcal{P}_{LP}^{3}$ | 3506864 | 00:35 | 3497558 | 00:52 | 3491685 | 04:15 | 4602283 | 01:09 | 4589952 | 02:14 | 4585457 | 05:56
$\mathcal{P}_{IP}^{3}$ | 3659201 | 00:20 | 3566092 | 00:11 | 3578774 | 00:20 | 4818918 | 00:20 | 4736313 | 00:20 | 4678596 | 00:20
4 | $\mathcal{P}_{LP}^{4}$ | 3506644 | 00:32 | 3499237 | 01:26 | 3494201 | 02:29 | 4604535 | 00:52 | 4591145 | 02:36 | 4595692 | 03:34
$\mathcal{P}_{IP}^{4}$ | 3606381 | 00:20 | 3516807 | 00:01 | 3540972 | 00:01 | 4727106 | 00:20 | 4654627 | 00:20 | 4624747 | 00:01
5 | $\mathcal{P}_{LP}^{5}$ | 3507647 | 00:29 | 3500169 | 00:42 | 3494409 | 02:36 | 4603253 | 00:47 | 4592463 | 02:03 | |
$\mathcal{P}_{IP}^{5}$ | 3559484 | 00:04 | 3517585 | 00:01 | 3527254 | 00:01 | 4683130 | 00:20 | 4632708 | 00:02 | |
6 | $\mathcal{P}_{LP}^{6}$ | 3507101 | 00:20 | 3501523 | 00:43 | 3496498 | 01:00 | 4603093 | 00:45 | 4593094 | 02:00 | |
$\mathcal{P}_{IP}^{6}$ | 3547304 | 00:02 | 3504085 | 00:01 | 3496498 | 00:01 | 4681335 | 00:20 | 4625993 | 00:01 | |
7 | $\mathcal{P}_{LP}^{7}$ | 3508166 | 00:18 | 3502118 | 00:31 | | | 4603638 | 00:46 | 4593431 | 01:04 | |
$\mathcal{P}_{IP}^{7}$ | 3517436 | 00:01 | 3502118 | 00:01 | | | 4651002 | 00:06 | 4619643 | 00:01 | |
8 | $\mathcal{P}_{LP}^{8}$ | 3508502 | 00:17 | | | | | 4604073 | 00:44 | 4594146 | 01:03 | |
$\mathcal{P}_{IP}^{8}$ | 3508502 | 00:01 | | | | | 4634316 | 00:02 | 4594146 | 00:01 | |
9 | $\mathcal{P}_{LP}^{9}$ | | | | | | | 4606250 | 00:28 | | | |
$\mathcal{P}_{IP}^{9}$ | | | | | | | 4614420 | 00:01 | | | |
10 | $\mathcal{P}_{LP}^{10}$ | | | | | | | 4607534 | 00:17 | | | |
$\mathcal{P}_{IP}^{10}$ | | | | | | | 4607534 | 00:01 | | | |
Final Solution | 3508502 | 05:45 | 3502118 | 11:05 | 3496498 | 23:28 | 4607534 | 12:02 | 4594146 | 22:27 | 4624747 | 30:17
∗All values in the “Cost” columns are in USD, and all the corresponding real
values are rounded-off to the next integer values. All values in the “Time”
columns are in HH:MM, and all the corresponding seconds’ values are rounded-
off to the next minute values.
Impact of Termination Settings of IPP-solutioning:
As mentioned before, integerization of an LPP solution using an MIP solver is
extremely time-consuming, particularly for large-scale CPOPs, and more so
those involving complex flight networks. Hence, from a practical perspective,
the $AirCROP$ framework imposes a threshold on the upper time limit for IPP-
solutioning (for any given $T$), namely $Th_{ipt}$, in case it does not self-
terminate a priori. To investigate the impact of $Th_{ipt}$ on
$AirCROP^{\prime}s$ performance, experiments are performed with three
different settings, including, 00:20 (one-third of an hour), 00:40 (two-third
of an hour), and 01:00 (an hour). The results are presented in Table 9, and
the key observations are as follows. In the case of TC-2, as the $Th_{ipt}$ is
raised, the run-time to obtain the final solution increases, while the cost of
the final solution decreases. However, there are exceptions to this trend in
the case of TC-5. Notably, the cost quality of the final solution
corresponding to $Th_{ipt}=$ 00:20 remains superior to that obtained for both
$Th_{ipt}=$ 00:40 and 01:00. For these two settings, the quality of LPP-
solution at $T=8$ turned worse compared to the case of $Th_{ipt}=$ 00:20, and
the gap could not be bridged even in the subsequent LPP-IPP interaction
($T=9$). The worsening of LPP-solution could be attributed to the fact that
LPP-solutioning relies on random number based heuristics, and the resulting
pairing combinations may not necessarily offer lower cost within the pre-
specified termination settings.
Based on the above, it may be inferred that despite the changes in the
termination parameter settings, $AirCROP$ is able to offer solutions with
reasonably close cost quality, though significant variations in run time may
be observed. It is also evident that even the lowest setting (desired from a
practical perspective) for $Th_{ipt}=$ 00:20 offers a good balance between
solution’s cost quality and run time, and this explains why it has been used
as the base setting for the experimental results presented in this paper.
Table 9: Performance assessment of $AirCROP$ on TC-2 and TC-5, against three
different termination settings ($Th_{ipt}=$ 00:20, 00:40 & 01:00) of the IPP-
solutioning∗
LPP-IPP | TC-2 | TC-5
---|---|---
Interactions | $\bm{Th_{ipt}=}$ 00:20 | $\bm{Th_{ipt}=}$ 00:40 | $\bm{Th_{ipt}=}$ 01:00 | $\bm{Th_{ipt}=}$ 00:20 | $\bm{Th_{ipt}=}$ 00:40 | $\bm{Th_{ipt}=}$ 01:00
$T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time
$\mathcal{P}_{IFS}$ | 129221508 | 00:08 | 129221508 | 00:08 | 129221508 | 00:08 | 131443284 | 00:09 | 131443284 | 00:09 | 131443284 | 00:09
1 | $\mathcal{P}_{LP}^{1}$ | 3495054 | 03:51 | 3495054 | 04:21 | 3495054 | 03:57 | 4584525 | 07:28 | 4584525 | 07:46 | 4584525 | 07:49
$\mathcal{P}_{IP}^{1}$ | 3769811 | 00:20 | 3744301 | 00:40 | 3760028 | 01:00 | 4974341 | 00:20 | 4958532 | 00:40 | 4987497 | 01:00
2 | $\mathcal{P}_{LP}^{2}$ | 3495311 | 01:57 | 3497483 | 01:49 | 3495562 | 02:19 | 4589091 | 02:25 | 4585347 | 05:00 | 4588371 | 03:56
$\mathcal{P}_{IP}^{2}$ | 3628514 | 00:20 | 3629401 | 00:40 | 3632875 | 01:00 | 4782657 | 00:20 | 4778465 | 00:40 | 4766924 | 01:00
3 | $\mathcal{P}_{LP}^{3}$ | 3497558 | 00:52 | 3497473 | 01:33 | 3494305 | 01:50 | 4589952 | 02:14 | 4589481 | 02:56 | 4588911 | 04:25
$\mathcal{P}_{IP}^{3}$ | 3566092 | 00:11 | 3566247 | 00:40 | 3579899 | 01:00 | 4736313 | 00:20 | 4699845 | 00:40 | 4713402 | 01:00
4 | $\mathcal{P}_{LP}^{4}$ | 3499237 | 01:26 | 3500607 | 01:06 | 3495273 | 01:13 | 4591145 | 02:36 | 4590618 | 01:56 | 4591028 | 02:09
$\mathcal{P}_{IP}^{4}$ | 3516807 | 00:01 | 3524672 | 00:01 | 3551863 | 00:05 | 4654627 | 00:20 | 4656611 | 00:40 | 4681015 | 01:00
5 | $\mathcal{P}_{LP}^{5}$ | 3500169 | 00:42 | 3501809 | 00:49 | 3496754 | 00:52 | 4592463 | 02:03 | 4591826 | 01:18 | 4591448 | 02:03
$\mathcal{P}_{IP}^{5}$ | 3517585 | 00:01 | 3501809 | 00:01 | 3528564 | 00:01 | 4632708 | 00:02 | 4644467 | 00:15 | 4639287 | 00:29
6 | $\mathcal{P}_{LP}^{6}$ | 3501523 | 00:43 | | | 3496342 | 00:53 | 4593094 | 02:00 | 4592492 | 02:21 | 4591372 | 02:06
$\mathcal{P}_{IP}^{6}$ | 3504085 | 00:01 | | | 3512692 | 00:01 | 4625993 | 00:01 | 4617694 | 00:01 | 4616944 | 00:01
7 | $\mathcal{P}_{LP}^{7}$ | 3502118 | 00:31 | | | 3497967 | 00:59 | 4593431 | 01:04 | 4594599 | 01:30 | 4594479 | 01:23
$\mathcal{P}_{IP}^{7}$ | 3502118 | 00:01 | | | 3519996 | 00:01 | 4619643 | 00:01 | 4607261 | 00:01 | 4608085 | 00:01
8 | $\mathcal{P}_{LP}^{8}$ | | | | | 3498726 | 01:24 | 4594146 | 01:03 | 4595739 | 01:08 | 4595424 | 01:03
$\mathcal{P}_{IP}^{8}$ | | | | | 3518299 | 00:01 | 4594146 | 00:01 | 4598624 | 00:01 | 4603634 | 00:01
9 | $\mathcal{P}_{LP}^{9}$ | | | | | 3499104 | 00:40 | | | 4595703 | 00:45 | 4596929 | 00:59
$\mathcal{P}_{IP}^{9}$ | | | | | 3504258 | 00:01 | | | 4595703 | 00:01 | 4596929 | 00:01
10 | $\mathcal{P}_{LP}^{10}$ | | | | | 3499117 | 01:10 | | | | | |
$\mathcal{P}_{IP}^{10}$ | | | | | 3509608 | 00:01 | | | | | |
11 | $\mathcal{P}_{LP}^{11}$ | | | | | 3499609 | 00:45 | | | | | |
$\mathcal{P}_{IP}^{11}$ | | | | | 3499609 | 00:01 | | | | | |
Final solution | 3502118 | 11:05 | 3501809 | 12:24 | 3499609 | 19:22 | 4594146 | 22:27 | 4595703 | 27:55 | 4596929 | 30:35
∗All values in the “Cost” columns are in USD, and all the corresponding real
values are rounded-off to the next integer values. All values in the “Time”
columns are in HH:MM, and all the corresponding seconds’ values are rounded-
off to the next minute values.
## 5 Conclusion and Future Research
For an airline, crew operating cost is the second largest expense, after the
fuel cost, making the crew pairing optimization critical for business
viability. Over the last three decades, CPOP has received an unprecedented
attention from the OR community, as a result of which numerous CPOP solution
approaches have been proposed. Yet, the emergent flight networks with conjunct
scale and complexity largely remain unaddressed in the available literature.
Such a scenario is all the more alarming, considering that the air traffic is
expected to scale up to double over the next 20 years, wherein, most airlines
may need to cater to multiple crew bases and multiple hub-and-spoke
subnetworks. This research has proposed an Airline Crew Pairing Optimization
Framework ($AirCROP$) based on domain-knowledge driven CG strategies for
efficiently tackling real-world, large-scale and complex flight networks. This
paper has presented not just the design of the $AirCROP$’s constitutive
modules, but has also shared insights on how these modules interact and how
sensitive the $AirCROP^{\prime}s$ performance is to the sources of
variability, choice of different methods and parameter settings.
Given a CPOP, $AirCROP$ first preprocesses the entire duty overnight-
connection network via its Legal Crew Pairing Generator777This module is
utilized again to facilitate legal crew pairings when required in real-time in
other modules of $AirCROP$ Subsequently, $AirCROP$ is initialized using an IFS
generated by the proposed method (IPDCH). Next, the $AirCROP$’s Optimization
Engine attempts to find a good-quality CPOP solution via intermittent
interactions of its submodules, namely, CG-driven LPP-solutioning and IPP-
solutioning. The efficacy of $AirCROP$ has been demonstrated on real-world
airline flight network characterized by an unprecedented (in reference to
available literature) conjunct scale-and-complexity, marked by over 4200
flights, 15 crew bases, multiple hub-and-spoke subnetworks, and billion-plus
pairings. The distinctive contribution of this paper is also embedded in its
empirical investigation of critically important questions relating to
variability and sensitivity, which the literature is otherwise silent on. In
that:
* •
first, the sensitivity analysis of $AirCROP$ is performed in the presence and
absence of sources of variability. It is empirically highlighted that
$AirCROP$ is capable of offering comparable cost solutions, both in the
presence or absence of the sources of variability. This endorses the
robustness of its constitutive modules.
* •
second, the sensitivity of $AirCROP$ with respect to the cost quality of the
initial solution and the associated run-time is investigated
vis-$\grave{a}$-vis three different initialization methods. Again, the
robustness of $AirCROP$ is endorsed, considering that it is found to be
capable of offering similar cost solutions, despite the significantly varying
cost and run-time of the initial solutions.
* •
last, the sensitivity of $AirCROP$ to the termination parameter settings
associated with the Optimization Engine’s submodules, is investigated. The
fact that with the variation in termination settings of both LPP-solutioning
and IPP-solutioning (independent of each other)- the $AirCROP$’s performance
strongly aligns with the logically expected trends, is a testimony to the
robustness of its constitutive modules.
Notably, $AirCROP$ has been implemented using Python scripting language,
aligned with the industrial sponsor’s preferences. However, a significant
reduction in run-time could be achieved by the use of compiled programming
languages such as C++, Java, etc. Moreover, employing the domain-knowledge
driven CG strategies during the IPP-solutioning phase too, may augment the
overall cost- and time-efficiency of the $AirCROP$. Furthermore, the emerging
trend of utilizing the Machine Learning capabilities for assisting
combinatorial optimization tasks, may also hold promise for the airline crew
pairing optimization, towards which an exploratory attempt has been made by
the authors (Aggarwal, Singh Saxena, 2020). Despite the scope for improvement,
the authors hope that with the emergent trend of evolving scale and complexity
of airline flight networks, this paper shall serve as an important milestone
for the affiliated research and applications.
## Acknowledgment
This research work is a part of an Indo-Dutch joint research project,
supported by the Ministry of Electronics and Information Technology (MEITY),
India [grant number 13(4)/2015-CC&BT]; Netherlands Organization for Scientific
Research (NWO), the Netherlands; and General Electric (GE) Aviation, India.
The authors thank GE Aviation, particularly, Saaju Paulose (Senior Manager),
Arioli Arumugam (Senior Director- Data & Analytics), and Alla Rajesh (Senior
Staff Data & Analytics Scientist) for providing real-world test cases, and
sharing their domain knowledge which has helped the authors significantly in
successfully completing this research work.
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* Virtanen . (2020) scipyVirtanen, P., Gommers, R., Oliphant, TE., Haberland, M., Reddy, T., Cournapeau, D.Contributors, S. 2020\. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods17261–272. https://doi.org/10.1038/s41592-019-0686-2
* Zeren Özkol (2012) zeren2012improvedZeren, B. Özkol, İ. 2012\. An improved genetic algorithm for crew pairing optimization An improved genetic algorithm for crew pairing optimization. Journal of Intelligent Learning Systems and Applications40170.
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|
2024-09-04T02:54:58.320151 | 2020-03-09T09:39:36 | 2003.03999 | {
"authors": "LHCb collaboration: R. Aaij, C. Abell\\'an Beteta, T. Ackernley, B.\n Adeva, M. Adinolfi, H. Afsharnia, C.A. Aidala, S. Aiola, Z. Ajaltouni, S.\n Akar, P. Albicocco, J. Albrecht, F. Alessio, M. Alexander, A. Alfonso Albero,\n G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, L. An,\n L. Anderlini, G. Andreassi, M. Andreotti, F. Archilli, A. Artamonov, M.\n Artuso, K. Arzymatov, E. Aslanides, M. Atzeni, B. Audurier, S. Bachmann, J.J.\n Back, S. Baker, V. Balagura, W. Baldini, A. Baranov, R.J. Barlow, S. Barsuk,\n W. Barter, M. Bartolini, F. Baryshnikov, J.M. Basels, G. Bassi, V.\n Batozskaya, B. Batsukh, A. Battig, A. Bay, M. Becker, F. Bedeschi, I.\n Bediaga, A. Beiter, V. Belavin, S. Belin, V. Bellee, K. Belous, I. Belyaev,\n G. Bencivenni, E. Ben-Haim, S. Benson, A. Berezhnoy, R. Bernet, D.\n Berninghoff, H.C. Bernstein, C. Bertella, E. Bertholet, A. Bertolin, C.\n Betancourt, F. Betti, M.O. Bettler, Ia. Bezshyiko, S. Bhasin, J. Bhom, M.S.\n Bieker, S. Bifani, P. Billoir, A. Bizzeti, M. Bj{\\o}rn, M.P. Blago, T. Blake,\n F. Blanc, S. Blusk, D. Bobulska, V. Bocci, O. Boente Garcia, T. Boettcher, A.\n Boldyrev, A. Bondar, N. Bondar, S. Borghi, M. Borisyak, M. Borsato, J.T.\n Borsuk, T.J.V. Bowcock, C. Bozzi, M.J. Bradley, S. Braun, A. Brea Rodriguez,\n M. Brodski, J. Brodzicka, A. Brossa Gonzalo, D. Brundu, E. Buchanan, A.\n B\\\"uchler-Germann, A. Buonaura, C. Burr, A. Bursche, A. Butkevich, J.S.\n Butter, J. Buytaert, W. Byczynski, S. Cadeddu, H. Cai, R. Calabrese, L.\n Calero Diaz, S. Cali, R. Calladine, M. Calvi, M. Calvo Gomez, P. Camargo\n Magalhaes, A. Camboni, P. Campana, D.H. Campora Perez, A.F. Campoverde\n Quezada, L. Capriotti, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, I.\n Carli, P. Carniti, K. Carvalho Akiba, A. Casais Vidal, G. Casse, M. Cattaneo,\n G. Cavallero, S. Celani, R. Cenci, J. Cerasoli, M.G. Chapman, M. Charles, Ph.\n Charpentier, G. Chatzikonstantinidis, M. Chefdeville, V. Chekalina, C. Chen,\n S. Chen, A. Chernov, S.-G. Chitic, V. Chobanova, S. Cholak, M. Chrzaszcz, A.\n Chubykin, V. Chulikov, P. Ciambrone, M.F. Cicala, X. Cid Vidal, G. Ciezarek,\n F. Cindolo, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, J.L.\n Cobbledick, V. Coco, J.A.B. Coelho, J. Cogan, E. Cogneras, L. Cojocariu, P.\n Collins, T. Colombo, A. Contu, N. Cooke, G. Coombs, S. Coquereau, G. Corti,\n C.M. Costa Sobral, B. Couturier, D.C. Craik, J. Crkovsk\\'a, A. Crocombe, M.\n Cruz Torres, R. Currie, C.L. Da Silva, E. Dall'Occo, J. Dalseno, C.\n D'Ambrosio, A. Danilina, P. d'Argent, A. Davis, O. De Aguiar Francisco, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, M. De Serio, P.\n De Simone, J.A. de Vries, C.T. Dean, W. Dean, D. Decamp, L. Del Buono, B.\n Delaney, H.-P. Dembinski, A. Dendek, V. Denysenko, D. Derkach, O. Deschamps,\n F. Desse, F. Dettori, B. Dey, A. Di Canto, P. Di Nezza, S. Didenko, H.\n Dijkstra, V. Dobishuk, F. Dordei, M. Dorigo, A.C. dos Reis, L. Douglas, A.\n Dovbnya, K. Dreimanis, M.W. Dudek, L. Dufour, P. Durante, J.M. Durham, D.\n Dutta, M. Dziewiecki, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev,\n S. Eidelman, S. Eisenhardt, S. Ek-In, L. Eklund, S. Ely, A. Ene, E. Epple, S.\n Escher, J. Eschle, S. Esen, T. Evans, A. Falabella, J. Fan, Y. Fan, N.\n Farley, S. Farry, D. Fazzini, P. Fedin, M. F\\'eo, P. Fernandez Declara, A.\n Fernandez Prieto, F. Ferrari, L. Ferreira Lopes, F. Ferreira Rodrigues, S.\n Ferreres Sole, M. Ferrillo, M. Ferro-Luzzi, S. Filippov, R.A. Fini, M.\n Fiorini, M. Firlej, K.M. Fischer, C. Fitzpatrick, T. Fiutowski, F. Fleuret,\n M. Fontana, F. Fontanelli, R. Forty, V. Franco Lima, M. Franco Sevilla, M.\n Frank, C. Frei, D.A. Friday, J. Fu, Q. Fuehring, W. Funk, E. Gabriel, A.\n Gallas Torreira, D. Galli, S. Gallorini, S. Gambetta, Y. Gan, M. Gandelman,\n P. Gandini, Y. Gao, L.M. Garcia Martin, J. Garc\\'ia Pardi\\~nas, B. Garcia\n Plana, F.A. Garcia Rosales, L. Garrido, D. Gascon, C. Gaspar, D. Gerick, E.\n Gersabeck, M. Gersabeck, T. Gershon, D. Gerstel, Ph. Ghez, V. Gibson, A.\n Giovent\\`u, P. Gironella Gironell, L. Giubega, C. Giugliano, K. Gizdov, V.V.\n Gligorov, C. G\\\"obel, E. Golobardes, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, I.V. Gorelov, C. Gotti, E. Govorkova, J.P. Grabowski, R. Graciani\n Diaz, T. Grammatico, L.A. Granado Cardoso, E. Graug\\'es, E. Graverini, G.\n Graziani, A. Grecu, R. Greim, P. Griffith, L. Grillo, L. Gruber, B.R. Gruberg\n Cazon, C. Gu, E. Gushchin, A. Guth, Yu. Guz, T. Gys, P. A. G\\\"unther, T.\n Hadavizadeh, G. Haefeli, C. Haen, S.C. Haines, P.M. Hamilton, Q. Han, X. Han,\n T.H. Hancock, S. Hansmann-Menzemer, N. Harnew, T. Harrison, R. Hart, C.\n Hasse, M. Hatch, J. He, M. Hecker, K. Heijhoff, K. Heinicke, A.M. Hennequin,\n K. Hennessy, L. Henry, J. Heuel, A. Hicheur, D. Hill, M. Hilton, P.H.\n Hopchev, J. Hu, J. Hu, W. Hu, W. Huang, W. Hulsbergen, T. Humair, R.J.\n Hunter, M. Hushchyn, D. Hutchcroft, D. Hynds, P. Ibis, M. Idzik, P. Ilten, A.\n Inglessi, K. Ivshin, R. Jacobsson, S. Jakobsen, E. Jans, B.K. Jashal, A.\n Jawahery, V. Jevtic, F. Jiang, M. John, D. Johnson, C.R. Jones, B. Jost, N.\n Jurik, S. Kandybei, M. Karacson, J.M. Kariuki, N. Kazeev, M. Kecke, F.\n Keizer, M. Kelsey, M. Kenzie, T. Ketel, B. Khanji, A. Kharisova, K.E. Kim, T.\n Kirn, V.S. Kirsebom, S. Klaver, K. Klimaszewski, S. Koliiev, A. Kondybayeva,\n A. Konoplyannikov, P. Kopciewicz, R. Kopecna, P. Koppenburg, M. Korolev, I.\n Kostiuk, O. Kot, S. Kotriakhova, L. Kravchuk, R.D. Krawczyk, M. Kreps, F.\n Kress, S. Kretzschmar, P. Krokovny, W. Krupa, W. Krzemien, W. Kucewicz, M.\n Kucharczyk, V. Kudryavtsev, H.S. Kuindersma, G.J. Kunde, T. Kvaratskheliya,\n D. Lacarrere, G. Lafferty, A. Lai, D. Lancierini, J.J. Lane, G. Lanfranchi,\n C. Langenbruch, O. Lantwin, T. Latham, F. Lazzari, R. Le Gac, S.H. Lee, R.\n Lef\\`evre, A. Leflat, O. Leroy, T. Lesiak, B. Leverington, H. Li, L. Li, X.\n Li, Y. Li, Z. Li, X. Liang, T. Lin, R. Lindner, V. Lisovskyi, G. Liu, X. Liu,\n D. Loh, A. Loi, J. Lomba Castro, I. Longstaff, J.H. Lopes, G. Loustau, G.H.\n Lovell, Y. Lu, D. Lucchesi, M. Lucio Martinez, Y. Luo, A. Lupato, E. Luppi,\n O. Lupton, A. Lusiani, X. Lyu, S. Maccolini, F. Machefert, F. Maciuc, V.\n Macko, P. Mackowiak, S. Maddrell-Mander, L.R. Madhan Mohan, O. Maev, A.\n Maevskiy, D. Maisuzenko, M.W. Majewski, S. Malde, B. Malecki, A. Malinin, T.\n Maltsev, H. Malygina, G. Manca, G. Mancinelli, R. Manera Escalero, D.\n Manuzzi, D. Marangotto, J. Maratas, J.F. Marchand, U. Marconi, S. Mariani, C.\n Marin Benito, M. Marinangeli, P. Marino, J. Marks, P.J. Marshall, G.\n Martellotti, L. Martinazzoli, M. Martinelli, D. Martinez Santos, F. Martinez\n Vidal, A. Massafferri, M. Materok, R. Matev, A. Mathad, Z. Mathe, V.\n Matiunin, C. Matteuzzi, K.R. Mattioli, A. Mauri, E. Maurice, M. McCann, L.\n Mcconnell, A. McNab, R. McNulty, J.V. Mead, B. Meadows, C. Meaux, G. Meier,\n N. Meinert, D. Melnychuk, S. Meloni, M. Merk, A. Merli, M. Mikhasenko, D.A.\n Milanes, E. Millard, M.-N. Minard, O. Mineev, L. Minzoni, S.E. Mitchell, B.\n Mitreska, D.S. Mitzel, A. M\\\"odden, A. Mogini, R.D. Moise, T. Momb\\\"acher,\n I.A. Monroy, S. Monteil, M. Morandin, G. Morello, M.J. Morello, J. Moron,\n A.B. Morris, A.G. Morris, R. Mountain, H. Mu, F. Muheim, M. Mukherjee, M.\n Mulder, D. M\\\"uller, K. M\\\"uller, C.H. Murphy, D. Murray, P. Muzzetto, P.\n Naik, T. Nakada, R. Nandakumar, T. Nanut, I. Nasteva, M. Needham, N. Neri, S.\n Neubert, N. Neufeld, R. Newcombe, T.D. Nguyen, C. Nguyen-Mau, E.M. Niel, S.\n Nieswand, N. Nikitin, N.S. Nolte, C. Nunez, A. Oblakowska-Mucha, V.\n Obraztsov, S. Ogilvy, D.P. O'Hanlon, R. Oldeman, C.J.G. Onderwater, J. D.\n Osborn, A. Ossowska, J.M. Otalora Goicochea, T. Ovsiannikova, P. Owen, A.\n Oyanguren, P.R. Pais, T. Pajero, A. Palano, M. Palutan, G. Panshin, A.\n Papanestis, M. Pappagallo, L.L. Pappalardo, C. Pappenheimer, W. Parker, C.\n Parkes, G. Passaleva, A. Pastore, M. Patel, C. Patrignani, A. Pearce, A.\n Pellegrino, M. Pepe Altarelli, S. Perazzini, D. Pereima, P. Perret, L.\n Pescatore, K. Petridis, A. Petrolini, A. Petrov, S. Petrucci, M. Petruzzo, B.\n Pietrzyk, G. Pietrzyk, M. Pili, D. Pinci, J. Pinzino, F. Pisani, A. Piucci,\n V. Placinta, S. Playfer, J. Plews, M. Plo Casasus, F. Polci, M. Poli Lener,\n M. Poliakova, A. Poluektov, N. Polukhina, I. Polyakov, E. Polycarpo, G.J.\n Pomery, S. Ponce, A. Popov, D. Popov, S. Poslavskii, K. Prasanth, L.\n Promberger, C. Prouve, V. Pugatch, A. Puig Navarro, H. Pullen, G. Punzi, W.\n Qian, J. Qin, R. Quagliani, B. Quintana, N.V. Raab, R.I. Rabadan Trejo, B.\n Rachwal, J.H. Rademacker, M. Rama, M. Ramos Pernas, M.S. Rangel, F. Ratnikov,\n G. Raven, M. Reboud, F. Redi, F. Reiss, C. Remon Alepuz, Z. Ren, V. Renaudin,\n S. Ricciardi, D.S. Richards, S. Richards, K. Rinnert, P. Robbe, A. Robert,\n A.B. Rodrigues, E. Rodrigues, J.A. Rodriguez Lopez, M. Roehrken, A. Rollings,\n V. Romanovskiy, M. Romero Lamas, A. Romero Vidal, J.D. Roth, M. Rotondo, M.S.\n Rudolph, T. Ruf, J. Ruiz Vidal, A. Ryzhikov, J. Ryzka, J.J. Saborido Silva,\n N. Sagidova, N. Sahoo, B. Saitta, C. Sanchez Gras, C. Sanchez Mayordomo, R.\n Santacesaria, C. Santamarina Rios, M. Santimaria, E. Santovetti, G. Sarpis,\n M. Sarpis, A. Sarti, C. Satriano, A. Satta, M. Saur, D. Savrina, L.G.\n Scantlebury Smead, S. Schael, M. Schellenberg, M. Schiller, H. Schindler, M.\n Schmelling, T. Schmelzer, B. Schmidt, O. Schneider, A. Schopper, H.F.\n Schreiner, M. Schubiger, S. Schulte, M.H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, S. Sellam, A. Semennikov, A. Sergi, N. Serra, J. Serrano, L.\n Sestini, A. Seuthe, P. Seyfert, D.M. Shangase, M. Shapkin, L. Shchutska, T.\n Shears, L. Shekhtman, V. Shevchenko, E. Shmanin, J.D. Shupperd, B.G. Siddi,\n R. Silva Coutinho, L. Silva de Oliveira, G. Simi, S. Simone, I. Skiba, N.\n Skidmore, T. Skwarnicki, M.W. Slater, J.G. Smeaton, A. Smetkina, E. Smith,\n I.T. Smith, M. Smith, A. Snoch, M. Soares, L. Soares Lavra, M.D. Sokoloff,\n F.J.P. Soler, B. Souza De Paula, B. Spaan, E. Spadaro Norella, P. Spradlin,\n F. Stagni, M. Stahl, S. Stahl, P. Stefko, O. Steinkamp, S. Stemmle, O.\n Stenyakin, M. Stepanova, H. Stevens, S. Stone, S. Stracka, M.E. Stramaglia,\n M. Straticiuc, S. Strokov, J. Sun, L. Sun, Y. Sun, P. Svihra, K. Swientek, A.\n Szabelski, T. Szumlak, M. Szymanski, S. Taneja, Z. Tang, T. Tekampe, F.\n Teubert, E. Thomas, K.A. Thomson, M.J. Tilley, V. Tisserand, S. T'Jampens, M.\n Tobin, S. Tolk, L. Tomassetti, D. Torres Machado, D.Y. Tou, E. Tournefier, M.\n Traill, M.T. Tran, E. Trifonova, C. Trippl, A. Tsaregorodtsev, G. Tuci, A.\n Tully, N. Tuning, A. Ukleja, A. Usachov, A. Ustyuzhanin, U. Uwer, A. Vagner,\n V. Vagnoni, A. Valassi, G. Valenti, M. van Beuzekom, H. Van Hecke, E. van\n Herwijnen, C.B. Van Hulse, M. van Veghel, R. Vazquez Gomez, P. Vazquez\n Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, A.\n Venkateswaran, M. Veronesi, M. Vesterinen, J.V. Viana Barbosa, D. Vieira, M.\n Vieites Diaz, H. Viemann, X. Vilasis-Cardona, G. Vitali, A. Vitkovskiy, A.\n Vollhardt, D. Vom Bruch, A. Vorobyev, V. Vorobyev, N. Voropaev, R. Waldi, J.\n Walsh, J. Wang, J. Wang, J. Wang, M. Wang, Y. Wang, Z. Wang, D.R. Ward, H.M.\n Wark, N.K. Watson, D. Websdale, A. Weiden, C. Weisser, B.D.C. Westhenry, D.J.\n White, M. Whitehead, D. Wiedner, G. Wilkinson, M. Wilkinson, I. Williams, M.\n Williams, M.R.J. Williams, T. Williams, F.F. Wilson, W. Wislicki, M. Witek,\n L. Witola, G. Wormser, S.A. Wotton, H. Wu, K. Wyllie, Z. Xiang, D. Xiao, Y.\n Xie, H. Xing, A. Xu, J. Xu, L. Xu, M. Xu, Q. Xu, Z. Xu, Z. Yang, Z. Yang, Y.\n Yao, L.E. Yeomans, H. Yin, J. Yu, X. Yuan, O. Yushchenko, K.A. Zarebski, M.\n Zavertyaev, M. Zdybal, M. Zeng, D. Zhang, L. Zhang, S. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, Y. Zheng, X. Zhou, Y. Zhou, X. Zhu, V. Zhukov, J.B.\n Zonneveld, S. Zucchelli",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26112",
"submitter": "Titus Momb\\\"acher",
"url": "https://arxiv.org/abs/2003.03999"
} | arxiv-papers | EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-EP-2020-023 LHCb-PAPER-2020-001 May 28, 2020
Search for the rare decays $B^{0}_{s}\rightarrow e^{+}e^{-}$ and
$B^{0}\rightarrow e^{+}e^{-}$
LHCb collaboration†††Authors are listed at the end of this Letter.
A search for the decays $B^{0}_{s}\rightarrow e^{+}e^{-}$ and
$B^{0}\rightarrow e^{+}e^{-}$ is performed using data collected with the LHCb
experiment in proton-proton collisions at center-of-mass energies of $7$, $8$
and $13\,\text{TeV}$, corresponding to integrated luminosities of $1$, $2$ and
$2\,\text{fb}^{-1}$, respectively. No signal is observed. Assuming no
contribution from $B^{0}\rightarrow e^{+}e^{-}$ decays, an upper limit on the
branching fraction $\mathcal{B}(B^{0}_{s}\rightarrow
e^{+}e^{-})<9.4\,(11.2)\times 10^{-9}$ is obtained at $90\,(95)\,\%$
confidence level. If no $B^{0}_{s}\rightarrow e^{+}e^{-}$ contribution is
assumed, a limit of $\mathcal{B}(B^{0}\rightarrow e^{+}e^{-})<2.5\,(3.0)\times
10^{-9}$ is determined at $90\,(95)\,\%$ confidence level. These upper limits
are more than one order of magnitude lower than the previous values.
Published in Phys. Rev. Lett. 124 (2020) 211802
© 2020 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.
Searches for rare particle decays provide ideal probes for contributions from
physics processes beyond the Standard Model (SM). Recent measurements of
decays involving $b\\!\rightarrow s\ell^{+}\ell^{-}$ transitions (the
inclusion of charge-conjugated processes is implied throughout this Letter)
hint at deviations from SM predictions in lepton-flavor universality tests [1,
2, 3, 4, 5, 6] and thus motivate measurements of decay rates into final states
involving leptons. Following the observation of the decay
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ [7, 8], the search for
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ decays provides an independent test of
lepton-flavor universality. According to SM predictions (calculated from Ref.
[9], neglecting QED corrections that are expected to be at the percent level),
${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ decays have branching fractions
of $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})=$(8.60\pm
0.36)\text{\times}{10}^{-14}$$ and
$\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})=$(2.41\pm
0.13)\text{\times}{10}^{-15}$$. With contributions beyond the SM, these
branching fractions could be significantly larger, reaching values of
$\mathcal{O}(10^{-8})$ for
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})$ and
$\mathcal{O}(10^{-10})$ for
$\mathcal{B}\mbox{(${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$)}$ [10]. These values
are close to the current experimental bounds of
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<$2.8\text{\times}{10}^{-7}$$
and
$\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<$8.3\text{\times}{10}^{-8}$$
at $90\text{\,}\mathrm{\char 37\relax}$ confidence level (CL) [11], set by the
CDF collaboration.
In this Letter a search for ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ decays is presented using data collected
with the LHCb experiment in proton-proton collisions at center-of-mass
energies of $7\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ in 2011,
$8\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ in 2012 and
$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ in 2015 and 2016, corresponding to
integrated luminosities of $1$, $2$ and $2\text{\,}\mathrm{f}\mathrm{b}^{-1}$,
respectively. The signal yields are determined from a fit to the data and
normalized to those of the
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decay, where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson decays to $e^{+}e^{-}$, which has a precisely measured branching
fraction [12] and a similar dielectron signature in the detector.
The LHCb detector [13, 14] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. Different types of charged hadrons are distinguished using
information from two ring-imaging Cherenkov detectors. Photons, electrons and
hadrons are identified by a calorimeter system consisting of scintillating-pad
and preshower detectors, an electromagnetic and a hadronic calorimeter. Muons
are identified by a system composed of alternating layers of iron and
multiwire proportional chambers.
The online event selection is performed by a trigger [15], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. At
the hardware trigger stage, events are required to have a high-energy deposit
in the calorimeters associated with a signal electron candidate, or a muon
candidate with high transverse momentum $p_{\mathrm{T}}$, or a photon,
electron or hadron candidate with high transverse energy from the decays of
other particles from the $pp$ collision. The software trigger requires a two-
track secondary vertex with a significant displacement from any primary $pp$
interaction vertex (PV). At least one charged particle must have high
$p_{\mathrm{T}}$ and be inconsistent with originating from a PV. A
multivariate algorithm [16, 17] is used in the trigger for the identification
of secondary vertices consistent with the decay of a $b$ hadron. Simulated
samples are used to optimize the candidate selection, estimate selection
efficiencies and describe the expected invariant-mass shapes of the signal
candidates and background decays. In the simulation, $pp$ collisions are
generated using Pythia [18, *Sjostrand:2007gs] with a specific LHCb
configuration [20]. Decays of unstable particles are described by EvtGen [21],
in which final-state radiation is generated using Photos [22]. The interaction
of the generated particles with the detector, and its response, are
implemented using the Geant4 toolkit [23, *Agostinelli:2002hh] as described in
Ref. [25]. The simulation is corrected for data-simulation differences in
$B$-meson production kinematics, detector occupancy and isolation criteria
[26] using ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ decays, with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{e^{+}e^{-}}$ and $\phi\\!\rightarrow{{K}^{+}}{{K}^{-}}$.
Particle identification variables are calibrated using data from
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ and ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays
[27]. The calibration data are binned in momentum and pseudorapidity of the
particle as well as detector occupancy to account for possible differences in
kinematics between the investigated decay and the calibration data.
The ${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ candidates are selected in
events passing the trigger requirements by combining two tracks that are
inconsistent with originating from any PV in the event and which form a good-
quality secondary vertex. The tracks are also required to have a momentum
larger than 3 $\text{\,Ge\kern-1.00006ptV\\!/}c$ and $p_{\mathrm{T}}$ greater
than 500 $\text{\,Me\kern-1.00006ptV\\!/}c$, and must be identified as
electrons using information from the Cherenkov detectors and calorimeters. The
dielectron candidate’s momentum must be aligned with the vector pointing from
a PV (the associated PV) to the two-track vertex and have a considerable
transverse component. The candidate must also have an invariant mass in the
range $[4166,6566]\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.
The measured electron momenta are corrected for losses due to bremsstrahlung
radiation by adding the momentum of photons consistent with being emitted
upstream of the magnet [28]. Candidates in data and simulation are separated
into three categories with either zero, one, or both electrons having a
bremsstrahlung correction applied. To avoid experimenters’ bias, the narrowest
dielectron invariant-mass region containing $90\text{\,}\mathrm{\char
37\relax}$ of simulated ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays,
corresponding to a range of [$4689$, $5588$]
$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, was removed from the data set until
the analysis procedure was finalized.
Candidates for the normalization mode,
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$, are constructed similarly, but require an additional track
consistent with being a kaon and originating from the same vertex as the
dielectron candidate. The dielectron candidate must have an invariant mass in
the range $[2450,3176]\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, consistent with
arising from a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson decay. In
addition, the reconstructed ${B}^{+}$ candidate mass, when the dielectron
candidate is constrained to the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ mass [12], must be above $5175\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$,
suppressing partially reconstructed decays.
A boosted decision tree (BDT) algorithm[29, 30, 31] is used to separate
${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ signal from random combinations
of two electrons (combinatorial background). The BDT is trained separately for
data taking periods 2011–2012 (Run 1) and 2015–2016 (Run 2) on simulated
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays as signal proxy and
dielectron candidates from data with a mass above
$5588\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ as background proxy. The split
between the data taking periods is done to account for changes in the center-
of-mass energies and trigger strategies, which significantly impact the data
distributions and improve the BDT and the particle identification algorithms
in Run 2. It is checked that the data behave consistently across the data-
taking periods. The BDT input variables comprise of the following: kinematic
information on the electron tracks and $B$ candidate, information on the
displacement of the electrons and $B$ candidate from the associated PV, and
isolation variables that quantify the compatibility of other tracks in the
event with originating from the same decay as the $B$ candidate [26, 32].
Candidates with a BDT response compatible with that of the background are
discarded, with the threshold chosen by maximizing the figure of merit
${\epsilon_{\text{signal}}}/{(\sqrt{N_{\text{background}}}+3/2)}$ [33], where
$\epsilon_{\text{signal}}$ is the signal efficiency and the expected
background yield in the signal region is $N_{\text{background}}$.
The final selected data set is separated by data-taking period and by category
of bremsstrahlung correction. The branching fraction
$\mathcal{B}({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ is measured
relative to that of the normalization channel via
$\displaystyle\mathcal{B}({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$
$\displaystyle=N({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})\times\alpha\times\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})\times\left(\frac{f_{d(s)}}{f_{u}}\right)^{-1},$ (1)
where
$\displaystyle\alpha$
$\displaystyle\equiv\frac{\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})}{\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})}\times\frac{1}{N({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})},$ (2)
$\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and
$\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$ denote the efficiencies of the signal and normalization
modes, and $N$(${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$) and
$N$(${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$) their yields. The normalization mode branching fraction
(including that for the decay ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{e^{+}e^{-}}$) is
$\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})=$(6.03\pm 0.17)\text{\times}{10}^{-5}$$, taken from
Ref.[12]. The $b$-hadron fragmentation fraction ratio $f_{d}/f_{u}$ is assumed
to be unity, while $f_{s}/f_{u}=$0.259\pm 0.015$$ [34] is used for the Run 1
data and is scaled by $1.068\pm 0.016$ for the Run 2 data, according to Ref.
[35], to account for center-of-mass energy differences. A measurement of
$f_{s}/f_{u}$ from Run 2 yields a consistent, but less precise, result [36].
The yield of the normalization mode is determined using an unbinned maximum-
likelihood fit to the ${K}^{+}$ $e^{+}e^{-}$ invariant mass separately for
each year of data taking and bremsstrahlung category. The fit model comprises
a Gaussian function with power-law tails [37] for the signal component, where
the tail parameters are fixed from simulation, and an exponential function to
describe combinatorial background. Summed over the bremsstrahlung categories,
the yield of the normalization mode is $20\,480\pm 140$ in the Run 1 data and
$33\,080\pm 180$ in the Run 2 data.
The selection efficiencies
$\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and
$\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$ are determined separately for each year of data taking and
bremsstrahlung category using simulated decays that are weighted to better
represent the data. Calibration data are used to evaluate particle-
identification efficiencies [27]. Trigger efficiencies are also estimated from
data, using the technique described in Ref. [38]. For simulated
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays, the mean ${B}^{0}_{s}$
lifetime [39] is assumed. The selection efficiency is assumed to be the same
for both ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays, which is consistent with
results from simulation. The normalization factors, $\alpha$, are combined
across the data-taking periods and given in Table 1, split by bremsstrahlung
category (for the selection efficiency ratio between normalization and signal
mode, see the Supplemental Material [40]).
Table 1: Normalization factors $\alpha$ for ${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$. The bremsstrahlung category denotes whether zero, one or both electrons are corrected for bremsstrahlung losses. The uncertainties include statistical uncertainties and uncertainties due to limited size of the simulated samples. Bremsstrahlung category | 2011–2012 $[10^{-5}]$ | 2015–2016 $[10^{-5}]$
---|---|---
No correction | $2.85\pm 0.24$ | $1.84\pm 0.08$
One electron corrected | $1.13\pm 0.08$ | $0.70\pm 0.03$
Both electrons corrected | $1.73\pm 0.20$ | $1.04\pm 0.06$
In addition to the combinatorial background, backgrounds due to
misidentification and partial reconstruction are present in the data. These
backgrounds differ significantly between the categories of bremsstrahlung
correction. Their invariant-mass shapes and relative contributions are
evaluated using simulation. In the lower mass region, partially reconstructed
backgrounds of the types ${B}\\!\rightarrow X{e^{+}e^{-}}$ and
${{B}^{+}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\rightarrow
Y^{+}{e^{-}}{{\overline{\nu}}_{e}}){e^{+}}{{\nu}_{e}}$ dominate, where $X$ and
$Y$ represent hadronic systems. The main source of background in the $B$-mass
region, however, stems from misidentified particles in the decays
${{B}^{0}}\\!\rightarrow{{\pi}^{-}}{e^{+}}{{\nu}_{e}}$ and
${B}\\!\rightarrow{h}^{+}{h}^{\prime-}$, where $h$ and $h^{\prime}$ are
hadrons. The latter has a peaking structure in the $B$-mass region.
Backgrounds involving misidentified particles contribute mostly to categories
in which at most one of the electrons has a bremsstrahlung correction applied.
The contribution from combinatorial background is evaluated from same-sign
lepton pairs in data and found to be small. The yields of the backgrounds are
Gaussian constrained to their expected values, estimated from simulation using
their known branching fractions [12].
The shape of the invariant mass of the
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ components is modeled using a Gaussian
function with power-law tails, where the parameters are obtained from
simulation and differ between each bremsstrahlung category and year of data
taking. The peak values and the widths of the functions are corrected for
data-simulation differences by a factor determined from the normalization
mode. The parameters of the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ line shapes are fixed to the same values
with the exception of the peak value, which is shifted according to the known
${B}^{0}_{s}$–${B}^{0}$ mass difference [12]. Due to the limited mass
resolution, arising from imperfect bremsstrahlung recovery, the line shapes
from ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ are highly overlapping. Therefore the
branching fraction of ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ is obtained by
performing a simultaneous fit to the dielectron invariant-mass distribution of
all six data sets while neglecting the contribution from
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$, and vice versa. In these fits, the only
shared parameters between categories are the branching fractions
$\mathcal{B}({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and
$\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$, and the ratio of the fragmentation fractions
$f_{s}/f_{u}$.
Systematic uncertainties are estimated separately for each data set. Dominant
sources of systematic uncertainties in the normalization arise from the
uncertainty on the fragmentation fraction ratio, the technique used to
evaluate the trigger efficiencies, and the determination of particle-
identification efficiencies; the systematic uncertainties from these sources
extend to $5.8\text{\,}\mathrm{\char 37\relax}$, $5.3\text{\,}\mathrm{\char
37\relax}$, and $5.3\text{\,}\mathrm{\char 37\relax}$ on the branching
fractions, respectively. The uncertainty on
$\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})$ of $2.8\text{\,}\mathrm{\char 37\relax}$ [12] is taken into
account. A difference of up to $4.1\text{\,}\mathrm{\char 37\relax}$ is found
between the efficiency of the BDT selection on simulated
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays and
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays in data, which is assigned as a systematic
uncertainty. The fraction of candidates in each bremsstrahlung-correction
category of the signal modes is taken from simulation. The difference between
simulation and data is investigated using
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays and its effect on the normalization, up to
$4.0\text{\,}\mathrm{\char 37\relax}$, is taken as a systematic uncertainty.
Systematic uncertainties on the invariant-mass resolution corrections are
determined by repeating the correction procedure with pseudoexperiments
obtained with the bootstrapping method [41], yielding up to
$1.1\text{\,}\mathrm{\char 37\relax}$. A difference between the total
selection efficiencies in the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ channels of up to
$2.5\text{\,}\mathrm{\char 37\relax}$ is assigned as a systematic uncertainty
on the ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ normalization factor. Due to the
presence of an additional kaon in the final state of the normalization mode,
the track-reconstruction efficiency is different between the signal and
normalization modes. An uncertainty of $1.1\text{\,}\mathrm{\char 37\relax}$
is assigned to the branching fraction as a systematic uncertainty on the kaon
reconstruction efficiency arising from the limited knowledge of the
interactions in the detector material [42]. Finally, an uncertainty of
$1.0\text{\,}\mathrm{\char 37\relax}$ is assigned to account for small
differences in detector occupancy between the signal and normalization mode
arising from the trigger selection. The dominant sources of systematic
uncertainties on the background composition are due to the imprecise knowledge
of the branching fractions of the background components. The largest
uncertainty of this type on the expected background yield in the $B$-mass
region is $14\text{\,}\mathrm{\char 37\relax}$, determined from refitting the
mass sidebands while varying the background components according to their
uncertainties. Taking all correlations into account, overall single event
sensitivities of $[4.71\pm 0.12\text{(stat.)}\pm 0.33\text{(syst.)}]\times
10^{-10}$ for ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and $[1.271\pm
0.034\text{(stat.)}\pm 0.063\text{(syst.)}]\times 10^{-10}$ for
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ are obtained.
The dielectron invariant-mass spectrum, summed over bremsstrahlung categories,
is shown in Fig. 1, with the result of the
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ fit. The individual categories are
shown in the Supplemental Material [40], as well as the distributions with the
result of the ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ fit. The measured
branching fractions are
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})=$(2.4\pm
4.4)\text{\times}{10}^{-9}$$ and
$\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})=$(0.30\pm
1.29)\text{\times}{10}^{-9}$$, where the uncertainties include both
statistical and systematic components. The results are in agreement with the
background-only hypothesis.
Figure 1: Simultaneous fit to the dielectron invariant-mass distribution, with
$\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})$ fixed to zero. The sum of
bremsstrahlung categories is shown for (left) Run 1 and (right) Run 2. The
relative proportions of background contributions change between Run 1 and Run
2 due to different performances of the particle identification algorithms and
BDT selections.
Upper limits on the branching fractions are set using the CLs method [43], as
implemented in the GammaCombo framework[44, 45] with a one-sided profile
likelihood ratio [46] as test statistic. The likelihoods are computed from
fits to the invariant-mass distributions. In the fits, the normalization
factor, normalization mode branching fraction, fragmentation fraction ratio,
and background yields are Gaussian constrained to their expected values within
statistical and systematic uncertainties. Pseudoexperiments, in which the
nuisance parameters are set to their fitted values from data, are used for the
evaluation of the test statistic.
The expected and observed CLs distributions are shown in Fig. 2. The upper
observed limits are
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<9.4\,(11.2)\times
10^{-9}$ and
$\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<2.5\,(3.0)\times 10^{-9}$ at
$90\,(95)\,\%$ confidence level. These are consistent with the expected upper
limits of
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<7.0\,(8.6)\times
10^{-9}$ and
$\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<2.0\,(2.5)\times 10^{-9}$ at
$90\,(95)\,\%$ confidence level, obtained as the median of limits determined
on background-only pseudoexperiments.
Figure 2: CLs values as a function of the branching fractions of the decays
(left) ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and (right)
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$. The red solid line (black solid line
with data points) corresponds to the distribution of the expected (observed)
upper limits, and the light blue (dark blue) band contains the $1\sigma$
$(2\sigma)$ uncertainties on the expected upper limits. Thresholds
corresponding to $90\,\%$ and $95\,\%$ confidence level are indicated with
dashed lines. The observed values are plotted for branching fractions greater
than the measured branching fraction in the data; the test statistic is
defined to be nonzero only in that region.
In conclusion, a search for the rare decays
${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ is performed using data from
proton-proton collisions recorded with the LHCb experiment, corresponding to a
total integrated luminosity of $5\text{\,}\text{\,fb}^{-1}$. No excess of
events is observed over the background. The resulting limits on the branching
fractions are
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<9.4\,(11.2)\times
10^{-9}$ and
$\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<2.5\,(3.0)\times 10^{-9}$ at
$90\,(95)\,\%$ confidence level, when neglecting the contribution from the
other decay. The mean ${B}^{0}_{s}$ lifetime is assumed for
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays. Assuming SM-like $C\\!P$-odd
($C\\!P$-even) ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays, an increase
(decrease) of $2.4\text{\,}\mathrm{\char 37\relax}$ with respect to the quoted
limit is found. The results improve the limits on these branching fractions
[11] by more than one order of magnitude and constrain contributions beyond
the SM, for example from scalar and pseudoscalar currents [10].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).
## Supplemental Material for LHCb-PAPER-2020-001
The individual categories of the simultaneous fit to the dielectron invariant-
mass using the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis are
presented in Fig. 3. The fit to the invariant dielectron mass including the
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis instead of the
${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis is shown in Fig. 4, where
the bremsstrahlung categories are summed. The individual categories of the
simultaneous fit to the dielectron invariant-mass using the
${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis are presented in Fig. 5.
Table 2 lists the inputs to the normalization factors: the ratio of
normalization and signal efficiencies and the normalization yield. The
efficiency of the normalization mode differs from the signal and causes the
efficiency ratio to decrease with bremsstrahlung category due to the slightly
different reconstruction and preselection and a different impact of the BDT
selection, where the differences mainly originate from the additional track in
the normalization mode.
Figure 3: Simultaneous fit to the dielectron invariant-mass distribution in
all categories, with $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})$ fixed
to zero. The top figures show the three bremsstrahlung categories in the Run 1
data set and the bottom figures show the Run 2 data set. From left to right,
the data sets correspond to the bremsstrahlung correction category with no
correction, correcting one electron and correcting both electrons. The
relative proportions of background contributions change between Run 1 and Run
2 due to different performances of the particle-identification algorithms and
BDT selections. Their relative fractions between bremsstrahlung categories
follow the expectation from simulation.
Figure 4: Simultaneous fit to the dielectron invariant-mass distribution, with
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})$ fixed to zero. The
bremsstrahlung categories are summed over the (left) Run 1 and (right) Run 2
data sets. The relative proportions of background contributions change between
Run 1 and Run 2 due to different performances of the particle-identification
algorithms and BDT selections.
Figure 5: Simultaneous fit to the dielectron invariant-mass distribution in all categories, with $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})$ fixed to zero. The top figures show the three bremsstrahlung categories in the Run 1 data set and the bottom figures show the Run 2 data set. From left to right, the data sets correspond to the bremsstrahlung correction category with no correction, correcting one electron and correcting both electrons. The relative proportions of background contributions change between Run 1 and Run 2 due to different performances of the particle-identification algorithms and BDT selections. Their relative fractions between bremsstrahlung categories follow the expectation from simulation. Table 2: Inputs for the normalization factors, the efficiency ratio $\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})/\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and normalization yield $N({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$. The bremsstrahlung category (Brem. cat.) denotes whether zero, one or both electrons are corrected for bremsstrahlung losses. The uncertainties on the efficiency ratios include statistical uncertainties from the calibration and uncertainties due to limited size of the simulated samples. Brem. cat. | 2011–2012 | 2015–2016
---|---|---
| Efficiency ratio | Norm. yield $[10^{3}]$ | Efficiency ratio | Norm. yield $[10^{3}]$
Brem. 0 | $0.144\pm 0.012$ | $5.05\pm 0.07$ | $0.148\pm 0.118$ | $7.96\pm 0.09$
Brem. 1 | $0.119\pm 0.008$ | $10.43\pm 0.11$ | $0.118\pm 0.005$ | $12.75\pm 0.05$
Brem. 2 | $0.086\pm 0.010$ | $4.95\pm 0.07$ | $0.085\pm 0.005$ | $8.306\pm 0.032$
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LHCb collaboration
R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala81, S. Aiola25, Z. Ajaltouni9, S. Akar66, P.
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L. An21, L. Anderlini21, G. Andreassi48, M. Andreotti20, F. Archilli16, A.
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Yuan67, O. Yushchenko43, K.A. Zarebski52, M. Zavertyaev15,c, M. Zdybal33, M.
Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, W.C. Zhang3,y, Y. Zhang47, A.
Zhelezov16, Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V. Zhukov13,39, J.B.
Zonneveld57, S. Zucchelli19,e.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27INFN Sezione di Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
75Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
76Universiteit Maastricht, Maastricht, Netherlands, associated to 31
77National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
78National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
79National Research University Higher School of Economics, Moscow, Russia,
associated to 41
80National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
81University of Michigan, Ann Arbor, United States, associated to 67
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
lDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain
mHanoi University of Science, Hanoi, Vietnam
nUniversità di Padova, Padova, Italy
oUniversità di Pisa, Pisa, Italy
pUniversità degli Studi di Milano, Milano, Italy
qUniversità di Urbino, Urbino, Italy
rUniversità della Basilicata, Potenza, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità di Modena e Reggio Emilia, Modena, Italy
uUniversità di Siena, Siena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xINFN Sezione di Trieste, Trieste, Italy
ySchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
zUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras
|
2024-09-04T02:54:58.337551 | 2020-03-09T10:05:41 | 2003.04013 | {
"authors": "Richard A. Fallows, Biagio Forte, Ivan Astin, Tom Allbrook, Alex\n Arnold, Alan Wood, Gareth Dorrian, Maaijke Mevius, Hanna Rothkaehl, Barbara\n Matyjasiak, Andrzej Krankowski, James M. Anderson, Ashish Asgekar, I. Max\n Avruch, Mark Bentum, Mario M. Bisi, Harvey R. Butcher, Benedetta Ciardi,\n Bartosz Dabrowski, Sieds Damstra, Francesco de Gasperin, Sven Duscha, Jochen\n Eisl\\\"offel, Thomas M.O. Franzen, Michael A. Garrett, Jean-Matthias\n Grie\\b{eta}meier, Andr\\'e W. Gunst, Matthias Hoeft, J\\\"org R. H\\\"orandel,\n Marco Iacobelli, Huib T. Intema, Leon V.E. Koopmans, Peter Maat, Gottfried\n Mann, Anna Nelles, Harm Paas, Vishambhar N. Pandey, Wolfgang Reich, Antonia\n Rowlinson, Mark Ruiter, Dominik J. Schwarz, Maciej Serylak, Aleksander\n Shulevski, Oleg M. Smirnov, Marian Soida, Matthias Steinmetz, Satyendra\n Thoudam, M. Carmen Toribio, Arnold van Ardenne, Ilse M. van Bemmel, Matthijs\n H.D. van der Wiel, Michiel P. van Haarlem, Ren\\'e C. Vermeulen, Christian\n Vocks, Ralph A.M.J. Wijers, Olaf Wucknitz, Philippe Zarka, Pietro Zucca",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26113",
"submitter": "Richard A. Fallows",
"url": "https://arxiv.org/abs/2003.04013"
} | arxiv-papers | 11institutetext: ASTRON - the Netherlands Institute for Radio Astronomy, Oude
Hoogeveensedijk 4, 7991 PD Dwingeloo, the Netherlands 22institutetext:
Department of Electronic and Electrical Engineering, University of Bath,
Claverton Down, Bath, BA2 7AY, UK 33institutetext: School of Science and
Technology, Nottingham Trent University, Clifton Lane, Nottingham, NG11 8NS,
UK 44institutetext: Space Environment and Radio Engineering, School of
Engineering, The University of
Birmingham, Edgbaston, Birmingham, B15 2TT, UK 55institutetext: Space Research
Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland
66institutetext: Space Radio-Diagnostics Research Centre, University of Warmia
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Space, UKRI STFC, Rutherford Appleton Laboratory, Harwell Campus, Oxfordshire,
OX11 0QX, UK
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2611, Australia 1313institutetext: Max Planck Institute for Astrophysics,
Karl-Schwarzschild-Str. 1, 85748 Garching, Germany 1414institutetext:
Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029, Hamburg,
Germany 1515institutetext: Thüringer Landessternwarte, Sternwarte 4, D-07778
Tautenburg, Germany 1616institutetext: Jodrell Bank Centre for Astrophysics
(JBCA), Department of Physics & Astronomy, Alan Turing Building, Oxford Road,
University of Manchester, Manchester M139PL, UK 1717institutetext: Leiden
Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, The
Netherlands 1818institutetext: LPC2E - Université d’Orléans / CNRS, 45071
Orléans cedex 2, France 1919institutetext: Station de Radioastronomie de
Nançay, Observatoire de Paris, PSL Research University, CNRS, Univ. Orléans,
OSUC, 18330 Nançay, France 2020institutetext: Radboud University, Department
of Astrophysics/IMAPP, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
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Netherlands 2222institutetext: Vrije Universiteit Brussel, Astronomy and
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Nettelbosje 1, 9747 AJ Groningen, The Netherlands 2828institutetext: Max-
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& USN, Observatoire de Paris, CNRS, PSL, SU/UP/UO, 92195 Meudon, France
# A LOFAR Observation of Ionospheric Scintillation from Two Simultaneous
Travelling Ionospheric Disturbances
R.A. Fallows111Corresponding author<EMAIL_ADDRESS>11 B. Forte 22 I. Astin
22 T. Allbrook222Now at BAE Systems (operation) Ltd. 22 A. Arnold333Now an
independent researcher 22 A. Wood 33 G. Dorrian 44 M. Mevius 11 H. Rothkaehl
55 B. Matyjasiak 55 A. Krankowski 66 J.M. Anderson 7788 A. Asgekar 99 I.M.
Avruch 1010 M.J.Bentum 11 M.M. Bisi 1111 H.R. Butcher 1212 B. Ciardi 1313 B.
Dabrowski 66 S. Damstra 11 F. de Gasperin 1414 S. Duscha 11 J. Eislöffel 1515
T.M.O Franzen 11 M.A. Garrett 16161717 J.-M. Grießmeier 18181919 A.W. Gunst 11
M. Hoeft 1515 J.R. Hörandel 202021212222 M. Iacobelli 11 H.T. Intema 1717
L.V.E. Koopmans 2323 P. Maat 11 G. Mann 2424 A. Nelles 25252626 H. Paas 2727
V.N. Pandey 112323 W. Reich 2828 A. Rowlinson 112929 M. Ruiter 11 D.J. Schwarz
3030 M. Serylak 31313232 A. Shulevski 2929 O.M. Smirnov 33333131 M. Soida 3434
M. Steinmetz 2424 S. Thoudam 3535 M.C. Toribio 3636 A. van Ardenne 11 I.M. van
Bemmel 3737 M.H.D. van der Wiel 11 M.P. van Haarlem 11 R.C. Vermeulen 11 C.
Vocks 2424 R.A.M.J. Wijers 2929 O. Wucknitz 2828 P. Zarka 3838 P. Zucca 11
(Received November 30, 2019)
###### Abstract
This paper presents the results from one of the first observations of
ionospheric scintillation taken using the Low-Frequency Array (LOFAR). The
observation was of the strong natural radio source Cassiopeia A, taken
overnight on 18-19 August 2013, and exhibited moderately strong scattering
effects in dynamic spectra of intensity received across an observing bandwidth
of 10-80 MHz. Delay-Doppler spectra (the 2-D FFT of the dynamic spectrum) from
the first hour of observation showed two discrete parabolic arcs, one with a
steep curvature and the other shallow, which can be used to provide estimates
of the distance to, and velocity of, the scattering plasma. A cross-
correlation analysis of data received by the dense array of stations in the
LOFAR “core” reveals two different velocities in the scintillation pattern: a
primary velocity of $\sim$20-40 m s-1 with a north-west to south-east
direction, associated with the steep parabolic arc and a scattering altitude
in the F-region or higher, and a secondary velocity of $\sim$110 m s-1 with a
north-east to south-west direction, associated with the shallow arc and a
scattering altitude in the D-region. Geomagnetic activity was low in the mid-
latitudes at the time, but a weak sub-storm at high latitudes reached its peak
at the start of the observation. An analysis of Global Navigation Satellite
Systems (GNSS) and ionosonde data from the time reveals a larger–scale
travelling ionospheric disturbance (TID), possibly the result of the
high–latitude activity, travelling in the north-west to south-east direction,
and, simultaneously, a smaller–scale TID travelling in a north-east to south-
west direction, which could be associated with atmospheric gravity wave
activity. The LOFAR observation shows scattering from both TIDs, at different
altitudes and propagating in different directions. To the best of our
knowledge this is the first time that such a phenomenon has been reported.
###### keywords:
ionospheric scintillation – travelling ionospheric disturbances – instability
mechanisms
## 1 Introduction
Radio waves from compact sources can be strongly affected by any ionised
medium through which they pass. Refraction through large-scale density
structures in the medium leads to strong lensing effects where the radio
source appears, if imaged, to focus, de-focus and change shape as the density
structures in the line of sight themselves move and change. Diffraction of the
wavefront by small-scale density structures leads to variations building up in
the intensity of the wavefront with distance from the scattering medium, due
to interference between the scattered waves, an effect known as scintillation.
Observations of all these effects thus contain a great deal of information on
the medium through which the radio waves have passed, including the large-
scale density, turbulence, and the movement of the medium across the line of
sight. Since the second world war, a large number of studies have shown the
effect of ionospheric density variations on radio signals, as reviewed by
Aarons (1982), and this can lead to disruption for applications using Global
Navigation Satellite Systems (GNSS, e.g., GPS), as thoroughly reviewed by,
e.g., Hapgood (2017). The Low-Frequency Array (LOFAR - van Haarlem et al.
(2013)) is Europe’s largest low-frequency radio telescope, operating across
the frequency band 10–250 MHz, and with a dense array of stations in the
Netherlands and, at the time of writing, 13 stations internationally from
Ireland to Poland. It was conceived and designed for radio astronomy but, at
these frequencies, the ionosphere can also have a strong effect on the radio
astronomy measurement (de Gasperin et al., 2018). Ionospheric scintillation,
which is rarely seen over the mid-latitudes on the high-frequency signals of
GNSS, is seen almost continually in observations of strong natural radio
sources by LOFAR.
The wide bandwidth available with LOFAR enables an easy and direct assessment
of scattering conditions and how they change in a given observation, including
whether scattering is weak or strong, or refractive effects dominate, and
enables further information to be gleaned from delay-Doppler spectra (the 2-D
FFT of a dynamic spectrum, termed variously as the “scattering function”,
“generalised power spectrum”, or “secondary spectrum” - here we use the term
“delay-Doppler” spectrum as this clearly describes what the spectrum shows).
In observations of interstellar scintillation these spectra can exhibit
discrete parabolic arcs which can be modelled to give information on the
distance to the scattering “screen” giving rise to the scintillation and its
velocity across the line of sight (Stinebring et al., 2001; Cordes et al.,
2006). Broadband observations of ionospheric scintillation are not common, but
such arcs have been observed using the Kilpisjärvi Atmospheric Imaging
Receiver Array (KAIRA, McKay-Bukowski et al. (2014) – an independent station
built using LOFAR hardware in arctic Finland) in a study by Fallows et al.
(2014). Model spectra produced by Knepp and Nickisch (2009) have also
illustrated parabolic arc structures, particularly in the case of scattering
from a thin scattering screen.
The wide spatial distribution of LOFAR stations also enables scintillation
conditions at these observing frequencies to be sampled over a large part of
western Europe. A dense “core” of 24 stations, situated near Exloo in the
north-east of the Netherlands, over an area with a diameter of $\sim$3.5 km
further provides a more detailed spatial view of the scintillation pattern in
its field of view.
LOFAR thus enables detailed studies of ionospheric scintillation to be
undertaken which can both reveal details which would be unavailable to
discrete-frequency observations such as those taken using GNSS receivers, and
act as a low-frequency complement to these observations to probe potentially
different scattering scales.
A number of different phenomena can lead to scattering effects in radio wave
propagation through the mid-latitude ionosphere: Ionisation structures due to
gradients in the spatial distribution of the plasma density can arise from a
southward expansion of the auroral oval or from large- to small- scale
travelling ionospheric disturbances (TIDs). Large-scale TIDs (LSTIDs) with
wavelengths of about 200 km typically propagate southward after forming in the
high-latitude ionosphere in response to magnetic disturbances (e.g. storms or
sub-storms, Tsugawa et al. (2004)). On the other hand, medium-scale TIDs
(MSTIDs) seem to form in response to phenomena occurring in the neutral
atmosphere triggering atmospheric gravity waves (AGWs), which then propagate
upwards to generate TIDs at ionospheric heights (Kelley, 2009). The morphology
of MSTIDs varies with local time, season, and magnetic longitude. Their
propagation shows irregular patterns that vary on a case-by-case basis,
although they commonly seem to propagate mainly equatorward during winter
daytime and westward during summer night-time (Hernández-Pajares et al., 2006,
2012; Tsugawa et al., 2007; Saito and Fukao, 1998; Emardson et al., 2013).
Smaller-scale ionisation gradients, likely associated with the Perkins
instability (Kelley, 2009, 2011), can then form as a consequence of the
presence of MSTIDs, potentially leading to scintillation at LOFAR frequencies.
In this paper, we perform an in-depth analysis of ionospheric scintillation
seen in an observation of the strong natural radio source Cassiopeia A (Cas A)
overnight on 18-19 August 2013. This observation was amongst the first of its
kind taken with LOFAR and exhibited quite strong scattering effects across the
10-80 MHz band. The purpose of this paper is both technical and scientific: We
first describe the observation itself, and then demonstrate several techniques
to analyse LOFAR data and show how these can bring out the details of
ionospheric structures. Finally, we use supporting data from GNSS and
ionosondes to get a broader picture of conditions in the ionosphere at the
time and how these give rise to the scintillation seen by LOFAR.
## 2 The LOFAR Observation
LOFAR observed Cas A (Right Ascension 23h23m24s, Declination +58°48’54”)
between 21:05 UT on 18 August 2013 and 04:05 UT on 19 August 2013, recording
dynamic spectra from each individual station with a sampling time of 0.083 s
over the band 2.24-97.55 MHz from each available station. The observing band
was sampled with 7808 channels of 12.207 kHz each, but averaged over each
successive 16-channel block to 488 subbands of 195.3125 kHz for the analyses
described in this paper. At the time of observation the available stations
were the 24 stations of the LOFAR “core”, 13 “remote” stations across the
north-east of the Netherlands, and the international stations at Effelsburg,
Unterweilenbach, Tautenburg, Potsdam, and Jülich (Germany), Nançay (France),
Onsala (Sweden), and Chilbolton (UK). The reader is referred to van Haarlem et
al. (2013) for full details of the LOFAR receiving system. The raw data for
this observation can be obtained from the LOFAR long-term archive
(lta.lofar.eu); observation ID L169059 under project “IPS”.
We first illustrate the data in a more traditional sense. Figure 1 shows time
series’ at three discrete observing frequencies of the data taken by LOFAR
station CS002, at the centre of the core, and their associated power spectra.
The power spectra show a fairly typical shape for intensity scintillation: An
initial flat section at the lowest spectral frequencies represents scattering
from larger-scale density structures which are close enough to the observer
that the scattered waves have not had the space to fully interfere to develop
a full intensity scintillation pattern; the turnover (often termed the
“Fresnel Knee”) indicates the largest density scales for which the intensity
scintillation pattern has fully formed; this is followed by a power-law in the
spectra illustrating the cascade from larger to smaller density scales, which
is cut off in these spectra by white noise due to the receiving system (the
flat section covering high spectral frequencies).
Figure 1: 1 Time series of intensity received at three discrete frequencies by
LOFAR station CS002 during the observation of Cas A on 18-19 August 2013,
plus, 1 and 1, power spectra of two 10-minute periods within these time
series’.
However, the advantage of observing a natural radio source with LOFAR is that
full dynamic spectra can be produced covering the full observed band. Dynamic
spectra of data taken by LOFAR station CS002 are presented in Figure 2, which
includes a dynamic spectrum of the full observation, alongside more detailed
views of three different single hours of the observation to illustrate the
range of scattering conditions seen. The strength of the scattering can be
seen much more clearly in this view, compared to time series’ from discrete
observing frequencies. In general, scattering appears weak in this observation
at the highest observing frequencies (where intensity remains highly
correlated across the observing band) with a transition to strong scattering
conditions as the observing frequency decreases. The frequency range displayed
in these dynamic spectra is restricted to exclude the radio–frequency
interference (RFI) which dominates below about 20 MHz and a fade in signal
strength at the higher frequencies due to the imposition of a hard filter to
exclude the FM waveband.
Figure 2: Dynamic spectra of normalised intensity data taken by LOFAR station
CS002 during the observation of Cas A on 18-19 August 2013. The dynamic
spectrum of the entire observing period is given at the top, with zooms into
three different hours of observation below to illustrate the range of
conditions seen. White areas within the plots indicate where RFI was
identified.
RFI is still visible as white areas within the plots. These were identified by
applying a median filter to the data using a window of (19.5 MHz $\times$ 4.2
s) to flatten out the scintillation pattern and then applying a threshold to
identify the RFI. This method appears to be quite successful at identifying
the RFI without also falsely identifying strong peaks in the scintillation as
RFI. For subsequent analysis the RFI data points are replaced by an
interpolation from nearby data, using the Python Astropy (Astropy
Collaboration et al., 2013; Price-Whelan et al., 2018) library routine,
“interpolate_replace_nans”. Normalisation of the data, to correct for long-
period temporal variations in the system (e.g., gain variations resulting from
the varying sensitivity of the receiving antenna array with source elevation),
is carried out after RFI excision by dividing the intensities for each single
frequency subband by a fitted 3rd-order polynomial.
When analysing the data, a variety of scattering conditions are observed
during the course of the observation, as indicated in Figure 2. Different
conditions also naturally occurred over the various international stations
compared to those observed over the Dutch part of LOFAR. In this paper we
therefore focus our analysis on only the first hour of observation and the
measurements taken by the 24 core stations. This allows us to demonstrate the
analysis techniques and to investigate the reason for the scintillation seen
over this interval. Observations from later in this dataset undoubtedly show
other effects and may be discussed in a subsequent publication.
## 3 LOFAR Data Analysis Methods and Results
### 3.1 Delay-Doppler Spectra
The first stage of analysis was the calculation of delay–Doppler spectra:
These were created from the dynamic spectra using five-minute time slices,
advancing every minute through the observation, following the methods
described in Fallows et al. (2014). To avoid regions more heavily contaminated
by RFI, the frequency band used was restricted to 28.5–64.1 MHz. Example
spectra from the first hour are presented in Figure 3.
Figure 3: Example delay-Doppler spectra from the first hour of observation,
taken using five-minute chunks of the dynamic spectrum from CS103 over the
frequency band 28.5-64.1 MHz.
The spectra show two clear arcs: the first is a steeper arc which varies in
curvature throughout the first hour (henceforth labelled for convenience as
the “primary arc”); the second is a very shallow arc (henceforth labelled as
the “secondary arc”) which remains stable for the first 40 minutes of the
observation before fading away. By the end of the first hour of observation
the primary arc also becomes less distinctive for a short while before the
delay–Doppler spectra again show distinctive structure, including a return of
the secondary arc.
The variability of the curvature of the primary arc appears to follow a
wave–like pattern during this part of the observation, as displayed in Figure
4. Here, simple parabolas involving only the square term ($y=Cx^{2}$ where $C$
is the curvature) were plotted with various curvatures until a reasonable
eyeball fit was achieved, and the resulting curvatures plotted for every
minute of observation for the first hour. It proved impossible to achieve
reasonable fits using least-squares methods due to confusion from non–arc
structure in the spectra: Fitting curvatures to these scintillation arcs is a
well–known problem in the interstellar scintillation field and new methods of
attempting this were presented at a recent workshop, but they are not easily
described and have yet to be published. Hence, we do not attempt their
application here.
Figure 4: Curvatures of the steeper arc seen in delay-Doppler spectra
calculated using data from CS103, from simple parabolas fitted by eye. The
grey bounds represent an estimated error.
The presence of two scintillation arcs likely indicates that scattering is
dominated by two distinct layers in the ionosphere. A simple analysis, as
described in Fallows et al. (2014), can be used to estimate the altitude of
the scattering region with a basic formula relating arc curvature $C$ to
velocity $V$ and distance $L$ along the line of sight to the scattering region
(Cordes et al., 2006):
$L=2CV^{2}$ (1)
The square term for the velocity illustrates the importance of gaining a good
estimate of velocity to be able to accurately estimate the altitude of the
scattering region via this method.
### 3.2 Scintillation Pattern Flow
The core area of LOFAR contains 24 stations within an area with a diameter of
$\sim$3.5 km. When viewing dynamic spectra from each of these stations it is
clear that the scintillation pattern is mobile over the core (i.e., temporal
shifts in the scintillation pattern are clear between stations) but does not
necessarily evolve significantly. Therefore, the flow of the scintillation
pattern over the core stations may be viewed directly by simply plotting the
intensity received, for a single subband, by each station on a map of
geographical station locations, for data from successive time steps. A movie
(CasA_20130818_NL.mp4) of the scintillation pattern flow through the
observation is published as an online supplement to this article. The result,
for 12 example time steps, is displayed in Figure 5, where a band of higher
intensities can be seen to progress from north-west to south-east over the
core. It should be noted that the data were integrated in time to 0.92 s for
this purpose, to reduce both flicker due to noise and the duration of the
movie. This does not average over any scintillation structure in this
observation; structure with periodicities shorter than one second would be
obvious in the delay–Doppler spectra as an extension of the arc(s) to greater
than 0.5 Hz along the Doppler frequency axis.
Figure 5: Normalised intensities received by all core stations at an observing
frequency of 44.13 MHz, plotted on a geographical map of the stations. The
intensities are colour-coded using a colour scale from yellow to purple with a
range of 0.8 to 1.3 respectively. Times are at $\sim$10 s intervals from
21:22:25 UT at top left to 21:24:15 UT at bottom right, and each plot uses
data samples with an integration time of 0.92 s. Plot diameter is $\sim$4.5
km.
However, this is not the entire picture because the lines of sight from radio
source to receivers are moving through the ionosphere as the Earth rotates,
meaning that the scintillation pattern flow observed is a combination of flow
due to the movement of density variations in the ionosphere and the movement
of the lines of sight themselves through the ionosphere. Since the speed with
which any single point on a line of sight passing through the ionosphere is
dependent on the altitude of that point (the so-called ionosphere “pierce
point”), this altitude needs to be either assumed or calculated to estimate a
correction to the overall flow speed to obtain the natural ionospheric
contribution. This introduces a natural uncertainty into estimates of
velocity. Figure 6 shows the track of an ionospheric pierce-point at an
assumed altitude of 200 km (an altitude chosen as representative of a typical
F-region altitude where large-scale plasma structures are commonly observed)
for the line of sight from core station CS002 to the radio source Cassiopeia A
through the 7-hour course of the observation to illustrate this movement.
Although not the subject of this paper, it is worth noting that an east to
west flow seen later in the observation appears to be solely due to the lines
of sight moving across a mostly static ionospheric structure (see the online
movie), if the 200 km pierce point is assumed, further illustrating the
necessity to take accurate care of the contribution from line of sight
movement when assessing ionospheric speeds.
Figure 6: Map showing the track of the 200 km pierce point of the line of
sight from core station CS002 to Cassiopeia A from 2013-08-18T21:05:00 to
2013-08-19T04:05:00 UT. The thicker orange part of the track enhances the
first hour of the observation. The black line winding a path across the centre
of the image is the location of the border between the Netherlands and
Germany. The location of CS002 is marked with a black star.
The movie of the scintillation pattern flow, assuming a 200 km pierce point,
shows a clear general north-west to south-east flow during the first hour of
the observation, but also indicates some short (minutes) periods of confusion
in which a north-east to south-west component might be just about discernable.
Any second flow is likely to be associated with a second ionospheric layer and
so warrants further investigation.
### 3.3 Estimating Velocities
The representation of the scintillation pattern flow in movie form gives a
direct and broad picture of the flow pattern and is very helpful in
discovering short time-scale changes in speed and direction. However a cross-
correlation analysis is still necessary to assess actual velocity(s).
Correlation functions are calculated as follows:-
* •
Time series’ of intensity received by each station are calculated by averaging
over the frequency band 55–65 MHz, with these frequencies chosen as the
scintillation pattern remains highly correlated over this band;
* •
For each three-minute data slice, advancing the start time of each successive
slice by 10 s:-
* –
Calculate auto- and cross- power spectra using intensities from every station
pair within the LOFAR core;
* –
Apply low- and high-pass filters to exclude the DC-component and any slow
system variation unlikely to be due to ionospheric effects, and white noise at
the high spectral frequencies. The white noise is also subtracted using an
average of spectral power over the high frequencies;
* –
Inverse–FFT the power spectra back to the time domain to give auto- and cross-
correlation functions.
In the analysis the high- and low-pass filter values were set to 0.01 Hz and
0.5 Hz respectively. This process results in a large set of cross-correlation
functions for each time slice, each of which has an associated station-station
baseline and a primary peak at, typically, a non-zero time delay from which a
velocity can be calculated. However, the direction of the scintillation
pattern flow still needs to be found for calculation of the actual velocity.
For this, directions were assumed for each degree in the full 360–degree range
of possible azimuth directions and the velocities re-calculated using the
components of all baselines aligned with each assumed direction. This results,
for each time slice, in 360 sets of velocities and from each set a median
velocity and standard deviation about the median can be calculated (the median
is used as this is less susceptible to rogue data points than the mean). The
actual flow direction corresponds to the azimuth with the maximum median
velocity and minimum standard deviation, as illustrated in Figure 7.
From this analysis the primary velocity of $\sim$20–40 m s-1 travelling from
north-west to south-east is found, illustrated in Figure 7, corresponding to
the obvious scintillation pattern flow seen in the movie. However, the
presence of a second flow is still not obvious, although a hint of it can be
seen in, for example, the second peak in the median velocity seen in Figure 7.
Figure 7: Plots for single 3-minute time slices of the median velocity and
standard deviation of velocities about the median versus azimuth direction,
calculated from the range of velocities found from all cross-correlation
functions with the baselines within each station pair re-calculated for each
assumed azimuth direction, in the usual form, counting clockwise from north. 7
Time slice commencing 21:05:00 UT using cross-correlations calculated after
applying a high-pass filter at 0.01 Hz; 7 Time slice commencing 21:15:00 UT
using cross-correlations calculated after applying a high-pass filter at 0.07
Hz. Note that the same y-axis is used for both velocity and standard
deviation.
A closer look at the auto- power spectra yielded the key to finding the second
flow. Many spectra show a “bump” which can be viewed as being a second
spectrum superposed on the main one. This is illustrated in Figure 8. To
isolate this part of the spectrum, the spectra were re-filtered with a high-
pass filter value of 0.07 Hz (the low-pass filter value remained the same),
and correlation functions re-calculated. After following the same analysis as
above to find median velocities and standard deviations, the second flow was
found, as illustrated in Figure 7.
Figure 8: Example power spectrum calculated from three minutes of intensity
data received by CS003. The black curve is the raw spectrum, the blue curve is
the filtered and noise-subtracted spectrum. The locations of the low-pass
filter and both high-pass filters used are illustrated.
The analysis, using both high-pass filter values, has been carried out for the
full data set. The velocities and associated directions in degrees azimuth for
the first hour of the observation are given in Figure 9. Error bounds in the
velocities are calculated as the standard deviation about the median of all
velocity values available for the calculated azimuth direction.
Figure 9: Top: Velocities calculated for the first hour of observation from
cross-correlations created after filtering using the two different high-pass
filter values. Bottom: Directions of these velocities, in degrees azimuth.
The higher velocity (henceforth labelled as the “secondary velocity”) shows
some scatter: Periods where the secondary velocity drops to around the primary
velocity values are due to the secondary velocity not being detected at these
times; in these cases, it can still be detected in short-duration drops of
velocity if correlation functions are re-calculated using an even higher high-
pass filter value (the bump in these spectra appears shifted to slightly
higher spectral frequencies). Values which decrease/increase towards/away from
the primary velocity values likely represent a mix between the two velocities.
The larger error bars seen in velocities may also be indicative in some
instances of the standard deviation being broadened by some velocity values
being more dominated by the other flow. The more extended period of scatter
around 21:40 to 22:00 UT is a period where the secondary velocity is less
apparent and the secondary scintillation arc fades from the delay-Doppler
spectra. This indicates that the secondary structure is restricted in either
space or time, either moving out of the field of view of the observation or
ceasing for a period around 21:40 UT. It gives a first indication that the
secondary velocity is associated with the secondary scintillation arc.
### 3.4 Estimating Scattering Altitudes
The velocities can now be used to estimate scattering altitudes, using the
curvatures of the scintillation arcs and the simple formula given in Equation
1. Initially the movement of the line of sight through the ionosphere is not
accounted for, since this correction also requires an estimate of the pierce-
point altitude to be reasonably calculated. Therefore an initial calculation
of the scattering altitudes is made based on velocity values which are not
corrected for this movement.
Using the primary velocities and combining these with the curvatures of the
primary arc (Figure 4) in Equation 1, a range of distances, L, along the line
of sight to the scattering region are found. These distances are converted to
altitudes by accounting for source elevation (Cas A increased in elevation
from 55 ∘ to 64 ∘ during the first hour of observation). This process resulted
in a range of altitudes to the scattering region of 200 to 900 km. Doing the
same for the secondary velocities and applying an arc curvature of 3.2$\pm$0.3
for the secondary scintillation arc gives estimated scattering altitudes of
only $\sim$70 km. If the primary/secondary velocities are combined vice-versa
with the secondary/primary arc curvatures respectively, then the resulting
scattering altitudes are clearly unreasonable (the secondary arc, primary
velocity combination gives estimated altitudes of only $\sim$10 km for
example), lending further credence to the secondary velocity being associated
with the secondary arc.
Velocity contributions from the line of sight movement are calculated as
follows: For each time slice, t, the geographical locations beneath the pierce
point of the line of sight through the ionosphere corresponding to the
estimated scattering altitude at t are calculated, for both t and t +
$\delta$t, where $\delta$t is taken as 3 minutes (the actual value is
unimportant for this calculation). A velocity and its direction are found from
the horizontal distance between these two locations and the direction of
travel from one to the other. The general direction of the movement of the
line of sight through the ionosphere is indicated by the orange line in Figure
6. Although the high scattering altitudes related to the primary scintillation
arc and primary scintillation velocities lead to line-of-sight movements of up
to $\sim$35 m s-1, this movement is almost perpendicular to the direction of
the primary scintillation velocity, limiting the actual contribution to $\sim$
5 m s-1. The line of sight movement is, however, in a very similar direction
to the secondary velocities but the low corresponding scattering altitudes
also limit the contribution in this case to $\sim$5 m s-1.
An iterative procedure is then followed to correct the scintillation
velocities for line-of-sight movement at the calculated scattering altitudes,
re-calculate these altitudes, and re-calculate the line-of-sight movement.
This procedure converges to a set of final scattering altitudes within 5
iterations. These are presented in Figure 10, with error bounds taken as the
lowest and highest possible altitudes resulting from applying this procedure
using the lower and upper limits of the arc curvature and scintillation
velocity error bounds.
Figure 10: Scattering altitudes estimated using Equation 1, the primary
velocities and primary scintillation arc curvatures (blue curve) and the
secondary velocities and the curvature of the secondary scintillation arc (red
dashed curves).
The range of scattering altitudes encompassed by the error bounds is quite
large in some instances, particularly where the calculated altitudes are
higher. Although the square term for the velocity in Equation 1 could lead to
the natural conclusion that the error in the velocity dominates the error in
scattering altitude, the errors in the velocity calculations are, for the most
part, relatively small. Nevertheless, the error in the secondary velocity does
appear to be the dominant error in the lower range of scattering altitudes
(the red curves in Figure 10. However, the dominant error for the higher range
of scattering altitudes appears to be the scintillation arc curvatures,
illustrating the importance of developing accurate fitting methods for these
curvatures. Despite these concerns, it is clear that scattering is seen from
two layers in the ionosphere; the primary scintillation arc arises from
scattering in the F-region and the secondary scintillation arc arises from
scattering much lower down in the D-region. Plasma decays by recombination
with neutral species. In the F-region these densities are lower and so plasma
lifetimes are longer than in the D-region. Typical plasma lifetimes in the
F-region are of the order of hours, while they are of the order of minutes in
the D-region. Hence the structures seen in each level may have a different
source and time history.
## 4 Conditions in the Ionosphere
We now investigate what the overall ionospheric conditions were at the time
and hence the possible cause(s) of the scintillation seen by LOFAR at the
time.
### 4.1 Geomagnetic Conditions
The overall geomagnetic conditions at the time are given in Figure 11, which
shows 24–hour traces of the H–component of magnetic field for a representative
set of magnetometers from the Norwegian magnetometer chain for 18 August 2013.
Activity can be described as unsettled, with a minor substorm at high
latitudes, peaking at the start of the LOFAR observation. However, geomagnetic
activity remains quiet further south, and Kp took a value of 1 at 21 UT on
18th August 2013, indicating that this is unlikely to be a direct cause of the
scintillation seen at LOFAR latitudes. We therefore investigate whether TIDs
were present at the time and whether these could be consistent with the
scintillation seen by LOFAR.
Figure 11: Traces of the H-component of the geomagnetic field recorded on 18
August 2013 by a selection of magnetometer stations from the Norwegian chain.
From top to bottom these are, along with their geomagnetic latitudes (2004,
altitude 100 km): Longyearbyen (75.31∘N), Bjørnøya (71.52∘N), Nordkapp
(67.87∘N), Tromsø(66.69∘N), Rørvik (62.28∘N), and Karmøy (56.43∘N).
### 4.2 Ionosonde Data
The presence of TIDs can be detected through the simultaneous appearance of
wave-like structures on multiple sounding frequencies recorded by an
ionosonde. This method is generally limited to a single point of observation
and detection. The spatial extent of TIDs can be attempted by comparing
multiple traces from different ionosondes, but this is limited by the low
density of ionosondes in a given region. Measurements from the ionosonde in
Chilton (UK) do indeed suggest the presence of wave-like patterns which, in
principle, could be due to a large-scale TID propagating southward and/or
MSTID triggered by a local Atmospheric Gravity Wave (Figure 12).
Figure 12: Multiple traces from the ionosonde in Chilton (UK) recorded between
20:00 18 August 2013 and 06:00 19 August 2013.
### 4.3 GNSS Data
However, measurements from ground-based GNSS receivers offer a more
comprehensive view of the characteristics of any MSTIDs present (Kelley,
2009). In the present study, we focus on perturbations in the slant Total
Electron Content (STEC) observed over the evening of 18 August 2013 from a
network of GNSS stations around the LOFAR core stations (see Figure 13). These
stations are sufficient to infer the presence of TIDs and to infer the upper
spatial scale-size limit of smaller-scale irregularities causing the intensity
scintillation seen at LOFAR wavelengths.
The presence of TID-induced perturbations can be deduced from the presence of
wave-like residuals on the STEC calculated for each satellite-receiver pair.
Figure 13: Map showing the locations of the GNSS stations used.
STEC was calculated and detrended following the methods of Hernández-Pajares
et al. (2006), with the detrending carried out according to:
$\Delta
STEC\left(t\right)=STEC\left(t\right)-\frac{STEC\left(t+\tau\right)+STEC\left(t-\tau\right)}{2}\left[TECu\right]$
(2)
where $\tau=300s$.
It is worth noting that the measured carrier phases $L_{1}$ and $L_{2}$ vary
with time as a consequence of the motion of GNSS satellites relative to a
given receiver on the Earth’s surface. As such, the spatial and temporal
variabilities of ionisation gradients (such as those connected with TIDs and
corresponding instabilities) become entangled. The various detrending methods
(similar to equation 2) lead to an estimate of ionisation gradients by
considering temporal gradients only, with spatial and temporal variabilities
intrinsically entangled in the GNSS observations.
Figure 14 shows examples of wave-like residuals on STEC for one pair of GNSS
stations (Dentergem and Bruxelles in Belgium) observing the same GNSS
satellite. The wave pattern is strongest over the first two hours shown (18:00
- 20:00 UT) but then weakens considerably by the start of the LOFAR
observation, although it remains evident. STEC from the observations of both
stations appears well–correlated, with the Bruxelles dataset lagging behind
that of Dentergem. Since Dentergem lies to the WNW of Bruxelles, this suggests
a strong westerly component in the direction of travel, which could correspond
with the secondary velocity seen by LOFAR.
Figure 14: Example of a satellite-station pair. 14 PRN01 as observed on 18
August 2013 from Dentergem (DENT, blue line) and Bruxelles (BRUX, red line),
both in Belgium, with baseline oriented from WNW to ESE; 14 azimuth/elevation
plot for PRN01 as observed from Dentergem.
Figure 15 shows hourly plots of the overall geographical distribution of the
STEC residuals calculated for all satellite passes seen within each hour by
the GNSS stations used. The patterns shown in Figure 15 suggest a spatially
and temporally varying propagation of MSTID wavefronts with components along
the NE-SW as well as the NW-SE directions. Furthermore, the examples shown in
Figure 15 also indicate the presence of smaller-scale ionisation structures in
proximity to the wavefronts of the MSTIDs. This suggests that the
scintillation seen by LOFAR is likely associated with the perpendicular
propagation of two MSTIDs. However, the STEC variations here are also seen to
fade by the start of the LOFAR observation.
Figure 15: Hourly geographical distribution of all STEC perturbations in the
evening of 18 August 2013: 15 18:00-19:00 UT, 15 19:00-20:00 UT, 15
20:00-21:00 UT, and 15 21:00-22:00 UT.
A further illustration looks at the overall power spectral densities for the
STEC residuals on all satellite–receiver pairs considered here over the hourly
periods 20:00 UT to 21:00] UT and 21:00 UT to 22:00 UT (Figure 16). The
earlier hour is chosen alongside the hour covering the LOFAR data period as
this better displays the components seen in the spectra The temporal
frequencies f can be converted into spatial scales L by assuming a given
velocity VREL for the motion of the ionospheric structures across a GNSS
raypath. That is:
$L=\frac{V_{REL}}{f}$ (3)
where VREL = VIONO-VSAT is the relative velocity between the velocity of the
ionospheric structures and the scan velocity of a single raypath (at the same
shell height). VSAT can be of the order of a few tens of m s-1 at 300 km.
Figure 16: Power Spectral Densities of all the TEC residuals considered during
the hours 16 20:00-21:00 UT and 16 21:00-22:00 UT. The arrows indicate the two
components considered in the text.
There appear to be two main components in the energy cascade from larger to
smaller ionisation scales: one with a period of $\sim$1666 s, and another
component with a period of $\sim$666 s. Taking VREL to be $\sim$100 m s-1 (the
secondary velocity seen by LOFAR as this is in a south-westerly direction and
the example GNSS data in Figure 14 indicate a westerly component), these
periodicities correspond to spatial scales of the order of 166 km and 66 km
respectively. Beyond these scales the STEC analysis is limited by the
sensitivity of the technique (Tsugawa et al., 2007), as the Power Spectral
Densities reach the noise floor (Figure 16). These orders of magnitudes
suggest the presence of a larger–scale TID together with a smaller–scale TID
(Kelley, 2009), while the energy cascade that can be observed through the
Power Spectral Densities indicates that the large–scale structure breaks down
into small–scale structures, likely owing to some instability mechanism.
### 4.4 Estimation of Scale Sizes of Plasma Structures
The scale sizes of the plasma structures causing the scintillation seen by
LOFAR can also be calculated. The variations in the intensity of the received
signal are caused by irregularities with a spatial scale size ranging from the
Fresnel dimension to an order of magnitude below this value (Basu et al.,
1998). The Fresnel length DF is related to the wavelength of the radio wave
$\lambda$ and the line of sight distance from the receiver to the scattering
region L:
$D_{F}=\sqrt{2\lambda L}$ (4)
The Fresnel length was calculated for plasma structures at altitudes of 70 km,
200 km, 350 km and 700 km, elevations of 55∘ and 64∘, and at frequencies of
25.19 MHz, 35.15 MHz and 60.15 MHz, and the results are shown in Table 1. The
altitudes were chosen to cover the range of altitudes identified for the
primary and secondary features in the LOFAR analysis, with the addition of 350
km as this altitude is commonly used within studies using GNSS satellites. The
elevations of the radio source at the start and the end of the first hour of
observation were used to establish the range of Fresnel scales for each
altitude. The frequencies were chosen to match Figure 1.
Table 1 shows that the Fresnel length ranges between $\sim$1 km and $\sim$5 km
and therefore the plasma structures causing the variations in signal intensity
are likely to have a spatial scale size between $\sim$100 m and $\sim$5 km.
The velocities calculated from the LOFAR data indicate that such structures
would take tens of seconds to pass through the source-to-receiver line and the
intensity variations in the observed signal occur on a similar timescale.
Altitude | 70 km | 200 km | 350 km | 700 km
---|---|---|---|---
Frequency | | | |
25.19 MHz | 1.4 | 2.3–2.4 | 3.0–3.2 | 4.3–4.5
35.15 MHz | 1.2 | 1.9–2.0 | 2.6–2.7 | 3.6–3.8
60.15 MHz | 0.9 | 1.5–1.6 | 2.0–2.1 | 2.8–2.9
Table 1: The Fresnel length at altitudes of 70 km, 200 km, 350 km and 700 km
for three different frequencies received by LOFAR station CS002. The ranges
represent calculation using the source elevation for the start and for the end
of the first hour of observation. Values are in km.
## 5 Further Discussion
Geomagnetic activity was low in the mid-latitudes at the time, so enhanced
activity was unlikely to be the direct cause of the scintillation observed.
However, a weak sub-storm was seen at high latitudes and this reached its peak
at the time of the start of the observation. An analysis of GNSS and ionosonde
data reveals the presence of an MSTID travelling in the north-west to south-
east direction. The larger-scale nature of this TID, and its direction of
travel, are strongly consistent with the primary velocity and F-region
scattering altitudes seen in the LOFAR observation. It is possible that this
TID was caused by the geomagnetic activity at high latitude, but this is not
confirmed. Simultaneously, an MSTID is also present travelling in a north-east
to south-west direction which would most likely be associated with an
atmospheric gravity wave propagating up from the neutral atmosphere. The
smaller–scale nature of it, its direction of travel, and likely low-altitude
source make it highly consistent with the secondary velocity and D–region
scattering altitudes observed by LOFAR.
The amplitude of TID activity observed through GNSS STEC residuals decreased
after 20:00 UT (as visible from Figure 14 as well as from the comparison of
hourly geographical maps in Figure 15). However, the LOFAR observation did not
start until 21:05 UT and the presence of scintillation on the radio
frequencies observed by LOFAR remained significant for much of the first hour
of observation. Whilst the presence of MSTIDs seems evident from the ionosonde
multiple traces and GNSS STEC residuals in the region considered, their
signatures do not appear simultaneously above the LOFAR core stations between
21:00 UT and 22:00 UT. This can be explained by the inability of GNSS to
detect smaller amplitudes in STEC residuals, as the noise floor is encountered
for observations with pierce points above the core LOFAR stations (Figures 15
and 16). The scale sizes of plasma structures calculated for the LOFAR data
indicate that these are an order of magnitude lower than those estimated from
GNSS STEC. Smaller ionisation scales developing, for example, through the
Perkins instability could induce scintillation on the VHF radio frequencies
received by LOFAR but not on the L-band frequencies of GNSS. Hence,
scintillation from these mid-latitude smaller-scale ionisation structures,
formed through the Perkins instability in conjuction with the presence of
TIDs, is likely to be what is detected through LOFAR.
## 6 Conclusions and Outlook
This paper presents the results from one of the first observations of
ionospheric scintillation taken using LOFAR, of the strong natural radio
source Cassiopeia A taken overnight on 18–19 August 2013. The observation
exhibited moderately strong scattering effects in dynamic spectra of intensity
received across an observing bandwidth of 10–80 MHz. Delay–Doppler spectra
from the first hour of observation showed two discrete parabolic arcs, one
with a steep and variable curvature and the other with a shallow and static
curvature, indicating that the scintillation was the result of scattering
through two distinct layers in the ionosphere.
A cross-correlation analysis of the data received by stations in the LOFAR
core reveals two different velocities in the scintillation pattern: A primary
velocity of $\sim$20-40 m s-1 is observed travelling in a north-west to south-
east direction, which is associated with the primary parabolic arc and
altitudes of the scattering layer varying in the range $\sim$200–700 km. A
secondary velocity of $\sim$110 m s-1 is observed travelling in a north-east
to south-west direction, which is associated with the secondary arc and a much
lower scattering altitude of $\sim$60–70 km. The latter velocity is associated
with a secondary “bump” seen at higher spectral frequencies in power spectra
calculated from time series’ of intensities, indicating that it is more
strongly associated with smaller–scale structure in the ionosphere.
GNSS and ionosonde data from the time suggest the presence of two MSTIDs
travelling in perpendicular directions. The F-region scattering altitudes
calculated from the LOFAR primary scintillation arc and primary velocity, and
the larger density scales associated with this, suggest that this is
associated with a larger–scale TID seen in GNSS data potentially resulting
from high–latitude geomagnetic activity. The D-region scattering altitudes of
the secondary arc and secondary velocity suggest an atmospheric gravity wave
source for a smaller-scale TID. These TIDs trigger an instability which leads
to the breakdown of the large-scale density structure into smaller scales,
giving rise to the scintillation observed. In the mid-latitude ionosphere the
Perkins mechanism is the most likely instability and the features of the
smaller-scale density variations observed seem consistent with this. To the
best of our knowledge this is the first time that two TIDs have been directly
observed simultaneously at different altitudes.
This observation demonstrates that LOFAR can be a highly valuable tool for
observing ionospheric scintillation in the mid–latitudes over Europe and
enables methods of analysis to be used which give greater insight into the
likely sources of scattering and could be used to improve modelling of them.
With a far greater range of frequencies (multi–octave if the LOFAR high–band
is also used) and fine sampling both across the frequency band and in time,
LOFAR observations offer a wider sensitivity than that available to GNSS
measurements. The analysis techniques shown in this paper also demonstrate
that LOFAR can observe ionospheric structures at different altitudes
simultaneously; a capability not commonly available for GNSS observations. It
also complements these measurements by probing potentially different
scintillation regimes to those observed by GNSS.
Since this observation was taken, many more have been carried out under a
number of projects, recording ionospheric scintillation data at times when the
telescope would otherwise be idle. These demonstrate a wide range of
scintillation conditions over LOFAR, some of which are seen only very
occasionally and perhaps by only one or two of the international stations,
illustrating the value to be had by monitoring the ionosphere at these
frequencies. A Design Study, LOFAR4SpaceWeather (LOFAR4SW – funded from the
European Community’s Horizon 2020 Programme H2020 INFRADEV-2017-1 under grant
agreement 777442) currently underway will design a possible upgrade to LOFAR
to enable, amongst other space weather observations, ionospheric monitoring in
parallel with the regular radio astronomy observations. Such a design, if
implemented, would enable a full statistical study of ionospheric
scintillation at these frequencies, alongside the advances in scintillation
modelling and our understanding of the ionospheric conditions causing it which
can be gleaned in focussed studies such as that presented here.
###### Acknowledgements.
This paper is based on data obtained with the International LOFAR Telescope
(ILT) under project code “IPS”. LOFAR (van Haarlem et al., 2013) is the Low
Frequency Array designed and constructed by ASTRON. It has observing, data
processing, and data storage facilities in several countries, that are owned
by various parties (each with their own funding sources), and that are
collectively operated by the ILT foundation under a joint scientific policy.
The ILT resources have benefitted from the following recent major funding
sources: CNRS-INSU, Observatoire de Paris and Université d’Orléans, France;
BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of
Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The
Science and Technology Facilities Council, UK; Ministry of Science and Higher
Education, Poland. The work carried out at the University of Bath was
supported by the Natural Environment Research Council [Grant Number
NE/R009082/1] and by the European Space Agency/Thales Alenia Space Italy
[H2020-MOM-TASI-016-00002]. We thank Tromsø Geophysical Observatory, UiT the
Arctic University of Norway, for providing the lyr, bjn, nor, tro, rvk, and
kar magnetometer data. The Kp index and the Chilton ionosonde data were
obtained from the U.K. Solar System Data Centre at the Rutherford Appleton
Laboratory. Part of the research leading to these results has received funding
from the European Community’s Horizon 2020 Programme
H2020-INFRADEV-2017-1under grant agreement 777442.
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|
2024-09-04T02:54:58.353153 | 2020-03-09T12:12:03 | 2003.04055 | {
"authors": "Dai Aoki, Ai Nakamura, Fuminori Honda, DeXin Li, Yoshiya Homma, Yusei\n Shimizu, Yoshiki J. Sato, Georg Knebel, Jean-Pascal Brison, Alexandre\n Pourret, Daniel Braithwaite, Gerard Lapertot, Qun Niu, Michal Valiska,\n Hisatomo Harima, and Jacques Flouquet",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26114",
"submitter": "Dai Aoki",
"url": "https://arxiv.org/abs/2003.04055"
} | arxiv-papers | September 16, 2019
# Spin-Triplet Superconductivity in UTe2 and Ferromagnetic Superconductors
Dai Aoki1,2 E-mail<EMAIL_ADDRESS>Ai Nakamura1 Fuminori Honda1 DeXin
Li1 Yoshiya Homma1 Yusei Shimizu1 Yoshiki J. Sato1 Georg Knebel2 Jean-Pascal
Brison2 Alexandre Pourret2 Daniel Braithwaite2 Gerard Lapertot2 Qun Niu2
Michal Vališka2 Hisatomo Harima3 and Jacques Flouquet2 1IMR1IMR Tohoku
University Tohoku University Oarai Oarai Ibaraki Ibaraki 311-1313 311-1313
Japan
2University Grenoble Alpes Japan
2University Grenoble Alpes CEA CEA IRIG-PHELIQS IRIG-PHELIQS F-38000 Grenoble
F-38000 Grenoble France
3Graduate School of Science France
3Graduate School of Science Kobe University Kobe University Kobe 657-8501 Kobe
657-8501 Japan Japan<EMAIL_ADDRESS>
###### Abstract
The spin-triplet state is most likely realized in uranium ferromagnetic
superconductors, UGe2, URhGe, UCoGe. The microscopic coexistence of
ferromagnetism and superconductivity means that the Cooper pair should be
realized under the strong internal field due the ferromagnetism. leading to
the spin-triplet state with equal spin pairing. The field-reinforced
superconductivity, which is observed in all three materials when the
ferromagnetic fluctuations are enhanced, is one of the strong evidences for
the spin-triplet superconductivity. We present here the results of a newly
discovered spin-triplet superconductor, UTe2, and compare those with the
results of ferromagnetic superconductors. Although no magnetic order is found
in UTe2, there are similarities between UTe2 and ferromagnetic
superconductors. For example, the huge upper critical field exceeding the
Pauli limit and the field-reentrant superconductivity for $H\parallel b$-axis
are observed in UTe2, URhGe and UCoGe. We also show the specific heat results
on UTe2 in different quality samples, focusing on the residual density of
states in the superconducting phase.
ferromagnetism, superconductivity, metamagnetism, reentrant superconductivity,
spin triplet, specific heat
The coexistence of ferromagnetism and superconductivity attracts much
attention because unconventional superconductivity with spin-triplet state is
realized. [1, 2] In general, ferromagnetism and superconductivity are
antagonistic because the large internal field due to the ferromagnetism easily
destroys the superconducting Cooper pairs in conventional superconductors.
Thus it is natural to consider that the spin-triplet superconductivity with
equal spin-pairing is realized in ferromagnetic superconductors. The
microscopic coexistence of ferromagnetism and superconductivity is established
only in uranium compounds so far, namely UGe2 [3], URhGe [4] and UCoGe [5].
All of these materials have fairly small ordered moments ($1$–$0.05\,\mu_{\rm
B}/{\rm U}$) in the ferromagnetic phase compared to that for the U free ion
($\sim 3\,\mu_{\rm B}$). Thus the $5f$ electrons for these compounds are
believed to be itinerant in the first principle, although the magnetic
anisotropy is rather strong, indicating the Ising properties. The
superconductivity occurs well below the Curie temperature, $T_{\rm Curie}$, in
the ferromagnetic state. One of the highlights in ferromagnetic
superconductors is the field-reentrant or field-reinforced superconductivity.
In URhGe, for example, the reentrant superconductivity appears when the field
is applied along the hard-magnetization $b$-axis in the orthorhombic
structure. [6] While the transition temperature $T_{\rm sc}$ is $0.25\,{\rm
K}$ at zero field, the reentrant superconducting phase has a maximum $T_{\rm
sc}$ of $0.4\,{\rm K}$ at $H_{\rm R}\sim 12\,{\rm T}$, indicating that the
superconductivity is indeed reinforced under magnetic field. The similar
field-reinforced superconductivity is also observed in UCoGe. [7]
Recently a new spin-triplet superconductor, namely UTe2 was discovered [8, 9].
UTe2 has the body-centered orthorhombic structure with the space group $Immm$
(#71, $D_{2h}^{25}$). The distance of the first nearest neighbor for U atom is
$3.78\,{\rm\AA}$, which is larger than the so-called Hill limit ($\sim
3.5\,{\rm\AA}$). Although no magnetic order was found down to $0.025\,{\rm
K}$, UTe2 is believed to be at the verge of ferromagnetic order. In fact, the
ferromagnetic fluctuations were observed in $\mu$SR [10] and NMR experiments
[11]. By substituting Te with Se, the ferromagnetic order appears at $69$ and
$33\,{\rm K}$ in UTe0.72Se1.28 and UTe0.24Se1.76, respectively, [12] although
the space group for these materials is $Pnma$, which is different from that in
UTe2. The superconducting transition occurs at $1.6\,{\rm K}$ with the sharp
and large specific heat jump. The large residual density of states nearly
$50\,{\%}$ may suggest the possibility for the spontaneous spin-polarization
and the “partially-gapped” superconductivity similar to the A1 state with non-
unitary state. However, it should be stressed that the direct transition from
the paramagnetic state to the non-unitary state at zero field is forbidden
from the symmetry restriction in this orthorhombic system. [13] Thus, the
hidden feature in the superconducting state is expected. No other transition
in the superconducting state is not observed yet at least at zero field. The
pressure study is definitely important to solve this problem. One of the
strongest support for the spin-triplet superconductivity in UTe2 is the huge
upper critical field, $H_{\rm c2}$. In all the field directions, $H_{\rm c2}$
extremely exceeds the Pauli limit ($\sim 3\,{\rm T}$) expected for the weak-
coupling BCS theory. The values of $H_{\rm c2}$ at $0\,{\rm K}$ are $7$ and
$11\,{\rm T}$ for $H\parallel a$ and $c$-axis, respectively. For $H\parallel
b$-axis, the spectacular field-reentrant superconductivity is observed. [14,
15] The transition temperature monotonously decreases with field down to
$0.4\,{\rm K}$ at $16\,{\rm T}$, then increases with field up to $0.9\,{\rm
K}$ at $35\,{\rm T}$. The first order metamagnetic transtion occurs at $H_{\rm
m}=35\,{\rm T}$ [16, 17], and the superconductivity is abruptly collapsed
above $H_{\rm m}$. The metamagnetic transition at $H_{\rm m}$ is connected to
the so-called $T_{\chi,\rm max}$ at low fields, where the magnetic
susceptibility shows a broad maximum for $H\parallel b$-axis. The magnetic
susceptibility shows the Curie-Weiss behavior at high temperatures. At low
temperatures, the anisotropic susceptibility is observed with the relation,
$\chi_{a}>\chi_{c}>\chi_{b}$, which is consistent with the anisotropy of
$H_{\rm c2}$.
In order to study more details on superconducting properties in UTe2, we have
grown single crystals of UTe2 with different quality, and measured the
specific heat at low temperatures. We compare the ($H,T$) phase diagrams for
$H\parallel b$-axis in UTe2, URhGe and UCoGe.
Figure 1: (Color online) Photographs of UTe2 single crystals grown by (a)
chemical vapor transport method and (b) Te-flux method. (c) Laue photograph of
UTe2 single crystal along $c$-axis. (d) U-Te phase diagram cited from Ref.18
Single crystals of UTe2 were grown using chemical vapor transport method. The
starting materials of U and Te were put into a quartz ampoule with the atomic
ratio, U : Te = 2 : 3, together with iodine as the transport agent to be the
density, $3\,{\rm mg/cm}^{3}$ in the inner volume of the quartz ampoule. The
ampoule was slowly heated and was kept at the temperature gradient of
$1060\,^{\circ}{\rm C}$/$1000\,^{\circ}{\rm C}$ for 10 days. Many single
crystals were obtained at lower temperature side, as shown in Fig. 1(a). The
obtained single crystals were checked by the single crystal X-ray analysis.
The lattice parameters and the atomic coordinates are in good agreement with
the values in the previous report. [19] The single crystals were oriented
using the Laue photograph, as shown in Fig. 1(c). The clear superconducting
transition was observed in resistivity and specific heat. The highest residual
resistivity ratio (RRR) is about 40. Note that the we also obtained single
crystals from the previous recipe, that is, a stoichiometric amount of
starting materials and lower temperature gradient $950\,^{\circ}{\rm
C}$/$850\,^{\circ}{\rm C}$. The single crystals were grown at high temperature
side in this case. However, the quality of the single crystal is lower with
the low RRR ($\sim 2$–$3$), and no superconductivity was observed down to
$0.1\,{\rm K}$.
As shown in Fig. 1(d), UTe2 is an incongruent melting compound in the U-Te
phase diagram, and single crystals of UTe2 can be grown using the Te-flux
method as well. The off-stoichiometric amounts of U and Te ($22$ and $78\,{\rm
at\%}$, respectively) were put into an alumina crucible, which was sealed in a
Ta-tube under Ar atmosphere gas. The Ta-tube was then sealed again in a quartz
ampoule. The quartz ampoule was slowly heated up to $1050\,^{\circ}{\rm C}$
and was cooled down to $960\,^{\circ}{\rm C}$. The Te-flux was removed at
$960\,^{\circ}{\rm C}$ in a centrifuge. The obtained single crystals were
large, as shown in Fig. 1(b). However the residual resistivity ratio is not
very large (${\rm RRR}\sim 3$). Although the superconductivity was confirmed
by the resistivity, it was not a bulk property as no anomaly was detected in
the specific heat. Hereafter, we show the results of single crystals obtained
by the chemical vapor transport method with off-stoichiometric amounts of
starting materials and the high temperature gradient.
Figure 2(a) shows the temperature dependence of the electronic specific heat
in UTe2. The part of the data is replotted from Ref. 8, 20. The data of sample
#1 show the highest $T_{\rm sc}$ with a sharp and large jump at $T_{\rm sc}$,
indicating the highest quality sample. The value of $T_{\rm sc}$ defined by
the entropy balance in #1 is $1.65\,{\rm K}$, and the residual $\gamma$-value,
$\gamma_{0}$, which is extrapolated from the fitting $C_{\rm
e}/T=\gamma_{0}+\alpha T^{2}$ at low temperatures assuming a point node gap,
is $\gamma_{0}=52\,{\rm mJ\,K^{-2}mol^{-1}}$. The residual $\gamma$-value is
equal to $44\,{\%}$ of the $\gamma$-value in the normal state. The lower
quality samples show the lower $T_{\rm sc}$ and the higher residual
$\gamma$-value. For example, $T_{\rm sc}$ and $\gamma_{0}$ in sample #4 are
$1.23\,{\rm K}$ and $89\,{\rm mJ\,K^{-2}mol^{-1}}$, respectively. In sample
#5, no superconductivity was observed down to $0.4\,{\rm K}$ in specific heat,
and it is confirmed by the resistivity measurement down $0.1\,{\rm K}$.
Figure 2(b) shows $T_{\rm sc}$ as a function of residual $\gamma$-value
normalized by the $\gamma$-value in the normal state. It is clear that $T_{\rm
sc}$ decreases with increasing the residual $\gamma$-value. It is known that
the decrease of $T_{\rm sc}$ can be described by the Abrikosov-Gor’kov pair-
breaking theory. On the basis of this model, the relation between $T_{\rm sc}$
and the residual density of states had been studied theoretically [21] and
experimentally [22] in high $T_{\rm c}$ cuprates and heavy fermion systems,
where the rapid increase of the residual density of states is reported,
compared to the decrease of $T_{\rm sc}$. This can be explained by the
unitarity scattering in unconventional superconductivity. The present result
in Fig. 2(b) supports the unconventional superconductivity in UTe2.
An important question is whether the residual density of states exists in the
ideal single crystal without impurities. In that case, the partial density of
states would be gapped, and the so-called A1 state should be realized, where
the time reversal symmetry must be broken at zero field. There is, however, no
experimental evidence for the breaking of time reversal symmetry in UTe2.
Figure 2: (Color online) (a) Electronic specific heat in the form of $C_{\rm
e}/T$ vs $T$ of UTe2 in different samples. The phonon contribution is
subtracted from the fitting at high temperature above $T_{\rm sc}$. Part of
the data is replotted from Ref. 8, 20. (b) $T_{\rm sc}$ as a function of the
normalized residual $\gamma$-value for different quality samples.
Next we show in Fig. 3 the ($H,T$) phase diagrams of UTe2 and two
ferromagnetic superconductors, URhGe and UCoGe for $H\parallel b$-axis,
corresponding to the hard-magnetization axis. The field-reentrant or field-
reinforced superconductivity is observed both in URhGe and in UCoGe. The
enhancement of superconductivity is clearly related to the suppression of
$T_{\rm Curie}$, where the ferromagnetic instabilities are realized.
In URhGe, the suppression of $T_{\rm Curie}$ leads to the spin-reorientation
at $H_{\rm R}$ in field sweep at low temperatures. The slope of magnetization
curve for $H\parallel b$-axis is larger than that for $c$-axis. The moment
gradually tilts from $c$ to $b$-axis, and finally it re-orients to $b$-axis at
$H_{\rm R}\sim 12\,{\rm T}$. The $\gamma$-value is enhanced with increasing
field, taking a maximum at $H_{\rm R}$. In NMR experiments, the spin-spin
relaxation rate, $1/T_{2}$ shows the diverging behavior around $H_{\rm R}$,
indicating the strong enhancement of the longitudinal ferromagnetic
fluctuations [23]. The 2nd order transition of $T_{\rm Curie}$ at low fields
changes into the weak 1st order transition at $H_{\rm R}$ through the
tricritical point (TCP). The reentrant superconductivity appears with the
maximum $T_{\rm sc}=0.4\,{\rm K}$ exactly at $H_{\rm R}$.
In UCoGe, the suppression of $T_{\rm Curie}$ with field is similar to the case
for URhGe. However, the spin reorientation is not observed in magnetization
curve, indicating the strong Ising property compared to URhGe. The
superconductivity shows an “S”-shaped curve, which is also connected to the
suppression of $T_{\rm Curie}$. The enhancement of $\gamma$-value and the
development of longitudinal fluctuation are observed in the field scan for
$H\parallel b$-axis.
In UTe2, the field reentrant superconductivity is also observed, while the
temperature range and field range are much wider, compared to those in
ferromagnetic superconductors, URhGe and UCoGe. The reentrant
superconductivity is again linked to the metamagnetic transition at $H_{\rm
m}$. The clear difference from ferromagnetic superconductors is that $H_{\rm
m}$ at high fields originates from the broad maximum of magnetic
susceptibility, $T_{\chi,\rm max}$, instead of $T_{\rm Curie}$. In the heavy
fermion system, it is well known that $H_{\rm m}$ is scaled with $T_{\chi,\rm
max}$ [24]. The value of $H_{\rm m}=35\,{\rm T}$ in UTe2 is consistent with
$T_{\chi,\rm max}=35\,{\rm K}$. The mass enhancement around $H_{\rm m}$ is
detected in the resistivity $A$ coefficient [16] and $\gamma$-value from the
Maxwell relation in magnetization [17] and the direct specific heat
measurements [25]. The crossover at $T_{\chi,\rm max}$ changes into the 1st
order transition at $H_{\rm m}$ through the critical end point (CEP). It
should be noted that the reentrant superconductivity is abruptly suppressed
above $H_{\rm m}$ in UTe2. On the other hand, the reentrant superconductivity
in URhGe still survives in the small field range above $H_{\rm m}$. This is
probably due to the abrupt change of $T_{\rm sc}$ as it is inferred from the
sharp increase of magnetoresistance at $H_{\rm m}$, implying the drastic
change of the electronic state at $H_{\rm m}$.
Figure 3: (Color online) ($H,T$) phase diagrams for $H\parallel b$-axis in
URhGe, UCoGe and UTe2. The data are taken from Ref. 1, 14, 16, 17
In summary, we presented the single crystal growth of the novel spin-triplet
superconductor, UTe2, and the results of specific heat in different quality
samples. The higher quality sample shows the higher $T_{\rm sc}$ and the lower
residual density of states. The rapid increase of the residual density of
states compared to the decrease of $T_{\rm sc}$ supports the unconventional
superconductivity in UTe2. The unusual field-reentrant superconductivity is a
common feature in ferromagnetic superconductors and UTe2. The precise high
field experiments from the microscopic point of views and pressure experiments
using a high quality single crystal are desired for the future studies.
## Acknowledgements
We thank Y. Tokunaga, S. Ikeda, Y. Ōnuki, K. Ishida, K. Izawa, K. Miyake, V.
Mineev, S. Ran, J. Ishizuka, Y. Yanase, K. Machida, C. Paulsen and K. Miyake
for fruitful discussion. This work was supported by ERC starting grant
(NewHeavyFermion), KAKENHI (JP15H05884, JP15H05882, JP15K21732, JP16H04006,
JP15H05745, JP19H00646), and ICC-IMR.
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|
2024-09-04T02:54:58.365392 | 2020-01-06T12:26:18 | 2003.04100 | {
"authors": "Lixin Ge, Xi Shi, Zijun Xu, and Ke Gong",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26115",
"submitter": "Lixin Ge",
"url": "https://arxiv.org/abs/2003.04100"
} | arxiv-papers | # Tunable Casimir equilibria with phase change materials: from quantum
trapping to its release
Lixin Ge<EMAIL_ADDRESS>School of Physics and Electronic Engineering,
Xinyang Normal University, Xinyang 464000, China Xi Shi Department of
physics, Shanghai Normal University, Shanghai, 200234, China Zijun Xu School
of Physics and Electronic Engineering, Xinyang Normal University, Xinyang
464000, China Ke Gong School of Physics and Electronic Engineering, Xinyang
Normal University, Xinyang 464000, China
###### Abstract
A stable suspension of nanoscale particles due to the Casimir force is of
great interest for many applications such as sensing, non-contract nano-
machines. However, the suspension properties are difficult to change once the
devices are fabricated. Vanadium dioxide (VO2) is a phase change material,
which undergoes a transition from a low-temperature insulating phase to a
high-temperature metallic phase around a temperature of 340 K. In this work,
we study Casimir forces between a nanoplate (gold or Teflon) and a layered
structure containing a VO2 film. It is found that stable Casimir suspensions
of nanoplates can be realized in a liquid environment, and the equilibrium
distances are determined, not only by the layer thicknesses but also by the
matter phases of VO2. Under proper designs, a switch from quantum trapping of
the gold nanoplate (“on” state) to its release (“off” state) as a result of
the metal-to-insulator transition of VO2, is revealed. On the other hand, the
quantum trapping and release of a Teflon nanoplate is found under the
insulator-to-metal transition of VO2. Our findings offer the possibility of
designing switchable devices for applications in micro-and nano-
electromechanical systems.
## I Introduction
Micro- and nano-electromechanical systems (MEMS and NEMS), which integrate
electrical and mechanical functionality on the micro- and nano-scales, have
attracted enormous attention Lyshevski (2018); Craighead (2000). Thanks to
small sizes, the MEMS and NEMS exhibit low mass, high mechanical resonance
frequencies and quantum effects, leading to a broad range of applications such
as biological/chemical detections Eom et al. (2011), accelerometers Xu et al.
(2011) and micro/nanomachines Wang (2013). One major problem in MEMS and NEMS
is the $stiction$ which makes the systems collapse and permanent adhesion
caused by the attractive Casimir forces Buks and Roukes (2001); Chan et al.
(2001). The Casimir force is a macroscopic quantum effect which arises from
quantum fluctuations of the electromagnetic field Casimir (1948). In most
cases, two neutral, parallel plates consisted of the same materials are
attractive to each other, and the magnitudes of the attraction depend on
several parameters such as separations, geometric thicknesses, finite
conductivities and temperatures (see, e.g., the review Klimchitskaya et al.
(2009) and Refs.Yampol’skii et al. (2008, 2010). Therefore, repulsive Casimir
forces are highly required for non-contact and low-friction MEMS and NEMS. The
repulsive Casimir forces have been intensively studied in many systems Woods
et al. (2016) including liquid-separated environments Munday et al. (2009);
van Zwol and Palasantzas (2010); Phan and Viet (2011); Dou et al. (2014),
meta-materials Rosa et al. (2008); Zhao et al. (2009, 2011); Song et al.
(2018), topological insulators Grushin and Cortijo (2011); Chen and Wan
(2012); Nie et al. (2013) and specific geometrics Tang et al. (2017); Levin et
al. (2010). In addition, the concept of Casimir equilibria was also
investigated, using the enclosed geometries Rodriguez et al. (2008); Rahi and
Zaheer (2010) and dispersive materials Rodriguez et al. (2010a). Lately,
stable Casimir equilibria of nanoplates above a Teflon-coated gold substrate
were reported by Zhao et al Zhao et al. (2019). However, the Casimir
equilibria of previous studies were mainly in passive systems. Once the
devices are fabricated, the trapping properties are difficult to change. Thus,
the tunable trapping or even the switching from the trapping to its release by
external stimuli (e.g., heating, electric fields or optical waves) is highly
desired in MEMS and NEMS.
In order to active modulate the Casimir effect, one straight way is to change
the dielectric properties of materials under external means Torricelli et al.
(2012); Sedighi et al. (2013); Torricelli et al. (2010). Vanadium dioxide
(VO2) Shao et al. (2018); Zylbersztejn and Mott (1975) is a phase change
material(PCM), which undergoes a transition from a low-temperature insulating
phase to a high-temperature metallic phase at critical temperature 340 K. The
phase transition of VO2 is accompanied by a structural transformation from the
monoclinic phase to the tetragonal one. Meanwhile, the dielectric function of
VO2 changes dramatically during the phase transition, leading to many
interesting applications Wu et al. (2017); Liu et al. (2017); Kats et al.
(2012); van Zwol et al. (2012). In general, the phase transition of VO2 can be
induced by changing the temperature of systems. Alternatively, the phase
transition can be driven by optical lasers Cavalleri et al. (2001); Rini et
al. (2008) or electrical gratings Qazilbash et al. (2008); Nakano et al.
(2012) on a sub-picosecond timescale. Recently, VO2 has been employed to study
the tunable Casimir effect in the vacuum Galkina et al. (2009); Pirozhenko and
Lambrecht (2008); Castillo-Garza et al. (2007). For a large separation (e.g.,
$>$1 $\mu$m), the contrast of Casimir forces due to the phase-transition is
quite large (e.g., over 2 times for two semi-infinite plates of VO2, this
value could be even larger for the case of finite thickness Galkina et al.
(2009); Pirozhenko and Lambrecht (2008)). As the separation is small (e.g.,
$\sim$100 nm), however, the modulation of Casimir forces owning to the phase
transition and finite-thickness decreases greatly Pirozhenko and Lambrecht
(2008); Castillo-Garza et al. (2007). Nonetheless, the Casimir forces are
always attractive and only magnitude modulations have been reported in a
vacuum-separated configuration. The influences of phase transition of VO2 on
the sign modulation of Casimir forces (e.g., from attraction to repulsion) are
yet less explored. In a liquid environment, the function of sign modulation
and the related phenomena such as tunable Casimir equilibria are expected
based on the phase transition of VO2.
Here, the Casimir forces between a nanoplate and a layered structure separated
by a liquid are investigated. The layered structure consists of two kinds of
materials, i.e., Vanadium dioxide (VO2) and Teflon. It is found that stable
Casimir equilibria of gold nanoplates can be realized when a VO2 film is
buried under a semi-infinite Teflon. The properties of Casimir equilibria are
determined, not only by the layer thicknesses but also by the matter phases of
VO2. For thick-film VO2, the Casimir equilibria and quantum traps can be
achieved for both the metallic and insulating phases. On the other hand, a
switch from quantum trapping of the gold nanoplate(“on” state) to its release
(“off” state) can be triggered by the metal-to-insulator phase transition when
the thickness of VO2 is thin (e.g., 20 nm). Finally, stable suspensions of
Teflon nanoplates are also proposed with a complementary design, where the
Teflon substrate is coated by a VO2 film. Unlike the case of gold nanoplates,
the quantum trapping of Teflon nanoplates and its release correspond to the
insulating and metallic phases of VO2. Moreover, the switching phenomena can
be realized only with a several-nanometers thickness of VO2.
Figure 1: (color online) (a) Schematic view of a gold nanoplate suspended in a
liquid environment. (b) The permittivity of different materials (gold, VO2,
bromobenzene and Teflon) as a function of imaginary frequency.
## II Theoretical models
The system in this work is schematically shown in Fig. 1(a), where a gold
nanoplate with thickness $L_{g}$ is suspended in a liquid of bromobenzene. The
separation between the nanoplate and the substrate is $d$. The substrate is
composed of a VO2 film buried under a semi-infinite plate of Teflon. The
thicknesses of the top-layer Teflon and VO2 are denoted as $L_{T}$ and
$L_{\mathrm{V}}$, respectively. The in-plane dimension of the gold nanoplate
is much larger than $L_{g}$ and $d$, and it is considered as a slab during our
calculations. The Casimir force is calculated by $F_{c}=-\partial
E_{c}(d)/\partial d$, where $E_{c}(d)$ is the Casimir energy between the gold
nanoplate and the substrate, having the form Nie et al. (2013); Zhao et al.
(2019)
$E_{c}(d)=A\hbar\int_{0}^{\infty}\frac{d\xi}{2\pi}\int\frac{d^{2}\mathbf{k_{\|}}}{(2\pi)^{2}}\log\det\left[1-\mathbf{R_{1}}\cdot\mathbf{R_{2}}e^{-2k_{3}d}\right],$
(1)
where $\hbar$ is the reduced Planck constant, $A$ is the in-plane area,
$\mathbf{k_{\parallel}}$ is the parallel wavevector,
$k_{3}=\sqrt{k_{\parallel}^{2}+\varepsilon_{liq}(i\xi)\xi^{2}/c^{2}}$ is the
vertical wavevector, $c$ is the speed of light in vacuum,
$\varepsilon_{liq}(i\xi)$ is the permittivity of the intervening liquid
evaluated with imaginary frequency $\omega=i\xi$, $\mathbf{R}_{1,2}$ is the
$2\times 2$ reflection matrix for layered structures, having the form
$\mathbf{R_{j}}=\left(\begin{array}[]{cc}r_{j}^{s}&0\\\
0&r_{j}^{p}\end{array}\right),$ (2)
where $r_{j}$ with $j$=1 and $j$=2 are the reflection coefficients for the
upper and lower layered structures, and the superscripts $s$ and $p$
correspond to the polarizations of transverse electric ($\mathbf{TE}$) and
transverse magnetic ($\mathbf{TM}$) modes, respectively. Note that the
temperature $T$ for Eq. (1) equals 0 K and it is an effective approximation as
the separation $d$ is smaller than 1 $\mu m$ for finite temperatures Milton
(2004). For a nanoplate suspended in a liquid, the reflection coefficients can
be given analytically as follows Zhao et al. (2011)
$r^{\alpha}=\frac{r_{0,j}^{\alpha}+r_{j,0}^{\alpha}e^{-2K_{j}L_{j}}}{1+r_{0,j}^{\alpha}r_{j,0}^{\alpha}e^{-2K_{j}L_{j}}},$
(3)
where $\alpha=s$ and $p$, $L_{j}$ is the thickness of the nanoplate,
$K_{j}=\sqrt{k_{\parallel}^{2}+\varepsilon_{j}(i\xi)\xi^{2}/c^{2}}$ is the
vertical wavevector, $\varepsilon_{j}(i\xi)$ is the permittivity of the
nanoplate. The subscripts of $r_{m,n}^{\alpha}$ represent the light is
incident from the medium $m$ to $n$ (0 means the liquid).
Alternatively, the reflection coefficients for layered structures can be
calculated by a transfer matrix method. The general form is given as
$r=M_{21}/M_{11}$, where $M_{21}$ and $M_{11}$ are the elements of the $M$
matrixZhan et al. (2013). The $M$ matrix is the multiplications of
transmission matrices across different interfaces and propagation matrices in
different layers. Considering an arbitrary $N$-layer system, the $M$-matrix is
given as :
$M=D_{0,1}P(L_{1})D_{1,2}P(L_{2})...D_{N-1,N}P(L_{N})D_{N,N+1},$ (4)
where the transmission matrix $D_{j,j+1}$ is given as:
$D_{j,j+1}=\frac{1}{2}\left[\begin{array}[]{cc}1+\eta&1-\eta\\\
1-\eta&1+\eta\end{array}\right],$ (5)
where $\eta=\varepsilon_{j}(i\xi)K_{j+1}/(\varepsilon_{j+1}(i\xi)K_{j})$ for
p-polarization and $\eta=K_{j+1}/K_{j}$ for s-polarization. The propagation
matric in the $j$-th layer (for both $s$ and $p$ polarizations) is written as:
$P(L_{j})=\left[\begin{array}[]{cc}e^{K_{j}L_{j}}&0\\\
0&e^{-K_{j}L_{j}}\end{array}\right].$ (6)
For example, we have $N=2$ for the multilayered substrate in Fig. 1. The $M$
matrix is given by $M=D_{0,1}P(L_{1})D_{1,2}P(L_{2})D_{2,3}$, where the
subscripts 0, 1, 2 and 3 represent the media of liquid, Teflon, VO2 and Teflon
(from top to down); the thicknesses $L_{1}=L_{T}$, $L_{2}=L_{V}$.
## III Results and discussions
Figure 1(b) shows the permittivity for different materials, where the used
models and parameters are given in the Appendixes. The dielectric function of
VO2 changes dramatically under different temperatures. For temperature
$T>T_{c}$, VO2 is in the metallic phase and it acts as a poor metal. For
$T<T_{c}$, it is in the insulating phase (or called semiconducting phase), and
the corresponding dielectric function nearly matches that of intrinsic silicon
at low frequency Pirozhenko and Lambrecht (2008). To create repulsive Casimir
forces between two dissimilar plates separated by a liquid, the permittivity
should satisfy
$\varepsilon_{1}(i\xi)>\varepsilon_{liq}(i\xi)>\varepsilon_{2}(i\xi)$ for a
vast range of frequency Munday et al. (2009). Clearly, the dielectric
functions of gold and VO2 (either metallic or insulating phase) are larger
than that of bromobenzene over a wide range of frequency. Therefore, the
Casimir force is always attractive for the layered structure of
gold/bromobenzene/VO2. While the Casimir force for the structure of
gold/bromobenzene/Teflon is repulsive instead. Nonetheless, the Casimir
equilibria can not be found for above two layered structures.
Figure 2: (color online) Casimir pressure via different thicknesses of VO2,
where the thickness $L_{T}$=45 nm and $L_{g}$=40 nm are fixed. (a) Thick
films. The solid and dashed lines represent the pressure for the metallic and
insulating phases of VO2, respectively. (b) Thin films. The positive
(negative) sign of the pressure corresponds to the repulsive (attractive)
force.
### III.1 Tunable Casimir equilibria for gold nanoplates
Now we consider the Casimir forces as the substrate is composed of a VO2 film
and Teflon (see Fig. 1(a)). The Casimir pressure ($P_{c}=F_{c}/A$) for the
thick film of VO2 is given in Fig. 2(a). The results show that the curves are
almost identical for $L_{\mathrm{V}}$=200, 500 and 1000 nm, indicating the
weak impact of the thickness for thick-film configurations. The pressure is
repulsive at small separation (e.g., $d<60$ nm), making the nanoplate stay
away from the substrate. As the separation increases further, the Casimir
equilibria (zero pressure) occur and quantum traps can be realized for both
metallic (solid lines) and insulating phases (dashed lines). In addition, the
equilibrium distance $d_{c}$ is shifted under the phase transition of VO2. On
the other hand, the thin-film thickness and the phase transition of VO2 can
play an important role in Casimir pressure as shown in Fig. 2(b). For the
thickness $L_{\mathrm{V}}$ =10 and 20 nm, quantum traps can be realized for
the metallic phase, whereas no trap is found for the insulating phase. Under
such configurations, a switch from quantum trapping of the nanoplate(“on”
state) to its release (“off” state) can be triggered by the metal-insulator
transition of VO2. However, the quantum trapping occurs for both metallic and
insulating phases as the thickness $L_{\mathrm{V}}$ increases to 30 nm, and
the “off” state disappears. Compared with the vacuum-separated configuration
Castillo-Garza et al. (2007), not only the magnitude of Casimir forces can be
modified in a liquid environment, but also the sign could be switched (e.g.,
from attraction to repulsion for $d$=100 nm, $L_{V}$=30 nm), due to the phase-
transition of VO2.
Figure 3: (color online) Casimir pressure contributed from different
frequencies and different parallel wavevectors. (a) and (b) $d$=30 nm; (c) and
(d) $d$=85 nm (close to critical separation); (e) and (f) $d$=150 nm. (a),
(c)and (e) VO2 in the metallic phase ($T>T_{c}$); (b), (d) and (f) VO2 in the
insulating phase ($T<T_{c}$). The layer thicknesses are set as
$L_{\mathrm{V}}$=20 nm and $L_{T}$=45 nm.
To understand the switch transition from the “on” to the “off” state, the
contour plots of Casimir pressure are shown in Fig. 3 under different
separations. The sign of pressure is determined by the competition of VO2 film
(attraction) and low-refractive-index Teflon (repulsion). For small separation
$d$=30 nm, the pressure is dominant by the repulsive component as shown in
Figs. 3(a) and 3(b). For the metallic phase, the attractive component
increases and it compensates the repulsive one as the separation becomes 85 nm
($d\approx d_{c}$), resulting in Casimir equilibrium (see Fig. 3(c)). While
the repulsion is still dominant for the insulating phase as shown in Fig.
3(d). As $d$ increases further to 150 nm, the Casimir pressure turns out to be
dominantly attractive in Fig. 3(e) for the metallic phase, resulting in a
restoring force for stable trapping. By contrast, the pressure is still
dominant by repulsion for the insulating phase as shown in Fig. 3(f). The
pressure maps between the metallic and insulating phases are almost identical
for large energy (e.g., $>$2 eV), whereas the discrepancy manifests at low
energy. The results indicate that the attractive component appears only at low
frequency and small $k$ vector for metallic VO2, where the field cannot
penetrate the metalZhao et al. (2019). Conversely, the field can penetrate the
thin-film of insulating VO2 easily, leading to repulsive Casimir forces.
Figure 4: (color online) (a)The equilibrium distances via the thicknesses of
VO2 under three different configurations (see the inset on the right). The
thickness $L_{T}$ is set as 45 nm. The solid (dashed) curves for type III
represent stable (unstable) equilibria. Contour plots of Casimir pressure via
the thicknesses of coating Teflon for (b) metallic VO2 and (c) insulating VO2,
where the thickness $L_{\mathrm{V}}$=20 nm is fixed. In (b) and (c), the gray
zones represent a strong repulsive pressure larger than 1 Pa. The colors of
the curves denote the same meaning as those in (a).
Practically, the influences of gravitation and buoyancy on the force balances
should be taken into account. The condition for the force equilibrium is
written as $\vec{n}\cdot(\mathbf{F}_{c}+\mathbf{F}_{\mathrm{GB}})$=0, where
$\vec{n}$ is the unit vector normal to the surface,
$F_{\mathrm{GB}}=(\rho_{g}-\rho_{liq})gL_{g}A$ is the sum of gravity and
buoyancy, $g$ is the gravitational acceleration, $\rho_{g}\approx$19.3 g/cm3
and $\rho_{liq}\approx$1.50 g/cm3 is the density of gold and liquid
bromobenzene, respectively. The magnitude of $F_{\mathrm{GB}}/A$ is about 7.0
mPa as the thickness $L_{g}$=40 nm. Three types of configurations are depicted
in the inset of Fig. 4(a) for the cross-section views. The type I
configuration corresponds to a zero-projection (or weightlessness in
aerospace), where the switching from quantum trapping (metallic state) to its
release (insulating state) can be obtained as $L_{\mathrm{V}}$ in a proper
range, from about 2 to 22 nm. For type II configuration, the attractive
$F_{\mathrm{GB}}$ can compensate the long-range repulsive Casimir force at
large $d$, leading to stable suspensions for both $T>T_{c}$ and $T<T_{c}$.
However, the equilibrium distances are different, and it can be inferred that
the stiffness of trapping for metallic phase is stronger than that of the
insulating phase. For type III configuration (a flipped down system), the
switching between trapping and its release can also be realized.
Interestingly, there are two equilibrium distances for this configuration. It
is not difficult to know that the smaller equilibrium distance (solid lines)
is stable, whereas the other one (dashed lines) with larger distance is
unstable to small perturbations in position. For both type II and III
configurations, the deviations from Type I become strong as $d_{c}$ is large.
In addition to the thickness of VO2 film, the top-layer Teflon can also play a
significant role in the Casimir effect. The plots of Casimir pressure via the
thicknesses of the coating Teflon $L_{T}$ are shown in Figs. 4(b) and 4(c),
where $L_{\mathrm{V}}$=20 nm is fixed. The results show that the switching
between quantum trapping and it release occurs only when $L_{T}$ is larger
than about 42 nm (no gravity). The larger the $L_{T}$, the larger of the
position for the Casimir equilibrium. As $L_{T}$ is smaller than 42 nm, the
equilibrium distance is also small, and quantum trappings can be realized for
both metallic and insulating phases. For comparison, the gravitation and
buoyancy are taken into account. Again, strong discrepancies among three
configurations occur as the equilibrium positions larger than about 150 nm,
resulting from the comparable magnitude of $F_{GB}$ and the Casimir force. The
impact of $F_{GB}$ can be further reduced by decreasing the thickness $L_{g}$
near the skin depth (about 22 nm) Lisanti et al. (2005).
Figure 5: (color online) Casimir pressure for a complementary design. A thin
film of VO2 with thickness $L_{V}$ is deposited on a Teflon substrate. (a)The
metallic VO2. (b)The insulating VO2. The thickness of the suspended nanoplate
is set as 100 nm.
Figure 6: (color online) Casimir pressure calculated for finite temperatures
and 0 K approximation from Eq. (1). (a)The trapping and release of a gold
nanoplate. The parameters for the substrate are $L_{T}$=45 nm and $L_{V}$=20
nm. (b)The trapping and release of a Teflon nanoplate. The thickness $L_{V}$
is set as 2 nm.
### III.2 Tunable Casimir equilibria for Teflon nanoplates
The active control of the low-refractive-index nanoplates can also be
significant in many applications. Inspiring by the work Zhao et al. (2019), a
complementary design is schematically shown in the inset of Fig. 5(a). A
Teflon nanoplate is suspended in a liquid of bromobenzene, and the substrate
is a semi-infinite plate of Teflon coated by a VO2 film (high refractive
index). Under such design, the Casimir force is repulsive at very short
separation, due to the dominant interaction between Teflon/bromobenzene/VO2.
As the separation increases, the attractive interaction from
Teflon/bromobenzene/Teflon can be dominant instead, resulting in a stable
Casimir trapping. To verify the design, the Casimir pressure is given
quantitatively in Figs. 5(a) and 5(b) as a function of separation.
Interestingly, the Casimir pressure shows a long-range repulsive behavior for
the metallic VO2, which corresponds to the “off” state. The repulsion pressure
becomes stronger as the thickness $L_{\mathrm{V}}$ enlarges from 2 to 6 nm.
For $L_{\mathrm{V}}$= 2 nm, a Casimir equilibria and strong restoring forces
can be found when VO2 is in the insulating phase. Therefore, the quantum
trapping and release of a Teflon nanoplate can be achieved under the
insulator-to-metal transition of VO2. As the thickness is 4 nm, the restoring
force decreases and the trapping stiffness drops considerably. The calculation
results indicate that the Casimir pressure is quite sensitive to the thickness
of VO2. Due to the low density of Teflon (2.1 g/cm3), the pressure $F_{GB}/A$
for the Teflon nanoplate is about 0.6 mPa, which is reduced significantly
compared with those of gold nanoplates.
### III.3 Finite temperatures effect
To achieve the phase transition of VO2, the temperatures of the devices need
to be changed. We assume that the dielectric functions of the gold and Teflon
are temperature-independent. For organic liquids, the change of refractive
index due to the temperature Li et al. (1994) is an order of $10^{-4}/$ K, and
the permittivity of bromobenzene is also treated as temperature-independent.
Nonetheless, it is interesting to check the finite temperature effect on
Casimir forces. The integral over frequency $\xi$ in Eq. (1) now is replaced
by a discrete summation Rahi et al. (2009):
$\frac{\hbar}{2\pi}\int_{0}^{\infty}d\xi\leftrightarrow
k_{b}T\overset{\infty}{\underset{n=0}{\sum}}^{\prime},$ (7)
where $\xi$ is replaced by discrete Matsubara frequencies
$\xi_{n}=2\pi\frac{k_{b}T}{\hbar}n(n=0,1,2,3\ldots),$ $k_{B}$ is the
Boltzmann’s constant and the prime denotes a prefactor 1/2 for the term $n$=0.
The Casimir pressures under different temperatures are shown in Figs. 6(a) and
6(b), where two different designs are demonstrated. It is found that the
curves for temperature 320 K (insulating phase) overlap with those calculated
from Eq. (1). For the temperature of 360 K, there is only a small deviation
between 0 K and 360 K. Overall, the calculation results from 320 and 360 K
confirm the accuracy of the 0 K approximation. Recently, the switching between
repulsive and attractive Casimir forces based on PCM has also been reported
Boström et al. (2018), where the equilibrium distances for switching occur
only at several nanometers. The equilibrium distances in our work are more
accessible to experiments, and it can be tuned by designing the geometric
thickness of VO2 and the Teflon.
Figure 7: (color online) The total energy of a suspended gold nanoplate (a)
and a Teflon nanoplate (b) under different types of gravity projection. The
solid and dashed lines represent the cases for the metallic VO2 ($T$=360 K)
and insulating VO2 ($T$=320 K), respectively. The in-plane area $A$ is set as
10 $\mu m\times$10 $\mu m$. Other parameters are kept the same as those in
Fig. 6.
### III.4 The effect of Brownian motion
In a real configuration, the position of a nanoplate has a fluctuation around
the equilibrium distances due to the Brownian motion. To evaluate the effect
of Brownian motion, the total energy of the suspended nanoplate should be
known, which are written as $U(d)=E_{c}+\Lambda\times(E_{g}+E_{b})$, where
$E_{c}$ is the Casimir energy given by Eq. [1], $E_{g}=\rho_{p}gL_{p}Ad$ and
$E_{b}=-\rho_{liq}gL_{p}Ad$ are respectively the energies caused by the
gravity and buoyancy Phan et al. (2012), $\rho_{p}$ and $L_{p}$ represent the
density and thickness of the suspended nanoplate. The coefficient $\Lambda$ is
the parameter depending on the gravity projection. For type I configuration
(see the inset of Fig. 4), $\Lambda$=0. While we have $\Lambda$=1 and -1 for
type II and type III configurations. The total energy of a gold and Teflon
nanoplate are shown in Figs. 7(a) and 7(b), respectively. The minimum of
$U(d)/k_{B}T$ corresponds to the equilibrium distance $d_{c}$. Clearly, stable
quantum trapping can be realized for a gold (Teflon) nanoplate when VO2 is in
the metallic (insulating) phase. Due to the balance of repulsive Casimir force
and gravity, stable trapping can also be realized for type II configuration.
Theoretically, the transition rate from the equilibrium distance to another
position due to the Brownian motion is proportional to $\exp(-\triangle
U/k_{B}T)$ Phan et al. (2012); Rodriguez et al. (2010b), where $\triangle U$
represents the energy barrier between these two positions. The calculated
results indicate that the transition rates from Casimir equilibria to stiction
are negligible since the energy barriers $\triangle U//k_{B}T$ are quite large
(e.g., over $10^{4}$) for the gold and Teflon nanoplates. For a flipped-down
system (type III), quantum trapping can be realized for gold (Teflon)
nanoplate when VO2 is in the metallic (insulating) phase. However, there is a
nonzero possibility that the nanoplates can escape from the equilibrium
distances to the free-liquid regime ($d\rightarrow\infty$). Fortunately, the
energy barrier $\triangle U/k_{B}T$ for such a transition is the order of
$10^{2}$ as shown in Figs. 7(a) and 7(b), and the transition rate of the
escape is also negligible.
## IV Conclusions
In summary, the Casimir forces between a nanoplate and a layered structure
containing VO2 films are investigated. In a liquid-separated environment, not
only the magnitude of Casimir forces can be modified, but also the sign could
be switched (e.g., from attraction to repulsion), due to the phase-transition
of VO2. Moreover, a stable Casimir suspension of nanoplates and its tunability
are revealed. For a gold nanoplate, a switch from the quantum trapping to its
release is obtained under the metal-to-insulator transition of VO2. In
addition, the quantum trapping and release of a Teflon nanoplate are
demonstrated with a complementary design. The switching performances due to
the layer thicknesses, gravitation and temperatures are discussed as well.
Theoretically, the bromobenzene can be substituted by other high-refractive-
index liquids (e.g., glycerol and styrene van Zwol and Palasantzas (2010)) as
long as the boiling points are larger than $T_{c}$. The Teflon can also be
replaced by other low-refractive-index materials (e.g., mesoporous silica Dou
et al. (2014)). This work offers the possibility of designing switchable
devices in MEMS/NEMS, resulting from the quantum fluctuations of the
electromagnetic field.
###### Acknowledgements.
This work is supported by the National Natural Science Foundation of China
(Grant No. 11804288, No. 11704254, No. 61571386 and No. 61974127), and the
Innovation Scientists and Technicians Troop Construction Projects of Henan
Province. The research of L.X. Ge is further supported by Nanhu Scholars
Program for Young Scholars of XYNU.
## Appendix A The permittivity of gold
Here, a generalized Drude-Lorentz model is applied for the permittivity of
gold Sehmi et al. (2017):
$\varepsilon(i\xi)=\varepsilon_{D}(i\xi)+\varepsilon_{L}(i\xi),$ (8)
where the Drude term is given by:
$\varepsilon_{D}(i\xi)=\varepsilon_{\infty}+\frac{\gamma\sigma}{\xi(\xi+\gamma)},$
(9)
where $\varepsilon_{\infty}=$0.83409, $\sigma=$3134.5 eV, and $\gamma=$0.02334
eV. The Lorentz term is described by four pairs of poles:
$\varepsilon_{L}(i\xi)=\overset{4}{\underset{j=1}{\sum}}\left(\frac{i\sigma_{j}}{i\xi-\Omega_{j}}+\frac{i\sigma_{j}^{\ast}}{i\xi+\Omega_{j}^{\ast}}\right)$
(10)
where $\sigma_{j}$ and $\Omega_{j}$ are the generalized conductivity and
resonant frequency of the $j$-th Lorentz pole. The star superscripts represent
the operation of complex conjugation. The generalized Drude-Lorentz model
respects causality, and it can represent the exact physical resonances in the
material. The parameters for the model are listed in the Table I.
Table 1: The fitted parameters for Lorentz poles of gold Sehmi et al. (2017). $j$-th | $\sigma_{j}(\mathrm{eV})$ | $\Omega_{j}(\mathrm{eV})$
---|---|---
1 | -0.01743+0.3059*I | 2.6905-0.16645*I
2 | 1.0349+1.2919*I | 2.8772-0.44473*I
3 | 1.2274+2.5605*I | 3.7911-0.81981*I
4 | 9.85+37.614*I | 4.8532-13.891*I
## Appendix B The permittivity of VO2
For temperature $T>T_{c}$, VO2 is in the metallic phase, and the permittivity
is given by Pirozhenko and Lambrecht (2008); Castillo-Garza et al. (2007)
$\displaystyle\varepsilon(i\xi)$ $\displaystyle=$ $\displaystyle
1+\frac{\omega_{p}^{2}}{\xi(\xi+\gamma)}+\frac{\varepsilon_{\infty}-1}{1+\xi^{2}/\omega_{\infty}^{2}}$
(11)
$\displaystyle+\underset{j=1}{\overset{4}{\sum}}\frac{s_{j}}{1+(\xi/\omega_{j})^{2}+\Gamma_{j}\xi/\omega_{j}},$
where $\varepsilon_{\infty}=3.95,\omega_{p}=3.33$ eV, and $\gamma=0.66$ eV.
The parameters $s_{j}$ and $\Gamma_{j}$ represent respectively the strength
and linewidth of the $j$-th oscillator (resonant frequency $\omega_{j}$).
For temperature $T<T_{c}$, VO2 is in the insulating phase, and the
permittivity is described as
$\varepsilon(i\xi)=1+\frac{\varepsilon_{\infty}-1}{1+\xi^{2}/\omega_{\infty}^{2}}+\underset{j=1}{\overset{7}{\sum}}\frac{s_{j}}{1+(\xi/\omega_{j})^{2}+\Gamma_{j}\xi/\omega_{j}},$
(12)
where $\varepsilon_{\infty}=4.26$ and $\omega_{\infty}=15$ eV. The above
equations for metallic and insulating VO2 are valid for a wide range of
frequency (up to about 10 eV)Castillo-Garza et al. (2007), which are modified
versions of Ref. Verleur et al. (1968). The parameters are listed in Table II.
Table 2: The parameters for the metallic and insulating VO2 Castillo-Garza et al. (2007). $j$-th ($T>T_{c}$) | $S_{j}$ | $\omega_{j}(\mathrm{eV})$ | $\Gamma_{j}$
---|---|---|---
1 | 1.816 | 0.86 | 0.95
2 | 0.972 | 2.8 | 0.23
3 | 1.04 | 3.48 | 0.28
4 | 1.05 | 4.6 | 0.34
$j$-th ($T<T_{c}$) | $S_{j}$ | $\omega_{j}(\mathrm{eV})$ | $\Gamma_{j}$
1 | 0.79 | 1.02 | 0.55
2 | 0.474 | 1.30 | 0.55
3 | 0.483 | 1.50 | 0.50
4 | 0.536 | 2.75 | 0.22
5 | 1.316 | 3.49 | 0.47
6 | 1.060 | 3.76 | 0.38
7 | 0.99 | 5.1 | 0.385
Table 3: The parameters for Teflon(left) and bromobenzene (right)van Zwol and Palasantzas (2010). $j$-th | $C_{j}$ | $\omega_{j}(\mathrm{eV})$ | $C_{j}$ | $\omega_{j}(\mathrm{eV})$
---|---|---|---|---
1 | 0.0093 | 0.0003 | 0.0544 | 0.00502
2 | 0.0183 | 0.0076 | 0.0184 | 0.0309
3 | 0.139 | 0.0557 | 0.0475 | 0.111
4 | 0.112 | 0.126 | 0.532 | 6.75
5 | 0.195 | 6.71 | 0.645 | 13.3
6 | 0.438 | 18.6 | 0.240 | 24.0
7 | 0.106 | 42.1 | 0.00927 | 99.9
8 | 0.0386 | 77.6 | |
## Appendix C The permittivity of Teflon and bromobenzene
The permittivity for the Teflon and bromobenzene are given by the oscillator
model van Zwol and Palasantzas (2010):
$\varepsilon(i\xi)=1+\underset{j}{\overset{}{\sum}}\frac{C_{j}}{1+(\xi/\omega_{j})^{2}},$
(13)
where $C_{j}$ corresponds to the oscillator strength for the $j$-th resonance,
and $\omega_{j}$ is the corresponding resonant frequency. The values of
$C_{j}$ and $\omega_{j}$ listed in Table III are fitted from the experimental
data in a wide range of frequency.
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|
2024-09-04T02:54:58.377245 | 2020-03-09T13:08:27 | 2003.04110 | {
"authors": "Sajid Ali, Georg Bergner, Henning Gerber, Istvan Montvay, Gernot\n M\\\"unster, Stefano Piemonte and Philipp Scior",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26116",
"submitter": "Sajid Ali",
"url": "https://arxiv.org/abs/2003.04110"
} | arxiv-papers | # MS-TP-20-17
Continuum extrapolation of Ward identities in $\mathbf{\mathcal{N}=1}$
supersymmetric SU(3) Yang-Mills theory
Sajid Ali<EMAIL_ADDRESS>University of Münster, Institute
for Theoretical Physics, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany
Government College University Lahore, Department of Physics, Lahore 54000,
Pakistan Georg Bergner<EMAIL_ADDRESS>University of Jena,
Institute for Theoretical Physics, Max-Wien-Platz 1, D-07743 Jena, Germany
University of Münster, Institute for Theoretical Physics, Wilhelm-Klemm-Str.
9, D-48149 Münster, Germany Henning Gerber<EMAIL_ADDRESS>University of Münster, Institute for Theoretical Physics, Wilhelm-Klemm-Str.
9, D-48149 Münster, Germany Istvan Montvay<EMAIL_ADDRESS>Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, D-22607 Hamburg, Germany Gernot
Münster11footnotemark: 1 University of Münster, Institute for Theoretical
Physics, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany Stefano Piemonte
<EMAIL_ADDRESS>University of Regensburg, Institute for Theoretical
Physics, Universitätsstr. 31, D-93040 Regensburg, Germany Philipp Scior
<EMAIL_ADDRESS>Universität Bielefeld, Fakultät für Physik,
Universitätsstr. 25, D-33615 Bielefeld, Germany
(14th May 2020)
###### Abstract
Abstract: In $\mathcal{N}=1$ supersymmetric Yang-Mills theory, regularised on
a space-time lattice, in addition to the breaking by the gluino mass term,
supersymmetry is broken explicitly by the lattice regulator. In addition to
the parameter tuning in the theory, the supersymmetric Ward identities can be
used as a tool to investigate lattice artefacts as well as to check whether
supersymmetry can be recovered in the chiral and continuum limits. In this
paper we present the numerical results of an analysis of the supersymmetric
Ward identities for our available gauge ensembles at different values of the
inverse gauge coupling $\beta$ and of the hopping parameter $\kappa$. The
results clearly indicate that the lattice artefacts vanish in the continuum
limit, confirming the restoration of supersymmetry.
## 1 Introduction
Supersymmetry (SUSY) is an elegant idea which relates fermions and bosons,
whose spin differs by 1/2, through supercharges [1]. SUSY provides dark matter
candidates, arising from the lightest supersymmetric particles [2]. In
addition to that, supersymmetric extensions of the Standard Model would
resolve the hierarchy problem [3]. $\mathcal{N}=1$ supersymmetric Yang-Mills
(SYM) theory, which is being considered in this article, provides an extension
of the pure gluonic part of the Standard Model [4]. It describes the strong
interactions between gluons and gluinos, the superpartners of the gluons.
Gluinos are Majorana particles that transform under the adjoint representation
of the gauge group. The on-shell Lagrangian of $\mathcal{N}=1$ SYM theory,
which consists of the gluon fields $A^{a}_{\mu}(x)$ and the gluino fields
$\lambda^{a}(x)$, where $a=1,\ldots,N^{2}_{c}-1$, can be written in Minkowski
space as
$\mathcal{L}_{\text{SYM}}=-\frac{1}{4}F^{a}_{\mu\nu}F^{a,\mu\nu}+\frac{\mathrm{i}}{2}\bar{\lambda}^{a}\gamma^{\mu}\left(\mathcal{D}_{\mu}\lambda\right)^{a}-\frac{m_{\tilde{g}}}{2}\bar{\lambda}^{a}\lambda^{a},$
(1)
where the first term, containing the field strength tensor $F^{a}_{\mu\nu}$,
is the gauge part, and $\mathcal{D}_{\mu}$ in the second term is the covariant
derivative in the adjoint representation of the gauge group SU($N_{c}$),
$N_{c}$ being the number of colors. The last part of the above Lagrangian is a
gluino mass term which breaks SUSY softly for $m_{\tilde{g}}\neq 0$, which
means that it does not affect the renormalisation properties of the theory and
that the spectrum of the theory depends on the gluino mass in a continuous
way. The physical spectrum of this theory is expected to consist of bound
states of gluons and gluinos, arranged in mass degenerate supermultiplets if
SUSY is not broken [5, 6].
In order to perform Monte-Carlo simulations of the theory, we discretise the
Euclidean action and put it onto a four-dimensional hypercubic lattice. We use
the Curci-Veneziano version [7] of the lattice action $S=S_{g}+S_{f}$, where
the gauge part $S_{g}$ is defined by the usual plaquette action
$S_{g}=-\frac{\beta}{N_{c}}\sum_{p}\mathrm{Re}\left[\mathrm{tr}\left(U_{p}\right)\right],$
(2)
with the inverse gauge coupling given by $\beta=2N_{c}/g^{2}$, and the
fermionic part
$S_{f}=\frac{1}{2}\sum_{x}\left\\{\bar{\lambda}^{a}_{x}\lambda_{x}^{a}-\kappa\sum_{\mu=1}^{4}\left[\bar{\lambda}^{a}_{x+\hat{\mu}}V_{ab,x\mu}(1+\gamma_{\mu})\lambda^{b}_{x}+\bar{\lambda}^{a}_{x}V^{T}_{ab,x\mu}(1-\gamma_{\mu})\lambda^{b}_{x+\hat{\mu}}\right]\right\\}$
(3)
implements the gluinos as Wilson fermions. Here the adjoint link variables are
defined by
$V_{ab,x\mu}=2\,\mathrm{tr}\,(U_{x\mu}^{\dagger}T_{a}U_{x\mu}T_{b})$, where
$T_{a}$ are the generators of the gauge group, and the hopping parameter
$\kappa$ is related to the bare gluino mass $m_{\tilde{g}}$ by
$\kappa=1/(2m_{\tilde{g}}+8)$. In order to approach the limit of vanishing
gluino mass, the hopping parameter has to be tuned properly. In our numerical
investigations the fermionic part is additionally $O(a)$ improved by adding
the clover term
$-(c_{sw}/4)\,\bar{\lambda}(x)\sigma_{\mu\nu}F^{\mu\nu}\lambda(x)$ [8].
In our previous investigations we have determined the low-lying mass spectrum
of the theory with gauge group SU(2) and SU(3) non-perturbatively from first
principles using Monte Carlo techniques [4, 9, 10, 11], and obtained mass
degenerate supermultiplets [12].
## 2 SUSY Ward identities
In classical physics, Noether’s theorem provides a relation between symmetries
and conservation laws. In the case of quantum field theories, symmetries are
translated to Ward identities, representing quantum versions of Noether’s
theorem. In $\mathcal{N}=1$ supersymmetric Yang-Mills theory a gluino mass
term breaks SUSY softly. The soft breaking effects vanish in the chiral limit,
a limit where theory is characterised by massless gluinos. In order to analyse
this breaking of supersymmetry and to identify the chiral limit, we employ the
Ward identities for supersymmetry. Moreover, on the lattice supersymmetry is
broken explicitly due to the introduction of the discretisation of space-time
lattice as a regulator of the theory. SUSY Ward identities can be used to
check whether supersymmetry is restored in the continuum limit.
In the Euclidean continuum, on-shell supersymmetry transformations of the
gauge and gluino fields are given by
$\delta
A_{\mu}^{a}=-2\,\mathrm{i}\,\overline{\lambda}^{a}\gamma_{\mu}\,\varepsilon\,,\quad\delta\lambda^{a}=-\sigma_{\mu\nu}F_{\mu\nu}^{a}\,\varepsilon\,,$
(4)
where the transformation parameter $\varepsilon$ is an anticommuting Majorana
spinor. From the variation of the action under a supersymmetry transformation
with a space-time-dependent parameter $\varepsilon(x)$ one derives the SUSY
Ward identities. For any suitable gauge invariant local operator $Q(y)$, they
read
$\left\langle\partial^{\mu}S_{\mu}(x)Q(y)\right\rangle=m_{\tilde{g}}\left\langle\chi(x)Q(y)\right\rangle-\left\langle\frac{\delta
Q(y)}{\delta\bar{\epsilon}(x)}\right\rangle,$ (5)
where $S_{\mu}(x)=(S_{\mu}^{\alpha}(x))$ is the supercurrent of spin 3/2, and
the term $m_{\tilde{g}}\left\langle\chi(x)Q(y)\right\rangle$ is due to the
gluino mass in the action of the theory. In the continuum the supercurrent
$S_{\mu}(x)$ and the operator $\chi(x)$ are given by
$\displaystyle S_{\mu}(x)$
$\displaystyle=-\frac{2\,\mathrm{i}}{g}\mathrm{tr}\left[F^{\nu\rho}(x)\sigma_{\nu\rho}\gamma_{\mu}\lambda(x)\right],$
(6) $\displaystyle\chi(x)$
$\displaystyle=+\frac{2\,\mathrm{i}}{g}\mathrm{tr}\left[F^{\mu\nu}(x)\sigma_{\mu\nu}\lambda(x)\right].$
(7)
The last term of Eq. (5) is a contact term, which contributes only if $x=y$,
and it can be avoided if $Q(y)$ is not localised at $x$. Therefore the contact
term is ignored in the following discussions.
The four-dimensional space-time lattice breaks SUSY explicitly. As a
consequence, the lattice versions of the Ward identities differ from their
continuum counter parts by an additional term $\left\langle
X_{S}(x)Q(y)\right\rangle$. The explicit form of this term is known, but need
not be displayed here. At tree level this term is proportional to the lattice
spacing $a$ and vanishes in the limit of zero lattice spacing. At higher
orders in perturbation theory, nevertheless, the contribution of this term is
finite in the continuum limit due to divergences proportional to 1/$a$ that
multiply the factor $a$. This plays a role for the renormalisation of the
supercurrent and of the gluino mass [7, 13]. In the renormalisation of SUSY
Ward identities, operators of dimensions $\leq 11/2$ have to be taken into
account. They lead to a modification of the gluino mass, and in addition a
current $T_{\mu}$, mixing with the supercurrent, appears, corresponding to an
operator of dimension $9/2$. Consequently, on the lattice the following Ward
identities are obtained
$Z_{S}\left\langle\nabla_{\mu}S_{\mu}(x)Q(y)\right\rangle+Z_{T}\left\langle\nabla_{\mu}T_{\mu}(x)Q(y)\right\rangle=m_{S}\left\langle\chi(x)Q(y)\right\rangle+O(a),$
(8)
where $Z_{S}$ and $Z_{T}$ are renormalisation coefficients. The subtracted
gluino mass is defined as $m_{S}=m_{\tilde{g}}-\bar{m}$, where $\bar{m}$ is
the mass subtraction coming from the operators of dimension $7/2$. The mixing
current is defined as
$T_{\mu}(x)=\frac{2\,\mathrm{i}}{g}\mathrm{tr}\left[F_{\mu\nu}(x)\gamma_{\nu}\lambda(x)\right].$
(9)
Regarding the local insertion operator $Q(y)$, our choice is the spinor
$Q(y)=\chi^{(\mathrm{sp})}(y)$, with
$\chi^{(\mathrm{sp})}(y)=\sum_{i<j}\mathrm{tr}\left[F_{ij}(y)\sigma_{ij}\lambda(y)\right],$
(10)
where the indices $i,j\in\\{1,2,3\\}$. The reason behind this choice is that
it gives the best signal [13].
## 3 Numerical analysis of SUSY Ward identities
We have analysed the SUSY Ward identities numerically, employing the
configurations produced in our project on $\mathcal{N}=1$ supersymmetric Yang-
Mills theory with gauge group SU(3). Numerically it is convenient to use
integrated Ward identities where integration or sum is performed over all
three spatial coordinates. The resulting identities will then hold for every
time-slice distance $t$. In the analysis the data from all time-slice
distances in an interval $t_{min}\leq t\leq t_{max}$ are included. The lower
limit $t_{min}$ is always taken to be larger or equal than 3 in order to avoid
contamination from contact terms. The choice of $t_{min}$ for the different
ensembles of configurations is discussed below. Since the correlation
functions are symmetric or antisymmetric in $t$, the upper limit $t_{max}$ is
chosen to be half of the time extent of the lattice. Each term in Eq. (8) is a
4$\times$4 matrix in spin-space and can be expanded in the basis of 16 Dirac
matrices, i. e.
$\left\\{\boldsymbol{1},\gamma_{5},\gamma_{\mu},\gamma_{\mu}\gamma_{5},\mathrm{i}\sigma_{\mu\nu}\right\\}$.
It can be shown, with the help of discrete symmetries, that only the following
two contributions are non-zero [13]:
$\hat{x}_{b,t,1}+A\hat{x}_{b,t,2}=B\hat{x}_{b,t,3},\qquad\text{with}\quad
b=1,2\,,$ (11)
where $A=Z_{T}Z^{-1}_{S}$, $B=am_{S}Z^{-1}_{S}$, and
$\displaystyle\hat{x}_{1,t,1}$
$\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}S_{4}(x)Q(0)\right\rangle,$
$\displaystyle\hat{x}_{2,t,1}$
$\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}S_{4}(x)\gamma_{4}Q(0)\right\rangle,$
$\displaystyle\hat{x}_{1,t,2}$
$\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}T_{4}(x)Q(0)\right\rangle,$
$\displaystyle\hat{x}_{2,t,2}$
$\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}T_{4}(x)\gamma_{4}Q(0)\right\rangle,$
(12) $\displaystyle\hat{x}_{1,t,3}$
$\displaystyle\equiv\sum_{\vec{x}}\left\langle\chi(x)Q(0)\right\rangle,$
$\displaystyle\hat{x}_{2,t,3}$
$\displaystyle\equiv\sum_{\vec{x}}\left\langle\chi(x)\gamma_{4}Q(0)\right\rangle.$
In these equations the Dirac indices of $S_{4}(x)$, $T_{4}(x)$, $\chi(x)$ and
of the insertion operator $Q(0)$ are not written, and sums over repeated
(hidden) Dirac indices are implied. Also, $O(a)$ terms that vanish in the
continuum limit are not written explicitly in these equations. Introducing a
double index $i=(b,t)$, running over $2T$ values, where $T$ is the time extent
of the lattice, and denoting $A_{1}=1,A_{2}=A,A_{3}=-B$, Eq. (11) is written
compactly
$\sum_{\alpha=1}^{3}A_{\alpha}\hat{x}_{i\alpha}=0\,.$ (13)
In these equations the $\hat{x}_{i\alpha}=\langle x_{i\alpha}\rangle$ are the
expectation values of random variables $x_{i\alpha}$, which themselves are
considered to be the results of a finite Markov chain. We compute the
estimators $x_{i\alpha}$ for the correlation functions $\hat{x}_{i\alpha}$
numerically using high performance facilities. The Eqs. (13), including all
time-slice distances $t$ from $t_{min}$ to $t_{max}$, are solved
simultaneously for $A_{\alpha}$ by means of minimal chi-squared methods. Two
methods, namely the so-called Local Method and Global Method, have been used
in the past by our collaboration [4, 13]. These methods, however, do not take
properly into account correlations between the different quantities appearing
in Eq. (13). For this purpose we have developed a new method based on a
generalised least squares fit, the so-called GLS Method [14], based on the
maximum likelihood. For fixed $A_{\alpha}$ ($\alpha=1,2,3$) and given
numerical data $x_{i\alpha}$, the probability distribution $P\sim\exp(-L)$ of
the quantities $\hat{x}_{i\alpha}$, subject to the constraints (13), has its
maximum at a point where $L=L_{min}$, with
$L_{min}=\frac{1}{2}\sum_{i,\alpha,j,\beta}(A_{\alpha}x_{i\alpha})(D^{-1})_{ij}(A_{\beta}x_{j\beta})\,,$
(14)
where
$D_{ij}=\sum_{\alpha,\beta}A_{\alpha}A_{\beta}(\langle
x_{i\alpha}x_{j\beta}\rangle-\langle x_{i\alpha}\rangle\langle
x_{j\beta}\rangle).$ (15)
Next, the desired coefficients $A_{\alpha}$ have to be found such that
$L_{min}$ as a function of $A_{2}$ and $A_{3}$ is minimised. This cannot be
solved analytically, and we find $A_{\alpha}$ numerically such that the global
minimum of $L_{min}(A_{2},A_{3})$ is reached; for details see Ref. [15]. In
particular, owing to $A_{3}=-am_{S}Z^{-1}_{S}$ this provides us with the
subtracted gluino mass $m_{S}$ up to the renormalisation factor. To estimate
the statistical uncertainties we employ the standard Jackknife procedure.
### 3.1 Discretisation effects
All terms in the Ward identity (8), including the $O(a)$ term $\left\langle
X_{S}(x)Q(y)\right\rangle$, are correlation functions of gauge invariant
operators. In the corresponding Eqs. (11) they are correlation functions of
operators localised on time slices or pairs of adjacent time slices at
distance $t$. As for any gauge invariant correlation function of this type,
they decay exponentially in $t$, with a decay rate given by the mass gap of
the theory. For very small $t$ the contributions of higher masses will affect
the impact of the $O(a)$ term on the Ward identities. Therefore we expect that
the value of the obtained gluino mass will depend on the minimal time slice
distance $t_{min}$. This effect should become negligible at sufficiently large
$t_{min}$. On the other hand, if $t_{min}$ is chosen too large, noise in the
data will dominate. The behaviour that can be observed in Fig. 1 is compatible
with these expectations.
Figure 1: The subtracted gluino mass $am_{S}Z^{-1}_{S}$ as a function of
$t_{min}$ calculated with the GLS Method at $\beta=5.6$. At small values of
$t_{min}$ the subtracted gluino mass is affected by contact terms and by
$O(a)$ terms. Data from $t_{min}=2$ and $t_{min}=3$ are shown, but do not
enter our final analysis.
An adequate choice of $t_{min}$ is therefore important for the quality of the
results. We cope with this in two ways.
In order to avoid perturbing effects at too small $t_{min}$ and a poor signal-
to-noise ratio at too large $t_{min}$, for each hopping parameter and inverse
gauge coupling, the value of $t_{min}$ is selected by finding an optimal
starting point where a plateau in the subtracted gluino mass begins. The
results are presented in Tab. 1.
$\beta=5.4$ | $\beta=5.4$ | $\beta=5.45$ | $\beta=5.5$ | $\beta=5.6$
---|---|---|---|---
$V=12^{3}\times 24$ | $V=16^{3}\times 32$ | $V=16^{3}\times 32$ | $V=16^{3}\times 32$ | $V=24^{3}\times 48$
$\kappa$ | $t_{min}$ | $\kappa$ | $t_{min}$ | $\kappa$ | $t_{min}$ | $\kappa$ | $t_{min}$ | $\kappa$ | $t_{min}$
0.1695 | 4 | 0.1692 | 4 | 0.1685 | 5 | 0.1667 | 5 | 0.1645 | 7
0.1700 | 4 | 0.1695 | 4 | 0.1687 | 5 | 0.1673 | 5 | 0.1650 | 7
0.1703 | 4 | 0.1697 | 4 | 0.1690 | 5 | 0.1678 | 5 | 0.1655 | 6
0.1705 | 4 | 0.1700 | 4 | 0.1692 | 5 | 0.1680 | 5 | 0.1660 | 7
- | - | 0.1703 | 4 | 0.1693 | 4 | 0.1683 | 5 | - | -
- | - | 0.1705 | 4 | - | - | - | - | - | -
Table 1: The values of $t_{min}$ for all available gauge ensembles, chosen
such that a plateau is formed.
In the second approach, we consider that our simulations of the theory are
done at different values of the lattice spacing $a$, which leads to different
$O(a)$ terms in the Ward identities. A fixed value of $t_{min}$ in lattice
units would mean a lower limit on the time-slice distances in physical units,
that is on the cutoff-scale and shrinks to zero in the continuum limit.
Instead it would be more appropriate to consider $t_{min}$ at constant
physical distance for all gauge ensembles. This is done in the following way.
At the coarsest lattice spacing, at inverse gauge coupling $\beta_{0}$, the
value of $t_{min}$ is selected according to the plateau criterion explained
above. For finer lattice spacings at inverse gauge couplings $\beta_{i}$ the
corresponding $t_{min}$ are then obtained by scaling with a physical scale. In
order to determine the physical scale we use the mass $m_{g\tilde{g}}$ of the
gluino-glue particle and the Wilson flow parameter $w_{0}$. Correspondingly,
$t_{min}$ is scaled according to
$\displaystyle t_{min,{\beta_{i}}}$
$\displaystyle=t_{min,\beta_{0}}\frac{m_{g\tilde{g},\beta_{0}}}{m_{g\tilde{g},\beta_{i}}}\,,$
(16) $\displaystyle\text{or}\quad t_{min,{\beta_{i}}}$
$\displaystyle=t_{min,\beta_{0}}\frac{w_{0,\beta_{i}}}{w_{0,\beta_{0}}}\,,$
(17)
where $\beta_{0}=5.4$, $\beta_{1}=5.45$, $\beta_{2}=5.5$, and $\beta_{3}=5.6$.
The resulting $t_{min}$ is rounded to the nearest integer value. The values
obtained by this method are collected in Tab. 2. In most points they are equal
or almost equal to those in Tab. 1.
$\beta$ | $t_{min}$ from $m_{g\tilde{g}}$ | $t_{min}$ from $w_{0}$
---|---|---
5.4 | 4 | 4
5.45 | 5 | 5
5.5 | 5 | 6
5.6 | 7 | 7
Table 2: The values of $t_{min}$ at fixed physical temporal distance from
scaling with the gluino-glue mass $m_{g\tilde{g}}$ and with the Wilson flow
parameter $w_{0}$.
### 3.2 Adjoint pion and remnant gluino mass
The chiral limit is defined by the vanishing of the subtracted gluino mass.
Its measured values can therefore be employed for the tuning of the hopping
parameter $\kappa$ to the chiral limit. On the other hand, we can also use the
vanishing of the adjoint pion mass $m_{\text{a-}\pi}$ for the tuning [16]. The
adjoint pion $\text{a-}\pi$ is an unphysical particle in the SYM theory, that
can be defined in partially quenched chiral perturbation theory [17]. In the
numerical simulations its correlation function can be computed as the
connected piece of the correlation function of the $\text{a-}\eta^{\prime}$
particle. Similar to the Gell-Mann-Oakes-Renner relation of QCD [5], in the
continuum limit there is a linear relation between the adjoint pion mass
squared and the gluino mass: $m^{2}_{\text{a-}\pi}\propto m_{\tilde{g}}$.
The numerical results for the subtracted gluino mass from the Ward identities
and the adjoint pion mass squared in lattice units are shown for $\beta=5.6$
in Fig. 2 together with their extrapolations towards the chiral limit.
(a) The subtracted gluino mass $am_{S}Z^{-1}_{S}$ and the adjoint pion mass
squared $(am_{\text{a-}\pi})^{2}$ as a function of $1/(2\kappa)$, and the
corresponding extrapolations towards the chiral limit ($\kappa_{c}$).
(b) The subtracted gluino mass $am_{S}Z^{-1}_{S}$ as a function of the adjoint
pion mass squared $(am_{\text{a-}\pi})^{2}$ in order to obtain the remnant
gluino mass $\Delta(am_{S}Z^{-1}_{S})$.
Figure 2: Chiral limit and determination of the remnant gluino mass at
$\beta=5.6$. All quantities are in lattice units.
In the continuum the subtracted gluino mass and the adjoint pion mass should
vanish at the same point. On the lattice, however, this is not the case due to
lattice artefacts. As an estimate for this discrepancy we determine the value
of the subtracted gluino mass at vanishing adjoint pion mass. This quantity is
called the remnant gluino mass $\Delta(am_{S}Z^{-1}_{S})$, and it is expected
to vanish in the continuum limit. The values of the remnant gluino mass,
obtained by taking an average of the values calculated using the procedures
explained above, are presented in Tab. 3.
$\beta$ | 5.4 | 5.45 | 5.5 | 5.6
---|---|---|---|---
$\Delta(am_{S}Z^{-1}_{S})$ | 0.0334(48) | 0.019(12) | 0.0099(88) | 0.0103(33)
Table 3: The values of the remnant gluino mass $\Delta(am_{S}Z^{-1}_{S})$
obtained at four different values of the inverse gauge coupling.
### 3.3 Continuum limit
The remnant gluino mass is a lattice artefact and should vanish in the
continuum limit $a\rightarrow 0$. It is therefore a quantity to check on
whether supersymmetry is recovered or not. Concerning the dependence of the
remnant gluino mass on the lattice spacing, arguments based on partially
quenched chiral perturbation theory suggest that the remnant gluino mass is of
order $a^{2}$ at $m^{2}_{\text{a-}\pi}=0$ [13]. In order to investigate this
relation, the remnant gluino mass has to be expressed in physical units. Our
choice for the scale is the Wilson flow parameter $w_{0}$, which is defined
through the gradient flow [10]. We use its values extrapolated to the chiral
limit, $w_{0,\chi}$. Similarly the lattice spacing is represented by
$a/w_{0,\chi}$. Our numerical results for the remnant gluino mass as a
function of the lattice spacing and its extrapolation towards the continuum
limit are shown in Fig. 3. The data points in Fig. 3(a) show the results from
separate chiral extrapolations for each lattice spacing and the corresponding
extrapolation to the continuum limit. The extrapolation to the continuum and
the error of this extrapolation are obtained by means of parametric bootstrap
with linear fits. On the other hand, Fig. 3(b) is obtained by means of a
simultaneous fit of the dependence on the hopping parameter and the lattice
spacing [18].
(a) The remnant gluino mass from separate extrapolations to the chiral limit
where $m^{2}_{\text{a-}\pi}$ is zero, and the extrapolation to the continuum
limit.
(b) The remnant gluino mass from a simultaneous chiral and continuum
extrapolation. By construction, in this method the data points coincide with
the error band.
Figure 3: The remnant gluino mass $\Delta{(w_{0}m_{S}Z^{-1}_{S})}$ in physical
units $w_{0}$ as a function of the lattice spacing squared, and its linear
extrapolation towards the continuum limit.
The remnant gluino mass in the continuum limit is compatible with zero within
one standard-deviation, confirming the preliminary results present in Ref.
[15] with only two data points. Lattice artefacts vanish in the continuum
limit as expected, and supersymmetry is recovered in the chiral and continuum
limits, in agreement with our findings from the mass spectrum [12].
## 4 Conclusion
In this paper we have presented numerical results of an analysis of SUSY Ward
identities in $\mathcal{N}=1$ supersymmetric Yang-Mills theory on the lattice
with gauge group SU(3). Contact terms and $O(a)$ lattice artefacts in the Ward
identities have been controlled by suitable choices of time-slice distances.
Ensembles of gauge configurations at four different values of the lattice
spacing and various hopping parameters have been analysed, allowing us for the
first time to perform an extrapolation to the continuum limit, where the
lattice artefacts vanish. The remnant gluino mass has been extrapolated in two
alternative ways, on the one hand by extrapolating to the chiral limit at each
lattice spacing separately and then to the continuum limit, and on the other
hand by means of a simultaneous extrapolation to the chiral and continuum
limit. With both extrapolations the lattice artefacts in the subtracted gluino
mass appear to scale to zero as of order $a^{2}$ in agreement with the
theoretical expectations. Our findings support the validity of SUSY Ward
identities and the restoration of supersymmetry in the continuum limit.
## Acknowledgments
The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V.
(www.gauss-centre.eu) for funding this project by providing computing time on
the GCS Supercomputer JUQUEEN and JURECA at Jülich Supercomputing Centre (JSC)
and SuperMUC at Leibniz Supercomputing Centre (LRZ). Further computing time
has been provided on the compute cluster PALMA of the University of Münster.
This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through
the Research Training Group “GRK 2149: Strong and Weak Interactions - from
Hadrons to Dark Matter”. G. Bergner acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG) Grant No. BE 5942/2-1. S. Ali acknowledges
financial support from the Deutsche Akademische Austauschdienst (DAAD).
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|
2024-09-04T02:54:58.388585 | 2020-03-09T13:16:15 | 2003.04121 | {
"authors": "Sean Prendiville",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26117",
"submitter": "Sean Prendiville",
"url": "https://arxiv.org/abs/2003.04121"
} | arxiv-papers | # The inverse theorem for the nonlinear Roth configuration: an exposition
Sean Prendiville Department of Mathematics and Statistics
Lancaster University
UK<EMAIL_ADDRESS>
###### Abstract.
We give an exposition of the inverse theorem for the cut-norm associated to
the nonlinear Roth configuration, established by Peluse and the author in [6].
###### Contents
1. 1 Introduction
2. 2 An outline of our argument
3. 3 PET induction
4. 4 An inverse theorem for the arithmetic box norm
5. 5 Quantitative concatenation
6. 6 Degree lowering
7. 7 Proof of the cut norm inverse theorem
8. A Basic theory of the Gowers norms
## 1\. Introduction
Peluse and the author recently obtained an effective bound on the density of
sets of integers lacking the configuration
$x,\ x+y,\ x+y^{2}\qquad(y\neq 0).$ (1.1)
We call this pattern the _nonlinear Roth configuration_ , after Bourgain and
Chang [1].
###### Theorem 1.1 (Peluse and Prendiville [6]).
There exists an absolute constant $c>0$ such that if
$A\subset\left\\{1,2,\dots,N\right\\}$ lacks the configuration (1.1), then
$|A|\ll N(\log\log N)^{-c}.$
We have since removed a logarithm from this bound.
###### Theorem 1.2 (Peluse and Prendiville [7]).
There exists an absolute constant $c>0$ such that if
$A\subset\left\\{1,2,\dots,N\right\\}$ lacks the configuration (1.1), then
$|A|\ll N(\log N)^{-c}.$
The main innovation behind both of these results is [6, Theorem 7.1], an
inverse theorem for the counting operator associated to this configuration. It
is the purpose of this note to give an exposition of this inverse theorem. The
approach is essentially the same as that in [6]. We hope that having two
distinct accounts is useful for those interested in utilising these ideas.
###### Definition 1.3 (Counting operator).
For positive integers $q\leq N$ write
$M:=\left\lfloor\sqrt{N/q}\right\rfloor.$ (1.2)
Given this, define the _counting operator_ on the functions
$f_{i}:\mathbb{Z}\to\mathbb{C}$ by
$\Lambda_{q,N}(f_{0},f_{1},f_{2}):=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+y)f_{2}(x+qy^{2}).$
(1.3)
When the $f_{i}$ all equal $f$ we simply write $\Lambda_{q,N}(f)$.
###### Definition 1.4 (Local function).
We call a function $\phi:\mathbb{Z}\to\mathbb{C}$ a _local function of
resolution $M$ and modulus $q$_ if there exists a partition of $\mathbb{R}$
into intervals of length $M$ such that $\phi$ is constant on the intersection
of every such interval with every congruence class mod $q$.
###### Definition 1.5 (Cut norm).
Define the _cut norm_ of $f:\mathbb{Z}\to\mathbb{C}$ by
$\left\|f\right\|_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,\
|\Lambda_{q,N}(g_{1},f,g_{2})|,\ |\Lambda_{q,N}(g_{1},g_{2},f)|\\},$ (1.4)
where the supremum is taken over all 1-bounded functions
$g_{i}:[N]\to\mathbb{C}$. We note that, in spite of our nomenclature, this is
not a norm but a seminorm. One could remedy this by summing over $y\geq 0$ in
the counting operator (1.3)
This seminorm is useful in [7]. However, it is too restrictive for the
approach developed in [6], where we (implicitly) only work with the following
quantities:
$\left\|f\right\|^{\sharp}_{q}:=\sup\\{|\Lambda_{q,N}(g_{0},g_{1},f)|:|g_{i}|\leq
1\ \text{ and }\ \mathrm{supp}(g_{i})\subset[N]\\}$ (1.5)
and
$\left\|f\right\|^{\flat}_{q}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,\
|\Lambda_{q,N}(g_{1},f,g_{2})|\ :\ |g_{i}|\leq 1\ \text{ and }\
\mathrm{supp}(g_{i})\subset[N]\\}.$ (1.6)
Here then is a re-formulation and slight generalisation of [6, Theorem 7.1].
###### Theorem 1.6 (Partial cut norm inverse theorem).
Let $q\leq N$ be positive integers, $\delta>0$, and
$f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$.
Suppose that
$\left\|f\right\|^{\flat}_{q,N}\geq\delta.$
Then either $N\ll(q/\delta)^{O(1)}$ or there exists a 1-bounded local function
$\phi$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, modulus $qq^{\prime}$ for
some $q^{\prime}\ll\delta^{-O(1)}$, and such that
$\sum_{x\in[N]}f(x)\phi(x)\gg\delta^{2^{66}}N.$
This exposition is organised as follows. In §2, we give a more detailed
outline of the proof of Theorem 1.6. In §§3–5 we develop an effective approach
to a (special case of a) so-called _concatenation_ theorem of Tao and Ziegler
[10]. This allows us to show that if our counting operator is large, then the
function weighting the nonlinear term must have large Gowers uniformity norm.
The drawback is that the degree of the resulting Gowers norm is large (in our
approach it is the $U^{5}$-norm). In §6 we give a _degree-lowering_ procedure,
which utilises properties specific to our configuration to show that one may
replace the $U^{5}$-norm with the $U^{1}$-norm. In §7 we combine the results
of the previous sections in order to prove Theorem 1.6.
### 1.1. Notation
#### 1.1.1. Standard conventions
We use $\mathbb{N}$ to denote the positive integers. For a real $X\geq 1$,
write $[X]=\\{1,2,\ldots,\left\lfloor X\right\rfloor\\}$. A complex-valued
function is _1-bounded_ if the modulus of the function does not exceed 1.
We use counting measure on $\mathbb{Z}$, so that for
$f,g:\mathbb{Z}\to\mathbb{C}$ we have
$\left\langle
f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)}\qquad\text{and}\qquad\left\|f\right\|_{L^{p}}:=\biggl{(}\sum_{x}|f(x)|^{p}\biggr{)}^{\frac{1}{p}}.$
Any sum of the form $\sum_{x}$ is to be interpreted as a sum over
$\mathbb{Z}$. We use Haar probability measure on
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, so that for measurable
$F:\mathbb{T}\to\mathbb{C}$ we have
$\left\|F\right\|_{L^{p}}:=\biggl{(}\int_{\mathbb{T}}|F(\alpha)|^{p}\mathrm{d}\alpha\biggr{)}^{\frac{1}{p}}=\biggl{(}\int_{0}^{1}|F(\alpha)|^{p}\mathrm{d}\alpha\biggr{)}^{\frac{1}{p}}$
For $\alpha\in\mathbb{T}$ we write $\left\|\alpha\right\|$ for the distance to
the nearest integer.
For a finite set $S$ and function $f:S\to\mathbb{C}$, denote the average of
$f$ over $S$ by
$\mathbb{E}_{s\in S}f(s):=\frac{1}{|S|}\sum_{s\in S}f(s).$
Given functions $f,g:G\to\mathbb{C}$ on an additive group with measure
$\mu_{G}$ we define their convolution by
$f*g(x):=\int_{G}f(x-y)g(y)\mathrm{d}\mu_{G},$ (1.7)
when this makes sense.
We define the Fourier transform of $f:\mathbb{Z}\to\mathbb{C}$ by
$\hat{f}(\alpha):=\sum_{x}f(x)e(\alpha x)\qquad(\alpha\in\mathbb{T}),$ (1.8)
again, when this makes sense. Here $e(\alpha)$ stands for $e^{2\pi i\alpha}$.
The difference function of $f:\mathbb{Z}\to\mathbb{C}$ is the function
$\Delta_{h}f:\mathbb{Z}\to\mathbb{C}$ given by
$\Delta_{h}f(x)=f(x)\overline{f(x+h)}.$
Iterating gives
$\Delta_{h_{1},\dots,h_{s}}f:=\Delta_{h_{1}}\dots\Delta_{h_{s}}f.$
This allows us to define the Gowers $U^{s}$-norm
$\left\|f\right\|_{U^{s}}:=\left(\sum_{x,h_{1},\dots,h_{s}}\Delta_{h_{1},\dots,h_{s}}f(x)\right)^{1/2^{s}}.$
(1.9)
If $\|\cdot\|$ is a seminorm on an inner product space, recall that its dual
seminorm $\|\cdot\|^{*}$ is defined by
$\|f\|^{*}:=\sup_{\|g\|\leq 1}|\langle f,g\rangle|.$
Hence
$\left|\left\langle
f,g\right\rangle\right|\leq\left\|f\right\|^{*}\left\|g\right\|.$ (1.10)
For a function $f$ and positive-valued function $g$, write $f\ll g$ or
$f=O(g)$ if there exists a constant $C$ such that $|f(x)|\leq Cg(x)$ for all
$x$. We write $f=\Omega(g)$ if $f\gg g$. We sometimes opt for a more explicit
approach, using $C$ to denote a large absolute constant, and $c$ to denote a
small positive absolute constant. The values of $C$ and $c$ may change from
line to line.
#### 1.1.2. Local conventions
Up to normalisation, all of the above are well-used in the literature. Next we
list notation specific to our paper. We have tried to minimise this in order
to aid the casual reader.
For a real parameter $H\geq 1$, we use $\mu_{H}:\mathbb{Z}\to[0,1]$ to
represent the following normalised Fejér kernel
$\mu_{H}(h):=\frac{1}{\left\lfloor
H\right\rfloor}\left(1-\frac{|h|}{\left\lfloor
H\right\rfloor}\right)_{+}=\frac{(1_{[H]}*1_{[H]})(h)}{\left\lfloor
H\right\rfloor^{2}}.$ (1.11)
For a multidimensional vector $h\in\mathbb{Z}^{d}$ we write
$\mu_{H}(h):=\mu_{H}(h_{1})\dotsm\mu_{H}(h_{d}).$ (1.12)
We observe that this is a probability measure on $\mathbb{Z}^{d}$ with support
in the interval $(-H,H)^{d}$.
## 2\. An outline of our argument
In this section we describe the ideas behind Theorem 1.6. In the hope of
making the ideas clearer, we make the simplification that $q=1$ in our
counting operator (1.3). Hence, for finitely supported functions
$f_{0},f_{1},f_{2}:\mathbb{Z}\to\mathbb{C}$, write
$\Lambda(f_{0},f_{1},f_{2}):=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/2}]}f_{0}(x)f_{1}(x+y)f_{2}(x+y^{2}).$
(2.1)
For this operator, Theorem 1.6 can be deduced from the following.
###### Lemma 2.1.
Let $f_{0},f_{1},f_{2}:\mathbb{Z}\to\mathbb{C}$ be $1$-bounded functions
supported in the interval $[N]$ and $\delta>0$. Suppose that
$|\Lambda(f_{0},f_{1},f_{2})|\geq\delta.$
Then either $N\ll\delta^{-O(1)}$ or there exist positive integers
$q\ll\delta^{-O(1)}$ and $N^{\prime}\gg\delta^{O(1)}N^{1/2}$ such that
$\sum_{x}\left|\sum_{y\in[N^{\prime}]}f_{1}(x+qy)\right|\gg\delta^{O(1)}NN^{\prime}.$
(2.2)
Using the notation (1.9), notice that the left-hand side of (2.2) is equal to
$\sum_{x}\left\|f_{1}\right\|_{U^{1}(x+q\cdot[N^{\prime}])}.$
### 2.1. Quantitative concatenation
To prove Lemma 2.1, we first prove that our counting operator (2.1) is
controlled by the $U^{5}$-norm of $f_{2}$. The purpose of this subsection is
to sketch how we do this with polynomial bounds.
By applying the Cauchy–Schwarz and van der Corput inequalities a number of
times, we show in §3 that, when $f_{0},f_{1},f_{2}:\mathbb{Z}\to\mathbb{C}$
are $1$-bounded functions supported in the interval $[N]$, largeness of the
counting operator (2.1) implies largeness of the sum
$\sum_{a,b\in[N^{1/2}]}\sum_{h_{1},h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{ah_{1},bh_{2},(a+b)h_{3}}f_{2}(x).$
(2.3)
This deduction is made following the PET induction scheme of Bergelson and
Leibman [2]. The gain in working with the counting operator (2.3) over (2.1)
is that univariate polynomials such as $y^{2}$, whose image constitute a
sparse set, have been replaced by bilinear forms such as $ah_{1}$, whose image
is much denser
In §§4–5, we show that largeness of (2.3) implies largeness of
$\|f_{2}\|_{U^{5}}$. If there were no dependence between the coefficients of
the $h_{i}$ in (2.3), then we could in fact bound (2.3) in terms of
$\|f_{2}\|_{U^{3}}$. Since the argument is informative, we illustrate why this
is the case for the sum
$\sum_{a,b,c\in[N^{1/2}]}\sum_{h_{1},h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{ah_{1},bh_{2},ch_{3}}f_{2}(x).$
(2.4)
The following fact is key, the formal version of which is Lemma 5.3.
###### Claim 2.2.
If $\displaystyle\sum_{a,h\in[N^{1/2}]}\sum_{x}\Delta_{ah}f(x)$ is large then
so is $\displaystyle\sum_{k\in(-N,N)}\sum_{x}\Delta_{k}f(x)$.
###### Sketch proof.
Apply the Cauchy–Schwarz inequality to double the $a$ and $h$ variables,
yielding a bound in terms of
$\sum_{a,a^{\prime}\in[N^{1/2}]}\sum_{h,h^{\prime}\in[N^{1/2}]}\sum_{x}\Delta_{ah-a^{\prime}h^{\prime}}f(x).$
(2.5)
For a random choice of $a,a^{\prime}\in[N^{1/2}]$, the progression
$a\cdot[N^{1/2}]-a^{\prime}\cdot[N^{1/2}]$ covers a large portion of the
interval $(-N,N)$ relatively smoothly. One can make this intuition rigorous
and thus deduce largeness of the sum
$\sum_{k\in(-N,N)}\sum_{x}\Delta_{k}f(x).$∎
Applying Claim 2.2 three times allows us to replace each of $ah_{1}$, $bh_{2}$
and $ch_{3}$ in (2.4) with $k_{1},k_{2},k_{3}\in(-N,N)$, yielding largeness of
$\left\|f_{2}\right\|_{U^{3}}$.
Since the PET induction scheme outputs (2.3), and not (2.4), the problem
remains of how to handle the dependency between the differencing parameters in
(2.3). If we were not concerned with quantitative bounds, we could apply a
‘concatenation’ theorem of Tao and Ziegler [10, Theorem 1.24] to obtain
largeness of the $U^{9}$-norm of $f_{2}$. However, the qualitative nature of
this argument means that it cannot be used to obtain bounds in the nonlinear
Roth theorem. In its place we prove Theorem 5.6, which is a special case of
[10, Theorem 1.24], using a very different argument that gives polynomial
bounds. We spend the remainder of this subsection sketching the argument.
We begin by viewing (2.3) as the average
$\sum_{a,h_{1}\in[N^{1/2}]}\left\|\Delta_{ah_{1}}f_{2}\right\|_{a},$ (2.6)
where
$\|f\|_{a}^{4}:=\sum_{b\in[N^{1/2}]}\sum_{h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{bh_{2},(a+b)h_{3}}f(x)$
(2.7)
One can view this as an average of 2-dimensional Gowers box norms where, for
fixed $b$, the inner sum corresponds to a box norm in the ‘directions’ $b$ and
$a+b$. Note that if we could bound the quantity $\|\Delta_{ah_{1}}f_{2}\|_{a}$
in terms of the $U^{4}$-norm of $\Delta_{ah_{1}}f_{2}$ for many pairs
$(a,h_{1})$, then by Claim 2.2 we deduce largeness of the $U^{5}$-norm of
$f_{2}$. We show that, on average, one can indeed control $\|\cdot\|_{a}$ in
terms of $\|\cdot\|_{U^{4}}$, with polynomial bounds. The following can be
extracted from the proof of (the more general) Theorem 5.6.
###### Lemma 2.3.
For each $a\in[N^{1/2}]$ let $f_{a}:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded
function supported in the interval $[N]$. Suppose that
$\mathbb{E}_{a\in[N^{1/2}]}\|f_{a}\|_{a}^{4}\geq\delta\left\|1_{[N]}\right\|_{a}^{4}.$
Then
$\mathbb{E}_{a\in[N^{1/2}]}\|f_{a}\|_{U^{4}}^{16}\gg\delta^{O(1)}\left\|1_{[N]}\right\|_{U^{4}}^{16}.$
To finish this subsection, we briefly discuss the proof of this key lemma. For
most choices of $a,b\in[N^{1/2}]$, the ‘directions’ $a$ and $a+b$ of the box
norm
$\sum_{h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{bh_{2},(a+b)h_{3}}f_{a}(x)$
(2.8)
are close to ‘independent’, in the sense that at least one of the directions
$a$ and $a+b$ is large and together they have small greatest common divisor.
The proof of Lemma 2.3 thus begins by viewing $\|\cdot\|_{a}$ as an average of
box norms
$\|f\|_{\square(X,Y)}^{4}:=\sum_{x_{1},x_{2}\in X,y_{1},y_{2}\in
Y}f(x_{1},y_{1})\overline{f(x_{1},y_{2})f(x_{2},y_{1})}f(x_{2},y_{2}).$ (2.9)
It is easy to show that largeness of $\|f\|_{\square(X,Y)}$ implies that $f$
correlates with a function of the form $(x,y)\mapsto l(x)r(y)$. We show,
analogously, that provided $b$ and $a+b$ are not too small and have greatest
common divisor not too large, then largeness of the arithmetic box norm (2.8)
implies that $f_{a}$ correlates with a product $g_{b}h_{a+b}$ of 1-bounded
functions, where $g_{b}$ is $b$-periodic and $h_{a+b}$ is almost periodic
under shifts by integer multiples of $a+b$. As a consequence, for most
$a\in[N^{1/2}]$, largeness of $\|f_{a}\|_{a}$ implies largeness of
$\sum_{b\in[N^{1/2}]}\sum_{x}f_{a}(x)g_{b}(x)h_{a+b}(x).$ (2.10)
In fact, an application of Cauchy–Schwarz allows us give an explicit
description of $h_{a+b}$ in terms of $f_{a}$, namely we may take it to be of
the form
$h_{a+b}(x)=\mathbb{E}_{k\in[N^{1/2}]}f_{a}(x+(a+b)k)g_{b}(x+(a+b)k).$ (2.11)
This presentation makes apparent the almost periodicity of $h_{a+b}$.
###### Claim 2.4.
Largeness of (2.10) implies that $\mathbb{E}_{b\in[N^{1/2}]}h_{a+b}$ has large
$U^{3}$-norm.
Let us first show why Claim 2.4 in turn implies that $f_{a}$ has large
$U^{4}$-norm, completing our sketch proof of Lemma 2.3. The expression (2.11)
and the triangle inequality for Gowers norms together imply that largeness of
$\mathbb{E}_{b\in[N^{1/2}]}\left\|h_{a+b}\right\|_{U^{3}}$ implies largeness
of $\mathbb{E}_{b\in[N^{1/2}]}\left\|f_{a}g_{b}\right\|_{U^{3}}$. Utilising
the $b$-periodicity of $g_{b}$ we have
$\left\|f_{a}g_{b}\right\|_{U^{3}}=\mathbb{E}_{k\in[N^{1/2}]}\left\|f_{a}(\cdot)g_{b}(\cdot+bk)\right\|_{U^{3}}.$
(2.12)
The product $f_{a}(\cdot)g_{b}(\cdot+bk)$ resembles a difference function in
the direction $b$. Indeed the Gowers–Cauchy–Schwarz inequality (see [9,
Exercise 1.3.19]) shows that if (2.12) is large (on average over
$b\in[N^{1/2}]$) then so is
$\mathbb{E}_{b,k\in[N^{1/2}]}\left\|\Delta_{bk}f_{a}\right\|_{U^{3}}$
Largeness of $\left\|f_{a}\right\|_{U^{4}}$ then follows from Claim 2.2.
Finally we sketch the proof of Claim 2.4. The Cauchy–Schwarz inequality allows
us to remove the weight $f_{a}(x)$ from (2.10) and deduce largeness of
$\sum_{x}\sum_{b,b^{\prime}\in[N^{1/2}]}\overline{g_{b}(x)h_{a+b}(x)}g_{b^{\prime}}(x)h_{a+b^{\prime}}(x).$
Using the periodicity properties of $g_{b}$, $g_{b^{\prime}}$ and $h_{a+b}$,
this is approximately equal to
$\sum_{x}\sum_{\begin{subarray}{c}b,b^{\prime}\in[N^{1/2}]\\\
k_{1},k_{2},k_{3}\in[N^{1/2}]\end{subarray}}\overline{g_{b}(x-bk_{1})h_{a+b}(x-(a+b)k_{2})}g_{b^{\prime}}(x-b^{\prime}k_{3})h_{a+b^{\prime}}(x).$
Changing variables in $x$, we obtain largeness of the sum
$\sum_{x}\sum_{\begin{subarray}{c}b,b^{\prime}\in[N^{1/2}]\\\
k_{1},k_{2},k_{3}\in[N^{1/2}]\end{subarray}}\overline{g_{b}(x+(a+b)k_{2}+b^{\prime}k_{3})h_{a+b}(x+bk_{1}+b^{\prime}k_{3})}\\\
g_{b^{\prime}}(x+bk_{1}+(a+b)k_{2})h_{a+b^{\prime}}(x+bk_{1}+(a+b)k_{2}+b^{\prime}k_{3}).$
The point here is that all but the last function have arguments depending on
at most two of the bilinear forms $bk_{1}$, $(a+b)k_{2}$ and
$b^{\prime}k_{1}^{\prime}$. This enables us to employ the
Gowers–Cauchy–Schwarz inequality (in the form of Lemma A.4) to deduce
largeness of a sum similar to
$\sum_{x}\sum_{\begin{subarray}{c}b,b^{\prime}\in[N^{1/2}]\\\
k_{1},k_{2},k_{3}\in[N^{1/2}]\end{subarray}}\Delta_{bk_{1},\,(a+b)k_{2},\,b^{\prime}k_{3}}h_{a+b^{\prime}}(x).$
The utility of this expression is that the directions of the differencing
parameters are all ‘independent’ of the direction of periodicity of
$h_{a+b^{\prime}}$. Indeed the approximate $(a+b^{\prime})$-periodicity of
$h_{a+b^{\prime}}$ means that one can replace $\Delta_{y}h_{a+b^{\prime}}$
with $\mathbb{E}_{k}\Delta_{y+(a+b^{\prime})k}h_{a+b^{\prime}}$ at the cost of
a small error. We thereby obtain largeness of
$\sum_{x}\sum_{b,b^{\prime}\in[N^{1/2}]}\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3}\in[N^{1/2}]\\\
k_{1}^{\prime},k_{2}^{\prime},k_{3}^{\prime}\in[N^{1/2}]\end{subarray}}\Delta_{bk_{1}+(a+b^{\prime})k_{1}^{\prime},\,(a+b)k_{2}+(a+b^{\prime})k_{2}^{\prime},\,b^{\prime}k_{3}+(a+b^{\prime})k_{3}^{\prime}}h_{a+b^{\prime}}(x).$
(2.13)
For a random triple $(a,b,b^{\prime})\in[N^{1/2}]$ the greatest common divisor
of the pairs $(b,a+b^{\prime})$, $(a+b,a+b^{\prime})$ and
$(b^{\prime},a+b^{\prime})$ are all small, and these are the pairs appearing
in the differencing parameters of (2.13). The argument used to treat (2.5) may
be therefore be employed to replace (2.13) with
$\sum_{x}\sum_{b^{\prime}\in[N^{1/2}]}\sum_{k_{1},k_{2},k_{3}\in[N]}\Delta_{k_{1},k_{2},k_{3}}h_{a+b^{\prime}}(x),$
and thereby yield Claim 2.4.
### 2.2. Degree lowering
After we have shown that $\Lambda(f_{0},f_{1},f_{2})$ is controlled by the
$U^{5}$-norm of $f_{2}$, we carry out a ‘degree lowering’ argument. This
technique originated in the work [5] in finite fields. The basic idea is that,
under certain conditions, one can combine $U^{s}$-control with understanding
of two-term progressions to deduce $U^{s-1}$-control. Repeating this gives a
sequence of implications
$U^{5}\text{-control}\implies U^{4}\text{-control}\implies
U^{3}\text{-control}\implies U^{2}\text{-control}\implies
U^{1}\text{-control}.$
Despite the appearance of the $U^{5}$-norm, $U^{4}$-norm, and $U^{3}$-norm,
the degree lowering argument, both in [5] and here, does not require the
$U^{s}$-inverse theorem for any $s\geq 3$. Instead it relies on Fourier
analysis in the place of these inverse theorems.
Adapting the degree lowering argument of [5] to the integer setting requires
several significant modifications. The first modification is that the
$U^{s}$-control described above is control in terms of the $U^{s}$-norm of the
dual function
$F(x):=\mathbb{E}_{y\in[N^{1/2}]}f_{0}(x-y^{2})f_{1}(x+y-y^{2}).$ (2.14)
Thus, to begin the degree lowering argument, we must show that largeness of
$\Lambda(f_{0},f_{1},f_{2})$ implies largeness of $\|F\|_{U^{5}}$. To do this,
we use a simple Hahn–Banach decomposition as described in [3, Proposition
3.6], for details see §7.
We conclude this section by sketching an instance of degree-lowering: how
$U^{3}$-control of the dual (2.14) implies $U^{2}$-control, starting from the
assumption that
$\|F\|_{U^{3}}^{8}\geq\delta\left\|1_{[N]}\right\|_{U^{3}}^{8}.$
Using the fact that $\|F\|_{U^{3}}^{8}=\sum_{h}\|\Delta_{h}F\|_{U^{2}}^{4}$
and applying the $U^{2}$-inverse theorem, we deduce the existence of a
function $\phi:\mathbb{Z}\to\mathbb{T}$ such that, for at least $\gg\delta N$
choices of differencing parameter $h$, we have
$\left|\sum_{x\in[N]}\Delta_{h}F(x)e(\phi(h)x)\right|\gg\delta N.$ (2.15)
Note that if, in the above inequality, we could replace the function $\phi(h)$
by a constant $\beta\in\mathbb{T}$ not depending on $h$, then we could easily
deduce largeness of $\|F\|_{U^{2}}$. Indeed, writing $g(h)$ for the phase of
the sum inside absolute values, this would give
$\sum_{x,h}\overline{g(h)}\overline{F(x+h)}F(x)e(\beta
x)\gg\delta^{O(1)}N^{3},$
and the usual argument111One can either use orthogonality and extraction of a
large Fourier coefficient, as in the proof of Lemma A.1, or use two
applications of Cauchy–Schwarz. showing $U^{2}$-control of the equation
$x+y=z$ implies that
$\|F\|_{U^{2}}^{4}\gg\delta^{O(1)}\left\|1_{[N]}\right\|_{U^{2}}$. It thus
remains to show that such a $\beta$ exists.
Expanding the definition of the difference and dual functions in (2.15), and
using the Cauchy–Schwarz inequality (as is done in greater generality in the
proof of Lemma 6.3), one can show that there exists $h^{\prime}$ such that for
many $h$ satisfying (2.15) we have
$\left|\sum_{x}\sum_{y\in[N^{1/2}]}\Delta_{h-h^{\prime}}f_{0}(x)\Delta_{h-h^{\prime}}f_{1}(x+y)e([\phi(h)-\phi(h^{\prime})][x+y^{2}])\right|\gg\delta^{O(1)}N^{3/2}$
Further application of Cauchy–Schwarz allows us to remove the difference
functions from the above inequality and deduce largeness of the exponential
sum
$\sum_{z\in[N^{1/2}]}\left|\sum_{y\in[N^{1/2}]}e(2\left[\phi(h)-\phi(h^{\prime})\right]yz)\right|.$
Summing the inner geometric progression and using a Vinogradov-type lemma then
shows that $\phi(h)-\phi(h^{\prime})$ is major arc. There are very few major
arcs, so the pigeonhole principle gives the existence of
$\beta_{0}\in\mathbb{T}$ such that $\phi(h)-\phi(h^{\prime})$ is very close to
$\beta_{0}$ for many $h\in(-N,N)$ that also satisfy (2.15). We may therefore
take $\beta=\beta_{0}+\phi(h^{\prime})$ in the argument following (2.15).
## 3\. PET induction
We prove Theorem 1.6 over the course of §§3–7. We begin in §§3–5 by showing
how our counting operator $\Lambda_{q,N}(f_{0},f_{1},f_{2})$, as defined in
(1.3), is controlled by the $U^{5}$-norm of $f_{2}$. This argument starts with
the PET induction scheme of Bergelson–Leibman [2], which in some sense
‘linearises’ a polynomial progression, replacing univariate polynomials such
as $y^{2}$ with bilinear forms $ah$. The outcome of this procedure is Lemma
3.3.
For the following, we recall our definition (1.11) of $\mu_{H}$.
###### Lemma 3.1 (van der Corput inequality).
Let $f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded and $M,H\geq 1$. Then we have the
estimate
$\biggl{|}\mathbb{E}_{y\in[M]}f(y)\biggr{|}^{2}\leq\frac{M+H}{M}\sum_{h}\mu_{H}(h)\mathbb{E}_{y\in[M]}\Delta_{h}f(y).$
###### Proof.
This is standard, see for instance [8, Lemma 3.1].∎
###### Lemma 3.2 (Difference functions control linear configurations).
Let $f_{i}:\mathbb{Z}\to\mathbb{C}$ be $1$-bounded functions with support in
an interval $I_{i}$ of size $|I_{i}|=N$. Then for any $a,b\in\mathbb{Z}$ and
$1\leq H\leq M$ we have
$\biggl{|}\mathbb{E}_{x\in
I_{0}}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\biggr{|}^{8}\\\
\ll\sum_{h}\mu_{H}(h)\mathbb{E}_{x\in
I_{3}}\Delta_{ah_{1},bh_{2},(a+b)h_{3}}f_{3}(x).$ (3.1)
###### Proof.
Applying Cauchy-Schwarz in the $x$ variable gives
$\biggl{|}\mathbb{E}_{x\in
I_{0}}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\biggr{|}^{2}\\\
\leq\frac{1}{N}\sum_{x}\bigg{|}\mathbb{E}_{y\in[M]}f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\bigg{|}^{2}.$
Bounding the inner sum using van der Corput’s inequality (Lemma 3.1) and
making the change of variables $x\mapsto x-ay$ (valid since $x$ is ranging
over $\mathbb{Z}$), the latter is at most
$2\sum_{h_{1}}\mu_{H}(h_{1})\mathbb{E}_{x\in
I_{1}}\mathbb{E}_{y\in[M]}\Delta_{ah_{1}}f_{1}(x)\Delta_{bh_{1}}f_{2}(x+(b-a)y)\Delta_{(a+b)h_{1}}f_{3}(x+by).$
Here we may restrict $x$ to $I_{1}$ on observing that the support of
$\Delta_{ah_{1}}f_{1}$ is contained in the support of $f_{1}$. Making use of
the fact that $\mu_{H}$ is a probability measure, we repeat the procedure of
applying Cauchy–Schwarz, van der Corput then a change of variables, to deduce
that
$\biggl{|}\mathbb{E}_{x\in
I_{0}}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\biggr{|}^{4}\\\
\leq 8\sum_{h_{1},h_{2}}\mu_{H}(h_{1})\mu_{H}(h_{2})\mathbb{E}_{x\in
I_{2}}\mathbb{E}_{y\in[M]}\Delta_{bh_{1},(b-a)h_{2}}f_{2}(x)\Delta_{(a+b)h_{1},bh_{2}}f_{3}(x+ay).$
A final iteration of the same procedure then yields (3.1). ∎
Before embarking on the following, we remind the reader of our convention
(1.2) regarding $M$.
###### Lemma 3.3 (Linearisation).
Let $f_{i}:\mathbb{Z}\to\mathbb{C}$ be $1$-bounded functions, each with
support in the interval $[N]$. Then for any $1\leq H\leq M$ we have
$\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|^{32}\ll\sum_{a,b,h}\mu_{M}(a)\mu_{M}(b)\mu_{H}(h)\mathbb{E}_{x\in[N]}\Delta_{2q(a+b)h_{1},\,2qbh_{2},\,2qah_{3}}f_{2}(x).$
(3.2)
###### Proof.
We repeat the procedure given in the proof of Lemma 3.2, applying Cauchy-
Schwarz, followed by van der Corput’s inequality and a change of variables. A
first application gives
$\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|^{2}\leq\\\
2\sum_{a}\mu_{M}(a)\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}\Delta_{a}f_{1}(x)f_{2}\bigl{(}x+qy^{2}-y\bigr{)}\overline{f_{2}\bigl{(}x+q(y+a)^{2}-y\bigr{)}}.$
A second application then gives
$\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|^{4}\ll\sum_{a,b}\mu_{M}(a)\mu_{M}(b)\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}f_{2}(x)\overline{f_{2}\bigl{(}x+2qay+qa^{2}\bigr{)}}\\\
\overline{f_{2}\bigl{(}x+2qby+qb^{2}-b\bigr{)}}f_{2}\bigl{(}x+2q(a+b)y+q(a+b)^{2}-b\bigr{)}.$
Applying Lemma 3.2 to bound the inner sum over $x$ and $y$, we obtain (3.2)
after a final change of variables ∎
## 4\. An inverse theorem for the arithmetic box norm
The objective in this section is to characterise those 1-bounded functions
$f:\mathbb{Z}\to\mathbb{C}$ with support in $[N]$ for which the following
quantity is large
$\sum_{h,x}\mu_{H}(h)\Delta_{ah_{1},bh_{2}}f(x).$ (4.1)
One can think of this as an arithmetic analogue of the two-dimensional ‘box
norm’ (2.9). In our eventual application we are able to ensure that $a$ and
$b$ are a generic pair of integers from the interval $[N^{1/2}]$. In
particular, at least one of them has size proportional to $N^{1/2}$ and their
highest common factor is small. One may think of this as a proxy for linear
independence.
We begin by characterising largeness of (4.1) when the directions are coprime.
###### Lemma 4.1 (Inverse theorem for the arithmetic box norm).
Let $a,b$ be positive integers with $\gcd(a,b)=1$. Suppose that
$f:\mathbb{Z}\to\mathbb{C}$ is $1$-bounded with support in the interval $[N]$
and satisfies
$\sum_{h,x}\mu_{H}(h)\Delta_{ah_{1},bh_{2}}f(x)\geq\delta N.$ (4.2)
Then there exist 1-bounded functions $g,h:\mathbb{Z}\to\mathbb{C}$ such that
* •
$g$ is $a$-periodic, in the sense that $g(x+a)=g(x)$ for all $x$;
* •
$h$ is approximately $b$-periodic, in the sense that for any $\varepsilon>0$
we have
$\\#\left\\{x\in[N]:h(x+by)\neq h(x)\text{ for some }|y|\leq\varepsilon
N/b\right\\}\leq\left(1+\tfrac{2\varepsilon
N}{b}\right)\left(1+\tfrac{N}{a}\right);$
and furthermore
$\biggl{|}\sum_{x}f(x)g(x)h(x)\biggr{|}\geq\delta\left\lfloor
H\right\rfloor^{2}-2\left(\tfrac{H}{a}+\tfrac{Hb}{N}\right)\left\lfloor
H\right\rfloor^{2}.$ (4.3)
###### Remark.
In parsing the above inequalities, it may be helpful to keep in mind that in
our application $a$, $b$ and $H$ are of order $\sqrt{N}$, with $H$
considerably smaller than $a$, in which case the lower bound in (4.3) becomes
$\Omega(\delta H^{2})$.
###### Proof.
The majority of our proof is concerned with manipulating (4.2) until we can
interpret it as a genuine box norm (2.9), and thereby apply the box norm
inverse theorem. The essential observation is that, since $\gcd(a,b)=1$, every
integer $x$ can be uniquely represented in the form
$x=ay+bz\qquad(y\in\mathbb{Z},\ z\in[a]).$
We note that if $x\in[N]$ then the constraint on $z$ forces $y$ to lie in the
range $-b<y<N/a$.
Defining $F:\mathbb{Z}\times\mathbb{Z}\to\mathbb{C}$ by $F(y,z):=f(ay+bz)$,
the left-hand side of (4.2) becomes
$\sum_{y,y^{\prime}\in\mathbb{Z}}\sum_{\begin{subarray}{c}z\in[a]\\\
z^{\prime}\in\mathbb{Z}\end{subarray}}F(y,z)\overline{F(y^{\prime},z)}\overline{F(y,z^{\prime})}F(y^{\prime},z^{\prime})\mu_{H}(y^{\prime}-y)\mu_{H}(z^{\prime}-z).$
If $z^{\prime}$ and $z$ contribute to the above sum then $z^{\prime}\in
z+(-H,H)\subset(-H+1,a+H).$ Hence we can restrict the range of summation of
$z^{\prime}$ to $[a]$, at the cost of perturbing the sum by at most
$2\left\lfloor H\right\rfloor(\frac{N}{a}+b).$ It follows that
$\biggl{|}\sum_{y,y^{\prime}}\sum_{z,z^{\prime}\in[a]}F(y,z)\overline{F(y^{\prime},z)}\overline{F(y,z^{\prime})}F(y^{\prime},z^{\prime})\mu_{H}(y^{\prime}-y)\mu_{H}(z^{\prime}-z)\biggr{|}\\\
\geq\delta N-2\left\lfloor H\right\rfloor\left(\tfrac{N}{a}+b\right).$
We remove the Fejér kernels by Fourier expansion:
$\sum_{\begin{subarray}{c}y,y^{\prime}\\\
z,z^{\prime}\in[a]\end{subarray}}F(y,z)\overline{F(y^{\prime},z)F(y,z^{\prime})}F(y^{\prime},z^{\prime})\mu_{H}(y^{\prime}-y)\mu_{H}(z^{\prime}-z)=\\\
\int_{\mathbb{T}^{2}}\sum_{\begin{subarray}{c}y,y^{\prime}\\\
z,z^{\prime}\in[a]\end{subarray}}F(y,z)\overline{F(y^{\prime},z)F(y,z^{\prime})}F(y^{\prime},z^{\prime})\hat{\mu}_{H}(\alpha)\hat{\mu}_{H}(\beta)e(\alpha(y^{\prime}-y)+\beta(z^{\prime}-z))\mathrm{d}\alpha\mathrm{d}\beta\\\
\leq\left(\int_{\mathbb{T}}|\hat{\mu}_{H}(\alpha)|\mathrm{d}\alpha\right)^{2}\sup_{\alpha,\beta\in\mathbb{T}}\biggl{|}\sum_{\begin{subarray}{c}y,y^{\prime}\\\
z,z^{\prime}\in[a]\end{subarray}}F(y,z)F_{2}(y^{\prime},z)F_{3}(y,z^{\prime})F_{4}(y^{\prime},z^{\prime})\biggr{|},$
where $F_{2}(y^{\prime},z):=\overline{F(y^{\prime},z)}e(-\beta z)$,
$F_{3}(y,z^{\prime}):=\overline{F(y,z^{\prime})}e(-\alpha y)$, and
$F_{4}(y^{\prime},z^{\prime})$ $:=F(y^{\prime},z^{\prime})e(\alpha
y^{\prime}+\beta z^{\prime})$.
We observe that
$\hat{\mu}_{H}(\alpha)=|\hat{1}_{[H]}(\alpha)|^{2}/\left\lfloor
H\right\rfloor^{2}$, which implies that
$\int_{\mathbb{T}}|\hat{\mu}(\alpha)|d\alpha=\left\lfloor
H\right\rfloor^{-1}$. Therefore
$\biggl{|}\sum_{\begin{subarray}{c}y,y^{\prime}\\\
z,z^{\prime}\in[a]\end{subarray}}F(y,z)F_{2}(y^{\prime},z)F_{3}(y,z^{\prime})F_{4}(y^{\prime},z^{\prime})\biggr{|}\geq\delta\left\lfloor
H\right\rfloor^{2}N-2\left\lfloor
H\right\rfloor^{3}\left(\tfrac{N}{a}+b\right),$ (4.4)
for $1$-bounded functions $F_{i}:\mathbb{Z}\times[a]\to\mathbb{C}$ of the form
$F_{i}(y,z)=f(ay+bz)e(\alpha_{1}y+\alpha_{2}z)$. Since $f$ is supported on
$[N]$, there are exactly $N$ pairs
$(y^{\prime},z^{\prime})\in\mathbb{Z}\times[a]$ for which
$F(y^{\prime},z^{\prime})\neq 0$. Thus, by pigeonholing in $y^{\prime}$ and
$z^{\prime}$ in (4.4) and setting $L(y):=F_{3}(y,z^{\prime})$ and
$R(z):=F_{2}(y^{\prime},z)F_{4}(y^{\prime},z^{\prime})$, we get that
$\biggl{|}\sum_{y}\sum_{z\in[a]}F(y,z)L(y)R(z)\biggr{|}\geq\delta\left\lfloor
H\right\rfloor^{2}-2\left\lfloor
H\right\rfloor^{3}\left(\tfrac{1}{a}+\tfrac{b}{N}\right).$
For each $x\in\mathbb{Z}$, define $l(x)\in\mathbb{Z}$ and $r(x)\in[a]$ by
$x=al(x)+br(x)$, and set $g(x):=R\circ r(x)$ and $h(x):=L\circ l(x)$. Then it
remains to check the invariance properties of $g$ and $h$. To see that
$g(x)=g(x+ay)$ for all $x,y\in\mathbb{Z}$, just note that $r(x)=r(x+ay)$ for
every $x,y\in\mathbb{Z}$.
Finally we establish that, for most $x\in[N]$, we have $h(x)=h(x+bz)$ for all
$|z|\leq\varepsilon N/b$. First note that $l(x)=l(x+bz)$ whenever $\varepsilon
N/b<r(x)\leq a-\varepsilon N/b$. Hence for this to fail, $x$ must lie in one
of at most $1+2\varepsilon N/b$ congruence classes modulo $a$. The number of
such $x$ lying in the interval $[N]$ is at most
$\left(1+\frac{2\varepsilon N}{b}\right)\left(1+\frac{N}{a}\right).$
∎
The lemma also yields a result in the situation in which $\gcd(a,b)>1$. In
proving this we take the opportunity to smooth out the $b$-invariance of $h$
slightly, whilst also giving an explicit description of $h$ in terms of $f$.
More concretely, we replace $h$ with a projection of $fg$ onto cosets of
$b\cdot\mathbb{Z}$.
###### Lemma 4.2.
There exists an absolute constant $c>0$ such that on assuming $1\leq H\leq
c\delta^{3}N^{1/2}$ and $1\leq K\leq c\delta^{2}H^{2}N^{-1/2}$ the following
holds. Let $a,b\in[N^{1/2}]$ with $\gcd(a,b)\leq\delta^{-1}$ and
$a,b\geq\delta N^{1/2}$. Suppose that $f:\mathbb{Z}\to\mathbb{C}$ is
$1$-bounded, supported on the interval $[N]$, and satisfies
$\biggl{|}\sum_{h,x}\mu_{H}(h)\Delta_{ah_{1},bh_{2}}f(x)\biggr{|}\geq\delta
N.$
Then there exists a 1-bounded $a$-periodic function $g$ such that
$\sum_{x}f(x)g(x)\sum_{k}\mu_{K}(k)\overline{f(x+bk)g(x+bk)}\gg\delta^{2}H^{4}/N.$
(4.5)
###### Proof.
Set $q:=\gcd(a,b)\leq\delta^{-1}$. For each $u\in[q]$, define a $1$-bounded
function $f_{u}:\mathbb{Z}\to\mathbb{C}$ by $f_{u}(x):=f(u+qx)$, and let
$I_{u}:=\left\\{x:u+qx\in[N]\right\\}$ denote the interval on which $f_{u}$ is
supported. By the pigeon-hole principle, for some $u$ we have
$\sum_{x,h_{1},h_{2}}\mu_{H}(h_{1})\mu_{H}(h_{2})\Delta_{\frac{a}{q}h_{1},\frac{b}{q}h_{2}}f_{u}(x)\geq\delta|I_{u}|.$
Note that $\gcd(a/q,b/q)=1$, so by the previous lemma, there exist 1-bounded
functions $g_{u},h_{u}:\mathbb{Z}\to\mathbb{C}$ such that
$\biggl{|}\sum_{x}f_{u}(x)g_{u}(x)h_{u}(x)\biggr{|}\geq\delta\left\lfloor
H\right\rfloor^{2}-2\left(\tfrac{Hq}{a}+\tfrac{Hb}{q|I_{u}|}\right)\left\lfloor
H\right\rfloor^{2}\gg\delta H^{2}.$
Furthermore, $g_{u}$ is $(a/q)$-periodic and
$\\#\left\\{x\in I_{u}:h_{u}(x)\neq h_{u}(x+yb/q)\text{ for some
}|y|\leq\varepsilon|I_{u}|q/b\right\\}\\\
\leq\left(1+\tfrac{2q\varepsilon|I_{u}|}{b}\right)\left(1+\tfrac{q|I_{u}|}{a}\right)\ll\tfrac{N}{a}+\tfrac{\varepsilon
N^{2}}{ab}.$
Defining $g_{u^{\prime}}$ and $h_{u^{\prime}}$ to be identically zero when
$u^{\prime}\neq u$, we set $g(u^{\prime}+qx):=g_{u^{\prime}}(x)$ and
$h(u^{\prime}+qx):=h_{u^{\prime}}(x)$. One can then check that $g$ is
$a$-invariant, that
$\biggl{|}\sum_{x}f(x)g(x)h(x)\biggr{|}\gg\delta H^{2},$
and that
$\\#\left\\{x\in[N]:h(x)\neq h(x+by)\text{ for some }|y|\leq\varepsilon
N/b\right\\}\ll\tfrac{N}{a}+\tfrac{\varepsilon N^{2}}{ab}.$
We may use the latter property to show that, provided $K\geq 1$, we have
$\biggl{|}\sum_{x}f(x)g(x)h(x)-\sum_{x}h(x)\mathbb{E}_{y\in[K]}g(x+by)f(x+by)\biggr{|}\ll\tfrac{NK}{a}.$
Provided that $K\leq c\delta^{2}H^{2}N^{-1/2}$ we deduce that
$\biggl{|}\sum_{x}h(x)\mathbb{E}_{y\in[K]}g(x+bk)f(x+bk)\biggr{|}\gg\delta
H^{2}.$
One can check that, as a function of $x$, the inner expectation is 1-bounded
with support in $[-2N,2N]$. Applying the Cauchy–Schwarz inequality and
changing variables then gives (4.5). ∎
Finally we observe that a function of the form
$h(x):=\sum_{k}\mu_{K}(k)f(x+by)$ (4.6)
has nice $b$-periodicity properties.
###### Lemma 4.3.
If $h$ is defined as in (4.6) for some 1-bounded $f$, then $h$ is
$O(K^{-1})$-Lipschitz along $b\cdot\mathbb{Z}$, in that for any
$x,y\in\mathbb{Z}$ we have $h(x+by)=h(x)+O(|y|/K)$.
###### Proof.
Recalling the definition (1.11), note that $\mu_{K}$ is $(2/\left\lfloor
K\right\rfloor)$-Lipschitz, in that $|\mu_{K}(k+y)-\mu_{K}(k)|\leq
2|y|/\left\lfloor K\right\rfloor$ for all $k,y\in\mathbb{Z}$. Hence, for
$|y|\leq K$, a change of variables gives
$|h(x+by)-h(x)|\leq\sum_{k}|\mu_{K}(k-y)-\mu_{K}(k)|\ll\frac{|y|}{K}\sum_{|k|<2K}1.$
∎
## 5\. Quantitative concatenation
The endpoint of this section is to show how our counting operator (1.3) is
controlled by the $U^{5}$-norm. We begin with four technical lemmas. The first
says that convolving Fejér kernels along progressions of coprime common
difference covers a substantial portion of an interval in a somewhat regular
manner, a fact that can be interpreted Fourier analytically in the following.
###### Lemma 5.1.
Let $K,L\geq 1$ and let $a,b$ be integers satisfying $a\geq\delta L$,
$b\geq\delta K$ and $\gcd(a,b)\leq\delta^{-1}$. Then
$\int_{\mathbb{T}}\bigl{|}\widehat{\mu}_{K}(a\beta)\bigr{|}\bigl{|}\widehat{\mu}_{L}(b\beta)\bigr{|}\mathrm{d}\beta\ll\frac{\delta^{-4}}{\left\lfloor
K\right\rfloor\left\lfloor L\right\rfloor}.$
###### Proof.
Expanding Fourier transforms, one can check that
$\int_{\mathbb{T}}\bigl{|}\widehat{\mu}_{H}(a\beta)\bigr{|}\bigl{|}\widehat{\mu}_{K}(b\beta)\bigr{|}\mathrm{d}\beta\\\
=\left\lfloor K\right\rfloor^{-2}\left\lfloor
L\right\rfloor^{-2}\\#\biggl{\\{}(x,y)\in[K]^{2}\times[L]^{2}:a(x_{1}-x_{2})=b(y_{1}-y_{2})\biggr{\\}}.$
Writing $d:=\gcd(a,b)$, the number of solutions to the equation is at most
$\left\lfloor K\right\rfloor\left\lfloor
L\right\rfloor\left(\tfrac{\left\lfloor
K\right\rfloor}{b/d}+1\right)\left(\tfrac{\left\lfloor
L\right\rfloor}{a/d}+1\right).$
∎
Our next lemma allows us to discard pairs of integers $a,b$ which are not
sufficiently coprime. We exploit this repeatedly.
###### Lemma 5.2.
For fixed integers $0\leq a_{1},a_{2}\leq M$. The number of pairs $(b,c)$ of
integers $0\leq b,c\leq M$ such that $\gcd(a_{1}+b,a_{2}+c)>\delta^{-1}$ is
$\ll\delta M^{2}$.
###### Proof.
Notice that if $d=\gcd(a_{1}+b,a_{2}+c)$ then $d\leq 2M$. Hence
$\displaystyle\sum_{\begin{subarray}{c}0\leq b,c\leq M\\\
\gcd(a_{1}+b,a_{2}+c)>\delta^{-1}\end{subarray}}1\leq\sum_{\delta^{-1}<d\leq
2M}\ \biggl{(}\ \sum_{0\leq m\leq 2M,\ d\mid m}1\biggr{)}^{2}$
$\displaystyle\leq\sum_{\delta^{-1}<d\leq 2M}\left(\frac{2M}{d}+1\right)^{2}$
$\displaystyle\ll M^{2}\sum_{d>\delta^{-1}}\frac{1}{d^{2}}\ll\delta M^{2}.$
∎
The following lemma says that, as $a$ and $h$ range over $[N^{1/2}]$, the
difference function $\Delta_{ah}f$ behaves like $\Delta_{k}f$ with $k\in[N]$,
at least on average.
###### Lemma 5.3.
Let $f:\mathbb{Z}\to\mathbb{C}$ be a 1-bounded function with support in $[N]$.
Suppose that $\delta N^{1/2}\leq H\leq N^{1/2}$ and
$\mathbb{E}_{a\in[N^{1/2}]}\sum_{h}\mu_{H}(h)\left\|\Delta_{ah}f\right\|_{U^{s}}^{2^{s}}\geq\delta\left\|1_{[N]}\right\|_{U^{s}}^{2^{s}}.$
Then
$\left\|f\right\|_{U^{s+1}}^{2^{s+1}}\gg\delta^{12}\left\|1_{[N]}\right\|_{U^{s+1}}^{2^{s+1}}$
###### Proof.
Expanding the definition of the $U^{s}$-norm
$\mathbb{E}_{a\in[N^{1/2}]}\sum_{h}\mu_{H}(h)\left\|\Delta_{ah}f\right\|_{U^{s}}^{2^{s}}\\\
=\sum_{h_{1},\dots,h_{s},x}\overline{\Delta_{h_{1},\dots,h_{s}}f(x)}\mathbb{E}_{a\in[N^{1/2}]}\sum_{h}\mu_{H}(h)\Delta_{h_{1},\dots,h_{s}}f(x+ah).$
Employing the Cauchy–Schwarz inequality to double the $a$ and $h$ variables
gives
$\mathbb{E}_{a,a^{\prime}\in[N^{1/2}]}\sum_{h_{i}}\sum_{x}\sum_{h,h^{\prime}}\mu_{H}(h)\mu_{H}(h^{\prime})\Delta_{h_{1},\dots,h_{s},ah-a^{\prime}h^{\prime}}f(x)\gg\delta^{2}N^{s+1}.$
By Lemma 5.2 and the pigeon-hole principle, we deduce the existence of
$a,a^{\prime}\gg\delta^{2}N^{1/2}$ with $\gcd(a,a^{\prime})\ll\delta^{-2}$
such that
$\sum_{h_{i}}\sum_{x}\sum_{h,h^{\prime}}\mu_{H}(h)\mu_{H}(h^{\prime})\Delta_{h_{1},\dots,h_{s},ah-a^{\prime}h^{\prime}}f(x)\gg\delta^{2}N^{s+1}.$
By Fourier inversion and extraction of a large Fourier coefficient, there
exists $\alpha\in\mathbb{T}$ such that the right-hand side above is at most
$\int_{\mathbb{T}}\left|\widehat{\mu}_{H}(a\beta)\right|\left|\widehat{\mu}_{H}(a^{\prime}\beta)\right|\mathrm{d}\beta\biggl{|}\sum_{h_{i}}\sum_{x}\Delta_{h_{1},\dots,h_{s},h_{s+1}}f(x)e(\alpha
h_{s+1})\biggr{|}.$
The result follows on employing Lemma 5.1 and Lemma A.3. ∎
We now prove a similar lemma, but with $\Delta_{ah}f$ replaced by $fg_{a}$
where $g_{a}$ is $a$-periodic. The moral is that these are similar quantities
(on average).
###### Lemma 5.4.
Let $f,g_{a}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions such that $g_{a}$
is $a$-periodic and $\mathrm{supp}(f)\subset[N]$. Suppose that
$\mathbb{E}_{a\in[N^{1/2}]}\left\|fg_{a}\right\|_{U^{s}}^{2^{s}}\geq\delta\left\|1_{[N]}\right\|_{U^{s}}^{2^{s}}.$
Then
$\left\|f\right\|_{U^{s+1}}^{2^{s+1}}\gg\delta^{24}\left\|1_{[N]}\right\|_{U^{s+1}}^{2^{s+1}}$
###### Proof.
Fix $a\in[N^{1/2}]$. By the periodicity of $g_{a}$ and a change of variables,
we have
$\sum_{h_{i}}\sum_{x}\Delta_{h_{1},\dots,h_{s}}g_{a}(x)\Delta_{h_{1},\dots,h_{s}}f(x)=\sum_{h_{i}}\sum_{x}\Delta_{h_{1},\dots,h_{s}}g_{a}(x)\mathbb{E}_{y\in[N^{1/2}]}\Delta_{h_{1},\dots,h_{s}}f(x+ay).$
Notice that the sum over $x$ is non-zero only if $|x|,|h_{i}|<N$, hence by
Cauchy–Schwarz and a change of variables
$\displaystyle\biggl{(}\mathbb{E}_{a\in[N^{1/2}]}\left\|fg_{a}\right\|_{U^{s}}^{2^{s}}\biggr{)}^{2}$
$\displaystyle\ll
N^{s+1}\mathbb{E}_{a\in[N^{1/2}]}\sum_{h_{i}}\sum_{x}\sum_{y}\mu_{N^{1/2}}(y)\Delta_{h_{1},\dots,h_{s},ay}f(x)$
$\displaystyle=N^{s+1}\mathbb{E}_{a\in[N^{1/2}]}\sum_{y}\mu_{N^{1/2}}(y)\left\|\Delta_{ay}f\right\|_{U^{s}}^{2^{s}}$
The result follows on employing Lemma 5.3. ∎
We are now ready to give the technical heart of this section. The (somewhat
lengthy) assumptions come from our eventual application of Lemma 4.2.
###### Lemma 5.5.
Fix $a\in\mathbb{N}$ and let $\delta N^{1/2}\leq K\leq N^{1/2}$. For each
$b\in[N^{1/2}]$ let $f,g_{b},h_{b}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded
functions such that $\mathrm{supp}(f),\mathrm{supp}(h_{b})\subset[N]$ and
where $g_{b}$ is $b$-periodic. Set
$\tilde{h}_{b}(x):=\sum_{k}\mu_{K}(k)h_{b}(x+(a+b)k)$
and suppose that
$\sum_{\begin{subarray}{c}\delta\sqrt{N}\leq b\leq\sqrt{N}\\\
\gcd(a,b)\leq\delta^{-1}\end{subarray}}\sum_{x}f(x)g_{b}(x)\tilde{h}_{b}(x)\geq\delta
N^{3/2}.$
Then
$\mathbb{E}_{b\in[N^{1/2}]}\big{\|}h_{b}\big{\|}_{U^{3}}^{8}\gg\delta^{208}\left\|1_{[N]}\right\|_{U^{3}}^{8}.$
###### Proof.
To ease notation, write
$\tilde{h}_{b}(x):=\sum_{k}\mu_{K}(k)h_{b}(x+(a+b)k)$
We apply Cauchy–Schwarz to remove the weight $f(x)$ and double the $b$
variable, yielding
$\sum_{\begin{subarray}{c}\delta\sqrt{N}\leq b,b^{\prime}\leq\sqrt{N}\\\
\gcd(a,b)\leq\delta^{-1}\end{subarray}}\sum_{x}g_{b}(x)\tilde{h}_{b}(x)\overline{g_{b^{\prime}}(x)\tilde{h}_{b^{\prime}}(x)}\geq\delta^{2}N^{2}.$
Employing Lemma 5.2, we may discard those $b,{b^{\prime}}$ for which one of
$\gcd(b^{\prime},a+{b})$ or $\gcd(a+b^{\prime},a+{b})$ is greater than
$C\delta^{-2}$. On combining this with the popularity principle, we deduce the
existence of $\mathcal{B}\subset[\delta N^{1/2},N^{1/2}]$ of size
$|\mathcal{B}|\gg\delta^{2}N^{1/2}$ such that for each $b\in\mathcal{B}$ there
exists $b^{\prime}\in[N^{1/2}]$ with all of $\gcd(b,a+{b})$,
$\gcd({b^{\prime}},a+{b})$, $\gcd(a+b^{\prime},a+{b})$ at most
$O(\delta^{-2})$ and satisfying
$\sum_{x}g_{b}(x)\overline{\tilde{h}_{b^{\prime}}(x)g_{b^{\prime}}(x)}\tilde{h}_{b}(x)\gg\delta^{2}N.$
(5.1)
Expanding the definition of $\tilde{h}_{b^{\prime}}$, using the invariance of
$g_{b}$ and changing variables gives
$\sum_{x}\mathbb{E}_{k_{1},k_{3}\in[K]}\sum_{k_{2}}\mu_{K}(k_{2})g_{b}(x+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\overline{h_{b^{\prime}}(x+bk_{1}+{b^{\prime}}k_{3})}\\\
\overline{g_{b^{\prime}}(x+bk_{1}+(a+b^{\prime})k_{2})}\
\tilde{h}_{b}(x+bk_{1}+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\gg\delta^{2}N.$
Since $h_{b^{\prime}}$ is supported on $[N]$ and $b,{b^{\prime}},K\leq
N^{1/2}$, there are at most $O(N)$ values of $x$ which contribute to the above
sum. Applying Hölder’s inequality then gives
$\sum_{x}\biggl{(}\mathbb{E}_{k_{1},k_{3}\in[K]}\sum_{k_{2}}\mu_{K}(k_{2})g_{b}(x+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\overline{h_{b^{\prime}}(x+bk_{1}+{b^{\prime}}k_{3})}\\\
\overline{g_{b^{\prime}}(x+bk_{1}+(a+b^{\prime})k_{2})}\
\tilde{h}_{b}(x+bk_{1}+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\biggr{)}^{8}\gg\delta^{16}N.$
The sum inside the 8th power corresponds to an integral with respect to three
probability measures on $\mathbb{Z}$, with integrand amenable to Lemma A.4.
Combining this with a change of variables gives
$\sum_{x}\sum_{k_{1},k_{2},k_{3}}\mu_{K}(k_{1})\nu_{K}(k_{2})\mu_{K}(k_{3})\Delta_{bk_{1},(a+b^{\prime})k_{2},{b^{\prime}}k_{3}}\
\tilde{h}_{b}(x)\gg\delta^{16}N,$
where we set
$\nu_{K}(k):=\sum_{k_{1}-k_{2}=k}\mu_{K}(k_{1})\mu_{K}(k_{2}).$
By Lemma 4.3, each $\tilde{h}_{b}$ is $O(K^{-1})$-Lipschitz along
$(a+b)\cdot\mathbb{Z}$. Hence, if $l_{i}\in[L]$, a telescoping identity shows
that
$|\Delta_{h_{1}+(a+{b})l_{1},h_{2}+(a+{b})l_{2},h_{3}+(a+{b})l_{3}}\tilde{h}_{b}(x)-\Delta_{h_{1},h_{2},h_{3}}\tilde{h}_{b}(x)|\ll
L/K.$
Taking $L:=c\delta^{16}K$ we obtain
$\sum_{x}\sum_{k_{1},k_{2},k_{3}}\mu_{K}(k_{1})\nu_{K}(k_{2})\mu_{K}(k_{3})\mathbb{E}_{l_{1},l_{2},l_{3}\in[L]}\\\
\Delta_{bk_{1}+(a+{b})l_{1},\,(a+b^{\prime})k_{2}+(a+{b})l_{2},\,{b^{\prime}}k_{3}+(a+{b})l_{3}}\
\tilde{h}_{b}(x)\gg\delta^{16}N.$
We may replace the uniform measure on the $l_{i}$ by Fejér kernels at the cost
of three applications of Cauchy–Schwarz; this gives
$\sum_{x}\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3}\\\
l_{1},l_{2},l_{3}\end{subarray}}\mu_{K}(k_{1})\nu_{K}(k_{2})\mu_{K}(k_{3})\mu_{L}(l_{1})\mu_{L}(l_{2})\mu_{L}(l_{3})\\\
\Delta_{bk_{1}+(a+{b})l_{1},\,(a+b^{\prime})k_{2}+(a+{b})l_{2},\,{b^{\prime}}k_{3}+(a+{b})l_{3}}\
\tilde{h}_{b}(x)\gg\delta^{128}N.$
Write
$\displaystyle\lambda_{1}(h):=\sum_{bk+(a+{b})l=h}$
$\displaystyle\mu_{K}(k)\mu_{L}(l),\qquad\lambda_{2}(h):=\sum_{(a+b^{\prime})k+(a+{b})l=h}\nu_{K}(k)\mu_{L}(l),$
$\displaystyle\lambda_{3}(h):=\sum_{{b^{\prime}}k+(a+{b})l=h}\mu_{K}(k)\mu_{L}(l).$
Then
$\sum_{x}\sum_{h_{1},h_{2},h_{3}}\lambda_{1}(h_{1})\lambda_{2}(h_{2})\lambda_{3}(h_{3})\\\
\Delta_{h_{1},h_{2},h_{3}}\ \tilde{h}_{b}(x)\gg\delta^{128}N.$
By Fourier inversion and extraction of a large Fourier coefficient, there
exist $\alpha_{i}\in\mathbb{T}$ such that
$\biggl{|}\sum_{x}\sum_{h_{1},h_{2},h_{3}}\Delta_{h_{1},h_{2},h_{3}}\
\tilde{h}_{b}(x)e(\underline{\alpha}\cdot\underline{h})\biggr{|}\prod_{i=1}^{3}\int_{\mathbb{T}}\bigl{|}\widehat{\lambda}_{i}(\beta)\bigr{|}\mathrm{d}\beta\gg\delta^{128}N.$
By our choice of $b$, $b^{\prime}$ (see the paragraph preceding (5.1)),
together with Lemma 5.1, for each $i$ we have
$\int_{\mathbb{T}}\bigl{|}\widehat{\lambda}_{i}(\alpha)\bigr{|}\mathrm{d}\alpha\ll\frac{\delta^{-8}}{KL}\ll\frac{\delta^{-26}}{N},$
(5.2)
the latter following from the fact that $L\gg c\delta^{16}K$ and $K\geq\delta
N^{1/2}$. On combining this with Lemma A.3 we obtain
$\big{\|}\tilde{h}_{b}\big{\|}_{U^{3}}^{8}\gg\delta^{206}N^{4}.$
Since $\tilde{h}_{b}$ is an average of translates of $h_{b}$, we may apply the
triangle inequality for the $U^{3}$-norm, together with the fact that Gowers
norms are translation invariant, and conclude that
$\left\|h_{b}\right\|_{U^{3}}^{8}\gg\delta^{206}N^{4}$. Summing over
$b\in\mathcal{B}$ gives our final bound. ∎
Finally we synthesise Lemmas 3.3, 4.2 and 5.5.
###### Theorem 5.6 (Global $U^{5}$-control).
Let $g_{0},g_{1},f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions, each with
support in $[N]$. Suppose that
$\left|\Lambda_{q,N}(g_{0},g_{1},f)\right|\geq\delta\Lambda_{q,N}(1_{[N]}).$
Then
$\sum_{u\in[q]}\left\|f\right\|_{U^{5}(u+q\mathbb{Z})}^{2^{5}}\gg\delta^{2^{25}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{5}(u+q\mathbb{Z})}^{2^{5}}.$
###### Proof.
We recall our convention (1.2) regarding $M$. We begin by applying the
linearisation procedure (Lemma 3.3) to deduce that
$\sum_{a,b\in(-2M,2M)}\
\biggl{|}\sum_{h}\mu_{H}(h)\sum_{x}\Delta_{q(a+b)h_{1},qbh_{2},qah_{3}}f(x)\biggr{|}\\\
\gg\delta^{32}NM^{2}.$
We note that the sum inside the absolute value is invariant under
$a\mapsto-a$. Hence we may restrict to $a,b\in[0,2M]$ at the cost of changing
the absolute constant. Applying Lemma 5.2 we may discard those $a,b$ for which
either $\gcd(a,b)>C\delta^{-32}$ or $b<c\delta^{32}M$. Partitioning the sum
over $x$ into congruence classes $u\bmod q$, the popularity principle gives:
* •
at least $\Omega(\delta^{32}q)$ residues $u\in[q]$;
* •
for each of which there is a subset of $h_{3}\in(-H,H)$ of
$\mu_{H}$-measure222i.e.
$\sum_{h_{3}\in\mathcal{H}}\mu_{H}(h_{3})\gg\delta^{32}$. at least
$\Omega(\delta^{32})$;
* •
for each of which there exist $\Omega(\delta^{32}M)$ values of $a\in[2M]$;
* •
for each of which there are $\Omega(\delta^{32}M)$ values of $b\in[2M]$
satisfying $\gcd(a,b)\ll\delta^{-32}$ and $b\gg\delta^{32}M$;
and together these satisfy
$\biggl{|}\sum_{h_{1},h_{2}}\mu_{H}(h_{1},h_{2})\sum_{x}\Delta_{(a+b)h_{1},bh_{2},ah_{3}}f(qx-u)\biggr{|}\\\
\gg\delta^{32}M^{2}.$
For fixed $u,h_{3},a$ write $\tilde{f}(x):=\Delta_{ah_{3}}f(qx-u),$ so that
$\tilde{f}$ has support in the interval $[(2M)^{2}]$ and
$\biggl{|}\sum_{h_{1},h_{2}}\mu_{H}(h_{1},h_{2})\sum_{x}\Delta_{(a+b)h_{1},bh_{2}}\tilde{f}(x)\biggr{|}\\\
\gg\delta^{32}M^{2}.$
Set
$H:=c\delta^{96}M\qquad\text{and}\qquad K:=c^{3}\delta^{160}M,$ (5.3)
with $c$ sufficiently small to ensure that we may apply Lemma 4.2. This gives
the existence of a 1-bounded $b$-periodic function $g_{b}$ such that on
setting
$\tilde{h}_{b}(x):=\sum_{k}\mu_{K}(k)\overline{\tilde{f}(x+(a+b)k)g_{b}(x+(a+b)k)}$
(5.4)
we have
$\sum_{x}\tilde{f}(x)g_{b}(x)\tilde{h}_{b}(x)\gg\delta^{448}M^{2}.$
Setting $\eta:=c\delta^{480}$ for some small absolute constant $c>0$, we may
sum over our set of permissible $b$ to deduce that
$\sum_{\begin{subarray}{c}\eta M\leq b\leq 2M\\\
\gcd(a,b)\leq\eta^{-1}\end{subarray}}\sum_{x}\tilde{f}(x)g_{b}(x)h_{b}(x)\geq\eta
M^{3}.$
The hypotheses of Lemma 5.5 having been met, we conclude that
$\mathbb{E}_{b\in[2M]}\big{\|}\tilde{f}g_{b}\big{\|}_{U^{3}}^{8}\gg\delta^{99,840}\left\|1_{[M^{2}]}\right\|_{U^{3}}^{8}.$
Applying Lemma 5.4 then gives
$\big{\|}\tilde{f}\big{\|}_{U^{4}}^{16}\gg\delta^{2,396,160}\left\|1_{[M^{2}]}\right\|_{U^{4}}^{16}.$
Recalling that $\tilde{f}(x)=\Delta_{ah_{3}}f_{u}(x)$ where
$f_{u}(x):=f(qx-u)$, we may integrate over the set of permissible $h_{3}$ and
$a$, utilising positivity to extend the range of summation, and deduce that
$\mathbb{E}_{a\in[2M]}\sum_{h}\mu_{H}(h_{3})\big{\|}\Delta_{ah_{3}}f_{u}\big{\|}_{U^{4}}^{16}\gg\delta^{2,396,224}\left\|1_{[M^{2}]}\right\|_{U^{4}}^{16}$
Using Lemma 5.3 and summing over the permissible range of $u$ we get that
$\mathbb{E}_{u\in[q]}\left\|f_{u}\right\|_{U^{5}}^{32}\gg\delta^{28,754,720}\left\|1_{[M^{2}]}\right\|_{U^{5}}^{32},$
and the result follows. ∎
## 6\. Degree lowering
So far, we have shown that $\Lambda_{q,N}(f_{0},f_{1},f_{2})$ is controlled by
$\mathbb{E}_{u\in[q]}\|f_{2}\|_{U^{5}(u+q\mathbb{Z})}^{2^{5}}$ whenever
$f_{0},f_{1},$ and $f_{2}$ are $1$-bounded complex-valued functions supported
on the interval $[N]$. The next step in our argument is to bound
$\Lambda_{q,N}(f_{0},f_{1},f_{2})$ in terms of the $U^{5}(u+q\mathbb{Z})$-norm
of the dual function
$F(x):=\mathbb{E}_{y\in[M]}f_{0}(x-qy^{2})f_{1}(x+y-qy^{2}).$ (6.1)
We postpone this deduction until §7. In this section we show how
$U^{5}$-control of the dual implies $U^{2}$-control.
Our argument combines three simple lemmas: Weyl’s inequality; what we call
‘dual–difference interchange’, which allows us to replace the difference
function of the dual by the dual of the difference functions; and the fact
that a function whose difference functions correlate with ‘low rank’ Fourier
coefficients must have a large uniformity norm of lower degree.
The following log-free variant of Weyl’s inequality can be found in [4, Lemma
A.11].
###### Lemma 6.1 (Weyl’s inequality).
There exists an absolute constant $C$ such that the following holds. Let
$\alpha,\beta\in\mathbb{T}$, $\delta\in(0,1)$ and let $I\subset\mathbb{Z}$ be
an interval with $|I|\geq C\delta^{-6}$ and
$\big{|}\mathbb{E}_{y\in I}e(\alpha y^{2}+\beta y)\big{|}\geq\delta.$
Then there exists a positive integer $q\ll\delta^{-4}$ such that
$\|q\alpha\|\ll\delta^{-14}|I|^{-2}.$
This has the following consequence, which uses our convention (1.2) regarding
$M$.
###### Lemma 6.2.
There exist an absolute constant $C$ such that for $N\geq C(q/\delta)^{C}$ the
following holds. Suppose that for $\alpha\in\mathbb{T}$ there are $1$-bounded
functions $g_{0},g_{1}:\mathbb{Z}\to\mathbb{C}$ supported on the interval
$[N]$ such that
$\left|\sum_{x}\sum_{y\in[M]}g_{0}(qx)g_{1}(qx+y)e(\alpha(x+y^{2}))\right|\geq\delta
MN/q.$
Then there exists a positive integer $q^{\prime}\ll\delta^{-4}$ such that
$\|q^{\prime}q^{2}\alpha\|\ll\delta^{-14}q^{3}/N$.
###### Proof.
We split the sum over $y\in[M]$ into arithmetic progressions modulo $q$ and
split the sum over $x$ into intervals of length $M/q$. Hence, by the pigeon-
hole principle, there exists $u\in[q]$ and an integer $m$ such that on
rounding the sum over $y$ we have
$\left|\sum_{x,y\in[M/q]}g_{0}(q(m+x))g_{1}(u+q(m+x+y))e\left(\alpha\left(x+(u+qy)^{2}\right)\right)\right|\\\
\gg\delta(M/q)^{2}.$
Define the functions
$\displaystyle h_{0}(x):=g_{0}(q(m+x))$ $\displaystyle e(\alpha
x)1_{[M/q]}(x),\qquad h_{1}(x):=g_{1}(u+q(m+x))1_{[2M/q]},$ $\displaystyle
h_{2}(x):=e\left(\alpha(u+qx)^{2}\right)1_{[M/q]}(x)$
Then by orthogonality, extraction of a large Fourier coefficient and Parseval
we have
$\displaystyle\delta
M^{2}/q^{2}\ll\left|\int_{\mathbb{T}}\hat{h}_{0}(\beta)\hat{h}_{1}(-\beta)\hat{h}_{2}(\beta)\mathrm{d}\alpha\right|\ll\big{\|}\hat{h}_{2}\big{\|}_{\infty}\big{\|}\hat{h}_{0}\big{\|}_{L^{2}}\big{\|}\hat{h}_{1}\big{\|}_{L^{2}}\ll\big{\|}\hat{h}_{2}\big{\|}_{\infty}M/q.$
It follows that there exists $\beta\in\mathbb{T}$ such that
$\left|\sum_{x\in[M/q]}e\left(\alpha(u+qx)^{2}+\beta x\right)\right|\gg\delta
M/q.$
Applying Weyl’s inequality, we deduce the existence of
$q^{\prime}\ll\delta^{-4}$ such that
$\left\|q^{\prime}q^{2}\alpha\right\|\ll\delta^{-14}(q/M)^{2}$. ∎
###### Lemma 6.3 (Dual–difference interchange).
For each $y\in[M]$, let $F_{y}:\mathbb{Z}\to\mathbb{C}$ be a 1-bounded
function with support in an interval of length $N$. Set
$F(x):=\mathbb{E}_{y\in[M]}F_{y}(x).$
Then for any function $\phi:\mathbb{Z}^{s}\to\mathbb{T}$ and finite set
$\mathcal{H}\subset\mathbb{Z}^{s}$ we have
$\left(N^{-s-1}\sum_{\underline{h}\in\mathcal{H}}\left|\sum_{x}\Delta_{\underline{h}}F(x)e\bigl{(}\phi(\underline{h})x\bigr{)}\right|\right)^{2^{s}}\ll_{s}\\\
N^{-2s-1}\sum_{\underline{h}^{0},\underline{h}^{1}\in\mathcal{H}}\left|\sum_{x}\mathbb{E}_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1})x\bigr{)}\right|,$
where
$\phi(\underline{h}^{0};\underline{h}^{1}):=\sum_{\omega\in\left\\{0,1\right\\}^{s}}(-1)^{|\omega|}\phi(\underline{h}^{\omega})\qquad\text{and}\qquad\underline{h}^{\omega}:=(h_{1}^{\omega_{1}},\dots,h_{s}^{\omega_{s}}).$
###### Proof.
We proceed by induction on $s\geq 0$, the base case being an identity. Suppose
then that $s\geq 1$. For $\underline{h}\in\mathbb{Z}^{s-1}$ and
$h\in\mathbb{Z}$, we note that
$\Delta_{(\underline{h},h)}F(x)=\Delta_{\underline{h}}\left(\mathbb{E}_{y,y^{\prime}\in[M]}F_{y}(x)\overline{F_{y^{\prime}}(x+h)}\right)$
Hence by the induction hypothesis
$\left(N^{-s-1}\sum_{h}\sum_{\begin{subarray}{c}\underline{h}\\\
(\underline{h},h)\in\mathcal{H}\end{subarray}}\left|\sum_{x}\Delta_{(\underline{h},h)}F(x)e\bigl{(}\phi(\underline{h})x\bigr{)}\right|\right)^{2^{s}}\ll_{s}\\\
\left(N^{-2s}\sum_{h}\sum_{\begin{subarray}{c}\underline{h}^{0},\underline{h}^{1}\\\
(\underline{h}^{i},h)\in\mathcal{H}\end{subarray}}\left|\sum_{x}\mathbb{E}_{y,y^{\prime}\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)\overline{F_{y^{\prime}}(x+h)}e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1};h)x\bigr{)}\right|\right)^{2},$
where
$\phi(\underline{h}^{0};\underline{h}^{1};h):=\sum_{\omega\in\left\\{0,1\right\\}^{s-1}}(-1)^{|\omega|}\phi(\underline{h}^{\omega},h).$
Letting $e(\psi(\underline{h}^{0};\underline{h}^{1};h))$ denote the phase of
the inner absolute, we take the sum over $h$ inside and apply Cauchy–Schwarz
to obtain
$\left(\sum_{\underline{h}^{0},\underline{h}^{1},x}\mathbb{E}_{y,y^{\prime}\in[M]}\sum_{\begin{subarray}{c}h\\\
(\underline{h}^{i},h)\in\mathcal{H}\end{subarray}}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)\overline{F_{y^{\prime}}(x+h)}e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1};h)x+\psi(\underline{h}^{0};\underline{h}^{1};h)\bigr{)}\right)^{2}\\\
\leq
N^{2s-1}\sum_{\underline{h}^{0},\underline{h}^{1}}\sum_{\begin{subarray}{c}h^{0},h^{1}\\\
(\underline{h}^{i},h^{j})\in\mathcal{H}\end{subarray}}\\\
\left|\sum_{x}\mathbb{E}_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)\overline{F_{y}(x+h^{0}-h^{1})}e\Bigl{(}\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1};h^{0})-\phi(\underline{h}^{0};\underline{h}^{1};h^{1})\bigr{)}x\Bigr{)}\right|.$
The result follows. ∎
If $\phi(h_{1},\dots,h_{s-1})$ is a function of $s-1$ variables we write
$\phi(h_{1},\dots,\hat{h}_{i},\dots,h_{s}):=\phi(h_{1},\dots,h_{i-1},h_{i+1},\dots,h_{s}).$
We say that $\phi(h_{1},\dots,h_{s})$ is _low rank_ if there exist functions
$\phi_{i}(h_{1},\dots,h_{s-1})$ such that
$\phi(h_{1},\dots,h_{s})=\sum_{i=1}^{s}\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s}).$
From the definition of the Gowers norm together with the $U^{2}$-inverse
theorem (Lemma A.1), one can show that largeness of the $U^{s+2}$-norm is
equivalent to the existence of $\phi:\mathbb{Z}^{s}\to\mathbb{T}$ such that
$\sum_{h_{1},\dots,h_{s}}\left|\sum_{x}\Delta_{h}f(x)e(\phi(h)x)\right|\gg
N^{s+1}.$
The following lemma says that if $\phi$ is low-rank, then the $U^{s+1}$-norm
must also be large.
###### Lemma 6.4 (Low rank correlation implies lower degree).
Let $f:\mathbb{Z}\to\mathbb{C}$ be a 1-bounded function with support in $[N]$.
Then for $\phi_{1},\dots,\phi_{m}:\mathbb{Z}^{s-1}\to\mathbb{T}$ with $m\leq
s$ we have
$\frac{1}{N^{s+1}}\sum_{h_{1},\dots,h_{s}}\left|\sum_{x}\Delta_{h}f(x)e\left(\sum_{i=1}^{m}\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s})x\right)\right|\\\
\ll_{m}\left(\frac{\left\|f\right\|_{U^{s+1}}^{2^{s+1}}}{N^{s+2}}\right)^{2^{-m-1}}.$
(6.2)
###### Proof.
We proceed by induction on $m\geq 0$, the base case corresponding to the
Cauchy–Schwarz inequality. Suppose then that $m\geq 1$ and the result is true
for smaller values of $m$. Letting $e(\psi(h))$ denote the phase of the inner-
most sum, the left-hand side of (6.2) is equal to
$\frac{1}{N^{s+1}}\sum_{h_{2},\dots,h_{s},x}\Delta_{h_{2},\dots,h_{s}}f(x)e\left(\phi_{1}(h_{2},\dots,h_{s})\right)\sum_{h_{1}}\Delta_{h_{2},\dots,h_{s}}\overline{f}(x+h_{1})\\\
e\left(\sum_{i=2}^{m}\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s})x+\psi(h_{1},\dots,h_{s})\right).$
By Cauchy–Schwarz, the square of this is at most
$\frac{1}{N^{s+2}}\sum_{h_{2},\dots,h_{s}}\
\sum_{h_{1},h_{1}^{\prime}\in(-N,N)}\\\
\left|\sum_{x}\Delta_{h_{1}-h_{1}^{\prime},h_{2},\dots,h_{s}}f(x)e\left(\sum_{i=2}^{m}\left(\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s})-\phi_{i}(h_{1}^{\prime},\dots,\hat{h}_{i},\dots,h_{s})\right)x\right)\right|.$
Taking a maximum over $h_{1}^{\prime}\in(-N,N)$ and changing variables in
$h_{1}$, the latter is at most an absolute constant times
$\frac{1}{N^{s+1}}\sum_{h_{1},h_{2},\dots,h_{s}}\Bigg{|}\sum_{x}\Delta_{h_{1},h_{2},\dots,h_{s}}f(x)\\\
e\left(\sum_{i=2}^{m}\left(\phi_{i}(h_{1}+h_{1}^{\prime},h_{2}\dots,\hat{h}_{i},\dots,h_{s})-\phi_{i}(h_{1}^{\prime},h_{2}\dots,\hat{h}_{i},\dots,h_{s})\right)x\right)\Bigg{|}.$
This phase has lower rank than the original, hence we may apply the induction
hypothesis to yield the lemma. ∎
###### Lemma 6.5 (Degree lowering).
There exists an absolute constant such that for $N\geq C(q/\delta)^{C}$ the
following holds. Let $f_{0},f_{1}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded
functions with support in $[N]$ and define the dual
$F(x):=\mathbb{E}_{y\in[M]}f_{0}(x-qy^{2})f_{1}(x+y-qy^{2}).$
If, for $s\geq 3$, we have
$\sum_{u\in[q]}\left\|F\right\|_{U^{s}(u+q\cdot\mathbb{Z})}^{2^{s}}\geq\delta\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{s}(u+q\cdot\mathbb{Z})}^{2^{s}},$
then
$\sum_{u\in[q]}\left\|F\right\|_{U^{s-1}(u+q\cdot\mathbb{Z})}^{2^{s-1}}\gg_{s}\delta^{4^{s+2}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{s-1}(u+q\cdot\mathbb{Z})}^{2^{s-1}},$
###### Proof.
Write $M:=\left\lfloor(N/q)^{1/2}\right\rfloor$. Given $u\in[q]$ let
$F_{u}(x):=F(u+qx)$, a function with support in the interval $[2N/q]$.
Applying the popularity principle, there exists a set of $\Omega(\delta q)$
residues $u\in[q]$ for which
$\left\|F_{u}\right\|_{U^{s}}^{2^{s}}\gg\delta(N/q)^{s+1}$. Expanding the
definition of the $U^{s}$-norm (1.9) we have
$\sum_{h_{1},\dots,h_{s-2}}\left\|\Delta_{h_{1},\dots,h_{s-2}}F_{u}\right\|_{U^{2}}^{4}\gg\delta(N/q)^{s+1}.$
Applying the $U^{2}$-inverse theorem (Lemma A.1), there exists
$\mathcal{H}\subset(-2N/q,2N/q)^{s-2}$ of size
$|\mathcal{H}|\gg\delta(N/q)^{s-2}$ and a function
$\phi:\mathbb{Z}^{s-2}\to\mathbb{T}$ such that for every
$\underline{h}\in\mathcal{H}$ we have
$\left|\sum_{x}\Delta_{\underline{h}}F_{u}(x)e\bigl{(}\phi(\underline{h})x\bigr{)}\right|\gg\delta
N/q.$ (6.3)
Set $T:=\left\lceil C\delta^{-1}N/q\right\rceil$, with $C$ an absolute
constant taken sufficiently large to ensure that, on rounding
$\phi(\underline{h})$ to the nearest fraction of the form $t/T$, the validity
of (6.3) remains. Summing over $\underline{h}\in\mathcal{H}$ and applying
Lemma 6.3, we deduce that
$\sum_{\underline{h}^{0},\underline{h}^{1}\in\mathcal{H}}\left|\sum_{x}\mathbb{E}_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{0}(u+qx-
qy^{2})\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{1}(u+qx+y-qy^{2})\right|\\\
e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1})x\bigr{)}\gg_{s}\delta^{2^{s-1}}(N/q)^{2s-1}.$
Applying the pigeon-hole and popularity principle, there exists
$\mathcal{H}^{\prime}\subset\mathcal{H}$ of size
$\Omega_{s}(\delta^{2^{s-1}}(N/q)^{s-2})$ and
$\underline{h}^{1}\in\mathcal{H}$ such that for every
$\underline{h}^{0}\in\mathcal{H}^{\prime}$ we have
$\left|\sum_{x}\sum_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{0}(u+qx-
qy^{2})\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{1}(u+qx+y-qy^{2})e\bigl{(}\phi(\underline{h}^{0},\underline{h}^{1})x\bigr{)}\right|\\\
\gg\delta^{2^{s-1}}MN/q.$
By Lemma 6.2, for each $\underline{h}^{0}\in\mathcal{H}^{\prime}$ there exists
$q^{\prime}\ll\delta^{-2^{s+1}}$ such that
$\left\|q^{\prime}q^{2}\phi(\underline{h}^{0},\underline{h}^{1})\right\|\ll\delta^{-2^{s}\times
7}q^{3}/N$
Notice that $\phi(\underline{h}^{0},\underline{h}^{1})$ is an element of the
additive group $\left\\{t/T:t\in[T]\right\\}\subset\mathbb{T}$. Moreover, for
any $Q_{i}$ we have the inclusion
$\left\\{\alpha\in\mathbb{T}:\exists q^{\prime}\leq Q_{1}\text{ with
}\left\|q^{\prime}q^{2}\alpha\right\|\leq
Q_{2}q^{3}/N\right\\}\subset\bigcup_{\begin{subarray}{c}1\leq a\leq q\leq
Q_{1}\\\
\mathrm{hcf}(a,q)=1\end{subarray}}\left[\frac{a}{q^{\prime}q^{2}}-\frac{Q_{2}}{N},\frac{a}{q^{\prime}q^{2}}+\frac{Q_{2}}{N}\right].$
By a volume packing argument, the number of $t/T$ lying in this union of
intervals is at most $O\left(Q_{1}^{2}(1+\tfrac{Q_{2}T}{N})\right)$. It
therefore follows from the pigeon-hole principle that there exists
$\mathcal{H}^{\prime\prime}\subset\mathcal{H}^{\prime}$ of size
$\Omega\left(\delta^{2^{s+3}+1-2^{s}}(N/q)^{s-2}\right)$ and $t_{0}\in[T]$
such that for any $\underline{h}^{0}\in\mathcal{H}^{\prime\prime}$ we have
$\phi(\underline{h}^{0},\underline{h}^{1})=t_{0}/T$. In particular, when
restricted to the set $\mathcal{H}^{\prime\prime}$, the function $\phi$
satisfies
$\phi(\underline{h}^{0})=t_{0}/T-\sum_{\omega\in\left\\{0,1\right\\}^{s}\setminus\left\\{0\right\\}}(-1)^{|\omega|}\phi(\underline{h}^{\omega}).$
The right-hand side of this identity is clearly _low rank_ according to the
terminology preceding Lemma 6.4.
Summing over $\underline{h}\in\mathcal{H}^{\prime\prime}$ in (6.3), we deduce
the existence of a low rank function $\psi:\mathbb{Z}^{s-2}\to\mathbb{T}$ such
that
$\sum_{\underline{h}}\left|\sum_{x}F_{u}(x)e\bigl{(}\psi(\underline{h})x\bigr{)}\right|\gg\delta^{2^{s+3}+1-2^{s}}(N/q)^{s-1}.$
Employing Lemma 6.4 then gives
$\left\|F_{u}\right\|_{U^{s-1}}^{2^{s-1}}\gg\delta^{(2^{s+3}+1-2^{s})2^{s+1}}(N/q)^{s}.$
Summing over permissible $u$, then extending to the full sum over $u\in[q]$ by
positivity, we obtain the bound claimed in the lemma. ∎
## 7\. Proof of the cut norm inverse theorem
In this section we complete our proof of Theorem 1.6. We first show how the
dual function is controlled by the $U^{5}$-norm, and hence by the degree
lowering of §6, the dual is controlled by the $U^{1}$-norm.
The following can be found in the discussion following [3, Proposition 3.6].
Although the statement therein is for norms, and not seminorms, one can check
that the (simple) argument remains valid in this greater generality333On
occasion the relevant results in [3] appear to assume that unit balls are
_bounded_ (if we take the definition of _convex body_ to be a compact convex
set with non-empty interior), which may not be true for the unit ball of a
seminorm. However, the boundedness assumption is not necessary in the
pertinent proofs. Moreover, one could quotient by the norm zero set to obtain
a genuine norm..
###### Lemma 7.1.
Let $\|\cdot\|$ be a seminorm on the space of complex-valued functions
supported on $[N]$. For any such function $f$ and $\varepsilon>0$ there exists
a decomposition $f=f_{str}+f_{unf}$ such that
$\left\|f_{str}\right\|^{*}\leq\varepsilon^{-1}\left\|f\right\|_{2}\quad\text{and}\quad\left\|f_{unf}\right\|\leq\varepsilon\left\|f\right\|_{2}.$
###### Lemma 7.2 ($U^{5}$-control of the dual).
There exists an absolute constant $C$ such that for $N\geq Cq\delta^{-C}$ the
following holds. Let $g_{0},g_{1},f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded
functions, each with support in $[N]$. Suppose that
$\left|\Lambda_{q,N}(g_{0},g_{1},f)\right|\geq\delta\Lambda_{q,N}(1_{[N]}).$
Then, on defining the dual
$G(x):=\mathbb{E}_{y\in[M]}g_{0}(x-qy^{2})g_{1}(x+y-qy^{2}),$ (7.1)
we have
$\sum_{u\in[q]}\left\|G\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}}\gg\delta^{2^{26}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}}.$
###### Proof.
Applying Lemma 7.1 to $f$ with
$\left\|\cdot\right\|:=\left\|\cdot\right\|^{\sharp}_{q}$ as defined in (1.5)
and $\varepsilon:=\tfrac{1}{2}\delta\Lambda_{q,N}(1_{[N]})N^{-1/2}$, we deduce
that
$|\Lambda_{q,N}(g_{0},g_{1},f_{str})|\geq\delta\Lambda_{q,N}(1_{[N]})-|\Lambda_{q,N}(g_{0},g_{1},f_{unf})|\\\
\geq\delta\Lambda_{q,N}(1_{[N]})-\left\|f_{unf}\right\|_{q,N}^{\sharp}\geq\tfrac{1}{2}\delta\Lambda_{q,N}(1_{[N]}).$
We note that our lower bound assumption on $N$ implies that
$\Lambda_{q,N}\left(1_{[N]}\right)\gg 1$. Hence the dual inequality (1.10)
gives
$\delta\ll N^{-1}|\left\langle
f_{str},G\right\rangle|\ll\delta^{-1}\left\|G\right\|^{\sharp}_{q}.$
Invoking Theorem 5.6 yields the result. ∎
Taken together, the work in §§3–6 gives the following.
###### Proof of Theorem 1.6.
Applying Lemma 7.2, we deduce that
$\sum_{u\in[q]}\left\|G\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}}\gg\delta^{2^{26}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}},$
where $G$ is defined as in (7.1).
We now apply Lemma 6.5 three times. The first application gives
$\sum_{u\in[q]}\left\|G\right\|_{U^{4}(u+q\cdot\mathbb{Z})}^{2^{4}}\gg\delta^{2^{40}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{4}(u+q\cdot\mathbb{Z})}^{2^{4}},$
a second replaces $U^{4}$ with $U^{3}$ at the cost of replacing
$\delta^{2^{40}}$ with $\delta^{2^{52}}$. With a final application, we obtain
$\sum_{u\in[q]}\left\|G\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}\gg\delta^{2^{62}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}.$
Let $\eta:=\delta^{2^{62}}$. By the popularity principle, there are at least
$\Omega(\eta q)$ values of $u\in[q]$ for which
$\left\|G\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}\gg\eta\left\|1_{[N]}\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}$.
The inverse theorem for the $U^{2}$-norm then gives the existence of
$\phi(u)\in\mathbb{T}$ for which
$\left|\sum_{x}G(u+qx)e(\phi(u)x)\right|\gg\eta^{1/2}N/q.$ (7.2)
Set $T:=\left\lceil C\eta^{-1/2}N/q\right\rceil$, with $C$ an absolute
constant taken sufficiently large to ensure that, on rounding $\phi(u)$ to the
nearest fraction of the form $t/T$, the inequality (7.2) remains valid.
By Lemma 6.2, for each $u$ satisfying (7.2), there exists a positive integer
$q^{\prime}\ll\eta^{2}$ such that
$\|q^{\prime}q^{2}\phi(h)\|\ll\eta^{-7}q^{3}/N$. By a volume packing argument
similar to that given in the proof of Lemma 6.5, the function $\phi$ is
constant on a proportion of at least $\Omega\bigl{(}\eta^{11}\bigr{)}$ of the
residues $u\in[q]$ satisfying (7.2). Summing over these $u$, then extending
the sum to all of $[q]$, we deduce the existence of $\alpha\in\mathbb{T}$ and
$q^{\prime}\ll\eta^{-2}$ such that
$\|q^{\prime}q^{2}\alpha\|\ll\eta^{-7}q^{3}/N$ and
$\sum_{u\in[q]}\left|\sum_{x}G(u+qx)e(\alpha x)\right|\gg\eta^{12}N.$ (7.3)
Expanding the dual function, there is a 1-bounded function $\psi(u\bmod q)$
such that the left-hand side of the above is equal to
$\sum_{u\in[q]}\psi(u\bmod q)\sum_{x\equiv
u(q)}\mathbb{E}_{y\in[M]}g_{0}(x-qy^{2})g_{1}(x+y-qy^{2})e(\alpha x/q)\\\
=\sum_{x}g_{0}(x)\psi(x\bmod q)e(\alpha
x/q)\mathbb{E}_{y\in[M]}g_{1}(x+y)e(\alpha y^{2}).$ (7.4)
Let us first suppose that $f=g_{0}$, we deal with the case $f=g_{1}$ shortly.
Setting
$\phi(x):=\psi(x\bmod q)e(\alpha x/q)\mathbb{E}_{y\in[M]}g_{1}(x+y)e(\alpha
y^{2}),$
we have $\left\langle f,\overline{\phi}\right\rangle\gg\eta^{12}N$. Our aim is
to show that $\phi$ can be approximated by a local function of the type
claimed in the lemma.
We begin by removing the phase from the expectation over $[M]$, at the cost of
passing to shorter progressions. Let $M^{\prime}\leq M/q^{\prime}q^{2}$ be a
quantity to be determined. If $y\in[M^{\prime}]$ then for any
$m\in[-M,M]\cap\mathbb{Z}$ we have
$\left|e(\alpha(m+q^{\prime}q^{2}y)^{2})-e(\alpha
m^{2})\right|\ll\left\|\alpha\left(2mq^{\prime}q^{2}y+(q^{\prime}q^{2}y)^{2}\right)\right\|\ll
q^{\prime}q^{4}\eta^{-7}M^{\prime}/M.$ (7.5)
Hence, partitioning $\mathbb{Z}$ into progressions $P$ of common difference
$q^{\prime}q^{2}$ and length $M^{\prime}$, there exist phases $\omega_{P}$
such that for any $x\in\mathbb{Z}$ we have
$\left|\mathbb{E}_{y\in[M]}g_{1}(x+y)e(\alpha
y^{2})-M^{-1}\sum_{P}\omega_{P}\sum_{y\in[M]\cap P}g_{1}(x+y)\right|\ll
q^{\prime}q^{4}\eta^{-7}M^{\prime}/M.$ (7.6)
Notice that there are at most $O(M/M^{\prime})$ progressions $P$ such that
$P\cap[M]\neq\emptyset$ (since we are assuming $M^{\prime}\leq
M/q^{\prime}q^{2}$).
Next we show how the phase $e(\alpha x/q)$ is approximately periodic. Suppose
that $z\in[M^{\prime\prime}]$, with $M^{\prime\prime}\leq M^{\prime}/q$ to be
determined. Then for any $x\in\mathbb{Z}$ we have
$\left|e\left(\alpha(x+q^{\prime}q^{3}z)/q\right)-e\left(\alpha
x\right)\right|\ll\left\|\alpha
q^{\prime}q^{2}\right\|M^{\prime\prime}\ll\eta^{-7}q^{3}M^{\prime\prime}/N$
and by a boundary estimate
$\left|\sum_{y\in[M]\cap P}g_{1}(x+q^{\prime}q^{3}z+y)-\sum_{y\in[M]\cap
P}g_{1}(x+y)\right|\ll qM^{\prime\prime}.$
It then follows from a telescoping identity that for all $x\in\mathbb{Z}$ and
$z\in[M^{\prime\prime}]$ we have
$\displaystyle\left|\phi(x+q^{\prime}q^{3}z)-\phi(x)\right|$
$\displaystyle\ll\frac{\eta^{-7}q^{3}M^{\prime\prime}}{N}+\frac{\eta^{-7}q^{\prime}q^{4}M^{\prime}}{M}+\frac{qM^{\prime\prime}}{M}\sum_{\begin{subarray}{c}P\\\
P\cap[M]\neq\emptyset\end{subarray}}1$
$\displaystyle\ll\frac{\eta^{-7}q^{\prime}q^{4}M^{\prime}}{M}+\frac{qM^{\prime\prime}}{M^{\prime}}.$
Taking $M^{\prime}:=c\eta^{19}M/q^{\prime}q^{4}$ and
$M^{\prime\prime}:=c\eta^{12}M^{\prime}/q$ for a sufficiently small absolute
constant $c>0$ we have
$\left|\phi(x+q^{\prime}q^{3}z)-\phi(x)\right|\leq\eta^{12}/C\quad\text{for
all }x\in\mathbb{Z}\text{ and }z\in[M^{\prime\prime}].$ (7.7)
Partitioning $\mathbb{Z}$ into translates $T$ of
$q^{\prime}q^{3}\cdot[M^{\prime\prime}]$ we deduce that
$\sum_{T}\biggl{|}\sum_{x\in T}f(x)\biggr{|}\gg\eta^{12}N.$
Write $\chi(x)$ for the phase of the inner sum when $x\in T$. Then $\chi$ is a
1-bounded local function of modulus $q^{\prime}q^{3}$ and resolution
$\Omega\left((\delta/q)^{O(1)}M\right)$ satisfying
$\sum_{x}f(x)\overline{\chi(x)}\gg\delta^{2^{66}}N,$
as required.
Next we give the argument for when $f=g_{1}$. Returning to (7.4) we have
$\sum_{x}\left|\mathbb{E}_{y\in[M]}f(x+y)e(\alpha y^{2})\right|\gg\eta^{12}N.$
Utilising (7.5) and (7.6), we may partition $\mathbb{Z}$ into progressions $P$
of common difference $q^{\prime}q^{2}$ and length
$M^{\prime}:=c\eta^{19}M/q^{\prime}q^{4}$ such that
$\sum_{x}\sum_{P}\left|\mathbb{E}_{y\in[M]\cap P}f(x+y)\right|\gg\eta^{12}N.$
Since $O(M/M^{\prime})$ of the $P$ intersect $[M]$, the pigeon-hole principle
gives $P^{\prime}:=P\cap[M]$ such that
$\sum_{x}\left|\sum_{y\in P^{\prime}}f(x+y)\right|\gg\eta^{12}NM^{\prime}.$
In particular $|P^{\prime}|\gg\eta^{12}M^{\prime}\gg(q/\delta)^{C}M$.
Partitioning $\mathbb{Z}$ into translates of $P^{\prime}$ of the form
$\mathbb{Z}=\bigsqcup_{i}(a_{i}+P^{\prime}),$
the pigeon-hole principle gives $z\in P^{\prime}$ such that
$\sum_{i}\left|\sum_{y\in P^{\prime}}f(a_{i}+y+z)\right|\gg\eta^{12}N.$
Writing $\chi(x)$ for the phase of the inner sum when $x\in a_{i}+P$ one sees
that $\chi$ is a local function of resolution $\gg(q/\delta)^{C}M$ and modulus
$q^{\prime}q^{2}$ which satisfies $\left\langle
f,\chi\right\rangle\gg\eta^{12}N$. The proof is complete on noting that a
local function of modulus $q^{\prime}q^{2}$ is also a local function of
modulus $q^{\prime}q^{3}$. ∎
## Appendix A Basic theory of the Gowers norms
###### Lemma A.1 (Inverse theorem for the $U^{2}$-norm).
Let $f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in
$[N]$. Then there exists $\alpha\in\mathbb{T}$ such that
$\|f\|_{U^{2}}^{4}\leq N\left|\sum_{x}f(x)e(\alpha x)\right|^{2}.$
###### Proof.
Using the definition of the Fourier transform (1.8), together with
orthogonality of additive characters, we have
$\left\|f\right\|_{U^{2}}^{4}=\int_{\mathbb{T}}\bigl{|}\hat{f}(\alpha)\bigr{|}^{4}\mathrm{d}\alpha\leq\big{\|}\hat{f}\big{\|}_{\infty}^{2}\int_{\mathbb{T}}\bigl{|}\hat{f}(\alpha)\bigr{|}^{2}\mathrm{d}\alpha\leq\big{\|}\hat{f}\big{\|}_{\infty}^{2}N.$
∎
For each $\omega\in\\{0,1\\}^{s}$, let $f_{\omega}:\mathbb{Z}\to\mathbb{C}$ be
a function with finite support. Then we define the _Gowers inner product_ by
$[f_{\omega}]_{U^{s}}:=\sum_{x,h_{1},\dots,h_{s}}\prod_{\omega\in\left\\{0,1\right\\}^{s}}\mathcal{C}^{|\omega|}f_{\omega}(x+\omega\cdot
h).$
Here $\mathcal{C}$ denotes the operation of complex conjugation. Notice that
$[f]_{U^{s}}=\left\|f\right\|_{U^{s}}^{2^{s}}$.
###### Lemma A.2 (Gowers–Cauchy–Schwarz).
For each $\omega\in\\{0,1\\}^{s}$, let $f_{\omega}:\mathbb{Z}\to\mathbb{C}$ be
a function with finite support. Then we have
$[f_{\omega}]_{U^{s}}\leq\prod_{\omega\in\\{0,1\\}^{s}}\|f_{\omega}\|_{U^{s}}.$
###### Proof.
See [9, Exercise 1.3.19]. ∎
###### Lemma A.3 (Phase invariance for $s\geq 2$).
Let $L\in\mathbb{R}[x,h_{1},\dots,h_{s}]$ be a linear form, with $s\geq 2$ and
let $f:\mathbb{Z}\to\mathbb{C}$. Then
$\biggl{|}\sum_{x,h_{1},\dots,h_{s}}\Delta_{h_{1},\dots,h_{s}}f(x)e(L(x,h_{1},\dots,h_{s}))\biggr{|}\leq\left\|f\right\|_{U^{s}}^{2^{s}}.$
###### Proof.
The linear form may be written as
$L(x,h_{1},\dots,h_{s})=\alpha x+\beta_{1}(x+h_{1})+\dots+\beta_{s}(x+h_{s}),$
for some real $\alpha$ and $\beta_{i}$. Write $f_{0}(x):=f(x)e(\alpha x)$,
$f_{e_{i}}(x):=f(x)e(-\beta_{i}x)$ for $i=1,\dots,s$, and for
$\omega\in\left\\{0,1\right\\}^{s}\setminus\left\\{0,e_{1},\dots,e_{s}\right\\}$
set $f_{\omega}:=f$. Then by Gowers–Cauchy–Schwarz we have
$\biggl{|}\sum_{x,h_{1},\dots,h_{s}}\Delta_{h_{1},\dots,h_{s}}f(x)e(L(x,h_{1},\dots,h_{s}))\biggr{|}\leq\prod_{\omega}\left\|f_{\omega}\right\|.$
It therefore suffice to prove that for a phase function $e_{\alpha}:x\mapsto
e(\alpha x)$ $\left\|fe_{\alpha}\right\|_{U^{s}}=\left\|f\right\|_{U^{s}}.$
The latter follows on observing that
$\Delta_{h_{1},\dots,h_{s}}(fe_{\alpha})=\left(\Delta_{h_{1},\dots,h_{s}}f\right)\left(\Delta_{h_{1},\dots,h_{s}}e_{\alpha}\right),$
and for any $x,h_{1},\dots,h_{s}$ with $s\geq 2$ we have
$\Delta_{h_{1},\dots,h_{s}}e_{\alpha}(x)=1.$ ∎
###### Lemma A.4 (Box Cauchy–Schwarz).
Let $\mu_{1},\mu_{2},\mu_{3}$ be probability measures on $\mathbb{Z}$ with the
discrete sigma algebra. If $F_{1},F_{2},F_{3}$ are 1-bounded function on
$\mathbb{Z}^{2}$ and $F$ is a 1-bounded function on $\mathbb{Z}^{3}$ then
$\left|\sum_{x\in\mathbb{Z}^{3}}F_{1}(x_{2},x_{3})F_{2}(x_{1},x_{3})F_{3}(x_{1},x_{2})F(x)\underline{\mu}(x)\right|^{8}\\\
\leq\sum_{x^{0},x^{1}\in\mathbb{Z}^{3}}\prod_{\omega\in\left\\{0,1\right\\}^{3}}\mathcal{C}^{|\omega|}F(x_{1}^{\omega_{1}},x_{2}^{\omega_{2}},x_{3}^{\omega_{3}})\mu_{1}(x_{1}^{0})\mu_{1}(x_{1}^{1})\mu_{2}(x_{2}^{0})\mu_{2}(x_{2}^{1})\mu_{3}(x_{3}^{0})\mu_{3}(x_{3}^{1}).$
## References
* BC [17] J. Bourgain and M.-C. Chang. Nonlinear Roth type theorems in finite fields. Israel J. Math., 221(2):853–867, 2017.
* BL [96] V. Bergelson and A. Leibman. Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc., 9(3):725–753, 1996.
* Gow [10] W. T. Gowers. Decompositions, approximate structure, transference, and the Hahn-Banach theorem. Bull. Lond. Math. Soc., 42(4):573–606, 2010.
* GT [08] B. Green and T. Tao. Quadratic uniformity of the Möbius function. Ann. Inst. Fourier (Grenoble), 58(6):1863–1935, 2008.
* Pel [19] S. Peluse. On the polynomial szemerédi theorem in finite fields. Duke Math. J., 168(5):749–774, 04 2019.
* PP [19] S. Peluse and S. Prendiville. Quantitative bounds in the non-linear Roth theorem. ArXiv e-prints, 2019.
* PP [20] S. Peluse and S. Prendiville. A polylogarithmic bound in the nonlinear Roth theorem. ArXiv e-prints, 2020.
* Pre [17] S. Prendiville. Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case. Discrete Anal., pages 34, Paper No. 5, 2017.
* Tao [12] T. Tao. Higher order Fourier analysis, volume 142 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.
* TZ [16] T. Tao and T. Ziegler. Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors. Discrete Anal., pages 60, Paper No. 13, 2016.
|
2024-09-04T02:54:58.404089 | 2020-03-09T13:16:29 | 2003.04122 | {
"authors": "Sarah Peluse and Sean Prendiville",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26118",
"submitter": "Sean Prendiville",
"url": "https://arxiv.org/abs/2003.04122"
} | arxiv-papers | # A polylogarithmic bound in the nonlinear Roth theorem
Sarah Peluse Mathematical Institute
University of Oxford
UK<EMAIL_ADDRESS>and Sean Prendiville Department of
Mathematics and Statistics
Lancaster University
UK<EMAIL_ADDRESS>
###### Abstract.
We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^{2}$
have at most polylogarithmic density.
###### Contents
1. 1 Introduction
2. 2 Iterating the density increment
3. 3 The cut norm inverse theorem
4. 4 A weak regularity lemma
5. 5 The density increment lemma
6. 6 Global control by major arc Fourier coefficients
7. 7 Longer progressions
## 1\. Introduction
### 1.1. Density bound
In [9] the authors obtained, for the first time, an effective bound for
subsets of $\left\\{1,\dots,N\right\\}$ lacking the nonlinear Roth
configuration $x$, $x+y$, $x+y^{2}$. There it was established that such sets
have cardinality at most $O(N/(\log\log N)^{c})$, where $c>0$ is an absolute
constant. The key breakthrough of [9] was a “local $U^{1}$-control” result,
from which a bound for sets lacking the nonlinear Roth configuration follows
via standard methods. Here, we combine this local $U^{1}$-control result with
a more sophisticated argument to remove a logarithm from the bound of [9].
###### Theorem 1.1 (Density bound).
There exists an absolute constant $c>0$ such that the following holds. Suppose
that $A\subset\left\\{1,\dots,N\right\\}$ lacks configurations of the form
$x,\ x+y,\ x+y^{2}\qquad(y\neq 0).$ (1.1)
Then
$|A|=O\left(N/(\log N)^{c}\right).$
A careful analysis shows that the exponent $c=2^{-150}$ is permissible, where
150 represents the combined number of times we utilise the Cauchy–Schwarz
inequality in [9] and this paper
### 1.2. Major arc correlation
The techniques which yield Theorem 1.1 also allow us to show, in a
quantitatively effective manner, that the major arc Fourier coefficients of a
set determine how many nonlinear Roth configurations (1.1) the set contains.
###### Theorem 1.2 (Major-arc control).
Let $\delta>0$ and $f,g,h:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with
support in $\left\\{1,\dots,N\right\\}$. Suppose that
$\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta
N^{3/2}.$
Then either $N\ll\delta^{-O(1)}$, or there is a frequency
$\alpha\in\mathbb{R}$ and a positive integer $q\ll\delta^{-O(1)}$ such
that111Here $\left\|\cdot\right\|$ denotes the distance to the nearest
integer, and $e(\alpha):=e^{2\pi i\alpha}$. For our conventions regarding
asymptotic notation see §1.5. $\left\|q\alpha\right\|\ll\delta^{-O(1)}/N$ and
$\left|\sum_{x\in\mathbb{Z}}h(x)e(\alpha x)\right|\gg\delta^{O(1)}N.$
In the nomenclature of [14], the major arc linear phases are the only
obstructions to uniformity for the nonlinear Roth configuration. We emphasise
that Theorem 1.2 is not used in the proof of Theorem 1.1.
The major arc Fourier coefficients of a subset of $\\{1,\dots,N\\}$
essentially measure its distribution in arithmetic progressions of common
difference $\ll 1$ and length $\gg N$. To illustrate this, the following
definition is useful.
###### Definition 1.3 (Local function).
We call a function $\phi:\mathbb{Z}\to\mathbb{C}$ a _local function of
resolution $M$ and modulus $q$_ if there exists a partition of $\mathbb{Z}$
into intervals of length $M$ such that $\phi$ is constant on the intersection
of every such interval with every congruence class mod $q$.
###### Corollary 1.4 (Local control of the nonlinear term).
Let $\delta>0$ and $f,g,h:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with
support in $\left\\{1,\dots,N\right\\}$. Suppose that
$\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta
N^{3/2}.$
Then either $N\ll\delta^{-O(1)}$, or there is a 1-bounded local function
$\phi$ of resolution $M\gg\delta^{O(1)}N$ and modulus $q\ll\delta^{-O(1)}$
such that
$\left|\sum_{x\in\mathbb{Z}}h(x)\phi(x)\right|\gg\delta^{O(1)}N.$
One cannot hope to prove that the functions $f$ and $g$ above also correlate
globally with local functions, as the following example illustrates. For any
positive integers $x_{1},x_{2}\leqslant N^{1/2}$, set
$f\left(x_{1}+(x_{2}-1)\left\lfloor
N^{1/2}\right\rfloor\right)=\begin{cases}1&\text{ if }x_{2}\equiv
0\pmod{4},\\\ 0&\text{ if }x_{2}\equiv 1\pmod{4},\\\ -1&\text{ if }x_{2}\equiv
2\pmod{4},\\\ 0&\text{ if }x_{2}\equiv 3\pmod{4};\end{cases}$
and set $f(x)=0$ everywhere else. Taking $g:=f$ and $h:=1_{\\{1,\dots,N\\}}$,
one can check that either $N\ll 1$ or
$\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\gg N^{3/2}.$
However, for any arithmetic progression $P\subset\\{1,\dots,N\\}$, we have
$\left|\sum_{x\in P}f(x)\right|\ll N^{1/2}.$
Hence, for any 1-bounded local function $\phi$ of resolution $\geqslant\delta
N$ and modulus $\leqslant\delta^{-1}$, the triangle inequality gives the
discorrelation
$\left|\sum_{x\in\mathbb{Z}}f(x)\phi(x)\right|\ll\delta^{-2}N^{1/2}.$
This example is a local obstruction coming from the real numbers: the nature
of our counting operator means that we cannot disentangle possible
correlations between the $f$ and $g$ functions on subintervals of length
$N^{1/2}$. We can, however, show that these are the only other possible
obstructions to uniformity.
###### Theorem 1.5 (Local control of all terms).
Let $\delta>0$ and $f_{1},f_{2},f_{3}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded
functions with support in $\left\\{1,\dots,N\right\\}$. Suppose that
$\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f_{1}(x)f_{2}(x+y)f_{3}(x+y^{2})\right|\geqslant\delta
N^{3/2}.$
Then either $N\ll\delta^{-O(1)}$, or for each $i=1,2,3$ there is a 1-bounded
local function $\phi_{i}$ of resolution $\gg\delta^{O(1)}N^{1/2}$ and modulus
$q_{i}\ll\delta^{-O(1)}$ such that
$\left|\sum_{x\in\mathbb{Z}}f_{i}(x)\phi_{i}(x)\right|\gg\delta^{O(1)}N.$
###### Proof.
This is an immediate consequence of Corollary 1.4 and Lemma 3.2. ∎
### 1.3. Longer polynomial progressions
In analogy with the first author’s generalisation [8] of [9], it is natural to
ask whether the methods of this paper yield polylogarithmic bounds for sets of
integers lacking longer progressions
$x,\ x+P_{1}(y),\ \dots,\ x+P_{m}(y),$ (1.2)
where the $P_{i}\in\mathbb{Z}[y]$ have zero constant term and $\deg
P_{1}<\dots<\deg P_{m}$.
As was mentioned above, the key input to this paper is the local
$U^{1}$-control result [9, Theorem 7.1]. Replacing this with [8, Theorem 3.3],
our argument generalises in a straightforward manner to yield polylogarithmic
bounds for subsets of $\\{1,\dots,N\\}$ lacking (1.2) when $m=2$, that is, for
all three-term polynomial progressions with distinct degrees and zero constant
term.
Obtaining polylogarithmic bounds for longer polynomial progressions requires
an additional idea. We sketch a strategy in §7, which relies on obtaining an
appropriate generalisation of [8, Theorem 3.3], a generalisation that would
require re-running the majority of the arguments therein.
### Acknowledgements
S. Peluse is supported by the NSF Mathematical Sciences Postdoctoral Research
Fellowship Program under Grant No. DMS-1903038
### 1.4. An outline of our argument
Effective Szemerédi-type theorems are commonly proved via a density increment
strategy, the prototypical example being the proof of Roth’s theorem [11] on
three-term arithmetic progressions. This strategy begins with a set
$A\subset\\{1,\dots,N\\}$ of density $\delta:=|A|/N$ that lacks the
configuration in question. It then proceeds to show that there is a
substructure $S\subset\\{1,\dots,N\\}$ on which $A$ has increased density
$\delta+\Omega_{\delta}(1)$. One then hopes to iterate the argument with
$A\cap S$ in place of $A$ and $S$ in place of $\\{1,\dots,N\\}$.
One avenue to obtaining polylogarithmic bounds in a Szemerédi-type theorem is
to obtain a constant proportion density increment $\delta+\Omega(\delta)$ on a
substructure $S$ of polynomial size $|S|\approx N^{\Omega(1)}$. This was
accomplished for three-term arithmetic progressions by Heath–Brown [7] and
Szemerédi [13] (in fact, they were able to handle a smaller lower bound on
$|S|$).
An alternative strategy for obtaining polylogarithmic bounds is to obtain the
weaker polynomial increment $\delta+\Omega(\delta^{O(1)})$, yet on a _dense_
or _global_ substructure $S$, that is, a substructure of size
$|S|\geqslant\exp(-O(\delta^{-O(1)}))N$. This was accomplished by Sárközy [12]
for the configuration $x,x+y^{2}$ and for three-term arithmetic progressions
by Bourgain [2].
Both of these strategies are achievable for the nonlinear Roth configuration.
The global structure strategy is perhaps the most natural, and may be
accomplished by utilising a generalisation of Theorem 1.2. In this note we do
not pursue this, and instead give details for a constant-proportion density
increment, as our argument is somewhat cleaner in this form.
More specifically, we show that if $A\subset\left\\{1,\dots,N\right\\}$ has
density $\delta$ and lacks nontrivial configurations of the form
$x,x+y,x+y^{2}$, then there exists an arithmetic progression $P$ of length
$|P|\gg\delta^{O(1)}N^{1/2}$ and common difference $q\ll\delta^{-O(1)}$ such
that we have the density increment
$\frac{|A\cap P|}{|P|}\geqslant(1+\Omega(1))\frac{|A|}{N}.$ (1.3)
As outlined in [9], the ‘almost bounded’ size of $q$ allows us to iterate this
procedure. (In [9], we obtain the weaker density increment
$(1+\Omega(\delta^{O(1)}))|A|/N$, which leads to the extra logarithm appearing
in the bound there.)
We obtain the constant-proportion increment (1.3) by combining the local
$U^{1}$-control result of [9] with a strategy of Heath–Brown [7] and Szemerédi
[13], which has a very robust formulation due to Green and Tao [6]. To
accomplish this, we first give a structural characterisation of sets lacking
the nonlinear Roth configuration (this is Lemma 3.3, whose essence is captured
in the weaker Theorem 1.5). These sets resemble the level sets of the product
of a function that is constant on intervals of length $N^{1/2}$ and a function
that is constant on congruence classes modulo a bounded $q$.
Having obtained such a structural characterisation, an energy increment
procedure closely following [6] allows us to approximate an arbitrary set of
integers by these level sets, up to an error that does not contribute
substantially to the count of nonlinear Roth configurations. A combinatorial
argument then allows us to deduce that our set must have a substantial density
increment on one of these level sets, of the form $\delta+\Omega(\delta)$. As
a result, our density increment procedure requires only
$\log(\delta^{-1})+O(1)$ iterations, compared with the $O(\delta^{-O(1)})$
required in [9], and this yields the polylogarithmic improvement over our
previous density increment iteration.
The remainder of this paper is organized as follows. We derive Theorem 1.1 in
§2 via a density increment iteration. Our deduction uses a density increment
lemma that is established in §§3–5. We prove Theorem 1.2 and Corollary 1.4 in
§6.
### 1.5. Notation
#### 1.5.1. Standard conventions
We use $\mathbb{N}$ to denote the positive integers. For a real number
$X\geqslant 1$, write $[X]=\\{1,2,\ldots,\left\lfloor X\right\rfloor\\}$. A
complex-valued function is said to be _1-bounded_ if the modulus of the
function does not exceed 1.
We use counting measure on $\mathbb{Z}$, so that for
$f,g:\mathbb{Z}\to\mathbb{C}$, we have
$\left\|f\right\|_{\ell^{p}}:=\biggl{(}\sum_{x}|f(x)|^{p}\biggr{)}^{\frac{1}{p}},\
\left\langle f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)},\ \text{and}\
(f*g)(x)=\sum_{y}f(y)g(x-y).$
Any sum of the form $\sum_{x}$ is to be interpreted as a sum over
$\mathbb{Z}$. The _support_ of $f$ is the set
$\mathrm{supp}(f):=\left\\{x\in\mathbb{Z}:f(x)\neq 0\right\\}$. We write
$\left\|f\right\|_{\infty}$ for $\sup_{x\in\mathbb{Z}}|f(x)|$.
We use Haar probability measure on $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, so
that for measurable $F:\mathbb{T}\to\mathbb{C}$, we have
$\left\|F\right\|_{L^{p}}:=\biggl{(}\int_{\mathbb{T}}|F(\alpha)|^{p}d\alpha\biggr{)}^{\frac{1}{p}}=\biggl{(}\int_{0}^{1}|F(\alpha)|^{p}d\alpha\biggr{)}^{\frac{1}{p}}.$
We write $\left\|\alpha\right\|_{\mathbb{T}}$ for the distance from
$\alpha\in\mathbb{R}$ to the nearest integer
$\min_{n\in\mathbb{Z}}|\alpha-n|.$ This remains well-defined on $\mathbb{T}$.
We define the Fourier transform of $f:\mathbb{Z}\to\mathbb{C}$ by
$\hat{f}(\alpha):=\sum_{x}f(x)e(\alpha x)\qquad(\alpha\in\mathbb{T}),$ (1.4)
when this makes sense. Here $e(\alpha)$ stands for $e^{2\pi i\alpha}$.
For a finite set $S$ and function $f:S\to\mathbb{C}$, denote the average of
$f$ over $S$ by
$\mathbb{E}_{s\in S}f(s):=\frac{1}{|S|}\sum_{s\in S}f(s).$
For a complex-valued function $f$ and positive-valued function $g$, write
$f\ll g$ or $f=O(g)$ if there exists a constant $C$ such that $|f(x)|\leq
Cg(x)$ for all $x$. We write $f=\Omega(g)$ if $f\gg g$. We subscript this
notation when the implicit constant may depend on the subscripted parameters.
#### 1.5.2. Local conventions
Up to normalisation, all of the above are widely used in the literature. Next,
we list notation specific to our paper. We have tried to minimise this in
order to aid the casual reader.
The quantity $(N/q)^{1/2}$ appears repeatedly, where $N$ and $q$ are integers
fixed throughout the majority of our paper. We therefore adopt the convention
that
$M:=\left\lfloor\sqrt{N/q}\right\rfloor.$ (1.5)
Assuming this, define the _counting operator_ on the functions
$f_{i}:\mathbb{Z}\to\mathbb{C}$ by
$\Lambda_{q,N}(f_{0},f_{1},f_{2}):=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+y)f_{2}(x+qy^{2}).$
(1.6)
When $f_{0}=f_{1}=f_{2}=f$, we simply write $\Lambda_{q,N}(f)$ for
$\Lambda_{q,N}(f_{0},f_{1},f_{2})$.
For a real parameter $H\geqslant 1$, we use $\mu_{H}:\mathbb{Z}\to[0,1]$ to
represent the following normalised Fejér kernel
$\mu_{H}(h):=\frac{1}{\left\lfloor
H\right\rfloor}\left(1-\frac{|h|}{\left\lfloor
H\right\rfloor}\right)_{+}=\frac{(1_{[H]}*1_{-[H]})(h)}{\left\lfloor
H\right\rfloor^{2}}.$ (1.7)
This is a probability measure on $\mathbb{Z}$ with support in the interval
$(-H,H)$.
## 2\. Iterating the density increment
In this section we prove Theorem 1.1 using the following lemma, which we will
devote §§3–5 to proving.
###### Lemma 2.1 (Density increment lemma).
Let $q\leqslant N$ be positive integers and $\delta>0$. Suppose that
$A\subset[N]$ satisfies $|A|\geqslant\delta N$ and lacks the configuration
$x,\ x+y,\ x+qy^{2}\qquad(y\neq 0).$ (2.1)
Then either $N\ll(q/\delta)^{O(1)}$ or there exists
$q^{\prime}\leqslant\exp\left(O\left(\delta^{-O(1)}\right)\right)$ and
$N^{\prime}\geqslant
q^{-O(1)}\exp\left(-O\left(\delta^{-O(1)}\right)\right)N^{1/2}$ such that, for
some $a\in\mathbb{Z}$, we have
$|A\cap(a+qq^{\prime}\cdot[N^{\prime}])|\geqslant(1+\Omega(1))\delta
N^{\prime}.$ (2.2)
###### Proof of Theorem 1.1 given Lemma 2.1.
This is the same as the proof of [9, Theorem 1.1], but using the improved
density increment lemma above in place of the density increment lemma of [9].
Note first that if $A$ lacks the configuration (2.1), then the set
$\\{x:a+qq^{\prime}x\in A\\},$
lacks configurations of the form
$x,\ x+y,\ x+q^{2}q^{\prime}y^{2}\qquad(y\neq 0).$
Let $A\subset[N]$ have size $\delta N$, and suppose that it has no non-linear
Roth configurations (1.1). Setting $A_{0}:=A$, $N_{0}:=N$ and $q_{0}=1$, let
us suppose we have a sequence of tuples $(A_{i},N_{i},q_{i})$ for
$i=0,1,\dots,n$ that each satisfy the following:
1. (i)
$A_{i}$ lacks configurations of the form
$x,\ x+y,\ x+q_{0}^{2^{i}}q_{1}^{2^{i-1}}\dotsm
q_{i-1}^{2}q_{i}y^{2}\qquad(y\neq 0).$
2. (ii)
$q_{i}\leqslant\exp\left(O\left(\delta^{-O(1)}\right)\right)$;
3. (iii)
$A_{i}\subset[N_{i}]$ and for $i\geqslant 1$ we have
$\frac{|A_{i}|}{N_{i}}\geqslant(1+c)\frac{|A_{i-1}|}{N_{i-1}},$
where $c=\Omega(1)$ is a positive absolute constant;
4. (iv)
for $i\geqslant 1$ we have the lower bound
$N_{i}\geqslant\frac{N_{i-1}^{1/2}}{\left(q_{0}^{2^{i-1}}\dotsm
q_{i-1}\exp\left(\delta^{-O(1)}\right)\right)^{O(1)}}.$
Applying Lemma 2.1 with $q=q_{0}^{2^{i}}q_{1}^{2^{i-1}}\dotsm
q_{i-1}^{2}q_{i}$, either
$N_{n}\ll\left(q_{0}^{2^{n}}q_{1}^{2^{n-1}}\dotsm
q_{n-1}^{2}q_{n}/\delta\right)^{O(1)},$ (2.3)
or we may obtain $(A_{n+1},N_{n+1},q_{n+1})$ satisfying conditions (i)–(iv).
If (2.3) holds, then our iterative process terminates at stage $n$.
If the number of iterations $n$ is at least $c^{-1}$, then the density of
$A_{n}$ on $[N_{n}]$ is at least $2\delta$. After an additional
$\tfrac{1}{2}c^{-1}$ iterations, the density is at least $4\delta$. Hence if
the number of iterations is at least
$\left\lceil
c^{-1}\right\rceil+\left\lceil\tfrac{1}{2}c^{-1}\right\rceil+\left\lceil\tfrac{1}{4}c^{-1}\right\rceil+\dots+\left\lceil\tfrac{1}{2^{m-1}}c^{-1}\right\rceil,$
then the density is at least $2^{m}\delta$. The density therefore exceeds one
if the number of iterations exceeds $2c^{-1}+\log_{2}(\delta^{-1})$. Since
this cannot happen, it follows that there exists
$n\leqslant\log_{2}(\delta^{-1})+O(1)$ such that the procedure terminates at
stage $n$.
At the point of termination, the smallness assumption (2.3) must hold, so that
$N_{n}\leqslant\exp\left(O\Bigl{(}\delta^{-O(1)}\Bigr{)}\right).$
On the other hand, iteratively applying the lower bound (iv), we have
$\begin{split}N_{n}&\geqslant\frac{N_{n-1}^{1/2}}{\left(q_{0}^{2^{n-1}}\dotsm
q_{n-1}\exp\left(\delta^{-O(1)}\right)\right)^{O(1)}}\\\ &\geqslant
N^{1/2^{n}}\left[q_{0}^{2^{n-1}}\dotsm
q_{n-1}\exp\left(\delta^{-O(1)}\right)\right]^{-O(1+\frac{1}{2}+\frac{1}{4}+\dots+2^{1-n})}\\\
&\gg\exp\left(-O\left(\delta^{-O(1)}\right)\right)N^{\Omega(\delta)},\end{split}$
where we use the upper bound (ii) on the $q_{i}$’s, together with
$n\leqslant\log_{2}(\delta^{-1})+O(1)$. Taking a logarithm and comparing upper
and lower bounds for $N_{n}$ gives $\log N\ll\delta^{-O(1)},$ which yields the
bound claimed in Theorem 1.1. ∎
## 3\. The cut norm inverse theorem
The first step of the proof of Lemma 2.1 is to use the main technical result
of [9] to prove an inverse theorem for the cut norm associated to
$\Lambda_{q,N}$, which we now define.
###### Definition 3.1 (Cut norm).
For positive integers $q\leqslant N$, we define the _cut norm_ of
$f:\mathbb{Z}\to\mathbb{C}$ by
$\left\|f\right\|_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,\
|\Lambda_{q,N}(g_{1},f,g_{2})|,\ |\Lambda_{q,N}(g_{1},g_{2},f)|\\},$ (3.1)
where the supremum is taken over all 1-bounded functions
$g_{i}:[N]\to\mathbb{C}$. We note that, in spite of our nomenclature, this is
not a norm, but a seminorm. One could remedy this by summing over $y\geqslant
0$ in the counting operator (1.6).
Initially, the cut norm is too restrictive for us, so we begin by working with
the weaker quantity
$\left\|f\right\|^{\flat}_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,|\Lambda_{q,N}(g_{1},f,g_{2})|:|g_{i}|\leqslant
1\text{ and }\mathrm{supp}(g_{i})\subset[N]\\},$ (3.2)
which we refer to as the _partial cut norm_.
The following lemma is simply a rephrasing of [9, Theorem 7.1], which is the
technical heart of that paper. See Definition 1.3 for the meaning of ‘local
function’.
###### Lemma 3.2 (Partial cut norm inverse theorem).
Let $q\leqslant N$ be positive integers, $\delta>0$, and
$f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$.
Suppose that
$\left\|f\right\|^{\flat}_{q,N}\geqslant\delta.$
Then either $N\ll(q/\delta)^{O(1)}$ or there exists a 1-bounded local function
$\phi$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, modulus $qq^{\prime}$ for
some $q^{\prime}\ll\delta^{-O(1)}$, and such that
$\sum_{x\in[N]}f(x)\phi(x)\gg\delta^{O(1)}N.$
###### Proof.
By compactness, there exist 1-bounded functions $g_{1},g_{2}:[N]\to\mathbb{C}$
such that either $|\Lambda_{q,N}(f,g_{1},g_{2})|\geqslant\delta$ or
$|\Lambda_{q,N}(g_{1},f,g_{2})|\geqslant\delta.$ In the latter case, we may
apply [9, Theorem 7.1] to deduce that there exist positive integers
$q^{\prime}\ll\delta^{-O(1)}$ and $N^{\prime}\gg(\delta/q)^{O(1)}N^{1/2}$ such
that
$\sum_{x}\left|\sum_{y\in[N^{\prime}]}f(x+qq^{\prime}y)\right|\gg\delta^{O(1)}NN^{\prime}.$
In the former case, the reader may check that the argument of [9, Theorem 7.1]
delivers the same conclusion222For details see the second author’s exposition
[10]..
To ease notation, write $Q:=qq^{\prime}$. Partitioning the integers into
arithmetic progressions of length $N^{\prime}$ and common difference $Q$ gives
$\delta^{O(1)}NN^{\prime}\ll\sum_{z\in[N^{\prime}]}\sum_{u\in[Q]}\sum_{x\in\mathbb{Z}}\left|\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\right|\\\
\leqslant
N^{\prime}\max_{z}\sum_{u\in[Q]}\sum_{x\in\mathbb{Z}}\left|\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\right|.$
Defining $\psi_{z}(u,x)$ to be the conjugate phase of the inner sum, we deduce
the existence of $z$ for which
$\displaystyle\delta^{O(1)}N\ll\sum_{u\in[Q]}\sum_{x}\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\psi_{z}(u,x).$
The result follows on noting that every integer has a unique representation of
the form $QN^{\prime}x+u+Qy$ with $u\in[Q]$, $x\in\mathbb{Z}$ and
$y\in[N^{\prime}]$. Hence the map
$Qz+QN^{\prime}x+u+Qy\mapsto\psi_{z}(u,x)$
is a local function of resolution $QN^{\prime}$ and modulus $Q$. ∎
Now we can prove an inverse theorem for the cut norm itself.
###### Lemma 3.3 (Full cut norm inverse theorem).
Let $q\leqslant N$ be positive integers, $\delta>0$, and
$f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$.
Suppose that
$\left\|f\right\|_{q,N}\geqslant\delta.$
Then either $N\ll(q/\delta)^{O(1)}$ or there exist 1-bounded local functions
$\phi_{1}$ and $\phi_{2}$, of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and
moduli $qq_{1}$ and $qq_{2}$, respectively, for some
$q_{1},q_{2}\ll\delta^{-O(1)}$ such that
$\left|\sum_{x\in[N]}f(x)\phi_{1}(x)\phi_{2}(x)\right|\gg\delta^{O(1)}N.$
(3.3)
###### Proof.
By the definition of the cut norm (3.1) and Lemma 3.2, we may assume that
there are 1-bounded functions $g,h:[N]\to\mathbb{C}$ such that
$|\Lambda_{q,N}(g,h,f)|\geqslant\delta.$ (3.4)
Recalling that $M:=\lfloor\sqrt{N/q}\rfloor$, define the dual function
$F(x):=\mathbb{E}_{y\in[M]}h(x+y)f(x+qy^{2}).$
Re-parametrising (3.4) and applying the Cauchy–Schwarz inequality, we have
that
$\delta^{2}\leqslant\mathbb{E}_{x\in[N]}F(x)^{2}=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}F(x)h(x+y)f(x+qy^{2}).$
Recalling the definition of the partial cut norm (3.2), we deduce that
$\left\|F\right\|_{q,N}^{\flat}\geqslant\delta^{2}.$
Applying the partial cut norm inverse theorem (Lemma 3.2), there exists a
1-bounded local function $\phi_{1}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{1}$ for some
$q_{1}\ll\delta^{-O(1)}$ such that
$\left|\sum_{x\in[N]}F(x)\phi_{1}(x)\right|\gg\delta^{O(1)}N.$
Thus
$|\Lambda_{q,N}(\phi_{1},h,f)|\gg\delta^{O(1)}.$
We now re-run our argument on $h$ instead of $f$, deducing the existence of a
1-bounded local function $\phi_{2}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{2}$ for some
$q_{2}\ll\delta^{-O(1)}$ such that
$|\Lambda_{q,N}(\phi_{1},\phi_{2},f)|\gg\delta^{O(1)}.$
Expanding the counting operator and taking a maximum over $y\in[M]$ gives
$\displaystyle\delta^{O(1)}NM$
$\displaystyle\ll\left|\sum_{y\in[M]}\sum_{x}f(x)\phi_{1}(x-qy^{2})\phi_{2}(x-qy^{2}+y)\right|$
$\displaystyle\leqslant
M\left|\sum_{x}f(x)\tilde{\phi}_{1}(x)\tilde{\phi}_{2}(x)\right|,$
where both $\tilde{\phi}_{i}$ are 1-bounded local functions of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and moduli $qq_{i}$ for some
$q_{i}\ll\delta^{-O(1)}$. ∎
## 4\. A weak regularity lemma
Much of the material is this section is standard, and closely follows the
expositions in Green [4] and Green–Tao [6]. To simplify the exposition of
later arguments, while the factors in [4] and [6] are $\sigma$-algebras, our
factors will be the set of atoms of certain $\sigma$-algebras (which can
obviously be recovered by taking the $\sigma$-algebra generated by the set of
atoms).
###### Definition 4.1 (Factor).
We define a _factor_ $\mathcal{B}$ of $[N]$ to be a partition of $[N]$, so
that $[N]=\sqcup_{B\in\mathcal{B}}B$. We say that a factor
$\mathcal{B}^{\prime}$ _refines_ $\mathcal{B}$ if every element of
$\mathcal{B}$ is a union of elements of $\mathcal{B}^{\prime}$. The _join_
$\mathcal{B}_{1}\vee\dots\vee\mathcal{B}_{d}$ of factors
$\mathcal{B}_{1},\dots,\mathcal{B}_{d}$ is the factor formed by taking the
$d$-fold intersections of the elements of $\mathcal{B}_{1}$, …,
$\mathcal{B}_{d}$, that is,
$\mathcal{B}_{1}\vee\dots\vee\mathcal{B}_{d}:=\\{B_{1}\cap\dots\cap
B_{d}:B_{i}\in\mathcal{B}_{i}\text{ for }i=1,\dots,d\\}.$
###### Definition 4.2 (Measurability, projection).
Given a factor $\mathcal{B}$, we say that a function $f:[N]\to\mathbb{C}$ is
_$\mathcal{B}$ -measurable_ if it is constant on the elements of
$\mathcal{B}$.
Define the _projection_ of any function $f:[N]\to\mathbb{C}$ onto
$\mathcal{B}$ by
$\Pi_{\mathcal{B}}f(x)=\mathbb{E}_{y\in B_{x}}f(y),$ (4.1)
where $B_{x}$ is the element of $\mathcal{B}$ that contains $x$. Notice that
$\Pi_{\mathcal{B}}f$ is $\mathcal{B}$-measurable, and is just the conditional
expectation of $f$ with respect to the $\sigma$-algebra generated by the
elements of $\mathcal{B}$.
We record some well-known properties of the projection operator
$\Pi_{\mathcal{B}}$ (that is, properties of conditional expectation) in the
next lemma.
###### Lemma 4.3 (Properties of the projection operator).
1. (i)
The operator $\Pi_{\mathcal{B}}$ linearly projects onto the space of
$\mathcal{B}$-measurable functions.
2. (ii)
$\Pi_{\mathcal{B}}$ is self-adjoint with respect to the inner product
$\left\langle
f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)}\qquad(f,g:[N]\to\mathbb{C}),$
so that $\left\langle
f,\Pi_{\mathcal{B}}g\right\rangle=\left\langle\Pi_{\mathcal{B}}f,g\right\rangle$.
3. (iii)
If $\mathcal{B}^{\prime}$ is a refinement of $\mathcal{B}$ then
$\Pi_{\mathcal{B}^{\prime}}\Pi_{\mathcal{B}}f=\Pi_{\mathcal{B}}f.$
4. (iv)
If $\mathcal{B}^{\prime}$ refines $\mathcal{B}$ then $\Pi_{\mathcal{B}}f$ is
orthogonal to $\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f$.
###### Proof.
Inspecting the formula (4.1) reveals that $\Pi_{\mathcal{B}}$ is linear, that
$\Pi_{\mathcal{B}}f$ is constant on elements of $\mathcal{B}$, and that if $f$
itself is constant on elements of $\mathcal{B}$, then $\Pi_{\mathcal{B}}f=f$.
This establishes (i).
Interchanging the order of summation gives
$\begin{split}\left\langle
f,\Pi_{\mathcal{B}}g\right\rangle=\sum_{B\in\mathcal{B}}|B|^{-1}\sum_{x,y\in
B}f(x)\overline{g(y)}=\left\langle\Pi_{\mathcal{B}}f,g\right\rangle.\end{split}$
This proves that $\Pi_{\mathcal{B}}$ is self-adjoint.
The first refinement property follows from the fact that $\Pi_{\mathcal{B}}f$
is $\mathcal{B}^{\prime}$-measurable.
We utilise self-adjointness of $\Pi_{\mathcal{B}}$ and the first refinement
property to conclude that
$\begin{split}\left\langle\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}}f-\Pi_{\mathcal{B}^{\prime}}f\right\rangle&=\left\langle\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}}f-f\right\rangle=\left\langle
f,\Pi_{\mathcal{B}}f-\Pi_{\mathcal{B}}f\right\rangle=0.\end{split}$
∎
Now we describe the particular type of factors that will be relevant to us.
###### Definition 4.4 (Local factor).
A _simple real factor_ of resolution $M$ is a factor of $[N]$ obtained by
partitioning $\mathbb{R}$ into intervals all of length $M$.
A _simple congruence factor_ of modulus $q$ is the factor of $[N]$ obtained by
partitioning into congruence classes mod $q$.
We say that $\mathcal{B}$ is a _simple local factor_ of resolution $M$ and
modulus $q$ if it is the join of a simple real factor of resolution $M$ and a
simple congruence factor of modulus $q$. Notice that $\mathcal{B}$ is a simple
local factor if and only if it consists of the level sets of a local function
(Definition 1.3) of resolution $M$ and modulus $q$.
A _local factor_ of dimension $d$, resolution $M$ and modulus $q$ is the join
of $d$ simple local factors $\mathcal{B}_{i}$, each of resolution $M_{i}$ and
modulus $q_{i}$, where $M_{i}\geqslant M$ and
$q=\mathrm{lcm}[q_{1},\dots,q_{d}]$.
Local factors of large resolution and small modulus and dimension necessarily
contain few sets. This fact will be useful later in the proof of Lemma 2.1.
###### Lemma 4.5 (Size of a local factor).
If $\mathcal{B}$ is a local factor of dimension $d$, resolution $M$, and
modulus $q$, then
$|\mathcal{B}|\leqslant qd\left(\frac{N}{M}+2\right).$
###### Proof.
By the definition of a local factor, it suffices to bound the size of the join
of $d$ simple real factors, and then bound the size of the join of $d$ simple
congruence factors. The product of these two numbers gives us our final bound.
Joining $d$ congruence simple factors with moduli $q_{1},\dots,q_{d}$ results
in another congruence simple factor of modulus
$q=\mathrm{lcm}[q_{1},\dots,q_{d}]$. The number of parts in such a partition
is $q$.
The join of $d$ simple real factors partitions $[N]$ into intervals. The upper
endpoint of each of these intervals is either equal to $N$ or is equal to an
endpoint of an interval in one of the original simple real factors. For a
simple real factor of resolution $M$, at most $1+N/M$ upper endpoints lie in
$[1,N)$. Hence the number of intervals in the join of $d$ simple real factors
of resolutions $M_{1}$, …, $M_{d}$ is at most
$2d+N(M_{1}^{-1}+\dots+M_{d}^{-1})$.∎
We now prove a weak regularity lemma for the cut norm via an energy increment
argument.
###### Lemma 4.6 (Weak regularity).
Let $q\leqslant N$ be positive integers and $\delta>0$. Either
$N\ll(q/\delta)^{O(1)}$, or for any function $f:[N]\to[0,1]$ there exists a
local factor $\mathcal{B}$ of dimension $d\ll\delta^{-O(1)}$, resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$, and modulus $qq^{\prime}$ for some
$q^{\prime}\leqslant O\left(1/\delta\right)^{O(d)}$ such that
$\left\|f-\Pi_{\mathcal{B}}f\right\|_{q,N}\leqslant\delta.$ (4.2)
###### Proof.
We run an energy increment argument, initialising at stage $0$ with the
trivial factor $\mathcal{B}_{0}:=\left\\{[N]\right\\}$. Suppose that at stage
$d$ of this iteration we have a local factor $\mathcal{B}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$, dimension at most $2d$, and modulus
$qq^{\prime}$ for some $q^{\prime}\leqslant O(1/\delta)^{O(d)}$. In addition,
suppose that we have the energy lower bound
$\left\|\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}\gg d\delta^{O(1)}N.$ (4.3)
With these assumptions in place, we query if the following holds
$\left\|f-\Pi_{\mathcal{B}}f\right\|_{q,N}\leqslant\delta.$ (4.4)
If so, then the process terminates. If not, we show how our iteration may
proceed to stage $d+1$.
Applying the cut norm inverse theorem (Lemma 3.3), we conclude that there
exist 1-bounded local functions $\phi_{i}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{i}$ for some
$q_{i}\leqslant\delta^{-O(1)}$ such that
$\left|\left\langle
f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|=\left|\sum_{x\in[N]}(f-\Pi_{\mathcal{B}}f)(x)\phi_{1}(x)\phi_{2}(x)\right|\gg\delta^{O(1)}N.$
Let $\mathcal{B}^{\prime}$ denote the join of $\mathcal{B}$ and the simple
local factors generated by $\phi_{1}$ and $\phi_{2}$, so that
$\mathcal{B}^{\prime}$ is a local factor of dimension at most $2(d+1)$,
resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq^{\prime\prime}$ for
some $q^{\prime\prime}\leqslant q^{\prime}q_{1}q_{2}\leqslant
O(1/\delta)^{O(d+1)}$. Since $\phi_{1}\phi_{2}$ is
$\mathcal{B}^{\prime}$-measurable, we can use the properties listed in Lemma
4.3 together with the Cauchy–Schwarz inequality to deduce that
$\begin{split}\left|\left\langle
f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|&=\left|\left\langle
f-\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}^{\prime}}(\phi_{1}\phi_{2})\right\rangle\right|=\left|\left\langle\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|\\\
&\leqslant
N^{1/2}\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}.\end{split}$
It follows that
$\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}\gg\delta^{O(1)}N^{1/2}.$
Lemma 4.3 (iv) tells us that $\Pi_{\mathcal{B}}f$ is orthogonal to
$\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f$, hence by Pythagoras’s
theorem
$\left\|\Pi_{\mathcal{B}^{\prime}}f\right\|_{\ell^{2}}^{2}=\left\|\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}+\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}.$
The energy bound (4.3) follows for $\mathcal{B}^{\prime}$, allowing us to
proceed to the next stage of our iteration.
Since the function $f$ is $1$-bounded, the projection $\Pi_{\mathcal{B}}f$ is
also 1-bounded, hence the energy (4.3) is always bounded above by $N$. It
follows that this energy increment must terminate at stage $d$ for some
$d\ll\delta^{-O(1)}$, yielding the lemma.∎
## 5\. The density increment lemma
In this section we prove Lemma 2.1, modelling our argument on that given by
Green and Tao [6, Corollary 5.8]. We first record, for the sake of
convenience, the following immediate consequence of the triangle inequality.
###### Lemma 5.1 ($\ell^{1}$-control).
Suppose that $N\geqslant q$. Then for any $f_{0},f_{1},f_{2}:[N]\to\mathbb{C}$
we have
$|\Lambda_{q,N}(f_{0},f_{1},f_{2})|\leqslant
N^{-1}\left\|f_{i}\right\|_{\ell^{1}}\prod_{j\neq
i}\left\|f_{j}\right\|_{\infty}.$
###### Proof.
We prove the result for $i=1$, the other cases being similar. A
reparametrisation gives
$\displaystyle\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|$
$\displaystyle=\left|\mathbb{E}_{x\in[N]}f_{1}(x)\mathbb{E}_{y\in[M]}f_{0}(x-y)f_{2}(x+qy^{2}-y)\right|$
$\displaystyle\leqslant\mathbb{E}_{x\in[N]}|f_{1}(x)|\mathbb{E}_{y\in[M]}|f_{0}(x-y)||f_{2}(x+qy^{2}-y)|.$
∎
We are now in a position to prove Lemma 2.1, and thereby complete our proof of
Theorem 1.1.
###### Proof of Lemma 2.1.
Let $A$ satisfy the assumptions of Lemma 2.1. Increasing $\delta$ only
strengthens our conclusion, so we may assume that $|A|=\delta N$. Since
$\Lambda_{q,N}(1_{A})=0$, we have that
$\left|\Lambda_{q,N}(1_{A})-\Lambda_{q,N}(\delta
1_{[N]})\right|=\delta^{3}\Lambda_{q,N}(1_{[N]})\gg\delta^{3}$.
Applying the weak regularity lemma (Lemma 4.6), there exists a local factor
$\mathcal{B}$ of dimension $d\ll\delta^{-O(1)}$, resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$, and modulus $qq^{\prime}$ for some
$q^{\prime}\leqslant O(1/\delta)^{O(d)}$ such that
$\left\|1_{A}-\Pi_{\mathcal{B}}1_{A}\right\|_{q,N}\leqslant\tfrac{1}{6}\delta^{3}\Lambda_{q,N}({1_{[N]}}).$
Setting $f:=\Pi_{\mathcal{B}}1_{A}$, a telescoping identity thus yields
$\left|\Lambda_{q,N}(f)-\Lambda_{q,N}(\delta
1_{[N]})\right|\geqslant\tfrac{1}{2}\delta^{3}\Lambda_{q,N}({1_{[N]}})\gg\delta^{3}.$
Define the $\mathcal{B}$-measurable set
$S:=\left\\{x\in[N]:f(x)\geqslant(1+c)\delta\right\\},$
where $c>0$ is a sufficiently small absolute constant that will be chosen to
make the following argument valid. By Lemma 5.1 and a telescoping identity, we
have $\left|\Lambda_{q,N}(f)-\Lambda_{q,N}(f1_{S^{c}})\right|\leqslant
3|S|/N$, so that
$\tfrac{|S|}{N}+\left|\Lambda_{q,N}(f1_{S^{c}})-\Lambda_{q,N}(\delta
1_{[N]})\right|\gg\delta^{3}.$
Yet another telescoping identity, in conjunction with Lemma 5.1, gives
$\displaystyle\left|\Lambda_{q,N}(f1_{S^{c}})-\Lambda_{q,N}(\delta
1_{[N]})\right|$
$\displaystyle\ll\tfrac{\delta^{2}}{N}\left\|f1_{S^{c}}-\delta
1_{[N]}\right\|_{\ell^{1}}\leqslant\tfrac{\delta^{2}}{N}\left\|f-\delta
1_{[N]}\right\|_{\ell^{1}}+\tfrac{|S|}{N},$
so that
$|S|+\delta^{2}\left\|f-\delta 1_{[N]}\right\|_{\ell^{1}}\gg\delta^{3}N.$
Since $f-\delta 1_{[N]}$ has mean zero, its $\ell^{1}$-norm is equal to twice
the $\ell^{1}$-norm of its positive part. The function $\left(f-\delta
1_{[N]}\right)_{+}$ can only exceed $c\delta$ on $S$, so taking $c$ small
enough gives $|S|\gg\delta^{3}N$. Letting $B$ denote the largest element of
$\mathcal{B}$ for which $B\subset S$, the bound in Lemma 4.5 yields
$|B|\gg q^{-O(1)}\delta^{O(d)}2^{-O(d)}N^{1/2}.$
By construction (see Definition 4.4), the set $B$ is an arithmetic progression
of common difference $qq^{\prime}$ with $q^{\prime}\leqslant
O(1/\delta)^{O(d)}$. Moreover, the density of $A$ on $B$ is equal to the value
of $f(x)$ for any $x\in B$, and this is at least $(1+c)\delta$ by the
definition of $S$. ∎
## 6\. Global control by major arc Fourier coefficients
The purpose of this section is to prove Theorem 1.2 and Corollary 1.4. We
begin with an alternative version of Lemma 3.2, replacing the rigid local
function found therein with something more continuous.
###### Definition 6.1 ($C$-Lipschitz).
We say that $\phi:\mathbb{Z}\to\mathbb{C}$ is _$C$ -Lipschitz along
$q\cdot\mathbb{Z}$_ if for any $x,y\in\mathbb{Z}$ we have
$|\phi(x+qy)-\phi(x)|\leqslant C|y|.$
Recalling our definition for the Fejér kernel (1.7), we observe that a
function of the form
$x\mapsto\sum_{h}\mu_{H}(h)f(x+qh)$ (6.1)
is Lipschitz along $q\cdot\mathbb{Z}$.
###### Lemma 6.2.
Let $q,H$ be positive integers and $f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded.
If $\phi$ is defined as in (6.1), then $\phi$ is $O(H^{-1})$-Lipschitz along
$q\cdot\mathbb{Z}$.
###### Proof.
Recalling (1.7), the triangle inequality for $|\cdot|$ and $\max\\{\cdot,0\\}$
show that $|\mu_{H}(h+y)-\mu_{H}(h)|\leqslant|y|/\left\lfloor
H\right\rfloor^{2}$ for all $h,y\in\mathbb{Z}$. Hence a change of variables
gives
$|\phi(x+qy)-\phi(x)|\leqslant\sum_{h}|\mu_{H}(h-y)-\mu_{H}(h)|\ll\frac{|y|}{H^{2}}\sum_{h\in(-H,H)\cup(y-H,y+H)}1.$
∎
Now we prove another partial cut norm inverse theorem, this time getting
correlation with functions that are Lipschitz along progressions with small
common difference.
###### Lemma 6.3 (Partial cut norm inverse theorem II).
Let $N$ be a positive integer, $\delta>0$, and $f,g,h:\mathbb{Z}\to\mathbb{C}$
be $1$-bounded functions with support in $[N]$. Suppose that
$\left|\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/2}]}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta.$
Then either $N\ll\delta^{-O(1)}$, or there exists $q\ll\delta^{-O(1)}$ and a
1-bounded function $\phi$ that is $O(\delta^{-O(1)}N^{-1/2})$-Lipschitz along
$q\cdot\mathbb{Z}$ such that
$\sum_{x\in[N]}g(x)\phi(x)\gg\delta^{O(1)}N.$
###### Proof.
Applying [9, Theorem 7.1], we obtain positive integers $q\ll\delta^{-O(1)}$
and $N^{1/2}\geqslant M\gg\delta^{O(1)}N^{1/2}$ such that
$\sum_{x}\left|\sum_{y\in[M]}g(x+qy)\right|\gg\delta^{O(1)}NM.$
By the Cauchy–Schwarz inequality and a change of variables, we have
$\sum_{x}g(x)\sum_{y_{1},y_{2}\in[M]}\overline{g(x+q(y_{1}-y_{2}))}\gg\delta^{O(1)}NM^{2}.$
Setting
$\phi(x):=\mathbb{E}_{y_{1},y_{2}\in[M]}\overline{g(x+q(y_{1}-y_{2}))},$
Lemma 6.2 shows this function has the required properties. ∎
Before proving Theorem 1.2, we record two standard facts.
###### Lemma 6.4.
There are at most $O(N^{4})$ solutions $x\in[N]^{6}$ to the equation
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=x_{4}^{2}+x_{5}^{2}+x_{6}^{2}.$
###### Proof.
There are a number of ways to prove this. Perhaps the most robust is via the
circle method, see [3]. The result can be read out of [1, Proposition 1.10]. ∎
###### Lemma 6.5 (Weyl’s inequality).
Let $P\subset\mathbb{Z}$ be an arithmetic progression with common difference
$q$ and let $0<\delta\leqslant 1$. Suppose that
$\left|\sum_{x\in P}e(\alpha x^{2})\right|\geqslant\delta|P|.$
Then either $|P|\ll\delta^{-O(1)}$ or there exists a positive integer
$q^{\prime}\ll\delta^{-O(1)}$ such that
$\|q^{\prime}q^{2}\alpha\|\ll\delta^{-O(1)}|P|^{-2}.$
###### Proof.
Let $P=x_{0}+q\cdot[N]$, so that our exponential sum becomes
$\sum_{x\in P}e(\alpha x^{2})=\sum_{y\in[N]}e(\alpha q^{2}y^{2}+2\alpha
qx_{0}y+\alpha x_{0}^{2}).$
Applying [5, Lemma A.11], either $N\ll\delta^{-O(1)}$ or the conclusion of our
lemma follows. ∎
###### Proof of Theorem 1.2.
Write $\Lambda_{N}$ for the counting operator $\Lambda_{1,N}$ (that is, the
average (1.6) with $q=1$). Let $f,g,h:[N]\to\mathbb{C}$ be 1-bounded functions
satisfying
$|\Lambda_{N}(f,g,h)|\geqslant\delta.$
Define the seminorm
$\left\|g\right\|:=\sup\left\\{|\Lambda_{N}(g_{1},g,g_{2})|:|g_{i}|\leqslant
1\text{ and }\mathrm{supp}(g_{i})\subset[N]\right\\}.$
and the dual function
$F(x):=\mathbb{E}_{y\in[N^{1/2}]}f(x-y)h(x+y^{2}-y).$
We follow the argument in the proof of Lemma 3.3 to deduce that
$\left\|F\right\|\geqslant\delta^{2}.$
Hence, by Lemma 6.3, there exists $q\ll\delta^{-O(1)}$ and a 1-bounded
function $\phi$ that is $O(\delta^{-O(1)}N^{-1/2})$-Lipschitz along
$q\cdot\mathbb{Z}$ and satisfies
$\sum_{x\in[N]}F(x)\phi(x)\gg\delta^{O(1)}N.$
Expanding the definition of the dual function, we have
$\sum_{x\in[N]}\sum_{y\in[N^{1/2}]}f(x)\phi(x+y)h(x+y^{2})\gg\delta^{O(1)}N^{3/2}.$
Let us partition $\mathbb{Z}$ into arithmetic progressions $P$ each of common
difference $q$ and length $M$, where $M$ will be chosen shortly. For each such
arithmetic progression $P$, fix an element $y_{P}\in P$. Using the Lipschitz
property of $\phi$, for any $x\in\mathbb{Z}$ and $y\in P$ we have
$|\phi(x+y_{P})-\phi(x+y)|\ll\delta^{-O(1)}MN^{-1/2}.$
Hence,
$\left|\sum_{P}\sum_{x\in[N]}\sum_{y\in
P\cap[N^{1/2}]}f(x)[\phi(x+y)-\phi(x+y_{P})]h(x+y^{2})\right|\ll\delta^{-O(1)}MN.$
We can therefore take $M$ sufficiently small to satisfy both
$M\gg\delta^{O(1)}N^{1/2}$ and
$\left|\sum_{P}\sum_{x}\sum_{y\in
P\cap[N^{1/2}]}f(x)\phi(x+y_{P})h(x+y^{2})\right|\gg\delta^{O(1)}N^{3/2}.$
Set $f_{P}(x):=f(x)\phi(x+y_{P})$. The number of progressions $P$ that
intersect $[N^{1/2}]$ is at most $O(N^{1/2}M^{-1}+q)=O(\delta^{-O(1)})$.
Therefore, the pigeon-hole principle gives a progression $P$ for which
$\left|\sum_{x}\sum_{y\in
P\cap[N^{1/2}]}f_{P}(x)h(x+y^{2})\right|\gg\delta^{O(1)}N^{3/2}.$ (6.2)
In particular, $|P\cap[N^{1/2}]|\gg\delta^{O(1)}N^{1/2}$.
Writing $S_{P}(\alpha)$ for $\sum_{y\in P\cap[N^{1/2}]}e\left(\alpha
y^{2}\right)$, the orthogonality relations allow us to reformulate (6.2) as
$\displaystyle\left|\int_{\mathbb{T}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\gg\delta^{O(1)}N^{3/2}.$
Let $\eta>0$ be a parameter to be determined shortly, and define the major
arcs
$\mathfrak{M}:=\left\\{\alpha\in\mathbb{T}:|S_{P}(\alpha)|\geqslant\eta
N^{1/2}\right\\}.$
Parseval’s identity then gives
$\left|\int_{\mathbb{T}\setminus\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\leqslant\eta
N^{1/2}\big{\|}\hat{f}_{P}\big{\|}_{2}\big{\|}\hat{h}\big{\|}_{2}\leqslant\eta
N^{3/2}.$
Hence we may take $\eta\gg\delta^{O(1)}$ and ensure that
$\displaystyle\left|\int_{\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\gg\delta^{O(1)}N^{3/2}.$
By Lemma 6.4 and orthogonality, we have $\left\|S_{P}\right\|_{6}\ll N^{1/3}$.
Thus, by Hölder’s inequality, we get that
$\left|\int_{\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\leqslant\big{\|}\hat{f}_{P}\big{\|}_{2}\big{\|}\hat{h}\big{\|}_{2}^{2/3}\big{\|}S_{P}\big{\|}_{6}\sup_{\alpha\in\mathfrak{M}}\bigl{|}\hat{h}(-\alpha)\bigr{|}^{1/3}.$
We therefore deduce that there exists $\alpha\in\mathfrak{M}$ such that
$\bigl{|}\hat{h}(-\alpha)\bigr{|}\gg\delta^{O(1)}N.$
Finally, an application of Weyl’s inequality (Lemma 6.5) shows that if
$-\alpha\in\mathfrak{M}$ then $\alpha$ has the required Diophantine
approximation property. ∎
###### Proof of Corollary 1.4.
Let $\alpha\in\mathbb{R}$ be the frequency and $q$ the positive integer
provided by Theorem 1.2. For any integer $a$ and positive integer $M$, if
$x,y\in a+q\cdot[M]$, then
$\left|e(\alpha x)-e(\alpha y)\right|\leqslant
2\pi\left\|\alpha(x-y)\right\|\ll\delta^{-O(1)}MN^{-1}.$
Partitioning $\mathbb{Z}$ into arithmetic progressions of common difference
$q$ and length $M$ then gives
$\delta^{O(1)}N\ll\sum_{P}\Bigl{|}\sum_{x\in P}h(x)\Bigr{|}+\delta^{-O(1)}M.$
We thus take $M\gg\delta^{O(1)}N$ sufficiently small to ensure that
$\delta^{O(1)}N\ll\sum_{P}\Bigl{|}\sum_{x\in P}h(x)\Bigr{|}.$
Write $\theta_{P}$ for the conjugate phase of the inner sum. Then the map
$x\mapsto\sum_{P}\theta_{P}1_{P}(x)$ is a local function of resolution
$\gg\delta^{O(1)}N$ and modulus $\ll\delta^{-O(1)}$, yielding the corollary. ∎
## 7\. Longer progressions
As mentioned in §1.3, the main obstacle to generalising our polylogarithmic
bound to longer configurations such as (1.2) is in obtaining an appropriate
generalisation of Lemma 3.3; in particular, showing that if the relevant
counting operator is large, then _all_ functions must correlate with a product
of a bounded number of local functions.
Let us demonstrate where the argument breaks down for $m>2$. Given polynomials
as in (1.2) and 1-bounded functions
$f_{0},f_{1},\dots,f_{m}:[N]\to\mathbb{C}$, define the counting operator
$\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m}):=\\\
\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/\deg
P_{m}}]}f_{0}(x)f_{1}(x+P_{1}(y))\dotsm f_{m}(x+P_{m}(y)).$
Using the main technical result of [8], [8, Theorem 3.3], one can show that if
$\left|\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m})\right|\geqslant\delta,$
then both $f_{0}$ and $f_{1}$ correlate with local functions $\phi_{0}$ and
$\phi_{1}$. Combining this with a dual function argument, as in our proofs of
Theorem 1.2 and Lemma 3.3, one may conclude that
$\left|\Lambda_{P_{1},\dots,P_{m}}^{N}(\phi_{0},\phi_{1},f_{2},\dots,f_{m})\right|\gg\delta^{O(1)},$
If $m=2$, one can then pigeon-hole in the smaller $y$ variable appearing in
the counting operator (as we do in the proof of Lemma 3.3) to conclude that
$f_{2}$ correlates with a product of two local functions. It is this simple
pigeon-holing argument that fails when $m>2$.
### 7.1. An alternative strategy for longer progressions
A more productive strategy is to follow our proof of Theorem 1.2 instead of
Theorem 1.1. In proving Theorem 1.2 we replace the counting operator
$\Lambda_{y,y^{2}}^{N}(f_{0},f_{1},f_{2})$ with
$\Lambda_{y,y^{2}}^{N}(f_{0},\phi,f_{2})$, where $\phi$ is a local function
that is constant on progressions of length $\approx N^{1/2}$ with common
difference of size $\approx O(1)$. Provided that we pass to appropriate
subprogressions in all of the variables appearing in our counting operator, we
can exploit the properties of this local function and ‘remove’ it from our
count. In effect (after passing to subprogressions of bounded common
difference), we replace the count $\Lambda_{y,y^{2}}^{N}(f_{0},f_{1},f_{2})$
with one of the form $\Lambda_{Q}^{N^{\prime}}(f_{0},f_{2})$, where $Q$ is a
quadratic polynomial and $N^{\prime}$ is slightly smaller than $N$.
Generalising this approach, one can use [8, Theorem 3.3] to replace the
counting operator $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m})$
with $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},\phi,f_{2},\dots,f_{m})$, where
$\phi$ is a local function. Provided that this local function has resolution
$\gg N^{\deg P_{1}/\deg P_{m}}$ and common difference $q\ll 1$, we have
$\phi(x+P_{1}(y))\approx\phi(x)$
for any $x\in\mathbb{Z}$ and any $y$ constrained to a subprogression of common
difference $q$ and length $\approx N^{\deg P_{1}/\deg P_{m}}$. Passing to
subprogressions in $x$ and $y$, one should then be able to replace the
operator
$\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},\phi,f_{2},\dots,f_{m})$
by one of the form
$\Lambda_{Q_{2},\dots,Q_{m}}^{N^{\prime}}(f_{0},f_{2},\dots,f_{m}).$
Applying induction on $m$ may then allow one to show that every function in
the original counting operator correlates with a local function.
The main impediment to carrying out this strategy is that the polynomials
$Q_{2}$, …, $Q_{m}$, which arise on passing to a subprogression, may not
satisfy the hypotheses required to reapply [8, Theorem 3.3]. It is likely that
the polynomials are sufficiently well-behaved for the arguments of [8] to
remain valid, but we leave this verification to the energetic reader.
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|
2024-09-04T02:54:58.416642 | 2020-03-09T13:24:21 | 2003.04127 | {
"authors": "Johannes Hillbrand, Nikola Opacak, Marco Piccardo, Harald Schneider,\n Gottfried Strasser, Federico Capasso, Benedikt Schwarz",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26119",
"submitter": "Johannes Hillbrand",
"url": "https://arxiv.org/abs/2003.04127"
} | arxiv-papers | Mode-locked ultrashort pulses from an 8 µm wavelength semiconductor laser
Johannes Hillbrand1,2,∗, Nikola Opačak1, Marco Piccardo2,3, Harald Schneider4,
Gottfried Strasser1, Federico Capasso2, Benedikt Schwarz1,2,†
1Institute of Solid State Electronics, TU Wien, Gußhausstraße 25, 1040 Vienna,
Austria
2Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, USA
3CNST – Fondazione Istituto Italiano di Tecnologia, Via Pascoli 70/3, 20133
Milano, Italy
4Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum
Dresden-Rossendorf, Germany
Quantum cascade lasers (QCL) have revolutionized the generation of mid-
infrared light. Yet, the ultrafast carrier transport in mid-infrared QCLs has
so far constituted a seemingly insurmountable obstacle for the formation of
ultrashort light pulses. Here, we demonstrate that careful quantum design of
the gain medium and control over the intermode beat synchronization enable
transform-limited picosecond pulses from QCL frequency combs. Both an
interferometric radio-frequency technique and second-order autocorrelation
shed light on the pulse dynamics and confirm that mode-locked operation is
achieved from threshold to rollover current. Being electrically pumped and
compact, mode-locked QCLs pave the way towards monolithically integrated non-
linear photonics in the molecular fingerprint region beyond 6 µm wavelength.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
The discovery of ultrashort light pulses has led to numerous breakthroughs in
science and technology, including frequency combs1, high-speed optical
telecommunication2 and refractive surgery in ophthalmology3. Nowadays, optical
pulses are routinely generated in mode-locked lasers operating in the visible
or near-infrared range4, 5. Currently, large efforts are aimed at bringing
ultrafast laser science in the mid-infrared (MIR) region to a similarly high
degree of maturity6. Due to the lack of suitable gain media, methods for the
generation of pulses in the molecular fingerprint region beyond 5 µm
wavelength have so far relied on non-linear downconversion of near-infrared
pulses 7. Established techniques such as optical parametric oscillators8 or
difference frequency generation9, 10 either require sophisticated optical
setups with tabletop dimensions or are restricted to mW-level of output power.
Quantum cascade lasers11 (QCL) have matured to become the dominant MIR laser
source. While being microchip-sized and electrically pumped, they are capable
of producing Watt-level average power12, 13. Quantum engineering of the active
region allows to tailor the emission wavelength throughout the entire mid-
infrared region. Hence, harnessing high-performance QCL technology for the
generation of MIR pulses represents a long-sought milestone in ultrafast laser
science. Mode-locked QCLs could serve as monolithic pump lasers for
microresonators and resonant supercontinuum generation14, paving the way
towards broadband and high-brightness frequency combs. So far, the sub-
picosecond carrier transport in QCL active regions has constituted a seemingly
insurmountable obstacle for the formation of short light pulses15, 16, 17. To
date, the only successful attempt of mode-locking in monolithic MIR QCLs was
observed using a specially designed active region with strongly enhanced
upper-state lifetime of the lasing transition17. However, the necessary design
modifications limited mode-locked operation to cryogenic temperatures and peak
power below 10 mW, thus impeding their practical use.
Figure 1: Bi-functional quantum cascade lasers for mode-locking. a: Scanning
electron microscope image of three adjacent laser ridges. Each laser consists
of a roughly 3 mm long gain section and a shorter (320-480 µm) modulation
section. b: Simulated gain and loss spectrum in a standard active region
design18 depending on the applied bias. Upon decreasing the bias, the
structure becomes almost transparent at the lasing wavelength $\lambda_{L}$,
limiting the maximally achievable modulation depth. c: Simulated gain and loss
spectrum in a bi-functional active region design12, allowing to tune the gain
at 10 V continuously to absorption (shown as negative gain) at 0 V. d:
Measured light-current-voltage (L-I-V) characteristics of an epi-up mounted
bi-functional QCL at 15$\,{}^{\circ}$C. e: Illustration of a system of coupled
oscillators. This system shows an in-phase and anti-phase synchronization
state, which oscillate at different frequencies depending on the coupling.
Without external stimulation, the anti-phase state is more favorable due to
the damped coupling. However, both synchronization frequencies can be probed
by exerting mechanical force on the platform coupling the oscillators. In the
QCL, the oscillators are represented by the intermode beatings, which tend to
synchronize in anti-phase due to gain damping19, 20. Both synchronization
frequencies are probed by applying modulation to the laser. f: Average optical
power depending on the modulation frequency and power. Two synchronization
states at $f_{\mathrm{rep}}^{0}$ and 60 MHz above are observed. g: Signal of a
2-QWIP sensitive to peak power as function of modulation frequency and power.
The strongly increased signal of the lobe at $f_{\mathrm{rep}}^{0}$+60 MHz
indicates in-phase synchronization.
In this work, we demonstrate the generation of transform limited picosecond
pulses in high-performance 8 µm wavelength QCLs at room temperature both
experimentally and theoretically. Mode-locking is achieved by electrically
modulating the intracavity loss using a short modulation section designed for
efficient radio-frequency (RF) injection (Fig. 1a). In order to achieve the
large modulation depth required for stable mode-locking, close attention has
to be paid to the band structure of the QCL active region. This effect is
illustrated in Figs. 1a,b. As the bias applied to a standard QCL structure is
decreased, it does not switch to absorption at the lasing wavelength, but
becomes nearly transparent for the intracavity light due to a bias-dependent
shift of the electronic levels, known as Stark effect (Fig. 1b). Hence, the
modulation depth is severely limited in standard QCL designs. For this reason,
we employ a bi-functional active region whose lasing wavelength and absorption
wavelength at zero bias were matched to each other 21, 12 (Fig. 1c). This
strategy allows to overcome the aforementioned limitations of the modulation
depth caused by the Stark shift. Most importantly, the bi-functional design
shows excellent overall performance, which is competitive with other state-of-
the-art designs. A 3.5 mm long device mounted epitaxial-side up emits more
than 130 mW average power in continuous wave at room temperature (Fig. 2d).
As a first step towards pulse generation, it is essential to determine the
optimal modulation frequency. For this purpose, mode-locking can be seen as
synchronization of coupled oscillators20 (Fig. 1e). Each pair of neighboring
cavity modes creates a beating at their difference frequency, which is equal
to the cavity roundtrip frequency. These beatings can be seen as oscillators
coupled by the non-linearity of the gain medium. Thanks to this coupling, the
cavity modes of a free-running QCL can be locked together without modulation,
thus giving rise to a self-starting frequency comb 19. Yet, this kind of
frequency comb does not emit isolated pulses, but rather a quasicontinuous
wave accompanied by a strong linear frequency chirp22. This corresponds to
anti-phase synchronization and will be called frequency modulated (FM) comb in
the following. In contrast, in-phase synchronization of the intermode beatings
leads to the formation of short pulses. It is well known from coupled
oscillators that the in-phase and anti-phase states synchronize at different
frequencies depending on the coupling. As a consequence, while the cavity
roundtrip frequency of the FM QCL comb $f_{\mathrm{rep}}^{0}$ may seem like a
reasonable choice, we expect the optimal modulation frequency for generating
pulses to differ from $f_{\mathrm{rep}}^{0}$.
In order to investigate these two synchronization states experimentally, we
start by operating the laser well above its threshold current. Subsequently,
the DC bias of the modulation section is decreased to 2.8 V, where the large
absorption caused by the bifunctional design (Fig. 1c) brings the QCL just
slightly below lasing threshold. In these conditions, modulation at the right
frequencies can provide enough additional gain to reach threshold. Fig. 1f
shows the laser power depending on modulation frequency and power. At 33 dBm
modulation power, the QCL reaches threshold when modulating close to
$f_{\mathrm{rep}}^{0}$. Strikingly, a second modulation frequency where lasing
occurs is observed almost 60 MHz higher than $f_{\mathrm{rep}}^{0}$, as
predicted by the picture of synchronized oscillators. Both the range around
the two synchronization frequencies, where lasing is observed, as well as the
optical power grow upon increasing the modulation power.
Figure 2: Mode-locked pulses from an 8 µm wavelength QCL. a: SWIFTS
characterization of the QCL operated close to lasing threshold. The laser is
modulated at the in-phase synchronization frequency and at 37 dBm power level.
b: Reconstructed time-domain signal of the QCL, showing a train of transform
limited pulses with 6.5 ps FWHM. c: Simulation of the QCL using the coherent
Master equation described in supp. section 1. d: Interferometric
autocorrelation (IAC) of the QCL pulses close to threshold. Red dots: envelope
of the IAC reconstructed using SWIFTS. e: IAC at higher current. f: IAC at the
rollover current, still displaying the 8:1 ratio. The second burst at a delay
equal to the cavity roundtrip time is due to the interference of subsequently
emitted pulses. Its peak value of 8 provides another proof for the coherence
of the pulses because phase-decoherence would smear out the fringes of the IAC
and thus decrease the peak value to smaller than 8. Inset: zoom on the
interferometric fringes.
Even more insight is provided by using a two-photon quantum well infrared
photodetector (2-QWIP) to detect the emitted light. The signal of the 2-QWIP
is proportional to the square of the intensity. This allows to identify which
modulation frequency leads to in-phase and which to anti-phase synchronization
(Fig. 1g). Again, two lobes appear around $f_{\mathrm{rep}}^{0}$ and 60 MHz
above. Yet, the 2-QWIP signal is more than an order of magnitude larger in the
lobe at higher $f_{\mathrm{mod}}$. At this frequency, the laser operates in
the in-phase synchronization regime and emits intense pulses, which leads to a
strongly increased 2-QWIP signal.
Figure 3: Synchronization under strong modulation. a: Schematic of a 3-section
QCL comprised of modulation, gain and high-speed detector sections. b: First
three harmonics of the beatnote of the free-running 7 mm long QCL FM comb
(red) compared to the actively mode-locked QCL (AM comb, blue). c: laser
beatnote while free-running (bottom), at
$f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}$ (middle) and at
$f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}{+}33\,$MHz (top). While a broad
pedestal is visible for $f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}$, the
beatnote is perfectly locked for
$f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}{+}33\,$MHz. d: RF spectrum around
$f^{0}_{\mathrm{rep}}{=}$6.196 GHz as the modulation frequency is varied
around $f^{0}_{\mathrm{rep}}$. The phase-noise of the RF spectrum disappears
abruptly at $f_{\mathrm{mod}}{\approx}f_{\mathrm{rep}}^{0}{+}20\,$MHz,
corresponding to in-phase synchronization. Here, the beatnote consists of a
single narrow peak, indicating that the laser is phase-locked to the
modulation. Furthermore, the sharp sidepeaks visible at
$f_{\mathrm{mod}}{=}6.18\,$GHz are attributed to a periodic modulation of the
QCL output, as previously observed in simulations16.
In order to unequivocally prove mode-locking, we employ two independent
methods to characterize the pulse dynamics at three points of operation from
threshold up to the rollover current. Firstly, an interferometric RF technique
called ’Shifted wave interference Fourier transform spectroscopy’ (SWIFTS)23,
24 is used to measure the phases of the QCL spectrum (details in Methods
section). This information not only enables the reconstruction of the temporal
waveform, but also allows to assess the phase-coherence of the pulses and
whether they form a frequency comb. Secondly, we measure the interferometric
autocorrelation (IAC) of the pulses using the 2-QWIP, which constitutes an
additional well-established proof for mode-locking and the pulse width. Fig.
2a shows the SWIFTS characterization of the QCL operated close to threshold.
In contrast to the free-running laser, the intensity spectrum consists of a
single Gaussian-shaped lobe. The SWIFTS spectrum represents the part of the
intensity spectrum which is beating exactly at the modulation frequency. Since
the SWIFTS spectrum has the same shape as the intensity spectrum over its
entire span, the QCL generates a frequency comb whose repetition frequency is
given by the modulation frequency. The intermodal difference phases
$\Delta\phi$, which correspond to the spectral group delay, are synchronized
almost perfectly in-phase. Hence, all parts of the spectrum have the same
group delay and form a pulse. Indeed, the reconstructed waveform in Fig. 2b
shows the emission of 6.5 ps short pulses. The full-width-half-maximum (FWHM)
of the reconstructed pulses is given by the transform limit of the spectrum in
Fig. 2b, indicating that there is negligible chirp in the pulses. In these
conditions, the peak power reaches almost 250 mW, which constitutes an
enhancement of more than 12 compared to the average power of 20 mW. In order
to model the cavity dynamics, we use a fully coherent master equation25 (supp.
section 1). This single equation for the complex field replaces the entire
Maxwell-Bloch system and reliably predicts the spectral shape, phase
relationship and pulse width observed experimentally (Fig. 2c). Furthermore,
it allows experimentally unavailable analyses, e.g. the influence of
dispersion and nonlinearities (supp. Figs. 2,3,7).
The IAC close to threshold (Fig. 2d) shows a prominent peak at zero path
difference caused by constructive interference of the pulses after the
Michelson interferometer. The ratio of this peak to the background at a delay
larger than the pulse width is 8:1, which is generally regarded as the smoking
gun of mode-locked pulses. Encouragingly, the measured IAC is in excellent
agreement with the expected IAC, which was calculated using the pulses
obtained by SWIFTS (red dots in Fig. 2d). This confirms successful mode-
locking and the retrieved pulse width.
The generation of pulses becomes increasingly challenging at higher gain
current. Due to gain saturation, the wings of a pulse experience more gain
than the peak. This effect leads to pulse broadening and can destabilize mode-
locking. Fortunately, the large modulation depth provided by the bi-functional
quantum design enables mode-locking over the entire lasing range from
threshold to rollover. The IAC traces at 3.7 kA/cm2 and at the rollover
current still show the required peak-to-background ratio of 8:1. The pulses at
rollover are slightly broadened to roughly 12 ps, which is attributed
partially to a slight chirp and partially to the gain saturation effect
mentioned above (supp. Fig. 6). Yet, the average power is greatly increased to
62 mW, which results in over 430 mW peak power and 5 pJ pulse energy - more
than an order of magnitude higher than recent reports of comparable mid-
infrared semiconductor lasers emitting at shorter wavelengths 26, 27.
Another fascinating aspect of bi-functional quantum design is the possibility
to monolithically integrate ultrafast photodetectors. While this is
particularly important in applications such as photonic integrated circuits,
it also provides a tool to measure the beatnote with very large signal-to-
noise ratio directly on the chip (Fig. 3a). This provides crucial information
about the type of synchronization state and about its stability. Fig. 3b shows
the first three harmonics of the beatnote in the free-running and the actively
mode-locked regime. In the latter conditions, the beatnote amplitudes increase
by 19 dB due to the much larger amplitude modulation. The zoom on the first
harmonic beatnote (Fig. 3c) allows to assess the phase-coherence and stability
of the frequency comb. The free-running QCL is operating in the anti-phase
state showing a weak beatnote at $f_{\mathrm{rep}}^{0}$. Previous work28, 29,
30 has shown that a weak electrical modulation can be used to lock and
stabilize the beatnote of the anti-phase state. However, the situation is very
different when applying strong modulation at $f_{\mathrm{rep}}^{0}$. In this
case, the modulation enforces an AM waveform, which is contrary to the natural
anti-phase behaviour of the laser. As a result, the beatnote of this waveform
is not phase-locked, as indicated by the pedestal around $f_{\mathrm{mod}}$.
The situation changes completely, when the modulation frequency is tuned to
the synchronization frequency of the in-phase state
($f_{\mathrm{rep}}^{0}{+}33\,$MHz). There, the strong modulation is in
consensus with the natural behavior of the laser, leading to a phase-locked
frequency comb with narrow beatnote. This can also be seen in Fig. 3d, which
shows the laser beatnote while tuning the modulation frequency across the in-
phase and anti-phase synchronization frequencies. While the frequency of the
beatnote is controlled by the modulation over the entire span, a phase-locked
comb is only generated around the in-phase synchronization frequency.
In conclusion, our experiments provide unambiguous proof for the generation of
mode-locked pulses in high-performance QCLs at room temperature - a goal which
remained elusive since the invention of the QCL - and confirm stunning
similarities to synchronization in coupled oscillators. These mode-locked QCLs
constitute the first compact and electrically pumped source for ultrashort
pulses beyond 5 µm wavelength, demonstrating that they are a highly promising
technology for ultrafast laser science in the long-wave infrared region. The
availability of such a source paves the way towards a semiconductor-based
platform for non-linear photonics, potentially enabling broadband mid-infrared
frequency combs and supercontinuum generation.
## Methods
QCLs optimized for RF modulation: The QCLs are processed as buried
heterostructures. The width of the QCL ridges is 12 µm and the facets are left
as cleaved. The area of the top contact of the modulation section is minimized
to decrease its parasitic capacitance, which increases the RF injection
efficiency. Ground contacts for the modulation are provided by etching through
the Fe-doped InP layer next to the laser ridges. The modulation signal is
provided by a HP8341B synthesized sweeper, amplified up to roughly 5 W and
injected via coplanar tips. The insertion loss at 12 GHz is 14 dB including
cables and bias-tee.
SWIFTS and IAC: The light emitted by the QCL is shone through a Bruker Vertex
70v FTIR spectrometer. In order to perform SWIFTS, the light is then detected
by a home-built fast QWIP at the exit of the FTIR. The optical beating
obtained from the QWIP is subsequently amplified and mixed down to roughly 10
MHz using a local oscillator. A Zurich Instruments HF2LI lock-in amplifier and
the trigger of the FTIR are used to record the SWIFTS and intensity
interferograms in rapid scan mode. The IAC is obtained by detecting the pulses
at the exit of the FTIR using the 2-QWIP and recording its photocurrent
depending on the path delay of the FTIR.
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## Acknowledgements
This work was supported by the Austrian Science Fund (FWF) in the framework of
”Building Solids for Function” (Project W1243), the projects ”NanoPlas”
(P28914-N27) and ”NextLite” (F4909-N23).
## Author contributions
J.H. processed the QCLs and carried out the experiments. B.S. and J.H. built
up the SWIFTS and IAC setups. N.O. and B.S. developed the simulation tool.
M.P. carried out the temporal reconstruction using the IAC data. H.S. provided
the 2-QWIP. J.H. wrote the manuscript with editorial input from N.O., B.S.,
G.S and F.C. All authors analysed the results and commented on the paper.
|
2024-09-04T02:54:58.433052 | 2020-03-04T23:32:28 | 2003.04133 | {
"authors": "Nils Ohlendorf, Wolf-Peter Schill",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26120",
"submitter": "Wolf-Peter Schill",
"url": "https://arxiv.org/abs/2003.04133"
} | arxiv-papers | # Frequency and duration of low-wind-power events in Germany
Nils Ohlendorf<EMAIL_ADDRESS>Wolf-Peter Schill<EMAIL_ADDRESS>Mercator Research Institute on Global Commons and Climate Change (MCC), EUREF
Campus 19, Torgauer Straße 12-15, 10829 Berlin, Germany German Institute for
Economic Research (DIW Berlin), Mohrenstrasse 58, 10117 Berlin, Germany
Energy Transition Hub, Climate & Energy College, The University of Melbourne
###### Abstract
In the transition to a renewable energy system, the occurrence of low-wind-
power events receives increasing attention. We analyze the frequency and
duration of such events for onshore wind power in Germany, based on 40 years
of reanalysis data and open software. We find that low-wind-power events are
less frequent in winter than in summer, but the maximum duration is
distributed more evenly between months. While short events are frequent, very
long events are much rarer. Every year, a period of around five consecutive
days with an average wind capacity factor below 10% occurs, and every ten
years a respective period of nearly eight days. These durations decrease if
only winter months are considered. The longest event in the data lasts nearly
ten days. We conclude that public concerns about low-wind-power events in
winter may be overrated, but recommend that modeling studies consider multiple
weather years to properly account for such events.
###### keywords:
Wind power; Low-wind-power events; Reanalysis data;
††journal: arXiv
## 1 Introduction
The Paris Agreement calls for an extensive decarbonization of the global
economy. A major strategy for achieving this goal is a massive expansion of
variable renewable energy sources, in particular solar photovoltaics (PV) and
wind power [de Coninck et al., 2018]. While power generation from solar PV
largely follows diurnal and seasonal cycles with annually repeating patterns,
wind power is subject to more irregular inter-annual as well as intra-annual
variations which are relevant from a security of supply perspective. In
countries with growing shares of wind power, the occurrence of low-wind-power
(LWP) events thus receives increasing attention.
This is particularly true in Germany. In the context of its energy transition,
Germany is one of the global front-runners in wind power deployment. In 2018,
a total capacity of 52.5 GW of onshore wind power was installed in Germany,
generating 90.5 TWh of electricity. This corresponds to 15% of German gross
electricity consumption [BMWi, 2019]. Given the government’s targets to expand
the share of renewables in electricity consumption to 65% by 2030 and at least
80% by 2050 [Bundesregierung, 2019], the dependence of the German energy
system on wind power is set to increase strongly in the future. Concerns about
LWP events have been discussed in German media [Wetzel, 2017, 2019] and in the
German parliament [Deutscher Bundestag, 2019a], and LWP events are also
mentioned in the government’s energy transition reporting [Deutscher
Bundestag, 2019b]. In this context, the term Dunkelflaute is increasingly
used. It refers to a persistent situation with very low power generation from
wind and solar PV, which would be especially challenging in the German winter
season where PV availability is low and electric load has its peak. Yet no
clear definition of this concept has been provided so far [Wissenschaftliche
Dienste, 2019], and quantitative evidence on the frequency and duration of
such events is missing. In Table $15$ of Deutscher Bundestag [2019b], an
independent expert commission generally assumes a no-wind-no-solar period of
two weeks.
Yet research on LWP events is sparse so far. In this paper, we contribute to
filling this gap, focusing on onshore wind power in Germany. We provide an in-
depth analysis of the frequency, duration, and magnitude of LWP events, making
use of reanalysis data for 40 full years (1980 to 2019) and power curves of
recently installed wind turbines. In doing so, we propose two definitions of
LWP events and investigate three different thresholds of capacity factors (2%,
5% and 10%). We also compare the spatial distributions of the most persistent
LWP event and the mean electricity generation. Parts of our analysis
explicitly focus on winter months: these are particularly relevant, as power
generation from solar PV is relatively low during this season, while the
German peak load also occurs in winter. In order to allow for the highest
degree of transparency and reproducibility, we provide the source code of our
analysis under a permissive open-source license [Ohlendorf, 2020].
There are only few dedicated analyses on the frequency and duration of LWP
events. Early contributions address reliability aspects of spatially dispersed
wind power in California [Kahn, 1979] or in the midwestern United States
[Archer and Jacobson, 2007]. Analyses explicitly focusing on LWP events only
recently emerged. Yet these differ from our work, amongst other factors, with
respect to geographical and temporal coverage, data sources used, and
methodologies applied. In particular, previous low-wind analyses mostly draw
on local measurement data and either evaluate wind speeds [Leahy and McKeogh,
2013, Patlakas et al., 2017] or wind power [Handschy et al., 2017, Kruyt et
al., 2017]. Leahy and McKeogh [2013] and Patlakas et al. [2017] investigate
low-wind events for Ireland and the North Sea area, respectively. Both studies
firstly evaluate low-wind events that are constantly below a given wind speed
threshold, and secondly determine annual minimum moving average wind speeds
for given durations, using extreme value distributions. Kruyt et al. [2017]
and Handschy et al. [2017] go one step further and calculate respective power
generation from wind speeds for Switzerland and the United States, using a
power curve. While the findings of these studies are necessarily idiosyncratic
to the specific geographical applications, some common findings emerge. First,
low-wind events are less frequent and less persistent if more, and spatially
more dispersed, measurement stations are used. Second, there are generally
less events in winter than in summer.
The measurement-based analyses face challenges related to their data sources.
In general, studies that draw on measured wind speeds are spatially biased,
have low measurement densities, and extrapolation from measurement height to
hub height is challenging because of distorting effects of terrain, elevations
or buildings [Sharp et al., 2015]. Measurement data may further be subject to
inconsistencies caused by changing equipment and measurement errors. Extreme
event analyses further require consistent measurements over large time periods
to sufficiently capture climatic variations.
These issues can be addressed by using long-term meteorological reanalysis
data. Such data is increasingly applied for onshore wind energy modelling.
Several studies focus on data accuracy and on validating models of wind power
generation [Decker et al., 2012, Sharp et al., 2015, Olauson and Bergkvist,
2015, Rose and Apt, 2015, Staffell and Pfenninger, 2016, González-Aparicio et
al., 2017, Germer and Kleidon, 2019]. Other analyses deal with variability
aspects of wind power, but do not focus on extreme low-wind events. For
example, Grams et al. [2017] explain longer-term fluctuations in European wind
power generation with different types of weather regimes, based on MERRA-2
data. With similar approaches, Collins et al. [2018] investigate inter-annual
variations of European wind and solar power, and Santos-Alamillos et al.
[2017] explore optimal allocations of renewable generation capacity in a
European super grid. For the contingent U.S. states, Shaner et al. [2018]
investigate the reliability of future power systems dominated by wind and/or
solar PV, and Kumler et al. [2019] explore inter-annual renewable variability
for Texas. Yet none of these studies explicitly focuses on the frequency and
duration of extreme low-wind-power events.
A notable reanalysis study that does focus on extreme wind events is conducted
by Cannon et al. [2015] for Great Britain. Using 33 years of MERRA as well as
ERA-Interim data, the authors conclude that the frequency and duration of low-
wind-power events can be approximated by a Poisson-like process. Weber et al.
[2019] also use ERA-Interim data for a superstatistical analysis of extreme
wind power events at nine specific European sites, including one German
onshore location. They find that the distribution of low-wind events has a
heavy tail, as low-wind events may result from a combination of different
weather and circulation patterns.111Weber et al. [2019] base their analysis on
wind speeds, not wind power generation, with a cut-off threshold of $4$ m/s.
In another analysis based on ERA-Interim reanalysis data and other sources,
Raynaud et al. [2018] define and investigate the occurrence of renewable
“energy droughts”, which are measured relative to average daily generation.
They find that wind power droughts are both relatively frequent and relatively
short in most European countries, compared to hydro power droughts.
We contribute to this emerging literature with a dedicated open-source,
reanalysis-based study that investigates LWP events in Germany in detail. To
the best of our knowledge, we are the first to use MERRA-2 data in this
context, i.e., spatially and temporally consistent reanalysis data covering 40
years at 50 m above surface. Compared to Cannon et al. [2015], we also make
use of not only one, but three recent power curves to represent different
types of wind turbines that are characteristic for different locations defined
by mean wind speeds. Complementary to Raynaud et al. [2018], we further
present an alternative approach to defining and evaluating LWPs by looking
either at hours that are constantly below a threshold, or at hours with a mean
below a threshold.
## 2 Methods and data
### 2.1 General approach
Based on wind speeds and power curves, we derive an hourly aggregated time
series of capacity factors for wind power in Germany. First, we take wind
speeds at 50 m above surface from the MERRA-2 reanalysis dataset, which covers
40 years from 1980 to 2019, and extrapolate to hub heights.222See Section A
for further information on the use of reanalysis data for energy modelling.
Second, capacity factors of each MERRA-2 grid cell are calculated based on
power curves of recently installed wind turbines. Third, we spatially
aggregate these capacity factors using a weighting scheme that considers the
current spatial distribution of onshore wind power capacity in Germany.
Finally, we investigate the resulting time series of hourly aggregated
capacity factors by applying a narrower and a wider definition of LWP events.
### 2.2 Wind speeds derived from reanalysis data
We use the MERRA-2 dataset provided by NASA [Gelaro et al., 2017]. Data is
available starting from the year 1980. In contrast to several other global
reanalysis datasets which have time resolutions of 3 to 6 hours and provide
wind speeds at 10 m above surface, MERRA-2 includes hourly wind speed data at
50 m, which allows better modelling of wind power generation.
Figure 1: MERRA-2 grid points (blue) and grid cells that intersect with
Germany.
The MERRA-2 grid consists of 576 longitudinal and 361 latitudinal horizontal
grid points, i.e., a resolution of $0.625^{\circ}$ x $0.5^{\circ}$ which for
Germany roughly corresponds to 50 x 50 km [Bosilovich et al., 2016]. Figure 1
shows the grid points in blue and all grid cells extrapolated from these
points that intersect with Germany. For each grid cell, MERRA-2 provides
hourly northward and eastward wind speed data at 50 m above surface. Our
dataset further includes surface roughness data for the year 2019.
### 2.3 Aggregated wind power derived from wind speeds using power curves
We calculate the magnitude of the horizontal wind speed ($U$) for each MERRA-2
grid point based on northward ($u$) and eastward components ($v$) at 50 m
(Equation 1).
$U=\sqrt{(u^{2}+v^{2})}$ (1)
In line with Kruyt et al. [2017], we use the logarithmic power law to
extrapolate wind speeds to hub-height ($h$) with $U_{hub}$ as the wind speed
at hub height and $z_{0}$ as the surface roughness data for every grid point
and each hour of the year 2019 (Equation 2).
$U_{hub}=w\frac{\ln\frac{h}{z_{0}}}{\ln\frac{50}{z_{0}}}$ (2)
Figure 2: Wind speed zones in Germany. Dark blue implies high mean wind
speeds, blue medium wind speeds, and light blue low mean wind speeds.
We define three types of wind zones, based on mean local wind speeds over 40
years for each MERRA-2 grid cell (Figure 2), and assign typical hub heights
for wind turbines. For high-wind-speed sites, we assign a hub height of 100 m,
for medium-wind-speed sites of 125 m, and for low-wind-speed sites of 139 m
[Wallasch et al., 2015]. These values reflect the mean hub heights of recently
installed wind power plants in respective German wind speed zones.
We calculate hourly capacity factors for each grid cell by applying power
curves characteristic for the three wind zones. The power curves are based on
manufacturer data of currently available wind turbines for low-, medium- and
high-wind sites, respectively. Both the low- and high-wind site power curves
represent an average of four wind turbines of similar diameters and
capacities. We consider turbines from six manufacturers (see B), among them
four large companies which cover 87% of the capacity installed in Germany in
2015 [Lüers, 2016].
Manufacturers generally provide discrete capacity factors ($CF_{disc}$) for
wind speed intervals of 1 m/s. For both the low- and high-wind curves, we
first calculate discrete mean capacity factors for each wind speed and then
calculate continuous capacity factors using a generalized logistic function
(Equation 3).
$CF_{cont}=A+\frac{C}{(1+Te^{-B(U_{hub}-M)})^{1/T}}$ (3)
Here, $CF_{cont}$ is the continuous capacity factor and $A$, $B$, $C$, $M$ and
$T$ are fitted coefficients based on minimising the squared deviations between
$CF_{disc}$ and $CF_{cont}$. For both the low- and the high-wind power curve,
cut-in wind speeds of around 3 m/s emerge, and the resulting capacity factors
are capped at 0% and 100%. The medium-wind power curve represents the average
of the low- and high-wind curves (Figure 3).
Figure 3: Power curves of three types of wind turbines
Aggregated hourly capacity factor time series for overall Germany are derived
by weighting all grid cells with the current distribution of installed wind
power generation capacity. The latter is extracted from Open Power System Data
[Open Power System Data, 2017, Wiese et al., 2019] and open-source GIS data.
The red points in Figure 4 indicate the installed wind capacity of locally
aggregated wind power plant sites in Germany and the blue squares show the
corresponding relative capacity distribution of the MERRA-2 grid cells. Grid
cells only partly intersecting with the German land area receive lower weights
according to the overlapping area. We implicitly assume that the transmission
infrastructure allows geographical balancing of wind power in Germany, which
is currently largely the case.333This assumptions is particularly valid for
low-wind periods. During high-wind, high-load periods, the German transmission
grid can be constrained in North-South direction.
Figure 4: Distribution of currently installed wind power capacity in Germany.
Darker colors indicate a larger share of total or relative installed capacity.
### 2.4 Definition of low-wind-power events
We propose two different measures of low-wind-power periods, a narrower and a
wider one (Figure 5). We further consider three alternative capacity factor
thresholds of 2%, 5%, and 10%.
As for the narrower definition, we consider LWP events to be consecutive hours
in which the aggregated capacity factors are Constantly Below the Threshold
(CBT). This concept bears some resemblance to the “runs analysis” by Leahy and
McKeogh [2013] or the “duration given intensity” method by Patlakas et al.
[2017]. Starting in the first hour, we list annual LWP events for durations
starting from five consecutive hours and report the number of hours constantly
below the given capacity factor threshold. We then increase the duration in
hourly steps and repeat until there are no further events listed.
To provide a wider definition, we consider LWP events to consist of
consecutive hours in which the moving average of capacity factors is under the
same threshold, i.e., Mean Below the Threshold (MBT). Again, we list all LWP
periods until we reach the threshold value, ensuring that LWP periods do not
overlap. By definition, the MBT method results in more low-wind-power events
for a given duration and also results in longer events for each threshold,
compared to CBT.
Figure 5: Illustration of the two LWP event definitions
The average annual amount of LWP events per duration over all 40 years equals
the expected value of events per year. Further, the reciprocal value of the
annual average provides the return period, that is the expected temporal
distance between two similar reoccurring events. Periods overlapping annually
or monthly are assigned to the year or month in which more than 50% of the
hours are located444 Accounting for annually overlapping periods requires
December data from the previous year, and January data from the subsequent
year. For the two boundary years 1980 and 2019, we substitute the missing data
for December 1979 (January 2020) with data from December 1980 (January 2019).
.
## 3 Results
### 3.1 Seasonal distribution and frequency of low-wind-power events
Figure 6 shows that LWP events are generally most frequent in summer (here
defined as June-August) and least frequent in winter (December-February). The
results for spring (March-May) and autumn (September-November) are mostly
close to the annual average. Accordingly, respective findings made for other
European countries [Leahy and McKeogh, 2013, Cannon et al., 2015, Kruyt et
al., 2017] are also valid for Germany.
Figure 6: Average seasonal duration (horizontal axis) and frequency (vertical
axis) of LWP events in Germany
The frequency of events for a given duration is about 1.5-3 times higher for
the wider MBT definition compared to the narrower CBT concept. For both
metrics, the frequency of LWP events increases substantially with the capacity
factor threshold value. For example, a 10-hour event below a capacity factor
of 2% occurs on average around 0.2 times per winter for CBT and slightly less
than once per winter for MBT. For a 10% capacity factor threshold, there are
on average around eight such winter events for CBT and 13 for MBT. In general,
we find that short LWP events with a duration of up to around half a day are
relatively frequent and may occur several times per year, especially under the
wider MBT definition. Longer LWP events, in contrast, are much less frequent.
To provide a complementary perspective, we calculate the return periods for
different durations of LWP events (Table 1). The return periods are the
reciprocal of the average (annual or seasonal) frequency of LWP events for
different durations, considering both definitions and all three thresholds
(cf. Figure 6). For example, an LWP event with an average frequency of 0.2 for
a given duration leads to a return period of 5 years for this specific
duration. The longer a given duration, the lower its average frequency and the
longer its return period.
For a return period of ten years, we find a duration of 17 hours (2% capacity
factor threshold), 41 hours (5%) and 77 hours (10%) under the narrower CBT
definition, and a duration of 34 hours (2%), 79 hours (5%) and 188 hours (10%)
under the wider MBT concept. In other words, every ten years the German energy
system has to deal with a period of nearly eight days of average wind power
generation (MBT) below 10% of the installed capacity.
Table 1: Duration in hours for LWP events in winter or in any season for different return periods | Constantly below threshold (CBT) | Mean below threshold (MBT)
---|---|---
| Winter | Any season | Winter | Any season
Return period | 2% | 5% | 10% | 2% | 5% | 10% | 2% | 5% | 10% | 2% | 5% | 10%
1 year | 5 | 15 | 29 | 11 | 23 | 45 | 8 | 30 | 63 | 18 | 58 | 122
2 years | 7 | 21 | 40 | 13 | 32 | 57 | 12 | 45 | 92 | 21 | 69 | 144
3 years | 8 | 23 | 44 | 14 | 33 | 60 | 14 | 52 | 101 | 23 | 71 | 161
4 years | 9 | 30 | 48 | 14 | 33 | 63 | 16 | 62 | 112 | 27 | 72 | 173
5 years | 10 | 32 | 57 | 15 | 35 | 65 | 22 | 68 | 113 | 28 | 75 | 178
6 years | 10 | 32 | 57 | 15 | 35 | 67 | 25 | 69 | 114 | 29 | 76 | 182
7 years | 12 | 33 | 60 | 15 | 36 | 67 | 27 | 70 | 114 | 31 | 76 | 186
8 years | 14 | 33 | 63 | 17 | 37 | 69 | 28 | 72 | 117 | 33 | 79 | 186
9 years | 14 | 33 | 63 | 17 | 37 | 69 | 28 | 72 | 117 | 33 | 79 | 186
10 years | 14 | 33 | 64 | 17 | 41 | 77 | 28 | 72 | 126 | 34 | 79 | 188
15 years | 17 | 36 | 67 | 18 | 41 | 77 | 31 | 76 | 129 | 38 | 82 | 189
20 years | 19 | 41 | 77 | 19 | 49 | 81 | 34 | 79 | 131 | 45 | 89 | 221
25 years | 19 | 41 | 77 | 19 | 49 | 81 | 34 | 79 | 131 | 45 | 89 | 221
30 years | 19 | 41 | 77 | 19 | 49 | 81 | 34 | 79 | 131 | 45 | 89 | 221
To better interpret these return periods, we provide an example for the German
onshore wind power capacity of 52.5 GW installed in 2018. For this wind
turbine fleet, average power generation is expected to not exceed around five
GW, i.e., 10% of capacity, during a period of around five consecutive days
every year (122 hours, MBT for ’Any Season’ in 1). Every ten years, this
period increases to nearly eight days, and every twenty years to more than
nine full days. Looking only at LWP events in winter, these durations decrease
to less than three days every winter, less than five days every tenth winter,
and around five and a half days every twentieth winter. The remaining load has
to be covered by other generators, energy storage or demand-side measures.
However, wind power still contributes some generation capacity above the 10%
threshold during some of these hours, as indicated by much lower CBT return
periods.
### 3.2 Magnitude of the most extreme low-wind-power events
The most extreme LWP events over the entire 40 years analyzed can be
interpreted as worst cases from an energy system planning perspective. In an
annual perspective, the most extreme events occurred in 1985 for all capacity
factor thresholds (Figure 7). Under the narrower CBT definition, there are
nearly four consecutive days with wind power generation constantly below 10%
in 1985, and still around two consecutive days with generation constantly
below 5%. Under the wider MBT definition, the duration of this most extreme
event increases to nearly ten days (10%) or around four days (5%).
Figure 7: Most extreme LWP events per year. The vertical axis shows the
duration of the longest event per year for the three capacity factor
thresholds.
While this 1985 event is the most extreme one under both CBT and MBT, the
ranking of the second most extreme yearly events differs between the LWP
definitions. For example, the second-longest event occurred in 1984 under the
CBT definition. Yet under MBT, the duration of the most extreme event in 1984
is only average. In general, the definition of LWP events and the chosen
thresholds have a substantial impact on quantitative results. Under MBT, the
most extreme annual events are generally around twice as high compared to CBT.
We further find very large inter-annual variations. Considering the 10%
threshold, the longest event for the MBT definition lasted for almost 10 days
in 1985, but in 2005 the longest duration was only three days for the same
threshold. The relative difference between the longest events for each year
increases with the threshold. These large variations of the most extreme
annual LWP events complement the findings made by Collins et al. [2018], who
determine large inter-annual variations of average renewable availability.
We next look at the most extreme LWP event in a monthly perspective,
irrespective of the year in which these occur (Figure 8). The most extreme
events for the 10% threshold occur in March for both definitions. This is the
1985 event discussed above, with durations of nearly four (CBT) or nearly ten
consecutive days (MBT).
Figure 8: Most extreme LWP events per month. The vertical axis shows the
duration of the longest event of all respective months for the three capacity
factor thresholds.
Considering all thresholds and both LWP definitions, there is no clear trend
of the most extreme monthly LWP events. That is, substantial extreme events
may occur throughout the year, and also in winter months. This contrasts the
previous finding that the frequency of LWP events is generally much higher in
summer than in winter, as shown in Section 3.1. Under CBT, the most extreme
events in each of the winter months are even longer than those in summer
months for the 10% capacity threshold. This finding is, however, not confirmed
under the MBT definition.
### 3.3 Spatial distribution of wind power during most extreme LWP event
To also explore the spatial dimension of LWP events, we compare the
distribution of capacity factors during the most extreme LWP of 1985 to the
distribution of annual mean capacity factors in the same year (Figure 9).
Figure 9: Spatial distribution of wind power. Left: Average wind power during
most extreme LWP event (10% capacity factor, MBT) in dataset in March 1985
(Scale: From 0% to 20% of mean capacity factors). Right: Mean wind power in
the entire year 1985 (Scale: From 5% to 50% of mean capacity factors).
The spatial pattern of annual mean capacity factors (Figure 9, right panel)
largely resembles that of average wind speeds in Germany (Figure 2). Mean
capacity factors are generally higher in Northern than in Southern Germany.
They are highest close to the Northern and the Baltic Sea, and lowest in the
southern Alpine region.
The spatial pattern of mean capacity factors during the most extreme LWP event
(Figure 9, left panel) substantially deviates from the distribution of the
means. In particular, capacity factors of the north-eastern region and parts
of the northern region are relatively low. The respective spatial
distributions of capacity factors for other thresholds under both the CBT and
MBT definitions of the same event also show substantial deviations from annual
means.
Accordingly, the spatial distribution of capacity factors during extreme LWP
events does not necessarily correspond to the annual mean pattern. This
indicates that low-wind events can be very pronounced even in regions with
very good average wind resources.
## 4 Conclusions
We analyze the seasonal distribution, frequency and magnitude of onshore low-
wind-power events in Germany, as well as spatial aspects of the most extreme
events, based on MERRA-2 reanalysis data and open software. We propose and
evaluate two definitions of low-wind-power events for three capacity factor
thresholds.
We synthesize three key results from the analysis. First, LWP events are
generally most frequent in summer and least frequent in winter. Nonetheless,
substantial events occur in all months of the year, and also in winter. The
most persistent LWP event in the dataset occurred in March.
Second, while short events with a duration of up to around half a day are
relatively frequent, very long events are much rarer.555Weber et al. [2019]
argue that low-wind event statistics do not follow a simple exponential
distribution, but have “heavy tails”, i.e. the probability decreases rather
slowly with increasing duration. Every year, the German energy system has to
deal with a period of around five consecutive days during which average wind
power generation is below 10% of the installed capacity. Every ten years, a
respective period of nearly eight days is to be expected. Looking only at
winter months, the durations of these expected events decrease to less than
three days every winter and less than five days every tenth winter. The most
persistent low-wind event in the entire dataset has a duration of nearly ten
consecutive days of average wind power generation below a 10% capacity factor.
Third, the spatial pattern of LWP events may be very different from the one of
average wind power resources. During the most persistent LWP event, we find
average generation to be particularly low in several regions which have some
of the best wind resources.
We conclude that energy modeling studies that only consider one historic
weather year are likely to substantially underestimate the occurrence of low-
wind-power events and related system implications. In particular, analyses
with an energy system planning perspective should take less frequent LWP
events into account, e.g., the discussed events with a return period of ten
years, or even the most extreme event identified here. This is particularly
important when the complementary role of other variable and dispatchable
generators, energy storage, or demand-side measures in highly-renewable energy
systems is to be explored.666This is demonstrated, for example, by Schill and
Zerrahn [2018] in an analysis of storage requirements for renewable energy
integration in a sensitivity analysis with one artificial no-wind week.
Further, analyses dealing with the pros and cons of either more decentralized
or more centralized renewable energy systems should consider the spatial
dimension of LWP events. Although not in the focus of our analysis, our
results indicate that LWP events are more pronounced for smaller geographic
areas.
From an energy policy perspective, our findings on LWP events occurring in
winter may be most relevant. Our analysis indicates that concerns about
frequent and persistent LWP events in German winters appear to be overrated,
considering that the longest event with an average capacity factor below 10%
and a ten-year return period in winter has a duration of less than five days.
We further recommend that policy makers or regulators develop a proper
definition of the Dunkelflaute term, which currently appears to be used in a
rather qualitative way. Our two definitions of LWP events proposed here may be
useful in this context.
While our analysis deliberately focuses on LWP events of onshore wind power in
Germany, we see an avenue for future research that would ideally combine the
analysis of low production periods of onshore and offshore wind power as well
as solar PV with time series of load, while expanding the geographic focus
beyond Germany. The open-source provision of the tool used for the present
analysis may be a useful starting point for such research.
## Acknowledgment
This analysis is a result of the research project P2X, funded by the German
Federal Ministry of Education and Research (FKZ 03SFK2B1). Wolf-Peter Schill
carried out parts of the work during a research stay at the University of
Melbourne. Nils Ohlendorf mainly worked on this project while employed at DIW
Berlin, and partly also after being employed at MCC. We thank the participants
of the DIW Sustainability Cluster Seminar in April 2017, Strommarkttreffen
Berlin November 2017, IAEE International Conference Groningen 2018 and Enerday
Dresden 2018 for valuable comments on earlier drafts.
## Data availability statement
The data that support the findings of this study have been created with
software that is openly available under an MIT license at
https://doi.org/10.5281/zenodo.3694373.
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## Appendix A Reanalysis data and its use for energy modelling
Reanalysis data is increasingly used for energy modelling as it provides
consistent global time series of long-term atmosphere data such as wind speed,
temperature and air pressure in regular spatial and temporal resolutions. The
underlying global circulation models extrapolate measurement station data on
wind speeds, temperature, moisture and surface pressure as well as data from
satellites and precipitation measurements [Decker et al., 2012]. Several
publicly available second-generation global reanalysis datasets have been
released since the early 2000s. We use MERRA-2, which builds on and improves
the previous MERRA dataset, using advanced models and data sources [Molod et
al., 2015].
Decker et al. [2012] evaluate the accuracy of several reanalysis datasets
(MERRA, NCEP, ERA-40, ERA-Interim, CFSR and GLDAS) using flux tower
measurements in the Northern Hemisphere. Almost all products overestimate the
monthly and 6-hourly wind speeds and their variability. MERRA and ERA-Interim
show the lowest values root-mean-square error and bias for diurnal cycles.
Sharp et al. [2015] review other data validation studies of different
reanalysis datasets. Three studies derive Pearson’s correlation coefficients
for MERRA between 0.75 and 0.89 based on measurement stations in Sweden,
Portugal, Norway and Denmark [Liléo and Petrik, 2011, Liléo et al., 2013,
Carvalho et al., 2014]. Staffell and Pfenninger [2016] propose country-
specific wind speed bias correction factors for MERRA and MERRA-2 to increase
the correlation with national capacity factors. Without such correction,
average capacity factors for Germany based on raw MERRA or MERRA-2 wind speeds
would be overestimated. Staffell and Green [2014] make a similar point for the
UK. In contrast, Cannon et al. [2015] do not use correction factors in a UK
application. Even if MERRA wind speeds turn out to be not particularly valid
for single measurement points, spatial aggregation of mean wind speed over all
stations results in a correlation coefficient of 0.94. This indicates a high
validity of MERRA data for large-scale wind patterns. Following Cannon et al.
[2015], we also refrain from introducing correction factors and instead make
use of the error-smoothing effect of spatial aggregation. In doing so, we also
avoid model artefacts, particularly as the usefulness of correction factors
has only been demonstrated for average wind speeds, but not for extreme
values.
## Appendix B Wind power turbines
The low- and high-wind power curves used in our analysis are based on data of
eight wind power turbines by six manufacturers, namely Nordex, Senvion,
Enercon, Vestas, Gamesa and Vensys. Specifically, we use the following high-
wind power turbines:
* 1.
Nordex N90-2.5MW
* 2.
Vestas V90-2.0MW
* 3.
Gamesa G97-2MW
* 4.
Vensys 100-2.5MW
Analogously, we use following low-wind power turbines:
* 1.
Nordex N131-3.3MW
* 2.
Senvion 3.2M122
* 3.
Enercon E126 EP4/4.2MW
* 4.
Vestas V126-3.3MW
## Appendix C Discussion of limitations
We briefly discuss some limitations of our analysis and how these may
qualitatively impact results.
First, there are general limitations of using reanalysis data which have been
discussed in the literature, for example spatial biases or issues with
upscaling to hub heights [Sharp et al., 2015, Olauson and Bergkvist, 2015,
Rose and Apt, 2015, Staffell and Pfenninger, 2016]. It is, however, not clear
if there are specific distortions with respect to extreme low-wind events
derived from reanalysis data. A limitation specific to the MERRA-2 dataset is
the relatively coarse 50x50 km grid cell size, which insufficiently represent
local impacts on wind speeds. Regional reanalysis data with more refined
geographical resolutions may resolve this issue, e.g. COSMO-REA2 with 2x2 km,
or COSMO-REA6 with 6x6 km [Hans Ertel Zentrum, 2019], yet these are only
available for shorter periods of time. The global coverage of MERRA-2 further
allows repeating our open-source analysis for other countries and world
regions.
Second, we use power curves of currently available wind turbines and assume
hub-heights of recently constructed plants. We may thus overestimate wind
power generation compared to the currently existing fleet of wind turbines in
Germany, which includes many older and smaller turbines, and in turn
underestimate the magnitude of current LWP events. Conversely, we may
underestimate power generation of future turbines, and accordingly
overestimate the magnitude of future low-wind-power events, assuming that
turbine efficiency and hub height increases further, with corresponding upward
shifts in the power curves. Once LWP events become more relevant for the
overall energy system, this may also trigger specific technology improvements
toward lower cut-in speeds and a steeper slope of the power curve on the very
left-hand side. Quantifying the potentially mitigating effects of such
developments on LWP periods is left for future research.
Third, we use the current spatial capacity distribution of German wind power
plants for deriving an aggregated capacity factor time series. We implicitly
assume that this distribution also persists in the future. In reality, a
relative increase of wind power deployment at sites with lower wind resources
may occur, for example in southern Germany. From the results presented in
Section 3.1, we infer that a more even spatial dispersion of wind turbines
could slightly mitigate LWP events.
Next, climate change has an impact on wind speeds. Future time series of wind
power capacity factors will thus differ from the historic ones investigated
here. Tobin et al. [2016] find that wind power variability in Europe may
generally increase, but Schlott et al. [2018] conclude that this has no
substantial effect on optimal deployment of onshore wind power in highly
renewable future scenarios. Moemken et al. [2018] find that climate change
will increase the occurrence of low wind speeds.
Finally, the focus of this analysis is a detailed but selective investigation
of onshore LWP events in Germany. This geographic focus helps to keep the
analysis tractable and avoids making implicit assumptions on continental
electricity transmission infrastructure. It is also relevant from an energy
policy perspective, which often includes national energy security
considerations. Yet expanding the geographic scope of the analysis would allow
raising complementary insights on larger-scale spatial patterns of extreme LWP
events. Focusing on onshore wind power, and not including other renewable
energy sources such as offshore wind power and solar PV, allows for
parsimonious model assumptions, and findings remain valid for any level of
installed capacity. Analyses that would combine periods of low production from
various renewable energy sources, and also explore their correlation with
electric load, appear to be a promising field for future research. The work of
Raynaud et al. [2018], albeit with lower temporal and spatial detail compared
to our analysis, can be considered as a first step in this direction.
|
2024-09-04T02:54:58.461125 | 2020-03-09T16:24:41 | 2003.04239 | {
"authors": "Debajyoti Choudhuri, Jiabin Zuo",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26121",
"submitter": "Debarjoyti Choudhuri",
"url": "https://arxiv.org/abs/2003.04239"
} | arxiv-papers | # A shadow of algebraic topology and variational method - Prandtl Batchelor
problem
Debajyoti Choudhuri111Corresponding author<EMAIL_ADDRESS>ORCID ID:
0000-0001-8744-9350
Department of Mathematics, National Institute of Technology Rourkela, India
###### Abstract
In this paper we study the existence of nontrivial weak solution to a Prandtl-
Batchelor type free boundary value elliptic problem involving a $p$-Laplacian
operator and a power nonlinearity. Topics from algebraic topology will be used
to establish the existence of a solution to the approximating problem,
whereas, the variational technique will be used to fix the claim of existence
of a solution to the main problem. In the process, a couple of classical
results were also improved to suit the purpose of establishing the existence
of a nontrivial solution.
Keywords: Dirichlet free boundary value problem, Sobolev space, Morse
relation, cohomology group.
AMS Classification: 35J35, 35J60.
## 1 Introduction
We will investigate the existence of solution to the following free boundary
value problem.
$\displaystyle\begin{split}-\Delta_{p}u&=\lambda\chi_{\\{u>1\\}}(u-1)_{+}^{q-1},~{}\text{in}~{}\Omega\setminus
H(u),\\\ |\nabla u^{+}|^{p}-|\nabla
u^{-}|^{p}&=\frac{p}{p-1},~{}\text{in}~{}H(u)\\\
u&=0,\text{on}~{}\partial\Omega.\end{split}$ (1.1)
Here, $\lambda>0$ is a parameter, $(u-1)_{+}=\max\\{u-1,0\\}$ and
$H(u)=\partial\\{u>1\\}.$
Also $\nabla u^{\pm}$ are the limits of $\nabla u$ from the sets $\\{u>1\\}$
and $\\{u\leq 1\\}^{\circ}$ respectively. The domain
$\Omega\subset\mathbb{R}^{N}(N\geq 2)$ is bounded with a sufficiently smooth
boundary $\partial\Omega$. The relation between the exponents are assumed in
the order $1<p\leq q-1$, with $q<p^{*}=\dfrac{Np}{N-p}$. The solution(s)
satisfy the free boundary condition in the following sense: for all
$\vec{\phi}\in C_{0}^{1}(\mathbb{R}^{N})$ such that $u\neq 1$ a.e. on the
support of $\vec{\phi}$,
$\displaystyle\underset{\epsilon^{+}\rightarrow
0}{\lim}\int_{u=1+\epsilon^{+}}\left(\frac{p}{p-1}-|\nabla
u|^{p}\right)\vec{\phi}\cdot\hat{n}dS-\underset{\epsilon^{-}\rightarrow
0}{\lim}\int_{u=1-\epsilon^{-}}|\nabla u|^{p}\vec{\phi}\cdot\hat{n}dS$
$\displaystyle=0,$ (1.2)
where $\hat{n}$ is the outward drawn normal to
$\\{1-\epsilon^{-}<u<1+\epsilon^{+}\\}$. Note that the sets
$\\{u=1\pm\epsilon^{\pm}\\}$ are smooth hypersurfaces for almost all
$\epsilon^{\pm}>0$ by the Sard’s theorem. The limit above in (1.2) is taken by
running such $\epsilon^{\pm}>0$ towards zero.
A rich literature survey has been done in the book due to Perera et al. [10]
where the author has discussed problems of several variety involving the
$p$-Laplacian operators which could be studied using the Morse theory. The
motivation for the current work has been drawn from the work due to Perera
[13]. The treatment used to address the existence of atleast one (or two)
solution(s) to the approximating problem may be classical (section $3$,
Theorems 3.3 and 3.5) but the result concerning the reguarity of the free
boundary is very new and the question of existence of solution to the problem
(1.1) has not been answered till now (section $4$, Lemma 4.1), to the best of
my knowledge. Two more results due to Alt-Caffarelli [1] (section $4$, Lemma
4.2) and Caffarelli et al. [6] (Appendix, Lemma 4.3) were improved to the best
possible extent to suit the purpose of the problem in this paper.
### 1.1 A physical motivation
Consider the problem
$\displaystyle\begin{split}-\Delta
u&=\lambda\chi_{\\{u>1\\}}(x),~{}\text{in}~{}\Omega\setminus H(u),\\\ |\nabla
u^{+}|^{2}-|\nabla u^{-}|^{2}&=2,~{}\text{in}~{}H(u)\\\
u&=0,\text{on}~{}\partial\Omega.\end{split}$ (1.3)
This is the well known Prandtl-Batchelor free boundary value problem, where
the phase $\\{u>1\\}$ is a representation of the vortex patch bounded by the
vortex line $u=1$ in a steady fluid flow for $N=2$ (refer Batchelor [2, 3]).
Thus the current problem is a more generalized version of (1.3). For a more
physical application to this problem we direct the reader’s attention to the
work due to Caflisch [4], Elcrat and Miller [7].
Another instance of occurrence of such a phenomena is in the non-equilibrium
system of melting of ice. In a given block of ice, the heat equation can be
solved with a given set of appropriate initial/boundary conditions in order to
determine the temperature. However, if there exists a region of ice in which
the temperature is greater than the melting point of ice, this subdomain will
be filled with water. The boundary thus formed due to the ice-water interface
is controlled by the solution of the heat equation.
Thus encountering a free boundary in the nature is not unnatural. The problem
in this paper is a large enough generalization to this physical phenomena
which besides being a new addition to the literature can also serve as a note
to bridge the problems in elliptic PDEs with algebraic topology.
## 2 Preliminaries
We begin by giving the relevant definitions and results besides defining the
function space which will be used very frequently in the article. Let $X$ be a
topological space and $A\subset X$ be a topological subspace. Roughly, a
homology group is an algebraic group constructed from a topological object or
a space. Following is the fundamental tool that will be used to work with,
namely the homology theory [12].
###### Definition 2.1.
A homology group on a family of pairs of spaces $(X,A)$ consists of:
1. 1.
A sequence $\\{H_{k}(X,A)\\}_{k\in\mathbb{N}_{0}}$ of abelian groups is known
as homology group for the pair $(X,A)$ (note that for the pair $(X,\phi)$, we
write $H_{k}(X),k\in\mathbb{N}_{0}$). Here
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$.
2. 2.
To every map of pairs $\varphi:(X,A)\rightarrow(Y,B)$ is associated a
homomorphism $\varphi^{*}:H_{k}(X,A)\rightarrow H_{k}(Y,B)$ for all
$k\in\mathbb{N}_{0}$.
3. 3.
To every $k\in\mathbb{N}_{0}$ and every pair $(X,A)$ is associated a
homomorphism $\partial:H_{k}(X,A)\rightarrow H_{k-1}(A)$ for all
$k\in\mathbb{N}_{0}$.
These items satisfy the following axioms.
($A_{1}$)
If $\varphi=id_{X}$, then $\varphi_{*}=id|_{H_{k}(X,A)}$.
($A_{2}$)
If $\varphi:(X,A)\rightarrow(Y,B)$ and $\psi:(Y,B)\rightarrow(Z,C)$ are maps
of pairs, then $(\psi\circ\varphi)_{*}=\psi_{*}\circ\varphi_{*}$.
($A_{3}$)
If $\varphi:(X,A)\rightarrow(Y,B)$ is a map of pairs, then
$\partial\circ\varphi_{*}=(\varphi|_{A})_{*}\circ\partial$.
($A_{4}$)
If $i:A\rightarrow X$ and $j:(X,\phi)\rightarrow(X,A)$ are inclusion maps,
then the following sequence is exact
$...\xrightarrow[]{\partial}H_{k}(A)\xrightarrow[]{i_{*}}H_{k}(X)\xrightarrow[]{j_{*}}H_{k}(X,A)\xrightarrow[]{\partial}H_{k-1}(A)\rightarrow...$
Recall that a chain
$...\xrightarrow[]{\partial_{K+1}}C_{k}(X)\xrightarrow[]{\partial_{k}}C_{K-1}(X)\xrightarrow[]{\partial_{k-1}}C_{k-2}(X)\xrightarrow[]{\partial_{k-2}}...$
is said to be exact if $im(\partial_{k+1})=ker(\partial_{k})$ for each
$k\in\mathbb{N}_{0}$.
($A_{5}$)
If $\varphi,\psi:(X,A)\rightarrow(Y,B)$ are homotopic maps of pairs, then
$\varphi_{*}=\psi_{*}$.
($A_{6}$)
(Excision): If $U\subseteq X$ is an open set with
$\bar{U}\subseteq\text{int}(A)$ and $i:(X\setminus U,A\setminus
U)\rightarrow(X,A)$ is the inclusion map, then $i_{*}:H_{k}(X\setminus
U,A\setminus U)\rightarrow H_{k}(X,A)$ is an isomorphism.
($A_{7}$)
If $X=\\{*\\}$, then $H_{k}({*})=0$ for all $k\in\mathbb{N}$.
###### Definition 2.2.
A continuous map $F:X\times[0,1]\to X$ is a deformation retraction of a space
$X$ onto a subspace $A$ if, for every $x\in X$ and $a\in A$, $F(x,0)=x$,
$F(x,1)\in A$, and $F(a,1)=a$.
A crucial notion in analysis is the idea of compactness and the Palais-Smale
condition is a special type of compactness which is given as follows.
###### Definition 2.3.
(S. Kesavan [11]) Let $V$ be a Banach space and $J:V\rightarrow\mathbb{R}$ a
$C^{1}$ functional. It is said to satisfy the Palais-Smale condition (PS) if
the following holds: whenever $(u_{n})$ is a sequence in $V$ such that
$(J(u_{n}))$ is bounded and $J^{\prime}(u_{n})\rightarrow 0$ in $V^{\prime}$
(the dual space of $V$), then $(u_{n})$ has a strongly convergent subsequence.
The following is a deformation lemma which will be quintessential in computing
the homology groups.
###### Lemma 2.4.
(S. Kesavan [11]) Let $J:V\rightarrow\mathbb{R}$ be a $C^{1}$ functional
satisfying the Palais-Smale condition. Let $c,a$ be real numbers. Define
$K_{J,c}=\\{v\in X:J(u)=c,J^{\prime}(v)=0\\}$, $K^{a}=\\{v\in X:J(v)\leq a\\}$
(likewise we define $K_{a}=\\{v\in X:J(v)\geq a\\}$). Let $K_{J,c}=\O$. Then
there exists $\epsilon^{\prime}>0$ and a continuous homotopy $\eta:[0,1]\times
V\rightarrow V$ such that $\forall~{}0<\epsilon\leq\epsilon^{\prime}$
1. 1.
$\eta(0,v)=v$ for all $v\in X$.
2. 2.
$\eta(t,v)=v$ for all $t\in[0,1]$, $v\neq J^{-1}([c-\epsilon,c+\epsilon])$.
3. 3.
$\eta(1,K^{c+\epsilon})\subset K^{c-\epsilon}$.
###### Definition 2.5.
Morse index of a functional $J:V\rightarrow\mathbb{R}$ is defined to be the
maximum subspace of $V$ such that $J^{\prime\prime}$, the second Fréchet
derivative, is negative definite on it.
### 2.1 Space description
We begin by defining the standard Lebesgue space $L^{p}(\Omega)$ for $1\leq
p<\infty$ as
$L^{p}(\Omega)=\left\\{u:\Omega\rightarrow\mathbb{R}:u\;{\text{is measurable
and}}\int_{\Omega}|u|^{p}dx<\infty\right\\}$
endowed with the norm $\|u\|_{p}=\left(\int_{\Omega}|\nabla
u|^{p}dx\right)^{\frac{1}{p}}$. We will define the Sobolev space as
$W^{1,p}(\Omega)=\\{u\in L^{p}(\Omega):\nabla u\in(L^{p}(\Omega)^{N}\\}$
with the norm $\|u\|_{1,p}^{p}=\|u\|_{p}+\|\nabla u\|_{p}$. We further define
$W_{0}^{1,p}(\Omega)=\\{u\in
W^{1,p}(\Omega):u=0~{}\text{on}~{}\partial\Omega\\}.$
The associated norm is $\|u\|^{p}=\|\nabla u\|_{p}$.
With these norms, $L^{p}(\Omega)$, $W^{1,p}(\Omega)$ and $W_{0}^{1,p}(\Omega)$
are separable, reflexive Banach spaces([11]). We now state the Hölder’s
inequality and embedding results in the following propositions.
###### Proposition 2.6.
For any $u\in L^{p}(\Omega)$ and $v\in L^{p^{\prime}}(\Omega)$, where
$L^{p^{\prime}}(\Omega)$ is the conjugate space of $L^{p}(\Omega)$ such that
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$,
$\big{|}\int_{\Omega}uv\;dx\big{|}\leq\|u\|_{p}\|v\|_{p^{\prime}}$
###### Proposition 2.7.
If $p<N$, then $W^{1,p}(\Omega)\hookrightarrow L^{r}(\Omega)$ is continuous
for $r\in[p,p^{*}]$ and compact for $r\in[p,p^{*})$. If $p=N$, then
$W^{1,p}(\Omega)\hookrightarrow L^{r}(\Omega)$ is continuous and compact for
$r\in[p,\infty)$. Further, if $p>N$, then $W^{1,p}(\Omega)\hookrightarrow
C^{1-\left[\frac{N}{p}\right]}(\bar{\Omega})$.
## 3 The way to tackle the problem using Morse theory
We at first define an energy functional associated to the problem in (1.1)
which is as follows.
$\displaystyle\begin{split}I(u)&=\int_{\Omega}\frac{|\nabla
u|^{p}}{p}dx+\int_{\Omega}\chi_{\\{u>1\\}}(x)dx-\lambda\int_{\Omega}\frac{(u-1)_{+}^{q}}{q}dx.\end{split}$
This functional is not even differentiable and hence poses serious issues as
far as the application of variational theorems are concerned. Thus we
approximate $I$ using the following functionals that varies with respect to a
parameter $\alpha>0$. This method is adapted from the work of Jerison-Perera
[9]. We define a smooth function $g:\mathbb{R}\rightarrow[0,2]$ as follows:
$g(t)=\begin{cases}0,&\text{if}~{}t\leq 0\\\ \text{a positive
function},&\text{if}~{}0<t<1\\\ 0,&\text{if}~{}t\geq 0\end{cases}$
and $\int_{0}^{1}g(t)dt=1$. We further let $G(t)=\int_{0}^{t}g(t)dt$. Clearly,
$G$ is smooth and nondecreasing function such that
$G(t)=\begin{cases}0,&\text{if}~{}t\leq 0\\\ \text{a positive
function}<1,&\text{if}~{}0<t<1\\\ 1,&\text{if}~{}t\geq 0.\end{cases}$
We thus define
$\displaystyle\begin{split}I_{\alpha}(u)&=\int_{\Omega}\frac{|\nabla
u|^{p}}{p}dx+\int_{\Omega}G\left(\frac{u-1}{\alpha}\right)dx-\lambda\int_{\Omega}\frac{(u-1)_{+}^{q}}{q}dx.\end{split}$
This functional $I_{\alpha}$, is of at least $C^{2}$ class and hence
$\displaystyle\langle I_{\alpha}^{\prime\prime}(u)v,w\rangle=$
$\displaystyle\int_{\Omega}[|\nabla u|^{p-2}\nabla v\cdot\nabla w+(p-2)|\nabla
u|^{p-4}(\nabla u\cdot\nabla v)(\nabla u\cdot\nabla w)]dx$
$\displaystyle+\int_{\Omega}\frac{1}{\alpha^{2}}g^{\prime}\left(\frac{u-1}{\alpha}\right)vwdx-\lambda\int_{\Omega}(u-1)_{+}^{q-2}vwdx.$
Following is an important result in Morse theory which explains the effect of
the associated Homology groups on the set $K_{J,(-\infty,a]}=\\{x\in
V:f(x)\leq a\\}$.
###### Theorem 3.1.
Let $J\in C^{2}(V)$ satisfy the Palais-Smale condition and let ‘$a$’ be a
regular value of $J$. Then if, $H_{*}(V,J^{a})\neq 0$, implies that
$K_{J,(-\infty,a]}\neq\emptyset$.
###### Remark 3.2.
Before we apply the Morse lemma we recall that for a Morse function the
following holds
1. 1.
$H_{*}(J^{c},f^{c}\setminus\text{Crit}(J,c))=\oplus_{j}H_{*}(J^{c}\cap
N_{j},J^{c}\cap N_{j}\setminus\\{x_{j}\\}),$
where $\text{Crit}(J,c)=\\{x\in V:J(x)=c,J^{\prime}(x)=0\\}$, $N_{j}$ is a
neighbourhood of $x_{j}$.
2. 2.
$H_{k}(J^{c}\cap N,J^{c}\cap
N\setminus\\{x\\})=\begin{cases}\mathbb{R},&k=m(x)\\\
0,&\text{otherwise}\end{cases}$
where $m(x)$ is a Morse index of $x$, a critical point of $J$.
3. 3.
Further
$H_{k}(J^{a},J^{b})=\oplus_{\\{i:m(x_{i})=k\\}}\mathbb{R}=\mathbb{R}^{m_{k}(a,b)}$
where $m_{k}(a,b)=n(\\{i:m(x_{i})=k,x_{i}\in K_{J,(a,b)}\\})$. Here $n(S)$
denotes the number of elements present in the set $S$.
4. 4.
Morse relation
$\sum_{u\in K_{J,[a,b]}}\sum_{k\geq 0}\text{dim}(C_{k}(J,u))t^{k}=\sum_{k\geq
0}\text{dim}(H_{k}(J^{a},J^{b}))t^{k}+(1+t)\mathcal{Q}_{t}$
for all $t\in\mathbb{R}$. Here $Q_{t}$ is a nonnegative polynomial in
$\mathbb{N}_{0}[t]$.
###### Theorem 3.3.
The functional $I_{\alpha}$ has at least one nontrivial critical point when
$0<\lambda\leq\lambda_{1}$, $\lambda_{1}$ being the first eigen value of
$(-\Delta)_{p}$.
###### Proof.
We observe that $I_{\alpha}(tu)\rightarrow-\infty$ as $t\rightarrow\infty$. A
key observation here is that there exists $R$ sufficiently small such that
$I_{\alpha}(u)\geq\alpha>0$
whenever $\|u\|=R$. We choose $\epsilon>0$ such that $c=\epsilon$ is a regular
value of $I_{\alpha}$. Thus, $I_{\alpha}^{\epsilon}$ is not path connected
since it has at least two path connected components namely in the form of a
neighbourhood of $0$ and a set $\\{u:\|u\|\geq R\\}$ for $R$ suffciently
large. From the theory of homology groups we get that
$\text{dim}(H_{0}(I_{\alpha}^{\epsilon}))\geq 2$, ‘dim’ denoting the dimension
of the Homology group. From the Definition 2.1, let us consider the following
exact sequence
$...\rightarrow
H_{1}(W_{0}^{1,p(x)}(\Omega),I_{\alpha}^{\epsilon})\xrightarrow[]{\partial_{1}}H_{0}(I_{\alpha}^{\epsilon},\emptyset)\xrightarrow[]{i_{0}}H_{0}(W_{0}^{1,p(x)}(\Omega),\emptyset)\rightarrow...$
Obviously $\text{dim}(H_{0}(W_{0}^{1,p(x)}(\Omega),\emptyset))=1$ and
$\text{dim}(H_{0}(I_{\alpha}^{\epsilon}))\geq 2$. Due to the exactness of the
sequence we conclude that
$\text{dim}H_{1}(W_{0}^{1,p(x)}(\Omega),I_{\alpha}^{\epsilon})\geq 1$. Thus by
the Remark we have $K_{I_{\alpha},(-\infty,\epsilon]}\neq\emptyset$.
Suppose that the only critical point to (1.1) is $u=0$ at which the energy of
the functional $I_{\alpha}$ is also $0$. Thus from the discussion above and
the Remark (3.2)-(4) we have from the Morse relation we have the following
identity over $\mathbb{R}$
$1=t+\mathcal{P}(t)+(1+t)\mathcal{Q}_{t},$
$g$ being a power series in $t$, $\mathcal{Q}_{t}\geq 0$. This is a
contradiction. Thus there exists at least one $u\neq 0$ which is a critical
point to $I_{\alpha}$ whenever $\lambda\leq\lambda_{1}$. ∎
###### Definition 3.4 (Krasnoselskii genus).
Let $V$ be a Banach space and $S\subset V$. A set $S$ is said to be symmetric
if $u\in S$ implies $-u\in S$. Let $S$ be a close, symmetric subset of $V$
such that $0\notin S$. We define a genus $\gamma(S)$ of $S$ by the smallest
integer $k$ such that there exists an odd continuous mapping from $S$ to
$\mathbb{R}^{k}\setminus\\{0\\}$. We define $\gamma(S)=\infty$, if no such $k$
exists.
To each closed and symmetric subsets $M$ of $W_{0}^{1,p}(\Omega)$ with the
Krasnoselskii genus $\gamma(M)\geq k$, define
$\lambda_{k}=\inf_{M\in\mathfrak{F}_{k}}\sup_{u\in M}I_{\alpha}(u).$
Here $\mathfrak{F}_{k}=\\{M\subset W_{0}^{1,p}(\Omega),~{}\text{closed and
symmetric}:\gamma(M)\geq k\\}$. A natural question at this point will be to
ask if the same conclusion can be drawn when
$\lambda_{k}<\lambda\leq\lambda_{k+1}$. We will define $\lambda_{0}=0$. The
next theorem answers this question.
###### Theorem 3.5.
The problem in (1.1) has at least one nontrivial solution when
$\lambda_{k}<\lambda\leq\lambda_{k+1}$, $\lambda_{k}$ being as defined above.
###### Proof.
We at first show that $H_{k}(W_{0}^{1,p}(\Omega),I_{\alpha}^{-a})=0$ for all
$k\geq 0$. Pick a $u\in\\{v:\|v\|=1\\}=\partial B^{\infty}$, where
$B^{\infty}=\\{v:\|v\|\leq 1\\}$. Then
$I_{\alpha}(tu)=\int_{\Omega}\frac{|\nabla(tu)|^{p}}{p}dx+\int_{\Omega}G\left(\frac{tu-1}{\alpha}\right)dx-\lambda\int_{\Omega}\frac{(tu-1)_{+}^{q}}{q}dx<-a<0$
for all $t\geq t_{0}$. It can be easily seen that for a fixed $u$, we have
$I^{\prime}(tu)>0$. Further, for any $t\geq t_{0}$ we have
$I_{\alpha}(tu)<-a<0$. Thus, there exists $t(u)$ such that
$I_{\alpha}^{\prime}(tu)=0$ by the continuity of $I_{\alpha}^{\prime}$. We can
thus say that there exists a $C^{1}$-function
$T:W_{0}^{1,p}(\Omega)\setminus\\{0\\}\rightarrow\mathbb{R}^{+}$. We now
define a standard deformation retract $\eta$ of $W_{0}^{1,p}(\Omega)\setminus
B_{R^{\prime}}(0)$ into $I_{\alpha}^{-a}$ as follows (refer Definition 2.2).
$\eta(s,u)=\begin{cases}(1-s)u+sT\left(\frac{u}{\|u\|}\right)\frac{u}{\|u\|},&\|u\|\geq
R^{\prime},I_{\alpha}(u)\geq-a\\\ u,&I_{\alpha}(u)\leq-a.\end{cases}$
It is not difficult to see that $\eta$ is a $C^{1}$ function over $[0,1]\times
W_{0}^{1,p}(\Omega)\setminus B_{R^{\prime}}(0)$. On using the map
$\delta(s,u)=\dfrac{u}{\|u\|}$, for $u\in W_{0}^{1,p}(\Omega)\setminus
B_{R^{\prime}}(0)$ we claim that
$H_{k}(W_{0}^{1,p}(\Omega),W_{0}^{1,p}(\Omega)\setminus
B_{r}(0))=H_{k}(B^{\infty},S^{\infty})$ for all $k\geq 0$. This is because,
$H_{k}(B^{\infty},S^{\infty})\cong H_{k}(*,0)$. From elementary computation of
homology groups with two $0$-dimensional simplices it is easy to see that
$H_{k}(*,0)=\\{0\\}$ for each $k\geq 0$. A result in [10] says that
$C_{m}(I,u)=\begin{cases}\mathbb{R},&\text{if}~{}m(u)=m\\\
0,&\text{otherwise}\end{cases}$
Therefore, from the Morse relation in the Remark (3.2)-4 and the result above,
we have for $b>0$
$\displaystyle\sum_{u\in K_{I,[-a,\infty)}}\sum_{k\geq
0}\text{dim}(C_{k}(I,u))t^{k}$ $\displaystyle=t^{m(u)}+p(t)$ (3.1)
where $m(u)$ is the Morse index of $u$ and $\mathcal{P}(t)$ contains the rest
of the powers of $t$ corresponding to the other critical points, if any. The
Morse index is finite because of the following reason. From the argument which
helped in establishing a ‘maxima’, say $u_{0}$, using the mountain pass
geometry around $0$, we had to assume $\lambda<C^{-q}\frac{q}{p}\|u\|^{p-q}$.
Owing to $u_{0}$ being a maxima, we have $I_{\alpha}^{\prime\prime}(u_{0})<0$
which necessarily requires $\lambda>C^{-q}\frac{p-1}{q-1}\|u\|^{p-q}$. Thus we
have
$C^{-q}\frac{p-1}{q-1}\|u\|^{p-q}<\lambda<C^{-q}\frac{q}{p}\|u\|^{p-q}.$
This implies that $\lambda_{i}<\lambda<\lambda_{j}$ for some
$i,j\in\mathbb{N}_{0}$. On further using the the Morse relation we obtain
$\displaystyle t^{m(u)}+\mathcal{P}(t)$ $\displaystyle=(1+t)\mathcal{Q}_{t}.$
(3.2)
This is because the $H_{k}$s are all trivial groups. Hence, $Q_{t}$ either
contains $t^{m(u)}$ or $t^{m(u)-1}$ or both. Thus there exists at least one
nontrivial $u\in K_{I_{\alpha},[-a,\infty)}$ with $m(u)\leq n+1$. ∎
###### Remark 3.6.
If $0<\lambda\leq\lambda_{k+1}$, then there exists at least $k$ solutions to
the equation (1.1).
## 4 Existence of solution to the main problem (1.1) and smoothness of the
boundary $\partial\\{u>1\\}$
###### Lemma 4.1.
Let $\alpha_{j}\rightarrow 0$ ($\alpha_{j}>0$) as $j\rightarrow\infty$ and
$u_{j}$ be a critical point of $I_{\alpha_{j}}$. If $(u_{j})$ is bounded in
$W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$, then there exists $u$, a
Lipschitz continuous function, on $\bar{\Omega}$ such that $u\in
W_{0}^{1,p}(\Omega)\cap C^{2}(\bar{\Omega}\setminus H(u))$ and a subsequence
(still denoted by $(u_{j})$) such that
1. (i)
$u_{j}\rightarrow u$ uniformly over $\bar{\Omega}$,
2. (ii)
$u_{j}\rightarrow u$ locally in $C^{1}(\bar{\Omega}\setminus\\{u=1\\})$,
3. (iii)
$u_{j}\rightarrow u$ strongly in $W_{0}^{1,p}(\Omega)$,
4. (iv)
$I(u)\leq\lim\inf I_{\alpha_{j}}(u_{j})\leq\lim\sup I_{\alpha_{j}}(u_{j})\leq
I(u)+|\\{u=1\\}|$, i.e. $u$ is a nontrivial function if $\lim\inf
I_{\alpha_{j}}(u_{j})<0$ or $\lim\sup I_{\alpha_{j}}(u_{j})>0$.
Furthemore, $u$ satisfies
$-\Delta_{p}u=\lambda\chi_{\\{u>1\\}}(x)(u-1)_{+}^{q-1}$
classically in $\Omega\setminus H(u)$, the free boundary condition is
satisfies in the generalized sense and vanishes continuously on
$\partial\Omega$. In the case of $u$ being nontrivial, then $u>0$ in $\Omega$,
the set $\\{u<1\\}$ is connected and the set $\\{u>1\\}$ is nonempty.
An important result that will be used to pass the limit in the proof of the
Lemma 4.1 is the following theorem which is in line to the theorem due to
Caffarelli et al. in [6, Theorem $5.1$].
###### Lemma 4.2.
Let $u$ be a Lipschitz continuous function on the unit ball
$B_{1}(0)\subset\mathbb{R}^{N}$ satisfying the distributional inequalities
$\pm\Delta_{p}u\leq
A\left(\dfrac{1}{\alpha}\chi_{\\{|u-1|<\alpha\\}}(x)+1\right)$
for constants $A>0$ and $0<\alpha\leq 1$. Then there exists a constant $C>0$
depending on $N,A$ and $\int_{{B_{1}}(0)}u^{p}dx$, but not on $\alpha$, such
that
$\underset{x\in B_{\frac{1}{2}}(0)}{\text{esssup}}\\{|\nabla u(x)|\\}\leq C.$
###### Proof.
Given that $u$ is a Lipschitz continuous function on the unit ball
$B_{1}(0)\subset\mathbb{R}^{N}$, so $u$ is also bounded in the unit ball say
by a constant $M_{0}$. Not just that, $u$ is also differentiable a.e. in
$B_{1}(0)$. We will prove the result stated in the lemma for $u_{+}$, as the
proof for $u_{-}$ will follow suit. Denote $v(x)=\frac{15}{\alpha}u_{-}(\alpha
x/15)$ and
$v_{1}=v+\underset{B_{1/4}}{\max}\\{v^{-}\\}.$
Therefore, $0\leq v_{1}\leq M_{1}$. Let us choose a test function $\eta\in
C_{0}^{\infty}(B_{1/4})$ which is such that $0\leq\eta\leq 1$ in $B_{3/4}$ and
$\eta=1$ in $B_{1/2}$. Thus
$\displaystyle\begin{split}\int_{\Omega}\eta^{p}|\nabla
v_{1}|^{p}=&-\int_{\Omega}(pv_{1}\eta^{p-1}|\nabla v_{1}|^{p-2}(\nabla
v_{1}\cdot\nabla\eta)+\eta^{p}v_{1}\Delta_{p}v_{1}dx)dx\\\
\leq&\frac{1}{p}\int_{\Omega}\eta^{p}|\nabla
v_{1}|^{p}dx+p\int_{\Omega}v_{1}^{p}|\nabla\eta|^{p}dx\\\
&+AM_{1}\int_{\Omega}\eta^{p}\left(\frac{1}{\alpha}\chi_{\\{|u-1|<\alpha\\}}(x)+1\right)dx\\\
\leq&\frac{1}{p}\int_{\Omega}\eta^{p}|\nabla
v_{1}|^{p}dx+pM_{1}^{p}\int_{\Omega}|\nabla\eta|^{p}dx\\\
&+AM_{1}\int_{\Omega}\eta^{p}\left(\frac{1}{\alpha}\chi_{\\{|u-1|<\alpha\\}}(x)+1\right)dx.\end{split}$
(4.1)
It is now established that
$\displaystyle\frac{p-1}{p}\int_{B_{1/2}}|\nabla v_{1}|^{p}dx$
$\displaystyle\leq M_{2}.$ (4.2)
However, $u$ being Lipschitz continuous, the gradient $\nabla u$ is bounded
a.e. in $B_{1}(0)$ and hence in $B_{1/2}(0)$. Thus
$\underset{B_{1/2}(0)}{\text{esssup}}\\{|\nabla u|\\}\leq C$, for some $C>0$.
∎
###### Proof of Lemma 4.1.
Let $0<\alpha_{j}<1$. Consider the problem sequence $(P_{j})$
$\displaystyle\begin{split}-\Delta_{p}u_{j}&=-\frac{1}{\alpha_{j}}g\left(\frac{(u_{j}-1)_{+}}{\alpha_{j}}\right)+\lambda(u-1)_{+}^{q-1}~{}\text{in}~{}\Omega\\\
u_{j}&>0~{}\text{in}~{}\Omega\\\
u_{j}&=0~{}\text{on}~{}\partial\Omega.\end{split}$ (4.3)
The nature of the problem being a sublinear one allows us to conclude by an
iterative technique that the sequence $(u_{j})$ is bounded in
$L^{\infty}(\Omega)$. Therefore, there exists $C_{0}$ such that $0\leq
g\left(\frac{(u_{j}-1)_{+}}{\alpha_{j}}\right)(u-1)_{+}^{q-1}\leq C_{0}$. Let
$\varphi_{0}$ be a solution of
$\displaystyle\begin{split}-\Delta_{p}\varphi_{0}&=\lambda
C_{0}~{}\text{in}~{}\Omega\\\
\varphi_{0}&=0~{}\text{on}~{}\partial\Omega.\end{split}$ (4.4)
Now since $g\geq 0$, we have that $-\Delta_{p}u_{j}\leq\lambda
C_{0}=-\Delta\varphi_{0}$ in $\Omega$. Therefore by the maximum principle,
$\displaystyle 0\leq u_{j}(x)\leq\varphi_{0}(x)~{}\forall x\in\Omega.$ (4.5)
Since $\\{u_{j}\geq 1\\}\subset\\{\varphi_{0}\geq 1\\}$, hence $\varphi_{0}$
gives a uniform lower bound, say $d_{0}$, on the distance from the set
$\\{u_{j}\geq 1\\}$ to $\partial\Omega$. Thus $(u_{j})$ is bounded with
respect to the $C^{2,a}$ norm. Therefore, it has a convergent subsequence in
the $C^{2}$-norm in a $\dfrac{d_{0}}{2}$ neighbourhood of the boundary
$\partial\Omega$. Obviously $0\leq g\leq 2\chi_{(-1,1)}$ and hence
$\displaystyle\begin{split}\pm\Delta
u_{j}&=\pm\frac{1}{\alpha_{j}}g\left(\frac{(u_{j}-1)_{+}}{\alpha_{j}}\right)\mp\lambda(u_{j}-1)_{+}^{q-1}\\\
&\leq\frac{2}{\alpha_{j}}\chi_{\\{|u_{j}-1|<\alpha_{j}\\}}(x)+\lambda
C_{0}.\end{split}$ (4.6)
Since, $(u_{j})$ is bounded in $L^{2}(\Omega)$ and by Lemma 4.2 it follows
that there exists $A>0$ such that
$\displaystyle\underset{x\in B_{\frac{r}{2}}(x_{0})}{\text{esssup}}\\{|\nabla
u_{j}(x)|\\}$ $\displaystyle\leq\frac{A}{r}$ (4.7)
for a suitable $r>0$ such that $B_{r}(0)\subset\Omega$. However, since
$(u_{j})$ is a sequence of Lipschitz continuous functions that are also
$C^{1}$, therefore
$\displaystyle\underset{x\in B_{\frac{r}{2}}(x_{0})}{\sup}\\{|\nabla
u_{j}(x)|\\}$ $\displaystyle\leq\frac{A}{r}.$ (4.8)
Thus $(u_{j})$ is uniformly Lipschitz continuous on the compact subsets of
$\Omega$ such that its distance from the boundary $\partial\Omega$ is at least
$\frac{d_{0}}{2}$ units.
Thus by the Ascoli-Arzela theorem applied to $(u_{j})$ we have a subsequence,
still named the same, such that it converges uniformly to a Lipschitz
continuous function $u$ in $\Omega$ with zero boundary values and with strong
convergence in $C^{2}$ on a $\frac{d_{0}}{2}$-neighbourhood of
$\partial\Omega$. By the Eberlein-Šmulian theorem we conclude that
$u_{j}\rightharpoonup u$ in $W_{0}^{1,p}(\Omega)$.
We now prove that $u$ satisfies
$\displaystyle-\Delta_{p}u$
$\displaystyle=\alpha\chi_{\\{u>1\\}}(x)(u-1)_{+}^{q-1}$ (4.9)
in the set $\\{u\neq 1\\}$. Let $\varphi\in C_{0}^{\infty}(\\{u>1\\})$ and
therefore $u\geq 1+2\delta$ on the support of $\varphi$ for some $\delta>0$.
On using the convergence of $u_{j}$ to $u$ uniformly on $\Omega$ we have
$|u_{j}-u|<\delta$ for any sufficiently large $j,\delta_{j}<\delta$. So
$u_{j}\geq 1+\delta_{j}$ on the support of $\varphi$. On testing (4.9) with
$\varphi$ yields
$\displaystyle\int_{\Omega}|\nabla u_{j}|^{p-2}\nabla u_{j}\cdot\nabla\varphi
dx$ $\displaystyle=\lambda\int_{\Omega}(u_{j}-1)_{+}^{q-1}\varphi dx.$ (4.10)
On passing the limit $j\rightarrow\infty$ to (4.9), we get
$\displaystyle\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi dx$
$\displaystyle=\lambda\int_{\Omega}(u_{j}-1)_{+}^{q-1}\varphi dx.$ (4.11)
To arrive at (4.11) we have used the weak convergence of $u_{j}$ to $u$ in
$W_{0}^{1,p}(\Omega)$ and the uniform convergence of the same in $\Omega$.
Hence $u$ is a weak solution of $-\Delta_{p}u=\lambda(u-1)_{+}^{q-1}$ in
$\\{u>1\\}$. Since $u$ is a Lipschitz continuous function, hence by the
Schauder estimates we conclude that it is also a classical solution of
$-\Delta_{p}u=\lambda(u-1)_{+}^{q-1}$ in $\\{u>1\\}$. Similarly on choosing
$\varphi\in C_{0}^{\infty}(\\{u<1\\})$ one can find a $\delta>0$ such that
$u\leq 1-2\delta$. Therefore, $u_{j}<1-\delta$.
Let us now analyze the nature of $u$ in the set $\\{u\leq 1\\}^{\circ}$. On
testing (4.9) with any nonnegative function and passing the limit
$j\rightarrow\infty$ and using the fact that $g\geq 0$, $G\leq 1$ we can show
that $u$ satisfies
$\displaystyle-\Delta_{p}u$
$\displaystyle\leq\lambda(u-1)_{+}^{q-1}~{}\text{in}~{}\Omega$ (4.12)
in the distributional sense. Furthermore, $\mu=\Delta_{p}u$ is a positive
Radon measure supported on $\Omega\cap\partial\\{u<1\\}$ (refer Lemma 4.3 in
Appendix). From (4.12), the positivity of the Radon measure $\mu$ and the
usage of Section $9.4$ in Gilbarg-Trudinger [8] we conclude that $u\in
W_{\text{loc}}^{2,p}(\\{u\leq 1\\}^{\circ})$, $1<p<\infty$. Thus $\mu$ is
supported on $\Omega\cap\partial\\{u<1\\}\cap\partial\\{u>1\\}$ and $u$
satisfies $-\Delta_{p}u=0$ in the set $\\{u\leq 1\\}^{\circ}$.
In order to prove $(ii)$, we will show that $u_{j}\rightarrow u$ locally in
$C^{1}(\Omega\setminus\\{u=1\\})$. Note that we have already proved that
$u_{j}\rightarrow u$ in the $C^{2}$ norm in a neighbourhood of
$\partial\Omega$ of $\bar{\Omega}$. Suppose $M\subset\subset\\{u>1\\}$. In
this set $M$ we have $u\geq 1+2\delta$ for some $\delta>0$. Thus for
sufficiently large $j$, with $\delta_{j}<\delta$, we have $|u_{j}-u|<\delta$
in $\Omega$ and hence $u_{j}\geq 1+\delta_{j}$ in $M$. From (4.3) we have
$-\Delta_{p}u_{j}=\lambda(u_{j}-1)_{+}^{q-1}~{}\text{in}~{}M.$
Clearly, $(u_{j}-1)_{+}^{q-1}\rightarrow(u-1)_{+}^{q-1}$ in $L^{p}(\Omega)$
for $1<p<\infty$ and $u_{j}\rightarrow u$ uniformly in $\Omega$. This analysis
says something more stronger - since
$(-\Delta_{p})u_{j}=\lambda(u_{j}-1)_{+}^{q-1}$ in $M$, we have that
$u_{j}\rightarrow u$ in $W^{2,p}(M)$. By the embedding
$W^{2,p}(M)\hookrightarrow C^{1}(M)$ for $p>2$, we have $u_{j}\rightarrow u$
in $C^{1}(M)$. This shows that $u_{j}\rightarrow u$ in $C^{1}(\\{u>1\\})$.
Working on similar lines we can also show that $u_{j}\rightarrow u$ in
$C^{1}(\\{u<1\\})$.
We will now prove $(iii)$. Since $u_{j}\rightharpoonup u$ in
$W_{0}^{1,p}(\Omega)$, we have that by the weak lower semicontinuity of the
norm $\|\cdot\|$ that
$\|u\|\leq\lim\inf\|u_{j}\|.$
It is sufficient to prove that $\lim\sup\|u_{j}\|\leq\|u\|$. To achieve this,
we multiply (4.3) with $(u_{j}-1)$ and then integrate by parts. We will also
use the fact that $tg\left(\frac{t}{\delta_{j}}\right)\geq 0$ for any
$t\in\mathbb{R}$. This gives,
$\displaystyle\begin{split}\int_{\Omega}|\nabla
u_{j}|^{p}dx&\leq\lambda\int_{\Omega}f(u_{j}-1)_{+}^{q}dx-\int_{\partial\Omega}\frac{\partial
u_{j}}{\partial\hat{n}}dS\\\
&\rightarrow\lambda\int_{\Omega}(u-1)_{+}^{q}dx-\int_{\partial\Omega}\frac{\partial
u}{\partial\hat{n}}dS\end{split}$ (4.13)
as $j\rightarrow\infty$. Here $\hat{n}$ is the outward drawn normal to
$\partial\Omega$. ∎
We choose $\vec{\varphi}\in C_{0}^{1}(\Omega,\mathbb{R}^{N})$ such that $u\neq
1$ a.e. on the support of $\vec{\varphi}$. On multiplying $\nabla
u_{n}\cdot\vec{\varphi}$ to the weak formulation of (4.3) and integrating over
the set $\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}$ gives
$\displaystyle\begin{split}\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}\left[-\Delta_{p}u_{n}+\frac{1}{\alpha_{n}}g\left(\frac{u_{n}-1}{\alpha_{n}}\right)\right]\nabla
u_{n}\cdot\vec{\varphi}dx\\\
=\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}(u_{n}-1)_{+}^{q-1}\nabla
u_{n}\cdot\vec{\varphi}dx.\end{split}$ (4.14)
The term on the left hand side of (4.14) can be expressed as follows.
$\displaystyle\nabla\cdot\left(\frac{1}{p}|\nabla
u_{n}|^{p}\vec{\varphi}-(\nabla u_{n}\cdot\vec{\varphi})|\nabla
u_{n}|^{p-2}\nabla u_{n}\right)+(\nabla\vec{\varphi}\cdot\nabla
u_{n})\cdot\nabla u_{n}|\nabla u_{n}|^{p-2}$ $\displaystyle-\frac{1}{p}|\nabla
u_{n}|^{p}\nabla\cdot\vec{\varphi}$ $\displaystyle+\nabla
G\left(\frac{u_{n}-1}{\alpha_{n}}\right)\cdot\vec{\varphi}.$ (4.15)
Using (4) and on integrating by parts we obtain
$\displaystyle\begin{split}\int_{\\{u_{n}=1+\epsilon^{+}\\}\cup\\{u_{n}=1-\epsilon^{-}\\}}\left[\frac{1}{p}|\nabla
u_{n}|^{p}\vec{\varphi}-(\nabla u_{n}\cdot\vec{\varphi})|\nabla
u_{n}|^{p-2}\nabla
u_{n}+G\left(\frac{u_{n}-1}{\alpha_{j}}\right)\hat{\varphi}\right]\cdot\hat{n}dS\\\
=\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}\left(\frac{1}{p}|\nabla
u_{n}|^{p}\nabla\cdot\vec{\varphi}-(\nabla\vec{\varphi}\cdot\nabla
u_{n})|\nabla u_{n}|^{p-2}\nabla u_{n}\right)dx\\\
+\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}\left[G\left(\frac{u_{n}-1}{\alpha_{n}}\right)\nabla\cdot\vec{\varphi}+\lambda(u_{n}-1)_{+}^{q-1}(\nabla
u_{n}\cdot\vec{\varphi})\right]dx.\end{split}$ (4.16)
The integral on the left of equation (4.16) converges to
$\displaystyle\int_{\\{u_{n}=1+\epsilon^{+}\\}\cup\\{u_{n}=1-\epsilon^{-}\\}}\left(\frac{1}{p}|\nabla
u|^{p}\vec{\varphi}-(\nabla u_{n}\cdot\vec{\varphi})|\nabla u_{n}|^{p-2}\nabla
u_{n}\right)\cdot\hat{n}dS+\int_{\\{u_{n}=1+\epsilon^{+}\\}}\vec{\varphi}\cdot\hat{n}dS$
(4.17)
$\displaystyle=\int_{\\{u_{n}=1+\epsilon^{+}\\}}\left[1-\left(\frac{p-1}{p}\right)|\nabla
u_{n}|^{p}\right]\vec{\varphi}\cdot\hat{n}dS-\int_{\\{u_{n}=1-\epsilon^{-}\\}}\left(\frac{p-1}{p}\right)|\nabla
u_{n}|^{p}\vec{\varphi}\cdot\hat{n}dS.$ (4.18)
Thus the equation (4.17) under the limit $\epsilon\rightarrow 0$ becomes
$\displaystyle 0=\underset{\epsilon\rightarrow
0}{\lim}\int_{\\{u=1+\epsilon^{+}\\}}\left[\left(\frac{p}{p-1}\right)-|\nabla
u|^{p}\right]\vec{\varphi}\cdot\hat{n}dS-\underset{\epsilon\rightarrow
0}{\lim}\int_{\\{u=1-\epsilon^{-}\\}}|\nabla
u|^{p}\vec{\varphi}\cdot\hat{n}dS$ (4.19)
This is because $\hat{n}=\pm\dfrac{\nabla u}{|\nabla u|}$ on the set
$\\{u=1+\epsilon^{+}\\}\cup\\{u=1-\epsilon^{-}\\}$. This proves that $u$
satisfies the free boundary condition. The solution cannot be trivial as it
satisfies the free boundary condition. Thus a solution to (1.1) exists that
obeys the free boundary condition besides the Dirichlet boundary condition.
## Appendix
###### Lemma 4.3.
$u$ is in $W_{\text{loc}}^{1,p}(\Omega)$ and the Radon measure
$\mu=\Delta_{p}u$ is nonnegative and supported on $\Omega\cap\\{u<1\\}$.
###### Proof.
We follow the proof due to Alt-Caffarelli [1]. Choose $\delta>0$ and a test
function $\varphi^{p}\chi_{\\{u<1-\delta\\}}$ where $\varphi\in
C_{0}^{\infty}(\Omega)$. Therefore,
$\displaystyle\begin{split}0&=\int_{\Omega}|\nabla u|^{p-2}\nabla
u\cdot\nabla(\varphi^{p}\min\\{u-1+\delta,0\\})dx\\\
&=\int_{\Omega\cap\\{u<1-\delta\\}}|\nabla u|^{p-2}\nabla
u\cdot\nabla(\varphi^{p}\min\\{u-1+\delta,0\\})dx\\\
&=\int_{\Omega\cap\\{u<1-\delta\\}}|\nabla
u|^{p}\varphi^{p}dx+p\int_{\Omega\cap\\{u<1-\delta\\}}\varphi^{p-1}(u-1+\delta)|\nabla
u|^{p-2}\nabla u\cdot\nabla\varphi dx,\end{split}$ (4.20)
and so by Caccioppoli like estimate we have
$\displaystyle\begin{split}\int_{\Omega\cap\\{u<1-\delta\\}}|\nabla
u|^{p}\varphi^{p}dx&=-p\int_{\Omega\cap\\{u<1-\delta\\}}\varphi^{p-1}(u-1+\delta)|\nabla
u|^{p-2}\nabla u\cdot\nabla\varphi dx\\\ &\leq
c\int_{\Omega}u^{p}|\nabla\varphi|^{p}dx.\end{split}$ (4.21)
Since $\int_{\Omega}|u|^{p}dx<\infty$, therefore on passing the limit
$\delta\rightarrow 0$ we conclude that $u\in W_{\text{loc}}^{1,p}(\Omega)$.
Furthermore, for a nonnegative $\zeta\in C_{0}^{\infty}(\Omega)$ we have
$\displaystyle\begin{split}-\int_{\Omega}|\nabla u|^{p-2}\nabla
u\cdot\nabla\zeta
dx=&\left(\int_{\Omega\cap\\{0<u<1-2\delta\\}}+\int_{\Omega\cap\\{1-2\delta<u<1-\epsilon\\}}+\int_{\Omega\cap\\{1-\delta<u<1\\}}\right.\\\
&\left.+\int_{\Omega\cap\\{u>1\\}}\right)\\\ &\left[|\nabla u|^{p-2}\nabla
u\cdot\nabla\left(\zeta\max\left\\{\min\left\\{2-\frac{1-u}{\delta},1\right\\},0\right\\}\right)\right]dx\\\
\geq&\int_{\Omega\cap\\{1-2\delta<u<1-\delta\\}}\left[|\nabla u|^{p-2}\nabla
u\cdot\left(2-\frac{1-u}{\delta}\right)\nabla\zeta+\frac{\zeta}{\delta}|\nabla
u|^{p}\right]dx\geq 0.\end{split}$ (4.22)
On passing the limit $\delta\rightarrow 0$ we obtain $\Delta_{p}(u-1)_{-}\geq
0$ in the distributional sense and hence there exists a Radon measure $\mu$
(say) such that $\mu=\Delta(u-1)_{-}\geq 0$. ∎
## Acknowledgement
The author thanks the community of the free boundary value problems for
injecting a new lease of life to the study of elliptic PDEs and the CSIR,
India (25(0292)/18/EMR-II) for the financial support.
## References
* [1] Alt, H.W., Caffarelli, L.A., Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325, 105-144, 1981.
* [2] Batchelor, G.K., On steady state laminar flow with closed streamlines at large Reynolds number, J. Fluid mech., 1, 177-190, 1956.
* [3] Batchelor, G.K., A proposal concerning laminar wakes behind bluff bodies at large Reynolds number, J. Fluid mech., 1, 388-398, 1956.
* [4] Caflisch, R.E., Mathematical analysis of vortex dynamics. In: Mathematical Aspects of Vortex Dynamics (Leesburg VA, 1988), pp 1-24, SIAM, Philadelphia, PA, 1989.
* [5] Caffarelli, L.A., Peral, I., On $W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51, 1-21, 1998.
* [6] Caffarelli, L.A., Jerison, D., Kenig, C.E., Some new monotonicity theorems with applications to free boundary problems, Ann. Math. (2), 155(2), 369-404, 2002.
* [7] Elcrat, A.R., Miller, K.G., Variational formulas on Lipschitz domains, Trans. Am. Math. Soc., 347(7), 2669-2678, 1995.
* [8] Gilbarg, D., Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg, 2001.
* [9] Jerison, D., Perera, K., A multiplicity result for the Prandtl-Batchelor free boundary problem (Preprint arXiv:2003.05921).
* [10] Kanishka Perera, Ravi P. Agarwal and Donald O’Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operators, Mathematical surveys and Monographs, Amer. Math. Soc., 161, 2010.
* [11] Kesavan, S., Topics in functional analysis and applications, New Age International (P) Ltd., 2003.
* [12] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš, Nonlinear Analysis - Theory and Methods, Springer, 2019.
* [13] Perera, K., On a class of elliptic free boundary problems with multiple solutions, Nonlinear Differ. Equ. Appl., 28, Art: 36, 2021.
|
2024-09-04T02:54:58.480708 | 2020-03-05T21:18:21 | 2003.04294 | {
"authors": "Peng Zhang, Jianbin Fang, Canqun Yang, Chun Huang, Tao Tang, Zheng\n Wang",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26122",
"submitter": "Zheng Wang",
"url": "https://arxiv.org/abs/2003.04294"
} | arxiv-papers | # Optimizing Streaming Parallelism on Heterogeneous Many-Core Architectures: A
Machine Learning Based Approach
Peng Zhang, Jianbin Fang, Canqun Yang, Chun Huang, Tao Tang, Zheng Wang Peng
Zhang, Jianbin Fang, Canqun Yang, Chun Huang, and Tao Tang are with National
University of Defense Technology, China.
E-mail: {zhangpeng13a, j.fang, canqun<EMAIL_ADDRESS>Zheng Wang is
with University of Leeds, United Kingdom.
E-mail<EMAIL_ADDRESS>
###### Abstract
As many-core accelerators keep integrating more processing units, it becomes
increasingly more difficult for a parallel application to make effective use
of all available resources. An effective way for improving hardware
utilization is to exploit spatial and temporal sharing of the heterogeneous
processing units by multiplexing computation and communication tasks – a
strategy known as heterogeneous streaming. Achieving effective heterogeneous
streaming requires carefully partitioning hardware among tasks, and matching
the granularity of task parallelism to the resource partition. However,
finding the right resource partitioning and task granularity is extremely
challenging, because there is a large number of possible solutions and the
optimal solution varies across programs and datasets. This article presents an
automatic approach to quickly derive a good solution for hardware resource
partition and task granularity for task-based parallel applications on
heterogeneous many-core architectures. Our approach employs a performance
model to estimate the resulting performance of the target application under a
given resource partition and task granularity configuration. The model is used
as a utility to quickly search for a good configuration at runtime. Instead of
hand-crafting an analytical model that requires expert insights into low-level
hardware details, we employ machine learning techniques to automatically learn
it. We achieve this by first learning a predictive model offline using
training programs. The learnt model can then be used to predict the
performance of any unseen program at runtime. We apply our approach to 39
representative parallel applications and evaluate it on two representative
heterogeneous many-core platforms: a CPU-XeonPhi platform and a CPU-GPU
platform. Compared to the single-stream version, our approach achieves, on
average, a 1.6x and 1.1x speedup on the XeonPhi and the GPU platform,
respectively. These results translate to over 93% of the performance delivered
by a theoretically perfect predictor.
###### Index Terms:
Heterogeneous computing; Parallelism; Performance Tuning; Machine learning
## 1 Introduction
Heterogeneous many-cores, as representative by GPGPUs and Intel’s XeonPhi, are
widely used for accelerating parallel applications [1, 2, 3]. As users demand
higher performance, many-core accelerators have become more powerful by
providing more and more processing units. While the abundant computing
resources offer the potential for higher performance, it becomes harder for a
parallel application to utilize all the available computing resources [4, 5].
As a result, many parallel applications fail to fully unlock the performance
potential of a many-core accelerator.
One way for improving heterogeneous many-core utilization is to exploit
spatial and temporal sharing of processing resources. This strategy is also
known as heterogeneous streaming [6]. The idea is to exploit the computation
and communication independency of task parallelism to improve hardware
utilization. It works by partitioning the processor cores to allow independent
communication and computation tasks (i.e. streams) to run concurrently on
different hardware resources, which effectively overlaps the concurrent kernel
execution with data movements. Representative heterogeneous streaming
implementations include CUDA Streams [7], OpenCL Command Queues [8], and Intel
heterogeneous streams library (hStreams) [9, 6]. These implementations allow a
parallel program to spawn more than one stream (or pipeline) so that the data
movement stage of one pipeline overlaps the kernel execution stage of another.
Prior work on heterogeneous streaming mainly targets GPUs [10, 11, 12].
Compared to GPU implementations, OS-enabled coprocessors, like the Intel
XeonPhi, provides some unique features that are currently unavailable on the
GPU. For example, besides specifying the number of streams, developers can
explicitly map streams to different groups of cores on XeonPhi to control the
number of cores of each hardware partition. This parameter is not exposed to
programmers on GPUs, making previous work on GPU-based parallel streaming
optimizations infeasible to fully exploit Xeon-Phi-like many-core accelerators
(see also Section 6.3). On the other hand, ample evidence is showing that
choosing the right stream configuration, i.e., the number of processor core
partitions and the number of concurrent tasks of a multi-stream application,
values, has a significant impact the application’s performance on many-core
architectures [13, 14, 15]. However, attempting to find the optimal values
through exhaustive profiling would be ineffective, because the range of the
possible values for the two parameters is huge. What we need is a technique
that automatically determines the optimal stream configuration for any
streamed application in a fast manner.
This article presents a novel approach to determine the right number of
processor core partitions and tasks for heterogeneous streams, targeting
heterogeneous many-core architectures. Our key insight is to use a performance
model to quickly search for the optimal stream configuration. The performance
model estimates the resulting performance of the target streamed application
when it runs under a given stream configuration. If the prediction can be
performed quickly with low overhead, we can then quickly explore a large
configuration space. Instead of hand-crafting the performance model that
requires human modification whenever the architecture evolves (i.e., when the
number and types of cores change), we employ machine learning techniques to
automatically construct a predictive model. Our predictor is first trained
_off-line_. Then, using code and dynamic runtime features of the program, the
model predicts performance for a _new_ , _unseen_ program under a given stream
configuration.
Our prior work [16] develops a machine learning based classifier to predict
the optimal stream configuration. However, this approach can only choose from
a limited set of configurations seen during the training phase. Unlike a
classification-based approach, the approach presented in the article allows us
to explore a larger number of stream configurations (including those that are
not seen during the training phase) with negligible runtime overhead. This
advantage significantly improves the generalization ability of the proposed
approach (Section 3).
Due to the newness of heterogeneous streaming execution model, there are very
few multi-stream benchmarks available. To evaluate our approach on a wide
range of applications, we have developed a compiler-based tool to
automatically translate standard OpenMP benchmarks into their streamed
variants for the backends of XeonPhi and GPU architectures (Section 4). With
the help of this code generator, we can apply our approach to 39 parallel
benchmarks. We argue that this tool can help generate more streamed code and
thus is an added value to the community.
We evaluate our approach on two representative heterogeneous many-core
platforms: a 57-core Intel XeonPhi and an NVIDIA 1080Ti GPU platforms. We
achieve, on average, a 1.6x and 1.1x speedup over the single-stream execution
on the XeonPhi and the GPU platforms, respectively. This translates to over
93% of the best available performance.
The core contribution of this paper is a novel machine-learning-guided
approach for automatically determining the optimal stream configuration on
heterogeneous many-cores. We show that our approach delivers good performance
across benchmarks and heterogeneous many-core platforms. While we do not seek
to advance the machine learning algorithm itself, our work shows how machine
learning can be used to address the challenging problem of tuning fine-grained
streaming parallelism on heterogeneous many-core architectures. In this work,
we demonstrate the usefulness of our approach on XeonPhi and an NVIDIA GPU,
but our approach is equally applicable on other heterogeneous platforms like
AMD GPUs.
## 2 Background and Overview
In this section, we first give a brief introduction of heterogeneous
streaming; we then define the scope of this work, before motivating the need
of our scheme and providing an overview of our approach.
### 2.1 Heterogeneous Streaming
The idea of heterogeneous streaming is to exploit spatial and temporal sharing
of computing resources to utilize the hardware resources to improve
application performance.
Spatial Sharing. Modern many-core accelerators offer a large number of
processing units. Since many applications cannot fully utilize all the cores
at a time, we can partition the computing units into multiple groups to
concurrently execute multiple tasks. In this way, the computing resource is
spatially shared across concurrently-running application tasks. The key to
spatial sharing is to determine the right number of partitions, because over-
provisioning of processing units would waste computing resources but under-
provisioning would lead to slowed down performance.
Temporal Sharing. Code written for heterogeneous computing devices typically
consists of several stages, such as host device communication and computation.
Using temporal sharing, one can overlap some of these stages to exploit
pipeline parallelism to improve performance by overlapping the host-device
communication and kernel execution.
⬇ 1//setting the partition-size and task granularity
hStreams_app_init(partition_size,streams_p_part); 3 //stream queue id
5stream_id = 0; for(…){ 7 //enquque host-device transfer to current stream
hStreams_app_xfer_memory(,,, stream_id, HSTR_SRC_TO_SINK,…); 9 … //enqueue
computation to the current stream 11 hStreams_EnqueueCompute(stream_id,
”kernel1”, …); … 13 //move to the next stream stream_id = (stream_id++) %
MAX_STR; 15} //transfer data back to host 17hStreams_app_xfer_memory(,,,
HSTR_SINK_TO_SRC,…);
Figure 1: Heterogeneous streaming using hStreams as an example.
(a) binomial
(b) prefixsum
Figure 2: Heatmaps show the resultant speedup (over single-stream) of binomial
and prefixsum under different stream configurations. The #partitions and
#tasks have a significant impact on the resultant performance, and the sweet
spots are sparse and vary across programs.
### 2.2 Problem Scope
Our work aims to improve the performance of a data parallel application by
exploiting spatial and temporal sharing of heterogeneous streams. We do so by
determining at runtime how many partitions should be used to group the cores
(_#partitions_) and how many data parallel tasks (_#tasks_) should be used to
run the application. Our current implementation is applicable to XeonPhi and
GPUs by using different runtime back-ends (hStream for XeonPhi, and CUDA or
OpenCL for GPUs).
Code Example. Figure 1 gives a simplified code example written with Intel’s
hStreams APIs that can run on the XeonPhi many-core. At line 2 we initialize
the stream execution by setting the number of partitions and tasks/streams per
partition. This initialization process essentially creates multiple processor
domains and determines how many logical streams can run on a partition. In the
_for_ loop (lines 7-14) we enqueue the communication and computation tasks to
a number of streams identified by the stream_id variable. In this way,
communication and computation of different streams can be overlapped during
execution (temporal sharing); and streams on different processor domains (or
partitions) can run concurrently (spatial sharing). Our predictive model
determines the #partitions and the #tasks before invoking the hStreams
initialization routine, hStreams_app_init().
Figure 3: Color table showing the speedups of best-performing configurations
across inputs for dct. Each cell shows the performance for one of the 16 best-
performing configurations, $Cn$, on a given input, $Dn$. The best
configuration varies across inputs and a good configuration on one input can
give poor performance on another dataset.
### 2.3 Motivating Examples
Consider Figure 2 which shows the resultant performance improvement given by
multi-stream parallelism over the single-stream version of the code for two
applications on a 57-core Intel XeonPhi system. We use two streamed programs
from prior work [13]: binomial computes the price evolution over a given
period and prefixSum calculates the prefix sum for a sequence of numbers.
It is observed from this example that not all multi-stream configurations give
improved performance. As can be seen from the diagrams, the search space of
multi-stream configurations is huge but good configurations are sparse. The
performance varies significantly over stream configurations (#partitions,
#tasks). The optimal #tasks for binomial ranges from 1 to 30, and the best
#partitions is between 1 and 40. In contrast to binomial, prefixsum benefits
from fine-grained parallelism when using a larger #tasks (220 to 224) and
#partitions (60 to 80). However, the stream configurations that are effective
for prefixsum give no speedup over the single-stream version for binomial.
Now consider Figure 3 that shows the speedups of dct under 16 multi-stream
configurations over the single-stream version, where each configuration is
found to give the best-performance for one of the 16 inputs. In the color
table, each cell shows the performance of a stream configuration
($C1,...,C16$) on a specific input dataset ($D1,...,D16$); and the values
along the diagonal line represent the best-available performance (found
through profiling) for an input. As can be seen from the figure, the best
stream configuration can vary across inputs for the same benchmark. For
example, while $C4$ gives a speedup of 1.33x over the baseline for dataset
$D4$, it delivers a poor performance for dataset $D14$ by doubling the
execution time over the single-stream version. This diagram also suggests that
no single configuration can give improved performance for all inputs.
Lesson Learned. These two examples show that choosing the stream configuration
has a great impact on performance and the best configuration must be
determined on a per-program and per-dataset basis. Later, we will show that
this observation is not unique to XeonPhi but also holds for GPUs. Attempting
to find the optimal configuration through means of an exhaustive search would
be ineffective, and the overhead involved would be far bigger than the
potential benefits. Online search algorithms, while can speed up the search
process, the overhead can still outweigh the benefit. For example, when
applying simulated annealing to binomial, the best-found configuration only
reaches 84% of the best-available performance after 310,728 iterations111In
Section 6.1, we show that our approach achieves 93% of the best-available
performance for binomial on XeonPhi.. Classical hand-written heuristics are
not ideal either, as they are not only complex to develop, but are likely to
fail due to the variety of programs and the ever-changing hardware
architecture. An alternate approach, and the one we chose to use, is to use
machine learning to automatically construct a performance model to estimate
the benefit of any candidate configuration, providing minimal runtime overhead
for searching for a good configuration, and having little development cost
when targeting new architectures.
### 2.4 Overview of Our Approach
Our library-based approach, depicted in Figure 4, is completely automated. To
determine the best streaming configuration, our approach follows a number of
steps described as follows. We use a set of information or _features_ to
capture the characteristics of the program. We develop a LLVM [17] compiler
pass to extract static code features at compile time, and a low-overhead
profiling pass to collect runtime information at execution time (i.e., during
the first few loop iterations). Because profiling also contributes to the
final program output, no computation cycle is wasted. At runtime, we search
for a good configuration through an offline trained performance model to
estimate the resulting performances for all candidate configurations. The
performance model takes in the feature values, a given configuration of
resource partition and task granularity and estimates the potential speedup
for the given configuration over the single-stream version. The overhead of
runtime feature collection and search is a few milliseconds, which is included
in all our experimental results. Since our training process can be performed
automatically, we can easily target our performance model for different
architectures.
Figure 4: Our machine learning based performance model (trained _offline_)
predicts the speedup based on the extracted feature values of the code and a
given stream configuration. We use the predictions to quickly rank candidate
configurations at runtime to choose the one with the best predicted
performance.
## 3 Performance Modeling
At the core of our approach is a machine learned performance model built upon
the Multi-layer Perceptron (MLP) artificial neural network (ANN). Our
prototype is implemented using the Python scikit-learn machine learning
package [18]. It is to note that our prior work [16] uses a Support Vector
Machine (SVM) based classifier. However, such an approach can only make
predictions on a limited set of configurations seen at the training time.
Unlike a classification-based approach, the new approach presented in this
article is a _regression-based_ model which can make predictions on any stream
configuration. This new approach thus has a better generalization ability for
various heterogeneous architectures. We have also evaluated a number of
alternative modeling techniques, including MLP, SVM, and decision trees. We
chose MLP because it gives the best performance and has modest training
overhead (see Section 6.6.1).
Our performance model takes as input the feature values and a given
configuration (e.g., #partitions and #tasks for XeonPhi and #tasks for GPUs).
It predicts the speedup for the given configuration. Building and using such a
model follows a 3-step process for supervised learning: (i) generate training
data (ii) train a performance model (iii) use the performance model, described
as follows.
Figure 5: The training process of our performance model.
### 3.1 Training the Performance Model
Our method for model training is shown in Figure 5. To learn a regression
model, we first need to profile the execution time (in order to calculate the
speedup over the single-stream version) of all candidate configurations for
each training program, and extract the feature values from the program. We
then use the feature values, configuration settings and speedups to train a
model.
#### 3.1.1 Generating Training Data
To generate training data, we apply _cross-validation_ to 39 benchmarks, i.e.,
by excluding the testing benchmarks from the training dataset (see also
Section 5.3.1). We execute each training program and benchmark a number of
times until the gap of the upper and lower confidence bounds is smaller than
5% under a 95% confidence interval setting. We then calculate the average
speedup for a given stream configuration over the single-stream version. We
exhaustively execute each training program across a wide range of stream
configurations, and record the performance of each. Next, we calculate the
speedup for each configuration, program and dataset. Finally, we extract the
values of our selected set of features from each program and dataset. We
stress that the trained model can be applied to stream configurations that are
not seen in the training phase.
#### 3.1.2 Profiling Configurations
During the training phase, we exhaustively execute each training program
across a set of streamed configurations. On XeonPhi, we profile each training
program using the _#partitions_ ranging from 1 to 224 (the maximum number of
physical threads on XeonPhi) and the _#tasks_ ranging from 1 to 256 222We
chose these values because configuration settings beyond these values give a
poor performance during our initial evaluation.. On GPUs, we cannot configure
the number of partitions currently, we set the _#partitions_ to the same as
_#tasks_ to be consistent with XenPhi. On this platform, we also set the
_#tasks_ to be range between $2^{0}$ and $2^{10}$, which is big enough to
include the optimal values according to our experiments. Note that these
parameter ranges can be configured by the user.
#### 3.1.3 Building The Model
Each evaluated configuration is appended to the feature value vector of a
training program to form a model input. The model inputs and the corresponding
speedups (i.e., ground truths) for all training programs are passed to a
learning algorithm. The algorithm finds a correlation between the input vector
and the desired prediction. The output of our learning algorithm is an MLP
model where the weights of the model are determined from the training data.
Model parameter tuning is performed on the training dataset for each targeting
hardware architecture, using cross-validation (see also Section 6.6.3). In our
case, the overall training process for all the 39 training programs (which is
dominated by training data generation) takes less than a week on a single
machine. Since training is performed only once “at the factory”, this is a
_one-off_ cost.
### 3.2 Features
Our performance models are based exclusively on code and dynamic features of
the target programs. Code features are extracted from the program source code,
and dynamic features are collected using hardware performance counters during
the initial profiling run of the target application. We restrict us in using
hardware performance counters that are commonly available on modern processors
such as the data cache misses to ensure that our approach can be applied to a
wide range of many-core architectures.
We considered 38 candidate raw features in this work. Some features were
chosen from our intuition based on factors that can affect the performance
such as dts (host-device data transfer size) and #xfer_mem, while other
features were chosen based on previous work [19, 20].
#### 3.2.1 Feature Selection
To build an accurate model through supervised learning, the training sample
size typically needs to be at least one order of magnitude greater than the
number of features. In this work, we start from 311 training samples and 38
raw features, so we would like to reduce the number of features in use. Our
process for feature selection is fully automatic, described as follows.
We first combine several raw features to form a set of combined normalized
features, which are able to carry more information than the individual parts.
For example, instead of reporting raw branch hit and miss counts, we use the
branch miss rate. Next, we removed raw features that carried similar
information which is already captured by chosen features. To find which
features are closely correlated, we constructed a correlation coefficient
matrix using the Pearson correlation coefficient [21]. The closer a
coefficient between two features is to +/-1, the stronger the correlation
between the two input features. We removed any feature which had a correlation
coefficient (taking the absolute value) greater than 0.7. Similar features
include the number of executed instructions and the number of E-stage cycles
that were successfully completed.
Our feature selection process reduces the number of features to 10 for XeonPhi
(see Table I) and 10 for the NVIDIA Titan 1080Ti GPU (see Table II), where
some features are shared. Since our approach for feature selection is
automatic, the approach can be applied to other sets of candidate features. It
is to note that feature selection is also performed using cross-validation
(see also Section 5.2).
Table I: Chosen features for XeonPhi performance model Feature | Description
---|---
loop nest | at which level the outermost parallelizable loop lies on
loop count | # of the parallel loop iterations
#xfer_mem | # of host-device transfer API calls
dts | total host-device transfer size
redundant transfer size | host-device transfer size among overlapping tasks
max blocks | the maximum number of tasks of the application
min task unit | the minimum task granularity for a partition
# instructions | the total number of instructions of the kernel
branch miss | branch miss rate
L1 DCR | L1 Data cache miss rate
Table II: Chosen features for GPU programs Feature | Description
---|---
Access type 1 | # array access, whose fastest varying index is an affine function of the block id
Access type 2 | #array accesses, whose second or higher dimensional index is an affine function of the block id
#xfer_mem | # of host-device transfer API calls
host to device transfer size | total host to device transfer size
device to host transfer size | total device to host transfer size
redundant transfer size | host-device transfer size among overlapping tasks
max blocks | the maximum number of tasks
# instructions | the total number of instructions of the kernel
divergent branches | # divergent branches
L2 read miss rate | L2 cache read miss rate
#### 3.2.2 Feature Standardization
Supervised learning typically requires the feature values to lie in a certain
range. Therefore, we scaled the value for each of our features between the
range of 0 and 1. We record the maximum and minimum value of each feature
found at the training phase, and use these values to scale features extracted
from a new application after deployment. We truncate a value during deployment
if the value is outside the minimum/maximum value range seen during training.
It is to note that we also use the same approach to normalize the model
predictions (speedups) to the range of 0 and 1. In this work, we choose
Z-score to standardize the training data, and the details of quantifying the
impact of feature engineering methods can be found in Section 6.6.2.
(a) XeonPhi
(b) NVIDIA GPU
Figure 6: Feature importance on (a) XeonPhi and (b) NVIDIA GPU.
#### 3.2.3 Feature Importance
To understand the usefulness333In Section 6.6.4, we give a further breakdown
of the impact of individual feature to the model performance on a per
benchmark basis. of each feature, we apply a factor analysis technique called
Varimax rotation [22] to the feature space transformed by the principal
component analysis (PCA). This technique quantifies the contribution of each
feature to the overall variance in each of the PCA dimensions. Intuitively,
the more variances a feature brings to the space, the more useful information
the feature carries.
As an example, Figure 6 shows the top features chosen for XeonPhi and NVIDIA
GPU architectures. For the XeonPhi platform, features that capture the
parallelism degree (e.g. max blocks), host-device communication (e.g.
redundant transfer size), and computation (e.g. #instructions) are found to be
important. Other features such as L1 DCR and loop nest are useful, but are
less important compared to others. On the NVIDIA GPU platform, we note that
the parallelism degree is important, and the other features are equally useful
(Figure 6b). This figure shows that prediction can accurately draw upon a
subset of aggregated feature values.
### 3.3 Runtime Deployment
Once we have built and trained our performance model as described above, we
can use it as a cost function to search for the best stream configuration for
any _new_ , _unseen_ program. Feature values are extracted from the single-
stream version of the code. Static code features (such as loop count) are
extracted from the program source at compile time. Dynamic features (such as
branch miss) are extracted by profiling the program without partitioning for a
few loop iterations (which typically translate to several microseconds). After
feature collection, we feed the feature values to the search engine to rank
all candidate configurations using the performance model. The top-ranked
stream configuration is then used for the target program. In Section 4.4, we
provide further details on how the performance model can be integrated with
the host code generation process.
#### 3.3.1 Adapt to Changing Program Phases
Our current implementation chooses a configuration for each kernel and does
not change the configuration throughout the kernel execution. Therefore, it
can adapt to different behaviors across kernels because predictions are
performed on a per-kernel basis. We found that this strategy is sufficient for
many data-parallel kernels targeted in this work.
Our approach can be extended to adapt phase or program behavior changes within
a kernel. One way of doing this is to first partition the input data into
groups and then perform configuration selection before launching the kernel
that performs on an input data group. To reduce the prediction and
configuration overhead, we can sample periodically to see if the performance
counter readings are significantly different from the ones used for the
current prediction to trigger re-configuration. Dynamic re-configuration of a
running kernel will require extending the underlying runtime (e.g., hStreams
or CUDA) to adjust thread mapping and having hardware support to stop and
resume the execution contexts. We leave this as future work.
## 4 OpenMP to Streamed Code Generator
Figure 7: Work flow for translating OpenMP programs to streamed programs using
our automatic code generator.
Currently, there are very few publicly available benchmarks for utilizing the
streaming capability of heterogeneous many-core architectures, in particular,
XeonPhi. To evaluate our approach on a diverse set of benchmarks, we have
developed a compiler-based code generator, autostreamer, to automatically
translate OpenMP programs onto streamed code depending on the target
architecture. Our code generator is open sourced444Available at:
https://github.com/wisdom-moon/autostreamer.. Our implementation currently
supports converting OpenMP code to hStreams, CUDA and OpenCL programs. While
we do not claim novelty on this as several works on source-to-source
translation from OpenMP to CUDA[23, 24, 25, 26] or OpenCL[20, 27] exist, we
believe the tool could serve as a useful utility for translating OpenMP
programs to exploit multi-stream performance on heterogeneous many-core
architectures.
### 4.1 Code Generator Overview
Figure 7 depicts our source to source code generator for translating OpenMP
code to streamed programs. We use LLVM’s Clang front-end to convert OpenMP
code into the abstract syntax tree (AST). We then traverse the AST to obtain
the information to generate candidate streamed kernels and host-device
management code. The generated kernel and host code make use of exiting
programming models for kernel launching and communication management. We use
hStreams for XeonPhi and CUDA or OpenCL for GPUs.
Our current implementation supports the translation of OpenMP parallel loops,
i.e., loops annotated with omp for or omp for reduction constructs. For each
parallel loop, we outline the loop body and translate it into an individual
kernel function. We then replace the original loop body with a function call
(running on the host CPU) to launch the generated kernel. We also generate
management code for streaming context initialization, data partitioning, data
movements between the host and the accelerator, etc.
Our code generator relies on the native host/device compiler to optimize the
generated code. We have also compared our automatically generated code against
the manually translated code used in our prior work [16] and found that there
is little difference in performance for the set of OpenMP benchmarks used in
this work.
### 4.2 Preprocessing
As an example, Figure 8 illustrates how an OpenMP parallel loop can be
translated into hStreams code for XeonPhi. Note that a similar code generation
process is implemented for GPUs, using CUDA for NVIDIA GPU architectures and
OpenCL for other GPU platforms.
For each OpenMP parallel loop, we extract information of loop iterations from
the loop head. In this work, partitioning is achieved by splitting the loop
iteration space. Furthermore, we collect all the variables needed by the
hStreams kernel. Because hStreams requires kernel parameters to be passed as
the uint64_t (lines 1-2 of Figure 8b), the kernel parameters will be cast into
this type. The kernel parameters need to be packed into an array (line 21 in
Figure 8c). Then the hStreams library will unpack kernel parameters from the
array and pass the parameters to kernel function.
During the preprocessing stage, we also extract the static code feature values
of each target parallel loop. The code feature values will be encoded into the
source code during host code generation. It is to note that our approach can
be easily applied to existing hStreams programs – by first gathering feature
values from an hStreams kernel, and then storing the extracted information in
an auxiliary file or source code through a compiler front-end pass.
⬇ 1// An OpenMP C code for vector addition float * hostOutput = (float *)
malloc(inputLength*sizeof(float)); 3… #pragma omp parallel for 5for(int i=0;
i<inputLength; i++) { 7 hostOutput[i] = hostInput1[i] + hostInput2[i]; } 9…
(a) OpenMP code.
⬇ 1COINATIVELIBEXPORT void kernel (uint64_t arg0, uint64_t arg1, … uint64_t
arg5) 3{ int _start = (int) arg0; 5 … float *hostInput2 = (float *) arg5; 7
#pragma omp parallel for 9 for(int i= _start; i< _end; i++) hostOutput[i] =
hostInput1[i] + hostInput2[i]; 11}
(b) hStreams kernel code.
⬇ 1//output buffer float * hostOutput = (float *)
malloc(inputLength*sizeof(float)); 3 //Feature update and prediction 5Stream
config; 7conf_search(&config, &kernel_1_features, kernel_1_profile_runs); int
partitions = config.partitions; 9int tasks = config.tasks; 11//hStreams
Initialization hStreams_app_init(partitions, 1); . 13…
hStreams_app_create_buf((float *)hostInput1, …); 15… 17//Work partition int
sub_blocks = inputLength / tasks; 19int remain_index = inputLength % tasks;
21//Initialize kernel arguments uint64_t args[6]; args[2] = (uint64_t)
inputLength; 23… for (int idx = 0; idx < tasks; idx++) { 25 args[0] =
(uint64_t) _start; _end = _start + sub_blocks; 27 if (idx < remain_index) 29
_end ++; 31 args[1] = (uint64_t) _end;
hStreams_app_xfer_memory(&hostInput1[_start], &hostInput1[_start], (_end-
_start)*sizeof(float), idx % partitions, HSTR_SRC_TO_SINK, NULL); 33
hStreams_app_xfer_memory(&hostInput2[_start], …); 35 //Kernel launch
hStreams_EnqueueCompute(idx % partitions, ”kernel_1”, 3, 3, args, …); 37
//Read back results 39 hStreams_app_xfer_memory(&hostOutput[_start], …);
_start = _end; 41} … 43//hStreams cleanup code hStreams_app_fini();
(c) hStreams host code.
Figure 8: A running example of translating (a) an OpenMP parallel loop to (b)
hStreams kernel and (c) host management code.
### 4.3 Kernel Code Generation
Generating a streamed kernel function is straightforward as much of the OpenMP
code can be re-used. Figure 8b gives an example of the automatically generated
kernel for the OpenMP loop given in Figure 8a for hStreams kernels.
For the example given in Figure 8, an hStreams kernel starts with a pre-
processor macro COINATIVELIBEXPORT (lines 1-2 in Figure 8b). The number and
the type of the kernel parameters are loop-specific and are automatically
determined by our code generator. Within the generated kernel, all the
function parameters are cast from uint64_t into an appropriate type before
they are used. Note that the OpenMP parallel for pragmas are kept in the
generated kernel code per hStreams requirement (line 8 in Figure 8b).
With our code generator, the original outer-most loop iteration space will be
partitioned among parallel streams. The amount of work given to a specific
stream is determined by the _start and _end variables, which define which part
of the loop iteration space a stream instance will work on. A similar kernel
code generation approach is implemented for GPUs using CUDA or OpenCL.
### 4.4 Host Code Generation
To generate host code, we replace the original OpenMP parallel loop with a
function call to invoke the generated kernel (e.g., hStreams_EnqueueCompute in
Figure 8c)) together with additional code to initialize the host context and
to manage data transfer.
#### 4.4.1 Feature Value Collection
Static code features, extracted by our code generator, will be encoded as a
feature vector of real values. The feature vector will be passed to our
configuration search engine to find the optimal stream configuration at
runtime. Dynamic feature values are automatically collected by running the
generated streamed kernel for 5 iterations under the single-stream
configuration. As some loop bounds are dependent on the input, we might be
unable to determine certain feature values at compile time. These features are
represented as static symbolic pre-computation of loop bound variables, which
will be updated using runtime values at runtime.
#### 4.4.2 Setting Stream Configurations
To partition tasks among streams, we break the loop iterations into a number
of chunks of an equal size of subtask. We then group the hardware processor
cores into partitions, where each partition contains a fixed set of streams.
Processor partitioning and streams creation are achieved by calling the
hStreams_app_init (line 12 in Figure 8c) function for XeonPhi (and
cudaStreamCreate and clCreateCommandQueue for CUDA and OpenCL programs
respectively) by passing the stream configuration given by our search engine.
To overlap host device communications, we further split the input/output data
arrays to multiple data blocks (lines 32-39 in Figure 8c) where each task
operates on one block at a time while another data block is transferring
between the host and the accelerator. The number of data blocks is determined
by the stream configuration chosen at program runtime. The amount of work per
task and the size of transferred data can be determined with kernel
parameters. For example, in _for-loop_ at line 24 of Figure 8c, we calculate
them with the starting position (_start) and the block size (sub_block).
Thereafter, we schedule tasks and transfer the corresponding data blocks onto
streams in a round-robin fashion.
#### 4.4.3 Runtime Prediction
When a streamed (e.g., hStreams or CUDA) kernel is invoked, the configuration
selection engine library will choose a stream configuration (line 7 in Figure
8c) for the kernel. It uses the performance model to rank the candidate stream
configurations and returns the optimal configuration (_#partitions_ and
_#tasks_ for the example shown in Figure 8). The returned values are then used
to initialize the streamed context (lines 8-9 of Figure 8c). The overhead of
prediction is negligible (a few milliseconds) and is included in the results.
#### 4.4.4 Supporting OpenMP Constructs
OpenMP variables may have additional type information specified by directives,
including default, share, private, firstprivate, lastprivate, copyin and
threadprivate. Our generator uses these directives to map data onto the
accelerator memory space. Each variable with the share or default directive
will be translated into a global variable shared by all parallel threads.
Variables declared as private and threadprivate are translated such that there
is a private copy for each streamed kernel; no memory transfer between the
host and the accelerator is needed. For each variable specified as copyin or
first private, we create a private copy for each streamed kernel but
initialize each copy using explicit memory transfers before its first use.
Similarly, we create a private copy of a last private variable and the
original variable is updated by a stream that executes the last iteration.
Our implementation also supports a number of synchronization and thread
constructs. Structured blocks identified with master, and single directives
are executed by one thread on the host multi-core. barrier is implemented by
splitting up the parallel loop into smaller tasks to create synchronization
points among multiple streams. critical is implemented by using a mutex lock
to restrict the execution of the associated structured blocks to a single
thread at a time. The atomic and flush directives are already supported by
hStreams, CUDA or OpenCL.
#### 4.4.5 Host-Accelerator Communication Optimization
For each buffer that is used by both the host and the accelerator, we manage
two copies: one on the host memory and the other on the accelerator memory.
Our runtime records the status of each variable and checks whether the copy on
a device memory space is valid or not. No memory transfer is needed as long as
the copy in the target memory space is valid. We currently use a conservative
approach: if an element of an buffer has been updated, the entire buffer needs
to be synchronized before it can be used by threads running on a different
device. We also avoid unnecessary device to host data transfer by tracking the
data dependence between the kernel and the host program. For example, when
there are data-dependencies between two kernels but the host does not access
this data in between the two kernels, we directly pass the memory address of
the buffer to the later kernel (without moving the data back to the host).
## 5 Experimental Setup
### 5.1 Hardware, Systems Software and Benchmarks
Table III: Our evaluation platforms | CPU-XeonPhi | CPU-GPU
---|---|---
CPU | 8-core Xeon CPU @ 2.6 GHz | Core i7-8700K CPU @ 3.7 GHz
Accelerator | Intel Xeon 31SP Phi | NVIDIA GeForce GTX 1080 Ti GPU
Platforms. We evaluate our approach on two heterogeneous many-core platforms:
one is a CPU-XeonPhi platform and the other is a CPU-GPU platform. Table III
gives details of our hardware platforms.
Systems software. On the CPU-XeonPhi platform, the host CPU and the
accelerator are connected through PCIe. The host runs Redhat Linux v7.0 (with
kernel v3.10). The coprocessor runs a customized uOS (v2.6.38.8). We use
Intel’s MPSS (v3.6) to communicate between the host and the coprocessor. We
use the Intel hStreams library (v3.6) and Intel ICC (v16.0.3) for compilation
(with -O3 as the compiler option). The CPU-GPU platform runs Ubuntu 16.04
(with kernel v4.15). We use CUDA v10.0 and gcc v5.3 as the host compiler with
option “-O3”.
Benchmarks. We use our code generator to translate 37 OpenMP applications from
commonly used benchmark suites into hStreams and CUDA programs. We have
excluded benchmarks where the data transfer cannot be overlapped with the
kernel execution, which do not benefit from streamed parallelization. Table IV
gives the full list of these benchmarks. Among them, convolutionFFT2d and
convolutionSeparable have algorithm-dependent parameters, which are regarded
as different benchmarks in the experiments. This setting gives us a total of
39 programs. We run the majority of the programs using over 25 different
datasets, except for some applications where we used around 10 datasets
because the algorithmic constraints prevent us from using a larger number of
inputs.
Table IV: Streamed benchmarks used in our experiments. Suite | Name | Acronym | Name | Acronym
---|---|---|---|---
| convol.Separable | convsepr1(8) | dotProduct | dotprod
| convolutionFFT2d | fftx1y1(4y3) | fwt | fwt
| MonteCarlo | montecarlo | matVecMul | mvmult
| scalarProd | scalarprod | transpose | transpose
NVIDIA SDK | vectorAdd | vecadd | |
AMD SDK | binomial | binomial | BlackScholes | blackscholes
dct | dct | prefixSum | prefix
| bfs | bfs | histo | histo
| lbm | lbm | mri-q | mri-q
| mri-gridding | mri-gridding | sad | sad
Parboil | sgemm | sgemm | spmv | spmv
POLY BENCH | 2mm | 2mm | 3mm | 3mm
adi | adi | correlation | correlation
covariance | covariance | deriche | deriche
gemm | gemm | gemver | gemver
gesummv | gesummv | heat-3d | heat-3d
jacobi-1d | jacobi-1d | jacobi-2d | jacobi-2d
mvt | mvt | syr2k | syr2k
syrk | syrk | |
### 5.2 Competitive Approaches
We compare our regression-based approach against our preliminary work that
employs an SVM-based classifier to predict the optimal stream configuration
[16]. We denote our prior approach as SVM-classifier. We also compare our
approach against two recent models for predicting the optimal stream
configuration on GPUs. As it is currently not possible to configure the number
of processor partitions on GPUs, the relevant GPU models can only predict the
number of tasks.
_Liu et al._ In [12], Liu _et al._ use linear regression models to search for
the optimal number of tasks for GPU programs [12]. The approach employs
several analytic models, described as follows.
For a task with an input data size of $m$, the transferring time between the
CPU and the accelerator, $T_{t}$, is determined as $T_{t}=\alpha\cdot
m+\beta$, and the computation time, $T_{c}$, is calculated as:
$T_{c}=\eta\cdot m+\gamma$ where the model coefficients, $\alpha$, $\beta$,
$\eta$ and $\gamma$, are determined through empirical experiments. For a given
kernel with $N$ input data elements running using $n$ streams, this approach
partitions the computation into $n$ tasks, where the data size for each task,
$m$, is equal to $N$/$n$. For the programs which kernel dominated, the total
execution time, $T_{total}$, can be determined by:
$T_{total}=T_{t}+nT_{c}=\alpha\cdot m+\frac{N\gamma}{m}+N\eta+\beta$
For the programs which data transfer dominated:
$T_{total}=\alpha\cdot N+2\frac{N}{m}\beta$
By calculating the partial differential and second-order partial differential
of $T_{total}$ with respect to $m$, we can obtain the optimal task-granularity
as $m=\sqrt{\frac{N\gamma}{\alpha}}$. Then we can calculate the number of
tasks ($n$).
Note that $m=N/2$ is the optimal parameter for programs which data transfer
dominated, i.e., the optimal number of tasks is 2. Another problem of this
model is that it does not consider scenarios where communications in different
direction (i.e., host to device and device to host) can overlap with each
other. Note that we set the #partitions to be the same as $n$ for XeonPhi.
_Werkhoven et al._ The work presented by Werkhoven _et al._ models the
performance of data transfers between the CPU and the GPU [10]. They use the
LogGP model to estimate the host-device data transfer time. Specifically, the
model estimates the data transfer time using five parameters: the
communication latency ($L$), overhead ($o$), the gap ($g$), the number of
processors ($P$), and the PCIe bandwidth ($G$).
Let $B_{hd}$ denotes the amount of data transferred from the host to the
device and $B_{dh}$ denotes vice versa, and $T_{kernel}$ donates the kernel
execution time. For the dominant transfer scenario, the optimal number of
tasks(i.e., _#tasks_), $N_{s}$, can be estimated by solving the following
equations:
$B_{dh}*G_{dh}+g*(N_{s}-1)=\begin{cases}\frac{T_{kernel}}{N_{s}}+\frac{B_{dh}}{N_{s}}*G_{dh},&\text{if}B_{dh}>B_{hd}\\\
\frac{B_{hd}}{N_{s}}*G_{hd}+\frac{T_{kernel}}{N_{s}},&\text{otherwise}\end{cases}$
This model does not consider the dominant kernel scenario, as it assumes the
kernel execution time will increase as the number of streams increases and can
not model the kernel execution time. Here, we use the same equation to
calculate the optimal number of tasks. For this model, we also set the
#partitions to be equal to the optimal $N_{s}$ value on XeonPhi.
### 5.3 Evaluation Methodology
#### 5.3.1 Model Evaluation
We use cross-validation to evaluate our machine learning models. To test the
portability of our approach, we apply _leave-one-out_ cross-validation,
described as follows. We exclude the target program for predictions from the
training program set, and learn a model using the _remaining_ programs. We
then apply the learned model to the testing program. We repeat this process
until each benchmark is tested once. This is a standard evaluation
methodology, providing an estimate of the generalization ability of a machine
learned model in predicting _unseen_ data. Note that we exclude both
convolutionFFT2d and convolutionSeparable from the training set when one of
the two is evaluated, and we make sure all approaches are trained on the same
benchmarks for fair comparisons.
#### 5.3.2 Performance Report
We run each program under a stream configuration multiple times and report the
_geometric mean_ of the runtime. Compared to the arithmetic mean, the
geometric mean is often considered as a more suitable metric for reporting
program performance, as it can better minimize the impact of outliers [28]. To
determine how many runs are needed, we calculated the confidence range using a
95% confidence interval and make sure that the difference between the upper
and lower confidence bounds is smaller than 5%.
## 6 Experimental Results
In this section, we first present the overall performance of our approach on
both platforms. We then compare our approach to that uses fixed stream
configurations, two prior analytical models and our previous work. We futher
discuss the benefit sources of the streaming parallelism and the working
mechanism of our approach. At last, we demonstrate the tunning process of our
model.
### 6.1 Overall Performance
(a) XeonPhi
(b) NVIDIA GPU
Figure 9: Overall performance of our approach over a single-stream version on
XeonPhi (a) and NVIDIA GPU (b). Our approach achieves, on average, 93.7% and
97.9% of the oracle performance on XeonPhi and NVIDIA GPU, respectively. The
min-max bars show the range of performance achieved across different inputs.
In this experiment, we exhaustively profiled each application with all
possible stream configurations and report the best-found performance as the
_Oracle_ performance. The Oracle gives an indication of how close our approach
is to a _theoretically perfect_ solution. The baseline used to calculate the
speedup is running the application using a single-stream without processor
core or task partitioning.
The overall result is shown in Figure 9. The min-max bar on the diagram shows
the range of speedups per application across all evaluated inputs. Overall,
our approach achieves an average speedup of 1.57$\times$ and 1.1$\times$ over
the single-stream configuration on XeonPhi and the GPU respectively. This
translates to 93.7% and 97.9% of the Oracle performance on XeonPhi and the GPU
respectively.
On XeonPhi, the performance improvement of our approach comes from two
factors. First, by predicting the right processor partition size, our approach
allows effective overlapping of the host-device communication and computation.
Second, by matching task parallelism to the number of available processor
cores, our approach can reduce the overhead of thread management, compared to
the single-stream execution. When the host-device communication time dominates
the streaming process, the performance improvement mainly comes from
computation-communication overlapping and the speedup from streaming is
consistently less than 2$\times$. When the kernel execution time dominates the
stream process, the application can benefit from the overhead reduction of
thread management. In this case, the speedup can be as large as 5$\times$. We
provide a further discussion on this later in Section 6.5.1.
On the GPU, we can exploit bidirectional data transfer between the host and
the device by using pined memory which is not supported by hStreams. The
support of bidirectional data transfer allows us to obtain further performance
gains by overlapping host-device data transfer and computation. The
theoretically up-bound speedup on the GPU platform is 3$\times$, when data
transfer is perfectly overlapped with computation. The representative sample
is fftx4y3 with the larges dataset, the data transfer time in the two
directions is the same, and the kernel execution time is 1.5 times of the data
transfer time. The oracle speedup is 2.3$\times$, and our approach achieves a
speedup of 2.2 $\times$. On the other hand, because the current GPU
implementation does not support processor core partition, the kernel execution
time benefits less from using multiple streams. Programs which the kernel
execution time dominated have no speedup using multiple streams, such as bfs,
MonteCarlo.
### 6.2 Comparison to Fixed Stream Configurations
Our approach predicts from a wide range of stream configurations, which
configuration is likely to give the best performance for a given program and
dataset. A natural question to ask is that: is there a fixed stream
configuration that gives reasonable good performance across benchmarks and
datasets? To answer this question, we compare our predictive modeling based
approach to two specific configurations on each of our evaluation platforms.
Our justification for why we selecting the fixed configurations are described
as follows. On XeonPhi, our initial results in Section 2 indicate that using
the stream configuration of $(4,16)$, i.e. partitioning the cores to 4 groups
and running 4 tasks on each partition (16 tasks in total), gives good
performance. The statistics obtained from the training data suggest that the
configuration of $(17,85)$ give the best average performance across training
samples. On the GPU, several programs support a maximum of 4 tasks. Thus we
select the two configurations $(2,2)$ and $(4,4)$. The results are shown in
Figure 10.
(a) XeonPhi
(b) NVIDIA GPU
Figure 10: Comparing the performance with two fixed configurations on XeonPhi
(a) and NVIDIA GPU (b): config. $(4,16)$ of 4 partitions and 4 tasks per
partition, config. $(17,85)$ of 17 partitions and 5 tasks per partition,
config. $(2,2)$ of 2 partitions and 1 tasks per partition, and config. $(4,4)$
of 4 partitions and 1 tasks per partition.
(a) XeonPhi
(b) NVIDIA GPU
Figure 11: Violin plot showing the distribution of speedups per scheme across
benchmarks and datasets on XeonPhi (a) and GPU (b). The shape of the violin
corresponds to the speedup distribution to the oracle performance. The thick
black line shows where 50% of the data lies.
#### 6.2.1 XeonPhi
On XeonPhi, we observe improved performance for several benchmarks such as
mri-gridding, transpose, sad, under both configurations, but slower
performance for dotprod, vecadd, blackscholes, lbm, and mir-q (Figure 10a).
For prefix, configuration $(17,85)$ delivers improved performance while
configuration $(4,16)$ leads to slowdown performance. Overall, none of the two
fixed configurations give an improved performance on average. On average, our
approach outperforms the two fixed configurations by a factor of 1.4, and
delivers consistently improved performance across benchmarks and datasets.
The violin plot in Figure 11a shows how far is each of the three schemes to
the Oracle performance across benchmarks and datasets. Our approach not only
delivers the closest performance to the Oracle, but also has the largest
number of samples whose performance is next to the Oracle. By contrast, the
performance given by the fixed configurations for many samples is farther from
the Oracle performance.
#### 6.2.2 GPU
On the GPU, in most cases, the performance of configuration $(2,2)$ is
moderate, not great, but not much worse than single-version, leading to an
average speedup 1.03$\times$ (Figure 10b). By contrast, although configuration
$(4,4)$ performs poorly on two programs, it delivers a slightly larger
averaged speedup of 1.04$\times$. By choosing the stream configuration on a
per-program basis, our approach outperforms the two fixed configurations,
achieving an averaged speedup 1.10$\times$. On only four programs, our
approach delivers slightly worse performance with a small margin.
The violin plot in Figure 11b also confirms the strengths of our approach by
presenting the distribution of performance improvement. The results on the
diagram are normalized to the Oracle (best-available) performance. For most of
the programs, the two fixed configurations deliver 80% to 100% to the Oracle
performance. However, configuration $(4,4)$ can lead to rather poor
performance (less than 40% to the best available performance) on some
programs. Compared to the fixed configurations, the performance distribution
given by our approach is centralized on a range between 90% to 100%, where
most programs are within this percentile range. Furthermore, compared to the
fixed configurations, our approach has a fewer number of performance outliers,
which have less serious performance slowdown. Therefore, our approach delivers
consistently better performance compared with the fixed configurations.
#### 6.2.3 Summary
This experiment confirms that a fixed configuration fails to deliver improved
performance across applications and datasets, and selecting a right stream
configuration on a per program, per dataset basis is thus required.
### 6.3 Comparison to Analytical Models
(a) XeonPhi
(b) NVIDIA GPU
Figure 12: Comparing against _Liu et al._ and _Werkhoven et al._ on XeonPhi
(a) and NVIDIA GPU (b).
In this experiment, we compare our approach to the two recent analytical
models described in Section 5.2. The results are shown in Figures 12 and 13.
On XeonPhi, both competitive models prefer using $2$ tasks across benchmarks
and datasets. This is because that many programs are kernel dominated, the
analytical models simply assume that task partition has no effect on kernel’s
performance, and do not consider the thread management overhead. On the GPU,
the model proposed by _Liu et al._ tends to use $2$ tasks across benchmarks
and datasets. This is due to the fact that most programs are data transfer
dominated and this model ignores the overlap of the bidirectional data
transfers between the host and the device.
XeonPhi. Figure 12a demonstrates that our approach gives better performance
for nearly all programs on XeonPhi. For the remaining handful programs, all
three approaches deliver comparable performance. Compared to the results
Figure 10, we can find the performance of the analytical models is similar to
fixed stream configurations. This is because the performance of the seven
programs, such as binomial, changes dramatically with different stream
configurations (see also Figure 2). The performance of the remaining programs
is not sensitive to the variation of stream configurations. From Figure 13a,
we can further see that _Liu et al._ and _Werkhoven et al._ deliver a speedup
within a range on 20% to 80%, while the performance of our approach is
centralized on a range between 80% to 100%. Thus, our approach delivers
consistently better performance compared with the alternative models.
GPU. Figure 12b shows that our approach delivers better performance for around
75% of the programs on the GPU. Since _Werkhoven et al._ and _Liu et al._ are
manually tuned for the GPUs, they give better performance on some benchmarks
over our approach. However, our approach has the advantages of being
automatically learned from training data, with little expert involvement. The
performance of our approach can be further improved by using more training
examples to better cover the program space. Figure 13b shows that _Liu et al._
and _Werkhoven et al._ delivers a speedup within a range of 5% to 80%, and 70%
to 100%, respectively. By contrast, the performance of our approach is
centralized within a range between 90% to 100% for more programs. Therefore,
overall, our approach delivers better average performance compared with the
alternative models.
(a) XeonPhi
(b) NVIDIA GPU
Figure 13: Violin plots showing the distribution of speedups across benchmarks
and datasets on XeonPhi (a) and GPU (b).
### 6.4 Comparison to Classification-based Approach
(a) XeonPhi
(b) NVIDIA GPU
Figure 14: Comparing against a classification based approach on XeonPhi (a)
and NVIDIA GPU (b).
Our prior work uses a SVM classifier to predict the configurations [16].
Compared with it, the regression-based model presented in this article has
several advantages.
A classification model predicts which of a set of predefined labels the input
belongs to. Using this strategy, we will need to label each unique stream
configuration. This will lead to a total of 175 labels for 311 profiling
samples on the XeonPhi, and 11 labels on the GPU. On the XeonPhi, the ratio of
samples to labels is too small to build an accurate model. As a result, we
have to merge labels in our prior work [16] at the cost of losing accuracy.
Classification is a constraint optimization problem where the model has to
know all the possible configurations during training. Our new regression-based
approach avoids this pitfall by directly modeling the impact of the stream
configuration; it thereby can be used on any stream configuration as the
configuration is the model’s input.
Figure 14a presents results obtained on the XeonPhi. Our regression-based
approach outperforms the SVM-classifier in 21 of the 39 programs and achieves
over 5% performance improvement for 13 programs. It is to note that the
overhead for ranking stream configurations is included in the experimental
results. Overall, our regression-based approach improves the SVM-classifier
by, on average, 3% (up to 46%). Unlike XeonPhi, we were able to obtain
sufficient training samples per label (because the optimization space is
smaller) on the GPU to build a more accurate classification model. As can be
seen from Figure 14b, the average speedup of SVM-classifier and the
regression-based approach is comparable.
Compared to a classifier, our regression-based approach has the advantage of
being able to be applied to configurations that were not seen during the
training phase. Therefore, our approach has a better generalization ability.
### 6.5 Further Analysis of Performance Results
We now take a closer look into the performance results, using XeonPhi as a
case study.
Figure 15: Reduction of kernel computation time over a single-stream execution
on XeonPhi. The performance improvement comes from the reduction of the
threading overhead. A stream configuration is annotated as (_#partitions_ ,
_#tasks_).
#### 6.5.1 High Speedup Cases
On XeonPhi, bidirectional data transfer between the host and the accelerator
cannot be overlapped, i.e., we can only issue data transfer from the host to
the device or vice versa at once but not simultaneously. As a result, the
theoretical up-bound speedup for overlapping computation and communication is
2$\times$, when the computation is perfectly overlapped with the data transfer
time. It is interesting to observe that several benchmarks achieve a speedup
of over 2$\times$ on XeonPhi (see Figure 9a). After having a closer
investigation, we notice that such performance is attributed to the reduction
in the kernel execution time in additional to the overlapping of communication
and computation.
To quantify the benefit of kernel time reduction, we measure the kernel
execution time with and without multiple streams and calculate the speedup
between them. Note that we _exclude the host-device communication time in this
case_ to isolate the contributing factors. The kernel time improvement for
transpose, binomial, and fftx1y1 is shown in Figure 15. As can be seen from
the diagram, choosing a good stream configuration can lead to more than 4x
reduction on the kernel execution time. This is because these benchmarks are
implemented by parallelizing the inner loop within a nested loop. During
runtime, the parallel threads working on the inner loop will be created,
synchronized, or destroyed for each outer loop iteration. Such threading
overhead could be significant when the outer loop iterates a large number of
times. With multiple streams, we divide the whole outer loop iteration space
into multiple smaller iterations. This allows multiple groups of threads to be
managed simultaneously, leading to a significant decrease in threading
overhead and faster kernel execution time. On the other hand, using too many
streams and partitions will lead to a performance decrease. This is because
stream management also comes at a cost, which increases as the number of
partitions grows. Nonetheless, for applications where the kernel computation
dominates the program execution time, by reducing the kernel time can lead to
additional improvement, yielding more than 2x speedups.
(a) XeonPhi
(b) NVIDIA GPU
Figure 16: Violin plot showing the distribution of speedups per benchmark
across datasets on XeonPhi (a) and NVIDIA GPU (b). The shape of the violin
corresponds to the speedup distribution. The thick black line shows where 50%
of the data lies.
#### 6.5.2 Speedup Distribution
Figure 16 gives the speedup per benchmark across datasets on XeonPhi and the
GPU. The shape of the violin plot corresponds to the speedup distribution.
On XeonPhi, we see that the speedups of montecarlo and prefix distribute
fairly uniformly while the data distribution of fftx1y1 and fftx4y3 is
multimodal (i.e. it has two peaks). Further, the input datasets have little
impact on the behavior of fwt and lbm, so the speedups remain constant across
datasets.
On the GPU, the speedups of dotprod, vecadd, blackscholes and mri-q distribute
fairly uniformly while the data distribution of convsepr1, convsepr8, fftx1y1,
fftx4y3 and dct is unimodal (i.e. it has one peak). Furthermore, the input
datasets have a very slight impact on the performance behaviors of montecarlo,
scalarprod, transpose and binomial. Thus, their speedups remain constant
across datasets.
To conclude, the streaming speedups of some applications are sensitive to
their input datasets whereas the others are not. And the distribution of
speedups on the GPU is more concentrated than XeonPhi. This is because the
current GPU implementation does not support processor core partition, the
kernel execution time benefits less from multiple streams than XeonPhi.
Figure 17: The relation between computation-communication ratio and the
speedup. The computation-communication ratio is normalized using the natural
logarithm function. Thus, the kernel computation time equals the host-device
communication time when $ratio=0$. In general, a higher computation-
communication ratio leads to a better speedup.
#### 6.5.3 Correlation Analysis
Figure 17 shows the relation between the computation-communication ratio and
the achieved speedup when using heterogeneous streams across all benchmarks
and datasets on XeonPhi. We see that the computation-communication ratio
varies over the benchmarks and the speedup changes accordingly, but in
general, a higher computation-to-communication ratio leads to a greater
speedup. As explained in Section 6.5.1, in addition to overlapping computation
and communication, our approach can also reduce the kernel computation time by
choosing the right stream configuration. Therefore, benchmarks with a high
computation-communication ratio also benefit from a reduction in the kernel
computation time.
To quantify the relation between the computation-communication ratio and the
speedup, we calculate the Pearson correlation coefficient of the two
variables. The calculation gives a correlation coefficient of 0.7, indicating
that the two variables (the computation-communication ratio and the speedup)
have a strong linear correlation. By carefully selecting the stream
configuration, our approach tries to maximize the overlap between
communication and computation, which thus leads to favourable performance.
#### 6.5.4 Impact of Streaming Parallelism
Figure 18: Breakdown of program execution time ($T$), host-device data
transfer time ($T_{m}$), kernel execution time ($T_{k}$), hStreams context
initialization overhead ($T_{c}$) and communication-computation overlapping
time ($T_{o}$) for single and best-performing multi-stream configurations.
Our earlier experiments show that by carefully exploiting streaming
parallelism, we can significantly improve application performance. We now take
a closer look at three representative benchmarks, fftx1y1, fwt and gesummv, to
get a better understanding of streaming performance on XeonPhi. These
benchmarks represent different degrees of benefits obtained from streamed
parallelism (with a speedup of 2$\times$, 1.5$\times$ and 1$\times$,
respectively).
We use the following analytical model to breakdown the execution time of a
multi-stream program:
$T=T_{m}+T_{k}+T_{c}-T_{o}$ (1)
where $T_{m}$ is host-device data transfer time, $T_{k}$ is kernel execution
time, $T_{c}$ is the overhead for initializing the context, and $T_{o}$ is
overlapping time between data transfer and kernel execution. We measure $T$,
$T_{m}$, $T_{k}$, and $T_{c}$, and use the measurements to calculate $T_{o}$.
Figure 18 gives the breakdown for the five components in Equation 1. For each
testing program, we compare the single-stream configuration against the best-
performing multi-stream configuration. The host-device data transfer time,
$T_{m}$, is nearly constant among a single and a multiple stream
configuration, but multi-streaming can reduce the kernel execution time,
$T_{k}$, by exploiting the spatial sharing of processing resources among
computation tasks. The overhead of initializing the hStreams context, $T_{c}$,
depends on the kernel execution time. For fftx1y1 and fwt, whose kernels run
for a sufficiently long time, this one-off runtime overhead is negligible.
However, for gesummv, this overhead cannot be ignored due to the relatively
short kernel running time. The contribution for overlapping host-device
communications with kernel execution, $T_{o}$, varies across programs. For
fftx1y1 and fwt, it accounts for around 50% of $T_{m}$, suggesting that by
exploiting temporal sharing to overlap communication with kernel execution can
amortize the host-device communication overhead. For gesummv, $T_{o}$ is small
due to little alignment between data transfer and kernel execution. As such,
there is little benefit for exploiting temporal sharing for this program.
This experiment gives a more detailed analysis for the benefits of exploiting
multiple streams. The results reinforce our claim that the benefit for
streaming parallelism depends on the computation kernel and hence an adaptive
scheme for choosing the optimal stream configuration is needed. Our work aims
to offer such a capability.
### 6.6 Analysis of Predictive Modeling Techniques
In this section, we analyse the working mechanism of our predictive model,
using XeonPhi as an evaluation platform.
#### 6.6.1 Comparison to Alternative Modeling Techniques
We compare our MLP-based model against four widely used regression methods:
the DCT (Decision Tree), the RF (Random Forest), the XGB (eXtreme Gradient
Boosting) and SVM (Support Vector Machine) as well as four classification
models: SVM, DCT, MLP and KNN (K-Nearest Neighbors). We use the Radial basis
function kernel for the SVM models.
For each technique, we follow the same training methodology and use the same
features and training examples to build a model. For classification models, we
apply the label merging process described in our prior work [16] to improve
the prediction accuracy. Table V compares the training overhead, average
prediction time and achieved average speedup for each model. We note that
training a regression-based SVM model has the largest overhead. Although
training a DCT has less overhead over our MLP-based regression model, MLP
gives better prediction performance. The RF and XGB models are based on DCT,
but they do not yield a better performance. Compared to regression models, a
classification model takes less time to train and make predictions. However,
classification models give worse performance over regression models as they
require more training data to cover the optimization space. Overall, we choose
to use a regression-based approach and employ MLP because it gives the best
overall prediction performance and has modest training overhead.
Table V: Comparison to alternative modeling techniques Technique | Training time | Avg. pred. time | Avg. speedup
---|---|---|---
SVM (regression) | 100 hours | 2280 ms | 1.56
DCT (regression) | 65.57 seconds | 0.74 ms | 1.51
RF (regression) | 317.89 seconds | 11.94 ms | 1.51
XGB (regression) | 28.46 seconds | 0.74 ms | 1.49
MLP (regression, ours) | 245.8 seconds | 0.76 ms | 1.57
SVM (classifier) | 1.28 seconds | 0.10 ms | 1.53
DCT (classifier) | 0.79 seconds | 0.05 ms | 1.38
MLP(classifier) | 46.45 seconds | 0.05 ms | 1.41
KNN (classifier) | 0.22 seconds | 0.23 ms | 1.43
#### 6.6.2 Feature Engineering
Feature engineering has a significant impact on the performance of a machine
learning model (Section 3.2). Here we quantify the impact of feature
engineering methods. In this work, we consider three standard feature
engineering approaches including standardization, normalization and dimension
reduction.
Standardization converts all features value to be in a common range, e.g.,
between 0 and 1. The idea is to prevent the feature value range to dominate
the importance of that feature. In this work we apply a commonly used
standardization method called _Z-score_ [29] to standardize the raw feature
values and the speedups (i.e., prediction targets) in the training data. We
found that feature standardization improves the achieved speedup by 3% on
average, and speedup standardization improves the achieved speedup by 5% on
average.
Normalization scales the feature values to make them follow the normal
distribution. We have tested a range of normalization methods including the
square root, the reciprocal of square root and the natural logarithm
transformation. However, we found that normalization does not improve our
model prediction accuracy.
Dimension reduction reduces the number of features, which is often useful when
the number of training examples is not proportional to the number of feature
dimensions. In this work, we apply factor analysis (FA) [30] and principal
component analysis (PCA) [31] to the raw features. Applying PCA and using 9
PCA components gives the best overall result, by improving the average speedup
by 17%. PCA outperforms FA which gives an average 3% improvement on the
achieved speedup.
#### 6.6.3 MLP Parameters Tuning
We now discuss the impact of the MLP parameter choices. There are four
configurable parameters for an MLP model: the activation function, the number
of hidden layers, the number of neurons, and the learning algorithm (i.e., the
solver). For activation functions, we consider identity, logistic, tanh and
relu. For hidden layers and neurons, we vary the number of hidden layers from
1 to 5 and the number of neurons per layer from 3 to 100. For the solver, we
consider three commonly used weight optimizers: lbfgs, sgd and and adam. We
use scikit-learn implementations for the activation function and the solver.
Our experimental results suggest that the best-performing activation function
and solver are tanh and adam respectively, and using three hidden layers with
9 neurons per layers gives the best overall results on our training data.
Overall, tuning MLP model parameter improves the average speedup by 5% over
the default parameter setting.
Figure 19: A Hinton diagram showing the impact of each feature used by the
performance model to the resultant application performance. The larger the
box, the more likely a feature has a greater impact on the performance of the
respective benchmark.
#### 6.6.4 Impact of Individual Feature
In this experiment, we consider the impact of a specific feature to the
resultant performance. Figure 19 presents a Hinton diagram illustrating how
important a feature contribution to the performance model prediction accuracy
(which in turns affects the resulting application performance). The larger the
box, the more significant a feature for a given program’s performance. Here,
the x-axis denotes the programs, and the y-axis denotes the features used by
our performance model. The impact of a feature is quantified by measuring how
much speedup improvement can be obtained if that feature is used by the
performance model. Note that this is a post-hoc analysis and, in general, we
cannot know in advance the importance of a feature on _unseen_ programs.
Figure 19 shows that all the features are important for the set of benchmarks
targeted in the work, but the importance of features varies across programs.
This diagram illustrates how hard it is to develop an analytical model to
capture the diverse behaviors and characteristics of streaming programs.
## 7 Related Work
Our work builds upon the following past foundation, while qualitatively
differing from each.
Task Scheduling. There is considerable work on distributing work across
heterogeneous processors to improve application performance [32, 33, 34].
Prior work in the area typically assumes that the processor configuration is
fixed and relies on the operating system to schedule parallel tasks across
processing units. Recent studies show that by partitioning the processing
units into groups it is possible to significantly improve the application
performance by overlapping the host-device communication and computation on
coprocessors like Intel XeonPhi [14, 6]. However, existing approaches rely on
static tuning to find the processor partition and the best number of streams
to run within a partition. As a result, previous approaches cannot adapt to
the change of program inputs. As a departure from prior work, we develop an
automatic approach to dynamically adjust the processor partition and task-
granularity during runtime, considering the characteristics of applications
and input datasets; our approach thereby can adapt to the change of program
inputs.
Domain-specific Optimizations There is considerable work on domain-specific
optimization on Intel XeonPhi. Cheng _et al._ [35] and Jha _et al._ [36] show
that in-memory database applications suffer from under-utilization of
processor resources and hence a fine-grained tuning approach is required.
Mrphi is a framework for optimizing MapReduce workload on the XeonPhi [37]. It
employs a set of techniques to improve the resource utilization to obtain
higher application performance. Other works look at performance optimization
for numerical solvers [38], sparse matrix vector multiplication [39, 40], and
dynamic stochastic economic models [39]. Ferrão _et al._ [41] and Memeti _et
al._ [42] develop a stream processing framework for XeonPhi to increase the
programming productivity. The runtime can automatically distribute workloads
across CPUs and accelerating devices. These approaches improve the processor
utilization by adjusting the algorithmic design, which are complementary to
our work on tuning multi-streaming parallelism for data parallel applications.
Multiple Streams Modeling. Gomez-Luna _et al._ [11] develop a set of models to
estimate the asynchronous data transfer overhead on different GPU
architectures. The models can be used to estimate the optimal number of
streams to use on a given GPU platform. Werkhoven _et al._ [10] present an
analytical model to determine when to apply an overlapping method on GPUs. Liu
_et al._ [12] also develop an analytical based approach to determine the
optimal number of streams to use on GPUs. However, none of these approaches
considers the processor partition. As we have shown in Section 6.3, ignoring
the processor partitioning parameter can lead to poor performance on Intel
XeonPhi. Furthermore, these hand-crafted models have the drawback of being not
portable across architectures as the model is tightly coupled to a specific
GPU architecture. Our work advances prior work by employing machine learning
to automatically learn the optimal processor partition and the number of
streams/tasks to use. Since our models are automatically learned from
empirical observations, one can easily re-learn a model for a new
architecture.
Predictive Modeling. Recent studies have shown that machine learning based
predictive modeling is effective in code optimization [43, 44], performance
predicting [45, 46], parallelism mapping [47, 48, 20, 49, 50], and task
scheduling [51, 52, 53, 54, 55, 56]. Its great advantage is its ability to
adapt to the ever-changing platforms as it has no prior assumption about their
behavior. The work presented by Wen _et al._ [57] employs SVMs to develop a
binary classifier to predict that if a given OpenCL kernel can achieve a large
speedup or not. Our work differs from [57] in that it targets a different
architecture and programming model, and it predicts from a larger number of
configurations instead of making a binary prediction. Our prior work developed
an SVM based classifier to predict the optimal stream configuration for Intel
XeonPhi [16]. However, it requires having sufficient training data samples to
cover all possible stream configurations. Our approach improves the prior work
by directly modeling the impact of the stream configuration. As a result, our
approach can make predictions for any stream configuration (even those are not
seen in the training data).
Autotuning Parallel Programs. Our approach is closely related to autotuning
that searches for the best-performing optimization configuration [58, 59].
This technique is demonstrated to be effective for choosing algorithmic
choices [60], tuning GPU code [61, 62, 63], optimizing structured parallel
programs [64, 65, 66] and non-uniform memory access (NUMA) architectures [67],
and more recently for deep neural networks [68]. Many of the prior works in
this area employ an evolutionary-based approach by applying and profiling
candidate optimization options to choose a good option to use. One of the key
changes of autotuning is how to avoid the profiling overhead which could be
prohibitively expensive. We do so by using a performance model to quickly
evaluate the profitability of a candidate optimization option. We show that
our approach has low runtime overhead, which thus permits us to apply it at
runtime to best match the optimization strategy to the program input.
Furthermore, our work is the first for tuning heterogeneous streaming
parallelism on heterogeneous many-cores (XeonPhis and GPUs).
Automatic Generation of Parallel Programs. The OpenMPC compiler [69]
translates OpenMP to CUDA programs. Wang _et al._ [24, 20, 70] translates
OpenMP to OpenCL programs and use machine learning to select the most suitable
device from the host CPU and the GPU to run the code. Rawat _et al._ presents
an automatic approach to generate GPU code from a domain-specific language
(DSL) for stencil programs [71]. All of the above approaches target GPUs, and
do not utilize the multi-streaming strategy.
## 8 Conclusion
This article has presented an automatic approach to exploit streaming
parallelism on heterogeneous many-cores. Central to our approach is a machine
learning-based model that predicts the resulting performance when running the
target application under a given streamed configuration. The performance
predictor is then used as a cost function to quickly rank candidate
configurations at runtime, to determine which stream configuration should be
used on a per-program per-dataset basis. We have evaluated our approach on an
Intel XeonPhi and an NVIDIA GTX 1080 Ti GPU, with 39 representative
benchmarks. Experimental results show that our approach delivers an average
speedup of 1.6x and 1.1x on XeonPhi and the GPU, respectively. These results
translate to over 93% of the best-available performance.
## Acknowledgment
This work was partially funded by the National Key Research and Development
Program of China under Grant No. 2018YFB0204301, the National Natural Science
Foundation of China under Grant agreements 61972408, 61602501 and 61872294;
For any correspondence, please contact Jianbin Fang (Email:
[email protected]).
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|
2024-09-04T02:54:58.500113 | 2020-03-09T18:36:38 | 2003.04352 | {
"authors": "LHCb collaboration: R. Aaij, C. Abell\\'an Beteta, T. Ackernley, B.\n Adeva, M. Adinolfi, H. Afsharnia, C.A. Aidala, S. Aiola, Z. Ajaltouni, S.\n Akar, P. Albicocco, J. Albrecht, F. Alessio, M. Alexander, A. Alfonso Albero,\n G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, L. An,\n L. Anderlini, G. Andreassi, M. Andreotti, F. Archilli, A. Artamonov, M.\n Artuso, K. Arzymatov, E. Aslanides, M. Atzeni, B. Audurier, S. Bachmann, J.J.\n Back, S. Baker, V. Balagura, W. Baldini, A. Baranov, R.J. Barlow, S. Barsuk,\n W. Barter, M. Bartolini, F. Baryshnikov, J.M. Basels, G. Bassi, V.\n Batozskaya, B. Batsukh, A. Battig, A. Bay, M. Becker, F. Bedeschi, I.\n Bediaga, A. Beiter, L.J. Bel, V. Belavin, S. Belin, V. Bellee, K. Belous, I.\n Belyaev, G. Bencivenni, E. Ben-Haim, S. Benson, S. Beranek, A. Berezhnoy, R.\n Bernet, D. Berninghoff, H.C. Bernstein, C. Bertella, E. Bertholet, A.\n Bertolin, C. Betancourt, F. Betti, M.O. Bettler, Ia. Bezshyiko, S. Bhasin, J.\n Bhom, M.S. Bieker, S. Bifani, P. Billoir, A. Bizzeti, M. Bj{\\o}rn, M.P.\n Blago, T. Blake, F. Blanc, S. Blusk, D. Bobulska, V. Bocci, O. Boente Garcia,\n T. Boettcher, A. Boldyrev, A. Bondar, N. Bondar, S. Borghi, M. Borisyak, M.\n Borsato, J.T. Borsuk, T.J.V. Bowcock, C. Bozzi, M.J. Bradley, S. Braun, A.\n Brea Rodriguez, M. Brodski, J. Brodzicka, A. Brossa Gonzalo, D. Brundu, E.\n Buchanan, A. B\\\"uchler-Germann, A. Buonaura, C. Burr, A. Bursche, A.\n Butkevich, J.S. Butter, J. Buytaert, W. Byczynski, S. Cadeddu, H. Cai, R.\n Calabrese, L. Calero Diaz, S. Cali, R. Calladine, M. Calvi, M. Calvo Gomez,\n P. Camargo Magalhaes, A. Camboni, P. Campana, D.H. Campora Perez, A.F.\n Campoverde Quezada, L. Capriotti, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, I. Carli, P. Carniti, K. Carvalho Akiba, A. Casais Vidal, G. Casse,\n M. Cattaneo, G. Cavallero, S. Celani, R. Cenci, J. Cerasoli, M.G. Chapman, M.\n Charles, Ph. Charpentier, G. Chatzikonstantinidis, M. Chefdeville, V.\n Chekalina, C. Chen, S. Chen, A. Chernov, S.-G. Chitic, V. Chobanova, S.\n Cholak, M. Chrzaszcz, A. Chubykin, P. Ciambrone, M.F. Cicala, X. Cid Vidal,\n G. Ciezarek, F. Cindolo, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier,\n J.L. Cobbledick, V. Coco, J.A.B. Coelho, J. Cogan, E. Cogneras, L. Cojocariu,\n P. Collins, T. Colombo, A. Comerma-Montells, A. Contu, N. Cooke, G. Coombs,\n S. Coquereau, G. Corti, C.M. Costa Sobral, B. Couturier, D.C. Craik, J.\n Crkovsk\\'a, A. Crocombe, M. Cruz Torres, R. Currie, C.L. Da Silva, E.\n Dall'Occo, J. Dalseno, C. D'Ambrosio, A. Danilina, P. d'Argent, A. Davis, O.\n De Aguiar Francisco, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda,\n L. De Paula, M. De Serio, P. De Simone, J.A. de Vries, C.T. Dean, W. Dean, D.\n Decamp, L. Del Buono, B. Delaney, H.-P. Dembinski, A. Dendek, V. Denysenko,\n D. Derkach, O. Deschamps, F. Desse, F. Dettori, B. Dey, A. Di Canto, P. Di\n Nezza, S. Didenko, H. Dijkstra, V. Dobishuk, F. Dordei, M. Dorigo, A.C. dos\n Reis, L. Douglas, A. Dovbnya, K. Dreimanis, M.W. Dudek, L. Dufour, G. Dujany,\n P. Durante, J.M. Durham, D. Dutta, M. Dziewiecki, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, R. Ekelhof, S.\n Ek-In, L. Eklund, S. Ely, A. Ene, E. Epple, S. Escher, S. Esen, T. Evans, A.\n Falabella, J. Fan, N. Farley, S. Farry, D. Fazzini, P. Fedin, M. F\\'eo, P.\n Fernandez Declara, A. Fernandez Prieto, F. Ferrari, L. Ferreira Lopes, F.\n Ferreira Rodrigues, S. Ferreres Sole, M. Ferrillo, M. Ferro-Luzzi, S.\n Filippov, R.A. Fini, M. Fiorini, M. Firlej, K.M. Fischer, C. Fitzpatrick, T.\n Fiutowski, F. Fleuret, M. Fontana, F. Fontanelli, R. Forty, V. Franco Lima,\n M. Franco Sevilla, M. Frank, C. Frei, D.A. Friday, J. Fu, Q. Fuehring, W.\n Funk, E. Gabriel, A. Gallas Torreira, D. Galli, S. Gallorini, S. Gambetta, Y.\n Gan, M. Gandelman, P. Gandini, Y. Gao, L.M. Garcia Martin, J. Garc\\'ia\n Pardi\\~nas, B. Garcia Plana, F.A. Garcia Rosales, L. Garrido, D. Gascon, C.\n Gaspar, D. Gerick, E. Gersabeck, M. Gersabeck, T. Gershon, D. Gerstel, Ph.\n Ghez, V. Gibson, A. Giovent\\`u, O.G. Girard, P. Gironella Gironell, L.\n Giubega, C. Giugliano, K. Gizdov, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, I.V. Gorelov, C. Gotti, E. Govorkova, J.P.\n Grabowski, R. Graciani Diaz, T. Grammatico, L.A. Granado Cardoso, E.\n Graug\\'es, E. Graverini, G. Graziani, A. Grecu, R. Greim, P. Griffith, L.\n Grillo, L. Gruber, B.R. Gruberg Cazon, C. Gu, E. Gushchin, A. Guth, Yu. Guz,\n T. Gys, P. A. G\\\"unther, T. Hadavizadeh, G. Haefeli, C. Haen, S.C. Haines,\n P.M. Hamilton, Q. Han, X. Han, T.H. Hancock, S. Hansmann-Menzemer, N. Harnew,\n T. Harrison, R. Hart, C. Hasse, M. Hatch, J. He, M. Hecker, K. Heijhoff, K.\n Heinicke, A.M. Hennequin, K. Hennessy, L. Henry, J. Heuel, A. Hicheur, D.\n Hill, M. Hilton, P.H. Hopchev, J. Hu, W. Hu, W. Huang, W. Hulsbergen, T.\n Humair, R.J. Hunter, M. Hushchyn, D. Hutchcroft, D. Hynds, P. Ibis, M. Idzik,\n P. Ilten, A. Inglessi, K. Ivshin, R. Jacobsson, S. Jakobsen, E. Jans, B.K.\n Jashal, A. Jawahery, V. Jevtic, F. Jiang, M. John, D. Johnson, C.R. Jones, B.\n Jost, N. Jurik, S. Kandybei, M. Karacson, J.M. Kariuki, N. Kazeev, M. Kecke,\n F. Keizer, M. Kelsey, M. Kenzie, T. Ketel, B. Khanji, A. Kharisova, K.E. Kim,\n T. Kirn, V.S. Kirsebom, S. Klaver, K. Klimaszewski, S. Koliiev, A.\n Kondybayeva, A. Konoplyannikov, P. Kopciewicz, R. Kopecna, P. Koppenburg, M.\n Korolev, I. Kostiuk, O. Kot, S. Kotriakhova, L. Kravchuk, R.D. Krawczyk, M.\n Kreps, F. Kress, S. Kretzschmar, P. Krokovny, W. Krupa, W. Krzemien, W.\n Kucewicz, M. Kucharczyk, V. Kudryavtsev, H.S. Kuindersma, G.J. Kunde, T.\n Kvaratskheliya, D. Lacarrere, G. Lafferty, A. Lai, D. Lancierini, J.J. Lane,\n G. Lanfranchi, C. Langenbruch, O. Lantwin, T. Latham, F. Lazzari, C.\n Lazzeroni, R. Le Gac, R. Lef\\`evre, A. Leflat, O. Leroy, T. Lesiak, B.\n Leverington, H. Li, L. Li, X. Li, Y. Li, Z. Li, X. Liang, R. Lindner, V.\n Lisovskyi, G. Liu, X. Liu, D. Loh, A. Loi, J. Lomba Castro, I. Longstaff,\n J.H. Lopes, G. Loustau, G.H. Lovell, Y. Lu, D. Lucchesi, M. Lucio Martinez,\n Y. Luo, A. Lupato, E. Luppi, O. Lupton, A. Lusiani, X. Lyu, S. Maccolini, F.\n Machefert, F. Maciuc, V. Macko, P. Mackowiak, S. Maddrell-Mander, L.R. Madhan\n Mohan, O. Maev, A. Maevskiy, D. Maisuzenko, M.W. Majewski, S. Malde, B.\n Malecki, A. Malinin, T. Maltsev, H. Malygina, G. Manca, G. Mancinelli, R.\n Manera Escalero, D. Manuzzi, D. Marangotto, J. Maratas, J.F. Marchand, U.\n Marconi, S. Mariani, C. Marin Benito, M. Marinangeli, P. Marino, J. Marks,\n P.J. Marshall, G. Martellotti, L. Martinazzoli, M. Martinelli, D. Martinez\n Santos, F. Martinez Vidal, A. Massafferri, M. Materok, R. Matev, A. Mathad,\n Z. Mathe, V. Matiunin, C. Matteuzzi, K.R. Mattioli, A. Mauri, E. Maurice, M.\n McCann, L. Mcconnell, A. McNab, R. McNulty, J.V. Mead, B. Meadows, C. Meaux,\n G. Meier, N. Meinert, D. Melnychuk, S. Meloni, M. Merk, A. Merli, M.\n Mikhasenko, D.A. Milanes, E. Millard, M.-N. Minard, O. Mineev, L. Minzoni,\n S.E. Mitchell, B. Mitreska, D.S. Mitzel, A. M\\\"odden, A. Mogini, R.D. Moise,\n T. Momb\\\"acher, I.A. Monroy, S. Monteil, M. Morandin, G. Morello, M.J.\n Morello, J. Moron, A.B. Morris, A.G. Morris, R. Mountain, H. Mu, F. Muheim,\n M. Mukherjee, M. Mulder, D. M\\\"uller, K. M\\\"uller, C.H. Murphy, D. Murray, P.\n Muzzetto, P. Naik, T. Nakada, R. Nandakumar, T. Nanut, I. Nasteva, M.\n Needham, N. Neri, S. Neubert, N. Neufeld, R. Newcombe, T.D. Nguyen, C.\n Nguyen-Mau, E.M. Niel, S. Nieswand, N. Nikitin, N.S. Nolte, C. Nunez, A.\n Oblakowska-Mucha, V. Obraztsov, S. Ogilvy, D.P. O'Hanlon, R. Oldeman, C.J.G.\n Onderwater, J. D. Osborn, A. Ossowska, J.M. Otalora Goicochea, T.\n Ovsiannikova, P. Owen, A. Oyanguren, P.R. Pais, T. Pajero, A. Palano, M.\n Palutan, G. Panshin, A. Papanestis, M. Pappagallo, L.L. Pappalardo, C.\n Pappenheimer, W. Parker, C. Parkes, G. Passaleva, A. Pastore, M. Patel, C.\n Patrignani, A. Pearce, A. Pellegrino, M. Pepe Altarelli, S. Perazzini, D.\n Pereima, P. Perret, L. Pescatore, K. Petridis, A. Petrolini, A. Petrov, S.\n Petrucci, M. Petruzzo, B. Pietrzyk, G. Pietrzyk, M. Pili, D. Pinci, J.\n Pinzino, F. Pisani, A. Piucci, V. Placinta, S. Playfer, J. Plews, M. Plo\n Casasus, F. Polci, M. Poli Lener, M. Poliakova, A. Poluektov, N. Polukhina,\n I. Polyakov, E. Polycarpo, G.J. Pomery, S. Ponce, A. Popov, D. Popov, S.\n Poslavskii, K. Prasanth, L. Promberger, C. Prouve, V. Pugatch, A. Puig\n Navarro, H. Pullen, G. Punzi, W. Qian, J. Qin, R. Quagliani, B. Quintana,\n N.V. Raab, R.I. Rabadan Trejo, B. Rachwal, J.H. Rademacker, M. Rama, M. Ramos\n Pernas, M.S. Rangel, F. Ratnikov, G. Raven, M. Reboud, F. Redi, F. Reiss, C.\n Remon Alepuz, Z. Ren, V. Renaudin, S. Ricciardi, D.S. Richards, S. Richards,\n K. Rinnert, P. Robbe, A. Robert, A.B. Rodrigues, E. Rodrigues, J.A. Rodriguez\n Lopez, M. Roehrken, S. Roiser, A. Rollings, V. Romanovskiy, M. Romero Lamas,\n A. Romero Vidal, J.D. Roth, M. Rotondo, M.S. Rudolph, T. Ruf, J. Ruiz Vidal,\n A. Ryzhikov, J. Ryzka, J.J. Saborido Silva, N. Sagidova, N. Sahoo, B. Saitta,\n C. Sanchez Gras, C. Sanchez Mayordomo, R. Santacesaria, C. Santamarina Rios,\n M. Santimaria, E. Santovetti, G. Sarpis, A. Sarti, C. Satriano, A. Satta, M.\n Saur, D. Savrina, L.G. Scantlebury Smead, S. Schael, M. Schellenberg, M.\n Schiller, H. Schindler, M. Schmelling, T. Schmelzer, B. Schmidt, O.\n Schneider, A. Schopper, H.F. Schreiner, M. Schubiger, S. Schulte, M.H.\n Schune, R. Schwemmer, B. Sciascia, A. Sciubba, S. Sellam, A. Semennikov, A.\n Sergi, N. Serra, J. Serrano, L. Sestini, A. Seuthe, P. Seyfert, D.M.\n Shangase, M. Shapkin, L. Shchutska, T. Shears, L. Shekhtman, V. Shevchenko,\n E. Shmanin, J.D. Shupperd, B.G. Siddi, R. Silva Coutinho, L. Silva de\n Oliveira, G. Simi, S. Simone, I. Skiba, N. Skidmore, T. Skwarnicki, M.W.\n Slater, J.G. Smeaton, A. Smetkina, E. Smith, I.T. Smith, M. Smith, A. Snoch,\n M. Soares, L. Soares Lavra, M.D. Sokoloff, F.J.P. Soler, B. Souza De Paula,\n B. Spaan, E. Spadaro Norella, P. Spradlin, F. Stagni, M. Stahl, S. Stahl, P.\n Stefko, O. Steinkamp, S. Stemmle, O. Stenyakin, M. Stepanova, H. Stevens, S.\n Stone, S. Stracka, M.E. Stramaglia, M. Straticiuc, S. Strokov, J. Sun, L.\n Sun, Y. Sun, P. Svihra, K. Swientek, A. Szabelski, T. Szumlak, M. Szymanski,\n S. Taneja, Z. Tang, T. Tekampe, F. Teubert, E. Thomas, K.A. Thomson, M.J.\n Tilley, V. Tisserand, S. T'Jampens, M. Tobin, S. Tolk, L. Tomassetti, D.\n Tonelli, D. Torres Machado, D.Y. Tou, E. Tournefier, M. Traill, M.T. Tran, E.\n Trifonova, C. Trippl, A. Trisovic, A. Tsaregorodtsev, G. Tuci, A. Tully, N.\n Tuning, A. Ukleja, A. Usachov, A. Ustyuzhanin, U. Uwer, A. Vagner, V.\n Vagnoni, A. Valassi, G. Valenti, M. van Beuzekom, H. Van Hecke, E. van\n Herwijnen, C.B. Van Hulse, M. van Veghel, R. Vazquez Gomez, P. Vazquez\n Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, A.\n Venkateswaran, M. Vernet, M. Veronesi, M. Vesterinen, J.V. Viana Barbosa, D.\n Vieira, M. Vieites Diaz, H. Viemann, X. Vilasis-Cardona, A. Vitkovskiy, A.\n Vollhardt, D. Vom Bruch, A. Vorobyev, V. Vorobyev, N. Voropaev, R. Waldi, J.\n Walsh, J. Wang, J. Wang, J. Wang, M. Wang, Y. Wang, Z. Wang, D.R. Ward, H.M.\n Wark, N.K. Watson, D. Websdale, A. Weiden, C. Weisser, B.D.C. Westhenry, D.J.\n White, M. Whitehead, D. Wiedner, G. Wilkinson, M. Wilkinson, I. Williams, M.\n Williams, M.R.J. Williams, T. Williams, F.F. Wilson, W. Wislicki, M. Witek,\n L. Witola, G. Wormser, S.A. Wotton, H. Wu, K. Wyllie, Z. Xiang, D. Xiao, Y.\n Xie, H. Xing, A. Xu, L. Xu, M. Xu, Q. Xu, Z. Xu, Z. Yang, Z. Yang, Y. Yao,\n L.E. Yeomans, H. Yin, J. Yu, X. Yuan, O. Yushchenko, K.A. Zarebski, M.\n Zavertyaev, M. Zdybal, M. Zeng, D. Zhang, L. Zhang, S. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, Y. Zheng, X. Zhou, Y. Zhou, X. Zhu, V. Zhukov, J.B.\n Zonneveld, S. Zucchelli",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26123",
"submitter": "Matthew Rudolph",
"url": "https://arxiv.org/abs/2003.04352"
} | arxiv-papers | EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-EP-2020-020 LHCb-PAPER-2019-043 9 March 2020
Search for the lepton flavour violating decay
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ using $B_{s2}^{*0}$
decays
LHCb collaboration†††Authors are listed at the end of this paper.
A search is presented for the lepton flavour violating decay
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ using a sample of
proton–proton collisions at centre-of-mass energies of 7, 8, and
$13\text{\,}\mathrm{Te\kern-1.00006ptV}$, collected with the LHCb detector and
corresponding to a total integrated luminosity of 9$\text{\,fb}^{-1}$. The
$\tau$ leptons are selected inclusively, primarily via decays with a single
charged particle. The four-momentum of the $\tau$ lepton is determined by
using ${B}^{+}$ mesons from ${B_{s2}^{*0}}\rightarrow{{B}^{+}}{{K}^{-}}$
decays. No significant excess is observed, and an upper limit is set on the
branching fraction
${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})<$3.9\text{\times}{10}^{-5}$\text{
at 90\% confidence level}.$
The obtained limit is comparable to the world-best limit.
Submitted to JHEP
© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.
## 1 Introduction
A number of experimental hints of lepton flavour universality violation in the
semileptonic transitions $b\\!\rightarrow s\ell^{+}\ell^{-}$ [1, 2, 3] and
$b\\!\rightarrow c\ell^{-}{\overline{\nu}}_{\ell}$ [4, 5, 6, 7, 8, 9] have
recently been found.111The inclusion of charge-conjugate processes is implied
throughout. In general, physics beyond the Standard Model that generates
lepton flavour non-universality is likely to also produce direct lepton
flavour violation [10]. Theoretical models seeking to simultaneously explain
all these anomalies, for example with a vector leptoquark, often lead to
relatively large branching fractions for the decays
${B}\\!\rightarrow{K}\mu^{\pm}\tau^{\mp}$ [11, 12, 13, 14, 15, 16].
The branching fractions for the two $\mu\tau$ charge combinations are not in
general the same, as they depend on the details of the physics mechanism
producing the decay. In this paper, we present a search for the decay
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$. From an experimental
point of view, this combination is preferred over
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}}{\tau^{-}}$ as it has a lower
background from semileptonic ${B}\\!\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptD}}X{\mu^{+}}{{\nu}_{\mu}}$ decays, because
Cabibbo-favoured decays of the charm meson are likely to lead to kaons of the
same charge as the muon. An upper limit on the branching fraction for the
signal decay has been previously set by the BaBar collaboration [17]
${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})<$2.8\text{\times}{10}^{-5}$$
at 90% confidence level (CL).
We reconstruct the full four-momentum of the $\tau$ lepton using ${B}^{+}$
mesons from the decay ${B_{s2}^{*0}}\\!\rightarrow{{B}^{+}}{{K}^{-}}$, which
amounts to about 1% of ${B}^{+}$ production. By reconstructing the decay
vertex of the ${B}^{+}$ meson from the ${{K}^{+}}{\mu^{-}}$ pair and the
momentum of the ${K}^{-}$ meson, it is possible to determine the momentum of
the ${B}^{+}$ meson up to a quadratic ambiguity by imposing mass constraints
on the $B_{s2}^{*0}$ and ${B}^{+}$ mesons [18]. This technique was first used
to study relative branching fractions in ${{B}^{+}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}X{\mu^{+}}\nu$ decays [19]. We
then search for a peak in the missing-mass squared distribution corresponding
to the $\tau$ mass squared, $m_{\tau}^{2}$. Even signal ${B}^{+}$ mesons not
coming from a $B_{s2}^{*0}$ decay show a peak at $m_{\tau}^{2}$. We account
for the contribution of these non-$B_{s2}^{*0}$ candidates in the analysis.
The $\tau$ leptons are selected inclusively, as we only require one additional
charged track near the ${{K}^{+}}{\mu^{-}}$ pair to help discriminate against
background. To normalise the branching fraction, we use the decay
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$, with
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. The
normalisation channel is also used to quantify the contributions from
$B_{s2}^{*0}$ decays, as well as non-$B_{s2}^{*0}$ candidates with nearby
kaons.
In addition to providing the missing-mass discriminating variable, this method
allows us to study the control sample composed of same-sign
${{B}^{+}}{{K}^{+}}$ decays, which does not include any $B_{s2}^{*0}$
component. We use this sample to optimise the signal selection, and motivate
our description of the background missing-mass shape.
## 2 Detector, data samples, and simulation
The LHCb detector [20, 21] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of the momentum, $p$,
of charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200 GeV.222Natural units with $c=1$ are used throughout.
The minimum distance of a track to a primary $pp$ interaction vertex (PV), the
impact parameter, is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in GeV. Different types of
charged hadrons are distinguished using information from two ring-imaging
Cherenkov detectors. Muons are identified by a system composed of alternating
layers of iron and multiwire proportional chambers. The online event selection
is performed by a trigger, which consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage, which applies a full event reconstruction. At the hardware trigger
stage, events are required to have a muon with high $p_{\mathrm{T}}$ or a
hadron, photon or electron with high transverse energy deposited in the
calorimeters. The software trigger requires a two-, three- or four-track
secondary vertex with a significant displacement from any primary vertex.
We use data samples collected from 2011 to 2018, at centre-of-mass energies of
7, 8, and $13\text{\,}\mathrm{Te\kern-1.00006ptV}$, corresponding to an
integrated luminosity of 9$\text{\,fb}^{-1}$. We model signal and
normalisation decays using simulation. In the simulation, $pp$ collisions are
generated using Pythia [22, *Sjostrand:2007gs] with a specific LHCb
configuration [24]. Decays of hadronic particles are described by EvtGen [25].
The interaction of the generated particles with the detector, and its
response, are implemented using the Geant4 toolkit [26, *Agostinelli:2002hh]
as described in Ref. [28].
For the signal, we consider both a phase space model and variations of the
decay kinematics with effective operators for the $b\\!\rightarrow
s{\mu^{+}}{\tau^{-}}$ interaction and their corresponding Wilson coefficients
using the distributions from Ref. [29] and the form factors from Ref. [30].
The branching fraction limit is determined for various hypotheses: for the
phase-space decay, for a decay via the vector or axial-vector operators
$\mathcal{O}_{9}^{(^{\prime})}$ or $\mathcal{O}_{10}^{(^{\prime})}$, and for a
decay using the scalar or pseudoscalar operators
$\mathcal{O}^{(^{\prime})}_{S}$ or $\mathcal{O}^{(^{\prime})}_{P}$ [29].
## 3 Selection and missing mass calculation
The selection of ${B}^{+}$ candidates begins with a ${{K}^{+}}{\mu^{-}}$ pair
with an invariant mass
$m_{{{K}^{+}}{\mu^{-}}}>$1800\text{\,}\mathrm{Me\kern-1.00006ptV}$$ to reduce
background from semileptonic charm decays. The ${K}^{+}$ and $\mu^{-}$
candidates are formed from high-quality tracks consistent with kaon and muon
hypotheses and inconsistent with being produced at any PV in the event. The
${{K}^{+}}{\mu^{-}}$ vertex must be of high quality and well separated from
any PV.
To better separate signal candidates with $\tau$ leptons from background, we
require an additional track, labelled $t^{+}$, with charge opposite to that of
the muon. By adding this third track, we also fully reconstruct the
normalisation mode
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$, with
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. Many
background candidates are expected to come from $B$-meson decays of the form
${B}\\!\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptD}}\quantity(\\!\rightarrow{{K}^{+}}X{\mu^{-}}){{K}^{+}}Y$,
where $X$ and $Y$ refer to any number of additional particles. In these cases
the kaon originating from the $\kern 1.79993pt\overline{\kern-1.79993ptD}$
meson is assigned as the additional track. Since only approximately 2% of
$\tau$ decays contain a charged kaon, we apply particle identification
requirements so that the track is unlikely to be a charged kaon. Events in
which a candidate
${\tau^{+}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\overline{\nu}}_{\tau}}$
decay is found are not used in this search to avoid overlap with ongoing
searches at LHCb exclusively using this decay channel. In addition, events in
which we find multiple candidates are not used in this analysis. These
requirements do remove signal with multi-prong $\tau$ decays, with an overall
loss of less than 3%. Multiple candidate events are more likely to come from
background, however. We split the data samples into signal and normalisation
regions based on the invariant mass of the ${{K}^{+}}{\mu^{-}}t^{+}$ triple,
using the muon hypothesis for the third track. Candidates with
$m_{K\mu\mu}<$4800\text{\,}\mathrm{Me\kern-1.00006ptV}$$ fall into the signal
region, while candidates with
$5180<m_{K\mu\mu}<$5380\text{\,}\mathrm{Me\kern-1.00006ptV}$$ and
$\absolutevalue{m_{\mu\mu}-m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi}}}}<$40\text{\,}\mathrm{Me\kern-1.00006ptV}$$
fall into the normalisation region.
The ${B}^{+}$ candidate direction is estimated using the PV and
${{K}^{+}}{\mu^{-}}$ vertex positions. We next consider prompt tracks, _i.e._
those that are consistent with being produced at that PV. Those tracks
identified as kaons, with a charge opposite to that of the kaon in the
${{K}^{+}}{\mu^{-}}$ pair and a small perpendicular momentum relative to the
${B}^{+}$ candidate direction, are combined with the ${B}^{+}$ candidates to
form $B_{s2}^{*0}$ candidates. We refer to this sample as the opposite-sign
kaon (OS$K$) sample. Additionally, we select a control sample, referred to as
same-sign kaon (SS$K$) sample, by adding prompt kaons of the same sign as the
kaon in the ${{K}^{+}}{\mu^{-}}$ pair.
From Ref. [19], the two $B$-meson energy solutions are
$\displaystyle E_{B}$
$\displaystyle=\frac{\Delta^{2}}{2E_{K}}\frac{1}{1-\quantity(p_{K}/E_{K})^{2}\cos^{2}\theta}\quantity[1\pm\sqrt{d}],\text{
where}$ (1) $\displaystyle d$
$\displaystyle=\frac{p_{K}^{2}}{E_{K}^{2}}\cos^{2}\theta-\frac{4m_{B}^{2}p_{K}^{2}\cos^{2}\theta}{\Delta^{4}}\quantity(1-\frac{p_{K}^{2}}{E_{K}^{2}}\cos^{2}\theta),$
(2) $\displaystyle\Delta^{2}$ $\displaystyle=m_{BK}^{2}-m_{B}^{2}-m_{K}^{2},$
(3)
where $m_{BK}=m_{{B_{s2}^{*0}}}$ is the assumed ${{B}^{+}}{{K}^{-}}$ mass,
$p_{K}$ and $E_{K}$ are the reconstructed prompt kaon momentum and energy, and
$\theta$ is the laboratory frame angle between the prompt kaon and $B$-meson
directions. The missing four-momentum of the $\tau$ lepton, $P_{\text{miss}}$,
is then reconstructed as $P_{B}-P_{{{K}^{+}}{\mu^{-}}}$, where $P_{B}$ and
$P_{{{K}^{+}}{\mu^{-}}}$ are the four-momenta of the $B$ meson and
${{K}^{+}}{\mu^{-}}$ pair. The missing mass squared is calculated using the
lowest energy, real solution for which the resulting missing energy is greater
than the reconstructed energy of the third track under a pion mass hypothesis.
With this choice, we correctly reconstruct the energy of signal decays in
simulation in more than 75% of cases. About 9% of all signal decays have no
such solution and are lost. Both signal and normalisation candidates, as well
as the SS$K$ control-sample candidates, are required to pass this procedure.
Candidates in the signal region are additionally required to have the residual
missing mass squared, defined as the four-momentum difference of the $B$ meson
and ${{K}^{+}}{\mu^{-}}t^{+}$ triple,
$\quantity(P_{B}-P_{{{K}^{+}}{\mu^{-}}}-P_{t})^{2}$, greater than
$-0.5\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$. This requirement removes
background and only poorly reconstructed signal candidates which do not peak
at the $\tau$ mass squared. The minimum mass difference, defined in Ref. [19]
as
$\Delta
m_{\mathrm{min}}=\sqrt{m_{B}^{2}+m_{K}^{2}+2m_{B}\sqrt{p_{K}^{2}\sin^{2}\theta+m_{K}^{2}}}-m_{B}-m_{K},$
(4)
is required to be greater than $30\text{\,}\mathrm{Me\kern-1.00006ptV}$. This
removes contributions from $B_{s1}^{0}$ and
${B_{s2}^{*0}}\\!\rightarrow{B}^{*+}{{K}^{-}}$ decays, as well as background
in which a kaon from the $B$ decay is wrongly associated to the primary
vertex.
Missing-mass distributions for the signal simulation and the full data sample
after the above selection are shown in Fig. 1. All signal decays, whether they
come from a $B_{s2}^{*0}$ meson or not, peak at the known $m_{\tau}^{2}$,
however the non-$B_{s2}^{*0}$ candidates have a much wider peak than the
$B_{s2}^{*0}$ ones. The data distributions are shown for both the OS$K$ and
SS$K$ samples. They have similar shapes with a broad hump centred near
$5\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$. We note that the OS$K$ sample
has a higher yield than the SS$K$; this excess has been observed in both fully
and partially reconstructed decays [31, 19].
Figure 1: Missing mass squared, $m_{\mathrm{miss}}^{2}$, distributions for
(left) simulated signal ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$
decays and (right) all selected candidates in data before applying the signal
optimisation described in Sect. 5.
## 4 Normalisation
We determine the yield of the normalisation decay, as well as the relative
efficiency of the signal modes with respect to the normalisation mode,
separately for each data-taking year. For the normalisation mode, we determine
the inclusive yield of
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays,
whether or not they originate from a $B_{s2}^{*0}$ meson, by a binned maximum-
likelihood fit to the ${{K}^{+}}{\mu^{-}}t^{+}$ mass distribution, where we
assign the muon mass hypothesis to the third track. The signal is described
with a Gaussian distribution, and the background with a linear model.
We determine the fraction of the normalisation candidates coming from
$B_{s2}^{*0}$ decays using a ${{K}^{+}}{\mu^{-}}t^{+}$ mass fit for the
combined-years data sample using the same model as the separated-years
samples, along with a binned maximum-likelihood fit to the measured mass-
difference distribution $m_{{{B}^{+}}{{K}^{-}}}-m_{{{B}^{+}}}-m_{{{K}^{-}}}$
around the $B_{s2}^{*0}$ peak. For the latter fit, we describe the signal peak
with a Gaussian core that transitions to an exponential tail on each side, and
we model the background with a third-degree polynomial. The results of these
fits are shown in Fig. 2. The total data sample contains $4240\pm 70$
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays;
the fraction originating from $B_{s2}^{*0}$ decays is
$f_{B_{s2}^{*0}}=$25.4\pm 1.8\text{\,}\mathrm{\char 37\relax}$$, where the
uncertainty combines the statistical and systematic uncertainties from the
choice of fit function. The year-to-year variation is not found to be
statistically significant, so we use the value obtained from the combined
dataset for all years.
Figure 2: Distributions of normalization candidates in (left) mass,
$m_{{{K}^{+}}{\mu^{-}}{\mu^{+}}}$, and (right) the mass difference,
$m_{{{B}^{+}}{{K}^{-}}}-m_{{{B}^{+}}}-m_{{{K}^{-}}}$. The result of each fit
is shown as a solid line, with the background component as a dashed line.
The relative efficiency of the signal and normalisation modes is determined
using simulation with corrections from data. For $B_{s2}^{*0}$ decays the
relative efficiencies in different years average around 30%, with an absolute
year-to-year variation of less than 3%. Different signal decay models change
the relative efficiency by approximately 10%, with the decays via scalar and
pseudoscalar operators having a lower overall efficiency. Signal events in
which the ${B}^{+}$ meson does not originate from a $B_{s2}^{*0}$ decay have a
lower selection efficiency, primarily because fewer of these candidates pass
the residual missing-mass requirement and fall into the missing-mass fit
range. Using simulation, we derive an additional efficiency factor for this
signal component of $r_{\text{non-{$B_{s2}^{*0}$}}}=0.849\pm 0.007$.
## 5 Multivariate signal selection
We further improve the signal selection using a Boosted Decision Tree (BDT)
classification with the Adaboost algorithm [32]. The BDT inputs are primarily
chosen to distinguish additional tracks coming from signal $\tau$ lepton
decays from various sources of background. Some examples are semileptonic
$b$-hadron decays to charm where the charm hadron produces a kaon with charge
opposite that of the muon, or $b$-hadron decays where the muon is produced in
the semileptonic decay of a child charm hadron. The background training sample
is taken from the SS$K$ sample in the $m_{\mathrm{miss}}^{2}$ region around
$m_{\tau}^{2}$. This focuses the training on the sources of background which
fall near the signal peak. We describe the signal with simulation samples that
include only $B_{s2}^{*0}$ decays; the effect of the BDT on non-$B_{s2}^{*0}$
signal simulation is then estimated separately. The training makes use of
different topological reconstructions of the ${{K}^{+}}{\mu^{-}}t^{+}$ triple:
in addition to the signal selection, we also first combine either the kaon and
the track or the muon and the track into a pair before adding the third
particle. The pair masses and the flight distance of the pair in each topology
help to distinguish the signal from background, for instance when the pair
comes from a charm hadron decay. We also include the flight distance of the
$\tau$, which we reconstruct as the distance along the $\tau$ trajectory found
in the missing-mass calculation from the ${{K}^{+}}{\mu^{-}}$ vertex to the
point of closest approach of the third track.
The result of a separate isolation discriminant is included to reduce
background with additional charged tracks; this discriminant is trained to
distinguish additional tracks belonging to the same $b$-hadron decay from
other tracks in the event based on kinematic and topological variables. We
perform the rest of the analysis in four bins of the signal optimisation BDT
output, keeping about 70% of all simulated $B_{s2}^{*0}$ signal candidates and
about 40% of non-$B_{s2}^{*0}$ signal candidates. The bins are chosen by
optimising the expected upper limit using a number of background events
derived from the OS$K$ and SS$K$ $m_{\mathrm{miss}}^{2}$ sidebands.
## 6 Background studies
The background in this analysis is composed of a large number of different
partially reconstructed $b$-hadron decays. None of them, however, produce a
narrow peak in $m_{\mathrm{miss}}^{2}$. Only ${B}^{+}$ mesons produced from
$B_{s2}^{*0}$ decays have a resolution comparable to the signal. Furthermore,
if there is more than one missing particle then the true missing-mass
distribution will be much wider than the expected signal peak. Charm hadrons
have masses close to the $\tau$ mass, however there is no Standard Model decay
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{{D}^{+}}$. Because of their low
branching fraction, we are not sensitive to decays such as
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{D}^{+}}$, where the pion is
misidentified as a muon. We expect that the missing-mass distribution, summed
over many different background components, is smooth, and we model it as a
polynomial.
These assumptions are tested using simulation and data. We produce fast
simulation samples with RapidSim [33] of a number of potential exclusive
background sources from ${B}^{+}$, ${B}^{0}$, ${B}^{0}_{s}$, and ${\mathchar
28931\relax}^{0}_{b}$ hadrons; the true missing-mass distributions for these
decays are smeared to estimate their shapes in data. No sign of any sharply
peaking component is found. In data we consider a number of different control
samples, namely all possible $K\mu t$ charge combinations in both OS$K$ and
SS$K$ samples, excluding the signal selection of ${{K}^{+}}{\mu^{-}}t^{+}$ in
the OS$K$ sample. There is no sign of any narrow peak in any of the
distributions, even after applying a tight requirement on the BDT output.
Maximum-likelihood fits to the SS$K$ sample using polynomials of different
degrees in the restricted $m_{\mathrm{miss}}^{2}$ range from
$16\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$ are used to study the
background shape in more detail. The optimal number of free polynomial
parameters in the most signal-like BDT output bin, based on the best-fit value
of $-2\log\mathcal{L}$, penalised by one for each additional parameter, is
four. We further study the effect of background modelling by performing a
large number of pseudoexperiments, both background-only and with injected
signal at branching fractions of $1\text{\times}{10}^{-5}$ and
$2\text{\times}{10}^{-5}$. In these studies, we first fit a background model
of some polynomial degree to one of the control samples. From this background
model we generate many pseudodatasets that we fit with a model of a different
degree. Based on these studies, we take into account the systematic
uncertainty due to the background modelling by reporting the weakest limit
using background descriptions of third, fourth, or fifth degree polynomials,
all of which well describe the background shapes in the pseudoexperiments.
## 7 Fit description
We search for the ${{K}^{+}}{\mu^{-}}{\tau^{+}}$ missing-mass peak with an
unbinned maximum-ikelihood fit simultaneously in four bins of BDT output in
the OS$K$ ${{K}^{+}}{\mu^{-}}t^{+}$ signal channel. The fit is performed in
the missing-mass range
$1<m_{\mathrm{miss}}^{2}<$6\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$$. The
parameter of interest is the branching fraction
${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})$.
We describe the $m_{\mathrm{miss}}^{2}$ shape for the signal component with a
generalized hyperbolic distribution with shape parameters obtained from
simulation. Two signal shapes are used: one for $B_{s2}^{*0}$ decays, and one
for the wider non-$B_{s2}^{*0}$ contribution. We determine the shapes
separately in each bin of BDT response. The signal decay model does not
significantly affect the signal missing-mass shape. The background is
described by polynomial functions which vary independently in each BDT output
bin.
We base the normalization of the signal components on the yields of the
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays
determined in data year-by-year. We combine this together with the relative
efficiencies, $\varepsilon_{\text{rel}}$; the known
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ with
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ combined
branching fraction, abbreviated as
${\mathcal{B}}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}})$; and the
parameter of interest to derive a total number of
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ signal decays. This
total is divided between $B_{s2}^{*0}$ and non-$B_{s2}^{*0}$ decays based on
the observed fraction in the normalization channel, and then distributed
across the four BDT bins. This gives yields in each BDT bin $j$ of
$\displaystyle
N_{j}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}|{B_{s2}^{*0}})={}$
$\displaystyle\varepsilon_{{B_{s2}^{*0}},j}\frac{{\mathcal{B}}\quantity({{K}^{+}}{\mu^{-}}{\tau^{+}})}{{\mathcal{B}}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}})}f_{{B_{s2}^{*0}}}\times{}$
$\displaystyle\sum_{i\in\text{years}}\varepsilon_{\text{rel},i}N_{i}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}),$
(5) $\displaystyle
N_{j}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}|\text{non-{$B_{s2}^{*0}$}})={}$
$\displaystyle\varepsilon_{\text{non-{$B_{s2}^{*0}$}},j}\frac{{\mathcal{B}}\quantity({{K}^{+}}{\mu^{-}}{\tau^{+}})}{{\mathcal{B}}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}})}\quantity(1-f_{{B_{s2}^{*0}}})\times{}$
$\displaystyle\sum_{i\in\text{years}}\varepsilon_{\text{rel},i}r_{\text{non-{$B_{s2}^{*0}$}}}N_{i}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}),$
(6)
where $\varepsilon_{{B_{s2}^{*0}},j}$ and
$\varepsilon_{\text{non-{$B_{s2}^{*0}$}},j}$ are the separate efficiencies for
each signal component to be found in BDT bin $j$. The main parameters of the
fit are thus the ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$
branching fraction, four parameters for the background normalisation in each
BDT bin, and up to five parameters describing the polynomial background shapes
in each BDT bin.
The largest systematic uncertainty comes from the choice of background model.
The fifth degree background description obtains the weakest limit among the
tested background models. We include the effects of other systematic
uncertainties using Gaussian-constrained nuisance parameters. These nuisance
parameters modify the normalisation yield, the relative efficiency of the
signal and normalisation channels, the signal yield in each BDT bin, and the
signal shapes. The largest effects come from the modelling of the kinematics
of $B_{s2}^{*0}$ decays in simulation, which results in 5% changes in the
relative efficiency and in the signal fractions in each bin of BDT response.
The relative statistical uncertainty of the $B_{s2}^{*0}$ fraction taken from
the normalisation channel is also approximately 5%. Altogether, the total
effect of these systematic uncertainties on the final limit is small, at the
$10^{-6}$ level.
## 8 Results and conclusion
The result at the best fit point is shown in Fig. 3. The obtained value for
the signal branching fraction from the maximum-likelihood fit is
${\quantity(1.9\pm 1.5)\times 10^{-5}}$. No significant excess is observed,
and we set upper limits on the branching fraction using the CLs method [34].
We perform a scan in the signal branching fraction, obtaining the signal and
background $p$-values from the distributions of a one-sided profile-
likelihood-ratio test statistic obtained with pseudoexperiments in which we
vary the constraints on the systematic uncertainties. The scan used to
determine the observed limits, compared to the expected one, is shown in Fig.
4. The expected upper limit at 90% CL is $2.3\text{\times}{10}^{-5}$. The
observed 90% and 95% CL limits, assuming a phase space signal decay model,
are:
$\displaystyle{\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})$
$\displaystyle<$3.9\text{\times}{10}^{-5}$\text{ at 90\% CL},$
$\displaystyle<$4.5\text{\times}{10}^{-5}$\text{ at 95\% CL}.$
An identical limit is obtained when the decay is generated from the effective
operators $\mathcal{O}_{9}^{(^{\prime})}$ or $\mathcal{O}_{10}^{(^{\prime})}$.
If instead it is produced from $\mathcal{O}^{(^{\prime})}_{S}$ or
$\mathcal{O}^{(^{\prime})}_{P}$, the obtained limit is
${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})<$4.4\text{\times}{10}^{-5}$$
at 90% CL and ${}<$5.0\text{\times}{10}^{-5}$$ at 95% CL.
Figure 3: Fits to the missing-mass-squared distribution OS$K$ signal sample in
each bin of BDT output included in the final fit. The best fit is overlaid.
BDT bin 1 is the most background-like. The fit is performed using a fifth
degree polynomial description of the background. Figure 4: Scan of the
$p$-value in the signal branching fraction used to determine the CLs upper
limits, compared to the expected one. The horizontal red line shows a
$p$-value of 0.1, used to define the 90% CL upper limit.
This is the first result from the LHCb experiment for the lepton-flavour
violating decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$. By
studying ${B}^{+}$ mesons from $B_{s2}^{*0}$ decays, we are able to make the
first analysis at LHCb of a $B$ hadron decay using inclusive $\tau$ decays.
This provides complementary information to searches for lepton-flavour
violation at LHCb with three-prong $\tau$ decays, for example
$B_{(s)}^{0}\\!\rightarrow\tau^{\pm}\mu^{\mp}$ decays [35]. We observe no
significant signal, and set an upper limit slightly above that obtained by the
BaBar collaboration [17].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).
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LHCb collaboration
R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala80, S. Aiola25, Z. Ajaltouni9, S. Akar66, P.
Albicocco22, J. Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44,
G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
L. An21, L. Anderlini21, G. Andreassi48, M. Andreotti20, F. Archilli16, A.
Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M. Atzeni49, B.
Audurier11, S. Bachmann16, J.J. Back55, S. Baker60, V. Balagura11,b, W.
Baldini20,47, A. Baranov41, R.J. Barlow61, S. Barsuk11, W. Barter60, M.
Bartolini23,47,h, F. Baryshnikov77, J.M. Basels13, G. Bassi28, V.
Batozskaya35, B. Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28,
I. Bediaga1, A. Beiter67, L.J. Bel31, V. Belavin41, S. Belin26, V. Bellee48,
K. Belous43, I. Belyaev38, G. Bencivenni22, E. Ben-Haim12, S. Benson31, S.
Beranek13, A. Berezhnoy39, R. Bernet49, D. Berninghoff16, H.C. Bernstein67, C.
Bertella47, E. Bertholet12, A. Bertolin27, C. Betancourt49, F. Betti19,e, M.O.
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Bifani52, P. Billoir12, A. Bizzeti21,u, M. Bjørn62, M.P. Blago47, T. Blake55,
F. Blanc48, S. Blusk67, D. Bobulska58, V. Bocci30, O. Boente Garcia45, T.
Boettcher63, A. Boldyrev78, A. Bondar42,x, N. Bondar37, S. Borghi61,47, M.
Borisyak41, M. Borsato16, J.T. Borsuk33, T.J.V. Bowcock59, C. Bozzi20, M.J.
Bradley60, S. Braun16, A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A.
Brossa Gonzalo55, D. Brundu26, E. Buchanan53, A. Büchler-Germann49, A.
Buonaura49, C. Burr47, A. Bursche26, A. Butkevich40, J.S. Butter31, J.
Buytaert47, W. Byczynski47, S. Cadeddu26, H. Cai72, R. Calabrese20,g, L.
Calero Diaz22, S. Cali22, R. Calladine52, M. Calvi24,i, M. Calvo Gomez44,m, P.
Camargo Magalhaes53, A. Camboni44,m, P. Campana22, D.H. Campora Perez31, A.F.
Campoverde Quezada5, L. Capriotti19,e, A. Carbone19,e, G. Carboni29, R.
Cardinale23,h, A. Cardini26, I. Carli6, P. Carniti24,i, K. Carvalho Akiba31,
A. Casais Vidal45, G. Casse59, M. Cattaneo47, G. Cavallero47, S. Celani48, R.
Cenci28,p, J. Cerasoli10, M.G. Chapman53, M. Charles12,47, Ph. Charpentier47,
G. Chatzikonstantinidis52, M. Chefdeville8, V. Chekalina41, C. Chen3, S.
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Xu5, Z. Xu4, Z. Yang3, Z. Yang65, Y. Yao67, L.E. Yeomans59, H. Yin7, J.
Yu7,aa, X. Yuan67, O. Yushchenko43, K.A. Zarebski52, M. Zavertyaev15,c, M.
Zdybal33, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, W.C. Zhang3,z, Y.
Zhang47, A. Zhelezov16, Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V.
Zhukov13,39, J.B. Zonneveld57, S. Zucchelli19,e.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France , Orsay,
France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27INFN Sezione di Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
75Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
76National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
77National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
78National Research University Higher School of Economics, Moscow, Russia,
associated to 41
79National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
80University of Michigan, Ann Arbor, United States, associated to 67
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
yINFN Sezione di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras
|
2024-09-04T02:54:58.521393 | 2020-03-03T20:35:36 | 2003.04360 | {
"authors": "Vaishali Ingale, Pushpender Singh",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26124",
"submitter": "Pushpender Singh",
"url": "https://arxiv.org/abs/2003.04360"
} | arxiv-papers | # GenNet: Reading Comprehension with Multiple Choice Questions using
Generation and Selection model
Vaishali Ingale
Department of Information Technology
Army Institute of Technology, Pune
<EMAIL_ADDRESS>
Pushpender Singh
Department of Information Technology
Army Institute of Technology, Pune
<EMAIL_ADDRESS>
###### Abstract
Multiple-choice machine reading comprehension is difficult task as its
required machines to select the correct option from a set of candidate or
possible options using the given passage and question.Reading Comprehension
with Multiple Choice Questions task, required a human (or machine) to read a
given passage, question pair and select the best one option from n given
options. There are two different ways to select the correct answer from the
given passage. Either by selecting the best match answer to by eliminating the
worst match answer. Here we proposed GenNet model, a neural network-based
model. In this model first we will generate the answer of the question from
the passage and then will matched the generated answer with given answer, the
best matched option will be our answer. For answer generation we used S-net
(Tan et al.,, 2017) model trained on SQuAD and To evaluate our model we used
Large-scale RACE (ReAding Comprehension Dataset From Examinations) (Lai et
al.,, 2017).
## 1 Introduction
Reading comprehension is one of the fundamental skills for human, which one
learn systematically since the elementary school. Reading comprehension give
human the ability of reading texts, understanding their meanings,and with the
help of given context answering questions. When machines are required to
comprehend texts, they first need to understand the unstructured text and do
reasoning based on given text (Chen et al.,, 2016)(Wang et al., 2018b,
).Answering questions based a passage requires an individual unique skill set.
It requires ability to perform basic mathematical operations and logical
ability (e.g. to answer questions like how many times Amit visited sweet
shop?), look-up ability, ability to deduce, ability to gather information
contained in multiple sentences and passages. This diverse and unique skill
set makes question answering a challenging task.There are several variants of
this task, For example, if we have a given passage and a question, the answer
could either (i) be generated from the passage (ii) match some span in the
passage (iii) or could be one of the n number of given candidate answers. The
last variant is mostly used in various high school, quiz , middle school, and
different competitive examinations. This variant of Reading Comprehension
generally referred as Reading Comprehension with Multiple Choice Questions
(RC-MCQ).In the given figure 1 We have a passage and a question and 4
candidate answers. Task here defined is to find the most suitable answer from
the passage for given question. While answering such Multiple Choice Questions
(MCQs) figure 1, humans typically use a combination of option elimination and
option selection or sometimes they find answer from the passage i.e they
generate the answer of the question from passage and match the generated
answer with given options and they choose more close candidate as correct
answer.
Here we proposed model which mimic the answer generation and then matching
human process.First the span where possible answer in the passage is computed.
we first compute a question-aware representation of the passage (which
essentially tries to retain portions of the passage which are only relevant to
the question). Then we use answer generation using state-of-art S-Net model
(Tan et al.,, 2017)which extract and generate answer figure 2. After we have
answer generated from the passage now we weight every given candidate option
and select the best matched option. That best matched option was our answer
figure 3.
Figure 1: An example multiple-choice reading comprehension question. Figure
2: Overview of S-Net.(Tan et al.,, 2017) Figure 3: Overview of option matching
and selection.
## 2 Related Work
Datasets played an important role in machine reading comprehension, there were
different type of datasets designed to solve different variant of machine
reading comprehension. SQuAD dataset(Rajpurkar et al.,, 2016) was designed to
answer simple question answer reading comprehension that aims to answer a
question with exact text spans in a passage. Later MS-MACRO dataset(Nguyen et
al.,, 2016) was designed for multi-passage reading comprehension. CNN/
Dailymail (Chen et al.,, 2016) and Who did what dataset(Onishi et al.,, 2016)
designed for cloze variant reading comprehension. MCtest(Richardson et al.,,
2013) and RACE dataset(Lai et al.,, 2017) are released for Multiple choice
question variant reading comprehension.
Similar work in reading comprehension where Multiple choice variant of
Comprehension considered includes Hierarchical Attention Flow model(Zhu et
al.,, 2018), in this model the candidate options leverage to model the
interaction between question options and passage.This was a option selection
model which select the correct option from the given candidate options. Other
work relatable to this paper was eliminating options model(Parikh et al.,,
2019) which eliminate the wrong answer from the candidate answer.Multi
matching network(Tang et al.,, 2019) models interaction relationship between
passage, questions and candidate answer. It take different paradigm of
matching into account. Option comparison Network (Ran et al.,, 2019) compares
between options at word level and identify correlation to help buildup logic
and reasoning. Co-matching model (Wang et al., 2018a, ) is used to match
between answer and question and passage pair. It match for the relationship
between question and answer with the passage. Bidirectional co-matching based
model (Zhang et al.,, 2019) matched passage and question, answer
bidirectionally. The Convolutional Spatial Attention (CSA) model (Chen et
al.,, 2019) form the enriched representaion by fully extract the mutual
information among the passage, question, and the candidates.
To generate answer several models are there like QANet (Yu et al.,, 2018)
which combined local Convolution with Global Self-Attention and its encoder
consist exclusively of convolution and self-attention.Bidirectional Attention
Flow model (Seo et al.,, 2016) use to focus on large span of passage. BIDAF
network is a multi stage hierarchical process and use bidirection attention
flow to obtain a query-aware context representation. But the reason to use
S-Net model as answer generation model because S-Net not only find the answer
from the passage but it can also synthesise passage when required. Some
questions are tricky and there answer lies in different span of passage. In
such situation S-Net is useful as it remember the past context for longer time
as it have GRU as basic component.
## 3 Proposed model
There are two tasks needs to be performed in this model. First is Answer
extraction and Answer Synthesis/Generation and then option selection. Answer
extraction and Generation will be done using state-of-art S-NET model(Tan et
al.,, 2017). S-Net first pull out evidence snippets by matching the question
and passage respectively, and then generates the answer by filtering the
question, passage, and evidence snippets. consider a passage
$P=[p_{1},p_{2},p_{3},...p_{p}]$ of word length P, Question
$Q=[Q_{1},Q_{2},Q_{3},...Q_{q}]$ of word length Q, and n options
$Z_{n}=[z_{1},z_{2},z_{3},...z_{k}]$ where n > 1 and word length k. We first
convert the words to their word-level embedding and character-level embedding
using GLOVE(Pennington et al.,, 2014).The encoding and embedding layers take
in a series of tokens and represent it as a series of vectors. The character-
level embeddings are cause by taking the final hidden states of a bi-
directional GRU applied to embedding of characters in the token. They then use
a bi-directional Gated Recurrent Unit to give rise to new depiction
$u_{1}^{p},u_{2}^{p},u_{3}^{p},...u_{p}^{p}$ for questions as well as
$u_{1}^{q},u_{2}^{q},u_{3}^{q},...u_{q}^{q}$ for passages too and
$u_{1}^{z},u_{2}^{z},u_{3}^{z},...u_{z}^{z}$ for options as well. The
embedding matrix is boot only once and not trained in the entire learning
process. As shown in Figure 4 S-NET uses the series-to-series model to
incorporate the answer with the extracted evidences as features. They first
produce the depiction It first produce the depiction $h_{p}^{t}$ and
$h_{q}^{t}$ of all words in the question and passage respectively. When giving
out the answer depiction, it merge the basic word embedding $e_{p}^{t}$ with
some added features $f_{s}^{t}$ and $f_{e}^{t}$ to indicate the end and start
place of the evidence snippet given out by evidence extraction model.
$f_{s}^{t}$=1 and $f_{e}^{t}$=1 mean the position t is the start and end of
the evidence span, respectively.
$h_{t}^{p}=BiGRU(h_{t-1}^{p},[e_{t}^{p},f_{t}^{s},f_{t}^{e}])$ (1)
$h_{t}^{q}=BiGRU(h_{t-1}^{q},e_{t}^{q})$ (2)
On top of the encoder, S-Net uses GRU with attention as the decoder to produce
the answer. At each decoding time step t , the GRU reads the previous word
embedding $w_{t-1}$ and previous context vector $c_{t-1}$ and finally produced
answer.
Figure 4: Answer Synthesis/Generation Model(Tan et al.,, 2017)
The produced answer will be stored in Answer vector.
$A_{n}=[a_{1},a_{2},a_{3},...a_{a}]$ where a is length of the answer.Figure 3
shows the overview of selection module. The selection module will take the
refined answer representation $a_{t}$ and computes its bi-linear similarity
with each option representation.
$score(i)=a_{t}W_{att}z_{t_{i}}$ (3)
where i is the number of option, $a_{t}$ is generated answer vector,
$z_{t_{i}}$ is option vector and $W_{att}$ is a matrix which needs to be
learned. We select the option which gives the highest score as computed above.
We train the model using the cross entropy loss by normalizing the above
scores (using softmax) first to obtain a probability distribution.
## 4 Experimental Setup
Here we discussed about the dataset used to evaluate our model, Training
procedure, result comparison and future work.
### 4.1 Dataset
We evaluate our model on RACE dataset(Lai et al.,, 2017) Race is a large-scale
reading comprehension dataset with more than 28,000 passages and nearly
100,000 questions. The dataset is collected from English examinations in
China, which are designed for middle school and high school students. Each
passage is a JSON file. The JSON file contains fields (i) article: A string,
which is the passage (ii) questions: A string list. Each string is a query.
There are two types of questions. First one is an interrogative sentence.
Another one has a placeholder, which is represented by _. (iii)options: A list
of the options list. Each options list contains 4 strings, which are the
candidate option. (iv) answers: A list contains the golden label of each
query.(v) id: Each passage has a unique id in this dataset. RACE has wide
variety of questions like Summarization, Inference, Deduction and Context
matching.
Figure 5: Statistic information about Reasoning type in RACE dataset
### 4.2 Training Procedures and Hyper-parameter
We integrate two different model into once. First we train our model on S-Net.
To train model on S-Net we process dataset differently. We only consider
passage and question and correct option to train model on S-Net. Later we pass
the result on to next stage on our model where we train model using generated
answer and all candidate options. To train the model, we used stochastic
gradient descent with ADAM optimizer.(Kingma and Ba,, 2014) We initialize
learning rate with 0.005. Gradients are clipped in L2-norm to no larger than
10\. To update model parameter per step,we used A mini-batch of 32 samples. We
have created a vocabulary using top 65k words from passage and questions and
if a new out-of-vocabulary(OOV) word encountered we add a special token UNK.
We use the same vocabulary for the passage, question, and options vector
embedding. We tune all our models based on the accuracy achieved on the
validation set. We use 300 dimensional Glove embedding (Pennington et al.,,
2014) for word embedding and word and character encoding.We experiment with
both fine-tuning and not fine-tuning these word embedding. We train all our
models for upto 80 epochs as we do not see any benefit of training beyond 80
epochs as result were starting recurrence.The hidden state size of all GRU
network is 128. We apply dropout (Srivastava et al.,, 2014)to word embeddings
and BiGRU’s outputs with a drop rate of 0.45.
### 4.3 Results and Future Work
Model | RACE-Mid | RACE-High | RACE
---|---|---|---
Random* | 24.6 | 25.0 | 24.9
Sliding Window* | 37.3 | 30.4 | 32.2
GA Reader (100D)* | 43.7 | 44.2 | 44.1
Stanford AR (100D)* | 44.2 | 43.0 | 43.3
Sliding Window* | 37.3 | 30.4 | 32.2
GenNet | 79.6 | 75.4 | 77.3
Table 1: Accuracy on test set of RACE-M, RACE-H and RACE. * indicates the
results from (Lai et al.,, 2017) which are trained with 100D pre-trained Glove
word embeddings
The Human Ceiling Performance reported by CMU on RACE dataset is 94.2. Our
model gives accuracy of 79.6 % on RACE-M 75.4 % on RACE-H and 77.3% on RACE
FULL which outperform several other model. Since in this model first answer
are generated and then option is selected such model can be used to solve such
multiple choice question whose answer option is not present or MCQ with "none
of the above" or "No answer" type multiple choice questions.
## 5 Conclusion
In this paper, we present the GenNet model for multiple-choice reading
comprehension. Specifically, the model uses a combination of Generation and
selection to arrive at the correct option. This is achieved by first
generating the answer for the questions from the passage and then matching
generated answer with the options.At last, the proposed model achieves overall
sate-of-the-art accuracy on RACE and significantly outperforms neural network
baselines on RACE-M, RACE-H and RACE FULL.As future work, we would like to
work towards unanswerable questions or questions where no option matched.
## References
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|
2024-09-04T02:54:58.531454 | 2020-03-09T19:28:33 | 2003.04376 | {
"authors": "Kyle M. Whitcomb and Chandralekha Singh",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26125",
"submitter": "Kyle Whitcomb",
"url": "https://arxiv.org/abs/2003.04376"
} | arxiv-papers | # Not all disadvantages are equal: Racial/ethnic minority students have
largest disadvantage of all demographic groups in both STEM and non-STEM GPA
Kyle M. Whitcomb Department of Physics and Astronomy, University of
Pittsburgh, Pittsburgh, PA, 15260 Chandralekha Singh Department of Physics
and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260
###### Abstract
An analysis of institutional data to understand the outcome of the many
obstacles faced by students from historically disadvantaged backgrounds is
important in order to work towards promoting equity and inclusion for all
students. We use 10 years of institutional data at a large public research
university to investigate the grades earned (both overall and in STEM courses
only) by students categorized on four demographic characteristics: gender,
race/ethnicity, low-income status, and first-generation college student
status. We find that on average across all years of study and for all clusters
of majors, underrepresented minority students experience a larger penalty to
their mean overall and STEM GPA than even the most disadvantaged non-URM
students. Moreover, the underrepresented minority students with additional
disadvantages due to socioeconomic status or parental education level were
even further penalized in their average GPA. Furthermore, we also find that
while women in all demographic groups had a higher average overall GPA, these
gender differences are almost completely non-existent in STEM GPA except among
the most privileged students. These findings suggest that there is need to
provide support to bridge the gaps that emanate from historical disadvantages
to certain groups.
## I Introduction and Theoretical Framework
The importance of evidence-based approaches to improving student learning and
ensuring that all students have the opportunity to excel regardless of their
background is becoming increasingly recognized by Science, Technology,
Engineering, and Mathematics (STEM) departments across the US Johnson (2012);
Johnson _et al._ (2017); Maltese and Tai (2011); Borrego _et al._ (2008);
Borrego and Bernhard (2011); Borrego and Henderson (2014); Henderson and Dancy
(2008); Dancy and Henderson (2010); Henderson _et al._ (2012). With advances
in digital technology in the past few decades, institutions have been keeping
increasingly large digital databases of student records. We have now reached
the point where there is sufficient data available for robust statistical
analyses using data analytics that can provide valuable information useful for
transforming learning for all students Baker and Inventado (2014); Papamitsiou
and Economides (2014). This has lead to many recent studies utilizing many
years of institutional data to perform analyses that were previously limited
by statistical power Lord _et al._ (2009, 2015); Ohland and Long (2016); Matz
_et al._ (2017); Witherspoon and Schunn (2019). Therefore, here we focus on
harnessing institutional data to investigate the obstacles faced by students
with various disadvantages who must overcome obstacles in their pursuit of
higher education.
The theoretical framework for this study has two main foundations: critical
theory and intersectionality. Critical theories of race, gender, etc. identify
historical sources of inequity within society, that is, societal norms that
perpetuate obstacles to the success of certain groups of disadvantaged people
Crenshaw _et al._ (1995); Kellner (2003); Yosso (2005); Gutiérrez (2009);
Taylor _et al._ (2009); Tolbert _et al._ (2018); Schenkel and Calabrese
Barton (2020). Critical theory tells us that the dominant group in a society
perpetuates these norms, which are born out of their interests, and pushes
back against support systems that seek to subvert these norms Crenshaw _et
al._ (1995); Kellner (2003); Yosso (2005). These highly problematic societal
norms are founded in the historical oppression of various groups of people,
and manifest today in many ways including economic disadvantages, stereotypes
about who can succeed in certain career paths, and racist and/or sexist
barriers to opportunity, including educational advancement. While these norms
are, by definition, specific to a particular culture or even country, they are
nonetheless pervasive and oppressive and demand attention to rectify these
historical wrongs.
Much important work has been done on building critical race and/or gender
theories of STEM education Johnson (2012); Johnson _et al._ (2017); Solorzano
_et al._ (2000); Lewis _et al._ (2009); Bang and Medin (2010); Estrada _et
al._ (2018); Ong _et al._ (2018); Tolbert _et al._ (2018); Green _et al._
(2019); Mutegi _et al._ (2019); Sheth (2019); Schenkel and Calabrese Barton
(2020). In one study, Bancroft (2018) lays out a “critical capital theory,”
using varying forms of capital (economic, social, and cultural) to examine
persistence through graduation in STEM doctoral programs and to contextualize
the mechanisms behind racial inequities in STEM education Bancroft (2018). The
idea that race, gender, or another demographic characteristic alone cannot
fully explain the intricacies of the obstacles that students face is rooted in
the framework of intersectionality Crenshaw (1990); Cho _et al._ (2013);
Mitchell _et al._ (2014); Charleston _et al._ (2014); Morton and Parsons
(2018). In particular, the combination of different aspects of an individual’s
social identity (e.g., gender, race, first-generation college status, and
socioeconomic status) leads to unique levels of disadvantages that cannot be
explained by simply adding together the effects of the individual components
of identity Crenshaw (1990). For example, according to the framework of
intersectionality, in many STEM disciplines where the societal norm expects
that students are white men, the experience of a black woman is not a simple
sum of the experiences of white women and black men Charleston _et al._
(2014); Morton and Parsons (2018).
With an eye toward this intersectional approach to critical theory, we seek to
understand the relationship between four different aspects of student identity
that can lead to obstacles in STEM education: race/ethnicity, gender, low-
income status, and first-generation college student status. The students
disadvantaged by low-income or first-generation status are likely to
experience a lack of resources relative to their more privileged peers Lam
_et al._ (2005); Dika and D’Amico (2016); Katrevich and Aruguete (2017). Women
and underrepresented minority students are susceptible to additional stress
and anxiety from stereotype threat (i.e., the fear of confirming stereotypes
pertaining to their identity) which is not experienced by their majority group
peers Lewis _et al._ (2009); Johnson (2012); Green _et al._ (2019); Mutegi
_et al._ (2019); Sheth (2019); Astin (1993); Cross (1993); Felder _et al._
(1995, 1998); Bianchini _et al._ (2002); Britner and Pajares (2006);
Bianchini (2013); Basile and Lopez (2015); Cheryan _et al._ (2017); Hilts
_et al._ (2018). In summary, the different mechanisms by which students
belonging to each demographic characteristic can be disadvantaged are as
follows.
* •
Race/Ethnicity: Students belonging to underrepresented minority (URM) groups
may experience stereotype threat that causes anxiety and robs the students of
their cognitive resources, particularly during high-stakes testing.
* •
Gender: There are pervasive societal biases against women succeeding in many
STEM disciplines which can result in stereotype threat.
* •
Low-Income Status: Low-Income (LI) students are more likely to need to work to
support themselves, reducing their time and energy available to devote to
their studies, in addition to anxiety due to the financial burden of attending
college. These burdens are in addition to other factors that low-income
students may be more likely to face, such as lower quality preparation for
college.
* •
First-Generation Status: First-Generation (FG) students may lack the resources
of encouragement, advice, and support that are available more readily to
students with degree-holding parents. This lack of resources can make FG
students more susceptible to the stress of the unknown in college.
All of these mechanisms can produce an inequitable learning environment
wherein students belonging to any of these groups are forced to work against
obstacles that their peers do not have. The framework of intersectionality
asserts that for students that belong to more than one of these groups,
complex interactions between these different obstacles can result in
compounded disadvantages that are not a simple sum of the individual effects
Crenshaw (1990); Cho _et al._ (2013); Mitchell _et al._ (2014); Charleston
_et al._ (2014); Morton and Parsons (2018).
In order to measure the long-term effects of these systemic disadvantages, we
will investigate the academic achievement of students belonging to these
various demographic groups over the course of their studies at one large
public research university using 10 years of institutional data. By grouping
students according to their demographic background, we will be able to
investigate how different combinations of obstacles affect student grade point
averages.
## II Research Questions
Our research questions regarding the intersectional relationships between
demographic characteristics and academic achievement are as follows.
1. RQ1.
Are there differences in the overall or STEM grades earned by students
belonging to different demographic groups (i.e., underrepresented minority,
low-income status, and first-generation college student status)?
2. RQ2.
Do any patterns observed in RQ1 differ for men and women?
3. RQ3.
Do grades earned in STEM courses alone exhibit similar demographic patterns as
grades earned in all courses?
4. RQ4.
What are the trends over time in the mean GPA of these different demographic
groups among different clusters of majors (i.e., computer science,
engineering, mathematics, and physical science majors, other STEM majors, and
non-STEM majors)?
## III Methodology
### III.1 Sample
Using the Carnegie classification system, the university at which this study
was conducted is a public, high-research doctoral university, with balanced
arts and sciences and professional schools, and a large, primarily residential
undergraduate population that is full-time and reasonably selective with low
transfer-in from other institutions Indiana University Center for
Postsecondary Research (2018).
The university provided for analysis the de-identified institutional data
records of students with Institutional Review Board approval. In this study,
we examined these records for $N=24,567$ undergraduate students enrolled in
three colleges within the university: the colleges of Arts and Sciences,
Computing and Information, and Engineering. This sample of students includes
all of those from ten cohorts who met several selection criteria, namely that
the student had first enrolled at the university in a Fall semester from Fall
2005 to Fall 2014, inclusive, and the institutional data on the student was
not missing or unspecified for any of the following measures: gender,
race/ethnicity, parental education level, and family income. This sample of
students is $50\%$ female and had the following race/ethnicities: 79% White,
9% Asian, 7% Black, 3% Hispanic, and 2% other or multiracial. Further, this
sample is $16\%$ first-generation college students and $21\%$ “low-income”
students (to be defined in the following section).
We acknowledge that gender is not a binary construct, however in self-
reporting their gender to the university students were given the options of
“male” or “female” and so those are the two self-reported genders that we are
able to analyze. There were $39$ students who had met all other selection
criteria but who had not indicated any gender on the survey, these students
were removed from the sample and are not included in the reported sample size
or any analyses.
### III.2 Measures
#### III.2.1 Demographic Characteristics
Four primary measures are the demographic characteristics mentioned in the
previous section, namely gender, race/ethnicity, parental education level, and
family income. All of these were converted into binary categories intended to
distinguish between the most and least privileged students on each measure.
* •
Gender. Gender was reported as a binary category to begin with (either “male”
or “female”), therefore no further steps were required.
* •
First-generation. Students for whom both parents had a highest completed level
of education of high school or lower were grouped together as “first-
generation” (FG) college students and correspondingly students for whom at
least one parent had earned a college degree were labeled non-FG.
* •
Low-income. Students whose reported family Adjusted Gross Income (AGI) was at
or below 200% of the federal U.S. poverty line were categorized as “low-
income” (LI), and those above 200% of the poverty line as non-LI Cauthen and
Fass (2007); Jiang _et al._ .
* •
Underrepresented minority. All students who identified as any race or
ethnicity other than White or Asian were grouped together as “underrepresented
minority” (URM) students, including multiracial students who selected White
and/or Asian in addition to another demographic option. Students who only
identified as White and/or Asian students were categorized as non-URM
students.
#### III.2.2 Academic Performance
Measures of student academic performance were also included in the provided
data. High school GPA was provided by the university on a weighted scale from
0-5 that includes adjustments to the standard 0-4 scale for Advanced Placement
and International Baccalaureate courses. The data also include the grade
points earned by students in each course taken at the university. Grade points
are on a 0-4 scale with $\text{A}=4$, $\text{B}=3$, $\text{C}=2$,
$\text{D}=1$, $\text{F}=0$, where the suffixes “$+$” and “$-$” add or
subtract, respectively, $0.25$ grade points (e.g. $\text{B}-=2.75$), with the
exception of $\text{A}+$ which is reported as the maximum 4 grade points. The
courses were categorized as either STEM or non-STEM courses, with STEM courses
being those courses taken from any of the following departments: biological
sciences, chemistry, computer science, economics, any engineering department,
geology and environmental science, mathematics, neuroscience, physics and
astronomy, and statistics. We note that for the purposes of this paper, “STEM”
does not include the social sciences other than economics, which has been
included due to its mathematics-intensive content.
#### III.2.3 Year of Study
Finally, the year in which the students took each course was calculated from
the students’ starting term and the term in which the course was taken. Since
the sample only includes students who started in fall semesters, each “year”
contains courses taken in the fall and subsequent spring semesters, with
courses taken over the summer omitted from this analysis. For example, if a
student first enrolled in Fall 2007, then their “first year” occurred during
Fall 2007 and Spring 2008, their “second year” during Fall 2008 and Spring
2009, and so on in that fashion. If a student is missing both a fall and
spring semester during a given year but subsequently returns to the
university, the numbering of those post-hiatus years is reduced accordingly.
If instead a student is only missing one semester during a given year, no
corrections are made to the year numbering. In this study we consider up
through the students’ sixth year of study or the end of their enrollment at
the studied institution, whichever comes first.
### III.3 Analysis
The primary method by which we grouped students in this analysis was by their
set of binary demographic categories. This grouping was performed in two
different ways. First, use of all four binary categories (gender, FG, LI, URM)
resulted in sixteen mutually exclusive groups (e.g., “female, FG+URM” or
“male, LI”). Second, use of all categories except gender resulted in eight
mutually exclusive categories.
We calculated each student’s yearly (i.e., not cumulative) grade point average
(GPA) across courses taken in each year of study from the first to sixth
years. In addition, we calculated the student’s yearly STEM GPA, that is, the
GPA in STEM courses alone. Then, using the aforementioned grouping schemes, we
computed the mean GPA in each demographic group as well as the standard error
of the mean separately for each year of study Freedman _et al._ (2007).
Further, in the case of grouping by gender, we computed the effect size of the
gender differences within each demographic group using Cohen’s $d$, which is
typically interpreted using minimum cutoff values for “small” ($d=0.20$),
“medium” ($d=0.50$), and “large” ($d=0.80$) effect sizes Cohen (1988); Neter
_et al._ (2004); Montgomery _et al._ (2012).
All analyses were conducted using R R Core Team (2019), making use of the
package tidyverse Wickham (2017) for data manipulation and plotting.
## IV Results
### IV.1 GPA Trends by Demographic Group: “Dinosaur Plots”
In order to answer RQ1, we plotted in Fig. 1 the mean GPA earned by students
in each demographic group, including gender as a grouping characteristic. We
start with overall GPA, rather than STEM GPA alone, in order to provide
context for the results in STEM GPA and identify trends that may or may not be
present when viewing STEM grades alone. Groups are ordered from left to right
first by the ascending number of selected characteristics and then
alphabetically. Mean GPA is plotted separately (i.e., not cumulatively) for
each year of study from the first to sixth year. Setting aside the gender
differences for a moment, we note that the general GPA trends by demographic
group in Fig. 1 follow a shape resembling the neck, back, and tail of a
sauropod, and so accordingly we refer to the plots in Fig. 1 as “dinosaur
plots.” This shape is clearest in the plots for the first through fourth
years, as the sample size drops significantly in the fifth year as the
majority of students graduate.
Looking more closely at Fig. 1, particularly the first four years, we see that
the “neck” is consistently comprised of the group of students with the most
privileges, namely those students that are non-FG, non-LI, and non-URM.
Following this, the “back” is relatively flat across the next four groups,
namely students that are FG only, LI only, URM only, or FG and LI. Notably,
the URM group of students typically have the lowest mean GPA within this set
of demographic groups. Finally, the “tail” consists of the final three groups,
FG+URM, LI+URM, and FG+LI+URM. The mean GPA in this set of groups tends to
decrease from left to right in the plots. Notably, the four groups that
contain URM students are consistently in the lowest four or five mean GPAs.
Figure 1: Average GPA of each demographic group. Students are binned into
separate demographic groups based on their status as first-generation (FG),
low-income (LI), and/or underrepresented minority (URM) students. The men and
women in each demographic group are plotted separately. The mean GPA in all
courses taken by students in each demographic group is plotted along with the
standard error on the mean, with a separate plot for each of the (a) first,
(b) second, (c) third, (d) fourth, (e) fifth, and (f) sixth years. The sample
size is reported by each point, and Cohen’s $d$ Cohen (1988) measuring the
effect size of the gender difference in each group is reported.
### IV.2 Intersectionality with Gender
We now turn our attention to the differences between men and women in Fig. 1
in order to answer RQ2. We note in particular that across all demographic
groups women’s mean GPA is roughly 0.2 grade points higher than men’s. The
effect sizes (Cohen’s $d$) of this difference range from small to medium Cohen
(1988). This difference in mean GPA earned is substantial enough to indicate a
change in letter grade, given that the grading system at the studied
university uses increments of 0.25 grade points for letter grades containing
“$+$” or “$-$.” Further, this trend holds in the fifth year (Fig. 1e) and
sixth year (Fig. 1f), with some exceptions in demographic groups with
particularly low sample sizes after the fourth year.
### IV.3 STEM GPA Trends
In order to answer RQ3, Figure 2 plots students’ mean STEM GPA in a similar
manner to Fig. 1. We note that the general “dinosaur” pattern discussed in
Fig. 1 also holds at least for the first and second years (Figs. 2a and 2b,
respectively). In the third year and beyond, the general features of the trend
continue to hold, with the most privileged students having the highest mean
GPA, followed by those with one disadvantage as well as the first-generation
and low-income group, followed by the remaining groups of URM students with
one or more additional disadvantages. However, in these later years, the finer
details of the plots noted before fall away in favor of a sharper mean GPA
decrease for URM students with at least one additional disadvantage in the
third year (Fig. 2c) and a more gradual decrease across all groups in the
fourth year (Fig. 2d) and fifth year (Fig. 2e). When restricting the GPA
calculations to STEM courses, the sample size becomes too small in the sixth
year (Fig. 2f) to draw meaningful conclusions.
Figure 2: Average STEM GPA of each demographic group. Students are binned
into separate demographic groups based on their status as first-generation
(FG), low-income (LI), and/or underrepresented minority (URM) students. The
men and women in each demographic group are plotted separately. The mean GPA
in all courses taken by students in each demographic group is plotted along
with the standard error on the mean, with a separate plot for each of the (a)
first, (b) second, (c) third, and (d) fourth, (e) fifth, and (f) sixth years.
The sample size is reported by each point, and Cohen’s $d$ Cohen (1988)
measuring the effect size of the gender difference in each group is reported.
We further observe a trend of students earning higher grades on average in
later years, although the rise from the first to the fourth year is somewhat
lower in STEM GPA than in overall GPA. Notably, while in overall GPA this
trend seemed to be somewhat universal across demographic groups, in Fig. 2 we
see a quicker rise in mean STEM GPA over time for the more privileged students
than the less privileged students, particularly comparing the leftmost and
rightmost groups.
Regarding gender differences, Fig. 2 shows smaller gender differences in STEM
GPA than those observed in overall GPA in Fig. 1. While in overall GPA women
earned roughly 0.2 grade points more than men on average, in STEM GPA that
difference is much less consistent and typically ranges from 0 to 0.1 grade
points. For many demographic groups we see no significant differences between
men and women’s mean STEM GPA. We do see that there is still a consistent STEM
GPA gender difference, albeit smaller than in Fig. 1, among the group of the
most privileged students (i.e., those with “None” of the disadvantages). There
is also a STEM GPA gender difference among first-generation low-income but
non-URM students, however this difference is less consistent and in fact
briefly vanishes in the third year.
### IV.4 GPA Trends By Major Over Time
In order to better understand the trends over time in both overall and STEM
GPA and answer RQ4, we plotted the mean GPA by year in Fig. 3 and mean STEM
GPA by year in Fig. 4. In these plots, we have not separated men and women and
instead focus on the other demographic characteristics while further grouping
students into three different groups of majors in order to understand if these
trends differ for students in different areas of study. Further, since the
sample size becomes quite small in years five and six for many of the
demographic groups of interest, we plot only the mean GPA over the first four
years. In Figs. 3a and 4a, we plot the mean overall and STEM GPA,
respectively, of all students. In the other subfigures, we plot the mean GPA
earned by students majoring in different clusters of majors. In particular, we
plot the mean GPA of engineering (including computer science), mathematics,
and physical science (i.e., chemistry and physics) majors in Figs. 3b and 4b,
the remaining STEM majors in Figs. 3c and 4c, and non-STEM majors in Figs. 3d
and 4d.
Figure 3: Students are binned into separate demographic groups as in Fig. 1,
but not separated by gender. The mean GPA in all courses of each group is
plotted over time from year one to four, along with the standard error of the
mean. The plots show this for four subpopulations: (a) all students; (b)
chemistry, computer science, engineering, mathematics, and physics students;
(c) biology, economics, geology, neuroscience, and statistics students; and
(d) non-STEM students including psychology. Figure 4: Students are binned
into separate demographic groups as in Fig. 2, but not separated by gender.
The mean GPA in STEM courses of each group is plotted over time from year one
to four along with the standard error of the mean. The plots show this for
four subpopulations: (a) all students; (b) chemistry, computer science,
engineering, mathematics, and physics students; (c) biology economics,
geology, neuroscience, and statistics students; and (d) non-STEM students
including psychology.
These plots make clearer some of the trends noted earlier, especially the rise
in mean GPA over time from the first to the fourth year. However, we can now
see that this is not universally true since the first-generation URM students
have a drop in mean GPA in the second year for physical science majors (Fig.
3b), and in the third year for other STEM majors (Fig. 3c). This trend is even
more noticeable in STEM GPA (Fig. 4), where the mean STEM GPA of the group of
first-generation URM students drops in the third year for every subpopulation
by major.
## V Discussion
To start, we consider how much the current system disadvantages students who
are first-generation, low-income, or underrepresented minority but not a
combination of the two. Discussing these groups first is helpful in setting
the stage for a more complex discussion of the intersectionality of these
various demographic characteristics. We find in Figs. 1 and 2 that not all of
these disadvantages are equal. In particular, non-URM students who have one
disadvantage, namely the first-generation (but not low-income) and low-income
(but not first-generation) students, still earn slightly higher grades than
even the URM students who are not low-income or first-generation. Notably,
this trend (the “back” of the dinosaur plots) is similar in both overall
grades (Fig. 1) and in STEM grades alone (Fig. 2). The size of this mean grade
difference varies from year to year, but in STEM grades it can reach as high
as about 0.25 grade points, which at the studied institution is the difference
between, for example, a B and B$+$ or B$-$ grade.
The group with the grades most similar to these non-first-generation, non-low-
income URM students are the first-generation, low-income non-URM students, who
earn both overall (Fig. 1) and STEM (Fig. 2) grades similar to or very
slightly higher than the URM students. One explanation could be that the lack
of resources available due to being first-generation or low-income is not as
severe an obstacle as the stereotype threat experienced by URM students.
Turning then to the “tail” in the dinosaur plots, we find that consistently
the most disadvantaged students in both overall grades (Fig. 1) and STEM
grades (Fig. 2) are the URM students with at least one additional obstacle. In
this case, it appears that the intersection of being low-income and URM is the
most disadvantageous combination, with no notable difference in either Fig. 1
or Fig. 2 among these students whether or not they are also first-generation.
Meanwhile, the first-generation URM students who are not low-income sometimes
have a slightly higher mean GPA than the low-income URM students (Fig. 1).
Another avenue to investigate intersectionality is how gender interacts with
the other demographic groups. Interestingly, in overall GPA (Fig. 1), gender
appears to have about the same effect across all demographic groups. That is,
there does not appear to be an intersectional effect of gender identity with
other identities as measured by overall GPA. However, Fig. 2 shows that this
is a context-dependent effect, with the gender gap substantially and unevenly
reduced across all groups in mean STEM GPA. For most demographic groups in
Fig. 2, the higher overall GPA earned by women in Fig. 1 has vanished
completely in STEM GPA. This is consistent with stereotype threat being the
mechanism of disadvantage for women, where stereotypes surrounding STEM
disciplines unfairly cause stress and anxiety for women Astin (1993); Cross
(1993); Felder _et al._ (1995, 1998); Britner and Pajares (2006); Basile and
Lopez (2015); Cheryan _et al._ (2017); Hilts _et al._ (2018). Notably, while
the gender gap is reduced nearly to zero for most groups in Fig. 2, there does
remain a small consistent gender gap favoring women in the most privileged
group of students. In other groups the gender gap in Fig. 2 is inconsistent
across years. One explanation could be that the wealth of resources available
to them may help to alleviate the stereotype threat.
Taking a more temporal view of these GPA trends, Fig. 3 (overall GPA) and Fig.
4 (STEM GPA) have grouped men and women together in order to focus on the
other demographic characteristics more closely. In these plots, the most
noteworthy trend is again that, with the sole exception of the first year in
Fig. 3b, the four groups with the lowest mean GPA (Fig. 3) and STEM GPA (Fig.
4) across the first four years are always the four groups containing URM
students. Notably, this trend is true regardless of which group of majors we
investigate. The consistency of this result is particularly striking, showing
that the most otherwise disadvantaged non-URM students have fewer obstacles to
success than even the most privileged URM students among all students.
Focusing further on the STEM GPA of STEM majors in Figs. 4b and 4c, we see
that while non-URM students consistently rise in mean GPA over time, the same
is not true for all URM students. In particular, the first-generation URM
students who major in chemistry, computer science, engineering, mathematics,
or physics (Fig. 4b) experience a steady decline in mean STEM GPA from year
one to two and year two to three. While the standard error of those means is
quite large due to a relatively small sample size, that lack of representation
for these students could itself be what is hindering their coursework by
causing a stereotype threat.
Based upon the frameworks of critical theory and intersectionality, the main
implication of these findings is that many students who come from less
privileged backgrounds are not being adequately supported in college in order
to catch up with the privileged students Crenshaw _et al._ (1995); Kellner
(2003); Yosso (2005); Gutiérrez (2009); Taylor _et al._ (2009); Tolbert _et
al._ (2018); Schenkel and Calabrese Barton (2020); Johnson (2012); Johnson
_et al._ (2017); Crenshaw (1990); Cho _et al._ (2013); Mitchell _et al._
(2014); Charleston _et al._ (2014); Morton and Parsons (2018). The
disadvantages of these less privileged students manifest as lower mean overall
and STEM GPA for those demographic groups. In order to promote equity and
inclusion, it is crucial that these students are provided appropriate
mentoring, guidance, scaffolding, and support in college so that these
obstacles can be cleared for students who have been put at a disadvantage
relative to their peers through no fault of their own Birt _et al._ (2019).
We note that these demographic groups with more disadvantages are likely to
consist of students who had K-12 education from schools with fewer resources
and less well-prepared teachers than those of the more privileged students,
with high school being an especially important time for disadvantages related
to STEM learning increasing Bianchini _et al._ (2003); Maltese and Tai
(2011); Means _et al._ (2017); Bottia _et al._ (2018); Daley (2019); Dou
_et al._ (2019). Analyses such as those discussed here can help inform the
allocation of resources to support these students, with efforts to reduce the
classroom stereotype threat of URM students and creating a low-anxiety
environment in which all students have a high sense of belonging and can
participate fully without fear of being judged being clear priorities.
Additional resources to support low-income and/or first-generation students,
e.g., financial support and timely advising pertaining to various academic and
co-curricular opportunities, are also important in order to level the playing
field and work towards a goal of all students succeeding in college,
regardless of their race/ethnicity, socioeconomic status, and parental
education history.
## VI Acknowledgments
This research is supported by the National Science Foundation Grant
DUE-1524575 and the Sloan Foundation Grant G-2018-11183.
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|
2024-09-04T02:54:58.547629 | 2020-03-09T20:06:20 | 2003.04389 | {
"authors": "Adam Gaier, Alexander Asteroth, Jean-Baptiste Mouret",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26126",
"submitter": "Adam Gaier",
"url": "https://arxiv.org/abs/2003.04389"
} | arxiv-papers | # Discovering Representations for Black-box Optimization
Adam Gaier Inria, CNRS, Université de LorraineBonn-Rhein-Sieg University of
Applied Sciences<EMAIL_ADDRESS>, Alexander Asteroth Bonn-Rhein-Sieg
University of Applied SciencesSankt AugustinGermany53757
<EMAIL_ADDRESS>and Jean-Baptiste Mouret Inria, CNRS,Université
de LorraineNancyFrance54000<EMAIL_ADDRESS>
(2020; 2020)
###### Abstract.
The encoding of solutions in black-box optimization is a delicate, handcrafted
balance between expressiveness and domain knowledge — between exploring a wide
variety of solutions, and ensuring that those solutions are useful. Our main
insight is that this process can be automated by generating a dataset of high-
performing solutions with a quality diversity algorithm (here, MAP-Elites),
then learning a representation with a generative model (here, a Variational
Autoencoder) from that dataset. Our second insight is that this representation
can be used to scale quality diversity optimization to higher dimensions — but
only if we carefully mix solutions generated with the learned representation
and those generated with traditional variation operators. We demonstrate these
capabilities by learning an low-dimensional encoding for the inverse
kinematics of a thousand joint planar arm. The results show that learned
representations make it possible to solve high-dimensional problems with
orders of magnitude fewer evaluations than the standard MAP-Elites, and that,
once solved, the produced encoding can be used for rapid optimization of
novel, but similar, tasks. The presented techniques not only scale up quality
diversity algorithms to high dimensions, but show that black-box optimization
encodings can be automatically learned, rather than hand designed.
††copyright: rightsretained††doi: 10.1145/nnnnnnn.nnnnnnn††isbn: 978-x-xxxx-
xxxx-x/YY/MM††conference: the Genetic and Evolutionary Computation Conference
2020; July 8–12, 2020; Cancún, Mexico††journalyear: 2020††price:
15.00††journalyear: 2020††copyright: rightsretained††conference: Genetic and
Evolutionary Computation Conference; July 8–12, 2020; Cancún,
Mexico††booktitle: Genetic and Evolutionary Computation Conference (GECCO
’20), July 8–12, 2020, Cancún, Mexico††doi: 10.1145/3377930.3390221††isbn:
978-1-4503-7128-5/20/07 Figure 1. Data-Driven Encoding MAP-Elites (DDE-
Elites) searches the space of representations to search for solutions. A data-
driven encoding (DDE) is learned by training a VAE on the MAP-Elites archive.
High fitness solutions, which increase the bias of the DDE toward performance,
are found using the DDE. Novel solutions, which increase the range of
solutions which can be expressed, are found using mutation operators. UCB1, a
bandit algorithm, balances the mix of these explorative and exploitative
operators.
## 1\. Introduction
The method of encoding solutions is one of the most critical design decisions
in optimization, as the representation defines the way an algorithm can move
in the search space (Rothlauf, 2006). Work on representations tends to focus
on encoding priors or innate biases: aerodynamic designs evolved with splines
to encourage smooth forms (Olhofer et al., 2001), Compositional Pattern
Producing Networks (CPPNs) with biases for symmetry and repetition in images
and neural network weight patterns (Stanley, 2007; Stanley et al., 2009),
modularity induced in evolved neural networks (Mouret and Doncieux, 2008; Durr
et al., 2010; Doncieux and Meyer, 2004), or neural network structures which
encode strong enough biases to perform without training (Gaier and Ha, 2019).
The best representations balance a bias for high performing solutions, so they
can easily be discovered, and the ability to express a diversity of potential
solutions, so the search space can be widely explored. At the one extreme, a
representation which only encodes the global optimum is easy to search but
useless for finding any other solution. At the other, a representation which
can encode anything presents a difficult and dauntingly vast search space.
Given a large set of example solutions, representations could be learned from
data instead of being hand-tailored by trial-and-error: a learned
representation would replicate the same biases toward performance and the same
range of expressivity as the source data set. For instance, given a dataset of
face images, a Variational Autoencoder (VAE) (Kingma and Welling, 2014) or a
Generative Adversarial Network (GAN) (Goodfellow et al., 2014) can learn a
low-dimensional latent space, or encoding, that makes it possible to explore
the space of face images. In essence, the decoder which maps the latent space
to the phenotypic space learns the “recipe” of faces. Importantly, the
existence of such a low-dimensional latent space is possible because _the
dataset is a very small part of the set of all possible images_.
However, using a dataset of preselected high-performing solutions “traps” the
search within the distribution of solutions that are already known: a VAE
trained on white faces will never generate a black face. This limits the
usefulness of such data-driven representations for discovering _novel_
solutions to hard problems.
In this paper, we propose the use of the MAP-Elites algorithm (Mouret and
Clune, 2015) to automatically generate a dataset for representations using
only a performance function and a diversity space. Quality diversity (QD)
algorithms (Cully and Demiris, 2018; Pugh et al., 2016) like MAP-Elites are a
good fit for representation discovery: creating archives of diverse high-
performing solutions is precisely their purpose. Using the MAP-Elites archive
as a source of example solutions, we can capture the genetic distribution of
the highest performing solutions, or elites, by training a VAE and obtaining a
latent representation. As the VAE is only trained on elites, this learned
representation, or Data-Driven Encoding (DDE), has a strong bias towards
solutions with high fitness; and because the elites have varying phenotypes,
the DDE is able to express a range of solutions. Though the elites vary along
a phenotypic continuum, they commonly have many genotypic similarities
(Vassiliades and Mouret, 2018), making it more likely to find a well-
structured latent space.
Nonetheless, MAP-Elites will struggle to find high-performing solutions
without an adequate representation. Fortunately, the archive is produced by
MAP-Elites in an iterative, any-time fashion, so there is no “end state” to
wait for before a DDE can be trained — a DDE can be trained during
optimization. The DDE can then be used to enhance optimization. By improving
the quality of the archive the DDE improves the quality of its own source
data, establishing a virtuous cycle of archive and encoding improvement.
A DDE based on an archive will encounter the same difficulty as any learned
encoding: the DDE can only represent solutions that are already in the
dataset. How then, can we discover new solutions? Fundamentally, to search for
an encoding we need to both _exploit the best known representation_ , that is,
create better solutions according to the current best “recipes”, and also
_explore new representations_ — solutions which do not follow any “recipe”.
In this paper, we address this challenge by mixing solutions generated with
the DDE with solutions obtained using standard evolutionary operators. Our
algorithm applies classic operators, such as Gaussian mutation, to create
candidates which could not be captured by the current DDE. At the same time we
leverage the DDE to generalize common patterns across the map and create new
solutions that are likely to be high-performing. To avoid introducing new
hyper-parameters, we tune this exploration/exploitation trade-off optimally
using a multi-armed bandit algorithm (Garivier and Moulines, 2011).
This new algorithm, DDE-Elites, reframes optimization as a search for
representations (Figure 1). Integrating MAP-Elites with a VAE makes it
possible to apply quality diversity to high-dimensional search spaces, and to
find effective representations for future uses. We envision application to
domains that have straightforward but expansive low-level representations, for
instance: joints positions at 20Hz for a walking robot ($12\times 100=1200$
joint positions for a 5-second gait of a robot with $12$ degrees of freedom),
3D shapes in which each voxel is encoded individually (1000-dimensional for a
$10\times 10\times 10$ grid), images encoded in the pixel-space, etc.
Ideally, the generated DDE will capture the main regularities of the domain.
In robot locomotion, this could correspond to periodic functions, since we
already know that a $36$-dimensional controller based on periodic functions
can produce the numerous joint commands required every second to effectively
drive a 12-joint walking robot in many different ways (Cully et al., 2015). In
many domains the space of possible solutions can be vast, while the inherent
dimensionality of interesting solutions is still compact. By purposefully
seeking out a space of solutions, rather than the solutions themselves, we can
solve high-dimensional problems in a lower dimensional space.
## 2\. Background
### 2.1. Optimization of Representations
In his 30 year perspective on adaptation in evolutionary algorithms, Kenneth
De Jong identified representation adaptation as ”perhaps the most difficult
and least understood area of EA design.” (De Jong, 2007)
Despite the difficulty of creating adaptive encodings, the potential rewards
have lured researchers for decades. Directly evolving genotypes to increase in
complexity has a tradition going back to the eighties (Goldberg et al., 1989;
Altenberg, 1994). The strategy of optimizing a solution at low complexity and
then adding degrees of freedom has proved effective on problems from optimal
control (Gaier and Asteroth, 2014), to aerodynamic design (Olhofer et al.,
2001), to neural networks (Stanley and Miikkulainen, 2002). Evolving the
genome’s structure is particularly important when the structure itself is the
solution, such as in genetic programming (Koza, [n. d.]) or neural
architecture search (Elsken et al., 2019; Miikkulainen et al., 2019; Gaier and
Ha, 2019).
Recent approaches toward representation evolution have focused on genotype-
phenotype mappings (Bongard and Pfeifer, 2003). Neural networks, which map
between inputs and outputs, are a natural choice for such ‘meta-
representations’. These mappings can evolve with the genome (Scott and
Bassett, 2015; Simões et al., 2014), or fix the genome and evolve only the
mapping (Stanley et al., 2009; Stanley, 2007).
Supervised methods have been previously applied to learn encodings. These
approaches require a set of example solutions for training. Where large, well-
curated data sets are available this strategy has proven effective at creating
representations well suited to optimization (Volz et al., 2018; Bontrager et
al., 2018b; Bontrager et al., 2018a), but where a corpus of solutions does not
exist it must be created. In (Scott and De Jong, 2018; Moreno et al., 2018)
these solutions were collected by saving the champion solutions found after
repeatedly running an optimizer on the problem, with the hope that the learned
representation would then be effective in similar classes of problems.
### 2.2. MAP-Elites
MAP-Elites (Mouret and Clune, 2015) is a QD algorithm which uses a niching
approach to produce high-performing solutions which span a continuum of user-
defined phenotypic dimensions. These phenotypic dimensions, or behavior
descriptors, describe the way the problem is solved, and are often orthogonal
to performance. MAP-Elites has been used in such diverse cases as optimizing
the distance traveled by a walking robot using different legs (Cully et al.,
2015), the drag of aerodynamic designs with varied volumes and curvatures
(Gaier et al., 2017), and the win rate of decks composed of different cards in
deck-building games (Fontaine et al., 2019).
MAP-Elites is a steady-state evolutionary algorithm which maintains a
population in a discretized grid or ‘archive’. This grid divides the
continuous space of possible behaviors into bins, or ‘niches’ with each bin
holding a single individual, or ‘elite’. These elites act as parents, and are
mutated to form new individuals. These child individuals are evaluated and
assigned a niche based on their behavior. If the niche is empty the child is
placed inside; if the niche is already occupied, the individual with higher
fitness is stored in the niche and the other discarded. By repeating this
process, increasingly optimal solutions which cover the range of phenotype
space are found. The MAP-Elites algorithm is summarized in Algorithm 1.
Algorithm 1 MAP-Elites
1:function MAP-Elites($fitness()$, $variation()$, $\mathcal{X}_{initial}$)
2: $\mathcal{X}\leftarrow\emptyset$, $\mathcal{F}\leftarrow\emptyset$
$\triangleright$ Map of genomes $\mathcal{X}$, and fitnesses $\mathcal{F}$
3: $\mathcal{X}\leftarrow\mathcal{X}_{initial}$ $\triangleright$ Place initial
solutions in map
4: $\mathcal{F}\leftarrow fitness(\mathcal{X}_{initial})$
5: for iter = $1\to I$ do
6: $\mathbf{x^{\prime}}\leftarrow variation(\mathcal{X})$ $\triangleright$
Create new solution from elites
7: $\mathbf{p^{\prime}},\mathbf{b^{\prime}}\leftarrow
fitness(\mathbf{x^{\prime}})$ $\triangleright$ Get performance and behavior
8: if $\mathcal{F}(\mathbf{b^{\prime}})=\emptyset$ or
$\mathcal{F}(\mathbf{b^{\prime}})<\mathbf{f^{\prime}}$ then $\triangleright$
Replace if better
9: $\mathcal{F}(\mathbf{b^{\prime}})\leftarrow\mathbf{f^{\prime}}$
10: $\mathcal{X}(\mathbf{b^{\prime}})\leftarrow\mathbf{x^{\prime}}$
11: end if
12: end for
13: return $(\mathcal{X}$, $\mathcal{F})$ $\triangleright$ Return illuminated
map
14:end function
Though phenotypically diverse the elites are often genotypically similar,
existing in an “elite hypervolume”, a high performing region of genotype space
(Vassiliades and Mouret, 2018). Just as in nature, where species as diverse as
fruit flies and humans share nearly 60 percent of their genome (Adams et al.,
2000), the “recipe” for high performance is often composed of many of the same
ingredients.
This insight was leveraged in (Vassiliades and Mouret, 2018) to create a new
variation operator which considers the correlation among elites. Genes which
vary little across the elites, and so are likely common factors that produce
high performance, are also subject to the smallest amount of perturbation —
lowering the chance their children stray from the elite hypervolume. Biasing
mutation in this way ensures that exploration is focused on factors which
induce phenotypic variation without drifting into regions of poor performance.
### 2.3. Variational Autoencoders
Autoencoders (AEs) (Hinton and Salakhutdinov, 2006) are neural networks
designed to perform dimensionality reduction. AEs are composed of two
components: an encoder, which maps the input to a lower dimensional latent
space; and a decoder, which maps the latent space back to the original space.
The decoder is trained to reconstruct the input through this lower dimensional
latent “bottleneck”. The encoder component can be viewed as a generalization
of Principal Component Analysis (Wold et al., 1987), with the latent space
approximating principal components.
Though the AE is able to represent the data at a lower dimensionality, and
reproduce it with minimal loss, it can still be a poor representation for
optimization. An important quality of representations is ‘locality’, that a
small change in the genotype induces a small change in the phenotype
(Rothlauf, 2006). When AEs are trained only to minimize reconstruction error
they may overfit the distribution of the training data and create an irregular
latent space. The low-locality of such latent spaces limits their usefulnesses
in optimization: nearby points in latent space may decode to very different
solutions, meaning even a small mutation could have a large effect.
Variational autoencoders (VAEs) (Kingma and Welling, 2014) are AEs whose
training is regularized to ensure a high-locality latent space. The
architecture is broadly the same: an encoder and decoder mediated by a
bottleneck, but rather than encoding the input as a single point it is encoded
as a normal distribution in the latent space. When training the model a point
from this input distribution is sampled, decoded, and the reconstruction error
computed. By encoding the input as a normal distribution we induce the
distributions produced by the encoder to be closer to normal. VAEs are trained
by minimizing two terms: (1) the reconstruction error, and (2) the Kullback-
Liebler (KL) divergence (Kullback and Leibler, 1951) of the latent space to a
unit Gaussian distribution, giving the loss function:
(1)
$loss=\|x-\hat{x}\|^{2}+KL\left[N\left(\mu_{x},\sigma\right),N(0,1)\right]$
Inducing solutions to be encoded in the form of a normal distribution
structures the latent space in a continuous and overlapping way, creating a
local encoding better suited to optimization.
## 3\. DDE-Elites
Figure 2. DDE-Elites Algorithm
(1) A VAE is trained on the archive, and used to create a ‘reconstructive
crossover’ operator which creates new solutions by averaging the parameters of
an individual with its own reconstruction; (2) the mix of exploitative and
explorative variation operators predicted to have the most success is chosen
by the multi-armed bandit algorithm UCB1 and used to create new solutions; (3)
the new solutions are added to the archive and the success rate of the applied
variation operator is updated.
Every representation biases optimization in some way, improving optimization
by limiting the range of solutions that can be expressed to those which are
valid or high-performing (Rothlauf, 2006). But finding a balance between
expressivity and bias is an arduous task requiring considerable domain
expertise. Our method, DDE-Elites, automates the process of representation
design and learns new encodings in tandem with search — allowing optimization
and representation learning to improve each other in a self-reinforcing cycle.
DDE-Elites learns an encoding from examples of high performing solutions. To
create these examples we use MAP-Elites, which produces a variety of high
performing solutions rather than converging to a single optima. The variety
produced by MAP-Elites is critical — the expressivity of any learned encoding
is limited by the variety of examples. That MAP-Elites not only produces a
variety of solutions, but allows us to define the nature of that variety,
makes it particularly powerful for crafting useful representations. By
defining the type of variety we want to explore we are defining the biases and
expressivity we encode in our representation.
DDE-Elites is a variant of the MAP-Elites algorithm. The core component of
competition within a niched archive is maintained, but novel methods of
producing child solutions are introduced. Child solutions are created using an
encoding learned from the archive. This encoding is refined as the archive
improves, which in turn improves the optimization process. DDE-Elites
optimizes an archive of varied solutions by reframing optimization as a search
for the best representation, rather than the best solution.
The DDE-Elites algorithm proceeds as follows (see Figure 2 and Algorithm 2):
(1) a DDE and reconstructive crossover operator is created by training a VAE
on the archive; (2) the probability of using each variation operator is
determined by the UCB1 bandit algorithm; (3) MAP-Elites is run with the chosen
variation operator probabilities. The success rate of the variation operators
to create solutions is used to update the bandit and the improved archive is
used to create a new DDE and reconstructive crossover operator.
#### Data Driven Encoding
The MAP-Elites archive is a record of the highest-performing solutions yet
found in each bin. When the archive is updated the VAE is trained to
reconstruct the individuals in the archive. Reconstruction is a mapping from
one phenotype to another, mediated through latent space; and the mapping from
latent space to phenotype space analogous to a genotype-phenotype mapping,
which we refer to as a Data-Driven Encoding (DDE).
Features common in high performing solutions will be the most successfully
compressed and reconstructed — and features widely shared by high performing
solutions are likely to lead to high performance. Critically, by training the
encoding only on high-performing solutions we bias the space of solutions the
DDE can express to those with high performance.
#### Reconstructive Crossover
By limiting the range of solutions which can be expressed by a representation,
we are able to bias the solutions found during search. When a solution is
reconstructed with the VAE it is mapped onto the restricted space of solutions
expressible by the DDE — a space characterized by high performance.
Reconstructing individuals with the VAE can create new solutions with higher
fitness than the originals, but cannot create novel solutions. Solutions
created by the DDE are based on those already in the archive, so cannot reach
solutions which lie outside of the encoded distribution. At early stages of
optimization when there are few example solutions, using only reconstruction
to create new solutions would doom our encoding to a small region of
expression.
Rather than completely replacing individuals with their reconstructions we
instead shift them closer to forms expressible by the DDE with a new variation
operator, reconstructive crossover. Child solutions are created by performing
crossover with two parents: a parent chosen from the archive and its
reconstruction. Crossover takes the form of an element-wise mean of the
parameter vectors.
(2)
$\mathbf{x}_{i}^{(t+1)}=\frac{1}{2}*(\mathbf{x}_{i}^{(t)}+VAE.Decode(VAE.Encode(\mathbf{x}_{i}^{(t)})))$
The reconstructive crossover operator slows the loss of diversity by only
moving an individual toward the distribution of solutions encoded by the DDE,
not directly into it. By only shifting solutions rather than replacing them,
we allow exploration outside of the distribution to continue. Even when there
is little gain in fitness, solutions that are the result of reconstructive
crossover have a lower inherent dimensionality, on the account of having
parents pass through the compressive bottleneck of the VAE. In this way the
reconstructive crossover operator not only spreads globally advantageous genes
throughout the archive, but also pulls the archive towards more easily
compressed solutions.
#### Line Mutation
Reconstructive crossover enables effective optimization within the range of
solutions that the DDE can express, but explorative operators are required to
widen the pool of example solutions and improve the DDE. So when creating new
solutions we choose to either produce them through reconstructive crossover,
or through random mutation.
In addition to isometric Gaussian mutation commonly used in MAP-Elites, we
apply the line mutation operator proposed in (Vassiliades and Mouret, 2018).
Line mutation imposes a directional component on the Gaussian perturbations.
During mutation the parent genome is compared to a random genome from the
archive. The variance of mutation in each dimension is then scaled by the
difference in each gene:
(3)
$\mathbf{x}_{i}^{(t+1)}=\mathbf{x}_{i}^{(t)}+\sigma_{1}\mathcal{N}(0,\mathbf{I})+\sigma_{2}\left(\mathbf{x}_{j}^{(t)}-\mathbf{x}_{i}^{(t)}\right)\mathcal{N}(0,1)$
where $\sigma_{1}$ and $\sigma_{2}$ are hyperparameters which define the
relative strength of the isometric and directional mutations. Intuitively,
when two genes have similar values the spread of mutation will be small, when
the values are very different the spread will be large.
In many cases certain parameter values will be correlated to high fitness,
regardless of the individual’s place in behavior space. The line operator is a
simple way of exploiting this similarity, but in contrast to reconstructive
crossover does not limit expressivity – allowing it to be used as a method of
exploring new solutions. Though both the reconstructive crossover and line
mutation operators take advantage of the similarities between high performing
individuals, their differing approaches allow them to be effectively combined
as explorative and exploitative operators.
#### Parameter Control
DDE-Elites explores the space of representations with the exploitative
operator of reconstructive crossover, which finds high performing solutions
similar to those already encoded by the DDE, and explorative operators of
mutation, which expand the space of solutions beyond the range of the DDE.
The optimal ratio to use these operators is not only domain dependent, but
dependent on the stage of the algorithm. When the archive is nearly empty, it
makes little sense to base a representation on a few randomly initialized
solutions; once the behavior space has been explored, it is beneficial to
continue optimization through the lens of the DDE; and when the archive is
full of solutions produced by the DDE it is more useful to expand the range of
possible solutions with mutation. These stages are neither predictable nor
clear cut, complicating the decision of when to use each operator.
Faced with a trade-off between exploration and exploitation we frame the
choice of operators as a multi-armed bandit problem (Auer et al., 2002).
Multi-armed bandits imagine sets of actions as levers on a slot machine, each
with their own probability of reward. The goal of a bandit algorithm is to
balance exploration, trying new actions, and exploitation, repeating actions
that yield good rewards. Bandit approaches are straightforward to implement
and have been previously used successfully to select genetic operators
(DaCosta et al., 2008).
We define a set of possible actions as usage ratios between reconstructive
crossover, line mutation, and isometric mutation. The ratio of
$[\frac{1}{4},\frac{3}{4},0]$, for example, would have solutions created by
reconstructive crossover with a probability of $\frac{1}{4}$, line mutation
with a probability of $\frac{3}{4}$, and never with isometric mutation. Each
action is used to create a batch of child solutions and a reward is assigned
in proportion to the number of children who earned a place in the archive. At
each generation a new action is chosen, and the reward earned for that action
recorded.
Actions are chosen based on UCB1 (Auer et al., 2002), a simple and effective
bandit algorithm which minimizes regret. Actions with the greatest potential
reward are chosen, calculated as:
(4) $Q(a)+\sqrt{(2\log t)/(N_{t}(a))}$
where $Q(a)$ is the reward for an action $a$, $t$ is the total number of
actions that have been performed, and $N_{t}(a)$ the number of times that
action has been performed. UCB1 is an optimistic algorithm which rewards
uncertainty — given two actions with the same mean reward, the action which
has been tried fewer times will be chosen.
Our archive is in constant flux, and so the true reward of each mix of
operators changes from generation to generation. To handle the non-stationary
nature of the problem we use a sliding window (Garivier and Moulines, 2011),
basing our predictions only on the most recent generations.
Algorithm 2 DDE-Elites
1:function DDE-Elites($fitness()$ $\mathcal{X}_{initial}$)
2: $\mathcal{X}\leftarrow\mathcal{X}_{initial}$
3: $\mathcal{V}$: Possible Variation Operator Probabilities (vector)
4: (e.g., [0,0.5,0.5], [0.8,0.0,0.2], [1.0,0.0,0.0] for [xover,line,iso])
5: successes $\leftarrow zeros(len(\mathcal{V}))$ $\triangleright$ # successes
for each option
6: selection $\leftarrow zeros(len(\mathcal{V}))$ $\triangleright$ #
selections for each option
7: for iter = $1\to I$ do
8: — Train VAE on Current Archive —
9: VAE.Train ($\mathcal{X}$)
10: — Choose Variation Based on UCB1 —
11: $i\leftarrow\arg\max\left(\frac{\text{successes}[s]}{\text{ selected
}[s]}+\sqrt{\frac{2\ln(\text{sum}(\text{successes}))}{\text{selected}[s]}}\right)$
12: — Run MAP-Elites Using Chosen Variation —
13: $variation()\leftarrow\mathcal{V}[i]$
14: $\mathcal{X^{\prime}}\leftarrow$MAP-
Elites$(fitness(),variation(),\mathcal{X}$)
15: — Track Performance of Chosen Variation —
16: $selection[i]\leftarrow selection[i]+1$
17: $successes[i]\leftarrow
successes[i]+nImproved(\mathcal{X^{\prime}},\mathcal{X})$
18: end for
19: DDE $\leftarrow$ VAE.Decode()
20: return $\mathcal{~{}X}$, DDE
21:end function
1:function Isometric Mutation($\mathcal{X}$)
2: $\mathbf{x~{}}\leftarrow random\\_selection(\mathcal{X})$
3: return $\mathbf{x}+\sigma\mathcal{N}(0,\mathbf{I})$
4:end function
1:function Line Mutation($\mathcal{X}$)
2: $\mathbf{x~{},y~{}}\leftarrow random\\_selection(\mathcal{X})$
3: return
$\mathbf{x}+\sigma_{1}\mathcal{N}(0,\mathbf{I})+\sigma_{2}(\mathbf{x}-\mathbf{y})\mathcal{N}(0,1)$
4:end function
1:function Reconstructive Crossover($\mathcal{X}$)
2: $\mathbf{x~{}}\leftarrow random\\_selection(\mathcal{X})$
3: $\mathbf{y~{}}\leftarrow VAE.Decode(VAE.Encode(\mathbf{x}))$
$\triangleright$ VAE Reconstruction
4: return $(\mathbf{x}+\mathbf{y})/2$
5:end function
Figure 3. Archive Illumination
Archive illumination performance of MAP-Elites with different variation
operators: standard isometric mutation (MAP-Elites), line mutation (ME-Line),
reconstructive crossover (DDE-XOver) and DDE-Elites, which uses the UCB1
bandit algorithm to choose between the three at every generation. We measure
fitness as the mean fitness of all solutions in the archive; coverage as the
fraction of behavior space bins which contain solutions. Results over 20
replicates with lines indicating medians and quartile bounds shaded. The
median of DDE-Elites, our approach, is additionally noted with black dots. All
final results are significantly different ($p<0.01$ Mann-Whitney U) in fitness
and coverage. Progress is shown in evaluations (0 to 1 million); a batch size
of 100 evaluations per generation was used, so this scale corresponds to
generations from 0 to 10,000.
## 4\. Experiments
#### Planar Arm Inverse Kinematics
111see Figure 5 for a visualization of this domain
We demonstrate the effectiveness of DDEs and DDE-Elites on in the inverse
kinematics (IK) problem of a 2D robot arm, a common QD benchmark problem
(Cully and Demiris, 2018; Vassiliades and Mouret, 2018). Given target
coordinates a configuration of joint angles should be found to place the end
effector at the target. To solve this task, a discretized behavior space is
defined over the x,y plane and MAP-Elites finds a configuration of joint
angles which places the end effector in each bin. The location of the end
effector is derived for an arm with $n$ joints with angles $y$ with using the
forward kinematics equation:
$\mathbf{b}(\mathbf{y})=\left[\begin{array}[]{c}{l_{1}\cos(y_{1})+l_{2}\cos(y_{1}+y_{2})+\cdots+l_{n}\cos(y_{1}+\cdots+y_{n})}\\\
{l_{1}\sin(y_{1})+l_{2}\sin(y_{1}+y_{2})+\cdots+l_{n}\sin(y_{1}+\cdots+y_{n})}\end{array}\right]$
There are many solutions to this IK problem, but solutions with lower joint
variance are preferred to allow for smoother transitions between
configurations. We define fitness as the negative joint variance:
$-\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\mu)^{2}$ where $(\mu=\sum_{i=1}^{n}y_{i})$.
To summarize: the phenotype is the angle of each joint, the behavior is the
x,y coordinates of the end effector, and the fitness the negative variance of
the joint angles. The difficulty of the problem can be easily scaled up by
increasing the number of joints in the arm: we solve this task with 20, 200,
and 1000 joints. When a DDE is used 10 latent dimensions are used for the 20D
arm, and 32 dimensions for the 200 and 1000D arms. The same archive structure
is used for all domains. A unit circle is divided into 1950 bins, with each
bin defined by the Voronoi cell (Vassiliades et al., 2017) with centers placed
in a ring formation222See supplementary material for a visualization of this
structure.
Figure 4. Archive Recreation with Data-Driven Encoding
Performance of MAP-Elites algorithm when run with direct or data-driven
encoding. When using the direct encoding, MAP-Elites was given one order of
magnitude more evaluations (note logarithmic scale of evaluations). Fitness is
measured as the mean fitness of all solutions in the archive, coverage as the
fraction of behavior space bins which contain solutions. Results over 50
replicates with dotted lines indicating medians and quartile bounds shaded.
### 4.1. Archive Illumination
We first demonstrate the ability of DDE-Elites to scale up illumination to
high-dimensional problems. The performance of DDE-Elites is compared to three
algorithmic variants: the canonical MAP-Elites algorithm using isometric
mutation (MAP-Elites); MAP-Elites using line, or directional, mutation (ME-
Line); and MAP-Elites using the reconstructive crossover (DDE-XOver). Our
proposed approach DDE-Elites uses all operators at a ratio determined by the
UCB1 bandit algorithm. These treatments are summarized in Table 1.
| | Isometric
---
Mutation
| Line
---
Mutation
| Reconstructive
---
Crossover
MAP-Elites | X | |
ME-Line | | X |
DDE-XOver | | | X
DDE-Elites | X | X | X
Table 1. Algorithm variants. DDE-Elites is our approach.
These variants are compared based on the quality of the archive at each
generation (Figure 3). Archives are judged based on two metrics: (1) coverage,
the number of bins filled, and (2) performance, the mean fitness of
solutions.333Sixty-four core machines were used to evaluate 100 individuals in
parallel, requiring $\sim$0.2s, $\sim$0.8s, $\sim$1.6s, for the arm at 20d,
200d, and 1000D arm respectively. In every case the VAE required $\sim$2.4s to
train on a single CPU core.
In the 20-dimensional case ME-Line quickly fills the map with high performing
solutions. In only a one hundred thousand evaluations ME-Line creates an
archive unmatched by MAP-Elites even after one million evaluations. When only
the reconstructive crossover operator is used, despite promising early
progress, a chronic lack of exploration results in archives which are worse
than the standard MAP-Elites. DDE-Elites, with access to all operators,
explores as quickly as ME-Line and creates archives of similar quality.
When the dimensionality of the arm is scaled up to 200D, we see the
convergence rate of ME-Line slow down considerably. While still reaching high
levels of performance it does so only after one million evaluations, a tenth
of the evaluations required in in the 20D case — suggesting that the
effectiveness of ME-Line scales linearly with the dimensionality of the
problem. In contrast DDE-Elites is barely affected by a ten-fold increase in
parameters — exploration is only slightly slowed, and high-performing
solutions are found from the very earliest iterations. The effects of scaling
can be observed even more clearly in the 1000D case: ME-Line illuminates the
archive only very slowly, while the performance of DDE-Elites is marked by the
same burst of exploration and consistently high fitness solutions that
characterized its performance in lower dimensions.
The line mutation operator is clearly able to leverage the similarities in
high performing solutions across the archive — in every case performing far
better than the isometric mutation operator. The mechanism for doing this,
adjusting the range of parameter mutations, does not appear to scale well
enough to handle very high dimensional problems. The reconstructive crossover
operator is able to rapidly find high-performing solutions even in high-
dimensional spaces, but is poor at exploring. Search with reconstructive
crossover is confined to the distribution of genes that already exist in the
archive, if used exclusively that distribution of genes is limited to the
initial population. By combining these operators — expanding the range of
genes in the archive with mutation, and spreading high performing genes with
reconstructive crossover — DDE-Elites is able to create high-performing
archives even in high-dimensional problems.
Figure 5. Optimization with Direct and Data-Driven Encodings
CMA-ES is given a set budget to find a solution with a target behavior, and
searches with either a direct encoding or a DDE.
Left: Example solutions for target matching with the direct and data driven
encodings. End effectors in yellow, targets in red.
Top: Optimization over time of median distance (dotted line) to the 18 targets
over 50 replicates (quartiles shaded).
Bottom: The final distance to the targets, and a characteristic of the
solution. These characteristics were not optimized by CMA-ES, but optimized
during the creation of the DDE, biasing the solutions produced.
### 4.2. Archive Recreation
DDE-Elites is as much a method of optimizing representations as solutions. By
learning a representation from the archive, we create an encoding that is
biased towards high performance and has a range of expression matching the
defined behavior space. In these experiments, our DDE encodes smooth joint
configurations which place an arm’s end effector anywhere in its reach. To
demonstrate that DDE-Elites does more than guide search, but learns a
representation, we search the space again, using the found DDE in place of the
direct encoding.
We run the standard MAP-Elites algorithm, with isometric mutation only, using
a learned DDE444The decoder network of the VAE found in the highest coverage
replicate of DDE-Elites. acting as our genome. In the 20D arm this DDE has 10
parameters, in the 200D and 1000D arms the DDE has 32 parameters. No previous
solutions are maintained, only the trained DDE. For reference we compare to
the MAP-Elites algorithm using the direct encoding. An order of magnitude
fewer evaluations were budgeted when using the DDE.
In every case the DDE far outperforms the direct encoding, reaching the same
levels of fitness and coverage with several orders of magnitude fewer
evaluations (Figure 4). The DDE can express the same range of solutions as
were found in the original archive, and finds them rapidly. Archives were
recreated after only 10,000 evaluations — a rate of about 5 evaluations per
bin.55510,000 individuals/1950 bins $\approx$ 5 evaluations/bin discovered.
The found solutions are also high performing.
Such improvement cannot be explained away by the decrease in dimensionality of
the search. In both low and high dimensional cases the bias toward high
performance is also apparent: the mean fitness curve is nearly flat at the
optima, indicating that when new solutions are added to the map they are
already near optimal. The contrast with the direct encoding is stark, with the
direct encoding considerable effort is taken to search for good solutions, the
DDE finds little else. DDE-Elites not only produces solutions, but learns
domain-specific representation.
### 4.3. Optimization with Learned Encodings
Beyond its place in the DDE-Elites optimization loop, the produced DDE is a
powerful representation with high expressivity and built in biases. Though
created by MAP-Elites, the DDE is not tied to it. Once discovered, a DDE can
be used as a representation for any black box optimization algorithm.
We illustrate this generality by using again solving the arm inverse
kinematics problem with the black-box optimizer CMA-ES (Hansen and Ostermeier,
2001). A set of target positions for the end effector is defined (Figure 5,
left), and CMA-ES used to find a joint configuration which reaches each
target. In one case optimization is performed using the DDE; in the other the
direct encoding is used.
When optimizing with the DDE, CMA-ES quickly finds solutions to the target
hitting problems with a precision never matched with the direct encoding
(Figure 5, top). Moreover, a bias for how the problem is solved is built into
the representation (Figure 5, bottom). As the DDE was trained only on
solutions with low joint variance, this same property is found in the
solutions found by CMA-ES with the DDE — even without searching for them. With
the DDE CMA-ES not only finds solutions to the IK problem, the built-in priors
of the DDE ensures we find kind of solutions we want.
## 5\. Discussion
Learning representations by combining quality diversity (here, MAP-Elites) and
generative models (here, a VAE) opens promising research avenues for domains
in which optimizations of the same cost function are launched continuously.
This is, for example, the case of Model Predictive Control (Mayne et al.,
2000), in which the sequence of actions for the next seconds is optimized at
every time-step of the control loop, or the case of shape optimization in
interactive design tools (Hoyer et al., 2019; Bendsøe and Sigmund, 1995), in
which each modification by the user requires a novel optimization.
In preliminary experiments, we searched for an encoding to describe action
sequences for a walking robot. The results show that using MAP-Elites to
generate a diversity of sequences, then using a VAE to learn a representation
leads to an encoding that can accelerate future optimizations by several
orders of magnitude. Nevertheless, using the representation during
optimization, as described in this paper, did not accelerate the quality
diversity optimization as much as in the high-dimensional arm used here. One
hypothesis is that the regularities in action sequences are harder to
recognize than in the arm experiments, especially at the beginning of the
process. For instance, it might help to use an auto-encoder that is especially
designed for sequences (Vaswani et al., 2017; Co-Reyes et al., 2018).
For other tasks, appropriate generative models could be explored, for example
convolutional models for tasks with spatial correlations (Salimans et al.,
2015). In addition, though the latent spaces created by VAEs are easier to
navigate than those created by normal autoencoders, even better models offer
the opportunities for further improvements. Much work has been done to create
VAEs which have even better organized latent spaces (Higgins et al., 2017;
Burgess et al., 2018; Chen et al., 2018; Kim and Mnih, 2018), ideally with
each dimension responsible for a single phenotypic feature such as the
lighting or color of an image.
A second research avenue is to improve the bandit algorithm used to balance
between operators. In theory, it should ensure that adding new operators can
only aid optimization, since useless or detrimental operators would rarely be
selected. However, we observed that it is not always effective: in some cases,
using only the line mutation outperformed DDE-Elites, whereas DDE-Elites could
revert to using only line mutation with a perfect bandit. Our hypothesis is
that this is a sign that “successes” — child solutions which discover new bins
or improve on existing solutions — is not the perfect measure of utility for a
QD algorithm. In the case of our experiments, it may be that reconstructive
crossover consistently improves solutions, but may only do so slightly.
According to the “success” metric, a tiny improvement is worth the same as a
large one. To best utilize the bandit, other methods of judging performance in
QD algorithms should be explored.
Beyond performance advantages, for both the current and future optimizations,
these “disentangled” representations offer even more interesting
opportunities. Reducing the dimensionality of the search space into meaningful
components would allow rapid model-based optimization of single solutions
(Shahriari et al., 2015), or entire archives (Gaier et al., 2018). Engineers
could interactively explore and understand such encodings, laying bare the
underlying properties responsible for performance and variation — and so from
encodings receive, rather than provide, insight and domain knowledge.
## Acknowledgements
This work received funding from the European Research Council (ERC) under the
EU Horizon 2020 research and innovation programme (grant agreement number
637972, project ”ResiBots”) and the German Federal Ministry of Education and
Research (BMBF) under the Forschung an Fachhochschulen mit Unternehmen
programme (grant agreement number 03FH012PX5, project ”Aeromat”).
## Source Code
The source code used to produce the results in this paper is available at
https://github.com/resibots/2020_gaier_gecco
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## Supplemental Material
### A.. Example Maps
Arm20 | Arm200 | Arm1000
---|---|---
| |
Table 2. Example Maps
Final archives colored by fitness value for each cell for each domain. In the
20D Arm both MAP-Elites and DDE-Elites converge on similar optimal solutions.
In the 200D and 1000D Arm MAP-Elites is unable to reach the levels of
performance of DDE-Elites in any region.
### B.. Hyperparameters of DDE Experiments
Hyperparameter | Value
---|---
Isometric Mutation Strength | 0.003
Line Mutation Strength | 0.1
Batch Size | 100
Bandit Options, | | [0.00:0.00:1.00], [0.25:0.00:0.75],
---
[0.50:0.00:0.50], [0.75:0.00:0.25],
[1.00:0.00:0.00], [0.00:0.25:0.75],
[0.00:0.50:0.50], [0.00:0.75:0.25],
[0.00:1.00:0.00]
Bandit Window Length | 1000
Generations per VAE Training | 1
Epochs per VAE Training | 5
Mutation Strength when Searching DDE | 0.15
Latent Vector Length [Arm20] | 10
Latent Vector Length [Arm200] | 32
Latent Vector Length [Arm1000] | 32
|
2024-09-04T02:54:58.573582 | 2020-03-10T00:10:22 | 2003.04470 | {
"authors": "V.M. Ngo, N.A. Le-Khac, and M.T. Kechadi",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26127",
"submitter": "Vuong M. Ngo",
"url": "https://arxiv.org/abs/2003.04470"
} | arxiv-papers | Int. J. Business Process Integration and Management
Ngo, V.M., Le-Khac, N.A. and Kechadi M.T. Int. J. Business Process Integration
and Management 10 1 2020
Vuong M. Ngo E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Nhien-An Le-Khac E-mail<EMAIL_ADDRESS>
M-Tahar Kechadi E-mail<EMAIL_ADDRESS>
Ho Chi Minh City Open University, HCMC, Vietnam
University College Dublin, Belfield, Dublin 4, Ireland
# Data Warehouse and Decision Support on
Integrated Crop Big Data
###### Abstract
In recent years, precision agriculture is becoming very popular. The
introduction of modern information and communication technologies for
collecting and processing Agricultural data revolutionise the agriculture
practises. This has started a while ago (early 20th century) and it is driven
by the low cost of collecting data about everything; from information on
fields such as seed, soil, fertiliser, pest, to weather data, drones and
satellites images. Specially, the agricultural data mining today is considered
as Big Data application in terms of volume, variety, velocity and veracity.
Hence it leads to challenges in processing vast amounts of complex and diverse
information to extract useful knowledge for the farmer, agronomist, and other
businesses. It is a key foundation to establishing a crop intelligence
platform, which will enable efficient resource management and high quality
agronomy decision making and recommendations. In this paper, we designed and
implemented a continental level agricultural data warehouse (ADW). ADW is
characterised by its (1) flexible schema; (2) data integration from real
agricultural multi datasets; (3) data science and business intelligent
support; (4) high performance; (5) high storage; (6) security; (7) governance
and monitoring; (8) consistency, availability and partition tolerant; (9)
cloud compatibility. We also evaluate the performance of ADW and present some
complex queries to extract and return necessary knowledge about crop
management.
Data warehouse, decision support, crop Big Data, smart agriculture.
to this paper should be made as follows: Ngo, V.M., Le-Khac, N.A. and Kechadi,
M.T. (2020) ‘Data Warehouse and Decision Support on Integrated Crop Big Data’,
Int. J. Business Process Integration and Management, Vol. 10, No. 1, pp.
17–28.
Vuong M. Ngo received the B.E, M.E and PhD degrees in computer science at HCMC
University of Technology in 2004, 2007 and 2013 respectively. He is currently
a Senior Researcher at UCD and HCMC Open University. His research interests
include information retrieval, sentiment analysis, data mining, graph matching
and data
Nhien-An Le-Khac is currently a Lecturer at the School of Computer Science,
UCD and a Programme Director of MSc programme in forensic computing and
cybercrime investigation. He obtained his PhD in computer science in 2006 at
the Institut National Polytechnique Grenoble, France. His research interest
spans the area of cybersecurity and digital forensics, data mining/distributed
data mining for security, grid and high performance computing.
M-Tahar Kechadi was awarded PhD and Master degrees in computer science from
University of Lille 1, France. He joined the UCD School of Computer Science in
1999. He is currently Professor of Computer Science at UCD. His research
interests span the areas of data mining, data analytics, distributed data
mining, heterogeneous distributed systems, grid and cloud Computing,
cybersecurity, and digital forensics. He is a Principal Investigator at
Insight Centre for Data Analytics and CONSUS project. He is a member of IEEE
and ACM.
## 1 Introduction
Annual world cereal productions were $2,608$ million tons and $2,595$ million
tons in $2017$ and $2018$, respectively (USDA report, 2018; FAO-CSDB report,
2018). However, there were also around $124$ million people in $51$ countries
faced food crisis and food insecurity (FAO-FSIN report, 2018). According to
United Nations (UN document, 2017), we need an increase $60\%$ of cereal
production to meet $9.8$ billion people needs by $2050$. To satisfy the huge
increase demand for food, crop yields must be significantly increased using
modern farming approaches, such as smart farming also called precision
agriculture. As highlighted in the European Commission report (EC report,
2016), precision agriculture is vitally important for the future and can make
a significant contribution to food security and safety.
The precision agriculture’s current mission is to use the decision-support
system (DSS) based on Big Data approaches to provide precise information for
more control of waste and farming efficiency, such as soil nutrient (Rogovska
and et al., 2019), early warning (Rembold and et al., 2019), forecasting
(Bendre and et al., 2015), irrigation systems (Huang and et al., 2013),
evapotranspiration prediction (Paredes and et al., 2014), soil and herbicide,
insecticide optimisation (Ngo and Kechadi, 2020), awareness (Lokers and et
al., 2016), supply chain (Protopop and Shanoyan, 2016) and financial services
(Ruan and et al., 2019). Normally, the DSSs implement a knowledge discovery
process also called data mining process, which consists of data collection and
data modelling, data warehousing, data analysis (using machine learning or
statistical techniques), and knowledge deployment (Dicks and et al., 2014).
Hence, designing and implementing an efficient agricultural data warehouse
(ADW) is one of the key steps of this process, as it defines a uniform data
representation through its schema model and stores the derived datasets so
that they can be analysed to extract useful knowledge. However, currently,
this step was not given much attention. Therefore, there are very few reports
in the literature that focus on the design of efficient ADWs with the view to
enable Agricultural Big Data analytics and mining. The design of large scale
ADWs is very challenging. Because, the agricultural data is spatial, temporal,
complex, heterogeneous, non-standardised, high dimensional, collected from
multi-sources, and very large. In particular, it has all the features of Big
Data; volume, variety, velocity and veracity. Moreover, the precision
agriculture system can be used by different kinds of users at the same time,
for instance by farmers, policymakers, agronomists, and so on. Every type of
user needs to analyse different information, sets thus requiring specific
analytics.
Unlike in any other domains; health-care, financial data, etc, the data and
its warehousing in precision agriculture are unique. This is because, there
are very complex relationships between the agricultural data dimensions. The
data sources are very diversified and varying levels of quality. Precision
agriculture (PA) warehousing has many decision-making processes and each needs
different levels of data access and different needs of analysis. Finally,
there are many stakeholders involved in the data ownership and exploitation.
So, the data has significant number of uncertainties. For examples, the
quality of data collected by farmers depends directly on their knowledge,
routines and frequency of information recording, and support tools, etc. All
these issues make the PA data unique when it becomes to its storage, access,
and analysis. These issues may exist in other domains, but not at the same
scale and as in agriculture practices.
In this research, we firstly analyse real-world agricultural Big Data to build
the effective constellation schema. From this schema, some simple questions
can be easily answered directly from the modelled data. These questions
include: (1) For a given field, what kind of crops are suitable to grow? (2)
Which companies can purchase a specific crop with the highest price in the
past season? (3) List the history of soil texture and applied fertilisers for
a given field; (4) List costs of production for wheat and barley in the last 5
years, and so on. Secondly, the proposed ADW has enough main features and
characteristics of Big Data Warehouse (BDW). These are (1) high storage
capacity, high performance and cloud computing compatibility; (2) flexible
schema and integrated storage structure; (3) data ingestion, monitoring, and
security to deal with the data veracity. Besides, an experimental evaluation
is conducted to study the performance of ADW storage.
The rest of this paper is organised as follows: in the next Section, we
reviewed the related work about decision support systems and data warehouses
in agriculture. In Sections 3, 4 and 5, we presented big data aspects of PA,
our ADW architecture and its modules. In Sections 6, 7, 8 and 9, the quality
criteria, implementation, performance analysis and decision-making
applications of the proposed ADW are presented respectively. Section 10 gives
some concluding remarks and future research directions. Finally, a concrete
example about the ADW and its operational average run-times are shown in the
appendix.
## 2 Related Work
In precision agriculture, DSSs are designed to support different stakeholders
such as farmers, advisers and policymakers to optimise resources, farms’
management and improve business practices (Gutierreza and et al., 2019). For
instance, DSSs were built to 1) manage microbial pollution risks in dairy
farming (Oliver and et al., 2017); 2) analyse nitrogen fertilisation from
satellite images (Lundstrom and Lindblom, 2018); 3) control pest and disease
under uncertainty in climate conditions (Devitt and et al., 2017); 4) manage
drip irrigation and its schedule (Friedman and et al., 2016); 5) predict and
adopt climate risks (Han and et al., 2017). However, the datasets that were
used in the mentioned studies are small. Besides, they focused on using
visualisation techniques to assist end-users understand and interpret their
data.
Recently, many papers have been published on how to exploit intelligent
algorithms on sensor data to improve agricultural economics Pantazi (2016),
Park and et al. (2016), Hafezalkotob and et al. (2018), Udiasa and et al.
(2018) and Rupnik and et al. (2019). In Pantazi (2016), the authors predicted
crop yield by using self-organising-maps; namely supervised Kohonen networks,
counter-propagation artificial networks and XY-fusion. In Park and et al.
(2016), one predicted drought conditions by using three rule-based machine
learning; namely random forest, boosted regression trees, and Cubist. To
select the best olive harvesting machine, the authors in Hafezalkotob and et
al. (2018) applied the target-based techniques on the main criteria, which are
cost, vibration, efficiency, suitability, damage, automation, work capacity,
ergonomics, and safety. To provide optimal management of nutrients and water,
the paper Udiasa and et al. (2018) exploited the multi-objective genetic
algorithm to implement an E-Water system. This system enhanced food crop
production at river basin level. Finally, in Rupnik and et al. (2019) the
authors predicted pest population dynamics by using time series clustering and
structural change detection which detected groups of different pest species.
However, the proposed solutions are not scalable enough to handle agricultural
Big Data; they present weaknesses in one of the following aspects: data
integration, data schema, storage capacity, security and performance.
From a Big Data point of view, the papers Kamilaris and et al. (2018) and
Schnase and et al. (2017) have proposed “smart agricultural frameworks”. In
Kamilaris and et al. (2018), the authors used Hive to store and analyse sensor
data about land, water and biodiversity which can help increase food
production with less environmental impact. In Schnase and et al. (2017), the
authors moved toward a notion of climate analytics-as-a-service, by building a
high-performance analytics and scalable data management platform, which is
based on modern cloud infrastructures, such as Amazon web services, Hadoop,
and Cloudera. However, the two papers did not discuss how to build and
implement a DW for a precision agriculture.
The proposed approach, inspired from Schulze and et al. (2007), Schuetz and et
al. (2018), Nilakanta and et al. (2008) and Ngo and et al. (2018), introduces
ways of building agricultural data warehouse (ADW). In Schulze and et al.
(2007), the authors extended entity-relationship concept to model operational
and analytical data; called multi-dimensional entity-relationship model. They
also introduced new representation elements and showed how can be extended to
an analytical schema. In Schuetz and et al. (2018), a relational database and
an RDF triple store were proposed to model the overall datasets. The data is
loaded into the DW in RDF format, and cached in the RDF triple store before
being transformed into relational format. The actual data used for analysis
was contained in the relational database. However, as the schemas used in
Schulze and et al. (2007) and Schuetz and et al. (2018) were based on entity-
relationship models, they cannot deal with high-performance, which is the key
feature of a data warehouse.
In Nilakanta and et al. (2008), a star schema model was used. All data marts
created by the star schemas are connected via some common dimension tables.
However, a star schema is not enough to present complex agricultural
information and it is difficult to create new data marts for data analytics.
The number of dimensions of the DW proposed in Nilakanta and et al. (2008) is
very small; only 3-dimensions – Species, Location, and Time. Moreover, the DW
concerns livestock farming. Overcoming disadvantages of the star schema, the
authors of Ngo and et al. (2018) and Ngo and Kechadi (2020) proposed a
constellation schema for an agricultural DW architecture in order to satisfy
the quality criteria. However, they did not describe how to design and
implement their DW.
## 3 Crop Big Data
### 3.1 Crop Datasets
The datasets were primarily obtained from an agronomy company, which extracted
it from them operational data storage systems, research results, and field
trials. Especially, we were given real-world agricultural datasets on iFarms,
Business-to-Business (B2B) sites, technology centres and demonstration farms.
Theses datasets were collected from several European countries and they are
presented in Figures 1 and 2 (Origin report, 2018). These datasets describe
more than $112$ distribution points, $73$ demonstration farms, $32$
formulation and processing facilities, $12.7$ million hectares of direct farm
customer footprint and $60,000$ trial units.
Figure 1: Data from UK and Ireland. Figure 2: Data in Continental Europe.
There is a total of 29 datasets. On average, each dataset contains $18$ tables
and is about $1.4$ GB in size. Each dataset focuses on a few information that
impact the crop. For instance, the weather dataset includes information on
location of weather stations, temperature, rainfall and wind speed over time.
Meanwhile, soil component information in farm sites, such as mineral, organic
matter, air, water and micro-organisms, were stored in the soil dataset. The
fertiliser dataset contains information about field area and geographic
position, crop name, crop yield, season, fertiliser name and quantity.
### 3.2 Big Data Challenges
Raw and semi-processed agricultural datasets are usually collected through
various sources: Internet of Thing (IoT) devices, sensors, satellites, weather
stations, robots, farm equipment, farmers and agronomists, etc. Besides,
agricultural datasets are very large, complex, unstructured, heterogeneous,
non-standardised, and inconsistent. Hence, it has all the features of Big
Data.
1. 1.
Volume: The amount of agricultural data is increasing rapidly and is
intensively produced by endogenous and exogenous sources. The endogenous data
is collected from operational systems, experimental results, sensors, weather
stations, satellites, and farming equipment. The systems and devices in the
agricultural ecosystem can be connected through IoT. The exogenous data
concerns the external sources, such as government agencies, retail
agronomists, and seed companies. They can help with information about local
pest and disease outbreak tracking, crop monitoring, food security, products,
prices, and knowledge.
2. 2.
Variety: Agricultural data has many different forms and formats, structured
and unstructured data, video, imagery, chart, metrics, geo-spatial, multi-
media, model, equation, text, etc.
3. 3.
Velocity: The collected data increases at very high rate, as sensing and
mobile devices are becoming more efficient and cheaper. The datasets must be
cleaned, aggregated and harmonised in real-time.
4. 4.
Veracity: The tendency of agronomic data is uncertain, inconsistent, ambiguous
and error prone because the data is gathered from heterogeneous sources,
sensors and manual processes.
### 3.3 ADW Schema
Figure 3: A part of ADW schema for Precision Agriculture
The DW uses schema to logically describe the entire datasets. A schema is a
collection of objects, including tables, views, indexes, and synonyms which
consist of some fact and dimension tables (Oracle document, 2017). The DW
schema can be designed based on the model of source data and the user
requirements. There are three kind of models, namely star, snowflake and fact
constellation. With the its various uses, the ADW schema needs to have more
than one fact table and should be flexible. So, the constellation schema, also
known galaxy schema should be used to design the ADW schema.
Figure 4: Field and Crop dimension tables Figure 5: Soil and Pest dimension
tables
We developed a constellation schema for ADW and it is partially described in
Figure 3. It includes few fact tables and many dimension tables. FieldFact
fact table contains data about agricultural operations on fields. Order and
Sale fact tables contain data about farmers’ trading operations. The key
dimension tables are connected to their fact table. There are some dimension
tables connected to more than one fact table, such as Crop and Farmer.
Besides, CropState, Inspection, Site, and Weather Reading dimension tables are
not connected to any fact table. CropState and Inspection tables are used to
support Crop table. While, Site and Weather Reading tables support Field and
WeatherStation tables. FieldFact fact table saves the most important facts
about teh field; yield, water volume, fertiliser quantity, nutrient quantity,
spray quantity and pest number. While, in Order and Sale tables, the important
facts needed by farm management are quantity and price.
Table 1: Descriptions of other dimension tables No. | Dim. tables | Particular attributes
---|---|---
1 | Business | BusinessID, Name, Address, Phone, Mobile, Email
2 | CropState | CropStateID, CropID, StageScale, Height, MajorStage, MinStage, MaxStage, Diameter, MinHeight, MaxHeight, CropCoveragePercent
3 | Farmer | FarmerID, Name, Address, Phone, Mobile, Email
4 | Fertiliser | FertiliserID, Name, Unit, Status, Description, GroupName
5 | Inspection | InspectionID, CropID, Description, ProblemType, Severity, ProblemNotes, AreaValue, AreaUnit, Order, Date, Notes, GrowthStage
6 | Nutrient | NutrientID, NutrientName, Date, Quantity
7 | Operation Time | OperationTimeID, StartDate, EndDate, Season
8 | Plan | PlanID, PName, RegisNo, ProductName, ProductRate, Date, WaterVolume
9 | Product | ProductID, ProductName, GroupName
10 | Site | SiteID, FarmerID, SiteName, Reference, Country, Address, GPS, CreatedBy
11 | Spray | SprayID, SprayProductName, ProductRate, Area,Date, WaterVol, ConfDuration, ConfWindSPeed, ConfDirection, ConfHumidity, ConfTemp, ActivityType
12 | Supplier | SupplierID, Name, ContactName, Address, Phone, Mobile, Email
13 | Task | TaskID, Desc, Status, TaskDate, TaskInterval, CompDate, AppCode
14 | Trans Time | TransTimeID, OrderDate, DeliverDate, ReceivedDate, Season
15 | Treatment | TreatmentID, TreatmentName, FormType, LotCode, Rate, ApplCode, LevlNo, Type, Description, ApplDesc, TreatmentComment
16 | Weather Reading | WeatherReadingID, WeatherStationID, ReadingDate, ReadingTime, AirTemperature, Rainfall, SPLite, RelativeHumidity, WindSpeed, WindDirection, SoilTemperature, LeafWetness
17 | Weather Station | WeatherStationID, StationName, Latitude, Longitude, Region
The dimension tables contain details on each instance of an object involved in
a crop yield or farm management. Figure 4 describes attributes of Field and
Crop dimension tables. Field table contains information about name, area, co-
ordinates (being longitude and latitude of the centre point of the field),
geometric (being a collection of points to show the shape of the field) and
site identify the site that the field it belongs to. While, Crop table
contains information about name, estimated yield of the crop (estYield), BBCH
Growth Stage Index (BbchScale), harvest equipment and its weight. These
provide useful information for crop harvesting.
Figure 5 describes attributes of Soil and Pest dimension tables. Soil table
contains information about PH value (a measure of the acidity and alkalinity),
minerals (nitrogen, phosphorus, potassium, magnesium and calcium), its texture
(texture label and percentage of Silt, Clay and Sand), cation exchange
capacity (CEC) and organic matter. Besides, information about recommended
nutrient and testing dates ware also included in this table. In Pest table
contains name, type, density, coverage and detected dates of pests. For the
remaining dimension tables, their main attributes are described in Table 1.
## 4 ADW Architecture
A DW is a federated repository for all the data that an enterprise can collect
through multiple heterogeneous data sources; internal or external. The authors
in Golfarelli and Rizzi (2009) and Inmon (2005) defined DW as a collection of
methods, techniques, and tools used to conduct data analyses, make decisions
and improve information resources. DW is defined around key subjects and
involves data cleaning, data integration and data consolidations. Besides, it
must show its evolution over time and is not volatile.
The general architecture of a typical DW system includes four separate and
distinct modules; Raw Data, Extraction Transformation Loading (ETL),
Integrated Information and Data Mining (Kimball and Ross, 2013), which is
illustrated in Figure 6. In that, Raw Data (source data) module is originally
stored in various storage systems (e.g. SQL, sheets, flat files, …). The raw
data often requires cleansing, correcting noise and outliers, dealing with
missing values. Then it needs to be integrated and consolidated before loading
it into a DW storage through ETL module.
Figure 6: Agricultural Data Warehouse Architecture.
The Integrated Information module is a logically centralised repository, which
includes the DW storage, data marts, data cubes and OLAP engine. The DW
storage is organised, stored and accessed using a suitable schema defined by
the metadata. It can be either directly accessed or used to create data marts,
which is usually oriented to a particular business function or an enterprise
department. A data mart partially replicates DW storage’s contents and is a
subset of DW storage. Besides, the data is extracted in a form of data cube
before it is analysed in the data mining module. A data cube is a data
structure that allows advanced analysis of data according to multiple
dimensions that define a given problem. The data cubes are manipulated by the
OLAP engine. The DW storage, data mart and data cube are considered as
metadata, which can be applied to the data used to define other data. Finally,
Data Mining module contains a set of techniques, such as machine learning,
heuristic, and statistical methods for data analysis and knowledge extraction
at multiple level of abstraction.
## 5 ETL and OLAP
The ETL module contains Extraction, Transformation, and Loading tools that can
merge heterogeneous schemata, extract, cleanse, validate, filter, transform
and prepare the data to be loaded into a DW. The extraction operation allows
to read, retrieve raw data from multiple and different types of data sources
systems and store it in a temporary staging. During this operation, the data
goes through multiple checks – detect and correct corrupted and/or inaccurate
records, such as duplicate data, missing data, inconsistent values and wrong
values. The transformation operation structures, converts or enriches the
extracted data and presents it in a specific DW format. The loading operation
writes the transformed data into the DW storage. The ETL implementation is
complex, and consuming significant amount of time and resources. Most DW
projects usually use existing ETL tools, which are classified into two groups.
The first is a commercial and well-known group and includes tools such as
Oracle Data Integrator, SAP Data Integrator and IBM InfoSphere DataStage. The
second group is famous for it open source tools, such as Talend, Pentaho and
Apatar.
OLAP is a category of software technology that provides the insight and
understanding of data in multiple dimensions through fast, consistent,
interactive access, management and analysis of the data. By using roll-up
(consolidation), drill-down, slice-dice and pivot (rotation) operations, OLAP
performs multidimensional analysis in a wide variety of possible views of
information that provides complex calculations, trend analysis and
sophisticated data modelling quickly. The OLAP systems are divided into three
categories: 1) Relational OLAP (ROLAP), which uses relational or extended-
relational database management system to store and manage the data warehouse;
2) Multidimensional OLAP (MOLAP), which uses array-based multidimensional
storage engines for multidimensional views of data, rather than in a
relational database. It often requires pre-processing to create data cubes. 3)
Hybrid OLAP (HOLAP), which is a combination of both ROLAP and MOLAP. It uses
both relational and multidimensional techniques to inherit the higher
scalability of ROLAP and the faster computation of MOLAP.
In the context of agricultural Big Data, HOLAP is more suitable than both
ROLAP and MOLAP because: 1) ROLAP has quite slow performance and does not meet
all the users’ needs, especially when performing complex calculations; 2)
MOLAP is not capable of handling detailed data and requires all calculations
to be performed during the data cube construction; 3) HOLAP inherits
advantages of both ROLAP and MOLAP, which allow the user to store large data
volumes of detailed information and perform complex calculations within
reasonable response time.
## 6 Quality Criteria
The accuracy of data mining and analysis techniques depends on the quality of
the DW. As mentioned in Adelman and Moss (2000) and Kimball and Ross (2013),
to build an efficient ADW, the quality of the DW should meet the following
important criteria:
1. 1.
Making information easily accessible.
2. 2.
Presenting consistent information.
3. 3.
Integrating data correctly and completely.
4. 4.
Adapting to change.
5. 5.
Presenting and providing right information at the right time.
6. 6.
Being a secure bastion that protects the information assets.
7. 7.
Serving as the authoritative and trustworthy foundation for improved decision
making. The analytics tools need to provide right information at the right
time.
8. 8.
Achieving benefits, both tangible and intangible.
9. 9.
Being accepted by DW users.
The above criteria must be formulated in a form of measurements. For example,
with the 8th criterion, it needs to determine quality indicators about
benefits, such as improved fertiliser management, cost containment, risk
reduction, better or faster decision, and efficient information transaction.
In the last criterion, a user satisfaction survey should be used to find out
how a given DW satisfies its user’s expectations.
## 7 ADW Implementation
Currently, there are many popular large-scale database types that can
implement DWs. Redshift (Amazon document, 2018), Mesa (Gupta and et al.,
2016), Cassandra (Hewitt and Carpenter, 2016; Neeraj, 2015), MongoDB
(Chodorow, 2013; Hows and et al., 2015) and Hive (Du, 2018; Lam and et al.,
2016). In Ngo and et al. (2019), the authors analysed the most popular no-sql
databases, which fulfil most of the aforementioned criteria. The advantages,
disadvantages, as well as similarities and differences between Cassandra,
MongoDB and Hive were investigated carefully in the context of ADW. It was
reported that Hive is a better choice as it can be paired with MongoDB to
implement the proposed ADW for the following reasons:
1. 1.
Hive is based on Hadoop which is the most powerful cloud computing platform
for Big Data. Besides, HQL is similar to SQL which is popular for the majority
of users. Hive supports well high storage capacity, business intelligent and
data science more than MongoDB or Cassandra. These Hive features are useful to
implement ADW.
2. 2.
Hive does not have real-time performance so it needs to be combined with
MongoDB or Cassandra to improve its performance.
3. 3.
MongoDB is more suitable than Cassandra to complement Hive because: 1) MongoDB
supports joint operation, full text search, ad-hoc query and second index
which are helpful to interact with the users. Cassandra does not support these
features; 2) MongoDB has the same master – slave structure with Hive that is
easy to combine. While the structure of Cassandra is peer - to - peer; 3) Hive
and MongoDB are more reliable and consistent. So the combination of both Hive
and MongoDB adheres to the CAP theorem.
Figure 7: Agricultural Data Warehouse Implementation
The ADW implementation is illustrated in Figure 7 which contains three
modules, namely Integrated Information, Products and Raw Data. The Integrated
Information module includes two components; MongoDB and Hive. MongoDB receives
real-time data; as user data, logs, sensor data or queries from Products
module, such as web application, web portal or mobile app. Besides, some
results which need to be obtained in real-time will be transferred from the
MongoDB to Products. Hive stores the online data and sends the processed data
to MongoDB. Some kinds of queries having complex calculations will be sent
directly to Hive.
In the Raw Data module, almost data in Operational Databases or External Data
components, is loaded into Cassandra. It means that we use Cassandra to
represent raw data storage. Hence, with the diverse formats of raw data;
image, video, natural language and sql data, Cassandra is better to store them
than SQL databases. In the idle times of the system, the updated raw data in
Cassandra will be imported into Hive through the ELT tool. This improves the
performance of ETL and helps us deploy ADW on cloud or distributed systems.
## 8 Performance Analysis
The performance analysis was conducted using MySQL 5.7.22, JDK 1.8.0_171,
Hadoop 2.6.5 and Hive 2.3.3 which run on Bash, on Ubuntu 16.04.2, and on
Windows 10. All experiments were run on a desktop with an Intel Core i7 CPU
(2.40 GHz) and 16 GB memory. We only evaluate the performance of reading
operation as ADW is used for reporting and data analysis. The database of ADW
is duplicated into MySQL to compare performance. By combining popular HQL/SQL
commands, namely Where, Group by, Having, Left (right) Join, Union and Order
by, we created 10 groups for testing. Every group has 5 queries and uses one,
two or more commands (see Table 2). Moreover, every query uses operators; And,
Or, $\geq$, Like, Max, Sum and Count, to express complex queries.
Table 2: Command combinations of queries Group | Commands
---|---
$G_{1}$ | Where
$G_{2}$ | Where, Group by
$G_{3}$ | Where, Left (right) Join
$G_{4}$ | Where, Union
$G_{5}$ | Where, Order by
$G_{6}$ | Where, Left (right) Join, Order by
$G_{7}$ | Where, Group by, Having
$G_{8}$ | Where, Group by, Having, Order by
$G_{9}$ | Where, Group by, Having, Left (right) Join,
| Order by
$G_{10}$ | Where, Group by, Having, Union, Order by
$0$$10$$20$$30$$40$$50$$0$$10$$20$$30$1Queries ($q_{i}$)Different times
($Times_{q_{i}}$)Group 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group
9Group 10 Figure 8: Different times between MySQL and
ADW in runtime of every Query
All queries were executed three times and we took the average value of the
their execution timess. The difference in runtime between MySQL and ADW for a
query $q_{i}$ is calculated as
$Times_{q_{i}}=RT^{mysql}_{q_{i}}/RT^{ADW}_{q_{i}}$. Where,
$RT^{mysql}_{q_{i}}$ and $RT^{ADW}_{q_{i}}$ are average runtimes of query
$q_{i}$ on MySQL and ADW, respectively. Moreover, with each group $G_{i}$, the
difference in runtime between MySQL and ADW is
$Times_{G_{i}}=RT^{mysql}_{G_{i}}/RT^{ADW}_{G_{i}}$. Where,
$RT_{G_{i}}=Average(RT_{q_{i}})$ is average runtime of group $G_{i}$ on MySQL
or ADW.
Figure 8 describes the time difference between MySQL and ADW for every query.
Although running on one computer, but with large data volume, ADW is faster
than MySQL on 46 out of 50 queries. MySQL is faster for three queries
$12^{th}$, $13^{th}$ and $18^{th}$ belonging to groups $3^{rd}$ and $4^{th}$.
The two systems returned the same time for query $24^{th}$ from group
$5^{th}$. Within each query group, for fair performance comparison, the
queries combine randomly fact tables and dimensional tables. This makes
complex queries taking more time and the time difference is significant. When
varying the sizes and structures of the tables, the difference is very
significant; see Figure 8.
$0$$2$$4$$6$$8$$10$$2$$4$$6$Mean$6.24$$2.92$$1.22$$2.86$$2.27$$4.66$$3.36$$4.63$$3.16$$1.56$$3.19$Groups
($G_{i}$)Different times ($Times_{G_{i}}$) Figure 9: Different times between
MySQL and
ADW in runtime of every group
Beside comparing runtime in every query, we aslo compare runtime of every
group presented in Figure 9. Comparing to MySQL, ADW is more than at most
(6.24 times) at group $1^{st}$ which uses only Where command, and at least
(1.22 times) at group $3^{rd}$ which uses Where and Joint commands.
12345678910Mean$0$$500$$1{,}000$$1{,}081.5$$599.7$$111.7$$790.4$$776.6$$1{,}109.2$$483$$1{,}057.3$$297.9$$571.1$$687.8$$173.4$$205.2$$91.2$$276.4$$342.8$$238$$143.7$$228.3$$94.2$$366.4$$216.1$Groups
($G_{i}$)Average runtimes (seconds)MySQLADW Figure 10: Average Runtimes of
MySQL and
ADW in every Groups
Figure 10 presents the average runtime of the 10 query groups on MySQL and
ADW. Mean, the run time of a reading query on MySQL and ADW is 687.8 seconds
and 216.1 seconds, respectively. It means that ADW is faster 3.19 times. In
the future, by deploying ADW solution on cloud or distributed systems, we
believe that the performance will be even much better than MySQL.
## 9 Application for Decision Making
The proposed ADW and study its performance on real agricultural data, we
illustrated some queries examples to show how to extract information from ADW.
These queries incorporate inputs on crop, yield, pest, soil, fertiliser,
inspection, farmer, businessman and operation time to reduce labour and
fertiliser inputs, farmer services, disease treatment and also increase
yields. These query information could not be extracted if the Origin’s
separate 29 datasets have not been integrated into ADW. The data integration
through ADW is actually improve the value of a crop management data over time
to better decision-making.
Example 1: List fields, crops in the fields, yield and pest in the field with
conditions: (1) the fields do not used ’urea’ fertilizer; (2) the crops has
’yellow rust’ or ’brown rust’ diseases; (3) the crops were grown in 2015.
select CR.CropName, FI.FieldName, FF.Yield,
PE.CommonName, FF.PestNumber, PE.Description
from FieldFact FF, Crop CR, Field FI, Pest PE,
Fertiliser FE, Inspection INS, OperationTime OP
where FF.CropID = CR.CropID and
FF.FieldID = FI.FieldID and
FF.PestID = PE.PestID and
FF.FertiliserID = FE.FertiliserID and
CR.CropID = INS.CropID and
FF.OperationTimeID = OP.OperationTimeID and
FE.FertiliserName <> ’urea’ and
(INS.Description = ’Yellow Rust’ or
INS.Description = ’Brown Rust’) and
Year(INS.Date) = ’2015’ and
Year(OP.StartDate) = ’2015’ and
Year(OP.EndDate) = ’2015’
Example 2: List farmers and their crop quantities were sold by Ori Agro
company in 08/2016.
select FA.FarmerID, FA.FarmerName, CR.CropName,
SF.Unit, SUM(SF.Quantity)
from Salefact SF, business BU, farmer FA, crop CR
where SF.BusinessID = BU.BusinessID and
SF.FarmerID = FA.FarmerID and
SF.CropID = CR.CropID and
Month(SF.SaleDate) = ’08’ and
Year(SF.SaleDate) = ’2016’ and
BU.BusinessName = ’Ori Agro’
group by CR.CropName
Example 3: List Crops and their fertiliser and treatment information. In that,
crops were cultivated and harvested in 2017, Yield $>$ 10 tons/ha and attached
by ’black twitch’ pest. Besides, the soil in field has PH $>6$ and Silt $<=50$
mg/l.
Select CR.CropName, FE.FertiliserName,
FF.FertiliserQuantity, TR.TreatmentName,
TR.Rate, TR.TreatmentComment
From FieldFact FF, Crop CR, OperationTime OT,
Soil SO, PEST PE, Fertiliser FE, Treatment TR
Where FF.CropID = CR.CropID and
FF.OperationTimeID = OT.OperationTimeID and
FF.SoildID = SO.SoilID and
FF.PestID = PE.PestID and
FF.FertiliserID = FE.FertiliserID and
FF.TreatmentID = TR.TreatmentID and
Year(OT.StartDate) = ’2017’ and
Year(OT.EndDate) = ’2017’ and
FF.Yield > 10 and
SO.PH > 6 and SO.Silt <= 50 and
PE.CommonName = ’Black twitch’
Example 4: List crops, fertilisers, corresponding fertiliser quantities in
spring, 2017 in every field and site of 10 farmers (crop companies) who used
the large amount of $P_{2}O_{5}$ in winter, 2016.
To execute this request, the query needs to exploit data in the FieldFact fact
table and the six dimension tables, namely Crop, Field, Site, Farmer,
Fertiliser and OperationTime. The query consists of two subqueries which
return 10 farmers (crop companies) that used the largest amount of Urea in
spring, 2016.
Select FI.FieldName, SI.SiteName, FA.FarmerName,
CR.CropName, FE.FertiliserName,
FF.FertiliserQuantity, FE.Unit, OT.StartDate
From FieldFact FF, Crop CR, Field FI, Site SI,
Farmer FA, Fertiliser FE, Operationtime OT
Where FF.CropID = CR.CropID and
FF.FieldID = FI.FieldID and
FF.FertiliserID = FE.FertiliserID and
FF.OperationTimeID = OT.OperationTimeID and
FI.SiteID = SI.SiteID and
SI.FarmerID = FA.FarmerID and
OT.Season = ’Spring’ and
YEAR(OT.StartDate) = ’2017’ and
FA.FarmerID IN(
Select FarmerID
From
(Select SI.FarmerID as FarmerID,
SUM(FF.FertiliserQuantity) as SumFertiliser
From FieldFact FF, Field FI, Site SI,
Fertiliser FE, OperationTime OT
Where FF.FieldID = FI.FieldID and
FF.FertiliserID = FE.FertiliserID and
FF.OperationTimeID =
OT.OperationTimeID and
SI.SiteID = FI.SiteID and
FE.FertiliserName = ’SO3’ and
OT.Season = ’Spring’ and
YEAR(OT.StartDate) = ’2016’
Group by SI.FarmerID
Order by SumFertiliser DESC
Limit 10
)AS Table1
)
## 10 Conclusion and Future Work
In this paper, we presented a schema herein optimised for the real
agricultural datasets that were made available to us. The schema been designed
as a constellation so it is flexible to adapt to other agricultural datasets
and quality criteria of agricultural Big Data. Based on some existing popular
open source DWs, We designed and implemented the agricultural DW by combining
Hive, MongoDB and Cassandra DWs to exploit their advantages and overcome their
limitations. ADW includes necessary modules to deal with large scale and
efficient analytics for agricultural Big Data. Moreover, through particular
reading queries using popular HQL/SQL commands, ADW storage outperforms MySQL
by far. Finally, we outlined some complex HQL queries that enabled knowledge
extraction from ADW to optimize of agricultural operations.
In the future work, we shall pursue the deployment of ADW on a cloud system
and implement more functionalities to exploit this DW. The future developments
will include: (1) experimentation and analyzation the performance of MongoDB
and the affectation between MongoDB and Hive; (2) The sophisticated the data
mining and the spreading activation algorithms (Ngo, 2014) to determine crop
data characteristics and combine with expected outputs to extract useful
knowledge; (3) Predictive models based on machine learning algorithms; (4) An
intelligent interface and graph representation (Helmer and et al., 2015) for
data access; (5) Combination with the ontology to extract knowledge (Ngo and
et al., 2011; Cao and et al., 2012).
## Appendix
The followings are HQL/SQL scripts of 10 queries which are representative of
10 query groups. The average runtimes of these queries on MySQL and ADW are
shown in Figure 11.
1) The query $5^{th}$ belongs to the group $1^{st}$:
SELECT fieldfact.FieldID, crop.cropname,
fieldfact.yield
FROM fieldfact, crop
WHERE fieldfact.cropid = crop.cropid and
SprayQuantity = 7 and
(crop.CropName like ’P\%’ or
crop.CropName like ’R\%’ or
crop.CropName like ’G\%’);
2) The query $10^{th}$ belongs to the group $2^{nd}$:
SELECT soil.PH, count(*)
FROM fieldfact, soil
WHERE fieldfact.SoildID = soil.SoilID and
fieldfact.sprayquantity = 2
GROUP by soil.PH;
5101520253035404550$0$$1{,}000$$2{,}000$$97.9$$754.8$$52.7$$2{,}297$$1{,}192$$2{,}188.4$$95.4$$265.9$$439.5$$892.4$$3$$233.2$$3.6$$479$$422.6$$226.7$$5.2$$7.6$$212.3$$472.1$Queries
($q_{i}$)Average runtimes (seconds)MySQLADW Figure 11: Average runtimes of
MySQL and
ADW in 10 typical queries
3) The query $15^{th}$ belongs to the group $3^{rd}$:
SELECT fieldfact.yield,
fertiliser.fertiliserName,
fertiliser.fertiliserGroupName
FROM fieldfact
RIGHT JOIN fertiliser on
fieldfact.fertiliserID = fertiliser.fertiliserID
WHERE fieldfact.fertiliserQuantity = 10 and
fertiliser.fertiliserName like ’%slurry%’;
4) The query $20^{th}$ belongs to the group $4^{th}$:
SELECT sprayproductname
FROM fieldfact, spray
WHERE fieldfact.sprayid = spray.sprayid and
fieldfact.watervolumn > 5 and
fieldfact.watervolumn < 20
UNION
SELECT productname
FROM product, orderfact
WHERE product.ProductID = orderfact.ProductID
and (orderfact.Quantity = 5 or
orderfact.Quantity = 6);
5) The query $25^{th}$ belongs to the group $5^{th}$:
SELECT fieldfact.fieldID, field.FieldName,
field.FieldGPS, spray.SprayProductName
FROM fieldfact, field, spray
WHERE fieldfact.FieldID = field.FieldID and
fieldfact.SprayID = spray.SprayID and
fieldfact.PestNumber = 6
ORDER BY field.FieldName;
6) The query $30^{th}$ belongs to the group $6^{th}$:
SELECT fieldfact.FieldID, nutrient.NutrientName,
nutrient.Quantity, nutrient.‘Year‘
FROM fieldfact
RIGHT JOIN nutrient on
fieldfact.NutrientID = nutrient.NutrientID
WHERE fieldfact.NutrientQuantity = 3 and
fieldfact.fertiliserquantity = 3
ORDER BY nutrient.NutrientName
LIMIT 10000;
7) The query $35^{th}$ belongs to the group $7^{th}$:
SELECT crop.cropname,
sum(fieldfact.watervolumn) as sum1
FROM fieldfact, crop
WHERE fieldfact.cropid = crop.cropid and
fieldfact.sprayquantity = 8 and
crop.EstYield >= 1 and crop.EstYield <=10
GROUP BY crop.cropname
HAVING sum1 > 100;
8) The query $40^{th}$ belongs to the group $8^{th}$:
SELECT crop.cropname,
sum(fieldfact.fertiliserquantity) as sum1
FROM fieldfact, crop
WHERE fieldfact.cropid = crop.cropid and
fieldfact.nutrientquantity= 5 and
crop.EstYield <=1
GROUP by crop.cropname
HAVING sum1 > 30
ORDER BY crop.cropname;
9) The query $45^{th}$ belongs to the group $9^{th}$:
SELECT nutrient.NutrientName,
sum(nutrient.Quantity) as sum1
FROM fieldfact
LEFT JOIN nutrient on
fieldfact.NutrientID = nutrient.NutrientID
WHERE nutrient.nutrientName like ’%tr%’ and
(fieldfact.pestnumber = 16 or
fieldfact.pestnumber = 15)
GROUP by nutrient.NutrientName
HAVING sum1 <300
ORDER BY nutrient.NutrientName;
10) The query $50^{th}$ belongs to the group $10^{th}$:
SELECT sprayproductname as name1,
sum(fieldfact.watervolumn) as sum1
FROM fieldfact, spray
WHERE fieldfact.sprayid = spray.sprayid and
fieldfact.Yield > 4 and fieldfact.Yield < 8
GROUP by sprayproductname
HAVING sum1 > 210
UNION
SELECT productname as name1,
sum(orderfact.Quantity) as sum2
FROM product, orderfact
WHERE product.ProductID = orderfact.ProductID and
(orderfact.Quantity = 5 or
orderfact.Quantity = 6)
GROUP by productname
HAVING sum2 > 50
ORDER BY name1;
## Acknowledgment
This research is an extended work of Ngo and et al. (2019) being part of the
CONSUS research program. It is funded under the SFI Strategic Partnerships
Programme (16/SPP/3296) and is co-funded by Origin Enterprises Plc.
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|
2024-09-04T02:54:58.587189 | 2020-03-10T00:25:25 | 2003.04472 | {
"authors": "Jie Ren, Wen-Long You, and Xiaoqun Wang",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26128",
"submitter": "Jie Ren",
"url": "https://arxiv.org/abs/2003.04472"
} | arxiv-papers | # Entanglements and correlations of one-dimensional quantum spin-1/2 chain
with anisotropic power-law long range interactions
Jie Ren<EMAIL_ADDRESS>Department of Physics, Changshu Institute of
Technology, Changshu 215500, China Wen-Long You College of Science, Nanjing
University of Aeronautics and Astronautics, Nanjing, 211106, China School of
Physical Science and Technology, Soochow University, Suzhou, Jiangsu 215006,
China Xiaoqun Wang<EMAIL_ADDRESS>Key Laboratory of Artificial
Structures and Quantum Control of MOE, Shenyang National Laboratory for
Materials Science, School of Physics and Astronomy,
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
Collaborative Innovation Center for Advanced Microstructures, Nanjing
University, Nanjing 210093, China Beijing Computational Science Research
Center, Beijing 100084, China
###### Abstract
The correlations, entanglement entropy, and fidelity susceptibility are
calculated for a one-dimensional spin-1/2 XXZ chain with anisotropic power-law
long range interactions by employing the density matrix renormalization group
method. In particular, this long-range interaction is assigned to
ferromagnetic for transversal components, while it can be either ferro- or
antiferromagnetic for the longitudinal spin component. Two ground-state phase
diagrams are established versus the anisotropy of the interactions which not
only changes the phase boundaries of the counterparts with short-range
interactions, but also leads to the emergence of exotic phases. We found that
the long-range interactions of the $z$-component results in a Wigner crystal
phase, whereas the transversal one may break a continuous symmetry, resulting
in a continuous symmetry breaking phase.
###### pacs:
03.67.-a,05.30.Jp
## I Introduction
The quantum phase transition (QPT) and quantum critical phenomena are
generally important in understanding novel properties involved in strongly
correlated systems, such as quantum magnetic materials. Usually, short-range
interactions, e.g., nearest neighbor and next nearest neighbor interactions,
are considered to be sufficient for appropriate descriptions on the major
magnetic properties of those systems Sachdev ; XWang2000 ; Luo2017 ; You19 ;
WN19 ; Luo2019 . However, there actually exist several types of long range
interactions such as the Coulomb interaction $1/r$ Saffman , the dipole-dipole
interaction $1/r^{3}$ Lahaye ; Deng ; Yan , and the van der Waals interaction
$1/r^{6}$ Saffman in some complicated compounds, where relevant electrons are
in higher orbits of atoms with lower symmetries subject to crystal field
effects. Moreover, in recent years, some long-range interactions have been
generated in ultracold atomic systems with the optical lattices or trapped
ions. For instance, a power-law Ising interaction $1/r^{\alpha}$ with an
adjustable exponent $0<\alpha<3$ has been realized in trapped ions Britton ;
Islam ; Gorshkov ; Jurcevic . This kind of experimental progress has greatly
stimulated theoretical studies on possible novel effects particularly
resulting from long-range interactions W ; Koffel ; Sun01 ; Zhu ; gong16 ;
gong17 ; gong17L ; gong16R ; Frerot ; Vanderstraeten . In particular, a
transition was revealed by the calculation of the entanglement for a long-
range ($\sim r^{-\alpha}$) antiferromagnetic Ising chain Koffel , and is
affirmed further by the fidelity susceptibility, being second-order for all
$\alpha$ Sun01 ; Zhu . Moreover, by combining the linear spin-wave theory,
field theory approach and density-matrix renormalization-group (DMRG) white ;
KWHP ; U01 ; U02 ; McCulloch , effects of the long range interactions on local
correlation functions, entanglement entropy and central charge are
investigated for both spin-1/2 gong17 and spin-1 gong16 to await
experimental observation. In addition, one also finds that long-range
interactions and long-range hopping may lead to drastic effects on the many-
body localization in a one-dimensional (1D) spinless fermion system Nag2019 ,
which essentially corresponds to a $XY$ type of long range spin interaction.
In this regard, the anisotropic long-range spin interaction can be anticipated
to give rise to more effects on quantum transitions.
In this paper, we study a ID spin-1/2 XXZ system with anisotropic power-law
long range interactions in terms of the entanglement entropy, fidelity
susceptibility, and correlation functions by performing DMRG calculations.
Phase diagrams are established with respect to the power exponents and the
anisotropy of interactions. In the following, Sec II presents the Hamiltonian
in our studies. The details on DMRG calculations and the definitions of those
calculated quantities are discussed in Sec. III. Numerical results are shown
in Sec IV with further discussions given in the last section.
## II Hamiltonian
In the paper, we consider the following spin-$1/2$ chain with anisotropic
long-range interactions, and its Hamiltonian is given by:
$\displaystyle
H=\sum_{j>i}\\{\frac{J_{xy}}{|i-j|^{\alpha}}(S^{x}_{i}S^{x}_{j}+S^{y}_{i}S^{y}_{j})+\frac{J_{z}}{|i-j|^{\beta}}S^{z}_{i}S^{z}_{j}\\},$
(1)
where $i$ and $j$ are the sites of one dimensional lattice, and
$S^{\gamma}=\sigma^{\gamma}/2$ with $\gamma=x,y$, or $z$, setting $\hbar=1$
and $\sigma^{\gamma}$ being the Pauli matrices. Interactions between two spins
separated by a distance of $r=|i-j|$ decay as $r^{-\alpha}$ for both $x$ and
$y$ components of spins, but as $r^{-\beta}$ for the $z$ direction. As usual,
the parameters $\alpha,\beta$ are both taken positive, while $J_{xy}=-1$ is
set up for the simplicity so that $J_{z}$ readily stands for an anisotropic
parameter involved in the establishment of the phase diagram.
For this system, in the limit of $\alpha,\beta\rightarrow+\infty$, the
Hamiltonian is reduced to describe a spin-1/2 anisotropic chain with the
nearest-neighbor interaction. It turns out that the system involves a
ferromagnetic (FM) phase for $J_{z}<-1$, whereas a gapful antiferromagnetic
(AFM) phase can be shown for $J_{z}>1$. Furthermore, in the region of
$-1<J_{z}\leq 1$, the system displays an $XY$ phase where quantum fluctuations
exclude the existence of any long-range order but correlation functions behave
as a power-law decay of the distance characterized as in the Luttinger liquid.
For more general values of $\alpha$ and $\beta$, long range interactions may
result in different features for those phases, which are expected also to be
properly characterized by long-distance correlation functions as exploited
below.
## III Measurements and Method
Thanks to the DMRG methodwhite ; KWHP ; U01 , the ground state properties of
quasi-one-dimensional systems can be calculated with very high accuracy. For
the present studies of Hamiltonian (1), we adopt both infinite-size DMRG
(iDMRG) McCulloch and finite-size DMRG, which are based on matrix product
states U02 . The number of eigenstates for the reduced matrix is kept up to
$m=400$ in the truncation of bases, which allows the truncation error to be
smaller than $10^{-9}$. In our calculations where finite-size DMRG algorithm,
we handle the long range interaction with directly using as a summation over
matrix product of operators (MPOs) rather than the summation of finite
exponential terms with MPOs Vidal , which inevitably introduces additional
systematic error otherwise. Our codes are mainly based on iTensor C++ library
tesnor .
Since the $z$-component of the total spins for the present system commutes
with the Hamiltonian (1), the ground-state energy is obtained by comparing the
lowest energies for each subspace of $S^{z}_{t}=\sum_{i=1}^{L}\langle
S^{z}_{i}\rangle$. We found that the ground state resides in the sector of
either $S^{z}_{t}=0$ or $S^{z}_{t}=L/2$. To examine the reliability of our
numerics, we also perform the finite-size DMRG with varying the number of
states in the truncated bases. Once the ground state energy and the
corresponding ground state are identified accurately, the first excited state
and the corresponding energy (gap) can be determined similarly as
orthonormalized to the ground state.
For a quantum many-body system, the entanglement entropy (EE) can be extracted
from the ground state wavefunction $|\psi_{0}\rangle$ properly to characterize
the quantum phase transition induced by the interaction or external fields.
Usually, one may separate a given Hamiltonian into two subsystems $A$ and $B$,
and compute the reduced density matrix for part $A$ by partially tracing over
the degree of freedom of the subsystem $B$, which can be written formally as
$\rho_{A}=\textrm{Tr}_{B}(|\psi_{0}\rangle\langle\psi_{0}|).$
Then, the entanglement entropy measuring the entanglement between parts $A$
and $B$ is given by
$\displaystyle S_{A}=-\textrm{Tr}(\rho_{A}\ln\rho_{A}).$ (2)
which is evaluated in terms of the eigenvalues of $\rho_{A}$ feasibly in DMRG
calculations. For a one-dimensional short-range interacting system with an
open boundary condition (OBC), the conformal field theory (CFT) suggests that
the entanglement entropy for the subsystem $A$ with size $l$ possesses the
following finite-size $L$ scaling behavior Cardy
$\displaystyle S_{l}=\frac{c}{6}\ln[\frac{L}{\pi}\sin(\frac{\pi
l}{L})]+S_{0},$ (3)
where $c$ is the central charge which usually has different values for
different phases and $S_{0}$ is a non-universal constant. This scaling
behavior has been employed to explore the critical entanglement of defects
Zhao2006 and Gaussian transitionHu2011 . In this paper, we will show that
this scaling behavior is applicable to a case associated with long-range
interactions.
## IV Results
### IV.1 $1/\alpha=0$
Now we first consider the case of $\alpha=\infty$, which implies that only the
nearest-neighbor term of $xy-$interaction survives. It turn out that the long-
range interaction for $z-$component governed by $\beta$ may result in novel
properties in competition with the $xy-$components. In this case, Hamiltonian
(1) can be recast to describe a one-dimensional interacting spinless fermionic
chain via the Jordan-Wigner transformation:
$\displaystyle S^{z}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-c_{i}^{\dagger}c_{i},$ $\displaystyle S^{+}_{i}$
$\displaystyle=$ $\displaystyle
e^{i\pi\sum_{j=1}^{i-1}c_{i}^{\dagger}c_{i}}c_{i},$ $\displaystyle S^{-}_{i}$
$\displaystyle=$ $\displaystyle
e^{i\pi\sum_{j=1}^{i-1}c_{i}^{\dagger}c_{i}}c_{i}^{\dagger},$
where $S^{\pm}_{i}$=$S^{x}_{i}$ $\pm$ $iS^{y}_{i}$ are the raising and
lowering spin operators. Subsequently, the ferromagnetic $J_{xy}-$term thus
simply represents the hopping of fermions, while the $J_{z}-$term stands for
the density-density interactions of fermions, which can be either attractive
for $J_{z}<0$ or repulsive for $J_{z}>0$. One may expect that this density-
density interaction results in quantum transitions for different $\alpha$ and
$\beta$.
To explore this, we compute the correlation functions between two spins at $i$
and $j$ with a distance of $r=|i-j|$ and for $\beta$ = 2 with using the iDMRG
algorithm. Figure 1 shows results for $r=99$. One can see that when
$J_{z}<-0.636$, the transverse correlation $\langle
S^{+}_{i}S^{-}_{i+99}\rangle=0$ and the longitudinal correlation $\langle
S^{z}_{i}S^{z}_{i+99}\rangle=1/4$, implying that the system is in the FM
phase, and then $\langle S^{+}_{i}S^{-}_{i+99}\rangle$ suddenly jumps to a
positive value at $J_{z}=-0.636$ and $\langle S^{z}_{i}S^{z}_{i+99}\rangle$
drops to zero simultaneously. This discontinuity indicates that the ground
state undergoes a first order transition from the FM phase into the $XY$
phase. This discontinuous feature is thus utilized here to determine the
critical values of $\beta$ and $J_{z}$ for the quantum phase transition
between the $XY$ and FM phases.
Figure 1: (Color online) Correlation functions $\langle
S^{+}_{i}S^{-}_{i+r}\rangle$ and $\langle S^{z}_{i}S^{z}_{i+r}\rangle$ are
plotted as a function of $z$-component interaction $J_{z}$ for
$\alpha=\infty$, $\beta=2$ and $r=99$. Inset: a log-log plot for $\langle
S^{+}_{i}S^{-}_{i+r}\rangle$ as a function of $r$ when $J_{z}=\pm 0.5$. Figure
2: (Color online) (a) Entanglement entropies are plotted as a function of
$z$-component interaction $J_{z}$ for various system sizes with
$\alpha=\infty$ and $\beta=2$. (b) The peak positions of $S_{L/2}$ versus
system sizes $L$.
Moreover, as $J_{z}$ further increases, the transverse correlation $\langle
S^{+}_{i}S^{-}_{i+99}\rangle$ gradually reduce to zero, while the longitudinal
correlation $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ turns to negative for
$J_{z}\gtrsim 3/2$, which signals that the system is driven into a AFM phase.
A little scrutiny reveals that the transverse correlation $\langle
S^{+}_{i}S^{-}_{i+r}\rangle$ satisfies a power-law decay with the distance $r$
gong17 , as manifested in the inset of Fig. 1. To determine the critical point
at the transition between the $XY$ phase and AFM phase more precisely, we also
calculate the von Neumann entropy, i.e. entanglement entropy, for the right
part apart from the rest for the chain with using the finite-size DMRG
algorithm. The entanglement entropy is shown in Fig. 2 as a function of
$J_{z}$ with $\beta=2$ for different sizes of the chain. With increasing
$J_{z}$, the EE increases first and then declines. The peak becomes more
pronounced for a larger size $L$ and the location of the peak moves to a lower
value of $J_{z}$, characterizing a transition between the $XY$ phase and the
AFM phase Wang . According to the finite-size scaling theory Fisher ; Barber83
, it is expected that the position of the pseudo-critical point for a finite-
size system approaches the true critical point as $L$ $\to$ $\infty$. For
relevant operators in the driving Hamiltonian on sufficiently large-size
systems, i.e., $\nu$$d$$<$2, where $\nu$ is the critical exponent of the
correlation length and $d$ the dimensionality of the system, the leading term
in the expansion of pseudo-critical point obeys
$\displaystyle|J_{z}^{c}(L)-J_{z}^{c}(\infty)|\propto L^{-1/\nu},$ (4)
where $J_{z}^{c}(\infty)$ is the critical value for the thermodynamic limit.
Such algebraic convergence can be accelerated considerably by some elaborated
strategies Roncaglia . We obtain that $J_{z}^{c}=1.520$ and $\nu=1.695$ for
the present case consistent with the inflection point of the correlations
shown in Fig. 1. We note that the scaling behavior of Eq. (4) with $L$ is also
valid for the maximum of fidelity susceptibility defined in Eq.(6) You2011
(see below).
Figure 3: (Color online) Finite size scaling of the energy gap $\Delta$ with
various $\beta$ and $J_{z}$. Symbols show numerical results obtained by DMRG
calculations and solid lines are fits of the data by quadratic polynomials in
$1/L$. The results for $J_{z}=0$ is also plotted as for comparison.
Low-lying excitation energy often reveals perspective features of different
phases in the quantum many-body interacting systems. As mentioned previously,
the system involves the gapless $XY$ phase for $-1<J_{z}\leq 1$ in the limit
of $\beta=\infty$, which has the central charge $c_{\rm eff}=1$ owing to the
conformal symmetry Vidal03 . In the Jordan-Wigner representation of the
Hamiltonian (1), the spinless interacting fermion has a linear
$1/L-$dependence for the finite-size energy gap as a relativistic spectrum at
the Fermi point or the low-lying property of the spectrum for the Luttinger
liquid. When $\beta\neq\infty$, however, it is clearly of great interest
whether such a $XY$ phase can be robust against a strong long-range repulsive
interaction. For $J_{z}=1$ and $\beta=1$ Schulz ; Li , it was suggested that
the ground state would be a quasi-Wigner crystal (WC), which results from the
dominant long-range repulsive interaction over the kinetic energy. We
calculated the finite-size gap energy $\Delta(L)$ between the ground state
and the first excitation energies as a function of system sizes for various
cases as illustrated in Fig.3, one can see that the energy gap
$\Delta(\beta,J_{z})$ can be either zero, including the case of $J_{z}=1$ and
$\beta=1$, or finite in the thermodynamic limit, which can be assigned to $XY$
and gapped quasi-WC phases, respectively. However, for given $J_{z}$, when
$\beta$ approaches its critical values $\beta_{c}$ from either $XY$ phase or
WC phase where $\Delta(L)=\Delta(\beta,J_{z})+A_{1}/L+O(1/L^{2})$ You14 , it
becomes rather difficult to accurately determine the phase boundary between
these two phases due to limited precisions on tiny values of $\Delta(L)$.
Instead, we adopt the effective center charge $c_{\rm eff}$ deducted from the
scaling behavior of the entanglement entropy given in Eq.(3) which enable us
more accurately to allocate the phase boundary. We note that this scaling
behavior is valid in the presence of the long range interaction as
demonstrated numerically in Fig. 4, although it was originally derived for the
short range interacting cases with conformal symmetries Cardy ; Laflorencie .
Figure 4: (Color online) The Scaling behavior of entanglement entropy versus
$\ln(x)=\ln[L/\pi\sin(\pi l/L)]$ for different values of $\beta^{-1}$ with
$L=300$. Inset shows the fitted coefficients as a function of $\beta^{-1}$ for
system sizes $L=200$ (square) and $L=300$ (circle).
Figure 4 shows the entanglement entropy as a function of $\ln[L/\pi\sin(\pi
l/L)]$ for various values of $\beta$ and positive $J_{z}$. It is instructive
that the entanglement entropy still follows up the scaling behavior of Eq.
(3), although conformal symmetries are not yet known here in general.
Subsequently, the slope of the linear behavior gives rise to an effective
central charge $c_{\rm eff}$ which varies with $\beta$ as illustrated for
system sizes $L=200$ and $300$ at $J_{z}=1$ in the inset of Fig. 4. One can
see that finite-size effects for small $1/\beta$ is small but still visible,
resulting in the correction to $c^{0}_{\rm eff}=1$ for the thermodynamic
limit, but diminishes with increasing $1/\beta$. The curves for these two
sizes cross with a horizontal line corresponding to $c_{eff}=c^{0}_{\rm eff}$
at $1/\beta_{c}=0.756$, where irrelevant corrections vanish to Eq. (3). The
finite-size effect then becomes negligible for $\beta\leq\beta_{c}$. This
provides alternative way with higher accuracy to determine transition points
between the $XY$ (critical) and WC (noncritical) phases Alet ; gong16 ; gong17
.
Figure 5: (Color online) Phase diagram of Hamiltonian (1) as a functions of
the interaction $J_{z}$ and $1/\beta$ with $\alpha\rightarrow+\infty$.
In addition, we note that the FM phase is formed owing to the instability of
effectively attractive density-density interaction for $J_{z}\leq 0$ upon
changing $1/\beta$. Accordingly, the central charge is zero for the FM phase,
but it has the value of 3/2 on its phase boundary with the $XY$ phase for the
thermodynamic limitsChen ; Olalla ; Alba . To this end, the phase diagram is
depicted in Fig. 5 for $\alpha=\infty$. One can see that the critical points
between the $XY$ phase and the FM phase asymptotically approach $J_{z}=0$,
while the critical points between the AFM phase and the $XY$ phase mounts up
with increasing $1/\beta$. Moreover, it is worthwhile to mention that at
$\beta=0$ with $J_{z}>0$, $J_{z}$ term effectively results in one sort of
long-range frustrations and has the same strength for all the sites, among
which diagonal elements cancel each other in the ground state in
correspondence to $S^{z}_{total}=0$ subspaceZerobeta . In this case, the
ground state again becomes gapless and the central charge equals to one.
Particularly, the energy gap $\Delta(L)$ is scaled to zero in the limit of
$L\rightarrow\infty$ independent of $J_{z}$ as illustrated for both $J_{z}=1$
and $J_{z}=2$ in Fig. 3. Moreover, the entanglement entropy behaves as same
between $J_{z}=1,2$, resulting in $c_{\rm eff}\simeq 1.02$, as seen in Fig. 4.
As connected to $J_{z}=0$, it is natural to consider that the system is indeed
in the $XY$ phase, i.e. the transition between the FM and $XY$ phases takes
place at $J_{z}=0$ for $1/\beta=\infty$.
### IV.2 $1/\beta=0$
In this section, we turn to the case of $\beta\rightarrow+\infty$. In this
case, only the nearest neighbor interaction survives in the $J_{z}-$terms of
the Hamiltonian Eq. (1). The exponent $\alpha$ of the $XY-$long range
interaction can be considered a tunable parameter to explore the quantum phase
transition for various values of $J_{z}$.
Figure 6: (Color online) Correlation functions $\langle
S^{+}_{i}S^{-}_{i+99}\rangle$ and $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ are
plotted as a function of the interaction $J_{z}$ for (a) $\alpha=4$ and (b)
$\alpha=2$. Inset: A log-log plot of $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ as
a function of the distance $r$ with $J_{z}=\pm 0.5$.
Figure 6 shows the dependence of two-spin correlations on $J_{z}$ with a
distance of $|i-j|=99$ for different $\alpha$, calculated by using the iDMRG
algorithm. When $J_{z}$ is negatively large enough, $\langle
S^{+}_{i}S^{-}_{i+99}\rangle=0$, $\langle S^{z}_{i}S^{z}_{i+99}\rangle=1/4$,
suggesting that the system is in the FM phase. When $J_{z}$ is sufficiently
large, the transverse correlations remain zero, whereas $\langle
S^{z}_{i}S^{z}_{i+99}\rangle$ becomes negative so that the ground state is a
AFM state. Analogous to the case of $\alpha=\infty$, here we again utilize the
discontinuity of the correlation functions to allocate the critical points for
$\alpha$ and $J_{z}$ at the boundary of the FM phase, while the boundary of
the AFM phase is also determined in terms of the entanglement entropy (see
below).
In an intermediate range of $J_{z}$, one can further see that the transverse
correlations $\langle S^{+}_{i}S^{-}_{i+99}\rangle$ is positive but
longitudinal correlations $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ vanish.
Interestingly, we find that $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ is a concave
function of $J_{z}$ for $\alpha=2$, but becomes a convex one for $\alpha=4$.
Moreover, when $J_{z}=\pm 0.5$, $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ behaves
as a power-law of $1/r$, vanishing in the limit of $r\rightarrow\infty$ as
illustrated for $\alpha=4$ in the inset of Fig. 6(a), but
${\lim_{r\to+\infty}}\langle S^{+}_{i}S^{-}_{i+r}\rangle$ approaches a finite
constant as seen for $\alpha=2$ from the inset of Fig. 6(b). Therefore, the
ground states for $\alpha=2$ in the intermediate range of $J_{z}$ is different
that for $\alpha=4$.
In this range of $J_{z}$, it is natural to assign the large$-\alpha$ phase to
the $XY$ phase, since this phase contains a special case where $\alpha=\infty$
and $J_{z}=0$ such that the Hamiltonian (1) is reduced to describe a standard
$XY$ chain, as already shown Fig. (5). Moreover, when $\alpha$ is small or
even not too large, one can show that a $U(1)$ symmetry in the ground state is
spontaneously broken at $J_{z}=0$ with using the conformal field analysis and
perturbation calculationgong17 . It turns out that one can expect the
emergence of a continuous symmetry breaking (CSB) phase with gapless
excitations for a small$-\alpha$ phase. It has been shown that a Berezinskii-
Kosterlitz-Thouless like transition happens between the CSB phase and the $XY$
phase at $1/\alpha_{c}\simeq 0.34$, at which the central charge is numerically
increased by $4\%$ from unit. However, the criteria of the $4\%$ addition to
the central charge might be invalid for the determination of the critical
points with general values of $J_{z}$. To address this issue, we calculate the
fidelity susceptibility which has been proposed for the identification of the
critical points of continuous quantum phase transitionsGu2010 and even
deconfined quantum critical points Sun19 , and successfully applied to various
strongly correlated systems You15 ; You17 ; Ren18 ; Luo18 .
As a quantum information metric Gu2010 ; You , the fidelity measures the
similarity between the two closest ground states when the parameter $\alpha$
is tuned tiny for the Hamiltonian (1), which is defined as
$F=|\langle\psi_{0}(\alpha)|\psi_{0}(\alpha+\delta\alpha)\rangle|,$ (5)
where $\delta\alpha$ denotes a tiny deviation. Subsequently, we obtain the
derivatives of interactions $\delta
J_{i,j}=-\frac{J_{xy}}{|i-j|^{\alpha}}\ln|i-j|\delta\alpha$, where $J_{i,j}$
is the interaction strength between two spins at sites $i$ and $j$. The
average derivatives of interactions per site are practically considered as an
effective tuning parameter $\delta J=\frac{\sum_{i<j}\delta J_{i,j}}{L}$.
Therefore, the fidelity susceptibility per site can be calculated numerically
by
$\chi=\lim_{\delta J\rightarrow 0}\frac{-2\textrm{ln}F}{L(\delta J)^{2}},$ (6)
whose peak is thus used to identify the critical value of $\alpha$ and to
separate the CSB phase from the $XY$ phase for each $J_{z}$.
In our numerical calculations, we take $\delta\alpha=0.005$. For the case of
$L=100$ and $\alpha=3$, the effective tuning parameter $\delta J\simeq 0.001$.
The ground-state fidelity susceptibility per site $\chi$ is shown for
$J_{z}=0,1$ as a function of the parameter $\alpha$ for different sizes in
Fig. 7 (a) and (b), respectively. For each $J_{z}$, one can see that the peaks
of $\chi$ grow with respect to increasing the system size so that a divergence
peak would be expected for the $L\rightarrow\infty$ limit to signal the
appearance of a quantum phase transition. In order to locate the quantum
critical point $\alpha_{c}$ for the thermodynamic limit, we uses the finite-
size scaling analysis to obtain $\alpha_{c}=2.83$ and $\nu=1$ at $J_{z}=0$ as
seen in the inset of Fig. 7(a). This value of $\alpha_{c}$ is good consistent
with that determined by the central charge and the perturbation theory
calculation gong17 . Similarly, we can determine critical points at other
values of $J_{z}$ for the boundary between the CSB and $XY$ pases. In
particular, the critical value of $\alpha_{c}=2.45$ for $J_{z}=1.0$ is
obtained from the results shown in Fig. 7(b).
Figure 7: (Color online) Fidelity susceptibility per site is plotted as a
function of parameter $\alpha$ for various system sizes with (a) $J_{z}=0$ and
(b) $J_{z}=1.0$. Inset: Scaling behavior of the fidelity susceptibility peak
points with respect to $1/L$.
Now we turn to quantum phase transitions between the intermediate and AFM
phases, which are characterized by the peaks of the entanglement entropies as
demonstrated for $\alpha=2,4$ in Fig. 8. One can see that the peaks for both
cases in (a) and (c) of Fig. 8 move to lower values of $J_{z}$ when $L$
increases. Fitting the locations of peaks with the formula (4) as shown in (b)
and (d) of Fig. 8, one can obtain that $J_{z}^{c}=1.35$ and $2.21$,
respectively. Such fitted results agree very well with the inflexion points of
the correlations shown in Fig. 6. In the same manner, we allocate more
critical values of $J_{z}$ and $\alpha$ for the boundary of the AFM phase with
both $XY$ and CSB phases.
Figure 8: (Color online) Entanglement entropy is plotted as a function of the
interaction $J_{z}$ on different system sizes for (a) $\alpha=4$ and (c)
$\alpha=2$. The peak positions of $S_{L/2}$ versus the system size $L$ for (b)
$\alpha=4$ and (d) $\alpha=2$.
Based on the above analysis on the properties of the correlation functions,
the fidelity susceptibility and the entanglement entropy, we establish the
ground-state phase diagram for the Hamiltonian (1) with $\alpha=\infty$ as
shown in Fig. 9.
Figure 9: (Color online) Phase diagram of Hamiltonian (1) as a functions of
the interaction $J_{z}$ and $\alpha$ with $\beta\rightarrow+\infty$.
## V Discussion
In this paper, we study quantum phase transitions for a quantum spin-$1/2$
chain with anisotropic power-law-decaying long-range interactions, which are
characterized by exponent parameters $\alpha$ for $xy-$term and $\beta$ for
$z-$term, by employing density-matrix renormalization-group method. With
numerically analyzing the effects of $\alpha$ and $\beta$ on the spin-spin
correlation functions, the entanglement entropy and the central charge, and
the fidelity susceptibility, we establish two phase diagrams for
$\alpha=\infty$ and $\beta=\infty$, respectively.
Both cases involve a ferromagnetic phase and an antiferromagnetic phase
corresponding to sufficiently negative and positive $J_{z}$, respectively.
However, in the intermediate regime of $J_{z}$, the former involves not only a
usual $XY$ phase effectively equivalent to a short range repulsive density-
density interaction, but also a Wigner-crystal phase which essentially results
from for a sufficient strong long-range $J_{z}$ term; for the later, the
gapped Wigner crystal phase is replaced by a continuous $U(1)$ symmetry
breaking phase. Moreover, it is interesting to notice that the WC and CSB
phases actually reveal two different mechanisms, which intrinsically result
from either two-body processes of the strong long-range repulsive interaction
or one-body kinetic processes of the long-range hoping in the fermion
representation.
From this study, we found that the entanglement entropy and the central charge
can be used efficiently to extract critical values of the quantum phase
transition between two phases when one of them possesses a well-defined
central charge but another one is gapful Luo2019 . However, when one is
encountered with a quantum phase transition between two gapless phases, the
fidelity susceptibility alternatively provides a more feasible way to allocate
the critical points as applied to the transition between the $XY$ and
continuous $U(1)$ symmetry breaking phases.
We so far focus on the ground state phase diagrams only for $\alpha=\infty$
and $\beta=\infty$. There are actually a couple of important aspects beyond
the above two cases for the Hamiltonian (1), such as ground phase diagrams
with $\alpha=\beta$ and $J_{xy}>0$, extensions to two leg-ladders and even two
dimensions, etc. The emergence of any non-trivial gapless phase, corresponding
novel low-lying excitation spectra or exotic collective excitations with
special symmetries, and thermodynamic and dynamic properties would be very
interesting questions for the presence of long range interactions but are
certainly open for further studies in the future.
###### Acknowledgements.
This work is supported by the National Program on Key Research Project (Grant
No. 2016YFA0300501) and the National Natural Science Foundation of China under
Grants No. 11104021, 11474211, 61674110 and 11974244. W.L.Y is appreciative of
support from the start-up fund of Nanjing University of Aeronautics and
Astronautics. X.W. also acknowledges additional supports from a Shanghai
talent program.
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|
2024-09-04T02:54:58.599700 | 2020-03-10T03:43:55 | 2003.04520 | {
"authors": "Yongtao Li",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26129",
"submitter": "Yongtao Li",
"url": "https://arxiv.org/abs/2003.04520"
} | arxiv-papers | # Extensions of some matrix inequalities related to trace and partial
traces††thanks: This paper was firstly announced in March, 2020, and was later
published on Linear Algebra and its Applications 639 (2022) 205–224. See
https://doi.org/10.1016/j.laa.2022.01.006. E-mail addresses:
<EMAIL_ADDRESS>(Yǒngtāo Lǐ).
Yongtao Li∗
School of Mathematics, Hunan University
Changsha, Hunan, 410082, P.R. China
###### Abstract
We first present a determinant inequality related to partial traces for
positive semidefinite block matrices. Our result extends a result of Lin
[Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear
Algebra 66 (2018)]. Moreover, we provide a unified treatment of a result of
Ando [ILAS Conference (2014)] and a recent result of Li, Liu and Huang
[Operators and Matrices 15 (2021)]. Furthermore, we also extend some
determinant inequalities involving partial traces to a larger class of
matrices whose numerical ranges are contained in a sector. In addition, some
extensions on trace inequalities for positive semidefinite $2\times 2$ block
matrices are also included.
Dedicated to Prof. Weijun Liu on his 60th birthday
Key words: Partial traces; Trace inequalities; Fiedler and Markham; Numerical
range in a sector;
2010 Mathematics Subject Classification. 15A45, 15A60, 47B65.
## 1 Introduction
Throughout the paper, we use the following standard notation. The set of
$n\times n$ complex matrices is denoted by $\mathbb{M}_{n}(\mathbb{C})$, or
simply by $\mathbb{M}_{n}$, and the identity matrix of order $n$ by $I_{n}$,
or $I$ for short. We write $\lambda_{i}(A)$ and $\sigma_{i}(A)$ for the $i$-th
largest eigenvalue and singular value of $A$, respectively. By convention, if
$A\in\mathbb{M}_{n}$ is positive semidefinite, we write $A\geq 0$. For
Hermitian matrices $A$ and $B$ with the same size, $A\geq B$ means that $A-B$
is positive semidefinite, i.e., $A-B\geq 0$. If $A=[a_{i,j}]$ is of order
$m\times n$ and $B$ is of order $s\times t$, the tensor product of $A$ with
$B$, denoted by $A\otimes B$, is an $ms\times nt$ matrix that partitioned into
$m\times n$ block matrices with the $(i,j)$-block being the $s\times t$ matrix
$a_{i,j}B$. In this paper, we are interested in complex block matrices. Let
$\mathbb{M}_{n}(\mathbb{M}_{k})$ be the set of complex matrices partitioned
into $n\times n$ blocks with each block being $k\times k$. The element of
$\mathbb{M}_{n}(\mathbb{M}_{k})$ is usually written as
${H}=[H_{i,j}]_{i,j=1}^{n}$, where $H_{i,j}\in\mathbb{M}_{k}$ for all $i,j$.
Now we introduce the definition of partial traces, which comes from Quantum
Information Theory [32, p. 12]. For $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$, the
first partial trace (map) $H\mapsto\mathrm{tr}_{1}H\in\mathbb{M}_{k}$ is
defined as the adjoint map of the embedding map $X\mapsto I_{n}\otimes
X\in\mathbb{M}_{n}\otimes\mathbb{M}_{k}$. Correspondingly, the second partial
trace (map) $H\mapsto\mathrm{tr}_{2}H\in\mathbb{M}_{n}$ is defined as the
adjoint map of the embedding map $Y\mapsto Y\otimes
I_{k}\in\mathbb{M}_{n}\otimes\mathbb{M}_{k}$. Therefore, we have
$\langle I_{n}\otimes X,H\rangle=\langle
X,\mathrm{tr}_{1}H\rangle,\quad\forall X\in\mathbb{M}_{k},$ (1)
and
$\langle Y\otimes I_{k},H\rangle=\langle
Y,\mathrm{tr}_{2}H\rangle,\quad\forall Y\in\mathbb{M}_{n},$
where $\langle\cdot,\cdot\rangle$ stands for the Hilbert-Schmidt inner
product, i.e., $\langle A,B\rangle={\rm tr}(A^{*}B)$. The above definition of
partial traces is implicit. Assume that $H=[H_{i,j}]_{i,j=1}^{n}$ is an
$n\times n$ block matrix with $H_{i,j}\in\mathbb{M}_{k}$, the visualized
version of the partial traces is equivalently given in [4, pp. 120–123] as
$\mathrm{tr}_{1}{H}=\sum\limits_{i=1}^{n}H_{i,i},$ (2)
and
$\mathrm{tr}_{2}{H}=\bigl{[}\mathrm{tr}H_{i,j}\bigr{]}_{i,j=1}^{n}.$
It is easy to see that both ${\rm tr}_{1}H$ and ${\rm tr}_{2}H$ are positive
semidefinite whenever ${H}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ is positive
semidefinite; see, e.g. [36, p. 237] or [37] for more details. The first or
second partial trace is a source for matrix inequalities and extensively
studied in recent years; see [2, 7, 11, 22, 29] for related topics.
Let $A=[A_{i,j}]_{i,j=1}^{n}$ be an $n\times n$ block matrix with each block
being a $k\times k$ matrix. The usual transpose of $A$ is defined as
$A^{T}=[A_{j,i}^{T}]_{i,j=1}^{n}$. We define the partial transpose of $A$ by
$A^{\tau}=[A_{j,i}]_{i,j=1}^{n}$, that is, the partial transpose of $A$ is the
matrix obtained by transposing blocks of $A$ independently. More precisely,
$A^{T}=\begin{bmatrix}A_{1,1}^{T}&\cdots&A_{n,1}^{T}\\\
\vdots&\ddots&\vdots\\\
A_{1,n}^{T}&\cdots&A_{n,n}^{T}\end{bmatrix}~{}~{}\text{and}~{}~{}A^{\tau}=\begin{bmatrix}A_{1,1}&\cdots&A_{n,1}\\\
\vdots&\ddots&\vdots\\\ A_{1,n}&\cdots&A_{n,n}\end{bmatrix}.$
Although $A$ and $A^{\tau}$ have the same trace, they may have different
eigenvalues, so they are not necessarily similar. Moreover, it is known that
$A\geq 0$ does not necessarily imply $A^{\tau}\geq 0$. For example, taking
$A=\begin{bmatrix}A_{1,1}&A_{1,2}\\\
A_{2,1}&A_{2,2}\end{bmatrix}=\left[\begin{array}[]{cc;{2pt/2pt}cc}1&0&0&1\\\
0&0&0&0\\\ \hdashline[2pt/2pt]0&0&0&0\\\ 1&0&0&1\end{array}\right].$ (3)
We can see from the definition that
$A^{\tau}=\begin{bmatrix}A_{1,1}&A_{2,1}\\\
A_{1,2}&A_{2,2}\end{bmatrix}=\left[\begin{array}[]{cc;{2pt/2pt}cc}1&0&0&0\\\
0&0&1&0\\\ \hdashline[2pt/2pt]0&1&0&0\\\ 0&0&0&1\end{array}\right].$
One could easily observe that $A$ is positive semidefinite, but $A^{\tau}$ is
not positive semidefinite since it contains a principal submatrix
$\left[\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right]\ngeq 0$.
Moreover, the eigenvalues of $A$ are $2,0,0,0$, and the eigenvalues of
$A^{\tau}$ are $1,1,1,-1$, so $A$ and $A^{\tau}$ are not similar. In addition,
replacing $A_{1,1}$ in the above matrix by $\left[\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right]$ also gives a well example. From this discussion,
we say that $A$ is positive partial transpose (or PPT for short) if both $A$
and $A^{\tau}$ are positive semidefinite. We recommend [10, 19, 26, 27] for
recent progress.
The paper is organized as follows. In Section 2, we shall review some
preliminaries for a class of matrices whose numerical ranges are contained in
a sector (known as the sector matrices). This is a natural extension of the
class of positive definite matrices. In Section 3, we shall study the recent
results involving the Fiedler–Markham inequality. We provide an extension of a
result of Lin [29], and our result is also an improvement of a result of Kuai
[17]; see Theorem 3.5. Moreover, we shall extend a result of Choi [6] to the
so-called sector matrices; see Theorem 3.7. In Section 4, we give a unified
treatment of a result of Ando [2] (or see [30]) as well as a recent result of
Li, Liu and Huang [22]. Our new treatment is more concise than original proof.
Moreover, we also present some Ando type determinant inequalities for partial
traces, and then we extend these inequalities to sector matrices; see Theorems
4.7 and 4.8. In Section 5, we shall prove some inequalities for positive
semidefinite $2\times 2$ block matrices; see Theorems 5.2, 5.3 and 5.4. Our
result extend slightly the recent elegant work on trace inequalities that
proved by Kittaneh and Lin [18] and Lin [26] as well.
## 2 Preliminaries
Recall that $\sigma_{i}(A)$ denotes $i$-th largest singular value of $A$. When
$A$ is Hermitian, we know that all eigenvalues of $A$ are real numbers, and we
write $\lambda_{i}(A)$ for the $i$-th largest eigenvalue. The numerical range
of $A\in\mathbb{M}_{n}$ is defined by
$W(A)=\\{x^{*}Ax:x\in\mathbb{C}^{n},x^{*}x=1\\}.$
For $\alpha\in[0,{\pi}/{2})$, let $S_{\alpha}$ be the sector on complex plane
defined as
$S_{\alpha}=\\{z\in\mathbb{C}:\Re z>0,|\Im z|\leq(\Re
z)\tan\alpha\\}=\\{re^{i\theta}:r>0,|\theta|\leq\alpha\\}.$
For $A\in\mathbb{M}_{n}$, the Cartesian (Toeptliz) decomposition is given as
$A=\Re A+i\cdot\Im A$, where
$\Re A=\frac{1}{2}(A+A^{*})$ and $\Im A=\frac{1}{2i}(A-A^{*})$.
We know from the definition that if $W(A)\subseteq S_{0}$, then $A$ is
positive definite. Moreover, it is easy to verify that if $W(A)\subseteq
S_{\alpha}$ for some $\alpha\in[0,{\pi}/{2})$, then $\Re(A)$ is positive
definite. Such class of matrices whose numerical ranges are contained in a
sector is called the sector matrices class. Clearly, the concept of sector
matrices is an extension of positive definite matrices. Over the past few
years, various studies on sector matrices have been obtained in the
literature; see, e.g., [8, 16, 17, 28, 34, 38].
Before starting our results, we now summarise the following lemmas.
###### Lemma 2.1
[28] Let $0\leq\alpha<{\pi}/{2}$ and $A\in\mathbb{M}_{n}$ with $W(A)\subseteq
S_{\alpha}$. Then
$|\det A|\leq(\sec\alpha)^{n}\det(\Re A).$
###### Lemma 2.2
[14, p. 510] Let $X$ be an $n$-square complex matrix. Then
$\lambda_{i}(\Re X)\leq\sigma_{i}(X),\quad i=1,2,\ldots,n.$
Moreover, if $\Re X$ is positive definite, then
$\det\Re X+|\det\Im X|\leq|\det X|.$
The following lemma is called the Fischer inequality, which gives an upper
bound for the determinant of a positive semidefinite block matrix in terms of
the determinants of its principal diagonal blocks. In particular, when all
blocks have order $1\times 1$, this inequality is also known as the Hadamard
inequality; see, e.g., [14, p. 506] and [36, p. 217].
###### Lemma 2.3
Let $H=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive
semidefinite. Then
$\det H\leq\prod_{i=1}^{n}\det H_{i,i}.$
###### Lemma 2.4
If $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ satisfies $W(H)\\!\subseteq
S_{\alpha}$, then $W({\rm tr}_{1}H)\\!\subseteq S_{\alpha}$ and $W({\rm
tr}_{2}H)\\!\subseteq S_{\alpha}$, i.e., if $H$ is a sector matrix with angle
$\alpha\in[0,\pi/2)$, then so are ${\rm tr}_{1}H$ and ${\rm tr}_{2}H$.
We remark that this lemma was partially proved in [17, Proposition 3.2] for
the case ${\rm tr}_{2}H$. Motivated by [17], we here include a detailed proof
for the remaining case ${\rm tr}_{1}H$.
Proof. Consider the Cartesian decomposition $H=\Re H+i\cdot\Im H$, then
${\rm tr}_{1}H={\rm tr}_{1}(\Re H)+i\cdot{\rm tr}_{1}(\Im H).$
For every $x\in\mathbb{C}^{k}$ with $x^{*}x=1$, as $\Re H$ is positive
definite, we get
$\Re\bigl{(}x^{*}({\rm tr}_{1}H)x\bigr{)}=x^{*}\bigl{(}\Re({\rm
tr}_{1}H)\bigr{)}x=x^{*}\bigl{(}{\rm tr}_{1}(\Re H)\bigr{)}x>0.$
On the other hand, by a direct computation,
$\frac{\left|\Im\bigl{(}x^{*}({\rm
tr}_{1}H)x\bigr{)}\right|}{\Re\bigl{(}x^{*}({\rm
tr}_{1}H)x\bigr{)}}=\frac{\left|x^{*}({\rm tr}_{1}(\Im H))x\right|}{x^{*}({\rm
tr}_{1}(\Re H))x}=\frac{\left|\langle xx^{*},{\rm tr}_{1}(\Im
H)\rangle\right|}{\langle xx^{*},{\rm tr}_{1}(\Re H)\rangle}.$
Note that $I_{n}\otimes(xx^{*})$ is positive semidefinite. We consider the
spectral decomposition
$I_{n}\otimes(xx^{*})=\sum_{i=1}^{nk}\lambda_{i}u_{i}u_{i}^{*},$
where $\lambda_{i}\geq 0$ and $u_{i}$ are unit vectors in $\mathbb{C}^{nk}$.
By the definition in (1), it follows that
$\displaystyle\frac{\left|\langle xx^{*},{\rm tr}_{1}(\Im
H)\rangle\right|}{\langle xx^{*},{\rm tr}_{1}(\Re H)\rangle}$
$\displaystyle=\frac{\left|\langle I_{n}\otimes(xx^{*}),\Im
H\rangle\right|}{\langle I_{n}\otimes(xx^{*}),\Re
H\rangle}=\frac{\left|\sum_{i=1}^{nk}\lambda_{i}\langle u_{i}u_{i}^{*},\Im
H\rangle\right|}{\sum_{i=1}^{nk}\lambda_{i}\langle u_{i}u_{i}^{*},\Re
H\rangle}$
$\displaystyle\leq\frac{\sum_{i=1}^{nk}\lambda_{i}\left|u_{i}^{*}(\Im
H)u_{i}\right|}{\sum_{i=1}^{nk}\lambda_{i}u_{i}^{*}(\Re
H)u_{i}}\leq\max_{1\leq i\leq nk}\frac{\left|u_{i}^{*}(\Im
H)u_{i}\right|}{u_{i}^{*}(\Re H)u_{i}}=\max_{1\leq i\leq
nk}\frac{\left|\Im(u_{i}^{*}Hu_{i})\right|}{\Re(u_{i}^{*}Hu_{i})}.$
This completes the proof.
Remark. Based on the second equivalent definition (2), one could also give
other ways to prove Lemma 2.4. We leave the details for the interested reader.
## 3 Extensions on Fiedler–Markham’s inequality
Let ${H}=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive
semidefinite. Recall that both ${\rm tr}_{1}H$ and ${\rm tr}_{2}H$ are
positive semidefinite; see, e.g., [37]. In 1994, Fiedler and Markham [9,
Corollary 1] proved a celebrated determinant inequality involving the second
partial trace.
###### Theorem 3.1
[9] Let ${H}=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be
positive semidefinite. Then
$\left(\frac{\det\bigl{(}{\rm tr}_{2}H\bigr{)}}{k}\right)^{k}\geq\det{H}.$
In 2016, Lin [29] revisited this inequality using some terminology from
quantum information theory, and gave an alternative proof of Theorem 3.1 by
applying an important identity connecting ${\rm tr}_{2}H$ and $H$. Moreover, a
natural question is that whether an analogous result corresponding to the
Fiedler–Markham inequality holds for ${\rm tr}_{1}H$. Lin [29] answered this
question and proved the following counterpart.
###### Theorem 3.2
[29] Let ${H}=[H_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be
positive semidefinite. Then
$\left(\frac{\det({\rm tr}_{1}H)}{n}\right)^{n}\geq\det H.$
It is clear that in the proof of both Theorem 3.1 and Theorem 3.2, Fiedler and
Markham, and Lin used the superadditivity of determinant functional, which
states that
$\det\left(\sum_{i=1}^{n}H_{i,i}\right)\geq\sum_{i=1}^{n}\det H_{i,i}\geq
n\left(\prod_{i=1}^{n}\det H_{i,i}\right)^{1/n}.$
This inequality can be improved by the Fan-Ky determinant inequality (see [14,
p. 488]), i.e., the log-concavity of the determinant over the cone of positive
semidefinite matrices:
$\det\left(\frac{1}{n}\sum_{i=1}^{n}H_{i,i}\right)\geq\left(\prod_{i=1}^{n}\det
H_{i,i}\right)^{1/n}.$ (4)
In addition, we mention here that a careful examination of the new proof of
Theorem 3.1 in [29] can also reveal this improvement. This improvement was
also pointed out in [6, 31]. Next, we state the strong version of Theorem 3.1
and Theorem 3.2.
###### Theorem 3.3
Let ${H}=[H_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive
semidefinite. Then
$\left(\frac{\det\bigl{(}{\rm tr}_{2}H\bigr{)}}{k^{n}}\right)^{k}\geq\det{H},$
and
$\left(\frac{\det({\rm tr}_{1}H)}{n^{k}}\right)^{n}\geq\det H.$
We observe in Theorem 3.3 that the second inequality seems easier to prove
than the first inequality because it is more convenient to build inequalities
on ${\rm tr}_{1}H=\sum_{i=1}^{n}H_{i,i}$. In [20], the authors showed that
both inequalities can be deduced mutually.
In 2018, Kuai [17] (or see [34]) further extended Theorem 3.3 to sector
matrices and showed that if $0\leq\alpha<{\pi}/{2}$ and
$H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ satisfies $W(H)\subseteq S_{\alpha}$,
then
$\left|\frac{\det({\rm tr}_{2}H)}{k^{n}}\right|^{k}\geq(\cos\alpha)^{nk}|\det
H|,$ (5)
and
$\left|\frac{\det({\rm tr}_{1}H)}{n}\right|^{n}\geq(\cos\alpha)^{(3n-2)k}|\det
H|.$ (6)
Our first goal in this section is to improve Kuai’s result (6). The key step
in our improvement is the following identity connecting ${\rm tr}_{1}(H)$ and
$H$, which has been applied to quantum information theory, such as the sub-
additivity of $q$-entropies. This identity can be found in [15, eq.(26)] or
[5, Lemma 2].
###### Lemma 3.4
Let $X$ and $Y$ be generalized Pauli matrices on $\mathbb{C}^{n}$; these
operators act as $Xe_{j}=e_{j+1}$ and $Ye_{j}=e^{2\pi j\sqrt{-1}/n}e_{j}$,
where $e_{j}$ is the $j$-th column of the identity matrix $I_{n}$ and
$e_{n+1}=e_{1}$. Then
$\frac{1}{n}\sum_{l,j=1}^{n}(X^{l}Y^{j}\otimes I_{k})H(X^{l}Y^{j}\otimes
I_{k})^{*}=I_{n}\otimes({\rm tr}_{1}H).$
Remark. The identity in this lemma can yield an alternative proof of Lemma
2.4. Moreover, the analogous identity for ${\rm tr}_{2}H$ can be seen in [15]
or [33, eq.(14)].
Now, we are ready to present an improvement on inequality (6).
###### Theorem 3.5
Let $0\leq\alpha<{\pi}/{2}$ and $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be such
that $W(H)\subseteq S_{\alpha}$. Then
$\left|\frac{\det({\rm tr}_{1}H)}{n^{k}}\right|^{n}\geq(\cos\alpha)^{nk}|\det
H|.$
Proof. Note that both $X$ and $Y$ in Lemma 3.4 are unitary, so are
$X^{l}Y^{j}\otimes I_{k}$ for all $l,j$. Moreover, we have $\Re(UHU^{*})=U(\Re
H)U^{*}$ for every unitary $U$. Thus,
$\displaystyle|\det H|\\!\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\prod_{l,j=1}^{n}\left|\det(X^{l}Y^{j}\otimes
I_{k})H(X^{l}Y^{j}\otimes I_{k})^{*}\right|^{1/n^{2}}$ (7)
$\displaystyle\overset{\text{Lemma \ref{lem22}}}{\leq}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!(\sec\alpha)^{nk}\prod_{l,j=1}^{n}\left(\det(X^{l}Y^{j}\otimes
I_{k})(\Re H)(X^{l}Y^{j}\otimes I_{k})^{*}\right)^{1/n^{2}}$
$\displaystyle\overset{\text{Fan-Ky ineq.(\ref{eqfk})}}{\leq}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!(\sec\alpha)^{nk}\det\Bigg{(}\frac{1}{n^{2}}\sum_{l,j=1}^{n}(X^{l}Y^{j}\otimes
I_{k})(\Re H)(X^{l}Y^{j}\otimes I_{k})^{*}\Bigg{)}$
$\displaystyle\overset{\text{Lemma \ref{lem24}}}{=}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!(\sec\alpha)^{nk}\det\left(\frac{1}{n}\Bigl{(}I_{n}\otimes{\rm
tr}_{1}(\Re H)\Bigr{)}\right)$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{(\sec\alpha)^{nk}}{n^{nk}}\det\Bigl{(}I_{n}\otimes{\rm
tr}_{1}(\Re H)\Bigr{)}.$
Clearly, we have ${\rm tr}_{1}(\Re H)=\Re({\rm tr}_{1}H)$. For
$X\in\mathbb{M}_{n}$ and $Y\in\mathbb{M}_{k}$, it is well-known that
$\det(X\otimes Y)=(\det X)^{k}(\det Y)^{n}$; see, e.g., [35, Chapter 2]. It
follows that
$\det\Bigl{(}I_{n}\otimes{\rm tr}_{1}(\Re H)\Bigr{)}=(\det
I_{n})^{k}\bigl{(}\det({\rm tr}_{1}\Re H)\bigr{)}^{n}=\bigl{(}\det\Re({\rm
tr}_{1}H)\bigr{)}^{n}.$
By Proposition 2.4, we have $W({\rm tr}_{1}H)\subseteq S_{\alpha}$, which
implies that $\Re({\rm tr}_{1}H)$ is positive definite. Therefore, by Lemma
2.2, we get
$\bigl{(}\det\Re({\rm tr}_{1}H)\bigr{)}^{n}\leq\bigl{(}|\det({\rm
tr}_{1}H)|-|\det\Im({\rm tr}_{1}H)|\bigr{)}^{n}\leq|\det({\rm tr}_{1}H)|^{n},$
which together with (7) yields the desired result.
Remark. By applying the techniques from [20], we know that Kuai’s inequality
(5) can also be deduced from the inequality in Theorem 3.5 and vice versa.
In the sequel, we shall focus our attention on some recent results which are
similar with the Fiedler–Markham inequality. Let
${H}=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be a block matrix
with $H_{i,j}=[h_{l,m}^{i,j}]_{l,m=1}^{k}$. We define an $n\times n$ matrix
$G_{l,m}$ as below.
$G_{l,m}:=\bigl{[}h_{l,m}^{i,j}\bigr{]}_{i,j=1}^{n}\in\mathbb{M}_{n}.$
A direct computation yields
${\rm
tr}_{1}H=\sum_{i=1}^{n}H_{i,i}=\sum_{i=1}^{n}\bigl{[}h_{l,m}^{i,i}\bigr{]}_{l,m=1}^{k}=\left[\begin{matrix}\sum\limits_{i=1}^{n}h_{l,m}^{i,i}\end{matrix}\right]_{l,m=1}^{k}=\bigl{[}{\rm
tr}\,G_{l,m}\bigr{]}_{l,m=1}^{k}.$
For notational convenience, we denote
$\widetilde{H}=\bigl{[}G_{l,m}\bigr{]}_{l,m=1}^{k}\in\mathbb{M}_{k}(\mathbb{M}_{n}).$
We can see that $\widetilde{H}$ is obtained from $H$ by rearranging the
entries in an appropriate order. The above observation yields ${\rm
tr}_{1}H={\rm tr}_{2}\widetilde{H}$. Moreover, it is not hard to check that
$\widetilde{H}$ and $H$ are unitarily similar; see, e.g., [6, Theorem 7] or
[20, Theorem 4]. Motivated by these relations, Choi [6] introduced recently
the definition of partial determinants corresponding to partial traces. For
$H=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$, the partial
determinants are defined as
$\mathrm{det}_{1}H:=\bigl{[}\det G_{l,m}\bigr{]}_{l,m=1}^{k},$
and
$\mathrm{det}_{2}H:=\bigl{[}\det H_{i,j}\bigr{]}_{i,j=1}^{n}.$
To some extent, the partial determinants share some common properties relative
to partial traces. For instance, it is easy to see that if
${H}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ is positive semidefinite, then both
$\mathrm{det}_{1}H$ and $\mathrm{det}_{2}H$ are positive semidefinite; see,
e.g. [36, p. 221]. Moreover, it was proved in [6] that
$\mathrm{det}({\rm tr}_{1}H)\geq{\rm tr}(\mathrm{det}_{2}H),$
and
$\mathrm{det}({\rm tr}_{2}H)\geq{\rm tr}(\mathrm{det}_{1}H).$
Additionally, Choi [6] proved two analogues of Theorem 3.1 and Theorem 3.2 for
partial determinants.
###### Theorem 3.6
[6] Let ${H}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive semidefinite. Then
$\left(\frac{{\rm tr}(\mathrm{det}_{1}H)}{k}\right)^{k}\geq\det H,$
and
$\left(\frac{{\rm tr}(\mathrm{det}_{2}H)}{n}\right)^{n}\geq\det H.$
Next, we will extend Theorem 3.6 to sector matrices. We write $|A|$ for the
nonnegative matrix whose entries are the absolute of the entries of $A$. This
notation is only used in the following theorem.
###### Theorem 3.7
Let $0\leq\alpha<{\pi}/{2}$ and $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be such
that $W(H)\subseteq S_{\alpha}$. Then
$\left(\frac{{\rm
tr}|\mathrm{det}_{1}H|}{k}\right)^{k}\geq(\cos\alpha)^{nk}|\det H|,$
and
$\left(\frac{{\rm
tr}|\mathrm{det}_{2}H|}{n}\right)^{n}\geq(\cos\alpha)^{nk}|\det H|.$
Proof. First of all, we shall prove the second inequality. We observe that
$\Re H_{1,1}$, $\ldots,\Re H_{n,n}$ are the diagonal block matrices of $\Re
H$. By Lemma 2.1 and Lemma 2.3, we obtain
$\displaystyle|\det H|$ $\displaystyle\leq(\sec\alpha)^{nk}\det(\Re
H)\leq(\sec\alpha)^{nk}\prod_{i=1}^{n}\det(\Re H_{i,i})$
$\displaystyle\leq(\sec\alpha)^{nk}\prod_{i=1}^{n}|\det
H_{i,i}|\leq(\sec\alpha)^{nk}\left(\frac{1}{n}\sum_{i=1}^{n}|\det
H_{i,i}|\right)^{n},$
where the third inequality follows from Lemma 2.2 and the last one follows
from the arithmetic mean-geometric mean inequality.
We now prove the first desired inequality by employing the relations between
$\mathrm{det}_{1}$ and $\mathrm{det}_{2}$. Recall that
$\widetilde{H}=[G_{l,m}]_{l,m=1}^{k}\in\mathbb{M}_{k}(\mathbb{M}_{n})$ and
$\mathrm{det}_{1}H=\mathrm{det}_{2}\widetilde{H}$. Since $\widetilde{H}$ and
$H$ are unitarily similar, we can get $\det\widetilde{H}=\det H$ and
$W(\widetilde{H})\subseteq S_{\alpha}$. Moreover, $\widetilde{H}$ is also
positive semidefinite. By applying the second inequality to $\widetilde{H}$,
we get
$\left(\frac{{\rm tr}|\mathrm{det}_{1}H|}{k}\right)^{k}=\left(\frac{{\rm
tr}|\mathrm{det}_{2}\widetilde{H}|}{k}\right)^{k}\geq(\cos\alpha)^{kn}|\det\widetilde{H}|=(\cos\alpha)^{kn}|\det
H|.$
This completes the proof.
## 4 Extensions on Ando’s inequality
To make our statements more transparent and compatible with the previous works
in the literature. In this section, we assume that $A$ is an $m\times m$ block
matrix with each block being an $n\times n$ matrix. Let
${A}=[A_{i,j}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive
semidefinite. We know that both ${\rm tr}_{1}A$ and ${\rm tr}_{2}A$ are
positive semidefinite; see, e.g., [36, p. 237] and [37, Theorem 2.1]. To some
degree, these two partial traces are closely related and mutually affect each
other. We write $\lVert
A\lVert_{q}=\left(\sum_{i}\sigma_{i}(A)^{q}\right)^{1/q}$ for the Schatten
$q$-norm of $A$. In 2007, Audenaert [1] proved the following norm inequality,
${\rm tr}\,A+\lVert A\lVert_{q}\geq\lVert{\rm tr}_{1}A\rVert_{q}+\lVert{\rm
tr}_{2}A\rVert_{q}.$ (8)
A straightforward argument exploiting Audenaert’s result leads to a proof of
the subadditivity of $q$-entropies (Tsallis entropies) for finite-dimensional
bipartite quantum states; see [1, 5] and references therein. In 2014, Ando [2]
(or see [30, Proposition 2.2] for an alternative proof) established the
following remarkable inequality in the sense of the Löwner ordering.
###### Theorem 4.1
[2, 30] Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite.
Then
$({\rm tr}A)I_{mn}+A\geq
I_{m}\otimes(\mathrm{tr}_{1}A)+(\mathrm{tr}_{2}A)\otimes I_{n}.$
Ando’s result reveals closely the interplay between the first and second
partial trace. Equivalently, this inequality can be rewritten as
$({\rm tr}A)I_{mn}-(\mathrm{tr}_{2}A)\otimes I_{n}\geq
I_{m}\otimes(\mathrm{tr}_{1}A)-A.$ (9)
We observe that the positivity of $A$, together with the identity ${\rm
tr}\,A=\sum_{i=1}^{m}{\rm tr}A_{i,i}={\rm tr}({\rm tr}_{2}A)$, leads to $({\rm
tr}A)I_{m}\geq\lambda_{\max}({\rm tr}_{2}A)I_{m}\geq{\rm tr}_{2}A$, which
guarantees that in (9) the left hand side $({\rm
tr}A)I_{mn}-(\mathrm{tr}_{2}A)\otimes I_{n}$ is positive semidefinite.
However, the two matrices of the right hand side in (9) might be incomparable.
For instance, the matrix $A$ in (3) gives an example. Motivated by this
observation, Li, Liu and Huang [22] presented a further generalization.
###### Theorem 4.2
[22] Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then
$({\rm tr}A)I_{mn}-({\rm tr}_{2}A)\otimes I_{n}\geq A-I_{m}\otimes({\rm
tr}_{1}A),$
and
$({\rm tr}A)I_{mn}+({\rm tr}_{2}A)\otimes I_{n}\geq A+I_{m}\otimes({\rm
tr}_{1}A).$
A map (not necessarily linear) $\Phi:\mathbb{M}_{n}\to\mathbb{M}_{k}$ is
called positive if it maps positive semidefinite matrices to positive
semidefinite matrices. A map $\Phi:\mathbb{M}_{n}\to\mathbb{M}_{k}$ is said to
be $m$-positive if for every $m\times m$ block matrix
$[A_{i,j}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$,
$[A_{i,j}]_{i,j=1}^{m}\geq 0\Rightarrow[\Phi(A_{i,j})]_{i,j=1}^{m}\geq 0.$
Clearly, being $1$-positive is equivalent to being positive. The map $\Phi$ is
said to be completely positive if it is $m$-positive for every integer $m\geq
1$. It is well-known that both the trace map and determinant map are
completely positive; see, e.g., [36, p. 221, p. 237] or [37]. On the other
hand, a map $\Phi$ is said to be $m$-copositive if for every
$[A_{i,j}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$,
$[A_{i,j}]_{i,j=1}^{m}\geq 0\Rightarrow[\Phi(A_{j,i})]_{i,j=1}^{m}\geq 0,$
and $\Phi$ is said to be completely copositive if it is $m$-copositive for
every integer $m\geq 1$. Furthermore, a map $\Phi$ is called completely PPT if
it is both completely positive and completely copositive; see [26, 10, 39] for
related topics.
Both Theorem 4.1 and Theorem 4.2 illustrated the implicit interaction and
connection between the first trace and second trace. The proof of Theorem 4.1
depends mainly on the 2-copositivity of $\Psi(X)=({\rm tr}X)I-X$; see e.g.,
[2] and [30] for more details. Correspondingly, the proof of Theorem 4.2
relies similarly on the 2-copositivity of $\Phi(X)=({\rm tr}X)I+X$; see [22].
For more application of these two maps, we refer readers to papers [26, 21].
In this section, we give a unified treatment of both Theorem 4.1 and Theorem
4.2. Our treatment is more concise than the original proof. We need to use a
recent result of Choi [6, 7], which investigates more relations between the
partial traces and the partial transpose.
###### Lemma 4.3
[6, 7] Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then
$({\rm tr}_{2}A^{\tau})\otimes I_{n}\geq\pm A^{\tau}$,
and
$I_{m}\otimes{\rm tr}_{1}A^{\tau}\geq\pm A^{\tau}.$
Now, we present a unified treatment of Theorems 4.1 and 4.2 as well.
New proof of Theorem 4.1. We define the map
$\Phi:\mathbb{M}_{m}(\mathbb{M}_{n})\to\mathbb{M}_{m}(\mathbb{M}_{n})$ as
$\Phi_{2}^{-}(X):=({\rm tr}_{2}X^{\tau})\otimes I_{n}-X^{\tau}.$
On the other hand, we define
$\Phi_{1}^{-}(X):=I_{m}\otimes{\rm tr}_{1}X^{\tau}-X^{\tau}.$
Lemma 4.3 implies that both $\Phi_{2}^{-}$ and $\Phi_{1}^{-}$ are positive
linear maps on $\mathbb{M}_{m}(\mathbb{M}_{n})$. Let $A$ be a positive
semidefinite block matrix. Thus, we have
$\Phi_{2}^{-}(A)=({\rm tr}_{2}A^{\tau})\otimes I_{n}-A^{\tau}\geq 0.$
Acting the map $\Phi_{1}^{-}$ to the matrix $\Phi_{2}^{-}(A)$, we can obtain
$\Phi_{1}^{-}\bigl{(}\Phi_{2}^{-}(A)\bigr{)}=I_{m}\otimes{\rm
tr}_{1}{\Phi_{2}^{-}(A)}^{\tau}-{\Phi_{2}^{-}(A)}^{\tau}\geq 0.$ (10)
By a directed computation, we can get ${\Phi_{2}^{-}(A)}^{\tau}=({\rm
tr}_{2}A)\otimes I_{n}-A$ and
${\rm tr}_{1}{\Phi_{2}^{-}(A)}^{\tau}={\rm tr}_{1}\bigl{(}({\rm
tr}_{2}A)\otimes I_{n}-A\bigr{)}=\sum_{i=1}^{m}({\rm tr}A_{i,i})I_{n}-{\rm
tr}_{1}A=({\rm tr}A)I_{n}-{\rm tr}_{1}A.$
Therefore, inequality (10) yields the desired result in Theorem 4.1.
$\blacksquare$
Remarks. In the above proof, we can see that Theorem 4.1 is just a direct
consequence of Lemma 4.3. To our surprise, Theorem 4.1 can also be proved by
using the positivity of $\Phi_{1}^{-}$ first, and then applying the positivity
of $\Phi_{2}^{-}$ later. More precisely, we first derive
${\Phi_{1}^{-}(A)}\geq 0$, and then we have $\Phi_{2}^{-}(\Phi_{1}^{-}(A))\geq
0$. Upon simplification, one can immediately get Theorem 4.1 again. We
summarize this observation as the following proposition.
###### Proposition 4.4
For every $X\in\mathbb{M}_{m}(\mathbb{M}_{n})$, we have
$\Phi_{1}^{-}(\Phi_{2}^{-}(X))=\Phi_{2}^{-}(\Phi_{1}^{-}(X))$.
Correspondingly, we can present an alternative proof of Theorem 4.2 similarly.
New proof of Theorem 4.2. We define the maps $\Phi_{2}^{+}$ and $\Phi_{1}^{+}$
on $\mathbb{M}_{m}(\mathbb{M}_{n})$ as
$\Phi_{2}^{+}(X):=({\rm tr}_{2}X^{\tau})\otimes I_{n}+X^{\tau},$
and
$\Phi_{1}^{+}(X):=I_{m}\otimes{\rm tr}_{1}X^{\tau}+X^{\tau}.$
We can see from Lemma 4.3 that both $\Phi_{2}^{+}$ and $\Phi_{1}^{+}$ are
positive linear maps. Similar to the lines of the previous proof, we get
$\Phi_{1}^{-}(\Phi_{2}^{+}(A))=\Phi_{2}^{+}(\Phi_{1}^{-}(A))\geq 0$, which
leads to
$({\rm tr}A)I_{mn}-({\rm tr}_{2}A)\otimes I_{n}\geq A-I_{m}\otimes({\rm
tr}_{1}A).$
Moreover, we have
$\Phi_{1}^{+}(\Phi_{2}^{-}(A))=\Phi_{2}^{-}(\Phi_{1}^{+}(A))\geq 0$. It
follows that
$({\rm tr}A)I_{mn}+({\rm tr}_{2}A)\otimes I_{n}\geq A+I_{m}\otimes({\rm
tr}_{1}A).$
We mention that the positivity of $\Phi_{1}^{+}(\Phi_{2}^{+}(A))$ yields a
trivial result. $\blacksquare$
In the remaining of this section, we shall pay attention to determinant
inequalities of sector matrices involving partial traces. Motivated by
Audenaert’s result (8), Lin [29] recently obtained a determinantal inequality
for partial traces, which states that if $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$
is positive semidefinite, then
$({\rm tr}A)^{mn}+\det A\geq\det({\rm tr}_{1}A)^{m}+\det({\rm tr}_{2}A)^{n}.$
(11)
We remark here that Fu, Lau and Tam [11, Corollary 2.2] recently improved (11)
when $A$ is a density matrix, i.e., a positive semidefinite matrix with trace
equal to $1$.
The key step in the proof of (11) attributes to Theorem 4.1 together with the
following interesting lemma. It is worth noting that Lemma 4.5 is graceful and
useful in deriving matrix inequalities; see, e.g., [23, 24, 25] for
applications on Oppenheim type inequalities.
###### Lemma 4.5
[30] Let $X,Y,W$ and $Z$ be positive semidefinite matrices of the same order.
If $X\geq W,X\geq Z$ and $X+Y\geq W+Z$, then
$\det X+\det Y\geq\det W+\det Z.$
Remark. We observe that Lemma 4.5 implies the determinant inequality:
$\det(A+B+C)+\det C\geq\det(A+C)+\det(B+C),$
where $A,B$ and $C$ are positive semidefinite matrices.
With the help of Lemma 4.5, we can easily present two analogues of (11).
###### Proposition 4.6
Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then
$({\rm tr}A)^{mn}+\det({\rm tr}_{1}A)^{m}\geq\det A+\det({\rm tr}_{2}A)^{n},$
and
$({\rm tr}A)^{mn}+\det({\rm tr}_{2}A)^{n}\geq\det A+\det({\rm tr}_{1}A)^{m}.$
Proof. We prove the first inequality only, since the second one can be proved
in exactly the same way. Let $X=({\rm tr}A)I_{mn},Y=I_{m}\otimes({\rm
tr}_{1}A),W=A$ and $Z=({\rm tr}_{2}A)\otimes I_{n}$. It is easy to see that
$({\rm tr}A)I_{m}=\sum_{i=1}^{m}({\rm tr}A_{i,i})I_{m}=\bigl{(}{\rm tr}({\rm
tr}_{2}A)\bigr{)}I_{m}\geq\lambda_{\max}({\rm tr}_{2}A)I_{m}\geq{\rm
tr}_{2}A,$
which implies that $X\geq Z\geq 0$, and clearly $X\geq W\geq 0$. Moreover,
Theorem 4.2 says that $X+Y\geq W+Z$. That is, all conditions in Lemma 4.5 are
satisfied. Therefore,
$\displaystyle({\rm tr}A)^{mn}+\det\bigl{(}I_{m}\otimes({\rm
tr}_{1}A)\bigr{)}\geq\det A+\det\bigl{(}({\rm tr}_{2}A)\otimes I_{n}\bigr{)}.$
It is well-known [35, p. 37] that for every $X\in\mathbb{M}_{m}$ and
$Y\in\mathbb{M}_{n}$,
$\det(X\otimes Y)=(\det X)^{n}(\det Y)^{m}.$
Thus, we complete the proof of the required result.
We next give an improvement on Proposition 4.6.
###### Theorem 4.7
Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then
$({\rm tr}A)^{mn}+\det({\rm tr}_{1}A)^{m}\geq m^{nm}\bigl{(}\det A+\det({\rm
tr}_{2}A)^{n}\bigr{)},$
and
$({\rm tr}A)^{mn}+\det({\rm tr}_{2}A)^{n}\geq n^{mn}\bigl{(}\det A+\det({\rm
tr}_{1}A)^{m}\bigr{)}.$
Proof. We only prove the second inequality. Invoking Theorem 3.3, we get
$\left(\frac{\det({\rm tr}_{2}A)}{n^{m}}\right)^{n}\geq\det A.$
Equivalently, we have $\det({\rm tr}_{2}A)^{n}\geq n^{mn}\det A$. It suffices
to show that
$({\rm tr}A)^{n}\geq n^{n}\det({\rm tr}_{1}A).$
Note that
${\rm tr}A=\sum_{i=1}^{m}{\rm tr}(A_{i,i})={\rm
tr}\left(\sum_{i=1}^{m}A_{i,i}\right)={\rm tr}({\rm tr}_{1}A).$
We denote $X:={\rm tr}_{1}A$, which is a positive semidefinite matrix of order
$n$. So we need to prove that $({\rm tr}X)^{n}\geq n^{n}\det X$. This is
equivalent to showing
$\left(\sum_{i=1}^{n}\lambda_{i}(X)\right)^{n}\geq
n^{n}\prod_{i=1}^{n}\lambda_{i}(X),$
which is a direct consequence of the AM-GM inequality.
Surprisingly, the proof of Theorem 4.7 seems simpler than that of Proposition
4.6 since it does not rely on Theorem 4.2 and Lemma 4.5. However, it allows us
to provide a great improvement on Proposition 4.6 whenever $m,n$ are large
integers.
In the sequel, we shall denote $|A|=(A^{*}A)^{1/2}$, which is called the
modulus of $A$. We remark that this notation is different from that in Theorem
3.7. Note that $|A|$ is positive semidefinite, and the eigenvalues of $|A|$
are called the singular values of $A$. In 2019, Yang, Lu and Chen [34]
extended (11) to sector matrices.
$({\rm tr}|A|)^{mn}+\det|A|\geq(\cos\alpha)^{mn}|\det({\rm
tr}_{1}A)|^{m}+(\cos\alpha)^{mn}|\det({\rm tr}_{2}A)|^{n}.$
Now, we are ready to present an extension on Theorem 4.7.
###### Theorem 4.8
Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be such that $W(A)\subseteq
S_{\alpha}$. Then
$({\rm tr}|A|)^{mn}+\left|{\det({\rm
tr}_{1}A)}\right|^{m}\geq(m\cos\alpha)^{mn}\bigl{(}\det|A|+|\det({\rm
tr}_{2}A)|^{n}\bigr{)},$
and
$({\rm tr}|A|)^{mn}+\left|{\det({\rm
tr}_{2}A)}\right|^{n}\geq(n\cos\alpha)^{mn}\bigl{(}\det|A|+|\det({\rm
tr}_{1}A)|^{m}\bigr{)}.$
Proof. We only prove the first inequality. According to the definition of
$S_{\alpha}$, if $W(A)\subseteq S_{\alpha}$, then $\Re A$ is positive definite
and its trace is positive. By Lemma 2.2, we have
${\rm tr}|A|=\sum_{i=1}^{mn}\sigma_{i}(A)\geq\sum_{i=1}^{mn}\lambda_{i}(\Re
A)={\rm tr}(\Re A)\geq 0.$
It is noteworthy by Lemma 2.4 that $W({\rm tr}_{1}A)\subseteq S_{\alpha}$ and
$W({\rm tr}_{2}A)\subseteq S_{\alpha}$. Clearly, we have $\Re({\rm
tr}_{1}A)={\rm tr}_{1}(\Re A)$ and $\Re({\rm tr}_{2}A)={\rm tr}_{2}(\Re A)$.
By setting $X={\rm tr}_{1}A$ in Lemma 2.2, we get
$|\det({\rm tr}_{1}A)|\geq\det\bigl{(}\Re({\rm
tr}_{1}A)\bigr{)}=\det\bigl{(}{\rm tr}_{1}(\Re A)\bigr{)}.$
Note that $\Re A$ is positive semidefinite. By applying Theorem 4.7, we can
obtain
$\displaystyle({\rm tr}|A|)^{mn}+\left|{\det({\rm tr}_{1}A)}\right|^{m}$
$\displaystyle\geq({\rm tr}\,\Re A)^{mn}+\bigl{(}{\det{\rm tr}_{1}(\Re
A)}\bigr{)}^{m}$ $\displaystyle\geq m^{nm}\bigl{(}\det(\Re
A)+\bigl{(}\det\Re({\rm tr}_{2}A)\bigr{)}^{n}\bigr{)}$
$\displaystyle\geq(m\cos\alpha)^{mn}|\det A|+(m\cos\alpha)^{mn}|\det({\rm
tr}_{2}A)|^{n},$
where the last inequality holds from Lemma 2.2 by setting $X=A$ and ${\rm
tr}_{2}A$ respectively.
## 5 Trace inequalities for two by two block matrices
Positive semidefinite $2\times 2$ block matrices are extensively studied, such
a partition yields a great deal of versatile and elegant matrix inequalities;
see, e.g., [13, 18, 21, 12] for details. Recently, Kittaneh and Lin [18] (or
see [26]) proved the following trace inequalities.
###### Theorem 5.1
[18, 26] Let $\begin{bmatrix}A&B\\\
B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive
semidefinite. Then
${\rm tr}A\,{\rm tr}C-{\rm tr}B^{*}\,{\rm tr}B\geq\bigl{|}{\rm tr}AC-{\rm
tr}B^{*}B\bigr{|},$
and
${\rm tr}A\,{\rm tr}C+{\rm tr}B^{*}\,{\rm tr}B\geq{\rm tr}AC+{\rm tr}B^{*}B.$
In this section, we present some inequalities related to trace for $2\times 2$
block matrices, which are slight extensions of the result of Kittaneh and Lin.
We now need to introduce some notations. Let
$\otimes^{r}A:=A\otimes\cdots\otimes A$ be the $r$-fold tensor power of $A$.
###### Theorem 5.2
Let $\begin{bmatrix}A&B\\\
B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive
semidefinite. Then for $r\in\mathbb{N}^{*}$,
$(\mathrm{tr}A\,\mathrm{tr}C)^{r}-({\rm
tr}B^{*}\,\mathrm{tr}B)^{r}\geq\bigl{|}(\mathrm{tr}AC)^{r}-(\mathrm{tr}B^{*}B)^{r}\bigr{|},$
and
$(\mathrm{tr}A\,\mathrm{tr}C)^{r}+({\rm
tr}B^{*}\,\mathrm{tr}B)^{r}\geq(\mathrm{tr}AC)^{r}+(\mathrm{tr}B^{*}B)^{r}.$
Proof. Note that $\begin{bmatrix}\\!\\!\\!\otimes^{r}A&\otimes^{r}B\\\
\otimes^{r}B^{*}&\otimes^{r}C\end{bmatrix}$ is a principal submatrix of
$\otimes^{r}\begin{bmatrix}A&B\\\ B^{*}&C\end{bmatrix}$. Thus
$\begin{bmatrix}\\!\\!\\!\otimes^{r}A&\otimes^{r}B\\\
\otimes^{r}B^{*}&\otimes^{r}C\end{bmatrix}$
is again positive semidefinite. By applying Theorem 5.1 to this block matrix,
we get
$\bigl{|}{\rm tr}(\otimes^{r}A)(\otimes^{r}C)-{\rm
tr}(\otimes^{r}B^{*})(\otimes^{r}B)\bigr{|}\leq{\rm
tr}\otimes^{r}\\!\\!A\,{\rm tr}\otimes^{r}\\!\\!C-{\rm
tr}\otimes^{r}\\!B^{*}{\rm tr}\otimes^{r}\\!\\!B,$
and
${\rm tr}(\otimes^{r}A)(\otimes^{r}C)+{\rm
tr}(\otimes^{r}B^{*})(\otimes^{r}B)\leq{\rm tr}\otimes^{r}\\!\\!A\,{\rm
tr}\otimes^{r}\\!\\!C+{\rm tr}\otimes^{r}\\!B^{*}{\rm tr}\otimes^{r}\\!\\!B.$
Invoking the well-known facts [35, Chapter 2]:
$(\otimes^{r}X)(\otimes^{r}Y)=\otimes^{r}(XY)$ and ${\rm
tr}(\otimes^{r}X)=({\rm tr}\,X)^{r}$, the desired inequalities follow
immediately.
Remark. Theorem 5.2 was proved in the first version of our manuscript
(announced on March 10, 2020, arXiv: 2003.04520v1). We remark that this result
was recently and independently rediscovered by Fu and Gumus in [12] using a
quite different method.
Let $e_{t}(X)$ denote the $t$-th elementary symmetric function of the
eigenvalues of the square matrix $X$.
$e_{t}(X):=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{t}\leq
n}\prod_{j=1}^{t}\lambda_{i_{j}}(X).$
In particular, we know that $e_{1}(X)={\rm tr}(X)$. We can get the following
theorem.
###### Theorem 5.3
Let $\begin{bmatrix}A&B\\\
B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive
semidefinite. Then for $t\in\\{1,2,\ldots,k\\}$,
$e_{t}(A)e_{t}(C)-e_{t}(B^{*})e_{t}(B)\geq|e_{t}(AC)-e_{t}(B^{*}B)|,$
and
$e_{t}(A)e_{t}(C)+e_{t}(B^{*})e_{t}(B)\geq e_{t}(AC)+e_{t}(B^{*}B).$
Proof. The first inequality can be found in [18, Corollary 2.7]. We next give
the outline of the proof of the second one. Note that
$\begin{bmatrix}\\!\\!\\!\otimes^{t}A&\otimes^{t}B\\\
\otimes^{t}B^{*}&\otimes^{t}C\end{bmatrix}$ is positive semidefinite. By
restricting this block matrix to the symmetric class of tensor product (see,
e.g., [3, pp. 16–20]), we know that
$\begin{bmatrix}\\!\\!\\!\wedge^{t}A&\wedge^{t}B\\\
\wedge^{t}B^{*}&\wedge^{t}C\end{bmatrix}$
is still positive semidefinite. Note that $e_{t}(X)={\rm tr}(\wedge^{t}X)$ and
$(\wedge^{t}X)(\wedge^{t}Y)=\wedge^{t}(XY)$. Applying Theorem 5.1 to this
block matrix yields the required result.
Let $s_{t}(X)$ be the $t$-th complete symmetric polynomial of eigenvalues of
$X$, i.e.,
$s_{t}(X):=\sum_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{t}\leq
n}\prod_{j=1}^{t}\lambda_{i_{j}}(X).$
Clearly, we have $s_{1}(X)={\rm tr}(X)$. We can get the following slight
extension similarly.
###### Theorem 5.4
Let $\begin{bmatrix}A&B\\\
B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive
semidefinite. Then for $t\in\\{1,2,\ldots,k\\}$,
$s_{t}(A)s_{t}(C)-s_{t}(B^{*})s_{t}(B)\geq|s_{t}(AC)-s_{t}(B^{*}B)|,$
and
$s_{t}(A)s_{t}(C)+s_{t}(B^{*})s_{t}(B)\geq s_{t}(AC)+s_{t}(B^{*}B).$
Proof. Note that $\begin{bmatrix}\\!\\!\\!\otimes^{t}A&\otimes^{t}B\\\
\otimes^{t}B^{*}&\otimes^{t}C\end{bmatrix}$ is positive semidefinite. By
restricting this block matrix to the symmetric class of tensor product (see,
e.g., [3, pp. 16–20]), we know that
$\begin{bmatrix}\\!\\!\\!\vee^{t}A&\vee^{t}B\\\
\vee^{t}B^{*}&\vee^{t}C\end{bmatrix}$
is still positive semidefinite. Similarly, we know that ${\rm
tr}(\vee^{t}X)=s_{t}(X)$ and $(\vee^{t}X)(\vee^{t}Y)=\vee^{t}(XY)$. Applying
Theorem 5.1 to this block matrix leads to the desired result.
## Acknowledgments
This paper is dedicated to Prof. Weijun Liu (Central South University) on his
60th birthday, October 22 of the lunar calendar in 2021. I would like to thank
Prof. Yuejian Peng for reading carefully through an earlier version of this
paper. This work was supported by NSFC (Grant No. 11931002).
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|
2024-09-04T02:54:58.633886 | 2020-03-10T10:46:12 | 2003.04628 | {
"authors": "Ygor Gallina, Florian Boudin, B\\'eatrice Daille",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26130",
"submitter": "Ygor Gallina",
"url": "https://arxiv.org/abs/2003.04628"
} | arxiv-papers | # Large-Scale Evaluation of Keyphrase Extraction Models
Ygor Gallina<EMAIL_ADDRESS>LS2N, Université de
NantesNantesFrance , Florian Boudin<EMAIL_ADDRESS>LS2N,
Université de NantesNantesFrance and Béatrice Daille beatrice.daille@univ-
nantes.fr LS2N, Université de NantesNantesFrance
(2020)
###### Abstract.
Keyphrase extraction models are usually evaluated under different, not
directly comparable, experimental setups. As a result, it remains unclear how
well proposed models actually perform, and how they compare to each other. In
this work, we address this issue by presenting a systematic large-scale
analysis of state-of-the-art keyphrase extraction models involving multiple
benchmark datasets from various sources and domains. Our main results reveal
that state-of-the-art models are in fact still challenged by simple baselines
on some datasets. We also present new insights about the impact of using
author- or reader-assigned keyphrases as a proxy for gold standard, and give
recommendations for strong baselines and reliable benchmark datasets.
Keyphrase generation, natural language processing, evaluation
††copyright: acmcopyright††journalyear: 2020††doi:
10.1145/1122445.1122456††conference: JCDL ’20: ACM/IEEE Joint Conference on
Digital Libraries; August 01–05, 2020; Xi’an, Shaanxi, China††booktitle: JCDL
’20: ACM/IEEE Joint Conference on Digital Libraries, August 01–05, 2020,
Xi’an, Shaanxi, China††isbn: 978-1-4503-XXXX-X/20/06††ccs: Information systems
Digital libraries and archives††ccs: Information systems Information
retrieval††ccs: Computing methodologies Information extraction
## 1\. Introduction
Keyphrases are single or multi-word lexical units that represent the main
concepts in a document (Evans and Zhai, 1996). They are particularly useful
for indexing, searching and browsing digital libraries (Barker et al., 1972;
Zhai, 1997; Gutwin et al., 1999; Witten et al., 2009), and have proven
themselves as effective features in many downstream natural language
processing tasks (Hulth and Megyesi, 2006; Litvak and Last, 2008; Berend,
2011). Still, most documents do not have assigned keyphrases, and manual
annotation is simply not a feasible option (Mao and Lu, 2017). There is
therefore a great need for automated methods to assign relevant keyphrases to
documents.
Automatic keyphrase extraction111Also referred to as keyphrase generation or
keyphrase annotation. – that is, the task of extracting keyphrases either from
the content of the document or from a controlled vocabulary – has received
much attention from the research community (Kim et al., 2010; Gollapalli et
al., 2015; Augenstein et al., 2017). Thus, many keyphrase extraction models
were proposed over the last years, ranging from early statistics-based models
(Witten et al., 1999), to popular graph-based ranking models (Mihalcea and
Tarau, 2004), and recent neural models (Meng et al., 2017). However, because
of the great discrepancies in experimental setups among past studies, it is
very difficult to compare and contrast the effectiveness of these models, and
even more so to assess the progress of the field as a whole.
More specifically, we observe striking differences in how models are
parameterized, evaluated and compared in previous work. To name just a few
examples, experiments are most often conducted on different benchmark
datasets, all of which differ in domain, size, language or quality of the gold
standard (that is, reference keyphrases supplied by authors, readers or
professional indexers). This not only makes the reported results hard to
contrast, but also has a profound impact on trained model performance (Gallina
et al., 2019). In addition, and since there is no consensus as to which
evaluation metric is most reliable for keyphrase extraction (Zesch and
Gurevych, 2009; Hussey et al., 2012; Hasan and Ng, 2014), diverse measures are
commonly seen in the literature, thus preventing any further direct
comparisons. Moreover, the evaluation of missing keyphrases – that is, gold
keyphrases that do not occur in the content of the document – is still an open
question and there is little agreement on whether they should be included or
not (Kim et al., 2010).
We strongly believe that this lack of empirical rigor is a real hindrance to
progress on keyphrase extraction, and that a systematic comparison of existing
models under the same conditions is needed to fully understand how they
actually perform. In this work, we resolve this issue by conducting the first
large-scale study on automatic keyphrase extraction. More precisely, we
present an extensive comparative analysis of state-of-the-art keyphrase
extraction models involving 9 benchmark datasets from various domains. To
ensure controlled, fair and reliable experiments, we embarked upon the
difficult process of re-implementing all of the models presented in this
paper222Link to the code will appear here after the review period. and pre-
processing the datasets in a unified and systematic way333Link to the datasets
will appear here after the review period..
Using these new large-scale experimental results, we seek to better understand
how well state-of-the-art models perform across sources, domains and
languages. We also go further than prior work and investigate the following
research questions:
1. (1)
How much progress have we made on keyphrase extraction since early models?
2. (2)
What is the impact of using non-expert gold standards, that is, author- or
reader-assigned keyphrases, when training and evaluating keyphrase extraction
models?
3. (3)
Which baselines and benchmark datasets should be included in future work for a
better understanding of the pros and cons of a newly proposed model?
## 2\. Benchmark Datasets
Benchmark datasets for evaluating automatic keyphrase extraction cover a wide
range of sources ranging from scientific articles and web pages to twitter and
email messages. We collected 9 of the most widely used datasets which we
believe are representative of the different sources and domains found in
previous work. Detailed statistics for each selected dataset are shown in
Table 2. They are grouped into three categories that are outlined below:
Scientific articles:
Among the selected datasets, three are composed of full-text scientific
publications: ACM (Krapivin et al., 2009) and SemEval (Kim et al., 2010) about
computer science, and PubMed (Schutz, 2008) from the medical domain. Not
surprisingly, they contain only a small number of documents due to copyright
reasons. These datasets provide author-assigned keyphrases which serve as a
reasonable, but far from perfect, proxy for expert annotations. In the case of
SemEval, student annotators were hired to extend gold annotation labels.
Paper abstracts:
Scientific abstracts, often referred to as bibliographic records, are arguably
the most prevalent documents for benchmarking keyphrase extraction. They are
readily available in great quantities and come with author-assigned keyphrases
that can be used as gold standard. We gathered three datasets, all dealing
with the computer science domain: Inspec (Hulth, 2003), WWW (Caragea et al.,
2014) and KP20k (Meng et al., 2017). It is worth noting that with more than
half a million documents, KP20k is the largest dataset to date and one of the
few that is large enough to train neural models.
News articles:
News texts are the last source of documents present among the collected
datasets. Similar to paper abstracts, online news are available in large
quantities and can be easily mined from the internet. We selected the
following three datasets: DUC-2001 (Wan and Xiao, 2008), 500N-KPCrowd (Marujo
et al., 2012) and KPTimes (Gallina et al., 2019). The first two datasets
provide reader-assigned keyphrases, while KPTimes supplies indexer-assigned
key-phrases extracted from metadata and initially intended for search engines.
It is interesting to observe that only two datasets in our study, namely
Inspec and KPTimes, provide gold keyphrases annotated by professional
indexers.
Dataset | Ann. | Train | Test | #words | #kp | %abs
---|---|---|---|---|---|---
5pt. PubMed (Schutz, 2008) | $A$ | - | 1 320 | 5 323 | 5.4 | 16.9
ACM (Krapivin et al., 2009) | $A$ | - | 2 304 | 9 198 | 5.3 | 16.3
SemEval (Kim et al., 2010) | $A\cup R$ | 144 | 100 | 7 961 | 14.7 | 19.7
Scientific articles (avg.) | 7 494 | 8.5 | 17.6
5pt. Inspec (Hulth, 2003) | $I$ | 1 000 | 500 | 135 | 9.8 | 22.4
WWW (Caragea et al., 2014) | $A$ | - | 1 330 | 164 | 4.8 | 52.0
KP20k (Meng et al., 2017) | $A$ | 530K | 20K | 176 | 5.3 | 42.6
Paper abstracts (avg.) | 158 | 6.6 | 39.0
5pt. DUC-2001 (Wan and Xiao, 2008) | $R$ | - | 308 | 847 | 8.1 | 3.7
KPCrowd (Marujo et al., 2012) | $R$ | 450 | 50 | 465 | 46.2 | 11.2
KPTimes (Gallina et al., 2019) | $I$ | 260K | 10K | 921 | 5.0 | 54.7
News articles (avg.) | 744 | 19.8 | 23.2
Table 1. Statistics of the datasets. Gold annotation is supplied by authors
($A$), readers ($R$) or professional indexers ($I$). The number of documents
in the training and testing splits are shown. The average number of keyphrases
(#kp) and words (#words) per document, and the ratio of missing keyphrases
(%abs) are computed on the test set.
Datasets containing scientific articles or abstracts rely primarily on author-
assigned keyphrases as gold standard. They therefore exhibit similar
properties for the average number of ground truth keyphrases per document
($\approx 5$). On the other hand, articles are on average significantly longer
than abstracts ($\approx 7500$ words vs. $\approx 160$ words respectively) and
consequently reveal a much smaller fraction of missing keyphrases ($\approx
18\%$ vs. $\approx 39\%$ respectively). Datasets with reader-assigned
keyphrases exhibit the lowest numbers of missing keyphrases, which can be
explained by the fact that readers appear to produce gold-standard annotations
in an extractive fashion (Wang et al., 2015). We also confirmed this
empirically by computing the ratio of missing keyphrases in the author-
assigned ($24\%$) and reader-assigned ($17.5\%$) gold annotations of the
SemEval dataset.
In contrast, the opposite trend is observed for KPTimes that comes with gold
standards annotated by professional indexers and that shows the highest
percentage of missing keyphrases ($54.7\%$). This indicates the the more
abstractive nature of indexer-assigned keyphrases. Put differently, it is
known that non-expert annotations are less constrained and may include seldom-
used variants or misspellings (Sood et al., 2007), whereas indexers strive to
rely on a consistent terminology and assign the same keyphrase to all
documents for a given topic, even when it does not occur in these documents.
To investigate this further, we looked at how many variants of an index term,
in this case “artificial neural network”, could be found in the author-
assigned keyphrases of KP20k. All in all, we found dozens of variants for this
term, including “neural network”, “neural network (nns)”, “neural net”,
“artificial neural net” or “nn”. This apparent lack of annotation consistency
intuitively has two consequences: 1) it makes it harder for supervised
approaches to learn a good model, 2) it makes automatic evaluation much less
reliable as it is based on exact string matching.
It is important to stress that datasets containing scientific articles may
contain noisy texts. Indeed, most articles were automatically converted from
PDF format to plain text and thus are likely to contain irrelevant pieces of
text (e.g. muddled sentences, equations). Previous work show that noisy inputs
undermine the overall performance of keyphrase extraction models (Boudin et
al., 2016). In this study, we do not insist on a perfect input and we are
aware that reported results may be improved with an increase in pre-processing
effort.
## 3\. Models
Roughly speaking, previous works on keyphrase extraction can be divided into
two groups depending on whether they adopt a supervised learning procedure or
not. This section starts by introducing the baselines we will use in our
experiments, and then proceeds to describe the state-of-the-art keyphrase
extraction models we re-implemented sorted into the aforementioned two groups.
### 3.1. Baselines
Having strong baselines to compare with is a prerequisite for contrasting the
results of proposed models. In previous studies, various baselines were
considered, complicating the analysis and interpretation of the reported
results. Our stance here is to establish three baselines, each associated with
a particular feature that is commonly used in keyphrase extraction models. All
baselines are also unsupervised, allowing their use and performance analysis
on any of the benchmark datasets
Keyphrase position is a strong signal for both unsupervised and supervised
models, simply because texts are usually written so that the most important
ideas go first (Marcu, 1997). In single document summarization for example,
the lead baseline –that is, the first sentences from the document–, while
incredibly simple, is still a competitive baseline (Kedzie et al., 2018).
Similar to the lead baseline, we propose the FirstPhrases baseline that
extracts the first $N$ keyphrase candidates from a document. We are not aware
of any previous work reporting that baseline, yet, as we will see in §5, it
achieves remarkably good results.
Graph-based ranking models for keyphrase extraction are, perhaps, the most
popular models in the literature. Therefore, as a second baseline, we use
TextRank (Mihalcea and Tarau, 2004), which weights keyphrase candidates using
a random walk over a word-graph representation of the document. In a nutshell,
TextRank defines the importance of a word in terms of how it relates to other
words in the document, and ranks candidates according to the words they
contain.
The third baseline, TF$\times$IDF (Salton and Buckley, 1988), have been
repeatedly used in previous comparative studies (Kim et al., 2010; Meng et
al., 2017, inter alia). In contrast with the other two baselines that do no
require any resources whatsoever (beyond the document itself), TF$\times$IDF
makes use of the statistics collected from unlabelled data to weight keyphrase
candidates. As such, it often gives better results, in some cases even on par
with state-of-the-art models (Ye and Wang, 2018).
### 3.2. Unsupervised models
Annotated data are not always available or easy to obtain, which motivates the
further development of unsupervised models for keyphrase extraction. Besides,
looking back at previous work, most attempts to address this problem employ
unsupervised approaches. In this study, we selected three recent state-of-the-
art models based on their reported performance.
The first model we investigate is PositionRank (Florescu and Caragea, 2017), a
graph-based model that incorporates two features (position and frequency) into
a biased PageRank algorithm. This model operates at the word level, and
assigns a score to each candidate using the sum of its individual word scores.
As such, it suffers from over-generation errors444These errors occur when a
model correctly outputs a keyphrase because it contains an important word, but
at the same time erroneously predicts other keyphrases because they contain
the same word. (Hasan and Ng, 2014), but still achieves good performance on
short texts.
The second model we consider, MPRank (Boudin, 2018), relies on a multipartite
graph representation to enforce topical diversity while ranking keyphrase
candidates. It includes a mechanism to incorporate keyphrase selection
preferences in order to introduce a bias towards candidates occurring first in
the document. MultipartiteRank was shown to consistently outperform other
unsupervised graph-based ranking models.
Both aforementioned models only exploit the document itself to extract
keyphrases. The third model we include, EmbedRank (Bennani-Smires et al.,
2018), leverages sentence embeddings for ranking keyphrase candidates.
Candidates are weighted according to their cosine distance to the document
embedding, while diversity in the selected keyphrases is promoted using
Maximal Marginal Relevance (MMR) (Goldstein and Carbonell, 1998). Despite its
simplicity, this model was shown to outperform other unsupervised models on
short texts (abstracts and news).
### 3.3. Supervised models
Supervised models can be further divided into two categories, depending on
whether they rely on a neural network or not.
Traditional supervised models treat the keyphrase extraction problem as a
binary classification task. Here, we include such a model, namely Kea (Witten
et al., 1999), in order to precisely quantify the performance gap with recent
neural-based models. KEA uses a Naive Bayes classifier trained on a set of
only two handcrafted features we have elected as baseline features: the
TF$\times$IDF score of the candidate and the normalized position of its first
occurrence in the document. Previous work has reported confusing and
conflicting results555On SemEval, (Meng et al., 2017) report an F@10 score of
$2.6$ while (Boudin, 2016) report a score of $19.3$. for Kea, raising
questions about how it actually performs.
Neural models for keyphrase extraction rely on an encoder-decoder architecture
(Cho et al., 2014; Sutskever et al., 2014) with an attention mechanism
(Bahdanau et al., 2014; Luong et al., 2015). Training these models require
large amounts of annotated training data, and is therefore only possible on
the KP20k and KPTimes datasets. The second supervised model we include in this
study is CopyRNN (Meng et al., 2017), an encoder-decoder model that
incorporates a copying mechanism (Gu et al., 2016) in order to be able to
predict phrases that rarely occur. When properly trained, this model was shown
to be very effective in extracting keyphrases from scientific abstracts.
The third supervised model we use, CorrRNN (Chen et al., 2018), extends the
aforementioned model by introducing correlation constraints. It employs a
coverage mechanism (Tu et al., 2016) that diversifies attention distributions
to increase topic coverage, and a review mechanism to avoid generating
duplicates. As such, it produces more diverse and less redundant keyphrases.
Note that only neural models have the ability to generate missing keyphrases,
which in theory gives them a clear advantage over the other models.
## 4\. Experimental settings
In addition to the variation in the choice of benchmark datasets and
baselines, there are also major discrepancies in parameter settings and
evaluation metrics between previous studies. For example, there is no point in
contrasting the results in (Meng et al., 2017), (Florescu and Caragea, 2017)
and (Teneva and Cheng, 2017), three papers about keyphrase extraction
published in the same year at ACL, since neither benchmark datasets, parameter
settings nor evaluation metrics are comparable. To address this problem, we
use the same pre-processing tools, parameter settings and evaluation procedure
across all our experiments.
| Scientific articles | Paper abstracts | News articles
---|---|---|---
| PubMed | ACM | SemEval | Inspec | WWW | KP20k | DUC-2001 | KPCrowd | KPTimes
Model | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP
FirstPhrases | 15.4 | 14.7 | 13.6 | 13.5 | 13.8 | 10.5 | 29.3 | 27.9 | 10.2 | 09.8 | 13.5 | 12.6 | 24.6 | 22.3 | 17.1 | 16.5 | 09.2 | 08.4
TextRank | 01.8 | 01.8 | 02.5 | 02.4 | 03.5 | 02.3 | 35.8 | 31.4 | 08.4 | 05.6 | 10.2 | 07.4 | 21.5 | 19.4 | 07.1 | 09.5 | 02.7 | 02.5
TF$\times$IDF | 16.7 | 16.9 | 12.1 | 11.4 | 17.7 | 12.7 | 36.5 | 34.4 | 09.3 | 10.1 | 11.6 | 12.3 | 23.3 | 21.6 | 16.9 | 15.8 | 09.6 | 09.4
PositionRank | 04.9 | 04.6 | 05.7 | 04.9 | 06.8 | 04.1 | 34.2 | 32.2 | 11.6† | 08.4 | 14.1† | 11.2 | 28.6† | 28.0† | 13.4 | 12.7 | 08.5 | 06.6
MPRank | 15.8 | 15.0 | 11.6 | 11.0 | 14.3 | 10.6 | 30.5 | 29.0 | 10.8† | 10.4 | 13.6† | 13.3† | 25.6 | 24.9† | 18.2 | 17.0 | 11.2† | 10.1†
EmbedRank | 03.7 | 03.2 | 02.1 | 02.1 | 02.5 | 02.0 | 35.6 | 32.5 | 10.7† | 07.7 | 12.4 | 10.0 | 29.5† | 27.5† | 12.4 | 12.4 | 04.0 | 03.3
Kea | 18.6† | 18.6† | 14.2† | 13.3 | 19.5† | 14.7† | 34.5 | 33.2 | 11.0† | 10.9† | 14.0† | 13.8† | 26.5† | 24.5† | 17.3 | 16.7 | 11.0† | 10.8†
CopyRNN | 24.2† | 25.4† | 24.4† | 26.3† | 20.3† | 13.8 | 28.2 | 26.4 | 22.2† | 24.9† | 25.4† | 28.7† | 10.5 | 07.2 | 08.4 | 04.2 | 39.3† | 50.9†
CorrRNN | 20.8† | 19.4† | 21.1† | 20.5† | 19.4 | 10.9 | 27.9 | 23.6 | 19.9† | 20.3† | 21.8† | 22.7 | 10.5 | 06.5 | 07.8 | 03.2 | 20.5† | 20.3†
Table 2. Performance of keyphrase extraction models. † indicates significance
over the baselines.
### 4.1. Parameter settings
We pre-process all the texts using the Stanford CoreNLP suite (Manning et al.,
2014) for tokenization, sentence splitting and part-of-speech (POS) tagging.
All non-neural models operate on a set of keyphrase candidates, extracted from
the input document. Selecting appropriate candidates is particularly important
since it determines the upper bound on recall, and the amount of irrelevant
candidates that models will have to deal with. For a fair and meaningful
comparison, we use the same candidate selection heuristic across models. We
follow the recommendation by Wang et al. (2014) and select the sequences of
adjacent nouns with one or more preceding adjectives of length up to five
words. Candidates are further filtered by removing those shorter than 3
characters or containing non-alphanumeric symbols.
We implemented the neural models in PyTorch (Paszke et al., 2017) using
AllenNLP (Gardner et al., 2018), and the non-neural models using the pke
toolkit (Boudin, 2016). As neural models require large amounts of annotated
data to be trained, we trained our models on the KP20k dataset for both
scientific papers and abstracts, and on KPTimes for news texts. We compute
Document Frequency (DF) counts and learn Kea models on training sets. For
datasets without training splits, we apply a leave-one-out cross-validation
procedure on the test sets for calculating DF counts and training models. We
use the optimal parameters suggested by the authors for each model, and
leverage pre-trained sentence embeddings666https://github.com/epfml/sent2vec
for EmbedRank. We also found out that the training set of KP20k contains a
non-negligible number of documents from the test sets of other datasets. We
removed those documents prior to training.
### 4.2. Evaluation metrics
Although there is no consensus as to which metric is the most reliable for
keyphrase extraction, a popular evaluation strategy is to compare the top $k$
extracted keyphrases against the gold standard. We adopt this strategy and
report the f-measure at the top 10 extracted keyphrases. In previous work, we
often see differences in how gold standards are handled during evaluation. For
example, some studies evaluate their models on the present and missing
portions of the gold standard separately (Meng et al., 2017; Ye and Wang,
2018; Chen et al., 2018, inter alia), whereas other work use the entire gold
standard (Florescu and Caragea, 2017; Boudin, 2018, inter alia). We chose the
latter because recent models, in addition to extracting keyphrases from the
content of the document, are able to generate missing keyphrases. Following
common practice, gold standard and output keyphrases are stemmed to reduce the
number of mismatches. One issue with the f-measure is that the ranks of the
correct keyphrases are not taken into account. To evaluate the overall ranking
performance of the models, we also report the Mean Average Precision (MAP)
scores of the ranked lists of keyphrases. We use the Student’s paired t-test
to assess statistical significance at the $0.05$ level.
### 4.3. Replicability of results
In Table 3, we compare the results of our re-implementations against those
reported in the original papers. We note that all models show comparable
results. We observe the largest differences with original scores for CopyRNN
($+2$) and CorrRNN ($-4.3$) that can be easily explained by minor differences
in training parameters.
Model | Dataset (metric) | Orig. | Ours
---|---|---|---
PositionRank | WWW (F$@$8) | 12.3 | 11.7
MPRank | SemEval-2010 (F$@$10) | 14.5 | 14.3
EmbedRank | Inspec (F$@$10) | 37.1 | 35.6
CopyRNN | KP20k (F$@$10 on present) | 26.2 | 28.2
CorrRNN | ACM (F$@$10 on present) | 27.8 | 23.5
Table 3. Original vs. re-implementation scores.
## 5\. Results
Results are presented in Table 2. First of all, we notice that no model
significantly outperforms the baselines on all datasets. This is rather
surprising, as one would expect that neural models would be consistently
better than a simple TF$\times$IDF model for example. Rather, we see that the
TF$\times$IDF baseline is very competitive on long documents, while the
FirstPhrases baseline performs remarkably well, especially on news texts.
Still, overall, CopyRNN achieves the best performance with, in the case of
KPTimes, MAP scores exceeding 50%. When we look at only unsupervised models,
MPRank achieves the best results across datasets. Also, it comes as no
surprise that Kea exhibits strong performance across datasets because it
combines two effective features, as demonstrated by the results of the
TF$\times$IDF and FirstPhrases baselines. Conversely, despite the addition of
mechanisms for promoting diversity in the output, CorrRNN is almost always
outperformed by CopyRNN, suggesting that the added correlation constraints are
not effective at filtering out spurious keyphrases.
In light of the above, we can now answer the following question: “How much
progress have we made since early models?”. It is clear that neural-based
models are the new state-of-the-art for keyphrase extraction, achieving F@10
scores up to three times that of previous models. That being said, CopyRNN,
which is the best overall model, fails to consistently outperform the
baselines on all datasets. One reason for that is the limited generalization
ability of neural-based models (Meng et al., 2017; Chen et al., 2018; Gallina
et al., 2019), which means that their performance degrades on documents that
differ from the ones encountered during training. This is besides confirmed by
the extremely low performance of these models on DUC-2001 and KPCrowd. Much
more work needs to be done in tackling this issue if neural models are to
substitute for older supervised models. Perhaps most disappointing is the fact
that state-of-the-art unsupervised models are still challenged by the
TF$\times$IDF baseline. Here, we suspect the reasons are twofold. First, the
models we have investigated do not use in-domain data which may not only limit
their performance, but also, as in the case of EmbedRank that uses out-of-
domain (Wikipedia) data, be detrimental to their performance. Second, unlike
neural generative models, they are not able to produce keyphrases that do not
occur in the source document, further limiting their potential effectiveness.
As outlined in §2, gold standards provided by lay annotators, such as authors
and readers, exhibit strong inconsistency issues. One might therefore wonder
“What is the impact of non-expert annotations on training and evaluating
keyphrase extraction models?”. Intuitively, models evaluated against these
annotations are likely to receive lower scores because they make training more
difficult (that is, assigning different keyphrases to documents about the same
topic may confuse the model) while increasing the number of false negatives
during evaluation. This is exactly what we observe in Table 2 where the best
scores for Inspec and KPTimes, whose gold standards are provided by
professional indexers, are higher in magnitude than those of the other
datasets. Precisely quantifying how much impact lay annotations have on
performance is no easy task as it implies a double-annotation process by both
expert and non-expert annotators. Luckily enough, a small sample of documents
from Inspec are also found in KP20k, allowing us to compare the performance of
keyphrases models between both annotation types. Results are shown in Table 4.
First, we see that overall performance is nearly cut in half when evaluating
against author-provided gold standard, suggesting that reported scores in
previous studies are arguably underestimated. Second, neural models again do
not show their superiority against indexer-assigned keyphrases, which
advocates the need for more experiments on datasets that include expert
annotations.
| $\text{F}@10$ | MAP
---|---|---
Model | I | A | I | A
FirstPhrases | 25.8 | 13.7 | 26.1 | 13.2
TextRank | 33.4 | 12.2 | 29.6 | 09.3
TF$\times$IDF | 34.6 | 14.2 | 33.3 | 16.1
PositionRank | 32.9 | 15.9 | 31.0 | 13.0
MPRank | 26.4 | 13.8 | 27.6 | 13.6
EmbedRank | 34.3 | 15.3 | 31.3 | 11.5
Kea | 32.5 | 15.2 | 31.9 | 15.9
CopyRNN | 33.7 | 28.9‡ | 29.8 | 33.8‡
CorrRNN | 28.6 | 25.3 | 24.2 | 28.2
Avg. | 31.3 | 17.2 | 29.4 | 17.2
Table 4. Results on a subset of 55 documents from Inspec for indexer (I) and
author (A) gold annotations. ‡ indicates significance over every other model.
Figure 1. Average number of keyphrases in common between model outputs.
The third question we want to address in this study is “Which baselines and
benchmark datasets should be included in future work for a better
understanding of the pros and cons of a newly proposed model?”. Having strong
baselines to compare with is of utmost importance, and our results give an
indication of which model is relevant. When properly trained, neural models
drastically outperform all other models and represent the state-of-the-art.
Since CopyRNN achieve the best results, it should be included in future work
for comparison. In an unsupervised setting, or in a data-sparse scenario where
neural models can not be applied, the picture is less clear. To help us
understand which model is worth investigating, we conducted an additional set
of experiments aimed at comparing the outputs from all models in a pairwise
manner. The motivation behind these experiments is that including multiple
models that behave similarly is of limited interest. Similarities between
model outputs, viewed in terms of the number of keyphrases in common, are
graphed as a heatmap in Figure 1. Overall, we observe different patterns for
each source of documents. The shorter the document is, the more similar
outputs are, which is mostly due to a smaller search space (that is, a smaller
number of keyphrase candidates). We note that the three best unsupervised
models, namely FirstPhrases, MPRank and TF$\times$IDF, generate very similar
keyphrases (up to 42% identical). Considering this, and given their reported
performances (Table 2), we argue that TF$\times$IDF (or KEA if seed training
data is available) should be considered as strong unsupervised baseline in
subsequent work.
These recommendations of baselines also affect the choice of which benchmark
datasets one has to use. As neural models are data-hungry, KP20k and KPTimes
are the default options for paper abstracts and news articles. For scientific
articles, we recommend using SemEval for two reasons: 1) it is widely used by
existing studies; and 2) it provides a double-annotated gold standard (author-
and reader-assigned keyphrases) that alleviates annotation inconsistencies to
some extent.
Our experiments highlight several issues in evaluating keyphrase extraction
models with existing benchmark datasets. Another way of assessing the
effectiveness of these models would be to explore their impact on other tasks
as an extrinsic evaluation. To the best of our knowledge, there is no
previously published research on that matter despite many downstream tasks
that already benefit from keyphrase information such as article recommendation
(Collins and Beel, 2019) or browsing interfaces (Gutwin et al., 1999) in
digital libraries. This points to an interesting future direction that allows
for a deeper understanding of the limitations of current models.
## 6\. Conclusion
This paper presents a large scale evaluation of keyphrase extraction models
conducted on multiple benchmark datasets from different sources and domains.
Results indicate that keyphrase extraction is still an open research question,
with state-of-the-art neural-based models still challenged by simple baselines
on some datasets. We hope that this work will serve as a point of departure
for more rigorous analysis and evaluation of proposed keyphrase extraction
models. We provide all the code and data on a public repository777Link to the
repository will appear here after the review period., as well as a public
leaderboard to facilitate the comparison between models.
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|
2024-09-04T02:54:58.652265 | 2020-03-10T11:56:55 | 2003.04654 | {
"authors": "O.V. Kancheli",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26131",
"submitter": "O. V. Kancheli",
"url": "https://arxiv.org/abs/2003.04654"
} | arxiv-papers | Parton models and frame independence of high-energy cross-sections.
O.V. Kancheli 111 Email<EMAIL_ADDRESS>
Institute for Theoretical and Experimental Physics,
117218, Moscow, Russia
###### Abstract
We describe some ambiguities which take place when on calculates the cross-
sections in parton models at high energies and the connected limitations on
the asymptotic of high energy amplitudes that follows from the conditions of
boost-invariance of cross-sections.
It turns out that the resulting constraints are of the same type as the
following from the t-channel unitarity conditions. So that on can suppose that
this similarity, by their nature, has much more general grounds.
## 1 Introduction
There are two main theoretical approaches to a study of the behavior of high
energy amplitudes and cross-sections.
In one approach, we directly calculate the amplitudes by summing the
contributions of the Feynman diagrams of the corresponding field theory or use
some effective theory like reggeon diagrams or various string-like dual
models.
In the other - parton like approach to high-energy collisions 222 A few useful
reviews ( [1]\- [7]) we usually consider separately three main stages of the
system evolution in the process of particles collision. Firstly, one
constructs the quantum states $\Psi(P)$ of high energy particle with momentum
$P\gg m$ in terms of superposition
$\Psi(P)~{}=~{}\sum_{n}\int_{\\{k_{i}\\}}~{}f_{n}(P,\\{k_{i}\\})~{}|n,\\{k_{i}\\}>$
(1)
of the n-particle states $|n,\\{k_{i}\\}>$ of some “primary” constituents -
partons with 3-momenta $\\{k_{i}\\}$. The “choice” of these partons is not
unique, and partons can be bare point like particles, particles with varying
virtuality, QCD color dipoles, fast string configurations, distributions of
the Coulomb-like fields, etc.
The state $\Psi(\vec{P})$ must fulfill the Schroedinger equation
$\hat{H}~{}\Psi(\vec{P})=\sqrt{\vec{P}^{2}+m^{2}}~{}~{}\Psi(\vec{P})~{},$
where the Hamiltonian $\hat{H}$ is the function of parton fields, so that
$\Psi(\vec{P})$ is the eigenfunction of $\hat{H}$ with eigenvalues defining
the particles physical mass $m$.
After that one can use such a state $\Psi(P_{1})$ to calculate its interaction
with some low energy target or with other fast particle in the state
$\Psi(P_{2})$ in terms of “simple” amplitudes of parton interaction.
There is also the third stage corresponding to an evolution in the final state
when moving away partons transform and combine into physical particles
(hadronization…). But often this stage is not very restrictive, especially
when we calculate various integrated cross-sections. And we will not consider
it in this article.
It is essential that with the energy growing in most parton descriptions the
structure of the parton state becomes more and more complicated for all the
theories containing vector (like QCD) and tensor fields (gravity) and mean
parton number in states $\Psi(P)$and the average transverse size of the region
they occupy grow with $P$.
When we consider the collision of two fast particles in the parton states
$\Psi(P_{1})$ and $\Psi(P_{2})$ at some large $s=(P_{1}+P_{2})^{2}\gg m^{2}$
we can choose for this any longitudinal Lorentz system. But the resulting
values of cross-sections of various processes must not depend from this choice
of frame. And this is nontrivial condition in parton approach, because in
different longitudinal systems (that is for various $P_{1}$ and $P_{2}$ at the
same value of $s$) the different parton configurations firstly meet one
another at the moment of particles collision. And, moreover, by choosing a
different system we also can move the dynamics, from stage one to two and vice
versa.
If we make all calculation precisely - with hermitian Hamiltonian we probably
can be sure that all restrictions coming from Lorentz-invariance and the
unitarity conditions will be satisfied. But if we make some approximations,
especially dictated by phenomenological or pictorial arguments, the unitarity
conditions itself can probably be the only general way to check that the
results are not contradictory.
Various restrictions from the t-channel unitarity are very essential for the
amplitudes describing high energy hadron interactions, and they are directly
taken into account in reggeon amplitudes [10]. But in parton approaches it is
not evident how to take them into account.
In the reggeon field theory and in the dual (string) models the t-unitarity
conditions are automatically fulfilled. But at high reggeon (pomeron) density
such un approach can become unreliable. The parton approach has no problems
with high parton density, but here there is no direct way how to control
possible restrictions coming from the t-unitarity.
One can hope that the longitudinal Lorentz (boost) invariance of all cross-
sections calculated in a parton approach is in some sense equivalent to the
mean form of the t-unitarity for multiparticle amplitudes. So, if we calculate
any cross-section using the partonic wave functions $\Psi(P_{a})$ and
$\Psi(P_{b})$ of fast colliding hadrons with momenta $P_{a},$ $P_{b}$ then we
expect that this cross-section must be the same in all longitudinal Lorentz
frames \- that is if we calculate the cross-sections using
$\Psi(L(\vartheta)P_{a})$ and $\Psi(L^{-1}(\vartheta)P_{b})$, where
$L(\vartheta)$ is a longitudinal boost. It is essential, that in a parton
picture such boosts $L(\vartheta)$ act on hadrons Fock state very nontrivial
changing the number of partons, etc.
No precise arguments for such general propositions (the boost invariance for
parton cross-sections $\simeq$ t-unitarity) are known. Although it is by
itself natural that the calculations of cross-sections in the parton picture
must give a frame independent answer. Also this is, in particular, confirmed
in if we give the partonic interpretation to reggeon diagrams, by t-cutting
them at various intermediate rapidities, as if we calculate various
multiparticle inclusive cross-sections.
In this article 333The material of this paper partially intercepts with the
article of the author [8]. we consider some examples illustrating how the
requirement of boost-invariance essentially restricts the structure of high
energy collision dynamics. We see that it restricts in the same way as it
follows from the conditions of t-unitarity.
## 2 Restrictions on a parton states from the boost invariance of high-energy
collision cross-sections.
Simple Examples
In this section we illustrate how the requirement of the frame independence
(boost-invariance - BI) restricts the behavior of high-energy cross-sections
calculated in the parton approach.
We suppose that partons are point like particles with perturbative interaction
and consider here some examples which show how BI condition works. Also we
choose very high energy interactions, where the mean number of parton in HE
state is large, so one can consider firstly only states with mean number of
partons and only after that take into account corrections from other
components of the Fock wave function of a fast particle. So, the picture of
interaction is almost quasiclassical.
We consider the behavior at a boost-transformation of the inelastic cross-
sections $\sigma_{in}$ or of the connected quantity - the transparency
$T=1-\sigma_{in}=|S|^{2}$, which is often more sensitive to the breaking of
BI. We choose some frame where the colliding particles have rapidities
$y_{1}=y$ and $y_{2}=Y-y$, where $Y=\ln(s/m^{2})$, and require that calculated
cross-sections do not depend on $y$
We begin from the simplest parton models of a fast hadron - the rare parton
gas state and of the black disk state.
### 2.1 Collision of a rare gas like parton states
Let us consider the collision of two particles which can be represented as the
partonic clouds that are in a state of a very rare gas. This is the case
usually described by reggeon diagrams, that, by their construction, include
t-unitarity requirements. Let the mean number of partons in colliding hadrons
be $n(y)$, $n(Y-y)$ and the mean transverse radii of regions occupied by these
partons are $R(y)$, $R(Y-y)$, respectively. Then the total inelastic cross-
section can be expressed as:
$\displaystyle\sigma_{in}(Y)~{}=~{}\sigma_{0}~{}n(y)~{}n(Y-y)~{}~{}-~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}-~{}c_{1}~{}\sigma_{0}~{}n(y)n(Y-y)~{}\Big{(}~{}\frac{\sigma_{0}~{}n(y)}{R^{2}(y)}~{}+~{}\frac{\sigma_{0}~{}n(Y-y)}{R^{2}(Y-y)}~{}+~{}$
(2)
$\displaystyle~{}+~{}\frac{\sigma_{0}~{}n(y)~{}n(Y-y)}{R^{2}(y)+R^{2}(Y-y)}~{}\Big{)}~{}+...~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
where $\sigma_{0}$ is the parton-parton cross-section, $c_{1}\sim 1$. The
first term in (2.1) corresponds to a collision of at least one pair of
partons. The next terms describe corrections from screening and multiple
collisions 444The cross-sections of local interactions of point-like particles
decrease as a function of their relative energy as $\sigma_{0}(s)\sim 1/s$. As
a result in (2.1) enter, in fact, only the numbers of low energy partons
$n(y),n(Y-y)$ of the colliding particles in this coordinate system. .
For the rare parton gas one can at first approximation neglect multiple
collisions and screening, that is to leave only the first term in (2.1). Then,
from the requirement of the independence of $\sigma_{0}~{}n(y)n(Y-y)$ on $y$
follows the unique solution for
$n(y)=~{}n_{0}e^{y\Delta_{0}}$ (3)
with some real constants $n_{0}$, $\Delta_{0}$. The following from (3)
behavior of
$\sigma_{in}(Y)=\sigma_{0}~{}n_{0}^{2}~{}e^{Y\Delta_{0}}$ (4)
in the elastic amplitude corresponds to a regge pole in the complex angular
momentum plane (and not to a cut or some more complicated regge singularity ).
And this condition follows [9] in a relativistic Regge approach only from the
2-particle t-unitarity of the elastic amplitude.
Note that the coefficient in (4) is in fact factorized - for the collision of
different particles a+b one must $n^{2}\rightarrow n_{a}n_{b}$ . This
factorization in regge approach also follows from t-unitarity.
Moreover, it is interesting to consider [6] the behavior cross-section
$\sigma_{in}$ with the definite impact parameter $B$, normalized so, that
$\sigma_{in}(Y)~{}=~{}\int~{}d^{2}B~{}\sigma_{in}(Y,y,B)~{}~{}.$ (5)
In this case the analog of the first term in (2.1) can be represented as
$\sigma_{in}(Y,y,B)~{}=~{}\sigma_{0}\int
d^{2}x_{\bot}~{}\rho(y,|x_{\bot}|)~{}\rho(Y-y,|B-x_{\bot}|)~{}~{}~{}~{},~{}~{}~{}~{}$
(6)
where $\rho(y,x_{\bot})$ is the transverse parton density $(~{}n(y)=\int
d^{2}x_{\bot}\rho(y,x_{\bot})~{})$ . Then from the frame independence of the
$\sigma_{in}(Y,y,B)$ the form of transverse parton density $n(y,x_{\perp})$
can be essentially restricted. The condition of y-independence can be writhen
as
$\frac{\partial}{\partial y}~{}\sigma_{in}(Y,y,B)~{}=~{}0~{}.$ (7)
Going here to conjugate to $x_{\bot}$ variable
$\rho(y,x_{\bot})=\int d^{2}k\cdot e^{ikx_{\bot}}~{}\tilde{\rho}(y,k)$
we come from (7) to the equation
$\frac{\partial}{\partial
y}~{}\big{(}~{}\tilde{\rho}(y,k)~{}\tilde{\rho}(Y-y,k)~{}\big{)}~{}=~{}0~{},$
which has the solution
$\tilde{\rho}(y,k)=f_{1}(k)\cdot e^{yf_{2}(k)}$
and then as a result
$\rho(y,x_{\bot})\sim\int
d^{2}k~{}e^{ikx_{\bot}}~{}f_{1}(k)~{}e^{yf_{2}(k)}~{},$ (8)
where $f_{1}$, $f_{2}$ are arbitrary functions of $k$. For
$y\rightarrow\infty$ the integral in (8) can be taken by the steepest decent
method, so that only the neighborhoods of zeros of $\partial f_{2}(k)/\partial
k$ are essential. Then from the positivity of the parton density $\rho$ it
follows that $f_{2}$ is positive and so the dominant contribution must come
from the region $k\sim 0$, otherwise $\rho(y,x_{\bot})$ will oscillate in
$x_{\bot}$. So in the essential region $f_{2}(k)\simeq
c_{1}-c_{2}k^{2},~{}~{}c_{2}>0$, and estimating the integral (8) we come to
the expression for the density of low energy partons
$\rho(y,x_{\bot})~{}\sim~{}y^{-1}~{}e^{\big{(}c_{1}y-x_{\bot}^{2}/4c_{2}r_{0}^{2}y\big{)}}~{}~{},~{}~{}~{}c_{2}>0~{}.$
(9)
The expression (9) corresponds to the Gauss form of parton distribution in
$x_{\bot}$ which usually results from the diffusion of partons in $x_{\bot}$
plane during the parton cascading. The mean radius of a low energy parton
cloud $R(y)~{}\sim r_{0}\sqrt{~{}y}$ is also fixed here only from the
condition of the frame independence. In the elastic amplitude the Eq. (9)
corresponds to the contribution of the regge pole with the trajectory
$\alpha(t)=1+\Delta+\alpha^{\prime}t$ , where
$\Delta=c_{1},~{}\alpha^{\prime}=c_{2}$.
If we make the next step and impose the condition of $y$ independence on the
sum of two terms in the right side of (2.1) and assume that the correction to
(3) is small, we become instead of (3) the corrected expression
$n(y)=n_{0}~{}e^{\Delta_{0}y}-~{}a_{2}~{}n_{0}^{2}\frac{\sigma_{0}}{R^{2}(y)}~{}e^{2\Delta_{0}y}~{}+...~{}~{}~{}~{}$
(10)
From here it is simple to conclude that
$\sigma_{in}(Y)=n_{0}^{2}~{}\sigma_{0}e^{(\Delta_{0}Y)}~{}-~{}~{}n_{0}^{4}\frac{a_{2}\sigma_{0}^{2}}{R^{2}(Y)}~{}e^{2(\Delta_{0}Y)}~{}~{}.$
(11)
The second term in (11) corresponds to the the contribution of two reggeon
cuts, whose structure is almost complectly fixed here from the boost-
invariance. The arbitrary coefficient $a_{2}>1$, depends on the weight of the
diffractive amplitudes entering in the two regeon emission vertex. The
possible next terms in (10), corresponding to higher regge cuts, can be found
in the same way by iterative applying the boost-invariance condition to the
combinations of screening terms in the expression (2.1) for $\sigma_{in}$.
Thus, it can be seen that for rare parton states we come to the restrictions
on there structure that arise from the reggeon diagrams and are defined by the
t-unitarity.
At the end of this section note that at at all currently available energies
the dominant high energy hadron interactions are well described by the regge
approach with the soft pomeron exchange and the respective cuts. This directly
corresponds to the Gauss-like parton distribution consistent with the parton
frame independence.
### 2.2 Collision of a black disks
Now let us consider the opposite limiting case of colliding parton clouds,
when the mean parton density is very high and partons fill a transverse disk
with the radius $R(y)$ depending on particles energy $E=me^{y}$. Then the
total inelastic cross-section can be determined from purely geometrical
conditions - it is defined by the area of an impact parameter space,
corresponding to the overlapping of the colliding black disks :
$\sigma_{in}(Y)~{}=~{}\pi~{}\Big{(}R(y)+R(Y-y)\Big{)}^{2}~{}.$ (12)
From the condition of independence of the right side of Eq.(12) on $y$
evidently follows the unique solution for
$R(y)~{}=~{}r_{0}\cdot y+r_{1}$ (13)
It is interesting that in this case we immediately come directly to
asymptotically constant cross-sections (when $r_{1}=0$), or to the Froissart
type behavior of cross-sections
$\sigma_{in}(Y)\simeq\pi r_{0}^{2}Y^{2}+\pi r_{0}r_{1}Y+\pi r_{1}^{2}~{}~{}.$
(14)
Here, in the Froissart case the elastic cross-section is diffractive and
$\sigma_{el}=\sigma_{in}$. Also the terms $~{}\pi r_{0}r_{1}Y$ in (14)
correspond to a diffraction generation as is natural in the Froissart case.
### 2.3 Collision of the parton grey disks
The real parton disk (even at $Y\gg 1$) cannot be absolutely black because the
parton density at every particles energy is finite. Besides that the local
parton density fluctuations also can lower the parton density in the
individual events and this leads to the grow of the locale transparency of
such disks. For such parton disks the conditions of BI can lead to rather
strong restrictions on the structure of “grey” parton states and their
interactions.
Firstly, consider the collisions of grey disks with some constant grayness -
when the mean transverse parton density is stabilized at some fixed value and
do not grow with energy i.e. the local disk transparency also does not change
with energy 555One can expect this type of the behavior in the (2+1)D QCD,
which is soft and if here the parton saturation takes place [8].. Then it is
easy to see that the condition of the boost invariance can at all not be
fulfilled for such models.
In the lab.frame of one particle the transparency
$T_{lab}(Y,B)=const(Y)~{}~{}~{}~{}~{}at~{}~{}~{}~{}~{}Y\rightarrow\infty~{}~{},$
because at all $B$ only a finite number ($\sim 1$) of partons must penetrate
through the grey parton disk of the other fast particle.
And in the center of the mass system at the same impact parameter the large
number of partons $N_{12}$ must penetrate. For the grey disk $N_{12}\sim
S_{12}(y,Y,B)$ \- the transverse area of two disks overlapping region. And for
growing with $Y$ disk radius the $S_{12}$ also grows. For the Froisart type
growth we have $S_{12}\sim Y^{2}$ at $B\ll R(Y)$. Then in systems close to the
center of mass
$T_{scm}(Y,B)\sim e^{(-cN_{12})}\sim\exp{(-cY^{2})}\rightarrow
0~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}c\sim 1~{}~{}~{}.$
Therefore, the case of a grey disk with a constant (or a slowly ($\sim 1$)
varying ) parton density should be probably excluded.
In all more or less realistic situations, such, for example that one can
expect in the QCD, the parton disk can have a grey parton border, even when
the inner parts of the disk become almost black. In this case the average
parton density can be roughly represented as
$\rho(y,x_{\bot})~{}\simeq~{}\rho_{d}(x_{\bot})~{}\theta(R(y)-x_{\bot})~{}+~{}\rho_{0}~{}\theta(x_{\bot}-R(y))f(x_{\bot})~{}~{},$
(15)
where $\rho_{d}(x_{\bot})$ describes the behavior of the parton density in the
inner part of the disk, and where the grey border has the width $\lambda(y)\ll
R(y)$. In this border the parton density varies from the high value (almost
black) to small one. For example, it can have the form
$f(x_{\bot})~{}\simeq~{}e^{-(x_{\bot}-R(y))/\lambda(y)}~{}~{}.$ (16)
For collisions with an impact parameter $B<R(Y)$, when colliding disks stuck
with their almost black parts, we possibly can have a boost-invariant behavior
of $\sigma_{in}$. But for collisions with $B-R(Y)\sim\lambda(Y))$, when the
discs collide with their grey edges, the situation is different.
In the Lab frame of one of particle the transparency $T_{lab}=const\sim 1$,
because here only some ($\sim 1$) partons must penetrate without interaction
through the grey edge of the large disc.
And in the arbitrary system at the same impact parameter the transparency is
$T(y,Y-y,B)\sim e^{-N_{12}(y,Y-y,B)}\sim
e^{-S_{12}(y,Y-y,B)/r_{0}^{2}}~{}~{},$
where $N_{12}(y,Y-y,B)$ is the average number of parton interactions during
the collision and $S_{12}(y,Y-y,B)$ is the area of the two disks intersection
region. This region has a form of elongated figure whose width is
$\sim\lambda(y)$ and the length $l(y)\sim\sqrt{R(y)\lambda(y)}$ for $y\lesssim
Y-y$. So, for such B the two disks intersection area is
$S_{12}(y,Y-y,B\simeq
R(Y)+\lambda)~{}~{}\sim~{}~{}R(y)^{1/2}*\lambda(y)^{3/2}~{}~{}.$
In the center of mass system this gives for $Y\gg 1$
$T_{scm}\sim\exp(-c(R(y)/r_{0})^{1/2})\rightarrow 0$
even for parton disks with $\lambda\sim const(Y)$, although the width of the
grey border can also grows together with the disk size 666One can expect [8]
that for a realistic parton disc due to border shape fluctuations the mean
width of the grey zone grows with Y as $\lambda\sim\sqrt{Y}$. Therefore, for
the border collisions with such $B$ and $Y$ we have no boost-invariance of T ,
and this conclusion in fact almost does not depend on the explicit form of the
border distribution $f(x_{\bot})$. Probably, the only exception is the Gauss
type distribution of the parton density when the whole disk has the “structure
of border”.
### 2.4 Particle to heavy nuclei interaction
A slightly different type of restriction on the parton structure follows from
the boost invariance if we consider the high energy collision of a particle p
(for example a proton, a pion or any test color dipole) with heavy nucleus
$(A\gg 1)$.
To see this we compare the estimate of transparency T in Lab frame of nuclei
and the Lab frame of p. Also we choose Y not very large, but so that $Y\gg\ln
A$, and consider a collision at $B=0$. In fact, in this case we have a
collision of p with a long ($\sim A^{1/3}$) tube of nucleons, and we want to
calculate the probability of the passage of p through A without interaction.
First, consider the p $\bigotimes$ A collision in the Lab frame of p. For such
an Y due to the Lorentz contraction of the moving nuclei all soft partons of
the fast nuclei are placed in a tiny transverse region of the longitudinal
size $\sim 1/m$. And if the parton saturation take place, the number of soft
partons $N_{A}$ interacting with p should almost not depend on A, because all
“additional” soft partons coming from different nucleons in the A-tube are
absorbed one by another. Therefore, one can expect that the transparency in
the p-lab. frame is
$T_{p}~{}\sim~{}e^{-N_{p}(Y)}~{}~{}.$ (17)
On the other hand, to calculate T in the Lab frame of A at the same B and Y we
must find the probability that fast particle p penetrate without interaction
trough the $A^{1/3}$ long tube of nucleons. Here one can expect that
$T_{A}~{}\sim~{}e^{-c(Y)A^{1/3}}$ (18)
Because in such a “thought experiment” we can arbitrary choose Y and A and the
distance between the nucleons in the tube, we come to an apparent
contradiction with the frame independence. This means that some constrains
must be imposed on the parton dynamics. The simplest way is to suppose that
there is almost no parton saturation in the A-tube. Or on the contrary - that
some kind of the mechanism works, which makes the interaction of a fast p
particle with the nucleus otherwise dependent on A.
Possibly some indications on the causes of this inconsistency can be found if
we consider the regge description of this reaction, where we can calculate
$\sigma_{in}(Y,B)=1-T$ for a large A and not to a large Y.
If we take for a single pomeron exchange in the p $\bigotimes$ A reaction the
amplitude
$v(y,b)\sim ig^{2}A^{1/3}\exp{(\Delta y-b^{2}/4\alpha^{\prime}y)}$
and consider firstly the simple eiconal case which corresponds to a situation
without parton saturation we become for the corresponding S-matrix
$S(Y,B)=\exp{(iv(Y,B))}$, and this gives for the transparency
$T(Y,B=0)~{}=~{}|~{}S(Y,B=0~{}|^{2}~{}\sim~{}\exp{\Big{(}-2g^{2}A^{1/3}e^{\Delta
Y}\Big{)}}~{}~{}.$ (19)
The simplest way to take into account something similar to the parton
saturation is to include into the single pomeron exchange amplitude the
pomeron cascading from the side of A-vertices. So that from the p-side the
pomeron line joins to p and from the A side the pomeron line branching joins
to many nucleons. This corresponds to the new amplitude
$v~{}\rightarrow~{}\tilde{v}~{}=~{}\frac{v}{1-i\frac{r}{g\Delta}v}~{}~{},$
(20)
were $r$ is the 3-pomeron vertex. In this case for large $A^{1/3}$ and
$B~{}=~{}0$ the amplitude $\tilde{v}$ is stabilized at the value
$|\tilde{v}|=g\Delta/r$. And, therefore, the corresponding transparency
approaches to
$T(Y,B=0)~{}=~{}\exp{\big{(}-2|\tilde{v}|~{}\big{)}}~{}=~{}\exp{\Big{(}-2g\Delta/r\Big{)}}~{}~{}.$
(21)
Comparing the expressions ( 17 ) with (21) and (18) with (19) we see their
similarity, but this unfortunately does not help to find the right answer,
because the simple expression like (20) dos not take into account various
pomeron interactions in the $\tilde{v}$-cascade and also the other pomeron
interactions in eiconal multipomeron diagrams 777 Note that approximately the
same inconsistency appears if we consider the heavy A $\bigotimes$ A
interaction end compare the estimates of T in the Lab frame and in the CM
system .
### 2.5 Possible boost-invariant parton density distributions in a grey disk
In fact, in the case of asymptotically growing cross-section all parton
distributions corresponding to real theories like QCD will, probably, lead to
the grey dick. And it is interesting to find the sensible examples of parton
distributions that correspond to the boost-invariant T.
Let us consider collisions of particles with some parton distribution
$\rho(y,x_{\bot})$ and try to find the minimal conditions on the form of
$\rho(y,x_{\bot})$ for which cross-sections are boost-invariant
With the exponential precision the transparency can be expressed as:
$T(Y,y,B)~{}\sim~{}\exp\Big{(}~{}-N(y,Y-y,B)~{}\Big{)}~{}~{}~{},$ (22)
where
$N(y,Y-y,B)=\sigma_{0}\cdot\int d^{2}b\cdot\rho(y,|b|)\cdot\rho(Y-y,|B-b|)$
(23)
is proportional to the mean number of the parton scattering when two $\cal F$
d penetrate one through another during their collision at the impact parameter
B.
Because the expression (23) has the same structure as (6) one can repeat here
the calculation given above. Then we find that the expression ( 23 ) can be
boost invariant only for some very special Gaussian form of parton density
$\rho$ inside the disk :
$\rho(y,x_{\bot})~{}\sim~{}\rho_{0}~{}\frac{1}{y}~{}e^{\Delta
y-x_{\bot}^{2}/yr_{0}^{2}}~{}~{},~{}~{}$ (24)
This corresponds to the distribution arising in the parton cascade when
partons only split and do not join. The same answer (Eq (9)) for
$\rho(y,x_{\bot})$ was found for the rare parton systems - but here the
density can be arbitrary high. In the connected elastic amplitude it
corresponds to a regge pole exchange with the intercept $\Delta$. In fact, the
expression (24) for $\Delta>0$ corresponds again to almost black disk (but
without parton saturation !) of the radius $r_{0}\Delta~{}y$ with a thin grey
border, because the parton density changes here very fast from a small to a
big values at the distances $\delta x_{\bot}\sim r_{0}/\sqrt{\Delta}$.
In general case one must take into account that partons in the colliding disks
can have different virtualities $u\sim\ln k^{2}_{\bot}/m^{2}$ , where the
parton density $\rho(y,b,u)$ has now nontrivial dependence on $u$. Partons
with large $u$ are more strongly localized in transverse coordinates and their
interaction cross-sections $\sigma(u_{1},u_{2})$ usually decrease for large
$u_{i}$. The expression for the transparency in the process of collision of
two parton disks has again the form (22), where the mean number of parton
interactions during the collision is given by the following generalization of
(23)
$N(y,Y-y,B)=\int d^{2}b\int
du_{1}du_{2}~{}\sigma(u_{1},u_{2})\cdot\rho(y,|b|,u_{1})\rho(Y-y,|B-b|,u_{2})~{}.$
(25)
In this case the restrictions on the form of $\rho(y,b,u)$ coming from the
frame independence condition $(\partial/\partial y)N(y,Y-y,B)=0$ are not so
strong as for (23 \- 24).
If the parton cross-sections that enter (25) can be approximately factorized
as
$\sigma(u_{1},u_{2})\sim\ell(u_{1})\cdot\ell(u_{2})~{}~{},$
then the condition for the boost invariance of $N$ can be reduced to the more
simple equation
$\int du~{}\ell(u)\rho(y,b,u)~{}=\rho_{0}\frac{1}{y}~{}e^{\Delta
y-b^{2}/yr_{0}^{2}}$ (26)
In this case the form of $\rho(y,b,u)$ for some interesting models are again
almost completely fixed. For example, so is the superposition of grey
saturated disks with different virtualities
$\rho(y,b,u)\sim\varphi(y,u)~{}\theta(r_{1}\chi(y)-bu^{a})~{},$
so that the mean radii of these disks $r_{1}\chi(y)/u^{a}$ decrease 888In QCD
the radii of hard subdisks can grow as $\sim y/\sqrt{u}$, and this corresponds
to $a=2$ with growth of u. Here, from equation(26), one can find the explicit
expression for
$\varphi(y,u)~{}=~{}\varphi_{0}~{}\frac{e^{\Delta
y}}{\ell(u)~{}u^{2a}~{}y}~{}\exp{\Big{(}-c_{2}\frac{\chi^{2}(y)}{y~{}u^{2a}}~{}~{},\Big{)}}$
(27)
where $\varphi_{0},~{}a,~{}\Delta,~{}c_{2}$ and functions $\ell(u),~{}\chi(y)$
can be chosen arbitrary. If we choose $\chi(y)=~{}\chi_{0}y$ so to have the
Froissart type of the growth of disk radius we will have from (27) for the
disk density
$\varphi(y,u)~{}\sim~{}~{}\frac{e^{\Delta
y}}{\ell(u)u^{2a}y}~{}\exp{\Big{(}-\tilde{c_{2}}\frac{y}{u^{2a}}~{}~{}.\Big{)}}$
(28)
For large $u$ it is natural to expect that $\ell(u)\sim e^{-cu}$ and therefore
the mean density of hard subdisks will grow with u and y.
### 2.6 Corrections to the mean picture from a big fluctuations in the
colliding states
To discuss if the boost-invariance can be somehow restored also when the mean
parton density $\rho(y,x_{\bot})$ is not of the Gauss form (24) on must take
into account all essential parton configurations, and also these ones that are
very far from the mean one. In this case, one can hope that in different
frames the main contribution into cross-sections comes from some different
parton components so to compensate the variation of the contribution of the
mean states. Here especially interesting can be the rare components of
$\Psi(P)$. In the Fock state of a fast particle such rare parton
configurations contain a relatively small number of partons and therefore it
can give large contribution to the transparency and compensate the boost non-
invariance of $T$ and other quantities in the mean density states. Such
configurations can mainly arise due to large fluctuations in the initial
stages of the patron cascade. CM one can ask for such a parton component
$|~{}bare>$ for a fast hadron that does not contain a black disk at all and
interacts slowly (or does not interact at all). We can schematically represent
such a state of fast particle :
$\Psi(P\rightarrow\infty)~{}\simeq~{}f_{d}~{}|disk>+~{}f_{b}~{}|~{}bare>~{},~{}~{}~{}f_{d}\gg
f_{b}~{},$
where $f_{b}$ is the amplitude of the rare component $|bare>$ and $|disk>$ is
the superposition of “big” parton components that gives the main contributions
in a various cross-sections.
The probability for a fast hadron to be in the rare state is
$w(y)\sim|f_{b}|^{2}$. In this case the expression for the transparency can be
generalized to :
$\displaystyle
T(y,Y-y)~{}~{}\simeq~{}~{}T_{mean}(y,Y-y)+~{}\tau_{bd}\cdot\big{(}w(y)+w(Y-y)\big{)}~{}~{}+$
$\displaystyle+~{}\tau_{bb}\cdot w(Y-y))\cdot
w(y)~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (29)
where the transparencies of the rare component $\tau_{bd}$ and $\tau_{bb}$ can
be finite and do not decrease with growth of $y$.
The term $T_{mean}(y,Y-y)$ coming from the $|disk>\bigotimes|disk>$
interaction is not boost-invariant and it can be $const(Y\rightarrow\infty$)
in the lab.frame for a saturated (grey) disk, and very small in csm.
The two last terms in (2.6) coming from the $|bare>\cdot~{}|disk>$ and
$|bare>\cdot~{}|bare>$ components can dominate and so can make $T$ boost
invariant. But it is possible only if $w(y)$ is approximately constant for
high $y$. Various estimates of $w(y)$ lead to a decreasing function of the
type
$w(y)\sim\exp(-\gamma\cdot y)$
for the case of the growing total cross-section. It corresponds to the choice
at every rapidity stage of such an evolution direction, that does not increase
the parton number 999Such a behavior of $w(y)$ can also be found from the
boost-invariance condition applied to the behavior of some hard cross-
sections. See Eq.(31 \- 32) . Such a behavior of $w$ leads to the expression
$T(y,Y-y)~{}\sim~{}\tau_{bd}\cdot\big{(}~{}e^{-\gamma~{}(Y-y)}~{}+~{}e^{-\gamma~{}y}~{}\big{)}+\tau_{bb}\cdot
e^{-\gamma~{}Y}~{},$ (30)
corresponding to the collision of the rare state $|~{}bare>$ with other
particle. Such contribution to $T$ is $y$ dependent, and therefore on this way
the frame indeprndence can also not be restored.
### 2.7 Collision of parton disks in the case of particles moving in the same
direction
When we impose the condition of the independence of the cross-sections
$\sigma_{in}(Y,y,b)$ on the choice of system (i.e. on $y$), we can choose the
values of $y$ not only in the interval $0<y<Y$, i.e. between Lab and center of
mass ( CM ) systems. But let us also consider systems with $y<0$ and $y>Y$ ,
when both parton disks move in one direction.
This, in principle, can lead to additional constraints on the amplitudes. But
in this case, at first glance, paradoxes may also arise when estimating the
probability of the interaction. Especially this is seen in the case of the
growing cross-section.
To illustrate this let us consider the case of colliding Froissart type disks
when their radii $R(y)$ and $R(Y-y)$ grow with the particles rapidity as
$R(y)=r_{0}y$ and $R(Y-y)=r_{0}(Y-y)$, and estimate the behavior of the
inelastic cross-section with the definite impact parameter
$\sigma_{in}(Y,y,B)$ .
We chose $B>r_{0}Y$. In this case, when $0<y<Y$, the parton disks pass one by
another without interaction. But this is only if they move towards each other
because here $B>R(y)+R(Y-y)$ and therefore $\sigma_{in}=0$. But if we at the
same $B$ choose the system so that disks move in the same direction and so
that $y\gg Y\gg 1$ then disks will overlap when one disk will go through
another. And therefore partons from one disk can interact with partons from
another disk.
But it is essential that in such disk interaction no new particles can be
created. Indeed, if in this case a particles can be created, their momenta
will be small ($\sim m$) in this system. And the creation of such a particle
in CM system would correspond to a creation of a particle with energy $\sim
me^{y}$, where $y\gg Y$ and this is forbidden by the energy-momentum
conservation ; so $\sigma_{in}=0$.
From the other hand, the exchange of particles between these discs with an
approximate momentum conservation (or, with the exchange of small transverse
momenta) can give a contribution to their elastic scattering and which comes
here also from large transverse distances ( $B>R(Y)$ ).
The parton wave functions of these “ intersecting disks” can be entangled one
by another by such a mechanism, and also the conversion of a pure state to a
mixed one for every disk can in principal take place. There is here probably
no contradiction with the parton picture, since there is no way to distinguish
between low energy partons in the wave function (1) and the close energy
partons from vacuum fluctuations.
The entanglement between states of disks in such a collisions is proportional
to their area. This suggests that these discs have entropy $\sim$ their area
($\sim$ the number of low energy partons), i.e. $\sim y^{2}$ in this case of
the Froissart type growth of cross-sections.
### 2.8 Limitations on the dynamic of a hard elastic scattering
In the field theory the high energy hard elastic scattering of point-like
particles leads usually to the power behavior of elastic cross sections
$d\sigma_{1}^{el}(s,t\simeq-s/2)/dt\sim 1/s^{a}~{},~{}~{}~{}~{}~{}$
For the scattering of particles composed from n constituents with
approximately equal momenta we have
$d\sigma_{n}^{el}(s,t\sim-s/2)/dt\sim\mu^{-4(n-1)}(d\sigma_{1}(s/n^{2},t\sim-s/n^{2})/dt)^{n}~{}~{}.$
But the mean state can contain the growing number of partons and the direct
application of this expression leads to a small contribution. In this case the
main contribution to $d\sigma/dt$ can come from the rare parton configurations
containing the minimal number of partons (when both particles are in a “bare”
state). Then, the cross-section of particles in the system, where
$~{}s=m^{2}e^{Y}$ and the energies of colliding particles are
$me^{y},~{}me^{Y-u}$, can be represented as
$d\sigma^{el}(s,t\sim-s/2)/dt\sim\big{(}~{}d\sigma_{0}(s,t\sim-s/2)/dt~{}\big{)}^{n_{0}}~{}w(y)w(Y-y)~{}~{},$
(31)
where $w(y)$ is the probability that particle with energy $me^{y}$ is in the
bare state, and $n_{0}$ \- the number of “valent” components in the bare state
($n_{0}\simeq 2\div 3$ for meson $\div$ baryon). It follows from the boost-
invariance of (31) that
$w(y)\sim e^{-2cy}$ (32)
This condition essentially restricts the behavior of the asymptotic of hard
scattering and, in particular, gives the information about the amplitude (
$\sim\sqrt{w(y)}$ ) of the bare component of $\Psi(P)$.
The similar limitation follows from the consideration of the asymptotic cross-
sections of two particle reactions with exchange of quantum numbers (such as
$\pi^{-}+p~{}\rightarrow~{}\pi^{0}+n$). Here again, the dominant parton
configuration contributing to such reactions must contain the minimum number
of partons. So again, we have the factor $w(y)w(Y-y)$ in the cross-section.
Additionally, there is the factor of type $e^{-2gy}$ connected with the
probability that this parton configuration contains also the small energy
parton with “needed” quantum numbers. Therefore, from the frame independence
of amplitudes of such reactions we also come to the condition (32). And, if
interpreting in terms of the exchange of some nonvacuum reggeon we come to
estimate their intercept as $\alpha(0)\simeq 1-c-g$ .
## 3 Summary
The main aim of this note was to illustrate that the condition of boost-
invariants essentially restricts the behavior of high energy cross-sections
calculated in parton approaches. And the form of resulting constrains is of
the same type as coming from the t-channel unitarity condition. So that one
can suppose that this similarity, by their nature, has much more general
grounds.
Such a condition works especially effectively in the case of growing with
energy cross-section, that is, just when the t-unitarity conditions for
amplitudes is complicated to apply - because here the multiparticle exchange
becomes important. In this case the resulting restrictions on the asymptotic
behavior are rather strong and can, in principle, exclude some popular models.
## References
* [1] V.N. Gribov, arXiv:hep-ph/0006158
* [2] S.Brodsky, H-C Pauli, S.Pinsky, 9705477, Phys.Rept 301 (1998) 299
* [3] M.Perry, arXiv:hep-ph/9612244
* [4] T.Heinzl, arXiv hep-th/0008096
* [5] A.Harindranath, arXiv:hep-ph/9612244
* [6] A.B.Kaidalov, ITEP School of Physics 1983
* [7] Y. Kovchegov, E. Levin , Quantum Chromodynamics at High Energy,
Cambridge University Press, 2012
* [8] O.V. Kancheli arXiv 1609.07657
* [9] V.N. Gribov, I.Ya. Pomeranchuk, Phys.Rev.Lett. 8,343,412
* [10] V.N. Gribov, I.Ya. Pomeranchuk and K.A. Ter-Martirosyan, Phys. Rev. 139B (1965) 184 ;
V.N. Gribov, Soviet Phys. JETP 26, 414, (1968) .
|
2024-09-04T02:54:58.663665 | 2020-03-10T12:35:38 | 2003.04662 | {
"authors": "Isaac Alonso Asensio, Claudio Dalla Vecchia, Yannick M. Bah\\'e, David\n J. Barnes and Scott T. Kay",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26132",
"submitter": "Isaac Alonso Asensio",
"url": "https://arxiv.org/abs/2003.04662"
} | arxiv-papers | # The intra-cluster light as a tracer of the total matter density
distribution: a view from simulations
Isaac Alonso Asensio,1,2 Claudio Dalla Vecchia,1,2 Yannick M. Bahé,3David J.
Barnes4 and Scott T. Kay5
1Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, E-38205 La Laguna,
Tenerife, Spain
2Departamento de Astrofísica, Universidad de La Laguna, Av. Astrofísico
Francisco Sánchez s/n, E-38206 La Laguna, Tenerife, Spain
3Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The
Netherlands
4Department of Physics, Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
5Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
School of Natural Sciences,
The University of Manchester, Manchester M13 9PL, UK E-mail<EMAIL_ADDRESS>(IAA)
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
By using deep observations of clusters of galaxies, it has been recently found
that the projected stellar mass density closely follows the projected total
(dark and baryonic) mass density within the innermost $\sim 140$ kpc. In this
work, we aim to test these observations using the Cluster-EAGLE simulations,
comparing the projected densities inferred directly from the simulations. We
compare the iso-density contours using the procedure of Montes & Trujillo
(2019), and find that the shape of the stellar mass distribution follows that
of the total matter even more closely than observed, although their radial
profiles differ substantially. The ratio between stellar and total matter
density profiles in circular apertures, shows a slope close to $-1$, with a
small dependence on the cluster’s total mass. We propose an indirect method to
calculate the halo mass and mass density profile from the radial profile of
the intra-cluster stellar mass density.
###### keywords:
galaxies: clusters: general – methods: numerical
††pubyear: 2020††pagerange: The intra-cluster light as a tracer of the total
matter density distribution: a view from simulations–References
## 1 Introduction
Ten to more than thirty percent of the stellar light of clusters of galaxies
comes from a diffuse distribution of stars emitting the so called intra-
cluster light (ICL), the inferred fraction depending on the definition of the
border between the brightest central galaxy and the diffuse stellar component,
the radial extent at which the stellar mass distribution is integrated and the
relaxation state of the clusters (e.g., Krick & Bernstein, 2007; Gonzalez et
al., 2013; Mihos et al., 2017; Jiménez-Teja et al., 2018; Zhang et al., 2019).
This distribution is produced by the stripping of stars from galaxies
undergoing mergers and tidal interactions during their evolution in the
cluster environment (see Mihos, 2015, for a review). Due to its low surface
brightness, the observational study of the stellar population producing the
intra-cluster light has been challenging. An increasing effort towards deep
imaging of clusters of galaxies in the recent years, both through individual
cluster imaging, up to $z\simeq 1.5$, (e.g., Mihos et al., 2005; Montes &
Trujillo, 2014; Burke et al., 2015; Morishita et al., 2017; Ko & Jee, 2018;
Jiménez-Teja et al., 2018; Montes & Trujillo, 2018; DeMaio et al., 2018;
DeMaio et al., 2020), and by stacking observations of multiple clusters (e.g.,
Zibetti et al., 2005; Zhang et al., 2019) has allowed new insights into the
ICL.
Figure 1: Projected density of stars (top) and matter (bottom) for three
different cluster of the C-EAGLE simulations at $z=0.352$. The secondary
density peak in the Cluster CE-21 (middle panel) can be due to a recent major
merger event which has stripped stars from the interacting galaxies, or
numerical artifacts produced by SUBFIND not being able to correctly assign
stellar particles to substructures.
One recent, remarkable result achieved with deep imaging is the tight
correlation between the distribution of the stellar surface density, inferred
from its surface brightness, and the surface density of the total mass,
measured by modelling the gravitational lensing signal (Montes & Trujillo,
2019, hereafter MT19). MT19 proposed that the surface density of the stellar
mass not bound to galaxies should settle in the potential well of the cluster
similarly to the dark matter. This could be used to trace the total matter
distribution of clusters within a cluster-centric distance set by the depth of
the observations. They also compared their result with total mass surface
densities inferred from the X-ray emission of the intra-cluster medium, and
concluded that this method is limited by the misalignment of the gaseous
component with respect to the dark matter and stellar mass in non-relaxed
clusters. Their quantitative analysis made use of the Modified Haussdorf
Distance (MHD) (Dubuisson & Jain, 1994) to quantify the deviation between iso-
density contours of stars and total matter. They found that, in general, the
stellar surface density has smaller MHD values than that of the intra-cluster
medium where both are compared with the iso-density contours of total mass.
In this Letter, we test this observational result with state-of-the-art
cosmological, hydrodynamic simulations of the Cluster-EAGLE project (C-EAGLE,
Barnes et al., 2017; Bahé et al., 2017). We give a brief description of the
simulations in the next section and a description of the analysis in section
3. The main results of this work are shown in section 4 and discussed in
section 5, along with some concluding remarks.
## 2 Simulations
We have used the set of 30 zoom-in cluster simulations performed within the
C-EAGLE project. The simulated clusters are uniformly distributed in the mass
range $10^{14}<M_{200}/\mathrm{M}_{\odot}<10^{15.4}$, where $M_{200}$ is the
halo mass.111$M_{200}$ is the mass enclosed in a sphere of radius $r_{200}$,
whose mean density equals 200 times the critical density of the Universe. The
simulations were performed with the EAGLE model for galaxy formation and
evolution, with the AGNdT9 calibration (Schaye et al., 2015). They provide a
physical spatial resolution of $\epsilon=0.7~{}\mathrm{kpc}$ (at $z<2.8$) and
baryonic mass resolution of $m_{\mathrm{gas}}\approx 1.81\times
10^{6}~{}\mathrm{M}_{\odot}$. For more information on the EAGLE model and its
comparison with global relations of the observed galaxy population, the reader
is referred to Schaye et al. (2015) and Crain et al. (2015).
For more details on the numerical algorithms describing photo-ionization
equilibrium cooling, star formation, stellar evolution, stellar feedback,
black hole growth and feedback, and the hydrodynamic scheme we refer the
reader to Wiersma et al. (2009a), Schaye & Dalla Vecchia (2007), Wiersma et
al. (2009b), Dalla Vecchia & Schaye (2012), Rosas-Guevara et al. (2015), and
Schaller et al. (2015), respectively.
Figure 2: Fraction of stellar mass contributing to the ICL,
$f_{\mathrm{ICL}}$, as function of halo mass, $M_{200}$, for all clusters in
the sample. There is no evidence for a correlation with halo mass. The solid
line marks the average value of $f_{\mathrm{ICL}}$, and the dashed lines the
spread around it.
For the results presented here, we have used the particle data, friends-of-
friends and SUBFIND (Dolag et al., 2009) groups at $z=0.352$ to match the
average redshift of the Hubble Frontier-Fields clusters (Lotz et al., 2017).
Furthermore, the same analysis was performed at $z=0$, and we found no
significant difference. Throughout the paper we assume the cosmological
parameters of the C-EAGLE simulations,
$(\Omega_{0},\Omega_{\Lambda},h,n_{\mathrm{s}},\sigma_{8})=(0.307,0.693,0.6777,0.961,0.8288)$
(Planck Collaboration et al., 2014), where $\Omega_{0}$ and $\Omega_{\Lambda}$
are the matter and dark energy fractions, $h$ is the Hubble constant in units
of $100~{}\mathrm{km}\,\mathrm{Mpc}^{-1}\,\mathrm{s}^{-1}$, $n_{\mathrm{s}}$
and $\sigma_{8}$ are the spectral index and the power spectrum normalisation
used to generate the initial conditions.
In the analysis, we have used all particles belonging to the main halo of the
largest friends-of-friends group in each simulation, i.e., we excluded all
particles bound to satellite galaxies and substructures within the same
friends-of-friends group. Maps of projected stellar and total matter density
were produced with a spatial resolution of $5~{}\mathrm{kpc}$, in order to
mimic the spatial resolution employed in the analysis of the observational
data ($3\times 3~{}\mathrm{arcsec}^{2}$ at $z\simeq 0.35$). We have repeated
the analysis with higher ($3.75~{}\mathrm{kpc}$) and lower
($7.5~{}\mathrm{kpc}$) resolution without finding any remarkable difference.
The main advantage with respect to observations is that there is no need of
masking the light of satellite galaxies. However, debris from tidal
interactions between galaxies will be included in the projected matter
density. Furthermore, there are biases due to SUBFIND failing to assign
stellar particles to satellites (Bahé et al., in prep).
Figure 3: Isodensity contours of the inner ($R\leq 140~{}\mathrm{kpc}$) and
outer ($R>140~{}\mathrm{kpc}$) regions (top and bottom, respectively) of total
matter (blue dotted lines) and stars (red dashed lines) for three different
clusters. Lighter colours indicate larger distances (lower densities) from the
centre.
Examples of the projected stellar and total mass density are shown in Fig. 1
for three simulated clusters of increasing virial mass. The top row
corresponds to the projected density of stars, while the bottom row shows the
density of total matter (dark and baryonic).
Uncertainties on the amount of ICL mass produced and its radial distribution
may arise from the modelling of the star formation rate and the spatial and
mass resolution of numerical simulations. The EAGLE model matches quite
accurately the observed stellar mass and luminosity functions (Schaye et al.,
2015; Trayford et al., 2015). Moreover, it reproduces the evolution of the
stellar mass function and the observationally inferred density of stars in the
universe up to high redshift ($z=7$) (Furlong et al., 2015). However, while
the reference simulation matches the observed sizes of galaxies over several
decades in stellar mass, the AGNdT9 calibration yields an offset in the
relation towards more compact galaxies (Schaye et al., 2015). This last point
seems to be relevant in the interpretation of the ICL mass fractions described
in the next section, where the inferred values are on the low side of the
distribution of those derived from observations (see references in section 1):
compact galaxies are less prone to stripping. On the other hand, Henden et al.
(2019) noted that having too large in size galaxies in their simulations
boosts the effect of tidal stripping, increasing the fraction of stellar mass
in the ICL, and that uncertainties in galaxy sizes are the major contributors
to the uncertainty in the determining the fraction of mass in the ICL in
simulations.
## 3 Analysis
Before describing the methodology used in the analysis of the simulation data,
we briefly discuss a consistency check for the simulated clusters. We computed
the fraction of stellar mass in the ICL, $f_{\mathrm{ICL}}$, and compared it
with expected observational and theoretical values. For the sake of ease, we
adopted the methodology of (Rudick et al., 2011). The mass fraction has been
computed as the stellar mass with projected stellar density below some
threshold surface brightness, $\mu$, with respect to the total stellar mass
within $r_{200}$. As in (Rudick et al., 2011), we have converted the stellar
surface density into surface brightness assuming a constant mass-to-light
ratio of $5~{}\mathrm{M}_{\odot}\,\mathrm{L}^{-1}_{\odot}$, and set
$\mu=26.5~{}\mathrm{mag}\,\mathrm{arcsec}^{-2}$ as the threshold.
We show in figure 2 the computed $f_{\mathrm{ICL}}$ as function of halo mass.
We find that $f_{\mathrm{ICL}}=0.091\pm 0.013$ (solid and dashed lines), with
no significant correlation with the total mass of the clusters (the Pearson
correlation coefficient is $0.0063$). The result is consistent with that of
(Rudick et al., 2011). Although the range of halo masses in our sample is
rather narrower, similar fractions and the lack of correlation have been
reported by (Pillepich et al., 2018), when using a definition of the ICL
related to the size of the central galaxy. The result is consistent with
previous simulations (Rudick et al., 2011; Contini et al., 2014), where they
applied semi-analytical models to N-body simulations, and hydrodynamical
cosmological simulations (Pillepich et al., 2018; Henden et al., 2019).
Finally, observations using similar thresholds have reported as well similar
mass fractions (Krick & Bernstein, 2007; Montes & Trujillo, 2014).
We have followed a methodology similar to MT19 to extract iso-density
contours. We computed circularly averaged radial profiles of the density of
the stellar and total mass. For this, we take the position of the minimum of
the potential energy as centre of the cluster (McAlpine et al., 2016). The
projected densities for drawing the contours222We used the contour function of
matplotlib to compute the contours. were selected interpolating the profiles
at radii of 50, 75, 100, 125, $140~{}\mathrm{kpc}$ (the distances used by
MT19) for the inner part, and of 170, 220, 300, 460, 620, 780, 940,
$1100~{}\mathrm{kpc}$ for the outer regions, and only up to $r_{200}$. At
large distance from the centre of the clusters ($r>140~{}\mathrm{kpc}$), we
down-sample the images merging $4\times 4$ pixels, thus degrading the spatial
resolution to $20~{}\mathrm{kpc}$, to smooth the otherwise very noisy
contours. The contours of the projected densities are shown in Fig. 3, for the
same three clusters as depicted in Fig. 1. The projected total mass density
contours are drawn with blue dotted lines, and the projected stellar density
contours with red dashed lines, where a darker colour indicates a smaller
radius. The top row is a close-up view of the contours near the centre of the
clusters, out to $140~{}\mathrm{kpc}$, whereas the contours at larger
distances are shown in the bottom row.
We measured projected radial distances from the centre of the cluster instead
of elliptical distances to the centre of the brightest central galaxy, as
usually done in observations. This simplification is not crucial to derive the
iso-density contours, as it only changes the values of density at which the
contours will be drawn. In practice, this means that the distances we use are
systematically different from those of MT19, the difference depending on the
eccentricity of the brightest central galaxy, or the presence of more than one
central galaxy, that we excluded from the analysis, or both. As this is only
an exploratory analysis we ignore these differences.
As in MT19, to compare the shape of the contours, we estimated the Modified
Hausdorff distance (MHD) defined by Dubuisson & Jain (1994):
$d_{\mathrm{MH}}(X,Y)=\max\left(d(X,Y),d(Y,X)\right),\\\ $ (1)
where
$d(X,Y)=\frac{1}{N_{X}}\sum_{\textbf{x}\in X}\min_{\textbf{y}\in
Y}\|\textbf{x}-\textbf{y}\|.$ (2)
The two samples,
$X\equiv\\{\textbf{x}_{1},\textbf{x}_{2},\dots,\textbf{x}_{N_{x}}\\}$ and
$Y\equiv\\{\textbf{y}_{1},\textbf{y}_{2},\dots,\textbf{y}_{N_{y}}\\}$,
contains the points defining two contours, and $\|\cdot\|$ is the Euclidean
norm. As we may have different closed contours for the same density value, we
select for each distance the contour composed by the largest number of
segments. The selected contours are shown in Fig. 3.
Figure 4: Left panel. Comparison of the MHD of MT19 (in green, signle
measurements with error bars) and that from the C-EAGLE simulations (in blue,
solid line). The shadows indicate the 1-$\sigma$ region for each method. A
small scatter in the radial distance of the MT19 data has been added for
clarity. Right panel. Histogram of $\zeta$ computed from all the contours
taken inside the virial radius of each C-EAGLE cluster. The vertical (green)
solid line represents the mean value of $\zeta$ obtained by MT19, embedded in
its 1-$\sigma$ region. The dotted, vertical line indicates their lowest value.
When measuring the MHD close to the virial radius of the clusters, we would
expect an increase of its value, as the outskirts of clusters are not
dynamically relaxed and fewer stellar particles are populating it, producing
noisier contours. In order to compare the MHD across different distances, we
define the relative MHD as
$\zeta=\frac{d_{\mathrm{MH}}(r)}{r}\,,$ (3)
where $r$ is the distance at which the iso-density contours have been
computed. This way, we are measuring deviations as fraction of the distance.
We find that this definition removes almost entirely the correlation with
distance.
## 4 Results
In the left panel of Fig. 4, we show with the blue, solid line the mean value
of $d_{\mathrm{MH}}$, the shaded area depicts the 1-$\sigma$ confidence
interval. We overplotted the MHDs calculated by MT19, as well as their
1-$\sigma$ area, in green. For sake of clarity, observational points for
individual cluster are slightly displaced along the x-axis. From that panel,
we can highlight that:
1. 1.
the $d_{\mathrm{MH}}$ from both simulations and observations are of the same
order of magnitude;
2. 2.
they show the same trends with radius;
3. 3.
and simulations have a $\sim 50\%$ lower $d_{\mathrm{MH}}$ than observations,
with smaller scatter.
As $d_{\mathrm{MH}}$ increases monotonically with the distance at which it is
computed, we introduced the relative MHD, $\zeta$, to obtain a distance-free
similarity measurement. We show in Fig. 4 (right panel) the distribution of
$\zeta$ for all contours and clusters, in blue, and the $\zeta$ extracted from
MT19’s data, in green. Most of the values of $\zeta$ are lower than those
observed: 96 percent of the relative MHDs are below the mean observed value.
The shape of the distribution is remarkably close to a Gaussian distribution
in logarithmic space, with mean $\langle\zeta\rangle=0.107$ and dispersion
$\sigma_{\zeta}=0.080$, indicating that $\zeta$ is a solid, scale-free
estimate of the similarity of contours at any cluster-centric distance.
Figure 5: Left panel. Stellar (solid lines) and total matter(dashed lines)
surface density profiles from the particles of the main halo of the cluster.
We consider only the ICL mass (see text), including the particles not bounded
to any substructure. The dashed line is the threshold used for computing the
ICL mass fraction (i.e., $\mu=26.5$ mag arcsec-2, or $\Sigma_{*}\approx
1.4\times 10^{6}~{}\mathrm{M}_{\odot}\,\mathrm{kpc}^{-2}$). Right panel. The
ratio between the stellar and total matter density profiles for all the
clusters. The red, dashed line is the best-fit power law given in equation 4.
We would like to highlight two relevant issues with the definition of
$d_{\mathrm{MH}}$ that can bias the observational values towards higher
values. The $d_{\mathrm{MH}}$ is defined based on points and not continuous
segments. This obviously simplifies the computation, but it has to be taken
into account when dealing with coarse datasets, as two similar shapes can have
a non-negligible $d_{\mathrm{MH}}$. Second, each point’s contribution is
defined positive and with respect to the other set of points. This provides a
distance that increases monotonically with noise (Dubuisson & Jain, 1994),
thus special care must be taken when dealing with data with low signal-to-
noise or large uncertainties. Both these points could be driving the observed
$d_{\mathrm{MH}}$ towards higher values, as masking galaxies introduces non-
continuous contours and the spatial resolution of the lensing models is
limited.
In addition to the study of the similarity between the total matter and
stellar mass distribution, we have also compared the density profiles of the
stellar component. In Fig. 5 (left panel) we show circularly averaged density
profiles of the stellar particles. They follow a power-law behaviour up to
$\sim 500~{}\mathrm{kpc}$ for the lightest halos, and $\sim 1~{}\mathrm{Mpc}$
for the more massive ones. At such distances, the interactions between
substructures are weaker, and fewer particles get ejected to the intra-cluster
medium, thus they can no longer successfully trace the potential well. In the
right panel of Fig. 5 we show the ratio between the stellar and total matter
density profiles. This ratio is close to a power law with scatter of
$0.1~{}\mathrm{dex}$ and a slope of about $-1$. We have performed a fit to all
the profiles at once, with and without normalising the radial distance using
$r_{200}$, yielding the relations:
$\displaystyle\log_{10}\Sigma_{\mathrm{tot}}=$
$\displaystyle\log_{10}\Sigma_{*}+$ $\displaystyle(1.115\pm
0.005)\log_{10}r-(0.25\pm 0.01)\,,$ (4)
$\displaystyle\log_{10}\Sigma_{\mathrm{tot}}=$
$\displaystyle\log_{10}\Sigma_{*}+$ $\displaystyle(1.085\pm
0.004)\log_{10}(r/r_{200})+(3.144\pm 0.005)\,.$ (5)
The residuals of both fits have a similar scatter: $0.147$ and $0.127$ dex for
equations 4 and 5, respectively. We recall that the AGNdT9 feedback
calibration, used in the C-EAGLE simulations, yields more compact galaxies
than the reference model for stellar masses
$M_{\star}>10^{10}~{}\mathrm{M}_{\odot}$. The less efficient tidal stripping
may therefore deposit more stellar mass closer to the centre of the cluster,
resulting in a steeper density profile. However, this bias may be of secondary
importance, at least within the central $100~{}\mathrm{kpc}$ (Bahé et al., in
prep.).
We propose a new, indirect way of measuring a cluster’s mass knowing its
stellar density profile in the innermost region. First, via deep imaging as
that performed by MT19, the stellar density profile can be obtained and
extrapolated up to $r_{200}$ assuming a power law. Then, using equation 4 or
5, the total mass density profile can be computed. This profile can be
integrated to obtain an estimation of the cluster’s total mass.
This procedure would be similar to that proposed by Pillepich et al. (2018).
In that case, however, only the power law slope of the 3D stellar mass density
profile was used to infer the total halo mass, in our case we use more
information (the 2D stellar density profile and equation 4 or 5), expecting
less scatter in the mass estimate.
## 5 Discussion & Conclusions
We have studied the similarity of the projected stellar and total matter
distributions in the halos of massive galaxy clusters using the C-EAGLE set of
30 zoom-in simulations of clusters of galaxies. In the analysis, we considered
as constituents of the diffuse distribution of stellar mass only particles in
the friends-of-friends group that were not assigned to any substructure by the
SUBFIND algorithm.
We can summarise our results as follows:
1. 1.
we confirm the finding of MT19: the projected distribution of stars closely
follow the projected distribution of the total mass, although their radial
profiles differ substantially;
2. 2.
the ICL, approximated as those stars in the region where
$\mu>26.5~{}\mathrm{mag}\,\mathrm{arcsec}^{-2}$ ($\Sigma_{*}\approx 1.4\times
10^{6}~{}\mathrm{M}_{\odot}$), accounts for $\sim 10$ percent of the stellar
content of the cluster within $r_{200}$; this fraction does not show any
correlation with the mass of the cluster;
3. 3.
the ratio between the surface density profiles of the stellar to the total
matter follows a simple power-law up to the virial radius, equations 4 and 5;
as the slope and amplitude of the stellar surface density profile can be
extracted from observation, we proposed a method to estimate the total mass
surface density profile, thus the mass of the halo;
4. 4.
the similarity between the stellar and total matter distributions in the
cluster halo is even higher in the simulations than that observed by MT19
(Fig. 4); This indicates that stars closely trace the underlying gravitational
potential;
5. 5.
in order to show any self-similarity, we have introduced the relative measure,
$\zeta=d_{\mathrm{MH}}/r$, whose distribution resembles a log normal when
using all the clusters and contours pairs; the parameter $\zeta$ could be used
to study the relaxation state of a cluster; the maximum of this distribution
is located at $\zeta\sim 0.1$, thus the typical $d_{\mathrm{MH}}$ is about
$10\%$ of the distance at which it is computed.
The study of the spatial distribution of the ICL can be used to infer, in high
detail, the distribution of the underlying dark matter in clusters of
galaxies. Moreover, the average density profile of total matter can be
extracted, and extrapolated up to the virial radius, only by measuring the
slope of the stellar mass density profile and its normalisation close to the
centre of the cluster. This is complementary to the study of Pillepich et al.
(2018), where only the total halo mass was given as function of the slope of
the 3D stellar density profile, with larger uncertainty.
## Acknowledgements
We are very grateful to Ignacio Trujillo and Mireia Montes for supporting this
work with useful ideas and discussions. CDV acknowledges the support of the
Spanish Ministry of Science, Innovation and Universities (MCIU) through grants
RYC-2015-18078 and PGC2018-094975-B-C22. YMB acknowledges funding from the EU
Horizon 2020 research and innovation programme under Marie Skłodowska-Curie
grant agreement 747645 (ClusterGal) and the Netherlands Organisation for
Scientific Research (NWO) through VENI grant 639.041.751. This work used the
DiRAC@Durham facility managed by the Institute for Computational Cosmology on
behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was
funded by BEIS capital funding via STFC capital grants ST/K00042X/1,
ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC
operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure.
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|
2024-09-04T02:54:58.674905 | 2020-03-08T06:14:13 | 2003.04720 | {
"authors": "Rohit Pandey",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26133",
"submitter": "Rohit Pandey",
"url": "https://arxiv.org/abs/2003.04720"
} | arxiv-papers | # The mean and variance in coupons required to complete a collection
Rohit Pandey
###### Abstract
This paper is about the Coupon collector’s problem. There are some coupons, or
baseball cards, or other plastic knick-knacks that are put into bags of chips
or under soda bottles, etc. A collector starts collecting these trinkets and
wants to form a complete collection of all possible ones. Every time they buy
the product however, they don’t know which coupon they will “collect” until
they open the product. How many coupons do they need to collect before they
complete the collection? In this paper, we explore the mean and variance of
this random variable, $N$ using various methods. Some of them work only for
the special case with the coupons having equal probabilities of being
collected, while others generalize to the case where the coupons are collected
with unequal probabilities (which is closer to a real world scenario).
## Problems and expressions
### Problems
There are $n$ coupons in a collection. A collector has the ability to purchase
a coupon, but can’t choose the coupon he purchases. Instead, the coupon is
revealed to be coupon $i$ with probability $p_{i}=\frac{1}{n}$. Let $N$ be the
number of coupons he’ll need to collect before he has at least one coupon of
each type. Let’s call this random variable $N$. Now, we want to solve the
following problems:
P1
The expected value of $N$ when the coupons have equal probabilities of being
collected.
P2
The expected value of $N$ when the coupons have unequal probabilities of being
collected.
P3
The variance of $N$ when the coupons have equal probabilities of being
collected.
P4
The variance of $N$ when the coupons have unequal probabilities of being
collected.
P5
The density function of $N$ (meaning the entire distribution) when the coupons
have equal probabilities.
P6
The density function of $N$ (meaning the entire distribution) when the coupons
have unequal probabilities.
This paper will go over various solutions, some more powerful (can answer more
of the above questions) than others. It’s also clear that if we can solve the
even numbered problems (2,4,6) we can simply substitute
$p_{i}=\frac{1}{n}\;\;\forall i$ and solve the corresponding odd numbered
problems (1,3,5) respectively.
### Expressions
In this section, we provide the solutions to the problems, P1 through P6 and
devote the rest of the paper to their derivations.
###### Theorem 1 (Expression for P1).
The expected number of coupons a collector will need to complete the
collection when the probabilities of collecting each of the $n$ coupons is
$\frac{1}{n}$ is:
$E(N)=n\sum\limits_{m=1}^{n}\frac{1}{m}$
###### Theorem 2 (Expression for P2).
The variance in the number of coupons a collector will need to complete the
collection when the probabilities of collecting each of the $n$ coupons is
$\frac{1}{n}$ is:
$V(N)=n^{2}\sum\limits_{i=1}^{n}\frac{1}{i^{2}}-n\sum\limits_{k=1}^{n}\frac{1}{k}$
###### Theorem 3 (Expression for P3).
The expected number of coupons a collector will need to complete the
collection when the probabilities of collecting coupon $i$ is $p_{i}$
($\sum\limits_{i=1}^{n}p_{i}=1$) is:
$E(N)=\sum\limits_{j}\frac{1}{p}_{j}-\sum\limits_{i<j}\frac{1}{p_{i}+p_{j}}+\dots+(-1)^{m-1}\frac{1}{p_{1}+\dots+p_{m}}$
###### Theorem 4 (Expression for P4).
The variance in the number of coupons a collector will need to complete the
collection when the probabilities of collecting coupon $i$ is $p_{i}$
($\sum\limits_{i=1}^{n}p_{i}=1$) is:
$V(N)=\left(\sum\frac{1}{p_{j}^{2}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})^{2}}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})^{2}}\right)-\\\
\left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)^{2}-\\\
\left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)$
## 1 A sum of geometric random variables
### 1.1 Proof 1 of theorem 1
Consider a state where the collector has already collected $m$ coupons. How
many coupons does he need to collect to get to $m+1$? Let this be represented
by the random variable, $N_{m}$. Then, if the total coupons needed is $N$, we
have:
$N=\sum\limits_{m=1}^{n}N_{m}$
Every coupon collected from here is like a coin toss where with probability
$\frac{m}{n}$, the collector hits a coupon he already has and makes no
progress. With probability $\frac{n-m}{n}$, he collects a new coupon. So, this
becomes a geometric random variable with $p=\frac{n-m}{n}$. We know that a
geometric random variable has a mean $\frac{1}{p}$ and variance
$\frac{1-p}{p^{2}}$. Hence,
$E(N_{m})=\frac{n}{n-m}$
Taking expectation of equation (1) and substituting we have:
$E(N)=E(N_{m})=\sum\limits_{m=1}^{n}\frac{n}{n-m}=n\sum\limits_{m=1}^{n}\frac{1}{n-m}$
Substituting $m=n-m$ we get:
$E(N)=n\sum\limits_{m=1}^{n}\frac{1}{m}$
### 1.2 Proof 1 of theorem 3
Since the random variables $N_{m}$ are independent, the variance of their sum
is equal to the sum of their variances. So, proceeding similarly to section
1.1 the variance, $V(N)$ can be calculated.
$V(N)=n^{2}\sum\limits_{i=1}^{n}\frac{1}{i^{2}}-n\sum\limits_{k=1}^{n}\frac{1}{k}$
## 2 Maximum of minimums identity
With this approach, we can prove theorems 1 and 2
### 2.1 Proof 1 of theorem 3
Let $N_{j}$ be the number of coupons to be collected before we see the first
coupon of type $j$ and $N$ the number of coupons until all are collected. We
have:
$N=\max_{1\leq j\leq n}N_{j}$
In conjunction with the maximum of minimums identity we get:
$N=\sum N_{j}-\sum_{1\leq j\leq k\leq n}\min N_{j},N_{k}+\sum_{1\leq j\leq
k\leq i\leq n}\min N_{j},N_{k},N_{i}-\dots$ (1)
and the fact that $\min_{1\leq j\leq m}N_{j}$ is a geometric random variable
with parameter $p=\sum\limits_{j=1}^{m}p_{j}$ lead to the result of theorem 3
and from there, we can substitute $p_{j}=\frac{1}{n}\forall j$ to get the
result of theorem 1
$E(N)=n\sum\limits_{k=1}^{n}\frac{1}{k}$
Note that it’s not easy to get the variance, $V(N)$ with this approach because
the terms in equation 1 are not independent.
## 3 A recurrence
With this approach, we can prove theorems 1 and 3.
Consider a state where the collector has $m$ coupons in his collection. Let
$T_{m}$ be the number of coupons needed to complete the collection. If the
total coupons he needs to collect to complete the collection is $N$, we then
have:
$N=T_{0}$
Now, we could observe that (the $N_{m}$ are the variables defined in section
1):
$N_{m}=T_{m+1}-T_{m}$
and summing over all $m$ (and noting that $T_{n}=0$) leads us to:
$T_{0}=\sum_{m}N_{m}$
and this leads to the approach in section 1 which makes the problem much
easier to solve. Alternately, we can continue working with the $T_{m}$’s and
construct a recurrence. Consider what happens when the collector has $m$
coupons and he collects one more. With probability $\frac{m}{n}$, he fails to
add a new coupon and is back to where he started, making no progress. Let
$I(\frac{n}{m})$ be a Bernoulli random variable with $p=\frac{n}{m}$. We then
have the expression:
$T_{m}=1+I\left(\frac{m}{n}\right)T_{m}^{\prime}+\left(1-I\left(\frac{m}{n}\right)\right)T_{m+1}$
(2)
Where $T_{m}^{\prime}$ is i.i.d with $T_{m}$.
### 3.1 Proof 2 of theorem 1
Taking expectation to both sides,
$E(T_{m})=1+\frac{m}{n}E(T_{m})+\frac{n-m}{n}T_{m+1}$
$E(T_{m})\left(1-\frac{m}{n}\right)=1+\left(1-\frac{m}{n}\right)T_{m+1}$
$E(T_{m})-E(T_{m+1})=\frac{n}{n-m}$
As noted before, the L.H.S is simply $E(N_{m})$ as defined in A1. In general
we have,
$\sum\limits_{m=k}^{n-1}E(T_{m})-\sum\limits_{m=k}^{n-1}E(T_{m+1})=\sum\limits_{m=k}^{n-1}\frac{n}{n-m}$
Noting that $T_{n}=0$ we have,
$E(T_{k})=\sum\limits_{m=k}^{n-1}\frac{n}{n-m}$
And letting $m=n-k$
$E(T_{n-m})=n\sum\limits_{k=1}^{m}\frac{1}{k}$
We’re interested in $T_{0}$, so let’s substitute $m=n$ in equation (3).
$E(T_{0})=n\sum\limits_{k=1}^{n}\frac{1}{k}$
### 3.2 Proof 2 of theorem 3
Now, let’s try and find the variance, $V(N)=V(T_{0})$. Let’s square both sides
of equation (1). To make the algebra easier, let’s re-arrange and note that
$I(\frac{m}{n})(1-I(\frac{m}{n}))=I(\frac{m}{n})-I(\frac{m}{n})^{2}=0$.
$=>(T_{m}-1)^{2}=I\left(\frac{m}{n}\right)^{2}T_{m}^{\prime
2}+(1+I\left(\frac{m}{n}\right)^{2}-2I\left(\frac{m}{n}\right))T_{m+1}^{2}$
Now, note the following property of Bernoulli random variables:
$I(\frac{m}{n})^{2}=I(\frac{m}{n})$. This means:
$T_{m}^{2}-2T_{m}+1=I\left(\frac{m}{n}\right)T_{m}^{\prime
2}+(1-I\left(\frac{m}{n}\right))T_{m+1}^{2}$
We have to be careful here to note which random variables are i.i.d. and which
are identical. See here.
Taking expectation and doing some algebra gives us,
$\left(1-\frac{m}{n}\right)E(T_{m}^{2})=2E(T_{m})+\left(1-\frac{m}{n}\right)E(T_{m+1}^{2})-1$
$=>E(T_{m}^{2})-E(T_{m+1}^{2})=2E(T_{m})\frac{n}{n-m}-\frac{n}{n-m}$
$=>\sum\limits_{m=0}^{n-1}E(T_{m}^{2})-\sum\limits_{m=0}^{n-1}E(T_{m+1}^{2})=\sum\limits_{m=0}^{n-1}2E(T_{m})\frac{n}{n-m}-\sum\limits_{m=0}^{n-1}\frac{n}{n-m}$
$=>E(T_{0}^{2})-E(T_{n}^{2})=\sum\limits_{m=0}^{n-1}2E(T_{m})\frac{n}{n-m}-\sum\limits_{m=0}^{n-1}\frac{n}{n-m}$
But, $T_{n}=0$ and from equation (3),
$E(T_{m})=n\sum\limits_{k=1}^{n-m}\frac{1}{k}$. So we get:
$E(T_{0}^{2})=\sum\limits_{m=0}^{n-1}2E(T_{m})\frac{n}{n-m}-\sum\limits_{m=0}^{n-1}\frac{n}{n-m}$
$=>E(T_{0}^{2})=2n^{2}\sum\limits_{m=0}^{n-1}\frac{1}{n-m}\sum\limits_{k=1}^{n-m}\frac{1}{k}-n\sum\limits_{m=0}^{n-1}\frac{1}{n-m}$
Now, change variables $j=n-m$
$=>E(T_{0}^{2})=2n^{2}\sum\limits_{j=n}^{1}\frac{1}{j}\sum\limits_{k=1}^{j}\frac{1}{k}-n\sum\limits_{j=n}^{1}\frac{1}{j}$
$=>E(T_{0}^{2})=2n^{2}\sum\limits_{1\leq k\leq j\leq n}\frac{1}{jk}-E(T_{0})$
This can be used in conjunction with the result of theorem 1 to get the
variance.
$V(T_{0})=2n^{2}\sum\limits_{1\leq k\leq j\leq
n}\frac{1}{jk}-E(T_{0})-E(T_{0})^{2}$
Substituting the result of theorem 1,
$V(T_{0})=2n^{2}\sum\limits_{1\leq k\leq j\leq
n}\frac{1}{jk}-n\sum\limits_{i=1}^{n}\frac{1}{i}-\left(n\sum\limits_{i=1}^{n}\frac{1}{i}\right)^{2}$
(3)
Comparing equation 3 above with the result of theorem 3 we get the easily
verifiable identity:
$2\sum_{1\leq j\leq k\leq
n}\frac{1}{jk}=\sum\limits_{i=1}^{n}\frac{1}{i^{2}}+\left(\sum\limits_{i=1}^{n}\frac{1}{i}\right)^{2}$
## 4 Using a Poisson process to make dependence disappear
Using the Poisson process to magically concoct independent random variables.
This is the most powerful of all approaches since it’s the only one that
allows us to solve for both mean and variance for the coupon collector’s
problem for the general case of coupons having unequal probabilities (and
higher moments as well). It is hence able to solve problems P1 through P4.
In example 5.17 of [1], the Coupon collector’s problem is tackled for the
general case where the probability of drawing coupon $j$ is given by $p_{j}$
and of course, $\sum\limits_{j}p_{j}=1$.
Now, he imagines that the collector collects the coupons in accordance to a
Poisson process with rate $\lambda=1$. Furthermore, every coupon that arrives
is of type $j$ with probability $p_{j}$.
Now, he defines $X_{j}$ as the first time a coupon of type $j$ is observed, if
the $j$th coupon arrives in accordance to a Poisson process with rate $p_{j}$.
We’re interested in the time it takes to collect all coupons, $X$ (for now,
eventually, we’re interested in the number of coupons to be collected, $N$).
So we get:
$X=\max_{1\leq j\leq m}X_{j}$
Note that if we denote $N_{j}$ as the number of coupons to be collected before
the first coupon of type $j$ is seen, we also have for the number needed to
collect all coupons, $N$:
$N=\max_{1\leq j\leq m}N_{j}$
This equation is less useful since the $N_{j}$ are not independent. It can
still be used to get the mean (see section 2), but trying to get the variance
with this approach gets considerably more challenging due to this lack of
independence of the underlying random variables (the are positively
correlated).
But, the incredible fact that the $X_{j}$ are independent (discussion on that
here), allows us to get:
$F_{X}(t)=P(X<t)=P(X_{j}<t\;\forall\;j)=\prod\limits_{j=1}^{m}(1-e^{-p_{j}t})$
(4)
### 4.1 Proof 2 of theorem 2
Now, Ross uses the expression: $E(X)=\int\limits_{0}^{\infty}S_{X}(t)dt$,
where $S_{X}(t)$ is the survival function to get:
$E(X)=\int\limits_{0}^{\infty}\left(1-\prod\limits_{j=1}^{m}(1-e^{-p_{j}t})\right)dt$
$=\sum\limits_{j}\frac{1}{p}_{j}-\sum\limits_{i<j}\frac{1}{p_{i}+p_{j}}+\dots+(-1)^{m-1}\frac{1}{p_{1}+\dots+p_{m}}$
and this proves the result of theorem 2.
### 4.2 Proof 4 of theorem 1
In the special case of all coupons having equal probabilities of being
collected we have: $p_{j}=\frac{1}{n}\forall j$
Substituting in the equation above we get:
$E(X)=\sum\limits_{k=1}^{n}(-1)^{k}\frac{{n\choose k}}{k}$ (5)
Let’s solve a general version of the binomial sum in equation 5.
###### Proposition 5.
We have the following binomial sum:
$\sum_{k=1}^{n}(-1)^{k-1}\frac{{n\choose
k}}{k^{r}}=\sum_{i_{1}<i_{2}<\dots<i_{r}}\frac{1}{i_{1}i_{2}\dots i_{r}}$
###### Proof.
Using the Binomial theorem:
$\frac{1-(1-t)^{n}}{t}=\sum\limits_{k=1}^{n}(-1)^{k-1}{{n\choose k}}{t^{k-1}}$
Integrate both sides from $0$ to $x$.
$\int\limits_{0}^{x}\frac{1-(1-t)^{n}}{t}dx=\sum\limits_{k=1}^{n}(-1)^{k-1}{{n\choose
k}}\frac{x^{k}}{k}$
For the LHS, let $1-t=u$
$\int\limits_{1}^{1-x}\frac{1-(u)^{n}}{1-u}(-du)=\sum\limits_{k=1}^{n}(-1)^{k-1}{{n\choose
k}}\frac{x^{k}}{k}$
$\frac{\sum\limits_{k=1}^{n}\frac{1-(1-x)^{k}}{k}}{x}=\sum\limits_{k=1}^{n}(-1)^{k-1}\frac{{n\choose
k}}{k}x^{k-1}$
Integrate both sides from $0$ to $1$, we get:
$\sum\limits_{k=1}^{n}\frac{1}{k}\int\limits_{0}^{1}\frac{1-(1-x)^{k}}{x}dx=\sum\frac{{n\choose
k}}{k^{2}}(-1)^{k-1}$
Substituting $1-x=t$ in the integral and expanding the geometric series we
get:
$\sum\limits_{k=1}^{n}\frac{1}{k}\sum\limits_{j=1}^{k}\frac{1}{j}=\sum\frac{{n\choose
k}}{k^{2}}(-1)^{k-1}=\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{k}\frac{1}{jk}$
This can very easily be extended to $k^{r}$ in the denominator:
$\sum_{k=1}^{n}(-1)^{k-1}\frac{{n\choose
k}}{k^{r}}=\sum_{i_{1}<i_{2}<\dots<i_{r}}\frac{1}{i_{1}i_{2}\dots i_{r}}$ (6)
∎
Substituting $r=1$ in equation 6 and equation 5 we have,
$E(X)=n\sum\limits_{k=1}^{n}\frac{1}{k}$
Further, Ross shows that $E(N)=E(X)$ using the law of total expectation.
First, he notes,
$E(X|N=n)=nE(T_{i})$
where $T_{i}$ are the inter-arrival times for coupon arrivals. Since these are
assume to be exponential with rate 1,
$E(X|N)=N$
Taking expectations on both sides and using the law of total expectation we
get:
$E(X)=E(N)$
### 4.3 Proof 1 of theorem 4
This approach can easily be extended to find $V(N)$, the variance (not covered
by Ross). We can use the following expression to get $E(X^{2})$:
$E(X^{2})=\int\limits_{0}^{\infty}2tP(X>t)dt=\int\limits_{0}^{\infty}2t\left(1-\prod\limits_{j=1}^{n}(1-e^{-p_{j}t})\right)dt$
Using the fact that $\int\limits_{0}^{\infty}te^{-pt}=\frac{1}{p^{2}}$ and the
same algebra as for $E(X)$ we get:
$\frac{E(X^{2})}{2}=\sum\frac{1}{p_{j}^{2}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})^{2}}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})^{2}}$
(7)
Equation 7 has given us $E(X^{2})$ but remember that we’re interested in
finding $E(N^{2})$ and from there, $V(N)$. So, we need to relate the variances
of the two random variables. Using the law of total variance we get:
$V(X)=E(V(X|N))+V(E(X|N))$
So per equation (3) we have:
$V(X)=E(V(X|N))+V(N)$
Now,
$V(X|N)=NV(T_{i})$
And since $T_{i}\sim Exp(1)$, we have $V(T_{i})=1$ meaning, $V(X|N)=N$.
Substituting into (2),
$V(X)=E(N)+V(N)$
So,
$V(N)=E(X^{2})-E(N)-E(N)^{2}$ (8)
Substituting equation 7 and the result of theorem 2 into equation 8 we get:
$V(N)=\left(\sum\frac{1}{p_{j}^{2}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})^{2}}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})^{2}}\right)-\\\
\left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)^{2}-\\\
\left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)$
(9)
### 4.4 Proof 3 of theorem 2
Now, let’s consider the special case where all coupons have an equal
probability of being selected. In other words,
$p_{j}=\frac{1}{n}\;\forall\;j$.
We get:
$\frac{E(X^{2})}{2}=n^{2}\left(\sum\limits_{k=1}^{n}(-1)^{k-1}\frac{{n\choose
k}}{k^{2}}\right)$ (10)
We now solve a general version of the binomial summation in equation 10 above.
Using equations 6 and 10 we get:
$E(X^{2})=2n^{2}\left(\sum_{j=1}^{n}\sum_{k=1}^{j}\frac{1}{jk}\right)$ (11)
Using equations 11 and 8, we get the same result we got from the recurrence in
section 3, equation 3.
## Acknowledgements
I’d like to thank mathexchange user, Simon for encouraging me to convert the
Q&A page on this into a paper.
## References
* [1] Ross, S. (2010). Introduction to Probability Models, 10th ed. Elsevier.
|
2024-09-04T02:54:58.689254 | 2020-03-07T13:23:01 | 2003.04730 | {
"authors": "Rapha\\\"el Berthon, Bastien Maubert, Aniello Murano, Sasha Rubin, Moshe\n Vardi",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26134",
"submitter": "Bastien Maubert",
"url": "https://arxiv.org/abs/2003.04730"
} | arxiv-papers | # Strategy Logic with Imperfect Information
Raphaël Berthon École Normale Supérieure de RennesComputer Science and
TelecommunicationRennesFrance<EMAIL_ADDRESS>, Bastien Maubert
0000-0002-9081-2920 Università degli Studi di Napoli “Federico
II”DIETINaplesItaly<EMAIL_ADDRESS>, Aniello Murano Università
degli Studi di Napoli “Federico II”DIETINaplesItaly<EMAIL_ADDRESS>, Sasha
Rubin Università degli Studi di Napoli “Federico II”DIETINaplesItaly
<EMAIL_ADDRESS>and Moshe Y. Vardi Rice UniversityHoustonTexasUSA
<EMAIL_ADDRESS>
(September 2018)
###### Abstract.
We introduce an extension of Strategy Logic for the imperfect-information
setting, called $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, and study
its model-checking problem. As this logic naturally captures multi-player
games with imperfect information, this problem is undecidable; but we
introduce a syntactical class of “hierarchical instances” for which,
intuitively, as one goes down the syntactic tree of the formula, strategy
quantifications are concerned with finer observations of the model, and we
prove that model-checking $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
restricted to hierarchical instances is decidable. This result, because it
allows for complex patterns of existential and universal quantification on
strategies, greatly generalises the decidability of distributed synthesis for
systems with hierarchical information. It allows us to easily derive new
decidability results concerning strategic problems under imperfect information
such as the existence of Nash equilibria, or rational synthesis.
To establish this result we go through an intermediary, “low-level” logic much
more adapted to automata techniques. $\textnormal{{QCTL}}^{*}$ is an extension
of $\textnormal{{CTL}}^{*}$ with second-order quantification over atomic
propositions that has been used to study strategic logics with perfect
information. We extend it to the imperfect information setting by
parameterising second-order quantifiers with observations. The simple syntax
of the resulting logic, $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$, allows us to provide a conceptually neat reduction of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ that separates
concerns, allowing one to forget about strategies and players and focus solely
on second-order quantification. While the model-checking problem of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is, in general,
undecidable, we identify a syntactic fragment of hierarchical formulas and
prove, using an automata-theoretic approach, that it is decidable.
We apply our result to solve complex strategic problems in the imperfect-
information setting. We first show that the existence of Nash equilibria for
deterministic strategies is decidable in games with hierarchical information.
We also introduce distributed rational synthesis, a generalisation of rational
synthesis to the imperfect-information setting. Because it can easily be
expressed in our logic, our main result provides a solution to this problem in
the case of hierarchical information.
strategic reasoning, imperfect information, perfect recall, distributed
synthesis, hierarchical information, Nash equilibria, rational synthesis
††journal: TOCL††copyright: acmlicensed††doi: 0000001.0000001††ccs: Theory of
computation Logic and verification††ccs: Theory of computation Modal and
temporal logics††ccs: Theory of computation Automata over infinite objects
## 1\. Introduction
Temporal logics such as LTL (Pnueli, 1977) or $\textnormal{{CTL}}^{*}$
(Emerson and Halpern, 1986) are extremely successful logics that have been
studied in great detail and extended in many directions along the past
decades, notably in relation with the development of the model-checking
approach to program verification (Clarke et al., 1999). When considering
systems with multiple components such as multi-agent systems or distributed
programs, popular extensions of temporal logics are the family of so-called
_logics for strategic reasoning_ , or _strategic logics_ , which introduce
operators that can express the existence of strategies for components to
ensure that the system’s executions satisfy certain temporal properties.
A fundational logic in this family is Alternating-time Temporal Logic (ATL)
(Alur et al., 2002). It extends $\textnormal{{CTL}}^{*}$ with a coalition
operator $\langle A\rangle\varphi$, where $A$ is a subset of components/agents
of the system, which reads as “coalition $A$ has a strategy to enforce
property $\varphi$ no matter what the other components/agents do”. This logic
is thus quite expressive, as it allows for instance to express the existence
of winning strategies in games played on graphs. However it is not well suited
to reason about other important solution concepts in game theory, such as Nash
equilibria. To address this problem Strategy Logic (SL) was introduced
(Chatterjee et al., 2010a; Mogavero et al., 2014). In SL strategies are
treated as first-order objects, thanks to strategy variables $x$ that can be
quantified upon and bound to players: $\langle\\!\langle x\rangle\\!\rangle$
reads as “there exists a strategy $x$”, and $(a,x)$ reads as “strategy $x$ is
assigned to player $a$”. This leads to a very expressive logic that can
express many solution concepts from game-theory such as best response,
existence of Nash equilibria or subgame-perfect equilibria.
Imperfect information. An essential property of realistic multi-player games
is that players often have a limited view of the system. Such imperfect
information, or partial observation, is usually captured by equipping the
models with equivalence relations $o$ (called _observations_) over the state
space, that specify indistinguishable states. Strategies are then required to
be _uniform_ , i.e., they cannot assign different moves to indistinguishable
situations. Imperfect information is known to make games computationally
harder to solve. For two-player reachability games, Reif showed in (Reif,
1984) that deciding the existence of winning strategies is Exptime -complete
for imperfect information, while it is in Ptime for perfect information. This
result has later been generalised to omega-regular objectives (Berwanger et
al., 2010; Doyen and Raskin, 2011), and adapted to the setting of program
synthesis from temporal specifications (Pnueli and Rosner, 1989; Kupferman and
Vardi, 1999). In the case of multiple players/components/agents, which
interests us here, the situation is even worse: the existence of distributed
winning strategies is undecidable already for two players with incomparable
observation trying to enforce some reachability objective in the presence of
an adversarial third player (Peterson and Reif, 1979), and a similar result
was also proved in the framework of distributed synthesis (Pnueli and Rosner,
1990). Since then, the formal-methods community has spent much effort finding
restrictions and variations that ensure decidability (Kupferman and Vardi,
2001; Pnueli and Rosner, 1990; Gastin et al., 2009; Peterson et al., 2002;
Finkbeiner and Schewe, 2005; Pinchinat and Riedweg, 2005; Schewe and
Finkbeiner, 2007; Berwanger et al., 2018). The common thread in these
approaches is hierarchical information: players can be totally ordered
according to how well they observe the game. Another line of works establishes
that decidability can be retained by forbidding private communication, i.e.,
by considering variants around the idea that all new information should be
public (van der Meyden and Vardi, 1998; van der Meyden and Wilke, 2005;
Ramanujam and Simon, 2010; Belardinelli et al., 2017b, a; Bouyer, 2018).
Strategy Logic with imperfect information. We propose an extension of Strategy
Logic to the imperfect-information setting, which we call
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. The first step is to choose
how to introduce imperfect information in the logic. In the formal-methods
literature it is typical to associate observations to players. In
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, instead, we associate
observations to strategies: the strategy quantifier $\langle\\!\langle
x\rangle\\!\rangle{}$ from SL is now parameterised by observation $o$, written
$\langle\\!\langle x\rangle\\!\rangle^{o}$. This novelty allows one to
express, in the logic, that a player’s observation changes over time, to
capture for instance the loss of a sensor resulting in a diminished
observation power. We also add to our logic
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ the outcome quantifier ${\bf
A}$ from Branching-time Strategy Logic (BSL) (Knight and Maubert, 2019), which
quantifies on outcomes of strategies currently used by the agents, and the
unbinding operator $(a,\operatorname{?})$, which frees an agent from her
current strategy. This does not increase the expressivity of the logic but
presents advantages that we discuss in Section 2.2. For instance it allows us
to naturally consider nondeterministic strategies (Strategy Logic only
considers deterministic ones), which in turn allows us to capture module
checking, the extension of model checking to open systems (Kupferman et al.,
2001; Jamroga and Murano, 2014, 2015).
The logic $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is very powerful:
it is an extension of SL (which considers perfect information), and of the
imperfect-information strategic logics
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$ (Bulling and Jamroga,
2014) and $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$
(Laroussinie et al., 2015). As already mentioned,
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express the distributed
synthesis problem (Pnueli and Rosner, 1990). This problem asks whether there
are strategies for components $a_{1},\dots,a_{n}$ of a distributed system to
enforce some property given as an LTL formula $\psi$ against all behaviours of
the environment. This can be expressed by the
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula
$\Phi_{\textsc{Synth}}:=\langle\\!\langle
x_{1}\rangle\\!\rangle^{o_{1}}\dots\langle\\!\langle
x_{n}\rangle\\!\rangle^{o_{n}}(a_{1},x_{1})\dots(a_{n},x_{n}){\bf A}\psi$,
where $o_{i}$ represents the local view of component $a_{i}$. Also,
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express more complicated
specifications by alternating quantifiers, binding the same strategy to
different agents and rebinding (these are inherited from SL), as well as
changing observations. For instance, it can express the existence of Nash
equilibria.
Main result. Of course, the high expressivity of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ comes at a cost from a
computational complexity point of view. Its satisfiability problem is
undecidable (this is already true of SL), and so is its model-checking problem
(this is already true of $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize
i,R}}$ even for the single formula $\langle\\{a,b\\}\rangle{\bf F}p$ (Dima and
Tiplea, 2011), which means that agents $a$ and $b$ have a strategy profile to
reach a situation where $p$ holds). We mentioned that the two main settings in
which decidability is retrieved for distributed synthesis are hierarchical
information and public actions. We extend the first approach to the setting of
strategic logics by introducing a syntactic class of “hierarchical instances”
of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, i.e., formula/model
pairs, and proving that the model-checking problem on this class of instances
is decidable. Intuitively, an instance of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is hierarchical if, as one
goes down the syntactic tree of the formula, the observations annotating
strategy quantifications can only become finer. Although the class of
hierarchical instances refers not only to the syntax of the logic but also to
the model, the class is syntactical in the sense that it depends only on the
structure of the formula and the observations in the model. Moreover, it is
straightforward to check (in linear time) whether an instance is hierarchical
or not.
Applications. Because the syntax of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ allows for arbitrary
alternations of quantifiers in the formulas, our decidability result for
hierarchical instances allows one to decide strategic problems more involved
than module checking and distributed synthesis. For instance, we show in
Section 7 how one can apply our result to establish that the existence of Nash
equilibria is decidable in games with imperfect information, in the case of
hierarchical observations and deterministic strategies. This problem is
relevant as Nash equilibria do not always exist in games with imperfect
information (Filiot et al., 2018). We then consider the problem of rational
synthesis (Fisman et al., 2010; Kupferman et al., 2016; Condurache et al.,
2016; Filiot et al., 2018), both in its cooperative and non-cooperative
variants. We introduce the generalisations of these problems to the case of
imperfect information, and call them cooperative and non-cooperative _rational
distributed synthesis_. We then apply again our main result to establish that
they are decidable in hierarchical systems for deterministic strategies. For
the non-cooperative variant, we need the additional assumption that the
environment is at least as informed as the system. This is the case for
example when one ignores the actual observation power of the environment, and
considers that it plays with perfect information. Doing so yields systems that
are robust to any observation power the environment may have. As Reif puts it,
this amounts to synthesising strategies that are winning even if the opponent
“cheats” and uses information it is not supposed to have access to (Reif,
1984).
Approach. In order to solve the model-checking problem for
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ we introduce an intermediate
logic $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, an extension to
the imperfect-information setting of $\textnormal{{QCTL}}^{*}$ (Laroussinie
and Markey, 2014), itself an extension of $\textnormal{{CTL}}^{*}$ by second-
order quantifiers over atoms. This is a low-level logic that does not mention
strategies and into which one can effectively compile instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. States of the models of the
logic $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ have internal
structure, much like the multi-player game structures from (Peterson et al.,
2001) and distributed systems (Halpern and Vardi, 1989). Model-checking
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is also undecidable
(indeed, we show how to reduce from the MSO-theory of the binary tree extended
with the equal-length predicate, known to be undecidable (Läuchli and Savioz,
1987)). We introduce the syntactical class
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ of
hierarchical formulas as those in which innermost quantifiers observe more
than outermost quantifiers, and prove that model-checking is decidable using
an extension of the automata-theoretic approach for branching-time logics. We
provide a reduction from model checking
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to model checking
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ that preserves being
hierarchical, thus establishing our main contribution, i.e., that model
checking the hierarchical instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is decidable.
Complexity. To establish the precise complexity of the problems we solve, we
introduce a new measure on formulas called _simulation depth_. This measure
resembles the notion of alternation depth (see, e.g., (Mogavero et al.,
2014)), which counts alternations between existential and universal strategy
(or second-order) quantifications. But instead of merely counting alternations
between such operators, simulation depth reflects the underlying automata
operations required to treat formulas, while remaining a purely syntactical
notion. We prove that the model-checking problem for the hierarchical fragment
of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ and
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ are both $(k+1)$-Exptime
-complete for formulas of simulation depth at most $k$. Already for the
perfect-information fragment, this result is more precise than what was
previously known. Indeed, precise upper bounds based on alternation depth were
known for syntactic fragments of SL but not for the full logic (Mogavero et
al., 2014).
Related work. The literature on imperfect information in formal methods and
artificial intelligence is very vast. Imperfect information has been
considered in two-player games (Reif, 1984; Doyen and Raskin, 2011; Berwanger
et al., 2010), module checking (Kupferman et al., 2001; Jamroga and Murano,
2015), distributed synthesis of reactive systems (Pnueli and Rosner, 1990;
Kupferman and Vardi, 2001; Finkbeiner and Schewe, 2005) and strategies in
multiplayer games (Peterson and Reif, 1979; Peterson et al., 2002; Berwanger
et al., 2018), Nash equilibria (Ramanujam and Simon, 2010; Bouyer et al.,
2017; Bouyer, 2018), rational synthesis (Filiot et al., 2018; Gutierrez et
al., 2018), doomsday equilibria (Chatterjee et al., 2017), admissible
strategies (Brenguier et al., 2017), quantitative objectives (Degorre et al.,
2010; Pérez, 2017), and more, some of which we detail below.
Limited alternation of strategy quantification was studied in (Chatterjee and
Doyen, 2014a), in which several decidability results are proved for two and
three alternations of existential and universal quantifiers. Except for one
where the first player has perfect information, all the problems solved in
this work are hierarchical instances, and are thus particular cases of our
main result.
Quantified $\mu$-Calculus with partial observation is studied in (Pinchinat
and Riedweg, 2005), where the model-checking problem is solved by considering
a syntactic constraint based on hierarchical information, as we do for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. However they consider
asynchronous perfect recall, and the automata techniques they use to deal with
imperfect information cannot be used in the synchronous perfect-recall setting
that we consider in this work. Similarly the narrowing operation on tree
automata (see Section 4.1), which is crucial in our model-checking procedure,
considers synchronous perfect recall and does not seem easy to adapt to the
asynchronous setting.
A number of works have considered strategic logics with imperfect information.
Various semantics for ATL with imperfect information have been studied in,
e.g., (Jamroga and Bulling, 2011; Jamroga and van der Hoek, 2004). The model-
checking problem for these logics, which is undecidable for agents with
perfect recall (Dima and Tiplea, 2011), has been studied for agents with
bounded memory, for which decidability is recovered (Schobbens, 2004; Lomuscio
and Raimondi, 2006). An epistemic strategic logic with original operators
different from those of ATL and SL is proposed in (Huang and Van Der Meyden,
2014). It considers imperfect information strategies, but only for agents
without memory. Concerning perfect recall, which interest us in this work,
decidability results have also been obtained for ATL (Guelev et al., 2011) and
ATL with strategy context (Laroussinie et al., 2015) when agents have the same
information.
In (Knight and Maubert, 2019), a branching-time variant of SL is extended with
epistemic operators and agents with perfect recall. Strategies are not
required to be uniform in the semantics, but this requirement can be expressed
in the language. However no decidability result is provided. Another variant
of SL extended with epistemic operators and imperfect-information, perfect-
recall strategies is presented in (Belardinelli, 2015), but model checking is
not studied. The latter logic is extended in (Belardinelli et al., 2017a), in
which its model-checking problem is solved on the class of systems where all
agents’ actions are public, which is an assumption orthogonal to hierarchical
information.
The work closest to ours is (Finkbeiner and Schewe, 2010) which introduces a
logic CL in which one can encode many distributed synthesis problems. In this
logic, hierarchical information is a necessary consequence of the syntax and
semantics, and as a result its model-checking problem is decidable. However,
CL is close in spirit to our $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$, and its semantics is less intuitive than that of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. Furthermore, by means of a
natural translation we derive that CL is strictly included in the hierarchical
instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ (Section 6.2).
In particular, hierarchical instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express non-observable
goals, while CL cannot. When considering players that choose their own goals
it may be natural to assume that they can observe the facts that define
whether their objectives are satisfied or not. But when synthesising programs
for instance, it may be enough that their behaviours enforce the desired
properties, without them having the knowledge that it is enforced. Such non-
observable winning conditions have been studied in, e.g., (Chatterjee and
Doyen, 2010; Degorre et al., 2010; Berwanger et al., 2018).
Outline. In Section 2 we define $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ and hierarchical instances, and present some examples. In Section 3 we
define $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ and its
hierarchical fragment $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$. The proof that model checking
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ is
decidable, including the required automata preliminaries, is in Section 4. The
hierarchy-preserving translation of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ into
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is in Section 5. In
Section 6 we compare $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with
related logics, and in Section 7 we apply our main result to obtain
decidability results for various strategic problems under imperfect
information. Finally we conclude and discuss future work in Section 8.
## 2\. SL with imperfect information
In this section we introduce $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$, an extension of SL to the imperfect-information setting with
synchronous perfect-recall. Our logic presents several original features
compared to SL, which we discuss in detail in Section 2.3: we introduce an
_outcome quantifier_ akin to the path quantifier in branching-time temporal
logics, we allow for nondeterministic strategies and unbinding agents from
their strategies, and we annotate strategy quantifiers with observation
symbols which denote the information available to strategies. We first fix
some basic notations.
### 2.1. Notations
Let $\Sigma$ be an alphabet. A _finite_ (resp. _infinite_) _word_ over
$\Sigma$ is an element of $\Sigma^{*}$ (resp. $\Sigma^{\omega}$). Words are
written $w=w_{0}w_{1}w_{2}\ldots$, i.e., indexing begins with $0$. The
_length_ of a finite word $w=w_{0}w_{1}\ldots w_{n}$ is $|w|:=n+1$, and
$\mbox{last}(w):=w_{n}$ is its last letter. Given a finite (resp. infinite)
word $w$ and $0\leq i<|w|$ (resp. $i\in\mathbb{N}$), we let $w_{i}$ be the
letter at position $i$ in $w$, $w_{\leq i}$ is the prefix of $w$ that ends at
position $i$ and $w_{\geq i}$ is the suffix of $w$ that starts at position
$i$. We write $w\preccurlyeq w^{\prime}$ if $w$ is a prefix of $w^{\prime}$,
and $\textit{pref}\,(w)$ is the set of finite prefixes of word $w$. Finally,
the domain of a mapping $f$ is written $\textit{dom}(f)$, its codomain
$\textit{codom}(f)$, and for $n\in\mathbb{N}$ we let
$[n]:=\\{i\in\mathbb{N}:1\leq i\leq n\\}$.
### 2.2. Syntax
For the rest of the paper, for convenience we fix a number of parameters for
our logics and models: AP is a finite non-empty set of _atomic propositions_ ,
Ag is a finite non-empty set of _agents_ or _players_ , and Var is a finite
non-empty set of _variables_. The main novelty of our logic is that we specify
which information is available to a strategy, by annotating strategy
quantifiers $\langle\\!\langle x\rangle\\!\rangle$ with _observation symbols_
$o$ from a finite set Obs, that we also fix for the rest of the paper. When we
consider model-checking problems, these data are implicitly part of the input.
###### Definition 2.1 ($\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
Syntax).
The syntax of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is defined by
the following grammar:
$\displaystyle\varphi:=$
$\displaystyle\;p\mid\neg\varphi\mid\varphi\vee\varphi\mid\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi\mid(a,x)\varphi\mid(a,\operatorname{?})\varphi\mid{\bf
E}\psi$ $\displaystyle\psi:=$
$\displaystyle\;\varphi\mid\neg\psi\mid\psi\vee\psi\mid{\bf X}\psi\mid\psi{\bf
U}\psi$
where $p\in\textnormal{AP}$, $x\in\textnormal{Var}$, $o\in\textnormal{Obs}$
and $a\in\textnormal{Ag}$.
Formulas of type $\varphi$ are called _state formulas_ , those of type $\psi$
are called _path formulas_ , and $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ consists of all the state formulas defined by the grammar.
Boolean operators and temporal operators, ${\bf X}$ (read “next”) and ${\bf
U}$ (read “until”), have the usual meaning. The _strategy quantifier_
$\langle\\!\langle x\rangle\\!\rangle^{o}$ is a first-order-like
quantification on strategies: $\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi$ reads as “there exists a strategy $x$ that
takes decisions based on observation $o$ such that $\varphi$ holds”, where $x$
is a strategy variable. The _binding operator_ $(a,x)$ assigns a strategy to
an agent, and $(a,x)\varphi$ reads as “when agent $a$ plays strategy $x$,
$\varphi$ holds”. The _unbinding operator_ $(a,\operatorname{?})$ instead
releases agent $a$ from her current strategy, if she has one, and
$(a,\operatorname{?})\varphi$ reads as “when agent $a$ is not assigned any
strategy, $\varphi$ holds”. Finally, the _outcome quantifier_ ${\bf E}$
quantifies on outcomes of strategies currently in use: ${\bf E}\psi$ reads as
“$\psi$ holds in some outcome of the strategies currently used by the
players”.
We use abbreviations $\top:=p\vee\neg p$, $\perp:=\neg\top$,
$\varphi\to\varphi^{\prime}:=\neg\varphi\vee\varphi^{\prime}$,
$\varphi\leftrightarrow\varphi^{\prime}:=\varphi\to\varphi^{\prime}\wedge\varphi^{\prime}\to\varphi$
for boolean connectives, ${\bf F}\varphi:=\top{\bf U}\varphi$ (read
“eventually $\varphi$”), ${\bf G}\varphi:=\neg{\bf F}\neg\varphi$ (read
“globally $\varphi$”) for temporal operators,
$[\\![x]\\!]^{o}\varphi:=\neg\langle\\!\langle
x\rangle\\!\rangle^{o}\neg\varphi$ (read “for all strategies $x$ based on
observation $o$, $\varphi$ holds”) and ${\bf A}\psi:=\neg{\bf E}\neg\psi$
(read “all outcomes of the current strategies satisfy $\psi$”).
For every formula $\varphi\in\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$,
we let $\textit{free}\,(\varphi)$ be the set of variables that appear free in
$\varphi$, i.e., that appear out of the scope of a strategy quantifier. A
formula $\varphi$ is a _sentence_ if $\textit{free}\,(\varphi)$ is empty.
Finally, we let the _size_ $|\varphi|$ of a formula $\varphi$ be the number of
symbols in $\varphi$.
### 2.3. Discussion on the syntax
We discuss the syntactic differences between our logic and usual Strategy
Logic.
Outcome quantifier. This quantifier was introduced in Branching-time Strategy
Logic (BSL) (Knight and Maubert, 2019), which corresponds to the perfect-
information fragment of the logic we define here. It removes a quirk of
previous definitions, in which temporal operators could only be evaluated in
contexts where all agents were assigned a strategy. The outcome quantifier,
instead, allows for evaluation of temporal properties on partial assignments.
As a result, the notions of free agents and agent-complete assignments from
previous definitions of Strategy Logic are no longer needed (see, e.g.,
(Mogavero et al., 2014)). In addition, the outcome quantifier highlights the
inherent branching-time nature of Strategy Logic: indeed, in SL, branching-
time properties can be expressed by resorting to artificial strategy
quantifications for all agents. It will also make the correspondence with
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ tighter, which will
allow us to establish the precise complexity of the problem we solve, while
the exact complexity of model checking classic SL with perfect information is
still not known. Finally, since the usual definition of SL requires that the
current strategies define a unique outcome on which linear-time temporal
operators are evaluated, only deterministic strategies were considered. The
introduction of the outcome quantifier allows us to consider nondeterministic
strategies.
Unbinding. With the possibility to evaluate temporal operators even when some
agents are not bound to any strategy, it becomes interesting to include the
unbinding operator $(a,\operatorname{?})$, introduced in (Laroussinie and
Markey, 2015) for ATL with strategy context and also present in BSL. Note that
the outcome quantifier and unbinding operator do not increase the expressivity
of SL, at the level of sentences (Knight and Maubert, 2019).
Observations. In games with imperfect information and ATL-like logics with
imperfect information, a strategy is always bound to some player, and thus it
is clear with regards to what observations it should be defined. In SL on the
other hand, strategy quantification and binding are separate. This adds
expressive power with regards to ATL by allowing, for instance, to assign the
same strategy to two different players, but it also entails that when a
quantification is made on a strategy, one does not know with regards to which
observation this strategy should be defined. We know of three ways to solve
this. One is the approach followed here, which consists in associating with
strategy quantifiers an observation power. The second solution is to abandon
the separation between quantification and binding and to use instead
quantifiers of the form $\exists_{a}$, meaning “there exists a strategy for
player $a$”, like in (Chatterjee et al., 2010b; Belardinelli, 2014): with this
operator, the strategy is immediately bound to player $a$, which indicates
with regards to which observation the strategy should be compatible. The third
one, adopted in (Belardinelli et al., 2017a), consists in requiring that a
strategy be uniform for all agents to whom it will be bound in the formula. We
chose to adopt the first solution for its simplicity and expressiveness.
Indeed the second solution limits expressiveness by disallowing, for instance,
binding the same strategy to different agents. The third solution leads to a
logic that is more expressive than the second one, but less than the first
one. Indeed, the logic that we study here can capture the logic from
(Belardinelli et al., 2017a) (assuming that models contain observations
corresponding to unions of individual observations), and in addition
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express changes of
agents’ observation power.
### 2.4. Semantics
The models of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ are classic
concurrent game structures extended by an interpretation for observation
symbols in Obs.
###### Definition 2.2 ($\textrm{CGS}_{\textnormal{ii}}$ ).
A _concurrent game structure with imperfect information_ (or
$\textrm{CGS}_{\textnormal{ii}}$ for short) is a tuple
$\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$ where
* •
Ac is a finite non-empty set of _actions_ ,
* •
$V$ is a finite non-empty set of _positions_ ,
* •
$E:V\times\textnormal{Ac}^{\textnormal{Ag}}\to V$ is a _transition function_ ,
* •
$\ell:V\to 2^{\textnormal{AP}}$ is a _labelling function_ ,
* •
$v_{\iota}\in V$ is an _initial position_ , and
* •
$\mathcal{O}:\textnormal{Obs}\to 2^{V\times V}$ is an _observation
interpretation_.
For $o\in\textnormal{Obs}$, $\mathcal{O}(o)$ is an equivalence relation on
positions, that we may write $\sim_{o}$. It represents what a strategy with
observation $o$ can see: $\mathcal{O}(o)$-equivalent positions are
indistinguishable to such a strategy. Also, $\ell(v)$ is the set of atomic
propositions that hold in position $v$.
We define the size $|\mathcal{G}|$ of a $\textrm{CGS}_{\textnormal{ii}}$
$\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$ as the size of
an explicit encoding of the transition function:
$|\mathcal{G}|:=|V|\times|\textnormal{Ac}|^{|\textnormal{Ag}|}\times\lceil\log(|V|)\rceil$.
We may write $v\in\mathcal{G}$ for $v\in V$.
We now introduce a number of notions involved in the semantics of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. Consider a
$\textrm{CGS}_{\textnormal{ii}}$
$\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$.
Joint actions. In a position $v\in V$, each player $a$ chooses an action
$c_{a}\in\textnormal{Ac}$, and the game proceeds to position $E(v,\bm{c})$,
where $\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$ stands for the _joint
action_ $(c_{a})_{a\in\textnormal{Ag}}$. Given a joint action
$\bm{c}=(c_{a})_{a\in\textnormal{Ag}}$ and $a\in\textnormal{Ag}$, we let
$\bm{c}_{a}$ denote $c_{a}$.
Plays. A _finite_ (resp. _infinite_) _play_ is a finite (resp. infinite) word
$\rho=v_{0}\ldots v_{n}$ (resp. $\pi=v_{0}v_{1}\ldots$) such that
$v_{0}=v_{\iota}$ and for every $i$ such that $0\leq i<|\rho|-1$ (resp. $i\geq
0$), there exists a joint action $\bm{c}$ such that $E(v_{i},\bm{c})=v_{i+1}$.
Strategies. A (nondeterministic) _strategy_ is a function $\sigma:V^{+}\to
2^{\textnormal{Ac}}\setminus\emptyset$ that maps each finite play to a
nonempty finite set of actions that the player may play. A strategy $\sigma$
is _deterministic_ if for all $\rho$, $\sigma(\rho)$ is a singleton. We let
Str denote the set of all strategies.
Assignments. An _assignment_ is a partial function
$\chi:\textnormal{Ag}\cup\textnormal{Var}\rightharpoonup\mbox{\emph{Str}}$,
assigning to each player and variable in its domain a strategy. For an
assignment $\chi$, a player $a$ and a strategy $\sigma$,
$\chi[a\mapsto\sigma]$ is the assignment of domain
$\textit{dom}(\chi)\cup\\{a\\}$ that maps $a$ to $\sigma$ and is equal to
$\chi$ on the rest of its domain, and $\chi[x\mapsto\sigma]$ is defined
similarly, where $x$ is a variable; also, $\chi[a\mapsto\operatorname{?}]$ is
the restriction of $\chi$ to domain $\textit{dom}(\chi)\setminus\\{a\\}$. In
addition, given a formula
$\varphi\in\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, an assignment is
_variable-complete for $\varphi$_ if its domain contains all free variables of
$\varphi$.
Outcomes. For an assignment $\chi$ and a finite play $\rho$, we let
$\textnormal{Out}(\chi,\rho)$ be the set of infinite plays that start with
$\rho$ and are then extended by letting players follow the strategies assigned
by $\chi$. Formally, $\textnormal{Out}(\chi,\rho)$ is the set of plays of the
form $\rho\cdot v_{1}v_{2}\ldots$ such that for all $i\geq 0$, there exists
$\bm{c}$ such that for all $a\in\textit{dom}(\chi)\cap\textnormal{Ag}$,
$\bm{c}_{a}\in\chi(a)(\rho\cdot v_{1}\ldots v_{i})$ and
$v_{i+1}=E(v_{i},\bm{c})$, with $v_{0}=\mbox{last}(\rho)$.
Synchronous perfect recall. In this work we consider players with _synchronous
perfect recall_ , meaning that each player remembers the whole history of a
play, a classic assumption in games with imperfect information and logics of
knowledge and time. Each observation relation is thus extended to finite plays
as follows: $\rho\sim_{o}\rho^{\prime}$ if $|\rho|=|\rho^{\prime}|$ and
$\rho_{i}\sim_{o}\rho^{\prime}_{i}$ for every $i\in\\{0,\ldots,|\rho|-1\\}$.
Imperfect-information strategies. For $o\in\textnormal{Obs}$, a strategy
$\sigma$ is an _$o$ -strategy_ if $\sigma(\rho)=\sigma(\rho^{\prime})$
whenever $\rho\sim_{o}\rho^{\prime}$. The latter constraint captures the
essence of imperfect information, which is that players can base their
strategic choices only on the information available to them. For
$o\in\textnormal{Obs}$ we let $\mbox{\emph{Str}}_{o}$ be the set of all
$o$-strategies.
###### Definition 2.3 ($\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
semantics).
The semantics of a state formula is defined on a
$\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$, an assignment $\chi$ that is
variable-complete for $\varphi$, and a finite play $\rho$. For a path formula
$\psi$, the finite play is replaced with an infinite play $\pi$ and an index
$i\in\mathbb{N}$. The definition by mutual induction is as follows:
$\begin{array}[]{lcl}\mathcal{G},\chi,\rho\models p&\text{ if
}&p\in\ell(\mbox{last}(\rho))\\\\[1.0pt]
\mathcal{G},\chi,\rho\models\neg\varphi&\text{ if
}&\mathcal{G},\chi,\rho\not\models\varphi\\\\[1.0pt]
\mathcal{G},\chi,\rho\models\varphi\vee\varphi^{\prime}&\text{ if
}&\mathcal{G},\chi,\rho\models\varphi\;\text{ or
}\;\mathcal{G},\chi,\rho\models\varphi^{\prime}\\\\[1.0pt]
\mathcal{G},\chi,\rho\models\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi&\text{ if
}&\exists\,\sigma\in\mbox{\emph{Str}}_{o}\;\text{ s.t.
}\;\mathcal{G},\chi[x\mapsto\sigma],\rho\models\varphi\\\\[1.0pt]
\mathcal{G},\chi,\rho\models(a,x)\varphi&\text{ if
}&\mathcal{G},\chi[a\mapsto\chi(x)],\rho\models\varphi\\\\[1.0pt]
\mathcal{G},\chi,\rho\models(a,\operatorname{?})\varphi&\text{ if
}&\mathcal{G},\chi[a\mapsto\operatorname{?}],\rho\models\varphi\\\\[1.0pt]
\mathcal{G},\chi,\rho\models{\bf E}\psi&\text{ if }&\text{there exists
}\pi\in\textnormal{Out}(\chi,\rho)\text{ such that
}\mathcal{G},\chi,\pi,|\rho|-1\models\psi\\\\[5.0pt]
\mathcal{G},\chi,\pi,i\models\varphi&\text{ if }&\mathcal{G},\chi,\pi_{\leq
i}\models\varphi\\\\[1.0pt] \mathcal{G},\chi,\pi,i\models\neg\psi&\text{ if
}&\mathcal{G},\chi,\pi,i\not\models\psi\\\\[1.0pt]
\mathcal{G},\chi,\pi,i\models\psi\vee\psi^{\prime}&\text{ if
}&\mathcal{G},\chi,\pi,i\models\psi\;\text{ or
}\;\mathcal{G},\chi,\pi,i\models\psi^{\prime}\\\\[1.0pt]
\mathcal{G},\chi,\pi,i\models{\bf X}\psi&\text{ if
}&\mathcal{G},\chi,\pi,i+1\models\psi\\\\[1.0pt]
\mathcal{G},\chi,\pi,i\models\psi{\bf U}\psi^{\prime}&\text{ if
}&\exists\,j\geq i\mbox{ s.t. }\mathcal{G},\chi,\pi,j\models\psi^{\prime}\\\
&&\text{ and }\forall\,k\text{ s.t. }i\leq
k<j,\;\mathcal{G},\chi,\pi,k\models\psi\end{array}$
###### Remark 1.
Observe that because of the semantics of the outcome quantifier, and unlike
usual definitions of SL, the meaning of an
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ sentence depends on the
assignment in which it is evaluated. For instance the
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula ${\bf A}{\bf F}p$ is
clearly a sentence, but whether $\mathcal{G},\chi,\rho\models{\bf A}{\bf F}p$
holds or not depends on which agents are bound to a strategy in $\chi$ and
what these strategies are. However, as usual, a sentence does not require an
assignment to be evaluated, and for an
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ sentence $\varphi$ we let
$\mathcal{G},\rho\models\varphi$ if $\mathcal{G},\emptyset,\rho\models\varphi$
for the empty assignment $\emptyset$, and we write $\mathcal{G}\models\varphi$
if $\mathcal{G},v_{\iota}\models\varphi$.
SL is the fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
obtained by interpreting all observation symbols as the identity relation
(which models perfect information), restricting to deterministic strategies,
and considering only assignments in which each agent has a strategy (in this
case the outcome of an assignment consists of a single play; one can thus get
rid of the outcome quantifier and evaluate temporal operators in the unique
outcome of the current assignment, as usually done in SL). Also,
$\textnormal{{CTL}}^{*}$ is the fragment of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ which uses no binding,
unbinding or strategy quantification.
### 2.5. Discussion on the semantics
We now discuss some aspects of the semantics.
Evaluation on finite plays. Unlike previous definitions of Strategy Logic, we
evaluate formulas on finite plays (instead of positions), where the finite
play represents the whole history starting from the initial position of the
$\textrm{CGS}_{\textnormal{ii}}$ in which the formula is evaluated. There are
several reasons to do so. First, it allows us to define the semantics more
simply without having to resort to the notion of assignment translations.
Second, it makes it easier to see the correctness of the reduction to
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, that we present in
Section 5. In SL, a strategy only has access to the history of the game
starting from the point where the strategy quantifier from which it arises has
been evaluated. In contrast, in $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ strategies have access to the whole history, starting from the initial
position. However this does not affect the semantics, in the sense that the
perfect-information fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ with deterministic strategies corresponds to SL. Indeed, when agents
have perfect information, having access to the past or not does not affect the
existence of strategies to enforce temporal properties that only concern the
future.
Players not remembering their actions. Our definition of synchronous perfect
recall only considers the sequence of positions in finite plays, and forgets
about actions taken by players. In particular, it is possible in this
definition that a player cannot distinguish between two finite plays in which
she plays different actions. This definition is standard in games with
imperfect information (van der Meyden and Wilke, 2005; Berwanger et al., 2010;
Doyen and Raskin, 2011; Berwanger et al., 2018), since remembering one’s
actions or not is indifferent for the existence of distributed winning
strategies or Nash equilibria. However it makes a difference for some more
involved solution concepts that are expressible in strategic logics such as
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. For instance it is observed
in (Bouyer, 2017, Appendix A) that some games admit subgame-perfect equilibria
only if agents remember their own past actions. Nonetheless we consider the
setting where agents do not remember their actions, as it is the most general.
Indeed, as noted in (Chatterjee and Doyen, 2014b, Remark 2.1, p.8), one can
simulate agents that remember their own actions by storing in positions of the
game the information of the last joint move played (this may create
$|\textnormal{Ac}|^{|\textnormal{Ag}|}$ copies of each position, but the
branching degree is unchanged). One can then adapt indistinguishability
relations to take actions into account. For instance, for an observation
symbol $o$ and an agent $a$, one could consider a new observation symbol
$o_{a}$ that would be interpreted in the enriched game structure as the
refinement of $\sim_{o}$ that considers two positions indistinguishable if
they are indistinguishable for $\sim_{o}$ and contain the same last action for
agent $a$. Binding agent $a$ only to strategies that use observation of the
form $o_{a}$ for some $o$ captures the fact that agent $a$ remembers her
actions.
Agents changing observation. In $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ observations are not bound to agents but to strategies. And because
agents can change their strategy thanks to the binding operator, it follows
that they can change observation, or more precisely they can successively play
with strategies that have different observations. For instance consider a
controller that observes a system through a set of $n$ sensors
$S=\\{s_{1},\ldots,s_{n}\\}$ as in, e.g., (Bittner et al., 2012). Let $o_{i}$
be the observation power provided by the set of sensors
$S\setminus\\{s_{i}\\}$ (one can think of a system where states are tuples of
local states, each sensor observing one component). Also let $o$ be the
observation power provided by the full set $S$ of sensors, and let atom
$\text{fault}_{i}$ represent the fact that a fault occurs on sensor $s_{i}$.
The formula
$\varphi:=\langle\\!\langle x\rangle\\!\rangle^{o}(a,x){\bf A}{\bf
G}\left(\text{safe}\wedge\bigwedge_{i=1}^{n}\text{fault}_{i}\to\langle\\!\langle
x\rangle\\!\rangle^{o_{i}}(a,x){\bf A}{\bf G}\text{\,safe}_{i}\right)$
expresses that the controller $a$ has a strategy (which uses all sensors in
$S$) to maintain the system safe, and if a sensor is lost, it can respond by
switching to a strategy using the remaining sensors to maintain some
alternative, possibly weaker, security requirement $\text{safe}_{i}$.
### 2.6. Model checking and hierarchical instances
We now introduce the main decision problem of this paper, which is the model-
checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. An
_$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ -instance_ is a model
together with a formula, i.e., it is a pair $(\mathcal{G},\Phi)$ where
$\mathcal{G}$ is a $\textrm{CGS}_{\textnormal{ii}}$ and
$\Phi\in\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$.
###### Definition 2.4 (Model checking
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$).
The _model-checking problem_ for $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ is the decision problem that, given an
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance
$(\mathcal{G},\Phi)$, returns ‘Yes’ if $\mathcal{G}\models\Phi$, and ‘No’
otherwise.
It is well known that deciding the existence of winning strategies in multi-
player games with imperfect information is undecidable for reachability
objectives (Peterson et al., 2001). Since this problem is easily reduced to
the model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$, we get the following result.
###### Theorem 2.5.
The model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ is undecidable.
Hierarchical instances. We now isolate a sub-problem obtained by restricting
attention to _hierarchical instances_. Intuitively, an
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance
$(\mathcal{G},\Phi)$ is hierarchical if, as one goes down a path in the
syntactic tree of $\Phi$, the observations tied to quantifications become
finer.
###### Definition 2.6 (Hierarchical instances).
An $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance
$(\mathcal{G},\Phi)$ is _hierarchical_ if for every subformula
$\varphi_{1}=\langle\\!\langle y\rangle\\!\rangle^{o_{1}}\varphi^{\prime}_{1}$
of $\Phi$ and subformula $\varphi_{2}=\langle\\!\langle
x\rangle\\!\rangle^{o_{2}}\varphi^{\prime}_{2}$ of $\varphi^{\prime}_{1}$, it
holds that $\mathcal{O}(o_{2})\subseteq\mathcal{O}(o_{1})$.
If $\mathcal{O}(o_{2})\subseteq\mathcal{O}(o_{1})$ we say that $o_{2}$ is
_finer_ than $o_{1}$ in $\mathcal{G}$, and that $o_{1}$ is _coarser_ than
$o_{2}$ in $\mathcal{G}$. Intuitively, this means that a player with
observation $o_{2}$ observes game $\mathcal{G}$ no worse than, i.e., knows at
least as much as a player with observation $o_{1}$.
###### Remark 2.
If one uses the trick described in Section 2.5 to model agents that remember
their own actions, then for an agent $a$ to know at least as much as another
agent $b$ it needs to be the case that, in particular, agent $a$ observes all
actions played by agent $b$.
###### Example 2.7 (Fault-tolerant diagnosibility).
Consider the following formula from Section 2.5:
$\varphi:=\langle\\!\langle x\rangle\\!\rangle^{o}(a,x){\bf A}{\bf
G}\left(\text{safe}\wedge\bigwedge_{i=1}^{n}\text{fault}_{i}\to\langle\\!\langle
x\rangle\\!\rangle^{o_{i}}(a,x){\bf A}{\bf G}\text{\,safe}_{i}\right)$
As already discussed, it expresses that the controller can react to the loss
of a sensor to keep ensuring some property of the system. Clearly, the
controller’s observation $o_{i}$ after the loss of sensor $i$ is coarser than
its original observation $o$, and thus formula $\varphi$ in such a system does
not form a hierarchical instance.
We now give an example of scenario where hierarchical instances occur
naturally.
###### Example 2.8 (Security levels).
Consider a system with different “security levels”, where higher levels have
access to more data (i.e., can observe more). Assume that the
$\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ is such that
$\mathcal{O}(o_{n})\subseteq\mathcal{O}(o_{n-1})\subseteq\ldots\subseteq\mathcal{O}(o_{1})$:
in other words, level $n$ has the highest security clearance, while level $1$
has the lowest. Consider that agent $a$ wants to reach some objective marked
by atom “goal”, that it starts with the lowest observation clearance $o_{1}$,
and that atomic formula “$\text{promote}_{i}$” means that the agent is granted
access to level $i$ (observe that whenever we have $\text{promote}_{i}$, we
should also have $\text{promote}_{j}$ for all $j<i$). For every $i$ we let
$\varphi_{i}(\varphi^{\prime}):=\text{goal}\vee(\text{promote}_{i}\wedge\langle\\!\langle
x\rangle\\!\rangle^{o_{i}}(a,x){\bf A}{\bf F}\varphi^{\prime})$
Now the formula
$\varphi:=\varphi_{1}(\varphi_{2}(\ldots\varphi_{n-1}(\varphi_{n}(\text{goal}))\ldots))$
means that agent $a$ can enforce her goal, possibly by first getting access to
higher security levels and using this additional observation power to reach
the goal. Because the strategy quantifications that are deeper in the formula
have access to more information, this formula forms a hierarchical instance in
$\mathcal{G}$.
Here is the main contribution of this work:
###### Theorem 2.9.
The model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ restricted to the class of hierarchical instances is decidable.
We prove this result in Section 5 by reducing it to the model-checking problem
for the hierarchical fragment of a logic called $\textnormal{{QCTL}}^{*}$ with
imperfect information, which we now introduce and study in order to use it as
an intermediate, “low-level” logic between tree automata and
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. We then discuss some
applications of this theorem in Section 7.
## 3\. $\textnormal{{QCTL}}^{*}$ with imperfect information
In this section we introduce an imperfect-information extension of
$\textnormal{{QCTL}}^{*}$ (Sistla, 1983; Kupferman, 1999; Kupferman et al.,
2000a; French, 2001; Laroussinie and Markey, 2014), which is an extension of
$\textnormal{{CTL}}^{*}$ with second-order quantification on atomic
propositions. In order to introduce imperfect information, instead of
considering equivalence relations between states as in concurrent game
structures, we will enrich Kripke structures by giving internal structure to
their states, i.e., we see states as $n$-tuples of local states. This way of
modelling imperfect information is inspired from Reif’s multi-player game
structures (Peterson et al., 2001) and distributed systems (Halpern and Vardi,
1989), and we find it very suitable to application of automata techniques, as
discussed in Section 3.3.
The syntax of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is
similar to that of $\textnormal{{QCTL}}^{*}$, except that we annotate second-
order quantifiers by subsets $\textnormal{{o}}\subseteq[n]$. The idea is that
quantifiers annotated by o can only “observe” the local states indexed by
$i\in\textnormal{{o}}$. We define the tree-semantics of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$: this means that we
interpret formulas on trees that are the unfoldings of Kripke structures (this
will capture the fact that players in
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ have synchronous perfect
recall). We then define the syntactic class of _hierarchical formulas_ and
prove, using an automata-theoretic approach, that model checking this class of
formulas is decidable.
For the rest of the section we fix some natural number $n\in\mathbb{N}$ which
parameterises the logic $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$, and which is the number of components in states of the models.
### 3.1. $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ Syntax
The syntax of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is very
similar to that of $\textnormal{{QCTL}}^{*}$: the only difference is that we
annotate quantifiers by a set of indices that defines the “observation” of
that quantifier.
Concrete observations. A set $\textnormal{{o}}\subseteq[n]$ is called a
_concrete observation_ (to distinguish it from observations $o$ in the
definitions of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$).
###### Definition 3.1 ($\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$
Syntax).
The syntax of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is
defined by the following grammar:
$\displaystyle\varphi:=$
$\displaystyle\;p\mid\neg\varphi\mid\varphi\vee\varphi\mid{\bf
E}\psi\mid\exists^{\textnormal{{o}}}p.\,\varphi$ $\displaystyle\psi:=$
$\displaystyle\;\varphi\mid\neg\psi\mid\psi\vee\psi\mid{\bf X}\psi\mid\psi{\bf
U}\psi$
where $p\in\textnormal{AP}$ and $\textnormal{{o}}\subseteq[n]$.
Formulas of type $\varphi$ are called _state formulas_ , those of type $\psi$
are called _path formulas_ , and
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ consists of all the
state formulas defined by the grammar. We use standard abbreviation ${\bf
A}\psi:=\neg{\bf E}\neg\psi$. We also use $\exists p.\,\varphi$ as a shorthand
for $\exists^{[n]}p.\,\varphi$, and we let $\forall p.\,\varphi:=\neg\exists
p.\,\neg\varphi$.
Given a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula
$\varphi$, we define the set of _quantified propositions_
${\textnormal{AP}_{\exists}}(\varphi)\subseteq\textnormal{AP}$ as the set of
atomic propositions $p$ such that $\varphi$ has a subformula of the form
$\exists^{\textnormal{{o}}}p.\,\varphi$. We also define the set of _free
propositions_ $\textnormal{AP}_{f}(\varphi)\subseteq\textnormal{AP}$ as the
set of atomic propositions that have an occurrence which is not under the
scope of any quantifier of the form $\exists^{\textnormal{{o}}}p.\,$ Observe
that ${\textnormal{AP}_{\exists}}(\varphi)\cap\textnormal{AP}_{f}(\varphi)$
may not be empty, i.e., a proposition may appear both free and quantified in
(different places of) a formula.
### 3.2. $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ semantics
Several semantics have been considered for $\textnormal{{QCTL}}^{*}$, the two
most studied being the _structure semantics_ and the _tree semantics_ (see
(Laroussinie and Markey, 2014) for more details). For the semantics of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ we adapt the tree
semantics, and we explain the reasons for doing so in Section 3.3.
As already mentioned, for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ we consider structures whose states are tuples of local states. We now
define these structures and related notions.
###### Definition 3.2 (Compound Kripke structures).
A _compound Kripke structure_ , or CKS , over AP is a tuple
$\mathcal{S}=(S,R,\ell,s_{\iota})$ where
* •
$S\subseteq\prod_{i\in[n]}L_{i}$ is a set of _states_ , with
$\\{L_{i}\\}_{i\in[n]}$ a family of $n$ disjoint finite sets of _local states_
,
* •
$R\subseteq S\times S$ is a left-total111i.e., for all $s\in S$, there exists
$s^{\prime}$ such that $(s,s^{\prime})\in R$. _transition relation_ ,
* •
$\ell:S\to 2^{\textnormal{AP}}$ is a _labelling function_ and
* •
$s_{\iota}\in S$ is an _initial state_.
A _path_ in $\mathcal{S}$ is an infinite sequence of states
$\lambda=s_{0}s_{1}\ldots$ such that for all $i\in\mathbb{N}$,
$(s_{i},s_{i+1})\in R$. A _finite path_ is a finite non-empty prefix of a
path. We may write $s\in\mathcal{S}$ for $s\in S$, and we define the _size_
$|\mathcal{S}|$ of a CKS $\mathcal{S}=(S,R,s_{\iota},\ell)$ as its number of
states: $|\mathcal{S}|:=|S|$.
Since we will interpret $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ on unfoldings of CKS , we now define infinite trees.
Trees. In many works, trees are defined as prefix-closed sets of words with
the empty word $\epsilon$ as root. Here trees represent unfoldings of Kripke
structures, and we find it more convenient to see a node $u$ as a sequence of
states and the root as the initial state. Let $X$ be a finite set of
_directions_ (typically a set of states). An _$X$ -tree_ $\tau$ is a nonempty
set of words $\tau\subseteq X^{+}$ such that:
* •
there exists $r\in X$, called the _root_ of $\tau$, such that each $u\in\tau$
starts with $r$ ($r\preccurlyeq u$);
* •
if $u\cdot x\in\tau$ and $u\cdot x\neq r$, then $u\in\tau$,
* •
if $u\in\tau$ then there exists $x\in X$ such that $u\cdot x\in\tau$.
The elements of a tree $\tau$ are called _nodes_. If $u\cdot x\in\tau$, we say
that $u\cdot x$ is a _child_ of $u$. The _depth_ of a node $u$ is $|u|$. An
$X$-tree $\tau$ is _complete_ if for every $u\in\tau$ and $x\in X$, $u\cdot
x\in\tau$. A _path_ in $\tau$ is an infinite sequence of nodes
$\lambda=u_{0}u_{1}\ldots$ such that for all $i\in\mathbb{N}$, $u_{i+1}$ is a
child of $u_{i}$, and $Paths(u)$ is the set of paths that start in node $u$.
Labellings. An _AP -labelled $X$-tree_, or _$(\textnormal{AP},X)$ -tree_ for
short, is a pair $t=(\tau,\ell)$, where $\tau$ is an $X$-tree called the
_domain_ of $t$ and $\ell:\tau\rightarrow 2^{\textnormal{AP}}$ is a
_labelling_ , which maps each node to the set of propositions that hold there.
For $p\in\textnormal{AP}$, a _$p$ -labelling_ for a tree is a mapping
$\ell_{p}:\tau\to\\{0,1\\}$ that indicates in which nodes $p$ holds, and for a
labelled tree $t=(\tau,\ell)$, the $p$-labelling of $t$ is the $p$-labelling
$u\mapsto 1$ if $p\in\ell(u)$, 0 otherwise. The composition of a labelled tree
$t=(\tau,\ell)$ with a $p$-labelling $\ell_{p}$ for $\tau$ is defined as
$t\otimes\ell_{p}:=(\tau,\ell^{\prime})$, where
$\ell^{\prime}(u)=\ell(u)\cup\\{p\\}$ if $\ell_{p}(u)=1$, and
$\ell(u)\setminus\\{p\\}$ otherwise. A $p$-labelling for a labelled tree
$t=(\tau,\ell)$ is a $p$-labelling for its domain $\tau$. A _pointed labelled
tree_ is a pair $(t,u)$ where $u$ is a node of $t$.
If $u=w\cdot x$, the _subtree_ $t_{u}$ of $t=(\tau,\ell)$ is defined as
$t_{u}:=(\tau_{u},\ell_{u})$ with $\tau_{u}=\\{x\cdot w^{\prime}\mid w\cdot
x\cdot w^{\prime}\in\tau\\}$, and $\ell_{u}(x\cdot w^{\prime})=\ell(w\cdot
x\cdot w^{\prime})$. A labelled tree is _regular_ if it has finitely many
disctinct subtrees.
In the tree semantics of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ that we consider here, formulas are evaluated on tree unfoldings of CKS
, which we now define.
Tree unfoldings. Let $\mathcal{S}=(S,R,\ell,s_{\iota})$ be a compound Kripke
structure over AP. The _tree-unfolding of $\mathcal{S}$_ is the
$(\textnormal{AP},S)$-tree $t_{\mathcal{S}}:=(\tau,\ell^{\prime})$, where
$\tau$ is the set of all finite paths that start in $s_{\iota}$, and for every
$u\in\tau$, $\ell^{\prime}(u):=\ell(\mbox{last}(u))$.
Note that a labelled tree is regular if and only if it is the unfolding of
some finite Kripke structure.
Narrowing. Let $X$ and $Y$ be two finite sets, and let $(x,y)\in X\times Y$.
The _$X$ -narrowing_ of $(x,y)$ is ${(x,y)\\!\downarrow_{X}}:=x$. This
definition extends naturally to words and trees over $X\times Y$ (point-wise).
Given a family of (disjoint) sets of local states $\\{L_{i}\\}_{i\in[n]}$ and
a subset $I\subseteq[n]$, we let $L_{I}:=\prod_{i\in I}L_{i}$ if
$I\neq\emptyset$ and $L_{\emptyset}:=\\{\mathbf{0}\\}$, where $\mathbf{0}$ is
a special symbol. For $I,J\subseteq[n]$ and $z\in L_{I}$, we also define
${z\\!\downarrow_{J}}:=z\\!\downarrow_{L_{I\cap J}}$, where $z$ is seen as a
pair $z=(x,y)\in L_{I\cap J}\times L_{I\setminus J}$, i.e., we apply the above
definition with $X=L_{I\cap J}$ and $Y=L_{I\setminus J}$. This is well defined
because having taken sets $L_{i}$ to be disjoint, the ordering of local states
in $z$ is indifferent. We also extend this definition to words and trees. In
particular, for every $L_{I}$-tree $\tau$, $\tau\\!\downarrow_{\emptyset}$ is
the only $L_{\emptyset}$-tree, $\mathbf{0}^{\omega}$.
Quantification and uniformity. In
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$
$\exists^{\textnormal{{o}}}p.\,\varphi$ holds in a tree $t$ if there is some
o-uniform $p$-labelling of $t$ such that $t$ with this $p$-labelling satisfies
$\varphi$. Intuitively, a $p$-labelling of a tree is o-uniform if every two
nodes that are indistinguishable for observation o agree on $p$.
###### Definition 3.3 (o-indistinguishability and o-uniformity in $p$).
Fix $\textnormal{{o}}\subseteq[n]$ and $I\subseteq[n]$.
* •
Two tuples $x,x^{\prime}\in L_{I}$ are _o -indistinguishable_, written
$x\approx_{\textnormal{{o}}}x^{\prime}$, if
$x\\!\downarrow_{\textnormal{{o}}}=x^{\prime}\\!\downarrow_{\textnormal{{o}}}$.
* •
Two words $u=u_{0}\ldots u_{i}$ and $u^{\prime}=u^{\prime}_{0}\ldots
u^{\prime}_{j}$ over alphabet $L_{I}$ are _o -indistinguishable_, written
$u\approx_{\textnormal{{o}}}u^{\prime}$, if $i=j$ and for all
$k\in\\{0,\ldots,i\\}$ we have
$u_{k}\approx_{\textnormal{{o}}}u^{\prime}_{k}$.
* •
A $p$-labelling for a tree $\tau$ is _o -uniform_ if for all
$u,u^{\prime}\in\tau$, $u\approx_{\textnormal{{o}}}u^{\prime}$ implies
$\ell_{p}(u)=\ell_{p}(u^{\prime})$.
###### Definition 3.4 ($\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$
semantics).
We define by induction the satisfaction relation $\models$ of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. Let $t=(\tau,\ell)$
be an AP-labelled $L_{I}$-tree, $u$ a node and $\lambda$ a path in $\tau$:
$\displaystyle t,u\models$ $\displaystyle\,p$ if $\displaystyle\quad
p\in\ell(u)$ $\displaystyle t,u\models$ $\displaystyle\,\neg\varphi$ if
$\displaystyle\quad t,u\not\models\varphi$ $\displaystyle t,u\models$
$\displaystyle\,\varphi\vee\varphi^{\prime}$ if $\displaystyle\quad
t,u\models\varphi\mbox{ or }t,u\models\varphi^{\prime}$ $\displaystyle
t,u\models$ $\displaystyle\,{\bf E}\psi$ if
$\displaystyle\quad\exists\,\lambda\in Paths(u)\mbox{ s.t.
}t,\lambda\models\psi$ $\displaystyle t,u\models$
$\displaystyle\,\exists^{\textnormal{{o}}}p.\,\varphi$ if
$\displaystyle\quad\exists\,\ell_{p}\mbox{ a $\textnormal{{o}}$-uniform
$p$-labelling for $t$ such that }t\otimes\ell_{p},u\models\varphi$
$\displaystyle t,\lambda\models$ $\displaystyle\,\varphi$ if
$\displaystyle\quad t,\lambda_{0}\models\varphi$ $\displaystyle
t,\lambda\models$ $\displaystyle\,\neg\psi$ if $\displaystyle\quad
t,\lambda\not\models\psi$ $\displaystyle t,\lambda\models$
$\displaystyle\,\psi\vee\psi^{\prime}\quad$ if $\displaystyle\quad
t,\lambda\models\psi\mbox{ or }t,\lambda\models\psi^{\prime}$ $\displaystyle
t,\lambda\models$ $\displaystyle\,{\bf X}\psi$ if $\displaystyle\quad
t,\lambda_{\geq 1}\models\psi$ $\displaystyle t,\lambda\models$
$\displaystyle\,\psi{\bf U}\psi^{\prime}$ if $\displaystyle\quad\exists\,i\geq
0\mbox{ s.t. }t,\lambda_{\geq i}\models\psi^{\prime}\text{ and }\forall
j\text{ s.t. }0\leq j<i,\;t,\lambda_{\geq j}\models\psi$
We write $t\models\varphi$ for $t,r\models\varphi$, where $r$ is the root of
$t$. Given a CKS $\mathcal{S}$ and a
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\varphi$, we
also write $\mathcal{S}\models\varphi$ if
$\mathcal{S},s_{\iota}\models\varphi$.
###### Example 3.5.
Consider the following CTL formula:
$\mathbf{border}(p):={\bf A}{\bf F}p\wedge{\bf A}{\bf G}(p\rightarrow{\bf
A}{\bf X}{\bf A}{\bf G}\neg p).$
This formula holds in a labelled tree if and only if each path contains
exactly one node labelled with $p$. Now, consider the following
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula:
$\mathbf{level}(p):=\exists^{\emptyset}p.\,\mathbf{border}(p).$
For a blind quantifier, two nodes of a tree are indistinguishable if and only
if they have same depth. Therefore, this formula holds on a tree iff the $p$’s
label all and only the nodes at some fixed depth. This formula can thus be
used to capture the equal level predicate on trees. Actually, just as
$\textnormal{{QCTL}}^{*}$ captures MSO, one can prove that
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ with tree semantics
subsumes MSO with equal level (Elgot and Rabin, 1966; Läuchli and Savioz,
1987; Thomas, 1992). In Theorem 3.7 we make use of a similar observation to
prove that model-checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ is undecidable.
### 3.3. Discussion on the definition of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$
We now motivate in detail some aspects of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$.
Modelling of imperfect information. We model imperfect information by means of
local states (rather than equivalence relations) because this greatly
facilitates the use of automata techniques. More precisely, in our decision
procedure of Section 4 we use an operation on tree automata called _narrowing_
, which was introduced in (Kupferman and Vardi, 1999) to deal with imperfect-
information in the context of distributed synthesis for temporal
specifications. Given an automaton $\mathcal{A}$ that works on $X\times
Y$-trees, where $X$ and $Y$ are two finite sets, and assuming that we want to
model an operation performed on trees while observing only the $X$ component
of each node, this narrowing operation allows one to build from $\mathcal{A}$
an automaton $\mathcal{A}^{\prime}$ that works on $X$-trees, such that
$\mathcal{A}^{\prime}$ accepts an $X$-tree if and only if $\mathcal{A}$
accepts its widening to $X\times Y$ (intuitively, this widening is the
$X\times Y$-tree in which each node is labelled as its projection on the
original $X$-tree; see Section 4 for details).
With our definition of compound Kripke structures, their unfoldings are trees
over the Cartesian product $L_{[n]}$. To model a quantification
$\exists^{\textnormal{{o}}}p$ with observation $\textnormal{{o}}\subseteq[n]$,
we can thus use the narrowing operation to forget about components $L_{i}$,
for $i\in[n]\setminus\textnormal{{o}}$. We then use the classic projection of
nondeterministic tree automata to perform existential quantification on atomic
proposition $p$. Since the choice of the $p$-labelling is made directly on
$L_{\textnormal{{o}}}$-trees, it is necessarily o-uniform.
Choice of the tree semantics. The two most studied semantics for
$\textnormal{{QCTL}}^{*}$ are the _structure semantics_ , in which formulas
are evaluated directly on Kripke structures, and the _tree semantics_ , in
which Kripke structures are first unfolded into infinite trees. Tree semantics
thus allows quantifiers to choose the value of a quantified atomic proposition
in each _finite path_ of the model, while in structure semantics the choice is
only made in each state. When $\textnormal{{QCTL}}^{*}$ is used to express
existence of strategies, existential quantification on atomic propositions
labels the structure with strategic choices; in this kind of application,
structure semantics reflects so-called _positional_ or _memoryless_
strategies, while tree semantics captures _perfect-recall_ or _memoryful_
strategies. Since in this work we are interested in perfect-recall strategies,
we only consider the tree semantics.
### 3.4. Model checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$
We now define the model-checking problem studied in the rest of this section.
###### Definition 3.6 (Model checking
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$).
The _model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$_ is the following
decision problem: given an instance $(\mathcal{S},\Phi)$ where $\mathcal{S}$
is a CKS, and $\Phi$ is a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ formula, return ‘Yes’ if $\mathcal{S}\models\Phi$ and ‘No’ otherwise.
We now prove that the model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is undecidable. This
comes as no surprise since, as we will show in Section 5,
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ can express the
existence of distributed winning strategies in imperfect-information games.
However we propose a proof that shows the connection between
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ and MSO with equal-
level predicate (Elgot and Rabin, 1966; Läuchli and Savioz, 1987; Thomas,
1992). This proof also has the benefit of showing that
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is undecidable already
for formulas that involve only propositional quantifiers that observe either
everything or nothing.
###### Theorem 3.7.
The model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is undecidable.
###### Proof.
Let $\textnormal{{MSO}}_{\textnormal{eq}}$ denote the extension of the logic
MSO (without unary predicates) by a binary predicate symbol eq.
$\textnormal{{MSO}}_{\textnormal{eq}}$ is interpreted on the full binary tree,
and the semantics of $\text{eq}(x,y)$ is that $x$ and $y$ have the same depth
in the tree. We show how to effectively translate
$\textnormal{{MSO}}_{\textnormal{eq}}$ into
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, and our result
follows since the $\textnormal{{MSO}}_{\textnormal{eq}}$-theory of the binary
tree is undecidable (Läuchli and Savioz, 1987). The translation from
$\textnormal{{MSO}}_{\textnormal{eq}}$ to
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is obtained by
extending that from MSO to QCTL (Laroussinie and Markey, 2014), using the
formula $\mathbf{level}(\cdot)$ from Example 3.5 to help capture the equal-
length predicate.
We define a translation $\widehat{\quad}$ from
$\textnormal{{MSO}}_{\textnormal{eq}}$ to
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ such that for every
tree $t$ with root $r$, nodes $u_{1},\ldots,u_{i}\in t$ and sets of nodes
$U_{1},\ldots,U_{j}\subseteq t$, and every
$\textnormal{{MSO}}_{\textnormal{eq}}$ formula
${\varphi(x,x_{1},\ldots,x_{i},X_{1},\ldots,X_{j})}$, we have that
(1)
$t,r,u_{1},\ldots,u_{i},U_{1},\ldots,U_{j}\models\varphi(x,x_{1},\ldots,x_{i},X_{1},\ldots,X_{j})\text{\quad
if and only if \quad}\widehat{t},r\models\widehat{\varphi}$
where $\widehat{t}$ is obtained from $t$ by defining the labelling for fresh
atomic propositions $p_{x_{k}}$ and $p_{X_{k}}$, with $k\in[i]$, as follows:
$p_{x_{k}}\in\widehat{\ell}(u)$ if $u=u_{k}$ and
$p_{X_{k}}\in\widehat{\ell}(u)$ if $u\in U_{k}$.
The translation of MSO to $\textnormal{{QCTL}}^{*}$ from (Laroussinie and
Markey, 2014) can be extended to one from
$\textnormal{{MSO}}_{\textnormal{eq}}$ to
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ by adding rules for
the equal level predicate. Indeed, for
$\varphi(x,x_{1},\ldots,x_{i},X_{1},\ldots,X_{j})\in\textnormal{{MSO}}_{\textnormal{eq}}$,
we inductively define the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ formula $\widehat{\varphi}$ as follows, where $k\in[i]$:
$\begin{array}[]{rclcrcl}\widehat{x=x_{k}}&:=&p_{x_{k}}&&\widehat{x_{k}=x_{l}}&:=&{\bf
E}{\bf F}(p_{x_{k}}\wedge p_{x_{l}})\\\\[5.0pt] \widehat{x\in
X_{k}}&:=&p_{X_{k}}&&\widehat{x_{k}\in X_{l}}&:=&{\bf E}{\bf
F}(p_{x_{k}}\wedge p_{X_{l}})\\\\[5.0pt]
\widehat{\neg\varphi^{\prime}}&:=&\neg\widehat{\varphi^{\prime}}&&\widehat{\varphi_{1}\vee\varphi_{2}}&:=&\widehat{\varphi_{1}}\vee\widehat{\varphi_{2}}\\\\[5.0pt]
\widehat{\exists x_{k}.\varphi^{\prime}}&:=&\lx@intercol\exists
<EMAIL_ADDRESS>\widehat{\exists X_{k}.\varphi^{\prime}}&:=&\lx@intercol\exists
<EMAIL_ADDRESS>\widehat{S(x,x_{k})}&:=&{\bf E}{\bf
X}p_{x_{k}}&&\widehat{S(x_{k},x)}&:=&\perp\\\\[5.0pt]
\widehat{S(x_{k},x_{l})}&:=&\lx@intercol{\bf E}{\bf F}(p_{x_{k}}\wedge{\bf
E}{\bf X}p_{x_{l}})\hfil\lx@intercol\end{array}$
where $\mathrm{uniq}(p):={\bf E}{\bf F}p\wedge\forall q.\;\left({\bf E}{\bf
F}(p\wedge q)\rightarrow{\bf A}{\bf G}(p\rightarrow q)\right)$ holds in a tree
iff it has exactly one node labelled with $p$. To understand the $x=x_{k}$ and
$x\in X_{k}$ cases, consider that $x$ will be interpreted as the root. For the
$S(x_{k},x)$ case, observe that $x$ has no incoming edge since it is
interpreted as the root. Second-order quantification $\exists X_{k}$ is
translated into quantification on atomic proposition $p_{X_{k}}$, and first-
order quantification $\exists x_{k}$ is treated similarly, with the additional
constraint that quantification is limited to $p_{x_{k}}$-labellings that set
$p_{x_{k}}$ to true in one and only one node of the tree.
The rules for eq are as follows:
$\displaystyle\widehat{\text{eq}(x,x_{k})}$ $\displaystyle:=p_{x_{k}}$
$\displaystyle\widehat{\text{eq}(x_{k},x_{l})}$
$\displaystyle:=\exists^{\emptyset}p.\,\mathbf{border}(p)\wedge{\bf A}{\bf
G}(p_{x_{k}}\rightarrow p\wedge p_{x_{l}}\rightarrow p)$
To understand the first case, observe that since $x$ is interpreted as the
root, $x_{k}$ is on the same level as $x$ if and only if it is also assigned
the root. For the second case, recall from Example 3.5 that the
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula
$\exists^{\emptyset}p.\,\mathbf{border}(p)$ places one unique horizontal line
of $p$’s in the tree, and thus requiring that $x_{k}$ and $x_{l}$ be both on
this line ensures that they are on the same level. The correctness of the
translation follows from (1), which is proven by induction.
Now take an instance $(t,\varphi(x))$ of the model-checking problem for
$\textnormal{{MSO}}_{\textnormal{eq}}$ on the full binary tree $t$. Let
$\mathcal{S}$ be a CKS with two states $s_{0}$ and $s_{1}$ (local states are
irrelevant here), whose transition relation is the complete relation, and with
empty labelling function. Clearly, $t_{\mathcal{S}}=t$, and applying (1) we
get:
$t,s_{0}\models\varphi(x)\text{\quad
iff\quad}\widehat{t},s_{0}\models\widehat{\varphi}.$
Observe that in the previous line, because there are no free variables besides
$x$, which stands for the root, we have that $\widehat{t}=t=t_{\mathcal{S}}$,
hence we have indeed produced an instance of the model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. ∎
## 4\. A decidable fragment of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$: hierarchy on
observations
The main result of this section is the identification of an important
decidable fragment of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$.
###### Definition 4.1 (Hierarchical formulas).
A $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\varphi$ is
_hierarchical_ if for all subformulas
$\varphi_{1}=\exists^{\textnormal{{o}}_{1}}p_{1}.\,\varphi^{\prime}_{1}$ and
$\varphi_{2}=\exists^{\textnormal{{o}}_{2}}p_{2}.\,\varphi^{\prime}_{2}$ of
$\varphi$ where $\varphi_{2}$ is a subformula of $\varphi^{\prime}_{1}$, we
have $\textnormal{{o}}_{1}\subseteq\textnormal{{o}}_{2}$.
In other words, a formula is hierarchical if innermore propositional
quantifiers observe at least as much as outermore ones.
###### Example 4.2.
Formula $\exists^{\\{1,2\\}}p.\,\exists^{\\{1,2,4\\}}q.\,{\bf A}{\bf G}(p\vee
q)$ is hierarchical because $\\{1,2\\}\subseteq\\{1,2,4\\}$. On the other
hand, formula $\exists^{\\{1,2\\}}p.\,\big{(}\exists^{\\{1,2,4\\}}q.\,{\bf
A}{\bf G}(p\vee q)\wedge\exists^{\\{3\\}}q^{\prime}.\,{\bf E}{\bf F}(p\wedge
q^{\prime})\big{)}$ is not, because $\\{1,2\\}\not\subseteq\\{3\\}$. Note that
neither is it the case that $\\{3\\}\subseteq\\{1,2\\}$: the observation power
of quantifiers $\exists^{\\{1,2\\}}p.\,$ and $\exists^{\\{3\\}}q^{\prime}.\,$
are incomparable. Finally, formula
$\forall^{\\{1,2,3\\}}p.\,\exists^{\\{1,2\\}}q.\,.{\bf A}{\bf G}(p\vee q)$ is
not hierarchical even though $\\{1,2\\}\subseteq\\{1,2,3\\}$, as the
quantifier that observes best is _higher_ in the syntactic tree.
We let $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$ be the set of hierarchical
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formulas.
###### Theorem 4.3.
Model checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$ is non-elementary decidable.
Since our decision procedure for the hierarchical fragment of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is based on an
automata-theoretic approach, we recall some definitions and results for
alternating tree automata.
### 4.1. Alternating parity tree automata
We recall alternating parity tree automata. Because their semantics is defined
via acceptance games, we start with basic definitions for two-player turn-
based parity games, or simply parity games.
Parity games. A _parity game_ is a structure $\mathcal{G}=(V,E,v_{\iota},C)$,
where $V=V_{E}\uplus V_{A}$ is a set of _positions_ partitioned between
positions of Eve ($V_{E}$) and those of Adam ($V_{A}$), $E\subseteq V\times V$
is a set of _moves_ , $v_{\iota}$ is an initial position and
$C:V\to\mathbb{N}$ is a colouring function of finite codomain. In positions
$V_{E}$, Eve chooses the next position, while Adam chooses in positions
$V_{A}$. A play is an infinite sequence of positions $v_{0}v_{1}v_{2}\ldots$
such that $v_{0}=v_{\iota}$ and for all $i\geq 0$, $(v_{i},v_{i+1})\in E$
(written $v_{i}\to v_{i+1}$). We assume that for every $v\in V$ there exists
$v^{\prime}\in V$ such that $v\to v^{\prime}$. A strategy for Eve is a partial
function $V^{*}\rightharpoonup V$ that maps each finite prefix of a play
ending in a position $v\in V_{E}$ to a next position $v^{\prime}$ such that
$v\to v^{\prime}$. A play $v_{0}v_{1}v_{2}\ldots$ _follows_ a strategy
$\sigma$ of Eve if for every $i\geq 0$ such that $v_{i}\in V_{E}$,
$v_{i+1}=\sigma(v_{0}\ldots v_{i})$. A strategy $\sigma$ is winning if every
play that follows it satisfies the parity condition, i.e., the least colour
seen infinitely often along the play is even.
Parity tree automata. Because it is sufficient for our needs and simplifies
definitions, we assume that all input trees are complete trees. For a set $Z$,
$\mathbb{B}^{+}(Z)$ is the set of formulas built from the elements of $Z$ as
atomic propositions using the connectives $\vee$ and $\wedge$, and with
$\top,\perp\in\mathbb{B}^{+}(Z)$. An _alternating tree automaton (ATA ) on
$(\textnormal{AP},X)$-trees_ is a structure
$\mathcal{A}=(Q,\delta,q_{{\iota}},C)$ where $Q$ is a finite set of states,
$q_{{\iota}}\in Q$ is an initial state, $\delta:Q\times
2^{\textnormal{AP}}\rightarrow\mathbb{B}^{+}(X\times Q)$ is a transition
function, and $C:Q\to\mathbb{N}$ is a colouring function. To ease reading we
shall write atoms in $\mathbb{B}^{+}(X\times Q)$ between brackets, such as
$[x,q]$. A _nondeterministic tree automaton (NTA ) on
$(\textnormal{AP},X)$-trees_ is an ATA $\mathcal{A}=(Q,\delta,q_{{\iota}},C)$
such that for every $q\in Q$ and $a\in 2^{\textnormal{AP}}$, $\delta(q,a)$ is
written in disjunctive normal form and for every direction $x\in X$ each
disjunct contains exactly one element of $\\{x\\}\times Q$. An NTA is
_deterministic_ if for each $q\in Q$ and $a\in 2^{\textnormal{AP}}$,
$\delta(q,a)$ consists of a single disjunct.
Acceptance of a pointed labelled tree $(t,u_{\iota})$, where $t=(\tau,\ell)$,
by an ATA $\mathcal{A}=(Q,\delta,q_{\iota},C)$ is defined via the parity game
$\mathcal{G}(\mathcal{A},t,u_{\iota})=(V,E,v_{\iota},C^{\prime})$ where
$V=\tau\times Q\times\mathbb{B}^{+}(X\times Q)$, position $(u,q,\alpha)$
belongs to Eve if $\alpha$ is of the form $\alpha_{1}\vee\alpha_{2}$ or
$[x,q^{\prime}]$, and to Adam otherwise,
$v_{{\iota}}=(u_{\iota},q_{\iota},\delta(q_{\iota},u_{\iota}))$, and
$C^{\prime}(u,q,\alpha)=C(q)$. Moves in $\mathcal{G}(\mathcal{A},t,u_{\iota})$
are defined by the following rules:
$\begin{array}[]{ll}(u,q,\alpha_{1}\;\mbox{$\dagger$}\;\alpha_{2})\rightarrow(u,q,\alpha_{i})&\mbox{where
}\mbox{$\dagger$}\in\\{\vee,\wedge\\}\mbox{ and }i\in\\{1,2\\},\\\
\lx@intercol(u,q,[x,q^{\prime}])\rightarrow(u\cdot
x,q^{\prime},\delta(q^{\prime},\ell(u\cdot x)))\hfil\lx@intercol\end{array}$
Positions of the form $(u,q,\top)$ and $(u,q,\perp)$ are sinks, winning for
Eve and Adam respectively.
A pointed labelled tree $(t,u)$ is _accepted_ by $\mathcal{A}$ if Eve has a
winning strategy in $\mathcal{G}(\mathcal{A},t,u)$, and the _language_ of
$\mathcal{A}$ is the set of pointed labelled trees accepted by $\mathcal{A}$,
written $\mathcal{L}(\mathcal{A})$. We write $t\in\mathcal{L}(\mathcal{A})$ if
$(t,r)\in\mathcal{L}(\mathcal{A})$, where $r$ is the root of $t$. Finally, the
_size_ $|\mathcal{A}|$ of an ATA $\mathcal{A}$ is its number of states plus
the sum of the sizes of all formulas appearing in the transition function.
Word automata. When the set of directions $X$ is a singleton, directions can
be forgotten and infinite trees can be identified with infinite words. We thus
call _parity word automaton_ a parity tree automaton on
$(\textnormal{AP},X)$-trees where $X$ is a singleton. In the case of a
nondeterministic parity word automaton, transitions can be represented as
usual as a mapping $\Delta:Q\times 2^{\textnormal{AP}}\to 2^{Q}$ which, in a
state $q\in Q$, reading the label $a\in 2^{\textnormal{AP}}$ of the current
position in the word, indicates a set of states $\Delta(q,a)$ from which Eve
can choose to send in the next position of the word.
We recall four classic operations on tree automata.
Complementation. Given an ATA $\mathcal{A}=(Q,\delta,q_{{\iota}},C)$, we
define its _dual_
$\overline{\mathcal{A}}=(Q,\overline{\delta},q_{{\iota}},\overline{C})$ where,
for each $q\in Q$ and $a\in 2^{\textnormal{AP}}$, $\overline{\delta}(q,a)$ is
the dual of $\delta(q,a)$, i.e., conjunctions become disjunctions and vice
versa, and $C(q):=C(q)+1$.
###### Theorem 4.4 (Complementation (Muller and Schupp, 1995)).
For every labelled tree $t$ and node $u$ in $t$,
$(t,u)\in\mathcal{L}(\overline{\mathcal{A}})\mbox{ if, and only if,
}(t,u)\notin\mathcal{L}(\mathcal{A}).$
Projection. The second construction is a projection operation, used by Rabin
to deal with second-order monadic quantification:
###### Theorem 4.5 (Projection (Rabin, 1969)).
Given an NTA $\mathcal{N}$ on $(\textnormal{AP},X)$-trees and an atomic
proposition $p\in\textnormal{AP}$, one can build in linear time an NTA
$\mathcal{N}\\!\Downarrow_{p}$ on $(\textnormal{AP}\setminus\\{p\\},X)$-trees
such that
$(t,u)\in\mathcal{L}(\mathcal{N}\\!\Downarrow_{p})\mbox{\;\;\;iff\;\;\;}\mbox{
there exists a $p$-labelling $\ell_{p}$ for $t$ s.t.
}(t\otimes\ell_{p},u)\in\mathcal{L}(\mathcal{N}).$
Intuitively, ${\mathcal{N}\\!\Downarrow_{p}}$ is automaton $\mathcal{N}$ with
the only difference that when it reads the label of a node, it can choose to
run as if $p$ was either true or false: if $\delta$ is the transition function
of $\mathcal{N}$, that of ${\mathcal{N}\\!\Downarrow_{p}}$ is
$\delta^{\prime}(q,a)=\delta(q,a\cup\\{p\\})\vee\delta(q,a\setminus\\{p\\})$,
for any state $q$ and label $a\in 2^{\textnormal{AP}}$. Another way of seeing
it is that $\mathcal{N}\\!\Downarrow_{p}$ guesses a $p$-labelling for the
input tree, and simulates $\mathcal{N}$ on this modified input.
Simulation. To prevent $\mathcal{N}\\!\Downarrow_{p}$ from guessing different
labels for a same node in different executions, it is crucial that
$\mathcal{N}$ be nondeterministic, which is the reason why we need the
following result:
###### Theorem 4.6 (Simulation (Muller and Schupp, 1995)).
Given an ATA $\mathcal{A}$, one can build in exponential time an NTA
$\mathcal{N}$ such that $\mathcal{L}(\mathcal{N})=\mathcal{L}(\mathcal{A})$.
The last construction was introduced by Kupferman and Vardi to deal with
imperfect information aspects in distributed synthesis. To describe it we need
to define a widening operation on trees which expands the directions in a
tree.
Tree widening. We generalise the widening operation defined in (Kupferman and
Vardi, 1999). In the following definitions we fix a CKS
$\mathcal{S}=(S,R,s_{\iota},\ell)$, and for $I\subseteq[n]$ we let
$S_{I}:=\\{s\\!\downarrow_{I}\,\mid s\in S\\}\subseteq L_{I}$ (recall that
$L_{I}=\prod_{i\in I}L_{i}$). Let $J\subseteq I\subseteq[n]$. For every
$S_{J}$-tree $\tau$ rooted in $s_{J}$ and $s_{I}\in S_{I}$ such that
$s_{I}\\!\downarrow_{J}=s_{J}$, we define the _$I$ -widening_ of $\tau$ as the
$S_{I}$-tree
$\tau\\!\uparrow^{I}_{s_{I}}:=\\{u\in s_{I}\cdot S_{I}^{*}\mid
u\\!\downarrow_{J}\in\tau\\}.$
For an $(\textnormal{AP},S_{J})$-tree $t=(\tau,\ell)$ rooted in $s_{J}$ and
$s_{I}\in S_{I}$ such that $s_{I}\\!\downarrow_{J}=s_{J}$, we let
$t\\!\uparrow^{I}_{s_{I}}:=(\tau\\!\uparrow^{I}_{s_{I}},\ell^{\prime}),\mbox{
where }\ell^{\prime}(u):=\ell(u\\!\downarrow_{J}).$
When clear from the context we may omit the subscript $s_{I}$. It is the case
in particular when referring to _pointed_ widenings of trees:
$(t\\!\uparrow^{I},u)$ stands for $(t\\!\uparrow^{I}_{u_{0}},u)$.
Narrowing. We now state a result from (Kupferman and Vardi, 1999) in our
slightly more general setting (the proof can be adapted straightforwardly).
The rough idea of this narrowing operation on ATA is that, if one just
observes $S_{J}$, uniform $p$-labellings on $S_{I}$-trees can be obtained by
choosing the labellings directly on $S_{J}$-trees, and then lifting them to
$S_{I}$.
###### Theorem 4.7 (Narrowing (Kupferman and Vardi, 1999)).
Given an ATA $\mathcal{A}$ on $S_{I}$-trees one can build in linear time an
ATA ${\mathcal{A}\\!\downarrow_{J}}$ on $S_{J}$-trees such that for every
pointed $(\textnormal{AP},S_{J})$-tree $(t,u)$ and every $u^{\prime}\in
S_{I}^{+}$ such that $u^{\prime}\\!\downarrow_{J}=u$,
$(t,u)\in\mathcal{L}(\mathcal{A}\\!\downarrow_{J})\mbox{ iff
}(t\\!\uparrow^{I},u^{\prime})\in\mathcal{L}(\mathcal{A}).$
### 4.2. Translating $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$ to ATA
In order to prove Theorem 4.3 we need some more notations and a technical
lemma that contains the automata construction.
###### Definition 4.8.
For every $\varphi\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$,
we let
$I_{\varphi}:=\bigcap_{\textnormal{{o}}\in\textnormal{Obs}(\varphi)}\textnormal{{o}}\subseteq[n],$
where $\textnormal{Obs}(\varphi)$ is the set of concrete observations that
occur in $\varphi$, with the intersection over the empty set defined as $[n]$.
For a CKS $\mathcal{S}$ with state set $S\subseteq\prod_{i\in[n]}L_{i}$ we
also let $S_{\varphi}:=\\{s\\!\downarrow_{I_{\varphi}}\mid s\in S\\}$.
Elements of $S_{\varphi}$ will be the possible directions used by the
automaton we build for $\varphi$. In other words, the automaton for $\varphi$
will work on $S_{\varphi}$-trees. The intuition is that the observations in
$\varphi$ determine which components of the model’s states can be observed by
the automaton.
Our construction, that transforms a
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$
formula $\varphi$ and a CKS $\mathcal{S}$ into an ATA, builds upon the classic
construction from (Kupferman et al., 2000b), which builds ATA for
$\textnormal{{CTL}}^{*}$ formulas. In addition, we use projection of automata
to treat second-order quantification, and to deal with imperfect information
we resort to automata narrowing.
Moreover, we use tree automata in an original way that allows us to deal with
non-observable atomic propositions, which in turn makes it possible to
consider non-observable winning conditions in our decidable fragment of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. The classical approach to
model checking via tree automata is to build an automaton that accepts all
tree models of the input formula, and check whether it accepts the unfolding
of the model (Kupferman et al., 2000b). We instead encode the model in the
automata, using the input tree only to guess labellings for quantified
propositions.
Encoding the model in the automaton. Quantification on atomic propositions is
classically performed by means of automata projection (see Theorem 4.5). But
in order to obtain a labelling that is uniform with regards to the observation
of the quantifier, we need to make use of the narrowing operation (see Theorem
4.7). Intuitively, to check that a formula
$\exists^{\textnormal{{o}}}p.\,\varphi$ holds in a tree $t$, we would like to
work on its narrowing $t^{\prime}:=t\\!\downarrow_{\textnormal{{o}}}$, guess a
labelling for $p$ on this tree thanks to automata projection, thus obtaining a
tree $t^{\prime}_{p}$, take its widening
$t_{p}^{\prime\prime}:=t^{\prime}_{p}\\!\uparrow^{[n]}$, obtaining a tree with
an o-uniform labelling for $p$, and then check that $\varphi$ holds on
$t_{p}^{\prime\prime}$. The problem is that unless $t=(\tau,\ell)$ is
o-uniform in every atomic proposition in AP, there is no way to define the
labelling of $\tau\\!\downarrow_{\textnormal{{o}}}$ without losing
information. This implies that, unless we restrict to models where all atomic
propositions are observable for all observations o, we cannot pass the model
as input to our automata, which will work on narrowings of trees.
Therefore, to model check a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ formula $\varphi$ on a CKS $\mathcal{S}$, each state of the automaton
that we build for $\varphi$ will contain a state of $\mathcal{S}$. The
automaton can thus guess paths in $\mathcal{S}$, and evaluate free occurrences
of atomic propositions in $\mathcal{S}$ without reading the input tree. The
input tree no longer represents the model, but we use it to carry labellings
for quantified atomic propositions in ${\textnormal{AP}_{\exists}}(\varphi)$:
we provide the automaton with an input tree whose labelling is initially
empty, and the automaton, through successive narrowing and projection
operations, decorates it with uniform labellings for quantified atomic
propositions.
We remark that this technique allows one to go beyond Coordination Logic
(Finkbeiner and Schewe, 2010): by separating between quantified atomic
propositions (that need to be uniform and are carried by the input tree) and
free atomic propositions (that state facts about the model and are coded in
the automaton), we manage to remove the restriction present in CL, that
requires all facts about the model to be known to every strategy (see
Proposition 6.3 in Section 6.2). To do this we assume without loss of
generality that propositions that are quantified in $\varphi$ do not appear
free in $\varphi$, i.e.,
${\textnormal{AP}_{\exists}}(\varphi)\cap\textnormal{AP}_{f}(\varphi)=\emptyset$.
Finally, given a formula $\varphi$, a CKS $\mathcal{S}$ and a state
$s\in\mathcal{S}$, the truth value of $\varphi$ in $(\mathcal{S},s)$ does not
depend on the labelling of $\mathcal{S}$ for atoms in
${\textnormal{AP}_{\exists}}(\varphi)$, which can thus be forgotten. Thus,
from now on we will assume that an instance $(\mathcal{S},\Phi)$ of the model-
checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$
is such that
${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\Phi)=\emptyset$ and
$\mathcal{S}$ is a CKS over $\textnormal{AP}_{f}(\Phi)$.
Merging the decorated input tree and the model. To state the correctness of
our construction, we will need to merge the labels for quantified
propositions, carried by the input tree, with those for free propositions,
carried by CKS $\mathcal{S}$. Because, through successive widenings, the input
tree (represented by $t$ in the definition below) will necessarily be a
complete tree, its domain will always contain the domain of the unfolding of
$\mathcal{S}$ (represented by $t^{\prime}$ below), hence the following
definition.
###### Definition 4.9 (Merge).
Let $t=(\tau,\ell)$ be a complete $(\textnormal{AP},X)$-tree and
$t^{\prime}=(\tau^{\prime},\ell^{\prime})$ an
$(\textnormal{AP}\,^{\prime},X)$-tree with same root as $t$, where
$\textnormal{AP}\cap\textnormal{AP}\,^{\prime}=\emptyset$. We define the
_merge_ of $t$ and $t^{\prime}$ as the
$(\textnormal{AP}\cup\textnormal{AP}\,^{\prime},X)$-tree
$t\merge
t^{\prime}:=(\tau\cap\tau^{\prime}=\tau^{\prime},\ell^{\prime\prime}),$
where $\ell^{\prime\prime}(u)=\ell(u)\cup\ell^{\prime}(u)$.
We now describe our automata construction. Let $(\mathcal{S},\Phi)$ be an
instance of the model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$,
where $\mathcal{S}=(S,R,\ell_{\mathcal{S}},s_{\iota})$.
###### Lemma 4.10 (Translation).
For every subformula $\varphi$ of $\Phi$ and state $s$ of $\mathcal{S}$, one
can build an ATA $\mathcal{A}_{s}^{\varphi}$ on
$({\textnormal{AP}_{\exists}}(\Phi),S_{\varphi})$-trees such that for every
$({\textnormal{AP}_{\exists}}(\Phi),S_{\varphi})$-tree $t$ rooted in
$s_{\iota}\\!\downarrow_{I_{\varphi}}$, every $u\in t_{\mathcal{S}}$ ending in
$s$, it holds that
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})\mbox{\;\;\;iff\;\;\;}t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi.$
###### Proof.
Let ${\textnormal{AP}_{\exists}}={\textnormal{AP}_{\exists}}(\Phi)$ and
$\textnormal{AP}_{f}=\textnormal{AP}_{f}(\Phi)$, and recall that $\mathcal{S}$
is labelled over $\textnormal{AP}_{f}$. For each state $s\in S$ and each
subformula $\varphi$ of $\Phi$ (note that all subformulas of $\Phi$ are also
hierarchical), we define by induction on $\varphi$ the ATA
$\mathcal{A}_{s}^{\varphi}$ on
$({\textnormal{AP}_{\exists}},S_{\varphi})$-trees.
$\bm{\varphi=p:}$ First, by Definition 4.8, $S_{\varphi}=S_{[n]}=S$. We let
$\mathcal{A}_{s}^{p}$ be the ATA over $S$-trees with one unique state
$q_{\iota}$, with transition function defined as follows:
$\delta(q_{\iota},a)=\begin{cases}\top&\mbox{if
}\begin{array}[]{c}p\in\textnormal{AP}_{f}\mbox{ and
}p\in\ell_{\mathcal{S}}(s)\\\ \mbox{ or }\\\
p\in{\textnormal{AP}_{\exists}}\mbox{ and }p\in a\end{array}\\\ \perp&\mbox{if
}\begin{array}[]{c}p\in\textnormal{AP}_{f}\mbox{ and
}p\notin\ell_{\mathcal{S}}(s)\\\ \mbox{ or }\\\
p\in{\textnormal{AP}_{\exists}}\mbox{ and }p\notin a\end{array}\end{cases}$
$\bm{\varphi=\neg\varphi^{\prime}:}$ We let
$\mathcal{A}_{s}^{\varphi}:=\overline{\mathcal{A}_{s}^{\varphi^{\prime}}}$.
$\bm{\varphi=\varphi_{1}\vee\varphi_{2}:}$ Because
$I_{\varphi}=I_{\varphi_{1}}\cap I_{\varphi_{2}}$, and each
$\mathcal{A}_{s}^{\varphi_{i}}$ for $i\in\\{1,2\\}$ works on
$L_{\varphi_{i}}$-trees, we first narrow them so that they work on
$L_{\varphi}$-trees: for $i\in\\{1,2\\}$, we let
$\mathcal{A}_{i}:={\mathcal{A}_{s}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}}=(Q^{i},\delta^{i},q_{\iota}^{i},C^{i})$.
Letting $q_{\iota}$ be a fresh initial state we define
$\mathcal{A}_{s}^{\varphi}:=(\\{q_{\iota}\\}\cup Q^{1}\cup
Q^{2},\delta,q_{\iota},C)$, where $\delta$ and $C$ agree with $\delta^{i}$ and
$C^{i}$, respectively, on states from $Q^{i}$, and
$\delta(q_{\iota},a)=\delta^{1}(q_{\iota}^{1},a)\vee\delta^{2}(q_{\iota}^{2},a)$.
The colour of $q_{\iota}$ does not matter.
$\bm{\varphi={\bf E}\psi:}$ Let
$\max(\psi)=\\{\varphi_{1},\ldots,\varphi_{k}\\}$ be the set of maximal state
subformulas of $\psi$. In a first step we see these maximal state subformulas
as atomic propositions, we see $\psi$ as an LTL formula over $\max(\psi)$, and
we build a nondeterministic parity word automaton
$\mathcal{W}^{\psi}=(Q^{\psi},\Delta^{\psi},q^{\psi}_{\iota},C^{\psi})$ over
alphabet $2^{\max(\psi)}$ that accepts exactly the models of $\psi$ (and uses
two colours) (Vardi and Wolper, 1994). We define the ATA $\mathcal{A}$ that,
given as input a $(\max(\psi),S_{\varphi})$-tree $t$, nondeterministically
guesses a path $\lambda$ in $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$, or
equivalently a path in $\mathcal{S}$ starting from $s$, and simulates
$\mathcal{W}^{\psi}$ on it, assuming that the labels it reads while following
$\lambda\\!\downarrow_{I_{\varphi}}$ in its input $t$ correctly represent the
truth value of formulas in $\max(\psi)$ along $\lambda$. Recall that
$\mathcal{S}=(S,R,s_{\iota},\ell_{\mathcal{S}})$; we define
$\mathcal{A}:=(Q,\delta,q_{{\iota}},C)$, where
* •
$Q=Q^{\psi}\times S$,
* •
$q_{{\iota}}=(q^{\psi}_{{\iota}},s)$,
* •
for each $(q^{\psi},s^{\prime})\in Q$,
$C(q^{\psi},s^{\prime})=C^{\psi}(q^{\psi})$, and
* •
for each $(q^{\psi},s^{\prime})\in Q$ and $a\in 2^{\max(\psi)}$,
$\delta((q^{\psi},s^{\prime}),a)=\bigvee_{q^{\prime}\in\Delta^{\psi}(q^{\psi},a)}\bigvee_{s^{\prime\prime}\in
R(s^{\prime})}[s^{\prime\prime}\\!\downarrow_{I_{\varphi}},\left(q^{\prime},s^{\prime\prime}\right)].$
The intuition is that $\mathcal{A}$ reads the current label in
$2^{\max(\psi)}$, chooses nondeterministically a transition in
$\mathcal{W}^{\psi}$, chooses a next state $s^{\prime\prime}$ in $S$ and
proceeds in the corresponding direction
$s^{\prime\prime}\\!\downarrow_{I_{\varphi}}\in S_{\varphi}$.
Now from $\mathcal{A}$ we build the automaton $\mathcal{A}_{s}^{\varphi}$ over
$S_{\varphi}$-trees labelled with “real” atomic propositions in
${\textnormal{AP}_{\exists}}$. Intuitively, in each node it visits,
$\mathcal{A}_{s}^{\varphi}$ guesses what should be its labelling over
$\max(\psi)$, it simulates $\mathcal{A}$ accordingly, and checks that the
guess it made is correct. If, after having guessed a finite path $u\in
t_{\mathcal{S}}$ ending in state $s^{\prime}$, $\mathcal{A}_{s}^{\varphi}$
guesses that $\varphi_{i}$ holds, it checks this guess by starting a copy of
automaton $\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$ from node
$v=u\\!\downarrow_{I_{\varphi}}$ in its input $t$.
Formally, for each $s^{\prime}\in\mathcal{S}$ and each
$\varphi_{i}\in\max(\psi)$ we first build
$\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$, which works on
$S_{\varphi_{i}}$-trees. Observe that
$I_{\varphi}=\cap_{i=1}^{k}I_{\varphi_{i}}$, so that we need to narrow down
these automata222In the conference version of this work (Berthon et al., 2017)
we made a mistake here: we wrote that $I_{\varphi}=I_{\varphi_{i}}$, which is
not the case in general. As a consequence we do need to narrow down automata,
unlike what was written in the conference version.: We let
$\mathcal{A}^{i}_{s^{\prime}}:=\mathcal{A}_{s^{\prime}}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}=(Q^{i}_{s^{\prime}},\delta^{i}_{s^{\prime}},q^{i}_{s^{\prime}},C^{i}_{s^{\prime}})$.
We also let
$\overline{\mathcal{A}^{i}_{s^{\prime}}}=(\overline{Q^{i}_{s^{\prime}}},\overline{\delta^{i}_{s^{\prime}}},\overline{q^{i}_{s^{\prime}}},\overline{C^{i}_{s^{\prime}}})$
be the dualisation of $\mathcal{A}^{i}_{s^{\prime}}$, and we assume without
loss of generality all the state sets are pairwise disjoint. We define the ATA
$\mathcal{A}_{s}^{\varphi}=(Q\cup\bigcup_{i,s^{\prime}}Q^{i}_{s^{\prime}}\cup\overline{Q^{i}_{s^{\prime}}},\delta^{\prime},q_{{\iota}},C^{\prime}),$
where the colours of states are left as they were in their original automaton,
and $\delta^{\prime}$ is defined as follows. For states in
$Q^{i}_{s^{\prime}}$ (resp. $\overline{Q^{i}_{s^{\prime}}}$),
$\delta^{\prime}$ agrees with $\delta^{i}_{s^{\prime}}$ (resp.
$\overline{\delta^{i}_{s^{\prime}}}$), and for $(q^{\psi},s^{\prime})\in Q$
and $a\in 2^{{\textnormal{AP}_{\exists}}}$ we let
$\delta^{\prime}((q^{\psi},s^{\prime}),a)$ be the disjunction over
${a^{\prime}\in 2^{\max(\psi)}}$ of
(2)
$\displaystyle\Bigg{(}\delta\left((q^{\psi},s^{\prime}),a^{\prime}\right)\wedge\bigwedge_{\varphi_{i}\in
a^{\prime}}\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)\;\wedge\bigwedge_{\varphi_{i}\notin
a^{\prime}}\overline{\delta^{i}_{s^{\prime}}}(\overline{q^{i}_{s^{\prime}}},a)\Bigg{)}.$
Note that in general it is not possible to define a $\max(\psi)$-labelling of
$t$ that faithfully represents the truth values of formulas in $\max(\psi)$
for all nodes in $t_{\mathcal{S}}$, because a node in $t$ may correspond to
different nodes in $t_{\mathcal{S}}$ that have same projection on
$S_{\varphi}$ but satisfy different formulas of $\max(\psi)$. However this is
not a problem because different copies of $\mathcal{A}_{s}^{\varphi}$ that
visit the same node can guess different labellings, depending on the actual
state of $\mathcal{S}$ (which is part of the state of
$\mathcal{A}_{s}^{\varphi}$).
$\bm{\varphi=\exists}^{\bm{\textnormal{{o}}}}\bm{p.\,\varphi^{\prime}:}$ We
build automaton $\mathcal{A}_{s}^{\varphi^{\prime}}$ that works on
$S_{\varphi^{\prime}}$-trees; because $\varphi$ is hierarchical, we have that
$\textnormal{{o}}\subseteq I_{\varphi^{\prime}}$ and we can narrow down
$\mathcal{A}_{s}^{\varphi^{\prime}}$ to work on $S_{\textnormal{{o}}}$-trees
and obtain
$\mathcal{A}_{1}:={\mathcal{A}_{s}^{\varphi^{\prime}}\\!\downarrow_{\textnormal{{o}}}}$.
By Theorem 4.6 we can nondeterminise it to get $\mathcal{A}_{2}$, which by
Theorem 4.5 we can project with respect to $p$, finally obtaining
$\mathcal{A}_{s}^{\varphi}:=\mathcal{A}_{2}\\!\Downarrow_{p}$.
Correctness. We now prove by induction on $\varphi$ that the construction is
correct. In each case, we let $t=(\tau,\ell)$ be a complete
$({\textnormal{AP}_{\exists}},S_{\varphi})$-tree rooted in
$s_{\iota}\\!\downarrow_{I_{\varphi}}$.
$\bm{\varphi=p:}$ First, note that $I_{p}=[n]$, so that $t$ is rooted in
$s_{\iota}\\!\downarrow_{I_{\varphi}}=s_{\iota}$, and
$u\\!\downarrow_{I_{\varphi}}=u$. Also recall that $u$ ends in $s$. Let us
consider first the case where $p\in\textnormal{AP}_{f}$. By definition of
$\mathcal{A}_{s}^{p}$, we have that $(t,u)\in\mathcal{L}(\mathcal{A}_{s}^{p})$
if and only if $p\in\ell_{\mathcal{S}}(s)$. We also have
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models p$ if and only if
$p\in\ell^{\prime}(u)$, where $\ell^{\prime}$ is the labelling of tree
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$. By definition of unfolding and
merge, we have that $\ell^{\prime}(u)=\ell_{\mathcal{S}}(s)$, which concludes
this direction. Now if $p\in{\textnormal{AP}_{\exists}}$: by definition of
$\mathcal{A}_{s}^{p}$, we have $(t,u)\in\mathcal{L}(\mathcal{A}_{s}^{p})$ if
and only if $p\in\ell(u)$; also, by definition of the merge and unfolding, we
have that $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models p$ if and only
if $p\in\ell(u)$, and we are done.
$\bm{\varphi=\neg\varphi^{\prime}:}$ Correctness follows from the induction
hypothesis and Theorem 4.4.
$\bm{\varphi_{1}\vee\varphi_{2}:}$ We have
$\mathcal{A}_{i}=\mathcal{A}_{s}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}$, so
by Theorem 4.7 we have
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{i})$ if and only
if
$(t\\!\uparrow^{I_{\varphi_{i}}},u\\!\downarrow_{I_{\varphi_{i}}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi_{i}})$,
which by induction hypothesis holds if and only if
$(t\\!\uparrow^{I_{\varphi_{i}}})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi_{i}$,
i.e., $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi_{i}$. We
conclude by observing that
$\mathcal{L}(\mathcal{A}_{s}^{\varphi})=\mathcal{L}(\mathcal{A}_{1})\cup\mathcal{L}(\mathcal{A}_{2})$.
$\bm{\varphi={\bf E}\psi:}$ Suppose that
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models{\bf E}\psi$. There exists
an infinite path $\lambda$ in $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$
starting at $u$ such that
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda\models\psi$. Again, let
$\max(\psi)$ be the set of maximal state subformulas of $\varphi$, and let $w$
be the infinite word over $2^{\max(\psi)}$ that agrees with $\lambda$ on the
state formulas in $\max(\psi)$, i.e., for each node $\lambda_{k}$ of $\lambda$
and formula $\varphi_{i}\in\max(\psi)$, it holds that $\varphi_{i}\in w_{k}$
if and only if
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda_{k}\models\varphi_{i}$. To
show that
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})$ we
show that Eve can win the acceptance game
$\mathcal{G}(\mathcal{A}_{s}^{\varphi},t,u\\!\downarrow_{I_{\varphi}})$. In
this game, Eve can guess the path $\lambda$ while the automaton follows
$\lambda\\!\downarrow_{I_{\varphi}}$ in its input $t$, and she can also guess
the corresponding word $w$ on $2^{\max(\psi)}$. By construction of
$\mathcal{W}^{\psi}$, Eve has a winning strategy $\sigma_{\psi}$ in the
acceptance game of $\mathcal{W}^{\psi}$ on $w$. From $\lambda$, $w$ and
$\sigma_{\psi}$ we can easily define a strategy for Eve in
$\mathcal{G}(\mathcal{A}_{s}^{\varphi},t,u\\!\downarrow_{I_{\varphi}})$ on all
positions that can be reached while Adam does not choose to challenge her on a
guess she made for the truth value of some maximal state subformula, and on
such plays this strategy is winning because $\sigma_{\psi}$ is winning.
Now if Adam challenges her on one of these guesses: Let $\lambda_{k}\in
t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$ be a node along $\lambda$, let
$s^{\prime}$ be its last direction and let
$\lambda_{k}^{\prime}=\lambda_{k}\\!\downarrow_{I_{\varphi}}\in t$. Assume
that in node $\lambda^{\prime}_{k}$ of the input tree, in a state
$(q^{\psi},s^{\prime})\in Q$, Adam challenges Eve on some
$\varphi_{i}\in\max(\psi)$ that she assumes to be true in
$\lambda^{\prime}_{k}$, i.e., such that $\varphi_{i}\in w_{k}$. Formally, in
the evaluation game this means that Adam chooses the conjunct
$\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)$ in transition formula 2, where
$a=\ell(\lambda^{\prime}_{k})$, thus moving to position
$(\lambda^{\prime}_{k},(q^{\psi},s^{\prime}),\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$.
We want to show that Eve wins from this position. To do so we first show that
$(t,\lambda^{\prime}_{k})\in\mathcal{L}(\mathcal{A}^{i}_{s^{\prime}})$.
First, since
$\mathcal{A}^{i}_{s^{\prime}}=\mathcal{A}_{s^{\prime}}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}$,
by Theorem 4.7,
$(t,\lambda^{\prime}_{k})\in\mathcal{L}(\mathcal{A}^{i}_{s^{\prime}})$ if and
only if
$(t\\!\uparrow^{I_{\varphi_{i}}},\lambda_{k}\\!\downarrow_{I_{\varphi_{i}}})\in\mathcal{L}(\mathcal{A}_{s^{\prime}}^{\varphi_{i}})$.
Next, by applying the induction hypothesis we get that
$(t\\!\uparrow^{I_{\varphi_{i}}},\lambda_{k}\\!\downarrow_{I_{\varphi_{i}}})\in\mathcal{L}(\mathcal{A}_{s^{\prime}}^{\varphi_{i}})$
if and only if
$t\\!\uparrow^{I_{\varphi_{i}}}\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda_{k}\models\varphi_{i}$,
i.e.,
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda_{k}\models\varphi_{i}$. The
latter holds because $\varphi_{i}\in w_{k}$, and by assumption $w_{k}$ agrees
with $\lambda_{k}$ on $\varphi_{i}$. Thus
$(t,\lambda^{\prime}_{k})\in\mathcal{L}(\mathcal{A}^{i}_{s^{\prime}})$.
This means that Eve has a winning strategy from the initial position
$(\lambda^{\prime}_{k},q^{i}_{s^{\prime}},\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$
of the acceptance game of $\mathcal{A}^{i}_{s^{\prime}}$ on
$(t,\lambda^{\prime}_{k})$. Since
$(\lambda^{\prime}_{k},q^{i}_{s^{\prime}},\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$
and
$(\lambda^{\prime}_{k},(q^{\psi},s^{\prime}),\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$
contain the same node $\lambda^{\prime}_{k}$ and transition formula
$\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)$, the subgames that start in
these positions are isomorphic and a winning strategy in one of these
positions induces a winning strategy in the other, and therefore Eve wins
Adam’s challenge (recall that positional strategies are sufficient in parity
games (Zielonka, 1998)). With a similar argument, we get that also when Adam
challenges Eve on some $\varphi_{i}\in\max(\psi)$ assumed not to be true in
node $\lambda_{k}$, Eve wins the challenge. Finally, Eve wins the acceptance
game of $\mathcal{A}_{s}^{\varphi}$ on $(t,u\\!\downarrow_{I_{\varphi}})$, and
thus
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})$.
For the other direction, assume that
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})$,
i.e., Eve wins the evaluation game of $\mathcal{A}_{s}^{\varphi}$ on
$(t,u\\!\downarrow_{I_{\varphi}})$. A winning strategy for Eve describes a
path $\lambda$ in $t_{\mathcal{S}}$ from $s$, which is also a path in
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$ from $u$. This winning strategy
also defines an infinite word $w$ over $2^{\max(\psi)}$ such that $w$ agrees
with $\lambda$ on the formulas in $\max(\psi)$, and it also describes a
winning strategy for Eve in the acceptance game of $\mathcal{W}^{\psi}$ on
$w$. Hence $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda\models\psi$, and
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi$.
$\bm{\varphi=\exists}^{\bm{\textnormal{{o}}}}\bm{p.\,\varphi^{\prime}:}$
First, by definition we have $I_{\varphi}=\textnormal{{o}}\cap
I_{\varphi^{\prime}}$. Because $\varphi$ is hierarchical,
$\textnormal{{o}}\subseteq\textnormal{{o}}^{\prime}$ for every
$\textnormal{{o}}^{\prime}$ that occurs in $\varphi^{\prime}$, and thus
$\textnormal{{o}}\subseteq I_{\varphi^{\prime}}$. It follows that
$I_{\varphi}=\textnormal{{o}}$. Next, by Theorem 4.5 we have that
(3)
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})\mbox{\;\;\;iff\;\;\;}\exists\,\ell_{p}\mbox{
a $p$-labelling for $t$ such that
}(t\otimes\ell_{p},u)\in\mathcal{L}(\mathcal{A}_{2}).$
By Theorem 4.6, $\mathcal{L}(\mathcal{A}_{2})=\mathcal{L}(\mathcal{A}_{1})$,
and since
$\mathcal{A}_{1}=\mathcal{A}_{s}^{\varphi^{\prime}}\\!\downarrow_{\textnormal{{o}}}=\mathcal{A}_{s}^{\varphi^{\prime}}\\!\downarrow_{I_{\varphi}}$
we get by Theorem 4.7 that
(4)
$(t\otimes\ell_{p},u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{2})\mbox{\;\;\;iff\;\;\;}((t\otimes\ell_{p})\\!\uparrow^{L_{\varphi^{\prime}}},u\\!\downarrow_{I_{\varphi^{\prime}}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi^{\prime}}).$
By induction hypothesis,
(5)
$((t\otimes\ell_{p})\\!\uparrow^{L_{\varphi^{\prime}}},u\\!\downarrow_{I_{\varphi^{\prime}}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi^{\prime}})\mbox{\;\;\;iff\;\;\;}(t\otimes\ell_{p})\\!\uparrow^{L_{\varphi^{\prime}}}\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}.$
Now, by points (3), (4) and (5) and the fact that
$(t\otimes\ell_{p})\\!\uparrow^{I_{\varphi^{\prime}}}\\!\uparrow^{[n]}=(t\otimes\ell_{p})\\!\uparrow^{[n]}$,
we get that
(6)
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})\mbox{\;\;\;iff\;\;\;}\exists\,\ell_{p}\mbox{
a $p$-labelling for $t$ such that
}(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}.$
We now prove the following equation which, together with point (6), concludes
the proof:
(7) $\begin{array}[]{c}\exists\,\ell_{p}\mbox{ a $p$-labelling for $t$ such
that
}(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}\\\
\mbox{\;\;\;iff\;\;\;}\\\
t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\exists^{\textnormal{{o}}}p.\,\varphi^{\prime}\end{array}$
Assume that there exists a $p$-labelling $\ell_{p}$ for $t$ such that
$(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}$.
Let $\ell_{p}^{\prime}$ be the $p$-labelling of
$(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$. By definition of
the merge, $\ell_{p}^{\prime}$ is equal to the $p$-labelling of
$(t\otimes\ell_{p})\\!\uparrow^{[n]}$, which by definition of the widening is
$I_{\varphi}$-uniform, i.e., it is o-uniform. In addition, it is clear that
$(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=(t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}})\otimes\ell_{p}^{\prime}$,
which concludes this direction.
For the other direction, assume that
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\exists^{\textnormal{{o}}}p.\,\varphi^{\prime}$:
there exists a o-uniform $p$-labelling $\ell_{p}^{\prime}$ for
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$ such that
$(t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}})\otimes\ell_{p}^{\prime},u\models\varphi^{\prime}$.
We define a $p$-labelling $\ell_{p}$ for $t$ such that
$(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}$.
First, let us write
$t^{\prime}=t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=(\tau^{\prime},\ell^{\prime})$.
For each node $u$ of $t$, let
$\ell_{p}(u)=\begin{cases}\ell_{p}^{\prime}(u^{\prime})&\mbox{if there exists
}u^{\prime}\in\tau^{\prime}\mbox{ such that
}u^{\prime}\\!\downarrow_{\textnormal{{o}}}=u,\\\
0&\mbox{otherwise.}\end{cases}$
This is well defined because $\ell_{p}^{\prime}$ is o-uniform in $p$, so that
if two nodes $u^{\prime},v^{\prime}$ project on $u$, we have
$u^{\prime}\approx_{\textnormal{{o}}}v^{\prime}$ and thus
$\ell_{p}^{\prime}(u^{\prime})=\ell_{p}^{\prime}(v^{\prime})$. In case there
is no $u^{\prime}\in\tau^{\prime}$ such that
$u^{\prime}\\!\downarrow_{I_{\varphi}}=u$, the value of $\ell_{p}(u)$ has no
impact on $(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$.
Finally,
$(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=(t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}})\otimes\ell_{p}^{\prime}$,
hence the result.
∎
### 4.3. Proof of Theorem 4.3
We now prove Theorem 4.3. Let $\mathcal{S}$ be a CKS with initial state
$s_{\iota}$, and let $\Phi\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$. By Lemma 4.10 one can build an ATA
$\mathcal{A}_{s_{\iota}}^{\Phi}$ such that for every labelled
$S_{\varphi}$-tree $t$ rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}$, and
every node $u\in t_{\mathcal{S}}$,
$(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s_{\iota}}^{\varphi})$
if, and only if, $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\Phi$.
Let $\tau$ be the full $S_{\varphi}$-tree rooted in
$s_{\iota}\\!\downarrow_{I_{\varphi}}$, and let $t=(\tau,\ell_{\emptyset})$,
where $\ell_{\emptyset}$ is the empty labelling. Clearly,
$t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=t_{\mathcal{S}}$, and because $t$
is rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}$, we get that
$t\in\mathcal{L}(\mathcal{A}_{s_{\iota}}^{\varphi})$ if, and only if
$t_{\mathcal{S}}\models\Phi$, i.e., $\mathcal{S}\models\Phi$. It remains to
check whether tree $t$, which is regular, is accepted by
$\mathcal{A}_{s_{\iota}}^{\Phi}$. This can be done by solving a parity game
built from the product of $\mathcal{A}_{s_{\iota}}^{\Phi}$ with a finite
Kripke structure representing $t$ (Löding, 2011).
### 4.4. Complexity
To state a precise upper bound on the complexity of our procedure, we first
introduce a syntactic notion of _simulation depth_ for formulas of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. While alternation depth
(see, e.g., (Mogavero et al., 2014)) simply counts the number of alternations
between existential and universal strategy quantifications, simulation depth
reflects automata operations required to treat a formula, and counts the
maximum number of nested simulations of alternating tree automata that need to
be performed when applying our automata construction. However, like
alternation depth, it is a purely syntactic notion. Formally we define a
function $\mbox{sd}:\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}\to\mathbb{N}\times\\{\mbox{nd},\mbox{alt}\\}$ which returns, for each
formula $\varphi$, a pair $\mbox{sd}(\varphi)=(k,x)$ where $k$ is the
simulation depth of $\varphi$, and $x\in\\{\mbox{nd},\mbox{alt}\\}$ indicates
whether the automaton $\mathcal{A}_{s}^{\varphi}$ built from $\varphi$ and a
state $s$ of a CKS $\mathcal{S}$ is nondeterministic (nd) or alternating
(alt). If $\mbox{sd}(\varphi)=(k,x)$ we shall denote $k$ by
$\mbox{sd}_{k}(\varphi)$ and $x$ by $\mbox{sd}_{x}(\varphi)$. The inductive
definition for state formulas is as follows:
$\begin{array}[]{l}\mbox{sd}(p):=(0,\mbox{nd})\\\\[5.0pt]
\mbox{sd}(\neg\varphi):=(\mbox{sd}_{k}(\varphi),\mbox{alt})\\\\[5.0pt]
\mbox{sd}(\varphi_{1}\vee\varphi_{2}):=\left(\max_{i\in\\{1,2\\}}\mbox{sd}_{k}(\varphi_{i}),x\right),\\\
\hfill\mbox{where }x=\begin{cases}\mbox{nd}&\mbox{if
}\mbox{sd}_{x}(\varphi_{1})=\mbox{sd}_{x}(\varphi_{2})=\mbox{nd}\\\
\mbox{alt}&\mbox{otherwise}\end{cases}\\\\[15.0pt] \mbox{sd}({\bf
E}\psi):=\begin{cases}(0,\mbox{nd})&\mbox{if }\psi\in\textnormal{LTL}\\\
(\max_{\varphi\in\max(\psi)}\mbox{sd}_{k}(\varphi),\mbox{alt})&\mbox{otherwise}\end{cases}\\\\[15.0pt]
\mbox{sd}(\exists^{\textnormal{{o}}}p.\,\varphi):=(k,\mbox{nd}),\\\
\hfill\quad\quad\quad\quad\quad\mbox{where
}k=\begin{cases}\mbox{sd}_{k}(\varphi)&\mbox{if
}\mbox{sd}_{x}(\varphi)=\mbox{nd}\mbox{ and
}\textnormal{{o}}=I_{\varphi}\quad\mbox{(recall Definition~{}\ref{def-
Iphi})}\\\ \mbox{sd}_{k}(\varphi)+1&\mbox{otherwise}\end{cases}\end{array}$
We explain each case. For an atomic proposition $p$, the automaton
$\mathcal{A}_{s}^{p}$ is clearly nondeterministic and no simulation is
involved in its construction. For a formula $\neg\varphi$, the automaton
$\mathcal{A}_{s}^{\neg\varphi}$ is obtained by dualising
$\mathcal{A}_{s}^{\varphi}$, an operation that in general does not return a
nondeterministic automaton but an alternating one; also this dualisation does
not involve any simulation, hence the definition of the first component. Now
for the disjunction, the first component should be clear; for the second one,
observe that by construction of
$\mathcal{A}_{s}^{\varphi_{1}\vee\varphi_{2}}$, if both
$\mathcal{A}_{s}^{\varphi_{1}}$ and $\mathcal{A}_{s}^{\varphi_{2}}$ are
nondeterministic, then so is $\mathcal{A}_{s}^{\varphi_{1}\vee\varphi_{2}}$;
otherwise, it is alternating. For the path quantifier, by construction
$\mathcal{A}_{s}^{{\bf E}\psi}$ is alternating in the general case as it
starts copies of automata for each maximal state subformula in $\psi$; for the
first component, we recall that $\max(\psi)$ denotes the set of these maximal
state subformulas and we observe that no additional simulation is performed to
build $\mathcal{A}_{s}^{{\bf E}\psi}$ besides those needed to construct the
automata for the maximal state subformulas. If $\psi$ is an LTL formula, then
one can build the nondeterministic word automaton $\mathcal{W}^{\psi}$
directly working on “real” atomic propositions in
${\textnormal{AP}_{\exists}}\cup\textnormal{AP}_{f}$. The automaton
$\mathcal{A}$ can then be built working directly on
${\textnormal{AP}_{\exists}}$, with $\mathcal{W}^{\psi}$ reading valuations
for ${\textnormal{AP}_{\exists}}$ in the input tree and those for atoms in
$\textnormal{AP}_{f}$ in the current state of $\mathcal{S}$. Because we do not
need to guess valuations of maximal state subformulas and launch additional
automata to check that these guesses are correct, we obtain a nondeterministic
automaton. Finally, for a formula of the form
$\exists^{\textnormal{{o}}}p.\,\varphi$, to build automaton
$\mathcal{A}_{s}^{\exists^{\textnormal{{o}}}p.\,\varphi}$ we first build
$\mathcal{A}_{s}^{\varphi}$, which we then narrow down to work on
$L_{\textnormal{{o}}}$-trees. Since the narrowing operation introduces
alternation, we need to nondeterminise the resulting automaton before
projecting it with respect to $p$. Now observe that if
$I_{\varphi}=\textnormal{{o}}$ we do not need to perform this narrowing, and
thus if $\mathcal{A}_{s}^{\varphi}$ is a nondeterministic automaton we can
directly perform the projection. This justifies the definition of the first
component; for the second one, observe that the projection of a
nondeterministic automaton is also nondeterministic.
###### Example 4.11.
Assume that $n=3$, i.e., states of CKS have three components (recall that
$[3]=\\{1,2,3\\}$). Let us consider formula
$\varphi=\forall^{\\{1,3\\}}p.\,\forall^{[3]}q.\,\exists^{[3]}r.\,{\bf E}{\bf
G}(p\wedge q\vee r)$. We describe how its simulation depth is computed. First,
let us rewrite
$\varphi=\neg\exists^{\\{1,3\\}}p.\,\exists^{[3]}q.\,\neg\exists^{[3]}r.\,{\bf
E}{\bf G}(p\wedge q\vee r)$.
Since ${\bf G}(p\wedge q\vee r)$ is an LTL formula, $\mbox{sd}({\bf E}{\bf
G}(p\wedge q\vee r))=(0,\mbox{nd})$. Next, because $I_{{\bf E}{\bf G}(p\wedge
q\vee r)}=[3]$, it follows that $\mbox{sd}(\exists^{[3]}r.\,{\bf E}{\bf
G}(p\wedge q\vee r))=(0,\mbox{nd})$, and $\mbox{sd}(\neg\exists^{[3]}r.\,{\bf
E}{\bf G}(p\wedge q\vee r))=(0,\mbox{alt})$. Next we have that
$\mbox{sd}(\exists^{[3]}q.\,\neg\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee
r))=(1,\mbox{nd})$. This reflects the fact that the automaton obtained for
formula $\neg\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r)$, which is
alternating because of complementation, needs to be simulated before
projecting it over $q$. Then, because $\\{1,3\\}\neq[3]$, it holds that
$\mbox{sd}(\exists^{\\{1,3\\}}p.\,\exists^{[3]}q.\,\neg\exists^{[3]}r.\,{\bf
E}{\bf G}(p\wedge q\vee r))=(2,\mbox{nd})$: to project over $p$ we first need
to narrow down the previous automaton to make it see only components 1 and 3,
and because the narrowing operation introduces alternation, the resulting
automaton needs to be simulated before projecting it. Finally, we get that
$\mbox{sd}(\varphi)=(2,\mbox{alt})$
We now introduce two additional depth measures on
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formulas, which help
us establish more precise upper bounds on the sizes of the automata we build.
For every $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula
$\varphi$, we let ${\bf E}\mathrm{d}(\varphi)$ be the maximum number of nested
path quantifiers ${\bf E}$ in $\varphi$, and $\exists\mathrm{d}(\varphi)$ is
the maximum number of nested second-order quantifiers $\exists$ in $\varphi$.
We also inductively define the function $\mathrm{exp}\big{(}k\mid n\big{)}$,
for $k,n\in\mathbb{N}$, as follows: $\mathrm{exp}\big{(}0\mid n\big{)}:=n$ and
$\mathrm{exp}\big{(}k+1\mid n\big{)}:=2^{\mathrm{exp}\big{(}k\mid n\big{)}}$.
###### Proposition 4.12.
Let $\Phi$ be a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$ formula, $\mathcal{S}$ a CKS and $s\in\mathcal{S}$ a
state.
* •
If $\mbox{sd}_{k}(\Phi)=0$, $\mathcal{A}_{s}^{\Phi}$ has at most
$f_{\mathcal{S}}^{\Phi}$ states and 2 colours, and
* •
if $\mbox{sd}_{k}(\Phi)\geq 1$, $\mathcal{A}_{s}^{\Phi}$ has at most
$\mathrm{exp}\big{(}\mbox{sd}_{k}(\Phi)\mid f_{\mathcal{S}}^{\Phi}\log
f_{\mathcal{S}}^{\Phi}\big{)}$ states and its number of colours is at most
$\mathrm{exp}\big{(}\mbox{sd}_{k}(\Phi)-1\mid f_{\mathcal{S}}^{\Phi}\log
f_{\mathcal{S}}^{\Phi}\big{)}$,
where
$f_{\mathcal{S}}^{\Phi}=m_{1}^{\exists\mathrm{d}(\Phi)}|\Phi||\mathcal{S}|^{{\bf
E}\mathrm{d}(\Phi)}2^{m_{2}|\Phi|{\bf E}\mathrm{d}(\Phi)}$, with
$m_{1},m_{2}\in\mathbb{N}$ constants.
Also, if $\mathcal{A}_{s}^{\varphi}$ has state set $Q$ then for each $q\in Q$
and $a\in 2^{{\textnormal{AP}_{\exists}}(\Phi)}$ we have
$|\delta(q,a)|\leq|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\varphi|}$, where
$H=1+{\bf E}\mathrm{d}(\varphi)$.
Constants $m_{1}$ and $m_{2}$ are derived from constants in the complexity of,
respectively, the simulation procedure, and the procedure that builds a
nondeterministic word automaton for an LTL formula. For more detail, see the
proof of Proposition 4.12 in Appendix A.
From this we get the following complexity result.
###### Proposition 4.13.
The model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$
formulas of simulation depth at most $k$ is $(k+1)$-Exptime -complete.
###### Proof.
We start with the upper bounds. For an instance $(\Phi,\mathcal{S})$, our
decision procedure in Section 4.3 first builds automaton
$\mathcal{A}_{s_{\iota}}^{\Phi}$, and concludes by testing whether the full
$S_{\Phi}$-tree with empty labelling $t$ is accepted by
$\mathcal{A}_{s_{\iota}}^{\Phi}$. This can be done in time
$O((|\mathcal{A}_{s_{\iota}}^{\Phi}|\cdot|t|)^{l})$, where $|t|$ is the size
of a smallest Kripke structure representing the regular tree $t$,
$|\mathcal{A}_{s_{\iota}}^{\Phi}|$ is the sum of the number of states and
sizes of formulas in the transition function of
$\mathcal{A}_{s_{\iota}}^{\Phi}$, and $l$ the number of colours it uses
(Löding, 2011). Clearly $t$ can be represented by a Kripke structure of size
$|S_{\Phi}|$, so that $|t|\leq|S_{\Phi}|\leq|\mathcal{S}|$.
By Proposition 4.12, each formula in the transition function of
$\mathcal{A}_{s_{\iota}}^{\Phi}$ is of size at most
$|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\Phi|}$, where $Q$ is the set of states
in $\mathcal{A}_{s_{\iota}}^{\Phi}$ and $H=1+{\bf E}\mathrm{d}(\Phi)$. There
are at most $|Q|2^{|{\textnormal{AP}_{\exists}}(\Phi)|}$ such formulas333In
fact the final automaton $\mathcal{A}_{s_{\iota}}^{\Phi}$ does not read
anything in its input, hence the alphabet could be considered to be a
singleton. We thus have only $|Q|$ different formulas in the transition
function, at most. and $|{\textnormal{AP}_{\exists}}(\Phi)|\leq|\Phi|$, so
that
$|\mathcal{A}_{s_{\iota}}^{\Phi}|\leq|Q|+|Q|2^{|{\textnormal{AP}_{\exists}}(\Phi)|}|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\Phi|}\leq
2|\mathcal{S}||Q|^{|\mathcal{S}|+1}2^{(H+1)|\Phi|}$. Also $H+1\leq|\Phi|$, so
we finally have $|\mathcal{A}_{s_{\iota}}^{\Phi}|\leq
2|\mathcal{S}||Q|^{|\mathcal{S}|+1}2^{|\Phi|^{2}}$.
If $k=0$, by Proposition 4.12 $\mathcal{A}_{s_{\iota}}^{\Phi}$ has at most
$f_{\mathcal{S}}^{\Phi}$ states and 2 colours, and $f_{\mathcal{S}}^{\Phi}$ is
polynomial in $|\mathcal{S}|$ but exponential in $|\Phi|$. Therefore
$|\mathcal{A}_{s_{\iota}}^{\Phi}|$ is exponential in $|\Phi|$ and in
$|\mathcal{S}|$, and so is the complexity of checking that $t$ is accepted by
$\mathcal{A}_{s_{\iota}}^{\Phi}$.
If $k\geq 1$, by Proposition 4.12, $|Q|$ is $k$-exponential in
$f_{\mathcal{S}}^{\Phi}\log f_{\mathcal{S}}^{\Phi}$, and
$f_{\mathcal{S}}^{\Phi}\log f_{\mathcal{S}}^{\Phi}$ itself is polynomial in
$|\mathcal{S}|$ but exponential in $|\Phi|$. As a result,
$|\mathcal{A}_{s_{\iota}}^{\Phi}|$ is $(k+1)$-exponential in $|\Phi|$ and
$k$-exponential in $|\mathcal{S}|$. Finally, still by Proposition 4.12, the
number of colours $l$ is $(k-1)$-exponential in $f_{\mathcal{S}}^{\Phi}\log
f_{\mathcal{S}}^{\Phi}$, hence $k$-exponential in $|\Phi|$. Checking that $t$
is accepted by $\mathcal{A}_{s_{\iota}}^{\Phi}$ can thus be done in time
$(k+1)$-exponential in $|\Phi|$, and $k$-exponential in $|\mathcal{S}|$, which
finishes to establish the upper bounds.
For the lower bounds, consider the fragment
$\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ of $\textnormal{{QCTL}}^{*}$
(with perfect information) which consists in formulas in prenex normal form,
i.e., with all second-order quantifications at the beginning, with at most $k$
alternations between existential and universal quantifiers, counting the first
quantifier as one alternation (see (Laroussinie and Markey, 2014, p.8) for a
formal definition). Clearly, $\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ is
a fragment of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ (with
$n=1$), and formulas of $\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ have
simulation depth at most $k$. It is proved in (Laroussinie and Markey, 2014)
that model checking $\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ is
$(k+1)$-Exptime -hard. ∎
###### Remark 3.
One may wonder why we do not get our lower bounds from the distributed
synthesis problem in systems with hierarchical information. The reason is that
this problem is $k$-Exptime -complete for LTL or $\textnormal{{CTL}}^{*}$
specifications (Pnueli and Rosner, 1990; Kupferman and Vardi, 2001) and can be
expressed with formulas of simulation depth $k$, and thus would only provide
$k$-Exptime lower-bounds for simulation depth $k$, while our problem is
$k+1$-Exptime -complete. This may seem surprising, but we point out that
thanks to alternation of existential and universal quantifiers,
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formulas with
simulation depth $k$ can express more complex problems than classic
distributed synthesis, such as existence of Nash equilibria (see Section 7.1).
Improved upper bound. We now refine the previous result by observing that some
subformulas can be model-checked independently in a bottom-up labelling
algorithm which uses the above model-checking procedure as a subroutine. The
height of exponential of the overall procedure for a formula $\Phi$ is thus
determined by the maximal simulation-depth of the successive independent
subformulas $\varphi$ treated by the labelling algorithm, instead of the
simulation depth of the full formula $\Phi$. To make this precise we define
the _simulation number_ of a sentence, akin to the alternation number
introduced in (Mogavero et al., 2014).
Let $\Phi\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, and assume
without loss of generality that
${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\Phi)=\emptyset$. A
state subformula $\varphi$ of $\Phi$ is a _subsentence_ if no atom quantified
in $\Phi$ appears free in $\varphi$, i.e., $\varphi$ is a subsentence of
$\Phi$ if
${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\varphi)=\emptyset$.444Observe
that since we always assume that
${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\Phi)=\emptyset$,
$\Phi$ is a subsentence of itself. The _simulation number_ $\mbox{sn}(\Phi)$
of a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\Phi$ is
the maximal simulation depth of $\Phi$’s subsentences, where the simulation
depth is computed by considering strict subsentences as atoms.
Note that because temporal operators of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can only talk about the
future, the truth value of a subsentence in a node $u$ of an unfolding
$t_{\mathcal{S}}$ only depends on the current state $\mbox{last}(u)$. The
bottom-up labelling algorithm for an instance $(\Phi,\mathcal{S})$ thus
consists in iteratively model checking innermore subsentences of $\Phi$ in all
states of $\mathcal{S}$, marking the states where they hold with fresh atomic
propositions with which the corresponding subsentences are replaced in $\Phi$.
###### Proposition 4.14.
The model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$
formulas of simulation number at most $k$ is $(k+1)$-Exptime -complete.
## 5\. Model-checking hierarchical instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
In this section we establish that the model-checking problem for
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ restricted to the class of
hierarchical instances is decidable (Theorem 2.9).
### 5.1. Reduction to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$
We build upon the proof in (Laroussinie and Markey, 2015) that establishes the
decidability of the model-checking problem for
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ by reduction to the
model-checking problem for $\textnormal{{QCTL}}^{*}$. The main difference is
that we reduce to the model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ instead, using
quantifiers on atomic propositions parameterised with observations that
reflect the ones used in the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
model-checking instance.
Let $(\mathcal{G},\Phi)$ be a hierarchical instance of the
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ model-checking problem, and
assume without loss of generality that each strategy variable is quantified at
most once in $\Phi$. We define an equivalent instance of the model-checking
problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$.
Constructing the CKS $\mathcal{S}_{\mathcal{G}}$. We define
$\mathcal{S}_{\mathcal{G}}$ so that (indistinguishable) nodes in its tree-
unfolding correspond to (indistinguishable) finite plays in $\mathcal{G}$. The
CKS will make use of atomic propositions $\textnormal{AP}_{v}:=\\{p_{v}\mid
v\in V\\}$ (that we assume to be disjoint from AP). The idea is that $p_{v}$
allows the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula
$(\Phi)_{s}^{\,\emptyset}$ to refer to the current position $v$ in
$\mathcal{G}$. Later we will see that $(\Phi)_{s}^{\,\emptyset}$ will also
make use of atomic propositions $\textnormal{AP}_{c}:=\\{p_{c}^{x}\mid
c\in\textnormal{Ac}\mbox{ and }x\in\textnormal{Var}\\}$ that we assume, again,
are disjoint from $\textnormal{AP}\cup\textnormal{AP}_{v}$. This allows the
formula to use $p_{c}^{x}$ to refer to the actions $c$ advised by strategies
$x$.
Suppose $\textnormal{Obs}=\\{o_{1},\ldots,o_{n}\\}$, and let
$\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$. For $i\in[n]$,
define the local states $L_{i}:=\\{[v]_{o_{i}}\mid v\in V\\}$ where $[v]_{o}$
is the equivalence class of $v$ for relation $\sim_{o}$. Since we need to know
the actual position of the $\textrm{CGS}_{\textnormal{ii}}$ to define the
dynamics, we also let $L_{n+1}:=V$.
Define the CKS $\mathcal{S}_{\mathcal{G}}:=(S,R,s_{{\iota}},\ell^{\prime})$
where
* •
$S:=\\{s_{v}\mid v\in V\\}$,
* •
$R:=\\{(s_{v},s_{v^{\prime}})\mid\exists\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}\mbox{
s.t. }E(v,\bm{c})=v^{\prime}\\}\subseteq S^{2}$,
* •
$s_{{\iota}}:=s_{v_{{\iota}}}$,
* •
$\ell^{\prime}(s_{v}):=\ell(v)\cup\\{p_{v}\\}\subseteq\textnormal{AP}\cup\textnormal{AP}_{v}$,
and $s_{v}:=([v]_{o_{1}},\ldots,[v]_{o_{n}},v)\in\prod_{i\in[n+1]}L_{i}$.
For every finite play $\rho=v_{0}\ldots v_{k}$, define the node
$u_{\rho}:=s_{v_{0}}\ldots s_{v_{k}}$ in $t_{\mathcal{S}_{\mathcal{G}}}$
(which exists, by definition of $\mathcal{S}_{\mathcal{G}}$ and of tree
unfoldings). Note that the mapping $\rho\mapsto u_{\rho}$ defines a bijection
between the set of finite plays and the set of nodes in
$t_{\mathcal{S}_{\mathcal{G}}}$.
Constructing the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$ formulas $(\varphi)_{s}^{\,f}$. We now describe how to
transform an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula
$\varphi$ and a partial function
$f:\textnormal{Ag}\rightharpoonup\textnormal{Var}$ into a
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula
$(\varphi)_{s}^{\,f}$ (that will also depend on $\mathcal{G}$). Suppose that
$\textnormal{Ac}=\\{c_{1},\ldots,c_{l}\\}$, and define $(\varphi)_{s}^{\,f}$
and $(\psi)_{p}^{\,f}$ by mutual induction on state and path formulas. The
base cases are as follows: $(p)_{s}^{\,f}:=p$ and
$(\varphi)_{p}^{\,f}:=(\varphi)_{s}^{\,f}$. Boolean and temporal operators are
simply obtained by distributing the translation:
$(\neg\varphi)_{s}^{\,f}:=\neg(\varphi)_{s}^{\,f}$,
$(\neg\psi)_{p}^{\,f}:=\neg(\psi)_{p}^{\,f}$,
$(\varphi_{1}\vee\varphi_{2})_{s}^{\,f}:=(\varphi_{1})_{s}^{\,f}\vee(\varphi_{2})_{s}^{\,f}$,
$(\psi_{1}\vee\psi_{2})_{p}^{\,f}:=(\psi_{1})_{p}^{\,f}\vee(\psi_{2})_{p}^{\,f}$,
$({\bf X}\psi)_{p}^{\,f}:={\bf X}(\psi)_{p}^{\,f}$ and $(\psi_{1}{\bf
U}\psi_{2})_{p}^{\,f}:=(\psi_{1})_{p}^{\,f}{\bf U}(\psi_{2})_{p}^{\,f}$.
We continue with the case of the strategy quantifier:
$\begin{array}[]{lrl}&(\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi)_{s}^{\,f}&:=\exists^{\widetilde{o}}p_{c_{1}}^{x}\ldots\exists^{\widetilde{o}}p_{c_{l}}^{x}.\varphi_{\text{str}}(x)\wedge(\varphi)_{s}^{\,f}\\\\[5.0pt]
\mbox{where}&\varphi_{\text{str}}(x)&:={\bf A}{\bf
G}\bigvee_{c\in\textnormal{Ac}}p_{c}^{x}\\\\[5.0pt]
\mbox{and}&\widetilde{o_{i}}&:=\\{j\mid\mathcal{O}(o_{i})\subseteq\mathcal{O}(o_{j})\\}.\end{array}$
The intuition is that for each possible action $c\in\textnormal{Ac}$, an
existential quantification on the atomic proposition $p_{c}^{x}$ “chooses” for
each node $u_{\rho}$ of the tree $t_{\mathcal{S}_{\mathcal{G}}}$ whether
strategy $x$ allows action $c$ in $\rho$ or not, and it does so uniformly with
regards to observation $\widetilde{o}$. $\varphi_{\text{str}}(x)$ checks that
at least one action is allowed in each node, and thus that atomic propositions
$p_{c}^{x}$ indeed define a strategy.
We define $\widetilde{o_{i}}$ as
$\\{j\mid\mathcal{O}(o_{i})\subseteq\mathcal{O}(o_{j})\\}$ instead of
$\\{i\\}$ in order to obtain a hierarchical instance. Note that including all
coarser observations does not increase the information accessible to the
quantifier: indeed, two nodes are $\\{i\\}$-indistinguishable if and only if
they are $\widetilde{o_{i}}$-indistinguishable.
Here are the remaining cases:
$\begin{array}[]{lrl}&((a,x)\varphi)_{s}^{\,f}&:=(\varphi)_{s}^{\,f[a\mapsto
x]}\quad\quad\text{for
}x\in\textnormal{Var}\cup\\{\operatorname{?}\\}\\\\[5.0pt] \mbox{and}&({\bf
E}\psi)_{s}^{\,f}&:={\bf
E}\,(\psi_{\text{out}}^{\,f}\wedge(\psi)_{p}^{\,f})\\\\[5.0pt]
\mbox{where}&\psi_{\text{out}}^{\,f}&:={\bf G}\bigvee_{v\in
V}\left(p_{v}\wedge\bigvee_{\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}}\bigwedge_{a\in\textit{dom}(f)}p_{\bm{c}_{a}}^{f(a)}\wedge{\bf
X}\,p_{E(v,\bm{c})}\right).\end{array}$
$\psi_{\text{out}}^{\,f}$ checks that each player $a$ in the domain of $f$
follows the strategy coded by the $p_{c}^{f(a)}$.
###### Remark 4.
If we consider the fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ that only allows for deterministic strategies, the translation can be
adapted by simply replacing formula $\varphi_{\text{str}}(x)$ above with its
deterministic variant
$\varphi_{\text{str}}^{\text{det}}(x):={\bf A}{\bf
G}\bigvee_{c\in\textnormal{Ac}}(p_{c}^{x}\wedge\bigwedge_{c^{\prime}\neq
c}\neg p_{c^{\prime}}^{x}),$
which ensures that _exactly one_ action is chosen for strategy $x$ in each
finite play, and thus that atomic propositions $p_{c}^{x}$ characterise a
deterministic strategy.
To prove correctness of the translation, given a strategy $\sigma$ and a
strategy variable $x$ we let $\ell_{\sigma}^{x}:=\\{\ell_{p_{c}^{x}}\mid
c\in\textnormal{Ac}\\}$ be the family of $p_{c}^{x}$-labellings for tree
$t_{\mathcal{S}_{\mathcal{G}}}$ defined as follows: for each finite play
$\rho$ in $\mathcal{G}$ and $c\in\textnormal{Ac}$, we let
$\ell_{p_{c}^{x}}(u_{\rho}):=1$ if $c\in\sigma(\rho)$, 0 otherwise. For a
labelled tree $t$ with same domain as $t_{\mathcal{S}_{\mathcal{G}}}$ we write
$t\otimes\ell_{\sigma}^{x}$ for
$t\otimes\ell_{p_{c_{1}}^{x}}\otimes\ldots\otimes\ell_{p_{c_{l}}^{x}}$.
Given an infinite play $\pi$ and a point $i\in\mathbb{N}$, we also let
$\lambda_{\pi,i}$ be the infinite path in $t_{\mathcal{S}_{\mathcal{G}}}$ that
starts in node $u_{\pi_{\leq i}}$ and is defined as
$\lambda_{\pi,i}:=u_{\pi_{\leq i}}u_{\pi_{\leq i+1}}u_{\pi_{\leq i+2}}\ldots$
Finally, for an assignment $\chi$ and a partial function
$f:\textnormal{Ag}\rightharpoonup\textnormal{Var}$, we say that $f$ is
_compatible_ with $\chi$ if
$\textit{dom}(\chi)\cap\textnormal{Ag}=\textit{dom}(f)$ and for all
$a\in\textit{dom}(f)$, $\chi(a)=\chi(f(a))$.
###### Proposition 5.1.
For every state subformula $\varphi$ and path subformula $\psi$ of $\Phi$,
finite play $\rho$, infinite play $\pi$, point $i\in\mathbb{N}$, for every
assignment $\chi$ variable-complete for $\varphi$ (resp. $\psi$) and partial
function $f:\textnormal{Ag}\rightharpoonup\textnormal{Var}$ compatible with
$\chi$, assuming also that no $x_{i}$ in
$\textit{dom}(\chi)\cap\textnormal{Var}=\\{x_{1},\ldots,x_{k}\\}$ is
quantified in $\varphi$ or $\psi$, we have
$\displaystyle\mathcal{G},\chi,{\rho}\models\varphi$ if and only if
$\displaystyle
t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models(\varphi)_{s}^{\,f}$
$\displaystyle\mathcal{G},\chi,{\pi},i\models\psi$ if and only if
$\displaystyle
t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},\lambda_{\pi,i}\models(\psi)_{p}^{\,f}$
In addition, $\mathcal{S}_{\mathcal{G}}$ is of size linear in $|\mathcal{G}|$,
and $(\varphi)_{s}^{\,f}$ and $(\psi)_{p}^{\,f}$ are of size linear in
$|\mathcal{G}|^{2}+|\varphi|$.
###### Proof.
The proof is by induction on $\varphi$. We detail the cases for binding,
strategy quantification and outcome quantification, the others follow simply
by definition of $\mathcal{S}_{\mathcal{G}}$ for atomic propositions and
induction hypothesis for remaining cases.
For $\varphi=(a,x)\varphi^{\prime}$, we have
$\mathcal{G},\chi,{\rho}\models(a,x)\varphi^{\prime}$ if and only if
$\mathcal{G},\chi[a\mapsto\chi(x)],{\rho}\models\varphi^{\prime}$. The result
follows by using the induction hypothesis with assignment $\chi[a\mapsto x]$
and function $f[a\mapsto x]$. This is possible because $f[a\mapsto x]$ is
compatible with $\chi[a\mapsto x]$: indeed $\textit{dom}(\chi[a\mapsto
x])\cap\textnormal{Ag}$ is equal to
$\textit{dom}(\chi)\cap\textnormal{Ag}\cup\\{a\\}$ which, by assumption, is
equal to $\textit{dom}(f)\cup\\{a\\}=\textit{dom}(f[a\mapsto x])$. Also by
assumption, for all $a^{\prime}\in\textit{dom}(f)$,
$\chi(a^{\prime})=\chi(f(a^{\prime}))$, and by definition
$\chi[a\mapsto\chi(x)](a)=\chi(x)=\chi(f[a\mapsto x](a))$.
For $\varphi=\langle\\!\langle x\rangle\\!\rangle^{o}\varphi^{\prime}$, assume
first that $\mathcal{G},\chi,{\rho}\models\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi^{\prime}$. There exists an $o$-uniform strategy
$\sigma$ such that
$\mathcal{G},\chi[x\mapsto\sigma],\rho\models\varphi^{\prime}.$
Since $f$ is compatible with $\chi$, it is also compatible with assignment
$\chi^{\prime}=\chi[x\mapsto\sigma]$. By assumption, no variable in
$\\{x_{1},\ldots,x_{k}\\}$ is quantified in $\varphi$, so that $x\neq x_{i}$
for all $i$, and thus $\chi^{\prime}(x_{i})=\chi(x_{i})$ for all $i$; and
because no strategy variable is quantified twice in a same formula, $x$ is not
quantified in $\varphi^{\prime}$, so that no variable in
$\\{x_{1},\ldots,x_{k},x\\}$ is quantified in $\varphi^{\prime}$. By induction
hypothesis
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi^{\prime}(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi^{\prime}(x_{k})}^{x_{k}}\otimes\ell_{\chi^{\prime}(x)}^{x},u_{\rho}\models(\varphi^{\prime})_{s}^{\,f}.$
Because $\sigma$ is $o$-uniform, each
$\ell_{p_{c}^{x}}\in\ell_{\sigma}^{x}=\ell_{\chi^{\prime}(x)}^{x}$ is
$\widetilde{o}$-uniform, and it follows that
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi^{\prime}(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi^{\prime}(x_{k})}^{x_{k}},u_{\rho}\models\exists^{\widetilde{o}}p_{c_{1}}^{x}\ldots\exists^{\widetilde{o}}p_{c_{l}}^{x}.\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}.$
Finally, since $\chi^{\prime}(x_{i})=\chi(x_{i})$ for all $i$, we conclude
that
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models(\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi^{\prime})_{s}^{\,f}.$
For the other direction, assume that
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models(\varphi)_{s}^{\,f},$
and recall that
$(\varphi)_{s}^{\,f}=\exists^{\widetilde{o}}p_{c_{1}}^{x}\ldots\exists^{\widetilde{o}}p_{c_{l}}^{x}.\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}$.
Write
$t=t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}}$.
There exist $\widetilde{o}$-uniform $\ell_{p_{c}^{x}}$-labellings such that
$t\otimes\ell_{p_{c_{1}}^{x}}\otimes\ldots\otimes\ell_{p_{c_{l}}^{x}}\models\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}.$
By $\varphi_{\text{str}}(x)$, these labellings code for a strategy $\sigma$,
and because they are $\widetilde{o}$-uniform, $\sigma$ is $o$-uniform. Let
$\chi^{\prime}=\chi[x\mapsto\sigma]$. For all $1\leq i\leq k$, by assumption
$x\neq x_{i}$, and thus $\chi^{\prime}(x_{i})=\chi(x_{i})$. The above can thus
be rewritten
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi^{\prime}(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi^{\prime}(x_{k})}^{x_{k}}\otimes\ell_{\chi^{\prime}(x)}^{x}\models\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}.$
By induction hypothesis we have
$\mathcal{G},\chi[x\mapsto\sigma],\rho\models\varphi^{\prime}$, hence
$\mathcal{G},\chi,\rho\models\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi^{\prime}$.
For $\varphi={\bf E}\psi$, assume first that
$\mathcal{G},\chi,{\rho}\models{\bf E}\psi$. There exists a play
$\pi\in\textnormal{Out}(\chi,\rho)$ such that
$\mathcal{G},\chi,\pi,|\rho|-1\models\psi$. By induction hypothesis,
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},\lambda_{\pi,|\rho|-1}\models(\psi)_{p}^{\,f}$.
Since $\pi$ is an outcome of $\chi$, each agent
$a\in\textit{dom}(\chi)\cap\textnormal{Ag}$ follows strategy $\chi(a)$ in
$\pi$. Because $\textit{dom}(\chi)\cap\textnormal{Ag}=\textit{dom}(f)$ and for
all $a\in\textit{dom}(f)$, $\chi(a)=\chi(f(a))$, each agent
$a\in\textit{dom}(f)$ follows the strategy $\chi(f(a))$, which is coded by
atoms $p_{c}^{f(a)}$ in the translation of $\Phi$. Therefore
$\lambda_{\pi,|\rho|-1}$ also satisfies $\psi_{\text{out}}^{\,\chi}$, hence
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},\lambda_{\pi,|\rho|-1}\models\psi_{\text{out}}^{\,\chi}\wedge(\psi)_{p}^{\,f}$,
and we are done.
For the other direction, assume that
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models{\bf
E}(\psi_{\text{out}}^{\,f}\wedge(\psi)_{p}^{\,f})$. There exists a path
$\lambda$ in
$t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}}$
starting in node $u_{\rho}$ that satisfies both $\psi_{\text{out}}^{\,f}$ and
$(\psi)_{p}^{\,f}$. By construction of $\mathcal{S}_{\mathcal{G}}$ there
exists an infinite play $\pi$ such that $\pi_{\leq|\rho|-1}=\rho$ and
$\lambda=\lambda_{\pi,|\rho|-1}$. By induction hypothesis,
$\mathcal{G},\chi,\pi,|\rho|-1\models\psi$. Because $\lambda_{\pi,|\rho|-1}$
satisfies $\psi_{\text{out}}^{\,f}$,
$\textit{dom}(\chi)\cap\textnormal{Ag}=\textit{dom}(f)$, and for all
$a\in\textit{dom}(f)$, $\chi(a)=\chi(f(a))$, it is also the case that
$\pi\in\textnormal{Out}(\chi,\rho)$, hence $\mathcal{G},\chi,\rho\models{\bf
E}\psi$.
The size of $\mathcal{S}_{\mathcal{G}}$, $(\varphi)_{s}^{\,f}$ and
$(\psi)_{p}^{\,f}$ are easily verified. ∎
Applying Proposition 5.1 to the sentence $\Phi$, $\rho=v_{\iota}$, any
assignment $\chi$, and the empty function $\emptyset$, we get:
$\mathcal{G}\models\Phi\quad\mbox{if and only if}\quad
t_{\mathcal{S}_{\mathcal{G}}}\models(\Phi)_{s}^{\,\emptyset}.$
Preserving hierarchy. To complete the proof of Theorem 2.9 it remains to check
that $(\Phi)_{s}^{\,\emptyset}$ is a hierarchical
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula, which is the
case because $\Phi$ is hierarchical in $\mathcal{G}$ and for every two
observations $o_{i}$ and $o_{j}$ in Obs such that
$\mathcal{O}(o_{i})\subseteq\mathcal{O}(o_{j})$, by definition of
$\widetilde{o_{k}}$ we have that
$\widetilde{o_{i}}\subseteq\widetilde{o_{j}}$.
### 5.2. Complexity
We now establish the complexity of model checking hierarchical instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. As we did for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, we first define the
simulation depth of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ state
formulas. In the following inductive definition, $\mathcal{O}_{\varphi}$
denotes the intersection of all indistinguishability relations used in
$\varphi$: $\mathcal{O}_{\varphi}:=\cap_{o\in\varphi}\mathcal{O}(o)$, with the
empty intersection being defined as the identity relation (perfect
information). Also, for a path formula $\psi$, $\max(\psi)$ is the set of
maximal state subformulas in $\psi$.
$\begin{array}[]{lcc}\mbox{sd}(p):=(0,\mbox{nd})&&\mbox{sd}(\neg\varphi):=(\mbox{sd}_{k}(\varphi),\mbox{alt})\\\\[7.0pt]
\lx<EMAIL_ADDRESS>\lx@intercol\hfil\mbox{where }x=\begin{cases}\mbox{nd}&\mbox{if
}\mbox{sd}_{x}(\varphi_{1})=\mbox{sd}_{x}(\varphi_{2})=\mbox{nd}\\\
<EMAIL_ADDRESS>\lx@intercol\mbox{sd}(\langle\\!\langle
<EMAIL_ADDRESS>\lx@intercol\hfil\mbox{where }k=\begin{cases}\mbox{sd}_{k}(\varphi)&\mbox{if
}\mbox{sd}_{x}(\varphi)=\mbox{nd}\mbox{ and
}\mathcal{O}(o)=\mathcal{O}_{\varphi}\\\
<EMAIL_ADDRESS>\mbox{sd}((a,x)\varphi):=\mbox{sd}(\varphi)\\\\[7.0pt]
\lx@intercol\mbox{sd}({\bf E}\psi):=\begin{cases}(0,\mbox{nd})&\mbox{if
}\psi\in\textnormal{LTL}\\\
(\max_{\varphi\in\max(\psi)}\mbox{sd}_{k}(\varphi),\mbox{alt})&\mbox{otherwise}\end{cases}\hfil\lx@intercol\end{array}$
###### Proposition 5.2.
The model-checking problem for hierarchical instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ of simulation depth at most
$k$ is $(k+1)$-Exptime -complete.
###### Proof.
The upper bounds follow from the fact that the translated formulas in our
reduction have essentially the same simulation depth as the original ones.
However this is not quite right, because in the case where
$\mbox{sd}_{x}(\varphi)=\mbox{nd}$ and $\mathcal{O}(o)=\mathcal{O}_{\varphi}$
we have $\mbox{sd}(\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi)=(\mbox{sd}_{k}(\varphi),\mbox{nd})$, while
$\mbox{sd}((\langle\\!\langle
x\rangle\\!\rangle^{o}\varphi)_{s}^{\,f})=(\mbox{sd}_{k}((\varphi)_{s}^{\,f})+1,\mbox{nd})$:
indeed, while it is the case that $\mathcal{O}(o)=\mathcal{O}_{\varphi}$
implies that $\widetilde{o}=I_{(\varphi)_{s}^{\,f}}$, the translation
introduces a conjunction with $\varphi_{\text{str}}(x)$, and even when
$\mbox{sd}_{x}((\varphi)_{s}^{\,f})=\mbox{nd}$, we have
$\mbox{sd}_{x}(\varphi_{\text{str}}(x)\wedge(\varphi)_{s}^{\,f})=\mbox{alt}$.
According to Proposition 4.13, this should thus induce an additional
exponential to check the translated formula. However, this can be avoided by
noticing that the fixed formula $\varphi_{\text{str}}(x)={\bf A}{\bf
G}\bigvee_{c\in\textnormal{Ac}}p_{c}^{x}$ can be checked by a simple
_deterministic_ tree automaton with two states $q_{\text{check}}$ and
$q_{\text{rej}}$: the automaton starts in state $q_{\text{check}}$, which is
accepting (it has parity zero); when it visits a node $u$ in state
$q_{\text{check}}$, if $\ell(u)$ satisfies
$\bigvee_{c\in\textnormal{Ac}}p_{c}^{x}$, then the automaton sends state
$q_{\text{check}}$ to all children of $u$, otherwise it sends the state
$q_{\text{rej}}$ to all children. State $q_{\text{rej}}$ is rejecting (it has
parity one) and is a sink: it sends itself to all children, independently on
the label of the visited node. If we restrict
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to deterministic strategies,
the same observation can be made: the automaton that checks formula
$\varphi_{\text{str}}^{\text{det}}(x)={\bf A}{\bf
G}\bigvee_{c\in\textnormal{Ac}}(p_{c}^{x}\wedge\bigwedge_{c^{\prime}\neq
c}\neg p_{c^{\prime}}^{x})$ is the same as the one described above, except
that it checks whether
$\bigvee_{c\in\textnormal{Ac}}(p_{c}^{x}\wedge\bigwedge_{c^{\prime}\neq c}\neg
p_{c^{\prime}}^{x})$ is satisfied by the label of the current node.
Given two tree automata $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, one
deterministic and one nondeterministic, one can easily build a
nondeterministic automaton $\mathcal{A}_{1}\cap\mathcal{A}_{2}$ of size
$|\mathcal{A}_{1}|\times|\mathcal{A}_{2}|$ that accepts the intersection of
their languages, so that in this case the conjunction does not introduce
alternation, and thus we do not need an additional simulation before
projecting to guess the strategy. We could refine the notion of simulation
depth to reflect this, but we find that it would become very cumbersome for
little added benefit, so we keep this observation in this proof.
The lower bounds are inherited from
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ thanks to the
polynomial reduction presented in Section 6.2.2, which preserves simulation
depth. ∎
We point out that all instances of the model-checking problem for the perfect-
information fragment are hierarchical, and thus this result provides improved
upper-bounds for SL, which was only known to be in $k$-Exptime for formulas of
length at most $k$ (Mogavero et al., 2014). Also the lower bounds for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ are inherited directly
from the perfect-information fragment $\textnormal{{QCTL}}^{*}$, which reduces
to the perfect-information fragment of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ following the construction
from Section 6.2.2. Therefore the lower bounds hold already for the perfect-
information fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$.
Note however that this does not provide lower bounds for the usual, linear-
time variant of Strategy Logic, where path quantifiers in
$\textnormal{{QCTL}}^{*}$ formulas must be simulated with strategy
quantifications which increase the simulation depth of the resulting Strategy
Logic formulas. The exact complexity of the linear-time variant is not known,
even in the perfect-information case.
Simulation number. The intuition behind the alternation number as considered
in (Mogavero et al., 2014) is to refine the classic alternation depth between
existential and universal quantifiers by observing that subsentences of a
sentence $\Phi$ to model-check can be treated independently thanks to a
bottom-up labelling algorithm: innermost sentences are evaluated in all states
of the model and replaced in $\Phi$ by atomic propositions that label the
states where they hold. The alternation number of $\Phi$ is the maximum
alternation depth of the successive subsentences that are treated by this
bottom-up procedure, and it determines the complexity of the overall model-
checking procedure.
However, as discussed in Remark 1, the semantics of the outcome quantifier
makes sentences sensitive to the assignment in which they are evaluated. As a
result, to define the notion of alternation number in our setting, we
introduce a notion of _independent subsentence_. Intuitively, a subsentence
$\varphi$ of a sentence $\Phi$ is _independent_ if it redefines or unbinds the
strategies of all players who are bound to a strategy when $\varphi$ is
reached in the evaluation of $\Phi$. More precisely, we say that an agent $a$
is _bound_ in a syntactic subformula $\varphi$ of $\Phi$ if the path that
leads to $\varphi$ in $\Phi$’s syntactic tree contains a binding operator
$(a,x)$ for $a$ which is not followed by an unbinding $(a,\operatorname{?})$
for her. A subsentence $\varphi$ of $\Phi$ is _independent_ if all agents that
are bound in $\varphi$ are either rebound by an operator $(a,x)$ or unbound by
an operator $(a,\operatorname{?})$ before any outcome quantifier is met in
$\varphi$. In an independent subsentence $\varphi$, the semantics of the
outcome quantifier does not depend on strategies that are quantified outside
$\varphi$, and in fact a subsentence $\varphi$ of $\Phi$ is independent if and
only if the formula that corresponds to $\varphi$ in
$(\Phi)_{s}^{\,\emptyset}$ is a subsentence of $(\Phi)_{s}^{\,\emptyset}$.
Similarly to what we did for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
ii}}$ we now define the _simulation number_ $\mbox{sn}(\Phi)$ of an
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ sentence $\Phi$ as the
maximum of the simulation depths for independent subsentences, where strict
independent subsentences are counted as atoms.
###### Lemma 5.3.
For every hierarchical instance $(\mathcal{G},\Phi)$ of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$,
$\mbox{sn}(\Phi)=\mbox{sn}((\Phi)_{s}^{\,\emptyset})$.
The following then follows from Proposition 5.1, Lemma 5.3 and Proposition
4.14.
###### Proposition 5.4.
The model-checking problem for hierarchical instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ of simulation number at most
$k$ is $(k+1)$-Exptime -complete.
We now compare the latter result with the complexity of model checking SL[NG],
the nested goal fragment of Strategy Logic with perfect information (we refer
the interested reader to (Mogavero et al., 2014) for a definition of this
fragment). It is established in (Chatterjee et al., 2010b; Mogavero et al.,
2014) that this problem is in $(k+1)$-Exptime for formulas of _alternation
number_ $k$. We remark that the simulation number of an SL[NG] formula
translated in our branching-time version of SL (this is done by adding outcome
quantifiers between bindings and temporal operators) is equal to its
alternation number plus one, and thus Proposition 5.4 gives a $(k+2)$-Exptime
upper bound for SL[NG] formulas of alternation number $k$. In (Chatterjee et
al., 2010b; Mogavero et al., 2014) the extra exponential is avoided by
resorting to universal and nondeterministic tree automata, depending on
whether the innermost strategy quantification is existential or universal, to
deal with temporal formulas. Thus, the innermost strategy quantification can
be dealt with without incurring an exponential blowup.
The same thing cannot be done for $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$, for two reasons. The first one is that in general the innermost
strategy quantification may have imperfect information and thus require a
narrowing of the automaton; this operation introduces alternation, which needs
to be removed at the cost of one exponential before dealing with strategy
quantification. The second reason is that even when the innermost strategy has
perfect information, the outcome quantifier that we introduce in Strategy
Logic allows the expression of $\textnormal{{CTL}}^{*}$ formulas which cannot
be dealt with by nondeterministic and universal automata as is done in
(Chatterjee et al., 2010b; Mogavero et al., 2014).
## 6\. Comparison with related logics
In this section we first show that $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ subsumes SL and the main imperfect-information extensions of ATL. Then
we show that model checking Coordination Logic (CL) reduces to model checking
hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
where the truth of all atomic propositions in the model is known by all agents
(or more precisely, all observations in the concurrent game structures are
fine enough to observe the truth value of all atomic propositions).
### 6.1. Comparison with ATL
The main difference between SL and ATL-like strategic logics is that in the
latter a strategy is always bound to some player, while in the former bindings
and quantifications are separated. This separation adds expressive power,
e.g., one can bind the same strategy to different players. Extending ATL with
imperfect-information is done by giving each player an indistinguishability
relation that its strategies must respect (Bulling and Jamroga, 2014). In
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ instead each strategy $x$ is
assigned an indistinguishability relation $o$ when it is quantified.
Associating observations to strategies rather than players allows us to obtain
a logic $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ that is a clean
generalisation of (perfect-information) SL, and subsumes imperfect-information
extensions of $\textnormal{{ATL}}^{*}$ that associate observations to players.
Concerning SL, it is rather easy to see that every sentence in SL has an
equivalent in the fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$ with deterministic strategies where all observation symbols are
interpreted as perfect information. We now prove that
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ also subsumes
$\textnormal{{ATL}}^{*}$ with imperfect information.
###### Proposition 6.1.
For every $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$
formula555See (Bulling and Jamroga, 2014) for the definition of
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$, where subscript i
refers to “imperfect information” and subscript R to “perfect recall”. Also,
we consider the so-called _objective semantics_ for
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$. $\varphi$ there is an
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula $\varphi^{\prime}$
such that for every $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ there is a
$\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}^{\prime}$ such that
$\mathcal{G}\models\varphi$ if, and only if,
$\mathcal{G}^{\prime}\models\varphi^{\prime}$.
We recall that an $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$
formula $\langle A\rangle\psi$ reads as “there are strategies for players in
$A$ such that $\psi$ holds whatever players in $\textnormal{Ag}\setminus A$
do”. Formula $\varphi^{\prime}$ is built from $\varphi$ by replacing each
subformula of the form $\langle A\rangle\psi$, where
$A=\\{a_{1},\ldots,a_{k}\\}\subset\textnormal{Ag}$ is a coalition of players
and $\textnormal{Ag}\setminus A=\\{a_{k+1},\ldots,a_{n}\\}$ with formula
$\langle\\!\langle x_{1}\rangle\\!\rangle^{o_{1}}\ldots\langle\\!\langle
x_{k}\rangle\\!\rangle^{o_{k}}(a_{1},x_{1})\ldots(a_{k},x_{k})(a_{k+1},\operatorname{?})\ldots(a_{n},\operatorname{?}){\bf
A}\,\psi^{\prime}$, where $\psi^{\prime}$ is the translation of $\psi$. Then
$\mathcal{G}^{\prime}$ is obtained from $\mathcal{G}$ by interpreting each
$o_{i}$ as the equivalence relation for player $i$ in $\mathcal{G}$, and
interpreting $o_{p}$ as the identity relation.
Third, $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ also subsumes the
imperfect-information extension of $\textnormal{{ATL}}^{*}$ with strategy
context (see (Laroussinie et al., 2015) for the definition of
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ with partial
observation, which we refer to as
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$).
###### Proposition 6.2.
For every $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$ formula
$\varphi$ there is an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
formula $\varphi^{\prime}$ such that for every
$\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ there is a
$\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}^{\prime}$ such that
$\mathcal{G}\models\varphi$ if, and only if,
$\mathcal{G}^{\prime}\models\varphi^{\prime}$.
The only difference between $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize
sc,i}}$ and $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$ is the
following: in $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$, when a
subformula of the form $\langle A\rangle\psi$ is met, we quantify
existentially on strategies for players in $A$ and quantify universally on
possible outcomes obtained by letting other players behave however they want.
Therefore, if any player in $\textnormal{Ag}\setminus A$ had previously been
assigned a strategy, it is forgotten. In
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$ on the other hand,
these strategies are stored in a _strategy context_ , which is a _partial_
assignment $\chi$, defined for the subset of players currently bound to a
strategy. A strategy context allows one to quantify universally only on
strategies of players who are not in $A$ and who are not already bound to a
strategy. It is then easy to adapt the translation presented for Proposition
6.1: it suffices not to unbind agents outside the coalition from their
strategies. $\mathcal{G}^{\prime}$ is defined as for Proposition 6.1.
### 6.2. Comparison with Coordination Logic
There is a natural and simple translation of instances of the model-checking
problem of CL (Finkbeiner and Schewe, 2010) into the hierarchical instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. Moreover, the image of this
translation consists of instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with a very restricted form:
atoms mentioned in the $\textnormal{{SL}}_{\textnormal{\scriptsize
ii}}$-formula are observable by all observations of the
$\textrm{CGS}_{\textnormal{ii}}$ , i.e., for all $o\in\textnormal{Obs}$ and
$p\in\textnormal{AP}$, $v\sim_{o}v^{\prime}$ implies that $p\in\ell(v)$ iff
$p\in\ell(v^{\prime})$.
###### Proposition 6.3.
There is an effective translation that, given a CL-instance
$(\mathcal{S},\varphi)$ produces a hierarchical
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance
$(\mathcal{G},\Phi)$ such that
1. (1)
$\mathcal{S}\models\varphi$ if, and only if, $\mathcal{G}\models\Phi$,
2. (2)
For all atoms $p\in\textnormal{AP}$ and observations $o\in\textnormal{Obs}$,
$v\sim_{o}v^{\prime}$ implies that $p\in\ell(v)$ iff $p\in\ell(v^{\prime})$.
To do this, one first translates CL into (hierarchical)
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, the latter is defined
in the next section. This step is a simple reflection of the semantics of CL
in that of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. Then one
translates $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ into
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ by a simple adaptation of
the translation of $\textnormal{{QCTL}}^{*}$ into
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ (Laroussinie and
Markey, 2015).
We briefly recall the syntax and semantics of CL, and refer to (Finkbeiner and
Schewe, 2010) for further details.
Notation for trees. Note that our definition for trees (see Section 3.2)
differs slightly from the one in (Finkbeiner and Schewe, 2010), where the root
is the empty word. Here we adopt this convention to stay closer to notations
in (Finkbeiner and Schewe, 2010). Thus, $(Y,X)$-trees in CL are of the form
$(\tau,l)$ where $\tau\subseteq X^{*}$ and $l:\tau\to 2^{Y}$.
For two disjoint sets $X$ and $Y$, we identify $2^{X}\times 2^{Y}$ with
$2^{X\cup Y}$. Let $X$ and $Y$ be two sets with $Z=X\cup Y$, and let $M$ and
$N$ be two disjoint sets. Given an ${M}$-labelled $2^{Z}$-tree
$t=(\tau,\ell_{M})$ and an ${N}$-labelled $2^{Z}$-tree
$t^{\prime}=(\tau^{\prime},\ell_{N})$ with same domain $\tau=\tau^{\prime}$,
we define $t\uplus t^{\prime}:=(\tau,\ell^{\prime})$, where for every
$u\in\tau$, $\ell^{\prime}(u)=\ell_{M}(u)\cup\ell_{N}(u)$. Now, given a
complete ${M}$-labelled $2^{X}$-tree $t=((2^{X})^{*},\ell_{M})$ and a complete
${N}$-labelled $2^{Y}$-tree $t^{\prime}=((2^{Y})^{*},\ell_{N})$, we define
$t\oplus t^{\prime}:=t\\!\uparrow^{2^{Z\setminus
X}}\uplus\,t^{\prime}\\!\uparrow^{2^{Z\setminus Y}}$.
CL Syntax. Let $\mathcal{C}$ be a set of _coordination variables_ , and let
$\mathcal{S}$ be a set of _strategy variables_ disjoint from $\mathcal{C}$.
The syntax of CL is given by the following grammar:
$\varphi::=x\mid\neg\varphi\mid\varphi\vee\varphi\mid{\bf
X}\varphi\mid\varphi{\bf U}\varphi\mid\Finv C\exists s.\,\varphi$
where $x\in\mathcal{C}\cup\mathcal{S}$, $C\subseteq\mathcal{C}$ and
$s\in\mathcal{S}$, and with the restriction that each coordination variable
appears in at most one _subtree quantifier_ $\Finv C\exists s.\,$, and
similarly for strategy variables.
The notion of free and bound (coordination or strategy) variables is as usual.
The set of free coordination variables in $\varphi$ is noted
$\mathcal{F}_{\varphi}$. A bound coordination variable $c$ is _visible_ to a
strategy variable $s$ if $s$ is in the scope of the quantifier that introduces
$c$, and $\textit{Scope}_{\varphi}(s)$ is the union of the set of bound
coordination variables visible to $s$ and the set of free coordination
variables (note that this union is disjoint). We will see, in the semantics,
that the meaning of a bound strategy variable $s$ is a strategy
$f_{s}:(2^{\textit{Scope}_{\varphi}(s)})^{*}\to 2^{\\{s\\}}$. Free strategy
variables are called _atomic propositions_ , and we denote the set of atomic
propositions in $\varphi$ by $\textnormal{AP}_{\varphi}$.
CL Semantics. A CL formula $\varphi$ is evaluated on a complete
$\textnormal{AP}_{\varphi}$-labelled $2^{\mathcal{F}_{\varphi}}$-tree $t$. An
$(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree $t=(\tau,\ell)$
satisfies a CL formula $\varphi$ if for every path $\lambda$ that starts in
the root we have $t,\lambda,0\models\varphi$, where the satisfaction of a
formula at position $i\geq 0$ of a path $\lambda$ is defined inductively as
follows:
$\displaystyle t,\lambda,i\models$ $\displaystyle\,p$ if $\displaystyle\quad
p\in\ell(\lambda_{i})$ $\displaystyle t,\lambda,i\models$
$\displaystyle\,\neg\varphi^{\prime}$ if $\displaystyle\quad
t,\lambda,i\not\models\varphi^{\prime}$ $\displaystyle t,\lambda,i\models$
$\displaystyle\,\varphi_{1}\vee\varphi_{2}$ if $\displaystyle\quad
t,\lambda,i\models\varphi_{1}\mbox{ or }t,\lambda,i\models\varphi_{2}$
$\displaystyle t,\lambda,i\models$ $\displaystyle\,{\bf X}\varphi^{\prime}$ if
$\displaystyle\quad t,\lambda,i+1\models\varphi^{\prime}$ $\displaystyle
t,\lambda,i\models$ $\displaystyle\,\varphi_{1}{\bf U}\varphi_{2}$ if
$\displaystyle\quad\exists\,j\geq i\mbox{ s.t.
}t,\lambda,j\models\varphi_{2}\text{ and }\forall k\text{ s.t. }i\leq
k<j,\;t,\lambda,k\models\varphi_{1}$ $\displaystyle t,\lambda,i\models$
$\displaystyle\,\Finv C\exists s.\,\varphi^{\prime}\quad$ if
$\displaystyle\quad\exists\,f:(2^{\textit{Scope}_{\varphi}(s)})^{*}\to
2^{\\{s\\}}\mbox{ s.t.
}t_{\lambda_{i}}\oplus((2^{\textit{Scope}_{\varphi}(s)})^{*},f)\models\varphi^{\prime},$
where $t_{\lambda_{i}}$ is the subtree of $t$ rooted in $\lambda_{i}$.
First, observe that in the last inductive case, $t_{\lambda_{i}}$ being a
$2^{\mathcal{F}_{\varphi}}$-tree,
$t_{\lambda_{i}}\oplus((2^{\textit{Scope}_{\varphi}(s)})^{*},f)$ is a
$2^{\mathcal{F}_{\varphi}\cup\textit{Scope}_{\varphi}(s)}$-tree. By
definition, $\textit{Scope}_{\varphi}(s)=\mathcal{F}_{\varphi}\cup
C=\mathcal{F}_{\varphi^{\prime}}$. It follows that
$\mathcal{F}_{\varphi}\cup\textit{Scope}_{\varphi}(s)=\textit{Scope}_{\varphi}(s)=\mathcal{F}_{\varphi^{\prime}}$,
hence $\varphi^{\prime}$ is indeed evaluated on a
$\mathcal{F}_{\varphi^{\prime}}$-tree.
###### Remark 5.
Note that all strategies observe the value of all atomic propositions.
Formally, for every CL-formula $\varphi$ of the form $\varphi=\Finv
C_{1}\exists s_{1}.\,\ldots,\Finv C_{i}\exists s_{i}.\,\varphi^{\prime}$
evaluated on a $2^{\mathcal{F}_{\varphi}}$-tree $t=(\tau,\ell)$,
$\varphi^{\prime}$ is evaluated on a $2^{\mathcal{F}_{\varphi}\cup
C_{1}\cup\ldots\cup C_{i}}$-tree $t^{\prime}=(\tau^{\prime},\ell^{\prime})$
such that for every $p\in\textnormal{AP}_{\varphi}$, for every pair of nodes
$u,u^{\prime}\in t^{\prime}$ such that
$u\\!\downarrow_{2^{\mathcal{F}_{\varphi}}}=u^{\prime}\\!\downarrow_{2^{\mathcal{F}_{\varphi}}}$,
it holds that $p\in\ell^{\prime}(u)$ iff $p\in\ell^{\prime}(u^{\prime})$.
Thus, in CL one cannot directly capture strategic problems where atomic
propositions are not observable to all players.
The input to the model-checking problem for CL consists of a CL formula
$\varphi$ and a finite representation of a
$(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree $t$. The standard
assumption is to assume $t$ is a regular tree, i.e., is the unfolding of a
finite structure. Precisely, a _finite representation_ of a
$(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree
$t=(\tau,\ell^{\prime})$ is a structure $\mathcal{S}=(S,R,\ell,s_{\iota})$
such that
* •
$S=2^{\mathcal{F}_{\varphi}}$,
* •
$R=S\times S$,
* •
$\ell:S\to 2^{\textnormal{AP}_{\varphi}}$,
* •
$s_{\iota}\in S$,
and $t=t_{\mathcal{S}}$ is the unfolding of $\mathcal{S}$.
Thus, an _instance_ of the model-checking problem for CL is a pair
$(\mathcal{S},\Phi)$ where $\mathcal{S}=(S,R,s_{{\iota}},\ell)$ is a finite
representation of an
$(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree and $\Phi$ is a
CL formula (over variables $\mathcal{S}\cup\mathcal{C}$). The _model-checking
problem for CL_ is the following decision problem: given an instance
$(\mathcal{S},\Phi)$, return ‘Yes’ if $t_{\mathcal{S}}\models\Phi$ and ‘No’
otherwise.
We now describe a natural translation of CL-instances to
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances. This translation:
1. (1)
reduces the model-checking problem of CL to that of the hierarchical fragment
of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$.
2. (2)
shows that CL only produces instances in which all atoms are uniform with
regard to all observations, i.e., instances $(\mathcal{G},\Phi)$ such that for
every $p\in\textnormal{AP}$ and $o\in\textnormal{Obs}$, $v\sim_{o}v^{\prime}$
implies $p\in\ell(v)\leftrightarrow p\in\ell(v^{\prime})$.
We will present the translation in two steps: first from CL-instances into
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$-instances, and then from
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$-instances to
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances such that
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$-instances translate to hierarchical
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances.
#### 6.2.1. Translating CL to
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$
Let $(\mathcal{S},\Phi)$ be an instance of the model-checking problem for CL,
where $\mathcal{S}=(S,R,\ell,s_{{\iota}})$. We will construct a
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$-instance
$(\widetilde{\mathcal{S}},\widetilde{\varphi})$ such that
$\mathcal{S}\models\Phi$ iff $\widetilde{\mathcal{S}}\models\widetilde{\Phi}$.
Let $\widetilde{\textnormal{AP}}$ be the set of all strategy variables
occurring in $\Phi$, let $\mathcal{C}(\Phi)$ be the set of coordination
variables that appear in $\Phi$, and assume, w.l.o.g., that
$\mathcal{C}(\varphi)=[n]$ for some $n\in\mathbb{N}$. Let
$\textit{hidden}(\Phi):=\mathcal{C}(\Phi)\setminus\mathcal{F}_{\varphi}$.
First, we define the CKS $\widetilde{\mathcal{S}}$ over
$\widetilde{\textnormal{AP}}$: the idea is to add in the structure
$\mathcal{S}$ the local states corresponding to coordination variables that
are not seen by all the strategies.
Formally,
$\widetilde{\mathcal{S}}:=(\widetilde{S},\widetilde{R},\widetilde{s_{{\iota}}},\widetilde{\ell})$
where
* •
$\widetilde{S}=\prod_{c\in\mathcal{C}(\Phi)}L_{c}$ where
$L_{c}=\\{c_{0},c_{1}\\}$,
* •
$\widetilde{R}=\widetilde{S}\times\widetilde{S}$,
* •
for every $s\in\widetilde{S}$,
$\widetilde{\ell}(s)=\ell(s\\!\downarrow_{\mathcal{F}_{\varphi}})$, and
* •
$\widetilde{s_{{\iota}}}\in\widetilde{S}$ is any state $s$ such that
$s\\!\downarrow_{\mathcal{F}_{\varphi}}=s_{{\iota}}$
Second, we define concrete observations corresponding to strategy variables in
$\Phi$. As explained in (Finkbeiner and Schewe, 2010), and as reflected in the
semantics of CL, the intuition is that a strategy variable $s$ in formula
$\Phi$ observes coordination variables $\textit{Scope}_{\varphi}(s)$.
Therefore, we simply define, for each strategy variable $s$ in $\Phi$, the
concrete observation $\textnormal{{o}}_{s}:=\textit{Scope}_{\varphi}(s)$.
Finally, we define the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$
formula $\widetilde{\Phi}$. This is done by induction on $\Phi$ as follows
(recall that we take for atomic propositions in
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ the set of all
strategy variables in $\Phi$):
$\displaystyle\widetilde{x}$ $\displaystyle:=x$
$\displaystyle\widetilde{\neg\varphi}$
$\displaystyle:=\neg\widetilde{\varphi}$
$\displaystyle\widetilde{\varphi_{1}\vee\varphi_{2}}$
$\displaystyle:=\widetilde{\varphi_{1}}\vee\widetilde{\varphi_{2}}$
$\displaystyle\widetilde{{\bf X}\varphi}$ $\displaystyle:={\bf
X}\,\widetilde{\varphi}$ $\displaystyle\widetilde{\varphi_{1}{\bf
U}\varphi_{2}}$ $\displaystyle:=\widetilde{\varphi_{1}}\,{\bf
U}\,\widetilde{\varphi_{2}}$ $\displaystyle\widetilde{\Finv C\exists
s.\,\varphi}$ $\displaystyle:=\exists^{\textnormal{{o}}_{s}}s.\,{\bf
A}\widetilde{\varphi}$
In the last case, note that
$C\subseteq\textnormal{{o}}_{s}=\textit{Scope}_{\varphi}(s)$.
Note that $\widetilde{\Phi}$ is a hierarchical
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$-formula. Also, one can
easily check that the following holds:
###### Lemma 6.4.
$t_{\mathcal{S}}\models\Phi\quad\mbox{iff}\quad
t_{\widetilde{\mathcal{S}}}\models{\bf A}\widetilde{\Phi}$.
Importantly, we notice that ${\bf
A}\widetilde{\Phi}\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$, and that:
###### Lemma 6.5.
For every $x\in\textnormal{AP}_{\varphi}$ and every $s$ quantified in $\Phi$,
$t_{\widetilde{\mathcal{S}}}$ is $\textnormal{{o}}_{s}$-uniform in $x$.
#### 6.2.2. Translation from
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ to
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$
We now present a translation of
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$-instances to
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances. It is a simple
adaptation of the reduction from the model-checking problem for
$\textnormal{{QCTL}}^{*}$ to the model-checking problem for
$\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ presented in
(Laroussinie and Markey, 2015).
Let $(\mathcal{S},\Phi)$ be an instance of the model-checking problem for
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, where
$\mathcal{S}=(S,R,\ell,s_{{\iota}})$ and $S\subseteq\prod_{i\in[n]}L_{i}$. We
assume, without loss of generality, that every atomic proposition is
quantified at most once, and that if it appears quantified it does not appear
free. Also, let ${\textnormal{AP}_{\exists}}(\Phi)=\\{p_{1},\ldots,p_{k}\\}$
be the set of atomic propositions quantified in $\Phi$, and for $i\in[k]$, let
$\textnormal{{o}}_{i}$ be the concrete observation associated to the
quantifier on $p_{i}$.
We build the $\textrm{CGS}_{\textnormal{ii}}$
$\mathcal{G}^{\mathcal{S}}:=(\textnormal{Ac},V,E,\ell^{\prime},v_{\iota},\mathcal{O})$
over agents $\textnormal{Ag}:=\\{a_{0},a_{1},\ldots,a_{k}\\}$, observations
$\textnormal{Obs}:=\\{o_{0},o_{1},\ldots,o_{k}\\}$ and atomic propositions
$\textnormal{AP}:={\textnormal{AP}_{\exists}}(\Phi)\cup\\{p_{S}\\}$, where
$p_{S}$ is a fresh atomic proposition. Intuitively, agent $a_{0}$ is in charge
of choosing transitions in $\mathcal{S}$, while agent $a_{i}$ for $i\geq 1$ is
in charge of choosing the valuation for
$p_{i}\in{\textnormal{AP}_{\exists}}(\Phi)$.
To this aim, we let
$V:=\begin{array}[]{l}\\{v_{s}\mid s\in S\\}\;\cup\\\ \\{v_{s,i}\mid s\in
S\mbox{ and }i\in[k]\\}\;\cup\\\ \\{v_{p_{i}}\mid 0\leq i\leq k\\}\;\cup\\\
\\{v_{\perp}\\}\end{array}$
and
$\textnormal{Ac}:=\\{c^{s}\mid s\in S\\}\cup\\{c^{i}\mid 0\leq i\leq k\\}.$
In positions of the form $v_{s}$ with $s\in S$, transitions are determined by
the action of agent $a_{0}$. First, she can choose to simulate a transition in
$\mathcal{S}$: for every joint action
$\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$ such that
$\bm{c}_{0}=c^{s^{\prime}}$,
$E(v_{s},\bm{c}):=\begin{cases}v_{s^{\prime}}&\text{if }R(s,s^{\prime})\\\
v_{\perp}&\text{otherwise}.\end{cases}$
She can also choose to move to a position in which agent $a_{i}$ will choose
the valuation for $p_{i}$ in the current node: for every joint action
$\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$ such that $\bm{c}_{0}=c^{i}$,
$E(v_{s},\bm{c}):=\begin{cases}v_{s,i}&\text{if }i\neq 0\\\
v_{\perp}&\text{otherwise}.\end{cases}$
Next, in a position of the form $v_{s,i}$, agent $a_{i}$ determines the
transition, which codes the labelling of $p_{i}$ in the current node: choosing
$c^{i}$ means that $p_{i}$ holds in the current node, choosing any other
action codes that $p_{i}$ does not hold. Formally, for every joint action
$\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$,
$E(v_{s,i},\bm{c}):=\begin{cases}v_{p_{i}}&\text{if }\bm{c}_{i}=c^{i}\\\
v_{\perp}&\text{otherwise}.\end{cases}$
Positions of the form $v_{p_{i}}$ and $v_{\perp}$ are sink positions.
The labelling function $\ell^{\prime}$ is defined as follows:
$\ell^{\prime}(v):=\begin{cases}\ell(s)\cup\\{p_{S}\\}&\mbox{if }v=v_{s}\mbox{
for some }s\in S\\\ \emptyset&\mbox{if }v\in\\{v_{s,i}\mid s\in
S,i\in[k]\\}\cup\,\\{v_{p_{0}},v_{\perp}\\}\\\ \\{p_{i}\\}&\mbox{if
}v=v_{p_{i}}\text{ with }i\in[k]\end{cases}$
Finally we let $v_{{\iota}}:=v_{s_{{\iota}}}$ and we define the observation
interpretation as follows:
$\mathcal{O}(o_{0}):=\\{(v,v)\mid v\in V\\},$
meaning that agent $a_{0}$ has perfect information, and for $i\in[k]$,
$\mathcal{O}(o_{i})$ is the smallest reflexive relation such that
$\mathcal{O}(o_{i})\supseteq\bigcup_{s,s^{\prime}\in
S}\\{(v_{s},v_{s^{\prime}}),(v_{s,i},v_{s^{\prime},i})\mid
s\approx_{\textnormal{{o}}_{i}}s^{\prime}\\}.$
We explain the latter definition. First, observe that for every finite play
$\rho$ in $\mathcal{G}^{\mathcal{S}}$ that stays in $V_{S}=\\{v_{s}\mid s\in
S\\}$, writing $\rho=v_{s_{0}}\ldots v_{s_{n}}$, one can associate a finite
path $\lambda_{\rho}=s_{0}\ldots s_{n}$ in $\mathcal{S}$. This mapping
actually defines a bijection between the set of finite paths in $\mathcal{S}$
that start in $s_{{\iota}}$ and the set of finite plays in
$\mathcal{G}^{\mathcal{S}}$ that remain in $V_{S}$.
Now, according to the definition of the transition function, a strategy
$\sigma_{i}$ for agent $i$ with $i\in[k]$ is only relevant on finite plays of
the form $\rho=\rho^{\prime}\cdot v_{s,i}$, where $\rho^{\prime}\in
V_{S}^{*}$, and $\sigma_{i}(\rho)$ is meant to determine whether $p_{i}$ holds
in $\lambda_{\rho^{\prime}}$. If $\sigma_{i}$ is $o_{i}$-uniform, by
definition of $\mathcal{O}(o_{i})$, it determines an
$\textnormal{{o}}_{i}$-uniform labelling for $p_{i}$ in $t_{\mathcal{S}}$.
Reciprocally, an $\textnormal{{o}}_{i}$-uniform labelling for $p_{i}$ in
$t_{\mathcal{S}}$ induces an $\mathcal{O}(o_{i})$-strategy for agent $a_{i}$.
It remains to transform $\Phi$ into an
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-formula.
We define the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula
$\widetilde{\Phi}$ by induction on $\Phi$ as follows:
$\displaystyle\widetilde{p}$ $\displaystyle:=\begin{cases}{\bf E}{\bf X}{\bf
X}p&\text{if }p=p_{i}\\\ p&\text{otherwise}\end{cases}$
$\displaystyle\widetilde{\neg\varphi}$
$\displaystyle:=\neg\widetilde{\varphi}$
$\displaystyle\widetilde{\varphi_{1}\vee\varphi_{2}}$
$\displaystyle:=\widetilde{\varphi_{1}}\vee\widetilde{\varphi_{2}}$
$\displaystyle\widetilde{{\bf E}\psi}$ $\displaystyle:={\bf E}({\bf
G}p_{S}\wedge\widetilde{\psi})$
$\displaystyle\widetilde{\exists^{\textnormal{{o}}_{i}}p_{i}.\,\varphi}$
$\displaystyle:=\langle\\!\langle
x_{i}\rangle\\!\rangle^{o_{i}}(a_{i},x_{i})\widetilde{\varphi}.$
The cases for path formulas are obtained by distributing over the operators.
Observe that player 0 is never bound to a strategy. In the case for atomic
propositions, the existential quantification on outcomes thus lets player 0
choose to move to a position where agent $i$ fixes the value for $p_{i}$
according to his strategy, fixed by the strategy quantifier in the translation
for formulas of the form $\exists^{\textnormal{{o}}_{i}}p_{i}.\,\varphi$. In
the translation of formulas of the form ${\bf E}\psi$, the existential
quantification on outcomes lets player 0 choose a path in the original CKS
$\mathcal{S}$.
We have the following:
###### Lemma 6.6.
$\mathcal{S}\models\Phi\quad\text{if and only
if}\quad\mathcal{G}^{\mathcal{S}}\models\widetilde{\Phi}$.
We observe that if $\Phi$ is hierarchical, then
$(\widetilde{\Phi},\mathcal{G}^{\mathcal{S}})$ is a hierarchical instance,
and:
###### Lemma 6.7.
For every $p\in\textnormal{AP}_{f}(\Phi)$ and for every $i\in[k]$, if
$t_{\mathcal{S}}$ is $\textnormal{{o}}_{i}$-uniform in $p$ then
$v\sim_{o_{i}}v^{\prime}$ implies that $p\in\ell(v)$ iff
$p\in\ell(v^{\prime})$.
Combining Lemma 6.4 with Lemma 6.6 we get a reduction from the model-checking
problem for CL to that for the hierarchical fragment of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, and Lemma 6.5 together with
Lemma 6.7 show that in the models produced by this reduction, all atomic
propositions are observable to all players. This implies that in CL one cannot
reason about strategic problems with unobservable objectives. As a result it
does not fully capture classic distributed synthesis (Pnueli and Rosner, 1990;
Kupferman and Vardi, 2001), where the specification can talk about all
variables, hidden and visible. It also shows that CL does not capture in a
natural way ATL with imperfect information as defined in (Alur et al., 2002,
Section 7.1), where imperfect information of agents is modelled by defining
which atomic propositions they can observe. This, as well as unobservable
objectives, can be naturally modelled in
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$.
## 7\. Applications
In this section we apply Theorem 2.9 to decide the existence of Nash
Equilibria in hierarchical games of imperfect information. We then use a
similar approach to obtain decidability results for the rational synthesis
problem. In this section, for a tuple of agents $\bm{a}=(a_{i})_{i\in[m]}$ and
tuple of strategy variables $\bm{x}=(x_{i})_{i\in[m]}$, we let
$(\bm{a},\bm{x})$ be a macro for $(a_{1},x_{1})\ldots(a_{m},x_{m})$, and
similarly for the unbinding operator $(\bm{a},\operatorname{?})$ which stands
for $(a_{1},\operatorname{?})\ldots(a_{m},\operatorname{?})$.
### 7.1. Existence of Nash Equilibria in games with hierarchical observations
A Nash equilibrium in a game is a tuple of strategies such that no player has
an incentive to deviate. Let $\textnormal{Ag}=\\{a_{i}:i\in[n]\\}$. Assuming
that agent $a_{i}$ has observation $o_{i}$ and LTL goal $\psi_{i}$, the
following $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula expresses
the existence of a Nash equilibrium:
$\displaystyle\Phi_{\textsc{NE}}:=$ $\displaystyle\langle\\!\langle
x_{1}\rangle\\!\rangle^{o_{1}}\dots\langle\\!\langle
x_{n}\rangle\\!\rangle^{o_{n}}(\bm{a},\bm{x})\bigwedge_{i\in[n]}\Big{[}\Big{(}\langle\\!\langle
y_{i}\rangle\\!\rangle^{o_{i}}(a_{i},y_{i})\,{\bf A}\psi_{i}\Big{)}\to{\bf
A}\psi_{i}\Big{]}$
where $\bm{a}=(a_{i})_{i\in[n]}$ and $\bm{x}=(x_{i})_{i\in[n]}$.
Nash equilibria do not always exist when one restricts attention to pure
strategies, as we do in this work. This is the case already in finite games,
and by extension also in the infinite concurrent games played on graphs that
we consider. This motivates the study of the Nash equilibria existence problem
in such games. In the perfect information case, the problem has been solved
for $\omega$-regular objectives, as well as more complex semi-quantitative
objectives (Bouyer et al., 2015). When moving to imperfect information, for
two players the problem is decidable for LTL objectives (Gutierrez et al.,
2018) and parity objectives (Filiot et al., 2018). However, as for distributed
synthesis, existence of Nash equilibria becomes undecidable for more than two
players. This result is proved in (Bouyer, 2018) for constrained Nash
equilibria (when one specifies for each player whether her objective is
satisfied or not), and in (Gutierrez et al., 2018) for unconstrained
equilibria. In both cases the proof proceeds by reduction from the distributed
synthesis problem (Peterson et al., 2001; Pnueli and Rosner, 1990).
The only known decidable cases for more than two players assume that all
players receive the same information. In (Bouyer, 2018) the problem is solved
on games where players observe the evolution of the game via _public signals_
and objectives are given by visible parity conditions or mean-payoff
functions. In (Belardinelli et al., 2017a), an epistemic extension of strategy
logic is used to solve the existence of Nash equilibria on games with
_broadcast actions_ for objectives given as formulas from epistemic temporal
logic. A stronger notion of Nash equilibria, called _locally consistent
equilibria_ , is studied in (Ramanujam and Simon, 2010). In a locally
consistent equilibrium, each player’s strategy has to be a best response not
only to other players’ strategies in the equilibrium, but also to all
strategies that are indistinguishable from those in the equilibrium. It is
proved in (Ramanujam and Simon, 2010) that the existence of such equilibria is
decidable on a model of games close in spirit to those with public signals
studied in (Bouyer, 2018).
Here we show that the existence of Nash equilibria is decidable for $n$
players when observations are hierarchical and objectives are given as LTL
formulas. Note that this result is orthogonal to those described above, which
all allow in a way or another some non-hierarchical information: in (Bouyer,
2018) players know their own actions in addition to the public signals, in
(Ramanujam and Simon, 2010) they know their private local state, and in
(Belardinelli et al., 2017a) they can have incomparable initial knowledge of
the situation.
###### Definition 7.1.
A $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ presents _hierarchical
observation_ (Berwanger et al., 2018) if the “finer-than” relation is a total
ordering, i.e., if for all $o,o^{\prime}\in\textnormal{Obs}$, either
$\mathcal{O}(o)\subseteq\mathcal{O}(o^{\prime})$ or
$\mathcal{O}(o^{\prime})\subseteq\mathcal{O}(o)$.
Let $\mathcal{G}$ be a $\textrm{CGS}_{\textnormal{ii}}$ with hierarchical
observation, and since all agents have symmetric roles in the problem
considered, assume without loss of generality that
$\mathcal{O}(o_{n})\subseteq\ldots\subseteq\mathcal{O}(o_{1})$.
Because of the nested strategy quantifiers $\langle\\!\langle
y_{i}\rangle\\!\rangle^{o_{i}}$, the instance
$(\mathcal{G},\Phi_{\textsc{NE}})$ is _not_ hierarchical even if $\mathcal{G}$
yields hierarchical observation (unless
$\mathcal{O}(o_{i})=\mathcal{O}(o_{j})$ for all $i,j\in[n]$). However,
considering the special observation symbol $o_{p}$ that is always interpreted
as the identity relation (and thus represents perfect observation), and
letting
$\displaystyle\Phi^{\prime}:=$ $\displaystyle\langle\\!\langle
x_{1}\rangle\\!\rangle^{o_{1}}\dots\langle\\!\langle
x_{n}\rangle\\!\rangle^{o_{n}}(\bm{a},\bm{x})\bigwedge_{i\in[n]}\Big{[}\Big{(}\langle\\!\langle
y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf E}\psi_{i}\Big{)}\to{\bf
E}\psi_{i}\Big{]},$
we have that $\Phi^{\prime}$ forms a hierarchical instance with any
$\textrm{CGS}_{\textnormal{ii}}$ that presents hierarchical observation.
Besides, we can prove that for deterministic strategies, $\Phi^{\prime}$ is
equivalent to $\Phi_{\textsc{NE}}$:
###### Lemma 7.2.
If we consider deterministic strategies, then
$\Phi_{\textsc{NE}}\equiv\Phi^{\prime}$.
###### Proof.
Concerning the universal versus existential quantification on outcomes, it is
enough to observe that assigning a deterministic strategy to each agent
determines a unique outcome. Next, to change each inner $o_{i}$ for $o_{p}$,
we exploit the fact that in a one-player game of partial observation (such a
game occurs when all but one player have fixed their strategies), the player
has a strategy enforcing some goal iff she has a uniform strategy enforcing
that goal.
To see this, it is enough to establish that for every
$\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ and position $v$,
$\mathcal{G},\chi,v\models\langle\\!\langle
y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf
E}\psi_{i}\leftrightarrow\langle\\!\langle
y_{i}\rangle\\!\rangle^{o_{i}}(a_{i},y_{i})\,{\bf E}\psi_{i},$
for every $i\in[n]$ and every assignment $\chi$ such that $\chi(a_{j})$ is
defined for all $j$.
To this end, fix $i$ and $\chi$. The right-to-left implication is immediate
(since $o_{p}$ is finer than $o_{i}$). For the converse, let $\sigma$ be an
$o_{p}$-strategy (i.e., a perfect-information strategy) such that
$\mathcal{G}^{\prime},\chi^{\prime},v_{{\iota}}\models\psi_{i}$, where
$\chi^{\prime}=\chi[y_{i}\mapsto\sigma,a_{i}\mapsto\sigma]$. Because we
consider deterministic strategies and $\chi^{\prime}$ assigns a strategy to
each agent, it defines a unique outcome $\pi$ from the initial position, i.e.,
$\textnormal{Out}(\chi^{\prime},v_{\iota})=\\{\pi\\}$. We construct an
$o_{i}$-strategy $\sigma^{\prime}$ such that if $a_{i}$ uses it instead of
$\sigma$, we obtain the same outcome $\pi$, i.e.,
$\textnormal{Out}(\chi^{\prime\prime},v_{\iota})=\\{\pi\\}$, where
$\chi^{\prime\prime}=\chi[y_{i}\mapsto\sigma^{\prime},a_{i}\mapsto\sigma^{\prime}]$.
This can be done as follows: if $\rho\sim_{o_{i}}\pi_{\leq|\rho|-1}$ then
define $\sigma^{\prime}(\rho):=\sigma(\pi_{\leq|\rho|-1})$, and otherwise let
$\sigma^{\prime}(\rho):=c$ for some fixed action $c\in\textnormal{Ac}$. It is
easy to see that $\sigma^{\prime}$ is an $o_{i}$-strategy and that
$\chi^{\prime\prime}$ produces the same outcome as $\chi$ from $v_{\iota}$. ∎
###### Corollary 7.3.
If we consider deterministic strategies, then the existence of Nash Equilibria
in games with hierarchical observation and $k$ different observations is in
$(k+1)$-Exptime .
###### Proof.
Deciding the existence of a Nash Equilibrium in a
$\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ amounts to model-checking
formula $\Phi_{\textsc{NE}}$ in $\mathcal{G}$, which by Lemma 7.2 is
equivalent to model-checking $\Phi^{\prime}$ in $\mathcal{G}$ if we restrict
to deterministic strategies. Because $\Phi^{\prime}$ forms hierarchical
instances with games that yield hierarchical observation, by Theorem 2.9 we
can model check it on such games. Now because each $\psi_{i}$ is an LTL
formula, we have that
$\displaystyle\mbox{sd}\left(\langle\\!\langle
y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf E}\psi_{i}\right)$
$\displaystyle=(0,\mbox{nd}),$
$\displaystyle\mbox{sd}\left(\bigwedge_{i\in[n]}\Big{[}\Big{(}\langle\\!\langle
y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf E}\psi_{i}\Big{)}\to{\bf
E}\psi_{i}\Big{]}\right)$ $\displaystyle=(0,\mbox{alt}),$
and finally we obtain that $\mbox{sd}(\Phi^{\prime})=(k,\mbox{nd})$, where $k$
is the number of different observations in $\mathcal{G}$, i.e.,
$k=|\\{\mathcal{O}(o_{1}),\ldots,\mathcal{O}(o_{n})\\}|$. By Proposition 5.2,
we can model check $\Phi^{\prime}$ on $\mathcal{G}$ in time
$(k+1)$-exponential, which concludes. ∎
We now show that, using the same trick, our main result can be applied to
solve a more general problem called _rational synthesis_.
### 7.2. Rational distributed synthesis in games with hierarchical
observations
In classic synthesis, the environment is considered monolithic and “hostile”,
in the sense that the system to be synthesised should be able to deal with all
possible behaviours of the environment, even the most undesirable ones. This
is a very strong requirement that can not always be met. When the environment
can be considered rational, and its objective is known, it is reasonable to
relax this requirement by asking that the system to synthesise behave well
against the _rational_ behaviours of the environment. This problem is known as
the _rational synthesis_ problem (Fisman et al., 2010; Kupferman et al., 2016;
Condurache et al., 2016; Filiot et al., 2018). In the setting considered in
the works above-mentioned, the system is seen as an agent $a$ and the
environment is composed of several components, say $\\{e_{1},\ldots,e_{m}\\}$,
that are assumed to be rational and follow individual objectives. While
(Condurache et al., 2016) and (Filiot et al., 2018) consider various types of
objectives such as reachability, safety or parity, here we consider LTL
objectives as is done in (Fisman et al., 2010; Kupferman et al., 2016): the
specification for the system is an LTL formula $\psi_{g}$, and the objective
of each component $e_{i}$ of the environment is an LTL formula $\psi_{i}$.
However note that the decidability results we establish would also hold for
arbitrary omega-regular objectives.
#### 7.2.1. Rational synthesis: state of the art
Two variants of the rational synthesis problem have been considered: the
_cooperative_ one, in which it is possible to tell the environment how to
behave, as long as the suggested behaviour for each component forms an
equilibrium, and the _non-cooperative_ one, in which the components of the
environment may have any behaviour that forms an equilibrium. The existence of
a solution to these problems can be expressed by the formulas
$\Phi_{\text{c-RS}}$ and $\Phi_{\text{nc-RS}}$, respectively, defined as
follows:
$\displaystyle\Phi_{\text{c-RS}}$ $\displaystyle:=\langle\\!\langle
x\rangle\\!\rangle^{o_{p}}\langle\\!\langle
y_{1}\rangle\\!\rangle^{o_{p}}\ldots\langle\\!\langle
y_{m}\rangle\\!\rangle^{o_{p}}(a,x)(\bm{e},\bm{y})\,\varphi_{\gamma}\wedge{\bf
A}\psi_{g}$ $\displaystyle\Phi_{\text{nc-RS}}$
$\displaystyle:=\langle\\!\langle
x\rangle\\!\rangle^{o_{p}}[\\![y_{1}]\\!]^{o_{p}}\ldots[\\![y_{m}]\\!]^{o_{p}}(a,x)(\bm{e},\bm{y})\,\varphi_{\gamma}\to{\bf
A}\psi_{g}$
where $\bm{e}=(e_{i})_{i\in[m]}$, $\bm{y}=(y_{i})_{i\in[m]}$, and
$\varphi_{\gamma}$ expresses that $\bm{y}$ forms an equilibrium for the
environment. Also, as in the previous section, $o_{p}$ represents the perfect-
information observation. Three different kinds of equilibria are considered in
(Kupferman et al., 2016): profiles of dominant strategies, Nash equilibria,
and subgame-perfect equilibria. Here we only consider Nash equilibria, because
subgames of games with imperfect information should start in situations where
all players have perfect information of the state, which we do not know how to
express in $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$; and for dominant
strategies, the natural formula to express them does not give rise to non-
trivial decidable cases in the imperfect-information setting that we introduce
later. The rational synthesis problem for Nash equilibria is obtained by
replacing $\varphi_{\gamma}$ in the above formula with:
$\displaystyle\varphi_{\text{NE}}$
$\displaystyle:=\bigwedge_{i\in[m]}\Big{[}\Big{(}\langle\\!\langle
y^{\prime}_{i}\rangle\\!\rangle^{o_{p}}(e_{i},y^{\prime}_{i})\,{\bf
A}\psi_{i}\Big{)}\to{\bf A}\psi_{i}\Big{]}$
It is proved in (Kupferman et al., 2016) that these problems are decidable for
perfect information. Concerning imperfect information, because the existence
of Nash equilibria is undecidable for three players, the problem is
undecidable when the environment consists of at least three components (Filiot
et al., 2018). Three decidable cases are known: when the environment consists
of a single component (Filiot et al., 2018), when actions of all components
are public (Belardinelli et al., 2017a), and when only the system has
imperfect information while the (finitely many) components of the environment
are perfectly informed (Filiot et al., 2018).
We now extend the latter result by defining a generalisation of the rational
synthesis problem that we call _rational distributed synthesis_ , and solving
it in the case of hierarchical information. The case where the environment is
perfectly informed and the system consists of a single component, solved in
(Filiot et al., 2018), is a particular case of our Corollary 7.5 below666We
only consider LTL objectives, but our automata construction can be adapted to
handle all $\omega$-regular objectives.. However the other decidability result
established in (Filiot et al., 2018) does not assume hierarchical information,
and thus cannot be derived from the results we now present.
#### 7.2.2. Rational distributed synthesis
While for perfect information, distributed synthesis amounts to synthesis for
a single meta-component which tells each component what to do, in the context
of imperfect information it makes sense to consider that the system to be
synthesised is composed of various components $\\{a_{1},\ldots,a_{n}\\}$ with
different observation power, say $o_{i}$ for component $a_{i}$. We also let
$o^{e}_{i}$ be the observation of the environment’s component $e_{i}$, for
$i\in[m]$.
We consider the imperfect-information variants of cooperative and non-
cooperative rational synthesis defined by the following
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formulas:
$\displaystyle\Phi^{\textnormal{\scriptsize ii}}_{\text{c-RS}}$
$\displaystyle:=\langle\\!\langle
x_{1}\rangle\\!\rangle^{o_{1}}\ldots\langle\\!\langle
x_{n}\rangle\\!\rangle^{o_{n}}\langle\\!\langle
y_{1}\rangle\\!\rangle^{o^{e}_{1}}\ldots\langle\\!\langle
y_{m}\rangle\\!\rangle^{o^{e}_{m}}(\bm{a},\bm{x})(\bm{e},\bm{y})\,\varphi_{\gamma}\wedge{\bf
A}\psi_{g}$ $\displaystyle\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$
$\displaystyle:=\langle\\!\langle
x\rangle\\!\rangle^{o_{1}}\ldots\langle\\!\langle
x_{n}\rangle\\!\rangle^{o_{n}}[\\![y_{1}]\\!]^{o^{e}_{1}}\ldots[\\![y_{m}]\\!]^{o^{e}_{m}}(\bm{a},\bm{x})(\bm{e},\bm{y})\,\varphi_{\gamma}\to{\bf
A}\psi_{g}$
The formula for Nash equilibrium is adapted as follows:
$\displaystyle\varphi^{\textnormal{\scriptsize ii}}_{\text{NE}}$
$\displaystyle:=\bigwedge_{i\in[m]}\Big{[}\Big{(}\langle\\!\langle
y^{\prime}_{i}\rangle\\!\rangle^{o^{e}_{i}}(e_{i},y^{\prime}_{i})\,{\bf
A}\psi_{i}\Big{)}\to{\bf A}\psi_{i}\Big{]}$
The only difference with the perfect-information case is that we use the
observation of the different components of the environment instead of the
perfect-information observation.
We call the problems expressed by formulas $\Phi^{\textnormal{\scriptsize
ii}}_{\text{c-RS}}$ and $\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$
_cooperative rational distributed synthesis_ and _non-cooperative rational
distributed synthesis_ , respectively. As in the previous section on the
existence of Nash equilibria, one can see that even if there is a total
hierarchy on all observations, these formula do not yield hierarchical
instances unless all observations are the same. However, the trick applied in
the proof of Corollary 7.3 also applies here, both for Nash equilibria and
subgame-perfect equilibria, i.e., we can replace each $o^{e}_{i}$ with $o_{p}$
in $\varphi^{\textnormal{\scriptsize ii}}_{\text{NE}}$ without affecting the
semantics of formulas $\Phi^{\textnormal{\scriptsize ii}}_{\text{c-RS}}$ and
$\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$. As a result, when there
is a hierarchy on observations
$o_{1},\ldots,o_{n},o^{e}_{1},\ldots,o^{e}_{m}$, the cooperative rational
distributed synthesis is decidable.
###### Corollary 7.4.
If we consider deterministic strategies and hierarchical observations, then
cooperative rational distributed synthesis is decidable.
For the non-cooperative variant, one cannot switch universal quantifications
on strategies for the environments with existential quantifications for the
system in order to obtain hierarchical instances, as the resulting formula
would then capture a different problem. As a consequence, in addition to a
hierarchy on observations $o_{1},\ldots,o_{n},o^{e}_{1},\ldots,o^{e}_{m}$, we
need to have that the components of the environment observe better than the
components of the system or, in other words, that the least informed component
of the environment observes better than the best informed component of the
system. When it is the case, we say that the environment is _more informed_
than the system.
###### Corollary 7.5.
Non-cooperative rational distributed synthesis is decidable for deterministic
strategies and hierarchical observations where the environment is more
informed than the system.
This result applies for instance when there is hierarchical information
amongst the components of the system, and the environment has perfect
information. Note that when the system consists of a single component, this
corresponds to the second decidability result in (Filiot et al., 2018). As we
mentioned in the introduction, considering that the opponent has perfect
information is something classic in two-player games with imperfect
information, as doing so ensures that the strategy one synthesises is winning
no matter how much the opponent observes. In Reif’s words, this amounts to
considering the possibility that the opponent may “cheat” and use information
that it normally does not have access to (Reif, 1984). The non-cooperative
rational synthesis problem is not precisely a two-player game, but it
resembles one in the sense that the system as a whole (composed of its various
components $a_{1},\ldots,a_{n}$) should win against any “rational” behaviour
of the environment as a whole. In this view, considering that the components
of the environment have perfect information thus yields a distributed system
that is robust to possible leaks of hidden information to the environment.
###### Remark 6.
When all components of the environment have perfect information,
$\Phi^{\textnormal{\scriptsize ii}}_{\text{c-RS}}$ and
$\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$ already form hierarchical
instances with games where there is hierarchical observation amongst the
system’s components, and one does not need to resort to the trick used in the
proof of Corollary 7.3. A consequence is that in that case, corollaries 7.4
and 7.5 also hold for nondeterministic strategies.
## 8\. Conclusion
We introduced $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, a logic for
reasoning about strategic behaviour in multi-player games with imperfect
information. The syntax specifies the observations with which strategies have
to work, and thus allows one to reason about strategic problems in settings
where agents can change observation power, for instance by being eventually
granted access to previously hidden information. Moreover our logic contains
an outcome quantifier and an unbinding operator which simplify the semantics,
make it easier to express branching-time properties, allow us to naturally
consider nondeterministic strategies, and make the correspondence with
$\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ tighter, enabling us
to derive precise complexity results for the model-checking of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$.
We isolated the class of hierarchical formula/model pairs $(\Phi,\mathcal{G})$
and proved that for such instances one can decide whether
$\mathcal{G}\models\Phi$. The proof reduces (hierarchical) instances of
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to (hierarchical) formulas
of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, a low-level logic
that we introduced, and that serves as a natural bridge between
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ and automata constructions.
We also studied in detail the complexity of the model-checking problems solved
in this work. To do so we introduced a new measure on formulas called
_simulation depth_. This measure, though being a purely syntactic notion,
reflects the complexity of automata constructions required to treat a given
formula.
Since one can alternate quantifiers in
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, our decidability result
goes beyond synthesis and can be used to easily obtain the decidability of
many strategic problems. In this work we applied it to the problem of
existence of Nash equilibria in games with hierarchical observation, and to
the imperfect-information generalisations of rational synthesis that we called
(cooperative and non-cooperative) _rational distributed synthesis_. Our result
has also been used to prove that the existence of admissible strategies in
games with hierarchical information is decidable (Brenguier et al., 2017).
An interesting direction for future work would be to try and adapt the notion
of hierarchical instances to allow for situations in which hierarchies can
change along a play, as done in (Berwanger et al., 2018). We would also like
to consider alternatives to the synchronous perfect recall setting considered
here, such as the classic asynchronous perfect recall setting (Fagin et al.,
1995; Puchala, 2010), or the more recent notion of causal knowledge (Genest et
al., 2015). Finally, it is often interesting in presence of imperfect
information to introduce epistemic operators to reason explicitely about what
agents know. We already generalised the main result of this work to an
extension of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with such
operators (Maubert and Murano, 2018); we would like to see if this can be used
to reason about subgame-perfect equilibria in games with imperfect
information, which do not seem to be easy to characterise in
$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, as mentioned in Section
7.2.1. Indeed, in games with imperfect information, the notion of subgame
specifies that the initial situation should be known to all players (Selten,
1965), a property that epistemic logics are meant to be able to express.
###### Acknowledgements.
We thank anonymous reviewers for their valuable comments on a previous version
of this work. This project has received funding from the Sponsor European
Union’s Horizon 2020 research and innovation programme
https://ec.europa.eu/programmes/horizon2020/en under the Marie Sklodowska-
Curie grant agreement No Grant #709188.
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## Appendix A Proof of Proposition 4.12
First, for every LTL formula $\psi$ one can build a parity word automaton
$\mathcal{W}^{\psi}$ with two colours and $2^{O(|\psi|)}$ states (Vardi and
Wolper, 1994). Let $K_{\psi}\in\mathbb{N}$ be such that the number of states
of $\mathcal{W}^{\psi}$ is bounded by $2^{K_{\psi}|\psi|}$.
We also state a more precise version of Theorem 4.6: for every ATA
$\mathcal{A}$ with $n$ states and $l$ colours, one can build an NTA
$\mathcal{N}$ with at most $2^{O(nl\log(nl))}$ states and $O(nl)$ colours such
that $\mathcal{L}(\mathcal{A})=\mathcal{L}(\mathcal{N})$ (Muller and Schupp,
1995; Löding, 2011). We let $K_{1},K_{2}\in\mathbb{N}$ be such that the number
of states of $\mathcal{N}$ is bounded by $2^{K_{1}nl\log(nl)}$ and the number
of colours by $K_{2}nl$.
Proposition 4.12 follows directly from the following.
###### Proposition A.1.
Let $\Phi$ be a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize
i,$\tiny{\subseteq}$}}$ formula, $\mathcal{S}$ a CKS , and let
${\textnormal{AP}_{\exists}}={\textnormal{AP}_{\exists}}(\Phi)$. For every
subformula $\varphi$ of $\Phi$ and state $s\in\mathcal{S}$, it holds that:
* •
if $\mbox{sd}_{k}(\varphi)=0$, $\mathcal{A}_{s}^{\varphi}$ has at most
$f_{\mathcal{S}}^{\varphi}$ states and 2 colours,
* •
if $\mbox{sd}_{k}(\varphi)\geq 1$, $\mathcal{A}_{s}^{\varphi}$ has at most
$\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid f_{\mathcal{S}}^{\varphi}\log
f_{\mathcal{S}}^{\varphi}\big{)}$ states, and its number of colours is at most
$\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid f_{\mathcal{S}}^{\varphi}\log
f_{\mathcal{S}}^{\varphi}\big{)}$,
with
$f_{\mathcal{S}}^{\varphi}=(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}2^{K_{\psi}|\varphi|{\bf E}\mathrm{d}(\varphi)}$.
In addition, if $\mathcal{A}_{s}^{\varphi}$ has state set $Q$, for each $q\in
Q$ and $a\in 2^{\textnormal{AP}_{\exists}}$, we have
$|\delta(q,a)|\leq|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\varphi|}$, where
$H=1+{\bf E}\mathrm{d}(\varphi)$.
###### Proof.
We prove the result by induction on $\varphi$.
$\bm{\varphi=p:}$ in this case
$\mbox{sd}_{k}(\varphi)=\exists\mathrm{d}(\varphi)={\bf
E}\mathrm{d}(\varphi)=0$. By construction, $\mathcal{A}_{s}^{\varphi}$ has one
state $q_{\iota}$ and two colours, so that the first part of the claim holds.
In addition, each formula of its transition function is of size one, so that
the second part of the claim also holds.
$\bm{\varphi=\neg\varphi^{\prime}:}$ Complementing an ATA does not change the
number of states, number of colours or size of formulas in the transition
function, so that the result follows by induction hypothesis and the fact that
$|\varphi^{\prime}|\leq|\varphi|$ and ${\bf E}\mathrm{d}(\varphi)={\bf
E}\mathrm{d}(\varphi^{\prime})$.
$\bm{\varphi=\varphi_{1}\vee\varphi_{2}:}$ To establish the claim about number
of states and colours we split cases. First we consider the case where
$\mbox{sd}_{k}(\varphi)=0$. In that case we also have
$\mbox{sd}_{k}(\varphi_{1})=\mbox{sd}_{k}(\varphi_{2})=0$. By induction
hypothesis, for $i\in\\{1,2\\}$, $\mathcal{A}_{s}^{\varphi_{i}}$ has at most
$f_{\mathcal{S}}^{\varphi_{i}}$ states and $2$ colours. These automata are
then narrowed down, but the narrowing operation leaves the size of formulas in
the transition function unchanged (in fact they may become smaller, but not
bigger, see (Kupferman and Vardi, 1999)). Therefore, by construction
$\mathcal{A}_{s}^{\varphi}$ has at most
$1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}$ states and two
colours.
Now we have that
$\displaystyle 1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}$
$\displaystyle=1+\sum_{i\in\\{1,2\\}}(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi_{i})}|\varphi_{i}||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi_{i})}2^{K_{\psi}|\varphi_{i}|{\bf
E}\mathrm{d}(\varphi_{i})}$
$\displaystyle=1+(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}\sum_{i\in\\{1,2\\}}2^{K_{\psi}|\varphi_{i}|{\bf
E}\mathrm{d}(\varphi)}$ $\displaystyle\leq
1+(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}2^{K_{\psi}(|\varphi_{1}|+|\varphi_{2}|){\bf
E}\mathrm{d}(\varphi)}$ $\displaystyle
1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}$
$\displaystyle\leq(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}2^{K_{\psi}(|\varphi_{1}|+|\varphi_{2}|+1){\bf
E}\mathrm{d}(\varphi)}$
We get that
(8) $1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}\leq
f_{\mathcal{S}}^{\varphi}$
which concludes the claim about the number of states.
Now for the case where $\mbox{sd}_{k}(\varphi)\geq 1$. By definition of
nondeterminisation depth, for at least one $i\in\\{1,2\\}$ we have
$\mbox{sd}_{k}(\varphi_{i})\geq 1$. Also, the number of colours used in
$\mathcal{A}_{s}^{\varphi}$ is the maximum between the number of colours used
in $\mathcal{A}_{s}^{\varphi_{1}}$ and those used in
$\mathcal{A}_{s}^{\varphi_{2}}$. By induction hypothesis it is the case that
$\mathcal{A}_{s}^{\varphi_{i}}$ has at most
$\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})-1\mid
f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$
colours if $\mbox{sd}_{k}(\varphi_{i})\geq 1$, or 2 if
$\mbox{sd}_{k}(\varphi_{i})=0$. Therefore, the number of colours in
$\mathcal{A}_{s}^{\varphi}$ is at most
$\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})-1\mid
f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$ for
some $i$, which is less than $\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid
f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$.
For the number of states $|Q|$ in $\mathcal{A}_{s}^{\varphi}$, we have that
$|Q|=1+|Q_{1}|+|Q_{2}|$, where $Q_{i}$ is the set of states of
$\mathcal{A}_{s}^{\varphi_{i}}$. By induction hypothesis we get
$\displaystyle|Q|$ $\displaystyle\leq
1+\sum_{i\in\\{1,2\\}}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})\mid
f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$
$\displaystyle\leq
1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid\sum_{i\in\\{1,2\\}}f_{\mathcal{S}}^{\varphi_{i}}\log
f_{\mathcal{S}}^{\varphi_{i}}\big{)}$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(\sum_{i\in\\{1,2\\}}f_{\mathcal{S}}^{\varphi_{i}}+1)\log
f_{\mathcal{S}}^{\varphi}\big{)}$ $\displaystyle|Q|$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid
f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}\mbox{\hskip
56.9055pt (using Equation~{}\eqref{eq-boum})}$
which concludes the claim about the number of states.
Concerning the size of formulas in the transition function, for all states
from $\mathcal{A}_{s}^{\varphi_{1}}$ and $\mathcal{A}_{s}^{\varphi_{2}}$ the
transition function is unchanged and the result thus holds by induction
hypothesis. For the remaining state $q_{\iota}$, we have by definition
$\delta(q_{\iota},a)=\delta^{1}(q_{\iota}^{1},a)\vee\delta^{2}(q_{\iota}^{2},a)$
and thus
$|\delta(q_{\iota},a)|=|\delta^{1}(q_{\iota}^{1},a)|+|\delta^{2}(q_{\iota}^{2},a)|+1$.
By induction hypothesis we get that
$\displaystyle|\delta(q_{\iota},a)|$
$\displaystyle\leq|\mathcal{S}||Q_{1}|^{|\mathcal{S}|}2^{H(\varphi_{1})|\varphi_{1}|}+|\mathcal{S}||Q_{2}|^{|\mathcal{S}|}2^{H(\varphi_{2})|\varphi_{2}|}+1$
$\displaystyle\leq|\mathcal{S}|2^{H(\varphi)(|\varphi_{1}|+|\varphi_{2}|)}(|Q_{1}|^{|\mathcal{S}|}+|Q_{2}|^{|\mathcal{S}|})$
$\displaystyle\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}(|Q_{1}|+|Q_{2}|)^{|\mathcal{S}|}$
And thus
$|\delta(q_{\iota},a)|\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}|Q|^{|\mathcal{S}|}$
as required.
$\bm{\varphi={\bf E}\psi:}$ The word automaton built for the LTL skeleton of
$\psi$ is in fact a Büchi automaton, and thus uses only two colours. The
number of colours used by $\mathcal{A}_{s}^{\varphi}$ is therefore the maximum
number of colours used by the automata $\mathcal{A}_{s}^{\varphi_{i}}$ built
for the maximal state subformulas $\varphi_{i}$ in $\psi$, and the result
follows by induction hypothesis.
Concerning the number of states, let $|Q_{\varphi}|$ (resp. $|Q_{i}|$,
$|Q_{\psi}|$) be the number of states in $\mathcal{A}_{s}^{\varphi}$ (resp.
$\mathcal{A}_{s}^{\varphi_{i}}$, $\mathcal{W}^{\psi}$). Note that the number
of states in $\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$ does not depend on
$s^{\prime}$. Recall that $\max(\psi)=\\{\varphi_{1},\ldots,\varphi_{n}\\}$ is
the set of maximal state subformulas of $\psi$, and let $\psi^{\prime}$ be the
LTL skeleton of $\psi$, i.e., the LTL formula obtained from $\psi$ by
replacing maximal state subformulas $\varphi_{i}$ with propositions
$p_{\varphi_{i}}$. We thus have
$\displaystyle|Q|$
$\displaystyle=|Q_{\psi}||\mathcal{S}|+2|\mathcal{S}|\sum_{i\in[n]}|Q_{i}|$
$\displaystyle\leq
2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|+2|\mathcal{S}|\sum_{i\in[n]}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})\mid
f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$
$\displaystyle|Q|$ $\displaystyle\leq
2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid\sum_{i\in[n]}f_{\mathcal{S}}^{\varphi_{i}}\log
f_{\mathcal{S}}^{\varphi_{i}}\big{)}\right)$
And thus
(9) $|Q|\leq
2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid\log
f_{\mathcal{S}}^{\varphi}\sum_{i\in[n]}f_{\mathcal{S}}^{\varphi_{i}}\big{)}\right)$
Now observe that for each $i\in[n]$ we have that ${\bf
E}\mathrm{d}(\varphi_{i})\leq{\bf E}\mathrm{d}(\varphi)-1$, and
$\exists\mathrm{d}(\varphi_{i})=\exists\mathrm{d}(\varphi)$. Therefore,
$\displaystyle\sum_{i\in[n]}f_{\mathcal{S}}^{\varphi_{i}}$
$\displaystyle=(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}\sum_{i\in[n]}|\varphi_{i}||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi_{i})}2^{K_{\psi}|\varphi_{i}|{\bf
E}\mathrm{d}(\varphi_{i})}$
$\displaystyle\leq(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)-1}(\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf
E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}$
Using this in Equation (9) we get
$\displaystyle|Q|$ $\displaystyle\leq
2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)-1}(\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf
E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\log
f_{\mathcal{S}}^{\varphi}\big{)}\right)$ $\displaystyle\leq
2^{K_{\psi}|\psi^{\prime}|}\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}(\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf
E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\log
f_{\mathcal{S}}^{\varphi}\big{)}\right)$ $\displaystyle\leq
2^{K_{\psi}|\psi^{\prime}|}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}(1+\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf
E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\log
f_{\mathcal{S}}^{\varphi}\big{)}$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}|\varphi|2^{K_{\psi}B}\log
f_{\mathcal{S}}^{\varphi}\big{)},$
where $B=({\bf
E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|+|\psi^{\prime}|$. To
conclude it only remains to show that $B\leq|\varphi|{\bf
E}\mathrm{d}(\varphi)$. Because $\varphi={\bf E}\psi$, it holds that ${\bf
E}\mathrm{d}(\varphi)\geq 1$. If ${\bf E}\mathrm{d}(\varphi)=1$, we have
$B=|\psi^{\prime}|\leq|\varphi|{\bf E}\mathrm{d}(\varphi)$. Now if ${\bf
E}\mathrm{d}(\varphi)\geq 2$, we have
$B=({\bf
E}\mathrm{d}(\varphi)-2)\sum_{i\in[n]}|\varphi_{i}|+|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|$
Clearly, $\sum_{i\in[n]}|\varphi_{i}|\leq|\varphi|$, and
$|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|\leq 2|\varphi|$, and the result
follows. Note that it could seem that
$|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|\leq|\varphi|$. It is true if one
defines the size of a formula as the number of connectors, but not if one also
counts atomic propositions, as we do here. However it is true that
$|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|\leq 2|\varphi|$, independently of
the definition of formulas’ size.
It remains to establish the claim about the size of transition formulas. By
definition, for every state $q$ of $\mathcal{A}_{s}^{\varphi}$ that comes from
some $\mathcal{A}^{i}_{s^{\prime}}$ or
$\overline{\mathcal{A}^{i}_{s^{\prime}}}$, the transition function is
unchanged and thus the result follows by induction hypothesis and the fact
that narrowing and complementation do not increase the size of formulas in
transition functions. Now for the remaining states, for each
$(q^{\psi},s^{\prime})\in Q$ and every $a\in
2^{{\textnormal{AP}_{\exists}}(\Phi)}$, we have
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$
$\displaystyle\leq\sum_{a^{\prime}\in
2^{\max(\psi)}}\left(|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|+1+\sum_{\varphi_{i}\in
a^{\prime}}(|\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)|+1)+\sum_{\varphi_{i}\notin
a^{\prime}}(|\overline{\delta^{i}_{s^{\prime}}}(\overline{q^{i}_{s^{\prime}}},a)|+1)\right)$
Now by induction hypothesis, and because complementation does not increase the
size of formulas, we get:
(10) $|\delta((q^{\psi},s^{\prime}),a)|\leq\sum_{a^{\prime}\in
2^{\max(\psi)}}\left(|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|+2\sum_{i\in[n]}|\mathcal{S}|2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}\right)+2^{|\max(\psi)|}+2|\max(\psi)|2^{|\max(\psi)|},$
where $|Q_{i}|$ is the number of states in automaton
$\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$. Now by definition,
$\displaystyle|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|$
$\displaystyle=\left(\sum_{q^{\prime}\in\Delta^{\psi}(q^{\psi},a^{\prime})}\sum_{s^{\prime\prime}\in
R(s^{\prime})}1\right)+|\Delta^{\psi}(q^{\psi},a^{\prime})||R(s^{\prime})|-1$
$\displaystyle|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|$
$\displaystyle\leq 2|\Delta^{\psi}(q^{\psi},a^{\prime})||R(s^{\prime})|-1$
We thus have
(11) $|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|\leq
2|Q_{\psi^{\prime}}||\mathcal{S}|-1$
where $Q_{\psi^{\prime}}$ is the set of states of the word automaton
$\mathcal{W}^{\psi}$. Using this in Equation 10 we get:
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq
2^{|\max(\psi)|}\left(2|Q_{\psi^{\prime}}||\mathcal{S}|-1+2\sum_{i\in[n]}|\mathcal{S}|2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}\right)+2^{|\max(\psi)|}+2|\max(\psi)|2^{|\max(\psi)|}$
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq
2^{|\max(\psi)|+1}|\mathcal{S}|\left(|Q_{\psi^{\prime}}|+\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}\right)+2|\max(\psi)|2^{|\max(\psi)|}$
But for natural numbers $\\{a_{i},b_{i}\\}_{i\in[n]}$, it holds that
$\sum_{i\in[n]}2^{a_{i}}b_{i}=2^{\sum_{i\in[n]}a_{i}}\sum_{i\in[n]}b_{i}-\sum_{i\in[n]}2^{a_{i}}(2^{\sum_{j\neq
i}a_{j}}-1)b_{i}$
Applying this to $a_{i}=H(\varphi_{i})|\varphi_{i}|$ and
$b_{i}=|Q_{i}|^{|\mathcal{S}|}$ we obtain
$\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}=2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}-\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq
i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}$
We thus get that
$|\delta((q^{\psi},s^{\prime}),a)|\leq
2^{|\max(\psi)|+1}|\mathcal{S}|\left(|Q_{\psi^{\prime}}|+2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}\right)+C,$
with
$\displaystyle C$
$\displaystyle=2|\max(\psi)|2^{|\max(\psi)|}-2^{|\max(\psi)|+1}|\mathcal{S}|\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq
i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}$
$\displaystyle=2^{|\max(\psi)|}\left(2|\max(\psi)|-2|\mathcal{S}|\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq
i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}\right)$
If $n=|\max(\psi)|>1$, i.e., there are at least two maximal state subformulas,
then $\sum_{j\neq i}H(\varphi_{j})|\varphi_{j}|>0$, hence
$2|\mathcal{S}|\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq
i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}\geq
4n=4|\max(\psi)|$, which implies that $C\leq 0$, and thus
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq
2^{|\max(\psi)|+1}|\mathcal{S}|\left(|Q_{\psi^{\prime}}|+2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}\right)$
$\displaystyle\leq
2^{|\max(\psi)|+1}|\mathcal{S}|2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\left(|Q_{\psi^{\prime}}|^{|\mathcal{S}|}+\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}\right)$
$\displaystyle\leq|\mathcal{S}|2^{|\max(\psi)|+1+(H(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\left(|Q_{\psi^{\prime}}|+\sum_{i\in[n]}|Q_{i}|\right)^{|\mathcal{S}|}$
$\displaystyle\leq|\mathcal{S}|2^{|\varphi|+(H(\varphi)-1)|\varphi|}|Q|^{|\mathcal{S}|}$
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$
$\displaystyle\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}|Q|^{|\mathcal{S}|}$
It remains to consider the case where $\max(\psi)=\\{\varphi_{1}\\}$. In that
case there are only two letters in the alphabet $2^{\max(\psi)}$, which are
$\emptyset$ and $\\{\varphi_{1}\\}$. The transition formulas then simplify and
one gets that
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$
$\displaystyle\leq|\delta_{\psi}((q^{\psi},s^{\prime}),\emptyset)|+1+|\overline{\delta^{1}_{s^{\prime}}}(\overline{q^{1}_{s^{\prime}}},a)|+1+|\delta_{\psi}((q^{\psi},s^{\prime}),\\{\varphi_{1}\\})|+1+|\delta^{1}_{s^{\prime}}(q^{1}_{s^{\prime}},a)|$
Using Equation (11) and the induction hypothesis we get
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq
4|Q_{\psi^{\prime}}||\mathcal{S}|-2+2|\mathcal{S}|2^{H(\varphi_{1})|\varphi_{1}|}|Q_{1}|^{|\mathcal{S}|}+3$
$\displaystyle\leq
1+2|\mathcal{S}|(2|Q_{\psi^{\prime}}|+2^{H(\varphi_{1})|\varphi_{1}|}|Q_{1}|^{|\mathcal{S}|})$
$\displaystyle\leq
1+2|\mathcal{S}|2^{(H(\varphi)-1)|\varphi_{1}|}(|Q_{\psi^{\prime}}|^{|\mathcal{S}|}+|Q_{1}|^{|\mathcal{S}|})$
$\displaystyle\leq
1+|\mathcal{S}|2^{H(\varphi)|\varphi|}(|Q_{\psi^{\prime}}|^{|\mathcal{S}|}+|Q_{1}|^{|\mathcal{S}|})$
$\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$
$\displaystyle\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}|Q|^{|\mathcal{S}|}$
$\bm{\varphi=\exists}^{\bm{\textnormal{{o}}}}\bm{p.\,\varphi^{\prime}:}$ We
first establish the claim for states and colours, and we start with the case
$\mbox{sd}_{k}(\varphi)=\mbox{sd}_{k}(\varphi^{\prime})$. By definition we
necessarily have that $\mbox{sd}_{x}(\varphi^{\prime})=\mbox{nd}$, i.e.,
$\mathcal{A}_{s}^{\varphi^{\prime}}$ is nondeterministic, and
$\textnormal{{o}}=I_{\varphi^{\prime}}$, therefore there is no need to use
narrowing or nondeterminisation here. $\mathcal{A}_{s}^{\varphi}$ is obtained
by directly projecting $\mathcal{A}_{s}^{\varphi^{\prime}}$, an operation that
does not change the number of states or colours, so that the claim for states
and colours follows directly by induction hypothesis.
Now we consider the case where
$\mbox{sd}_{k}(\varphi)\neq\mbox{sd}_{k}(\varphi^{\prime})$, which implies
that $\mbox{sd}_{k}(\varphi)\geq 1$. Let $n$ be the number of states and $l$
the number of colours in $\mathcal{A}_{s}^{\varphi^{\prime}}$. In this case
$\mathcal{A}_{s}^{\varphi^{\prime}}$ is first narrowed down, which does not
change number of states or colours. The resulting automaton is then
nondeterminised, yielding an automaton with at most $2^{K_{1}nl\log nl}$
states and $K_{2}nl$ colours.
Again, we split cases: if $\mbox{sd}_{k}(\varphi^{\prime})=0$, by induction
hypothesis, $n\leq f_{\mathcal{S}}^{\varphi^{\prime}}$ and $l=2$. For the
number of colours, observing that
$\exists\mathrm{d}(\varphi)=\exists\mathrm{d}(\varphi^{\prime})+1$, we have
$\displaystyle K_{2}nl\leq 2K_{2}f_{\mathcal{S}}^{\varphi^{\prime}}$
$\displaystyle=2K_{2}(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi^{\prime})}|\varphi^{\prime}||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi^{\prime})}2^{K_{\psi}|\varphi^{\prime}|{\bf
E}\mathrm{d}(\varphi^{\prime})}$
$\displaystyle\leq(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf
E}\mathrm{d}(\varphi)}2^{K_{\psi}|\varphi|{\bf E}\mathrm{d}(\varphi)}$
$\displaystyle K_{2}nl$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid
f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$
For the number of states, we have that
$\displaystyle 2^{K_{1}nl\log nl}$ $\displaystyle\leq
2^{2K_{1}f_{\mathcal{S}}^{\varphi^{\prime}}\log(2f_{\mathcal{S}}^{\varphi^{\prime}})}\leq\mathrm{exp}\big{(}\mbox{sd}_{k}{\varphi}\mid
f_{\mathcal{S}}^{\varphi}\log(f_{\mathcal{S}}^{\varphi})\big{)}$
Now for the final case, if
$\mbox{sd}_{k}(\varphi)=\mbox{sd}_{k}(\varphi^{\prime})+1$ and
$\mbox{sd}_{k}(\varphi^{\prime})\geq 1$, by induction hypothesis
$n\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid
f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ and
$l\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid
f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$. For the number of colours in
$\mathcal{A}_{s}^{\varphi}$ we thus get
$\displaystyle K_{2}nl$ $\displaystyle\leq
K_{2}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid
f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}2^{f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}}\big{)}$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid
2K_{2}f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ $\displaystyle K_{2}nl$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid
f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$
Concerning the number of states, we observe that
$\displaystyle nl$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid
f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}2^{f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}}\big{)}$ $\displaystyle nl$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid
2f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ $\displaystyle K_{1}nl\log nl$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid
2K_{1}f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}2^{2f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}}\big{)}$ $\displaystyle K_{1}nl\log nl$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid
4K_{1}f_{\mathcal{S}}^{\varphi^{\prime}}\log
f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ $\displaystyle K_{1}nl\log nl$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid
f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$ $\displaystyle
2^{K_{1}nl\log nl}$
$\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid
f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$
It only remains to establish the claim for the size of transition formulas.
Since $\mathcal{A}_{s}^{\varphi}$ is nondeterministic, formulas $\delta(q,a)$
are written in disjunctive normal form and for every direction $x\in
S_{\varphi}$ each disjunct contains exactly one element of $\\{x\\}\times Q$,
where $Q$ is the set of states in $\mathcal{A}_{s}^{\varphi}$. As a result,
each formula $\delta(q,a)$ is of size
$\displaystyle|\delta(q,a)|$
$\displaystyle\leq|Q|^{|S_{\varphi}|}(2|S_{\varphi}|-1)+|Q|^{|S_{\varphi}|}-1$
$\displaystyle\leq 2|S_{\varphi}||Q|^{|S_{\varphi}|}$
$\displaystyle|\delta(q,a)|$ $\displaystyle\leq
2^{H(\varphi)|\varphi|}|\mathcal{S}||Q|^{|\mathcal{S}|}$
∎
|
2024-09-04T02:54:58.721052 | 2020-03-06T13:14:33 | 2003.04738 | {
"authors": "Jens Hesse",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26135",
"submitter": "Jens Hesse",
"url": "https://arxiv.org/abs/2003.04738"
} | arxiv-papers | # EKOR strata on Shimura varieties with parahoric reduction
Jens Hesse Technische Universität Darmstadt<EMAIL_ADDRESS>
###### Abstract
We investigate the geometry of the special fiber of the integral model of a
Shimura variety with parahoric level at a given prime place.
To be more precise, we deal with the EKOR stratification which interpolates
between the Ekedahl-Oort and Kottwitz-Rapoport stratifications. In the Siegel
case we give a geometric description by suitably generalizing the theory of
$G$-zips of Moonen, Wedhorn, Pink and Ziegler to our context.
###### Contents
1. 1 Background
1. 1.1 Shimura data of Hodge type
2. 1.2 Bruhat-Tits buildings
3. 1.3 Bruhat-Tits group schemes
4. 1.4 Siegel integral models
5. 1.5 Local structure of the integral model
1. 1.5.1 Generizations and irreducible components
6. 1.6 The local model
1. 1.6.1 The Siegel case
2. 1.6.2 The relation between the integral and the local model
3. 1.6.3 The Pappas-Zhu construction
2. 2 EKOR strata and zips in the case of parahoric reduction
1. 2.1 The Ekedahl-Oort, Kottwitz-Rapoport and EKOR stratifications
1. 2.1.1 Iwahori-Weyl group and the admissible subset
2. 2.1.2 Kottwitz-Rapoport stratification
3. 2.1.3 Ekedahl-Oort stratification
4. 2.1.4 EKOR stratification
2. 2.2 $\overline{\mathcal{G}}_{K}$-zips in the Siegel case
1. 2.2.1 Preliminaries
2. 2.2.2 Lattice chains, zips, admissibility
3. 2.2.3 An explicit description of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$
4. 2.2.4 $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ in the Siegel case
5. 2.2.5 The example of $\operatorname{GSp}(4)$
Introduction
Shimura varieties are objects of arithmetic geometry (namely varieties over
number fields) that naturally arise in the search for generalized, non-abelian
reciprocity laws (i.e., in the Langlands program) and as moduli spaces of
abelian varieties (with certain extra structures on them). One way of
approaching these objects is to try to understand their mod-$p$ reduction
(which has to be carefully defined first). Insofar as a moduli interpretation
in the above sense exists and continues to exist likewise for the mod-$p$
reduction111There need not be a _literal_ moduli interpretation, but in any
event the stratifications in question derive from a close connection to moduli
problems., it allows us to stratify the moduli space according to several
invariants of the abelian varieties parametrized, e.g., the isomorphism
classes of their $p$-torsion. (An important observation is that these
stratifications genuinely live in the characteristic $p$ world, making use of
Frobenius endomorphisms and so on.) This, very roughly, is the general theme
everything in this article revolves around.
More precisely, we will be dealing with Shimura varieties of Hodge type and
parahoric level structure, at some fixed prime $v\mid p$ of the number field
over which the Shimura variety is defined. Under some reasonably mild
assumptions, cf. 1.17, Kisin and Pappas [KP15] constructed a canonical
integral model for such a Shimura variety. We try to understand some aspects
of the geometry of the special fiber of said integral model, namely the EKOR
strata (an interpolation between the Ekedahl-Oort strata, which in the case of
hyperspecial level are roughly the patches where the isomorphism class of the
$p$-torsion associated with the abelian variety is constant, and the Kottwitz-
Rapoport strata, which roughly are the patches where the Hodge filtration
looks constant) and defining them in a geometrical way.
Let us now go into more detail.
On the integral model $\mathscr{S}_{K}$ ($K$ parahoric level) we have a
“universal” abelian scheme (the quotation marks indicating that it is not
really universal for some moduli problem on $\mathscr{S}_{K}$, but it comes
from a universal abelian scheme via pullback) and we have various kinds of
Hodge tensors. We also have a “universal” isogeny chain of abelian schemes
tightly connected to the “universal” abelian scheme.
The overarching goal (and what we meant above by “defining the EKOR strata in
a geometrical way”) is to construct a “nice” algebraic stack
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ and a “nice” morphism
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$
from the mod-$p$ reduction of the Shimura variety to it, such that the fibers
are the EKOR strata. Shen, Yu and Zhang [SYZ19] solved this problem on
individual Kottwitz-Rapoport strata and globally after perfection, but not in
the form stated here (i.e., globally without passing to perfections). In the
Siegel case we propose a solution which specializes to that of Shen, Yu and
Zhang on Kottwitz-Rapoport strata, and should not be difficult to generalize
to many (P)EL cases. We show that
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$
is surjective. However, we have to leave the question of whether
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$
is smooth (which would be part of “nice”) an open conjecture.
For hyperspecial level, the EKOR stratification agrees with the Ekedahl-Oort
stratification, and the goal just set out is achieved by the stack of
$\overline{\mathcal{G}}_{K}$-zips, first defined in special cases by Moonen
and Wedhorn in [MW04] and then generally by Pink, Wedhorn and Ziegler in
[PWZ11, PWZ15]; the relation to Shimura varieties being established in
increasing generality in [MW04], by Viehmann and Wedhorn in [VW13], and
finally by Zhang in [Zha15].
One way of looking at the transition from hyperspecial level to general
parahoric level (at the very least in nice enough (P)EL cases) is from the
point of view of moduli problems of abelian varieties with extra structure,
where in the hyperspecial case we are really dealing just with that and in the
general case we are dealing with isogeny chains of abelian varieties with
extra structure, indexed by lattice chains coming from the Bruhat-Tits
building of the reductive $p$-adic Lie group in question. The basic idea in
generalizing zips from the hyperspecial to the general parahoric case then is
that one should be dealing with chains of zips in the old sense.
The zip of an abelian variety encodes the following information: the Hodge
filtration, the conjugate filtration, and the Cartier isomorphism relating the
two. In the general case, every abelian variety in the isogeny chain has a
Hodge filtration, a conjugate filtration and a Cartier isomorphism. Problems
now arise because we are dealing with $p$-primary isogenies on $p$-torsion
points, implying that the transition morphisms in these chains have non-
vanishing kernels. This introduces additional difficulty compared to the
hyperspecial case; there is a naive way of defining a zip stack, but
eventually we need to consider a certain admissible locus in it, which so far
suffers from the absence of a nice moduli description. Passing to perfections
however simplifies things and allows us to prove that the admissible locus is
closed. From here we arrive at the stack that we are really interested in by
dividing out a certain group action involving the unipotent radical of the
special fiber of the parahoric group scheme. A careful inspection shows that
on Kottwitz-Rapoport strata we arrive at the same result as in [SYZ19].
To sum up the results,
###### Theorem A:
In the Siegel case, there is an algebraic stack
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ and a surjective morphism
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$,
whose fibers are the EKOR strata and such that on Kottwitz-Rapoport strata,
one gets the stack and map constructed in [SYZ19].
For $\operatorname{GSp}(4)$ we do some calculations to illustrate the theory;
section 2.2.5.
### Acknowledgements
This article essentially is an extract of my doctoral thesis [Hes20] (another
extract222In particular, there is a large overlap between the “Background”
sections of the two articles., dealing with the foliation into central leaves,
is [Hes20a]). I thank Torsten Wedhorn for suggesting the topic of the
dissertation, his support and patiently answering my questions. Moreover I
thank Eva Viehmann and Paul Hamacher for their hospitality and helpful
discussions during a month-long stay in Munich at the TU München. I am also
grateful to Timo Richarz and Timo Henkel for numerous helpful discussions.
This research was supported by the Deutsche Forschungsgemeinschaft (DFG),
project number WE 2380/5.
## 1 Background
### 1.1 Shimura data of Hodge type
This article deals with aspects of the geometry of Shimura varieties (of Hodge
type), which are the (systems of) varieties associated with Shimura data (of
Hodge type).
###### Definition 1.1.
A _Shimura datum of Hodge type_ is a pair $(G,X)$, where $G$ is a reductive
algebraic group over $\mathbb{Q}$ and
$X\subseteq\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})$
is a $G(\mathbb{R})$-conjugacy class
($\mathbb{S}:=\operatorname{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}$
being the Deligne torus) subject to the following conditions:
1. (1)
For $h\in X$, the induced Hodge structure
$\mathbb{S}\xrightarrow{h}G_{\mathbb{R}}\xrightarrow{\mathrm{Ad}}\operatorname{GL}(\operatorname{Lie}(G_{\mathbb{R}}))$
satisfies
$\operatorname{Lie}(G_{\mathbb{C}})=\operatorname{Lie}(G_{\mathbb{C}})^{-1,1}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{0,0}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{1,-1}$.
2. (2)
$\operatorname{int}(h(i))\colon G^{\mathrm{ad}}_{\mathbb{R}}\to
G^{\mathrm{ad}}_{\mathbb{R}}$ is a Cartan involution, i.e., $\\{g\in
G^{\mathrm{ad}}(\mathbb{C})\;|\;gh(i)=h(i)\overline{g}\\}$ is compact. Another
way of phrasing this condition: Every finite-dimensional real representation
$V$ of $G^{\mathrm{ad}}_{\mathbb{R}}$ carries a
$G^{\mathrm{ad}}_{\mathbb{R}}$-invariant bilinear form $\varphi$ such that
$(u,v)\mapsto\varphi(u,h(i)v)$ is symmetric and positive definite. It is
enough to show that this holds for one _faithful_ finite-dimensional real
representation $V$.
3. (3)
$G^{\mathrm{ad}}$ _cannot_ be non-trivially written as $G^{\mathrm{ad}}\cong
H\times I$ over $\mathbb{Q}$ with $\mathbb{S}\to
G_{\mathbb{R}}\xrightarrow{\mathrm{proj}}H_{\mathbb{R}}$ trivial.
4. (4)
There exists an embedding
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$, where
$(\operatorname{GSp}(V),S^{\pm})$ is the Shimura datum associated with a
finite-dimensional symplectic $\mathbb{Q}$-vector space $V$ (see below). That
is, we have an embedding $G\hookrightarrow\operatorname{GSp}(V)$ of
$\mathbb{Q}$-group schemes such that the induced map
$\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})\hookrightarrow\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},\operatorname{GSp}(V_{\mathbb{R}}))$
restricts to a map $X\hookrightarrow S^{\pm}$.
###### Example 1.2.
Let $W$ be a finite-dimensional $\mathbb{R}$-vector space.
$\mathbb{R}$-group homomorphisms $\mathbb{S}\to\operatorname{GL}(W)$ then
correspond to Hodge decompositions of $W$, i.e., to decompositions
$W_{\mathbb{C}}=\oplus_{(p,q)\in\mathbb{Z}^{2}}W_{\mathbb{C}}^{p,q}$, such
that $W_{\mathbb{C}}^{p,q}$ is the complex conjugate of $W_{\mathbb{C}}^{q,p}$
for all $(p,q)\in\mathbb{Z}^{2}$. Under this correspondence,
$h\colon\mathbb{S}\to\operatorname{GL}(W)$ corresponds to the Hodge
decomposition $W_{\mathbb{C}}^{p,q}=\\{w\in W_{\mathbb{C}}\;|\;\forall
z\in\mathbb{S}(\mathbb{R})=\mathbb{C}^{\times}\colon
h(z)w=z^{-p}\bar{z}^{-q}w\\}$. Hodge decompositions of $W$ of type
$(-1,0)+(0,-1)$ correspond to complex structures on $W$: If
$h\colon\mathbb{S}\to\operatorname{GL}(W)$ yields such a Hodge decomposition,
then $h(i)$ gives an $\mathbb{R}$-endomorphism $J$ of $W$ with $J\circ
J=-\operatorname{id}_{W}$.
Let $V=(V,\psi)$ be a finite-dimensional symplectic $\mathbb{Q}$-vector space.
We say that a complex structure $J$ on $V_{\mathbb{R}}$ is positive (resp.
negative) if $\psi_{J}:=\psi_{\mathbb{R}}(\\_,J\\_)$ is a positive definite
(resp. negative definite) symmetric bilinear form on $V_{\mathbb{R}}$. Define
$S^{+}$ (resp. $S^{-}$) to be the set of positive (resp. negative) complex
structures on $(V_{\mathbb{R}},\psi_{\mathbb{R}})$ and $S^{\pm}:=S^{+}\sqcup
S^{-}$.
We can make this more concrete: A symplectic basis of
$(V_{\mathbb{R}},\psi_{\mathbb{R}})$ is a basis $e_{1},\dotsc,\allowbreak
e_{g},e_{-g},\dotsc,\allowbreak e_{-1}$, such that $\psi_{\mathbb{R}}$ is of
the form
$\begin{pmatrix}&\tilde{I}_{g}\\\ -\tilde{I}_{g}&\end{pmatrix}$
with respect to this basis, where $\tilde{I}_{g}=\begin{pmatrix}&&1\\\
&\iddots&\\\ 1&&\end{pmatrix}$ is the antidiagonal identity
matrix.333Occasionally (in particular when doing concrete matrix
calculations), it is more convenient to number the basis vectors
$1,\dotsc,g,-1,\dotsc,-g$ instead of $1,\dotsc,g,-g,\dotsc,-1$. Then the
standard symplectic form is given by $\left(\begin{smallmatrix}&I_{g}\\\
-I_{g}&\end{smallmatrix}\right)$, $I_{g}$ being the $g\times g$ identity
matrix.
Let $J$ be the endomorphism of $V_{\mathbb{R}}$ of the form
$\begin{pmatrix}&-\tilde{I}_{g}\\\ \tilde{I}_{g}&\end{pmatrix}$
with respect to this basis. Then $J\in S^{+}$ and what we have described is a
surjective map
$\\{\text{symplectic bases of
}(V_{\mathbb{R}},\psi_{\mathbb{R}})\\}\twoheadrightarrow S^{+}.$
In particular we see that
$\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=\psi_{\mathbb{R}}\\}$
(by virtue of acting simply transitively on the symplectic bases) acts
transitively on
$S^{+}\cong\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})/\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J)$
(where we define
$\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J):=\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})\cap
O(V_{\mathbb{R}},\psi_{J})=U((V_{\mathbb{R}},J),\psi_{J})$ for a fixed choice
of $J\in S^{+}$) and therefore the general symplectic group
$\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=c\cdot\psi_{\mathbb{R}}\text{
for some }c\in\mathbb{R}^{\times}\\}$ acts transitively on $S^{\pm}$ (note
that the element of the form $e_{\pm i}\mapsto e_{\mp i}$ of
$\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}})$ for any given choice of
symplectic basis $\left(e_{i}\right)_{i}$ permutes $S^{+}$ and $S^{-}$).
###### Definition 1.3.
Condition (1) of Definition 1.1 implies that the action of
$\mathbb{G}_{m,\mathbb{R}}$ (embedded in $\mathbb{S}$ in the natural way) on
$\operatorname{Lie}(G_{\mathbb{R}})$ is trivial, so that $h$ induces a
homomorphism
${w\colon\mathbb{G}_{m,\mathbb{R}}\to\operatorname{Cent}(G_{\mathbb{R}})}$.
This homomorphism is independent of the choice of $h\in X$ and is called the
_weight homomorphism_ of $(G,X)$.
Moreover, we denote by $\\{\mu\\}$ the the $G(\mathbb{C})$-conjugacy class of
the cocharacter
$\mu_{h}:=h\circ(\operatorname{id}_{\mathbb{G}_{m,\mathbb{C}}},1)\colon\mathbb{G}_{m,\mathbb{C}}\to\mathbb{G}_{m,\mathbb{C}}^{2}\cong\mathbb{S}_{\mathbb{C}}\to
G_{\mathbb{C}}$, where $h$ is as above. Obviously, the conjugacy class
$\\{\mu\\}$ is independent of the particular choice of $h\in X$.
###### Remark 1.4.
Let $L/\mathbb{Q}$ be a field extension such that $G_{L}$ contains a split
maximal torus $T$. Let $W:=\operatorname{Norm}_{G(L)}(T)/T$ be the Weyl group.
Then the natural map
$W\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},T)\to
G(L)\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},G_{L})$
is bijective.
Since the left hand side remains unchanged if we go from $L=\bar{\mathbb{Q}}$
(where as usual $\bar{\mathbb{Q}}$ denotes an algebraic closure of
$\mathbb{Q}$) to $L=\mathbb{C}$, we see that $\\{\mu\\}$ contains a
cocharacter defined over $\bar{\mathbb{Q}}$ and that we may then also consider
$\\{\mu\\}$ as a $G(\bar{\mathbb{Q}})$-conjugacy class.
###### Definition 1.5.
The _reflex field_ $\mathbf{E}=\mathbf{E}(G,X)$ of $(G,X)$ is the field of
definition of $\\{\mu\\}$, i.e., the fixed field in $\bar{\mathbb{Q}}$ of
$\\{\gamma\in\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\;|\;\gamma(\\{\mu\\})=\\{\mu\\}\\}$.
###### Example 1.6.
The reflex field of the Shimura datum
$(\operatorname{GSp}_{2g,\mathbb{Q}},S^{\pm})$ of Example 1.2 is $\mathbb{Q}$.
To wit, one of the cocharacters in the conjugacy class $\\{\mu\\}$ is
$\mu(z)=\left(\begin{smallmatrix}z&&&&&\\\ &\ddots&&&&\\\ &&z&&&\\\ &&&1&&\\\
&&&&\ddots&\\\ &&&&&1\end{smallmatrix}\right).$
###### Notation 1.7.
We denote the ring of (rational) adeles by
$\mathbb{A}:=\mathbb{A}_{\mathbb{Q}}$, the subring of finite adeles by
$\mathbb{A}_{f}:=\mathbb{A}_{\mathbb{Q},f}$ and the subring of finite adeles
away from some fixed prime $p$ by $\mathbb{A}_{f}^{p}$.
###### Definition and Remark 1.8.
Let $K\subseteq G(\mathbb{A}_{f})$ be a compact open subgroup. The _Shimura
variety of level $K$ associated with $(G,X)$_ is the double coset space
$\operatorname{Sh}_{K}(G,X):=G(\mathbb{Q})\backslash(X\times(G(\mathbb{A}_{f})/K)).$
A priori, this is just a set, but if $K$ is sufficiently small (i.e., “neat”
in the sense of [Bor69, Pin90]), $\operatorname{Sh}_{K}(G,X)$ can be
canonically written as a finite disjoint union of hermitian symmetric
domains.444If $K$ fails to be sufficiently small, one might very reasonably
argue that our definition of the Shimura variety of level $K$ really is the
definition of the _coarse_ Shimura variety and that one should be working with
stacks instead. Since we will only be interested in sufficiently small level,
this is inconsequential for us. In particular, this gives
$\operatorname{Sh}_{K}(G,X)$ the structure of a complex manifold.
In fact, by the theorem of Baily-Borel, this complex manifold attains the
structure of a quasi-projective complex variety in a canonical way.
By work of Deligne, Milne and Borovoi, this variety is defined already (and
again in a canonical way) over the reflex field $\mathbf{E}$. So in
particular, it is defined over a number field independent of $K$. This is
important when varying $K$ and it is the reason why we consider the whole
Shimura variety instead of its connected components over $\mathbb{C}$ on their
own. It is possible for the Shimura variety to have multiple connected
components over $\mathbb{C}$ while being connected over $\mathbf{E}$.
More detailed explanations may be found in [Mil05].
### 1.2 Bruhat-Tits buildings
Let $K$ be a complete discrete valuation field with ring of integers
$\mathcal{O}$, uniformizer $\varpi$ and perfect residue field
$\kappa:=\mathcal{O}/\varpi$.
###### Notation 1.9.
For a (connected) reductive group $G$ over $K$, we denote by
$\mathcal{B}(G,K)$ the extended (or enlarged) and by
$\mathcal{B}^{\mathrm{red}}(G,K)$ the reduced (i.e., non-extended) Bruhat-Tits
building of $G$ over $K$ [BT84]. Moreover,
$\mathcal{B}^{\mathrm{abstract}}(G,K)$ denotes the underlying abstract
simplicial complex.
###### Remark 1.10.
Let $V$ be a finite-dimensional $K$-vector space.
As described in [KP15, 1.1.9] (originally in [BT84a]), the points of
$\mathcal{B}(\operatorname{GL}(V),K)$ correspond to graded periodic lattice
chains $(\mathcal{L},c)$, i.e.,
* •
$\emptyset\neq\mathcal{L}$ is a totally ordered set of full
$\mathcal{O}$-lattices in $V$ stable under scalar multiplication (i.e.,
$\Lambda\in\mathcal{L}\iff\varpi\Lambda\in\mathcal{L}$),
* •
$c\colon\mathcal{L}\to\mathbb{R}$ is a strictly decreasing function such that
$c(\varpi^{n}\Lambda)=c(\Lambda)+n$.
###### Remark 1.11.
Fix such an $\mathcal{L}$ and let $\Lambda^{0}\in\mathcal{L}$. Then every
homothety class of lattices has a unique representative $\Lambda$ such that
$\Lambda\subseteq\Lambda^{0}$ and $\Lambda\not\subseteq\varpi\Lambda^{0}$.
Consider such representatives $\Lambda^{i}$ for all of the distinct homothety
classes of lattices that make up $\mathcal{L}$. Because $\mathcal{L}$ is
totally ordered and $\Lambda^{i}\not\subseteq\varpi\Lambda^{0}$, it follows
that $\Lambda^{i}\supseteq\varpi\Lambda^{0}$ for all $i$ and that
$\left\\{\Lambda^{i}/\varpi\Lambda^{0}\right\\}_{i}$ is a flag of non-trivial
linear subspaces of $\Lambda^{0}/\varpi\Lambda^{0}\cong\kappa^{n}$, where
$n:=\dim V$. Consequently, the number $r$ of homothety classes is in
$\\{1,\dotsc,n\\}$; it is called the _period length_ (or _rank_) of
$\mathcal{L}$. Numbering the $\Lambda^{i}$ in descending order we hence obtain
$r$ lattices $\Lambda^{0},\Lambda^{1},\dotsc,\Lambda^{r-1}$ such that
$\Lambda^{0}\supsetneqq\Lambda^{1}\supsetneqq\dotsb\supsetneqq\Lambda^{r-1}\supsetneqq\varpi\Lambda^{0}$
(1.12)
and $\mathcal{L}$ is given by the the strictly descending sequence of lattices
$\Lambda^{qr+i}=\varpi^{q}\Lambda^{i},\quad q\in\mathbb{Z},\;0\leq i<r.$
###### Remark 1.13.
Let $V$ be a finite-dimensional symplectic $K$-vector space.
$\mathcal{B}(\operatorname{GSp}(V),K)$ embeds into the subset of
$\mathcal{B}(\operatorname{GL}(V),K)$ consisting of those $(\mathcal{L},c)$
such that $\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}$.
Passing to the underlying abstract simplicial complexes means forgetting about
the grading $c$ and
$\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)=\\{\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GL}(V),K)\;|\;\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}\\}.$
If $\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)$
and $\\{\Lambda^{i}\\}_{i}$ is as in Remark 1.11, then there is an involution
$t\colon\mathbb{Z}\to\mathbb{Z}$ with
$\left(\Lambda^{i}\right)^{\vee}=\Lambda^{t(i)}$, $t(i+qr)=t(i)-qr$, and
$i<j\implies t(i)>t(j)$. So $-a:=t(0)>t(1)>\dotsb>t(r)=-a-r$, which implies
$t(i)=-i-a$. Thus $i_{0}-t(i_{0})=2i_{0}+a\in\\{0,1\\}$ for some unique
$i_{0}\in\mathbb{Z}$. Hence, upon renumbering the $\Lambda^{i}$, we may assume
that $a\in\\{0,1\\}$.
We therefore have
$\displaystyle\varpi\Lambda^{0}\subsetneqq\Lambda^{r-1}\subsetneqq\Lambda^{r-2}\subsetneqq\dotsb\subsetneqq\Lambda^{0}\subseteq\left(\Lambda^{0}\right)^{\vee}=\Lambda^{-a}\subsetneqq\left(\Lambda^{1}\right)^{\vee}=\Lambda^{-1-a}$
$\displaystyle\subsetneqq\dotsb\subsetneqq\left(\Lambda^{r-1}\right)^{\vee}=\Lambda^{-r+1-a}\subseteq\Lambda^{-r}=\varpi^{-1}\Lambda^{0}.$
###### Example 1.14.
See also section 2.2.5 for some elaborations on the building of
$\operatorname{GSp}_{4}(\mathbb{Q}_{p})$.
### 1.3 Bruhat-Tits group schemes
###### Notation 1.15.
Let $E$ be a finite field extension of $\mathbb{Q}_{p}$.
Denote by $\breve{E}$ the completion of the maximal unramified extension of
$E$ (hence $\breve{E}=E\cdot\breve{\mathbb{Q}}_{p}$).
###### Remark 1.16.
If $E/\mathbb{Q}_{p}$ is unramified, then ${{\cal
O}_{\breve{E}}}=W(\bar{\mathbb{F}}_{p})$, $\bar{\mathbb{F}}_{p}$ denoting an
algebraic closure of $\mathbb{F}_{p}$ and
$W\colon\mathrm{Ring}\to\mathrm{Ring}$ being the ($p$-adic) Witt vectors
functor. This generalizes to the ramified case using _ramified Witt vectors_
instead, see e.g. [Haz78, Chap. IV, (18.6.13)] or [Ahs11, Chapter 1].
Let $(G,X)$ be a Shimura datum of Hodge type, let
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$ be an embedding as in
Definition 1.1 (4), and let $x\in\mathcal{B}(G,\mathbb{Q}_{p})$ be a point in
the Bruhat-Tits building of $G$ over $\mathbb{Q}_{p}$.
We consider the associated Bruhat-Tits scheme ${\cal G}_{x}$, i.e., the affine
smooth model of $G_{\mathbb{Q}_{p}}$ over $\mathbb{Z}_{p}$ such that ${\cal
G}_{x}(\breve{\mathbb{Z}}_{p})\subseteq G(\breve{\mathbb{Q}}_{p})$ is the
stabilizer of the facet of $x$ in ${\cal
B}(G,\breve{\mathbb{Q}}_{p})\overset{\text{\cite[cite]{[\@@bibref{}{landvogt}{}{},
Prop.\leavevmode\nobreak\
2.1.3]}}}{=}\mathcal{B}(G,\mathbb{Q}^{\mathrm{ur}}_{p})$. Let $K_{p}:={\cal
G}_{x}(\mathbb{Z}_{p})\subseteq G(\mathbb{Q}_{p})$ and let $K^{p}\subseteq
G(\mathbb{A}_{f}^{p})$ be a sufficiently small open compact subgroup. Define
$K:=K_{p}K^{p}\subseteq G(\mathbb{A}_{f})$.
###### Assumptions 1.17.
From now on, we will always make the following assumptions:
* •
$\mathcal{G}_{x}=\mathcal{G}_{x}^{\circ}$ is connected.
* •
$G$ splits over a tamely ramified extension of $\mathbb{Q}_{p}$.
* •
$p\nmid\\#\pi_{1}(G^{\mathrm{der}})$.
###### Notation 1.18.
In order not to make notation overly cumbersome, we usually denote the base
change $G_{\mathbb{Q}_{p}}$ of $G$ to $\mathbb{Q}_{p}$ by $G$ again. (Later,
we will almost exclusively be dealing with $G_{\mathbb{Q}_{p}}$.)
### 1.4 Siegel integral models
With notation as above let
$\displaystyle N_{p}$
$\displaystyle:=\operatorname{Stab}_{\operatorname{GSp}(V)(\mathbb{Q}_{p})}(\mathcal{L})\quad\text{(as
before)},$ $\displaystyle J_{p}$
$\displaystyle:=\operatorname{Stab}_{\operatorname{GL}(V^{\S})(\mathbb{Q}_{p})}(\Lambda^{\S})\cap\operatorname{GSp}(V^{\S})(\mathbb{Q}_{p}).$
Let $N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$ and
$J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ be sufficiently
small open compact subgroups, and $N:=N_{p}N^{p}$, $J:=J_{p}J^{p}$.
In this subsection, we are going to describe integral models of
$\operatorname{Sh}_{N}(\operatorname{GSp}(V),S^{\pm})$ and of
$\operatorname{Sh}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ over
$\mathbb{Z}_{(p)}$ and relate the two.
###### Remark 1.19.
By [RZ96, Definition 6.9], the integral model
$\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})$ is given by the moduli
problem $(\mathbb{Z}_{(p)}\text{-scheme})\ni
S\mapsto\left\\{(A,\bar{\lambda},\eta^{p})\right\\}/{\scriptstyle\cong}$,
where:
1. (a)
$A=\left(A_{\Lambda}\right)_{\Lambda\in\mathcal{L}}$ is an $\mathcal{L}$-set
of abelian schemes, i.e.,
* •
for every $\Lambda\in\mathcal{L}$, an abelian $S$-scheme up to
$\mathbb{Z}_{(p)}$-isogeny $A_{\Lambda}$ (i.e., $A_{\Lambda}$ is an object of
the category $(\text{abelian }S\text{-schemes})\otimes\mathbb{Z}_{(p)}$, where
the category $\mathcal{A}\otimes R$ for $\mathcal{A}$ an preadditive category
and $R$ a ring has the same objects as $\mathcal{A}$ and
$\operatorname{Hom}_{\mathcal{A}\otimes
R}(X,Y)=\operatorname{Hom}(X,Y)\otimes_{\mathbb{Z}}R$ for all objects $X,Y$),
* •
for every inclusion $\Lambda_{1}\subseteq\Lambda_{2}$ a
$\mathbb{Z}_{(p)}$-isogeny $\rho_{\Lambda_{2},\Lambda_{1}}\colon
A_{\Lambda_{1}}\to A_{\Lambda_{2}}$,
* •
$\rho_{\Lambda_{3},\Lambda_{1}}=\rho_{\Lambda_{3},\Lambda_{2}}\circ\rho_{\Lambda_{2},\Lambda_{1}}$
if $\Lambda_{1}\subseteq\Lambda_{2}\subseteq\Lambda_{3}$ in $\mathcal{L}$,
* •
the height of $\rho_{\Lambda_{2},\Lambda_{1}}$ is
$\log_{p}|\Lambda_{2}/\Lambda_{1}|$. Here $\rho_{\Lambda_{2},\Lambda_{1}}$
gives rise to a well-defined homomorphism of $p$-divisible groups, and what we
mean is that the kernel of this homomorphism (which is a finite locally free
commutative group scheme, which we also refer to simply as the kernel of
$\rho_{\Lambda_{2},\Lambda_{1}}$) is to have order
$|\Lambda_{2}/\Lambda_{1}|$.
* •
For every $\Lambda\in\mathcal{L}$, there is an isomorphism (called
_periodicity isomorphism_) $\theta_{\Lambda}\colon A_{\Lambda}\to
A_{p\Lambda}$ such that
$\rho_{\Lambda,p\Lambda}\circ\theta_{\Lambda}=[p]\colon A_{\Lambda}\to
A_{\Lambda}$ is the multiplication-by-$p$ isogeny.
2. (b)
$\bar{\lambda}\colon A\to\tilde{A}$ is a $\mathbb{Q}$-homogeneous principal
polarization, i.e., a $\underline{\mathbb{Q}^{\times}}$-orbit of a principal
polarization $\lambda\colon A\to\tilde{A}$. Here $\tilde{A}$ is the
$\mathcal{L}$-set of abelian schemes over $S$ up to prime-to-$p$ isogeny given
by $\tilde{A}_{\Lambda}:=(A_{\Lambda^{\vee}})^{\vee}$. And being a
polarization $\lambda$ means being a quasi-isogeny of $\mathcal{L}$-sets
$\lambda\colon A\to\tilde{A}$ such that
$A_{\Lambda}\xrightarrow{\lambda_{\Lambda}}\tilde{A}_{\Lambda}=(A_{\Lambda^{\vee}})^{\vee}\xrightarrow{\varrho_{\Lambda^{\vee},\Lambda}^{\vee}}(A_{\Lambda})^{\vee}$
is a polarization of $A_{\Lambda}$ for all $\Lambda$. If $\lambda_{\Lambda}$
can be chosen to be an isomorphism up to prime-to-$p$ isogeny for all
$\Lambda$, then we speak of a principal polarization. In that case, when
referring to $\lambda_{\Lambda}$, we mean a $\lambda_{\Lambda}$ which is an
isomorphism up to prime-to-$p$ isogeny.
3. (c)
$\eta^{p}$ is a level-$N^{p}$-structure, i.e. (if $S$ is connected), it is a
$\pi_{1}(S,s)$-invariant $N^{p}$-orbit of symplectic similitudes
$\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to H_{1}(A_{s},\mathbb{A}_{f}^{p})$
(where $s$ is some geometric basepoint and $H_{1}(A_{s},\mathbb{A}_{f}^{p})$
with its $\pi_{1}(S,s)$-action corresponds to the Tate
$\mathbb{A}_{f}^{p}$-module of $A$ (cf. [RZ96, 6.8]), which is a smooth
$\mathbb{A}_{f}^{p}$-sheaf). Note that this forces the abelian schemes
$A_{\Lambda}$ to be $(\dim_{\mathbb{Q}}V)$-dimensional.
###### Definition 1.20.
Set $\Lambda^{\S}_{\mathbb{Z}_{(p)}}:=\Lambda^{\S}_{\mathbb{Z}_{p}}\cap
V^{\S}_{\mathbb{Q}}=\prod_{i=-(r-1)-a}^{r-1}\Lambda_{\mathbb{Z}_{(p)}}^{i}$.
We choose a lattice $\Lambda^{\S}_{\mathbb{Z}}\subseteq V^{\S}$ such that
$\Lambda^{\S}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}=\Lambda^{\S}_{\mathbb{Z}_{(p)}}$
and $\Lambda^{\S}_{\mathbb{Z}}\subseteq(\Lambda^{\S}_{\mathbb{Z}})^{\vee}$.
###### Remark 1.21.
Set
$d:=\bigl{|}\left(\Lambda_{\mathbb{Z}}^{\S}\right)^{\vee}/\Lambda_{\mathbb{Z}}^{\S}\bigr{|}$.
By [Kis10, 2.3.3, 3.2.4], the integral model
$\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ is given by the
moduli problem $(\mathbb{Z}_{(p)}\text{-schemes})\ni
S\mapsto\left\\{(A^{\S},\lambda^{\S},\epsilon^{p})\right\\}/{\scriptstyle\cong}$,
where
1. (a)
$A^{\S}$ is an abelian scheme over $S$ up to $\mathbb{Z}_{(p)}$-isogeny,
2. (b)
$\lambda^{\S}\colon A^{\S}\to\left(A^{\S}\right)^{\vee}$ is a polarization of
degree $d$ (i.e., the polarization of the (well-defined) associated
$p$-divisible group has degree $d$),
3. (c)
$\epsilon^{p}$ is a level-$J^{p}$-structure, i.e. (if $S$ is connected), it is
a $\pi_{1}(S,s)$-invariant $J^{p}$-orbit of symplectic similitudes
$\epsilon^{p}\colon V^{\S}_{\mathbb{A}_{f}^{p}}\to
H_{1}(A^{\S}_{s},\mathbb{A}_{f}^{p})$. Note that this forces the abelian
schemes $A^{\S}$ to be $(\dim_{\mathbb{Q}}V^{\S})$-dimensional.
This completes the descriptions of the moduli problems, and we turn to the
question of the relationship between the two. Consider (for appropriate
$N^{p},J^{p}$; see below) the morphism
$\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$
given on $S$-valued points by sending $(A,\bar{\lambda},\eta^{p})$ to
$(A^{\S},\lambda^{\S},\epsilon^{p})$, where
1. (a)
$\displaystyle A^{\S}:=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$,
2. (b)
$\displaystyle\lambda^{\S}:=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{\Lambda^{i}}\right)$,
3. (c)
$\epsilon^{p}$ is the product $\prod_{i=-(r-1)-a}^{r-1}\eta^{p}$, to be
interpreted as the product over $\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to
H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong
H_{1}(A_{s},\mathbb{A}_{f}^{p})$, where the isomorphism
$H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong
H_{1}(A_{s},\mathbb{A}_{f}^{p})$ is by definition the identity for some fixed
$i=i_{0}$ and otherwise induced by the transition map
$\rho_{\Lambda^{i},\Lambda^{i_{0}}}$. We need that $N^{p}$ is mapped into
$J^{p}$ by $\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$
for this to make sense.
###### Lemma 1.22.
Let $S$ be a scheme, $\ell\neq p$ prime numbers. If $\ell$ does not appear as
a residue characteristic of $S$, then the Tate module functors
$\displaystyle H_{1}(\\_,\mathbb{Z}_{\ell})$
$\displaystyle\colon(\text{abelian }S\text{-schemes})\to(\text{\~{A}©tale
}\mathbb{Z}_{\ell}\text{-local systems on }S),$ $\displaystyle
H_{1}(\\_,\mathbb{Q}_{\ell})$ $\displaystyle\colon(\text{abelian
}S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{Q}_{\ell}\text{-local systems
on }S)$
(cf. [Gro74, III, 5.4 and 6.2] for precise definitions) are faithful.
If only $p$ and $0$ appear as residue characteristics of $S$, then the Tate
module functor
$H_{1}(\\_,\mathbb{A}_{f}^{p})\colon(\text{abelian
}S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{A}_{f}^{p}\text{-local systems
on }S)$
is faithful.
###### Proof:
First note that the statements about $H_{1}(\\_,\mathbb{Q}_{\ell})$ and
$H_{1}(\\_,\mathbb{A}_{f}^{p})$ follows from the statement about
$H_{1}(\\_,\mathbb{Z}_{\ell})$, which is why it is enough to only look at
$H_{1}(\\_,\mathbb{Z}_{\ell})$.
A homomorphism of abelian $S$-schemes $f\colon A\to B$ vanishes if and only if
it vanishes over every (geometric) fiber of $S$: Indeed, if it vanishes
fiberwise, then it is flat by the fiber criterion for flatness. Applying that
criterion again we see that the closed immersion and fiberwise isomorphism
$\ker(f)\hookrightarrow A$ is flat, which means that is an isomorphism.
This way we are reduced to the case where $R$ is an (algebraically closed)
field of characteristic different from $\ell$. In this setting the
faithfulness is well-known (the salient point being that the $\ell$-primary
torsion is dense). □
###### Lemma 1.23.
Let $H$ be a totally disconnected locally compact555By (our) definition,
locally compact implies Hausdorff. group (i.e., a locally profinite group) and
let $N\subseteq H$ be a compact subgroup. Then
$N=\bigcap_{\begin{subarray}{c}N\subseteq J\\\ J\subseteq H\text{ open compact
subgrp.}\end{subarray}}J.$
Note that this is (a variant of) a well-known theorem by van Dantzig if
$N=\\{1\\}$ [Dan36].
###### Proof:
We make use of the following fact [AT08, Prop. 3.1.7]: A Hausdorff space is
locally compact and totally disconnected if and only if the open compact sets
form a basis of the topology. (Van Dantzig’s theorem is the group version of
this, which talks only about a neighborhood basis of the identity and open
compact _subgroups_.)
First we show that $N$ is contained in some open compact subset $K\subseteq
H$. For every $x\in N$ choose a compact open neighborhood $x\in K_{x}\subseteq
H$. This is possible by the fact cited above. Then there is a finite subset
$I\subseteq N$ such that $N\subseteq\bigcup_{x\in I}K_{x}=:K$.
Next, for every $x\in N$ choose an open neighborhood of the identity $U_{x}$
such that $xU_{x}K\subseteq K$. With $N\subseteq U:=\bigcup_{x\in N}xU_{x}$ we
obtain $UK\subseteq K$. Replacing $U$ by $U\cap U^{-1}$, we may moreover
assume it is symmetric. The subgroup generated by $U$ is open (hence closed)
and contained in $K$, hence is an open compact subgroup.
Thus $N$ even is contained in an open compact sub _group_ ; in other words, we
may assume that $H$ is compact, i.e., is a profinite group.
Then $H/N$ is compact666Hausdorff quotient spaces of compact spaces are
compact again, but for “locally compact” the analogous statement is not true
in general! and totally disconnected777Take $x,y\in H$ such that $xN\neq yN$.
We show that any subspace $S\subseteq H/N$ containing both $xN$ and $yN$ is
disconnected. Let $U\subseteq H/N$ be a neighborhood of $xN$ not containing
$yN$. Let $x\in V\subseteq\pi^{-1}(U)$ be open and compact, where $\pi\colon
H\to H/N$ is the projection. Then $yN\notin\pi(V)\subseteq H/N$ is open and
compact (hence closed) and we have $S=(\pi(V)\cap S)\sqcup S\setminus\pi(V)$
where both $\pi(V)\cap S$ and $S\setminus\pi(V)$ are open in $S$. This shows
that $S$ is disconnected. (i.e., is a Stone space). By the fact cited above,
$H/N\supseteq\\{1\\}=\bigcap_{L\subseteq H/N\text{ open compact subset}}L.$
Observe that the quotient map $H\to H/N$ is proper to deduce
$N=\bigcap_{\begin{subarray}{c}N\subseteq M\\\ M\subseteq H\text{ open compact
subset}\end{subarray}}M.$
Say $M$ is an open and compact subset of $H$ containing $N$. As we have shown
above, there is an open compact subgroup $J\subseteq H$ in between $N$ and
$M$, and this is all we need to complete the proof. □
###### Proposition 1.24.
For every compact open subgroup
$N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$
$\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$
is a well-defined morphism for all compact open subgroups $N^{p}\subseteq
J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ and is a closed
immersion for all sufficiently small compact open subgroups $N^{p}\subseteq
J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$.
###### Proof:
The fact that it’s well-defined is clear from the construction.
To show the second statement, as in [Del71, Prop. 1.15], it is enough to show
that
$\mathscr{S}_{N_{p}N^{p}}(\operatorname{GSp}(V),S^{\pm})\to\varprojlim_{J^{p}}\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$
is a closed immersion, i.e., a proper monomorphism.
We begin by proving that it is a monomorphism, i.e., injective on $S$-valued
points ($S$ arbitrary $\mathbb{Z}_{(p)}$-scheme). So, say
$(A_{1},\lambda_{1},\eta_{1}^{p})$ and $(A_{2},\lambda_{2},\eta_{2}^{p})$ both
map to $(A^{\S},\lambda^{\S},\epsilon_{J^{p}}^{p})$. That means precisely that
there is an isomorphism of abelian $S$-schemes up to
$\mathbb{Z}_{(p)}$-isogeny
$\phi\colon\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\cong}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}$
such that
$\phi^{\vee}\circ\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{2,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{2,\Lambda^{i}}\right)\circ\phi=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{1,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{1,\Lambda^{i}}\right)$
and
$H_{1}(\phi,\mathbb{A}_{f}^{p})\circ\epsilon_{1,J^{p}}^{p}=\epsilon_{2,J^{p}}^{p}\mod{J^{p}}.$
We claim that $\phi$ comes from isomorphisms
$\phi_{i}\colon A_{1,\Lambda^{i}}\xrightarrow{\cong}A_{2,\Lambda^{i}}.$
Certainly there is but one candidate for $\phi_{i}$: define $\phi_{i}$ to be
the composition
$A_{1,\Lambda^{i}}\xrightarrow{\mathrm{incl}}\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\phi}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}\xrightarrow{\mathrm{proj}}A_{2,\Lambda^{i}}.$
Our claim then is that
$\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}.$
Apply $H^{1}(\\_,\mathbb{A}_{f}^{p})$ on both sides. For the left hand side,
we have
$H_{1}(\phi,\mathbb{A}_{f}^{p})=\epsilon_{2,J^{p}}^{p}\circ\left(\epsilon_{1,J^{p}}^{p}\right)^{-1}\mod{J^{p}}.$
and the right hand side of this equation is block diagonal. So
$H_{1}(\phi,\mathbb{A}_{f}^{p})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{A}_{f}^{p})\mod{J^{p}}.$
Since (by Lemma 1.23)
$N^{p}=\bigcap_{\begin{subarray}{c}N_{\ell}\subseteq J_{\ell}\\\
J_{\ell}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{Q}_{\ell})\text{ cpt. open
subgrp.}\end{subarray}}J_{\ell},$
it follows that (with $\ell\neq p$)
$H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})\mod{N_{\ell}},$
hence (since $N_{\ell}$ acts block-diagonally) that
$H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})$.
Since $H_{1}(\\_,\mathbb{Q}_{\ell})$ is faithful (Lemma 1.22), this implies
$\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}$, as desired.
Next, consider the extension by zero of
$\left(H_{1}(\rho_{1/2,\Lambda^{j},\Lambda^{i}},\mathbb{A}_{f}^{p})\right)_{i,j}$
(where for “$1/2$” either “$1$” or “$2$” can be plugged in) to a map
$H_{1}(A^{\S},\mathbb{A}_{f}^{p})\to H_{1}(A^{\S},\mathbb{A}_{f}^{p})$. Under
the isomorphism given by the $J^{p}$-level structure this corresponds, up to
the $J^{p}$-action, to the map $V^{\S}_{\mathbb{A}_{f}^{p}}\to
V^{\S}_{\mathbb{A}_{f}^{p}}$ given by mapping the $i$’th copy of
$V_{\mathbb{A}_{f}^{p}}$ identically to the $j$’th copy and the rest to zero.
Thus $\rho_{1/2,i,j}$ yield the same up to $J^{p}$ after applying
$H_{1}(\\_,\mathbb{A}_{f}^{p})$, hence they are equal in the
$\mathbb{Z}_{(p)}$-isogeny category.
Consequently, $\chi$ is a monomorphism.
For properness, we will use the valuative criterion. Let $R$ be a discrete
valuation ring with field of fractions $K$ and assume that a $K$-point
$A^{\S}=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$ with its additional
structures coming from $(A_{\Lambda^{i}})_{i}$ extends to an $R$-point
$\mathcal{A}^{\S}$. Consider the map $A^{\S}\to A_{\Lambda^{i_{0}}}\to A^{\S}$
where the first map is a projection and the second an inclusion. By the Néron
mapping property, this extends to a map $\mathcal{A}^{\S}\to\mathcal{A}^{\S}$.
Define $\mathcal{A}_{\Lambda^{i_{0}}}$ to be the image of this map.
The Néron mapping property also allows us to extend the transition isogenies
$\rho_{\Lambda^{i_{0}},\Lambda^{j_{0}}}\colon\allowbreak{A_{\Lambda^{j_{0}}}\to
A_{\Lambda^{i_{0}}}}$, $i_{0}\leq j_{0}$, the periodicity isomorphisms, and
the polarization.
Since $\pi_{1}(\operatorname{Spec}K)$ surjects onto
$\pi_{1}(\operatorname{Spec}R)$ (see [Stacks, Tag 0BQM]), extending the level
structure away from $p$ is trivial. □
### 1.5 Local structure of the integral model
#### 1.5.1 Generizations and irreducible components
Let $\mathscr{X}\to\operatorname{Spec}\mathcal{O}_{\breve{E}}$ be a flat
scheme locally of finite type; denote the special fiber by
$X\to\operatorname{Spec}\bar{\mathbb{F}}_{p}$ and the generic fiber by
$\mathcal{X}\to\operatorname{Spec}\breve{E}$. We assume that $\mathcal{X}$ is
locally integral (e.g. smooth).
For example, we can consider
$(\mathscr{X},X,\mathcal{X})=(\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}},\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}\otimes_{{{\cal
O}_{\breve{E}}}}\bar{\mathbb{F}}_{p},\allowbreak{\operatorname{Sh}_{K}(G,X)\otimes_{E}\breve{E}})$.
Let $\bar{x}\in X(\bar{\mathbb{F}}_{p})$.
###### Lemma 1.25.
There is a generization $x$ of $\bar{x}$ which lies in the generic fiber
$\mathcal{X}$, and is a closed point in there, i.e., $x\in\mathcal{X}(L)$ for
a finite extension $L/\breve{E}$.
###### Definition 1.26.
We shall call such a point $x$ a _closed point generization_ of $\bar{x}$ for
short.
###### Proof:
Due to flatness (going-down) there is _some_ generization in the generic
fiber; call it $x_{0}$.
By [Stacks, Tag 053U] the following set is dense (and in particular non-empty)
in the closure of $\\{x_{0}\\}$ in $\mathcal{X}$:
$\left\\{x\in\mathscr{X}\;|\;x\text{ is a specialization of }x_{0}\text{ and a
closed point generization of }\bar{x}\right\\}.$
□
###### Lemma 1.27.
Notation as in the preceding lemma.
The specialization $x\leadsto\bar{x}$ can be realized by an ${\cal
O}_{L}$-valued point of $\mathscr{X}$.
###### Proof:
First off, by [EGA2, 7.1.9], it can be realized by a morphism
$\operatorname{Spec}R=\\{\eta,s\\}\to\mathscr{X}$ of ${{\cal
O}_{\breve{E}}}$-schemes, where $R$ is a discrete valuation ring such that
$L\cong\kappa(\eta)=\operatorname{Quot}(R)$ as field extensions of
$\kappa(x)$.
We hence get local homomorphisms of local rings ${{\cal
O}_{\breve{E}}}\to{\cal O}_{\mathscr{X},\bar{x}}\to R$.
Thus the discrete valuation on $L$ defined by $R$ extends the discrete
valuation on $\breve{E}$. But there is but one such extension and its
valuation ring is ${\cal O}_{L}$ (by definition). □
###### Lemma 1.28.
Mapping $x$ to the unique irreducible component of $\mathscr{X}$ that contains
$x$ establishes a surjection from the set of closed point generizations $x$ of
$\bar{x}$ to the set of irreducible components of $\mathscr{X}$ containing
$\bar{x}$.
###### Proof:
If $x_{0}\in\mathcal{X}$ is a generization of $\bar{x}$, then $x_{0}$ lies in
a unique irreducible component of $\mathscr{X}$ because $\mathcal{X}$ is
locally irreducible. Hence the map described above is well-defined.
Now for surjectivity: Given an irreducible component $C$ of $\mathscr{X}$
containing $\bar{x}$, let $x_{0}\in C$ be the generic point. Then $x_{0}$ must
be in the generic fiber (else we would be able to find a generization in the
generic fiber by going-down). Now go through the proof of Lemma 1.25 with this
particular choice of $x_{0}$. □
### 1.6 The local model
To give a very rough idea of what the _local model_ to be discussed in this
section is supposed to accomplish: It should be an $\mathcal{O}_{E}$-scheme
that is étale-locally isomorphic to $\mathscr{S}_{K}(G,X)$, but easier to
understand by virtue of being of a more “linear-algebraic flavor”. In
actuality however, the theory of local models quickly gets quite complicated
once one departs from the simplest examples.
#### 1.6.1 The Siegel case
We do start with the simplest example.
We consider the standard Iwahori subgroup
$I_{p}\subseteq\operatorname{GSp}_{2g}(\mathbb{Z}_{p})$, defined as the
preimage of the standard Borel subgroup of
$\operatorname{GSp}_{2g}(\mathbb{F}_{p})$. In terms of the building (cf.
Remark 1.13), it corresponds to the lattice chain
$\mathcal{L}_{\mathrm{full}}$ given by
$\displaystyle\Lambda^{0}=\mathbb{Z}_{p}^{2g}$
$\displaystyle\supsetneqq\Lambda^{1}=\mathbb{Z}_{p}^{2g-1}\oplus
p\mathbb{Z}_{p}\supsetneqq\Lambda^{2}=\mathbb{Z}_{p}^{2g-2}\oplus
p\mathbb{Z}_{p}^{2}$ (1.29)
$\displaystyle\supsetneqq\dotsb\supsetneqq\Lambda^{2g-1}=\mathbb{Z}_{p}\oplus
p\mathbb{Z}_{p}^{2g-1}\supsetneqq p\Lambda^{0}=p\mathbb{Z}_{p}^{2g}$
of period length $2g$.
Consider a subset $J=\\{j_{0}>\dotsb>j_{m-1}\\}\subseteq\\{1,\dotsc,2g\\}$
such that for each $j\in J$ with $1\leq j\leq 2g-1$ also $2g-j\in J$, and let
$K_{p}$ be the parahoric subgroup associated with the partial lattice chain
$\mathcal{L}\subseteq\mathcal{L}_{\mathrm{full}}$ obtained from
$\left\\{\Lambda^{j}\;|\;j\in J\right\\}$.
Define a scheme $\tilde{\mathscr{S}}_{K}(G,X)$ over $\mathscr{S}_{K}(G,X)$ as
follows:
$\tilde{\mathscr{S}}_{K}(G,X)(S)=\left\\{(A,\bar{\lambda},\eta^{p},\tau)\;\middle|\;\begin{tabular}[]{@{}l@{}}$(A,\bar{\lambda},\eta^{p})\in\mathscr{S}_{K}(\operatorname{GSp}_{2g},S^{\pm})(S)$,\\\
$\tau\colon
H_{\mathrm{dR}}^{1}(A)\xrightarrow{\sim}\mathcal{L}\otimes\mathcal{O}_{S}$
isomorphism of lattice chains\end{tabular}\right\\}$
for every $\mathbb{Z}_{p}$-scheme $S$.
By [RZ96, Appendix to Chap. 3],
$\tilde{\mathscr{S}}_{K}(G,X)\to\mathscr{S}_{K}(G,X)$ is a Zariski torsor
under the automorphism group of $\mathcal{L}$, i.e., the Iwahori group scheme.
This motivates the definition of the local model
$M^{\mathrm{loc}}_{K_{p}}\to\operatorname{Spec}\mathbb{Z}_{p}$ as the “moduli
space of Hodge filtrations”; more precisely:
###### Remark 1.30.
(See [Gör03, 91].) $M^{\mathrm{loc}}_{K_{p}}(S)$ is the set of isomorphism
classes of commutative diagrams
$\Lambda^{j_{0}}_{S}$$\mathcal{F}^{j_{0}}$$\Lambda^{j_{1}}_{S}$$\mathcal{F}^{j_{1}}$$\dotsb$$\dotsb$$\Lambda^{j_{0}}_{S}$$\mathcal{F}^{j_{0}}$$\Lambda^{j_{m-1}}_{S}$$\mathcal{F}^{j_{m-1}}$$\cdot
p$
with $\Lambda^{j}_{S}:=\Lambda^{j}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{S}$,
$\mathcal{F}^{j}\subseteq\Lambda^{j}_{S}$ locally direct summand of rank $g$,
such that for all $j\in J$,
$\mathcal{F}^{j}\to\Lambda^{j}_{S}\overset{\psi}{\cong}(\Lambda^{2g-j}_{S})^{*}\to(\mathcal{F}^{2g-j})^{*}$
vanishes, $\psi$ being the symplectic pairing.
By Grothendieck-Messing theory, one obtains a diagram
$\tilde{\mathscr{S}}_{K}(G,X)$$\mathscr{S}_{K}(G,X)$$M^{\mathrm{loc}}_{K}$smooth
of rel. dim.
$\dim\operatorname{Aut}(\mathcal{L})$$\operatorname{Aut}(\mathcal{L})$-torsor
Since both morphisms in this diagram are smooth of the same dimension, it
follows that for every finite field extension $\mathbb{F}_{q}/\mathbb{F}_{p}$
and every point $x\in\mathscr{S}_{K}(G,X)(\mathbb{F}_{q})$, there exists a
point $y\in M^{\mathrm{loc}}_{K}(\mathbb{F}_{q})$ and an isomorphism
$\mathcal{O}_{\mathscr{S}_{K}(G,X),x}^{h}\cong\mathcal{O}_{M^{\mathrm{loc}}_{K},y}^{h}$
of henselizations.
In many (P)EL situations one has similar descriptions with the obvious extra
structures. Sometimes however the so-called “naive” local models so obtained
additionally need to be flattened, which leaves one without any self-evident
moduli interpretation.
#### 1.6.2 The relation between the integral and the local model
Generalizing the Siegel example, we axiomatically characterize the
relationship between the integral model of the Shimura variety and its local
model: One wants a _local model diagram_ , i.e., a diagram of
$\mathcal{O}_{E}$-schemes functorial in $K$
$\tilde{\mathscr{S}}_{K}(G,X)$$\mathscr{S}_{K}(G,X)$$M^{\mathrm{loc}}_{K}$equivariant
and smooth of rel. dim.
$\dim\mathcal{G}_{\mathcal{O}_{E}}$$\mathcal{G}_{\mathcal{O}_{E}}$-torsor
(1.31)
where $M^{\mathrm{loc}}_{K}$ is a projective flat $\mathcal{O}_{E}$-scheme
with an action of $\mathcal{G}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{E}$ and
generic fiber the canonical model of $G_{\bar{\mathbb{Q}}_{p}}/P_{\mu^{-1}}$
over $E$.
By Kisin-Pappas [KP15] we do actually have such a diagram in our situation.
#### 1.6.3 The Pappas-Zhu construction
In [PZ13], Pappas and Zhu give a construction of the local model in quite a
general context, in particular with no assumptions going beyond our running
assumptions 1.17.
###### Remark 1.32.
To this end, they construct an affine smooth group scheme
$\underline{\mathcal{G}}_{K}\to\mathbb{A}^{1}_{\mathbb{Z}_{p}}=\operatorname{Spec}\mathbb{Z}_{p}[t]$
with the following key properties:
1. (1)
$\underline{\mathcal{G}}_{K}$ has connected fibers,
2. (2)
$\underline{\mathcal{G}}_{K}$ is reductive over
$\operatorname{Spec}\mathbb{Z}_{p}[t^{\pm 1}]$,
3. (3)
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto
p}\mathbb{Z}_{p}\cong\mathcal{G}_{K}$, in particular
* •
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto
p}\mathbb{Q}_{p}\cong G_{\mathbb{Q}_{p}}$ and
* •
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{F}_{p}:=\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto
0}\mathbb{F}_{p}\cong\mathcal{G}_{K}\otimes\mathbb{F}_{p}$,
4. (4)
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{Q}_{p}[\mkern-2.0mu[t]\mkern-2.0mu]$
is parahoric for
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{Q}_{p}(\mkern-4.0mu(t)\mkern-4.0mu)$,
5. (5)
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{F}_{p}[\mkern-2.0mu[t]\mkern-2.0mu]$
is parahoric for
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{F}_{p}(\mkern-4.0mu(t)\mkern-4.0mu)$.
###### Definition and Remark 1.33.
Let $X_{\mu}$ be the canonical model of
$G_{\bar{\mathbb{Q}}_{p}}/P_{\mu^{-1}}$ over $E$, where for a cocharacter
$\nu$ one defines $P_{\nu}:=\\{g\in G\;|\;\lim_{t\to
0}\nu(t)g\nu(t)^{-1}\text{ exists}\\}$.
Let $S_{\mu}$ be the closed subvariety of
$\operatorname{Gr}_{G}\times_{\mathbb{Q}_{p}}E$ with
$S_{\mu}(\bar{\mathbb{Q}}_{p})=G(\bar{\mathbb{Q}}_{p}[\mkern-2.0mu[t]\mkern-2.0mu])\mu(t)G(\bar{\mathbb{Q}}_{p}[\mkern-2.0mu[t]\mkern-2.0mu])/G(\bar{\mathbb{Q}}_{p}[\mkern-2.0mu[t]\mkern-2.0mu]).$
Then $S_{\mu}$ can be $G_{E}$-equivariantly identified with $X_{\mu}$.
###### Definition 1.34.
The local model $M^{\mathrm{loc}}_{G,\mu,K}$ now is defined to be the Zariski
closure of $X_{\mu}\subseteq\operatorname{Gr}_{G}\times_{\mathbb{Q}_{p}}E$ in
$\operatorname{Gr}_{\underline{\mathcal{G}}_{K},\mathbb{Z}_{p}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{E}$,
where
$\operatorname{Gr}_{\underline{\mathcal{G}}_{K},\mathbb{Z}_{p}}:=\operatorname{Gr}_{\underline{\mathcal{G}}_{K},\mathbb{A}_{\mathbb{Z}_{p}}^{1}}\otimes_{\mathbb{A}^{1}_{\mathbb{Z}_{p}},\,u\mapsto
p}\mathbb{Z}_{p}$ is a base change of the global affine Graßmannian as defined
in [PZ13].
## 2 EKOR strata and zips in the case of parahoric reduction
###### Notation 2.1.
We still fix a Shimura datum $(G,X)$ of Hodge type, a parahoric subgroup
$K_{p}\subseteq G(\mathbb{Q}_{p})$ (associated with a Bruhat-Tits group scheme
$\mathcal{G}=\mathcal{G}_{K}=\mathcal{G}_{K_{p}}\to\operatorname{Spec}\mathbb{Z}_{p}$
associated with a facet $\mathfrak{f}$) and a sufficiently small open compact
subgroup $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$. Define
$\overline{\mathcal{G}}_{K}:=\mathcal{G}_{K}\otimes_{\mathbb{Z}_{p}}\kappa$.
We also keep up our standard assumptions 1.17.
We now want to discuss the EKOR stratification on the special fiber of the
integral model with parahoric level structure. The EKOR stratification
interpolates between the Ekedahl-Oort (EO) and the Kottwitz-Rapoport (KR)
stratification (see Remark 2.22 below for a precise formulation). We begin by
explaining the basics about these stratifications and the combinatorics
involved in the first section of this chapter.
### 2.1 The Ekedahl-Oort, Kottwitz-Rapoport and EKOR stratifications
#### 2.1.1 Iwahori-Weyl group and the admissible subset
###### Notation 2.2.
1. (1)
We fix an Iwahori subgroup $I_{p}\subseteq K_{p}$, i.e., $I_{p}$ is the group
of $\mathbb{Z}_{p}$-points of the parahoric group scheme $\mathcal{I}$
associated with an alcove $\mathfrak{a}$ (facet of maximal dimension) such
that $\mathfrak{f}\subseteq\overline{\mathfrak{a}}$. As usual, we also define
$\breve{I}:=\mathcal{I}(\breve{\mathbb{Z}}_{p})\subseteq\breve{K}$.
2. (2)
Let $T\subseteq G$ be a maximal torus such that $T_{\breve{\mathbb{Q}}_{p}}$
is contained in a Borel subgroup of $G_{\breve{\mathbb{Q}}_{p}}$888Note that
by Steinberg’s theorem, $G_{\breve{\mathbb{Q}}_{p}}$ is quasi-split. [Ser97,
Chap. III, § 2] and let $S$ be the maximal $\breve{\mathbb{Q}}_{p}$-split
torus contained in $T_{\breve{\mathbb{Q}}_{p}}$. We can and do choose $T$ such
that the alcove $\mathfrak{a}$ is contained in the apartment associated with
$S$. By $N$ we denote the normalizer of $T$.
3. (3)
Let $(V,R)$ be the relative root system of
$(G_{\breve{\mathbb{Q}}_{p}},T_{\breve{\mathbb{Q}}_{p}})$, i.e., $V$ is the
$\mathbb{R}$-vector space
$X^{*}_{\breve{\mathbb{Q}}_{p}}(T_{\breve{\mathbb{Q}}_{p}})\otimes_{\mathbb{Z}}\mathbb{R}$
and $R\subseteq X^{*}_{\breve{\mathbb{Q}}_{p}}(T_{\breve{\mathbb{Q}}_{p}})$ is
(as usual) such that we have a decomposition
$\mathfrak{g}:=\operatorname{Lie}(G_{\bar{\mathbb{Q}}_{p}})=\operatorname{Lie}(T_{\bar{\mathbb{Q}}_{p}})\oplus\bigoplus_{\alpha\in
R}\mathfrak{g}_{\alpha}.$
Contrary to the absolute situation, $\dim\mathfrak{g}_{\alpha}$ may be greater
than $1$.
###### Definition 2.3.
1. (1)
The _(finite relative) Weyl group_ of $G$ (over $\breve{\mathbb{Q}}_{p}$) is
$W:=N(\breve{\mathbb{Q}}_{p})/T(\breve{\mathbb{Q}}_{p})$. It is the Weyl group
of the root system $(V,R)$, i.e., the group generated by the orthogonal
reflections through the hyperplanes defined by the elements of $R$.
2. (2)
As described in [Lan00, 1.2.3], one defines a set of affine roots
$R_{\mathrm{aff}}\supseteq R$ on $V$ using the valuation on
$\breve{\mathbb{Q}}_{p}$. By
$W_{a}\subseteq\mathrm{Aff}(V^{*})=\operatorname{GL}(V^{*})\ltimes V^{*}$ we
denote the _affine Weyl group_ of the affine root system
$(V,R_{\mathrm{aff}})$, i.e., the group generated by the orthogonal
reflections through the affine hyperplanes defined by the elements of
$R_{\mathrm{aff}}$.
3. (3)
$\widetilde{W}:=N(\breve{\mathbb{Q}}_{p})/(T(\breve{\mathbb{Q}}_{p})\cap\breve{I})$
is the _Iwahori-Weyl group_.
4. (4)
$W_{K}:=(N(\breve{\mathbb{Q}}_{p})\cap\breve{K})/(T(\breve{\mathbb{Q}}_{p})\cap\breve{I})\subseteq\widetilde{W}$.
(Recall that $\breve{K}=\mathcal{G}(\breve{\mathbb{Z}}_{p})$.)
###### Remarks 2.4.
1. (1)
We have $W\subseteq W_{a}$. With the systems of generators indicated above,
$W$ and $W_{a}$ become (affine) Coxeter groups; in particular we can talk
about reduced words and have length functions, cf. [BB05].
2. (2)
$W_{I}$ is the trivial group.
###### Proposition 2.5.
[HR08, Prop. 8] The Bruhat-Tits decomposition
$G(\breve{\mathbb{Q}}_{p})=\bigcup_{w\in\widetilde{W}}\breve{K}w\breve{K}$
identifies
$\breve{K}\backslash G(\breve{\mathbb{Q}}_{p})/\breve{K}\cong
W_{K}\backslash\widetilde{W}/W_{K}.$
###### Proposition 2.6.
[HR08, Prop. 13] Let $\breve{K}$ be the maximal parahoric subgroup of
$G(\breve{\mathbb{Q}}_{p})$ associated with a special vertex in the apartment
corresponding to $S$. Then $W_{K}\to W$ is an isomorphism and
$\widetilde{W}\cong W\ltimes
X_{*}(T)_{\operatorname{Gal}(\bar{\mathbb{Q}}_{p}/\breve{\mathbb{Q}}_{p})}$.999Notation:
Let $\Gamma$ be a group and $M$ a $\mathbb{Z}[\Gamma]$-module. Then
$M_{\Gamma}:=\mathbb{Z}\otimes_{\mathbb{Z}[\Gamma]}M=M/\langle\gamma
m-m\;|\;\gamma\in\Gamma,\;m\in M\rangle$ is the module of
$\Gamma$-coinvariants of $M$.
###### Notation 2.7.
We denote the map
$X_{*}(T)_{\operatorname{Gal}(\bar{\mathbb{Q}}_{p}/\breve{\mathbb{Q}}_{p})}\to\widetilde{W}$
of the proposition by $\nu\mapsto t_{\nu}$.
###### Proposition 2.8.
[HR08, Lemma 14] Let $\Omega\subseteq\widetilde{W}$ be the subgroup consisting
of those elements that preserve the base alcove $\mathfrak{a}$.
There is an exact sequence
$1\to W_{a}\to\widetilde{W}\to\Omega\to 1,$
with a canonical right splitting (namely the inclusion
$\Omega\hookrightarrow\widetilde{W}$), i.e., $\widetilde{W}\cong
W_{a}\rtimes\Omega$.
###### Definition 2.9.
The semidirect product decomposition of the preceding proposition means that
$\widetilde{W}$ is a “quasi-Coxeter” group. In practical terms, this means:
1. (1)
We define a length function $\ell$ on $\widetilde{W}$ as follows:
$\ell(w_{a},\omega):=\ell(w_{a})$ for all $w_{a}\in W_{a}$ and
$\omega\in\Omega$, where on the right hand side we use the length function of
the affine Coxeter group $W_{a}$.
Note that $\Omega=\ell^{-1}(0)$.
2. (2)
Likewise, we extend the Bruhat partial order from $W_{a}$ to $\widetilde{W}$
by defining
$(w_{a,1},\omega_{1})\leq(w_{a,2},\omega_{2})\;:\Longleftrightarrow\;w_{a,1}\leq
w_{a,2}\text{ and }\omega_{1}=\omega_{2}.$
Note that $w_{1}\leq w_{2}$ ($w_{1},w_{2}\in\widetilde{W}$) implies
$\ell(w_{1})\leq\ell(w_{2})$.
###### Definition 2.10.
1. (1)
Let $\\{\mu\\}$ be a $W_{\mathrm{abs}}$-conjugacy class of geometric
cocharacters of $T$ (cf. Remark 1.4),
$W_{\mathrm{abs}}:=N(\bar{\mathbb{Q}}_{p})/T(\bar{\mathbb{Q}}_{p})$ being the
absolute Weyl group. Let $\bar{\mu}\in
X_{*}(T)_{\operatorname{Gal}(\bar{\mathbb{Q}}_{p}/\breve{\mathbb{Q}}_{p})}$ be
the image of a cocharacter in $\\{\mu\\}$ whose image in
$X_{*}(T)\otimes_{\mathbb{Z}}\mathbb{R}$ is contained in the closed Weyl
chamber corresponding to some Borel subgroup of $G$ containing $T$ and defined
over $\breve{\mathbb{Q}}_{p}$.
2. (2)
$\mathrm{Adm}(\mu):=\mathrm{Adm}(\\{\mu\\}):=\\{w\in\widetilde{W}\;|\;w\leq
qt_{\bar{\mu}}q^{-1}=t_{q\bar{\mu}}\text{ for some }q\in W\\}$ is the
_$\\{\mu\\}$ -admissible subset_ of $\widetilde{W}$.
3. (3)
$\mathrm{Adm}(\\{\mu\\})^{K}:=W_{K}\mathrm{Adm}(\\{\mu\\})W_{K}\subseteq\widetilde{W}$.
4. (4)
$\mathrm{Adm}(\\{\mu\\})_{K}:=\mathrm{KR}(K,\\{\mu\\}):=W_{K}\backslash\mathrm{Adm}(\\{\mu\\})^{K}/W_{K}\subseteq
W_{K}\backslash\widetilde{W}/W_{K}$.
5. (5)
Define ${}^{K}\widetilde{W}\subseteq\widetilde{W}$ to be the set of
representatives of minimal length for the quotient
$W_{K}\backslash\widetilde{W}$.
6. (6)
${}^{K}\mathrm{Adm}(\\{\mu\\}):=\mathrm{EKOR}(K,\\{\mu\\}):=\mathrm{Adm}(\\{\mu\\})^{K}\cap{}^{K}\widetilde{W}\subseteq{}^{K}\widetilde{W}$.
###### Lemma 2.11.
(See [SYZ19, Thm. 1.2.2].)
${}^{K}\mathrm{Adm}(\\{\mu\\})=\mathrm{Adm}(\\{\mu\\})\cap{}^{K}\widetilde{W}$.
#### 2.1.2 Kottwitz-Rapoport stratification
Recall from Section 1.6.2 that we have an integral model and a local model
diagram
$\mathscr{S}_{K}\leftarrow\tilde{\mathscr{S}}_{K}\to M^{\mathrm{loc}}_{K}$
or, equivalently, a (smooth) morphism of stacks
$\mathscr{S}_{K}\to[\mathcal{G}_{K}\backslash M^{\mathrm{loc}}_{K}]$ (over
$\mathcal{O}_{E}$).
As explained in Section 1.6.3, by the construction in [PZ13], the special
fiber $M^{\mathrm{loc}}_{K}\otimes\kappa$ of $M^{\mathrm{loc}}_{K}$ is a
closed subscheme of the affine flag variety
$\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\kappa}=\mathcal{F}l_{\underline{\mathcal{G}}_{K}\otimes\kappa[\mkern-2.0mu[t]\mkern-2.0mu]}$,
which is the ind-projective ind-scheme over $\kappa$ given as the fpqc
sheafification (which exists in this case!) of the presheaf
$R\mapsto\underline{\mathcal{G}}_{K}(R(\mkern-4.0mu(t)\mkern-4.0mu))/\underline{\mathcal{G}}_{K}(R[\mkern-2.0mu[t]\mkern-2.0mu])$.
###### Definition 2.12.
Define
$L^{+}(\underline{\mathcal{G}}_{K}\otimes\kappa[\mkern-2.0mu[t]\mkern-2.0mu])$
to be the $\kappa$-functor sending a $\kappa$-algebra $R$ to
$\underline{\mathcal{G}}_{K}(R[\mkern-2.0mu[t]\mkern-2.0mu])$.
We let
$L^{+}(\underline{\mathcal{G}}_{K}\otimes\kappa[\mkern-2.0mu[t]\mkern-2.0mu])$
act on
$\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\kappa}$
from the left and call this action $a$ (within this subsection). The orbits of
this action on
$\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\bar{\kappa}}$
are the _Schubert cells_.
###### Remarks 2.13.
1. (1)
The Schubert cells can be indexed by $W_{K}\backslash\widetilde{W}/W_{K}$ by
Proposition 2.5 with the following in mind: Strictly speaking, using the
Bruhat-Tits decomposition here, we arrive at something involving the Iwahori-
Weyl group of
$\underline{\mathcal{G}}_{K}\otimes\bar{\kappa}(\mkern-4.0mu(t)\mkern-4.0mu)$.
However, by [PZ13, 9.2.2], this is isomorphic to the Iwahori-Weyl group of
$G_{\breve{\mathbb{Q}}_{p}}$.
2. (2)
$M^{\mathrm{loc}}_{K}\otimes\bar{\kappa}$ is a union of Schubert cells, namely
of those indexed by
$\mathrm{KR}(K,\\{\mu\\}):=W_{K}\backslash(W_{K}\mathrm{Adm}(\\{\mu\\})W_{K})/W_{K}$,
cf. [PZ13, Theorem 9.3].
###### Remark 2.14.
By construction, $M^{\mathrm{loc}}_{K}$ has an action $b$ of
$\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto
p}\mathcal{O}_{E}\cong\mathcal{G}_{K}\otimes\mathcal{O}_{E}$.
For $w\in\widetilde{W}$ choose a representative $\dot{w}\in
L\underline{\mathcal{G}}_{K}(\bar{\kappa})$ (with Remark 2.13 (1) in mind) and
let
$e_{0}\in\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\kappa}$
be the distinguished base point (associated with the identity). For $w\in
W_{K}\mathrm{Adm}(\\{\mu\\})W_{K}$, the orbit map of $\dot{w}\cdot e_{0}$ for
the action $a$ factors through the homomorphism
$L^{+}(\underline{\mathcal{G}}_{K}\otimes\bar{\kappa}[\mkern-2.0mu[t]\mkern-2.0mu])\to\mathcal{G}_{K}\otimes\kappa\cong\underline{\mathcal{G}}_{K}\otimes\bar{\kappa}$.
The orbits associated with the two $\mathcal{G}_{K}\otimes\kappa$-actions $a$
and $b$ on $M^{\mathrm{loc}}_{K}\otimes\kappa$ agree. The orbits of the
$\mathcal{G}_{K}\otimes\kappa$-action on $M^{\mathrm{loc}}_{K}\otimes\kappa$
are indexed by $\mathrm{KR}(K,\\{\mu\\})$.
###### Definition 2.15.
The stratifications thus obtained on $M^{\mathrm{loc}}_{K}\otimes\kappa$ and
$\mathscr{S}_{K}\otimes\kappa$ are called _Kottwitz-Rapoport stratifications_.
That is to say that Kottwitz-Rapoport strata on $\mathscr{S}_{K}\otimes\kappa$
are by definition pullbacks of Kottwitz-Rapoport strata on
$M^{\mathrm{loc}}_{K}$, which in turn are
$\mathcal{G}_{K}\otimes\kappa$-orbits.
#### 2.1.3 Ekedahl-Oort stratification
The Ekedahl-Oort stratification is only defined in the case of good reduction,
i.e., if $K_{p}$ is hyperspecial or, equivalently, if $\mathcal{G}_{K}$ is a
_reductive_ model of $G_{\mathbb{Q}_{p}}$. Then $G_{\mathbb{Q}_{p}}$ splits
over $\breve{\mathbb{Q}}_{p}$ (by definition of “hyperspecial”, cf. [Tit79,
1.10.2]).
We therefore put ourselves in the situation of good reduction for this
subsection.
###### Remark 2.16.
Then $W$ as defined in Definition 2.3 (1) agrees with the absolute Weyl group
of $G_{\mathbb{Q}_{p}}=\mathcal{G}_{K}\otimes\mathbb{Q}_{p}$, which in turn
agrees with the absolute Weyl group of
$\overline{\mathcal{G}}_{K}:=\mathcal{G}_{K}\otimes\kappa$, cf. [VW13, App.
A.5].
###### Definition 2.17.
Define $I$ to be the type (interpreted as a subset of simple reflections) of
the parabolic subgroup of $G_{\mathbb{Q}_{p}}$ defined by $\mu^{-1}$ (cf.
Remark 1.33), and ${}^{I}W\subseteq W$ to be the system of representatives of
the quotient group $W_{I}\backslash W$ containing the element of least length
of every coset.
###### Theorem 2.18.
[MW04, PWZ15, PWZ11, Zha18] There is a smooth algebraic stack
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{\kappa}:=\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\mu}_{\kappa}$
over $\kappa$ with underlying topological space ${}^{I}W$ together with a
smooth morphism
$\mathscr{S}_{K}\otimes\kappa\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\mu}_{\kappa}.$
The stratification of $\mathscr{S}_{K}\otimes\kappa$ thus obtained is the
_Ekedahl-Oort stratification_.
#### 2.1.4 EKOR stratification
###### Definition 2.19.
Let $L$ be a valued field extension of $\mathbb{Q}_{p}$ with ring of integers
$\mathcal{O}$, maximal ideal $\mathfrak{m}$ and residue field $\lambda$.
The _pro-unipotent radical_ of $\mathcal{G}_{K}(\mathcal{O})$ is
$\mathcal{G}_{K}(\mathcal{O})_{1}:=\\{g\in\mathcal{G}_{K}(\mathcal{O})\;|\;(g\mod\mathfrak{m})\in\bar{R}_{K}(\lambda)\\},$
where $\bar{R}_{K}$ is the unipotent radical of
$\mathcal{G}_{K}\otimes_{\mathbb{Z}_{p}}\lambda$.
In particular, if $K$ is hyperspecial, then
$\mathcal{G}_{K}(\mathcal{O})_{1}=\ker(\mathcal{G}_{K}(\mathcal{O})\to\mathcal{G}_{K}(\lambda))$.
Also,
$\overline{\breve{K}}:=\breve{K}/\breve{K}_{1}\cong\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}(\bar{\mathbb{F}}_{p})$,
where $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is the maximal reductive
quotient of $\overline{\mathcal{G}}_{K}:=\mathcal{G}_{K}\otimes\kappa$.
###### Remark 2.20.
[HR17, after Cor. 6.2] We have a commutative diagram
$G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$${}^{K}\widetilde{W}$$W_{K}\backslash\widetilde{W}/W_{K}.$$\breve{K}\backslash
G(\breve{\mathbb{Q}}_{p})/\breve{K}$
Consider the map
$v_{K}\colon\mathscr{S}_{K}\otimes\kappa\to
G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1}),$
which is the composition of the central leaves map
$\Upsilon_{K}\colon\mathscr{S}_{K}\otimes\kappa\to
G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$ (see [Hes20a]) with the
projection $G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}\to
G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$.
The Kottwitz-Rapoport map
$\lambda_{K}\colon{\mathscr{S}_{K}\otimes\kappa}\to\breve{K}\backslash
G(\breve{\mathbb{Q}}_{p})/\breve{K}$ factors through this map.
The fibers of $v_{K}$ are called _EKOR strata_. By [HR17, Thm. 6.15], they are
locally closed subsets of $\mathscr{S}_{K}\otimes\kappa$.
###### Remarks 2.21.
1. (1)
One can explicitly express the image of a EKOR stratum under a change-of-
parahoric map as a union of EKOR strata on the target [HR17, Prop. 6.11].
2. (2)
The closure of an EKOR stratum is a union of EKOR strata and one can
explicitly describe the associated order relation [HR17, Thm. 6.15].
###### Remark 2.22.
In the hyperspecial case, the EKOR stratification agrees with the Ekedahl-Oort
stratification. In the Iwahori case, it agrees with the Kottwitz-Rapoport
stratification
(${}^{K}\widetilde{W}=\widetilde{W}=W_{K}\backslash\widetilde{W}/W_{K}$ in
that case).
By definition, the EKOR stratification always is a refinement of the Kottwitz-
Rapoport stratification. So one way of approaching the EKOR stratification is
by looking at a fixed Kottwitz-Rapoport stratum and trying to understand how
it is subdivided into EKOR strata.
To get this started, let us recall some calculations from the proof of [HR17,
Thm. 6.1].
Fixing a Kottwitz-Rapoport stratum means restricting our view to
$\breve{K}w\breve{K}/\breve{K}_{\sigma}$ rather than the whole of
$G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$, for some fixed
$w\in\mathrm{KR}(K,\\{\mu\\})$. The EKOR strata in the Kottwitz-Rapoport
stratum associated with $w$ are therefore indexed by
$\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$.
Define $\sigma^{\prime}:=\sigma\circ\operatorname{Ad}(w)$ and consider the
bijection
$\displaystyle\breve{K}/(\breve{K}\cap w^{-1}\breve{K}w)_{\sigma^{\prime}}$
$\displaystyle\xrightarrow{\sim}\breve{K}w\breve{K}/\breve{K}_{\sigma},$
$\displaystyle k$ $\displaystyle\mapsto wk,$ $\displaystyle
k_{2}\sigma(k_{1})$ $\displaystyle\mapsfrom k_{1}wk_{2}.$
Let $J$ be the set of simple affine reflections in $W_{K}$, let $\bar{B}$ be
the image of $\breve{I}$ in $\overline{\breve{K}}$ and
$\bar{T}\subseteq\bar{B}$ the maximal torus. Set $J_{1}:=J\cap w^{-1}Jw$.
###### Proposition 2.23.
(See [Mor93, Lemma 3.19].) The image of $\breve{K}\cap w^{-1}\breve{K}w$ in
$\overline{\breve{K}}$ is $\bar{P}_{J_{1}}$, i.e., the standard parabolic
subgroup of $\overline{\breve{K}}$ associated with $J_{1}$.
###### Remark 2.24.
He and Rapoport invoke Carter’s book [Car93] at this point, which primarily
pertains to the case of (usual) BN-pairs attached to reductive groups. Morris
[Mor93] shows that the relevant results carry over likewise to the case of
generalized (or affine) BN-pairs.
Then we get a map
$\displaystyle\breve{K}w\breve{K}/\breve{K}_{\sigma}\to\breve{K}/(\breve{K}\cap
w^{-1}\breve{K}w)_{\sigma^{\prime}}$
$\displaystyle\to\overline{\breve{K}}/(\bar{P}_{J_{1}})_{\sigma^{\prime}}$
$\displaystyle\to\overline{\breve{K}}/(\bar{L}_{J_{1}})_{\sigma^{\prime}}(\bar{U}_{J_{1}})_{\sigma^{\prime}}\to\overline{\breve{K}}/(\bar{L}_{J_{1}})_{\sigma^{\prime}}(\bar{U}_{J_{1}}\times\bar{U}_{\sigma^{\prime}(J_{1})}),$
which factors through a bijection
$\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})\xrightarrow{\sim}\overline{\breve{K}}/(\bar{L}_{J_{1}})_{\sigma^{\prime}}(\bar{U}_{J_{1}}\times\bar{U}_{\sigma^{\prime}(J_{1})})\cong\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}(\bar{\mathbb{F}}_{p})/{{\scriptstyle\cong}}.$
Here, $\mathcal{Z}_{w}$ is the (connected) algebraic zip datum
$\mathcal{Z}_{w}=(\overline{\mathcal{G}}^{\mathrm{rdt}},\bar{P}_{J_{1}},\bar{P}_{\sigma^{\prime}(J_{1})},\sigma^{\prime})$,
as described in [SYZ19]. In [SYZ19], Shen, Yu and Zhang show that this
observation “globalizes” (with the drawback that “global” here still just
refers to the Kottwitz-Rapoport stratum101010They also give another
“globalization”; the drawback there being that it only works after
perfection.) in a pleasant way. To wit, one gets a smooth morphism [SYZ19,
Theorem A]
$\zeta_{w}\colon\overline{\mathscr{S}}_{K}^{w}\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}_{\kappa}$
(the source being a Kottwitz-Rapoport stratum).
### 2.2 $\overline{\mathcal{G}}_{K}$-zips in the Siegel case
Here we work with the Siegel Shimura datum, cf. Example 1.2.
#### 2.2.1 Preliminaries
###### Notation 2.25.
Fix $p\neq 2$,111111As in [RZ96], the principal reason for this restriction is
our use of the equivalence between alternating and skew-symmetric. See
Definition 2.30 (e). $g\in\mathbb{Z}_{\geq 1}$ and a subset
$J\subseteq\mathbb{Z}$ with $J=-J$ and $J+2g\mathbb{Z}=J$. Associated with $J$
is the partial lattice chain $\left\\{\Lambda^{j}\;|\;j\in J\right\\}$, where
$\Lambda^{j}$ are defined as in equation (1.29). Let $K_{p}$ be the
corresponding parahoric subgroup of $\operatorname{GSp}_{2g}(\mathbb{Q}_{p})$,
i.e., the stabilizer of said lattice chain. It contains the Iwahori subgroup
$I_{p}$ associated with the full lattice chain (1.29). For the maximal torus
$T$ we take the usual diagonal (split) torus.
###### Remark 2.26.
The Weyl group is
$\displaystyle W$ $\displaystyle=\\{\pi\in S_{2g}=\operatorname{Aut}(\\{\pm
1,\pm 2,\dotsc,\pm g\\})\;|\;\pi(-n)=-\pi(n)\text{ for }n=1,2,\dotsc,g\\}$
$\displaystyle\cong S_{g}\ltimes\\{\pm 1\\}^{g}.$
Here the transposition $(n\quad m)$ of
$S_{g}=\operatorname{Aut}(\\{1,2,\dotsc,g\\})$ corresponds to the element
${(n\quad m)(-n\quad{-m})}$ of $\operatorname{Aut}(\\{\pm 1,\pm 2,\dotsc,\pm
g\\})$ and the element of $\\{\pm 1\\}^{g}$ which has a $-1$ in position $i$
and $1$ everywhere else corresponds to $(i\quad{-i})$.
The affine Weyl group is $W_{a}=W\ltimes Y_{0}$ and the Iwahori-Weyl group
$\widetilde{W}=W\ltimes Y$ with
$\displaystyle\mathbb{Z}^{g+1}$ $\displaystyle\cong
Y=\\{(\nu_{1},\dotsc,\nu_{g},\nu_{-g},\dotsc,\nu_{-1})\in\mathbb{Z}^{2g}:\nu_{1}+\nu_{-1}=\dotsb=\nu_{g}+\nu_{-g}\\}$
$\displaystyle\supseteq
Y_{0}=\\{(\nu_{1},\dotsc,\nu_{g},\nu_{-g},\dotsc,\nu_{-1})\in\mathbb{Z}^{2g}:0=\nu_{1}+\nu_{-1}=\dotsb=\nu_{g}+\nu_{-g}\\}\cong\mathbb{Z}^{g}.$
The simple affine roots (whose walls bound the base alcove $\mathfrak{a}$) are
$\displaystyle 1-2e_{-1}+e_{0}=1+2e_{1}-e_{0},$ $\displaystyle
e_{-1}-e_{-2}=e_{2}-e_{1},e_{-2}-e_{-3},\dotsc,e_{-(g-1)}-e_{-g},$
$\displaystyle 2e_{-g}-e_{0}=e_{0}-2e_{g},$
where $e_{1},\dotsc,e_{g},e_{-g},\dotsc,e_{-1}\colon T\to\mathbb{G}_{m}$ are
the obvious cocharacters and $e_{0}=e_{1}+e_{-1}=\dotsb=e_{g}+e_{-g}$.
The reflections corresponding to the simple affine roots are
$((1\quad{-1}),\left(\begin{smallmatrix}-1\\\ 0\\\ \vdots\\\ 0\\\
1\end{smallmatrix}\right)),(-1\quad{-2})(1\quad
2),\dotsc,(-g\quad{-(g-1)})(g\quad{g-1}),(g\quad{-g}).$
The length zero subgroup $\Omega\subseteq\widetilde{W}$ is generated by
$((w_{0},\epsilon),y)\in(S_{g}\ltimes\\{\pm 1\\}^{g})\ltimes Y$, where
$w_{0}\in S_{g}$ is the longest element, $\epsilon=(-1,-1,\dotsc,-1)$ and
$y=(0^{g},1^{g})$.
###### Remark 2.27.
One also can choose $\dotsb\subseteq
p\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}^{2g-1}\subseteq\mathbb{Z}_{p}^{2g}\subseteq\dotsb$
instead of $\dotsb\subseteq\mathbb{Z}_{p}^{2g-1}\oplus
p\mathbb{Z}_{p}\subseteq\mathbb{Z}_{p}^{2g}\subseteq\dotsb$ as the standard
lattice chain. Then the simple affine roots would be
$1-2e_{1}+e_{0},e_{1}-e_{2}=e_{2}-e_{1},e_{2}-e_{3},\dotsc,e_{g-1}-e_{g},2e_{g}-e_{0}.$
###### Remark 2.28.
$\widetilde{W}=W\ltimes Y=N(\mathbb{Q}_{p})/T(\mathbb{Z}_{p})$ and
$N(\mathbb{Q}_{p})\to W\ltimes Y$ has a section $W\ltimes Y\to
N(\mathbb{Q}_{p})$, which sends $(\pi,\underline{\nu})\in W\ltimes Y$ to
$T_{\underline{\nu}}P_{w}$, where
$T_{\underline{\nu}}=\left(\begin{smallmatrix}p^{\nu_{1}}&&&&\\\
&p^{\nu_{2}}&&&\\\ &&\ddots&&\\\ &&&p^{\nu_{-2}}&\\\
&&&&p^{\nu_{-1}}\end{smallmatrix}\right)$ and $P_{w}$ is the permutation
matrix with $P_{w}(e_{i})=e_{w(i)}$.
###### Remark 2.29.
Using the results of [KR00] we also easily can compute
$\mathrm{Adm}(\\{\mu\\})$. One potential source of confusion at this point is
that, due to our choice of the base alcove (cf. Remark 2.27), in our setup we
need to use $\omega_{i}:=(0^{2g-i},1^{i})$ instead of
$\omega_{i}:=(1^{i},0^{2g-i})$ (notation of [KR00]), cf. [Yu08, 1268]. With
that convention in place, we have that $x\in\widetilde{W}$ is
$\\{\mu\\}$-admissible ($\mu=(1^{g},0^{g})$) if and only if
$(0,\dotsc,0)\leq x(\omega_{i})-\omega_{i}\leq(1,\dotsc,1)\quad\text{for all
}0\leq i<2g$
(component-wise comparison).
#### 2.2.2 Lattice chains, zips, admissibility
###### Definition 2.30.
Let $S$ be a $\mathbb{Z}_{p}$-scheme.
A _Siegel lattice chain in the weak sense on $S$ of type $J$_ is a tuple
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$,
where
1. (a)
for all $j\in J$, $\mathcal{V}^{j}$ is a vector bundle on $S$ of rank $2g$,
2. (b)
$\mathcal{L}$ is a line bundle on $S$,
3. (c)
for all $i,j\in J$ with $j>i$,
$\alpha_{j,i}\colon\mathcal{V}^{j}\to\mathcal{V}^{i}$ is a vector bundle
homomorphism, such that the $\bigl{(}\alpha_{j,i}\bigr{)}$ satisfy the obvious
cocycle condition (and we also define $\alpha_{i,i}:=\operatorname{id}$),
4. (d)
for all $j\in J$,
$\theta_{j}\colon\mathcal{V}^{j}\xrightarrow{\sim}\mathcal{V}^{j-2g}$ is a
vector bundle isomorphism such that the $\bigl{(}\theta_{j}\bigr{)}$ are
compatible with the $\bigl{(}\alpha_{j,i}\bigr{)}$ in that
$\theta_{i}\circ\alpha_{j,i}=\alpha_{j-2g,i-2g}\circ\theta_{j}$ and
$\alpha_{j,j-2g}=p\cdot\theta_{j}$,
5. (e)
for all $j\in J$ a vector bundle isomorphism
$\psi_{j}\colon\mathcal{V}^{j}\xrightarrow{\sim}(\mathcal{V}^{-j})^{*}\otimes\mathcal{L}$
compatible with $\bigl{(}\theta_{j}\bigr{)}$ and
$\bigl{(}\alpha_{j,i}\bigr{)}$, such that $-\psi_{j}(x,y)=\psi_{-j}(y,x)$ for
all $(x,y)\in\mathcal{V}^{j}\times\mathcal{V}^{-j}$.121212By
“$(x,y)\in\mathcal{V}^{j}\times\mathcal{V}^{-j}$” we of course mean that there
is an open subset $U\subseteq S$ such that
$(x,y)\in(\mathcal{V}^{j}\times\mathcal{V}^{-j})(U)$.
We also have a _standard_ Siegel lattice chain in the weak sense on
$\operatorname{Spec}\mathbb{Z}_{p}$ (and hence by base change on every
$\mathbb{Z}_{p}$-scheme $S$) of type $J$, namely the one given by the lattice
chain $\left\\{\Lambda^{j}\;|\;j\in J\right\\}$. We can think of the standard
Siegel lattice chain as either having varying $\mathcal{V}^{j}$ with the
$\alpha_{j,i}$ being the obvious inclusion maps (e.g. (if $\\{0,1\\}\subseteq
J$), $\mathcal{V}^{1}={\mathbb{Z}_{p}^{2g-1}\oplus
p\mathbb{Z}_{p}}\xrightarrow{\alpha_{1,0}=\mathrm{inclusion}}\mathbb{Z}_{p}^{2g}=\mathcal{V}^{0}$)
or as having constant $\mathcal{V}^{j}=\mathbb{Z}_{p}^{2g}$ with the
$\alpha_{j,i}$ being diagonal matrices with all entries either $p$ or $1$
(e.g.,
$\mathcal{V}^{1}=\mathbb{Z}_{p}^{2g}\xrightarrow{\alpha_{1,0}=\operatorname{diag}(1,1,\dotsc,1,p)}\mathbb{Z}_{p}^{2g}=\mathcal{V}^{0}$).
Usually the latter point of view is more convenient.
A _Siegel lattice chain on $S$ of type $J$_ (or _Siegel lattice chain in the
strong sense on $S$ of type $J$_) then is a Siegel lattice chain in the weak
sense on $S$ of type $J$ that Zariski-locally on $S$ is isomorphic to the
standard chain.
###### Remarks 2.31.
1. (1)
Let
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$
be a Siegel lattice chain in the weak sense on $S$ of type $J$. Then
$\tilde{\psi_{j}}:=(\tilde{\alpha}_{j,-j}^{*}\otimes\operatorname{id}_{\mathcal{L}})\circ\psi_{j}\colon\mathcal{V}^{j}\otimes\mathcal{V}^{j}\to\mathcal{L}$
is alternating.
Here $\tilde{\alpha}_{j,-j}$ is defined as follows: Let $n\in\mathbb{Z}$ be
maximal with $j-2gn\geq-j$. Then
$\tilde{\alpha}_{j,-j}:=\alpha_{j-2gn,-j}\circ\theta_{j-2g(n-1)}\circ\dotsb\circ\theta_{j}$.
2. (2)
Note that this means that $\tilde{\psi}_{j}$ is (twisted) symplectic if $-j\in
j+2g\mathbb{Z}$, i.e., if $j\in g\mathbb{Z}$.
###### Proof:
(of (1)) Let $x,y\in\mathcal{V}^{j}$. Then
$\displaystyle\tilde{\psi}_{j}(x,y)$
$\displaystyle=\psi_{j}(x,\tilde{\alpha}_{j,-j}(y))$
$\displaystyle=\psi_{j}(x,(\alpha_{j-2gn,-j}\circ\theta_{j-2g(n-1)}\circ\dotsb\circ\theta_{j})(y))$
$\displaystyle=\psi_{2gn-j}(\alpha_{j,2gn-j}(x),(\theta_{j-2g(n-1)}\circ\dotsb\circ\theta_{j})(y))$
$\displaystyle=\psi_{2g(n-1)-j}((\theta_{2gn-j}\circ\alpha_{j,2gn-j})(x),(\theta_{j-2g(n-2)}\circ\dotsb\circ\theta_{j})(y))$
$\displaystyle=\dotsb$
$\displaystyle=\psi_{-j}((\theta_{-j+2g}\circ\dotsb\circ\theta_{2gn-j}\circ\alpha_{j,2gn-j})(x),y)$
$\displaystyle=-\psi_{j}(y,(\theta_{-j+2g}\circ\dotsb\circ\theta_{2gn-j}\circ\alpha_{j,2gn-j})(x))$
$\displaystyle=-\tilde{\psi}_{j}(y,x).$
□
###### Reminder 2.32.
$\mathcal{G}_{K}$ is the automorphism group of the standard Siegel lattice
chain.
The following definition is a generalization of [VW13, Definition 3.1] in the
Siegel case.
###### Definition 2.33.
Let $S$ be an $\mathbb{F}_{p}$-scheme.
A $\overline{\mathcal{G}}_{K}$-zip over $S$ is a tuple
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet},\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet},\varphi_{\mathcal{L}})$,
where
1. (a)
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$
is a Siegel lattice chain on $S$ of type $J$,
2. (b)
for all $j\in J$, $\mathcal{C}^{j}\subseteq\mathcal{V}^{j}$ are locally direct
summands of rank $g$ compatible with
$\alpha_{\bullet\bullet},\theta_{\bullet}$, such that
$\mathcal{C}^{j}\hookrightarrow\mathcal{V}^{j}\overset{\psi_{j}}{\cong}(\mathcal{V}^{-j})^{*}\otimes\mathcal{L}\to(\mathcal{C}^{-j})^{*}\otimes\mathcal{L}$
vanishes. (cf. Remark 1.30 for the origins of this condition.)
3. (c)
$\mathcal{D}^{\bullet}\subseteq\mathcal{V}^{\bullet}$ satisfies the same
conditions as $\mathcal{C}^{\bullet}\subseteq\mathcal{V}^{\bullet}$,
4. (d)
$\varphi_{0}^{j}\colon(\mathcal{C}^{j})^{(p)}\xrightarrow{\sim}\mathcal{V}^{j}/\mathcal{D}^{j}$
and
$\varphi_{1}^{j}\colon(\mathcal{V}^{j}/\mathcal{C}^{j})^{(p)}\xrightarrow{\sim}\mathcal{D}^{j}$
are isomorphisms of vector bundles compatible with $\alpha_{\bullet\bullet}$
and $\theta_{\bullet}$ and
$\varphi_{\mathcal{L}}\colon\mathcal{L}^{(p)}\xrightarrow{\sim}\mathcal{L}$ is
an isomorphism of line bundles, such that
$(\mathcal{C}^{j})^{(p)}$$(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{*,(p)}\otimes\mathcal{L}^{(p)}$$(\mathcal{D}^{-j})^{*}\otimes\mathcal{L}$$\mathcal{V}^{j}/\mathcal{D}^{j}$$\varphi_{0}^{j}$$(\varphi_{1}^{-j})^{*}\otimes\varphi_{\mathcal{L}}^{-1}$$\psi_{j}^{(p)}$$\psi_{j}$
commutes, i.e.,
${\psi_{j}(\varphi_{0}^{j}(\\_),\varphi_{1}^{-j}(\\_))=\varphi_{\mathcal{L}}\circ\psi_{j}^{(p)}(\\_,\\_)\colon}{(\mathcal{C}^{j})^{(p)}\times(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{(p)}\to\mathcal{L}^{(p)}\to\mathcal{L}}.$
Since $\varphi_{\mathcal{L}}$ evidently is uniquely determined by the other
data, we sometimes leave it out.
We obtain a fibered category
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}\to\mathrm{Sch}_{\mathbb{F}_{p}}$.
###### Remark 2.34.
$\psi_{j}$ gives rise to isomorphisms
$\displaystyle\mathcal{C}^{j}$
$\displaystyle\xrightarrow{\sim}(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{*}\otimes\mathcal{L},$
$\displaystyle\mathcal{V}^{j}/\mathcal{C}^{j}$
$\displaystyle\xrightarrow{\sim}(\mathcal{C}^{-j})^{*}\otimes\mathcal{L},$
$\displaystyle\mathcal{D}^{j}$
$\displaystyle\xrightarrow{\sim}(\mathcal{V}^{-j}/\mathcal{D}^{-j})^{*}\otimes\mathcal{L},$
$\displaystyle\mathcal{V}^{j}/\mathcal{D}^{j}$
$\displaystyle\xrightarrow{\sim}(\mathcal{D}^{-j})^{*}\otimes\mathcal{L}.$
This way
$\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ and
$\mathcal{D}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{D}^{\bullet}$
become Siegel lattice chains in the weak(!) sense of type $J$. The Cartier
isomorphism then is an isomorphism in the category of Siegel lattice chains in
the weak sense of type $J$. Over an algebraically closed field, we call the
isomorphism type of the Siegel lattice chain in the weak sense
$\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ the
_Kottwitz-Rapoport type of the $\overline{\mathcal{G}}_{K}$-zip_.
We also define a linearly rigidified version of
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ as follows.
###### Definition 2.35.
We define the fibered category
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}\to\mathrm{Sch}_{\mathbb{F}_{p}}$
just like $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ but with the extra
condition that
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$
be the standard Siegel lattice chain (rather than just locally isomorphic to
it).
###### Lemma 2.36.
We always have a closed embedding of
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ into a product of
(classical) $\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$’s, and
therefore $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ is a scheme.
###### Proof:
Set $J^{\prime}:=J\cap\\{0,\dotsc,2g-1\\}$. Let
$\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}=E_{(1^{g},0^{g})}\backslash(\operatorname{GL}_{2g}\times\operatorname{GL}_{2g})$
be the $\mathbb{F}_{p}$-scheme of trivialized $\operatorname{GL}_{2g}$-zips
(so that
$[\operatorname{GL}_{2g}\backslash\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}]=\operatorname{GL}_{2g}\text{-}\mathrm{Zip}$)
with respect to the the cocharacter $(1^{g},0^{g})$, and $\prod_{j\in
J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ the product of
$\\#J^{\prime}$ copies of this scheme. On $J^{\prime}$ we define $-j:=2g-j$
for $1\leq j\leq 2g-1$ and $-0:=0$.
Then we get a monomorphism
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}\hookrightarrow\prod_{j\in
J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ (2.37)
by sending
$(\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet})$
to
$\left(\mathcal{C}^{j},\mathcal{D}^{j},\varphi_{0}^{j},\varphi_{1}^{j}\right)_{j\in
J^{\prime}}$.
The extra conditions for an element of $\prod_{j\in
J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ to be in
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ are as in Definition
2.33:
1. (1)
$\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet}$
are compatible with the transition maps (or, to put it differently,
${(\mathcal{C}^{j}\oplus\mathcal{V}^{j}/\mathcal{C}^{j})^{(p)}}\xrightarrow[\cong]{\varphi_{0}^{j}\oplus\varphi_{1}^{j}}{\mathcal{V}^{j}/\mathcal{D}^{j}\oplus\mathcal{D}^{j}}$
is compatible with the transition maps),
2. (2)
$\mathcal{C}^{j}\hookrightarrow\mathcal{V}^{j}\overset{\psi_{j}}{\cong}(\mathcal{V}^{-j})^{*}\to(\mathcal{C}^{-j})^{*}$
vanishes.
3. (3)
$\mathcal{D}^{j}\hookrightarrow\mathcal{V}^{j}\overset{\psi_{j}}{\cong}(\mathcal{D}^{-j})^{*}\to(\mathcal{D}^{-j})^{*}$
vanishes.
4. (4)
There is a (necessarily unique) isomorphism
$\varphi_{\mathcal{L}}\colon\mathcal{L}^{(p)}\xrightarrow{\sim}\mathcal{L}=\mathcal{O}_{S}$
of line bundles, such that
$(\mathcal{C}^{j})^{(p)}$$(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{*,(p)}$$(\mathcal{D}^{-j})^{*}$$\mathcal{V}^{j}/\mathcal{D}^{j}$$\varphi_{0}^{j}$$(\varphi_{1}^{-j})^{*}\otimes\varphi_{\mathcal{L}}^{-1}$$\psi_{j}^{(p)}$$\psi_{j}$
commutes.
We claim that the conditions are closed on $\prod_{j\in
J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ (hence the
monomorphism is a closed immersion).
To see this, we recall the construction of the scheme
$\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ as executed in [MW04,
(3.10), (3.11), (4.3)].
Recall our notational convention regarding the parabolic subgroup associated
with a cocharacter $\chi$ from Definition 1.33. As in [MW04], we denote by
$\mathrm{Par}_{\chi}$ the scheme of parabolic subgroups of type $\chi$.
There is a group scheme $H$ defined by the cartesian diagram
$H$$\mathcal{P}_{((-1)^{g},0^{g})}/\mathcal{U}_{((-1)^{g},0^{g})}$$\square$$\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$$\mathrm{Par}_{((-1)^{g},0^{g})}$$(\;)^{(p)}\circ\operatorname{pr}_{1}$
where $\mathcal{P}_{((-1)^{g},0^{g})}\to\mathrm{Par}_{((-1)^{g},0^{g})}$ is
the universal parabolic group scheme and $\mathcal{U}_{((-1)^{g},0^{g})}$ its
unipotent radical, such that
$\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ is an $H$-Zariski torsor
over $\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$,
where
$\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}\to\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$
is given by $(C,D,\varphi_{0},\varphi_{1})\mapsto(C,D)$.
Clearly, compatibility of $\mathcal{C}^{\bullet},\mathcal{D}^{\bullet}$ with
the transition maps is a closed condition on $\prod_{j\in
J^{\prime}}\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$
and then also on $\prod_{j\in
J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$. Similar for the
conditions (2) and (3).
Locally, we can choose complements (not necessarily compatible with the
transition maps) and then $\varphi_{\bullet}^{j}$ yield sections $g^{j}$ of
$\operatorname{GL}_{2g}$ as in [MW04, definition of $g\in G(S)$ in the proof
of (4.3)]. The $g^{j}$ are well-defined up to
$\mathcal{U}_{((-1)^{g},0^{g})}^{(p)}\times\mathcal{U}_{(1^{g},0^{g})}$, and
we want them to be compatible with the transition maps coming from the Siegel
lattice chains in the weak sense
$\mathcal{C}^{j}\oplus\mathcal{V}^{j}/\mathcal{C}^{j}$ and
$\mathcal{V}^{j}/\mathcal{D}^{j}\oplus\mathcal{D}^{j}$, respectively. With our
complements in place, these transition maps correspond to maps
$\mathcal{V}^{j}\to\mathcal{V}^{j-n}$. The question of whether $g^{j}$ is
compatible with these maps is independent of the choice of complements
(basically because the transition maps $\mathcal{V}^{j}\to\mathcal{V}^{j-n}$
depend on the choice of complements similar to how $g^{j}$ depends on that
choice).
So in effect we can view the conditions on
$\varphi_{0}^{\bullet},\varphi_{1}^{\bullet}$ of (1) as closed conditions on
$\prod_{j\in
J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim\sim}$, where
$\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim\sim}\to\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$
(an fpqc quotient map) additionally comes with complementary spaces of $C,D$
($\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim\sim}=\tilde{X}_{\tau}$ in
the notation of [MW04, proof of (4.3)]).
We also can reformulate condition (4) in those terms. □
###### Corollary 2.38.
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ is the algebraic quotient
stack
$[\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}]$.
Here by definition an element $\phi\in\overline{\mathcal{G}}_{K}$ acts on
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ by replacing
$(\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet},\varphi_{\mathcal{L}})$
by
$(\phi\mathcal{C}^{\bullet},\phi\mathcal{D}^{\bullet},\phi\varphi_{0}^{\bullet}\phi^{-(p)},\phi\varphi_{1}^{\bullet}\phi^{-(p)},\varphi_{\mathcal{L}})$.
###### Definition 2.39.
We let an element
$(X,Y)\in\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ act on
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ by replacing
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet},\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet})$
by
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet},X\mathcal{C}^{\bullet},Y\mathcal{D}^{\bullet},Y\varphi_{0}^{\bullet}X^{-(p)},Y\varphi_{1}^{\bullet}X^{-(p)}).$
###### Notation 2.40.
Let $\mathscr{S}_{K}\to\operatorname{Spec}\mathbb{Z}_{p}$ be the integral
model of the Siegel Shimura variety of level $K$ (where $K=K_{p}K^{p}$ with
$K^{p}$ sufficiently small), and recall $\tilde{\mathscr{S}}_{K}$ from Section
1.6.1. Moreover, define
$\overline{\mathscr{S}}_{K}:=\mathscr{S}_{K}\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}$
and
$\tilde{\overline{\mathscr{S}}}_{K}:=\tilde{\mathscr{S}}_{K}\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}$.
So $\tilde{\overline{\mathscr{S}}}_{K}\to\overline{\mathscr{S}}_{K}$ is a
$\overline{\mathcal{G}}_{K}$-torsor.
###### Remark 2.41.
We have morphisms
$\tilde{\overline{\mathscr{S}}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$
(take first de Rham cohomology with Frobenius and Verschiebung) and
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}\to\overline{M}^{\mathrm{loc}}_{K}$
(take the $\mathcal{C}^{\bullet}$-filtration) and therefore
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}_{K}].$
###### Remark 2.42.
In particular, $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ has a
Kottwitz-Rapoport stratification, which agrees with the notion of Kottwitz-
Rapoport type as defined in Remark 2.34.
For $w\in\mathrm{KR}(K,\\{\mu\\})$ denote the associated Kottwitz-Rapoport
stratum by $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{w}$, i.e., we
interpret $w$ as a $\bar{\mathbb{F}}_{p}$-valued point of
$[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}_{K}]$ and
form $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{w}$ as a fiber product.
###### Construction 2.43.
Fix $w\in\mathrm{Adm}(\\{\mu\\})^{K}\subseteq\widetilde{W}$ (so that
$W_{K}wW_{K}\in\mathrm{KR}(K,\\{\mu\\})$). We define a standard
$\overline{\mathcal{G}}_{K}$-zip of KR type $W_{K}wW_{K}$.
Using Remark 2.28, we interpret $w$ as an element of
$N(\mathbb{Q}_{p})\subseteq G(\mathbb{Q}_{p})$. The admissibility condition
implies that we can interpret it as an endomorphism $w^{\bullet}$ of the
standard lattice chain $\mathcal{V}^{\bullet}$ over
$\mathbb{Z}_{p}$.131313Take up the second point of view described in
Definition 2.30 regarding $\mathcal{V}^{\bullet}$. Define
$\underline{\nu}^{(0)}:=\underline{\nu}$,
$\underline{\nu}^{(1)}:=\underline{\nu}+\left(\begin{smallmatrix}0\\\
\vdots\\\ 0\\\ 0\\\ -1\end{smallmatrix}\right)+w\left(\begin{smallmatrix}0\\\
\vdots\\\ 0\\\ 0\\\ 1\end{smallmatrix}\right)$,
$\underline{\nu}^{(2)}:=\underline{\nu}+\left(\begin{smallmatrix}0\\\
\vdots\\\ 0\\\ -1\\\ -1\end{smallmatrix}\right)+w\left(\begin{smallmatrix}0\\\
\vdots\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)$, and so on. Then
$w^{j}=T_{\underline{\nu}^{(j)}}P_{w}$ for $0\leq j<2g$. From the formulation
of the admissibility condition as in Remark 2.29, we see that
$w\in\mathrm{Adm}(\\{\mu\\})^{K}$ is equivalent to the condition that
$\underline{\nu}^{(j)}$ be a permutation of $(1^{g},0^{g})$ for all relevant
$j$.
We denote the standard Siegel lattice chain over $\mathbb{Z}_{p}$ by
$\mathscr{V}^{\bullet}$ and its base change to $\mathbb{F}_{p}$ by
$\mathcal{V}^{\bullet}$. Define
$\mathscr{C}_{w}^{\bullet}:=pw^{\bullet,-1}\mathscr{V}^{\bullet}$ and
$\mathscr{D}_{w}^{\bullet}:=\sigma(w^{\bullet})\mathscr{V}^{\bullet}$. Then
$\mathcal{C}_{w}^{\bullet}:=\mathscr{C}_{w}^{\bullet}\otimes\mathbb{F}_{p}=\ker(w^{\bullet}\colon\mathcal{V}^{\bullet}\to\mathcal{V}^{\bullet})$,
so
$(\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet})^{(p)}\xrightarrow{\sim}\mathcal{D}_{w}^{\bullet}:=\mathscr{D}_{w}^{\bullet}\otimes\mathbb{F}_{p}$
via $\sigma(w^{\bullet})$ and
$(\mathcal{C}_{w}^{\bullet})^{(p)}\xrightarrow{\sim}\mathcal{V}^{\bullet}/\mathcal{D}_{w}^{\bullet}$
via $p^{-1}\sigma(w^{\bullet})$.
This defines a standard element $\widetilde{\mathrm{Std}}(w)$ of
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}(\mathbb{F}_{p})$ and a
standard element $\mathrm{Std}(w)$ of
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{w}(\mathbb{F}_{p})$.
###### Definition and Remark 2.44.
(See also [SYZ19, Lemma 3.3.2].)
$\mathcal{G}_{w}:=\operatorname{Aut}(\mathscr{C}_{w}^{\bullet}\subseteq\mathscr{V}^{\bullet})$
is a Bruhat-Tits group scheme with generic fiber $G_{\mathbb{Q}_{p}}$ and
$\breve{\mathbb{Z}}_{p}$-points $\breve{K}\cap w^{-1}\breve{K}w$; and
similarly for
$\mathcal{G}_{\sigma(w)^{-1}}:=\operatorname{Aut}(\mathscr{D}_{w}^{\bullet}\subseteq\mathscr{V}^{\bullet})$
with $\breve{K}\cap\sigma(w)\breve{K}\sigma(w)^{-1}$.
###### Definition 2.45.
We keep $w\in\mathrm{Adm}(\\{\mu\\})^{K}\subseteq\widetilde{W}$ fixed and
define
$\widetilde{E}_{w}\subseteq\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$
to be the stabilizer of $\widetilde{\mathrm{Std}}(w)$.
So $\widetilde{E}_{w}$ consists of those
$(X^{\bullet},Y^{\bullet})\in\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$
such that $X^{\bullet}\mathcal{C}_{w}^{\bullet}=\mathcal{C}_{w}^{\bullet}$,
$Y^{\bullet}\mathcal{D}_{w}^{\bullet}=\mathcal{D}_{w}^{\bullet}$, and
$Y^{\bullet}\circ\varphi_{j}^{\bullet}\circ
X^{\bullet,-(p)}=\varphi_{j}^{\bullet}$ for $j=0,1$.
In the notation of [SYZ19, Lemma 3.3.2] we have
$\widetilde{E}_{w}=\overline{\mathcal{G}}_{w}\times_{\overline{\mathcal{G}}_{w}^{L,(p)}}\overline{\mathcal{G}}_{\sigma(w)^{-1}}.$
(2.46)
Here $\overline{\mathcal{G}}_{w}^{L}$ is the image of
$\overline{\mathcal{G}}_{w}$ in
$\mathrm{DiagAut}(\mathcal{C}_{w}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet})$
(the automorphisms of
$\mathcal{C}_{w}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet}$
respecting both $\mathcal{C}_{w}^{\bullet}$ and
$\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet}$).
The orbit of $\widetilde{\mathrm{Std}}(w)$ in
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ is the fppf quotient
$(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})/\widetilde{E}_{w}$,
cf. [DG80, II, § 5, no. 3].
###### Lemma 2.47.
We have commutative diagrams
$\overline{\mathcal{G}}_{w}$$\overline{\mathcal{G}}$$\overline{\mathcal{G}}^{\mathrm{rdt}}$$\bar{P}_{J_{1}}$
and
$\overline{\mathcal{G}}_{\sigma(w)^{-1}}$$\overline{\mathcal{G}}$$\overline{\mathcal{G}}^{\mathrm{rdt}}$$\bar{P}_{\sigma^{\prime}(J_{1})}$
and
$\overline{\mathcal{G}}_{w}^{L}$$\overline{\mathcal{G}}$$\overline{\mathcal{G}}^{\mathrm{rdt}}$$\bar{L}_{J_{1}}$.
###### Proof:
This follows from Proposition 2.23. □
###### Lemma 2.48.
The image of $\widetilde{E}_{w}$ under
$\overline{\mathcal{G}}\times\overline{\mathcal{G}}\to\overline{\mathcal{G}}^{\mathrm{rdt}}\times\overline{\mathcal{G}}^{\mathrm{rdt}}$
is $E_{\mathcal{Z}_{w}}$.
###### Proof:
This follows from Lemma 2.47. □
###### Lemma 2.49.
Assume $0\in J$. The
$\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbit of
$\widetilde{\mathrm{Std}}(w)$ for $w\in\mathrm{Adm}(\\{\mu\\})^{K}$ depends
only on $W_{K}wW_{K}$.
###### Proof:
Let $x,y\in W_{K}\subseteq W$. As above we get endomorphisms
$x^{\bullet},y^{\bullet}$ of $\mathcal{V}^{\bullet}$, which in this case are
in fact automorphisms. Now
$\widetilde{\mathrm{Std}}(w)=((y^{\bullet})^{-1},\sigma(x^{\bullet}))\cdot\widetilde{\mathrm{Std}}(w)$.
□
###### Definition 2.50.
Define $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$ to be the
union of the
$\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbits of the
standard zips $\widetilde{\mathrm{Std}}(w)$ for
$w\in\mathrm{Adm}(\\{\mu\\})^{K}$. Here an orbit by definition is the image of
the orbit map endowed with the reduced subscheme structure, and—as we prove
just below—the union of orbits just referred to is a closed subset, which we
again endow with the reduced subscheme structure.
Define
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}:=[\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}]\subseteq[\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}]=\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$.
###### Lemma 2.51.
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$ is a closed subset
of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$.
###### Proof:
This being a purely topological question, we may freely pass to perfections,
which will be convenient since Dieudonné theory is simpler over perfect
rings. By “perfection” we mean the inverse perfection in the terminology of
[BG18, Section 5].
Consider therefore
$(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}$ as a
sheaf on $\mathrm{Perf}_{\mathbb{F}_{p}}$, the fpqc site of affine perfect
$\mathbb{F}_{p}$-schemes. Again denoting the standard Siegel lattice chain
over $\mathbb{Z}_{p}$ by $\mathscr{V}^{\bullet}$ and its base change to
$\mathbb{F}_{p}$ by $\mathcal{V}^{\bullet}$, we can describe the elements of
$(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}(R)=\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}(R)$,
where $R$ is a perfect $\mathbb{F}_{p}$-algebra as being given by
> homomorphisms
> $\mathcal{V}_{R}^{\bullet,(p)}\xrightarrow{F^{\bullet}}\mathcal{V}_{R}^{\bullet}\xrightarrow{V^{\bullet}}\mathcal{V}_{R}^{\bullet,(p)}$
> such that
> $\ker(F^{\bullet})=:\mathcal{C}^{\bullet,(p)}=\operatorname{im}(V^{\bullet})$
> and
> $\operatorname{im}(F^{\bullet})=:\mathcal{D}^{\bullet}=\ker(V^{\bullet})$
> and $\psi_{j}(F^{j}\\_,\\_)=u\sigma(\psi_{j}(\\_,V^{-j}\\_))$ for some $u\in
> R^{\times}$ and $\mathcal{C}^{\bullet,(p)},\mathcal{D}^{\bullet}$ have the
> same rank (namely $g$).
To see that $\mathcal{C}^{\bullet,(p)},\mathcal{D}^{\bullet}$ are direct
summands of $\mathcal{V}_{R}^{\bullet,(p)},\mathcal{V}_{R}^{\bullet}$ (which
makes the last part of the characterization given above meaningful), one
argues as in [Lau14, Lemma 2.4] (since both are finitely presented, it is
enough to show flatness and to that end, one looks at the fiber dimensions).
Define a presheaf $\mathcal{X}$ on $\mathrm{Sch}_{\mathbb{Z}_{p}}$ in the same
way but for the following changes: $\mathcal{V}^{\bullet}$ is replaced by
$\mathscr{V}^{\bullet}$, and we impose the condition that both compositions
$F^{\bullet}\circ V^{\bullet}$ and $V^{\bullet}\circ F^{\bullet}$ are
multiplication by $p$, and the $\ker=\operatorname{im}$-conditions are only
required to hold modulo $p$. We also slightly reformulate these
$\ker=\operatorname{im}$-conditions: We impose the condition that the
reductions $\bar{F}^{\bullet},\bar{V}^{\bullet}$ be fiberwise of rank $g$ over
$R/p$. (Note that the argument that
$\mathcal{C}^{\bullet,(p)},\mathcal{D}^{\bullet}$ are direct summands only
works over reduced rings.)
Then $\mathcal{X}$ is a separated $\mathbb{Z}_{p}$-scheme. To see this, we
build it up from scratch as follows. $\operatorname{End}(\mathscr{V}^{j})$
obviously is a $\mathbb{Z}_{p}$-scheme (an affine space), hence so is
$\operatorname{Hom}(\mathscr{V}^{j,(p)},\mathscr{V}^{j})$ since
$\mathscr{V}_{j}^{(p)}\cong\mathscr{V}_{j}$.
$\operatorname{Hom}(\mathscr{V}^{\bullet,(p)},\mathscr{V}^{\bullet})$ is a
locally closed subscheme of a finite product of such schemes. Homomorphisms
$\mathscr{V}^{\bullet,(p)}\xrightarrow{F^{\bullet}}\mathscr{V}^{\bullet}\xrightarrow{V^{\bullet}}\mathscr{V}^{\bullet,(p)}$
such that both compositions are multiplication by $p$ form a closed subscheme
$\mathcal{X}^{\prime}$ of
$\operatorname{Hom}(\mathscr{V}^{\bullet,(p)},\mathscr{V}^{\bullet})\times\operatorname{Hom}(\mathscr{V}^{\bullet},\mathscr{V}^{\bullet,(p)})$.
In the special fiber $\mathcal{X}^{\prime}_{\mathbb{F}_{p}}$ we now consider
the $\ker=\operatorname{im}$-conditions and show that they define an open
subscheme $\bar{\mathcal{X}}^{\prime\prime}$. Then
$\mathcal{X}=\mathcal{X}^{\prime}\times_{\mathcal{X}^{\prime}_{\mathbb{F}_{p}}}\bar{\mathcal{X}}^{\prime\prime}$.
Indeed, the extra conditions are that all $F^{\bullet},V^{\bullet}$ have some
non-vanishing $g$-minor—evidently open conditions.
The upshot is that we defined a $\mathbb{Z}_{p}$-scheme $\mathcal{X}$ such
that
$(\mathcal{X}\times_{\mathbb{Z}_{p}}\mathbb{F}_{p})^{\mathrm{perf}}=(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}$
and such that we have an obvious morphism
$\tilde{\mathscr{S}}_{K}\to\mathcal{X}$, which takes a principally polarized
isogeny chain of abelian schemes to the evaluation of the Dieudonné crystal
on the trivial thickening.141414This makes use of the crystalline-de Rham
comparison to make a trivialization of the de Rham cohomology into a
trivialization of the crystalline cohomology.
Observe that $\mathcal{X}$ also has a natural
$\mathcal{G}_{K}\times\mathcal{G}_{K}$-action: We interpret $\mathcal{G}_{K}$
as $\operatorname{Aut}(\mathscr{V}^{\bullet})$ and the action of
$(X^{\bullet},Y^{\bullet})$ transforms $(F^{\bullet},V^{\bullet})$ into
$(Y^{\bullet}\circ F^{\bullet}\circ X^{\bullet,-(p)},X^{(p)}\circ
V^{\bullet}\circ Y^{\bullet,-1})$. The identity
$(\mathcal{X}\times_{\mathbb{Z}_{p}}\mathbb{F}_{p})^{\mathrm{perf}}=(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}$
is an identity of
$\overline{\mathcal{G}}_{K}^{\mathrm{perf}}\times\overline{\mathcal{G}}_{K}^{\mathrm{perf}}$-varieties.
Now we claim that
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}=\left(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}}\right)^{\mathrm{perf}}$
topologically, where $\overline{\mathcal{X}_{\mathbb{Q}_{p}}}$ is the flat
closure of the generic fiber in $\mathcal{X}$. This of course implies
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}\subseteq\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$
being closed.
Both sets are constructible, so it suffices to check it on a very dense
subset, say the $\bar{\mathbb{F}}_{p}$-valued points.
Using Lemmas 1.25 and 1.27, we see that
$(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}})(\bar{\mathbb{F}}_{p})$
consists precisely of those elements
$\bar{x}\in\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}(\bar{\mathbb{F}}_{p})$
such that there exists a finite field extension $L/\breve{\mathbb{Q}}_{p}$ and
a point $x\in\mathcal{X}(\mathcal{O}_{L})$ lifting $\bar{x}$. (We’ll also say
that $\bar{x}$ is _liftable_ in this situation.)
Since $\mathcal{G}_{K}$ is flat over $\mathbb{Z}_{p}$, this liftability
condition for $\mathcal{G}_{K}$ (in lieu of $\mathcal{X}$) is always
satisfied. Consequently,
$(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}})(\bar{\mathbb{F}}_{p})$
is stable under the
$\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-action.
Also, the standard zips clearly are liftable. Thus,
$(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}})(\bar{\mathbb{F}}_{p})\supseteq\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}(\bar{\mathbb{F}}_{p})$.
For the converse inclusion, there are injective maps from
$\mathcal{X}(\mathcal{O}_{L})$ to $\mathcal{X}(L)$ to $\mathcal{G}_{K}(L)$
such that the corresponding Schubert cell (in the local model) is indexed by
the image mod
$\mathcal{G}_{K}(\mathcal{O}_{L})\times\mathcal{G}_{K}(\mathcal{O}_{L})^{\mathrm{op}}$,
cf. Proposition 2.5.151515Note that
$\mathcal{G}_{K}(\mathcal{O}_{L})\backslash\mathcal{G}_{K}(L)/\mathcal{G}_{K}(\mathcal{O}_{L})\cong
W_{K}\backslash\widetilde{W}/W_{K}$ for every strictly henselian discretely
valued field $L$ by [HR08, Prop. 8]. (And also in the construction of
$\widetilde{W}$ and $W_{K}$ any such field, not just
$L=\breve{\mathbb{Q}}_{p}$, can be used.) This proves it since we know which
Schubert cells belong to the local model. □
###### Remark 2.52.
Regarding the orbit closure relations for
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$, let us point out
that
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}\to\bar{M}^{\mathrm{loc}}_{K}$
is $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-equivariant,
where the action of
$\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ on
$M^{\mathrm{loc}}_{K}$ factors through the first projection map, and this map
is a bijection on orbits. Writing $w^{\prime}\preceq w$ if
$(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\mathrm{Std}(w^{\prime})\subseteq\overline{(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\mathrm{Std}(w)}$,
it follows from these observations that $w^{\prime}\leq w$ implies
$w^{\prime}\preceq w$. Here $\leq$ is the Bruhat order on
$W_{K}\backslash\widetilde{W}/W_{K}$ as explained in [PRS13, section 4.2].
It appears reasonable to suspect that $\preceq$ and $\leq$ in fact agree.
###### Conjecture 2.53.
The closure of
${(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\widetilde{\mathrm{Std}}(w)}$
is given by the disjoint union of
${(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\widetilde{\mathrm{Std}}(w^{\prime})}$
for $w^{\prime}\leq w$.
###### Lemma 2.54.
The map
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$
factors through $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}$.
###### Proof:
It is sufficient to check this on $k=\bar{\mathbb{F}}_{p}$-valued points.
The map
$\overline{\mathscr{S}}_{K}(k)\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}(k)$
factors through
$\Upsilon_{K}\colon\overline{\mathscr{S}}_{K}(k)\to\bigcup_{w\in\mathrm{KR}(K,\\{\mu\\})}\breve{K}w\breve{K}/\breve{K}_{\sigma}$
with
$\breve{K}w\breve{K}/\breve{K}_{\sigma}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}(k)$
given by sending $xwy$ to $(\bar{y}^{-1},\sigma(\bar{x}))\cdot\mathrm{Std}(w)$
(similar to Lemma 2.49). □
#### 2.2.3 An explicit description of
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$
In order to get a better feeling for the passage from
$\overline{\mathcal{G}}_{K}$ to the maximal reductive quotient
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}=\overline{\mathcal{G}}_{K}/R_{u}\overline{\mathcal{G}}_{K}$
(with $R_{u}\overline{\mathcal{G}}_{K}$ being the unipotent radical of
$\overline{\mathcal{G}}_{K}$), which is key in the definition of the EKOR
stratification, we describe $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ in
explicit, linear-algebraic terms in the Siegel case.
Let
$(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$
be the standard Siegel lattice chain on $S$ of type $J$. Assume $0\in J$. In
what follows, we sometimes use $j$ as a shorthand for $\mathcal{V}^{j}$.
By a _symmetric transition map_ , we mean a transition map from $j^{\prime}$
to $j^{\prime\prime}$, where $n\in\mathbb{Z}$, $j^{\prime},j^{\prime\prime}\in
J$, $ng\geq j^{\prime}\geq j^{\prime\prime}>(n-2)g$, and
$j^{\prime}+j^{\prime\prime}\in 2g\mathbb{Z}$. We will also call this the
symmetric transition map of $(j^{\prime},n)$ (or of $j^{\prime}$ if $n$
doesn’t matter).
By a _one-sided transition map_ , we mean a transition map from $j^{\prime}$
to $j^{\prime\prime}$, where $n\in\mathbb{Z}$, $j^{\prime},j^{\prime\prime}\in
J$, $ng\geq j^{\prime}\geq j^{\prime\prime}\geq(n-1)g$. Call it right-anchored
if $j^{\prime}=ng$ and left-anchored if $j^{\prime\prime}=(n-1)g$. We then
also speak of the right-anchored transition map of $j^{\prime\prime}$ and the
left-anchored transition map of $j^{\prime}$, respectively.
The kernels of the symmetric transition maps are symplectic subbundles of
$\mathcal{O}_{S}^{2g}$ (even of the form $\mathcal{O}_{S}^{I}$, where
$I\subseteq\\{\pm 1,\dotsc,\pm g\\}$ is symmetric (i.e., $-I=I$)), and the
kernels of the one-sided transition maps are totally isotropic subbundles
(even of the form $\mathcal{O}_{S}^{I}$, where $I\subseteq\\{1,\dotsc,g\\}$ or
$I\subseteq\\{-1,\dotsc,-g\\}$).
Let $\mathcal{O}_{S}^{I_{j}}$ be the kernel of the symmetric transition map of
$j$. Then $I_{j}\sqcup I_{-j}=\\{\pm 1,\dotsc,\pm g\\}$.
Every kernel of a one-sided transition map is a subbundle of a kernel of an
anchored transition map inside of which it is complemented by the kernel of
another one-sided transition map.
The kernel of the left-anchored transition map of $j$ is a subbundle of the
kernel of the symmetric transition map of $-j$ inside of which it is
complemented by the kernel of the right-anchored transition map of $-j$.
Likewise, the kernel of the right-anchored transition map of $j$ is a
subbundle of the kernel of the symmetric transition map of $j$ inside of which
it is complemented by the kernel of the left-anchored transition map of $-j$.
Now consider the standard symplectic bundle $\mathcal{O}_{S}^{2g}$ together
with the kernels of all the symmetric transition maps and all the one-sided
transition maps. So we have a symplectic bundle with a bunch of symplectic
subbundles coming in complementary pairs, some of which come with a further
decomposition into complementary Lagrangians, some of which come with further
decompositions into complementary subbundles (of course still totally
isotropic). We will also call these kernels _distinguished subspaces_.
Below we prove that $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is the
automorphism group scheme $\mathcal{A}$ of these data. Clearly, $\mathcal{A}$
is reductive; in fact it is a Levi subgroup of a parabolic of
$\operatorname{GSp}_{2g}$.
We have a map $\overline{\mathcal{G}}_{K}\to\mathcal{A}$; the image of an
$S$-point $f^{\bullet}$ under $\overline{\mathcal{G}}_{K}\to\mathcal{A}$ on
the kernel of a transition map starting at $j$ is given by $f^{j}$. Note that
$f^{j}=\tau\circ f^{j}$ on $\ker(\tau)$ for every transition map $\tau$
starting at $j$.
$\overline{\mathcal{G}}_{K}\to\mathcal{A}$ has a natural section
$\mathcal{A}\to\overline{\mathcal{G}}_{K}$, where in the image all the $f^{j}$
are the same as automorphisms of $\mathcal{O}_{S}^{2g}$. (This is well-
defined!)
###### Proposition 2.55.
$\mathcal{A}=\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$.
###### Proof:
Let us show that $\mathcal{K}:=\ker(\overline{\mathcal{G}}_{K}\to\mathcal{A})$
is unipotent. Consider $\overline{\mathcal{G}}_{K}$ as a subgroup of
$\prod_{j\in
J/2g\mathbb{Z}}\operatorname{GL}_{2g}\subseteq\operatorname{GL}_{N}$. We claim
that said kernel is contained in $\prod_{j\in J/2g\mathbb{Z}}U^{(j)}$,
$U^{(j)}$ being a conjugate of the standard unipotent subgroup
$\left(\begin{smallmatrix}1&\ast&\ast&\dotsb&\ast\\\ &1&\ast&\dotsb&\ast\\\
&&\ddots&\dotsb&\vdots\end{smallmatrix}\right)$ of $\operatorname{GL}_{2g}$.
Indeed, say $f^{\bullet}$ is in the kernel. Then $f^{j}$ acts as the identity
on the kernel of the symmetric transition map of $j$ and $f^{-j}$ acts as the
identity on the kernel of the symmetric transition map of $-j$. On the image
of the symmetric transition map $\tau_{j}$ of $j$, $f^{-j}$ agrees with
$\tau_{j}\circ f^{j}$. Note that
$\operatorname{im}(\tau_{j})=\ker(\tau_{-j})$. So $\tau_{j}\circ f^{j}$ is the
identity on $\ker(\tau_{-j})$. Hence, if $x\in\ker(\tau_{-j})$, then
$x=\tau_{j}(x)$ and $f^{j}(x)\equiv x\mod\ker(\tau_{j})$. Thus with respect to
the decomposition $\ker(\tau_{j})\oplus\ker(\tau_{-j})$, $f^{j}$ is of the
form $\begin{pmatrix}1&\ast\\\ &1\end{pmatrix}$.
Now we have $\overline{\mathcal{G}}_{K}=\mathcal{A}\ltimes\mathcal{K}$, in
particular
$\overline{\mathcal{G}}_{K}\cong\mathcal{A}\times_{\mathbb{F}_{p}}\mathcal{K}$
as schemes. Since both $\overline{\mathcal{G}}_{K}$ and $\mathcal{A}$ are
reduced and connected, so is $\mathcal{K}$.
All in all, we see that $\mathcal{A}$ is indeed
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ and
$\mathcal{K}=R_{u}\overline{\mathcal{G}}_{K}$ is the unipotent radical of
$\overline{\mathcal{G}}_{K}$. □
###### Example 2.56.
* •
If $J=\mathbb{Z}$, then
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}=\mathbb{G}_{m}^{g+1}$ is the
standard maximal torus of $\operatorname{GSp}_{2g}$.
* •
If $g=2$ and $J=2\mathbb{Z}$, then $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$
is the automorphism group of the standard twisted symplectic space
$\mathbb{F}_{p}^{4}$ with its standard Lagrangian decomposition, i.e.,
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\cong\operatorname{GL}_{2}\times\mathbb{G}_{m}$.
* •
If $g=2$ and $J/2g\mathbb{Z}=\\{-1,0,1\\}$, then
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is the automorphism group of the
standard twisted symplectic space $\mathbb{F}_{p}^{4}$ with its standard
decomposition in twisted symplectic subspaces and the totally isotropic
rank-$1$ subbundles generated by $e_{\pm 1}$, i.e.,
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\cong\operatorname{GL}_{2}\times\mathbb{G}_{m}$.
* •
Let $g=8$. We have the local Dynkin diagram
where we labelled the simple affine roots as follows: $1-2e_{-1}+e_{0}$ is
labelled 0, $e_{-i}-e_{-(i+1)}$ is labelled $i$ for $1\leq i\leq 7$, and
$2e_{-8}-e_{0}$ is labelled 8.
Consider $J/2g\mathbb{Z}=\\{0,\pm 3,\pm 5\\}$. Then the Dynkin diagram of
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ should (according to [Tit79,
3.5.1]) be the one we get by removing $0,3,5$ and the adjacent edges. So we
expect something along the lines161616i.e., having the same Dynkin diagram as
of
$\operatorname{GSp}(6)\times\operatorname{GL}(2)\times\operatorname{GL}(3)$.
We have the following (bases of) kernels of symmetric transition maps:
$\\{\pm 1,\pm 2,\pm 3\\},\\{\pm 4,\pm 5,\pm 6,\pm 7,\pm 8\\},\\{\pm 1,\pm
2,\pm 3,\pm 4,\pm 5\\},\\{\pm 6,\pm 7,\pm 8\\},$
and the following kernels of one-sided transition maps:
$\displaystyle\\{-3,-2,-1\\},\\{-5,-4\\},\\{-5,-4,-3,-2,-1\\},$
$\displaystyle\\{4,5\\},\\{1,2,3,4,5\\},\\{1,2,3\\}.$
So an element $A$ of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is given by
specifying linear automorphisms $A_{123}$ of $\langle 1,2,3\rangle$ and
$A_{45}$ of $\langle 4,5\rangle$ and a symplectic similitude $A_{\pm 6,\pm
7,\pm 8}$ of $\langle\pm 6,\pm 7,\pm 8\rangle$, such that
$\left.A\right|_{\langle 1,2,3\rangle}=A_{123}$, $\left.A\right|_{\langle
4,5\rangle}=A_{45}$, $\left.A\right|_{\langle\pm 6,\pm 7,\pm 8\rangle}=A_{\pm
6,\pm 7,\pm 8}$, where $\left.A\right|_{\langle-1,-2,-3\rangle}$ is uniquely
determined by $A_{123}$, $c(A_{\pm 6,\pm 7,\pm 8})$ ($c$ being the multiplier
character) and the imposition that $A$ be a symplectic similitude, and
similarly for $\left.A\right|_{\langle-4,-5\rangle}$.
If for example we consider $J/2g\mathbb{Z}=\\{0,\pm 2,\pm 3,\pm 5\\}$ instead,
we expect something along the lines of
$\operatorname{GSp}(6)\times\operatorname{GL}(2)\times\operatorname{GL}(2)$
and indeed we additionally get the subbundles
$\displaystyle\\{-2,-1\\},\\{1,2\\},\\{-3\\},\\{-5,-4,-3\\},$
$\displaystyle\\{3,4,5\\},\\{3\\},\\{3,4,5,6,7,8,-8,-7,-6,-5,-4,-3\\},\\{1,2,-2,-1\\}.$
So an element $A$ of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is given by
specifying linear automorphisms $A_{12}$ of $\langle 1,2\rangle$ and $A_{45}$
of $\langle 4,5\rangle$ and a symplectic similitude $A_{\pm 6,\pm 7,\pm 8}$ of
$\langle\pm 6,\pm 7,\pm 8\rangle$ in a similar way to above.
#### 2.2.4 $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ in the Siegel
case
Recall that we denote the unipotent radical of $\overline{\mathcal{G}}_{K}$ by
$R_{u}\overline{\mathcal{G}}_{K}$.
We divide out of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$
the action of the smooth normal subgroup
$R_{u}\overline{\mathcal{G}}_{K}\times
R_{u}\overline{\mathcal{G}}_{K}\subseteq\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$
and observe that $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$
still acts on $[R_{u}\overline{\mathcal{G}}_{K}\times
R_{u}\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}]=:\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$
(not a scheme).
We also define
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}:=[(\Delta({\overline{\mathcal{G}}_{K}})\cdot(R_{u}\overline{\mathcal{G}}_{K}\times
R_{u}\overline{\mathcal{G}}_{K}))\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}]$.
###### Proposition 2.57.
We have well-defined morphisms
$\displaystyle(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})/\widetilde{E}_{w}$
$\displaystyle\to(\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\times\overline{\mathcal{G}}_{K}^{\mathrm{rdt}})/E_{\mathcal{Z}_{w}},$
$\displaystyle\quad(X,Y)$
$\displaystyle\mapsto(X^{\mathrm{rdt}},Y^{\mathrm{rdt}}),$
$\displaystyle\overline{\mathcal{G}}_{K}/\widetilde{E}_{w}$
$\displaystyle\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}/E_{\mathcal{Z}_{w}},$
$\displaystyle\quad X$ $\displaystyle\mapsto X^{\mathrm{rdt}},$
and a bijectiion
$(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})/(\widetilde{E}_{w}\cdot(R_{u}\overline{\mathcal{G}}_{K}\times
R_{u}\overline{\mathcal{G}}_{K}))\to(\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\times\overline{\mathcal{G}}_{K}^{\mathrm{rdt}})/E_{\mathcal{Z}_{w}}.$
###### Proof:
The first assertion follows from the definition of $E_{\mathcal{Z}_{w}}$ and
equation (2.46). The second then follows from Lemma 2.48. □
###### Lemma 2.58.
Assume $0\in J$. The underlying topological spaces of the stacks in
consideration are as follows:
1. (1)
$|[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}]|=\mathrm{KR}(K,\\{\mu\\})\overset{\text{def.}}{=}W_{K}\backslash(W_{K}\mathrm{Adm}(\\{\mu\\})W_{K})/W_{K}$.
2. (2)
$|\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}|=\mathrm{EKOR}(K,\\{\mu\\})=\mathrm{Adm}(\\{\mu\\})^{K}\cap{}^{K}\widetilde{W}$
$\overset{\text{\ref{iw-
diagr}}}{\cong}\bigcup_{w\in\mathrm{KR}(K,\\{\mu\\})}\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$.
###### Proof:
(1) is well-known as explained in Section 2.1.2.
(2): By Lemma 2.49, the
$\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbits in
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$ are indexed by
$\mathrm{Adm}(\\{\mu\\})_{K}=\mathrm{KR}(K,\\{\mu\\})$.
Let us further investigate the
$\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbit of
$\widetilde{\mathrm{Std}}(w)$ in
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$ for some fixed
$w\in\mathrm{Adm}(\\{\mu\\})^{K}$.
By Proposition 2.57, its underlying topological space agrees with that of
$\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\sim,\mathcal{Z}_{w}}$.
By [SYZ19] we know that
$|\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}|\cong\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$,
whence the lemma. □
###### Corollary 2.59.
We have a morphism
$\left(\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}_{w}\right)_{\mathrm{red}}=\text{orbit
of
}\mathrm{Std}(w)\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}.$
This defines the EKOR stratification on
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}_{w}$. All in all, we get an
EKOR stratification on $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}$.
The morphism factors through
$(\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}_{w})_{\mathrm{red}}$, and
$(\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}_{w})_{\mathrm{red}}\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}$
is an isomorphism.
###### Corollary 2.60.
For every point of $[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$,
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$
is smooth as a map between the associated reduced fiber of
$\overline{\mathscr{S}}_{K}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$
and the associated reduced fiber of
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$.
###### Proof:
This follows from the preceding corollary by [SYZ19, Theorem A] (which says
that the map
$\overline{\mathscr{S}}_{K}^{w}\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}$
is smooth, cf. subsection 2.1.4). □
The key obstacle in going forward toward proving smoothness of
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$
now is that we do not know whether the fibers of
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$
are reduced.
###### Conjecture 2.61.
We conjecture that the answer is affirmative. In fact, we conjecture that
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$
is smooth.
###### Corollary 2.62.
$\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$
is surjective.
###### Proof:
This follows from the description of the topological space and what is already
known from [HR17, first paragraph of section 6.3]. □
We get a commutative diagram
$\tilde{\overline{\mathscr{S}}}_{K}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$$\overline{\mathscr{S}}_{K}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$$[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}_{K}]$
###### Remark 2.63.
Since $R_{u}\overline{\mathcal{G}}_{K}$ is smooth,
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$
is smooth.
###### Remark 2.64.
Another open question at this point is: what is the relationship between
$\overline{\mathcal{G}}_{K}\text{-EKORZip}^{\mathrm{perf}}$ and the shtuka
approach of [SYZ19, Section 4]?
###### Remark 2.65.
It should be straightforward to generalize (taking into account the extra
structure) our constructions to those (P)EL cases where the local model is the
“naive” local model of Rapoport-Zink [RZ96].
#### 2.2.5 The example of $\operatorname{GSp}(4)$
To illustrate some aspects, we look at the example $2g=4$.
##### The apartment.
We describe the (extended) apartment. We follow the general outline of
[Lan00], in particular as far as notation is concerned.
The roots are
$\pm(2e_{1}-e_{0}),\pm(2e_{2}-e_{0}),\pm(e_{1}-e_{2}),\pm(e_{1}+e_{2}-e_{0})$.
The simple affine roots and the (various variants of the) Weyl group are as
described in Remark 2.26. The root one-parameter subgroups171717The parameter
being additive here; i.e., we’re talking about homomorphisms
$\mathbb{G}_{a}\to G$. are given as follows:
$\displaystyle u_{e_{1}-e_{2}}(x)$ $\displaystyle=\begin{pmatrix}1&x&&\\\
&1&&\\\ &&1&-x\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{2}-e_{1}}(x)$
$\displaystyle=\begin{pmatrix}1&&&\\\ x&1&&\\\ &&1&\\\ &&-x&1\end{pmatrix},$
$\displaystyle u_{2e_{1}-e_{0}}(x)$ $\displaystyle=\begin{pmatrix}1&&&x\\\
&1&&\\\ &&1&\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{0}-2e_{1}}(x)$
$\displaystyle=\begin{pmatrix}1&&&\\\ &1&&\\\ &&1&\\\ x&&&1\end{pmatrix},$
$\displaystyle u_{2e_{2}-e_{0}}(x)$ $\displaystyle=\begin{pmatrix}1&&&\\\
&1&x&\\\ &&1&\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{0}-2e_{2}}(x)$
$\displaystyle=\begin{pmatrix}1&&&\\\ &1&&\\\ &x&1&\\\ &&&1\end{pmatrix},$
$\displaystyle u_{e_{1}+e_{2}-e_{0}}(x)$
$\displaystyle=\begin{pmatrix}1&&x&\\\ &1&&x\\\ &&1&\\\ &&&1\end{pmatrix},$
$\displaystyle u_{e_{0}-e_{1}-e_{2}}(x)$ $\displaystyle=\begin{pmatrix}1&&&\\\
&1&&\\\ x&&1&\\\ &x&&1\end{pmatrix}$
For $a\in R$ define $w_{a}(x):=u_{a}(x)u_{-a}(-x^{-1})u_{a}(x)$.
###### Remark 2.66.
$N(\mathbb{Q}_{p})$ is generated by $T(\mathbb{Q}_{p})$ and all $w_{a}(x)$ as
above.
###### Remark 2.67.
$w_{a}(x)=m(u_{-a}(-x^{-1}))$ in Landvogt’s notation [Lan00].
We have
$V_{1}:=X_{*}(T)\otimes\mathbb{R}=\\{(x_{1},x_{2},x_{-2},x_{-1})\in\mathbb{R}^{4}\;|\;x_{1}+x_{-1}=x_{2}+x_{-2}\\}$
and
$\nu_{1}\colon T(\mathbb{Q}_{p})\to V_{1},\;\begin{pmatrix}d_{1}&&&\\\
&d_{2}&&\\\ &&cd_{2}^{-1}&\\\
&&&cd_{1}^{-1}\end{pmatrix}\mapsto\begin{pmatrix}-v_{p}(d_{1})\\\
-v_{p}(d_{2})\\\ -v_{p}(cd_{2}^{-1})\\\ -v_{p}(cd_{1}^{-1})\end{pmatrix}.$
Also, $V_{0}=\\{v\in V_{1}\;|\;a(v)=0\;\forall
a\in\Phi\\}=\mathbb{R}(1,1,1,1)$, $V:=V_{1}/V_{0}$.
The extended apartment $A=A^{\mathrm{ext}}$ now is an affine $V_{1}$-space
together with the map $\nu_{1}\colon
N(\mathbb{Q}_{p})\to\operatorname{Aff}(A)=\operatorname{GL}(V_{1})\ltimes
V_{1}$, whose restriction to $T(\mathbb{Q}_{p})$ is given as above and (cf.
Remark 2.66)
$\displaystyle\nu_{1}(w_{2e_{1}-e_{0}}(x))$
$\displaystyle=(\left(\begin{smallmatrix}&&&1\\\ &1&&\\\ &&1&\\\
1&&&\end{smallmatrix}\right),\left(\begin{smallmatrix}-v_{p}(x)\\\ 0\\\ 0\\\
v_{p}(x)\end{smallmatrix}\right)),$
$\displaystyle\nu_{1}(w_{2e_{2}-e_{0}}(x))$
$\displaystyle=(\left(\begin{smallmatrix}1&&&\\\ &&1&\\\ &1&&\\\
&&&1\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\ -v_{p}(x)\\\
v_{p}(x)\\\ 0\end{smallmatrix}\right)),$
$\displaystyle\nu_{1}(w_{e_{1}-e_{2}}(x))$
$\displaystyle=(\left(\begin{smallmatrix}&1&&\\\ 1&&&\\\ &&&1\\\
&&1&\end{smallmatrix}\right),\left(\begin{smallmatrix}-v_{p}(x)\\\ v_{p}(x)\\\
-v_{p}(x)\\\ v_{p}(x)\end{smallmatrix}\right)),$
$\displaystyle\nu_{1}(w_{e_{1}+e_{2}-e_{0}}(x))$
$\displaystyle=(\left(\begin{smallmatrix}&&1&\\\ &&&1\\\ 1&&&\\\
&1&&\end{smallmatrix}\right),\left(\begin{smallmatrix}-v_{p}(x)\\\
-v_{p}(x)\\\ v_{p}(x)\\\ v_{p}(x)\end{smallmatrix}\right)),$
etc. (Recipe: Write $w_{a}(x)$ as a product of a diagonal matrix
$\operatorname{diag}(d_{1},d_{2},d_{-2},d_{-1})$ and a permutation matrix $P$
(this need not be a factorization in $\operatorname{GSp}(4)$); then
$\nu_{1}(w_{a}(x))=(P,\left(\begin{smallmatrix}-v_{p}(d_{1})\\\
-v_{p}(d_{2})\\\ -v_{p}(d_{-2})\\\ -v_{p}(d_{-1})\end{smallmatrix}\right)).)$
The reduced apartment $A^{\mathrm{red}}$ is the affine $V$-space together with
$\nu\colon
N(\mathbb{Q}_{p})\to\operatorname{Aff}(A^{\mathrm{red}})=\operatorname{GL}(V)\ltimes
V$ given by the same formulas.
The walls (or rather, wall conditions) are given as follows
($n\in\mathbb{Z}$):
$\displaystyle 2e_{1}-e_{0}$ $\displaystyle:n=x_{0}-2x_{1},$ $\displaystyle
2e_{2}-e_{0}$ $\displaystyle:n=x_{0}-2x_{2},$ $\displaystyle e_{1}-e_{2}$
$\displaystyle:n=x_{2}-x_{1},$ $\displaystyle e_{1}+e_{2}-e_{0}$
$\displaystyle:n=x_{0}-x_{1}-x_{2}.$
Figure 1: The reduced apartment with the base alcove highlighted.
##### Lattice chains and parahoric subgroups.
By [BT84a], the extended building
$\mathcal{B}(\operatorname{GL}(X),\mathbb{Q}_{p})$ is in bijection with
norms181818Defining conditions for a norm:
$\alpha(tx)=\alpha(x)+\operatorname{ord}_{p}(t)$,
$\alpha(x+y)\geq\min(\alpha(x),\alpha(y))$, $\alpha(x)=\infty\iff x=0$
$\alpha\colon X\to\mathbb{R}\cup\\{\infty\\}$. Norms in turn are in bijection
with graded lattice chains (cf. Remark 1.10). Indeed, if $\alpha$ is a norm,
define $\Delta_{\alpha}$ to be the set of its balls centered around zero and
$c_{\alpha}(\Lambda):=\inf_{\lambda\in\Lambda}\alpha(\lambda)$. Conversely,
given a graded lattice chain $(\Delta,c)$, define a norm $\alpha$ by
$\alpha(x):=c(\Lambda)$ for the smallest $\Lambda\in\Delta$ with
$x\in\Lambda$.
To go from the extended apartment of $\operatorname{GL}(X)$, an affine
$\mathbb{R}^{n}$-space, where $n=\dim X$, to norms, fix a basis
$e_{1},\dotsc,e_{n}$ of $X$. Then $v\in\mathbb{R}^{n}$ corresponds to the norm
$\alpha_{v}$ with
$\alpha_{v}(\sum t_{i}e_{i})=\min_{i}(\operatorname{ord}_{p}(t_{i})-v_{i}).$
There are seven types of points in the extended apartment (in each case we
choose one in the base alcove to represent all of its type) corresponding to
the vertices, edges and interior of the base alcove:
* •
standard hyperspecial: $x_{\mathrm{hs}}=(0,0,0,0)$
* •
paramodular: $x_{\mathrm{paramod}}=(-1/2,0,0,1/2)$
* •
Klingen: $x_{\mathrm{Klingen}}=(-1/4,0,0,1/4)$
* •
Siegel: $x_{\mathrm{Siegel}}=(-1/4,-1/4,1/4,1/4)$
* •
Iwahori: $x_{\mathrm{Iwahori}}=(-1/4,-1/8,1/8,1/4)$
* •
another hyperspecial: $x=(-1/2,-1/2,1/2,1/2)$
* •
another parahoric: $x=(-1/2,-1/4,1/4,1/2)$
The last two are conjugates (by the Atkin-Lehner element) of the standard
hyperspecial and the Klingen parahoric, respectively (see e.g. [Rös18, 151]);
therefore we will neglect them in the sequel.
For a set of lattices $S$ denote by $\langle S\rangle$ the closure under
homotheties, i.e., $\langle S\rangle:=\\{p^{n}s\;|\;n\in\mathbb{Z},\;s\in
S\\}$.
Then:
* •
$\Delta_{\mathrm{hs}}=\langle\mathbb{Z}_{p}^{4}\rangle$
and $c_{\mathrm{hs}}(\mathbb{Z}_{p}^{4})=0$.
* •
$\Delta_{\mathrm{paramod}}=\langle\mathbb{Z}_{p}^{3}\oplus
p\mathbb{Z}_{p},\;\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}\rangle$
and $c_{\mathrm{paramod}}(\mathbb{Z}_{p}^{3}\oplus
p\mathbb{Z}_{p})=-\frac{1}{2}$, $c_{\mathrm{paramod}}(\mathbb{Z}_{p}\oplus
p\mathbb{Z}_{p}^{3})=0$.
* •
$\Delta_{\mathrm{Klingen}}=\langle\mathbb{Z}_{p}^{4},\mathbb{Z}_{p}^{3}\oplus
p\mathbb{Z}_{p},\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}\rangle$
and $c_{\mathrm{Klingen}}(\mathbb{Z}_{p}^{4})=-1/4$,
$c_{\mathrm{Klingen}}(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p})=0$,
$c_{\mathrm{Klingen}}(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3})=1/4$.
* •
$\Delta_{\mathrm{Siegel}}=\langle\mathbb{Z}_{p}^{4},\mathbb{Z}_{p}^{2}\oplus
p\mathbb{Z}_{p}^{2}\rangle$
and $c_{\mathrm{Siegel}}(\mathbb{Z}_{p}^{4})=-1/4$,
$c_{\mathrm{Siegel}}(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2})=1/4$.
* •
$\Delta_{\mathrm{Iwahori}}=\langle\mathbb{Z}_{p}^{4},\mathbb{Z}_{p}^{3}\oplus
p\mathbb{Z}_{p},\mathbb{Z}_{p}^{2}\oplus
p\mathbb{Z}_{p}^{2},\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}\rangle$
and $c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}^{4})=-1/4$,
$c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p})=-1/8$,
$c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2})=1/8$,
$c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3})=1/4$.
The associated parahoric subgroups are
* •
hyperspecial: $\operatorname{GSp}_{4}(\mathbb{Z}_{p})$
* •
paramodular:
$\operatorname{GSp}_{4}(\mathbb{Q}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&p^{-1}\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$
* •
Klingen:
$\operatorname{GSp}_{4}(\mathbb{Z}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$
* •
Siegel:
$\operatorname{GSp}_{4}(\mathbb{Z}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$
* •
Iwahori:
$\operatorname{GSp}_{4}(\mathbb{Z}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$
###### Remark 2.68.
Dualizing with respect to the symplectic form, we have
$\displaystyle(\mathbb{Z}_{p}^{4})^{\vee}$ $\displaystyle=\mathbb{Z}_{p}^{4},$
$\displaystyle(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3})^{\vee}$
$\displaystyle=p^{-1}(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p}),$
$\displaystyle(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p})^{\vee}$
$\displaystyle=p^{-1}(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}),$
$\displaystyle(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2})^{\vee}$
$\displaystyle=p^{-1}(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2}).$
##### Admissible set.
We compute the admissible set in the way outlined in Remark 2.29. The
cocharacter $\mu$ is $(1,1,0,0)$.
We obtain
$\displaystyle\mathrm{Adm}(\\{\mu\\})=\bigl{\\{}$
$\displaystyle\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}1\\\ 1\\\
0\\\
0\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(2\quad{-2}),\left(\begin{smallmatrix}1\\\
0\\\ 1\\\
0\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}1\\\
0\\\ 1\\\ 0\end{smallmatrix}\right)\Bigr{)},$
$\displaystyle\Bigl{(}(1\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\
1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(1\quad
2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\
1\end{smallmatrix}\right)\Bigr{)},$ $\displaystyle\Bigl{(}(1\quad
2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\
1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}0\\\
1\\\ 0\\\ 1\end{smallmatrix}\right)\Bigr{)},$
$\displaystyle\Bigl{(}(1\quad{-1})(2\quad{-2}),\left(\begin{smallmatrix}0\\\
0\\\ 1\\\
1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(1\quad{-1}),\left(\begin{smallmatrix}0\\\
0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)},$
$\displaystyle\Bigl{(}(1\quad{-2})(2\quad{-1}),\left(\begin{smallmatrix}0\\\
0\\\ 1\\\
1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(1\quad{-2}\quad{-1}\quad
2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)},$
$\displaystyle\Bigl{(}(2\quad{-2}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\
1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}0\\\
0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)}\bigr{\\}},$
or, in terms of Frobenii (cf. Construction 2.43)
$\displaystyle\Bigl{\\{}$ $\displaystyle\left(\begin{smallmatrix}p&0&0&0\\\
0&p&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}p&0&0&0\\\ 0&0&1&0\\\
0&p&0&0\\\ 0&0&0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}p&0&0&0\\\
0&1&0&0\\\ 0&0&p&0\\\
0&0&0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&1\\\ 0&p&0&0\\\
0&0&1&0\\\ p&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&1&0\\\
p&0&0&0\\\ 0&0&0&1\\\ 0&p&0&0\end{smallmatrix}\right),$
$\displaystyle\left(\begin{smallmatrix}0&1&0&0\\\ p&0&0&0\\\ 0&0&0&1\\\
0&0&p&0\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0&0&0\\\ 0&p&0&0\\\
0&0&1&0\\\ 0&0&0&p\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&1\\\
0&0&1&0\\\ 0&p&0&0\\\
p&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&1\\\ 0&1&0&0\\\
0&0&p&0\\\ p&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&1&0\\\
0&0&0&1\\\ p&0&0&0\\\ 0&p&0&0\end{smallmatrix}\right),$
$\displaystyle\left(\begin{smallmatrix}0&1&0&0\\\ 0&0&0&1\\\ p&0&0&0\\\
0&0&p&0\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0&0&0\\\ 0&0&1&0\\\
0&p&0&0\\\ 0&0&0&p\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&p&0\\\ 0&0&0&p\end{smallmatrix}\right)\Bigr{\\}}.$
##### Siegel level.
From now on, we consider the Siegel level structure. Denote the Siegel
parahoric by $K$ and the standard hyperspecial subgroup by $H$. Here $W_{K}$
is generated by $({-1}\quad{-2})(1\quad 2)$, while $W_{H}$ is generated by
$W_{K}$ and $(2\quad{-2})$.
Recalling Remark 2.31 (2), we note that one has a natural morphism
$\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}\to\overline{\mathcal{G}}_{H}\text{-}\mathrm{Zip}\times\overline{\mathcal{G}}_{H}\text{-}\mathrm{Zip}$.
We have
$\displaystyle\mathrm{KR}(K,\\{\mu\\})$
$\displaystyle=\Bigl{\\{}(\operatorname{id},\left(\begin{smallmatrix}1\\\ 1\\\
0\\\
0\end{smallmatrix}\right)),((1\quad{-2})(2\quad{-1}),\left(\begin{smallmatrix}0\\\
0\\\ 1\\\ 1\end{smallmatrix}\right)),((1\quad
2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\
1\end{smallmatrix}\right)),$
$\displaystyle(\operatorname{id},\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\
1\end{smallmatrix}\right)),((1\quad{-2}\quad{-1}\quad
2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)),((1\quad
2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\
1\end{smallmatrix}\right))\Bigr{\\}},$
$\displaystyle\mathrm{EKOR}(K,\\{\mu\\})$
$\displaystyle=\mathrm{KR}(K,\\{\mu\\})\cup\Bigl{\\{}(\operatorname{id},\left(\begin{smallmatrix}0\\\
1\\\ 0\\\
1\end{smallmatrix}\right)),\quad((2\quad{-2}),\left(\begin{smallmatrix}0\\\
0\\\ 1\\\
1\end{smallmatrix}\right)),\quad((1\quad{-1}),\left(\begin{smallmatrix}0\\\
1\\\ 0\\\ 1\end{smallmatrix}\right))\Bigr{\\}}.$
In the following table, $w^{j}$ is the isomorphism type of the
$\overline{\mathcal{G}}_{H}$-zip at position $j$. For
$\mathcal{C}^{\bullet},\mathcal{D}^{\bullet}$ we give (indices of) basis
vectors. “$\leftarrow$” means “same as in the column adjacent to the left”.
$\alpha_{0}\colon\bar{\mathbb{F}}_{p}^{4}\to\bar{\mathbb{F}}_{p}^{4}$ is the
projection onto the plane spanned by the $1,2$-coordinates, $\alpha_{2}$ the
projection onto the plane spanned by the $-2,-1$-coordinates. By
$\alpha_{j,\mathcal{C}^{\bullet}/\mathcal{D}^{\bullet}}$ we denote the induced
maps on
$\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ and
$\mathcal{D}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{D}^{\bullet}$,
respectively. Each $\mathcal{C}^{j}\subseteq\mathcal{V}^{j}$ has a canonical
complement in terms of standard basis vectors. Importantly however, we will
not always have a complementary _chain_ of linear subspaces. In any event,
below we say what the $\alpha_{j,\mathcal{C}^{\bullet}/\mathcal{D}^{\bullet}}$
are the projection onto if interpreted as described. For instance, the
projection onto $\emptyset$ is the zero map. So in that case
$\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ (or
$\mathcal{D}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{D}^{\bullet}$) is a
chain of vector spaces with zero transition maps.
$w$ | KR-type | $\mathcal{C}^{0}$ | $\mathcal{D}^{0}$ | $\mathcal{C}^{2}$ | $\mathcal{D}^{2}$ | $w^{0}$ | $w^{2}$ | $\alpha_{2,\mathcal{C}^{\bullet}}$ | $\alpha_{0,\mathcal{C}^{\bullet}}$ | $\alpha_{2,\mathcal{D}^{\bullet}}$ | $\alpha_{0,\mathcal{D}^{\bullet}}$
---|---|---|---|---|---|---|---|---|---|---|---
$(\operatorname{id},\left(\begin{smallmatrix}1\\\ 1\\\ 0\\\ 0\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{-2,-1\\}$ | $(-2\quad 1)({-1}\quad 2)$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$
$((1\quad{-2})(2\quad{-1}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\operatorname{id}$ | $\leftarrow$ | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$
$((1\quad 2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\\{-2,-1\\}$ | $(-2\quad 2)$ | $\leftarrow$ | $\\{1\\}$ | $\\{-1\\}$ | $\\{2\\}$ | $\\{-2\\}$
$(\operatorname{id},\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{-2,-1\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{1,2\\}$ | $(-2\quad 1)({-1}\quad 2)$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$
$((1\quad{-2}\quad{-1}\quad 2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{-2,1\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,1\\}$ | $(-2\quad 2)$ | $\leftarrow$ | $\\{2\\}$ | $\\{-2\\}$ | $\\{1\\}$ | $\\{-1\\}$
$((1\quad 2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\operatorname{id}$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$
$(\operatorname{id},\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $((1\quad 2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\\{-1,2\\}$ | $\\{-1,2\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $(-2\quad 1)({-1}\quad 2)$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$
$((2\quad{-2}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $((1\quad{-2}\quad{-1}\quad 2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\\{-1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,1\\}$ | $(-2\quad{-1}\quad 2\quad 1)$ | $\leftarrow$ | $\\{1\\}$ | $\\{-1\\}$ | $\\{1\\}$ | $\\{-1\\}$
$((1\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $((1\quad 2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\\{1,2\\}$ | $\\{-1,2\\}$ | $\\{-2,1\\}$ | $\\{-2,-1\\}$ | $(-2\quad{-1}\quad 2\quad 1)$ | $\leftarrow$ | $\\{2\\}$ | $\\{-2\\}$ | $\\{2\\}$ | $\\{-2\\}$
###### Observations 2.69.
* •
We always have $w^{0}=w^{2}$. This is explained by the fact that the Ekedahl-
Oort stratification in this case agrees with the Newton stratification (and
isogenous abelian varieties by definition lie in the same Newton stratum).
* •
Consider the Kottwitz-Rapoport strata containing more than one EKOR stratum
(i.e., containing two EKOR strata). Then we can distinguish among the EKOR
strata by looking at the Ekedahl-Oort stratum. In other words, the EKOR
stratification is in this case the coarsest common refinement of the Kottwitz-
Rapoport and Ekedahl-Oort stratifications.
## References
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|
2024-09-04T02:54:58.744700 | 2020-03-07T09:19:04 | 2003.04739 | {
"authors": "Gioele Zardini, Nicolas Lanzetti, Mauro Salazar, Andrea Censi, Emilio\n Frazzoli, Marco Pavone",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26136",
"submitter": "Gioele Zardini",
"url": "https://arxiv.org/abs/2003.04739"
} | arxiv-papers | # On the Co-Design of AV-Enabled Mobility Systems
Gioele Zardini1,3, Nicolas Lanzetti2,3, Mauro Salazar3,4, Andrea Censi1,
Emilio Frazzoli1, and Marco Pavone3 1Institute for Dynamic Systems and
Control, ETH Zürich<EMAIL_ADDRESS>Control Laboratory, ETH Zürich<EMAIL_ADDRESS>of Aeronautics and
Astronautics, Stanford University<EMAIL_ADDRESS>Systems
Technology Group, Eindhoven University of Technology<EMAIL_ADDRESS>preliminary version of this paper was presented at the 99th Annual Meeting of
the Transportation Research Board [1].This research was supported by the
National Science Foundation under CAREER Award CMMI-1454737, the Toyota
Research Institute (TRI), and ETH Zürich. This article solely reflects the
opinions and conclusions of its authors and not NSF, TRI, or any other entity.
###### Abstract
The design of autonomous vehicles (AVs) and the design of AV-enabled mobility
systems are closely coupled. Indeed, knowledge about the intended service of
AVs would impact their design and deployment process, whilst insights about
their technological development could significantly affect transportation
management decisions. This calls for tools to study such a coupling and co-
design AVs and AV-enabled mobility systems in terms of different objectives.
In this paper, we instantiate a framework to address such co-design problems.
In particular, we leverage the recently developed theory of co-design to frame
and solve the problem of designing and deploying an intermodal Autonomous
Mobility-on-Demand system, whereby AVs service travel demands jointly with
public transit, in terms of fleet sizing, vehicle autonomy, and public transit
service frequency. Our framework is modular and compositional, allowing one to
describe the design problem as the interconnection of its individual
components and to tackle it from a system-level perspective. To showcase our
methodology, we present a real-world case study for Washington D.C., USA. Our
work suggests that it is possible to create user-friendly optimization tools
to systematically assess costs and benefits of interventions, and that such
analytical techniques might gain a momentous role in policy-making in the
future.
## I Introduction
Arguably, the current design process for AVs largely suffers from the lack of
clear, specific requirements in terms of the service such vehicles will be
providing. Yet, knowledge about their intended service (e.g., last-mile versus
point-to-point travel) might dramatically impact how the AVs are designed,
and, critically, significantly ease their development process. For example, if
for a given city we knew that for an effective on-demand mobility system
autonomous cars only need to drive up to 25 mph and only on relatively easy
roads, their design would be greatly simplified and their deployment could
certainly be accelerated. At the same time, from the system-level perspective
of transportation management, knowledge about the trajectory of technology
development for AVs would certainly impact decisions on infrastructure
investments and provision of service. In other words, the design of the AVs
and the design of a mobility system leveraging AVs are intimately coupled.
This calls for methods to reason about such a coupling, and in particular to
_co-design_ the AVs and the associated AV-enabled mobility system. A key
requirement in this context is the ability to account for a range of
heterogeneous objectives that are often not directly comparable (consider, for
instance, travel time and emissions).
Accordingly, the goal of this paper is to lay the foundations for a framework
through which one can co-design future AV-enabled mobility systems.
Specifically, we show how one can leverage the recently developed mathematical
theory of co-design [2, 3, 4], which provides a general methodology to co-
design complex systems in a modular and compositional fashion. This tool
delivers the set of rational design solutions lying on the Pareto front,
allowing one to reason about costs and benefits of the individual design
options. The framework is instantiated in the setting of co-designing
intermodal AMoD systems [5], whereby fleets of self-driving vehicles provide
on-demand mobility jointly with public transit. Aspects subject to co-design
include fleet size, AV-specific characteristics, and public transit service
frequency.
### I-A Literature Review
Our work lies at the interface of the design of urban public transportation
services and the design of AMoD systems. The first research stream is reviewed
in [6, 7], and comprises _strategic_ long-term infrastructure modifications
and _operational_ short-term scheduling. The joint design of traffic network
topology and control infrastructure has been presented in [8]. Public
transportation scheduling has been solved jointly with the design of the
transit network in a passengers’ and operators’ cost-optimal fashion in [9],
using demand-driven approaches in [10], and in an energy-efficient way in
[11]. However, these works only focus on the public transit system and do not
consider its joint design with an AMoD system. The research on the design of
AMoD systems is reviewed in [12] and mainly pertains their fleet sizing. In
this regard, studies range from simulation-based approaches [13, 14, 15, 16]
to analytical methods [17]. In [18], the authors jointly design the fleet size
and the charging infrastructure, and formulate the arising design problem as a
mixed integer linear program. The authors of [19] solve the fleet sizing
problem together with the vehicle allocation problem. Finally, [20] co-designs
the AMoD fleet size and its composition. More recently, the joint design of
multimodal transit networks and AMoD systems was formulated in [21] as a
bilevel optimization problem and solved with heuristics. Overall, the problem-
specific structure of existing design methods for AMoD systems is not amenable
to a modular and compositional problem formulation. Moreover, previous work
does not capture important aspects of AV-enabled mobility systems, such as
other transportation modes and AV-specific design parameters (e.g., the level
of autonomy).
### I-B Statement of Contribution
In this paper we lay the foundations for the systematic study of the design of
AV-enabled mobility systems. Specifically, we leverage the mathematical theory
of co-design [2] to devise a framework to study the design of intermodal AMoD
(I-AMoD) systems in terms of fleet characteristics and public transit service,
enabling the computation of the _rational_ solutions lying on the Pareto front
of minimal travel time, transportation costs, and emissions. Our framework
allows one to structure the design problem in a modular way, in which each
different transportation option can be “plugged in” in a larger model. Each
model has minimal assumptions: Rather than properties such as linearity and
convexity, we ask for very general monotonicity assumptions. For example, we
assume that the cost of automation increases monotonically with the speed
achievable by the AV. We are able to obtain the full Pareto front of
_rational_ solutions, or, given policies, to weigh incomparable costs (such as
travel time and emissions) and to present actionable information to the
stakeholders of the mobility ecosystem. We showcase our methodology through a
real-world case study of Washington D.C., USA. We show how, given the model,
we can easily formulate and answer several questions regarding the
introduction of new technologies and investigate possible infrastructure
interventions.
### I-C Organization
The remainder of this paper is structured as follows: Section II reviews the
mathematical theory of co-design. Section III presents the co-design problem
for AV-enabled mobility systems. We showcase our approach with real-world case
studies for Washington D.C., USA, in Section IV. Section V concludes the paper
with a discussion and an overview on future research directions.
## II Background
This paper builds on the mathematical theory of co-design, presented in [2].
In this section, we present a review of the main contents needed for this
work.
### II-A Orders
We will use basic facts from order theory, which we review in the following.
###### Definition II.1 (Poset).
A partially ordered set (poset) is a tuple
$\langle\mathcal{P},\preceq_{\mathcal{P}}\rangle$, where $\mathcal{P}$ is a
set and $\preceq_{\mathcal{P}}$ is a partial order, defined as a reflexive,
transitive, and antisymmetric relation.
Given a poset, we can formalize the idea of “Pareto front” through antichains.
###### Definition II.2 (Antichains).
A subset $S\subseteq\mathcal{P}$ is an antichain iff no elements are
comparable: For $x,y\in S$, $x\preceq y$ implies $x=y$. We denote by
$\textsf{A}\mathcal{P}$ the set of all antichains in $\mathcal{P}$.
###### Definition II.3 (Directed set).
A subset $S\subseteq\mathcal{P}$ is directed if each pair of elements in $S$
has an upper bound: For all $a,b\in S$, there exists a $c\in S$ such that
$a\preceq c$ and $b\preceq c$.
###### Definition II.4 (Completeness).
A poset is a complete partial order (CPO) if each of its directed subsets has
a supremum and a least element.
For instance, the poset $\langle\mathbb{R}_{+},\leq\rangle$, with
$\mathbb{R}_{+}\coloneqq\\{x\in\mathbb{R}\,|\,x\geq 0\\}$, is not complete, as
its directed subset $\mathbb{R}_{+}\subseteq\mathbb{R}_{+}$ does not have an
upper bound (and therefore a supremum). Nonetheless, we can make it complete
by artificially adding a top element $\top$, i.e., by defining
$\langle\overline{\mathbb{R}}_{+},\leq\rangle$ with
$\overline{\mathbb{R}}_{+}\coloneqq\mathbb{R}_{+}\cup\\{\top\\}$ and
$a\leq\top$ for all $a\in\mathbb{R}_{+}$. Similarly, we can complete
$\mathbb{N}$ to $\overline{\mathbb{N}}$.
In this setting, Scott-continuous maps will play a key role. Intuitively,
Scott-continuity can be understood as a stronger notion of monotonicity.
###### Definition II.5 (Scott continuity).
A map $f:\mathcal{P}\rightarrow\mathcal{Q}$ between two posets
$\langle\mathcal{P},\preceq_{\mathcal{P}}\rangle$ and
$\langle\mathcal{Q},\preceq_{\mathcal{Q}}\rangle$ is Scott-continuous iff for
each directed set $D\subseteq\mathcal{P}$ the image $f(D)$ is directed and
$\sup f(D)=f(\sup D)$.
### II-B Mathematical Theory of Co-Design
We start by presenting design problems with implementation (DPIs), which can
then be composed and interconnected to form a co-design problem with
implementation (CDPI).
###### Definition II.6 (DPI).
A DPI is a tuple
$\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$:
* •
$\mathcal{F}$ is a poset, called functionality space;
* •
$\mathcal{R}$ is a poset, called resource space;
* •
$\mathcal{I}$ is a set, called implementation space;
* •
the map $\textsf{{exe}}:\mathcal{I}\to\mathcal{F}$ maps an implementation to
the functionality it provides;
* •
the map $\textsf{{eva}}:\mathcal{I}\to\mathcal{R}$, maps an implementation to
the resources it requires.
Given a DPI we can define a map which, given a functionality
$\textsf{{f}}\in\mathcal{F}$, returns all the non-comparable resources (i.e.,
the antichain) which provide f.
###### Definition II.7 (Functionality to resources map).
Given a DPI
$\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$
define the map $h:\mathcal{F}\to\textsf{{A}}\mathcal{R}$ as
$\displaystyle h:$ $\displaystyle\mathcal{F}$ $\displaystyle\to$
$\displaystyle\textsf{{A}}\mathcal{R}$ (1) f $\displaystyle\mapsto$
$\displaystyle\min_{\preceq_{\mathcal{R}}}\\{\textsf{{eva}}(\textsf{{i}})\,|\,\textsf{{i}}\in\mathcal{I}\wedge\textsf{{f}}\preceq\textsf{{exe}}(\textsf{{i}})\\}.$
In particular, if a functionality is infeasible, then
$h(\textsf{{f}})=\emptyset$. We now turn our attention to “monotone” DPIs.
###### Definition II.8 (Monotone DPI).
We say a DPI
$\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$
is monotone if:
1. 1.
The posets $\mathcal{F}$ and $\mathcal{R}$ are CPOs.
2. 2.
The map $h$ (see Definition II.7) is Scott-continuous.
Individual DPIs can be composed in series (i.e., the functionality of a DPI is
the resource of a second DPI) and in parallel (i.e., two DPIs share the same
resource or functionality) to obtain a CDPI. Notably, such compositions
preserve monotonicity and, thus, all related algorithmic properties. For
further details we refer to [2].
## III Co-Design of AV-enabled Mobility Systems
### III-A Intermodal AMoD Framework
#### III-A1 Multi-Commodity Flow Model
The transportation system and its different modes are modeled using the
digraph $\mathcal{G}=\left(\mathcal{V},\mathcal{A}\right)$, shown in Footnote
2.
Figure 1: The I-AMoD network consists of a road, a walking, and a public
transportation digraph. The coloured circles represent stops or intersections
and the black arrows denote road links, pedestrian pathways, or public transit
arcs. Dashed lines are nodes which are close geographically, while grey arrows
denote the mode-switching arcs connecting them222We thank Ms. Sonia Monti for
the illustration..
It is described through a set of nodes $\mathcal{V}$ and a set of arcs
$\mathcal{A}\subseteq\mathcal{V}\times\mathcal{V}$. Specifically, it contains
a road network layer
$\mathcal{G}_{\mathrm{R}}=\left(\mathcal{V}_{\mathrm{R}},\mathcal{A}_{\mathrm{R}}\right)$,
a public transportation layer
$\mathcal{G}_{\mathrm{P}}=\left(\mathcal{V}_{\mathrm{P}},\mathcal{A}_{\mathrm{P}}\right)$,
and a walking layer
$\mathcal{G}_{\mathrm{W}}=\left(\mathcal{V}_{\mathrm{W}},\mathcal{A}_{\mathrm{W}}\right)$.
The road network is characterized through intersections
$i\in\mathcal{V}_{\mathrm{R}}$ and road segments
$(i,j)\in\mathcal{A}_{\mathrm{R}}$. Similarly, public transportation lines are
modeled through station nodes $i\in\mathcal{V}_{\mathrm{P}}$ and line segments
$(i,j)\in\mathcal{A}_{\mathrm{P}}$. The walking network contains walkable
streets $(i,j)\in\mathcal{A}_{\mathrm{W}}$, connecting intersections
$i\in\mathcal{V}_{\mathrm{W}}$. Our model allows mode-switching arcs
$\mathcal{A}_{\mathrm{C}}\subseteq\mathcal{V}_{\mathrm{R}}\times\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{W}}\times\mathcal{V}_{\mathrm{R}}\cup\mathcal{V}_{\mathrm{P}}\times\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{W}}\times\mathcal{V}_{\mathrm{P}}$,
connecting the road and the public transportation layers to the walking layer.
Consequently,
$\mathcal{V}=\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{R}}\cup\mathcal{V}_{\mathrm{P}}$
and
$\mathcal{A}=\mathcal{A}_{\mathrm{W}}\cup\mathcal{A}_{\mathrm{R}}\cup\mathcal{A}_{\mathrm{P}}\cup\mathcal{A}_{\mathrm{C}}$.
Consistently with the structural properties of road and walking networks in
urban environments, we assume the graph $\mathcal{G}$ to be strongly
connected. We represent customer movements by means of travel requests. A
travel request refers to a customer flow starting its trip at a node
$o\in\mathcal{V}$ and ending it at a node $d\in\mathcal{V}$.
###### Definition III.1 (Travel request).
A travel request $\rho$ is a triple
$(o,d,\alpha)\in\mathcal{V}\times\mathcal{V}\times\mathbb{R}_{+}$, described
by an origin node $o\in\mathcal{V}$, a destination node $d\in\mathcal{V}$, and
the request rate $\alpha>0$, namely, the number of customers who want to
travel from $o$ to $d$ per unit time.
To ensure that a customer is not forced to use a given transportation mode, we
assume all requests to lie on the walking digraph, i.e.,
$o_{m},d_{m}\in\mathcal{V}_{\mathrm{W}}$ for all
$m\in\mathcal{M}\coloneqq\\{1,\ldots,M\\}$.
The flow $f_{m}\left(i,j\right)\geq 0$ represents the number of customers per
unit time traversing arc $(i,j)\in\mathcal{A}$ and satisfying a travel request
$m$. Furthermore, $f_{0}\left(i,j\right)\geq 0$ denotes the flow of empty AVs
on road arcs $(i,j)\in\mathcal{A}_{\mathrm{R}}$. This accounts for rebalancing
flows of AVs between a customer’s drop-off and the next customer’s pick-up.
Assuming AVs to carry one customer at a time, the flows satisfy
$\displaystyle\sum_{i:(i,j)\in\mathcal{A}}f_{m}\left(i,j\right)+\mathds{1}_{j=o_{m}}\cdot\alpha_{m}=\sum_{k:(j,k)\in\mathcal{A}}f_{m}\left(j,k\right)+\mathds{1}_{j=d_{m}}\cdot\alpha_{m}$
$\displaystyle\hskip 136.5733pt\forall m\in\mathcal{M},\,j\in\mathcal{V}$ (2a)
$\displaystyle\sum_{i:(i,j)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(i,j\right)=\sum_{k:(j,k)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(j,k\right)\quad\forall
j\in\mathcal{V}_{\mathrm{R}},$ (2b)
where $\mathbb{1}_{j=x}$ denotes the boolean indicator function and
$f_{\mathrm{tot}}\left(i,j\right)\coloneqq
f_{0}\left(i,j\right)+\sum_{m\in\mathcal{M}}f_{m}\left(i,j\right)$.
Specifically, (2a) guarantees flows conservation for every transportation
demand, and (2b) preserves flow conservation for AVs on every road node.
Combining conservation of customers (2a) with the conservation of AVs (2b)
guarantees rebalancing AVs to match the demand.
### III-B Travel Time and Travel Speed
The variable $t_{ij}$ denotes the time needed to traverse an arc $(i,j)$ of
length $s_{ij}$. We assume a constant walking speed on walking arcs and infer
travel times on public transportation arcs from the public transit schedules.
Assuming that the public transportation system at node $j$ operates with
frequency $\varphi_{j}$, switching from a pedestrian vertex
$i\in\mathcal{V}_{\mathrm{W}}$ to a public transit station
$j\in\mathcal{V}_{\mathrm{P}}$ takes, on average,
$t_{ij}=t_{\mathrm{WS}}+0.5\cdot
1/\varphi_{j}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{W}},$ (3)
where $t_{\mathrm{WS}}$ is a constant sidewalk-to-station travel time. We
assume that the average waiting time for AMoD vehicles is $t_{\mathrm{WR}}$,
and that switching from the road graph and the public transit graph to the
walking graph takes the transfer times $t_{\mathrm{RW}}$ and
$t_{\mathrm{SW}}$, respectively. While each road arc
$(i,j)\in\mathcal{A}_{\mathrm{R}}$ is characterized by a speed limit
$v_{\mathrm{L,V},ij}$, AVs safety protocols impose a maximum achievable
velocity $v_{\mathrm{V,a}}$. In order to prevent too slow and therefore
dangerous driving behaviours, we only consider road arcs through which the AVs
can drive at least at a fraction $\beta$ of the speed limit: Arc
$(i,j)\in\mathcal{A}_{\mathrm{R}}$ is kept in the road network iff
$v_{\mathrm{V,a}}\geq\beta\cdot v_{\mathrm{L,V},ij},$ (4)
where $\beta\in(0,1]$. We set the velocity of all arcs fulfilling condition
(4) to $v_{\mathrm{V},ij}=\min\\{v_{\mathrm{V,a}},v_{\mathrm{L,V},ij}\\}$ and
compute the travel time to traverse them as
$t_{ij}=s_{ij}/v_{\mathrm{V},ij}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$
(5)
### III-C Road Congestion
We capture congestion effects with a threshold model. The total flow on each
road arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$, given by the sum of the AVs flow
$f_{\mathrm{tot}}\left(i,j\right)$ and the baseline usage $u_{ij}$ (e.g.,
private vehicles), must remain below the nominal capacity $c_{ij}$ of the arc:
$f_{\mathrm{tot}}\left(i,j\right)+u_{ij}\leq
c_{ij}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$ (6)
### III-D Energy Consumption
We compute the energy consumption of AVs for each road link considering an
urban driving cycle, scaled so that the average speed $v_{\mathrm{avg,cycle}}$
matches the free-flow speed on the link. The energy consumption is then scaled
as
$e_{ij}=e_{\mathrm{cycle}}\cdot
s_{ij}/s_{\mathrm{cycle}}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$ (7)
For the public transportation system, we assume a constant energy consumption
per unit time. This approximation is reasonable in urban environments, as the
operation of the public transportation system is independent from the number
of customers serviced, and its energy consumption is therefore customer-
invariant.
### III-E Fleet Size
We consider a fleet of $n_{\mathrm{V,max}}$ AVs. In a time-invariant setting,
the number of vehicles on arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$ is expressed
as the product of the total vehicles flow on the arc and its travel time.
Therefore, we constrain the number of used AVs as
$n_{\mathrm{V,u}}=\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(i,j\right)\cdot
t_{ij}\leq n_{\mathrm{V,max}}.$ (8)
### III-F Discussion
A few comments are in order. First, we assume the demand to be time-invariant
and allow flows to have fractional values. This assumption is in line with the
mesoscopic and system-level planning perspective of our study. Second, we
model congestion effects using a threshold model. This approach can be
interpreted as a municipality preventing AVs to exceed the critical flow
density on road arcs. AVs can be therefore assumed to travel at free-flow
speed [22]. This assumption is realistic for an initial low penetration of
AMoD systems in the transportation market, especially when the AV fleet is of
limited size. Finally, we allow AVs to transport one customer at the time
[23].
### III-G Co-Design Framework
We integrate the I-AMoD framework presented in Section III-A in the co-design
formalism, allowing one to decompose the CDPI of a complex system in the DPIs
of its individual components in a modular, compositional, and systematic
fashion. We aim at computing the antichain of resources, quantified in terms
of costs, average travel time per trip, and emissions required to provide the
mobility service to a set of customers. In order to achieve this, we decompose
the CDPI in the DPIs of the individual AVs (Section III-G1), of the AV fleet
(Section III-G3), and of the public transportation system (Section III-G2).
The interconnection of the presented DPIs is presented in Section III-G4.
#### III-G1 The Autonomous Vehicle Design Problem
The AV DPI consists of selecting the maximal speed of the AVs. Under the
rationale that driving safely at higher speed requires more advanced sensing
and algorithmic capabilities, we model the achievable speed of the AVs
$v_{\mathrm{V,a}}$ as a monotone function of the vehicle fixed costs
$C_{\mathrm{V,f}}$ (resulting from the cost of the vehicle $C_{\mathrm{V,v}}$
and the cost of its automation $C_{\mathrm{V,a}}$) and of the mileage-
dependent operational costs $C_{\mathrm{V,o}}$ (accounting for maintenance,
cleaning, energy consumption, depreciation, and opportunity costs [24]). In
this setting, the AV DPI provides the functionality $v_{\mathrm{V,a}}$ and
requires the resources $C_{\mathrm{V,f}}$ and $C_{\mathrm{V,o}}$.
Consequently, the functionality space is
$\mathcal{F}_{\mathrm{V}}=\overline{\mathbb{R}}_{+}$, and the resources space
is
$\mathcal{R}_{\mathrm{V}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.
#### III-G2 The Subway Design Problem
We design the public transit infrastructure by means of the service frequency
introduced in Section III-B. Specifically, we assume that the service
frequency $\varphi_{j}$ scales linearly with the size of the train fleet
$n_{\mathrm{S}}$ as
$\varphi_{j}/\varphi_{j,\mathrm{base}}=n_{\mathrm{S}}/n_{\mathrm{S,base}}.$
(9)
We relate a train fleet of size $n_{\mathrm{S}}$ to the fixed costs
$C_{\mathrm{S,f}}$ (accounting for train and infrastructural costs) and to the
operational costs $C_{\mathrm{S,o}}$ (accounting for energy consumption,
vehicles depreciation, and train operators’ wages). Given the passenger-
independent public transit operation in today’s cities, we reasonably assume
the operational costs $C_{\mathrm{S,o}}$ to be mileage independent and to only
vary with the size of the fleet. Formally, the number of acquired trains
$n_{\mathrm{S,a}}=n_{\mathrm{S}}-n_{\mathrm{S,base}}$ is a functionality,
whereas $C_{\mathrm{S,f}}$ and $C_{\mathrm{S,o}}$ are resources. The
functionality space is $\mathcal{F}_{\mathrm{S}}=\overline{\mathbb{N}}$ and
the resources space is
$\mathcal{R}_{\mathrm{S}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.
#### III-G3 The I-AMoD Framework Design Problem
The I-AMoD DPI considers demand satisfaction as a functionality. Formally,
$\mathcal{F}_{\mathrm{O}}=2^{\mathcal{V}\times\mathcal{V}\times\overline{\mathbb{R}}_{+}}$
with the partial order $\preceq_{\mathcal{F}_{\mathrm{O}}}$ defined by
$\mathcal{D}_{1}\coloneqq\\{(o^{1}_{i},d^{1}_{i},\alpha^{1}_{i})\\}_{i=1}^{M_{1}}\preceq_{\mathcal{F}_{\mathrm{O}}}\\{(o^{2}_{i},d^{2}_{i},\alpha^{2}_{i})\\}_{i=1}^{M_{2}}\eqqcolon\mathcal{D}_{2}$
iff for all $(o^{1},d^{1},\alpha^{1})\in\mathcal{D}_{1}$ there is some
$(o^{2},d^{2},\alpha^{2})\in\mathcal{D}_{2}$ with $o^{1}=o^{2}$,
$d^{1}=d^{2}$, and $\alpha^{2}_{i}\geq\alpha^{1}_{i}$. In other words,
$\mathcal{D}_{1}\preceq_{\mathcal{F}_{\mathrm{O}}}\mathcal{D}_{2}$ if every
travel request in $\mathcal{D}_{1}$ is in $\mathcal{D}_{2}$ too. To
successfully satisfy a given set of travel requests, we require the following
resources: (i) the achievable speed of the AVs $v_{\mathrm{V,a}}$, (ii) the
number of available AVs per fleet $n_{\mathrm{V,max}}$, (iii) the number of
trains $n_{\mathrm{S,a}}$ acquired by the public transportation system, and
(iv) the average travel time of a trip
$t_{\mathrm{avg}}\coloneqq\frac{1}{\alpha_{\mathrm{tot}}}\cdot\sum_{m\in\mathcal{M},(i,j)\in\mathcal{A}}t_{ij}\cdot
f_{m}\left(i,j\right),$ (10)
with $\alpha_{\mathrm{tot}}\coloneqq\sum_{m\in\mathcal{M}}\alpha_{m}$, (v) the
total distance driven by the AVs per unit time
$s_{\mathrm{V,tot}}\coloneqq\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}s_{ij}\cdot
f_{\mathrm{tot}}\left(i,j\right),$ (11)
(vi) the total AVs CO2 emissions per unit time
$m_{\mathrm{CO_{2},V,tot}}\coloneqq\gamma\cdot\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}e_{ij}\cdot
f_{\mathrm{tot}}\left(i,j\right),$ (12)
where $\gamma$ relates the energy consumption and the CO2 emissions. We assume
that customers’ trips and AMoD rebalancing strategies are chosen to maximize
customers’ welfare, defined through the average travel time
$t_{\mathrm{avg}}$. Hence, we link the functionality and resources of the
I-AMoD DPI through the following optimization problem:
$\begin{split}\min_{\begin{subarray}{c}f_{m}\left(\cdot,\cdot\right)\geq 0,\\\
f_{0}\left(\cdot,\cdot\right)\geq
0\end{subarray}}t_{\mathrm{avg}}&=\frac{1}{\alpha_{\mathrm{tot}}}\sum_{m\in\mathcal{M},(i,j)\in\mathcal{A}}t_{ij}\cdot
f_{m}\left(i,j\right)\\\ &\mathrm{s.t.\ Eq.}\eqref{eq:flowconstotal},\
\mathrm{Eq.}\eqref{eq:capacity},\
\mathrm{Eq.}\eqref{eq:fleetsizecar}.\end{split}$ (13)
Formally, $\mathcal{F}_{\mathrm{O}}=\overline{\mathbb{R}}_{+}$, and
$\mathcal{R}_{\mathrm{O}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{N}}\times\overline{\mathbb{N}}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.
###### Remark.
In general, the optimization problem (13) might possess multiple optimal
solutions, making the relation between resources and functionality ill-posed.
To overcome this subtlety, if two solutions share the same average travel
time, we select the one incurring in the lowest mileage.
#### III-G4 The Monotone Co-Design Problem
The functionality of the system is to provide mobility service to the
customers. Formally, the functionality provided by the CDPI is the set of
travel requests. To provide the mobility service, the following three
resources are required. First, on the customers’ side, we require an average
travel time, defined in (10). Second, on the municipality side, the resource
is the total transportation cost of the intermodal mobility system. Assuming
an average vehicles’ life $l_{\mathrm{V}}$, an average trains’ life
$l_{\mathrm{S}}$, and a baseline subway fleet of $n_{\mathrm{S,base}}$ trains,
we express the total costs as
$C_{\mathrm{tot}}=C_{\mathrm{V}}+C_{\mathrm{S}},$ (14)
where $C_{\mathrm{V}}$ is the AV-related cost
$C_{\mathrm{V}}=\frac{C_{\mathrm{V,f}}}{l_{\mathrm{V}}}\cdot
n_{\mathrm{V,max}}+C_{\mathrm{V,o}}\cdot s_{\mathrm{V,tot}},$ (15)
and $C_{\mathrm{S}}$ is the public transit-related cost
$C_{\mathrm{S}}=\frac{C_{\mathrm{S,f}}}{l_{\mathrm{S}}}\cdot
n_{\mathrm{S,a}}+C_{\mathrm{S,o}}.$ (16)
Third, on the environmental side, the resources are the total CO2 emissions
$m_{\mathrm{CO_{2},tot}}=m_{\mathrm{CO_{2},V,tot}}+m_{\mathrm{CO_{2},S}}\cdot
n_{\mathrm{S}},$ (17)
where $m_{\mathrm{CO_{2},S}}$ represents the CO2 emissions of a single train.
Formally, the set of travel requests $\\{\rho_{m}\\}_{m\in\mathcal{M}}$ is the
CDPI functionality, whereas $t_{\mathrm{avg}}$, $C_{\mathrm{tot}}$, and
$m_{\mathrm{CO_{2},tot}}$ are its resources. Consistently, the functionality
space is $\mathcal{F}=\overline{\mathbb{R}}_{+}$ and the resources space is
$\mathcal{R}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.
Note that the resulting CDPI (Figure 2) is indeed monotone, since it consists
of the interconnection of monotone DPIs [2].
#### III-G5 Discussion
A few comments are in order. First, we lump the autonomy functionalities in
its achievable velocity. We leave to future research more elaborated AV
models, accounting for instance for accidents rates [25] and for safety
levels. Second, we assume the service frequency of the subway system to scale
linearly with the number of trains. We inherently rely on the assumption that
the existing infrastructure can homogeneously accommodate the acquired train
cars. To justify the assumption, we include an upper bound on the number of
potentially acquirable trains in our case study design in Section IV. Third,
we highlight that the I-AMoD framework is only one of the many feasible ways
to map total demand to travel time, costs, and emissions. Specifically,
practitioners can easily replace the corresponding DPI with more sophisticated
models (e.g., simulation-based frameworks like AMoDeus [26]), as long as the
monotonicity of the DPI is preserved. In our setting, we conjecture the
customers’ and vehicles’ routes to be centrally controlled by the municipality
in a socially-optimal fashion. Implicitly, we rely on the existence of
effective incentives aligning private and societal interests. The study of
such incentives represents an avenue for future research. Fourth, we assume a
homogenous fleet of AVs. Nevertheless, our model is readily extendable to
capture heterogeneous fleets. Finally, we consider a fixed travel demand, and
compute the antichain of resources providing it. Nonetheless, our
formalization can be readily extended to arbitrary demand models preserving
the monotonicity of the CDPI (accounting for instance for elastic effects). We
leave this topic to future research.
I-AMoDVehicleSubway$\preceq$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\preceq$$+$$\preceq$$+$$\preceq$$\times$$\preceq$$\preceq$$+$$+$$\preceq$$+$$\preceq$$v_{\mathrm{V,a}}$$n_{\mathrm{S,a}}$$s_{\mathrm{V,tot}}$$C_{\mathrm{V,o}}$$C_{\mathrm{V,f}}$co-
design
constraint$C_{\mathrm{S,o}}$$C_{\mathrm{S,f}}$$n_{\mathrm{V,max}}$$C_{\mathrm{tot}}$$\alpha_{\mathrm{tot}}$$m_{\mathrm{CO_{2},V,tot}}$$n_{\mathrm{S}}$$m_{\mathrm{CO_{2},S}}$$m_{\mathrm{CO_{2},tot}}$$t_{\mathrm{avg}}$$l_{\mathrm{V}}$$l_{\mathrm{S}}$total
costaverage travel timetotal emissionstotal request rate
Figure 2: Schematic representation of the CDPI. In solid green the provided
functionalities and in dashed red the required resources. The edges represent
co-design constraints: The resources required by a first design problem are
the lower bound for the functionalities provided by the second one.
## IV Results
In this section, we leverage the framework presented in Section III to perform
a real-world case study of Washington D.C., USA. Section IV-A details the case
study. We then present numerical results in Sections IV-B and IV-C.
### IV-A Case Study
We base our studies on a real-world case of the urban area of Washington D.C.,
USA. We import the road network and its features from OpenStreetMap [27]. The
public transit network and its schedules are extracted from the GTFS data
[28]. Demand data is obtained by merging the origin-destination pairs of the
morning peak of May 31st 2017 provided by taxi companies [29] and the
Washington Metropolitan Area Transit Authority (WMATA) [23]. Given the lack of
reliable demand data for the MetroBus system, we focus our studies on the
MetroRail system and its design, inherently assuming MetroBus commuters to be
unaffected by our design methodology. To conform with the large presence of
ride-hailing companies, we scale the taxi demand rate by factor of 5 [30].
Overall, the demand dataset includes 15,872 travel requests, corresponding to
a demand rate of 24.22 $\nicefrac{{requests}}{{s}}$. To account for congestion
effects, we compute the nominal road capacity as in [31] and assume an average
baseline road usage of 93%, in line with [32]. We summarize the main
parameters together with their bibliographic sources in Table I. In the
remainder of this section, we tailor and solve the co-design problem presented
in Section III through the PyMCDP solver [33], and investigate the influence
of different AVs costs on the design objectives and strategies.
| Parameter | Name | Value | Units | Source
---|---|---|---|---|---
| Road usage | | $u_{ij}$ | 93 | % | [32]
Vehicle | | | | C1 | C2.1 | C2.2 | |
Operational cost | $C_{\mathrm{V,o}}$ | 0.084 | 0.084 | 0.062 | $\nicefrac{{USD}}{{mile}}$ | [34, 35]
Cost | $C_{\mathrm{V}}$ | 32,000 | 32,000 | 26,000 | $\nicefrac{{USD}}{{car}}$ | [34]
Automation cost | 20 mph | $C_{\mathrm{V,a}}$ | 15,000 | 20,000 | 3,700 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39]
25 mph | 15,000 | 30,000 | 4,400 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39]
30 mph | 15,000 | 55,000 | 6,200 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39]
35 mph | 15,000 | 90,000 | 8,700 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39]
40 mph | 15,000 | 115,000 | 9,800 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39]
45 mph | 15,000 | 130,000 | 12,000 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39]
50 mph | 15,000 | 150,000 | 13,000 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39]
Vehicle life | $l_{\mathrm{V}}$ | 5 | 5 | 5 | years | [34]
CO2 per Joule | $\gamma$ | 0.14 | 0.14 | 0.14 | $\nicefrac{{g}}{{kJ}}$ | [40]
Time $\mathcal{G}_{\mathrm{W}}$ to $\mathcal{G}_{\mathrm{R}}$ | $t_{\mathrm{WR}}$ | 300 | 300 | 300 | s | -
Time $\mathcal{G}_{\mathrm{R}}$ to $\mathcal{G}_{\mathrm{W}}$ | $t_{\mathrm{RW}}$ | 60 | 60 | 60 | s | -
| Speed fraction | $\beta$ | $\frac{1}{1.3}$ | $\frac{1}{1.3}$ | $\frac{1}{1.3}$ | - | -
Public transit | Operational cost | 100 % | $C_{\mathrm{S,o}}$ | 148,000,000 | $\nicefrac{{USD}}{{year}}$ | [41]
133 % | 197,000,000 | $\nicefrac{{USD}}{{year}}$ | [41]
200 % | 295,000,000 | $\nicefrac{{USD}}{{year}}$ | [41]
Fixed cost | $C_{\mathrm{S,f}}$ | 14,500,000 | $\nicefrac{{USD}}{{train}}$ | [42]
Train life | $l_{\mathrm{S}}$ | 30 | years | [42]
Emissions/train | $m_{\mathrm{CO_{2},S}}$ | 140,000 | $\nicefrac{{kg}}{{year}}$ | [43]
Fleet baseline | $n_{\mathrm{S,base}}$ | 112 | trains | [42]
Service frequency | $\varphi_{j,\mathrm{base}}$ | $1/6$ | $\nicefrac{{1}}{{min}}$ | [44]
Time $\mathcal{G}_{\mathrm{W}}$ to $\mathcal{G}_{\mathrm{P}}$ | $t_{\mathrm{WS}}$ | $60$ | s | -
Time $\mathcal{G}_{\mathrm{P}}$ to $\mathcal{G}_{\mathrm{W}}$ | $t_{\mathrm{SW}}$ | $60$ | s | -
TABLE I: Parameters, variables, numbers, and units for the case studies.
### IV-B Case 1 - Constant Cost of Automation
In line with [35, 36, 37, 38, 39], we first assume an average achievable-
velocity-independent cost of automation. As discussed in Section III, we
design the system by means of subway service frequency, AV fleet size, and
achievable free-flow speed. Specifically, we allow the municipality to (i)
increase the subway service frequency $\varphi_{j}$ by a factor of 0%, 33%, or
100%, (ii) deploy an AVs fleet of size
$n_{\mathrm{V,max}}\in\\{0,500,1000,\ldots,6000\\}$ vehicles, and (iii) design
the single AV achievable velocity
$v_{\mathrm{V,a}}\in\\{20\,\mathrm{mph},25\,\mathrm{mph},\ldots,50\,\mathrm{mph}\\}$.
We assume the AVs fleet to be composed of battery electric BEV-250 mile AVs
[34]. In Figure 3(a), we show the solution of the co-design problem by
reporting the antichain consisting of the total transportation cost, average
travel time, and total CO2 emissions. These solutions are _rational_ (and not
comparable) in the sense that there exists no instance which simultaneously
yields lower cost, average travel time, and emissions.
(a) Left: Three-dimensional representation of antichain elements and their
projection in the cost-time space. Right: Two-dimensional projections.
(b) Results for constant automation costs. On the left, the two-dimensional
representation of the antichain elements: In red are the unfeasible
strategies, in orange the feasible but irrational solutions, and in green the
Pareto front. On the right, the implementations corresponding to the
highlighted antichain elements, quantified in terms of achievable vehicle
speed, AV fleet size, and train fleet size.
Figure 3: Solution of the CDPI: state-of-the art case.
For the sake of clarity, we opt for a two-dimensional antichain
representation, by translating and including the emissions in the total cost.
To do so, we consider the conversion factor 40 $\nicefrac{{USD}}{{kg}}$ [45].
Note that since this transformation preserves the monotonicity of the CDPI it
smoothly integrates in our framework. Doing so, we can conveniently depict the
co-design strategies through the two-dimensional antichain (Figure 3(b), left)
and the corresponding municipality actions (Figure 3(b), right). Generally, as
the municipality budget increases, the average travel time per trip required
to satisfy the given demand decreases, reaching a minimum of about 17.1 min
with an expense of around 43 $\nicefrac{{MilUSD}}{{month}}$. This
configuration corresponds to a fleet of 5,500 AVs able to drive at 50 mph and
to the doubling of the current MetroRail train fleet. On the other hand, the
smallest rational investment of 12.9 $\nicefrac{{MilUSD}}{{month}}$ leads to a
42 % higher average travel time, corresponding to a inexistent autonomous
fleet and an unchanged subway infrastructure. Notably, an expense of 23
$\nicefrac{{MilUSD}}{{month}}$ (48 % lower than the highest rational
investment) only increases the minimal required travel time by 9 %, requiring
a fleet of 4,000 vehicles able to drive at 35 mph and no acquisition of
trains. Conversely, an investment of 15.6 $\nicefrac{{MilUSD}}{{month}}$ (just
2 $\nicefrac{{MilUSD}}{{month}}$ more than the minimal rational investment)
provides a 3 min shorter travel time. Remarkably, the design of AVs able to
exceed 40 mph only improves the average travel time by 6 %, and it is rational
just starting from an expense of 22.8 $\nicefrac{{MilUSD}}{{month}}$. This
suggests that the design of faster vehicles mainly results in higher emission
rates and costs, without substantially contributing to a more time-efficient
demand satisfaction. Finally, it is rational to improve the subway system only
starting from a budget of 28.5 $\nicefrac{{MilUSD}}{{month}}$, leading to a
travel time improvement of just 4 %. This trend can be explained with the high
train acquisition and increased operation costs, related to the subway
reinforcement. We expect this phenomenon to be more marked for other cities,
considering the moderate operation costs of the MetroRail subway system due to
its automation [44] and related benefits [46].
### IV-C Case 2 - Speed-Dependent Automation Costs
To relax the potentially unrealistic assumption of a velocity-independent
automation cost, we consider a performance-dependent cost structure. The large
variance in sensing technologies and their reported performances [47] suggests
that this rationale is reasonable. Indeed, the technology required today to
safely operate an autonomous vehicle at 50 mph is substantially more
sophisticated, and therefore more expensive, than the one needed at 20 mph. To
this end, we adopt the cost structure reported in Table I. Furthermore, the
frenetic evolution of automation techniques intricates their monetary
quantification. Therefore, we perform our studies with current (2020) costs as
well as with their projections for the upcoming decade (2025) [48, 34].
#### IV-C1 Case 2.1 - 2020
We study the hypothetical case of an immediate AV fleet deployment. We
introduce the aforementioned velocity-dependent automation cost structure and
obtain the results reported in Figure 4(a). Comparing these results with the
state-of-the-art parameters presented in Figure 3 confirms the previously
observed trend concerning high vehicle speeds. Indeed, spending 24.9
$\nicefrac{{MilUSD}}{{month}}$ (55 % lower than the highest rational expense)
only increases the average travel time by 10 %, requiring a fleet of 3,000 AVs
at 40 mph and no subway interventions. Nevertheless, the comparison shows two
substantial differences. First, the budget required to reach the minimum
travel time of 17.1 min is 28 % higher compared to the previous case, and
consists of the same strategy for the municipality, i.e., doubling the train
fleet and having a fleet of 5,500 AVs at 50 mph. Second, the higher vehicle
costs result in an average AV fleet growth of 5 %, an average velocity
reduction of 9 %, and an average train fleet growth of 7 %. This trend
suggests that, compared to Case 1, rational design strategies foster larger
fleets and less performing AVs.
(a) Results for speed-dependent automation costs in 2020.
(b) Results for speed-dependent automation costs in 2025.
Figure 4: Results for the speed-dependent automation costs. On the left, the
two-dimensional representation of the antichain elements: In red are the
unfeasible strategies, in orange the feasible but irrational solutions, and in
green the Pareto front. On the right, the implementations corresponding to the
highlighted antichain elements.
#### IV-C2 Case 2.2 - 2025
Experts forecast a substantial decrease of automation costs (up to 90 %) in
the next decade, mainly due to mass-production of the AVs sensing technology
[48, 49]. In line with this prediction, we inspect the futuristic scenario by
solving the CDPI for the adapted automation costs, and report the results in
Figure 4(b). Two comments are in order. First, the maximal rational budget is
25 % lower than in the immediate adoption case. Second, the reduction in
autonomy costs clearly eases the acquisition of more performant AVs,
increasing the average vehicle speed by 10 %. As a direct consequence, the AV
and train fleets are reduced in size by 5 % and 10 %, respectively.
### IV-D Discussion
We conclude the analysis of our case study with two final comments. First, the
presented case studies illustrate the ability of our framework to extract the
set of rational design strategies for an AV-enabled mobility system. This way,
stakeholders such as AVs companies, transportation authorities, and policy
makers can get transparent and interpretable insights on the impact of future
interventions. Second, we perform a sensitivity analysis through the variation
of the autonomy cost structures. On the one hand, this reveals a clear
transition from small fleets of fast AVs (in the case of low autonomy costs)
to slow fleets of numerous AVs (in the case of high autonomy costs). On the
other hand, our studies highlight that investments in the public transit
infrastructure are rational only when large budgets are available. Indeed, the
onerous train acquisition and operation costs lead to a comparative advantage
of AV-based mobility.
## V Conclusion
In this paper, we leveraged the mathematical theory of co-design to propose a
design framework for AV-enabled mobility systems. Specifically, the nature of
our framework allows both for the modular and compositional interconnection of
the DPIs of different mobility options and for multiple objectives. Starting
from the multi-commodity flow model of an I-AMoD system, we optimize the
design of AVs and public transit both from a vehicle-centric and fleet-level
perspective. In particular, we studied the problem of deploying a fleet of AVs
providing on-demand mobility in cooperation with public transit, optimizing
the speed achievable by the vehicles, the fleet size, and the service
frequency of the subway lines. Our framework allows the stakeholders involved
in the mobility ecosystem, from vehicle developers all the way to mobility-as-
a-service companies and governmental authorities, to characterize rational
trajectories for technology and investment development. We showcased our
methodology on a real-world case study of Washington D.C., USA. Notably, our
problem formulation allows for a systematic analysis of incomparable
objectives, providing stakeholders with analytical insights for the socio-
technical design of AV-enabled mobility systems. This work opens the field for
the following future research streams:
_Modeling:_ First, we would like to extend the presented framework to capture
additional modes of transportation, such as micromobility, and heterogeneous
fleets with different self-driving infrastructures, propulsion systems, and
passenger capacity. Second, we would like to investigate variable demand
models. Finally, we would like to analyze the interactions between multiple
stakeholders, characterizing the equilibrium arising from their conflicting
interests.
_Algorithms:_ It is of interest to tailor co-design algorithmic frameworks to
the particular case of transportation DPIs, possibly leveraging their specific
structure.
_Application:_ Finally, we would like to devise a user-friendly web interface
which supports mobility stakeholders to reason about strategic interventions
in urban areas.
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|
2024-09-04T02:54:58.766292 | 2020-02-29T21:59:52 | 2003.04805 | {
"authors": "Razvan V. Marinescu",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26137",
"submitter": "Razvan Marinescu",
"url": "https://arxiv.org/abs/2003.04805"
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|
2024-09-04T02:54:58.774191 | 2020-03-10T15:54:53 | 2003.04819 | {
"authors": "Benedek Rozemberczki, Oliver Kiss, Rik Sarkar",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26138",
"submitter": "Benedek Rozemberczki",
"url": "https://arxiv.org/abs/2003.04819"
} | arxiv-papers |
Karate Club: An API Oriented Open-Source Python Framework for Unsupervised Learning on Graphs
Benedek Rozemberczki
The University of Edinburgh
United Kingdom
Oliver Kiss
Central European University
Rik Sarkar
The University of Edinburgh
United Kingdom
Graphs encode important structural properties of complex systems. Machine learning on graphs has therefore emerged as an important technique in research and applications. We present Karate Club – a Python framework combining more than 30 state-of-the-art graph mining algorithms. These unsupervised techniques make it easy to identify and represent common graph features. The primary goal of the package is to make community detection, node and whole graph embedding available to a wide audience of machine learning researchers and practitioners.
Karate Club is designed with an emphasis on a consistent application interface, scalability, ease of use, sensible out of the box model behaviour, standardized dataset ingestion, and output generation. This paper discusses the design principles behind the framework with practical examples. We show Karate Club's efficiency in learning performance on a wide range of real world clustering problems and classification tasks along with supporting evidence of its competitive speed.
§ ACKNOWLEDGEMENTS
Benedek Rozemberczki was supported by the Centre for Doctoral Training in Data Science, funded by EPSRC (grant EP/L016427/1).
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|
2024-09-04T02:54:58.783178 | 2020-03-10T16:16:15 | 2003.04827 | {
"authors": "David I. Spivak, David Jaz Myers",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26139",
"submitter": "David Spivak",
"url": "https://arxiv.org/abs/2003.04827"
} | arxiv-papers | 0pt0pt *34pc* **1 1in1in* 2 *1.5* subsubsection subsection
# Dirichlet Polynomials
David I. Spivak David Jaz Myers
# Dirichlet Polynomials form a Topos
David I. Spivak David Jaz Myers
###### Abstract
One can think of power series or polynomials in one variable, such as
$P(\mathcal{y})=2\mathcal{y}^{3}+\mathcal{y}+5$, as functors from the category
$\mathsf{Set}$ of sets to itself; these are known as polynomial functors.
Denote by $\mathsf{Poly}_{\mathsf{Set}}$ the category of polynomial functors
on $\mathsf{Set}$ and natural transformations between them. The constants
$0,1$ and operations $+,\times$ that occur in $P(\mathcal{y})$ are actually
the initial and terminal objects and the coproduct and product in
$\mathsf{Poly}_{\mathsf{Set}}$.
Just as the polynomial functors on $\mathsf{Set}$ are the copresheaves that
can be written as sums of representables, one can express any Dirichlet
series, e.g. $\sum_{n=0}^{\infty}n^{\mathcal{y}}$, as a coproduct of
representable presheaves. A Dirichlet polynomial is a finite Dirichlet series,
that is, a finite sum of representables $n^{\mathcal{y}}$. We discuss how both
polynomial functors and their Dirichlet analogues can be understood in terms
of bundles, and go on to prove that the category of Dirichlet polynomials is
an elementary topos.
## Chapter 0 Introduction
Polynomials $P(\mathcal{y})$ and finite Dirichlet series $D(\mathcal{y})$ in
one variable $\mathcal{y}$, with natural number coefficients
$a_{i}\in\mathbb{N}$, are respectively functions of the form
$\displaystyle P(\mathcal{y})$
$\displaystyle=a_{n}\mathcal{y}^{n}+\cdots+a_{2}\mathcal{y}^{2}+a_{1}\mathcal{y}^{1}+a_{0}\mathcal{y}^{0},$
(1) $\displaystyle D(\mathcal{y})$
$\displaystyle=a_{n}n^{\mathcal{y}}+\cdots+a_{2}2^{\mathcal{y}}+a_{1}1^{\mathcal{y}}+a_{0}0^{\mathcal{y}}.$
The first thing we should emphasize is that the algebraic expressions in (1)
can in fact be regarded as _objects in a category_ , in fact two categories:
$\mathsf{Poly}$ and $\mathsf{Dir}$. We will explain the morphisms later, but
for now we note that in $\mathsf{Poly}$,
$\mathcal{y}^{2}=\mathcal{y}\times\mathcal{y}$ is a product and
$2\mathcal{y}=\mathcal{y}+\mathcal{y}$ is a coproduct, and similarly for
$\mathsf{Dir}$. The operators—in both the polynomial and the Dirichlet
case—are not just algebraic, they are category-theoretic. Moreover, these
categories have a rich structure.
The category $\mathsf{Poly}$ is well studied (see [GK12]). In particular, the
following are equivalent:
###### Theorem 1.
[GK12] For a functor $P\colon\mathsf{Fin}\to\mathsf{Fin}$, the following are
equivalent:
1. 1.
$P$ is polynomial.
2. 2.
$P$ is a sum of representables.
3. 3.
$P$ preserves connected limits – or equivalently, wide pullbacks.
In Theorem 8 we prove an analogous result characterizing Dirichlet
polynomials:
###### Theorem 2.
For a functor $D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$, the
following are equivalent:
1. 1.
$D$ is a Dirichlet polynomial.
2. 2.
$D$ is a sum of representables.
3. 3.
$D$ sends connected colimits to limits – or equivalently, $D$ preserves wide
pushouts.
We will also show that $\mathsf{Dir}$ is equivalent to the arrow category of
finite sets,
$\mathsf{Dir}\simeq\mathsf{Fin}^{\to},$
and in particular that $\mathsf{Dir}$ is an elementary topos.
If one allows _arbitary_ sums of functors represented by finite sets, one gets
_analytic_ functors in the covariant case—first defined by Joyal in his
seminal paper on combinatorial species [Joy81]—and _Dirichlet_ functors in the
contravariant case—first defined by Baez and Dolan and appearing in Baez’s
_This Week’s Finds_ blog [BD]. Baez and Dolan also drop the traditional
negative sign in the exponent (that is, they use $n^{s}$ where $n^{-s}$
usually appears), but also find a nice way to bring it back by moving to
groupoids. Here, we drop the negative sign and work with finite sets to keep
things as simple as possible. Similar considerations hold with little extra
work for infinite Dirichlet series or power series, and even more generally,
by replacing $\mathsf{Fin}$ with $\mathsf{Set}$.
## Chapter 1 Polynomial and Dirichlet functors
Recall that a _co-representable functor_ $\mathsf{Fin}\to\mathsf{Fin}$ is one
of the form $\mathsf{Fin}(k,-)$ for a finite set
$k=\\{`1\text{'},`2\text{'},\ldots,`k\text{'}\\}.$
We denote this functor by $\mathcal{y}^{k}$ and say it is _represented by_
$k\in\mathsf{Fin}$. Similarly, a _(contra-) representable functor_
$\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ is contravariant functor of
the form $\mathsf{Fin}(-,k)$; we denote this functor by $k^{\mathcal{y}}$. The
functors $\mathcal{y}^{-}$ and $-^{\mathcal{y}}$ are the contravariant and
covariant Yoneda embeddings,
$\mathcal{y}^{k}\coloneqq\mathsf{Fin}(k,-)\qquad\text{and}\qquad
k^{\mathcal{y}}\coloneqq\mathsf{Fin}(-,k).$
For example $\mathcal{y}^{3}(2)\cong 8$ and $3^{\mathcal{y}}(2)\cong 9$.
Note that the functor $0^{\mathcal{y}}\not\cong 0$ is not the initial object
in $\mathsf{Dir}$; it is given by
$0^{\mathcal{y}}(s)=\begin{cases}1&\textnormal{ if }s=0\\\ 0&\textnormal{ if
}s\geq 1.\end{cases}$
The coefficient $a_{0}$ of $1=\mathcal{y}^{0}$ in a polynomial $P$ is called
its _constant_ term. We refer to the coefficient $D_{\text{zc}}\coloneqq
a_{0}$ of $0^{\mathcal{y}}$ in a Dirichlet series $D$ as its _zero-content_
term. Rather than having no content, the content of the functor
$D_{\text{zc}}{\cdot}0^{\mathcal{y}}$ becomes significant exactly when it is
applied to zero.
###### Example 1.
The reader can determine which Dirichlet polynomial
$D(\mathcal{y})\in\mathsf{Dir}$ as in Eq. 1 has the following terms
$\begin{array}[]{c|ccccccc}\mathcal{y}&\cdots&5&4&3&2&1&0\\\ \hline\cr
D(\mathcal{y})&\cdots&96&48&24&12&6&7\end{array}$
Hint: its zero-content term is $D_{\text{zc}}=4$.
The set $P(1)$ (resp. the set $D(0)$) has particular importance; it is the set
of pure-power terms $\mathcal{y}^{k}$ in $P$ (resp. the pure-exponential terms
$k^{\mathcal{y}}$ in $D$). For example if $P=\mathcal{y}^{2}+4\mathcal{y}+4$
and $D=2^{\mathcal{y}}+4+4{\cdot}0^{\mathcal{y}}$ then $P(1)=D(0)=9$.
###### Definition 2.
A _polynomial functor_ is a functor $P\colon\mathsf{Fin}\to\mathsf{Fin}$ that
can be expressed as a sum of co-representable functors. Similarly, we define a
_Dirichlet functor_ to be a functor
$D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ that can be expressed
as a sum of representable presheaves (contravariant functors):
$P=\sum_{i=1}^{P(1)}\mathcal{y}^{p_{i}}\qquad\text{and}\qquad
D=\sum_{i=1}^{D(0)}(d_{i})^{\mathcal{y}}.$ (1)
That is, $P(X)=\sum_{i=1}^{P(1)}\mathsf{Fin}(p_{i},X)$ and
$D(X)=\sum_{i=1}^{D(0)}\mathsf{Fin}(X,d_{i})$ as functors applied to
$X\in\mathsf{Fin}$.
See Theorem 1 above for well-known equivalent conditions in $\mathsf{Poly}$
and Theorem 8 below for a Dirichlet analogue.
## Chapter 2 The categories $\mathsf{Poly}$ and $\mathsf{Dir}$
For any small category $C$, let $\mathsf{Fin}^{C}$ denote the category whose
objects are the functors $C\to\mathsf{Fin}$ and whose morphisms are the
natural transformations between them.
###### Definition 1.
The _category of polynomial functors_ , denoted $\mathsf{Poly}$, is the
(skeleton of the) full subcategory of $\mathsf{Fin}^{\mathsf{Fin}}$ spanned by
sums $P$ of representable functors. The _category of Dirichlet functors_ ,
denoted $\mathsf{Dir}$, is the (skeleton of the) full subcategory of
$\mathsf{Fin}^{(\mathsf{Fin}^{\textnormal{op}})}$ spanned by the sums $D$ of
representable presheaves.
While we will not pursue it here, one can take $\mathsf{Poly}_{\mathsf{Set}}$
to be the full subcategory of functors $\mathsf{Set}\to\mathsf{Set}$ spanned
by small coproducts of representables, and similarly for
$\mathsf{Dir}_{\mathsf{Set}}$.
###### Lemma 2.
The set of polynomial maps $P\to Q$ and Dirichlet maps $D\to E$ are given by
the following formulas:
$\mathsf{Poly}(P,Q)\coloneqq\prod_{i\in
P(1)}Q(p_{i})\qquad\text{and}\qquad\mathsf{Dir}(D,E)\coloneqq\prod_{i\in
D(0)}E(d_{i}).$
###### Example 3.
Let $P=2\mathcal{y}^{2}$, $Q=\mathcal{y}+1$, and let $D=2\cdot
2^{\mathcal{y}}$ and $E=1+0^{\mathcal{y}}$. Then there are nine ($9$)
polynomial morphisms $P\to Q$, zero ($0$) polynomial morphisms $Q\to P$, one
($1$) Dirichlet morphism $D\to E$, and eight ($8$) Dirichlet morphisms $E\to
D$.
###### Remark 4.
Sums and products of polynomials in the usual algebraic sense agree exactly
with sums and products in the categorical sense: if $P$ and $Q$ are
polynomials, i.e. objects in $\mathsf{Poly}$, then their coproduct is the
usual algebraic sum $P+Q$ of polynomials, and similarly their product is the
usual algebraic product $PQ$ of polynomials. The same is true for
$\mathsf{Dir}$: sums and products of Dirichlet polynomials in the usual
algebraic sense agree exactly with sums and products in the categorical sense.
#### Formal structures.
We review some formal structures of the categories $\mathsf{Poly}$ and
$\mathsf{Dir}$; all are straightforward to prove. There is an adjoint
quadruple and adjoint 5-tuple as follows, labeled by where they send objects
$n\in\mathsf{Fin}$, $P\in\mathsf{Poly}$, $D\in\mathsf{Dir}$:
${\mathsf{Fin}}$${\mathsf{Poly}}$$\scriptstyle{n}$$\scriptstyle{n\mathcal{y}}$$\scriptstyle{P(0)}$$\scriptstyle{P(1)}$${\scriptstyle\top}$${\scriptstyle\top}$${\scriptstyle\top}$
${\mathsf{Fin}}$${\mathsf{Dir}}$$\scriptstyle{n\cdot
0^{\mathcal{y}}}$$\scriptstyle{n}$$\scriptstyle{n^{\mathcal{y}}}$$\scriptstyle{D(0)}$$\scriptstyle{D(1)}$${\scriptstyle\bot}$${\scriptstyle\bot}$${\scriptstyle\bot}$${\scriptstyle\bot}$
(1)
All five of the displayed functors out of $\mathsf{Fin}$ are fully faithful.
For each $k:\mathsf{Fin}$ the functors $P\mapsto P(k)$ and $D\mapsto D(k)$
have left adjoints, namely $n\mapsto n\mathcal{y}^{k}$ and $n\mapsto
n{\cdot}k^{\mathcal{y}}$ respectively. These are functorial in $k$ and in fact
extend to two-variable adjunctions
$\mathsf{Fin}\times\mathsf{Poly}\to\mathsf{Poly}$ and
$\mathsf{Fin}\times\mathsf{Dir}\to\mathsf{Dir}$. Indeed, for
$n\in\mathsf{Fin}$ and $P,Q\in\mathsf{Poly}$ (respectively
$D,E\in\mathsf{Dir}$), we have
$\displaystyle\mathsf{Poly}(nP,Q)\cong\mathsf{Poly}(P,Q^{n})\cong\mathsf{Fin}(n,\mathsf{Poly}(P,Q)),$
$\displaystyle\mathsf{Dir}(nD,E)\cong\mathsf{Dir}(D,E^{n})\cong\mathsf{Fin}(n,\mathsf{Dir}(D,E)),$
where $nP$ and $nD$ denote $n$-fold coproducts and $P^{n}$ and $D^{n}$ denote
$n$-fold products.
Consider the unique function $0\to 1$. The natural transformation induced by
it, denoted $\pi_{D}\colon D(1)\to D(0)$, is equivalent to two natural
transformations on $\mathsf{Dir}$ via the adjunctions in Eq. 1:
$n{\cdot}0^{\mathcal{y}}\to n,\qquad D(1)\xrightarrow{\pi_{D}}D(0),\qquad n\to
n^{\mathcal{y}}.$ (2)
The one labeled $\pi_{D}$ is also $D(0!)$, where $0!\colon 0\to 1$ is the
unique function of that type.
The composite of two polynomial functors $\mathsf{Fin}\to\mathsf{Fin}$ is
again polynomial, $(P\circ Q)(n)\coloneqq P(Q(n))$; this gives a nonsymmetric
monoidal structure on $\mathsf{Poly}$. The monoidal unit is $\mathcal{y}$.
Day convolution for the cartesian product monoidal structure provides
symmetric monoidal structure
$\otimes\colon\mathsf{Poly}\times\mathsf{Poly}\to\mathsf{Poly}$, for which the
monoidal unit is $\mathcal{y}$. This monoidal structure—like the Cartesian
monoidal structure—distributes over $+$ We can write an explicit formula for
$P\otimes Q$, with $P,Q$ as in Eq. 1:
$P\otimes Q=\sum_{i=1}^{P(1)}\sum_{j=1}^{Q(1)}\mathcal{y}^{p_{i}q_{j}}$ (3)
We call this the _Dirichlet product_ of polynomials, for reasons we will see
in Remark 1.
The Dirichlet monoidal structure is closed; that is, for any
$A,Q:\mathsf{Poly}$ we define
$[A,Q]\coloneqq\prod_{i:A(1)}Q\circ(a_{i}\mathcal{y}),$ (4)
for example $[n\mathcal{y},\mathcal{y}]\cong\mathcal{y}^{n}$ and
$[\mathcal{y}^{n},\mathcal{y}]\cong n\mathcal{y}$. For any polynomial $A$
there is an $(-\otimes A)\dashv[A,-]$ adjunction
$\mathsf{Poly}(P\otimes A,Q)\cong\mathsf{Poly}(P,[A,Q]).$ (5)
In particular we recover Lemma 2 using Eqs. 4 and 1. The cartesian monoidal
structure on $\mathsf{Poly}$ is also closed, $\mathsf{Poly}(P\times
A,Q)\cong\mathsf{Poly}(P,Q^{A})$, and the formula for $Q^{A}$ is similar to
Eq. 4:
$Q^{A}\coloneqq\prod_{i:A(1)}Q\circ(a_{i}+\mathcal{y}).$
If we define the _global sections_ functor
$\Gamma\colon\mathsf{Poly}\to\mathsf{Fin}^{\textnormal{op}}$ by $\Gamma
P\coloneqq\mathsf{Poly}(P,\mathcal{y})$, or explicitly
$\Gamma(P)=[P,\mathcal{y}](1)=\prod_{i}p_{i}$, we find that it is left adjoint
to the Yoneda embedding
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Each of the categories $\mathsf{Poly}$ and $\mathsf{Dir}$ has pullbacks, which
we denote using “fiber product notation” $A\times_{C}B$. We can use pullbacks
in combination with monad units $\eta_{P}\colon P\to P(1)$ and $\eta_{D}\colon
D\to D(0)$ arising from Eq. 1 to recover Eq. 1:
$P=\sum_{i=1}^{P(1)}P\times_{P(1)}`i\text{'}\qquad\text{and}\qquad
D=\sum_{i=1}^{D(0)}D\times_{D(0)}`i\text{'}.$
###### Remark 5.
By a result of Rosebrugh and Wood [RW94], the category of finite sets is
characterized amongst locally finite categories by the existence of the five
left adjoints to its Yoneda embedding $k\mapsto
y^{k}\colon\mathsf{Fin}\to\mathsf{Fin}^{\mathsf{Fin}^{\textnormal{op}}}$. The
adjoint sextuple displayed in (1) is just the observation that five of these
six functors restrict to the subcategory $\mathsf{Dir}$.
## Chapter 3 $\mathsf{Poly}$ and $\mathsf{Dir}$ in terms of bundles
There is a bijection between the respective object-sets of these two
categories
$\displaystyle\operatorname{Ob}(\mathsf{Poly})$
$\displaystyle\xrightarrow{\cong}\operatorname{Ob}(\mathsf{Dir})$
$\displaystyle\sum_{i=1}^{n}\mathcal{y}^{k_{i}}$
$\displaystyle\mapsto\sum_{i=1}^{n}(k_{i})^{\mathcal{y}}.$ (1)
We call this mapping the _Dirichlet transform_ and denote it using an over-
line $P\mapsto\overline{P}$. We will see in Theorem 6 that this bijection
extends to an equivalence
$\mathsf{Poly}_{\textnormal{cart}}\cong\mathsf{Dir}_{\textnormal{cart}}$
between the subcategories of cartesian maps.
###### Remark 1.
With the Dirichlet transform in hand, we see why $P\otimes Q$ can be called
the Dirichlet product, e.g. in Eq. 3. Namely, the Dirichlet transform is
strong monoidal with respect to $\otimes$ and the cartesian monoidal structure
$\times$ in $\mathsf{Dir}$:
$\overline{P\otimes Q}=\overline{P}\times\overline{Q}.$
###### Proposition 2.
There is a one-to-one correspondence between the set of polynomials in one
variable, the set of Dirichlet polynomials, and the set of (isomorphism
classes of) functions $\pi\colon s\to t$ between finite sets.
###### Proof.
We already established a bijection $P\mapsto\overline{P}$ between polynomials
and finite Dirichlet series in Eq. 1.
Given a finite Dirichlet series $D$, we have a function $\pi_{D}\colon D(1)\to
D(0)$ as in Eq. 2. And given a function $\pi\colon s\to t$, define
$D_{\pi}\coloneqq\sum_{i=1}^{t}(d_{i})^{\mathcal{y}}$, where
$d_{i}\coloneqq\pi^{-1}(i)$ for each $1\leq i\leq t$. (N.B. Rather than
constructing $D_{\pi}$ from $\pi$ by hand, one could instead use a certain
orthogonal factorization system on $\mathsf{Dir}$.)
It is easy to see that the roundtrip on Dirichlet series is identity, and that
the round-trip for functions is a natural isomorphism. ∎
We will upgrade Proposition 2 to an equivalence
$\mathsf{Poly}_{\textnormal{cart}}\simeq\mathsf{Dir}_{\textnormal{cart}}$
between certain subcategories of $\mathsf{Poly}$ and $\mathsf{Dir}$ in Theorem
6.
###### Example 3.
Under the identification from Proposition 2, both the polynomial
$2\mathcal{y}^{3}+\mathcal{y}^{2}+3$ and the Dirichlet series
$2{\cdot}3^{\mathcal{y}}+1{\cdot}2^{\mathcal{y}}+3{\cdot}0^{\mathcal{y}}$
correspond to the function
$\bullet$$1$$\bullet$$2$$\bullet$$3$$\bullet$$4$$\bullet$$5$$\bullet$$6$$6\cong
D(0)\cong$$\bullet$$(1,1)$$\bullet$$(1,2)$$\bullet$$(1,3)$$\bullet$$(2,1)$$\bullet$$(2,2)$$\bullet$$(2,3)$$\bullet$$(3,1)$$\bullet$$(3,2)$$8\cong
D(1)\cong$$\pi$ (2)
We can think of a function $\pi\colon s\to t$, e.g. that shown in (2), as a
_bundle_ of fibers $\pi^{-1}(`i\text{'})$, one for each element $`i\text{'}\in
t$. In Definition 4 we define two different notions of morphism between
bundles. We will see in Theorem 6 that they correspond to morphisms in the
categories $\mathsf{Poly}$ and $\mathsf{Dir}$.
For any function $\pi^{\prime}\colon s^{\prime}\to t^{\prime}$ and function
$f\colon t\to t^{\prime}$, denote by $f^{*}(\pi^{\prime})$ the pullback
function as shown
${s\times_{t^{\prime}}s^{\prime}}$${s^{\prime}}$${t}$${t^{\prime}}$$\scriptstyle{f^{*}(\pi^{\prime})}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f}$${\lrcorner}$
###### Definition 4.
Let $\pi\colon s\to t$ and $\pi^{\prime}\colon s^{\prime}\to t^{\prime}$ be
functions between finite sets.
* •
a _bundle morphism_ consists of a pair $(f,f_{\sharp})$ where $f\colon t\to
t^{\prime}$ is a function and $f_{\sharp}\colon\pi\to f^{*}(\pi^{\prime})$ is
a morphism in the slice category over $t$;
* •
a _container morphism_ consists of a pair $(f,f^{\sharp})$ where $f\colon t\to
t^{\prime}$ is a function and $f^{\sharp}\colon f^{*}(\pi^{\prime})\to\pi$ is
a morphism in the slice category over $t$.
We say a bundle morphism $(f,f_{\sharp})$ (resp. a container morphism
$(f,f^{\sharp})$) is _cartesian_ if $f_{\sharp}$ (resp. $f^{\sharp})$ is an
isomorphism.
${s}$${t\times_{t^{\prime}}s^{\prime}}$${s^{\prime}}$${t}$${t^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{f^{*}\pi^{\prime}}$$\scriptstyle{f_{\sharp}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f}$${\lrcorner}$
${s}$${t\times_{t^{\prime}}s^{\prime}}$${s^{\prime}}$${t}$${t^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{f^{*}\pi^{\prime}}$$\scriptstyle{f^{\sharp}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f}$${\lrcorner}$
Figure 1: The categories $\mathsf{Bun}$ and $\mathsf{Cont}$ have the same
objects, functions $\pi\colon s\to t$. Here a morphism
$(f,f_{\sharp})\colon\pi\to\pi^{\prime}$ in $\mathsf{Bun}$ and a morphism
$(f,f^{\sharp})\colon\pi\to\pi^{\prime}$ in $\mathsf{Cont}$ are shown.
Define $\mathsf{Bun}$ (resp. $\mathsf{Cont}$) to be the category for which an
object is a function between finite sets and a morphism is a bundle morphism
(resp. container morphism); see Fig. 1. Denote by
$\mathsf{Bun}_{\textnormal{cart}}$ (resp. $\mathsf{Cont}_{\textnormal{cart}}$)
the subcategory of cartesian bundle morphisms.
One may note that $\mathsf{Bun}$ is the Grothendieck construction of the self-
indexing
$\mathsf{Fin}_{/(-)}\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Cat}$,
while $\mathsf{Cont}$ is the Grothendieck construction of its point-wise
opposite
$(\mathsf{Fin}_{/(-)})^{\textnormal{op}}\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Cat}$.
The name _container_ comes from the work of Abbot, Altenkirch, and Ghani
abbott2003categoriesabbott2005containersabbot2003categoriesthesis (see Remark
2.18 in [GK12] for a discussion of the precise relationship between the notion
of container and the notion of polynomial and polynomial functor).
###### Remark 5.
By the universal property of pullbacks, $\mathsf{Bun}\simeq\mathsf{Fin}^{\to}$
is equivalent (in fact isomorphic) to the category of morphisms and commuting
squares in $\mathsf{Fin}$. Furthermore, $\mathsf{Bun}_{\textnormal{cart}}$ is
equivalent to the category of morphisms and pullback squares in
$\mathsf{Fin}$, and
$\mathsf{Bun}_{\textnormal{cart}}\simeq\mathsf{Cont}_{\textnormal{cart}}$ (as
in both cases a cartesian morphism $(f,f_{\sharp})$ or $(f,f^{\sharp})$ is
determined by $f$ alone).
Next we show that $\mathsf{Bun}\simeq\mathsf{Dir}$ is also equivalent to the
category of Dirichlet functors, from Definition 1. Recall that a natural
transformation is called _cartesian_ if its naturality squares are pullbacks.
###### Theorem 6.
We have equivalences of categories
$\mathsf{Poly}\simeq\mathsf{Cont}\qquad\text{and}\qquad\mathsf{Dir}\simeq\mathsf{Bun}.$
In particular, this gives an equivalence
$\mathsf{Poly}_{\textnormal{cart}}\simeq\mathsf{Dir}_{\textnormal{cart}}$
between the category of polynomial functors and cartesian natural
transformations and the category of Dirichlet functors and cartesian natural
transformations.
###### Proof.
The functors $P_{-}\colon\mathsf{Cont}\to\mathsf{Poly}$ and
$D_{-}\colon\mathsf{Bun}\to\mathsf{Dir}$ are defined on each object, i.e.
function $\pi\colon s\to t$, by the formula $\pi\mapsto P_{\pi}$ and
$\pi\mapsto D_{\pi}\coloneqq\overline{P_{\pi}}$ as in Proposition 2. For each
$1\leq i\leq t$, denote the fiber of $\pi$ over $i$ by
$k_{i}\coloneqq\pi^{-1}(i)$.
For any finite set $X$, consider the unique map $X!\colon X\to 1$. Applying
$P_{-}$ and $D_{-}$ to it, we obtain the corresponding representable:
$P_{X!}\cong\mathcal{y}^{X}$ and $D_{X!}\cong X^{\mathcal{y}}$. We next check
that there are natural isomorphisms
$\displaystyle\mathsf{Poly}(P_{X!},P_{\pi})\cong P_{\pi}(X)=\sum
X^{k_{i}}\cong\mathsf{Cont}(X!,\pi),$
$\displaystyle\mathsf{Dir}(D_{X!},D_{\pi})\cong
D_{\pi}(X)=\sum_{i=1}^{t}(k_{i})^{X}\cong\mathsf{Bun}(X!,\pi).$ (3)
In both lines, the first isomorphism is the Yoneda lemma and the second is a
computation using Definition 4 (see Fig. 1). Thus we define $P_{-}$ on
morphisms by sending $f\colon\pi\to\pi^{\prime}$ in $\mathsf{Cont}$ to the
“compose-with-$f$” natural transformation, i.e. having $X$-component
$\mathsf{Cont}(X!,f)\colon\mathsf{Cont}(X!,\pi)\to\mathsf{Cont}(X!,\pi^{\prime})$,
which is clearly natural in $X$. We define $D_{-}$ on morphisms similarly: for
$f$ in $\mathsf{Bun}$, use the natural transformation $\mathsf{Bun}(-!,f)$.
By definition, every object in $\mathsf{Poly}$ and $\mathsf{Dir}$ is a
coproduct of representables, so to prove that we have the desired
equivalences, one first checks that coproducts in $\mathsf{Cont}$ and
$\mathsf{Bun}$ are taken pointwise:
$(\pi\colon s\to t)+(\pi^{\prime}\colon s^{\prime}\to
t^{\prime})\cong(\pi+\pi^{\prime})\colon(s+s^{\prime})\to(t+t^{\prime}),$
and then that $P_{\pi+\pi^{\prime}}=P_{\pi}+P_{\pi^{\prime}}$ and
$D_{\pi+\pi^{\prime}}=D_{\pi}+D_{\pi^{\prime}}$; see Remark 4.
By Remark 5, we know that
$\mathsf{Bun}_{\textnormal{cart}}\simeq\mathsf{Cont}_{\textnormal{cart}}$, and
we have just established the equivalences $\mathsf{Poly}\simeq\mathsf{Cont}$
and $\mathsf{Dir}\simeq\mathsf{Bun}$. It thus remains to check that the latter
equivalences identify cartesian natural transformations in $\mathsf{Poly}$
with cartesian morphisms in $\mathsf{Cont}$, and similarly for $\mathsf{Dir}$
and $\mathsf{Bun}$. For polynomial functors, we may refer to [GK12, Section
2].
Turning to Dirichlet functors, we want to show that for any $f\colon D\to
D^{\prime}$ the square
${D(1)}$${D^{\prime}(1)}$${D(0)}$${D^{\prime}(0)}$$\scriptstyle{f_{1}}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f_{0}}$
(4)
is a pullback in $\mathsf{Set}$ iff for all functions $g\colon X\to
X^{\prime}$, the naturality square
${D(X^{\prime})}$${D^{\prime}(X^{\prime})}$${D(X)}$${D^{\prime}(X)}$$\scriptstyle{f_{X^{\prime}}}$$\scriptstyle{D(g)}$$\scriptstyle{D^{\prime}(g)}$$\scriptstyle{f_{X}}$
(5)
is a pullback in $\mathsf{Set}$; we will freely use the natural isomorphism
$D_{\pi}(X)\cong\mathsf{Bun}(X!,\pi)$ from Eq. 3. The square in Eq. 4 is a
special case of that in Eq. 5, namely for $g\coloneqq 0!$ the unique function
$0\to 1$; this establishes the only-if direction.
To complete the proof, suppose that Eq. 4 is a pullback, take an arbitrary
$g\colon X\to X^{\prime}$, and suppose given a commutative solid-arrow diagram
as shown:
${X}$${X^{\prime}}$${D(1)}$${D^{\prime}(1)}$${1}$${1}$${D(0)}$${D^{\prime}(0)}$$\scriptstyle{g}$
We can interpret the statement that Eq. 5 is a pullback as saying that there
are unique dotted arrows making the diagram commute, since
$DX\cong\mathsf{Bun}(X!,D0!)$ and similarly for the other corners of the
square in Eq. 5. So, we need to show that if the front face is a pullback,
then there are unique diagonal dotted arrows as shown, making the diagram
commute. This follows quickly from the universal property of the pullback. ∎
###### Corollary 7.
$\mathsf{Dir}$ is an elementary topos.
###### Proof.
For any finite category $C$, the functor category $\mathsf{Fin}^{C}$ is an
elementary topos. The result now follows from Remarks 5 and 6, noting that
$\mathsf{Dir}\simeq\mathsf{Fin}^{\to}$. ∎
As we mentioned in the introduction, this all goes through smoothly when one
drops all finiteness conditions. The general topos of Dirichlet functors is
the category of (arbitrary) sums of representables
$\mathsf{Set}^{\textnormal{op}}\to\mathsf{Set}$, and this is equivalent to the
arrow category $\mathsf{Set}^{\to}$ and so is itself a topos.
We conclude with the equivalence promised in Dirichlet Polynomials.
###### Theorem 8.
A functor $D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ is a
Dirichlet polynomial if and only if it preserves connected limits, or
equivalently wide pullbacks.
###### Proof.
Let $D(\mathcal{y})=\sum_{i:D(0)}(d_{i})^{\mathcal{y}}$, and suppose that $J$
is any connected category. Then for any diagram $X\colon J\to\mathsf{Fin}$, we
have
$\displaystyle D(\operatorname*{colim}X_{j})$
$\displaystyle=\sum_{i:D(0)}(d_{i})^{\operatorname*{colim}X_{j}}$
$\displaystyle\cong\sum_{i:D(0)}\lim(d_{i})^{X_{j}}$
$\displaystyle\cong\lim\sum_{i:D(0)}(d_{i})^{X_{j}}$ $\displaystyle=\lim
D(X_{j})$
since connected limits commute with sums in any topos (in particular
$\mathsf{Set}$).
Now suppose $D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ is any
functor that preserves connected limits; in particular, it sends wide pushouts
to wide pullbacks. Every finite set $X$ can be expressed as the wide pushout
${X}$${1}$${1}$${\cdots}$${1}$${1}$${0}$
of its elements. Therefore, we have the following limit diagram:
${D(X)}$${D(1)}$${D(1)}$${\cdots}$${D(1)}$${D(1)}$${D(0)}$
That is, an element of $D(X)$ is a family of elements $a_{x}\in D(1)$, one for
each $x\in X$, such that the $D(0!)(a_{x})$ are all equal in $D(0)$. But this
is just a bundle map, i.e.
$D(X)\cong\mathsf{Bun}(X!,D(0!))$
where $X!\colon X\to 1$ and $D(0!)\colon D(1)\to D(0)$. Thus by Theorem 6, the
functor $D$ is the Dirichlet polynomial associated to the bundle $D(0!)$. ∎
### Acknowledgments
The authors thank Joachim Kock, André Joyal, and Brendan Fong for helpful
comments that improved the quality of this note. Spivak also appreciates
support by Honeywell Inc. as well as AFOSR grants FA9550-17-1-0058 and
FA9550-19-1-0113. Jaz Myers appreciates support by his advisor Emily Riehl and
the National Science Foundation grant DMS-1652600.
## References
* [AAG03] Michael Gordon Abbott, Thorsten Altenkirch and Neil Ghani “Categories of Containers” In _FoSSaCS_ , 2003
* [AAG05] Michael Abbott, Thorsten Altenkirch and Neil Ghani “Containers: Constructing strictly positive types” Applied Semantics: Selected Topics In _Theoretical Computer Science_ 342.1, 2005, pp. 3–27
* [Abb03] Michael Gordon Abbott “Categories of Containers”, 2003
* [BD] John Baez and James Dolan “This Week’s Finds 300” Accessed: 2020-02-16 URL: http://math.ucr.edu/home/baez/week300.html
* [GK12] Nicola Gambino and Joachim Kock “Polynomial functors and polynomial monads” In _Mathematical Proceedings of the Cambridge Philosophical Society_ 154.1 Cambridge University Press (CUP), 2012, pp. 153–192
* [Joy81] André Joyal “Une théorie combinatoire des séries formelles” In _Advances in Mathematics_ 42.1, 1981, pp. 1–82
* [RW94] Robert Rosebrugh and R.. Wood “An Adjoint Characterization of the Category of Sets” In _Proc. Amer. Math. Soc_ 122, 1994, pp. 409–413
|
2024-09-04T02:54:58.798419 | 2020-03-10T17:09:00 | 2003.04857 | {
"authors": "Yuqian Zhou, David Ren, Neil Emerton, Sehoon Lim, Timothy Large",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26140",
"submitter": "Yuqian Zhou",
"url": "https://arxiv.org/abs/2003.04857"
} | arxiv-papers | # Image Restoration for Under-Display Camera
Yuqian Zhou1, David Ren2, Neil Emerton3, Sehoon Lim3, Timothy Large3
1IFP, UIUC, 2CIL, UC Berkeley, 3Microsoft
###### Abstract
The new trend of full-screen devices encourages us to position a camera behind
a screen. Removing the bezel and centralizing the camera under the screen
brings larger display-to-body ratio and enhances eye contact in video chat,
but also causes image degradation. In this paper, we focus on a newly-defined
Under-Display Camera (UDC), as a novel real-world single image restoration
problem. First, we take a 4k Transparent OLED (T-OLED) and a phone Pentile
OLED (P-OLED) and analyze their optical systems to understand the degradation.
Second, we design a Monitor-Camera Imaging System (MCIS) for easier real pair
data acquisition, and a model-based data synthesizing pipeline to generate
Point Spread Function (PSF) and UDC data only from display pattern and camera
measurements. Finally, we resolve the complicated degradation using
deconvolution-based pipeline and learning-based methods. Our model
demonstrates a real-time high-quality restoration. The presented methods and
results reveal the promising research values and directions of UDC.
## 1 Introduction
Under-display Camera (UDC) is a new imaging system that mounts display screen
on top of a traditional digital camera lens, as shown in Figure 1. Such a
system has mainly two advantages. First, it brings a new product trend of
full-screen devices [11] with larger screen-to-body ratio, which can provide
better user perceptive and intelligent experience [12]. Without seeing the
bezel and extra buttons, users can easily access more functions by directly
touching the screen. Second, it provides better human computer interaction. By
putting the camera in the center of the display, it enhances teleconferencing
experiences with perfect gaze tracking, and it is increasingly relevant for
larger display devices such as laptops and TVs.
Unlike pressure or fingerprint sensors that can be easily integrated into a
display, it is relatively difficult to retain full functionality of an imaging
sensor after mounting it behind a display. The imaging quality of a camera
will be severely degraded due to lower light transmission rate and diffraction
effects. As a result, images captured will be noisy and blurry. Therefore,
while bringing better user experience and interaction, UDC may sacrifice the
quality of photography, face processing [35] and other downstream vision
tasks. Restoring and enhancing the images captured by UDC system will be
desired.
Figure 1: The newly proposed imaging system named Under-Display Camera (UDC).
We mount display screen on top of a traditional digital camera lens. The
design brings new trend of full-screen devices.
Traditional image restoration approaches form the task as an inverse problem
or an optimization problem like Maximum-a-Posterior (MAP). For the UDC
problem, for practical purposes, the proposed image restoration algorithm and
system are expected to work in real-time. Therefore, deconvolutional-based
methods like Wiener Filter [14] should be preferred. Deconvolution is the
inverse process of convolution and recovers the original signal from the
point-spread-function (PSF)-convolved image. The fidelity of the deconvolution
process is dependent on the space-invariance of the PSF over the image field
of-view (FOV) and on a low condition number for the inverse of the PSF [19].
For strongly non-delta-function-like PSFs such as those encountered when
imaging through a display, the value of condition number can be large. For
such PSFs an additional denoising step may be essential.
Another option is the emerging discriminative learning-based image restoration
model. Data-driven discriminative learning-based image restoration models
usually outperform traditional methods in specific tasks like image de-noising
[42, 48, 47, 26, 43, 3], de-bluring [21, 28],de-raining [40, 39], de-hazing
[13, 33], super-resolution [22, 37], and light-enhancement [9]. However,
working on synthesis data with single degradation type, existing models can be
hardly utilized to enhance real-world low-quality images with complicated or
combined degradation types. To address complicated real degradation like the
UDC problem, directly collecting real paired data or synthesizing near-
realistic data after fully understanding the degradation model is necessary.
In this paper, we present the first study to define and analyze the novel
Under-Display Camera (UDC) system from both optics and image restoration
viewpoints. For optics, we parse the optical system of the UDC pipeline and
analyze the characteristics of light transmission. Then we relate the obtained
intuitions and measurements to an image restoration pipeline, and propose two
ways of resolving the single-image restoration: A deconvolution-based Wiener
Filter [29] pipeline (DeP) and a data-driven learning-based approach.
Specifically, we regard UDC restoration as a combination of tasks such as low-
light enhancement, de-blurring, and de-noising.
Without loss of generality, our analysis focuses on two types of displays, a
4K Transparent Organic Light-Emitting Diode (T-OLED) and a phone Pentile OLED
(P-OLED), and a single camera type, a 2K FLIR RGB Point Grey research camera.
To obtain the real imaging data and measure the optical factors of the system,
we also propose a data acquisition system using the above optical elements.
In summary, the main contributions of our paper are: (1) A brand new imaging
system named Under-Display Camera (UDC) is defined, measured and analyzed.
Extensive experiments reveal the image degradation process of the system,
inspiring better approaches for restoring the captured images. (2) As a
baseline, two practical and potential solutions are proposed, including
conventional Wiener Filter and the recent learning-based method. (3) Adopting
the newly-assembled image acquisition system, we collect the first Under-
Display Camera (UDC) dataset which will be released and evaluated by the
public.
## 2 Related Work
#### Real-world Image Reconstruction and Restoration
Image restoration for UDC [46, 24, 23, 49] can be categorized into the problem
of Real-world restoration[3, 45]. It is becoming a new concept in low-level
vision. In the past decades, low-level vision works on synthetic data
(denoising on AWGN and SR on Bicubic), but the models are not efficient for
images with real degradation such as real noises or blur kernels. Making
models perform better on real-world inputs usually requires new problem
analysis and a more challenging data collection. Recently, researchers also
worked on challenging cases like lensless imaging problems [30, 27, 20], or
integrating optics theory with High Dynamic Range imaging [34]. Previously,
there has been two common ways to prepare adaptive training data for real-
world problems: real data collection and near-realistic data synthesis.
Recently, more real noise datasets such as DND [31], SIDD [2, 28], and RENOIR
[5], have been introduced to address practical denoising problems. Abdelrahman
et al. [3] proposed to estimate ground truth from captured smartphone noise
images, and utilized the paired data to train and evaluate the real denoising
algorithms. In addition to noise, Chen et al. first introduced the SID dataset
[9] to resolve extreme low-light imaging. In the area of Single Image Super
Resolution (SISR), researchers considered collecting optical zoom data [45,
10] to learn better computational zoom. Other restoration problems including
reflection removal [36, 32] also follow the trend of real data acquisition.
Collecting real data suffers from limitation of scene variety since most
previous models acquire images of postcards, static objects or color boards.
In this paper, we propose a novel monitor-camera imaging system, to add real
degradation to the existing natural image datasets like DIV2K [4].
A realistic dataset can be synthesized if the degradation model is fully
understood and resolved. One good practice of data synthesis is generating
real noises on raw sensors or RGB images. CBDNet [17] and Tim et al. [8]
synthesized realistic noise by unfolding the in-camera pipeline, and
Abdelhamed et al. [1] better fitted the real noise distribution with flow-
based generative models. Zhou et al. [48] adapted the AWGN-RVIN noise into
real RGB noise by analyzing the demosacing process. Other physics-based
synthesis was also explored in blur[7] or hazing[6]. For the UDC problem in
this paper, we either collected real paired data, or synthesized near-
realistic data from model simulation. In particular, we applied the theory of
Fourier optics to simulate the diffraction effects, and further adjusted the
data with other camera measurements. Our data synthesizing pipeline
demonstrates a promising performance for addressing real complicated
degradation problem.
(a)
(b)
Figure 2: Image formation pipeline of under-display camera (UDC) problem. (a)
Image Formation Pipeline. (b)Optics characteristics of UDC. The structure of
the 4K T-OLED has a grating-like pixel layout. P-OLED differs from T-OLED in
sub-pixel design. From left to right: Micrography of display patterns, PSFs
(red light only) and MTFs (red, green, and blue).
## 3 Formulation
In this section, we discuss the optical system and image formation process of
the proposed UDC imaging system. We analyze the degradation type, light
transmission rate and visualize the Point Spread Function (PSF). Moreover, we
formulate the image formation pipeline to compute simulated PSF from
measurements.
### 3.1 Optical System Analysis
Optical Elements. In our experiments, we focus on the Organic Light-Emitting
Diode (OLED) displays [38] as they have superior optical properties compared
to the traditional LCDs (Liquid Crystal Display). Due to confidentiality
reasons it is often difficult to obtain the sample materials used for demos
from commercial companies. In this case, we select the displays with different
transparencies to improve the generalization. Note that all the displays are
non-active in our experiments, since in real scenario, the display can be
turned off locally by setting black pixels on local regions of the OLED
display when the camera is in operation to 1) reduce unnecessary difficulty
from display contents while not affecting user experience and 2) provide users
with the status of the device and thus ensure privacy.
Owing to transparent materials being used in OLED display panels, visible
lights can be better transmitted through the OLEDs than LCDs. In the meantime,
pixels are also arranged such that open area is maximized. In particular, we
focus on 4k Transparent OLED (T-OLED) and a phone Pentile OLED (P-OLED).
Figure 2 is a micrograph illustration of the pixel layout in the two types of
OLED displays. The structure of the 4K T-OLED has a grating-like pixel layout.
P-OLED differs from T-OLED in sub-pixel design. It follows the basic structure
of RGBG matrix.
Table 1: Comparison of two displays in terms of light transmission rate and
physical pixel layout and open areas.
Metrics | T-OLED | P-OLED
---|---|---
Pixel Layout Type | Stripe | Pentile
Open Area | 21$\%$ | 23$\%$
Transmission Rate | 20$\%$ | 2.9$\%$
Major Degradation | Blur, Noise | Low-light, Color Shift, Noise
Light Transmission Rate. We measure the transmission efficiency of the OLEDs
by using a spectrophotometer and white light source. Table 1 compares the
light transmission rate of the two displays. For T-OLED, the open area
occupies about 21$\%$, and the light transmission rate is around 20$\%$. For
P-OLED, although the open area can be as large as 23$\%$, the light
transmission rate is only 2.9$\%$.
The loss of photons can be attributed mainly to the structure of P-OLED.
First, the P-OLED has a finer pixel pitch, so photos are scattered to higher
angles comparing to the T-OLED. As a result, high angle photons are not
collected by the lens. Second, P-OLED is a flexible/bendable display, which
has a poly-amide substrate on which the OLED is formed. Such a substrate has
relatively low transmission efficiency, causing photons to be absorbed. The
absorbed light with certain wavelengths may make the images captured through a
polyamide-containing display panel by a UDC appear yellow. As a result,
imaging through a P-OLED results in lower signal-to-noise ratio (SNR)
comparing to using a T-OLED, and has a color shift issue. One real imaging
example is shown in Figure 4.
Diffraction Pattern and Point Spread Function (PSF). Light diffracts as it
propagates through obstacles with sizes that are similar to its wavelength.
Unfortunately, the size of the openings in the pixel layout is on the order of
wavelength of visible light, and images formed will be degraded due to
diffraction.
Here we characterize our system by measuring the point spread function (PSF).
We do so by pointing a collimated red laser beam ($\lambda=$ 650nm) at the
display panel and recording the image formed on the sensor, as demonstrated in
Figure 1 and 2. An ideal PSF shall resemble a delta function, which then forms
a perfect image of the scene. However, light greatly spreads out in UDC. For
T-OLED, light spreads mostly across the horizontal direction due to its nearly
one dimensional structure in the pixel layout, while for P-OLED, light is more
equally distributed as the pixel layout is complex. Therefore, images captured
by UDC are either blurry (T-OLED) or hazy (P-OLED).
Modulation Transfer Function (MTF) Modulation Transfer Function (MTF) is
another important metric for an imaging system, as it considers the effect of
finite lens aperture, lens performance, finite pixel size, noise, non-
linearities, quantization (spatial and bit depth), and diffraction in our
systems. We characterize the MTF of our systems by recording sinusoidal
patterns with increasing frequency in both lateral dimensions, and we report
them in Figure 2. For T-OLED, contrasts along the horizontal direction are
mostly lost in the mid-band frequency due to diffraction. This phenomenon is
due to the nearly one-dimensional pixel layout of the T-OLED. Figure 4 shows
severe smearing horizontally when putting T-OLED in front of the camera. While
for P-OLED, the MTF is almost identical to that of display-free camera, except
with severe contrast loss. Fortunately, however, nulls have not been observed
in any particular frequencies.
### 3.2 Image Formation Pipeline
In this section, we derive the image formation process of UDC based on the
analysis in the previous sections. Given a calibrated pixel layout and
measurements using a specific camera, degraded images can be simulated from a
scene. From the forward model, we can compute the ideal PSF and consequently
synthesize datasets from ground truth images.
Given an object in the scene $\mathbf{x}$, the degraded observation
$\mathbf{y}$ can be modeled by a convolution process,
$\mathbf{y}=(\gamma\mathbf{x})\otimes\mathbf{k}+\mathbf{n},$ (1)
where $\gamma$ is the intensity scaling factor under the current gain setting
and display type, $\mathbf{k}$ is the PSF, and $\mathbf{n}$ is the zero-mean
signal-dependent noise. Notice that this is a simple noise model that
approximately resembles the combination of shot noise and readout noise of the
camera sensor, and it will be discussed in a later section.
Intensity Scaling Factor ($\gamma$) The intensity scaling factor measures the
changing ratio of the average pixel values after covering the camera with a
display. It simultaneously relates to the physical light transmission rate of
the display, as well as the digital gain $\delta$ setting of the camera.
$\gamma$ can be computed from the ratio of $\delta$-gain amplified average
intensity values $I_{d}(\delta,s)$ at position $s$ captured by UDC, to the
0-gain average intensity values $I_{nd}(0,s)$ by naked camera within an
enclosed region $S$. It is represented by,
$\gamma=\frac{\int_{S}I_{d}(\delta,s)ds}{\int_{S}I_{nd}(0,s)ds}$ (2)
Diffraction Model We approximate the blur kernel $\mathbf{k}$, which is the
Point Spread Function (PSF) of the UDC. As shown in Figure 1, in our model, we
assume the display panel is at the principle plane of the lens. We also assume
the input light is monochromatic plane wave with wavelength $\lambda$ (i.e.
perfectly coherent), or equivalently light from a distance object with unit
amplitude. Let the display pattern represented by transparency with complex
amplitude transmittance be $g(m,n)$ at the Cartesian co-ordinate $(m,n)$, and
let the camera aperture/pupil function $p(m,n)$ be 1 if $(m,n)$ lies inside
the lens aperture region and 0 otherwise, then the display pattern inside the
aperture range $g_{p}(m,n)$ becomes,
$g_{p}(m,n)=g(m,n)p(m,n).$ (3)
At the focal plane of the lens (i.e. 1 focal length away from the principle
plane), the image measured is the intensity distribution of the complex field,
which is proportional to the Fourier transform of the electric field at the
principle plane [16]:
$I(u,v)\propto\left|{\iint}^{\infty}_{-\infty}g_{p}(m,n)\exp\left[-j\frac{2\pi}{\lambda
f}(mu+nv)\right]\text{d}m\text{d}n\right|^{2}.$ (4)
Suppose $G_{p}(v_{m},v_{n})=\mathscr{F}(g_{p}(m,n))$, where
$\mathscr{F}(\cdot)$ is the Fourier transform operator, then
$I(u,v)\propto\left|G_{p}(v_{m},v_{n})\right|^{2}=\left|G_{p}(\frac{u}{\lambda
f},\frac{v}{\lambda f})\right|^{2},$ (5)
which performs proper scaling on the Fourier transform of the display pattern
on the focal plane.
Therefore, to compute the PSF $\mathbf{k}$ for image $\mathbf{x}$, we start
from computing Discrete Fourier Transform (DFT) with squared magnitude
$M(a,b)=|\hat{G_{p}}(a,b)|^{2}$ of the $N\times N$ microscope transmission
images $\hat{g_{p}}$ of the display pattern and re-scaling it. Then, the
spatial down-sampling factor $r$ (denoted by $\downarrow r$) becomes,
$r=\frac{1}{\lambda f}\cdot{\delta_{N}N}\cdot{\rho},$ (6)
where $\delta_{N}$ is the pixel size of the $\hat{g_{p}}$ images, and $\rho$
is the pixel size of the sensor. Finally, $\mathbf{k}$ can be represented as
$k(i,j)=\frac{M_{\downarrow r}(i,j)}{\sum_{(\hat{i},\hat{j})}M_{\downarrow
r}(\hat{i},\hat{j})}.$ (7)
$k$ is a normalized form since we want to guarantee that it represents the
density distribution of the intensity with diffraction effect. Note that only
PSF for a single wavelength is computed for simplicity. However, scenes in the
real-world are by no means monochromatic. Therefore, in order to calculate an
accurate color image from such UDC systems, PSF for multiple wavelengths shall
be computed. More details will be shown in Section 4.2.
Adding Noises We follow the commonly used shot-read noise model [8, 18, 25] to
represent the real noise on the imaging sensor. Given the dark and blur signal
$w=(\gamma\mathbf{x})\otimes\mathbf{k}$, the shot and readout noise can be
modeled by a heteroscedastic Gaussian,
$\mathbf{n}\sim\mathcal{N}(\mu=0,\sigma^{2}=\lambda_{read}+\lambda_{shot}w),$
(8)
where the variance $\sigma$ is signal-dependent, and $\lambda_{read}$ ,
$\lambda_{shot}$ are determined by camera sensor and gain values.
## 4 Data Acquisition and Synthesis
We propose an image acquisition system called Monitor-Camera Imaging System
(MCIS). In particular, we display natural images with rich textures on high-
resolution monitor and capture them with a static camera. The method is more
controllable, efficient, and automatic to capture a variety of scene contents
than using mobile set-ups to capture limited static objects or real scenes.
### 4.1 Monitor-Camera Imaging System
Figure 3: Monitor-Camera Imaging System (MCIS). MCIS consists of a 4K LCD
monitor, the 2K FLIR RGB Point-Grey research camera, and a panel that is
either T-OLED, P-OLED or Glass(i.e. no display). The camera is mounted on the
center line of the 4K monitor, and adjusted to cover the full monitor range.
(a) Display-free
(b) TOLED
(c) POLED
Figure 4: Real samples collected by the proposed MCIS. Images captured by
T-OLED are blur and noisy, while those captured by P-OLED are low-light,
color-shifted and hazy.
The system architecture is shown in Figure 3. MCIS consists of a 4K LCD
monitor, the 2K FLIR RGB Point-Grey research camera, and a panel that is
either T-OLED, P-OLED or Glass(i.e. no display). The camera is mounted on the
center line of the 4K monitor, and adjusted to cover the full monitor range.
We calibrate the camera gain by measuring a $256\times 256$ white square shown
on the monitor and matching the RGB histogram. For fair comparison and
simplicity, we adjust the focus and fix the aperture to f/1.8. It guarantees a
reasonable pixel intensity range avoiding saturation while collecting data
with no gain. Suppose we develop a real-time video system, the frame rate has
to be higher than 8 fps. So the lowest shutter speed is 125 ms for the better
image quality and the higher Signal-to-Noise Ratio (SNR).
Table 2: Camera Settings for different set of collected data
Parameteres | No-Display | T-OLED | P-OLED
---|---|---|---
Aperture | f/1.8
FPS/Shutter | 8/125ms
Brightness | 0
Gamma | 1
Gain | 1 | 6 | 25(Full)
White-balance | Yes | None | None
We split 300 images from DIV2K dataset [4], and take turns displaying them on
a 4K LCD in full screen mode. We either rotate or resize the images to
maintain the Aspect Ratio. For training purposes, we capture two sets of
images, which are the degraded images $\\{y_{i}\\}$, and the degradation-free
set $\\{x_{i}\\}$.
To capture $\\{x_{i}\\}$, we first cover the camera with a thin glass panel
which has the same thickness as a display panel. This process allows us to
avoid the pixel misalignment issues caused by light refraction inside the
panel. To eliminate the image noises in $\\{x_{i}\\}$, we average the 16
repeated captured frames. Then we replace the glass with a display panel
(T-OLED or P-OLED), calibrate the specific gain value avoiding saturation, and
capture $\\{y_{i}\\}$. For each set, we record both the 16-bit 1-channel
linear RAW CMOS sensor data as well as the 8-bit 3-channel linear RGB data
after in-camera pipeline that includes demosaicing. The collected pairs are
naturally well spatially-aligned in pixel-level. They can be directly used for
deep model training without further transformations.
Due to the yellow substrate inside the P-OLED, certain light colors,
especially blue, are filtered out and changes the white balance significantly.
We therefore did not further alter the white balance. The light transmission
ratio of P-OLED is extremely low, so we set up the gain value to be the
maximum (25) for higher signal values. All the detailed camera settings for
the two display types are shown in Table 2. One real data sample is shown in
Figure 4. As discussed and analyzed in Section 3.1, images captured by T-OLED
are blur and noisy, while those captured by P-OLED are low-light, color-
shifted and hazy.
Table 3: Measured parameters for data synthesis
Parameteres | T-OLED | P-OLED
---|---|---
| R | G | B | R | G | B
$\gamma$ | 0.97 | 0.97 | 0.97 | 0.34 | 0.34 | 0.20
$\lambda$ (nm) | 640 | 520 | 450 | 640 | 520 | 450
r | 2.41 | 2.98 | 3.44 | 2.41 | 2.98 | 3.44
### 4.2 Realistic Data Synthesis Pipeline
We follow the image formation pipeline to simulate the degradation on the
collected $\\{x_{i}\\}$. A model-based data synthesis method will benefit
concept understanding and further generalization. Note that all the camera
settings are the same as the one while collecting real data. We first
transform the 16-bit raw sensor data $\\{x_{i}\\}$ into four bayer channels
$x_{r}$, $x_{gr}$, $x_{gl}$, and $x_{b}$. Then, we multiply the measured
intensity scaling factor $\gamma$, compute the normalized and scaled PSF $k$,
and add noises to the synthesize degraded data.
Measuring $\gamma$: To measure $\gamma$ for each channel using the MCIS, we
select the region of interest $S$ to be a square region of size $256\times
256$, and display the intensity value input from 0 to 255 with stride 10 on
the monitor. We then record the average intensity both with and without the
display for each discrete intensity value, and plot the relationship between
display-covered values and no-display-covered ones. Using linear regression,
we obtain the ratios of lines for different RGGB channel. For T-OLED, the
measured $\gamma$ is 0.97, same for all the channels. For P-OLED,
$\gamma=0.20$ for the blue channel, and $\gamma=0.34$ for the other three
channels.
Computing PSF: Following equation 3, we acquire the transmission microscope
images of the display pattern and crop them with the approximated circular
aperture shape with diameter $3333\mu m$, the size of the camera aperture. In
equation 6, the $\delta_{N}N$ is $3333\mu m$. $\rho$ equals to $1.55\mu
m/pixel$ in Sony sensor. However, after re-arranging the raw image into four
RGGB channels, $\rho$ becomes 3.1 for each channel. The focal length is
$6000\mu m$. $\lambda=(640,520,450)$ for R, G, B channel, which are the
approximated center peaks of the R, G, B filters respectively on the sensor.
It yields the down-sampling ratio $r=(2.41,2.98,3.44)$ for the R, G, and B
channels.
Adding Noises: We measure $\lambda_{read}$ and $\lambda_{shot}$ to estimate
the noise statistics. We display random patterns within the $256\times 256$
window on the monitor, collect the paired noisy and noise-free RAW data, and
compute their differences. For each of the RGGB channel, we linearly regress
the function of noise variance to the intensity value, and obtain the ratio as
the shot noise variance, and the y-intersection as the readout noise variance.
We then repeat the process 100 times and collect pairs of data points.
Finally, we estimate the distribution and randomly sample $\lambda_{read}$ and
$\lambda_{shot}$. All the measurements are listed in Table 3.
Figure 5: Network structure of the proposed UNet. It takes a 4-channel RAW
sensor data observation $y$, and outputs the restored 3-channel RGB image $x$.
## 5 Image Restoration Baselines
We use the collected real paired data, synthetic paired data, simulated PSF,
and all the necessary measurements to perform image restoration. We split the
300 pairs of images in the UDC dataset into 200 for training, 40 for
validation and 60 images in the testing partition. All the images have a
resolution of $1024\times 2048$.
### 5.1 Deconvolution Pipeline (DeP)
The DeP is a general-purpose conventional pipeline concatenating denoising and
deconvolution (Wiener Filter), which is an inverse process of the analyzed
image formation. To better utilize the unsupervised Wiener Filter (WF) [29],
we first apply the BM3D denoiser to each RAW channel separately, afterwards we
linearly divide the measured $\gamma$ with the outputs for intensity scaling.
After that, WF is applied to each channel given the pre-computed PSF
$\mathbf{k}$. Finally, RAW images with bayer pattern are demosaiced by linear
interpolation. The restored results are evaluated on the testing partition of
the UDC dataset.
### 5.2 Learning-based Methods
UNet. We propose a learning-based restoration network baseline as shown in
Figure 5. The proposed model takes a 4-channel RAW sensor data observation
$y$, and outputs the restored 3-channel RGB image $x$. The model conducts
denoising, debluring, white-balancing, intensity scaling, and demosaicing in a
single network, whose structure is basically a UNet. We split the encoder into
two sub-encoders, one of which is for computing residual details to add, and
the other one learns content encoding from degraded images. By splitting the
encoder, compared with doubling the width of each layer, we will have fewer
parameters, and make the inference and learning more efficient. To train the
model from paired images, we apply the $L_{1}$ loss, which will at large
guarantee the temporal stability compared with adversarial loss [15]. Besides,
we also apply $SSIM$ and Perception Loss (VGG Loss) for ablation study.
We crop patches of $256\times 256$, and augment the training data using the
raw image augmentation [26] while preserving the RGGB bayer pattern. We train
the model for 400 epochs using Adam optimizer ($\beta_{1}=0.9$,
$\beta_{2}=0.999$ and $\epsilon=10^{-8}$) with learning rate $10^{-4}$ and
decay factor 0.5 after 200 epoches. We also train the same structure using the
synthetic data (denoted as UNet(Syn)) generated by the pipeline proposed in
section 4.2.
ResNet. Additionally, a data-driven ResNet trained with the same data is
utilized for evaluation. To our knowledge, UNet and ResNet-based structures
are two widely-used deep models for image restoration. We use 16 residual
blocks with a feature width of 64 for our ResNet architecture, just as Lim et
al. do for EDSR [22]. The model also takes 4-channel RAW data, and outputs
3-channel RGB images. The data-driven models cannot be directly adaptive to
UDC inputs if only trained with bi-cubic degradation. We did not compare with
their model structures because model novelty is not our main claim, and the
presented two methods are the most general ways which can achieve real-time
inference as the baselines. Other model variants can be further explored in
future work.
(a) T-OLED
(b) DeP
(c) UNet(Syn)
(d) UNet
(e) GT
Figure 6: Restoration Results Comparison for T-OLED. GT: Ground Truth.
(a) P-OLED
(b) DeP
(c) UNet(Syn)
(d) UNet
(e) GT
Figure 7: Restoration Results Comparison for P-OLED. GT: Ground Truth. Table
4: Pipeline Comparison
. 4K T-OLED P-OLED Pipeline Structure $\\#$P $\downarrow$ GFLOPs $\downarrow$
T $\downarrow$ PSNR/SSIM $\uparrow$ LPIPS $\downarrow$ PSNR/SSIM $\uparrow$
LPIPS $\downarrow$ DeP - - - 28.50/0.9117 0.4219 16.97/0.7084 0.6306 ResNet
1.37M 721.76 92.92 36.26/0.9703 0.1214 27.42/0.9176 0.2500 UNet(Syn) 8.93M
124.36 21.37 32.42/0.9343 0.1739 25.88/0.9006 0.3089 UNet 8.93M 124.36 21.37
36.71/0.9713 0.1209 30.45/0.9427 0.2219
Table 5: Ablation Study on UNet alternatives.
Alternatives | | | | 4K T-OLED | P-OLED
---|---|---|---|---|---
| $\\#$P $\downarrow$ | GFLOPs $\downarrow$ | T $\downarrow$ | PSNR/SSIM $\uparrow$ | LPIPS $\downarrow$ | PSNR/SSIM $\uparrow$ | LPIPS $\downarrow$
UNet Basseline | 8.93M | 124.36 | 21.37 | 36.71/0.9713 | 0.1209 | 30.45/0.9427 | 0.2219
Double Width | 31.03M | 386.37 | 40.42 | 37.00/0.9730 | 0.1171 | 30.37/0.9425 | 0.2044
Single Encoder | 7.76M | 97.09 | 15.85 | 36.47/0.9704 | 0.1288 | 30.26/0.9387 | 0.2318
$L_{1}\rightarrow L_{1}+SSIM$ | 8.93M | 124.36 | 21.37 | 36.69/0.9714 | 0.1246 | 30.37/0.9403 | 0.2131
$L_{1}\rightarrow L_{1}+VGG$ | 8.93M | 124.36 | 21.37 | 36.31/0.9711 | 0.1130 | 30.37/0.9403 | 0.2130
## 6 Experimental Results
### 6.1 Qualitative and Quantitative Comparisons
The qualitative restoration results are shown in Figure 6 and 7. As shown,
image Deconvolution Pipeline (DeP) successfully recovers image details but
still introduces some artifacts, and suffers from the inaccuracy of the
computed ideal PSF. The UNet-based model achieves better visual quality and
denoising performance. The results of UNet trained with the synthetic data are
visually better than DeP.
The quantitative results are listed in Table 4. We report the performance in
PSNR, SSIM, a perceptual metric LPIPS [44], inference time T (ms/MPixels) and
GFLOPs. The inference time is tested with one single Titan X, and the GFLOPs
is computed by input size of $512\times 1024\times 4$. ResNet achieves a
comparable performance to UNet, but it requires more computation operations
and longer inference time. The proposed UNet-based structure is efficient and
effective, which can therefore be deployed for real-time inference for high-
resolution inputs with a single GPU. In Table 4, we demonstrate that synthetic
data still has gaps with the real data, though it has already greatly out-
performed the DeP for the two display types. The domain gap mainly comes from
the following aspects. First, due to the existing distances between display
and lens, in real data there appears visible patterns of the display on the
image plane. We recall in the assumption of the diffraction model, the display
panel is exactly at the principle plane of the lens system. The cause of the
visible bands are illustrated in the supplementary material. Second, the
approximated light transmission rate may not be accurate, the measured values
may be influenced by other environment light sources. Third, impulse noise
caused by dead pixels or over-exposure in the camera sensors widely exist in
the real dataset. Those factors provide more improvement space for this work.
Figure 8: Face detection performance before and after applying restoration.
Without display, the original face recall rate is 60$\%$. Covering the camera
with T-OLED or P-OLED will decrease the recall rate to 8$\%$ and 0$\%$. After
image restoration, the recall rates recovered back to 56$\%$ and 39$\%$.
### 6.2 Ablation Study
For the best-performed UNet structure, we compare different UNet alternatives
in Table 5. We increase the parameter size by splitting the original encoders
into two sub-encoders, so the performance is also increased. The increment
parameter size and inference time is far less than doubling the width of each
layer of UNet, but the performance improvement is comparable (T-OLED), even
better (P-OLED). We claim that the proposed UNet structure will both maintain
a small number of parameters and operations, and achieve a real-time high-
quality inference. To try alternative loss functions, we add $SSIM$ or $VGG$
loss in additional to $L_{1}$ loss with 1:1 ratio. However, the performance
gains on either $SSIM$ or perceptual metric LPIPS are not significant enough,
and are not visually distinctive. Adversarial loss is not implemented due to
its temporal instability of GAN-based training.
### 6.3 Downstream Applications
The proposed image restoration also enhances the performance of downstream
applications including face detection. Figure 8 shows an example of detecting
faces using MTCNN [41]. Without display, the original face recall rate is
60$\%$. Covering the camera with T-OLED or P-OLED will decrease the recall
rate to 8$\%$ and 0$\%$. After image restoration, the recall rates are
recovered to 56$\%$ and 39$\%$.
## 7 Conclusion and Limitations
This paper defined and presented a novel imaging system named Under-Display-
Camera (UDC). Deploying UDC to full-screen devices improves the user
interaction as well as teleconferencing experience, but does harm to imaging
quality and other downstream vision applications. We systematically analyzed
the optical systems and modelled the image formation pipeline of UDC, and both
collected real data using a novel acquisition system and synthesized realistic
data and the PSF of the system using optical model. We then proposed to
address the image restoration of UDC using a Deconvolution-based Pipeline
(DeP) and data-driven learning-based methods. Our experiments showed that the
former achieved basic restoration and the latter demonstrated an efficient
high-quality restoration. The model trained with synthetic data also achieved
a remarkable performance indicating the potential generalization ability.
UDC problem has its promising research values in complicated degradation
analysis. In real-world applications, other factors like an active display,
reflection, lens flare etc. are still very challenging and complicated. Future
work can be exploring UDC-specific restoration models and working with
aperture and display researchers to analyze the influential factors of image
degradation. It will make the restoration model generalized better for mass
production, or helpful for down-stream tasks, as an ultimate goal.
## Appendix A Appendices
(a) Display-free
(b) T-OLED
(c) P-OLED
Figure A.1: More real data samples acquried by our MCIS set-up. (a) The image
captured with camera covered by thin glass, (b)T-OLED, and (c) P-OLED.
### A.1 Real Data
More examples in 8-bit RGB version in the UDC real dataset are shown in Fig.
A.1. Each image has a high resolution of $1024\times 2048\times 3$. Images
captured by T-OLED demonstrate a blur effect along the horizontal direction.
Some spatial frequencies (i.e. vertical bands) are missing due to diffraction
effects. Images captured by P-OLED are yellow-shifted, dark, and noisy. We
also stored the 16-bit raw sensor data, which is mainly used for training and
testing in the paper.
### A.2 Synthetic Data
Figure A.2: Real and the computed point spread function (kernel).
We follow the image formation pipeline to synthesize the near-realistic data.
Given only the display pattern, and some specific measurements of the cameras,
we could generate the blur kernels as shown in Fig. A.2 along with the
degraded images for training. Fig. A.3 compares the synthetic data with the
real data. Perceptually, two sets of data samples have similar visual
characteristics.
### A.3 Visible Bands for T-OLED
(a) Real data samples.
(b) Synthetic data samples.
Figure A.3: Comparison of real data and synthetic data. First row: T-OLED.
Second row: P-OLED.
(a) Synthetic data.
(b) Real data with bands.
Figure A.4: Visible bands in real data.
In addition to the degradation formulated in the paper, there is another minor
image artifact caused by the periodic grating-like pixel structure (i.e.
T-OLED): superposition of periodic bands over the image at low to moderate
visibility levels. As shown in the Fig. A.4, periodic bands are visible in the
real data, but not in the synthetic data. We regard it as the main gap of data
synthesis. Those bands are caused by the imperfect adhesion of the display to
the camera lens. In the degradation model, we assume the display pattern or
objects are exactly placed against the lens, while in practical set-up of our
experiments, there is still a small distance between them. We can consider the
grating as being imaged very out-of-focus on the sensor plane. There will be
an image on the image sensors consisting of the grating convoluted with the
very-out-of-focus point spread function – a circle. This problem can be
mitigated by real industrial manufacturing process, so we did not resolve it
explicitly in the paper with experimental settings. However, it still forms an
interesting problem regarding eliminating real periodic noises left for future
works.
### A.4 More Restoration Results
We show more restoration results in Fig. A.5.
(a) Display
(b) DeP
(c) UNet(Syn)
(d) UNet
(e) GT
Figure A.5: More restoration results. For each two-row group, the first row is
for T-OLED, and the second one is for P-OLED.
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|
2024-09-04T02:54:58.814672 | 2020-03-10T17:17:01 | 2003.04866 | {
"authors": "Ivan Vuli\\'c, Simon Baker, Edoardo Maria Ponti, Ulla Petti, Ira\n Leviant, Kelly Wing, Olga Majewska, Eden Bar, Matt Malone, Thierry Poibeau,\n Roi Reichart, Anna Korhonen",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26141",
"submitter": "Ivan Vuli\\'c",
"url": "https://arxiv.org/abs/2003.04866"
} | arxiv-papers | 12020
# Multi-SimLex: A Large-Scale Evaluation of Multilingual and Cross-Lingual
Lexical Semantic Similarity
https://multisimlex.com/
Ivan Vulić ♠ ♠Equal contribution; English Faculty Building, 9 West Road
Cambridge CB3 9DA, United Kingdom. E-mail:
<EMAIL_ADDRESS>LTL, University of Cambridge Simon
Baker ∗♠ LTL, University of Cambridge Edoardo Maria Ponti ∗♠ LTL, University
of Cambridge Ulla Petti ∗ LTL, University of Cambridge Ira Leviant Technion
City, Haifa 3200003, Israel. E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Faculty of
Industrial Engineering and Management, Technion, IIT Kelly Wing ∗ LTL,
University of Cambridge Olga Majewska ∗ LTL, University of Cambridge Eden
Bar ∗∗ Faculty of Industrial Engineering and Management, Technion, IIT Matt
Malone ∗ LTL, University of Cambridge Thierry Poibeau Rue Maurice Arnoux,
92120 Montrouge, France. E-mail<EMAIL_ADDRESS>LATTICE Lab, CNRS and
ENS/PSL and Univ. Sorbonne nouvelle/USPC Roi Reichart ∗∗ Faculty of
Industrial Engineering and Management, Technion, IIT Anna Korhonen ∗ LTL,
University of Cambridge
###### Abstract
We introduce Multi-SimLex, a large-scale lexical resource and evaluation
benchmark covering datasets for 12 typologically diverse languages, including
major languages (e.g., Mandarin Chinese, Spanish, Russian) as well as less-
resourced ones (e.g., Welsh, Kiswahili). Each language dataset is annotated
for the lexical relation of semantic similarity and contains 1,888
semantically aligned concept pairs, providing a representative coverage of
word classes (nouns, verbs, adjectives, adverbs), frequency ranks, similarity
intervals, lexical fields, and concreteness levels. Additionally, owing to the
alignment of concepts across languages, we provide a suite of 66 cross-lingual
semantic similarity datasets. Due to its extensive size and language coverage,
Multi-SimLex provides entirely novel opportunities for experimental evaluation
and analysis. On its monolingual and cross-lingual benchmarks, we evaluate and
analyze a wide array of recent state-of-the-art monolingual and cross-lingual
representation models, including static and contextualized word embeddings
(such as fastText, M-BERT and XLM), externally informed lexical
representations, as well as fully unsupervised and (weakly) supervised cross-
lingual word embeddings. We also present a step-by-step dataset creation
protocol for creating consistent, Multi-Simlex -style resources for additional
languages. We make these contributions - the public release of Multi-SimLex
datasets, their creation protocol, strong baseline results, and in-depth
analyses which can be be helpful in guiding future developments in
multilingual lexical semantics and representation learning - available via a
website which will encourage community effort in further expansion of Multi-
Simlex to many more languages. Such a large-scale semantic resource could
inspire significant further advances in NLP across languages.
††issue: 1
## 1 Introduction
The lack of annotated training and evaluation data for many tasks and domains
hinders the development of computational models for the majority of the
world’s languages Snyder and Barzilay (2010); Adams et al. (2017); Ponti et
al. (2019a). The necessity to guide and advance multilingual and cross-lingual
NLP through annotation efforts that follow cross-lingually consistent
guidelines has been recently recognized by collaborative initiatives such as
the Universal Dependency (UD) project Nivre et al. (2019). The latest version
of UD (as of March 2020) covers more than 70 languages. Crucially, this
resource continues to steadily grow and evolve through the contributions of
annotators from across the world, extending the UD’s reach to a wide array of
typologically diverse languages. Besides steering research in multilingual
parsing Zeman et al. (2018); Kondratyuk and Straka (2019); Doitch et al.
(2019) and cross-lingual parser transfer Rasooli and Collins (2017); Lin et
al. (2019); Rotman and Reichart (2019), the consistent annotations and
guidelines have also enabled a range of insightful comparative studies focused
on the languages’ syntactic (dis)similarities Bjerva and Augenstein (2018);
Ponti et al. (2018a); Pires, Schlinger, and Garrette (2019).
Inspired by the UD work and its substantial impact on research in
(multilingual) syntax, in this article we introduce Multi-SimLex, a suite of
manually and consistently annotated semantic datasets for 12 different
languages, focused on the fundamental lexical relation of semantic similarity
Budanitsky and Hirst (2006); Hill, Reichart, and Korhonen (2015). For any pair
of words, this relation measures whether their referents share the same
(functional) features, as opposed to general cognitive association captured by
co-occurrence patterns in texts (i.e., the distributional information).
Datasets that quantify the strength of true semantic similarity between
concept pairs such as SimLex-999 Hill, Reichart, and Korhonen (2015) or
SimVerb-3500 Gerz et al. (2016) have been instrumental in improving models for
distributional semantics and representation learning. Discerning between
semantic similarity and relatedness/association is not only crucial for
theoretical studies on lexical semantics (see §2), but has also been shown to
benefit a range of language understanding tasks in NLP. Examples include
dialog state tracking Mrkšić et al. (2017); Ren et al. (2018), spoken language
understanding Kim et al. (2016); Kim, de Marneffe, and Fosler-Lussier (2016),
text simplification Glavaš and Vulić (2018); Ponti et al. (2018b); Lauscher et
al. (2019), dictionary and thesaurus construction Cimiano, Hotho, and Staab
(2005); Hill et al. (2016).
Despite the proven usefulness of semantic similarity datasets, they are
available only for a small and typologically narrow sample of resource-rich
languages such as German, Italian, and Russian Leviant and Reichart (2015),
whereas some language types and low-resource languages typically lack similar
evaluation data. Even if some resources do exist, they are limited in their
size (e.g., 500 pairs in Turkish Ercan and Yıldız (2018), 500 in Farsi
Camacho-Collados et al. (2017), or 300 in Finnish Venekoski and Vankka (2017))
and coverage (e.g., all datasets which originated from the original English
SimLex-999 contain only high-frequent concepts, and are dominated by nouns).
This is why, as our departure point, we introduce a larger and more
comprehensive English word similarity dataset spanning 1,888 concept pairs
(see §4).
Most importantly, semantic similarity datasets in different languages have
been created using heterogeneous construction procedures with different
guidelines for translation and annotation, as well as different rating scales.
For instance, some datasets were obtained by directly translating the English
SimLex-999 in its entirety Leviant and Reichart (2015); Mrkšić et al. (2017)
or in part Venekoski and Vankka (2017). Other datasets were created from
scratch Ercan and Yıldız (2018) and yet others sampled English concept pairs
differently from SimLex-999 and then translated and reannotated them in target
languages Camacho-Collados et al. (2017). This heterogeneity makes these
datasets incomparable and precludes systematic cross-linguistic analyses. In
this article, consolidating the lessons learned from previous dataset
construction paradigms, we propose a carefully designed translation and
annotation protocol for developing monolingual Multi-SimLex datasets with
aligned concept pairs for typologically diverse languages. We apply this
protocol to a set of 12 languages, including a mixture of major languages
(e.g., Mandarin, Russian, and French) as well as several low-resource ones
(e.g., Kiswahili, Welsh, and Yue Chinese). We demonstrate that our proposed
dataset creation procedure yields data with high inter-annotator agreement
rates (e.g., the average mean inter-annotator agreement for Welsh is 0.742).
The unified construction protocol and alignment between concept pairs enables
a series of quantitative analyses. Preliminary studies on the influence that
polysemy and cross-lingual variation in lexical categories (see §2.3) have on
similarity judgments are provided in §5. Data created according to Multi-
SimLex protocol also allow for probing into whether similarity judgments are
universal across languages, or rather depend on linguistic affinity (in terms
of linguistic features, phylogeny, and geographical location). We investigate
this question in §5.4. Naturally, Multi-SimLex datasets can be used as an
intrinsic evaluation benchmark to assess the quality of lexical
representations based on monolingual, joint multilingual, and transfer
learning paradigms. We conduct a systematic evaluation of several state-of-
the-art representation models in §7, showing that there are large gaps between
human and system performance in all languages. The proposed construction
paradigm also supports the automatic creation of 66 cross-lingual Multi-SimLex
datasets by interleaving the monolingual ones. We outline the construction of
the cross-lingual datasets in §6, and then present a quantitative evaluation
of a series of cutting-edge cross-lingual representation models on this
benchmark in §8.
Contributions. We now summarize the main contributions of this work:
1) Building on lessons learned from prior work, we create a more comprehensive
lexical semantic similarity dataset for the English language spanning a total
of 1,888 concept pairs balanced with respect to similarity, frequency, and
concreteness, and covering four word classes: nouns, verbs, adjectives and,
for the first time, adverbs. This dataset serves as the main source for the
creation of equivalent datasets in several other languages.
2) We present a carefully designed and rigorous language-agnostic translation
and annotation protocol. These well-defined guidelines will facilitate the
development of future Multi-SimLex datasets for other languages. The proposed
protocol eliminates some crucial issues with prior efforts focused on the
creation of multi-lingual semantic resources, namely: i) limited coverage; ii)
heterogeneous annotation guidelines; and iii) concept pairs which are
semantically incomparable across different languages.
3) We offer to the community manually annotated evaluation sets of 1,888
concept pairs across 12 typologically diverse languages, and 66 large cross-
lingual evaluation sets. To the best of our knowledge, Multi-SimLex is the
most comprehensive evaluation resource to date focused on the relation of
semantic similarity.
4) We benchmark a wide array of recent state-of-the-art monolingual and cross-
lingual word representation models across our sample of languages. The results
can serve as strong baselines that lay the foundation for future improvements.
5) We present a first large-scale evaluation study on the ability of encoders
pretrained on language modeling (such as bert Devlin et al. (2019) and xlm
Conneau and Lample (2019)) to reason over word-level semantic similarity in
different languages. To our own surprise, the results show that monolingual
pretrained encoders, even when presented with word types out of context, are
sometimes competitive with static word embedding models such as fastText
Bojanowski et al. (2017) or word2vec Mikolov et al. (2013). The results also
reveal a huge gap in performance between massively multilingual pretrained
encoders and language-specific encoders in favor of the latter: our findings
support other recent empirical evidence related to the “curse of
multilinguality” Conneau et al. (2019); Bapna and Firat (2019) in
representation learning.
6) We make all of these resources available on a website which facilitates
easy creation, submission and sharing of Multi-Simlex-style datasets for a
larger number of languages. We hope that this will yield an even larger
repository of semantic resources that inspire future advances in NLP within
and across languages.
In light of the success of Universal Dependencies Nivre et al. (2019), we hope
that our initiative will instigate a collaborative public effort with
established and clear-cut guidelines that will result in additional Multi-
SimLex datasets in a large number of languages in the near future. Moreover,
we hope that it will provide means to advance our understanding of
distributional and lexical semantics across a large number of languages. All
monolingual and cross-lingual Multi-SimLex datasets–along with detailed
translation and annotation guidelines–are available online at:
https://multisimlex.com/.
## 2 Lexical Semantic Similarity
### 2.1 Similarity and Association
The focus of the Multi-SimLex initiative is on the lexical relation of pure
semantic similarity. For any pair of words, this relation measures whether
their referents share the same features. For instance, graffiti and frescos
are similar to the extent that they are both forms of painting and appear on
walls. This relation can be contrasted with the cognitive association between
two words, which often depends on how much their referents interact in the
real world, or are found in the same situations. For instance, a painter is
easily associated with frescos, although they lack any physical commonalities.
Association is also known in the literature under other names: relatedness
Budanitsky and Hirst (2006), topical similarity (McKeown et al., 2002), and
domain similarity (Turney, 2012).
Semantic similarity and association overlap to some degree, but do not
coincide Kiela, Hill, and Clark (2015); Vulić, Kiela, and Korhonen (2017). In
fact, there exist plenty of pairs that are intuitively associated but not
similar. Pairs where the converse is true can also be encountered, although
more rarely. An example are synonyms where a word is common and the other
infrequent, such as to seize and to commandeer. Hill, Reichart, and Korhonen
(2015) revealed that while similarity measures based on the WordNet graph (Wu
and Palmer, 1994) and human judgments of association in the University of
South Florida Free Association Database (Nelson, McEvoy, and Schreiber, 2004)
do correlate, a number of pairs follow opposite trends. Several studies on
human cognition also point in the same direction. For instance, semantic
priming can be triggered by similar words without association (Lucas, 2000).
On the other hand, a connection with cue words is established more quickly for
topically related words rather than for similar words in free association
tasks De Deyne and Storms (2008).
A key property of semantic similarity is its gradience: pairs of words can be
similar to a different degree. On the other hand, the relation of synonymy is
binary: pairs of words are synonyms if they can be substituted in all contexts
(or most contexts, in a looser sense), otherwise they are not. While synonyms
can be conceived as lying on one extreme of the semantic similarity continuum,
it is crucial to note that their definition is stated in purely relational
terms, rather than invoking their referential properties (Lyons, 1977; Cruse,
1986; Coseriu, 1967). This makes behavioral studies on semantic similarity
fundamentally different from lexical resources like WordNet Miller (1995),
which include paradigmatic relations (such as synonymy).
### 2.2 Similarity for NLP: Intrinsic Evaluation and Semantic Specialization
The ramifications of the distinction between similarity and association are
profound for distributional semantics. This paradigm of lexical semantics is
grounded in the distributional hypothesis, formulated by Firth (1957) and
Harris (1951). According to this hypothesis, the meaning of a word can be
recovered empirically from the contexts in which it occurs within a collection
of texts. Since both pairs of topically related words and pairs of purely
similar words tend to appear in the same contexts, their associated meaning
confounds the two distinct relations Hill, Reichart, and Korhonen (2015);
Schwartz, Reichart, and Rappoport (2015); Vulić et al. (2017b). As a result,
distributional methods obscure a crucial facet of lexical meaning.
This limitation also reflects onto word embeddings (WEs), representations of
words as low-dimensional vectors that have become indispensable for a wide
range of NLP applications (Collobert et al., 2011; Chen and Manning, 2014;
Melamud et al., 2016, inter alia). In particular, it involves both static WEs
learned from co-occurrence patterns Mikolov et al. (2013); Levy and Goldberg
(2014); Bojanowski et al. (2017) and contextualized WEs learned from modeling
word sequences (Peters et al., 2018; Devlin et al., 2019, inter alia). As a
result, in the induced representations, geometrical closeness (measured e.g.
through cosine distance) conflates genuine similarity with broad relatedness.
For instance, the vectors for antonyms such as sober and drunk, by definition
dissimilar, might be neighbors in the semantic space under the distributional
hypothesis. Turney (2012), Kiela and Clark (2014), and Melamud et al. (2016)
demonstrated that different choices of hyper-parameters in WE algorithms (such
as context window) emphasize different relations in the resulting
representations. Likewise, Agirre et al. (2009) and Levy and Goldberg (2014)
discovered that WEs learned from texts annotated with syntactic information
mirror similarity better than simple local bag-of-words neighborhoods.
The failure of WEs to capture semantic similarity, in turn, affects model
performance in several NLP applications where such knowledge is crucial. In
particular, Natural Language Understanding tasks such as statistical dialog
modeling, text simplification, or semantic text similarity Mrkšić et al.
(2016); Kim et al. (2016); Ponti et al. (2019c), among others, suffer the
most. As a consequence, resources providing clean information on semantic
similarity are key in mitigating the side effects of the distributional
signal. In particular, such databases can be employed for the intrinsic
evaluations of specific WE models as a proxy of their reliability for
downstream applications (Collobert and Weston, 2008; Baroni and Lenci, 2010;
Hill, Reichart, and Korhonen, 2015); intuitively, the more WEs are misaligned
with human judgments of similarity, the more their performance on actual tasks
is expected to be degraded. Moreover, word representations can be specialized
(a.k.a. retrofitted) by disentangling word relations of similarity and
association. In particular, linguistic constraints sourced from external
databases (such as synonyms from WordNet) can be injected into WEs (Faruqui et
al., 2015; Wieting et al., 2015; Mrkšić et al., 2017; Lauscher et al., 2019;
Kamath et al., 2019, inter alia) in order to enforce a particular relation in
a distributional semantic space while preserving the original adjacency
properties.
### 2.3 Similarity and Language Variation: Semantic Typology
In this work, we tackle the concept of (true) semantic similarity from a
multilingual perspective. While the same meaning representations may be shared
by all human speakers at a deep cognitive level, there is no one-to-one
mapping between the words in the lexicons of different languages. This makes
the comparison of similarity judgments across languages difficult, since the
meaning overlap of translationally equivalent words is sometimes far less than
exact. This results from the fact that the way languages ‘partition’ semantic
fields is partially arbitrary (Trier, 1931), although constrained cross-
lingually by common cognitive biases Majid et al. (2007). For instance,
consider the field of colors: English distinguishes between green and blue,
whereas Murle (South Sudan) has a single word for both (Kay and Maffi, 2013).
In general, semantic typology studies the variation in lexical semantics
across the world’s languages. According to (Evans, 2011), the ways languages
categorize concepts into the lexicon follow three main axes: 1) granularity:
what is the number of categories in a specific domain?; 2) boundary location:
where do the lines marking different categories lie?; 3) grouping and
dissection: what are the membership criteria of a category; which instances
are considered to be more prototypical? Different choices with respect to
these axes lead to different lexicalization patterns.111More formally,
colexification is a phenomenon when different meanings can be expressed by the
same word in a language François (2008). For instance, the two senses which
are distinguished in English as time and weather are co-lexified in Croatian:
the word vrijeme is used in both cases. For instance, distinct senses in a
polysemous word in English, such as skin (referring to both the body and
fruit), may be assigned separate words in other languages such as Italian
pelle and buccia, respectively (Rzymski et al., 2020). We later analyze
whether similarity scores obtained from native speakers also loosely follow
the patterns described by semantic typology.
## 3 Previous Work and Evaluation Data
Word Pair Datasets. Rich expert-created resources such as WordNet Miller
(1995); Fellbaum (1998), VerbNet Kipper Schuler (2005); Kipper et al. (2008),
or FrameNet Baker, Fillmore, and Lowe (1998) encode a wealth of semantic and
syntactic information, but are expensive and time-consuming to create. The
scale of this problem gets multiplied by the number of languages in
consideration. Therefore, crowd-sourcing with non-expert annotators has been
adopted as a quicker alternative to produce smaller and more focused semantic
resources and evaluation benchmarks. This alternative practice has had a
profound impact on distributional semantics and representation learning Hill,
Reichart, and Korhonen (2015). While some prominent English word pair datasets
such as WordSim-353 Finkelstein et al. (2002), MEN Bruni, Tran, and Baroni
(2014), or Stanford Rare Words Luong, Socher, and Manning (2013) did not
discriminate between similarity and relatedness, the importance of this
distinction was established by Hill, Reichart, and Korhonen (2015, see again
the discussion in §2.1) through the creation of SimLex-999. This inspired
other similar datasets which focused on different lexical properties. For
instance, SimVerb-3500 Gerz et al. (2016) provided similarity ratings for
3,500 English verbs, whereas CARD-660 Pilehvar et al. (2018) aimed at
measuring the semantic similarity of infrequent concepts.
Semantic Similarity Datasets in Other Languages. Motivated by the impact of
datasets such as SimLex-999 and SimVerb-3500 on representation learning in
English, a line of related work focused on creating similar resources in other
languages. The dominant approach is translating and reannotating the entire
original English SimLex-999 dataset, as done previously for German, Italian,
and Russian Leviant and Reichart (2015), Hebrew and Croatian Mrkšić et al.
(2017), and Polish Mykowiecka, Marciniak, and Rychlik (2018).
Venekoski:2017nodalida apply this process only to a subset of 300 concept
pairs from the English SimLex-999. On the other hand, Camacho-Collados et al.
(2017) sampled a new set of 500 English concept pairs to ensure wider topical
coverage and balance across similarity spectra, and then translated those
pairs to German, Italian, Spanish, and Farsi (SEMEVAL-500). A similar approach
was followed by Ercan and Yıldız (2018) for Turkish, by Huang et al. (2019)
for Mandarin Chinese, and by Sakaizawa and Komachi (2018) for Japanese.
Netisopakul, Wohlgenannt, and Pulich (2019) translated the concatenation of
SimLex-999, WordSim-353, and the English SEMEVAL-500 into Thai and then
reannotated it. Finally, Barzegar et al. (2018) translated English SimLex-999
and WordSim-353 to 11 resource-rich target languages (German, French, Russian,
Italian, Dutch, Chinese, Portuguese, Swedish, Spanish, Arabic, Farsi), but
they did not provide details concerning the translation process and the
resolution of translation disagreements. More importantly, they also did not
reannotate the translated pairs in the target languages. As we discussed in §
2.3 and reiterate later in §5, semantic differences among languages can have a
profound impact on the annotation scores; particulary, we show in §5.4 that
these differences even roughly define language clusters based on language
affinity.
A core issue with the current datasets concerns a lack of one unified
procedure that ensures the comparability of resources in different languages.
Further, concept pairs for different languages are sourced from different
corpora (e.g., direct translation of the English data versus sampling from
scratch in the target language). Moreover, the previous SimLex-based
multilingual datasets inherit the main deficiencies of the English original
version, such as the focus on nouns and highly frequent concepts. Finally,
prior work mostly focused on languages that are widely spoken and do not
account for the variety of the world’s languages. Our long-term goal is
devising a standardized methodology to extend the coverage also to languages
that are resource-lean and/or typologically diverse (e.g., Welsh, Kiswahili as
in this work).
Multilingual Datasets for Natural Language Understanding. The Multi-SimLex
initiative and corresponding datasets are also aligned with the recent efforts
on procuring multilingual benchmarks that can help advance computational
modeling of natural language understanding across different languages. For
instance, pretrained multilingual language models such as multilingual bert
Devlin et al. (2019) or xlm Conneau and Lample (2019) are typically probed on
XNLI test data Conneau et al. (2018b) for cross-lingual natural language
inference. XNLI was created by translating examples from the English MultiNLI
dataset, and projecting its sentence labels Williams, Nangia, and Bowman
(2018). Other recent multilingual datasets target the task of question
answering based on reading comprehension: i) MLQA Lewis et al. (2019) includes
7 languages ii) XQuAD Artetxe, Ruder, and Yogatama (2019) 10 languages; iii)
TyDiQA Clark et al. (2020) 9 widely spoken typologically diverse languages.
While MLQA and XQuAD result from the translation from an English dataset,
TyDiQA was built independently in each language. Another multilingual dataset,
PAWS-X Yang et al. (2019), focused on the paraphrase identification task and
was created translating the original English PAWS Zhang, Baldridge, and He
(2019) into 6 languages. We believe that Multi-SimLex can substantially
contribute to this endeavor by offering a comprehensive multilingual benchmark
for the fundamental lexical level relation of semantic similarity. In future
work, Multi-SimLex also offers an opportunity to investigate the correlations
between word-level semantic similarity and performance in downstream tasks
such as QA and NLI across different languages.
## 4 The Base for Multi-SimLex: Extending English SimLex-999
In this section, we discuss the design principles behind the English (eng)
Multi-SimLex dataset, which is the basis for all the Multi-SimLex datasets in
other languages, as detailed in §5. We first argue that a new, more balanced,
and more comprehensive evaluation resource for lexical semantic similarity in
English is necessary. We then describe how the 1,888 word pairs contained in
the eng Multi-SimLex were selected in such a way as to represent various
linguistic phenomena within a single integrated resource.
Construction Criteria. The following criteria have to be satisfied by any
high-quality semantic evaluation resource, as argued by previous studies
focused on the creation of such resources (Hill, Reichart, and Korhonen, 2015;
Gerz et al., 2016; Vulić et al., 2017a; Camacho-Collados et al., 2017, inter
alia):
(C1) Representative and diverse. The resource must cover the full range of
diverse concepts occurring in natural language, including different word
classes (e.g., nouns, verbs, adjectives, adverbs), concrete and abstract
concepts, a variety of lexical fields, and different frequency ranges.
(C2) Clearly defined. The resource must provide a clear understanding of which
semantic relation exactly is annotated and measured, possibly contrasting it
with other relations. For instance, the original SimLex-999 and SimVerb-3500
explicitly focus on true semantic similarity and distinguish it from broader
relatedness captured by datasets such as MEN Bruni, Tran, and Baroni (2014) or
WordSim-353 Finkelstein et al. (2002).
(C3) Consistent and reliable. The resource must ensure consistent annotations
obtained from non-expert native speakers following simple and precise
annotation guidelines.
In choosing the word pairs and constructing eng Multi-SimLex, we adhere to
these requirements. Moreover, we follow good practices established by the
research on related resources. In particular, since the introduction of the
original SimLex-999 dataset Hill, Reichart, and Korhonen (2015), follow-up
works have improved its construction protocol across several aspects,
including: 1) coverage of more lexical fields, e.g., by relying on a diverse
set of Wikipedia categories Camacho-Collados et al. (2017), 2) infrequent/rare
words Pilehvar et al. (2018), 3) focus on particular word classes, e.g., verbs
Gerz et al. (2016), 4) annotation quality control Pilehvar et al. (2018). Our
goal is to make use of these improvements towards a larger, more
representative, and more reliable lexical similarity dataset in English and,
consequently, in all other languages.
The Final Output: English Multi-SimLex. In order to ensure that the criterion
C1 is satisfied, we consolidate and integrate the data already carefully
sampled in prior work into a single, comprehensive, and representative
dataset. This way, we can control for diversity, frequency, and other
properties while avoiding to perform this time-consuming selection process
from scratch. Note that, on the other hand, the word pairs chosen for English
are scored from scratch as part of the entire Multi-SimLex annotation process,
introduced later in §5. We now describe the external data sources for the
final set of word pairs:
1) Source: SimLex-999. Hill, Reichart, and Korhonen (2015). The English Multi-
SimLex has been initially conceived as an extension of the original SimLex-999
dataset. Therefore, we include all 999 word pairs from SimLex, which span 666
noun pairs, 222 verb pairs, and 111 adjective pairs. While SimLex-999 already
provides examples representing different POS classes, it does not have a
sufficient coverage of different linguistic phenomena: for instance, it
contains only very frequent concepts, and it does not provide a representative
set of verbs (Gerz et al., 2016).
2) Source: SemEval-17: Task 2 (henceforth SEMEVAL-500; Camacho-Collados et
al., 2017). We start from the full dataset of 500 concept pairs to extract a
total of 334 concept pairs for English Multi-SimLex a) which contain only
single-word concepts, b) which are not named entities, c) where POS tags of
the two concepts are the same, d) where both concepts occur in the top 250K
most frequent word types in the English Wikipedia, and e) do not already occur
in SimLex-999. The original concepts were sampled as to span all the 34
domains available as part of BabelDomains Camacho-Collados and Navigli (2017),
which roughly correspond to the main high-level Wikipedia categories. This
ensures topical diversity in our sub-sample.
3) Source: CARD-660 Pilehvar et al. (2018). 67 word pairs are taken from this
dataset focused on rare word similarity, applying the same selection criteria
a) to e) employed for SEMEVAL-500. Words are controlled for frequency based on
their occurrence counts from the Google News data and the ukWaC corpus Baroni
et al. (2009). CARD-660 contains some words that are very rare (logboat),
domain-specific (erythroleukemia) and slang (2mrw), which might be difficult
to translate and annotate across a wide array of languages. Hence, we opt for
retaining only the concept pairs above the threshold of top 250K most frequent
Wikipedia concepts, as above.
4) Source: SimVerb-3500 Gerz et al. (2016) Since both CARD-660 and SEMEVAL-500
are heavily skewed towards noun pairs, and nouns also dominate the original
SimLex-999, we also extract additional verb pairs from the verb-specific
similarity dataset SimVerb-3500. We randomly sample 244 verb pairs from
SimVerb-3500 that represent all similarity spectra. In particular, we add 61
verb pairs for each of the similarity intervals:
$[0,1.5),[1.5,3),[3,4.5),[4.5,6]$. Since verbs in SimVerb-3500 were originally
chosen from VerbNet Kipper, Snyder, and Palmer (2004); Kipper et al. (2008),
they cover a wide range of verb classes and their related linguistic
phenomena.
5) Source: University of South Florida (USF; Nelson, McEvoy, and Schreiber,
2004) norms, the largest database of free association for English. In order to
improve the representation of different POS classes, we sample additional
adjectives and adverbs from the USF norms following the procedure established
by Hill, Reichart, and Korhonen (2015); Gerz et al. (2016). This yields
additional 122 adjective pairs, but only a limited number of adverb pairs
(e.g., later – never, now – here, once – twice). Therefore, we also create a
set of adverb pairs semi-automatically by sampling adjectives that can be
derivationally transformed into adverbs (e.g. adding the suffix -ly) from the
USF, and assessing the correctness of such derivation in WordNet. The
resulting pairs include, for instance, primarily – mainly, softly – firmly,
roughly – reliably, etc. We include a total of 123 adverb pairs into the final
English Multi-SimLex. Note that this is the first time adverbs are included
into any semantic similarity dataset.
Fulfillment of Construction Criteria. The final eng Multi-SimLex dataset spans
1,051 noun pairs, 469 verb pairs, 245 adjective pairs, and 123 adverb
pairs.222There is a very small number of adjective and verb pairs extracted
from CARD-660 and SEMEVAL-500 as well. For instance, the total number of verbs
is 469 since we augment the original 222 SimLex-999 verb pairs with 244
SimVerb-3500 pairs and 3 SEMEVAL-500 pairs; and similarly for adjectives. As
mentioned above, the criterion C1 has been fulfilled by relying only on word
pairs that already underwent meticulous sampling processes in prior work,
integrating them into a single resource. As a consequence, Multi-SimLex allows
for fine-grained analyses over different POS classes, concreteness levels,
similarity spectra, frequency intervals, relation types, morphology, lexical
fields, and it also includes some challenging orthographically similar
examples (e.g., infection – inflection).333Unlike SEMEVAL-500 and CARD-660, we
do not explicitly control for the equal representation of concept pairs across
each similarity interval for several reasons: a) Multi-SimLex contains a
substantially larger number of concept pairs, so it is possible to extract
balanced samples from the full data; b) such balance, even if imposed on the
English dataset, would be distorted in all other monolingual and cross-lingual
datasets; c) balancing over similarity intervals arguably does not reflect a
true distribution “in the wild” where most concepts are only loosely related
or completely unrelated. We ensure that the criteria C2 and C3 are satisfied
by using similar annotation guidelines as Simlex-999, SimVerb-3500, and
SEMEVAL-500 that explicitly target semantic similarity. In what follows, we
outline the carefully tailored process of translating and annotating Multi-
SimLex datasets in all target languages.
## 5 Multi-SimLex: Translation and Annotation
We now detail the development of the final Multi-SimLex resource, describing
our language selection process, as well as translation and annotation of the
resource, including the steps taken to ensure and measure the quality of this
resource. We also provide key data statistics and preliminary cross-lingual
comparative analyses.
Language Selection. Multi-SimLex comprises eleven languages in addition to
English. The main objective for our inclusion criteria has been to balance
language prominence (by number of speakers of the language) for maximum impact
of the resource, while simultaneously having a diverse suite of languages
based on their typological features (such as morphological type and language
family). Table 1 summarizes key information about the languages currently
included in Multi-SimLex. We have included a mixture of fusional,
agglutinative, isolating, and introflexive languages that come from eight
different language families. This includes languages that are very widely used
such as Chinese Mandarin and Spanish, and low-resource languages such as Welsh
and Kiswahili. We hope to further include additional languages and inspire
other researchers to contribute to the effort over the lifetime of this
project.
The work on data collection can be divided into two crucial phases: 1) a
translation phase where the extended English language dataset with 1,888 pairs
(described in §4) is translated into eleven target languages, and 2) an
annotation phase where human raters scored each pair in the translated set as
well as the English set. Detailed guidelines for both phases are available
online at: https://multisimlex.com.
Language | ISO 639-3 | Family | Type | # Speakers
---|---|---|---|---
Chinese Mandarin | cmn | Sino-Tibetan | Isolating | 1.116 B
Welsh | cym | IE: Celtic | Fusional | 0.7 M
English | eng | IE: Germanic | Fusional | 1.132 B
Estonian | est | Uralic | Agglutinative | 1.1 M
Finnish | fin | Uralic | Agglutinative | 5.4 M
French | fra | IE: Romance | Fusional | 280 M
Hebrew | heb | Afro-Asiatic | Introflexive | 9 M
Polish | pol | IE: Slavic | Fusional | 50 M
Russian | rus | IE: Slavic | Fusional | 260 M
Spanish | spa | IE: Romance | Fusional | 534.3 M
Kiswahili | swa | Niger-Congo | Agglutinative | 98 M
Yue Chinese | yue | Sino-Tibetan | Isolating | 73.5 M
Table 1: The list of 12 languages in the Multi-SimLex multilingual suite along
with their corresponding language family (IE = Indo-European), broad
morphological type, and their ISO 639-3 code. The number of speakers is based
on the total count of L1 and L2 speakers, according to ethnologue.com.
### 5.1 Word Pair Translation
Translators for each target language were instructed to find direct or
approximate translations for the 1,888 word pairs that satisfy the following
rules. (1) All pairs in the translated set must be unique (i.e., no duplicate
pairs); (2) Translating two words from the same English pair into the same
word in the target language is not allowed (e.g., it is not allowed to
translate car and automobile to the same Spanish word coche). (3) The
translated pairs must preserve the semantic relations between the two words
when possible. This means that, when multiple translations are possible, the
translation that best conveys the semantic relation between the two words
found in the original English pair is selected. (4) If it is not possible to
use a single-word translation in the target language, then a multi-word
expression (MWE) can be used to convey the nearest possible semantics given
the above points (e.g., the English word homework is translated into the
Polish MWE praca domowa).
Satisfying the above rules when finding appropriate translations for each
pair–while keeping to the spirit of the intended semantic relation in the
English version–is not always straightforward. For instance, kinship
terminology in Sinitic languages (Mandarin and Yue) uses different terms
depending on whether the family member is older or younger, and whether the
family member comes from the mother’s side or the father’s side. In Mandarin,
_brother_ has no direct translation and can be translated as either: 哥哥
(_older brother_) or 弟弟 (_younger brother_). Therefore, in such cases, the
translators are asked to choose the best option given the semantic context
(relation) expressed by the pair in English, otherwise select one of the
translations arbitrarily. This is also used to remove duplicate pairs in the
translated set, by differentiating the duplicates using a variant at each
instance. Further, many translation instances were resolved using near-
synonymous terms in the translation. For example, the words in the pair: _wood
– timber_ can only be directly translated in Estonian to _puit_ , and are not
distinguishable. Therefore, the translators approximated the translation for
timber to the compound noun _puitmaterjal_ (literally: _wood material_) in
order to produce a valid pair in the target language. In some cases, a direct
transliteration from English is used. For example, the pair: _physician_ and
_doctor_ both translate to the same word in Estonian (arst); the less formal
word _doktor_ is used as a translation of _doctor_ to generate a valid pair.
Languages: | cmn | cym | est | fin | fra | heb | pol | rus | spa | swa | yue | Avg
---|---|---|---|---|---|---|---|---|---|---|---|---
Nouns | 84.5 | 80.0 | 90.0 | 87.3 | 78.2 | 98.2 | 90.0 | 95.5 | 85.5 | 80.0 | 77.3 | 86.0
Adjectives | 88.5 | 88.5 | 61.5 | 73.1 | 69.2 | 100.0 | 84.6 | 100.0 | 69.2 | 88.5 | 84.6 | 82.5
Verbs | 88.0 | 74.0 | 82.0 | 76.0 | 78.0 | 100.0 | 74.0 | 100.0 | 74.0 | 76.0 | 86.0 | 82.5
Adverbs | 92.9 | 100.0 | 57.1 | 78.6 | 92.9 | 100.0 | 85.7 | 100.0 | 85.7 | 85.7 | 78.6 | 87.0
Overall | 86.5 | 81.0 | 82.0 | 82.0 | 78.0 | 99.0 | 85.0 | 97.5 | 80.5 | 81.0 | 80.5 | 84.8
Table 2: Inter-translator agreement (% of matched translated words) by
independent translators using a randomly selected 100-pair English sample from
the Multi-SimLex dataset, and the corresponding 100-pair samples from the
other datasets.
We measure the quality of the translated pairs by using a random sample set of
100 pairs (from the 1,888 pairs) to be translated by an independent translator
for each target language. The sample is proportionally stratified according to
the part-of-speech categories. The independent translator is given identical
instructions to the main translator; we then measure the percentage of matched
translated words between the two translations of the sample set. Table 2
summarizes the inter-translator agreement results for all languages and by
part-of-speech subsets. Overall across all languages, the agreement is 84.8%,
which is similar to prior work Camacho-Collados et al. (2017); Vulić,
Ponzetto, and Glavaš (2019).
### 5.2 Guidelines and Word Pair Scoring
Across all languages, 145 human annotators were asked to score all 1,888 pairs
(in their given language). We finally collect at least ten valid annotations
for each word pair in each language. All annotators were required to abide by
the following instructions:
1\. Each annotator must assign an integer score between 0 and 6 (inclusive)
indicating how semantically similar the two words in a given pair are. A score
of 6 indicates very high similarity (i.e., perfect synonymy), while zero
indicates no similarity.
2\. Each annotator must score the entire set of 1,888 pairs in the dataset.
The pairs must not be shared between different annotators.
3\. Annotators are able to break the workload over a period of approximately
2-3 weeks, and are able to use external sources (e.g. dictionaries, thesauri,
WordNet) if required.
4\. Annotators are kept anonymous, and are not able to communicate with each
other during the annotation process.
The selection criteria for the annotators required that all annotators must be
native speakers of the target language. Preference to annotators with
university education was given, but not required. Annotators were asked to
complete a spreadsheet containing the translated pairs of words, as well as
the part-of-speech, and a column to enter the score. The annotators did not
have access to the original pairs in English.
To ensure the quality of the collected ratings, we have employed an
adjudication protocol similar to the one proposed and validated by
Pilehvar:2018emnlp. It consists of the following three rounds:
Round 1: All annotators are asked to follow the instructions outlined above,
and to rate all 1,888 pairs with integer scores between 0 and 6.
Round 2: We compare the scores of all annotators and identify the pairs for
each annotator that have shown the most disagreement. We ask the annotators to
reconsider the assigned scores for those pairs only. The annotators may chose
to either change or keep the scores. As in the case with Round 1, the
annotators have no access to the scores of the other annotators, and the
process is anonymous. This process gives a chance for annotators to correct
errors or reconsider their judgments, and has been shown to be very effective
in reaching consensus, as reported by Pilehvar et al. (2018). We used a very
similar procedure as Pilehvar et al. (2018) to identify the pairs with the
most disagreement; for each annotator, we marked the $i$th pair if the rated
score $s_{i}$ falls within: $s_{i}\geq\mu_{i}+1.5$ or $s_{i}\leq\mu_{i}-1.5$,
where $\mu_{i}$ is the mean of the other annotators’ scores.
Round 3: We compute the average agreement for each annotator (with the other
annotators), by measuring the average Spearman’s correlation against all other
annotators. We discard the scores of annotators that have shown the least
average agreement with all other annotators, while we maintain at least ten
annotators per language by the end of this round. The actual process is done
in multiple iterations: (S1) we measure the average agreement for each
annotator with every other annotator (this corresponds to the APIAA measure,
see later); (S2) if we still have more than 10 valid annotators and the lowest
average score is higher than in the previous iteration, we remove the lowest
one, and rerun S1. Table 3 shows the number of annotators at both the start
(Round 1) and end (Round 3) of our process for each language.
We measure the agreement between annotators using two metrics, average
pairwise inter-annotator agreement (APIAA), and average mean inter-annotator
agreement (AMIAA). Both of these use Spearman’s correlation ($\rho$) between
annotators scores, the only difference is how they are averaged. They are
computed as follows:
Languages: | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue
---|---|---|---|---|---|---|---|---|---|---|---|---
R1: Start | 13 | 12 | 14 | 12 | 13 | 10 | 11 | 12 | 12 | 12 | 11 | 13
R3: End | 11 | 10 | 13 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 11
Table 3: Number of human annotators. R1 = Annotation Round 1, R3 = Round 3.
$\displaystyle{1)\textsc{apiaa}}=\frac{2\sum_{i,j}\rho(s_{i},s_{j})}{N(N-1)}\hskip
17.07164pt{2)\textsc{amiaa}}=\frac{\sum_{i}\rho(s_{i},\mu_{i})}{N}\,\text{,
where:}\;\mu_{i}=\frac{\sum_{j,j\neq i}s_{j}}{N-1}$ (1)
where $\rho(s_{i},s_{j})$ is the Spearman’s correlation between annotators $i$
and $j$’s scores ($s_{i}$,$s_{j}$) for all pairs in the dataset, and $N$ is
the number of annotators. APIAA has been used widely as the standard measure
for inter-annotator agreement, including in the original SimLex paper Hill,
Reichart, and Korhonen (2015). It simply averages the pairwise Spearman’s
correlation between all annotators. On the other hand, AMIAA compares the
average Spearman’s correlation of one held-out annotator with the average of
all the other $N-1$ annotators, and then averages across all $N$ ‘held-out’
annotators. It smooths individual annotator effects and arguably serves as a
better upper bound than APIAA (Gerz et al., 2016; Vulić et al., 2017a;
Pilehvar et al., 2018, inter alia).
Languages: | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue
---|---|---|---|---|---|---|---|---|---|---|---|---
Nouns | 0.661 | 0.622 | 0.659 | 0.558 | 0.647 | 0.698 | 0.538 | 0.606 | 0.524 | 0.582 | 0.626 | 0.727
Adjectives | 0.757 | 0.698 | 0.823 | 0.695 | 0.721 | 0.741 | 0.683 | 0.699 | 0.625 | 0.64 | 0.658 | 0.785
Verbs | 0.694 | 0.604 | 0.707 | 0.58 | 0.644 | 0.691 | 0.615 | 0.593 | 0.555 | 0.588 | 0.631 | 0.76
Adverbs | 0.699 | 0.593 | 0.695 | 0.579 | 0.646 | 0.595 | 0.561 | 0.543 | 0.535 | 0.563 | 0.562 | 0.716
Overall | 0.68 | 0.619 | 0.698 | 0.583 | 0.646 | 0.697 | 0.572 | 0.609 | 0.53 | 0.576 | 0.623 | 0.733
Table 4: Average pairwise inter-annotator agreement (APIAA). A score of $0.6$ and above indicates strong agreement. Languages: | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue
---|---|---|---|---|---|---|---|---|---|---|---|---
Nouns | 0.757 | 0.747 | 0.766 | 0.696 | 0.766 | 0.809 | 0.68 | 0.717 | 0.657 | 0.71 | 0.725 | 0.804
Adjectives | 0.800 | 0.789 | 0.865 | 0.79 | 0.792 | 0.831 | 0.754 | 0.792 | 0.737 | 0.743 | 0.686 | 0.811
Verbs | 0.774 | 0.733 | 0.811 | 0.715 | 0.757 | 0.808 | 0.72 | 0.722 | 0.69 | 0.71 | 0.702 | 0.784
Adverbs | 0.749 | 0.693 | 0.777 | 0.697 | 0.748 | 0.729 | 0.645 | 0.655 | 0.608 | 0.671 | 0.623 | 0.716
Overall | 0.764 | 0.742 | 0.794 | 0.715 | 0.76 | 0.812 | 0.699 | 0.723 | 0.667 | 0.703 | 0.71 | 0.792
Table 5: Average mean inter-annotator agreement (AMIAA). A score of $0.6$ and
above indicates strong agreement.
We present the respective APIAA and AMIAA scores in Table 4 and Table 5 for
all part-of-speech subsets, as well as the agreement for the full datasets. As
reported in prior work Gerz et al. (2016); Vulić et al. (2017a), AMIAA scores
are typically higher than APIAA scores. Crucially, the results indicate
‘strong agreement’ (across all languages) using both measurements. The
languages with the highest annotator agreement were French (fra) and Yue
Chinese (yue), while Russian (rus) had the lowest overall IAA scores. These
scores, however, are still considered to be ‘moderately strong agreement’.
### 5.3 Data Analysis
Similarity Score Distributions. Across all languages, the average score (mean
$=1.61$, median$=1.1$) is on the lower side of the similarity scale. However,
looking closer at the scores of each language in Table 6, we indicate notable
differences in both the averages and the spread of scores. Notably, French has
the highest average of similarity scores (mean$=2.61$, median$=2.5$), while
Kiswahili has the lowest average (mean$=1.28$, median$=0.5$). Russian has the
lowest spread ($\sigma=1.37$), while Polish has the largest ($\sigma=1.62$).
All of the languages are strongly correlated with each other, as shown in
Figure 1, where all of the Spearman’s correlation coefficients are greater
than 0.6 for all language pairs. Languages that share the same language family
are highly correlated (e.g, cmn-yue, rus-pol, est-fin). In addition, we
observe high correlations between English and most other languages, as
expected. This is due to the effect of using English as the base/anchor
language to create the dataset. In simple words, if one translates to two
languages $L_{1}$ and $L_{2}$ starting from the same set of pairs in English,
it is higly likely that $L_{1}$ and $L_{2}$ will diverge from English in
different ways. Therefore, the similarity between $L_{1}$-eng and $L_{2}$-eng
is expected to be higher than between $L_{1}$-$L_{2}$, especially if $L_{1}$
and $L_{2}$ are typologically dissimilar languages (e.g., heb-cmn, see Figure
1). This phenomenon is well documented in related prior work (Leviant and
Reichart, 2015; Camacho-Collados et al., 2017; Mrkšić et al., 2017; Vulić,
Ponzetto, and Glavaš, 2019). While we acknowledge this as a slight artifact of
the dataset design, it would otherwise be impossible to construct a
semantically aligned and comprehensive dataset across a large number of
languages.
Lang: cmn cym eng est fin fra heb pol rus spa swa yue Interval $[0,1)$ 56.99
52.01 50.95 35.01 47.83 17.69 28.07 49.36 50.21 43.96 61.39 57.89 $[1,2)$ 8.74
19.54 17.06 30.67 21.35 20.39 35.86 17.32 22.40 22.35 11.86 7.84 $[2,3)$ 13.72
11.97 12.66 16.21 12.02 22.03 16.74 11.86 11.81 14.83 9.11 11.76 $[3,4)$ 11.60
8.32 8.16 10.22 10.17 17.64 8.47 8.95 8.10 9.38 7.10 12.98 $[4,5)$ 6.41 5.83
6.89 6.25 5.61 12.55 6.62 7.57 5.88 6.78 6.30 6.89 $[5,6]$ 2.54 2.33 4.29 1.64
2.97 9.64 4.24 4.93 1.59 2.70 4.24 2.65
Table 6: Fine-grained distribution of concept pairs over different rating
intervals in each Multi-SimLex language, reported as percentages. The total
number of concept pairs in each dataset is 1,888. Figure 1: Spearman’s
correlation coefficient ($\rho$) of the similarity scores for all languages in
Multi-SimLex.
We also report differences in the distribution of the frequency of words among
the languages in Multi-SimLex. Figure 2 shows six example languages, where
each bar segment shows the proportion of words in each language that occur in
the given frequency range. For example, the 10K-20K segment of the bars
represents the proportion of words in the dataset that occur in the list of
most frequent words between the frequency rank of 10,000 and 20,000 in that
language; likewise with other intervals. Frequency lists for the presented
languages are derived from Wikipedia and Common Crawl corpora.444Frequency
lists were obtained from fastText word vectors which are sorted by frequency:
https://fasttext.cc/docs/en/crawl-vectors.html While many concept pairs are
direct or approximate translations of English pairs, we can see that the
frequency distribution does vary across different languages, and is also
related to inherent language properties. For instance, in Finnish and Russian,
while we use infinitive forms of all verbs, conjugated verb inflections are
often more frequent in raw corpora than the corresponding infinitive forms.
The variance can also be partially explained by the difference in monolingual
corpora size used to derive the frequency rankings in the first place:
absolute vocabulary sizes are expected to fluctuate across different
languages. However, it is also important to note that the datasets also
contain subsets of lower-frequency and rare words, which can be used for rare
word evaluations in multiple languages, in the spirit of Pilehvar:2018emnlp’s
English rare word dataset.
Figure 2: A distribution over different frequency ranges for words from Multi-
SimLex datasets for selected languages. Multi-word expressions are excluded
from the analysis.
Cross-Linguistic Differences. Table 7 shows some examples of average
similarity scores of English, Spanish, Kiswahili and Welsh concept pairs.
Remember that the scores range from 0 to 6: the higher the score, the more
similar the participants found the concepts in the pair. The examples from
Table 7 show evidence of both the stability of average similarity scores
across languages (_unlikely – friendly_ , _book – literature_ , and _vanish –
disappear_), as well as language-specific differences (_care – caution_). Some
differences in similarity scores seem to group languages into clusters. For
example, the word pair _regular – average_ has an average similarity score of
4.0 and 4.1 in English and Spanish, respectively, whereas in Kiswahili and
Welsh the average similarity score of this pair is 0.5 and 0.8. We analyze
this phenomenon in more detail in §5.4.
Word Pair | POS | eng | spa | swa | cym
---|---|---|---|---|---
Similar average rating | | | | |
unlikely – friendly | ADV | 0 | 0 | 0 | 0
book – literature | N | 2.5 | 2.3 | 2.1 | 2.3
vanish – disappear | V | 5.2 | 5.3 | 5.5 | 5.3
Different average rating | | | | |
regular – average | ADJ | 4 | 4.1 | 0.5 | 0.8
care – caution | N | 4.1 | 5.7 | 0.2 | 3.1
One language higher | | | | |
large – big | ADJ | 5.9 | 2.7 | 3.8 | 3.8
bank – seat | N | 0 | 5.1 | 0 | 0.1
sunset - evening | N | 1.6 | 1.5 | 5.5 | 2.8
purely – completely | ADV | 2.3 | 2.3 | 1.1 | 5.4
One language lower | | | | |
woman – wife | N | 0.9 | 2.9 | 4.1 | 4.8
amazingly – fantastically | ADV | 5.1 | 0.4 | 4.1 | 4.1
wonderful – terrific | ADJ | 5.3 | 5.4 | 0.9 | 5.7
promise – swear | V | 4.8 | 5.3 | 4.3 | 0
Table 7: Examples of concept pairs with their similarity scores from four
languages. For brevity, only the original English concept pair is included,
but note that the pair is translated to all target languages, see §5.1.
There are also examples for each of the four languages having a notably higher
or lower similarity score for the same concept pair than the three other
languages. For example, _large – big_ in English has an average similarity
score of 5.9, whereas Spanish, Kiswahili and Welsh speakers rate the closest
concept pair in their native language to have a similarity score between 2.7
and 3.8. What is more, _woman – wife_ receives an average similarity of 0.9 in
English, 2.9 in Spanish, and greater than 4.0 in Kiswahili and Welsh. The
examples from Spanish include _banco – asiento_ (_bank – seat_) which receives
an average similarity score 5.1, while in the other three languages the
similarity score for this word pair does not exceed 0.1. At the same time, the
average similarity score of _espantosamente – fantásticamente_ (_amazingly –
fantastically_) is much lower in Spanish (0.4) than in other languages (4.1 –
5.1). In Kiswahili, an example of a word pair with a higher similarity score
than the rest would be _machweo – jioni_ (_sunset – evening_), having an
average score of 5.5, while the other languages receive 2.8 or less, and a
notably lower similarity score is given to _wa ajabu - mkubwa sana_
(_wonderful – terrific_), getting 0.9, while the other languages receive 5.3
or more. Welsh examples include _yn llwyr - yn gyfan gwbl_ (_purely –
completely_), which scores 5.4 among Welsh speakers but 2.3 or less in other
languages, while _addo – tyngu_ (_promise – swear_) is rated as 0 by all Welsh
annotators, but in the other three languages 4.3 or more on average.
There can be several explanations for the differences in similarity scores
across languages, including but not limited to cultural context, polysemy,
metonymy, translation, regional and generational differences, and most
commonly, the fact that words and meanings do not exactly map onto each other
across languages. For example, it is likely that the other three languages do
not have two separate words for describing the concepts in the concept pair:
_big – large_ , and the translators had to opt for similar lexical items that
were more distant in meaning, explaining why in English the concept pair
received a much higher average similarity score than in other languages. A
similar issue related to the mapping problem across languages arose in the
Welsh concept pair _yn llwye – yn gyfan gwbl_ , where Welsh speakers agreed
that the two concepts are very similar. When asked, bilingual speakers
considered the two Welsh concepts more similar than English equivalents
_purely – completely_ , potentially explaining why a higher average similarity
score was reached in Welsh. The example of _woman – wife_ can illustrate
cultural differences or another translation-related issue where the word
‘wife’ did not exist in some languages (for example, Estonian), and therefore
had to be described using other words, affecting the comparability of the
similarity scores. This was also the case with the _football – soccer_ concept
pair. The pair _bank – seat_ demonstrates the effect of the polysemy mismatch
across languages: while ‘bank’ has two different meanings in English, neither
of them is similar to the word ‘seat’, but in Spanish, ‘ _banco_ ’ can mean
‘bank’, but it can also mean ‘bench’. Quite naturally, Spanish speakers gave
the pair _banco – asiento_ a higher similarity score than the speakers of
languages where this polysemy did not occur.
An example of metonymy affecting the average similarity score can be seen in
the Kiswahili version of the word pair: _sunset – evening_ (_machweo –
jioni_). The average similarity score for this pair is much higher in
Kiswahili, likely because the word ‘sunset’ can act as a metonym of ‘evening’.
The low similarity score of _wonderful – terrific_ in Kiswahili (_wa ajabu -
mkubwa sana_) can be explained by the fact that while ‘ _mkubwa sana_ ’ can be
used as ‘terrific’ in Kiswahili, it technically means ‘very big’, adding to
the examples of translation- and mapping-related effects. The word pair
_amazingly – fantastically_ (_espantosamente – fantásticamente_) brings out
another translation-related problem: the accuracy of the translation. While ‘
_espantosamente_ ’ could arguably be translated to ‘amazingly’, more common
meanings include: ‘frightfully’, ‘terrifyingly’, and ‘shockingly’, explaining
why the average similarity score differs from the rest of the languages.
Another problem was brought out by _addo – tyngu_ (_promise – swear_) in
Welsh, where the ‘ _tyngu_ ’ may not have been a commonly used or even a known
word choice for annotators, pointing out potential regional or generational
differences in language use.
Language | Word Pair | POS | Rating all participants agree with
---|---|---|---
eng | trial – test | N | 4-5
swa | archbishop – bishop | N | 4-5
spa, cym | start – begin | V | 5-6
eng | smart – intelligent | ADJ | 5-6
eng, spa | quick – rapid | ADJ | 5-6
spa | circumstance – situation | N | 5-6
cym | football – soccer | N | 5-6
swa | football – soccer | N | 6
swa | pause – wait | V | 6
swa | money – cash | N | 6
cym | friend – buddy | N | 6
Table 8: Examples of concept pairs with their similarity scores from four
languages where all participants show strong agreement in their rating.
Table 8 presents examples of concept pairs from English, Spanish, Kiswahili,
and Welsh on which the participants agreed the most. For example, in English
all participants rated the similarity of _trial – test_ to be 4 or 5. In
Spanish and Welsh, all participants rated _start – begin_ to correspond to a
score of 5 or 6. In Kiswahili, _money – cash_ received a similarity rating of
6 from every participant. While there are numerous examples of concept pairs
in these languages where the participants agreed on a similarity score of 4 or
higher, it is worth noting that none of these languages had a single pair
where all participants agreed on either 1-2, 2-3, or 3-4 similarity rating.
Interestingly, in English all pairs where all the participants agreed on a 5-6
similarity score were adjectives.
### 5.4 Effect of Language Affinity on Similarity Scores
Based on the analysis in Figure 1 and inspecting the anecdotal examples in the
previous section, it is evident that the correlation between similarity scores
across languages is not random. To corroborate this intuition, we visualize
the vectors of similarity scores for each single language by reducing their
dimensionality to 2 via Principal Component Analysis (Pearson, 1901). The
resulting scatter plot in Figure 3 reveals that languages from the same family
or branch have similar patterns in the scores. In particular, Russian and
Polish (both Slavic), Finnish and Estonian (both Uralic), Cantonese and
Mandarin Chinese (both Sinitic), and Spanish and French (both Romance) are all
neighbors.
Figure 3: PCA of the language vectors resulting from the concatenation of
similarity judgments for all pairs.
In order to quantify exactly the effect of language affinity on the similarity
scores, we run correlation analyses between these and language features. In
particular, we extract feature vectors from URIEL (Littell et al., 2017), a
massively multilingual typological database that collects and normalizes
information compiled by grammarians and field linguists about the world’s
languages. In particular, we focus on information about geography (the areas
where the language speakers are concentrated), family (the phylogenetic tree
each language belongs to), and typology (including syntax, phonological
inventory, and phonology).555For the extraction of these features, we employed
lang2vec: github.com/antonisa/lang2vec Moreover, we consider typological
representations of languages that are not manually crafted by experts, but
rather learned from texts. Malaviya, Neubig, and Littell (2017) proposed to
construct such representations by training language-identifying vectors end-
to-end as part of neural machine translation models.
The vector for similarity judgments and the vector of linguistic features for
a given language have different dimensionality. Hence, we first construct a
distance matrix for each vector space, such that both columns and rows are
language indices, and each cell value is the cosine distance between the
vectors of the corresponding language pair. Given a set of L languages, each
resulting matrix $S$ has dimensionality of $\mathbb{R}^{|L|\times|L|}$ and is
symmetrical. To estimate the correlation between the matrix for similarity
judgments and each of the matrices for linguistic features, we run a Mantel
test (Mantel, 1967), a non-parametric statistical test based on matrix
permutations that takes into account inter-dependencies among pairwise
distances.
The results of the Mantel test reported in Table 3 show that there exist
statistically significant correlations between similarity judgments and
geography, family, and syntax, given that $p<0.05$ and $z>1.96$. The
correlation coefficient is particularly strong for geography ($r=0.647$) and
syntax ($r=0.649$). The former result is intuitive, because languages in
contact easily borrow and loan lexical units, and cultural interactions may
result in similar cognitive categorizations. The result for syntax, instead,
cannot be explained so easily, as formal properties of language do not affect
lexical semantics. Instead, we conjecture that, while no causal relation is
present, both syntactic features and similarity judgments might be linked to a
common explanatory variable (such as geography). In fact, several syntactic
properties are not uniformly spread across the globe. For instance, verbs with
Verb–Object–Subject word order are mostly concentrated in Oceania (Dryer,
2013). In turn, geographical proximity leads to similar judgment patterns, as
mentioned above. On the other hand, we find no correlation with phonology and
inventory, as expected, nor with the bottom-up typological features from
Malaviya, Neubig, and Littell (2017).
Features | Dimension | Mantel r | Mantel p | Mantel z
---|---|---|---|---
geography | 299 | 0.647 | 0.007* | 3.443
family | 3718 | 0.329 | 0.023* | 2.711
syntax | 103 | 0.649 | 0.007* | 3.787
inventory | 158 | 0.155 | 0.459 | 0.782
phonology | 28 | 0.397 | 0.046 | 1.943
Malaviya, Neubig, and Littell (2017) | 512 | -0.431 | 0.264 | -1.235
Table 9: Mantel test on the correlation between similarity judgments from
Multi-SimLex and linguistic features from typological databases.
## 6 Cross-Lingual Multi-SimLex Datasets
A crucial advantage of having semantically aligned monolingual datasets across
different languages is the potential to create cross-lingual semantic
similarity datasets. Such datasets allow for probing the quality of cross-
lingual representation learning algorithms Camacho-Collados et al. (2017);
Conneau et al. (2018a); Chen and Cardie (2018); Doval et al. (2018); Ruder,
Vulić, and Søgaard (2019); Conneau and Lample (2019); Ruder, Søgaard, and
Vulić (2019) as an intrinsic evaluation task. However, the cross-lingual
datasets previous work relied upon Camacho-Collados et al. (2017) were limited
to a homogeneous set of high-resource languages (e.g., English, German,
Italian, Spanish) and a small number of concept pairs (all less than 1K
pairs). We address both problems by 1) using a typologically more diverse
language sample, and 2) relying on a substantially larger English dataset as a
source for the cross-lingual datasets: 1,888 pairs in this work versus 500
pairs in the work of Camacho:2017semeval. As a result, each of our cross-
lingual datasets contains a substantially larger number of concept pairs, as
shown in Table 11. The cross-lingual Multi-Simlex datasets are constructed
automatically, leveraging word pair translations and annotations collected in
all 12 languages. This yields a total of 66 cross-lingual datasets, one for
each possible combination of languages. Table 11 provides the final number of
concept pairs, which lie between 2,031 and 3,480 pairs for each cross-lingual
dataset, whereas Table 10 shows some sample pairs with their corresponding
similarity scores.
The automatic creation and verification of cross-lingual datasets closely
follows the procedure first outlined by Camacho:2015acl and later adopted by
Camacho:2017semeval (for semantic similarity) and Vulic:2019acl (for graded
lexical entailment). First, given two languages, we intersect their aligned
concept pairs obtained through translation. For instance, starting from the
aligned pairs attroupement – foule in French and rahvasumm – rahvahulk in
Estonian, we construct two cross-lingual pairs attroupement – rahvaluk and
rahvasumm – foule. The scores of cross-lingual pairs are then computed as
averages of the two corresponding monolingual scores. Finally, in order to
filter out concept pairs whose semantic meaning was not preserved during this
operation, we retain only cross-lingual pairs for which the corresponding
monolingual scores $(s_{s},s_{t})$ differ at most by one fifth of the full
scale (i.e., $\mid s_{s}-s_{t}\mid\leq 1.2$). This heuristic mitigates the
noise due to cross-lingual semantic shifts Camacho-Collados et al. (2017);
Vulić, Ponzetto, and Glavaš (2019). We refer the reader to the work of
Camacho:2015acl for a detailed technical description of the procedure.
Pair | Concept-1 | Concept-2 | Score | Pair | Concept-1 | Concept-2 | Score
---|---|---|---|---|---|---|---
cym-eng | rhyddid | liberty | 5.37 | cmn-est | 可能 | optimistlikult | 0.83
cym-pol | plentynaidd | niemądry | 2.15 | fin-swa | psykologia | sayansi | 2.20
swa-eng | kutimiza | accomplish | 5.24 | eng-fra | normally | quotidiennement | 2.41
cmn-fra | 有弹性 | flexible | 4.08 | fin-spa | auto | bicicleta | 0.85
fin-spa | tietämättömyys | inteligencia | 0.55 | cmn-yue | 使灰心 | 使气馁 | 4.78
spa-fra | ganador | candidat | 2.15 | cym-swa | sefyllfa | mazingira | 1.90
est-yue | takso | 巴士 | 2.08 | est-spa | armee | legión | 3.25
eng-fin | orange | sitrushedelmä | 3.43 | fin-est | halveksuva | põlglik | 5.55
spa-pol | palabra | wskazówka | 0.55 | cmn-cym | 学生 | disgybl | 4.45
pol-swa | prawdopodobnie | uwezekano | 4.05 | pol-eng | grawitacja | meteor | 0.27
Table 10: Example concept pairs with their scores from a selection of cross-
lingual Multi-SimLex datasets.
To assess the quality of the resulting cross-lingual datasets, we have
conducted a verification experiment similar to Vulic:2019acl. We randomly
sampled 300 concept pairs in the English-Spanish, English-French, and English-
Mandarin cross-lingual datasets. Subsequently, we asked bilingual native
speakers to provide similarity judgments of each pair. The Spearman’s
correlation score $\rho$ between automatically induced and manually collected
ratings achieves $\rho\geq 0.90$ on all samples, which confirms the viability
of the automatic construction procedure.
| cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue
---|---|---|---|---|---|---|---|---|---|---|---|---
cmn | 1,888 | – | – | – | – | – | – | – | – | – | – | –
cym | 3,085 | 1,888 | – | – | – | – | – | – | – | – | – | –
eng | 3,151 | 3,380 | 1,888 | – | – | – | – | – | – | – | – | –
est | 3,188 | 3,305 | 3,364 | 1,888 | – | – | – | – | – | – | – | –
fin | 3,137 | 3,274 | 3,352 | 3,386 | 1,888 | – | – | – | – | – | – | –
fra | 2,243 | 2,301 | 2,284 | 2,787 | 2,682 | 1,888 | – | – | – | – | – | –
heb | 3,056 | 3,209 | 3,274 | 3,358 | 3,243 | 2,903 | 1,888 | – | – | – | – | –
pol | 3,009 | 3,175 | 3,274 | 3,310 | 3,294 | 2,379 | 3,201 | 1,888 | – | – | – | –
rus | 3,032 | 3,196 | 3,222 | 3,339 | 3,257 | 2,219 | 3,226 | 3,209 | 1,888 | – | – | –
spa | 3,116 | 3,205 | 3,318 | 3,312 | 3,256 | 2,645 | 3,256 | 3,250 | 3,189 | 1,888 | – | –
swa | 2,807 | 2,926 | 2,828 | 2,845 | 2,900 | 2,031 | 2,775 | 2,819 | 2,855 | 2,811 | 1,888 | –
yue | 3,480 | 3,062 | 3,099 | 3,080 | 3,063 | 2,313 | 3,005 | 2,950 | 2,966 | 3,053 | 2,821 | 1,888
Table 11: The sizes of all monolingual (main diagonal) and cross-lingual
datasets.
(a) Rating distribution
(b) Distribution over POS classes
Figure 4: (a) Rating distribution and (b) distribution of pairs over the four
POS classes in cross-lingual Multi-SimLex datasets averaged across each of the
66 language pairs ($y$ axes plot percentages as the total number of concept
pairs varies across different cross-lingual datasets). Minimum and maximum
percentages for each rating interval and POS class are also plotted.
Score and Class Distributions. The summary of score and class distributions
across all 66 cross-lingual datasets are provided in Figure 4(a) and Figure
4(b), respectively. First, it is obvious that the distribution over the four
POS classes largely adheres to that of the original monolingual Multi-SimLex
datasets, and that the variance is quite low: e.g., the eng-fra dataset
contains the lowest proportion of nouns (49.21%) and the highest proportion of
verbs (27.1%), adjectives (15.28%), and adverbs (8.41%). On the other hand,
the distribution over similarity intervals in Figure 4(a) shows a much greater
variance. This is again expected as this pattern resurfaces in monolingual
datasets (see Table 6). It is also evident that the data are skewed towards
lower-similarity concept pairs. However, due to the joint size of all cross-
lingual datasets (see Table 11), even the least represented intervals contain
a substantial number of concept pairs. For instance, the rus-yue dataset
contains the least highly similar concept pairs (in the interval $[4,6]$) of
all 66 cross-lingual datasets. Nonetheless, the absolute number of pairs (138)
in that interval for rus-yue is still substantial. If needed, this makes it
possible to create smaller datasets which are balanced across the similarity
spectra through sub-sampling.
## 7 Monolingual Evaluation of Representation Learning Models
After the numerical and qualitative analyses of the Multi-SimLex datasets
provided in §§ 5.3–5.4, we now benchmark a series of representation learning
models on the new evaluation data. We evaluate standard static word embedding
algorithms such as fastText Bojanowski et al. (2017), as well as a range of
more recent text encoders pretrained on language modeling such as multilingual
BERT (Devlin et al., 2019). These experiments provide strong baseline scores
on the new Multi-SimLex datasets and offer a first large-scale analysis of
pretrained encoders on word-level semantic similarity across diverse
languages. In addition, the experiments now enabled by Multi-SimLex aim to
answer several important questions. (Q1) Is it viable to extract high-quality
word-level representations from pretrained encoders receiving subword-level
tokens as input? Are such representations competitive with standard static
word-level embeddings? (Q2) What are the implications of monolingual
pretraining versus (massively) multilingual pretraining for performance? (Q3)
Do lightweight unsupervised post-processing techniques improve word
representations consistently across different languages? (Q4) Can we
effectively transfer available external lexical knowledge from resource-rich
languages to resource-lean languages in order to learn word representations
that distinguish between true similarity and conceptual relatedness (see the
discussion in §2.3)?
### 7.1 Models in Comparison
Static Word Embeddings in Different Languages. First, we evaluate a standard
method for inducing non-contextualized (i.e., static) word embeddings across a
plethora of different languages: fastText (ft) vectors Bojanowski et al.
(2017) are currently the most popular and robust choice given 1) the
availability of pretrained vectors in a large number of languages Grave et al.
(2018) trained on large Common Crawl (CC) plus Wikipedia (Wiki) data, and 2)
their superior performance across a range of NLP tasks Mikolov et al. (2018).
In fact, fastText is an extension of the standard word-level CBOW and skip-
gram word2vec models Mikolov et al. (2013) that takes into account subword-
level information, i.e. the contituent character n-grams of each word Zhu,
Vulić, and Korhonen (2019). For this reason, fastText is also more suited for
modeling rare words and morphologically rich languages.666We have also trained
standard word-level CBOW and skip-gram with negative sampling (SGNS) on full
Wikipedia dumps for several languages, but our preliminary experiments have
verified that they under-perform compared to fastText. This finding is
consistent with other recent studies demonstrating the usefulness of subword-
level information Vania and Lopez (2017); Mikolov et al. (2018); Zhu, Vulić,
and Korhonen (2019); Zhu et al. (2019). Therefore, we do not report the
results with CBOW and SGNS for brevity.
We rely on $300$-dimensional ft word vectors trained on CC+Wiki and available
online for 157 languages.777https://fasttext.cc/docs/en/crawl-vectors.html The
word vectors for all languages are obtained by CBOW with position-weights,
with character n-grams of length 5, a window of size 5, 10 negative examples,
and 10 training epochs. We also probe another (older) collection of ft
vectors, pretrained on full Wikipedia dumps of each
language.888https://fasttext.cc/docs/en/pretrained-vectors.html. The vectors
are 300-dimensional, trained with the skip-gram objective for 5 epochs, with 5
negative examples, a window size set to 5, and relying on all character
n-grams from length 3 to 6. Following prior work, we trim the vocabularies for
all languages to the 200K most frequent words and compute representations for
multi-word expressions by averaging the vectors of their constituent words.
Unsupervised Post-Processing. Further, we consider a variety of unsupervised
post-processing steps that can be applied post-training on top of any
pretrained input word embedding space without any external lexical semantic
resource. So far, the usefulness of such methods has been verified only on the
English language through benchmarks for lexical semantics and sentence-level
tasks Mu, Bhat, and Viswanath (2018). In this paper, we assess if unsupervised
post-processing is beneficial also in other languages. To this end, we apply
the following post-hoc transformations on the initial word embeddings:
1) Mean centering (mc) is applied after unit length normalization to ensure
that all vectors have a zero mean, and is commonly applied in data mining and
analysis Bro and Smilde (2003); van den Berg et al. (2006).
2) All-but-the-top (abtt) Mu, Bhat, and Viswanath (2018); Tang, Mousavi, and
de Sa (2019) eliminates the common mean vector and a few top dominating
directions (according to PCA) from the input distributional word vectors,
since they do not contribute towards distinguishing the actual semantic
meaning of different words. The method contains a single (tunable) hyper-
parameter $dd_{A}$ which denotes the number of the dominating directions to
remove from the initial representations. Previous work has verified the
usefulness of abtt in several English lexical semantic tasks such as semantic
similarity, word analogies, and concept categorization, as well as in
sentence-level text classification tasks Mu, Bhat, and Viswanath (2018).
3) uncovec Artetxe et al. (2018) adjusts the similarity order of an arbitrary
input word embedding space, and can emphasize either syntactic or semantic
information in the transformed vectors. In short, it transforms the input
space $\bm{X}$ into an adjusted space $\bm{X}\bm{W}_{\alpha}$ through a linear
map $\bm{W}_{\alpha}$ controlled by a single hyper-parameter $\alpha$. The
$n^{\text{th}}$-order similarity transformation of the input word vector space
$\bm{X}$ (for which $n=1$) can be obtained as
$\bm{M}_{n}(\bm{X})=\bm{M}_{1}(\bm{X}\bm{W}_{(n-1)/2})$, with
$\bm{W}_{\alpha}=\bm{Q}\bm{\Gamma}^{\alpha}$, where $\bm{Q}$ and $\bm{\Gamma}$
are the matrices obtained via eigendecomposition of
$\bm{X}^{T}\bm{X}=\bm{Q}\bm{\Gamma}\bm{Q}^{T}$. $\bm{\Gamma}$ is a diagonal
matrix containing eigenvalues of $\bm{X}^{T}\bm{X}$; $\bm{Q}$ is an orthogonal
matrix with eigenvectors of $\bm{X}^{T}\bm{X}$ as columns. While the
motivation for the uncovec methods does originate from adjusting discrete
similarity orders, note that $\alpha$ is in fact a continuous real-valued
hyper-parameter which can be carefully tuned. For more technical details we
refer the reader to the original work of Artetxe et al. (2018).
As mentioned, all post-processing methods can be seen as unsupervised
retrofitting methods that, given an arbitrary input vector space $\bm{X}$,
produce a perturbed/transformed output vector space $\bm{X}^{\prime}$, but
unlike common retrofitting methods Faruqui et al. (2015); Mrkšić et al.
(2017), the perturbation is completely unsupervised (i.e., self-contained) and
does not inject any external (semantic similarity oriented) knowledge into the
vector space. Note that different perturbations can also be stacked: e.g., we
can apply uncovec and then use abtt on top the output uncovec vectors. When
using uncovec and abtt we always length-normalize and mean-center the data
first (i.e., we apply the simple mc normalization). Finally, we tune the two
hyper-parameters $d_{A}$ (for abtt) and $\alpha$ (uncovec) on the English
Multi-SimLex and use the same values on the datasets of all other languages;
we report results with $dd_{A}=3$ or $dd_{A}=10$, and $\alpha=-0.3$.
Contextualized Word Embeddings. We also evaluate the capacity of unsupervised
pretraining architectures based on language modeling objectives to reason over
lexical semantic similarity. To the best of our knowledge, our article is the
first study performing such analyses. State-of-the-art models such as bert
Devlin et al. (2019), xlm Conneau and Lample (2019), or roberta Liu et al.
(2019b) are typically very deep neural networks based on the Transformer
architecture Vaswani et al. (2017). They receive subword-level tokens as
inputs (such as WordPieces Schuster and Nakajima (2012)) to tackle data
sparsity. In output, they return contextualized embeddings, dynamic
representations for words in context.
To represent words or multi-word expressions through a pretrained model, we
follow prior work Liu et al. (2019a) and compute an input item’s
representation by 1) feeding it to a pretrained model in isolation; then 2)
averaging the $H$ last hidden representations for each of the item’s
constituent subwords; and then finally 3) averaging the resulting subword
representations to produce the final $d$-dimensional representation, where $d$
is the embedding and hidden-layer dimensionality (e.g., $d=768$ with bert). We
opt for this approach due to its proven viability and simplicity Liu et al.
(2019a), as it does not require any additional corpora to condition the
induction of contextualized embeddings.999We also tested another encoding
method where we fed pairs instead of single words/concepts into the pretrained
encoder. The rationale is that the other concept in the pair can be used as
disambiguation signal. However, this method consistently led to sub-par
performance across all experimental runs. Other ways to extract the
representations from pretrained models Aldarmaki and Diab (2019); Wu et al.
(2019); Cao, Kitaev, and Klein (2020) are beyond the scope of this work, and
we will experiment with them in the future.
In other words, we treat each pretrained encoder enc as a black-box function
to encode a single word or a multi-word expression $x$ in each language into a
$d$-dimensional contextualized representation
$\mathbf{x}_{\textsc{enc}}\in\mathbb{R}^{d}=\textsc{enc}(x)$ (e.g., $d=768$
with bert). As multilingual pretrained encoders, we experiment with the
multilingual bert model (m-bert) Devlin et al. (2019) and xlm (Conneau and
Lample, 2019). m-bert is pretrained on monolingual Wikipedia corpora of 102
languages (comprising all Multi-SimLex languages) with a 12-layer Transformer
network, and yields $768$-dimensional representations. Since the concept pairs
in Multi-SimLex are lowercased, we use the uncased version of
m-bert.101010https://github.com/google-
research/bert/blob/master/multilingual.md m-bert comprises all Multi-SimLex
languages, and its evident ability to perform cross-lingual transfer Pires,
Schlinger, and Garrette (2019); Wu and Dredze (2019); Wang et al. (2020) also
makes it a convenient baseline model for cross-lingual experiments later in
§8. The second multilingual model we consider,
xlm-100,111111https://github.com/facebookresearch/XLM is pretrained on
Wikipedia dumps of 100 languages, and encodes each concept into a
$1,280$-dimensional representation. In contrast to m-bert, xlm-100 drops the
next-sentence prediction objective and adds a cross-lingual masked language
modeling objective. For both encoders, the representations of each concept are
computed as averages over the last $H=4$ hidden layers in all experiments, as
suggested by Wu:2019arxiv.121212In our preliminary experiments on several
language pairs, we have also verified that this choice is superior to: a)
using the output of only the last hidden layer (i.e., $H=1$) and b) averaging
over all hidden layers (i.e., $H=12$ for the bert-base architecture).
Likewise, using the special prepended ‘[CLS]’ token rather than the
constituent sub-words to encode a concept also led to much worse performance
across the board.
Besides m-bert and xlm, covering multiple languages, we also analyze the
performance of “language-specific” bert and xlm models for the languages where
they are available: Finnish, Spanish, English, Mandarin Chinese, and French.
The main goal of this comparison is to study the differences in performance
between multilingual “one-size-fits-all” encoders and language-specific
encoders. For all experiments, we rely on the pretrained models released in
the Transformers repository Wolf et al.
(2019).131313github.com/huggingface/transformers. The full list of currently
supported pretrained encoders is available here: huggingface.co/models.
Unsupervised post-processing steps devised for static word embeddings (i.e.,
mean-centering, abtt, uncovec) can also be applied on top of contextualized
embeddings if we predefine a vocabulary of word types $V$ that will be
represented in a word vector space $\mathbf{X}$. We construct such $V$ for
each language as the intersection of word types covered by the corresponding
CC+Wiki fastText vectors and the (single-word or multi-word) expressions
appearing in the corresponding Multi-SimLex dataset.
Finally, note that it is not feasible to evaluate a full range of available
pretrained encoders within the scope of this work. Our main intention is to
provide the first set of baseline results on Multi-SimLex by benchmarking a
sample of most popular encoders, at the same time also investigating other
important questions such as performance of static versus contextualized word
embeddings, or multilingual versus language-specific pretraining. Another
purpose of the experiments is to outline the wide potential and applicability
of the Multi-SimLex datasets for multilingual and cross-lingual representation
learning evaluation.
Languages: cmn cym eng est fin fra heb pol rus spa swa yue fastText (CC+Wiki)
(272) (151) (12) (319) (347) (43) (66) (326) (291) (46) (222) (–) (1) ft:init
.534 .363 .528 .469 .607 .578 .450 .405 .422 .511 .439 – (2) ft:+mc .539 .393
.535 .473 .621 .584 .480 .412 .424 .516 .469 – (3) ft:+abtt (-3) .557 .389
.536 .495 .642 .610 .501 .427 .459 .523 .473 – (4) ft:+abtt (-10) .583 .384
.551 .476 .651 .623 .503 .455 .500 .542 .462 – (5) ft:+uncovec .572 .387 .550
.465 .642 .595 .501 .435 .437 .525 .437 – (1)+(2)+(5)+(3) .574 .386 .549 .476
.655 .604 .503 .442 .452 .528 .432 – (1)+(2)+(5)+(4) .577 .376 .542 .455 .652
.613 .510 .466 .491 .540 .424 – fastText (Wiki) (429) (282) (6) (343) (345)
(73) (62) (354) (343) (57) (379) (677) (1) ft:init .315 .318 .436 .400 .575
.444 .428 .370 .359 .432 .332 .376 (2) ft:+mc .373 .337 .445 .404 .583 .463
.447 .383 .378 .447 .373 .427 (3) ft:+abtt (-3) .459 .343 .453 .404 .584 .487
.447 .387 .394 .456 .423 .429 (4) ft:+abtt (-10) .496 .323 .460 .385 .581 .494
.460 .401 .400 .477 .406 .399 (5) ft:+uncovec .518 .328 .469 .375 .568 .483
.449 .389 .387 .469 .386 .394 (1)+(2)+(5)+(3) .526 .323 .470 .369 .564 .495
.448 .392 .392 .473 .388 .388 (1)+(2)+(5)+(4) .526 .307 .471 .355 .548 .495
.450 .394 .394 .476 .382 .396 m-bert (0) (0) (0) (0) (0) (0) (0) (0) (0) (0)
(0) (0) (1) m-bert:init .408 .033 .138 .085 .162 .115 .104 .069 .085 .145 .125
.404 (2) m-bert:+mc .458 .044 .256 .122 .173 .183 .128 .097 .123 .203 .128
.469 (3) m-bert:+abtt (-3) .487 .056 .321 .137 .200 .287 .144 .126 .197 .299
.135 .492 (4) m-bert:+abtt (-10) .456 .056 .329 .122 .164 .306 .121 .126 .183
.315 .136 .467 (5) m-bert:+uncovec .464 .063 .317 .144 .213 .288 .164 .144
.198 .287 .143 .464 (1)+(2)+(5)+(3) .464 .083 .326 .130 .201 .304 .149 .122
.199 .295 .148 .456 (1)+(2)+(5)+(4) .444 .086 .326 .112 .179 .305 .135 .127
.187 .285 .119 .447
Table 12: A summary of results (Spearman’s $\rho$ correlation scores) on the
full monolingual Multi-SimLex datasets for 12 languages. We benchmark fastText
word embeddings trained on two different corpora (CC+Wiki and only Wiki) as
well the multilingual m-bert model (see §7.1). Results with the initial word
vectors are reported (i.e., without any unsupervised post-processing), as well
as with different unsupervised post-processing methods, described in §7.1. The
language codes are provided in Table 1. The numbers in the parentheses (gray
rows) refer to the number of OOV concepts excluded from the computation. The
highest scores for each language and per model are in bold.
### 7.2 Results and Discussion
The results we report are Spearman’s $\rho$ coefficients of the correlation
between the ranks derived from the scores of the evaluated models and the
human scores provided in each Multi-SimLex dataset. The main results with
static and contextualized word vectors for all test languages are summarized
in Table 12. The scores reveal several interesting patterns, and also pinpoint
the main challenges for future work.
State-of-the-Art Representation Models. The absolute scores of CC+Wiki ft,
Wiki ft, and m-bert are not directly comparable, because these models have
different coverage. In particular, Multi-SimLex contains some out-of-
vocabulary (OOV) words whose static ft embeddings are not available.141414We
acknowledge that it is possible to approximate word-level representations of
OOVs with ft by summing the constituent n-gram embeddings as proposed by
Bojanowski:2017tacl. However, we do not perform this step as the resulting
embeddings are typically of much lower quality than non-OOV embeddings Zhu,
Vulić, and Korhonen (2019). On the other hand, m-bert has perfect coverage. A
general comparison between CC+Wiki and Wiki ft vectors, however, supports the
intuition that larger corpora (such as CC+Wiki) yield higher correlations.
Another finding is that a single massively multilingual model such as m-bert
cannot produce semantically rich word-level representations. Whether this
actually happens because the training objective is different—or because the
need to represent 100+ languages reduces its language-specific capacity—is
investigated further below.
Languages: cmn cym eng est fin fra heb pol rus spa swa yue fastText (CC+Wiki)
ft:init nouns (1,051) .561 .497 .592 .627 .709 .641 .560 .538 .526 .583 .544
.426 verbs (469) .511 .265 .408 .379 .527 .551 .458 .384 .464 .499 .391 .252
adj (245) .448 .338 .564 .401 .546 .616 .467 .284 .349 .401 .344 .288 adv
(123) .622 .187 .482 .378 .547 .648 .491 .266 .514 .423 .172 .103 fastText
(CC+Wiki) ft:+abtt (-3) nouns .601 .512 .599 .621 .730 .653 .592 .585 .578
.605 .553 .431 verbs .583 .305 .454 .379 .575 .602 .520 .390 .475 .526 .381
.314 adj .526 .372 .601 .427 .592 .646 .483 .316 .409 .411 .402 .312 adv .675
.150 .504 .397 .546 .695 .491 .230 .495 .416 .223 .081 m-bert m-bert:+abtt
(-3) nouns .517 .091 .446 .191 .210 .364 .191 .188 .266 .418 .142 .539 verbs
.511 .005 .200 .039 .077 .248 .038 .107 .181 .266 .091 .503 adj .227 .050 .226
.028 .128 .193 .044 .046 .002 .099 .192 .267 adv .282 .012 .343 .112 .173 .390
.326 .036 .046 .207 161 .049 xlm-100 xlm:+abtt (-3) all .498 .096 .270 .118
.203 .234 .195 .106 .170 .289 .130 .506 nouns .551 .132 .381 .193 .238 .234
.242 .184 .292 .378 .165 .559 verbs .544 .038 .169 .006 .190 .132 .136 .073
.095 .243 .047 .570 adj .356 .140 .256 .081 .179 .185 .150 .046 .022 .100 .220
.291 adv .284 .017 .040 .086 .043 .027 .221 .014 .022 .315 .095 .156
Table 13: Spearman’s $\rho$ correlation scores over the four POS classes
represented in Multi-SimLex datasets. In addition to the word vectors
considered earlier in Table 12, we also report scores for another
contextualized model, xlm-100. The numbers in parentheses refer to the total
number of POS-class pairs in the original eng dataset and, consequently, in
all other monolingual datasets.
The overall results also clearly indicate that (i) there are differences in
performance across different monolingual Multi-SimLex datasets, and (ii)
unsupervised post-processing is universally useful, and can lead to huge
improvements in correlation scores for many languages. In what follows, we
also delve deeper into these analyses.
Impact of Unsupervised Post-Processing. First, the results in Table 12 suggest
that applying dimension-wise mean centering to the initial vector spaces has
positive impact on word similarity scores in all test languages and for all
models, both static and contextualized (see the +mc rows in Table 12). Mimno
and Thompson (2017) show that distributional word vectors have a tendency
towards narrow clusters in the vector space (i.e., they occupy a narrow cone
in the vector space and are therefore anisotropic Mu, Bhat, and Viswanath
(2018); Ethayarajh (2019)), and are prone to the undesired effect of hubness
Radovanović, Nanopoulos, and Ivanović (2010); Lazaridou, Dinu, and Baroni
(2015).151515Hubness can be defined as the tendency of some points/vectors
(i.e., “hubs”) to be nearest neighbors of many points in a high-dimensional
(vector) space Radovanović, Nanopoulos, and Ivanović (2010); Lazaridou, Dinu,
and Baroni (2015); Conneau et al. (2018a) Applying dimension-wise mean
centering has the effect of spreading the vectors across the hyper-plane and
mitigating the hubness issue, which consequently improves word-level
similarity, as it emerges from the reported results. Previous work has already
validated the importance of mean centering for clustering-based tasks Suzuki
et al. (2013), bilingual lexicon induction with cross-lingual word embeddings
Artetxe, Labaka, and Agirre (2018a); Zhang et al. (2019); Vulić et al. (2019),
and for modeling lexical semantic change Schlechtweg et al. (2019). However,
to the best of our knowledge, the results summarized in Table 12 are the first
evidence that also confirms its importance for semantic similarity in a wide
array of languages. In sum, as a general rule of thumb, we suggest to always
mean-center representations for semantic tasks.
The results further indicate that additional post-processing methods such as
abtt and uncovec on top of mean-centered vector spaces can lead to further
gains in most languages. The gains are even visible for languages which start
from high correlation scores: for instance., cmn with CC+Wiki ft increases
from 0.534 to 0.583, from 0.315 to 0.526 with Wiki ft, and from 0.408 to 0.487
with m-bert. Similarly, for rus with CC+Wiki ft we can improve from 0.422 to
0.500, and for fra the scores improve from 0.578 to 0.613. There are
additional similar cases reported in Table 12.
Overall, the unsupervised post-processing techniques seem universally useful
across languages, but their efficacy and relative performance does vary across
different languages. Note that we have not carefully fine-tuned the hyper-
parameters of the evaluated post-processing methods, so additional small
improvements can be expected for some languages. The main finding, however, is
that these post-processing techniques are robust to semantic similarity
computations beyond English, and are truly language independent. For instance,
removing dominant latent (PCA-based) components from word vectors emphasizes
semantic differences between different concepts, as only shared non-
informative latent semantic knowledge is removed from the representations.
In summary, pretrained word embeddings do contain more information pertaining
to semantic similarity than revealed in the initial vectors. This way, we have
corroborated the hypotheses from prior work Mu, Bhat, and Viswanath (2018);
Artetxe et al. (2018) which were not previously empirically verified on other
languages due to a shortage of evaluation data; this gap has now been filled
with the introduction of the Multi-SimLex datasets. In all follow-up
experiments, we always explicitly denote which post-processing configuration
is used in evaluation.
POS-Specific Subsets. We present the results for subsets of word pairs grouped
by POS class in Table 13. Prior work based on English data showed that
representations for nouns are typically of higher quality than those for the
other POS classes Schwartz, Reichart, and Rappoport (2015, 2016); Vulić et al.
(2017b). We observe a similar trend in other languages as well. This pattern
is consistent across different representation models and can be attributed to
several reasons. First, verb representations need to express a rich range of
syntactic and semantic behaviors rather than purely referential features
Gruber (1976); Levin (1993); Kipper et al. (2008). Second, low correlation
scores on the adjective and adverb subsets in some languages (e.g., pol, cym,
swa) might be due to their low frequency in monolingual texts, which yields
unreliable representations. In general, the variance in performance across
different word classes warrants further research in class-specific
representation learning Baker, Reichart, and Korhonen (2014); Vulić et al.
(2017b). The scores further attest the usefulness of unsupervised post-
processing as almost all class-specific correlation scores are improved by
applying mean-centering and abtt. Finally, the results for m-bert and xlm-100
in Table 13 further confirm that massively multilingual pretraining cannot
yield reasonable semantic representations for many languages: in fact, for
some classes they display no correlation with human ratings at all.
Differences across Languages. Naturally, the results from Tables 12 and 13
also reveal that there is variation in performance of both static word
embeddings and pretrained encoders across different languages. Among other
causes, the lowest absolute scores with ft are reported for languages with
least resources available to train monolingual word embeddings, such as
Kiswahili, Welsh, and Estonian. The low performance on Welsh is especially
indicative: Figure 1 shows that the ratings in the Welsh dataset match up very
well with the English ratings, but we cannot achieve the same level of
correlation in Welsh with Welsh ft word embeddings. Difference in performance
between two closely related languages, est (low-resource) and fin (high-
resource), provides additional evidence in this respect.
The highest reported scores with m-bert and xlm-100 are obtained for Mandarin
Chinese and Yue Chinese: this effectively points to the weaknesses of
massively multilingual training with a joint subword vocabulary spanning 102
and 100 languages. Due to the difference in scripts, “language-specific”
subwords for yue and cmn do not need to be shared across a vast amount of
languages and the quality of their representation remains unscathed. This
effectively means that m-bert’s subword vocabulary contains plenty of cmn-
specific and yue-specific subwords which are exploited by the encoder when
producing m-bert-based representations. Simultaneously, higher scores with
m-bert (and xlm in Table 13) are reported for resource-rich languages such as
French, Spanish, and English, which are better represented in m-bert’s
training data. We also observe lower absolute scores (and a larger number of
OOVs) for languages with very rich and productive morphological systems such
as the two Slavic languages (Polish and Russian) and Finnish. Since Polish and
Russian are known to have large Wikipedias and Common Crawl data Conneau et
al. (2019) (e.g., their Wikipedias are in the top 10 largest Wikipedias
worldwide), the problem with coverage can be attributed exactly to the
proliferation of morphological forms in those languages.
Finally, while Table 12 does reveal that unsupervised post-processing is
useful for all languages, it also demonstrates that peak scores are achieved
with different post-processing configurations. This finding suggests that a
more careful language-specific fine-tuning is indeed needed to refine word
embeddings towards semantic similarity. We plan to inspect the relationship
between post-processing techniques and linguistic properties in more depth in
future work.
Multilingual vs. Language-Specific Contextualized Embeddings. Recent work has
shown that—despite the usefulness of massively multilingual models such as
m-bert and xlm-100 for zero-shot cross-lingual transfer Pires, Schlinger, and
Garrette (2019); Wu and Dredze (2019)—stronger results in downstream tasks for
a particular language can be achieved by pretraining language-specific models
on language-specific data.
In this experiment, motivated by the low results of m-bert and xlm-100 (see
again Table 13), we assess if monolingual pretrained encoders can produce
higher-quality word-level representations than multilingual models. Therefore,
we evaluate language-specific bert and xlm models for a subset of the Multi-
SimLex languages for which such models are currently available: Finnish
Virtanen et al. (2019) (bert-base architecture, uncased), French Le et al.
(2019) (the FlauBERT model based on xlm), English (bert-base, uncased),
Mandarin Chinese (bert-base) Devlin et al. (2019) and Spanish (bert-base,
uncased). In addition, we also evaluate a series of pretrained encoders
available for English: (i) bert-base, bert-large, and bert-large with whole
word masking (wwm) from the original work on BERT Devlin et al. (2019), (ii)
monolingual “English-specific” xlm Conneau and Lample (2019), and (iii) two
models which employ parameter reduction techniques to build more compact
encoders: albert-b uses a configuration similar to bert-base, while albert-l
is similar to bert-large, but with an $18\times$ reduction in the number of
parameters Lan et al. (2020).161616All models and their further specifications
are available at the following link: https://huggingface.co./models.
From the results in Table 5, it is clear that monolingual pretrained encoders
yield much more reliable word-level representations. The gains are visible
even for languages such as cmn which showed reasonable performance with m-bert
and are substantial on all test languages. This further confirms the validity
of language-specific pretraining in lieu of multilingual training, if
sufficient monolingual data are available. Moreover, a comparison of
pretrained English encoders in Figure 5(b) largely follows the intuition: the
larger bert-large model yields slight improvements over bert-base, and we can
improve a bit more by relying on word-level (i.e., lexical-level)
masking.Finally, light-weight albert model variants are quite competitive with
the original bert models, with only modest drops reported, and albert-l again
outperforms albert-b. Overall, it is interesting to note that the scores
obtained with monolingual pretrained encoders are on a par with or even
outperform static ft word embeddings: this is a very intriguing finding per se
as it shows that such subword-level models trained on large corpora can
implicitly capture rich lexical semantic knowledge.
(a) Monolingual vs multilingual
(b) Pretrained eng encoders
Figure 5: (a) A performance comparison between monolingual pretrained language
encoders and massively multilingual encoders. For four languages (cmn, eng,
fin, spa), we report the scores with monolingual uncased bert-base
architectures and multilingual uncased m-bert model, while for fra we report
the results of the multilingual xlm-100 architecture and a monolingual French
FlauBERT model Le et al. (2019), which is based on the same architecture as
xlm-100. (b) A comparison of various pretrained encoders available for
English. All these models are post-processed via abtt (-3).
Similarity-Specialized Word Embeddings. Conflating distinct lexico-semantic
relations is a well-known property of distributional representations Turney
and Pantel (2010); Melamud et al. (2016). Semantic specialization fine-tunes
distributional spaces to emphasize a particular lexico-semantic relation in
the transformed space by injecting external lexical knowledge Glavaš, Ponti,
and Vulić (2019). Explicitly discerning between true semantic similarity (as
captured in Multi-SimLex) and broad conceptual relatedness benefits a number
of tasks, as discussed in §2.1.171717For an overview of specialization methods
for semantic similarity, we refer the interested reader to the recent tutorial
Glavaš, Ponti, and Vulić (2019). Since most languages lack dedicated lexical
resources, however, one viable strategy to steer monolingual word vector
spaces to emphasize semantic similarity is through cross-lingual transfer of
lexical knowledge, usually through a shared cross-lingual word vector space
Ruder, Vulić, and Søgaard (2019). Therefore, we evaluate the effectiveness of
specialization transfer methods using Multi-SimLex as our multilingual test
bed.
We evaluate a current state-of-the-art cross-lingual specialization transfer
method with minimal requirements, put forth recently by
Ponti:2019emnlp.181818We have also evaluated other specialization transfer
methods, e.g., Glavaš and Vulić (2018); Ponti et al. (2018b), but they are
consistently outperformed by the method of Ponti:2019emnlp. In a nutshell,
their li-postspec method is a multi-step procedure that operates as follows.
First, the knowledge about semantic similarity is extracted from WordNet in
the form of triplets, that is, linguistic constraints $(w_{1},w_{2},r)$, where
$w_{1}$ and $w_{2}$ are two concepts, and $r$ is a relation between them
obtained from WordNet (e.g., synonymy or antonymy). The goal is to “attract”
synonyms closer to each other in the transformed vector space as they reflect
true semantic similarity, and “repel” antonyms further apart. In the second
step, the linguistic constraints are translated from English to the target
language via a shared cross-lingual word vector space. To this end, following
Ponti:2019emnlp we rely on cross-lingual word embeddings (CLWEs) Joulin et al.
(2018) available online, which are based on Wiki ft
vectors.191919https://fasttext.cc/docs/en/aligned-vectors.html; for target
languages for which there are no pretrained CLWEs, we induce them following
the same procedure of Joulin:2018emnlp. Following that, a constraint
refinement step is applied in the target language which aims to eliminate the
noise inserted during the translation process. This is done by training a
relation classification tool: it is trained again on the English linguistic
constraints and then used on the translated target language constraints, where
the transfer is again enabled via a shared cross-lingual word vector
space.202020We again follow Ponti:2019emnlp and use a state-of-the-art
relation classifier Glavaš and Vulić (2018). We refer the reader to the
original work for additional technical details related to the classifier
design. Finally, a state-of-the-art monolingual specialization procedure from
Ponti:2018emnlp injects the (now target language) linguistic constraints into
the target language distributional space.
The scores are summarized in Table 14. Semantic specialization with li-
postspec leads to substantial improvements in correlation scores for the
majority of the target languages, demonstrating the importance of external
semantic similarity knowledge for semantic similarity reasoning. However, we
also observe deteriorated performance for the three target languages which can
be considered the lowest-resource ones in our set: cym, swa, yue. We
hypothesize that this occurs due to the inferior quality of the underlying
monolingual Wikipedia word embeddings, which generates a chain of error
accumulations. In particular, poor distributional word estimates compromise
the alignment of the embedding spaces, which in turn results in increased
translation noise, and reduced refinement ability of the relation classifier.
On a high level, this “poor get poorer” observation again points to the fact
that one of the primary causes of low performance of resource-low languages in
semantic tasks is the sheer lack of even unlabeled data for distributional
training. On the other hand, as we see from Table 13, typological
dissimilarity between the source and the target does not deteriorate the
effectiveness of semantic specialization. In fact, li-postspec does yield
substantial gains also for the typologically distant targets such as heb, cmn,
and est. The critical problem indeed seems to be insufficient raw data for
monolingual distributional training.
Languages: cmn cym eng est fin fra heb pol rus spa swa yue fastText (Wiki)
(429) (282) (6) (343) (345) (73) (62) (354) (343) (57) (379) (677) ft:init
.315 .318 – .400 .575 .444 .428 .370 .359 .432 .332 .376 li-postspec .584 .204
– .515 .619 .601 .510 .531 .547 .635 .238 .267
Table 14: The impact of vector space specialization for semantic similarity.
The scores are reported using the current state-of-the-art specialization
transfer li-postspec method of Ponti:2019emnlp, relying on English as a
resource-rich source language and the external lexical semantic knowledge from
the English WordNet.
## 8 Cross-Lingual Evaluation
Similar to monolingual evaluation in §7, we now evaluate several state-of-the-
art cross-lingual representation models on the suite of 66 automatically
constructed cross-lingual Multi-SimLex datasets. Again, note that evaluating a
full range of cross-lingual models available in the rich prior work on cross-
lingual representation learning is well beyond the scope of this article. We
therefore focus our cross-lingual analyses on several well-established and
indicative state-of-the-art cross-lingual models, again spanning both static
and contextualized cross-lingual word embeddings.
### 8.1 Models in Comparison
Static Word Embeddings. We rely on a state-of-the-art mapping-based method for
the induction of cross-lingual word embeddings (CLWEs): vecmap Artetxe,
Labaka, and Agirre (2018b). The core idea behind such mapping-based or
projection-based approaches is to learn a post-hoc alignment of independently
trained monolingual word embeddings Ruder, Vulić, and Søgaard (2019). Such
methods have gained popularity due to their conceptual simplicity and
competitive performance coupled with reduced bilingual supervision
requirements: they support CLWE induction with only as much as a few thousand
word translation pairs as the bilingual supervision Mikolov, Le, and Sutskever
(2013); Xing et al. (2015); Upadhyay et al. (2016); Ruder, Søgaard, and Vulić
(2019). More recent work has shown that CLWEs can be induced with even weaker
supervision from small dictionaries spanning several hundred pairs Vulić and
Korhonen (2016); Vulić et al. (2019), identical strings Smith et al. (2017),
or even only shared numerals Artetxe, Labaka, and Agirre (2017). In the
extreme, fully unsupervised projection-based CLWEs extract such seed bilingual
lexicons from scratch on the basis of monolingual data only (Conneau et al.,
2018a; Artetxe, Labaka, and Agirre, 2018b; Hoshen and Wolf, 2018; Alvarez-
Melis and Jaakkola, 2018; Chen and Cardie, 2018; Mohiuddin and Joty, 2019,
inter alia).
Recent empirical studies Glavaš et al. (2019); Vulić et al. (2019); Doval et
al. (2019) have compared a variety of unsupervised and weakly supervised
mapping-based CLWE methods, and vecmap emerged as the most robust and very
competitive choice. Therefore, we focus on 1) its fully unsupervised variant
(unsuper) in our comparisons. For several language pairs, we also report
scores with two other vecmap model variants: 2) a supervised variant which
learns a mapping based on an available seed lexicon (super), and 3) a
supervised variant with self-learning (super+sl) which iteratively increases
the seed lexicon and improves the mapping gradually. For a detailed
description of these variants, we refer the reader to recent work Artetxe,
Labaka, and Agirre (2018b); Vulić et al. (2019). We again use CC+Wiki ft
vectors as initial monolingual word vectors, except for yue where Wiki ft is
used. The seed dictionaries of two different sizes (1k and 5k translation
pairs) are based on PanLex Kamholz, Pool, and Colowick (2014), and are taken
directly from prior work Vulić et al.
(2019),212121https://github.com/cambridgeltl/panlex-bli or extracted from
PanLex following the same procedure as in the prior work.
Contextualized Cross-Lingual Word Embeddings. We again evaluate the capacity
of (massively) multilingual pretrained language models, m-bert and xlm-100, to
reason over cross-lingual lexical similarity. Implicitly, such an evaluation
also evaluates “the intrinsic quality” of shared cross-lingual word-level
vector spaces induced by these methods, and their ability to boost cross-
lingual transfer between different language pairs. We rely on the same
procedure of aggregating the models’ subword-level parameters into word-level
representations, already described in §7.1.
As in monolingual settings, we can apply unsupervised post-processing steps
such as abtt to both static and contextualized cross-lingual word embeddings.
| cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue
---|---|---|---|---|---|---|---|---|---|---|---|---
cmn | | .076 | .348 | .139 | .154 | .392 | .190 | .207 | .227 | .300 | .049 | .484
cym | .041 | | .087 | .017 | .049 | .095 | .033 | .072 | .085 | .089 | .002 | .083
eng | .565 | .004 | | .168 | .159 | .401 | .171 | .182 | .236 | .309 | .014 | .357
est | .014 | .097 | .335 | | .143 | .161 | .100 | .113 | .083 | .134 | .025 | .124
fin | .049 | .020 | .542 | .530 | | .195 | .077 | .110 | .111 | .157 | .029 | .167
fra | .224 | .015 | .662 | .559 | .533 | | .191 | .229 | .297 | .382 | .038 | .382
heb | .202 | .110 | .516 | .465 | .445 | .469 | | .095 | .154 | .181 | .038 | .185
pol | .121 | .028 | .464 | .415 | .465 | .534 | .412 | | .139 | .183 | .013 | .205
rus | .032 | .037 | .511 | .408 | .476 | .529 | .430 | .390 | | .248 | .037 | .226
spa | .546 | .048 | .498 | .450 | .490 | .600 | .462 | .398 | .419 | | .055 | .313
swa | -.01 | .116 | .029 | .006 | .013 | -.05 | .033 | .052 | .035 | .045 | | .043
yue | .004 | .047 | .059 | .004 | .002 | .059 | .001 | .074 | .032 | .089 | -.02 |
Table 15: Spearman’s $\rho$ correlation scores on all 66 cross-lingual
datasets. 1) The scores below the main diagonal are computed based on cross-
lingual word embeddings (CLWEs) induced by aligning CC+Wiki ft in all
languages (except for yue where we use Wiki ft) in a fully unsupervised way
(i.e., without any bilingual supervision). We rely on a standard CLWE mapping-
based (i.e., alignment) approach: vecmap Artetxe, Labaka, and Agirre (2018b).
2) The scores above the main diagonal are computed by obtaining
768-dimensional word-level vectors from pretrained multilingual BERT (m-bert)
following the procedure described in §7.1. For both fully unsupervised vecmap
and m-bert, we report the results with unsupervised postprocessing enabled:
all $2\times 66$ reported scores are obtained using the +abbt (-10) variant.
(a) Average scores
(b) Scores on eng-fra
Figure 6: Further performance analyses of cross-lingual Multi-SimLex datasets.
(a) Spearman’s $\rho$ correlation scores averaged over all 66 cross-lingual
Multi-SimLex datasets for two pretrained multilingual encoders (m-bert and
xlm). The scores are obtained with different configurations that exclude
(init) or enable unsupervised post-processing. (b) A comparison of various
pretrained encoders available for the English-French language pair, see the
main text for a short description of each benchmarked pretrained encoder.
### 8.2 Results and Discussion
Main Results and Differences across Language Pairs. A summary of the results
on the 66 cross-lingual Multi-SimLex datasets are provided in Table 15 and
Figure 6(a). The findings confirm several interesting findings from our
previous monolingual experiments (§7.2), and also corroborate several
hypotheses and findings from prior work, now on a large sample of language
pairs and for the task of cross-lingual semantic similarity.
First, we observe that the fully unsupervised vecmap model, despite being the
most robust fully unsupervised method at present, fails to produce a
meaningful cross-lingual word vector space for a large number of language
pairs (see the bottom triangle of Table 15): many correlation scores are in
fact no-correlation results, accentuating the problem of fully unsupervised
cross-lingual learning for typologically diverse languages and with fewer
amounts of monolingual data Vulić et al. (2019). The scores are particularly
low across the board for lower-resource languages such as Welsh and Kiswahili.
It also seems that the lack of monolingual data is a larger problem than
typological dissimilarity between language pairs, as we do observe reasonably
high correlation scores with vecmap for language pairs such as cmn-spa, heb-
est, and rus-fin. However, typological differences (e.g., morphological
richness) still play an important role as we observe very low scores when
pairing cmn with morphologically rich languages such fin, est, pol, and rus.
Similar to prior work of Vulic:2019we and doval2019onthe, given the fact that
unsupervised vecmap is the most robust unsupervised CLWE method at present
Glavaš et al. (2019), our results again question the usefulness of fully
unsupervised approaches for a large number of languages, and call for further
developments in the area of unsupervised and weakly supervised cross-lingual
representation learning.
The scores of m-bert and xlm-100222222The xlm-100 scores are not reported for
brevity; they largely follow the patterns observed with m-bert. The aggregated
scores between the two encoders are also very similar as indicated by Figure
6(a). lead to similar conclusions as in the monolingual settings. Reasonable
correlation scores are achieved only for a small subset of resource-rich
language pairs (e.g., eng, fra, spa, cmn) which dominate the multilingual
m-bert training. Interestingly, the scores indicate a much higher performance
of language pairs where yue is one of the languages when we use m-bert instead
of vecmap. This boils down again to the fact that yue, due to its specific
language script, has a good representation of its words and subwords in the
shared m-bert vocabulary. At the same time, a reliable vecmap mapping between
yue and other languages cannot be found due to a small monolingual yue corpus.
In cases when vecmap does not yield a degenerate cross-lingual vector space
starting from two monolingual ones, the final correlation scores seem
substantially higher than the ones obtained by the single massively
multilingual m-bert model.
Finally, the results in Figure 6(a) again verify the usefulness of
unsupervised post-processing also in cross-lingual settings. We observe
improved performance with both m-bert and xlm-100 when mean centering (+mc) is
applied, and further gains can be achieved by using abtt on the mean-centered
vector spaces. A similar finding also holds for static cross-lingual word
embeddings232323Note that vecmap does mean centering by default as one of its
preprocessing steps prior to learning the mapping function Artetxe, Labaka,
and Agirre (2018b); Vulić et al. (2019)., where applying abbt (-10) yields
higher scores on 61/66 language pairs.
Fully Unsupervised vs. Weakly Supervised Cross-Lingual Embeddings. The results
in Table 15 indicate that fully unsupervised cross-lingual learning fails for
a large number of language pairs. However, recent work Vulić et al. (2019) has
noted that these sub-optimal non-alignment solutions with the unsuper model
can be avoided by relying on (weak) cross-lingual supervision spanning only
several thousands or even hundreds of word translation pairs. Therefore, we
examine 1) if we can further improve the results on cross-lingual Multi-SimLex
resorting to (at least some) cross-lingual supervision for resource-rich
language pairs; and 2) if such available word-level supervision can also be
useful for a range of languages which displayed near-zero performance in Table
15. In other words, we test if recent “tricks of the trade” used in the rich
literature on CLWE learning reflect in gains on cross-lingual Multi-SimLex
datasets.
First, we reassess the findings established on the bilingual lexicon induction
task Søgaard, Ruder, and Vulić (2018); Vulić et al. (2019): using at least
some cross-lingual supervision is always beneficial compared to using no
supervision at all. We report improvements over the unsuper model for all 10
language pairs in Table 16, even though the unsuper method initially produced
strong correlation scores. The importance of self-learning increases with
decreasing available seed dictionary size, and the +sl model always
outperforms unsuper with 1k seed pairs; we observe the same patterns also with
even smaller dictionary sizes than reported in Table 16 (250 and 500 seed
pairs). Along the same line, the results in Table 17 indicate that at least
some supervision is crucial for the success of static CLWEs on resource-leaner
language pairs. We note substantial improvements on all language pairs; in
fact, the vecmap model is able to learn a more reliable mapping starting from
clean supervision. We again note large gains with self-learning.
| cmn-eng | eng-fra | eng-spa | eng-rus | est-fin | est-heb | fin-heb | fra-spa | pol-rus | pol-spa
---|---|---|---|---|---|---|---|---|---|---
unsuper | .565 | .662 | .498 | .511 | .510 | .465 | .445 | .600 | .390 | .398
super (1k) | .575 | .602 | .453 | .376 | .378 | .363 | .442 | .588 | .399 | .406
+sl (1k) | .577 | .703 | .547 | .548 | .591 | .513 | .488 | .639 | .439 | .456
super (5k) | .587 | .704 | .542 | .535 | .518 | .473 | .585 | .631 | .455 | .463
+sl (5k) | .581 | .707 | .548 | .551 | .556 | .525 | .589 | .645 | .432 | .476
Table 16: Results on a selection of cross-lingual Multi-SimLex datasets where the fully unsupervised (unsuper) CLWE variant yields reasonable performance. We also show the results with supervised vecmap without self-learning (super) and with self-learning (+sl), with two seed dictionary sizes: 1k and 5k pairs; see §8.1 for more detail. Highest scores for each language pair are in bold. | cmn-fin | cmn-rus | cmn-yue | cym-fin | cym-fra | cym-pol | fin-swa
---|---|---|---|---|---|---|---
unsuper | .049 | .032 | .004 | .020 | .015 | .028 | .013
super (1k) | .410 | .388 | .372 | .384 | .475 | .326 | .206
+sl (1k) | .590 | .537 | .458 | .471 | .578 | .380 | .264
Table 17: Results on a selection of cross-lingual Multi-SimLex datasets where
the fully unsupervised (unsuper) CLWE variant fails to learn a coherent shared
cross-lingual space. See also the caption of Table 16.
Multilingual vs. Bilingual Contextualized Embeddings. Similar to the
monolingual settings, we also inspect if massively multilingual training in
fact dilutes the knowledge necessary for cross-lingual reasoning on a
particular language pair. Therefore, we compare the 100-language xlm-100 model
with i) a variant of the same model trained on a smaller set of 17 languages
(xlm-17); ii) a variant of the same model trained specifically for the
particular language pair (xlm-2); and iii) a variant of the bilingual xlm-2
model that also leverages bilingual knowledge from parallel data during joint
training (xlm-2++). We again use the pretrained models made available by
Conneau:2019nips, and we refer to the original work for further technical
details.
The results are summarized in Figure 6(b), and they confirm the intuition that
massively multilingual pretraining can damage performance even on resource-
rich languages and language pairs. We observe a steep rise in performance when
the multilingual model is trained on a much smaller set of languages (17
versus 100), and further improvements can be achieved by training a dedicated
bilingual model. Finally, leveraging bilingual parallel data seems to offer
additional slight gains, but a tiny difference between xlm-2 and xlm-2++ also
suggests that this rich bilingual information is not used in the optimal way
within the xlm architecture for semantic similarity.
In summary, these results indicate that, in order to improve performance in
cross-lingual transfer tasks, more work should be invested into 1) pretraining
dedicated language pair-specific models, and 2) creative ways of leveraging
available cross-lingual supervision (e.g., word translation pairs, parallel or
comparable corpora) Liu et al. (2019a); Wu et al. (2019); Cao, Kitaev, and
Klein (2020) with pretraining paradigms such as bert and xlm. Using such
cross-lingual supervision could lead to similar benefits as indicated by the
results obtained with static cross-lingual word embeddings (see Table 16 and
Table 17). We believe that Multi-SimLex can serve as a valuable means to track
and guide future progress in this research area.
## 9 Conclusion and Future Work
We have presented Multi-SimLex, a resource containing human judgments on the
semantic similarity of word pairs for 12 monolingual and 66 cross-lingual
datasets. The languages covered are typologically diverse and include also
under-resourced ones, such as Welsh and Kiswahili. The resource covers an
unprecedented amount of 1,888 word pairs, carefully balanced according to
their similarity score, frequency, concreteness, part-of-speech class, and
lexical field. In addition to Multi-Simlex, we release the detailed protocol
we followed to create this resource. We hope that our consistent guidelines
will encourage researchers to translate and annotate Multi-Simlex -style
datasets for additional languages. This can help and create a hugely valuable,
large-scale semantic resource for multilingual NLP research.
The core Multi-SimLex we release with this paper already enables researchers
to carry out novel linguistic analysis as well as establishes a benchmark for
evaluating representation learning models. Based on our preliminary analyses,
we found that speakers of closely related languages tend to express equivalent
similarity judgments. In particular, geographical proximity seems to play a
greater role than family membership in determining the similarity of judgments
across languages. Moreover, we tested several state-of-the-art word embedding
models, both static and contextualized representations, as well as several
(supervised and unsupervised) post-processing techniques, on the newly
released Multi-SimLex. This enables future endeavors to improve multilingual
representation learning with challenging baselines. In addition, our results
provide several important insights for research on both monolingual and cross-
lingual word representations:
1) Unsupervised post-processing techniques (mean centering, elimination of top
principal components, adjusting similarity orders) are always beneficial
independently of the language, although the combination leading to the best
scores is language-specific and hence needs to be tuned.
2) Similarity rankings obtained from word embeddings for nouns are better
aligned with human judgments than all the other part-of-speech classes
considered here (verbs, adjectives, and, for the first time, adverbs). This
confirms previous generalizations based on experiments on English.
3) The factor having the greatest impact on the quality of word
representations is the availability of raw texts to train them in the first
place, rather than language properties (such as family, geographical area,
typological features).
4) Massively multilingual pretrained encoders such as m-bert (Devlin et al.,
2019) and xlm-100 (Conneau and Lample, 2019) fare quite poorly on our
benchmark, whereas pretrained encoders dedicated to a single language are more
competitive with static word embeddings such as fastText (Bojanowski et al.,
2017). Moreover, for language-specific encoders, parameter reduction
techniques reduce performance only marginally.
5) Techniques to inject clean lexical semantic knowledge from external
resources into distributional word representations were proven to be effective
in emphasizing the relation of semantic similarity. In particular, methods
capable of transferring such knowledge from resource-rich to resource-lean
languages (Ponti et al., 2019c) increased the correlation with human judgments
for most languages, except for those with limited unlabelled data.
Future work can expand our preliminary, yet large-scale study on the ability
of pretrained encoders to reason over word-level semantic similarity in
different languages. For instance, we have highlighted how sharing the same
encoder parameters across multiple languages may harm performance. However, it
remains unclear if, and to what extent, the input language embeddings present
in xlm-100 but absent in m-bert help mitigate this issue. In addition,
pretrained language embeddings can be obtained both from typological databases
(Littell et al., 2017) and from neural architectures (Malaviya, Neubig, and
Littell, 2017). Plugging these embeddings into the encoders in lieu of
embeddings trained end-to-end as suggested by prior work (Tsvetkov et al.,
2016; Ammar et al., 2016; Ponti et al., 2019b) might extend the coverage to
more resource-lean languages.
Another important follow-up analysis might involve the comparison of the
performance of representation learning models on multilingual datasets for
both word-level semantic similarity and sentence-level Natural Language
Understanding. In particular, Multi-SimLex fills a gap in available resources
for multilingual NLP and might help understand how lexical and compositional
semantics interact if put alongside existing resources such as XNLI Conneau et
al. (2018b) for natural language inference or PAWS-X Yang et al. (2019) for
cross-lingual paraphrase identification. Finally, the Multi-SimLex annotation
could turn out to be a unique source of evidence to study the effects of
polysemy in human judgments on semantic similarity: for equivalent word pairs
in multiple languages, are the similarity scores affected by how many senses
the two words (or multi-word expressions) incorporate?
In light of the success of initiatives like Universal Dependencies for
multilingual treebanks, we hope that making Multi-SimLex and its guidelines
available will encourage other researchers to expand our current sample of
languages. We particularly encourage creation and submission of comparable
Multi-SimLex datasets for under-resourced and typologically diverse languages
in future work. In particular, we have made a Multi-Simlex community website
available to facilitate easy creation, gathering, dissemination, and use of
annotated datasets: https://multisimlex.com/.
###### Acknowledgements.
This work is supported by the ERC Consolidator Grant LEXICAL: Lexical
Acquisition Across Languages (no 648909). Thierry Poibeau is partly supported
by a PRAIRIE 3IA Institute fellowship ("Investissements d’avenir" program,
reference ANR-19-P3IA-0001).
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2024-09-04T02:54:58.838230 | 2020-03-10T17:42:28 | 2003.04875 | {
"authors": "Manuel Schilling, \\'Etienne Wodey, Ludger Timmen, Dorothee Tell, Klaus\n H. Zipfel, Dennis Schlippert, Christian Schubert, Ernst M. Rasel, J\\\"urgen\n M\\\"uller",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26142",
"submitter": "Manuel Schilling",
"url": "https://arxiv.org/abs/2003.04875"
} | arxiv-papers | # Gravity field modelling for the Hannover $10\text{\,}\mathrm{m}$ atom
interferometer
Manuel Schilling German Aerospace Center (DLR), Institute for Satellite
Geodesy and Inertial Sensing, c/o Leibniz Universität Hannover, DLR-Institut,
Welfengarten 1, 30167 Hannover, Germany Leibniz Universität Hannover,
Institut für Erdmessung, Schneiderberg 50, 30167 Hannover, Germany Étienne
Wodey Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1,
30167 Hannover, Germany Ludger Timmen Leibniz Universität Hannover, Institut
für Erdmessung, Schneiderberg 50, 30167 Hannover, Germany Dorothee Tell
Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167
Hannover, Germany Klaus H. Zipfel Leibniz Universität Hannover, Institut für
Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Dennis Schlippert
Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167
Hannover, Germany Christian Schubert German Aerospace Center (DLR),
Institute for Satellite Geodesy and Inertial Sensing, c/o Leibniz Universität
Hannover, DLR-Institut, Welfengarten 1, 30167 Hannover, Germany Leibniz
Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167
Hannover, Germany Ernst M. Rasel Leibniz Universität Hannover, Institut für
Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Jürgen Müller Leibniz
Universität Hannover, Institut für Erdmessung, Schneiderberg 50, 30167
Hannover, Germany
(This is a post-peer-review, pre-copyedit version of an article published in
Journal of Geodesy 94:122. The final authenticated version is available online
at: https://dx.doi.org/10.1007/s00190-020-01451-y)
Absolute gravimeters are used in geodesy, geophysics, and physics for a wide
spectrum of applications. Stable gravimetric measurements over timescales from
several days to decades are required to provide relevant insight into
geophysical processes. Users of absolute gravimeters participate in
comparisons with a metrological reference in order to monitor the temporal
stability of the instruments and determine the bias to that reference.
However, since no measurement standard of higher-order accuracy currently
exists, users of absolute gravimeters participate in key comparisons led by
the International Committee for Weights and Measures. These comparisons
provide the reference values of highest accuracy compared to the calibration
against a single gravimeter operated at a metrological institute. The
construction of stationary, large scale atom interferometers paves the way
towards a new measurement standard in absolute gravimetry used as a reference
with a potential stability up to
$1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at $1\text{\,}\mathrm{s}$
integration time. At the Leibniz University Hannover, we are currently
building such a very long baseline atom interferometer with a
$10\text{\,}\mathrm{m}$ long interaction zone. The knowledge of local gravity
and its gradient along and around the baseline is required to establish the
instrument’s uncertainty budget and enable transfers of gravimetric
measurements to nearby devices for comparison and calibration purposes. We
therefore established a control network for relative gravimeters and
repeatedly measured its connections during the construction of the atom
interferometer. We additionally developed a 3D model of the host building to
investigate the self-attraction effect and studied the impact of mass changes
due to groundwater hydrology on the gravity field around the reference
instrument. The gravitational effect from the building 3D model is in
excellent agreement with the latest gravimetric measurement campaign which
opens the possibility to transfer gravity values with an uncertainty below the
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level.
Keywords: atom interferometry, gravity acceleration, absolute gravimetry,
gravimeter reference
## Introduction
A variety of applications in geodesy, geophysics and physics require the
knowledge of local gravity g [67]. These applications include observing
temporal variations of the mass distribution in the hydrosphere, atmosphere
and cryosphere and furthermore the establishment and monitoring of height and
gravity reference frames, the determination of glacial isostatic adjustment,
and the realisation of SI111Système International d’unités units, e. g., of
force and mass [32, 28, 54]. The absolute value of gravity g is usually
measured by tracking the free-fall of a test mass using a laser interferometer
[33]. The operation of an absolute gravimeter (AG), especially the combination
of several instruments in a project, requires special consideration of the
offset to _true g_ and the change thereof. In addition, the long-term
stability of absolute gravimeters is of particular relevance when measuring
small gravity trends. For example, the determination of the glacial isostatic
adjustment (GIA) on regional scales of around $1000\text{\,}\mathrm{km}$ [63]
requires an instrument stable to the
$20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level over several years.
Extending this effort by deploying several AGs also requires the knowledge of
the biases of all the instruments involved [36]. The lack of a calibration
service with a
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$
uncertainty requires the participation in key comparisons [9, KC, e. g. ]
where the reference values are determined with an uncertainty of approximately
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. This uncertainty level
requires the participation of multiple gravimeters and cannot be achieved by
comparison against a single gravimeter operated at a metrological institute.
However, the development of stationary atom interferometers, which can be
operated as gravimeters, so-called quantum gravimeters (QG), may result in
such a superior reference in the future available for regular comparisons or
on demand by the user. A major requirement in this respect is the control of
systematic effects like wavefront aberration or the Coriolis effect. In this
paper, we focus on the modelling and measurement of the local gravity field.
We start by discussing the typical approaches for monitoring the long-term
stability of an AG and tracing the measurements back to the SI (section 2).
Then, after briefly describing the working principle of atomic gravimeters and
the case for very long baseline atom interferometry (section 3), we present a
gravity model for the Hannover Very Long Baseline Atom Interferometry
(Hannover-VLBAI) facility, a new $10\text{\,}\mathrm{m}$-scale baseline atom
interferometer in commissioning at the Leibniz University Hannover (section
4). Finally, we present the micro-gravimetric surveys performed at the
instrument’s site (section 5) to assess the accuracy of the gravity model
(section 6). This paves the way towards control of the systematics in the atom
interferometer and accurate transfers of measured g values between the VLBAI
operating as a gravimeter and transportable AGs in a nearby laboratory.
## Gravimeter bias and SI traceability
Micro-g LaCoste FG5(X) [34] instruments represent the current state of the art
in absolute gravimetry. They track the trajectories of a free-falling test
mass with corner cubes by means of laser interferometry to determine the local
acceleration of gravity g. These types of absolute gravimeters are referred to
as _classical absolute gravimeters_ in the following text.
As described by the 2015 CCM-IAG222Consultative Committee for Mass and related
quantities – International Association of Geodesy Strategy for Metrology in
Absolute Gravimetry [5], there are two complementary paths for the
traceability of absolute gravity measurements: a) calibration of incorporated
frequency generators and b) additional gravimeter comparisons against a
reference. The direct way of tracing absolute gravity measurements back to the
SI goes through the calibration of their incorporated laser and oscillator to
standards of length and time [68]. In high-accuracy instruments, the laser
frequency is typically locked to a standard transition of molecular iodine [6,
44]. The time reference is usually given by a rubidium oscillator which needs
to be regularly compared with a reference oscillator to ensure its accuracy as
external higher-accuracy time sources are typically not available at
measurement sites. In most cases, the oscillator’s frequency drift is linear
$($<0.5\text{\,}\mathrm{mHz}\text{/}\mathrm{month}$\text{ or
}$<1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$/$\mathrm{month}$)$ and a
few calibrations per year are sufficient. However, [30] and [53] report on
sudden jumps in frequency333Current publications refer to the Microsemi
(formerly Symmetricon) SA.22c rubidium oscillator equivalent to several tens
of $\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ due to increased concentrations of
gaseous helium [43] when measuring near superconducting gravimeters. Such
higher concentrations might occur after installation, maintenance, or repair
of a superconducting gravimeter and are unlikely during normal operation. The
frequency drift changes to an exponential decrease after the helium event and
may remain this way for years [51].
Figure 1: Degree of Equivalence (DoE) of joint participants of EURAMET.M.G-K1
[10, ], CCM.G-K2 [11, ], EURAMET.M.G-K2 [37, ] and EURAMET.M.G-K3 [9, ]. The
participants are sorted by DoE of the first KC. The expanded uncertainty is
given only for the last KC. Pilot Study (PS) indicates instruments of non
NMI/DI institutions. All AGs shown are laser interferometers of which eight
are FG5(X) type instruments.
The equivalence of gravity measurement standards and the definition of the
gravity reference are established by international comparisons in the
framework of the CIPM MRA444Mutual Recognition Agreement of the Comité
International des Poids et Mesures. Since no higher-order reference instrument
is available, key comparisons are held in an approximately two-year interval,
alternating between CIPM key comparisons and regional comparisons. There, the
instruments operated by National Metrology Institutes (NMI) and Designated
Institutes (DI) are used to determine the Key Comparison Reference Value
(KCRV). The bias to the KCRV, or Degree of Equivalence (DoE) is then
calculated for all individual instruments, including those without NMI/DI
status participating in the so-called pilot study (PS), and serves as
validation for their uncertainty.
Figure 1 shows the common participants, out of a total number of 35
gravimeters participating in the comparisons, to the last four KC held in
Europe [10, 11, 37, 9]. One observes that the spread of DoE over all
instruments is around $\pm 75\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$,
and at a similar level for the most extreme cases of individual instruments.
Even though the DoEs of the instruments in these comparisons are typically
within the uncertainties declared by the participants, figure 1 also shows the
necessity of determining these biases of gravimeters, classical and quantum
alike, to monitor an instrument’s stability in time. Biases can then be taken
into account in gravimetric projects. The variation of the bias of an
instrument can be explained by a variety of factors. For example, [35] show
that a permanent change in the bias of a classical AG can occur during
manufacturer service or unusual transport conditions (e. g. aviation
transport). Also, [25, 26] identified, characterised and partially removed
biases originating in the signal processing chain of FG5 gravimeters, e. g.
due to cable length and fringe signal amplitude.
Regional KCs are linked to a CIPM KC by a small number of common NMI/DI
participants applying the so-called linking converter [22, typically around
$\pm 10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$,]. The underlying
assumption is that instrumental biases of the NMI/DI instruments remain stable
[8]. Otherwise, this would introduce an additional shift in the bias of all
participating instruments of the regional KC and PS.
Quantum gravimeters, based on matter wave interferometry with cold atoms,
offer a fully independent design. They have demonstrated stabilities and
accuracies at levels comparable to those from state of the art classical AGs
by participating in KCs [16, 23] or common surveys with other instruments at
various locations [13, 51]. The availability of improved QGs as gravity
references provides an opportunity to enhance the stability of reference
values obtained during key comparisons and therefore lead to an international
gravity datum of better stability in time. Just by that alone QGs could become
a serious alternative to classical absolute gravimeters.
## Very long baseline atomic gravimetry
### Atominterferometric gravimetry
Most atomic gravimeters use cold matter waves as free-falling test masses to
measure absolute gravity. They exploit the coherent manipulation of the
external degrees of freedom of these atomic test masses with light pulses to
realise interferometers sensitive to inertial quantities and other forces.
These techniques are for example used to perform precision measurements of
fundamental constants [49, 3, 39], test fundamental physics [55, 48, 21],
sense small forces [2] and perform gravimetry, gravity-gradiometry, and
measure rotations with record instabilities and inaccuracies [31, 12, 15, 72,
50, 57].
Figure 2: Mach–Zehnder light-pulse atom interferometer geometry in a uniform
acceleration field $\mathbf{a}$. At time $t_{0}$, the atomic matterwave is put
in a superposition of momenta $p$ ( ) and $p+\hbar k_{\mathrm{eff}}$ ( ). The
momenta are reversed at time $t_{0}+T$ to recombine the wave packets with a
last light pulse at time $t_{0}+2T$. The populations in the two momentum
classes after the last light pulse allow extracting the interferometric phase
$\Delta\phi$.
Atomic gravimeters typically realise the Mach–Zehnder light-pulse atom
interferometer geometry [24] depicted in figure 2. In this analogon to the
eponymous configuration for optical interferometers, the leading-order
interferometric phase $\Delta\phi$ scales with the space-time area enclosed by
the interferometer:
$\Delta\phi=\mathbf{k}_{\mathrm{eff}}\cdot\mathbf{a}T^{2}$ (1)
where $\hbar\mathbf{k}_{\mathrm{eff}}$ is the recoil transfered to the atomic
wave packets by the atom-light interaction processes (cf. figure 2, $\hbar$ is
the reduced Planck constant and $\mathbf{k}_{\mathrm{eff}}$ the effective
optical wave vector), $\mathbf{a}$ the uniform acceleration experienced by the
atoms during the interferometric sequence, and $T$ the pulse separation time.
The full interferometer has a duration of $2T$. The knowledge of the
instrument’s scale factor $k_{\mathrm{eff}}T^{2}$ and the measurement of the
phase $\Delta\phi$ allow determining the projection of the acceleration
$\mathbf{a}$ along $\mathbf{k}_{\mathrm{eff}}$. When
$\mathbf{k}_{\mathrm{eff}}$ is parallel to $\mathbf{g}$, such an instrument
can therefore be used as a gravimeter, measuring the total vertical
acceleration of the matter waves used as test masses.
The Mach–Zehnder light-pulse atom interferometer works as follows. For each
interferometric sequence, a sample of cold atoms is prepared in a time
$T_{p}$. Then, at time $t=t_{0}$, the first atom-light interaction pulse puts
the matter wave in a superposition of quantum states with different momenta
$\mathbf{p}$ and $\mathbf{p}+\hbar\mathbf{k}_{\mathrm{eff}}$, thus effectively
creating two distinct semi-classical trajectories. At time $t=t_{0}+T$, a
second atom-light interaction process redirects the two atomic trajectories to
allow closing the interferometer at time $t=t_{0}+2T$ with a third light
pulse. Counting the population of atoms in the two momentum states provides an
estimation of the interferometric phase $\Delta\phi$. Finally, the cycle of
preparation of the cold atoms, coherent manipulation of the matter waves, and
detection is repeated. Since the atom-light interaction imprints the local
phase of the light on the matter waves, the above measurement principle can be
interpreted as measuring the successive positions of a free-falling matter
wave at known times $t_{0}$, $t_{0}+T$, and $t_{0}+2T$ with respect to the
light field. The inertial reference frame for the measurement system, similar
to the superspring in FG5(X) gravimeters, is usually realised by a mirror
retro-reflecting the light pulses, creating well-defined equiphase fronts.
Practically, the interferometric phase $\Delta\phi$ is scanned by accelerating
the optical wave fronts at a constant rate $\alpha$, effectively continuously
tuning the differential velocity between the matter waves and the optical
equiphase fronts. Assuming that $\mathbf{k}_{\mathrm{eff}}$ and $\mathbf{a}$
are parallel, the interferometric phase reads:
$\Delta\phi=k_{\mathrm{eff}}\left(a-\frac{\alpha}{k_{\mathrm{eff}}}\right)T^{2}\
.$ (2)
When $\alpha=k_{\mathrm{eff}}a$, the interferometric phase vanishes
independently of the interferometer’s duration $2T$, allowing to unambiguously
identify this operation point. Physically, $\alpha=k_{\mathrm{eff}}a$ exactly
compensates the Doppler effect experienced by the atomic matter waves due to
the acceleration $a$. Therefore, the measurement of the acceleration $a$
amounts to a measurement of the acceleration rate $\alpha$ which can be traced
back to the SI since it corresponds to frequency generation in the radio-
frequency domain.
Assuming white noise at a level $\delta\phi$ for the detection of the
interferometric phase, the instrument’s instability is given by:
$\delta
a(\tau)=\sqrt{2T+T_{p}}\cdot\frac{\delta\phi}{k_{\mathrm{eff}}T^{2}}\cdot\frac{1}{\sqrt{\tau}}\
.$ (3)
where $\tau$ is the measurement’s integration time. This expression reveals
the three levers for reducing the measurement instability: decreasing the
single shot noise level $\delta\phi$, increasing the scale factor
$k_{\mathrm{eff}}T^{2}$, and minimising the sample preparation time $T_{p}$,
as it contributes to the total cycle time without providing phase information.
In transportable devices, record instabilities have been achieved by [12] with
$\delta a=$96\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$$ at
$\tau=$1\text{\,}\mathrm{s}$$. Commercial instruments like the Muquans AQG
[31] reached instabilities of
$500\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at
$\tau=$1\text{\,}\mathrm{s}$$ with sample rates up to $2\text{\,}\mathrm{Hz}$.
The dominant noise source is vibrations of the mirror realising the reference
frame for the measurements.
The accuracy of such quantum gravimeters stems from the well-controlled
interaction between the test masses and their environment during the
measurement sequence. The main sources of inaccuracy in such instruments
originate from uncertainties in the atom-light interaction parameters (e. g.
imperfections of the equiphase fronts of the light wave), stray
electromagnetic field gradients creating spurious forces, thus breaking the
free-fall assumption, and knowledge of the inhomogeneous gravity field along
the trajectories. Extensive characterisation of these effects led to
uncertainties in QGs below $40\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$,
consistent with the results from CIPM key comparisons [15] or common surveys
with classical AGs [12].
### Very Long Baseline Atom Interferometry
Very Long Baseline Atom Interferometry (VLBAI) represents a new class of
ground-based atom interferometric platforms which extends the length of the
interferometer's baseline from tens of centimetres like in typical
transportable instruments [12, 15] to multiple meters. According to equation
(1), the vertical acceleration sensitivity of a Mach–Zehnder type atom
interferometer scales linearly with the length of the baseline ($\sim
aT^{2}$). Therefore, an increase in the length of the baseline potentially
enables a finer sensitivity for the atomic gravimeter through an increased
scale factor $k_{\mathrm{eff}}T^{2}$. A $10\text{\,}\mathrm{m}$-long baseline
instrument can for example extend the interferometric time $2T$ to around
$1\text{\,}\mathrm{s}$ if the atoms are simply dropped along the baseline or
up to $2.4\text{\,}\mathrm{s}$ if they are launched upwards in a fountain-like
fashion. In the simple drop case, the velocity acquired by the atoms between
their release from the source and the start of the interferometer leads to an
interferometer duration shorter than half of the one for the launch case. For
our apparatus, the distance between the top source chamber and the region of
interest is around $2\text{\,}\mathrm{m}$ (see figure 3), constraining
$T<$400\text{\,}\mathrm{ms}$$ for simple drops.
Using realistic parameters ($T_{p}=$3\text{\,}\mathrm{s}$$,
$\delta\phi=$10\text{\,}\mathrm{mrad}$$), equation (3) yields potential short-
term instabilities for VLBAIs ($\tau=$1\text{\,}\mathrm{s}$$ integration
time):
$\begin{array}[]{ll}T=$400\text{\,}\mathrm{ms}$\text{:\leavevmode\nobreak\
}&\delta a=$8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$\\\
T=$1.2\text{\,}\mathrm{s}$\text{:\leavevmode\nobreak\ }&\delta
a=$1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$\end{array}$ (4)
competing with the noise level of superconducting gravimeters [45, 46] while
providing absolute values of the gravity acceleration g.
Nevertheless, the increased scale factor $k_{\mathrm{eff}}T^{2}$ gained by the
expanded baseline comes at the price of a stationary device with added
complexity due to its size, and a vibration noise sensitivity magnified by the
same scale factor as the gravitational acceleration for frequencies below
$\nicefrac{{1}}{{(2T)}}$. Hence, the use of VLBAIs as ultra stable gravimeters
requires new developments in the control of environmental vibrations [19].
Also, time- and space-varying electromagnetic and gravity fields along the
free-fall trajectories of the matter waves have a direct impact on the
accuracy and stability of the instrument, as the corresponding spurious forces
depart from the assumptions of equation (1), therefore leading to biases [7]
and impacting the instrument’s effective height [60].
### Effective height
In order to compare measurements of a VLBAI gravimeter with other instruments,
it is crucial to determine the effective height $z_{\mathrm{eff}}$ defined by:
$g_{0}-\gamma
z_{\mathrm{eff}}=\frac{\Delta\phi_{\mathrm{tot}}}{k_{\mathrm{eff}}T^{2}}$ (5)
where $g_{0}\approx$9.81\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ is the
value of gravity at $z=0$, $\gamma\approx$3\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{m}}^{-1}$$ the magnitude of the
linear gravity gradient, and $\Delta\phi_{\mathrm{tot}}$ the phase shift
measured by the interferometer. The right-hand side is the value of gravity
measured by the atom interferometer, including all bias sources. Restricting
to first order in the gravity-gradient $\gamma$, and applying a path-integral
formalism, one gets [40]:
$z_{\mathrm{eff}}=z_{0}-\dfrac{\Delta g}{\gamma}\quad\text{with}\quad\Delta
g=\frac{7}{12}\gamma g_{0}T^{2}-\gamma\bar{v}_{0}T$ (6)
where $z_{0}$ is the height of the start of the interferometer and
$\bar{v}_{0}=v_{0}+\nicefrac{{\hbar k_{\mathrm{eff}}}}{{(2m)}}$ the mean
atomic velocity just after the interferometer opens ($v_{0}$ is the atomic
velocity before the first beamsplitter, and $m$ is the atomic mass). This
expression for $z_{\mathrm{eff}}$ is compatible with the one given for FG5
gravimeters by [38]. In particular, it only depends on the value of the
gradient $\gamma$ through $v_{0}$ and $z_{0}$. Indeed, the interferometer is
controlled in time and the initial position and velocity $z_{0}$ and $v_{0}$
are therefore given by the free-fall motion of the atoms between the source
chamber and the region of interest. In general, $z_{\mathrm{eff}}$ depends on
the geometry of the atom interferometer. For the simple drop case in the
Hannover VLBAI facility (see section 3.4),
$z_{\mathrm{eff}}\approx$9.2\text{\,}\mathrm{m}$$.
Corrections to equation (6) must be taken into account to constrain the
uncertainty on gravity at $z_{\mathrm{eff}}$ below
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. On the one hand, terms of
order $\gamma^{2}$ and higher in $\Delta\phi_{\mathrm{tot}}$ contribute at the
sub-$\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level. On the other hand, one can
use perturbation theory [66] to estimate the effect of the non-homogeneous
gravity gradient along the interferometer’s baseline. Using the data discussed
here, we evaluate this effect below
$5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$, therefore lying within the
model’s uncertainty (see section 6) and similar to the known contribution for
FG5(X) gravimeters [60].
Finally, when using multiple concurrent interferometers at different heights,
the effect of a homogeneous gravity gradient can be mitigated by measuring it
simultaneously with the acceleration value [4]. In this case, the effective
height corresponds to the position of the mirror giving the inertial
reference. Detailed modelling is however still necessary to push the
uncertainty budget in the
sub-$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ and calibrate the
instrument to the level of its instability.
### The Hannover VLBAI facility
We introduce the Hannover Very Long Baseline Atom Interferometry facility, an
instrument developed at the newly founded Hannover Institute of Technology
(HITec) of the Leibniz University Hannover, Germany. It builds on the concepts
outlined in section 3.2 to provide a platform to tackle challenges in extended
baseline atom interferometry. In the long term, it aims at tests of our
physical laws and postulates like for example the universality of free fall
[20], searches for new forces or phenomena, and the development of new methods
for absolute gravimetry and gravity gradiometry [56].
Upper atomic sourceLower atomic sourceBaselineultra-high vacuum chamberand
magnetic shieldInertial referencevibration isolated mirrorRegion of
interestfor precisionatom
interferometry$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$Height /
$\mathrm{m}$ Figure 3: The Hannover Very Long Baseline Atom Interferometry
(VLBAI) facility and its three main elements: source chambers , baseline , and
inertial reference system and vacuum vessel (VTS) . The baseline and upper
source chambers are supported by an aluminium structure (VSS, dark blue). The
region of interest for atom interferometry is shaded in light blue.
The Hannover VLBAI facility is built around three main elements shown in
figure 3:
1. 1.
Ultra cold samples of rubidium and ytterbium atoms are prepared in the two
_source chambers_ , allowing for both drop (max $T=$400\text{\,}\mathrm{ms}$$)
and launch (max $T=$1.2\text{\,}\mathrm{s}$$) modes of operation. Advanced
atom-optics promise enhanced free-fall times by relaunching the wave packets
during the interferometric sequence [1];
2. 2.
The reference frame for the inertial measurements is realised by a
_seismically isolated mirror_ at the bottom of the apparatus. The seismic
attenuation system (SAS) uses geometric anti-spring filters [69] to achieve
vibration isolation above its natural resonance frequency of
$320\text{\,}\mathrm{mHz}$. The isolation platform is operated under high
vacuum conditions to reduce acoustic and thermal coupling. The vacuum vessel
containing the SAS is denoted VTS in sections 4–6;
3. 3.
The $10.5\text{\,}\mathrm{m}$-long _baseline_ consists of a
$20\text{\,}\mathrm{cm}$ diameter cylindrical aluminium vacuum chamber and a
high-performance magnetic shield [71]. The interferometric sequences take
place along this baseline, in the $8\text{\,}\mathrm{m}$-long central _region
of interest_ where the longitudinal magnetic field gradients fall below
$2.5\text{\,}\mathrm{nT}\text{/}\mathrm{m}$.
In order to decouple the instrument from oscillations of the walls of the
building, the apparatus is only rigidly connected to the foundations of the
building. The VTS (and SAS) and lower source chamber are mounted on a
baseplate directly connected to the foundation. The baseline and upper source
chamber are supported by a $10\text{\,}\mathrm{m}$ high aluminium tower,
denoted as VLBAI support structure (VSS) in the following sections. The
footprint of the device on the floor is
$$2.5\text{\,}\mathrm{m}$\times$2.5\text{\,}\mathrm{m}$$. Traceability to the
SI is ensured by locking the instrument’s frequency references to standards at
the German NMI (PTB Braunschweig) via an optical link [42]. All heights are
measured from the instrument’s baseplate. The altitude of this reference point
in the German height datum is $50.545\text{\,}\mathrm{m}$.
## Environmental model
(a) HITec cross-section (not to scale)
(b) HITec top view
Figure 4: Views of HITec: cross-section (4(a)) of the VLBAI laboratories with
the gravimetric network of 2019 along two vertical profiles and region of
interest (blue). The indicated groundwater variation (thick bar) refers to an
average annual amplitude of $0.3\text{\,}\mathrm{m}$. The thin bar indicates
extreme low and high levels. The height $z=$0\text{\,}\mathrm{m}$$ refers to
the top of the baseplate. The top view of HITec (4(b)) shows the orientation
of our coordinate system, the location of the VLBAI facility (blue) and the
gravimetry lab including piers for gravimeters (light grey).
The VLBAI facility is implemented in the laboratory building of the Hannover
Institute of Technology. The building consists of three floors (one basement
level, two above street level) and is divided into a technical part mainly
containing the climate control systems, and a section with the laboratories
(see figure 4). In the laboratory part, a so-called backbone gives
laboratories access to the technical infrastructure and divides the building
in two parts along its long axis. The backbone and southern row of
laboratories have a footprint of
$$13.4\text{\,}\mathrm{m}$\times$55.4\text{\,}\mathrm{m}$$ and extend
approximately $5\text{\,}\mathrm{m}$ below surface level. The northern row of
laboratories is fully above ground except for the gravimetry laboratory which
is on an intermediate level, around $1.5\text{\,}\mathrm{m}$ below street
level and $3.4\text{\,}\mathrm{m}$ above basement level (see figure 4(a)). The
foundation of the building is $0.5\text{\,}\mathrm{m}$ thick except beneath
the gravimetry laboratory, which has a separate and $0.8\text{\,}\mathrm{m}$
thick one. Figure 4(a) also shows the measurement points for the relative
gravimeters along the VLBAI main axis and a second validation profile,
occupied using tripods, next to the VLBAI which were used for the measurements
presented in section 5.
### Physical model
Following the methods described by [27], we discretise the HITec building into
a model of rectangular prisms that accounts for more than $500$ elements. The
geometry is extracted from the construction plans, and we verified all the
heights by levelling, also including a benchmark with a known elevation in the
German height datum. The building is embedded in a sedimentary ground of sand,
clay, and marl ($2050\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$). For the
edifice itself, we include all walls and floors made of reinforced concrete
($2500\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$), the
$7\text{\,}\mathrm{cm}13\text{\,}\mathrm{cm}$ thick liquid flow screed
covering the concrete floors in the labs
($2100\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$), and the gypsum drywalls
($800\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$). We also incorporate the
insulation material ($150\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$) and
gravel on the roof ($1350\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$). We
use a simplified geometry to model the large research facilities in the
surroundings. This is for example the case for the Einstein-Elevator [29], a
free-fall simulator with a weight of $165\text{\,}\mathrm{t}$ and horizontal
distances of $32\text{\,}\mathrm{m}$ and $16\text{\,}\mathrm{m}$ to the VLBAI
facility and gravimetry laboratory, respectively. Finally, we account for
laboratory equipment, e. g. optical tables ($550\text{\,}\mathrm{kg}$ each)
according to the configuration at the time of the gravimetric measurement
campaigns.
During the first measurements (2017), the interior construction was still in
progress, and the laboratories were empty. By the time of the second campaign
(2019), the building was fully equipped. The VLBAI support structure (VSS) and
the vacuum tank (VTS) for the seismic attenuation system were in place. The
VLBAI instrument (atomic sources, magnetic shield, $10\text{\,}\mathrm{m}$
vacuum tube) and seismic attenuation system were completed after the second
campaign.
Due to their inclined or rounded surfaces, the VLBAI experimental apparatus
and its support structure require a more flexible method than rectangular
prisms to model their geometry. We apply the method described by [41] and
divide the surface of the bodies to be modelled into polygonal faces to
calculate the gravitational attraction from surface integrals. Contrary to the
rectangular prisms method, there are only few restrictions on the underlying
geometry. Most notably, all vertices of a face must lie in one plane and the
normal vectors of all surfaces must point outward of the mass. For example,
normal vectors of faces describing the outside surface of a hollow sphere must
point away from the sphere and normal vectors on the inside surface must point
towards the centre, away from the mass of the wall of the sphere. We extract
the geometry of the VLBAI facility components from their tridimensional CAD
model through an export in STL555Stereolithography or standard triangulation
language format [47]. This divides the surface of the bodies into triangular
faces, therefore ensuring planar faces by default. Moreover, the STL format
encodes normal vectors pointing away from the object. Both prerequisites for
the polygonal method by [41] are thus met. Using this method, the VSS
(aluminium, $2650\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$, total weight
$5825\text{\,}\mathrm{kg}$) consists of roughly $86000$ faces and the VTS and
corresponding baseplates (stainless steel,
$8000\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$, total weight
$2810\text{\,}\mathrm{kg}$) contain $187000$ faces, mostly due to the round
shape and fixtures of the VTS. As the overall computation time to extract the
attraction of these components with a $\mathrm{c}\mathrm{m}$-resolution on
both vertical profiles remains in the range of minutes on a desktop PC, we do
not need to simplify the models. The Monte Carlo simulations described in
section 6 nevertheless require the computing cluster of the Leibniz University
Hannover (LUH).
We use MATLAB666MATLAB Version 9.4.0.813654 (R2018a) to perform the numerical
calculations. As a cross-check, we implemented both the rectangular prisms and
polyhedral bodies methods for the calculation of the attraction effect of the
main frame of the HITec building. Both approaches agree within floating point
numerical accuracy.
### Time variable gravity changes
Mostly for the benefit of the future operations of the VLBAI, we include the
effects of groundwater level changes, atmospheric mass change, and Earth’s
body and ocean tides in our modelling. This is necessary for the individual
gravimetry experiment (and other physics experiments as well) in the VLBAI on
one hand, and for comparing measurements from different epochs, e. g. with
different groundwater levels, on the other hand. Previous investigations in
the gravimetry lab of a neighbouring building showed a linear coefficient of
$170\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ per meter change in the
local groundwater table [64]. This corresponds to a porosity of
$>30\text{\,}\mathrm{\char 37\relax}$ of the soil [17]. For our model, we
adapt a porevolume of $30\text{\,}\mathrm{\char 37\relax}$, which has to be
verified by gravimetric measurements and correlation with local groundwater
measurements. Two automatic groundwater gauges are available around the
building: one installed during the construction work and a second with records
dating back several decades also used by [64]. The effect of atmospheric mass
changes is calculated using the ERA5 atmospheric model provided by the
European Centre for Medium-Range Weather Forecasts777https://www.ecmwf.int and
the methods described by [51]. Tidal parameters are extracted from
observational time series [58, 52]. Other temporal gravity changes are not in
the scope of this work.
Currently, time variable gravity is also monitored with the gPhone-98
gravimeter of the Institute of Geodesy (IfE) at the LUH. In the long term, we
consider the addition of a superconducting gravimeter for this purpose when
the VLBAI facility is fully implemented and the experimental work is
beginning. The support of a superconducting gravimeter is also vital in the
characterisation of new gravimeters [12].
### Self-attraction results
Figure 5 shows the vertical component of the gravitational acceleration
generated by the building, equipment, VSS and VTS. The VLBAI main axis is in
the centre of the left plot ($x=$0\text{\,}\mathrm{m}$$). The large structures
around $5\text{\,}\mathrm{m}$ and $10\text{\,}\mathrm{m}$ correspond to the
floor levels. Smaller structures are associated to, for example, optical
tables or the VSS. The right panel of figure 5 highlights the attraction
calculated for the main axis ($x=$0\text{\,}\mathrm{m}$$) and for a second
profile along $x=$-1.8\text{\,}\mathrm{m}$$ and $y=$0\text{\,}\mathrm{m}$$.
The first profile shows a smooth curve except for the bottom
$2\text{\,}\mathrm{m}$, which are affected by the VTS. In this model, the part
above $2\text{\,}\mathrm{m}$ on the main axis is empty space. The second
profile, chosen as a sample from the xz-plane, passes through the floors,
hence the zig-zag features around $5\text{\,}\mathrm{m}$ and
$10\text{\,}\mathrm{m}$. While the main axis will later be occupied by the
instrument’s baseline, this second profile, similar to the validation profile,
represents a location that will always remain accessible to gravimeters.
Figure 5: Calculated gravitational attraction from the building, large
laboratory equipment, VSS and VTS in the xz-plane (left) and exemplarily on
two profiles (right).
### Effect of groundwater level changes
Based on the extensive groundwater level recordings from the gauge nearby the
HITec building, we study the impact of groundwater level changes [67, see
also] on gravitational attraction inside the building, specifically along the
VLBAI main and validation profiles, as well as in the gravimetry laboratory.
Due to the layout of the different basement levels in the building (see figure
4(a)), a change of the groundwater table affects gravity in the VLBAI
laboratories differently than in the gravimetry lab. Depending on the
groundwater level, the foundation beneath the VLBAI laboratories can be
partially within the groundwater table, whereas this is never the case for the
gravimetry laboratory. As shown on figure 4(a), the mean groundwater table is
nevertheless below the level of the foundation below the VLBAI laboratories.
Therefore, at certain points of the average annual cycle of amplitude
$0.3\text{\,}\mathrm{m}$, the groundwater table will rise only around the
foundation of the VLBAI laboratories, whereas its level will still increase
below the gravimetry laboratory. This effect is even more stringent for years
where the average cycle amplitude is exceeded (around one in four years).
Figure 6: Effect of groundwater variations (all heights in the height system
of the model, cf. figure 4(a)) on gravity in the gravimetry lab (left) and
along the VLBAI axis (right) with respect to the mean groundwater level
(dotted line ). The dashed line ( ) indicates the bottom of the foundation
below the VLBAI. The coloured lines indicate the change of gravity $\delta
g_{\mathrm{gw}}$ in various heights in the gravimetry and VLBAI laboratories.
The height of the gravimetry piers in the height system of the model is
$3.35\text{\,}\mathrm{m}$.
Figure 6 illustrates the different influence of the groundwater table level on
gravity in the VLBAI and gravimetry laboratories. The estimated change of
gravity $\delta g_{\mathrm{gw}}$ due to the attraction corresponding to
groundwater level variations is presented for different heights above the
gravimetry pier and along the VLBAI main axis. As the groundwater level is
always changing directly beneath the instrument piers in the gravimetry
laboratory, we expect an almost linear change of gravity with changing
groundwater level. The change of gravity is also almost independent of the
height above the pier, as shown by the almost identical lines for
$z=$3.35\text{\,}\mathrm{m}$$ directly on the pier and
$1.4\text{\,}\mathrm{m}$ above the pier, covering the instrumental heights of
transportable gravimeters. Therefore, AGs with various sensor heights, e. g.,
A-10 and FG5X, are affected in the same manner. The increase of $\delta
g_{\mathrm{gw}}$ is $32\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ in an
average year. This behaviour is different in the VLBAI laboratories. In
current records, the groundwater level never fell below the foundation of the
backbone (cf. figure 4(a)). This effect is seen in the small divergence (up to
$3\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$) for groundwater levels
below the foundation of the VLBAI (dashed line). Once the groundwater level
reaches the lower edge of the VLBAI foundation, gravity will not increase
linearly along the VLBAI main axis as the groundwater rises further. Moreover,
in this situation, the effect has a different magnitude depending on the
height in the room. In a year with the average amplitude of groundwater level
variation, ca. $\pm 0.15\text{\,}\mathrm{m}$ around the line indicating the
mean groundwater level, $\delta g_{\mathrm{gw}}$ will differ by
$5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ between basement and the top
floor. In years exceeding the average groundwater variation, the difference
between the basement and upper levels increases further. This effect is within
$\pm 2\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ on the validation
profile in the average groundwater cycle.
These observations will be crucial when comparing AGs in the gravimetry
laboratory to the VLBAI facility operated as a quantum gravimeter. Depending
on the geometry of a specific atom interferometer realisation, the
instrumental height of the VLBAI gravimeter changes and can introduce changes
in the measured value of g of more than
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ as a result of the
groundwater effect in years with a higher than usual groundwater level. The
magnitude of $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ is larger than
the targeted accuracy of the VLBAI and also a relevant size for classical AGs
in comparisons. It should also be noted, that the model only calculates the
gravitational attraction of the groundwater variation. A potential vertical
displacement of the ground itself is currently not taken into account, leading
to a possible underestimation of the effect.
In order to track the effect of groundwater level changes more accurately, we
plan to extend the findings of [64] by correlating periodic gravimetric
measurements on the validation profile in the VLBAI laboratories with the
recordings of the two groundwater level gauges around the building. This
should in particular allow us to take into account that, due to capillarity
effects, the groundwater level will probably not sink uniformly below the
foundation beneath the VLBAI laboratories once it reaches that level.
## Gravimetric measurements
In June 2017 and August 2019, we performed surveys using relative gravimeters
to verify our model from section 4 along the VLBAI main and validation
profiles. This approach was already demonstrated in [54], in which the gravity
field impact of a $200\text{\,}\mathrm{kN}$ force standard machine at the
Physikalisch-Technische Bundesanstalt in Braunschweig was modelled. That model
was verified with gravimetric measurements prior and after the installation of
the force machine. The difference between the modelled impact and the
measurement was within the uncertainty of the gravimeters used. For each
measurement point, we measured its connection to at least another point and
applied the step method with ten connections [65]. A connection corresponds to
one gravity difference observation between two points. Ten connections require
five occupations of a measurement point with a gravimeter. We measured most
connections with at least two different instruments, reducing the outcomes to
a mean instrumental height of $0.22\text{\,}\mathrm{m}$ above ground or
platform. We then performed a global least-squares adjustment using the
Gravimetry Net Least Squares Adjustment software from IfE [70, GNLSA,]. The
measurements are also calibrated in this process. We determined the individual
calibration factors of the gravimeters on the Vertical Gravimeter Calibration
Line in Hannover [61, 59] at least once in the week prior to the measurement
campaigns. The software also corrects Earth tides, applying our observed
parameters, and atmospheric mass changes by means of the linear factor of
$3\text{\,}\mathrm{nm}\text{/}{\mathrm{s}}^{2}\text{/}\mathrm{hPa}$ with
respect to normal air pressure at station elevation. In order to account for
instrumental drift in the global adjustment, we treat each day and each
instrument independently and use a variance component estimation to weight the
measurements in the global network adjustment. The specific groundwater effect
discussed in section 4.4, considering different magnitudes depending on
height, does not apply for either 2017 or 2019 because the groundwater levels
were below the foundation of the VLBAI in both years.
### 2017 Gravimetry campaign
We first mapped the gravity profile along the VLBAI profiles in June 2017,
when the HITec building was still under construction and the VLBAI
experimental apparatus not yet installed. Using the Scintrex CG3M-4492 (short
CG3M) and ZLS Burris B-144 (B-114) spring gravimeters [62, 52], we measured a
total of $147$ connections between seven positions spaced by ca.
$2\text{\,}\mathrm{m}$ along the VLBAI main axis, nine positions on the
validation profile, and two points outside of the building. We used a
scaffolding to access the measurement points on the main axis. However,
although the scaffold was anchored against the walls, the uppermost platforms
were too unstable to ensure reliable measurements. The B-114 was only able to
measure on the bottom three positions, because the feedback system was not
powerful enough to null the oscillating beam on the upper levels. The four
upper levels were only occupied by the CG3M. We connected each point on the
scaffold to another one on the same structure and to the closest fixed floor
level, at a point part of the validation profile. As shown in figure 4(a), the
validation profile included measurements on the floor and on different sized
tripods to determine the gradients.
The variance component estimation gives a posteriori standard deviations for a
single gravity tie observation of
$50\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for the B-114 and
$100\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for the CG3M. The standard
deviations for the adjusted gravity values range from
$15\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}42\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$
with a mean value of $28\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. The
standard deviations of the adjusted gravity differences vary from
$21\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ between fixed floor levels
to $59\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ between consecutive
levels on the scaffold. The transfer of height from the upper floor to the
basement through the intermediate levels on the scaffold showed a
$2\text{\,}\mathrm{mm}$ discrepancy compared to the heights from levelling. We
included the corresponding
$$2\text{\,}\mathrm{mm}$\cdot$3\text{\,}\mathrm{nm}\text{/}{\mathrm{s}}^{2}\text{/}\mathrm{mm}$=$6\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$$
as a systematic uncertainty for the adjusted gravity values for the values
measured on the scaffold. We also account for a $1\text{\,}\mathrm{mm}$
uncertainty on the determination of the relative gravimeter sensor height.
### 2019 Gravimetry campaign
Figure 7: Measurement at the VSS in 2019 with the B-64 (foreground) on the
validation profile and the CG6 (background) inside the VSS on a platform with
an operator wearing a security harness. The B-64 is operated on a small tripod
to raise the sensor height closer to the CG6 sensor height.
We mapped the gravity profile along the VLBAI axes in a more extensive manner
in summer and fall 2019. Most measurements were performed in one week of
August 2019, adding two days in October and November 2019. We used moveable
platforms inside the VSS, installed in June 2019, and could measure on $16$
levels on the main axis, spaced by
$0.45\text{\,}\mathrm{m}0.95\text{\,}\mathrm{m}$. The scheme for the
validation profile did not change. The layout of the network is depicted in
figure 4(a). For this campaign, we used the CG3M, the Scintrex CG6-0171 (CG6),
and ZLS Burris B-64 (B-64) spring gravimeters [62, 59, 52]. Owing to the high
mechanical stability of the VSS, measurements along the main axis were
unproblematic for all instruments and the measurement noise was at a similar
level on the moveable platforms and on the fixed floors (see figure 7). All
but one position were occupied with at least two gravimeters, amounting to
$439$ connections in the network adjustment.
The a posteriori standard deviations (single gravity tie measurement) of the
observations range from
$15\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}60\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$
with more than $50\text{\,}\mathrm{\char 37\relax}$ below
$30\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. The higher standard
deviations are a result of two days of measurements with the CG3M and
connections to two particular positions outside of the region of interest of
the VLBAI. The standard deviations of adjusted gravity values in the network
range from
$7\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}19\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$
with a mean of $9\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. This
improvement, compared to the previous campaign, can be attributed to the
stability of the VSS, the addition of the CG6 and the total number of
measurements performed. The height of the moveable platforms inside the VSS
was determined by a combination of levelling and laser distance
measurements888Leica Disto D210 to two fixed platforms and the ceiling. For
the height determination of the platforms, the uncertainty is
$1\text{\,}\mathrm{mm}$ due to the laser distance measurement. We also account
for an $1\text{\,}\mathrm{mm}$ uncertainty in the determination of the
instrumental height above the platforms.
## Combination of model and measurement
The measurement and model results along the VLBAI main and validation profiles
are presented in figure 8. Figure 8(a) shows the total variation of gravity
along the main axis. The plot is dominated by the normal decrease of gravity
with height. The effect of the building can be better seen when removing the
change of gravity with height and visualising only the attraction effect of
the building and laboratory equipment, as on figure 8(b). There, the model
corresponds to the configuration for the 2019 campaign and is identical to the
$x=$0\text{\,}\mathrm{m}$$, $y=$0\text{\,}\mathrm{m}$$ line in figure 5.
Figure 8(d) shows the model and measurements along the validation profile.
The models presented in figure 8 use the nominal values for the densities of
building elements (concrete floors and walls, drywalls, etc.). Since these can
have variations over the building, we performed a Monte Carlo simulation
($50000$ runs) varying the densities of the corresponding model elements by
$\pm 5\text{\,}\mathrm{\char 37\relax}$ according to a normal distribution.
This leads to a variation of attraction of $\pm
27\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}\pm
37\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for heights between
$4\text{\,}\mathrm{m}$ and $13\text{\,}\mathrm{m}$, as shown by the thin blue
lines on figures 88(b)–8(d). Using a uniform distribution of the density
parameters increases the variability by around
$20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. The VSS and VTS are not
part of the Monte Carlo simulation since their geometry and materials are well
known.
(a) central axis: gravity variation
(b) central axis: model
(c) central axis: residuals
(d) validation profile: model
Figure 8: Measurement and model results on the VLBAI central axis (8(a)–8(c))
and the validation profile (8(d)). The shaded area in (8(a)–8(c)) indicates
the region of interest. The total variation of gravity along the central axis
is shown in (8(a)). The modelled and measured attraction by the environment
(with the change of gravity with height removed) on the central and validation
profile is shown in (8(b)) and (8(d)). The errorbars indicate the standard
deviations from the network adjustment and the model simulations according to
equation (7). The maximum and minimum results of the $\pm
5\text{\,}\mathrm{\char 37\relax}$ density variations from Monte Carlo (MC)
simulation of model parameters are indicated by the thin blue lines. The
residuals of observations minus model $\delta g_{\mathrm{omc}}$ are given in
(8(c)) along with the standard deviation of the model $\sigma_{\text{mod}}$
according to equation (8).
The final location of the VLBAI facility and its main axis could only be
approximated to the $\mathrm{cm}$-level during the measurement campaigns
because of necessary installation tolerances. We estimated the effect of a
horizontal variation of $\pm 3\text{\,}\mathrm{cm}$ and a vertical variation
of $\pm 2\text{\,}\mathrm{mm}$ in a Monte Carlo simulation. The total
amplitude of the variations at the locations of the gravimetric measurements
is within $\pm 2\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ with a mean
standard deviation of $0.3\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for
the horizontal and $0.4\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for the
vertical component along the main axis.
The measurements, i. e. the markers in figure 8, are the result of the gravity
network adjustment. Additionally, we removed the effect of the change of
gravity with height for figures 88(b)–8(d). For this, the free air gradient is
modified with a model of the soil surrounding HITec. As the density is only
known to a certain degree, the Monte Carlo simulation also included the ground
around HITec. The standard deviation of the simulation results for each
gravimeter position is added to the measurements standard deviation by error
propagation. The simulations’ standard deviations range from
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at the height of
$4\text{\,}\mathrm{m}$ to $35\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$
at the topmost position. This is also reflected in the increase in the
standard deviations indicated by the errorbars in figure 8(c).
The uncertainty of the measurements now consists of the following components:
$\sigma_{\rm
obs}=\sqrt{\sigma_{g}^{2}+\sigma_{h,\mathrm{geo}}^{2}+\sigma_{z,\mathrm{mod}}^{2}+\sigma_{\mathrm{grad}}^{2}}\
.$ (7)
Here, the standard deviation of the network adjustment is $\sigma_{g}$. The
contribution of the determination of the height of the gravimeter is
$\sigma_{h,\mathrm{geo}}$. The result of the Monte Carlo simulations of the
vertical component of geometric position of the central axis
$\sigma_{z,\mathrm{mod}}$, and the modelling of the gravity gradient
$\sigma_{\mathrm{grad}}$ are also attributed to the measurements.
The standard deviation of the model consists of the following components:
$\sigma_{\mathrm{mod}}=\sqrt{\sigma_{\mathrm{MC}}^{2}+\sigma_{hz,\mathrm{mod}}^{2}}\
,$ (8)
where $\sigma_{\mathrm{MC}}$ is the standard deviation of the Monte Carlo
simulations of the model density, calculated in the heights of the gravimetric
measurements, and $\sigma_{hz,\mathrm{mod}}$ is the standard deviation of the
Monte Carlo simulations for the horizontal component of the geometric
positions along the VLBAI main axis. $\sigma_{\mathrm{mod}}$ is shown in
figure 8(c) with a range of
$6\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}11\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$
in the region of interest and about
$8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at
$z_{\mathrm{eff}}=$9.2\text{\,}\mathrm{m}$$ (see section 3.3).
Furthermore, a single parameter is estimated to reduce the gravity values from
the magnitude of $9.81\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-2}$ to the
order of magnitude of the model values for the attraction. This parameter is
the mean difference of observed minus computed results at the location of the
observation in the region of interest. The measurements of 2017 are also
corrected for the changes within the building with respect to 2019. No
additional parameters were estimated to fit the measurements to the model or
vice versa. The remaining signal should now contain the effect of the HITec
building on gravity.
In general, the 2017 measurements and the main axis model do not show a good
agreement [51, see also] due to the instability of the scaffolding used as a
platform [18, see also]. The agreement on the validation profile is better,
and only the two topmost points do not agree with the model and simulation.
These earlier measurements serve as a proof of concept and are given for the
sake of completeness. The following discussion concerns only the 2019
measurements.
The 2019 campaign provides a clear improvement considering the number of
stations along the VLBAI main axis, the stability of the platforms in the VSS
and therefore data quality. Consequently, the agreement between measurement
and model is significantly improved. The measurement scheme on the validation
profile remained unchanged compared to the 2017 campaign. Figure 8(c) shows
the difference between the measurements and the model on the central axis. The
region of interest for experiments in the VLBAI is approximately between
$4\text{\,}\mathrm{m}$ and $13\text{\,}\mathrm{m}$ (see figure 3). Within this
region, only the second-highest point is not within the simulation’s $\pm
5\text{\,}\mathrm{\char 37\relax}$ density variations. The two tailed
statistical test ($\alpha=0.05$) on the equality of model $\delta
g_{\mathrm{mod},i}$ and measurement $\delta g_{\mathrm{obs},i}$ at point $i$
according to
Null hypothesis: $\displaystyle\delta g_{\mathrm{omc},i}=$
$\displaystyle\delta g_{\mathrm{obs},i}-\delta g_{\mathrm{mod},i}=0$
Alternative hypothesis: $\displaystyle\delta g_{\mathrm{omc},i}\neq$
$\displaystyle 0$ Test statistics: $\displaystyle t_{i}=$
$\displaystyle\frac{\left|\delta
g_{\mathrm{omc},i}\right|}{\sqrt{\sigma_{\mathrm{obs},i}^{2}+\sigma_{\mathrm{mod},i}^{2}}}$
passes for all but three points. The null hypothesis, considering the symmetry
of the normal distribution, is rejected if
$t_{i}>N_{(0,1,1-\nicefrac{{\alpha}}{{2}})}$. The test fails for the points at
$z=$1.72\text{\,}\mathrm{m}5.55\text{\,}\mathrm{m}12.99\text{\,}\mathrm{m}$$.
The lowest point at $z=$1.72\text{\,}\mathrm{m}$$, directly on the VTS, was
challenging to measure, as the pump of the vacuum tank was active during the
measurements causing high-frequency vibrations. As this position is outside of
the experimental region of interest, no additional measurements were taken.
The cause for the significant deviation from the model at
$z=$12.99\text{\,}\mathrm{m}$$, which was measured with only one gravimeter,
is unknown. The height difference to the point above is only
$0.16\text{\,}\mathrm{m}$ of free space, so a real gravity variation appears
unlikely. Treating this point as an outlier, and repeating the test after
calculating the offset between adjusted gravity values and model without this
measurement, the test also passes for the point at
$z=$5.55\text{\,}\mathrm{m}$$. All points on the validation profile pass the
statistical test. The standard deviation of observations minus model is
$20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$
($31\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ if the second-highest
point is included) for the central axis in the region of interest and
$34\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ on the validation profile.
The density of the different model components, chosen initially from technical
documentation, are sufficient to generate a model which is identical to in
situ measurements at a $95\text{\,}\mathrm{\char 37\relax}$ confidence level.
Modelling a $5\text{\,}\mathrm{\char 37\relax}$ normally distributed variation
of these densities results in a narrow range of possible model variations,
which covers almost all measurements used to verify the model. We expect that
using individual densities for each floor instead of one common density value
for all concrete components in the building would improve the agreement
between model and observations on the validation profile. Such extra modelling
step should however be constrained not to deteriorate the model accuracy in
the experimental region of interest.
As a final step, the VLBAI magnetic shield and vacuum system [71] will be
added to the model. Similarly to the VSS and VTS, this component was designed
using CAD, built with known materials, and can be exported into the required
format for our model. While the assembly is significantly more complex, we
expect the octagonal symmetry of the magnetic shield to simplify the numerical
calculations and allow us to reach the same level of accuracy in the gravity
model as for the VSS and VTS. It will however only be possible to check the
quality of the extended model with measurements on the validation profile, as
the main axis is obstructed by the instrument’s vacuum chamber. Nevertheless,
the understanding of environmental variations (mostly hydrology) outlined in
section 4.4 will render this possible with good accuracy. Due to the work
associated with the installation of the VLBAI baseline components, this last
model extension and its corresponding validation have not been done yet.
Extending our model with the VLBAI baseline components will allow us to
connect gravimetric measurements along the validation profile and future data
acquired by a VLBAI quantum gravimeter along its main axis in our adjusted
gravimetric network. Since the measurement positions along the validation
profile will remain free during operation of the VLBAI facility, this will for
example enable comparisons of the VLBAI QG with FG5(X)-type classical AGs
positioned in the VLBAI laboratories. In this specific setup, contributions of
time variable gravity to the measurements are minimal for the VLBAI and
instrument under test. To further minimize the height dependency due to the
groundwater effect, the atom interferometer could be realised with an
effective height close to the instrumental height of the classical AG, e. g.
with the AG on the groundfloor. Taking into consideration the mean standard
deviation of the relative gravimeter network of
$9\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$, we expect to be able to
transfer g with an uncertainty of
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ and possibly below from the
VLBAI baseline. Furthermore, creating a similar network including stations
along the validation profile and in the HITec gravimetry laboratory would
permit gravimetric comparisons between the VLBAI QG and instruments operated
on the gravimetric piers. The estimates so far exclude the inevitable
contribution of the VLBAI gravity measurement. The determination and
validation of the VLBAI uncertainty budget will be published in a separate
study.
## Conclusions
We established a gravimetric control network for the Hannover VLBAI facility,
a novel $10\text{\,}\mathrm{m}$-scale atom interferometer. The network
consists of $439$ connections measured by relative gravimeters. A least
squares adjustment of the network results in a mean standard deviation of the
adjusted gravity values of $9\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$.
In addition, we developed a structural model of the building hosting the VLBAI
facility and its surroundings. When compared, the model and the measurements
agree with $95\text{\,}\mathrm{\char 37\relax}$ confidence, with standard
deviations of the residuals of
$20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ along the atom
interferometer’s baseline, and
$34\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ on a second, parallel
profile. Moreover, we gained insight on some dynamical aspects of the gravity
field around the instrument, namely the effect of groundwater level
variations.
We anticipate this gravimetric network to contribute to the assessment of the
quantum gravimeter’s uncertainty budget, which is currently not included in
our study. The current work is also essential to help determining the
effective instrumental height (g-value reference position) and enable
transfers of g values from the atom interferometer’s baseline to the
validation profile, accessible to mobile gravimeters for comparison and
possibly calibration purposes, at the
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ repeatability level
(relative to the VLBAI deduced g-values). Completing the model by including
the VLBAI baseline, refining the description of the soil surrounding the host
building, and including better estimates for the building material densities,
we expect to shift the possibility for gravity field measurement transfers and
mobile instrument calibration towards the
$5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level, improving the
temporal stability of the current state of the art, which is still largely
based on gravimeter comparisons. This paves the way for the realisation of a
new gravity standard based on atom interferometry. Finally, the knowledge of
the dynamical gravity field and its gradients is key to reaching new frontiers
in fundamental physics tests with very long baseline atom interferometry.
###### Acknowledgements.
The Hannover Very Long Baseline Atom Interferometry facility is a major
research equipment funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation). This work was supported by the DFG Collaborative
Research Center 1128 “geo-Q” (project A02, Contract Number 239994235) and is
supported by the CRC 1227 “DQ-mat” (project B07, Contract Number 274200144),
Germany's Excellence Strategy – EXC-2123 “QuantumFrontiers” – 390837967, and
the computing cluster of the Leibniz University Hannover under patronage of
the Lower Saxony Ministry of Science and Culture (MWK) and the DFG.
M. S., É. W., and C. S. acknowledge support from “Niedersächsisches Vorab”
through the “Quantum- and Nano-Metrology (QUANOMET)” initiative (project QT3),
and for initial funding of research in the DLR-SI institute. D. S.
acknowledges funding from the German Federal Ministry of Education and
Research (BMBF) through the funding program Photonics Research Germany
(contract number 13N14875).
The VLBAI support structure was conceived by the engineering office Heinz
Berlin (Wennigsen, Germany) in collaboration with the VLBAI science team, and
produced by Aljo Aluminium-Bau Jonuscheit GmbH (Berne, Germany).
We thank W. Ertmer for his vision and long lasting support on very long
baseline atom interferometry and the acquisition of funding for the Hannover
Institute of Technology. We are grateful to T. Froböse and A. Wanner for their
assistance during the installation of the vacuum tank and support structure.
We thank the three reviewers for their valuable input to improve this article.
###### Author contributions.
M.S., É.W., L.T. planned geometric and gravimetric measurements, evaluated the
data and prepared the initial draft. É.W., D.T., D.S., C.S., E.M.R.
conceptualised VSS, VTS. É.W., D.T., K.H.Z. designed and built measurement
platforms for VSS. M.S., É.W., L.T., D.T., K.H.Z. carried out the
measurements. M.S. developed and implemented the gravity model. D.T., K.H.Z.,
D.S., C.S., E.M.R., J.M. provided critical input to the manuscript and
approved the final version.
###### Data availability statement.
Data of absolute gravimeter key comparisons is available in the Key Comparison
Database (https://www.bipm.org/kcdb) and cited literature. Gravimetric
measurements in instrument specific ascii data formats and datasets generated
in this study are available from the corresponding author on reasonable
request.
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|
2024-09-04T02:54:58.853231 | 2020-03-10T17:48:47 | 2003.04878 | {
"authors": "Aritra De, Christopher Plumberg and Joseph I. Kapusta",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26143",
"submitter": "Aritra De",
"url": "https://arxiv.org/abs/2003.04878"
} | arxiv-papers | # Calculating Fluctuations and Self-Correlations Numerically for Causal Charge
Diffusion in Relativistic Heavy-Ion Collisions
Aritra De School of Physics & Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA Christopher Plumberg Department of Astronomy and
Theoretical Physics, Lund University, Sölvegatan 14A, SE-223 62 Lund, Sweden
Joseph I. Kapusta School of Physics & Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA
###### Abstract
We study the propagation and diffusion of electric charge fluctuations in the
Bjorken hydrodynamic model with both white and Catteneo noise using purely
numerical methods. We show that a global lattice of noise fluctuations is
required to fully calculate the two-point correlators of charge. We solve the
stochastic differential equations that arise from the charge conservation
equation on the lattice. We explicitly identify the self-correlation term in
the case of Catteneo noise and provide a physical interpretation. We provide a
numerical recipe to remove this contribution from the full two-point
correlators. Finally, we calculate the balance functions for charged hadrons.
By limiting the speed of signal propagation, we observe the expected narrowing
of the balance functions after removing the self-correlations.
## I Introduction
Relativistic hydrodynamics is used to study not only the equation of state but
also dynamical quantities, such as the transport coefficients, of the quark-
gluon plasma. The applicability of hydrodynamics is justified if the mean free
paths of the particles are small compared to the distances over which
thermodynamic quantities vary. It turns out that hydrodynamics is very
successful in modeling high energy nuclear collisions. There are experimental
facilities which produce and study quark-gluon plasma: the Relativistic Heavy
Ion Collider (RHIC) at Brookhaven National Laboratory and the Large Hadron
Collider (LHC) at CERN. Fluid equations describe conservation of energy,
momentum, baryon number, electric charge, and strangeness. Anisotropic
particle production, such as elliptic flow, in heavy ion collisions gives
credence to the use of hydrodynamics in simulating these collisions. It has
been successful in describing various properties like particle spectra,
particle correlations, and in obtaining values of transport quantities like
the ratio of shear viscosity to entropy density $\eta/s$. By comparing
particle spectra with experimental data, hydrodynamical simulations also help
in understanding the initial state, its fluctuations, and hence properties of
strongly interacting matter in general. Initially, the assumption of ideal
hydrodynamics worked very well in describing the data which indicated that the
system was strongly interacting. Later, important aspects like the lattice-
computed QCD equation of state and viscous properties were taken into account
to study transport properties of quark-gluon plasma with more precision.
The fluctuation-dissipation theorem relates the dissipative properties of a
system to its hydrodynamical fluctuations. In particular, it allows us to
infer quantities like shear and bulk viscosity, and electrical conductivity,
from the magnitude of fluctuations. Hydrodynamical fluctuations have also been
used to study static critical phenomena torres near a possible critical
point. More recently, there have been studies of dynamic critical phenomena
near a QCD critical point teaney ; yiyin ; nahrgang ; Bluhm . Critical points
are characterized by large fluctuations. This led to the suggestion to study
fluctuations in conserved quantities, such as electric charge, baryon number,
and strangeness on an event-by-event basis stephanov_shuryak . It has also
been suggested to study non-gaussianities (higher order cumulants) of the
fluctuations near critical points as they are more sensitive to large
correlation lengths stephanovprl . Correlation lengths theoretically diverge
near the critical points, but in the scenario of the heavy ion collisions are
limited by the finite system size stephanov_shuryak as well as by the finite
lifetime of the system berdnikov .
Thus it becomes imperative to study the hydrodynamics of fluctuations in the
context of heavy ion collisions. The relativistic theory of hydrodynamic
fluctuations in the context of heavy-ion collisions was introduced in Ref.
kapusta2012 . In the current work, we focus on calculating the two-point
correlations of charge fluctuations and the resulting balance functions of
pions. The balance function measures the difference in probability of finding
a particle of opposite charge in another fluid cell versus a particle of same
charge given a charged particle in a given fluid cell spratt . This problem
was studied analytically in the 1+1 dimensional Bjorken model in Ref.
plumbergkapusta . Analytic calculations are not possible for state of the art
3+1 dimensional, non-boost invariant, hydrodynamics. In preparation for
extensions to modern hydrodynamic models, we develop numerical methods to
solve the relevant stochastic differential equations numerically. Particular
attention is paid to the physical interpretation of self-correlations and how
they can be subtracted to make comparison to experimental data.
The outline of the paper is as follows. In Sec. II we review the normal
diffusion, the Cattaneo, and the Gurtin-Pipkin equations, and discuss how
self-correlations arise. In Sec. III we outline the application of the
relevant stochastic differential equations in the context of the Bjorken
hydrodynamic model for heavy-ion collisions. In Sec. IV we present solutions
to those equations. In Sec. V we show how self-correlations can be clearly
identified. In Sec. VI we calculate the balance functions that relate theory
to experiment. Our conclusions are presented in Sec. VII. Details of how the
stochastic differential equations are solved are presented in the Appendices.
The numerical method is readily transferrable to heavy-ion collisions which
have no spatial symmetries and thus useful for future calculations.
## II Noise, Fluctuations and Self-Correlations
Since the usual diffusion equation leads to instantaneous signal propagation,
which is inconsistent with special relativity, one needs a diffusion equation
which is the same order in spatial and temporal derivatives with
characteristic relaxation times and lengths. In this paper we solve the
simplest diffusion equation satisfying this condition, called the Catteneo
equation catteneo , numerically. The resulting differential equation is a
stochastic differential equation (SDE) because it contains random noise terms.
The way to solve SDEs is to solve the differential equation for a large number
of events (here on the order of 1 million or more) and study the correlation
functions. A finite difference method is used to solve the SDE.
Consider the ordinary diffusion equation with white noise. In the context of
the Bjorken model, which has boost invariance and no dependence on transverse
coordinates, the two variables are proper time $\tau=\sqrt{t^{2}-z^{2}}$ and
spatial rapidity $\xi={\textstyle{\frac{1}{2}}}\ln[(t+z)/(t-z)]$, where the
beam axis is along the $z$ direction. The noise $f$, appropriately defined
(see below), is a dimensionless random variable with correlator
$\langle
f(\tau_{1},\xi_{1})f(\tau_{2},\xi_{2})\rangle=\frac{N(\tau_{2})}{2\pi}\delta(\tau_{1}-\tau_{2})\delta(\xi_{1}-\xi_{2})\,,$
(1)
which is a product of Dirac $\delta$-functions in time and space with
normalization determined by the fluctuation-dissipation theorem
$N(\tau)=\frac{4\pi\sigma_{Q}(\tau)T(\tau)}{A\tau s^{2}(\tau)}\,.$ (2)
Here $\sigma_{Q}$ is the charge conductivity, $T$ is the temperature, $s$ is
the entropy density, and $A$ is the transverse area. To generate this
numerically on a discrete lattice with spacings $\Delta\xi$ and $\Delta\tau$,
we sample $f$ from a normal distribution with zero mean and variance
$N(\tau)/(2\pi\Delta\xi\Delta\tau)$. The analysis of how finite difference
methods work computationally for solving SDEs is discussed in Appendix A.
Consider the difference between white and colored noise. The standard two-
point function for white noise in frequency and momentum space is
$\displaystyle\langle\tilde{f}(\omega_{1},k_{1})\tilde{f}(\omega_{2},k_{2})\rangle$
$\displaystyle=$ $\displaystyle\int
d\tau_{1}\,d\tau_{2}\,d\xi_{1}\,d\xi_{2}\,e^{-i(k_{1}\xi_{1}+k_{2}\xi_{2})}\,e^{-i(\omega_{1}\tau_{1}+\omega_{2}\tau_{2})}\langle
f(\tau_{1},\xi_{1})f(\tau_{2},\xi_{2})\rangle$ (3) $\displaystyle=$
$\displaystyle\delta(k_{1}+k_{2})\tilde{N}(\omega_{1}+\omega_{2})$
where $\tilde{N}$ is the Fourier transform of $N$. Generalizing this to
Catteneo noise (which is an example of colored noise), we recall that the two-
point function for the noise obeys kapustayoung .
$\langle(\tau_{Q}\,\partial\tau_{1}+1)\tilde{f}(k_{1},\tau_{1})(\tau_{Q}\,\partial\tau_{2}+1)\tilde{f}(k_{2},\tau_{2})\rangle=N(\tau_{1})\delta(\tau_{1}-\tau_{2})\delta(k_{1}+k_{2})\,$
(4)
where $\tau_{Q}$ is a relaxation time. In frequency and momentum space this
becomes
$\langle\tilde{f}(\omega_{1},k_{1})\tilde{f}(\omega_{2},k_{2})\rangle=\frac{\delta(k_{1}+k_{2})\tilde{N}(\omega_{1}+\omega_{2})}{(i\tau_{Q}\omega_{1}+1)(i\tau_{Q}\omega_{2}+1)}\,.$
(5)
The noise correlator is no longer a Dirac $\delta$-function in time anymore;
instead, it is smeared out, hence the name colored noise.
The following three figures will help illustrate some of the physics to come.
Figure 1 shows a fluctuation, represented by a star, in a particular spacetime
cell. The signal, represented by bursts, is transmitted to the two adjacent
spatial cells in the next time step. Hence those two cells have correlated
fluctuations. This type of correlation can arise from either white or colored
noise. Figure 2 shows a fluctuation in one spacetime cell with its signal
transmitted to two spacetime cells two time steps later. This type of
correlation can also happen with either white or colored noise. Figure 3 shows
a situation that only happens with colored noise. The two stars are
correlated, and their signals lead to correlations between the same two cells
as shown in the previous figures.
Self-correlations arise from correlations in the same spatial cell. For white
noise this means the star and the burst are in the same spacetime cell. In
discretized spacetime this leads to a Kronecker $\delta$-function in $\xi$,
while in the continuum limit this leads to a Dirac $\delta$-function. The
latter is somewhat unphysical, since all correlations have some finite extent.
For colored noise, the self-correlation begins in the cell hosting the
original fluctuation, and then continues in subsequent time steps but always
in the same spatial cell due to the time-correlated nature of colored noise.
Noise generated at a previous time in the same spatial cell will
hydrodynamically evolve to a correlated charge fluctuation in a different
spatial fluid cell. Hence the self-correlation will be non-trivial for colored
noise.
Figure 1: An example of either white or colored noise. A fluctuation in one
cell, represented by a star, causes a correlation between two cells in the
next time step, represented by bursts, separated in space from each other and
from the original fluctuation. (color online) Figure 2: An example of either
white or colored noise. A fluctuation in one cell, represented by a star,
causes a correlation between two cells two time steps later, represented by
bursts, but only one is separated in space from the original fluctuation.
(color online) Figure 3: An example of colored noise. Fluctuations at the same
point in space but at different times are correlated, as represented by the
stars. This results in a correlation between the two cells, represented by
bursts. (color online)
Figure 4 shows another way to visualize the colored Cattaneo noise. At a fixed
spatial cell, correlations arise at different times due to $\tau_{Q}>0$.
Correlations also propagate to other spatial cells with increasing time via a
Green’s function. The mathematical formalism and details of how it is
implemented numerically with be presented in the following sections.
Figure 4: Schematic of the lattice setup for Catteneo noise.
The final charge correlations are determined at some $\tau_{f}$. One must
integrate over all prior times $\tau_{i}\leq\tau\leq\tau_{f}$ to obtain the
final time charge correlators. Thus one can define self-correlations as the
correlation of a charge fluctuation generated in $\xi_{1}$ at final time
$\tau_{f}$ with another charge fluctuation generated at the same $\xi_{1}$ but
at a previous time and hence had time to travel to a different $\xi_{2}$ at
$\tau_{f}$. It is non-trivial for colored noise because colored noise
generated in same $\xi$ are correlated in time.
One can go further and consider the Gurtin-Pipkin noise gurtin which
introduces a noise correlation in spatial rapidity in addition to the
correlation in proper time. Gurtin-Pipkin noise has been dealt with
analytically in Ref. kapustayoung . In Cartesian coordinates Gurtin-Pipkin
noise results in the following diffusion equation
$\left[\frac{\partial}{\partial
t}-D_{Q}\nabla^{2}+\tau_{Q}\frac{\partial^{2}}{\partial
t^{2}}+\tau_{2}^{2}\frac{\partial^{3}}{\partial
t^{3}}-\tau_{3}D_{Q}\frac{\partial}{\partial t}\nabla^{2}\right]n_{Q}=0\,.$
(6)
Numerical simulation of Gurtin-Pipkin noise is deferred to a future work.
## III Diffusion in Boost Invariant 1+1 Hydrodynamics
This section is a mini-review of the problem addressed previously in Ref.
plumbergkapusta to help setup the use of numerical methods for solving the
resulting SDE. We will work in 1+1 dimensional boost-invariant Bjorken
hydrodynamics. The longitudinal boost-invariance implies that the initial
conditions for local variables are only functions of the proper time $\tau$.
We neglect the bulk and shear viscosities in order to focus on charge
transport.
The energy-momentum tensor for an ideal fluid is
$T^{\mu\nu}=wu^{\mu}u^{\nu}-pg^{\mu\nu}\,.$ (7)
We take the Landau-Lifshitz approach where $u^{\mu}$ is the velocity of energy
transport. The electric current takes the form
$J_{Q}^{\mu}=n_{Q}u^{\mu}+\Delta J^{\mu}$ (8)
where $n_{Q}$ is the proper charge density and $\Delta J^{\mu}$ is the
dissipative part. In first-order viscous fluid dynamics $\Delta J^{\mu}$ takes
the form
$\Delta J^{\mu}=D_{Q}\Delta^{\mu}n_{Q}=\sigma_{Q}\Delta^{\mu}\mu_{Q}$ (9)
where $\mu_{Q}$ is the charge chemical potential, $\sigma_{Q}$ is the charge
conductivity and $\Delta^{\mu}$ is the transverse derivative
$\Delta^{\mu}=\partial^{\mu}-u^{\mu}(u\cdot\partial)\,.$ (10)
Conventional charge diffusion follows the usual diffusion equation
$\left(\frac{\partial}{\partial t}-D_{Q}\nabla^{2}\right)n_{Q}=0\,.$ (11)
The diffusion constant $D_{Q}$ and charge conductivity are related by the
Einstein relation $D_{Q}=\sigma_{Q}/\chi_{Q}$, where $\chi_{Q}$ is the
electric charge susceptibility defined by
$\chi_{Q}=\frac{\partial n_{Q}(T,\mu_{Q})}{\partial\mu_{Q}}\,.$ (12)
The diffusion equation leads to an infinite speed of propagation which is
unphysical and not suitable for hydrodynamic simulations of heavy-ion
collision. Therefore the usual diffusion equation is replaced by one with a
double derivative in time with a relaxation time factor $\tau_{Q}$.
$\left(\frac{\partial}{\partial
t}-D_{Q}\nabla^{2}+\tau_{Q}\frac{\partial^{2}}{\partial t^{2}}\right)n_{Q}=0$
(13)
This equation is called the Cattaneo equation catteneo . It is a combination
of the diffusion equation with the wave equation. The dissipative current gets
modified to
$\Delta
J^{\mu}=D_{Q}\Delta^{\mu}\left[\frac{1}{1+\tau_{Q}(u\cdot\partial)}\right]n_{Q}$
(14)
One can show that high frequency waves travel at a speed of
$v_{Q}=\sqrt{D_{Q}/\tau_{Q}}$ kapustayoung . The fluctuation-dissipation
theorem relates the two-point function, which provides a measure of the
variance of fluctuations, to the dissipation from diffusion. A stochastic
noise term $I^{\mu}$ is therefore added to the charge current.
$J^{\mu}=n_{Q}u^{\mu}+\Delta J^{\mu}+I^{\mu}$ (15)
One-point functions vanish and the two-point functions are determined by the
fluctuation-dissipation theorem. For the usual diffusion equation
$\langle I^{\mu}(x)\rangle=0\qquad\langle
I^{\mu}(x_{1})I^{\nu}(x_{2})\rangle=2\sigma_{Q}T\,h^{\mu\nu}\delta(x_{1}-x_{2})$
(16)
where $h^{\mu\nu}=u^{\mu}u^{\nu}-g^{\mu\nu}$ is the transverse projector. This
is white noise. In the Catteneo equation the fluctuations are
$\langle
I^{i}(x_{1})I^{j}(x_{2})\rangle=\frac{\sigma_{Q}T}{\tau_{Q}}\delta(\mbox{\boldmath$x$}_{1}-\mbox{\boldmath$x$}_{2})\,e^{-|t_{1}-t_{2}|/\tau_{Q}}\,\delta_{ij}$
(17)
The delta function in time is replaced by an exponential decay function. In
the limit $\tau_{Q}\rightarrow 0$ this two-point function becomes the Dirac
$\delta$\- function for white noise.
The following are the relations between the Cartesian coordinates and the
proper time and spatial rapidity appropriate for the Bjorken model.
$\displaystyle\begin{aligned} t&=&\tau\cosh\xi\qquad z&=\tau\sinh\xi\\\
\tau&=&\sqrt{t^{2}-z^{2}}\qquad\xi&=\tanh^{-1}\left(\frac{z}{t}\right)\\\
\end{aligned}$ (18)
The flow velocity is
$u^{0}=\cosh\xi\quad u^{z}=\sinh\xi\,.$ (19)
The transverse derivatives are
$\displaystyle\Delta^{0}=-\frac{\sinh\xi}{\tau}\frac{\partial}{\partial\xi}\qquad\Delta^{3}=-\frac{\cosh\xi}{\tau}\frac{\partial}{\partial\xi}\quad\text{with}\quad
u\cdot\partial=\frac{\partial}{\partial\tau}\,.$ (20)
The fluctuating contribution to the current is written as
$\displaystyle I^{0}$ $\displaystyle=$ $\displaystyle
s(\tau)f(\xi,\tau)\sinh\xi$ (21) $\displaystyle I^{3}$ $\displaystyle=$
$\displaystyle s(\tau)f(\xi,\tau)\cosh\xi\,.$ (22)
The entropy density $s$ is factored out to make $f$ dimensionless. The
background fluid equations for the proper charge density and entropy density
are
$\displaystyle\frac{ds}{d\tau}+\frac{s}{\tau}=0\;\;$
$\displaystyle\Rightarrow$
$\displaystyle\;\;s(\tau)=\frac{s_{i}\tau_{i}}{\tau}$ (23)
$\displaystyle\frac{dn_{Q}}{d\tau}+\frac{n_{Q}}{\tau}=0\;\;$
$\displaystyle\Rightarrow$
$\displaystyle\;\;n_{Q}(\tau)=\frac{n_{i}\tau_{i}}{\tau}\,.$ (24)
These are a manifestation of the conservation of entropy and charge,
respectively. The $s_{i}$ and $n_{i}$ are the densities at some initial time
$\tau_{i}$. We take the initial proper charge density $n_{i}$ to be zero,
hence the average charge density for subsequent times is zero as well.
Now let us look at the charge current conservation equation
$\partial_{\mu}J^{\mu}=0$. It is convenient to define the variable
$X=\tau\delta n$ because, in the absence of fluctuations, this quantity is
conserved during the hydrodynamic evolution. After a few steps of algebra the
full charge conservation equation becomes
$\left[\frac{\tau}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau}{D_{Q}\chi_{Q}T}\right)\right]\frac{\partial
X}{\partial\tau}+\frac{\tau_{Q}\tau}{D_{Q}\chi_{Q}T}\frac{\partial^{2}X}{\partial\tau^{2}}-\frac{1}{\tau\chi_{Q}T}\frac{\partial^{2}X}{\partial\xi^{2}}$
$+\left[\frac{\tau
s}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau
s}{D_{Q}\chi_{Q}T}\right)\right]\frac{\partial
f}{\partial\xi}+\frac{\tau_{Q}\tau
s}{D_{Q}\chi_{Q}T}\frac{\partial^{2}f}{\partial\xi\partial\tau}=0\,.$ (25)
For the case $\tau_{Q}=0$ (usual diffusion equation) this simplifies to
$\frac{\partial
X}{\partial\tau}-\frac{D_{Q}}{\tau^{2}}\frac{\partial^{2}X}{\partial\xi^{2}}+s\frac{\partial
f}{\partial\xi}=0\,.$ (26)
Due to boost invariance it is useful to use the Fourier transform
$X(\xi,\tau)=\int_{-\infty}^{\infty}\frac{dk}{2\pi}e^{ik\xi}\tilde{X}(k,\tau)\,,$
(27)
and similarly for $f$. Then the SDE for white noise is
$\frac{\partial}{\partial\tau}\tilde{X}+\frac{D_{Q}k^{2}}{\tau^{2}}\tilde{X}=-iks\tilde{f}$
(28)
and for colored Cattaneo noise
$\left[\frac{\tau}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau}{D_{Q}\chi_{Q}T}\right)\right]\frac{\partial\tilde{X}}{\partial\tau}+\frac{\tau_{Q}\tau}{D_{Q}\chi_{Q}T}\frac{\partial^{2}\tilde{X}}{\partial\tau^{2}}+\frac{k^{2}}{\tau\chi_{Q}T}\tilde{X}$
$=-ik\left[\frac{\tau
s}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau
s}{D_{Q}\chi_{Q}T}\right)\right]\tilde{f}-i\frac{k\tau_{Q}\tau
s}{D_{Q}\chi_{Q}T}\frac{\partial\tilde{f}}{\partial\tau}\,.$ (29)
For the sake of comparison and for definiteness, we follow Ref.
plumbergkapusta and assume both $D_{Q}$ and $\tau_{Q}$ are constant within
the range of temperature to be considered. This means that high frequency
waves propagate with a constant value of $v_{Q}$. For the same reasons we
assume that $s\sim T^{3}$ and $\chi\sim T^{2}$. Hence $T\sim\tau^{-1/3}$.
## IV Solving the Stochastic Differential Equations
We start by solving the stochastic differential equation for white noise. As
explained earlier, we will solve it on a spacetime lattice and choose spacings
$\Delta\xi=0.09$ and $\Delta\tau=10^{-4}$ fm/c. We set the parameters such
that $(\chi(\tau_{f})T_{f})/(\tau_{f}\Delta\xi)=0.5122$ MeV3 fm-3. We source
the noise function $f$ from a normal distribution with mean $0$ and variance
$1/\sqrt{\Delta t\Delta\xi}$. The density-density correlator arising from the
noise fluctuation which is a solution to the SDE in our discretized system,
evaluted at the final time $\tau_{f}$, has the analytical form
$\langle\delta n(\xi_{1},\tau_{f})\delta
n(\xi_{2},\tau_{f})\rangle=\frac{\chi_{Q}(\tau_{f})T_{f}}{A\tau_{f}}\left[\frac{\delta_{\xi_{1},\xi_{2}}}{\Delta\xi}-\frac{1}{\sqrt{\pi
w^{2}}}e^{-(\xi_{1}-\xi_{2})^{2}/w^{2}}\right]$ (30)
where
$w^{2}=8D_{Q}\left(\frac{1}{\tau_{i}}-\frac{1}{\tau_{f}}\right)\,.$ (31)
In the continuum limit
$\delta_{\xi_{1},\xi_{2}}/\Delta\xi\rightarrow\delta(\xi_{1}-\xi_{2})$. The
parameters chosen for this work are the same as in Ref. plumbergkapusta ,
namely $\tau_{i}=0.5$ fm/c, $\tau_{f}=6.352$ fm/c, $T_{i}=350$ MeV, and
$T_{f}=150$ MeV. We use diffusion constant $D_{Q}=0.162\>\text{fm}$ which is
an average over the temperature interval from 150 to 350 MeV taken from Ref.
Aarts . The equation of state used is the same as in Ref. torres , which is
$\chi_{Q}=\frac{2}{3}T^{2}$ (including up, down and strange quarks).
The details of how we solve an SDE are discussed in Appendix A. The solution
is presented in Fig. 5. The dots represent the result of the SDE simulation
for ten million random events. The solid curve is the Gaussian from Eq. (30);
it overlays the dots within the width of the line. The Kronecker
$\delta$-function at $\xi=0$ is clearly evident.
Figure 5: White noise density-density correlation function for 10 million
events. The solid curve is the Gaussian from Eq. 30. (color online)
Next we turn to colored noise. We have to generate a noise that has the
desired correlation in proper time but is uncorrelated in rapidity. The way we
do that is by solving another SDE which is called the Langevin equation.
$f+\tau_{Q}\frac{\partial f}{\partial\tau}=\zeta$ (32)
Here $\zeta$ is the regular white noise. The relaxation time $\tau_{Q}$
smoothens the Dirac $\delta$-correlation in proper time. The $\tau_{Q}$ also
introduces the maximum mode velocity to be $v_{Q}^{2}=D_{Q}/\tau_{Q}$, thereby
removing instantaneous signal propagation. The analytical solution to the
Langevin equation (with rapidity dependences suppressed) is
$\langle
f(\tau_{1})f(\tau_{2})\rangle=\frac{N(\tau_{2})}{4\pi\tau_{Q}}\left[e^{-|\tau_{1}-\tau_{2}|/\tau_{Q}}-e^{(2\tau_{i}-\tau_{1}-\tau_{2})/\tau_{Q}}\right]\equiv{\cal
N}(\tau_{1},\tau_{2})\,.$ (33)
The derivation is given in Appendix B. The numerically computed two-point
function is plotted in Fig. 6. The expected result (33) and the numerical
result are consistent for ten million simulated events.
Figure 6: Comparison of numerical and analytical results for $v_{Q}^{2}=0.16$
with rapidity dependences suppressed. (color online)
The grid sizes chosen ensures that they obey the Courant Friedrichs Lewy (CFL)
condition CFL . This condition states that the numerical domain of dependence
of any point in space and time must include the analytical domain of
dependence. Physically, this condition amounts to a signal propagating no more
than one spatial cell away during one time step. For speed $v_{Q}$ being a
constant, this amounts to the condition $\Delta\tau/\tau<\Delta\xi/v_{Q}$.
Figure 7 shows the dependence of the two-point correlator for two very
different values of the propogation speed, or equivalently the relaxation time
$\tau_{Q}$.
Figure 7: Variation of the density-density correlator with the propogation
speed $v^{2}_{Q}$. (color online)
## V Characterizing Self-Correlations
The self-correlations are trivial for the case of white noise; it’s a Dirac
$\delta$-function. Even the two-point correlation function of a free Boltzmann
gas has a $\delta$\- function term landau1 ; landau2 .
$\langle\delta n(\textbf{x}_{1})\delta n(\textbf{x}_{2})\rangle=\chi
T\delta^{3}(\textbf{x}_{1}-\textbf{x}_{2})+\cdots$ (34)
This is explained in the Ref. landau2 where $\langle(\Delta
N)^{2}\rangle=\chi TV$. One can see this in Eq. (30), where the denomintor
$\tau A$ is the Jacobian factor from the Bjorken expansion instead of
stationary Cartesian coordinates. Experiments measure just the two-particle
correlation and hence we have to subtract the self-correlation spratt . The
challenge is to characterize the self-correlations in the presence of colored
noise, since it is no longer a Dirac $\delta$-function.
Figure 8 shows the numerically computed self-correlation. If we subtract the
two-point correlation in Fig. 5 from that presented in this figure, we get the
expected Gaussian. This is shown in Fig. 9 where it is compared with the
analytical Gaussian function in Eq. (30) for 1 million events.
Figure 8: The self-correlation. Figure 9: The solid curve is the expected
Gaussian while the dots represent the result of the SDE simulation for 1
million random events. (color online)
Now we move on to the meaning of self-correlation for colored noise. Based on
the prescription of self-correlation that we discussed in the introduction, we
consider the schematic diagram in Fig. 10. We are interested in noise sources
at one particular $\xi$ because noise generated at any other $\xi$ would be
uncorrelated.
Figure 10: Schematic of the self-correlation. The star denotes a noise source
and the bursts are the charge fluctuations resulting from noise. (color
online)
Let us try to understand what the analytical formula for this would look like.
We start with the following expression for the charge fluctuation in
$k$-space.
$\delta\tilde{n}(k,\tau)=-\frac{1}{\tau}\int_{\tau_{i}}^{\tau}d\tau^{\prime}s(\tau^{\prime})\tilde{G}(k,\tau,\tau^{\prime})\tilde{f}(k,\tau^{\prime})\,.$
(35)
Here $\tilde{G}$ is the Green’s function for the homogeneous part of the SDE
(29), which can be written down in terms of Kummer’s function for the
temperature dependences listed after that equation plumbergkapusta . This
gives the full form of the two-point correlation function as in Eq. (49) of
Ref. plumbergkapusta .
$\displaystyle\langle\delta n(\xi_{1},\tau_{f})\delta
n(\xi_{2},\tau_{f})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{f}^{2}}\int\frac{dk}{2\pi}e^{ik(\xi_{1}-\xi_{2})}\int
d\tau^{\prime}s(\tau^{\prime})\int d\tau^{\prime\prime}s(\tau^{\prime\prime})$
(36) $\displaystyle\times$
$\displaystyle\tilde{G}(k,\tau_{f},\tau^{\prime})\;\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})\,{\cal
N}(\tau^{\prime},\tau^{\prime\prime})\,.$
Following Eqs. (54) and (55) of Ref. plumbergkapusta , we can write the self-
correlation term as
$\displaystyle\langle\delta n(\xi_{1},\tau_{f})$ $\displaystyle\delta n$
$\displaystyle(\xi_{2},\tau_{f})\rangle_{\text{self}}$ (37) $\displaystyle=$
$\displaystyle\frac{\chi_{Q}(\tau_{f})T_{f}}{A\tau_{Q}}\int\frac{d\tau^{\prime\prime}}{\tau^{\prime\prime}}\left[e^{-(\tau_{f}-\tau_{2})/\tau_{Q}}-e^{-(\tau_{f}+\tau_{2}-2\tau_{i})/\tau_{Q}}\right]\int\frac{dk}{2\pi}e^{ik(\xi_{1}-\xi_{2})}\frac{\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})}{ik}$
$\displaystyle=$
$\displaystyle\frac{s(\tau_{f})}{D_{Q}}\int\frac{dk}{2\pi}e^{ik(\xi_{1}-\xi_{2})}\int
d\tau^{\prime\prime}s(\tau^{\prime\prime})\;\frac{\tilde{G}(k,\tau_{f},\tau_{f})}{ik}\;\frac{\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})}{ik}\,{\cal
N}(\tau_{f},\tau^{\prime\prime})\,.$
Recall from Ref. plumbergkapusta that $\tilde{G}(k,\tau_{f},\tau_{f})=ik$. It
denotes a noise fluctuation that was generated at the final time and didn’t
have to move anywhere. The $\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})$ is a
noise fluctuation generated at a time $\tau^{\prime\prime}<\tau_{f}$ and then
moved to $\xi_{2}$ at $\tau_{f}$. For white noise, fluctuations generated at
two separate spacetime points can’t be correlated, so the fluctuation
generated at $\tau_{f}$ is only correlated to itself. Hence for white noise,
$\tau_{Q}\to 0$, and the self-correlation is just a Dirac $\delta$-function.
As $\tau_{Q}$ increases, the more backward in time the noise sources would be
correlated. Once generated the noise will travel and give rise to a correlated
electric charge fluctuation further away in spacetime rapidity. Hence we
expect the self-correlation term to be more spread out in spacetime rapidity.
One can use the same SDE solver for generating the self-correlation. The only
change is that the Green’s function $\tilde{G}$ is replaced by
$\tilde{G}/(ik)$ when solving for the charge density fluctuation.
$\displaystyle\langle$ $\displaystyle\delta n$
$\displaystyle(\xi_{2},\tau_{f})f(\xi_{1},\tau_{f})\rangle$ (38)
$\displaystyle=$
$\displaystyle\left\langle\int\frac{dk}{2\pi}e^{ik\xi_{2}}\delta\tilde{n}(k,\tau_{f})\int\frac{dk_{1}}{2\pi}e^{ik_{1}\xi_{1}}\tilde{f}(k,\tau_{f})\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\tau_{f}}\int^{\tau_{f}}_{\tau_{i}}d\tau^{\prime}s(\tau^{\prime})\int\frac{dk}{2\pi}e^{ik\xi_{2}}\frac{\tilde{G}(k,\tau^{\prime},\tau_{f})}{ik}\int\frac{dk_{1}}{2\pi}e^{ik_{1}\xi_{1}}\langle\tilde{f}(k,\tau^{\prime})\tilde{f}(k_{1},\tau_{f})\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\tau_{f}}\int^{\tau_{f}}_{\tau_{i}}d\tau^{\prime}s(\tau^{\prime})\int\frac{dk}{2\pi}e^{ik(\xi_{2}-\xi_{1})}\frac{\tilde{G}(k,\tau^{\prime},\tau_{f})}{ik}\mathcal{N}(\tau^{\prime},\tau_{f})$
$\displaystyle=$
$\displaystyle\frac{D_{Q}}{s_{f}\tau_{f}}\,\frac{\chi_{f}T_{f}}{A\tau_{Q}}\int^{\tau_{f}}_{\tau_{i}}\frac{d\tau^{\prime}}{\tau^{\prime}}\left[e^{-|\tau_{f}-\tau^{\prime}|/\tau_{Q}}-e^{(2\tau_{i}-\tau_{f}-\tau^{\prime})/\tau_{Q}}\right]\int\frac{dk}{2\pi}e^{ik(\xi_{2}-\xi_{1})}\frac{\tilde{G}(k,\tau^{\prime},\tau_{f})}{ik}$
Note that in going to the second step from the first step, we have used
$\tilde{G}/(ik)$ and not $\tilde{G}$. The implication is that for generating
the self-correlations, we will be using the differential equation whose
Green’s function is going to be $\tilde{G}/(ik)$, instead of $\tilde{G}$.
Thus, we arrive at the following relation for self-correlations.
$\langle\delta n(\xi_{1},\tau_{f})\delta
n(\xi_{2},\tau_{f})\rangle_{\text{self}}=\frac{s(\tau_{f})\tau_{f}}{D_{Q}}\langle\delta
n(\xi_{1},\tau_{f})f(\xi_{2},\tau_{f})\rangle\,.$ (39)
If $\tilde{G}/(ik)$ is our desired Green’s function, then $Z\equiv(\tau\delta
n(\xi,\tau))/(\tau_{f}s(\tau_{f}))=\delta n(\xi,\tau)/s(\tau)$ satisfies the
following equation
$\left(z^{2}+2z\frac{\tau_{Q}}{\tau_{i}}\right)\frac{\partial Z}{\partial
z}+z^{2}\frac{\tau_{Q}}{\tau_{i}}\frac{\partial^{2}Z}{\partial
z^{2}}-v_{Q}^{2}\frac{\tau_{Q}}{\tau_{i}}\frac{\partial^{2}Z}{\partial\xi^{2}}+\left(z+\frac{\tau_{Q}}{\tau_{i}}\right)f+z\frac{\tau_{Q}}{\tau_{i}}\frac{\partial
f}{\partial z}=0\,,$ (40)
where $z=\tau/\tau_{i}$. This is the same as Eq. (25) except that $\partial
f/\partial\xi$ is replaced by $f$ and $\partial^{2}f/\partial\xi\partial\tau$
is replaced by $\partial f/\partial\tau$. The justification is discussed in
the Appendix C.
In Fig. 11 we show the self-correlation at the final time $\tau_{f}$ for
various values of $\tau_{Q}$. As the speed of propagation decreases, the
height of the self-correlation decreases and widens. As a check, the limit
$\tau_{Q}\to 0$ is shown in Fig. 12.
Figure 11: Numerical results for self-correlations for colored noise. (color
online) Figure 12: Numerical results for self-correlation for white noise.
## VI Balance Functions
Balance functions are described in Ref. spratt . The width of a balance
function plotted against particle rapidity is a measure of the diffusion.
Balance functions have been experimentally studied by the ALICE and STAR
collaborations balance1 ; balance2 ; balance3 ; balance4 . Reference
ling_springer_stephanov studied the effect of white noise in balance
functions and compared their analytical results with experimental data.
Reference plumbergkapusta calculated balance functions for colored noise. We
will see how the widths of balance functions change if we vary the speed of
propagation of signals in case of Catteneo noise.
To see the effect of charge fluctuations in particle spectra we have to
calculate how the fluctuations freeze-out. The freeze-out happens when the
system has expanded and cooled to the extent that thermal equilibrium can no
longer be maintained. Then hadrons freeze-out and free-stream to the
detectors. The standard procedure to calculate freeze-out abundances of
particles is the Cooper-Frye prescription cooper_frye . This formula gives us
the distribution of emitted particles on a freeze-out hypersurface
$\Sigma_{f}$. This procedure has already been performed for this
hydrodynamical model in Refs. kapusta2012 ; plumbergkapusta ;
ling_springer_stephanov ; torres We will just give the salient features of
that calculation here.
$E\frac{dN}{d^{3}p}=d\int_{\Sigma_{f}}\frac{d^{3}\sigma_{\mu}}{(2\pi)^{3}}p^{\mu}f({\mbox{\boldmath$x$},\mbox{\boldmath$p$}})$
(41)
Here $d$ is the degeneracy of the particle species under consideration. We
take the distribution function to be the relativistic Boltzmann
$f({\mbox{\boldmath$x$},\mbox{\boldmath$p$}})=e^{-(u\cdot p-\mu)/T}\,,$ (42)
where $\mu$ is the chemical potential for that particle. The energy flux
through an infinitesimal freeze-out fluid cell is given by
$d^{3}\sigma_{\mu}p^{\mu}=\tau_{f}\,d\xi\,d^{2}x_{\perp}m_{\perp}\cosh(y-\xi)\,.$
(43)
The variable $y$ represents the particle rapidity
$p^{\mu}=(m_{\perp}\cosh y,p_{x},p_{y},m_{\perp}\sinh y)$ (44)
with $m_{\perp}=\sqrt{m^{2}+p_{\perp}^{2}}$ the transverse mass.
The average number of particles per unit rapidity at the final freeze-out time
is
$\left\langle\frac{dN}{dy}\right\rangle=\frac{dA\tau_{f}T_{f}^{3}}{4\pi^{2}}\int^{\infty}_{-\infty}\frac{dx}{\cosh^{2}x}\Gamma\left(3,\frac{m}{T_{f}}\cosh
x\right)\,.$ (45)
Reference plumbergkapusta calculates the fluctuation in this quantity due to
a $\mu$ around the freeze-out $\mu_{f}=0$. After a few more steps of algebra,
the fluctuation in $\frac{dN}{dy}$ reads
$\delta\left(\frac{dN}{dy}\right)=\frac{dA\tau_{f}T_{f}^{2}}{4\pi^{2}}\int
d\xi\,\delta n\,F_{n}(y-\xi)$ (46)
where $F_{n}$ is the smearing function
$F_{n}(x)=\frac{1}{\chi_{Q}\cosh^{2}x}\Gamma\left(3,\frac{m}{T_{f}}\cosh
x\right)\,.$ (47)
Using this in the definition of the charge balance function we arrive at the
Eq. (74) in Ref. plumbergkapusta .
$B(\Delta
y)=\frac{\langle\delta\left(dN/dy_{1}\right)\delta\left(dN/dy_{2}\right)\rangle}{\langle
dN/dy\rangle}=\frac{dA\tau_{f}T_{f}}{4\pi^{2}}\frac{C(\Delta
y)}{Q(m/T_{f})}\,.$ (48)
Here
$C(\Delta y)=2\pi\int
d\xi_{1}d\xi_{2}\,F_{n}(y_{1}-\xi_{1})\,F_{n}(y_{2}-\xi_{2})\,C_{nn}(\xi_{1}-\xi_{2},\tau_{f})\,.$
(49)
The two-point correlator for the charge fluctuation is
$C_{nn}(\xi_{1}-\xi_{2},\tau_{f})$ which is obtained from the solution of the
SDE. The function $Q$ is given by
$Q\left(\frac{m}{T_{f}}\right)=\int^{\infty}_{-\infty}\frac{dx}{\cosh^{2}x}\Gamma\left(3,\frac{m}{T_{f}}\cosh
x\right)\,.$ (50)
Let us first demonstrate the trivial self-correlation for white noise in terms
of the balance function for pions. Figure 13 shows the balance function for
the full unsubtracted correlation function for white noise. Notice the
positive and negative part of the curve; this is because the full two-point
correlation for white noise is composed of a positive self-correlation and a
negative piece which does not include any self-correlation. The balance
function for the self-correlation part only is shown in Fig. 14. When this is
subtracted from Fig. 13 one obtains the so-called subtracted balance function
shown in Fig. 15.
Figure 13: Balance function for the full white noise two-point function.
Figure 14: Balance function for the self-correlation of white noise. Figure
15: Balance function for the pure two-point function of white noise.
We follow the same procedure for carrying out cancellations of the
contributions arising from the self-correlations for colored noise to the
balance function. Figure 16 shows the full unsubtracted balance functions for
various values of $v_{Q}$ for colored noise. Figure 17 shows the self-
correlation part only, and Fig. 18 shows the subtracted balance functions. The
width of the subtracted balance function denotes the diffusion distance. That
width increases with increasing $v_{Q}$, as expected, since it represents the
rapidity interval over which the average charge pair has diffused to by freeze
out.
Figure 16: Balance function for the full unsubtracted two-point function.
(color online) Figure 17: Balance function for the self-correlation part of
two-point function. (color online) Figure 18: Balance function for the
subtracted two-point correlation function. (color online)
We estimated the error in our numerical simulations using the jackknife
method. This method estimates the error of statistics without making any
assumptions about the distribution that generated the data. It only uses the
sample provided. We create jackknife samples over the whole data set which are
“leave-one-out” data sets. In our case, we consider the two-point correlation
statistic $S$ on the original sample size of $10^{7}$ events. We leave out the
$i_{th}$ event to create the $i_{th}$ jackknife statistic $S_{i}$. The average
of the jackknife sample is $S_{\rm avg}=\sum_{i}S_{i}/n$. The jackknife error
is then estimated as
$\sigma_{\rm jack}=\sqrt{\frac{n-1}{n}\sum_{i}(S_{i}-S_{\rm avg})^{2}}$ (51)
The error we observe on $\langle\delta n\delta n\rangle$ is of the order of
$10^{-2}\;\text{MeV}^{3}\>\text{fm}^{-3}$. This amounts to
$\sigma_{\langle\delta n\delta n\rangle}/\langle\delta n\delta n\rangle\approx
10^{-3}$. We give a representative plot of the error bounds for
$v_{Q}^{2}=1/10$ in Fig. 19. The bounds are visible only when zoomed in. This
shows that for $10^{7}$ events, the statistical error in our simulations turn
out to be negligible.
Figure 19: Jackknife error bounds for $v_{Q}^{2}=1/10$. (color online)
## VII Conclusions
State-of-the-art modeling of high energy nuclear collisions uses relativistic
2nd order viscous hydrodynamics. The fluctuation-dissipation theorem says that
viscosity and thermal fluctuations are intricately connected. Although
thousands of particles are produced in these collisions, that is still
immensely smaller than Avogadro’s number. Therefore it has become apparent
that thermal fluctuations really ought to be part of the standard model for
heavy ion collisions kapusta2012 . Fully 3+1 dimensional hydrodynamic
simulations are required which presents a major challenge for implementation
of thermal noise. The goal of this paper is to understand the numerical
methods necessary to do this and, furthermore, how to subract self-
correlations from the numerically computed two-point correlators in order to
compare with experiment. We chose to study causal electric charge diffusion in
the boost-invariant 1+1 dimensional Bjorken model for two reasons. First, a
Cattaneo description of diffusion propagates signal at a finite speed which is
a necessity in heavy ion collisions. Second, this simple model was studied and
solved with essentially analytic methods plumbergkapusta against which we can
compare to verify the validity of the purely numerical approach.
We introduced the noise term in the dissipative charge conservation equation,
which in our case is the Catteneo noise. We simulated the stochastic
differential equations that arise from the electric charge conservation
equations. The way we solve the stochastic differential equations is by using
normal random number generators with a specific, well-defined variance and
then interpreting the derivatives of the noise in terms of what they mean when
integrating by parts. The whole machinery on how to handle the noise is
discussed in Appendix A. We used this methodology in simulating the white
noise charge conservation equation and obtained the expect result. Then we
generated colored Catteneo noise using a Langevin equation. We solved the full
colored noise charge conservation equation and again obtained the expected
result. The two-point charge correlator consists of two pieces. The first is
the self-correlation, which is a manifestation of the stochastic nature of the
dynamics. Once this piece is subtracted off, we are left with the physically
relevant two-point correlation function. The self-correlation is a trivial
Dirac $\delta$-function in the case of white noise, but is more complicated
for colored noise. In this work, we gave a physically insightful
interpretation of the meaning of self-correlation in the case of colored
noise. This interpretation allows us to use the stochastic differential
equation solver we developed to generate the self-correlations.
In the case of the white noise, we populated the whole spacetime lattice with
noise source terms uncorrelated to each other. It is obvious that all the
individual noise terms are not required to calculate the final two-point
correlation function, but more than a single noise term is necessary. Hence
Monte-Carlo simulations will be insufficient to reproduce the results for
colored noise. One can, however, speed up the stochastic differential equation
solving procedure by removing noise terms that are outside the causal past of
the spacetime points for which we want to calculate the twopoint correlations.
We used the results obtained to compute the balance functions for pions within
the context of this model. As one would expect, reducing the speed of
propagation of signal leads to narrowing of the balance functions and to a
corresponding increase in their height at small rapidities. As done previously
in Ref. plumbergkapusta we neglect the contributions from resonance decays to
the measured particle spectra used in the balance functions. Our results are
in good quantitative agreement with that previous study. The numerical method
used in this paper is verified.
Future work entails furthering the current methodology to a full 3+1
dimensional fluid dynamical models of heavy ion collisions such as MUSIC music
. Further, the prescription for self-correlations given in this paper for
Catteneo noise can be straightforwardly extended to the case of shear and bulk
viscosity and thermal conductivity, the details of which are deferred to a
future work. Since baryon charge conductivity diverges near a critical point,
this study can be extended to study charge fluctuations near the purported QCD
critical point, which is also deferred to future work. Another possible
direction of future work would be to study the higher order cumulants in the
presence of colored noise. Since two-point correlations and higher order
cumulants are expected to diverge near a QCD critical point, the ultimate
culmination of the present work would be to characterize the noisy
hydrodynamics of near-critical point behavior of heavy ion collisions.
## VIII Acknowledgements
A. D. thanks Gaurav Nirala for enlightening discussions. We thank Chun Shen
for suggesting the jackknife method. This work was supported by the U.S. DOE
Grant No. DE-FG02-87ER40328. C. P. acknowledges support from the CLASH project
(KAW 2017-0036). The authors acknowledge the Minnesota Supercomputing
Institute (MSI) at the University of Minnesota for providing resources that
contributed to the research results reported within this paper.
## Appendix A Numerical Simulation of SDEs
In this appendix, we show our procedure for representing the Dirac delta
function and its derivatives on a discrete lattice.
White noise is defined as $\langle
f(x)f(x^{\prime})\rangle=\delta(x-x^{\prime})$ and $\langle f(x)\rangle=0$.
This implies that
$\int dx\>g(x)\langle f(x)f(x^{\prime})\rangle=g(x^{\prime})$ (52)
On a discrete lattice, the integral becomes a sum over lattice points and $dx$
becomes the lattice spacing $\Delta x$. Hence
$g(x_{i})=\sum_{i^{\prime}}g(x_{i^{\prime}})\langle
f(x_{i})f(x_{i^{\prime}})\rangle\Delta
x=\sum_{i^{\prime}}g(x_{i^{\prime}})\left(\frac{\delta_{ii^{\prime}}}{\Delta
x}\right)\Delta x$ (53)
From the above, we can conclude
$\langle f(x_{i})f(x_{i^{\prime}})\rangle=\frac{\delta_{ii^{\prime}}}{\Delta
x}$ (54)
The $\delta_{ii^{\prime}}/\Delta x$ becomes a Dirac-delta function in the
limit $\Delta x\rightarrow 0$ which is the continuous case. Therefore we
sample the white noise function $f$ from a Normal distribution of mean $0$ and
standard deviation $1/\sqrt{\Delta x}$. We use a random number generator for a
large number of instances ($10^{6}$) and compute the two-point function. It
gives us the variance as the peak of a Kronecker delta at $x=0$. This is
illustrated in the following figure. We used $\Delta x=0.09$.
Figure 20: Two-point function of $f$ for 1 million events.
Next, we investigate the correlation between white noise $f$ and its
derivative $df/dx$. The two-point function $\langle
f(x)f^{\prime}(x^{\prime})\rangle$ must then satisfy
$\int dx\,g(x)\langle f(x)f^{\prime}(x^{\prime})\rangle=\int
dx\,g(x)\frac{\partial}{\partial
x^{\prime}}\delta(x-x^{\prime})=\frac{\partial}{\partial x^{\prime}}\int
dx\,g(x)\delta(x-x^{\prime})=g^{\prime}(x^{\prime})$ (55)
The derivative is $g^{\prime}(x)=(g_{i+1}-g_{i})/\Delta x$ in a discrete
lattice. Replacing the integral by the sum, we get
$g^{\prime}(x_{i})=\sum_{i^{\prime}}g(x_{i^{\prime}})\langle
f(x_{i})f^{\prime}(x_{i^{\prime}})\rangle\Delta x=\frac{g_{i+1}-g_{i}}{\Delta
x}=\sum_{i^{\prime}}g(x_{i^{\prime}})\left(\frac{\delta_{i+1,i^{\prime}}-\delta_{i,i^{\prime}}}{\Delta
x^{2}}\right)\Delta x$ (56)
Hence
$\langle
f(x_{i})f^{\prime}(x_{i^{\prime}})\rangle=\frac{\delta_{i+1,i^{\prime}}-\delta_{i,i^{\prime}}}{\Delta
x^{2}}$ (57)
If we again use the previous random number generator and calculate the two
point function we get the results shown in Fig. 21.
Figure 21: The correlation between $f$ and its derivative for 1 million
events.
Similarly, we can look into the correlation of the derivative of white noise
with itself.
$\langle
f^{\prime}(x)f^{\prime}(x^{\prime})\rangle=\frac{\partial^{2}}{\partial
x\partial x^{\prime}}\delta(x-x^{\prime})$ (58)
$\int dx\,g(x)\langle f^{\prime}(x)f^{\prime}(x^{\prime})\rangle=\int
dx\,g(x)\frac{\partial^{2}}{\partial x\partial
x^{\prime}}\delta(x-x^{\prime})=-\int dx\,g(x)\frac{\partial^{2}}{\partial
x^{2}}\delta(x-x^{\prime})=-g^{\prime\prime}(x^{\prime})$ (59)
In the second step we performed an integration by parts. The second derivative
is defined in the discrete case as
$g^{\prime\prime}(x)=(g_{i+1}+g_{i-1}-2g_{i})/\Delta x^{2}$. Substituting the
discrete sum in place of the integral, we get
$-g^{\prime\prime}(x^{\prime})=-\sum_{i}g_{i}\frac{\delta_{i,i^{\prime}+1}+\delta_{i,i^{\prime}-1}-2\delta_{i,i^{\prime}}}{\Delta
x^{3}}\Delta x=\sum_{i^{\prime}}g(x_{i^{\prime}})\langle
f^{\prime}(x_{i})f^{\prime}(x_{i^{\prime}})\rangle\Delta x$ (60) $\langle
f^{\prime}(x_{i})f^{\prime}(x_{i^{\prime}})\rangle=-\frac{\delta_{i,i^{\prime}+1}+\delta_{i,i^{\prime}-1}-2\delta_{i,i^{\prime}}}{\Delta
x^{3}}$ (61)
Figure 22 shows what we get numerically.
Figure 22: The two-point correlation of the derivative of $f$ for 1 million
events.
Integration of white noise is called a random walk $W(z)$ which is a
succession of random steps as a function of $z$. It is defined by
$W=\int_{z_{i}}^{z}f(z)dz$ (62)
We can easily calculate the variance of $W$:
$\langle
W^{2}(z)\rangle=\int_{z_{0}}^{z}dz^{\prime}\int_{z_{0}}^{z}dz^{\prime\prime}\langle
f(z^{\prime})f(z^{\prime\prime})\rangle=\int_{z_{0}}^{z}dz^{\prime}\int_{z_{0}}^{z}dz^{\prime\prime}\delta(z^{\prime}-z^{\prime\prime})=\int_{z_{0}}^{z}dz^{\prime}=z-z_{0}$
(63)
On a discrete lattice, $W_{i+1}=W_{i}+f\Delta z$ where we source $f$ from a
normal distribution of mean $0$ and standard deviation $1/\sqrt{\Delta z}$.
$W_{n}=\sum_{i=1}^{n}f\Delta z$ (64)
Hence the variance is
$\displaystyle\langle W^{2}\rangle=\langle(\sum_{i=1}^{n}\Delta
zf)^{2}\rangle=\langle\sum_{i=1}^{n}(\Delta
zf)^{2}\rangle=\sum_{i=1}^{n}(\Delta z)^{2}\langle
f^{2}\rangle=\sum_{i=1}^{n}(\Delta z)^{2}\frac{1}{\Delta
z}=\sum_{i=1}^{n}\Delta z=z-z_{0}$
Figure 23 shows the numerical results.
Figure 23: Two point correlation of $W$ for 1 million events with $z-z_{i}=5$.
We are ready to take up a simple stochastic differential equation to solve.
Consider
$\frac{dX}{dz}=-\frac{\partial f}{\partial\xi}$ (66)
where $z$ has dimensions of time and $\xi$ is dimensionless. Let us define the
following two-point function
$\langle
f(z_{1},\xi_{1})f(z_{2},\xi_{2})\rangle=M\delta(z_{1}-z_{2})\delta(\xi_{1}-\xi_{2})$
Here $M$ has dimensions of time to make $f$ dimensionless. We calculate the
two-point function in $\xi$.
$\displaystyle\langle X(z_{f},\xi_{1})X(z_{f},\xi_{2})\rangle$
$\displaystyle=$ $\displaystyle\left\langle\int^{z_{f}}_{z_{i}}\frac{\partial
f}{\partial\xi}(\xi_{1})dz\int^{z_{f}}_{z_{i}}\frac{\partial
f}{\partial\xi}(\xi_{2})dz^{\prime}\right\rangle$ (67) $\displaystyle=$
$\displaystyle\int^{z_{f}}_{z_{i}}\int^{z_{f}}_{z_{i}}dzdz^{\prime}\left\langle\frac{\partial
f}{\partial\xi}(z,\xi_{1})\frac{\partial
f}{\partial\xi}(z^{\prime},\xi_{2})\right\rangle$ $\displaystyle=$
$\displaystyle-\int^{z_{f}}_{z_{i}}\int^{z_{f}}_{z_{i}}dzdz^{\prime}M\left(\frac{\delta_{i+1}+\delta_{i-1}-2\delta_{i}}{\Delta\xi^{3}}\right)\delta(z-z^{\prime})$
$\displaystyle=$
$\displaystyle-M(z_{f}-z_{i})\left(\frac{\delta_{i+1}+\delta_{i-1}-2\delta_{i}}{\Delta\xi^{3}}\right)$
We used Eq. (61) in the above calculation. The two-point function has
dimensions of time-squared and so is the expression on the right. On a
discrete lattice,
$X(z+\Delta z)=X(z)-\Delta z\times\frac{\Delta f}{\Delta\xi}$ (68)
Figure 24 shows the numerical results using $z_{f}-z_{i}=10$.
Figure 24: Two point correlation of $X$ for a million events.
## Appendix B Analytical Solution of Langevin Equation
The Langevin equation can be written as
$\frac{df(\tau)}{d\tau}=-\frac{1}{\tau_{Q}}f(\tau)+\frac{1}{\tau_{Q}}\zeta(\tau)$
(69)
Here $\zeta$ is white noise and $f$ is the Catteneo noise.
$\langle\zeta(\tau)\rangle=0\qquad\langle\zeta(\tau_{1})\zeta(\tau_{2})\rangle=N(\tau_{1})\delta(\tau_{1}-\tau_{2})$
(70)
It does not matter whether we use $N(\tau_{1})$ or $N(\tau_{2})$ because of
the Dirac-delta function. Let us multiply both sides by the factor
$e^{\tau/\tau_{Q}}$.
$\int_{\tau_{i}}^{\tau}\frac{d}{d\tau}(e^{\tau/\tau_{Q}}f)d\tau=\int_{\tau_{i}}^{\tau}\frac{e^{\tau/\tau_{Q}}}{\tau_{Q}}\zeta
d\tau$ (71)
$e^{\tau/\tau_{Q}}f(\tau)-e^{\tau_{i}/\tau_{Q}}f(\tau_{i})=\frac{1}{\tau_{Q}}\int^{\tau}_{\tau_{i}}\zeta(\tau)e^{(\tau^{\prime}-\tau)/\tau_{Q}}d\tau^{\prime}$
(72)
Let us set $f(\tau_{i})=0$. Another way to see this is in an equilibrium
system, the system does not have any initial conditions to be sensitive to.
Any fluctuations in $f(\tau)$ will then be solely due to the action of
$\zeta(\tau)$. Now we consider two separate times $\tau_{1}$, $\tau_{2}$.
$\displaystyle\begin{aligned} \langle
f(\tau_{1})f(\tau_{2})\rangle&=\frac{N}{\tau_{Q}^{2}}\int_{\tau_{i}}^{\tau_{1}}e^{(\tau^{\prime\prime}-\tau_{1})/\tau_{Q}}d\tau^{\prime\prime}\int_{\tau_{i}}^{\tau_{2}}e^{(\tau^{\prime}-\tau_{2})/\tau_{Q}}d\tau^{\prime}\delta(\tau_{1}-\tau_{2})\\\
&=\frac{N}{\tau_{Q}^{2}}\int_{\tau_{i}}^{\text{min}(\tau_{1},\tau_{2})}e^{(2\tau^{\prime\prime}-\tau_{1}-\tau_{2})/\tau_{Q}}d\tau^{\prime\prime}=\frac{N}{2\tau_{Q}}\left[e^{|\tau_{1}-\tau_{2}|/\tau_{Q}}-e^{(2\tau_{i}-\tau_{1}-\tau_{2})/\tau_{Q}}\right]\end{aligned}$
(73)
## Appendix C Constructing the self-correlations
Self-correlations are defined by Eq. (37). As discussed above, their dynamics
can be modeled by an equation whose (Fourier transformed) Green’s function is
related to the original Green’s function by
$\tilde{G}_{\mathrm{self}}(k,\tau,\tau^{\prime})=\frac{\tilde{G}(k,\tau,\tau^{\prime})}{ik}$
(74)
The original Green’s function is defined schematically by the stochastic
differential equation
$D_{1}X(\tau,\xi)=D_{2}\frac{\partial f}{\partial\xi}(\tau,\xi),$ (75)
where $D_{1}$ and $D_{2}$ are differential operators which contain no explicit
$\xi$-dependence (other than $\xi$-derivatives) and $f$ is the noisy source.
Fourier transforming the $\xi$-dependence to $k$ as before, this equation
becomes
$\tilde{D}_{1}\tilde{X}(\tau,k)=ik\tilde{D}_{2}\tilde{f}(\tau,k)$ (76)
and its solution is written in terms of the original Green’s function as
$\tilde{X}(k,\tau)=-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\tilde{G}(k;\tau,\tau^{\prime})\tilde{f}(k,\tau^{\prime})$
(77)
We therefore seek an ‘unphysical’ field $X_{\mathrm{self}}$ whose two-point
function corresponds to the self-correlations which need to be subtracted out.
This field solution will be generated by the expression
$\displaystyle\tilde{X}_{\mathrm{self}}(k,\tau)$ $\displaystyle=$
$\displaystyle-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\tilde{G}_{\mathrm{self}}(k;\tau,\tau^{\prime})\tilde{f}(k,\tau^{\prime})$
(78) $\displaystyle=$
$\displaystyle-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\frac{\tilde{G}(k;\tau,\tau^{\prime})}{ik}\tilde{f}(k,\tau^{\prime})$
$\displaystyle=$
$\displaystyle-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\tilde{G}(k;\tau,\tau^{\prime})\left(\frac{\tilde{f}(k,\tau^{\prime})}{ik}\right)$
The self-correlations can be therefore straightforwardly generated by
replacing $\tilde{f}\to\tilde{f}/(ik)$ in $k$-space, which amounts to
discarding the $\xi$-derivative in Eq. (75). Thus, the unphysical self-
correlation field is generated by solving the modified equation
$D_{1}X(\tau,\xi)=D_{2}f(\tau,\xi)$ (79)
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|
2024-09-04T02:54:58.871503 | 2020-03-10T20:26:26 | 2003.04956 | {
"authors": "Bohan Wu, Feng Xu, Zhanpeng He, Abhi Gupta, and Peter K. Allen",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26144",
"submitter": "Bohan Wu",
"url": "https://arxiv.org/abs/2003.04956"
} | arxiv-papers | # SQUIRL: Robust and Efficient Learning from Video Demonstration of Long-
Horizon Robotic Manipulation Tasks
Bohan Wu, Feng Xu, Zhanpeng He, Abhi Gupta, and Peter K. Allen This work is
supported by NSF Grant CMMI-1734557. Authors are with Columbia University
Robotics Group, New York, NY, 10027, USA
###### Abstract
Recent advances in deep reinforcement learning (RL) have demonstrated its
potential to learn complex robotic manipulation tasks. However, RL still
requires the robot to collect a large amount of real-world experience. To
address this problem, recent works have proposed learning from expert
demonstrations (LfD), particularly via inverse reinforcement learning (IRL),
given its ability to achieve robust performance with only a small number of
expert demonstrations. Nevertheless, deploying IRL on real robots is still
challenging due to the large number of robot experiences it requires. This
paper aims to address this scalability challenge with a robust, sample-
efficient, and general meta-IRL algorithm, SQUIRL, that performs a new but
related long-horizon task robustly given only a single video demonstration.
First, this algorithm bootstraps the learning of a task encoder and a task-
conditioned policy using behavioral cloning (BC). It then collects real-robot
experiences and bypasses reward learning by directly recovering a Q-function
from the combined robot and expert trajectories. Next, this algorithm uses the
Q-function to re-evaluate all cumulative experiences collected by the robot to
improve the policy quickly. In the end, the policy performs more robustly
(90%+ success) than BC on new tasks while requiring no trial-and-errors at
test time. Finally, our real-robot and simulated experiments demonstrate our
algorithm’s generality across different state spaces, action spaces, and
vision-based manipulation tasks, e.g., pick-pour-place and pick-carry-drop.
## I Introduction
We aspire robots to become generalists who acquire new complex skills robustly
and quickly. The robotic system, whether planned or learned, needs to leverage
its existing knowledge to solve a new but related task in an efficient yet
high-performance manner. Thanks to recent advances in machine learning and
sim-to-real transfer mechanisms, short-horizon robotic manipulation such as
grasping has improved in performance. However, many real-world robotic
manipulation tasks are long-horizon, diverse, and abundant in volume. In the
absence of a scalable and systematic way to construct simulation environments
for a large number of tasks, the robot needs to learn a new task directly in
the physical world from only a handful of trials, due to the high cost of
collecting real-robot trial-and-errors and experiences.
Figure 1: Learning from a single video demonstration of a long-horizon
manipulation task via Soft Q-functioned Meta-IRL (SQUIRL). In the pick-pour-
place example above, the robot needs to approach, pick-up and carry the grey
bottle, pour the iron pebble inside the bottle into a specific container, and
finally place the bottle back on the table. During training, the robot is
given a single video demonstration for each of the 117 training tasks. After
learning from these 117 videos, the robot also practices 90 trial-and-errors
in total on these tasks. From such combined expert and robot trajectories, the
robot learns the general skills of pouring robustly. At test time, given a
single video demonstration of pouring into a new, unseen red container at a
new position, the robot successfully replicates this new task without the need
for any trial-and-errors.
We observe that real-world robotic skill acquisition can become more sample-
efficient in several important ways. First, we notice that humans learn tasks
quickly by watching others perform similar tasks. Among many forms of task
representations such as rewards, goal images, and language instructions, human
demonstrations guide exploration effectively and can lead to significant
sample efficiency gains. Furthermore, learning from video demonstrations
sidesteps hand-designing a proper reward function for every new task. In the
case of a vision-based task, video demonstrations also conveniently provide
the same pixel state space for the robot.
In learning from demonstrations (LfD), the robot should be sample-efficient in
two dimensions – it should use as few expert demonstrations (“demonstrations”
hereafter) as possible and take as few trial-and-errors (practices) as
possible on its own to learn a robust policy. Among LfD methods, behavioral
cloning (“BC” hereafter) is sample-efficient but susceptible to compounding
errors. Here, compounding errors refer to the problem in which every time a
behavioral-cloned robot makes a small error, it makes a larger error down the
road as it drifts away from the expert state distribution. In contrast, IRL
alleviates compounding errors by allowing the robot to try the tasks out in
the real world and measure its behavior against the expert. However, due to
the need to learn a reward function, IRL can require many trial-and-errors in
the real world, while BC does not require such robot experiences. We posit
that leveraging off-policy experiences of trial-and-errors is essential to
making IRL sample-efficient enough for real robots. Here, “off-policy
experiences” refer to the cumulative experiences that the robot has collected
thus far during training. In contrast, “on-policy experiences” are the most
recent experiences that the robot has collected using its current policy.
Humans leverage lifelong, cumulative experiences to learn quickly at present.
We envision robots to acquire new skills more quickly by learning from off-
policy (i.e., cumulative) experiences.
Finally, many real-world tasks are related and share structures and knowledge
that can be exploited to solve a new but similar task later. For example,
humans can quickly learn to pick and place a new object after learning to pick
and place many known objects. Meta-learning, explicitly utilizing this
property, aims to learn a new but related task quickly if it has already
learned several similar tasks in the past.
Figure 2: Fig.2: Pick-Pour-Place Robot Setup at Test Time. Given an RGB image
from the top-down (black) or 45°camera (also black), the UR5-Seed robot is
tasked to approach and pick-up the grey cylindrical bottle, pour the iron
pebble already inside the bottle into a specific container on the table and
finally place the bottle back on the table.
With these motivations, we introduce SQUIRL, a meta-IRL algorithm that learns
long-horizon tasks quickly and robustly by learning from 1) video
demonstrations, 2) off-policy robot experiences, and 3) a set of related
tasks. Fig.1 explains this algorithm using the example of a set of long-
horizon pick-pour-place tasks, using the UR5-Seed111Site:
www.seedrobotics.com/rh8d-dexterous-hand.html robot setup shown in Fig.2. In
this task, we have the containers (green, yellow, orange, and red), a
cylindrical bottle (grey), and an iron pebble inside the bottle. The robot
needs to first approach and pick-up the grey bottle, pour the iron pebble
inside the bottle into a specific container on the table, and then finally
place the bottle back on the table, as shown in each row of images in Fig.1.
At the beginning of each task, the bottle is not yet in hand, but the iron
pebble is already in the bottle. At training time, the robot is given a single
video demonstration for each of the 117 pick-pour-place tasks, as shown in the
first two rows of images in Fig.1. Every new combination of container
positions represents a different pick-pour-place task. Furthermore, the robot
only needs to pour into one of the containers in a single task. Therefore,
pouring into different containers also represents different tasks. After
learning from these 117 demonstrations, the robot also practices 90 trial-and-
errors on these tasks in total. From such a combination of expert and robot
trajectories, the robot learns the general skills of pick-pour-place robustly.
In all 117 training tasks, only two of the four containers appear on the
table: the green and yellow containers, as shown in the first two rows of
images in Fig.1. The orange and red containers are excluded during training
and only appear at test time, as shown in the last row of images in Fig.1. We
do so to evaluate our algorithm’s generalizability to unseen containers at
test time. As shown in the last row of images in Fig.1, the robot successfully
pours into a new container (red) at test time, at a new position never seen
before during training, without the need for any trials or practices.
To achieve such fast generalization to new tasks, our algorithm learns a task
encoder network and a task-conditioned policy. The task encoder generates a
32-dimensional task embedding vector that encodes task-specific information.
The policy network then learns to generalize to new tasks by accepting this
task embedding vector as input, thus becoming “task-conditioned”. During
training, our algorithm first bootstraps learning by training both the task
encoder and the policy jointly via the BC loss. The robot then collects 10
trials across 10 tasks using the warmed-up policy and the task encoder. Next,
using the combined experiences of the expert and the robot, our algorithm
bypasses reward learning by directly learning a task-conditioned Q-function.
Using this Q-function, our algorithm then reuses and re-evaluates all
cumulative experiences of the robot to improve the policy quickly. This cycle
repeats until the $90^{th}$ trial. Finally, at test time, the task encoder
generates a new task embedding from a single video demonstration of a new
task. This embedding is then inputted into the task-conditioned policy to
solve the new task without any trial-and-errors and yet in a high-performance
manner. In summary, our contributions are:
1. 1.
A robust meta-IRL algorithm that outperforms ($90\%$\+ success) its behavioral
cloning counterpart in real-robot and simulated vision-based long-horizon
manipulation;
2. 2.
A novel Q-functioned IRL formulation that circumvents reward learning and
improves IRL sample efficiency;
3. 3.
An efficient method that leverages off-policy robot experiences for training
and requires no trials at test time;
4. 4.
A general approach that tackles various long-horizon robotic manipulation
tasks and works with both vision and non-vision observations and different
action spaces.
## II Related Work
### II-A Inverse Reinforcement Learning (IRL) and Meta-IRL
Inverse reinforcement learning (IRL) models another agent’s (typically the
expert’s) reward function, given its policy or observed behavior. Previous
works have approached the IRL problem with maximum margin methods [1][2] and
maximum entropy methods [3][4][5]. In particular, maximum entropy methods
recover a distribution of trajectories that have maximum entropy among all
distributions and match the demonstrated policy’s behaviors. While these
methods have shown promising results in continuous control problems, they
suffer from low sample efficiency due to the need for evaluating the robot’s
policy, which can be alleviated by meta-learning (i.e., meta-IRL). SMILe [6]
and PEMIRL [7] are two meta-IRL algorithms based on AIRL [8] that leverage a
distribution of tasks to learn a continuous task-embedding space to encode
task information and achieve fast adaptation to a new but similar task. Our
work differs from [6][7] in four crucial ways. First, our meta-IRL algorithm
works with real robots and image observations. Second, instead of a reward
function, we directly model a Q-function that the policy can optimize, in
order to increase IRL sample efficiency. Third, we train the task encoder with
the behavioral cloning (BC) gradient as opposed to the IRL gradient for
stabler and more efficient learning. Lastly, we bootstrap policy and task
encoder learning using BC before training via meta-IRL.
### II-B Real-robot Learning from Demonstrations (LfD)
Our work is related to real-robot LfD [9], such as [10][11][12]. In
particular, [13] developed IRL on real robots without learning from raw
pixels. Other works (e.g., [14][15][16][17]) used BC for real-robot LfD.
Another work [18] developed goal-conditioned BC on a simulated robot to learn
long-horizon tasks by playing with objects in the scene. While enjoying
efficient learning by casting imitation learning into a supervised learning
problem, BC suffers from the covariate shift between the train and test data.
In comparison, IRL achieves robust performance by modeling the state-action
joint distribution instead of the conditional action distribution in BC [19].
Different from previous works, our meta-IRL algorithm works on real-robot
vision-based tasks, and its Q-functioned IRL policy gradient can be directly
combined with the BC gradient signal to approach both the sample efficiency of
BC and the robustness of IRL.
### II-C One-shot Meta-imitation Learning on Real Robots
Our algorithm attempts to enable robots to quickly and robustly imitate a
single unseen video demonstration by learning from a distribution of tasks
with shared structure, i.e., one-shot robot meta-imitation learning. For
example, [20] combines gradient-based meta-learning and BC on a real robot to
learn quickly from video demonstrations. [21] then extends [20] to enable
robots to learn from human-arm demonstrations directly. [22] then improves
[21] to meta-imitation-learn multi-stage real-robot visuomotor tasks in a
hierarchical manner. However, constrained by the covariate shift problem of
BC, these works show limited task performance (e.g., around $50\%$ success
rate for the training tasks). In contrast, our algorithm learns a vision-based
manipulation task robustly ($90\%+$ success rates) and efficiently (117
videos, 90 trials) by utilizing the generalization ability of task embeddings
[23] and a novel Q-functioned IRL formulation.
## III Preliminaries
### III-A Off-policy Reinforcement Learning via Soft Actor-Critic
Standard RL models a task $\mathcal{M}$ as an MDP defined by a state space
$\mathcal{S}$, an initial state distribution $\rho_{0}\in\Pi(\mathcal{S})$, an
action space $\mathcal{A}$, a reward function
$\mathcal{R}:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$, a dynamics model
$\mathcal{T}:\mathcal{S}\times\mathcal{A}\to\Pi(\mathcal{S})$, a discount
factor $\gamma\in[0,1)$, and a horizon $H$. Here, $\Pi(\cdot)$ defines a
probability distribution over a set. The robot acts according to stochastic
policy $\pi:\mathcal{S}\to\Pi(\mathcal{A})$, which specifies action
probabilities for each $s$. Each policy $\pi$ has a corresponding
$Q^{\pi}:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ function that defines the
expected discounted cumulative reward for taking an action $a$ from $s$ and
following $\pi$ onward.
Off-policy RL, particularly Soft Actor-Critic (SAC) [24], reuses historical
experiences to improve learning sample efficiency by recovering a “soft”
Q-function estimator $Q_{\theta}$. A policy can then be learned by minimizing
the KL divergence between the policy distribution and the exponential-Q
distribution: $\pi^{*}=\operatorname*{arg\,min}_{\pi\in\Pi}D_{KL}(\pi(a\mid
s)\;\|\;\frac{\exp(Q^{\pi_{old}}_{\theta}(s,a))}{Z(s)})$
### III-B Timestep-centric IRL as Adversarial Imitation Learning
The purpose of IRL is to learn the energy function $f_{\theta}$ implicit in
the provided expert demonstrations and use $f_{\theta}$ to learn a policy that
robustly matches the expert performance. In particular, timestep-centric IRL
aims to recover an energy function $f_{\theta}(s,a)$ to rationalize and match
the demonstrated expert’s action conditional distribution: $p_{\theta}(a\mid
s)=\frac{\exp(f_{\theta}(s,a))}{Z_{\theta}}\propto\exp(f_{\theta}(s,a))=\overline{p_{\theta}}(a\mid
s)$, where $Z_{\theta}$ is the partition function, an integral over all
possible actions given state $s$. In other words, IRL minimizes the KL
divergence between the actual and predicted expert conditional action
distributions: $\pi_{E}(a\mid s)$ and $p_{\theta}(a\mid s)$.
Adversarial IRL [8][25] provides a sampling-based approximation to MatEntIRL
[4] in an adversarial manner. Specifically, AIRL [8] learns a generative
policy $\pi_{\psi}$ and a binary discriminator $D_{\theta}$ derived from
energy function $f_{\theta}$:
$\displaystyle D_{\theta}(s,a)=P((s,a)\text{ is generated by expert})$
$\displaystyle=\frac{\overline{p_{\theta}}(a\mid
s)}{\overline{p_{\theta}}(a\mid s)+\pi_{\psi}(a\mid
s)}=\frac{\exp(f_{\theta}(s,a))}{\exp(f_{\theta}(s,a))+\pi_{\psi}(a\mid s)}$
(1)
and $\theta$ is trained to distinguish state-action pairs sampled from the
expert vs. the policy, using binary cross entropy loss:
$\displaystyle\mathcal{L}^{IRL}=-\mathbb{E}$
${}_{(s,a)\sim\pi_{\psi},\pi_{E}}[y(s,a)\log(D_{\theta}(s,a))$
$\displaystyle+(1-y(s,a))\log(1-D_{\theta}(s,a))]$ (2)
where $y(s,a)=\mathds{1}\\{(s,a)\text{ is generated by expert }\pi_{E}\\}$.
Meanwhile, the policy is trained to maximize the MaxEntIRL Objective [4], or
equivalently, to match the expert’s state-action joint distribution via
reverse KL divergence [19].
### III-C One-shot Meta-imitation Learning from A Single Video
In one-shot meta-imitation learning, the robot is trained to solve a large
number of tasks drawn from a task distribution $p(\mathcal{M})$. The total
number of tasks in this task distribution can be finite or infinite. Each
imitation task $\mathcal{M}_{train}^{i}$ consists of a single video
demonstration $\mathcal{D}^{i}_{\pi_{E}}$. During training, the robot can also
generate limited practice trajectories (e.g., 90). For example, in the Pick-
Pour-Place experiment in Fig.1, the robot receives a single video
demonstration for each of the 117 tasks. Each task is characterized by a
different combination of container positions, or pouring into the green vs.
the yellow container. At test time, the robot receives a single video of a new
task $\mathcal{M}_{test}^{i}$ drawn from $p(\mathcal{M})$. For example, a new
Pick-Pour-Place test task can be a new combination of container positions or
pouring into a new container (e.g., the red or orange container). The robot
then needs to solve this task the first time without trial-and-error.
### III-D Embedding-based Meta-learning
Embedding-based meta-learning [7][23] learns a task-specific embedding vector
$z$ that contains task-level abstraction to adapt to a new but related task
quickly. This method aims to learn a task-conditioned policy $\pi(a|s,z)$ that
maximizes task-conditioned expected returns:
$\max_{\pi}\mathbb{E}_{(s_{t},a_{t})\sim\pi,\rho_{0}}[\sum_{t=1}^{T}r(s_{t},a_{t}|c)+\alpha\mathcal{H}(\pi(a_{t}|s_{t},c))]$,
by learning an embedding space $Z$ that maximizes the mutual information
between $z$ and task context $c$. The goal is to make this learned embedding
space generalizable to new tasks so that at test time, the policy can quickly
adapt to unseen tasks with no or few practices. A key advantage of embedding-
based meta-learning is the ability to learn from off-policy experiences.
However, current methods are mostly if not only demonstrated in non-vision
tasks in simulation.
## IV Mathematical Formulation for SQUIRL
### IV-A SQUIRL: Timestep-centric IRL as Soft Q-Learning
Previous works in timestep-centric IRL such as [6][7][8] have interpreted the
energy function $f_{\theta}$ in Eq.III-B as a reward function $r_{\theta}$ and
later recover a Q or advantage function based on reward $r_{\theta}$ for
policy improvement. To improve IRL sample efficiency, we propose to bypass
this reward learning and directly interpret $f_{\theta}(s,a)$ as the soft
Q-function [24] $Q^{\pi_{mix}}_{\theta}(s,a)$. This soft Q-function models the
expert’s behavior as maximizing both the Q-value and its entropy (i.e.,
randomness) simultaneously. It also encourages the robot to explore the real
world to imitate the expert more robustly. Under this formulation,
approximating the expert’s conditional action distribution is equivalent to
recovering a soft Q-function under which the expert is soft Q-optimal:
$\displaystyle\operatorname*{arg\,min}_{\theta}D_{KL}(\pi_{E}(a\mid
s)\;\|\;p_{\theta}(a\mid s))$ $\displaystyle=$
$\displaystyle\operatorname*{arg\,max}_{\theta}\mathbb{E}_{a\sim\pi_{E}(a\mid
s)}[Q^{\pi_{mix}}_{\theta}(s,a)]-\log Z_{\theta}$ (3)
Eq.3 rationalizes the expert behavior intuitively because the expert should be
optimal with respect to the cumulative reward [3], not the immediate reward.
Here, $Q^{\pi_{mix}}_{\theta}$ is under a mixture policy $\pi_{mix}$ between
the robot and expert’s policies.
### IV-B SQUIRL as Expert Imitation and Adversarial Learning
Under SQUIRL, the policy learning objective (Eq.4) is also equivalent
(derivations on website) to matching: 1) the exponential-Q distribution of the
discriminator $\theta$ (Eq.5), 2) the generator’s objective in Generative
Adversarial Networks (GANs) [26] (Eq.6), and 3) the joint state-action
distribution of expert [19] (Eq.7):
$\pi^{*}=\operatorname*{arg\,min}_{\pi\in\Pi}\mathcal{L}^{RL}(\pi)$, where
$\displaystyle\mathcal{L}^{RL}(\pi)=D_{KL}(\pi_{\psi}(a\mid
s)\;\|\;\frac{\exp{Q^{\pi_{mix}}_{\theta}(s,a)}}{Z(s)})$ (4)
$\displaystyle=D_{KL}(\pi_{\psi}(a\mid s)\;\|\;p_{\theta}(a\mid s))$ (5)
$\displaystyle=\mathbb{E}_{(s,a)\sim\pi_{mix}}[\log(1-D_{\theta}(s,a))-\log(D_{\theta}(s,a))]$
(6) $\displaystyle=D_{KL}(\rho_{\pi_{\psi}}(s,a)\;\|\;\rho_{\pi_{E}}(s,a))$
(7)
Meanwhile, the discriminator $\theta$ is matching its Q-function to the log-
distribution of the expert’s conditional action distribution (Section III-B).
Therefore, when this Q-function is optimal:
$Q^{\pi_{mix}}_{\theta}=Q^{\pi_{mix}}_{\theta^{*}}$, the robot’s policy
objective (Eq.4) is also matching the expert’s conditional action
distribution:
$\psi^{*}=\operatorname*{arg\,min}_{\psi}E_{\rho_{\pi_{mix}}(s)}[D_{KL}(\pi_{\psi}(a\mid
s)\;\|\;\pi_{E}(a\mid s))]$ (8)
### IV-C Comparison to the Behavioral Cloning (BC) Objective
While BC attempts to learn a policy that also matches the expert’s conditional
action distribution [19], the fundamental difference is that the KL-divergence
in BC’s case is computed under the expert’s narrow state distribution
$\rho_{\pi_{E}}(s)$:
$\psi_{BC}^{*}=\operatorname*{arg\,min}_{\psi}E_{\rho_{\pi_{E}}(s)}[D_{KL}(\pi_{E}(a\mid
s)\;\|\;\pi_{\psi}(a\mid s))]$. In contrast, ours (Eq.8) is computed under
$\rho_{\pi_{mix}}(s)$: the state distribution of the combined cumulative
experience of the robot and the expert, which is a much wider distribution
than the expert distribution. We hypothesize that this, along with matching
the joint state-action distribution of the expert (Eq.7), makes our algorithm
less susceptible to compounding errors than BC, as experimentally tested in
Section VI.
Figure 3: SQUIRL: Soft Q-functioned Meta-IRL. To begin, our algorithm
bootstraps learning for the policy (orange) and the task encoder (yellow) via
behavioral cloning (the left third of Fig.3). Next, our algorithm uses the
warmed-up policy and task encoder to generate 10 trials in the physical world
(not in simulation). Using the combined expert and robot trajectories, our
algorithm learns a task-conditioned soft Q-function (green) that rationalizes
the expert’s behaviors as maximizing both cumulative reward and entropy (i.e.,
randomness). Using this Q-function, our algorithm then quickly improves the
policy using all cumulative robot and expert timesteps. This cycle repeats
until convergence, totaling 90 trials (the middle third of Fig.3). Finally, at
test time (the right third Fig.3), our algorithm generates a new embedding $z$
for the new task, and inputs this embedding into the task-conditioned policy
to solve the new task without any practices.
## V SQUIRL: Soft Q-functioned Meta-IRL
Shown in Fig.3, our algorithm learns three neural networks jointly – a task
encoder (yellow), a task-conditioned policy (orange), and a task-conditioned
soft Q-function (green):
1. 1.
$\Psi_{\phi}(c)$: a task encoder that encodes a sampled batch of $C=64$ expert
state-action pairs $c=\\{s^{i}_{1:C},a^{i}_{1:C}\\}$ from a task $i$ into a
single 32-dim embedding vector $z^{i}\in\mathbb{R}^{32}$ (by computing the
mean vector across 64 embeddings) that enables generalization to new tasks.
This batch of expert state-action pairs is randomly sampled and thus does not
encode time information. Both the policy and the Q-function accept this
embedding vector as input.
2. 2.
$\pi_{\psi}(s,z^{i})$: a task-conditioned policy the robot uses to perform a
task $i$ given state $s$ and the task embedding vector
$z^{i}\in\mathbb{R}^{32}$ outputted by the task encoder $\Psi_{\phi}(c)$.
3. 3.
$Q_{\theta}(s,a,z^{i})$: a task-conditioned soft Q-function used to train the
policy $\pi_{\psi}(s,z^{i})$ to more robustly mimic the expert’s behavior for
the robotic manipulation task $i$.
To begin, the robot is given an expert trajectory of state-action pairs
$\mathcal{D}_{\pi_{E}}$ for each of the 117 training tasks. The robot first
uses these expert trajectories to bootstrap training for both its policy
$\pi_{\psi}$, and the task encoder $\Psi_{\phi}$ via behavioral cloning
(Eq.9). This way, the robot can distinguish the train tasks better and learn
more quickly in the real world. Next, the robot generates 10 trials (state-
action pairs) $\overline{\mathcal{D}}_{\pi_{\psi}}$ in the physical world (not
simulation) using its warmed-up policy and task encoder. Then, the robot uses
both the expert’s and its state-action pairs to train a discriminator
$\theta$. This discriminator classifies which state-action pairs come from the
expert $\pi_{E}$ vs. the robot $\pi_{\psi}$. At first, the robot is
distinctively worse than the expert at performing the tasks. This makes it
easy for the discriminator to classify. By doing so, the discriminator learns
a Q-function $Q^{\pi_{mix}}_{\theta}$ using Eq.3.
Using the learned Q-function $Q^{\pi_{mix}}_{\theta}$, the robot trains its
policy $\pi_{\psi}$ via Eq.4. Meanwhile, the robot also has the option to
continue updating its task-conditioned policy and task encoder via behavioral
cloning (Eq.9). Since training the policy via Eq.4 is equivalent to indirectly
imitating the expert (Eq.7 and 8), as derived in Section IV-B, the
trajectories generated by the policy gradually become more similar to the
expert. This makes the state-action pairs more difficult for the discriminator
to classify. This difficulty, in turn, forces the discriminator to learn a
more precise Q-function, which then encourages the policy to mimic the expert
even more closely. This cycle repeats until convergence (90 trials in total),
at which point: 1) the policy matches the expert performance, 2) the task
encoder learns to generalize to new tasks, and 3) the discriminator continues
to struggle to distinguish state-action pairs correctly despite having learned
an accurate Q-function.
### V-A Rationale for Bypassing Reward Learning via SQUIRL
SQUIRL learns a Q-function without rewards because 1) the policy is ultimately
trained by the Q-function, not rewards, thus bypassing reward learning
improves IRL sample efficiency, and 2) circumventing reward learning avoids
off-policy Q-learning from a constantly changing reward function and makes
training easier and more stable empirically.
### V-B Architectures for Policy, Task Encoder, and Q-function
For all non-vision tasks, we parameterize $\pi_{\psi},\Psi_{\phi},Q_{\theta}$
with five fully-connected (FC) layers. For vision tasks, we use a 5-layer CNN
followed by a spatial-softmax activation layer for the RGB image. This
activation vector is then concatenated with the non-vision input vector and
together passed through five FC layers. Our algorithm is general and works
with many other network architectures, state, and action spaces.
### V-C Incorporating BC to Bootstrap and Accelerate Learning
Since our algorithm’s IRL objective (Eq.8) is compatible with BC, as explained
in Section IV-C, our algorithm can jointly be trained with BC to stabilize and
accelerate learning without conflicting gradient issues (line 16 in Algorithm
1):
$\mathcal{L}^{BC}=\mathbb{E}_{(s,a)\sim\pi_{E}}[\left\lVert\pi_{\psi}(s,\Psi_{\phi}(c))-a\right\rVert^{2}]$
(9)
This, combined with the off-policy nature of our algorithm, also allows the
robot to bootstrap learning by first “pre-training” via BC (Eq.9) using the
expert demonstrations, before improving performance further via meta-IRL
training.
Algorithm 1 SQUIRL: Soft Q-functioned Meta-IRL (Train)
Input: One expert video demonstration trajectory of state-action pairs
$\mathcal{D}^{i}_{\pi_{E}}=\\{s^{i}_{1:H},a^{i}_{1:H}\\}$ for each of the $n$
training tasks $i=1:n$, where $H$ is the horizon of the task (e.g.,
$n=117,H=100$)
1: Initialize soft Q-function $Q_{\theta}$, policy $\pi_{\psi}$, task encoder
$\Psi_{\phi}$, and an empty buffer of off-policy robot trajectories
$\mathcal{D}^{i}_{\pi_{\psi}}\leftarrow\\{\\}$ for each training task $i=1:n$
2: Warm-up policy and task encoder via $\mathcal{L}^{BC}$ (Eq.9)
3: while not converged do
4: Sample a batch of $m$ task indices $\\{i^{1:m}\\}$ from all training tasks
$i=1:n$, (e.g., $m=10$)
5: for $i=i^{1:m}$ do
6: Infer task embedding
$z^{i}=\mathbb{R}^{\mathcal{Z}}\leftarrow\Psi_{\phi}(c)$, where
$c=\\{s^{i}_{1:C},a^{i}_{1:C}\\}\sim\mathcal{D}^{i}_{\pi_{E}}$ (e.g.,
$\mathcal{Z}=32,C=64$)
7: Generate a robot trajectory of state-action pairs
$\overline{\mathcal{D}}^{i}_{\pi_{\psi}}=\\{s^{i}_{1:H},a^{i}_{1:H}\\}$ from
task $i$ using $\pi_{\psi},z^{i}$
8:
$\mathcal{D}^{i}_{\pi_{\psi}}\leftarrow\mathcal{D}^{i}_{\pi_{\psi}}\cup\overline{\mathcal{D}}^{i}_{\pi_{\psi}}$
9: end for
10: for $j=1:J$ (e.g., $J=400$) do
11: Sample another batch of $m$ task indices $\\{i^{1:m}\\}$
12: $\theta\leftarrow\theta-\nabla_{\theta}\mathcal{L}^{IRL}$ (Eq.III-B) using
a combined batch of $\mathcal{B}=128$ robot and expert timesteps:
$\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\cup\overline{\mathcal{D}}^{i}_{\pi_{E}}$
and $z^{i}$, where
$\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\sim\mathcal{D}^{i}_{\pi_{\psi}}$,
$\overline{\mathcal{D}}^{i}_{\pi_{E}}\sim\mathcal{D}^{i}_{\pi_{E}}$,
$i=\\{i^{1:m}\\}$
13: end for
14: for $k=1:K$ (e.g., $K=2000$) do
15: Sample another batch of $m$ task indices $\\{i^{1:m}\\}$
16: if necessary then
$\\{\psi,\phi\\}\leftarrow\\{\psi,\phi\\}-\nabla_{\psi,\phi}\mathcal{L}^{BC}$
(Eq.9) using a batch of $\mathcal{B}$ expert timesteps
$\overline{\mathcal{D}}^{i}_{\pi_{E}}\sim\mathcal{D}^{i}_{\pi_{E}},z^{i}$,
$i=\\{i^{1:m}\\}$ end if
17: $\psi\leftarrow\psi-\nabla_{\psi}\mathcal{L}^{RL}$ (Eq.4) using a combined
batch of $\mathcal{B}$ robot and expert timesteps:
$\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\cup\overline{\mathcal{D}}^{i}_{\pi_{E}}$
and $z^{i}$, where
$\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\sim\mathcal{D}^{i}_{\pi_{\psi}}$,
$\overline{\mathcal{D}}^{i}_{\pi_{E}}\sim\mathcal{D}^{i}_{\pi_{E}}$,
$i=\\{i^{1:m}\\}$
18: end for
19: end while
20: return soft Q-function $Q_{\theta}$, policy $\pi_{\psi}$, task encoder
$\Psi_{\phi}$
Algorithm 2 SQUIRL: Soft Q-functioned Meta-IRL (Test)
Input: $\pi_{\psi}$, $\Psi_{\phi}$, $Q_{\theta}$, and a single expert video
demonstration of state-action pairs
$\mathcal{D}^{i}_{\pi_{E}}=\\{s^{i}_{1:H}$, $a^{i}_{1:H}\\}$ from a new task
$i$ unseen during training
1: Infer task embedding vector
$z^{i}=\mathbb{R}^{\mathcal{Z}}\leftarrow\Psi_{\phi}(c)$, where
$c=\\{s^{i}_{1:C},a^{i}_{1:C}\\}\sim\mathcal{D}^{i}_{\pi_{E}}$ (e.g.,
$\mathcal{Z}=32,C=64$)
2: Rollout robot trajectory in the real world using $\pi_{\psi}$, $z^{i}$
Approach Box
Lower to Box
Grasp Box
Pick up Box
Carry Box
Drop Box
Figure 4: Pick-Carry-Drop Experiment. The robot needs to approach, lower to,
grasp, pick-up, carry, and drop the box to solve the task.
### V-D Using Expert Demonstration as Both the Input Task Context Variables
and Training Signal for the Task Encoder
Learning robust task embeddings enables robots to generalize to new tasks
quickly [23]. To this end, our algorithm proposes to use 64 expert timesteps
as the input task context variable $c$ into the task encoder, as opposed to 64
robot timesteps. This is because context variables should explore the task and
environment sufficiently well to expose the key information of the task, and
expert demonstration timesteps are an ideal candidate compared to the
timesteps from the robot’s suboptimal policy. As a result, the context
variable $c$ input into the task encoder only includes the states and actions
of the expert, but not the rewards or the next states.
In addition, we choose the BC loss $\mathcal{L}^{BC}$ in Eq.9 as the training
loss for learning the task encoder $\Psi_{\phi}$. This BC loss is stable since
the expert timesteps are fixed. In contrast, the IRL loss $\mathcal{L}^{IRL}$
(Eq.III-B) and the policy loss $\mathcal{L}^{RL}$ (Eq.4) are less stable
because the training data distribution for both losses are non-stationary.
This design choice also allows us to learn a robust task embeddings first via
BC pre-training before performing meta-IRL training via SQUIRL. We empirically
observe that such pre-training can improve the training stability and the
sample efficiency of SQUIRL, but the final policy performance is similar with
or without BC pre-training. In summary, our algorithm is detailed in Algorithm
1 (train) and Algorithm 2 (test), with hyperparameters detailed
here222Hyperparameters in Algorithm 1 and 2. Policy gradient batch size
$\mathcal{B}$: 1024 (non-vision), 128 (vision); task embedding batch size $C$:
64; all learning rates: $3e^{-4}$; starting SAC alpha: $1e^{-5}$; SAC target
entropy: $-300$; IRL updates per epoch $J$: $400$; policy updates per epoch
$K$: $2000$; task embedding size $\mathcal{Z}$: 32; meta-batch size $m$: 10;
discount rate $\gamma$: 0.99.
## VI Experiments and Results Analysis
We evaluate the generality and robustness of our algorithm across long-horizon
vision and non-vision tasks with continuous state and action spaces in both
simulation (Pick-Carry-Drop, a horizon of 1024 timesteps, 30 train tasks) and
real-world (Pick-Pour-Place, a horizon of 100 timesteps, 117 train tasks).
There is only a single expert demonstration for each of the train or test
tasks. We compare with the PEARL-BC baseline, which is the behavioral cloning
version of PEARL [23]. Evaluation: We evaluate real-robot and simulation
experiments on 50 and 500 trials respectively across 50 seen and unseen tasks.
We report mean and standard deviation (“stdev” hereafter). The performance
difference between different experiments is statistically significant if the
difference in mean is at least either standard deviation away. Experimental
video is at http://crlab.cs.columbia.edu/squirl.
Approach Bottle
Grasp Bottle
Carry Bottle
Pour Orange Cup
Carry Bottle
Place Bottle
Figure 5: Pick-Pour-Place at Test Time. To solve this task, the robot needs to
first approach, grasp and carry the grey bottle, pour the iron pebble inside
the bottle into a specific container, and carry and place the bottle back on
the table. At the beginning of each task, the bottle is not in hand, and but
the iron pebble is already in the bottle. Top row: top-down camera images.
Bottom row: 45°camera images.
### VI-A Simulation Experiment: Pick-Carry-Drop
Description. We modify the planar Stacker task [27] to create “Pick-Carry-
Drop”. Shown in Fig.4, a robot is tasked to approach, pick, carry, and drop
the black box into the stack marked in green. The task is successful if the
box is dropped into the stack within 1024 timesteps, and failed otherwise.
State Space. We evaluate our algorithm on both the vision and the non-vision
version of the task, to demonstrate that SQUIRL is general across different
state space modalities. The state space for the vision version includes 1) the
joint angles and velocities for its 5-DOFs, 2) a one-hot vector indicating the
current stage of the task, and 3) an RGB image shown in Fig.4. The non-vision
version’s state space replaces the RGB image with the position of the black
box.
Action Space. The robot controls its 5-DOF joint torques.
Task Definition. There are a total of 30 training tasks in this experiment,
each corresponding to a different drop location:
$x\in\\{-0.15,-0.14,\ldots,0.14\\}$. During test time, we randomly sample a
new, real-valued drop location from the maximum valid range:
$x\in[-0.25,0.25]$. The green drop location is invisible in both the vision
and the non-vision version of the task. Therefore, the robot needs to infer
the green drop location (i.e., task information) solely from the provided
expert video demonstration. On the other hand, the starting pose of the robot
and the location of the black box are all initialized randomly at the
beginning of each task.
Robot Trials. The robot uses 150 training trials in total.
Expert Demonstration. We trained an expert policy from scratch via RL to
provide expert demonstrations. The reward function used to train the expert
policy comprises of six stages, each stage with a reward of 10. Designing this
reward function has taken significant human effort, which exhibits the value
of directly learning from video demonstrations.
TABLE I: Pick-Carry-Drop Results (% Drop Success$\pm$Stdev) Tasks | Seen | Unseen | Seen | Unseen
---|---|---|---|---
| Vision | Non-Vision
SQUIRL (BC + IRL) | 95.8$\pm$1.7 | 95.0$\pm$1.5 | 97.3$\pm$3.0 | 96.9$\pm$2.0
Baseline (PEARL-BC) | 77.8$\pm$1.6 | 76.5$\pm$0.7 | 90.8$\pm$2.5 | 89.5$\pm$1.6
Ablation: No BC Joint Training or BC Pre-training
SQUIRL (IRL Only) | 93.8$\pm$1.8 | 93.2$\pm$1.6 | 94.7$\pm$1.7 | 93.9$\pm$1.4
Simulation Results and Analysis. As shown in Table I, our algorithm, “SQUIRL
(BC + IRL)”, pre-trains via BC and then trains the policy using both the BC
loss (Eq.9) and the IRL policy gradient loss (Eq.4). It statistically
significantly outperforms the PEARL-BC baseline in both the vision
(95.8%$\pm$1.7 vs. 77.8%$\pm$1.6) and non-vision (97.3%$\pm$3.0 vs.
90.8%$\pm$2.5) version of the task for seen tasks. For unseen tasks, we
observed similar outperformance (95.0%$\pm$1.5 vs. 76.5%$\pm$0.7 in the vision
case and 96.9%$\pm$2.0 vs. 89.5%$\pm$1.6 in the non-vision case).
Qualitatively, in the PEARL-BC’s case, the robot sometimes misses the drop
location as it attempts to drop the box or fails to pick up the box when the
box gets stuck by the walls of the stack (kindly see website). The performance
drop of the baseline from the non-vision version (90.8%$\pm$2.5 and
89.5%$\pm$1.6 for seen and unseen tasks) to the vision version (77.8%$\pm$1.6
and 76.5%$\pm$0.7 for seen and unseen tasks) is mainly because vision-based
manipulation tends to suffer from larger compounding errors. Nevertheless, as
evident in the statistical similarities between seen and unseen tasks for
SQUIRL (95.8%$\pm$1.7 vs. 95.0%$\pm$1.5 for vision) and PEARL-BC
(77.8%$\pm$1.6 vs. 76.5%$\pm$0.7 for vision), both algorithms can generalize
to unseen tasks, due to the generalizability of task embeddings.
Ablation: IRL Gradient Only. To compare the performance contribution of
SQUIRL’s meta-IRL core training procedure directly against PEARL-BC, we
created “SQUIRL (IRL only)”, which trains the policy using only the policy
gradient loss in Eq.4 (no BC joint training or pre-training). This ablated
version still outperforms the PEARL-BC baseline (93.8%$\pm$1.8 vs.
77.8%$\pm$1.6 for seen vision tasks, 93.2%$\pm$1.6 vs. 76.5%$\pm$0.7 for
unseen vision tasks). Nevertheless, by combining BC and IRL gradients, “SQUIRL
(BC + IRL)” improves performance slightly further (95.8%$\pm$1.7 and
95.0%$\pm$1.5). Intuitively, while BC only matches the expert’s conditional
action distribution under the expert’s state distribution, BC’s supervised
learning signal is stabler than IRL. Joint training with BC and IRL gradients
can be interpreted as combining the stability of BC and the robustness of
Q-functioned IRL, by matching the conditional action distribution of the
expert under the broader state distribution of the expert-robot mixture
experience (Eq.8), in addition to matching the expert’s joint state-action
distribution (Eq.7).
### VI-B Real-Robot Experiment: Pick-Pour-Place
Description. We evaluated our algorithm on the UR5-Seed robot (Fig.2) to
perform a set of long-horizon pick-pour-place tasks. As shown in Fig.2, in
each task, there is a grey cylindrical bottle, an iron pebble that is already
in the bottle, and more than one container on the table. The robot is tasked
to approach and pick-up the grey bottle, pour the iron pebble into a specific
container, and place the bottle back on the table. The task is a success only
if the pebble is poured into the correct container and the bottle is placed
upright on the table within $H=100$ timesteps, and a failure otherwise.
State Space. The state space contains a top-down or 45°camera’s RGB image
(Fig.5), and 2 binary indicators for whether the robot has poured or closed
the hand, respectively.
Action Space. The action space includes the Cartesian unit directional vector
for the end-effector movement. During each timestep, the robot can adjust the
end-effector by 2cm along any 3D direction. The action space also includes a
binary indicator to control the arm vs. the hand and a trinary indicator to
close, open, or rotate the hand for pouring.
Orthogonality to State and Action Representions. While Pick-Pour-Place can be
tackled by first localizing the correct container via object detection
(alternative state space) and then executing motion-planning trajectories to
pour (alternative action space), our algorithm is general across and
orthogonal to alternative state and action spaces.
Task Definition. As shown in each row of images in Fig.1, each task is defined
by the positions and colors of the containers, and by the correct container to
pour into. There are always only the green and yellow containers in the 117
train tasks. 25 of the 50 test tasks have the green and yellow containers at
new positions. The remaining 25 test tasks add the red and the orange unseen
containers, or either. Since there is always more than one container in the
RGB image, the robot will not know which container to pour into without the
expert demonstration. Therefore, the robot needs to depend solely on the task
encoder’s ability to extract the correct task information from the expert
demonstration.
Robot Trials. The robot collects 90 training trials in total.
Expert Demonstration. We collect demonstrations via teleoperation using a
Flock of Birds sensor333Flock of Birds is a 6D pose tracker from Ascension
Technologies Corp.. Using the human wrist pose detected by the sensor in real-
time, we move, open, close, or rotate the robot hand for pouring. We collected
$117$ video demonstrations across 117 tasks for training. It takes 1-2 minutes
to collect one demonstration.
TABLE II: Pick-Pour-Place Results (% Pour Success$\pm$Stdev) Tasks | RGB Image | Seen | Unseen
---|---|---|---
SQUIRL (BC + IRL) | Top-Down (90°) | 92.0$\pm$4.5 | 90.0$\pm$7.1
Baseline (PEARL-BC) | 70.0$\pm$7.1 | 68.0$\pm$11.0
Baseline (Standard-BC) | 60.0$\pm$10.0 | 56.0$\pm$11.4
SQUIRL (BC + IRL) | $45\degree$ (Ablation) | 90.0$\pm$7.1 | 88.0$\pm$8.4
Real-robot Results and Analysis. As shown in Table II, our algorithm
outperforms the PEARL-BC baseline statistically significantly in both seen
tasks (92.0%$\pm$4.5 vs. 70.0%$\pm$7.1) and unseen tasks (90.0%$\pm$7.1 vs.
68.0%$\pm$11.0). This observed outperformance mainly originates from our soft
Q-functioned IRL formulation, which forces the robot to imitate the expert
under a much wider state distribution provided by the expert-robot mixture
trajectories, instead of the narrow state distribution of the expert
demonstrations. This helps reduce compounding errors during task execution.
The low performance of the PEARL-BC baseline is mainly due to additional
compounding errors induced by real-world sensory noises such as unstable
lighting conditions and small perturbation to camera positions. Qualitatively,
the PEARL-BC baseline sometimes pours into the wrong container, misses the
target container by a few centimeters, or moves past the target container
while failing to pour in time (kindly see website for examples). Nevertheless,
from the statistical similarity between seen and unseen tasks for both our
algorithm (92.0%$\pm$4.5 vs. 90.0%$\pm$7.1) and PEARL-BC (70.0%$\pm$7.1 vs.
68.0%$\pm$11.0), we see that the learned task encoder is still effectively
generalizing to a new, related task.
Comparison to the “Standard-BC” Baseline. We also compared to “Standard-BC”
(60.0%$\pm$10.0 and 56.0%$\pm$11.4 for seen and unseen tasks), which performs
no meta-learning and learns every train or test task independently from
scratch via BC. As a result, the neural network overfits to the single
demonstration and fails to generalize to real-world sensory (camera) noises at
test time. Note that Standard-BC’s unseen-task performance is slightly lower
than seen tasks since the unseen tasks are more challenging with at most 4
containers on the table, compared to only 2 containers in seen tasks.
Ablation: Non-top-down Camera. We also tested our algorithm with a $45\degree$
RGB image (90.0%$\pm$7.1 and 88.0%$\pm$8.4 for seen and unseen tasks) against
a top-down RGB image (92.0%$\pm$4.5 and 90.0%$\pm$7.1 for seen and unseen
tasks). The statistical similarity between the two shows that SQUIRL is
general and can accept a non-top-down RGB input image.
## VII Conclusion
We introduced SQUIRL, a robust, efficient, and general Soft Q-functioned meta-
IRL algorithm, towards enabling robots to learn from limited expert (one per
task) and robot (90 in total) trajectories. This algorithm is statistically
significantly more robust than behavioral cloning and requires no trial-and-
errors at test time. Finally, this general algorithm has been tested to work
with various long-horizon manipulation tasks, and across vision and non-vision
state and action spaces. In the future, we will extend this algorithm to learn
from direct human-arm demonstrations instead of teleoperation. This will lower
the cost of collecting real-world expert demonstrations further. We also aim
to incorporate hierarchical learning into SQUIRL to solve much longer horizon
manipulation tasks by reusing low-level subpolicies.
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* [26] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,” in _Advances in neural information processing systems_ , 2014.
* [27] Y. Tassa, Y. Doron, A. Muldal, T. Erez, Y. Li, D. de Las Casas, D. Budden, A. Abdolmaleki, J. Merel, A. Lefrancq, T. Lillicrap, and M. Riedmiller, “DeepMind control suite,” Tech. Rep., Jan. 2018.
|
2024-09-04T02:54:58.885703 | 2020-03-10T20:41:24 | 2003.04960 | {
"authors": "Sanmit Narvekar and Bei Peng and Matteo Leonetti and Jivko Sinapov and\n Matthew E. Taylor and Peter Stone",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26145",
"submitter": "Sanmit Narvekar",
"url": "https://arxiv.org/abs/2003.04960"
} | arxiv-papers | # Curriculum Learning for Reinforcement Learning Domains: A Framework and
Survey
Sanmit Narvekar<EMAIL_ADDRESS>
Department of Computer Science
University of Texas at Austin Bei Peng<EMAIL_ADDRESS>
Department of Computer Science
University of Oxford Matteo Leonetti<EMAIL_ADDRESS>
School of Computing
University of Leeds Jivko Sinapov<EMAIL_ADDRESS>
Department of Computer Science
Tufts University Matthew E. Taylor<EMAIL_ADDRESS>
Alberta Machine Intelligence Institute
Department of Computing Science
University of Alberta Peter Stone<EMAIL_ADDRESS>
Department of Computer Science
University of Texas at Austin
and Sony AI
###### Abstract
Reinforcement learning (RL) is a popular paradigm for addressing sequential
decision tasks in which the agent has only limited environmental feedback.
Despite many advances over the past three decades, learning in many domains
still requires a large amount of interaction with the environment, which can
be prohibitively expensive in realistic scenarios. To address this problem,
transfer learning has been applied to reinforcement learning such that
experience gained in one task can be leveraged when starting to learn the
next, harder task. More recently, several lines of research have explored how
tasks, or data samples themselves, can be sequenced into a _curriculum_ for
the purpose of learning a problem that may otherwise be too difficult to learn
from scratch. In this article, we present a framework for curriculum learning
(CL) in reinforcement learning, and use it to survey and classify existing CL
methods in terms of their assumptions, capabilities, and goals. Finally, we
use our framework to find open problems and suggest directions for future RL
curriculum learning research.
Keywords: curriculum learning, reinforcement learning, transfer learning
## 1 Introduction
Curricula are ubiquitous throughout early human development, formal education,
and life-long learning all the way to adulthood. Whether learning to play a
sport, or learning to become an expert in mathematics, the training process is
organized and structured so as to present new concepts and tasks in a sequence
that leverages what has previously been learned. In a variety of human
learning domains, the quality of the curricula has been shown to be crucial in
achieving success. Curricula are also present in animal training, where it is
commonly referred to as shaping (Skinner, 1958; Peterson, 2004).
As a motivating example, consider the game of Quick Chess (shown in Figure 1),
a game designed to introduce children to the full game of chess, by using a
sequence of progressively more difficult “subgames.” For example, the first
subgame is played on a 5x5 board with only pawns, where the player learns how
pawns move, get promoted, and take other pieces. Next, in the second subgame,
the king piece is added, which introduces a new objective: keeping the king
alive. In each successive subgame, new elements are introduced (such as new
pieces, a larger board, or different configurations) that require learning new
skills and building upon knowledge learned in previous games. The final game
is the full game of chess.
The idea of using such curricula to train artificial agents dates back to the
early 1990s, where the first known applications were to grammar learning
(Elman, 1993; Rohde and Plaut, 1999), robotics control problems (Sanger,
1994), and classification problems (Bengio et al., 2009). Results showed that
the order of training examples matters and that generally, incremental
learning algorithms can benefit when training examples are ordered in
increasing difficulty. The main conclusion from these and subsequent works in
curriculum learning is that starting small and simple and gradually increasing
the difficulty of the task can lead to faster convergence as well as increased
performance on a task.
Recently, research in reinforcement learning (RL) (Sutton and Barto, 1998) has
been exploring how agents can leverage transfer learning (Lazaric et al.,
2008; Taylor and Stone, 2009) to re-use knowledge learned from a source task
when attempting to learn a subsequent target task. As knowledge is transferred
from one task to the next, the sequence of tasks induces a curriculum, which
has been shown to improve performance on a difficult problem and/or reduce the
time it takes to converge to an optimal policy.
Figure 1: Different subgames in the game of Quick Chess, which are used to
form a curriculum for learning the full game of Chess.
Many groups have been studying how such a curriculum can be generated
automatically to train reinforcement learning agents, and many approaches to
do so now exist. However, what exactly constitutes a curriculum and what
precisely qualifies an approach as being an example of curriculum learning is
not clearly and consistently defined in the literature. There are many ways of
defining a curriculum: for example, the most common way is as an ordering of
tasks. At a more fundamental level, a curriculum can also be defined as an
ordering of individual experience samples. In addition, a curriculum does not
necessarily have to be a simple linear sequence. One task can build upon
knowledge gained from multiple source tasks, just as courses in human
education can build off of multiple prerequisites.
Methods for curriculum generation have separately been introduced for areas
such as robotics, multi-agent systems, human-computer and human-robot
interaction, and intrinsically motivated learning. This body of work, however,
is largely disconnected. In addition, many landmark results in reinforcement
learning, from TD-Gammon (Tesauro, 1995) to AlphaGo (Silver et al., 2016) have
implicitly used curricula to guide training. In some domains, researchers have
successfully used methodologies that align with our definition of curriculum
learning without explicitly describing it that way (e.g., self-play). Given
the many landmark results that have utilized ideas from curriculum learning,
we think it is very likely that future landmark results will also rely on
curricula, perhaps more so than researchers currently expect. Thus, having a
common basis for discussion of ideas in this area is likely to be useful for
future AI challenges.
### Overview
The goal of this article is to provide a systematic overview of curriculum
learning (CL) in RL settings and to provide an over-arching framework to
formalize this class of methods. We aim to define classification criteria for
computational models of curriculum learning for RL agents, that describe the
curriculum learning research landscape over a broad range of frameworks and
settings. The questions we address in this survey include:
* •
What is a _curriculum_ , and how can it be represented for reinforcement
learning tasks? At the most basic level, a curriculum can be thought of as an
ordering over experience samples. However, it can also be represented at the
task level, where a set of tasks can be organized into a sequence or a
directed acyclic graph that specifies the order in which they should be
learned. We address this question in detail in Section 3.1.
* •
What is the _curriculum learning_ method, and how can such methods be
evaluated? We formalize this class of methods in Section 3.2 as consisting of
three parts, and extend metrics commonly used in transfer learning (introduced
in Section 2) to the curriculum setting to facilitate evaluation in Section
3.3.
* •
How can tasks be constructed for use in a curriculum? The quality of a
curriculum is dependent on the quality of tasks available to select from.
Tasks can either be generated in advance, or dynamically and on-the-fly with
the curriculum. Section 4.1 surveys works that examine how to automatically
generate good intermediate tasks.
* •
How can tasks or experience samples be sequenced into a curriculum? In
practice, most curricula for RL agents have been manually generated for each
problem. However, in recent years, automated methods for generating curricula
have been proposed. Each makes different assumptions about the tasks and
transfer methodology used. In Section 4.2, we survey these different automated
approaches, as well as describe how humans have approached curriculum
generation for RL agents.
* •
How can an agent transfer knowledge between tasks as it learns through a
curriculum? Curriculum learning approaches make use of transfer learning
methods when moving from one task to another. Since the tasks in the
curriculum can vary in state/action space, transition function, or reward
function, it’s important to transfer relevant and reusable information from
each task, and effectively combine information from multiple tasks. Methods to
do this are enumerated and discussed in Section 4.3.
The next section provides background in reinforcement learning and transfer
learning. In Section 3, we define the curriculum learning method, evaluation
metrics, and the dimensions along which we will classify curriculum learning
approaches. Section 4, which comprises the core of the survey, provides a
detailed overview of the existing state of the art in curriculum learning in
RL, with each subsection considering a different component of the overall
curriculum learning approach. Section 5 discusses paradigms related to
curriculum learning for RL, such as curriculum learning for supervised
learning and for human education. Finally, in Section 6, we identify gaps in
the existing literature, outline the limitations of existing CL methods and
frameworks, and provide a list of open problems.
## 2 Background
In this section, we provide background on Reinforcement Learning (RL) and
Transfer Learning (TL).
### 2.1 Reinforcement Learning
Reinforcement learning considers the problem of how an agent should act in its
environment over time, so as to maximize some scalar reward signal. We can
formalize the interaction of an agent with its environment (also called a
_task_) as a Markov Decision Process (MDP). In this article, we restrict our
attention to _episodic_ MDPs:111In continuing tasks, a discount factor
$\gamma$ is often included. For simplicity, and due to the fact that tasks
typically terminate in curriculum learning settings, we present the
undiscounted case. But unless otherwise noted, our definitions and discussions
can easily apply to the discounted case as well.
###### Definition 1
An episodic MDP $M$ is a 6-tuple $(\mathcal{S},\mathcal{A},p,r,\Delta
s_{0},\mathcal{S}_{f})$, where $\mathcal{S}$ is the set of states,
$\mathcal{A}$ is the set of actions, $p(s^{\prime}|s,a)$ is a transition
function that gives the probability of transitioning to state $s^{\prime}$
after taking action $a$ in state $s$, and $r(s,a,s^{\prime})$ is a reward
function that gives the immediate reward for taking action $a$ in state $s$
and transitioning to state $s^{\prime}$. In addition, we shall use $\Delta
s_{0}$ to denote the initial state distribution, and $\mathcal{S}_{f}$ to
denote the set of terminal states.
We consider time in discrete time steps. At each time step $t$, the agent
observes its state and chooses an action according to its _policy_ $\pi(a|s)$.
The goal of the agent is to learn an _optimal policy_ $\pi^{*}$, which
maximizes the expected _return_ $G_{t}$ (the cumulative sum of rewards $R$)
until the episode ends at timestep $T$:
$G_{t}=\sum_{i=1}^{T-t}R_{t+i}$
There are three main classes of methods to learn $\pi^{*}$: value function
approaches, policy search approaches, and actor-critic methods. In _value
function approaches_ , a value $v_{\pi}(s)$ is first learned for each state
$s$, representing the expected return achievable from $s$ by following policy
$\pi$. Through policy evaluation and policy improvement, this value function
is used to derive a policy better than $\pi$, until convergence towards an
optimal policy. Using a value function in this process requires a model of the
reward and transition functions of the environment. If the model is not known,
one option is to learn an action-value function instead, $q_{\pi}(s,a)$, which
gives the expected return for taking action $a$ in state $s$ and following
$\pi$ after:
$q_{\pi}(s,a)=\sum_{s^{\prime}}p(s^{\prime}|s,a)[r(s,a,s^{\prime})+q_{\pi}(s^{\prime},a^{\prime})]\textrm{
, where }a^{\prime}\sim\pi(\cdot|s^{\prime})$
The action-value function can be iteratively improved towards the optimal
action-value function $q_{*}$ with on-policy methods such as SARSA (Sutton and
Barto, 1998). The optimal action-value function can also be learned directly
with off-policy methods such as $Q$-learning (Watkins and Dayan, 1992). An
optimal policy can then be obtained by choosing action
$\text{argmax}_{a}q_{*}(s,a)$ in each state. If the state space is large or
continuous, the action-value function can instead be estimated using a
function approximator (such as a neural network), $q(s,a;\bm{w})\approx
q_{*}(s,a)$, where $\bm{w}$ are the weights of the network.
In contrast, _policy search methods_ directly search for or learn a
parameterized policy $\pi_{\bm{\theta}}(a|s)$, without using an intermediary
value function. Typically, the parameter $\bm{\theta}$ is modified using
search or optimization techniques to maximize some performance measure
$J(\bm{\theta})$. For example, in the episodic case, $J(\bm{\theta})$ could
correspond to the expected value of the policy parameterized by $\bm{\theta}$
from the starting state $s_{0}\sim\Delta s_{0}$: $v_{\pi_{\theta}}(s_{0})$.
A third class of methods, _actor-critic methods_ , maintain a parameterized
representation of both the current policy and value function. The actor is a
parameterized policy that dictates how the agent selects actions. The critic
estimates the (action-)value function for the actor using a policy evaluation
method such as temporal-difference learning. The actor then updates the policy
parameter in the direction suggested by the critic. An example of actor-critic
methods is Deterministic Policy Gradient (Silver et al., 2014).
### 2.2 Transfer Learning
In the standard reinforcement learning setting, an agent usually starts with a
random policy, and directly attempts to learn an optimal policy for the target
task. When the target task is difficult, for example due to adversarial
agents, poor state representation, or sparse reward signals, learning can be
very slow.
Transfer learning is one class of methods and area of research that seeks to
speed up training of RL agents. The idea behind transfer learning is that
instead of learning on the _target task_ tabula rasa, the agent can first
train on one or more _source task_ MDPs, and _transfer_ the knowledge acquired
to aid in solving the target. This knowledge can take the form of samples
(Lazaric et al., 2008; Lazaric and Restelli, 2011), options (Soni and Singh,
2006), policies (Fernández et al., 2010), models (Fachantidis et al., 2013),
or value functions (Taylor and Stone, 2005). As an example, in value function
transfer (Taylor et al., 2007), the parameters of an action-value function
$q_{source}(s,a)$ learned in a source task are used to initialize the action-
value function in the target task $q_{target}(s,a)$. This biases exploration
and action selection in the target task based on experience acquired in the
source task.
Some of these methods assume that the source and target MDPs either share
state and action spaces, or that a _task mapping_ (Taylor et al., 2007) is
available to map states and actions in the target task to known states and
actions in the source. Such mappings can be specified by hand, or learned
automatically (Taylor et al., 2008; Ammar et al., 2015). Other methods assume
the transition or reward functions do not change between tasks. The best
method to use varies by domain, and depends on the relationship between source
and target tasks. Finally, while most methods assume that knowledge is
transferred from one source task to one target task, some methods have been
proposed to transfer knowledge from several source tasks directly to a single
target (Svetlik et al., 2017). See Taylor and Stone (2009) or Lazaric (2012)
for a survey of transfer learning techniques.
### 2.3 Evaluation Metrics for Transfer Learning
There are several metrics to quantify the benefit of transferring from a
source task to a target task (Taylor and Stone, 2009). Typically, they compare
the learning trajectory on the target task for an agent after transfer, with
an agent that learns directly on the target task from scratch (see Figure 2a).
One metric is _time to threshold_ , which computes how much faster an agent
can learn a policy that achieves expected return $G_{0}\geq\delta$ on the
target task if it transfers knowledge, as opposed to learning the target from
scratch, where $\delta$ is some desired performance threshold. Time can be
measured in terms of CPU time, wall clock time, episodes, or number of actions
taken. Another metric is _asymptotic performance_ , which compares the final
performance after convergence in the target task of learners when using
transfer versus no transfer. The _jumpstart_ metric instead measures the
initial performance increase on the target task as a result of transfer.
Finally, the _total reward_ ratio compares the total reward accumulated by the
agent during training up to a fixed stopping point, using transfer versus not
using transfer.
$\begin{array}[]{cc}\includegraphics[height=113.81102pt]{img/weakTL.png}&\includegraphics[height=113.81102pt]{img/strongTL.png}\\\
(a)&(b)\end{array}$
Figure 2: Performance metrics for transfer learning using (a) weak transfer
and (b) strong transfer with offset curves.
An important evaluation question is whether to include time spent _learning in
source tasks_ into the cost of using transfer. The transfer curve in Figure 2a
shows performance on the target task, and starts at time 0, even though time
has already been spent learning one or more source tasks. Thus, it does not
reflect time spent training in source tasks before transferring to the target
task. This is known in transfer learning as the _weak transfer_ setting, where
time spent training in source tasks is treated as a sunk cost. On the other
hand, in the _strong transfer_ setting, the learning curves must account for
time spent in all source tasks. One way to do this is to offset the curves to
reflect time spent in source tasks, as shown in Figure 2b. Another option is
to freeze the policy while learning on source tasks, and plot that policy’s
performance on the target task.
## 3 The Curriculum Learning Method
A _curriculum_ serves to sort the experience an agent acquires over time, in
order to accelerate or improve learning. In the rest of this section we
formalize this concept and the methodology of _curriculum learning_ , and
describe how to evaluate the benefits and costs of using a curriculum.
Finally, we provide a list of attributes which we will use to categorize
curriculum learning approaches in the rest of this survey.
### 3.1 Curricula
A curriculum is a general concept that encompasses both schedules for
organizing past experiences, and schedules for acquiring experience by
training on tasks. As such, we first propose a fully general definition of
curriculum, and then follow it with refinements that apply to special cases
common in the literature.
We assume a _task_ is modeled as a Markov Decision Process, and define a
curriculum as follows:
###### Definition 2 (Curriculum)
Let $\mathcal{T}$ be a set of tasks, where
$m_{i}=(\mathcal{S}_{i},\mathcal{A}_{i},p_{i},r_{i})$ is a task in
$\mathcal{T}$. Let $\mathcal{D}^{\mathcal{T}}$ be the set of all possible
transition samples from tasks in $\mathcal{T}$:
$\mathcal{D}^{\mathcal{T}}=\\{(s,a,r,s^{\prime})\>|\>\exists\,m_{i}\in\mathcal{T}\;\mathrm{s.t.}\;s\in\mathcal{S}_{i},a\in\mathcal{A}_{i},s^{\prime}\sim
p_{i}(\cdot|s,a),r\leftarrow r_{i}(s,a,s^{\prime})\\}$. A _curriculum_
$C=(\mathcal{V},\mathcal{E},g,\mathcal{T})$ is a directed acyclic graph, where
$\mathcal{V}$ is the set of vertices,
$\mathcal{E}\subseteq\\{(x,y)\;|\;(x,y)\in\mathcal{V}\times\mathcal{V}\>\land
x\neq y\\}$ is the set of directed edges, and
$g:\mathcal{V}\to\mathcal{P}(\mathcal{D}^{\mathcal{T}})$ is a function that
associates vertices to subsets of samples in $\mathcal{D}^{\mathcal{T}}$,
where $\mathcal{P}(\mathcal{D}^{\mathcal{T}})$ is the power set of
$\mathcal{D}^{\mathcal{T}}$. A directed edge $\langle v_{j},v_{k}\rangle$ in
$C$ indicates that samples associated with $v_{j}\in\mathcal{V}$ should be
trained on before samples associated with $v_{k}\in\mathcal{V}$. All paths
terminate on a single sink node $v_{t}\in\mathcal{V}$.222In theory, a
curriculum could have multiple sink nodes corresponding to different target
tasks. For the purpose of exposition, we assume a separate curriculum is
created and used for each task.
A curriculum can be created online, where edges are added dynamically based on
the learning progress of the agent on the samples at a given vertex. It can
also be designed completely offline, where the graph is generated before
training, and edges are selected based on properties of the samples associated
with different vertices.
Creating a curriculum graph at the sample level can be computationally
difficult for large tasks, or large sets of tasks. Therefore, in practice, a
simplified representation for a curriculum is often used. There are 3 common
dimensions along which this simplification can happen. The first is the
single-task curriculum, where all samples used in the curriculum come from a
single task:
###### Definition 3 (Single-task Curriculum)
A _single-task curriculum_ is a curriculum $C$ where the cardinality of the
set of tasks considered for extracting samples $|\mathcal{T}|=1$, and consists
of only the target task $m_{t}$.
A single-task curriculum essentially considers how best to organize and train
on experience acquired from a single task. This type of curriculum is common
in experience replay methods (Schaul et al., 2016).
A second common simplification is to learn a curriculum at the task level,
where each vertex in the graph is associated with samples from a single task.
At the task level, a curriculum can be defined as a directed acyclic graph of
_intermediate_ tasks:
###### Definition 4 (Task-level Curriculum)
For each task $m_{i}\in\mathcal{T}$, let $\mathcal{D}^{\mathcal{T}}_{i}$ be
the set of all samples associated with task $m_{i}$:
$\mathcal{D}^{\mathcal{T}}_{i}=\\{(s,a,r,s^{\prime})\>|\>s\in\mathcal{S}_{i},a\in\mathcal{A}_{i},s^{\prime}\sim
p_{i}(\cdot|s,a),r\leftarrow r_{i}(s,a,s^{\prime})\\}$. A _task-level
curriculum_ is a curriculum $C=(\mathcal{V},\mathcal{E},g,\mathcal{T})$ where
each vertex is associated with samples from a single task in $\mathcal{T}$.
Thus, the mapping function $g$ is defined as
$g:\mathcal{V}\to\\{\mathcal{D}^{\mathcal{T}}_{i}\;|\;m_{i}\in\mathcal{T}\\}$.
In reinforcement learning, the entire set of samples from a task (or multiple
tasks) is usually not available ahead of time. Instead, the samples
experienced in a task depend on the agent’s behavior policy, which can be
influenced by previous tasks learned. Therefore, while generating a task-level
curriculum, the main challenge is how to order tasks such that the behavior
policy learned is useful for acquiring good samples in future tasks. In other
words, selecting and training on a task $m$ induces a mapping function $g$,
and determines the set of samples $\mathcal{D}_{i}^{\mathcal{T}}$ that will be
available at the next vertex based on the agent’s behavior policy as a result
of learning $m$. The same task is allowed to appear at more than one vertex,
similar to how in Definition 2 the same set of samples can be associated with
more than one vertex. Therefore, tasks can be revisited when the agent’s
behavior policy has changed. Several works have considered learning task-level
curricula over a graph of tasks (Svetlik et al., 2017; MacAlpine and Stone,
2018). An example can be seen in Figure 3b.
Finally, another simplification of the curriculum is the linear _sequence_.
This is the simplest and most common structure for a curriculum in existing
work:
###### Definition 5 (Sequence Curriculum)
A _sequence curriculum_ is a curriculum $C$ where the indegree and outdegree
of each vertex $v$ in the graph $C$ is at most 1, and there is exactly one
source node and one sink node.
These simplifications can be combined to simplify a curriculum along multiple
dimensions. For example, the sequence simplification and task-level
simplification can be combined to produce a task-level sequence curriculum.
This type of curriculum can be represented as an ordered list of tasks
$[m_{1},m_{2},...m_{n}]$. An example can be seen in Figure 3a (Narvekar et
al., 2017).
A final important question when designing curricula is determining the
stopping criteria: that is, how to decide _when_ to stop training on samples
or tasks associated with a vertex, and move on to the next vertex. In
practice, typically training is stopped when performance on the task or set of
samples has converged. Training to convergence is not always necessary, so
another option is to train on each vertex for a fixed number of episodes or
epochs. Since more than one vertex can be associated with the same
samples/tasks, this experience can be revisited later on in the curriculum.
### 3.2 Curriculum Learning
_Curriculum learning_ is a methodology to _optimize_ the order in which
experience is accumulated by the agent, so as to increase performance or
training speed on a set of final tasks. Through generalization, knowledge
acquired quickly in simple tasks can be leveraged to reduce the exploration of
more complex tasks. In the most general case, where the agent can acquire
experience from multiple intermediate tasks that differ from the final MDP,
there are 3 key elements to this method:
* •
Task Generation. The quality of a curriculum is dependent on the quality of
tasks available to choose from. Task generation is the process of creating a
good set of intermediate tasks from which to obtain experience samples. In a
task-level curriculum, these tasks form the nodes of the curriculum graph.
This set of intermediate tasks may either be pre-specified, or dynamically
generated during the curriculum construction by observing the agent.
* •
Sequencing. Sequencing examines how to create a partial ordering over the set
of experience samples $\mathcal{D}$: that is, how to generate the edges of the
curriculum graph. Most existing work has used manually defined curricula,
where a human selects the ordering of samples or tasks. However, recently
automated methods for curriculum sequencing have begun to be explored. Each of
these methods make different assumptions about the tasks and transfer
methodology used. These methods will be the primary focus of this survey.
* •
Transfer Learning. When creating a curriculum using multiple tasks, the
intermediate tasks may differ in state/action space, reward function, or
transition function from the final task. Therefore, transfer learning is
needed to extract and pass on reusable knowledge acquired in one task to the
next. Typically, work in transfer learning has examined how to transfer
knowledge from one or more source tasks directly to the target task.
Curriculum learning extends the transfer learning scenario to consider
training sessions in which the agent must repeatedly transfer knowledge from
one task to another, up to a set of final tasks.
$\begin{array}[]{cc}\includegraphics[height=156.49014pt]{img/sequences.png}&\includegraphics[height=156.49014pt]{img/dag.png}\\\
(a)&(b)\end{array}$
Figure 3: Examples of structures of curricula from previous work. (a) Linear
sequences in a gridworld domain (Narvekar et al., 2017) (b) Directed acyclic
graphs in block dude (Svetlik et al., 2017).
### 3.3 Evaluating Curricula
Curricula can be evaluated using the same metrics as for transfer learning
(cf. Section 2.3), by comparing performance on the target task after following
the complete curriculum, versus performance following no curriculum (i.e.,
learning from scratch). If there are multiple final tasks, the metrics can
easily be extended: for example, by comparing the average asymptotic
performance over a set of tasks, or the average time to reach a threshold
performance level over a set of tasks.
Similarly, it is possible to distinguish between weak and strong transfer.
However, in curriculum learning, there is the additional expense required to
_build_ the curriculum itself, in addition to training on intermediate tasks
in the curriculum, which can also be factored in when evaluating the cost of
the curriculum. As in the transfer learning case, cost can be measured in
terms of wall clock time, or data/sample complexity.
Most existing applications of curricula in reinforcement learning have used
curricula created by humans. In these cases, it can be difficult to assess how
much time, effort, and prior knowledge was used to design the curriculum.
Automated approaches to generate a curriculum also typically require some
prior knowledge or experience in potential intermediate tasks, in order to
guide the sequencing of tasks. Due to these difficulties, these approaches
have usually treated curriculum generation as a sunk cost, focusing on
evaluating the performance of the curriculum itself, and comparing it versus
other curricula, including those designed by people.
The best set of evaluation criteria to use ultimately depends on the specific
problem and settings being considered. For example, how expensive is it to
collect data on the final task compared to intermediate tasks? If intermediate
tasks are relatively inexpensive, we can treat time spent in them as sunk
costs. Is it more critical to improve initial performance, final performance,
or reaching a desired performance threshold? If designing the curriculum will
require human interaction, how will this time be factored into the cost of
using a curriculum? Many of these questions depend on whether we wish to
evaluate the utility of a specific curriculum (compared to another
curriculum), or whether we wish to evaluate the utility of using a curriculum
design approach versus training without one.
### 3.4 Dimensions of Categorization
We categorize curriculum learning approaches along the following seven
dimensions, organized by attributes (in bold) and the values (in italics) they
can take. We use these dimensions to create a taxonomy of surveyed work in
Section 4.
1. 1.
Intermediate task generation: _target / automatic / domain experts / naive
users_. In curriculum learning, the primary challenge is how to sequence a set
of tasks to improve learning speed. However, finding a good curriculum depends
on first having useful source tasks to select from. Most methods assume the
set of possible source tasks is fixed and given ahead of time. In the simplest
case, only samples from the _target_ task are used. When more than one
intermediate task is used, typically they are manually designed by humans. We
distinguish such tasks as designed by either _domain experts_ , who have
knowledge of the agent and its learning algorithm, or _naive users_ , who do
not have this information. On the other hand, some works consider
_automatically_ creating tasks online using a set of rules or generative
process. These approaches may still rely on some human input to control/tune
hyper-parameters, such as the number of tasks generated, or to verify that
generated tasks are actually solvable.
2. 2.
Curriculum representation: _single / sequence / graph_. As we discussed
previously, the most general form of a curriculum is a directed acyclic graph
over subsets of samples. However, in practice, simplified versions of this
representation are often used. In the simplest case, a curriculum is an
ordering over samples from a _single_ task. When multiple tasks can be used in
a curriculum, curricula are often created at the task-level. These curricula
can be represented as a linear chain, or _sequence_. In this case, there is
exactly one source for each intermediate task in the curriculum. It is up to
the transfer learning algorithm to appropriately retain and combine
information gathered from previous tasks in the chain. More generally, they
can be represented as a full directed acyclic _graph_ of tasks. This form
supports transfer learning methods that transfer from many-to-one, one-to-
many, and many-to-many tasks.
3. 3.
Transfer method: _policies / value function / task model / partial policies /
shaping reward / other / no transfer_. Curriculum learning leverages ideas
from transfer learning to transfer knowledge between tasks in the curriculum.
As such, the transfer learning algorithm used affects how the curriculum will
be produced. The type of knowledge transferred can be low-level knowledge,
such as an entire _policy_ , an _(action-)value function_ , or a full _task
model_ , which can be used to directly initialize the learner in the target
task. It can also be high-level knowledge, such as _partial policies_ (e.g.
options) or _shaping rewards_. This type of information may not fully
initialize the learner in the target task, but it could be used to guide the
agent’s learning process in the target task. We use partial policies as an
umbrella term to represent closely related ideas such as options, skills, and
macro-actions. When samples from a single task are sequenced, _no transfer_
learning algorithm is necessary. Finally, we use _other_ to refer to other
types of transfer learning methods. We categorize papers along this dimension
based on what is transferred between tasks in the curriculum in each paper’s
experimental results.
4. 4.
Curriculum sequencer: _automatic / domain experts / naive users_. Curriculum
learning is a three-part method, consisting of task generation, sequencing,
and transfer learning. While much of the attention of this survey is on
automated sequencing approaches, many works consider the other parts of this
method, and assume the sequencing is done by a human or oracle. Thus, we
identify and categorize the type of sequencing approach used in each work
similar to task generation: it can be done _automatically_ by a sequencing
algorithm, or manually by humans that are either _domain experts_ or _naive
users_.
5. 5.
Curriculum adaptivity: _static / adaptive_. Another design question when
creating a curriculum is whether it should be generated in its entirety before
training, or dynamically adapted during training. We refer to the former type
as _static_ and to the latter as _adaptive_. Static approaches use properties
of the domain and possibly of the learning agent, to generate a curriculum
before any task is learned. Adaptive methods, on the other hand, are
influenced by properties that can only be measured during learning, such as
the learning progress by the agent on the task it is currently facing. For
example, learning progress can be used to guide whether subsequent tasks
should be easier or harder, as well as how relevant a task is for the agent at
a particular point in the curriculum.
6. 6.
Evaluation metric: _time to threshold / asymptotic / jumpstart / total
reward_. We discussed four metrics to quantify the effectiveness of learned
curricula in Section 3.3. When calculating these metrics, one can choose
whether to treat time spent generating the curriculum and training on the
curriculum as a sunk cost, or whether to account for both of these for
performance. Specifically, there are three ways to measure the cost of
learning and training via a curriculum. 1) The cost of generating and using
the curriculum is treated as a sunk cost, and the designer is only concerned
with performance on the target task after learning. This case corresponds to
the weak transfer setting. 2) The cost of training on intermediate tasks in
the curriculum is accounted for, when comparing to training directly on the
target task. This case is most common when it is hard to evaluate the cost of
generating the curriculum itself, for example if it was hand-designed by a
human. 3) Lastly, the most comprehensive case accounts for the cost of
generating the curriculum as well as training via the curriculum. We will
refer to the last two as strong transfer, and indicate it by bolding the
corresponding metric. Note that achieving asymptotic performance improvements
implies strong transfer.
7. 7.
Application area: _toy / sim robotics / real robotics / video games / other_.
Curriculum learning methods have been tested in a wide variety of domains.
_Toy_ domains consist of environments such as grid worlds, cart-pole, and
other low dimensional environments. _Sim robotics_ environments simulate
robotic platforms, such as in MuJoCo. _Real robotics_ papers test their method
on physical robotic platforms. _Video games_ consist of game environments such
as Starcraft or the Arcade Learning Environment (Atari). Finally, _other_ is
used for custom domains that do not fit in these categories. We list these so
that readers can better understand the scalability and applicability of
different approaches, and use these to inform what methods would be suitable
for their own problems.
## 4 Curriculum Learning for Reinforcement Learning Agents
In this section, we systematically survey work on each of the three central
elements of curriculum learning: task generation (Section 4.1), sequencing
(Section 4.2), and transfer learning (Section 4.3). For each of these
subproblems, we provide a table that categorizes work surveyed according to
the dimensions outlined in Section 3. The bulk of our attention will be
devoted to the subproblem most commonly associated with curriculum learning:
sequencing.
### 4.1 Task Generation
Task generation is the problem of creating intermediate tasks specifically to
be part of a curriculum. In contrast to the life-long learning scenario, where
potentially unrelated tasks are constantly proposed to the agent (Thrun,
1998), the aim of task generation is to create a set of tasks such that
knowledge transfer through them is beneficial. Therefore, all the generated
tasks should be relevant to the final task(s) and avoid _negative transfer_ ,
where using a task for transfer hurts performance. The properties of the
research surveyed in this section are reported in Table 1.
Very limited work has been dedicated to formally studying this subproblem in
the context of reinforcement learning. All known methods assume the domain can
be parameterized using some kind of representation, where different
instantiations of these parameters create different tasks. For instance,
Narvekar et al. (2016) introduce a number of methods to create intermediate
tasks for a specific final task. The methods hinge on a definition of a domain
as a set of MDPs identified by a _task descriptor_ , which is a vector of
parameters specifying the _degrees of freedom_ in the domain. For example, in
the quick chess example (see Section 1), these parameters could be the size of
the board, number of pawns, etc. By varying the degrees of freedom and
applying task _restrictions_ , the methods define different types of tasks.
Methods introduced include: _task simplification_ , which directly changes the
degrees of freedom to reduce the task dimensions; _promising initialization_ ,
which modifies the set of initial states by adding states close to high
rewards; _mistake learning_ , which rewinds the domain to a state a few steps
before a mistake is detected and resumes learning from there; and several
other methods. The set of methods determine different kinds of possible tasks,
which form a space of tasks in which appropriate intermediate tasks can be
chosen.
Da Silva and Reali Costa (2018) propose a similar partially automated task
generation procedure in their curriculum learning framework, based on Object-
Oriented MDPs. Each task is assumed to have a class _environment_
parameterized by a number of attributes. A function, which must be provided by
the designer, creates simpler versions of the final task by instantiating the
attributes with values that make the tasks easier to solve. For example,
continuing the quick chess example, the attributes could be the types of
pieces, and the values are the number of each type of piece. The presence of
different kinds and numbers of objects provide a range of tasks with different
levels of difficulty. However, since the generation is mostly random, the
designer has to make sure that the tasks are indeed solvable.
Citation | Intermediate Task Generation | Curriculum Representation | Transfer Method | Curriculum Sequencer | Curriculum Adaptivity | Evaluation Metric | Application Area
---|---|---|---|---|---|---|---
Da Silva and Reali Costa (2018) | automatic | graph | value function | automatic | static | time to threshold, total reward | toy, video games
Narvekar et al. (2016) | automatic | sequence | value function | domain experts | adaptive | asymptotic | video games
Schmidhuber (2013) | automatic | sequence | partial policies | automatic | adaptive | asymptotic | other
Stone and Veloso (1994) | automatic | sequence | other | domain experts | adaptive | time to threshold | other
Table 1: The papers discussed in Section 4.1, categorized along the dimensions
presented in Section 3.4. Bolded values under evaluation metric indicate
strong transfer.
Generating auxiliary intermediate tasks is a problem that has been studied in
non-RL contexts as well. For instance, Stone and Veloso (1994) consider how to
semiautomatically create subproblems to aid in learning to solve difficult
_planning_ problems. Rather than using a static analysis of the domain’s
properties, they propose to use a partially completed search trajectory of the
target task to identify what makes a problem difficult, and suggest auxiliary
tasks. For example, if the task took too long and there are multiple goals in
the task, try changing the order of the goals. Other methods they propose
include reducing the number of goals, creating tasks to solve difficult
subgoals, and changing domain operators and objects available for binding.
Lastly, Schmidhuber (2013) introduced Powerplay, a framework that focuses on
inventing new problems to train a more and more general problem solver in an
unsupervised fashion. The system searches for both a new task and a
modification of the current problem solver, such that the modified solver can
solve all previous tasks, plus the new one. The search acts on a domain-
dependent encoding of the problem and the solver, and has been demonstrated on
pattern recognition and control tasks (Srivastava et al., 2013). The generator
of the task and new solver is given a limited computational budget, so that it
favors the generation of the simplest tasks that could not be solved before.
Furthermore, a possible task is to solve all previous tasks, but with a more
compact representation of the solver. The resulting iterative process makes
the system increasingly more competent at different tasks. The task generation
process effectively creates a curriculum, although in this context there are
no final tasks, and the system continues to generate pairs of problems and
solvers indefinitely, without any specific goal.
### 4.2 Sequencing
Given a set of tasks, or samples from them, the goal of sequencing is to order
them in a way that facilitates learning. Many different sequencing methods
exist, each with their own set of assumptions. One of the fundamental
assumptions of curriculum learning is that we can configure the environment to
create different tasks. For the practitioner attempting to use curriculum
learning, the amount of control one has to shape the environment affects the
type of sequencing methods that could be applicable. Therefore, we categorize
sequencing methods by the degree to which intermediate tasks may differ.
Specifically, they form a spectrum, ranging from methods that simply reorder
experience in the final task without modifying any property of the
corresponding MDP, to ones that define entirely new intermediate tasks, by
progressively adjusting some or all of the properties of the final task.
In this subsection, we discuss the different sequencing approaches. First, in
Section 4.2.1, we consider methods that reorder samples in the target task to
derive a curriculum. Experience replay methods are one such example. In
Section 4.2.2, we examine multi-agent approaches to curriculum generation,
where the cooperation or competition between two (typically evolving) agents
induces a sequence of progressively challenging tasks, like a curriculum.
Then, in Section 4.2.3, we begin describing methods that explicitly use
intermediate tasks, starting with ones that vary in limited ways from the
target task. In particular, these methods only change the reward function
and/or the initial and terminal state distributions to create a curriculum. In
Section 4.2.4, we discuss methods that relax this assumption, and allow
intermediate tasks that can vary in any way from the target task MDP. Finally,
in Section 4.2.5, we discuss work that explores how humans sequence tasks into
a curriculum.
#### 4.2.1 Sample Sequencing
First we consider methods that reorder samples from the final task, but do not
explicitly change the domain itself. These ideas are similar to curriculum
learning for supervised learning (Bengio et al., 2009), where training
examples are presented to a learner in a specific order, rather than
completely randomly. Bengio et al. (2009) showed that ordering these examples
from simple to complex can improve learning speed and generalization ability.
An analogous process can be used for reinforcement learning. For example, many
current reinforcement learning methods, such as Deep Q Networks (DQN) (Mnih et
al., 2015) use a replay buffer to store past state-action-reward experience
tuples. At each training step, experience tuples are sampled from the buffer
and used to train DQN in minibatches. The original formulation of DQN
performed this sampling uniformly randomly. However, as in the supervised
setting, samples can be reordered or “prioritized,” according to some measure
of usefulness or difficulty, to improve learning.
Citation | Intermediate Task Generation | Curriculum Representation | Transfer Method | Curriculum Sequencer | Curriculum Adaptivity | Evaluation Metric | Application Area
---|---|---|---|---|---|---|---
Sample Sequencing (Section 4.2.1)
Andrychowicz et al. (2017) | target | single | no transfer | automatic | adaptive | asymptotic | sim robotics
Fang et al. (2019) | target | single | no transfer | automatic | adaptive | asymptotic | sim robotics
Kim and Choi (2018) | target | single | no transfer | automatic | adaptive | asymptotic | toy, other
Lee et al. (2019) | target | single | no transfer | automatic | adaptive | time to threshold | toy, video games
Ren et al. (2018) | target | single | no transfer | automatic | adaptive | asymptotic | video games
Schaul et al. (2016) | target | single | no transfer | automatic | adaptive | asymptotic | video games
Co-learning (Section 4.2.2)
Baker et al. (2020) | automatic | sequence | policies | automatic | adaptive | asymptotic, time to threshold | other
Bansal et al. (2018) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics
Pinto et al. (2017) | automatic | sequence | policies | automatic | adaptive | time to threshold | sim robotics
Sukhbaatar et al. (2018) | automatic | sequence | policies | automatic | adaptive | time to threshold, asymptotic | toy, video games
Vinyals et al. (2019) | automatic | sequence | policies | automatic | adaptive | asymptotic | video games
Reward and Initial/Terminal State Distribution Changes (Section 4.2.3)
Asada et al. (1996) | domain experts | sequence | value function | automatic | adaptive | asymptotic | sim/real robotics
Baranes and Oudeyer (2013) | automatic | sequence | partial policies | automatic | adaptive | asymptotic | sim/real robotics
Florensa et al. (2017) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics
Florensa et al. (2018) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics
Ivanovic et al. (2019) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics
Racaniere et al. (2019) | automatic | sequence | policies | automatic | adaptive | asymptotic | toy, video games
Riedmiller et al. (2018) | domain experts | sequence | policies | automatic | adaptive | time to threshold | sim/real robotics
Wu and Tian (2017) | domain experts | sequence | task model | automatic | both | asymptotic | video games
No Restrictions (Section 4.2.4)
Bassich et al. (2020) | domain experts | sequence | policies | automatic | adaptive | asymptotic, time to threshold | toy
Da Silva and Reali Costa (2018) | automatic | graph | value function | automatic | static | time to threshold, total reward | toy, video games
Foglino et al. (2019a) | domain experts | sequence | value function | automatic | static | time to threshold, asymptotic, total reward | toy
Foglino et al. (2019b) | domain experts | sequence | value function | automatic | static | total reward | toy
Foglino et al. (2019c) | domain experts | sequence | value function | automatic | static | total reward | toy
Jain and Tulabandhula (2017) | domain experts | sequence | value function | automatic | adaptive | time to threshold, total reward | toy
Matiisen et al. (2017) | domain experts | sequence | policies | automatic | adaptive | asymptotic | toy, video games
Narvekar et al. (2017) | automatic | sequence | value function | automatic | adaptive | time to threshold | toy
Narvekar and Stone (2019) | domain experts | sequence | value function, shaping reward | automatic | adaptive | time to threshold | toy, video games
Svetlik et al. (2017) | domain experts | graph | shaping reward | automatic | static | asymptotic, time to threshold | toy, video games
Human-in-the-loop Curriculum Generation (Section 4.2.5)
Hosu and Rebedea (2016) | target | single | no transfer | automatic | adaptive | asymptotic | video games
Khan et al. (2011) | domain experts | sequence | no transfer | naive users | static | N/A | other
MacAlpine and Stone (2018) | domain experts | graph | policies | domain experts | static | asymptotic | sim robotics
Peng et al. (2018) | domain experts | sequence | task model | naive users | static | time to threshold | other
Stanley et al. (2005) | domain experts | sequence | partial policies | domain experts | adaptive | asymptotic | video games
Table 2: The papers discussed in Section 4.2, categorized along the dimensions
presented in Section 3.4. Bolded values under evaluation metric indicate
strong transfer.
The first to do this type of sample sequencing in the context of deep learning
were Schaul et al. (2016). They proposed Prioritized Experience Replay (PER),
which prioritizes and replays _important_ transitions more. Important
transitions are those with high expected learning progress, which is measured
by their temporal difference (TD) error. Intuitively, replaying samples with
larger TD errors allows the network to make stronger updates. As transitions
are learned, the distribution of important transitions changes, leading to an
implicit curriculum over the samples.
Alternative metrics for priority/importance have been explored as well. Ren et
al. (2018) propose to sort samples using a complexity index (CI) function,
which is a combination of a self-paced prioritized function and a coverage
penalty function. The self-paced prioritized function selects samples that
would be of appropriate difficulty, while the coverage function penalizes
transitions that are replayed frequently. They provide one specific
instantiation of these functions, which are used in experiments on the Arcade
Learning Environment (Bellemare et al., 2013), and show that it performs
better than PER in many cases. However, these functions must be designed
individually for each domain, and designing a broadly applicable domain-
independent priority function remains an open problem.
Kim and Choi (2018) consider another extension of prioritized experience
replay, where the weight/priority of a sample is jointly learned with the main
network via a secondary neural network. The secondary network, called
ScreenerNet, learns to predict weights according to the error of the sample by
the main network. Unlike PER, this approach is memoryless, which means it can
directly predict the significance of a training sample even if that particular
example was not seen. Thus, the approach could potentially be used to actively
request experience tuples that would provide the most information or utility,
creating an online curriculum.
Instead of using sample importance as a metric for sequencing, an alternative
idea is to restructure the training process based on trajectories of samples
experienced. For example, when learning, typically easy to reach states are
encountered first, whereas harder to reach states are encountered later on in
the learning cycle. However, in practical settings with sparse rewards, these
easy to reach states may not provide a reward signal. Hindsight Experience
Replay (HER) (Andrychowicz et al., 2017) is one method to make the most of
these early experiences. HER is a method that learns from “undesired
outcomes,” in addition to the desired outcome, by replaying each episode with
a goal that was actually achieved rather than the one the agent was trying to
achieve. The problem is set up as learning a Universal Value Function
Approximator (UVFA) (Schaul et al., 2015), which is a value function
$v_{\pi}(s,g)$ defined over states $s$ and goals $g$ . The agent is given an
initial state $s_{1}$ and a desired goal state $g$. Upon executing its policy,
the agent may not reach the goal state $g$, and instead land on some other
terminal state $s_{T}$. While this trajectory does not help to learn to
achieve $g$, it does help to learn to achieve $s_{T}$. Thus, this trajectory
is added to the replay buffer with the goal state substituted with $s_{T}$,
and used with an off-policy RL algorithm. HER forms a curriculum by taking
advantage of the implicit curriculum present in exploration, where early
episodes are likely to terminate on easy to reach states, and more difficult
to reach states are found later in the training process.
One of the issues with vanilla HER is that all goals in seen trajectories are
replayed evenly, but some goals may be more useful at different points of
learning. Thus, Fang et al. (2019) later proposed Curriculum-guided HER (CHER)
to adaptively select goals based on two criteria: curiosity, which leads to
the selection of diverse goals, and proximity, which selects goals that are
closer to the true goal. Both of these criteria rely on a measure of distance
or similarity between goal states. At each minibatch optimization step, the
objective selects a subset of goals that maximizes the weighted sum of a
diversity and proximity score. They manually impose a curriculum that starts
biased towards diverse goals and gradually shifts towards proximity based
goals using a weighting factor that is exponentially scaled over time.
Other than PER and HER, there are other works that reorder/resample
experiences in a novel way to improve learning. One example is the episodic
backward update (EBU) method developed by Lee et al. (2019). In order to speed
up the propagation of delayed rewards (e.g., a reward might only be obtained
at the end of an episode), Lee et al. (2019) proposed to sample a whole
episode from the replay buffer and update the values of all transitions within
the sampled episode in a backward fashion. Starting from the end of the
sampled episode, the $\max$ Bellman operator is applied recursively to update
the target $Q$-values until the start of the sampled episode. This process
basically reorders all the transitions within each sampled episode from the
last timestep of the episode to the first, leading to an implicit curriculum.
Updating highly correlated states in a sequence while using function
approximation is known to suffer from cumulative overestimation errors. To
overcome this issue, a diffusion factor $\beta\in(0,1)$ was introduced to
update the current $Q$-value using a weighted sum of the new bootstrapped
target value and the pre-existing $Q$-value estimate. Their experimental
results show that in 49 Atari games, EBU can achieve the same mean and median
human normalized performance of DQN by using significantly fewer samples.
Methods that sequence experience samples have wide applicability and found
broad success in many applications, since they can be applied directly on the
target task without needing to create intermediate tasks that alter the
environment. In the following sections, we consider sequencing approaches that
progressively alter how much intermediate tasks in the curriculum may differ.
#### 4.2.2 Co-learning
Co-learning is a multi-agent approach to curriculum learning, in which the
curriculum emerges from the interaction of several agents (or multiple
versions of the same agent) in the same environment. These agents may act
either cooperatively or adversarially to drive the acquisition of new
behaviors, leading to an implicit curriculum where both sets of agents improve
over time. Self-play is one methodology that fits into this paradigm, and many
landmark results such as TD-Gammon (Tesauro, 1995) and more recently AlphaGo
(Silver et al., 2016) and AlphaStar (Vinyals et al., 2019) fall into this
category. Rather than describing every work that uses self-play or co-
learning, we describe a few papers that focus on how the objectives of the
multiple agents can be set up to facilitate co-learning.
Sukhbaatar et al. (2018) proposed a novel method called asymmetric self-play
that allows an agent to learn about the environment without any external
reward in an unsupervised manner. This method considers two agents, a teacher
and a student, using the paradigm of “the teacher proposing a task, and the
student doing it.” The two agents learn their own policies simultaneously by
maximizing interdependent reward functions for goal-based tasks. The teacher’s
task is to navigate to an environment state that the student will use either
as 1) a goal, if the environment is resettable, or 2) as a starting state, if
the environment is reversible. In the first case, the student’s task is to
reach the teacher’s final state, while in the second case, the student starts
from the teacher’s final state with the aim of reverting the environment to
its original initial state. The student’s goal is to minimize the number of
actions it needs to complete the task. The teacher, on the other hand, tries
to maximize the difference between the actions taken by the student to execute
the task, and the actions spent by the teacher to set up the task. The
teacher, therefore, tries to identify a state that strikes a balance between
being the simplest goal (in terms of number of teacher actions) for itself to
find, and the most difficult goal for the student to achieve. This process is
iterated to automatically generate a curriculum of intrinsic exploration.
Another example of jointly training a pair of agents adversarially for policy
learning in single-agent RL tasks is Robust Adversarial RL (RARL) by Pinto et
al. (2017). Unlike asymmetric self-play (Sukhbaatar et al., 2018), in which
the teacher defines the goal for the student, RARL trains a protagonist and an
adversary, where the protagonist learns to complete the original RL task while
being robust to the disturbance forces applied by the adversarial agent. RARL
is targeted at robotic systems that are required to generalize effectively
from simulation, and learn robust policies with respect to variations in
physical parameters. Such variations are modeled as disturbances controlled by
an adversarial agent, and the adversarial agent’s goal is to learn the optimal
sequence of destabilizing actions via a zero-sum game training procedure. The
adversarial agent tries to identify the hardest conditions under which the
protagonist agent may be required to act, increasing the agent’s robustness.
Learning takes place in turns, with the protagonist learning against a fixed
antagonist’s policy, and then the antagonist learning against a fixed
protagonist’s policy. Each agent tries to maximize its own return, and the
returns are zero-sum. The set of “destabilizing actions” available to the
antagonist is assumed to be domain knowledge, and given to the adversary ahead
of time.
For multi-agent RL tasks, several works have shown how simple interaction
between multiple learning agents in an environment can result in emergent
curricula. Such ideas were explored early on in the context of evolutionary
algorithms by Rosin and Belew (1997). They showed that competition between 2
groups of agents, dubbed hosts and parasites, could lead to an “arms race,”
where each group drives the other to acquire increasingly complex skills and
abilities. Similar results have been shown in the context of RL agents by
Baker et al. (2020). They demonstrated that increasingly complex behaviors can
emerge in a physically grounded task. Specifically, they focus on a game of
hide and seek, where there are two teams of agents. One team must hide with
the help of obstacles and other items in the environment, while the other team
needs to find the first team. They were able to show that as one team
converged on a successful strategy, the other team was pressured to learn a
counter-strategy. This process was repeated, inducing a curriculum of
increasingly competitive agents.
A similar idea was explored by Bansal et al. (2018). They proposed to use
multi-agent curriculum learning as an alternative to engineering dense shaping
rewards. Their method interpolates between dense “exploration” rewards, and
sparse multi-agent competitive rewards, with the exploration reward gradually
annealed over time. In order to prevent the adversarial agent from getting too
far ahead of the learning agent and making the task impossible, the authors
propose to additionally sample older versions of the opponent. Lastly, in
order to increase robustness, the stochasticity of the tasks is increased over
time.
Curriculum learning approaches have also been proposed for cooperative multi-
agent systems (Wang et al., 2020; Yang et al., 2020). In these settings, there
is a natural curriculum created by starting with a small number of agents, and
gradually increasing them in subsequent tasks. The schedule with which to
increase the number of agents is usually manually defined, and the emphasis
instead is on how to perform transfer when the number of agents change.
Therefore, we discuss these approaches in more detail in Section 4.3.
Finally, while self-play has been successful in a wide variety of domains,
including solving games such as Backgammon (Tesauro, 1995) and Go (Silver et
al., 2016), such an approach alone was not sufficient for producing strong
agents in a complex, multi-agent, partially-observable game like Starcraft.
One of the primary new elements of Vinyals et al. (2019) was the introduction
of a Starcraft League, a group of agents that have differing strategies
learned from a combination of imitation learning from human game data and
reinforcement learning. Rather than have every agent in the league maximize
their own probability of winning against all other agents like in standard
self play, there were some agents that did this, and some whose goal was to
optimize against the main agent being trained. In effect, these agents were
trained to exploit weaknesses in the main agent and help it improve. Training
against different sets of agents over time from the league induced a
curriculum that allowed the main agents to achieve grandmaster status in the
game.
#### 4.2.3 Reward and Initial/Terminal State Distribution Changes
Thus far, the curriculum consisted of ordering experience from the target task
or modifying agents in the target environment. In the next two sections, we
begin to examine approaches that explicitly create different MDPs for
intermediate tasks, by changing some aspect of the MDP. First we consider
approaches that keep the state and action spaces the same, as well as the
environment dynamics, but allow the reward function and initial/terminal state
distributions to vary.
One of the earliest examples of this type of method was _learning from easy
missions_. Asada et al. (1996) proposed this method to train a robot to shoot
a ball into a goal based on vision inputs. The idea was to create a series of
tasks, where the agent’s initial state distribution starts close to the goal
state, and is progressively moved farther away in subsequent tasks, inducing a
curriculum of tasks. In this work, each new task starts one “step” farther
away from the goal, where steps from the goal is measured using a domain
specific heuristic: a state is closer to the terminal state if the goal in the
camera image gets larger. The heuristic implicitly requires that the state
space can be categorized into “substates,” such as goal size or ball position,
where the ordering of state transitions in a substate to a goal state is
known. Thus, each substate has a dimension for making the task simpler or more
complex. Source tasks are manually created to vary along these dimensions of
difficulty.
Recently, Florensa et al. (2017) proposed more general methods for performing
this reverse expansion. They proposed reverse curriculum generation, an
algorithm that generates a distribution of starting states that get
increasingly farther away from the goal. The method assumes at least one goal
state is known, which is used as a seed for expansion. Nearby starting states
are generated by taking a random walk from existing starting states by
selecting actions with some noise perturbation. In order to select the next
round of starting states to expand from, they estimate the expected return for
each of these states, and select those that produce a return between a
manually set minimum and maximum interval. This interval is tuned to expand
states where progress is possible, but not too easy. A similar approach by
Ivanovic et al. (2019) considered combining the reverse expansion phase for
curriculum generation with physics-based priors to accelerate learning by
continuous control agents.
An opposite “forward” expansion approach has also been considered by Florensa
et al. (2018). This method allows an agent to automatically discover different
goals in the state space, and thereby guide exploration of the space. They do
this discovery with a Generative Adversarial Network (GAN) (Goodfellow et al.,
2014), where the generator network proposes goal regions (parameterized
subsets of the state space) and the discriminator evaluates whether the goal
region is of appropriate difficulty for the current ability of the agent. Goal
regions are specified using an indicator reward function, and policies are
conditioned on the goal in addition to the state, like in a universal value
function approximator (Schaul et al., 2015). The agent trains on tasks
suggested by the generator. In detail, the approach consists of 3 parts: 1)
First, goal regions are labelled according to whether they are of appropriate
difficulty. Appropriate goals are those that give a return between
hyperparameters $R_{min}$ and $R_{max}$. Requiring at least $R_{min}$ ensures
there is a signal for learning progress. Requiring less than $R_{max}$ ensures
that it is not too easy. 2) They use the labeled goals to train a Goal GAN. 3)
Goals are sampled from the GAN as well as a replay buffer, and used for
training to update the policy. The goals generated by the GAN shift over time
to reflect the difficulty of the tasks, and gradually move from states close
to the starting state to those farther away.
Racaniere et al. (2019) also consider an approach to automatically generate a
curriculum of goals for the agent, but for more complex goal-conditioned tasks
in dynamic environments where the possible goals vary between episodes. The
idea was to train a “setter” model to propose a curriculum of goals for a
“solver” agent to attempt to achieve. In order to help the setter balance its
goal predictions, they proposed three objectives which lead to a combination
of three losses to train the setter model: goal validity (the goal should be
valid or achievable by the current solver), goal feasibility (the goal should
match the feasibility estimates for the solver with current skill), and goal
coverage (encourage the setter to choose more diverse goals to encourage
exploration in the space of goals). In addition, a “judge” model was trained
to predict the reward the current solver agent would achieve on a goal (the
feasibility of a goal) proposed by the setter. Their experimental results
demonstrate the necessity of all three criteria for building useful curricula
of goals. They also show that their approach is more stable and effective than
the goal GAN method (Florensa et al., 2018) on complex tasks.
An alternative to modifying the initial or terminal state distribution is to
modify the reward function. Riedmiller et al. (2018) introduce SAC-X
(Scheduled Auxiliary Control), an algorithm for scheduling and executing
auxiliary tasks that allow the agent to efficiently explore its environment
and also make progress towards solving the final task. Auxiliary tasks are
defined to be tasks where the state, action, and transition function are the
same as the original MDP, but where the reward function is different. The
rewards they use in auxiliary tasks correspond to changes in raw or high level
sensory input, similar to Jaderberg et al. (2017). However, while Jaderberg et
al. (2017) only used auxiliary tasks for improving learning of the state
representation, here they are used to guide exploration, and are sequenced.
The approach is a hierarchical RL method: they need to 1) learn intentions,
which are policies for the auxiliary tasks, and 2) learn the scheduler, which
sequences intention policies and auxiliary tasks. To learn the intentions,
they learn to maximize the action-value function of each intention from a
starting state distribution that comes as a result of following each of the
other intention policies. This process makes the policies compatible. The
scheduler can be thought of as a meta-agent that performs sequencing, whose
goal is to maximize the return on the target task MDP. The scheduler selects
intentions, whose policy is executed on the extrinsic task, and is used to
guide exploration.
Heuristic-based methods have also been designed to sequence tasks that differ
in their reward functions. One such approach is SAGG-RIAC (Self-Adaptive Goal
Generation - Robust Intelligent Adaptive Curiosity) (Baranes and Oudeyer,
2013). They define _competence_ as the distance between the achieved final
state and the goal state, and _interest_ as the change in competence over time
for a set of goals. A region of the task space is deemed more _interesting_
than others, if the latest tasks in the region have achieved a high increase
in competence. The approach repeatedly selects goals by first picking a region
with a probability proportional to its interest, and then choosing a goal at
random within that region. With a smaller probability the system also selects
a goal at random over the whole task set or a goal close to a previously
unsuccessful task. The bias towards interesting regions causes the goals to be
more dense in regions where the competence increases the fastest, creating a
curriculum. Because of the stochastic nature of the goal generating process,
however, not every task is necessarily beneficial in directly increasing the
agent’s ability on the target task, but contributes to updating the competence
and interest measures. Since the intermediate tasks are generated online as
the agent learns, in this approach both sequencing and generation result from
the same sampling process.
Finally, Wu and Tian (2017) also consider changing the transition dynamics and
the reward functions of the intermediate tasks. They propose a novel framework
for training an agent in a partially observable 3D Doom environment. Doom is a
First-Person Shooter game, in which the player controls the agent to fight
against enemies. In their experiment, they first train the agent on some
simple maps with several curricula. Each curriculum consists of a sequence of
progressively more complex environments with varying domain parameters (e.g.,
the movement speed or initial health of the agent). After learning a capable
initial task model, the agent is then trained on more complicated maps and
more difficult tasks with a different reward function. They also design an
adaptive curriculum learning strategy in which a probability distribution over
different levels of curriculum is maintained. When the agent performs well on
the current distribution, the probability distribution is shifted towards more
difficult tasks.
#### 4.2.4 No restrictions
Next, there is a class of methods that create a curriculum using intermediate
tasks, but make no restrictions on the MDPs of these intermediate tasks. We
categorize them in three ways by how they address the task sequencing problem:
treating sequencing 1) as an MDP/POMDP, 2) as a combinatorial optimization
over sequences, and 3) as learning the connections in a directed acyclic task
graph. Because there are no limitations on the types of intermediate tasks
allowed, some assumptions are usually made about the transfer learning
algorithm, and additional information about the intermediate tasks (such as
task descriptors) is typically assumed. Finally, we also discuss work on an
auxiliary problem to sequencing: how long to spend on each task.
#### MDP-based Sequencing
The first formalization of the sequencing problem is as a Markov Decision
Process. These methods formulate curriculum generation as an interaction
between 2 types of MDPs. The first is the standard MDP, which models a
_learning agent_ (i.e., the student) interacting with a task. The second is a
higher level meta-MDP for the _curriculum agent_ (i.e., the teacher), whose
goal is to select tasks for the learning agent.
Narvekar et al. (2017) denote the meta-MDP as a curriculum MDP (CMDP), where
the state space $\mathcal{S}$ is the set of policies the learning agent can
represent. These can be represented parametrically using the weights of the
learning agent. The action space $\mathcal{A}$ is the set of tasks the
learning agent can train on next. Learning a task updates the learning agent’s
policy, and therefore leads to a transition in the CMDP via a transition
function $p$. Finally, the reward function $r$ is the time in steps or
episodes that it took to learn the selected task. Under this model, a
curriculum agent typically starts in an initial state corresponding to a
random policy for the learning agent. The goal is to reach a terminal state,
which is defined as a policy that can achieve some desired performance
threshold on the target task, as fast as possible.
Matiisen et al. (2017) consider a similar framework, where the interaction is
defined as a POMDP. The state and action spaces of the meta-POMDP are the same
as in Narvekar et al. (2017), but access to the internal parameters of the
learning agent is not available. Instead, an observation of the current score
of the agent on each intermediate task is given. The reward is the change in
the score on the task from this timestep to the previous timestep when the
same task was trained on. Thus, while Narvekar et al. (2017) focused on
minimizing time to threshold performance on the target task, the design of
Matiisen et al. (2017) aims to maximize the sum of performance in all tasks
encountered.
While both approaches are formalized as POMDPs, learning on these POMDPs is
computationally expensive. Thus, both propose heuristics to guide the
selection of tasks. Narvekar et al. (2017) take a sample-based approach, where
a small amount of experience samples gathered on the target and intermediate
tasks are compared to identify relevant intermediate tasks. The task that
causes the greatest change in policy as evaluated on the target task samples
is selected. In contrast, Matiisen et al. (2017) select tasks where the
absolute value of the slope of the learning curve is highest. Thus it selects
tasks where the agent is making the most progress or where the agent is
forgetting the most about tasks it has already learned. Initially tasks are
sampled randomly. As one task starts making progress, it will be sampled more,
until the learning curve plateaus. Then another will be selected, and the
cycle will repeat until all the tasks have been learned.
Subsequently, Narvekar and Stone (2019) explored whether learning was possible
in a curriculum MDP, thus avoiding the need for heuristics in task sequencing.
They showed that you can represent a CMDP state using the weights of the
knowledge transfer representation. For example, if the agent uses value
function transfer, the CMDP state is represented using the weights of the
value function. By utilizing function approximation over this state space,
they showed it is possible to learn a policy over this MDP, termed a
curriculum policy, which maps from the current status of learning progress of
the agent, to the task it should learn next. In addition, the approach
addresses the question of how long to train on each intermediate task. While
most works have trained on intermediate tasks until learning plateaus, this is
not always necessary. Narvekar and Stone (2019) showed that training on each
intermediate task for a few episodes, and letting the curriculum policy
reselect tasks that require additional time, results in faster learning.
However, while learning a curriculum policy is possible, doing so
independently for each agent and task is still very computationally expensive.
#### Combinatorial Optimization and Search
A second way of approaching sequencing is as a combinatorial optimization
problem: given a fixed set of tasks, find the permutation that leads to the
best curriculum, where best is determined by one of the CL metrics introduced
in Section 3.3. Finding the optimal curriculum is a computationally difficult
black-box optimization problem. Thus, typically fast approximate solutions are
preferred.
One such popular class of methods are metaheuristic algorithms, which are
heuristic methods that are not tied to specific problem domains, and thus can
be used as black boxes. Foglino et al. (2019a) adapt and evaluate four
representative metaheuristic algorithms to the task sequencing problem: beam
search (Ow and Morton, 1988), tabu search (Glover and Laguna, 1998), genetic
algorithms (Goldberg, 1989), and ant colony optimization (Dorigo et al.,
1991). The first two are trajectory-based, which start at a guess of the
solution, and search the neighborhood of the current guess for a better
solution. The last two are population-based, which start with a set of
candidate solutions, and improve them as a group towards areas of increasing
performance. They evaluate these methods for 3 different objectives: time to
threshold, maximum return (asymptotic performance), and cumulative return.
Results showed that the trajectory-based methods outperformed their
population-based counterparts on the domains tested.
While metaheuristic algorithms are broadly applicable, it is also possible to
create specific heuristic search methods targeted at particular problems, such
as task sequencing with a specific transfer metric objective. Foglino et al.
(2019b) introduce one such heuristic search algorithm, designed to optimize
for the cumulative return. Their approach begins by computing transferability
between all pairs of tasks, using a simulator to estimate the cumulative
return attained by using one task as a source for another. The tasks are then
sorted according to their potential of being a good source or target, and
iteratively chained in curricula of increasing length. The algorithm is
anytime, and eventually exhaustively searches the space of all curricula with
a predefined maximum length.
Jain and Tulabandhula (2017) propose 4 different online search methods to
sequence tasks into a curriculum. Their methods also assume a simulator is
available to evaluate learning on different tasks, and use the learning
trajectory of the agent on tasks seen so far to select new tasks. The 4
approaches are: 1) Learn each source task for a fixed number of steps, and add
the one that gives the most reward. The intuition is that high reward tasks
are the easiest to make progress on. 2) Calculate a transferability matrix for
all pairs of tasks, and create a curriculum by chaining tasks backwards from
the target tasks greedily with respect to it. 3) Extract a feature vector for
each task (as in Narvekar et al., 2016), and learn a regression model to
predict transferability using the feature vector. 4) Extract pair wise feature
vectors between pairs of tasks, and learn a regression model to predict
transferability.
Finally, instead of treating the entire problem as a black box, it has also
been treated as a gray box. Foglino et al. (2019c) propose such an approach,
formulating the optimization problem as the composition of a white box
scheduling problem and black box parameter optimization. The scheduling
formulation partially models the effects of a given sequence, assigning a
utility to each task, and a penalty to each pair of tasks, which captures the
effect on the objective of learning two tasks one after the other. The white-
box scheduling problem is an integer linear program, with a single optimal
solution that can be computed efficiently. The quality of the solution,
however, depends on the parameters of the model, which are optimized by a
black-box optimization algorithm. This external optimization problem searches
the optimal parameters of the internal scheduling problem, so that the output
of the two chained optimizers is a curriculum that maximizes cumulative
return.
#### Graph-based Sequencing
Another class of approaches explicitly treats the curriculum sequencing
problem as connecting nodes with edges into a directed acyclic task graph.
Typically, the task-level curriculum formulation is used, where nodes in the
graph are associated with tasks. A directed edge from one node to another
implies that one task is a source task for another.
Existing work has relied on heuristics and additional domain information to
determine how to connect different task nodes in the graph. For instance,
Svetlik et al. (2017) assume the set of tasks is known in advance, and that
each task is represented by a task feature descriptor. These features encode
properties of the domain. For example, in a domain like Ms. Pac-Man, features
could be the number of ghosts or the type of maze. The approach consists of
three parts. First, a binary feature vector is extracted from the feature
vector to represent non-zero elements. This binary vector is used to group
subsets of tasks that share similar elements. Second, tasks within each group
are connected into subgraphs using a novel heuristic called _transfer
potential_. Transfer potential is defined for discrete state spaces, and
trades off the applicability of a source task against the cost needed to learn
it. Applicability is defined as the number of states that a value function
learned in the source can be applied to a target task. The cost of a source
task is approximated as the size of its state space. Finally, once subgraphs
have been created, they are linked together using directed edges from
subgraphs that have a set of binary features to subgraphs that have a superset
of those features.
Da Silva and Reali Costa (2018) follow a similar procedure, but formalize the
idea of task feature descriptors using an object-oriented approach. The idea
is based on representing the domain as an object-oriented MDP, where states
consist of a set of objects. A task OO-MDP is specified by the set of specific
objects in this task, and the state, action, transition, and reward functions
of the task. With this formulation, source tasks can be generated by selecting
a smaller set of objects from the target task to create a simpler task. To
create the curriculum graph, they adapt the idea of transfer potential to the
object-oriented setting: instead of counting the number of states that the
source task value function is applicable in, they compare the sets of objects
between the source and target tasks. While the sequencing is automated, human
input is still required to make sure the tasks created are solvable.
#### Auxiliary Problems
Finally, we discuss an additional approach that tackles an auxiliary problem
to sequencing: how long to spend on each intermediate task in the curriculum.
Most existing work trains on intermediate tasks until performance plateaus.
However, as we mentioned previously, Narvekar and Stone (2019) showed that
this is unnecessary, and that better results can be obtained by training for a
few episodes, and reselecting or changing tasks dynamically as needed.
Bassich et al. (2020) consider an alternative method for this problem based on
_progression_ functions. Progression functions specify the pace at which the
difficulty of the task should change over time. The method relies on the
existence of a task-generation function, which maps a desired complexity
$c_{t}\in[0,1]$ to a task of that complexity. The most complex task, for which
$c_{t}=1$, is the final task. After every episode, the progression function
returns the difficulty of the task that the agent should face at that time.
The authors define two types of progression functions: fixed progressions, for
which the learning pace is predefined before learning takes place; and
adaptive progressions, which adjust the learning pace online based on the
performance of the agent. Linear and exponential progressions are two examples
of fixed progression functions, and increase the difficulty of the task
linearly and exponentially, respectively, over a prespecified number of time
steps. The authors also introduce an adaptive progression based on a friction
model from physics, which increases $c_{t}$ as the agent’s performance is
increasing, and slows down the learning pace if performance decreases.
Progression functions allow the method to change the task at every episode,
solving the problem of deciding how long to spend in each task, while
simultaneously creating a continually changing curriculum.
#### 4.2.5 Human-in-the-Loop Curriculum Generation
Thus far, all the methods discussed in Section 4.2 create a curriculum
_automatically_ using a sequencing algorithm, which either reorders samples
from the final task or progressively alters how much intermediate tasks in the
curriculum may differ. Bengio et al. (2009) and Taylor (2009) both emphasize
the importance of better understanding how _humans_ approach designing
curricula. Humans may be able to design good curricula by considering which
intermediate tasks are “too easy” or “too hard,” given the learner’s current
ability to learn, similar to how humans are taught with the zone of proximal
development (Vygotsky, 1978). These insights could then be leveraged when
designing automated curriculum learning systems. Therefore, in this section,
we consider curriculum sequencing approaches that are done _manually_ by
humans who are either _domain experts_ , who have specialized knowledge of the
problem domain, or _naive users_ , who do not necessarily know about the
problem domain and/or machine learning.
One example of having domain experts manually generate the curriculum is the
work done by Stanley et al. (2005), in which they explore how to keep video
games interesting by allowing agents to change and to improve through
interaction with the player. They use the NeuroEvolving Robotic Operatives
(NERO) game, in which simulated robots start the game with no skills and have
to learn complicated behaviors in order to play the game. The human player
takes the role of a trainer and designs a curriculum of training scenarios to
train a team of simulated robots for military combat. The player has a natural
interface for setting up training exercises and specifying desired goals. An
ideal curriculum would consist of exercises with increasing difficulty so that
the agent can start with learning basic skills and gradually building on them.
In their experiments, the curriculum is designed by several NERO programmers
who are familiar with the game domain. They show that the simulated robots
could successfully be trained to learn different sophisticated battle tactics
using the curriculum designed by these domain experts. It is unclear whether
the human player who is not familiar with the game can design good curriculum.
A more recent example is by MacAlpine and Stone (2018). They use a very
extensive manually constructed curriculum to train agents to play simulated
robot soccer. The curriculum consists of a training schedule over 19 different
learned behaviors. It encompasses skills such as moving to different positions
on the field with different speeds and rotation, variable distance kicking,
and accessory skills such as getting up when fallen. Optimizing these skills
independently can lead to problems at the intersection of these skills. For
example, optimizing for speed in a straight walk can lead to instability if
the robot needs to turn or kick due to changing environment conditions. Thus,
the authors of this work hand-designed a curriculum to train related skills
together using an idea called overlapping layered learning. This curriculum is
designed using their domain knowledge of the task and agents.
While domain experts usually generate good curricula to facilitate learning,
most existing work does not explicitly explore their curriculum design
process. It is unclear what kind of design strategies people follow when
sequencing tasks into a curriculum. Published research on Interactive
Reinforcement Learning (Thomaz and Breazeal, 2006; Knox and Stone, 2009; Suay
and Chernova, 2011; Knox and Stone, 2012; Griffith et al., 2013; Subramanian
et al., 2016; Loftin et al., 2016; MacGlashan et al., 2017) has shown that RL
agents can successfully speed up learning using human feedback, demonstrating
the significant role can humans play in teaching an agent to learn a (near-)
optimal policy. This large body of work mainly focuses on understanding how
human teachers want to teach the agent and how to incorporate these insights
into the standard RL framework. Similarly, the way we define curriculum design
strategies still leaves a lot to be defined by human teachers. As pointed out
by Bengio et al. (2009), the notion of simple and complex tasks is often based
on human intuition, and there is value in understanding how humans identify
“simple” tasks. Along these lines, some work has been done to study whether
curriculum design is a prominent teaching strategy that naive users choose to
teach the agent and how they approach designing curricula.
$\begin{array}[]{cc}\includegraphics[height=79.6678pt]{img/dog_training_given_final_task.png}&\includegraphics[width=303.53267pt]{img/dog_training_curriculum.png}\\\
(a)&(b)\end{array}$
Figure 4: One example of curricula designed by human users. (a) Given final
task. (b) A curriculum designed by one human participant.
To study the teaching strategies followed by naive users, Khan et al. (2011)
conduct behavioral studies in which human participants need to teach a robot
the concept of whether an object can be grasped with one hand. In their
experiment, participants are provided with 31 cards with photos of common
objects (e.g., food, furniture, and animals) for them to select. The
experiment consists of two subtasks. In the first subtask, participants sort
the objects on the table based on their subjective ratings of their
graspability. In the second subtask, participants pick up the cards from the
table and show them to the robot while teaching the robot the concept of
graspability, using as few cards as possible. While teaching the robot the
object’s graspability, participants can either use any natural language or say
either “graspable” or “not graspable,” depending on one of the two conditions
they are randomly assigned. They observe that participants follow three
distinct teaching strategies, one of which is consistent with the curriculum
learning principle, i.e., starting simple and gradually increasing the
difficulty of the task. Furthermore, they propose a novel theoretical
framework as a potential explanation for the teaching strategy that follows
the curriculum learning principle, which shows that it is the result of
minimizing per-iteration expected error of the learner.
Peng et al. (2018) also explore how naive users design a curriculum of tasks
for an agent, but in a more complex sequential decision-making task.
Specifically, a simple simulated home environment is used, where the agent
must learn to perform tasks in a variety of environments. The tasks are
specified via text commands and the agent is trained to perform the task via
reinforcement and punishment feedback from a human trainer. It uses the goal-
directed Strategy-Aware Bayesian Learning (SABL) algorithm (Loftin et al.,
2016) for learning from human feedback. In the user study, participants are
asked to design a set of training assignments for the agent to help it quickly
learn to complete the given final assignment (shown in Figure 4a). A set of
source tasks are provided for human participants to select and sequence. One
example of curricula designed by human participants is shown in Figure 4b.
Their empirical results show that, compared to directly learning the pre-
specified final task from scratch, non-expert humans can successfully design
curricula that result in better overall agent performance on learning both the
entire curriculum and the final task. They also discover that humans are more
likely to select commands for intermediate tasks that include concepts that
are important for the final task, and that doing so results in curricula that
lead to better overall agent performance. Furthermore, they demonstrate that
by taking advantage of this type of non-expert guidance, their curriculum-
learning algorithm can be adapted to learn the human-generated curricula more
efficiently.
There is also some work that does not explicitly ask humans to design a
curriculum, but uses human data to help generate the curriculum. One example
is the work done by Hosu and Rebedea (2016), in which they propose a deep RL
method that combines online agent experiences with offline human experiences
to train the agent more efficiently. In some sparse-reward Atari games such as
Montezuma’s Revenge and Private Eye, the agent needs to execute a long
sequence of specific actions to receive the first positive reward from the
environment, which makes the exploration problem much harder. Thus, the
commonly used $\epsilon$-greedy strategy could not find any game paths to
reach a first state with positive reward, preventing the neural network from
learning relevant features to good states. Inspired by curriculum learning and
the human starts evaluation metric used for testing Atari agents, they use
checkpoints sampled from a human player’s game experience as starting points
for the learning process. The main intuition behind this approach is that at
least some of the checkpoints will be an “easier” starting point, which is
closer to some states with positive reward that the agent can benefit from.
While this method belongs to the class of sequencing approaches, as discussed
in Section 4.2.1, that reorders samples in the final task to derive a
curriculum, it additionally considers more informative sample data generated
by naive human users in order to build a more efficient curriculum.
We find that very limited work has been done on investigating how humans
design curricula. While the work discussed in this section enriches our
empirical understanding of human teaching and gives us some insights into the
development of new machine-learning algorithms and interfaces that can better
accommodate machine- or human-created curricula, we believe more work needs to
be done along this line.
### 4.3 Knowledge Transfer
While we view sequencing, as covered in Section 4.2, to be the core concept of
curriculum learning, the whole premise of CL depends on an agent’s ability to
transfer knowledge among tasks. While a full discussion of transfer learning
for RL is beyond the scope of this survey, this subsection is designed to
provide the reader a brief introduction to the area so that they can
effectively leverage it as part of their own explorations in curriculum
learning.
In curriculum learning, transfer learning methods are used to allow the agent
to reuse knowledge learned from one intermediate task to another within the
curriculum. It is worth noting that when creating a curriculum using only
samples from the target task (discussed in Section 4.2.1), there is no
transfer as there is only a single task (the target task) and correspondingly
no change in the environment. However, when creating a curriculum using
multiple intermediate tasks, which may differ in state/action space, reward
function, or transition function from the final task, transfer learning is
needed to extract and pass on reusable knowledge acquired in one intermediate
task to the next. The type of knowledge transferred also directly affects the
type of learner that is applicable to the learning process.
Transferred knowledge can be low-level, such as an entire policy, a value
function, a full task model, or some training instances, which can be directly
used to initialize the learner in the target task. The knowledge can also be
high-level, such as partial policies or options, skills, shaping rewards, or
subtask definitions. This type of information may not fully initialize the
learner in the target task, but it could be used to guide the agent’s learning
process in the target task. In this subsection, we discuss different transfer
learning approaches used in curricula.
In policy transfer, a policy learned in a source or intermediate task is used
to initialize the policy in the target task. When transferring policies
between different tasks, the tasks may differ in some aspect of the MDP, such
as starting states (Florensa et al., 2017), reward functions (Florensa et al.,
2018; Riedmiller et al., 2018), or transition functions (Clegg et al., 2017).
For instance, Clegg et al. (2017) demonstrate that an arm-like manipulator can
successfully learn the control policy for a simulated dressing task, by
transferring policies between tasks with different transition functions. In a
dressing task, the goal is to achieve a desired relative positioning of the
garment and the limb. To do this, they first train a sphere to move through a
funnel-like geometry to reach some target location. They then directly apply
the learned policy to a different scenario in which a manipulator with
arbitrary shape navigates through a simulated garment. The main trick is to
train multiple spheres using a curriculum learning strategy and then aggregate
them to control the manipulator in the dressing task.
Citation | Intermediate Task Generation | Curriculum Representation | Transfer Method | Curriculum Sequencer | Curriculum Adaptivity | Evaluation Metric | Application Area
---|---|---|---|---|---|---|---
Clegg et al. (2017) | domain experts | sequence | policies | domain experts | static | asymptotic, time to threshold | sim robotics
Fujii et al. (1998) | domain experts | sequence | partial policies | domain experts | static | asymptotic | real robotics
Karpathy and Van De Panne (2012) | domain experts/target | sequence/single | partial policies /no transfer | domain experts/automatic | static/adaptive | time to threshold | sim robotics
Rusu et al. (2016) | domain experts | sequence | policies | domain experts | static | asymptotic | video games
Shao et al. (2018) | domain experts | sequence | task model | domain experts | static | asymptotic, total reward | video games
Sinapov et al. (2015) | automatic | sequence | value function | automatic | static | jump start | video games
Tessler et al. (2017) | domain experts | sequence | partial policies | domain experts | static | asymptotic | video games
Vezhnevets et al. (2016) | automatic | sequence | partial policies | automatic | static | asymptotic, total reward | video games
Wang et al. (2020) | domain experts | sequence | policies | domain experts | static | asymptotic | video games
Yang and Asada (1996) | domain experts | sequence | partial policies | automatic | adaptive | asymptotic, time to threshold | real robotics
Yang et al. (2020) | domain experts | sequence | policies | domain experts | static | asymptotic, time to threshold | toy, other
Zimmer et al. (2018) | domain experts | sequence | partial policies | domain experts | static | asymptotic, total reward | sim robotics
Table 3: The papers discussed in Section 4.3, categorized along the dimensions
presented in Section 3.4. Bolded values under evaluation metric indicate
strong transfer.
In Shao et al. (2018), a learned task model is transferred between tasks,
which is used to initialize the policy network. Thus, it is similar to
transferring policies. Their work aims to solve the problem of multi-agent
decision making in StarCraft micromanagement, where the goal is to control a
group of units to destroy the enemy under certain terrain conditions. A
parameter sharing multi-agent gradient-descent Sarsa($\lambda$) (PS-MAGDS)
method is proposed to train the units to learn an optimal policy, which is
parametrized by a feed-forward neural network. PS-MAGDS simply extends the
traditional Sarsa($\lambda$) to multiple units by sharing parameters of the
policy network among units to encourage cooperative behaviors. A reward
function including small immediate rewards is also designed to accelerate the
learning process. When using transfer learning in their experiments, the
agents are first trained in some small scale source scenarios using PS-MAGDS.
The well-trained model is then used to initialize the policy network to learn
micromanagement in the target scenarios. To scale the combat to a large scale
scenario, they combine curriculum learning and transfer learning where the
agents are trained with a sequence of progressively more complex
micromanagement tasks. The difficulty of the micromanagement task is
controlled by changing the number and type of units.
Value function transfer is another common method for transferring low-level
knowledge between intermediate tasks within a curriculum. In most existing
work (Sinapov et al., 2015; Narvekar et al., 2017; Da Silva and Reali Costa,
2018), value function transfer is achieved by using the parameters of a value
function learned in one intermediate task to initialize the value function in
the next intermediate task in the curriculum, such that the agent learns the
final task with some initial policy that is better than random exploration.
For example, Sinapov et al. (2015) focus on addressing the task selection
problem in curriculum learning using value function transfer, under the
assumption that no samples from the final tasks are available. They propose to
use meta-data (i.e., a fixed-length feature vector that describes the task)
associated with each task to identify suitable intermediate tasks. The main
idea is to use such meta-data to learn the benefits of transfer between
different ‘source-target’ task pairs, and have this generalize to new unseen
task pairs to guide task selection.
When transferring low-level policies or value functions across tasks, there
are several challenges that arise, particularly in the modern context of deep
reinforcement learning. First is the problem of catastrophic forgetting, where
knowledge from previously learned tasks is lost as information on a new task
is incorporated. This effect occurs because the weights of the neural network
optimized for a first task must be changed to meet the objectives of a new
task, often resulting in poorer performance on the original task. Typically,
in the curriculum setting, we only care about performance in the final tasks.
However, if information from two orthogonal tasks needs to be combined (such
as two independent skills), this challenge needs to be addressed. One approach
is progressive neural networks (Rusu et al., 2016), which trains a new network
“column” for each new task, and leverages lateral connections to previously
learned network columns to achieve transfer. When training subsequent columns,
parameters from previous columns are frozen, which prevents catastrophic
forgetting. The limitation is that the number of parameters grows with the
number of tasks, and at inference time, the task label is needed to know which
column to extract output from.
A second problem is the case where the state and action spaces differ between
tasks. One alternative is to transfer higher-level knowledge across tasks,
such as partial policies or options. A partial policy is a policy that is not
necessarily defined for all states in the state space of an MDP. We use
partial policies as an umbrella term to represent closely related ideas such
as options, skills, and macro-actions. Yang and Asada (1996) transfer learned
control parameters between tasks, which are similar to partial policies. To
solve the impedance learning problem for high-speed robotic assembly, they
allow the system to learn impedance parameters associated with different
dynamic motions separately, rather than to learn all the control parameters
simultaneously. For instance, they first learn only the parameters associated
with quasistatic motion by driving the system slowly, leaving other parameters
unlearned. After the quasistatic parameters have been learned, they then
slightly increase the motion speed, and use the learned values to initialize
the quasistatic parameters when learning other parameters. Another example of
transferring partial policies between tasks is the work done by Zimmer et al.
(2018). Their main idea is to progressively increase the dimensionality of the
tackled problem by increasing the (continuous) state and action spaces of the
MDP, while an agent is learning a policy. The agent first learns to solve the
source task with reduced state and action spaces until the increase in
performance stagnates. Then, the partial policy learned by the agent is used
as an initialization to learn the full policy in the target task with full
state and action spaces. A developmental layer (like a dropout layer) is added
to the network to filter dimensions of the states and actions.
Similarly, Fujii et al. (1998) transfer options between tasks. To train mobile
robots to learn collision avoidance behaviors in multi-robot systems more
efficiently, they develop a multi-layered RL mechanism. Rather than gradually
increasing the level of task complexity based on the learner’s performance as
in Yang and Asada (1996), their learning process consists of four stages like
a curriculum in which each stage learns a pre-defined controller. Each
controller learns an option to solve a pre-defined sub-task. For instance, the
first controller learns to move toward a specific goal. Then the output (goal-
directed behavior) of the first controller is used as input for the second
controller, which aims to learn to avoid the collision to a single robot, and
so on.
Vezhnevets et al. (2016) also transfer high-level macro-actions between tasks,
which are simpler instances of options. In their experiment, the agent is
trained with a curriculum where the goal state is first set to be very close
to the start state and is then moved further away during learning process.
Although the task gets progressively harder, the temporally abstracted macro-
actions remain the same. The macro-actions learned early on can also be easily
adapted using their proposed architecture. Specifically, a deep recurrent
neural network architecture is used to maintain a multi-step action plan. The
network learns when to commit to the action plan to generate macro-actions and
when to update the plan based on observations.
Another mechanism for transfer are skills. Tessler et al. (2017) propose a
deep RL method that effectively retains and transfers learned skills to solve
lifelong learning in MineCraft. In their work, a set of $N$ skills are trained
a priori on various sub-tasks, which are then reused to solve the harder
composite task. In their MineCraft experiment, the agent’s action space
includes the original primitive actions as well as the set of pre-learned
skills (e.g., navigate and pickup). A hierarchical architecture is developed
to learn a policy that determines when to execute primitive actions and when
to reuse pre-learned skills, by extending the vanilla DQN architecture (Mnih
et al., 2015). The skills could be sub-optimal when they are directly reused
for more complex tasks, and this hierarchical architecture allows the agent to
learn to refine the policy by using primitive actions. They also show the
potential for reusing the pre-learned skill to solve related tasks without
performing any additional learning.
Rather than selectively reusing pre-learned skills, Karpathy and Van De Panne
(2012) focus on learning motor skills in an order of increasing difficulty.
They decompose the acquisition of skills into a two-level curriculum: a
_high-level_ curriculum specifies the order in which different motor skills
should be learned, while the _low-level_ curriculum defines the learning
process for a specific skill. The high-level curriculum orders the skills in a
way such that each skill is relatively easy to learn, using the knowledge of
the previously learned skills. For instance, the Acrobot first learns the Hop
(easy to learn from scratch) and Flip (similar to hopping very slowly) skills,
and then learns the more complex Hop-Flip skill. The learned skill-specific
task parameters for easier skills will highly constrain the states that the
Acrobat could be in, making it easier to learn more complex skills. For
example, the Hop-Flip skills begin from a hopping gait of some speed, which
can be reached by repeatedly executing the previously learned Hop skill.
In multi-agent settings, several specific methods have been designed for
curricula that progressively scale the number of agents between tasks. In
these settings, the state and action spaces often scale based on the number of
agents present. One common assumption in many of these methods is that the
state space can be factored into elements for the environment $s^{env}$, the
agent $s^{n}$, and all other agents $s^{-n}$. For example, Yang et al. (2020)
propose CM3, which takes a two-stage approach. In the first stage, a single
agent is trained without the presence of other agents. This is done by
inducing a new MDP that removes all dependencies on agent interactions (i.e.,
removing $s^{-n}$) and training a network on this subspace. Then in the second
stage, cooperation is learned by adding the parameters for the other agents
into the network.
Wang et al. (2020) propose 3 different approaches for multi-agent settings.
The first is buffer reuse, which saves the replay buffers from all previous
tasks, and samples experience from all of them to train in the current task.
Samples from lower dimensional tasks are padded with zeros. The second is
curriculum distillation, which adds a distillation loss based on KL divergence
between policies/q-values between tasks. The third is transferring the model
using a new network architecture called Dynamic Agent-number Network (DyAN).
In this architecture, the state space elements related to the agent and
environment go through a fully connected network, while the observations for
each teammate agent are passed through a graph neural network (GNN) and then
aggregated. These networks are subsequently combined to produce q-values or
policies.
## 5 Related Areas and Paradigms
Curriculum learning is an idea that has been studied in other areas of machine
learning and human education, and is similar to several existing paradigms in
reinforcement learning. In this section, we first relate curriculum learning
to approaches in reinforcement learning that aim to improve sample complexity,
and that consider learning multiple sets of tasks (Section 5.1). Then we
describe approaches to learn curricula in supervised learning (Section 5.2)
and for teaching and human education (Section 5.3). We include these
approaches with the idea that the insights discovered in these areas could be
adapted to apply to the reinforcement learning setting with autonomous agents.
### 5.1 Related Paradigms in Reinforcement Learning
One of the central challenges in applying reinforcement learning to real world
problems is sample complexity. Due to issues such as a sparse reward signal or
complex dynamics, difficult problems can take an RL agent millions of episodes
to learn a good policy, with many suboptimal actions taken during the course
of learning. Many different approaches have been proposed to deal with this
issue. To name a few, one method is imitation learning (Schaal, 1997), which
uses demonstrations from a human as labels for supervised learning to
bootstrap the learning process. Another example is off-policy learning (Hanna
et al., 2017), which uses existing data from an observed behavior policy, to
estimate the value of a desired target policy. Model-based approaches (Sutton
and Barto, 1998) first learn a model of the environment, which can then be
used for planning the optimal policy.
Each of these methods come with their advantages and disadvantages. For
imitation learning, the assumption is that human demonstrations are available.
However, these are not always easy to obtain, especially when a good policy
for the task is not known. In off-policy learning, in order to make full use
of existing data, it is assumed that the behavior policy has a nonzero
probability of selecting each action, and typically that every action to be
evaluated or the target policy has been seen at least once. Finally, model-
based approaches typically first learn a model of the environment, and then
use it for planning. However, any inaccuracies in the learned model can
compound as the planning horizon increases. Curriculum learning takes a
different approach, and makes a different set of assumptions. The primary
assumption is that the environment can be configured to create different
subtasks, and that it is easier for the agent to discover _on its own_
reusable pieces of knowledge in these subtasks that can be used for solving a
more challenging task.
Within reinforcement learning, there are also several paradigms that consider
learning on a set of tasks so as to make learning more efficient. Multitask
learning, lifelong/continuous learning, active learning, and meta-learning are
four such examples.
In _multitask learning_ , the goal is to learn how to solve _sets_ of
prediction or decision making tasks. Formally, given a set of tasks
$m_{1},m_{2},\ldots m_{n}$, the goal is to _co-learn_ all of these tasks, by
optimizing the performance over all $n$ tasks simultaneously. Typically, this
optimization is facilitated by learning over some shared basis space. For
example, Caruana (1997) considers multitask learning for supervised learning
problems, and shares layers of a neural network between tasks. In supervised
learning, these tasks are different classification or regression problems.
Similar ideas have been applied in a reinforcement learning context by Wilson
et al. (2007). In reinforcement learning, different tasks correspond to
different MDPs.
_Lifelong learning_ and _continual learning_ can be viewed as an online
version of multitask learning. Tasks are presented one at a time to the
learner, and the learner must use shared knowledge learned from previous tasks
to more efficiently learn the presented task. As in multitask learning,
typically the goal is to optimize performance over all tasks given to the
learner. Lifelong and continual learning have been examined in both the
supervised setting (Ruvolo and Eaton, 2013a) and the reinforcement learning
setting (Ring, 1997; Ammar et al., 2014). The distinguishing feature of
curriculum learning compared to these works is that in curriculum learning, we
have full control over the _order_ in which tasks are selected. Indeed, we may
have control over the _creation_ of tasks as well. In addition, the goal is to
optimize performance for a specific target task, rather than all tasks. Thus,
source tasks in curriculum learning are designed solely to improve performance
on the target task—we are not concerned with optimizing performance in a
source.
In _active learning_ , the learner chooses which task or example to learn or
ask about next, from a given set of tasks. Typically, active learning has been
examined in a semi-supervised learning setting: a small amount of labeled data
exists whereas a larger amount of unlabeled data is present. The labeled data
is used to learn a classifier to infer labels for unlabeled data. Unlabeled
data that the classifier is not confident about is requested for a label from
a human user. For example, Ruvolo and Eaton (2013b) consider active learning
in a lifelong learning setting, and show how a learner can actively select
tasks to improve learning speed for all tasks in a set, or for a specific
target task. The selection of which task to be learned next is similar to the
_sequencing_ aspect of curriculum learning. However, the full method of
curriculum learning is much broader, as it also encompasses creating the space
of tasks to consider. Ruvolo and Eaton (2013b) and similar active learning
work typically assume the set of tasks to learn and select from are already
given. In addition, typically active learning has been examined for supervised
prediction tasks, whereas we are concerned with reinforcement learning tasks.
Finally, in _meta-learning_ (Finn et al., 2017), the goal is to train an agent
on a variety of tasks such that it can quickly adapt to a new task within a
small number of gradient descent steps. Typically, the agent is not given
information identifying the task it is training on. In contrast, in curriculum
learning, the learning agent may or may not have information identifying the
task. However, the process that designs the curriculum by sequencing tasks
usually does have this information. Like in the lifelong setting, there is no
significance attached to the order in which tasks are presented to the
learner. In addition, the objective in meta-learning is to train for fast
adaptability, rather than for a specific final task as is the case in
curriculum learning.
### 5.2 Curricula in Supervised Machine Learning
In addition to reinforcement learning, curriculum learning has been examined
for supervised learning. While it is beyond the scope of this article to
extensively survey supervised CL methods, we would like to highlight a few
that could inspire ideas and draw parallels to the RL setting.
Bengio et al. (2009) first formalized the idea of curriculum learning in the
context of supervised machine learning. They conducted case studies examining
when and why training with a curriculum can be beneficial for machine learning
algorithms, and hypothesized that a curriculum serves as both a continuation
method and a regularizer. A continuation method is an optimization method for
non-convex criteria, where a smoothed version of the objective is optimized
first, with the smoothing gradually reduced over training iterations.
Typically, “easy” examples in a curriculum correspond to a smoother objective.
Using a simple shape recognition and language domain, they showed that
training with a curriculum can improve both learning speed and performance.
While many papers before Bengio et al. (2009) _used_ the idea of a curriculum
to improve training of machine learning algorithms, most work considering how
to systematically _learn_ a curriculum came after. One recent example is work
by Graves et al. (2017). They introduced measures of _learning progress_ ,
which indicate how well the learner is currently improving from the training
examples it is being given. They introduce 2 main measures based on 1) rate of
increase in prediction accuracy and 2) rate of increase of network complexity.
These serve as the reward to a non-stationary multi-armed bandit algorithm,
which learns a stochastic policy for selecting tasks. These signals of
learning progress could in theory be applied or adapted to the reinforcement
learning setting as well. Graves et al. (2017) also make an interesting
observation, which is that using a curriculum is similar to changing the step
size of the learning algorithm. Specifically, in their experiments, they found
that a random curriculum still serves as a strong baseline, because all tasks
in the set provide a gradient333Note however that in the reinforcement
learning setting, because the policy affects the distribution of states an
agent encounters, random training can be significantly worse.. Easier tasks
provide a stronger gradient while harder tasks provide a gradient closer to 0.
Thus, choosing easy, useful tasks allows the algorithm to take larger steps
and converge faster.
More recently, Fan et al. (2018) frame curriculum learning as “Learning to
Teach,” where a teacher agent learned to train a learning agent using a
curriculum. The process is formulated as an MDP between these two interacting
agents, similar to the MDP approaches discussed in Section 4.2.4: the teacher
agent selects the training data, loss function, and hypothesis space, while
the learning agent trains given the parameters specified by the teacher. The
state space of the MDP is represented as a combination of features of the
data, features of the student model, and features that represent the
combination of both data and learner models. The reward signal is the accuracy
on a held-out development set. Training a teacher agent can be computationally
expensive. They amortize this cost by using a learned teacher agent to teach a
new student with the same architecture. For example, they train the teacher
using the first half of MNIST, and use the learned teacher to train a new
student from the second half of MNIST. Another way they amortize the cost is
to train a new student with a different architecture (e.g., changing from
ResNet32 to ResNet110). Similar ideas have been explored in the reinforcement
learning setting. However, the test set distribution is different from the
training set distribution, which makes performing these kind of evaluations
more challenging. However, showing that the cost for training a teacher can be
amortized is an important direction for future work.
Finally, Jiang et al. (2015) explore the idea of self-paced curriculum
learning for supervised learning, which unifies and takes advantage of the
benefits of self-paced learning and curriculum learning. In their terminology,
curriculum learning uses prior knowledge, but does not adapt to the learner.
Specifically, a curriculum is characterized by a ranking function, which
orders a dataset of samples by priority. This function is usually derived by
predetermined heuristics, and cannot be adjusted by feedback from the learner.
In contrast, self-paced learning (SPL) adjusts to the learner, but does not
incorporate prior knowledge and leads to overfitting. In SPL, the curriculum
design is implicitly embedded as a regularization term into the learning
objective. However, during learning, the training loss usually dominates over
the regularization, leading to overfitting. This paper proposes a framework
that unifies these two ideas into a concise optimization problem, and
discusses several concrete implementations. The idea is to replace the
regularization term in SPL with a self-paced function, such that the weights
lie within a predetermined curriculum region. In short, the curriculum region
induces _a weak ordering_ over the samples, and the self-paced function
determines the actual learning scheme within that ordering. The idea has
parallels to a task-level curriculum for RL, where the curriculum induces a
weak ordering over samples from all tasks, and with the learning algorithm
determining the actual scheme within that ordering.
### 5.3 Algorithmically Designed Curricula in Education
Curriculum learning has also been widely used for building effective
Intelligent Tutoring Systems (ITS) for human education (Iglesias et al., 2003,
2009; Green et al., 2011; Brunskill and Russell, 2011; Doroudi et al., 2016).
An ITS system involves a student interacting with an intelligent tutor (a
computer-based system), with the goal of helping the student to master all
skills quickly, using as little learning content as possible. Given that
students have different learning needs, styles, and capabilities, the
intelligent tutor should be able to provide customized instructions to them.
To achieve this goal, one common strategy is called _curriculum sequencing_ ,
which aims to provide the learning materials in a meaningful order that
maximizes learning of the students with different knowledge levels. The main
problem this strategy must solve is to find the most effective lesson to
propose next, given the student’s current learning needs and capabilities.
Reinforcement learning is one of the machine learning techniques that has been
used with intelligent tutors to partially automate construction of the student
model and to automatically compute an optimal teaching policy (Woolf, 2007).
One advantage of using RL methods in tutoring is that the model can learn
adaptive teaching actions based on each individual student’s performance in
real time, without needing to encode complex pedagogical rules that the system
requires to teach effectively (e.g., how to sequence the learning content,
when and how to provide an exercise). Another advantage is that it is a
general domain-independent technique that can be applied in any ITS.
As a concrete example, Iglesias et al. (2003, 2009) adapt $Q$-learning
(Watkins, 1989) to an adaptive and intelligent educational system to allow it
to automatically learn how to teach each student. They formulate the learning
problem as an RL problem, where the state is defined as the description of the
student’s knowledge, indicating whether the student has learned each knowledge
item. The set of actions the intelligent tutor can execute includes selecting
and showing a knowledge item to the student. A positive reward is given when
all required content has been learned, otherwise no reward is given. The
system evaluates the student’s knowledge state through tests, which shows how
much the student knows about each knowledge item. The $Q$-value estimates the
usefulness of executing an action when the student is in a particular
knowledge state. Then, the tutoring problem can be solved using the
traditional $Q$-learning algorithm.
Green et al. (2011) propose using a multi-layered Dynamic Bayes Net (DBN) to
model the teaching problem in an ITS system. The main idea is to model the
dynamics of a student’s skill acquisition using a DBN, which is normally used
in RL to represent transition functions for state spaces. More specifically,
they formulate the problem as a factored MDP, where the state consists of one
factor for each skill, corresponding to the student’s proficiency on that
particular skill. The actions are to either provide a hint or to pose a
problem about a particular skill to the student. From a history of teacher-
student interaction, the teacher can model the student’s proficiency state,
with the goal of teaching the student to achieve the highest possible
proficiency value on each skill, using as few problems and hints as possible.
Subsequently, the learned DBN model is used by a planning algorithm to search
for the optimal teaching policy, mapping proficiency states of student
knowledge to the most effective problem or hint to pose next.
To allow the automated teacher to select a sequence of pedagogical actions in
cases where learner’s knowledge may be unobserved, a different problem
formulation is posed by Rafferty et al. (2016). They formulate teaching as a
partially observable Markov decision process (POMDP), where the learner’s
knowledge state is considered as a hidden state, corresponding to the
learner’s current understanding of the concept being taught. The actions the
automated teacher can select is a sequence of pedagogical choices, such as
examples or short quizzes. The learner’s next knowledge state is dependent on
her current knowledge state and the pedagogical action the teacher chooses.
Changes in the learner’s knowledge state reflect learning. In this framework,
the automated teacher makes some assumptions about student learning, which is
referred to as the learner model: it specifies the space of possible knowledge
states and how the knowledge state changes. Then the teacher can update its
beliefs about the learner’s current knowledge state based on new observations,
given this learner model. Using this POMDP framework, they explore how
different learner models affect the teacher’s selection of pedagogical
actions.
While most approaches seek to solely maximize overall learning gains,
Ramachandran and Scassellati (2014) propose an RL-based approach that uses a
personalized social robot to tutor children, that maximizes learning gains and
sustained engagement over the student-robot interaction. The main goal of the
social robot is to learn the ordering of questions presented to a child, based
on difficulty level and the child’s engagement level in real time. To
represent the idea that children with different knowledge levels need a
different curriculum, each child is categorized into a given group based on
knowledge level at the start of the one-on-one tutoring interaction. An
optimal teaching policy is then learned specific to each group. In particular,
their approach consists of a training phase and an interaction phase. In the
training phase, participants are asked to complete a tutoring exercise. A
pretest and post-test will be used to evaluate the participant’s relative
learning gains, which will also be used as the reward function to learn an
optimal policy during the training phase. Subsequently, in the interaction
phase, the child’s real-time engagement will be detected, serving as another
reward signal for the RL algorithm to further optimize the teaching policy.
Non-RL-based algorithms have been considered as well. Ballera et al. (2014)
leverage the roulette wheel selection algorithm (RWSA) to perform personalized
topic sequencing in e-learning systems. RWSA is typically used in genetic
algorithms to arrange the chromosomes based on their fitness function, such
that individuals with higher fitness value will have higher probability of
being selected (Goldberg, 1989). Similarly, in an e-learning system, a
chromosome is denoted by a lesson. Each lesson has a fitness value that
dynamically changes based on the student’s learning performance. This fitness
value indicates how well the topic was learned by the student, depending on
three performance parameters: exam performance, study performance, and review
performance of the learner. A lower fitness value means that the student has a
poorer understanding of the topic. Thus, a reversed mechanism of RWSA is
implemented, so as to select the lessons with lower fitness values more often
for reinforcement. Then, this reversed RWSA algorithm is combined with linear
ranking algorithm to sort the lessons.
## 6 Open Questions
Through our survey of the literature, we have identified several open problems
that have not been sufficiently studied in past work, and could be useful
avenues for future research.
### 6.1 Fully Automated Task Creation
Task creation is an important piece of the method of curriculum learning.
Whether tasks are created “on-demand” or all in advance, the quality of the
pool of tasks generated directly affects the quality of curricula that can be
produced. In addition, the _quantity_ of tasks produced affect the search
space and efficiency of curriculum sequencing algorithms. Despite this, very
limited work (see Section 4.1) has been done on the problem of automatically
generating tasks. Existing work either assumes the pool of tasks are manually
crafted and specified beforehand, or defines a set of rules for semi-
automatically creating tasks. However, these rules often have hyper-parameters
that control how many tasks are created, and are also usually manually tuned.
Reducing the amount of manual input required by these methods remains an
important area for future work.
### 6.2 Transferring Different Types of Knowledge
Between each pair of tasks in a curriculum, knowledge must be transferred from
one task to the subsequent task. In virtually all of the works surveyed, the
type of knowledge transferred has been fixed. For example, a value function
was always transferred between tasks by Narvekar et al. (2017) while a shaping
reward was always transferred by Svetlik et al. (2017). However, this
limitation opens the question of whether different tasks could benefit from
extracting different types of knowledge. For instance, it may be useful to
extract an option from one task, and a model from another. Thus, in addition
to deciding _which_ task to transfer from, we could also ask _what_ to extract
and transfer from that task. Past transfer learning literature has shown that
many forms of transfer are possible. The best type of knowledge to extract may
differ based on task, and techniques will need to be developed to effectively
combine these different types of knowledge.
### 6.3 Reusing Curricula and Sim-to-Real Curriculum Learning
Another limitation of many curriculum learning approaches is that the time to
generate a curriculum can be greater than the time to learn the target task
outright. This shortcoming stems from the fact that curricula are typically
learned independently for each agent and target task. However, in areas such
as human education, curricula are used to train multiple students in multiple
subjects. Thus, one way to amortize the cost would be to learn a curriculum to
train multiple different agents, or to solve multiple different target tasks
(Narvekar and Stone, 2020).
Another option for amortizing the cost is to learn curricula for a sim-to-real
setting on physical robots, where a curriculum is learned in simulation and
then used to train a physical robot. While the exact weights of the policy
learned in simulation would not apply in the real world, the semantics of the
curriculum tasks might. Therefore, the physical robot could go through the
same training regimen, but learn using the physics and dynamics of the real
world.
### 6.4 Combining Task Generation and Sequencing
The curriculum learning method can be thought of as consisting of 3 parts:
task generation, sequencing, and transfer learning. For the most part,
previous work has tackled each of these pieces independently. For example,
sequencing methods typically assume the tasks are prespecified, or a task
generation method exists. However, an interesting question is whether the task
generation and task sequencing phases can be done simultaneously, by directly
generating the next task in the curriculum. Some very preliminary work has
been done in this direction in the context of video game level generation. For
example, Green et al. (2019) used an evolutionary algorithm to generate maps
for a gridworld, where each tile had a different element. The generator was
optimized to maximize the loss of deep RL agent’s network, inducing a training
curriculum.
Combining task generation and sequencing has additional challenges, such as
specifying the space of possible maps, ensuring those maps are valid/solvable,
and creating maps that are challenging, but not too difficult to solve. In
addition, training the generator can be very expensive. However, it promises
an end-to-end solution that could reduce the amount of human intervention
needed to design curricula.
### 6.5 Theoretical Results
There have been many practical applications of curricula to speed up learning
in both supervised and reinforcement learning. However, despite empirical
evidence that curricula are beneficial, there is a lack of theoretical results
analyzing when and why they are useful, and how they should be created. An
initial analysis in the context of supervised learning was done by Weinshall
et al. (2018) and Weinshall and Amir (2018). They analyzed whether reordering
samples in linear regression and binary classification problems could improve
the ability to learn new concepts. They did this analysis by formalizing the
idea of an Ideal Difficulty Score (IDS), which is the loss of the example with
respect to the optimal hypothesis, and the Local Difficulty Score (LDS), which
is the loss of the example with respect to the current hypothesis. These are 2
ways to classify the difficulty of a sample, which can be used as a means to
sequence samples. They showed that the convergence of an algorithm like
stochastic gradient descent monotonically decreases with the IDS, and
monotonically increases with the LDS. An open question is whether similar
grounded metrics for difficulty of tasks can be identified in reinforcement
learning, and what kind of convergence guarantees we can draw from them.
### 6.6 Understanding General Principles for Curriculum Design
Determining the difficulty of a training example for an agent, and ensuring
that each example presented to the agent is suitable given its current
ability, is a major challenge in curriculum learning. In most existing work,
the curriculum is generated either automatically (see Section 4.2), by
ordering samples from the target tasks or iteratively selecting intermediate
tasks with increasing difficulty tailored to the current ability of the
learner; or manually by domain experts, who will typically have specialized
knowledge of the problem domain. Very limited work (see Section 4.2.5) has
been done to better understand how non-expert humans design curricula. The way
we define curriculum design strategies still leaves a lot to be defined by
human teachers.
Can non-expert humans design effective curricula for a given final task? What
kind of curriculum design strategies do they tend to follow when building
curricula? If we could find some general principles non-expert humans follow
for designing and/or sequencing more “interesting” intermediate tasks into a
curriculum, we could incorporate these insights into the automatic process of
generating useful source tasks for any task domain. Furthermore, can we adapt
curriculum learning algorithms to better take advantage of this type of non-
expert guidance to learn more efficiently? We believe a better understanding
of the curriculum-design strategies used by non-expert humans may help us to
1) understand the general principles that make some curriculum strategies work
better than others, and 2) inspire the design of new machine-learning
algorithms and interfaces that better accommodate the natural tendencies of
human trainers.
## 7 Conclusion
This survey formalized the concept of a curriculum, and the method of
curriculum learning in the context of reinforcement learning. Curriculum
learning is a 3-part approach consisting of 1) task generation, 2) sequencing,
and 3) transfer learning. We systematically surveyed existing work addressing
each of these parts, with a particular focus on sequencing methods. We broke
down sequencing methods into five categories, based on the assumptions they
make about intermediate tasks in the curriculum. The simplest of these are
sample sequencing methods, which reorder samples from the final task itself,
but do not explicitly change the domain. These were followed by co-learning
methods, where a curriculum emerges from the interaction of several agents in
the same environment. Next we considered methods that explicitly changed the
MDP to produce intermediate tasks. Some of these assumed that the environment
dynamics stay the same, but that the initial/terminal state distribution and
reward function can change. Others made no restrictions on the differences
allowed from the target task MDP. Finally, we also discussed how humans
approach sequencing, to shed light on manually designed curricula in existing
work. Our survey of the literature concluded with a list of open problems,
which we think will serve as worthwhile directions for future work. As a
budding area in reinforcement learning, we hope that this survey will provide
a common foundation and terminology to promote discussion and advancement in
this field.
Acknowledgments
We would like to sincerely thank Brad Knox, Garrett Warnell, and the anonymous
reviewers for helpful comments and suggestions that improved the presentation
of many ideas in this article. Part of this work has taken place in the
Learning Agents Research Group (LARG) at the Artificial Intelligence
Laboratory, The University of Texas at Austin. LARG research is supported in
part by grants from the National Science Foundation (CPS-1739964, IIS-1724157,
NRI-1925082), the Office of Naval Research (N00014-18-2243), Future of Life
Institute (RFP2-000), Army Research Office (W911NF-19-2-0333), DARPA, Lockheed
Martin, General Motors, and Bosch. The views and conclusions contained in this
document are those of the authors alone. Peter Stone serves as the Executive
Director of Sony AI America and receives financial compensation for this work.
The terms of this arrangement have been reviewed and approved by the
University of Texas at Austin in accordance with its policy on objectivity in
research. Part of this work has taken place in the Sensible Robots Research
Group at the University of Leeds, which is partially supported by the
Engineering and Physical Sciences Research Council of the UK (EP/R031193/1,
EP/S005056/1), and the British Council. Part of this work has taken place in
the Control, Robotics, Identification and Signal Processing (CRISP) Laboratory
at Tufts University which is partially supported by DARPA (W911NF-19-2-0006),
the Verizon Foundation, PTC Inc., and the Center for Applied Brain and
Cognitive Sciences (CABCS). Part of this work has taken place in the Whiteson
Research Lab at the University of Oxford, which is partially supported by the
European Research Council (ERC), under the European Union’s Horizon 2020
research and innovation programme (grant agreement number 637713). Part of
this work has taken place in the Intelligent Robot Learning (IRL) Lab at the
University of Alberta, which is supported in part by research grants from the
Alberta Machine Intelligence Institute.
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|
2024-09-04T02:54:58.913772 | 2020-02-15T06:15:17 | 2003.04978 | {
"authors": "Sairamvinay Vijayaraghavan, Ye Wang, Zhiyuan Guo, John Voong, Wenda\n Xu, Armand Nasseri, Jiaru Cai, Linda Li, Kevin Vuong, and Eshan Wadhwa",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26146",
"submitter": "Sairamvinay Vijayaraghavan",
"url": "https://arxiv.org/abs/2003.04978"
} | arxiv-papers | # Fake News Detection with Different Models
Sairamvinay Vijayaraghavan
<EMAIL_ADDRESS>
&Zhiyuan Guo
<EMAIL_ADDRESS>
&Ye Wang
<EMAIL_ADDRESS>
John Voong
<EMAIL_ADDRESS>
&Wenda Xu
<EMAIL_ADDRESS>
Armand Nasseri
<EMAIL_ADDRESS>
&Jiaru Cai
<EMAIL_ADDRESS>
Linda Li
<EMAIL_ADDRESS>
&Kevin Vuong
<EMAIL_ADDRESS>
&Eshan Wadhwa
<EMAIL_ADDRESS>
###### Abstract
Problem: The problem we intend to solve is modelled as a binary classification
problem. We intend to find the relation in the words and the context in which
the words appear within the text and how it could be used to classify texts as
real (negative cases) or fake (positive).
High-level description: Many news sources contain false information and are
therefore “fake news.” Because there is a lot of “fake news” articles and
fabricated, misleading information on the web, we would like to determine
which texts are legitimate (real) and which are illegitimate (fake). To solve
this as a binary classification problem, we investigate the effectiveness of
different Natural Language Processing models which are used to convert
character based texts into numeric representations such as TFIDF,
CountVectorizer and Word2Vec models and find out which model is able to
preserve most of the contextual information about the text used in a fake news
data set and how helpful and effective it is in detecting whether the text is
a fake news or not.
Results:We find that out of the three pre-training vectorizing algorithms,
Word2Vec performs comparatively the worst in general and the CountVectorizer
performs slightly better than the TF-IDF models in most of the cases. Out of
the five fine-tuning algorithms, neural networks (ANNs and LSTMs) perform
better. A combination of cv with LSTM achieves the best performance.
Contribution to the machine learning field: We presented a simple model which
can be used to classify a given text as “real” or “fake” mostly accurately.
This form of pre-training embedding algorithms and then fine-tuning on the
downstream supervised task (of binary classification) proves to be efficient
and effective in classifying susceptible news text.
## 1 Introduction
For this report, we are exploring the field of natural language processing,
which is the broad study of how computers and machines can understand human to
human communication and how texts are analyzed based on contextual information
by machines.
In particular, we are using natural language processing to classify news
articles as real news or “fake news”. Fake news is misinformation masked under
the guise of a real news article, and is used to deceptively influence
people’s beliefs.
For this report, we are classifying news articles as “real” or “fake”, which
will be a binary classification problem - classifying the samples as a
positive (with fake news) or negative (not fake news) sample. Many studies
have used machine learning algorithms and build classifiers based on features
like content, the author’s name and job-title, using lots of models like the
convolutional neural network (CNN), recurrent neural network (RNN), feed-
forward neural network (FFNN), long-short term memory (LSTM) and logistic
regression to find the most optimal model and return its results. In [1], the
author built a classifier using natural language processing and used models
like CNN, RNN, FFNN, and Logistic Regression and concluded that the CNN
classifiers could not be as competitive as the RNN classifiers. The authors in
[2] think that their study can be improved by having more features like
knowing the history of lies spoken by the news reporter or the speaker.
Moreover, apart from the traditional machine learning methods, new models have
also been developed. One of the newer models, TraceMiner, creates an LSTM-RNN
model inferring from the embedding of social media users in the social network
structure to propagate through the path of messages and has provided high
classification accuracy5. FAKEDETECTOR is another inference model developed to
detect the credibility of the fake news which is considered to be quite
reliable and accurate7.
There also have been studies that have a different approach. A paper surveys
the current state-of-the-art technologies that are imperative when adopting
and developing fake news detection and provides a classification of several
accurate assessment methods that analyze the text and detect anomalies3.
These previous approaches lack a clear contextual analysis used in NLP. We
considered the semantic meaning of each word and we feel that the presence of
particular words influence the meaning. We reckoned this important since we
felt the contextual meaning of the text needs to be preserved and analyzed for
better classification. Other studies emphasize the user and features related
to them. In [4], “45 features…[were used] for predicting accuracy…across four
types: structural, user, content, and temporal,” so features included
characteristics beyond the text. Article [6] "learn[s] the representations of
news articles, creators and subjects simultaneously." In our project, we
emphasize the content by working with articles whose labels only relate to the
text, nothing outside that scope, and have used SVM, Logistic Regression, ANN,
LSTM, and Random Forest.
We had devised this problem into 3 different phases: pre-processing, text-to-
numeric representation conversion using pre-trained algorithms, and then
evaluate the models using state-of-the-art machine learning algorithms. We had
analysed the data set and in particular the text part of the data explaining
how it is distributed and then we converted each text into numeric
representation using pre-training models such as TFIDF, CV and W2V for vector
representation. Finally, we evaluated our numeric conversion data using
significant machine learning algorithms such as neural networks,
classification algorithms etc to perform the classification.
## 2 Methods
### 2.1 The Dataset
The training data set has five features: ID, title, author, text, and label.
The ID uniquely identifies the news article. The title and author are the
title and author of the news article respectively. The text is the content of
the article, and may be incomplete. The label indicates whether the article is
reliable (real) or not (fake):
label = $\begin{cases}0&\textrm{if reliable news}\\\ 1&\textrm{if fake
news}\end{cases}$
The training data set contains 20800 odd number of samples.
The test data set does not have labels, so we do not use it. The test data set
will be selected from the training data set randomly when we are evaluating
our models.
In our project, since we hypothesized that the text and the words used within
the text are key to distinguish between real and fake news samples, we decided
to investigate only the text column.
### 2.2 Data Pre-processing
#### 2.2.1 Removed numbers
Within the context of a news article title or text, numbers simply quantify
claims and do not change the meaning of the text. Therefore it is best to
remove all numbers to minimize noise in our data. We use the string.digits
string constant in Python as well as the translate and maketrans methods from
Python’s string module to convert all numerical digits to an empty string,
effectively removing all digits.
#### 2.2.2 Removed punctuation and special characters
In addition of pre-processing the textual data, we removed all characters that
are not textual (not alphabets such as punctuation, extra delimiters etc.). We
used the string.punctuation module in Python to find all punctuation
characters. We remove all those punctuation characters from every word in the
texts, with the exception of the symbols ‘#’ and ‘@’. Because these are
characters used for Twitter hashtags and mentions, we handle these later.
Next, we removed an assortment of special characters that don’t appear on
traditional American keyboards and don’t contribute to the meaning of the
tweets. The long dash (“–”), single and double Asian quotations, ellipse
characters (…), and bullet points (•) all were removed for this reason.
After removing all special characters, there are still a couple of pre-
processing cases we account for. For these cases, we used regular expressions
to detect certain patterns we wish to remove. One of the patterns is Twitter
hashtags and mentions. In a news setting, Twitter hashtags and mentions are
often added to try to obtain more search results and relevance, but often
distract from the overall meaning of the news content itself. In our problem,
we are primarily concerned with words and mostly their contextual meanings
used in the text and we assumed that these unnecessary characters. To detect
the hashtags and mentions, we simply use regular expressions to remove all
text after a hashtag (#) or @ symbol, and stop removing text when we reach the
next space. We also use regular expressions to handle em dashes (—) and more
than two consecutive spaces. Em dashes are used in various linguistic contexts
like joining independent clauses. They do not add to the meaning of the text,
however they are surrounded by two words of different clauses, so we replaced
all em dashes with a single space to maintain the integrity of each phrase.
Lastly, we replace any set of two or more consecutive spaces with just one
space.
Proceeding further, we make all of our texts lowercase and then remove all
rows that have foreign language characters in their text, since we are only
interested in identifying fake news in English. To do this we used the package
langid in Python to identify the language of all texts, and removed all rows
with foreign characters. This finally ensures the text we preserve is only
with English words with no non-alpha character.
#### 2.2.3 Removed stop words
Stop words are a list of the most common words in a language, such as “a”,
“be”, “quite”, “should”…etc. They are often void of meaning, and does not add
anything to the content. They are also most frequently present in every text.
Hence, we presumed removal of stop words can have multiple advantages. For
once, it decreases memory overhead, since we cut down a huge amount of text
(and hence narrows down the number of features to train our models on).
Second, it reduces noise, since by eliminating stop words, we are able to
focus on more meaningful contents (the more distinct features between these
two classes). Although it is not often the case that removing stop words are
the most optimal, sometimes the information that we are looking for may be
included in the stop words that we removed. For example, in most cases of
language modeling, or translation, where it is important that we keep all the
stop words. However, in our circumstances, we are using the semantics of the
text to make a decision. In this case, we can safely remove stop words to
observe the more meaningful context words.
### 2.3 Data Distribution
We performed some data analysis on the text and wanted to understand how the
text is distributed. We had analyzed and represented our data (text)
distribution in a few different perspectives. We first analyzed the data
through graphing its sentiment polarity, most popular unigram and bigram, as
well as looking at the distribution of the word types. We will be comparing
the graphs before and after preprocessing, which includes, stop word removal,
removing punctuation and special characters, and numbers.
#### 2.3.1 Sentiment Polarity
Polarity Graphs before pre-processing
Polarity Graphs after pre-processing
For both before and after pre-processing, the distribution of the polarity of
fake news sentiment and real news sentiment are mostly the same. For both fake
news and real news, there are slightly more positive news than the negatives.
However, there is a noticeable difference between the polarity. We can see
that although not by much, fake news are a little bit more polar than real
news. There are more outliers, and the data are a little bit more spread out.
#### 2.3.2 Part of Speech Distribution
Part of Speech Graphs before pre-processing
Part of Speech Graphs after pre-processing
Although the differences are slight, there is a difference in part of speech
distribution between real and fake news. In fake news, there are a higher
percentage of adverbs and adjectives compared to all the other parts of
speech, while there is a lower percentage of proper pronoun; however, in real
news, there are a higher percentage of pronoun. We can interpret this as there
are more adverbs and adjectives in fakes new, and there are more pronoun in
real news. Perhaps, this is indicating that fake news are more likely to use
adverbs and adjectives to embellish their sentences, while real news use more
pronouns to establish as reference to their legitimacy.
#### 2.3.3 Unigram and Bigram
Unigrams
Real News Fake News Before After Before After the nt the nt to trump to Trump
of people of people and clinton and clinton in hillary in hillary that said
that said for like is like on new for new he time it time is World on world it
state as state was election with election said government are government mr
president this preseident with war by war as years before years his states was
states at american you american by obama have obama from media they media
Bigrams
Real News Fake News Before After Before After of the mr trump of the hillary
clinton in the united states in the donald trump to the new york to the united
states on the mr trumps on the white house mr trump white house and the new
york at the donald trump that the hillary clintons and the mrs clinton to be
clinton campaign that the said mr for the clinton foundation to be york times
it is secretary state he said islamic state with the nt know with the mr obama
from the american people from the breitbart news by the mainstream media by
the president trump at the foreign policy it was years ago hillary clinton
bill clinton
The comparison between the result of the top unigram and bigram before and
after preprocessing demonstrates that our decision to remove stop words is the
correct choice. The top unigram and bigram are all consisted of words, in
other words, filler words that does supply us with any explanation.
After removing the stop words, we can see that the top unigrams and bigrams
become much more specific.
### 2.4 Unsupervised Pre-training to encode our texts into numeric
representations
#### 2.4.1 Natural Language Processing Models
After text have been cleaned, they are mapped into numeric representations in
form of vectors of the textual data using three pre-training algorithms (i.e.
CountVectorizer, TF-IDFVectorizer, and Word2Vec). Each sample, originally
consisting of all text, is converted into a vector of features. Since only the
text is passed into these pre-training algorithm, this stage is unsupervised.
In the cases of CountVectorizer and TfidfVectorizer, the number of features is
clipped at 10000 to avoid memory overrun and overfitting (because of the large
number of features (the vocabulary)).
#### 2.4.2 CountVectorizer
The CountVectorizer provides a simple way to both tokenize a collection of
text documents and build a vocabulary of known distinct words, but also to
encode new documents using that vocabulary13.
Given a collection of text documents, $S$ , CountVectorizer will generate a
sparse matrix $A$ of size $m$ by $n$, where $m=$ total number of documents,
$n=$ total number of distinct words used in $S$.
$A=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\\
\vdots&\vdots&\vdots&\vdots\\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{pmatrix}$
This matrix is the one hot encoded representation of the different words
present in the corpus. Entry $a_{ij}=$ total number of times $j$th word
appears in the $i$th document.
We had converted the sparse matrix into a dense one since we found that there
are plenty of distinct words in the corpus which may not even be present in
some of the samples and hence they may be populated with zeros. Hence, we felt
that since zeros may be entirely populated, we decided to convert it to a
dense matrix using the todense() method call which a dense representation of
the sparse matrix.
#### 2.4.3 TF-IDFVectorizer
Although TF-IDF is an old algorithm, it is simple and effective to be used in
the phase of pre-training11. The computation of TfidfVectorizer involves
computing the product of term frequency and inverse document frequency. As the
term implies, TF-IDF calculates values for each word in a document through an
inverse proportion of the frequency of the word in a particular document to
the percentage of documents the word appears in12.
The term frequency $tf(t,d)$ calculates the proportion of times that the term
$t\in V(d)$ appears in the document $d$. The vocabulary $V(d)=\sum_{t}n(t,d)$
is constructed by the document $d$. Thus, if a word $w^{\prime}$ does not
appear in a document $d^{\prime}$, the term frequency
$tf(t^{\prime},d^{\prime})$ in this case would be zero. The idea of the term
frequency is essentially the same as CountVectorizer.
$tf(t,d)=\frac{n(t,d)}{V(d)}$ $n(t,d)=\textrm{ occurrence of the word
}t\textrm{ in the document }d$
Given a document collection $D$, the inverse document frequency $idf(t,D)$ is
the log of the number of documents $N$ divided by $df(t,D)$, the number of
documents $d\in D$ containing the term $t$. As a result, common words in $D$
will have a low term frequency score, while infrequent words will have a high
term frequency. Thus, the term frequency will be very likely to separate fake
news that often have less common words (even ungrammatical) from real news
that usually consist of common words.
$idf(t,D)=\log\Big{(}\frac{N}{df(t,D)}\Big{)}$
As a summary, TF-IDF score $w(t,d)$ for a word increases with its count, but
will be counteracted if the word appears in too many documents.
$w(t,d)=tf(t,d)\times idf(t,D)$
Similar to CountVectorizer, we found that most of the entries within the
matrix were 0. Hence, we used the dense (todense() call) to return the dense
representation of the sparse TFIDF matrix representation.
#### 2.4.4 Word2Vec
Word2Vec is another state of the art model used to represent words into
vectors. Word2Vec is a simple neural network which basically tries to predict
the next word within a context given a set of words provided. Word2Vec
basically represents a vector for each word within the context and the vector
representation is the weights of the particular connection from the input
layer node into one of the hidden layer neurons. This information is mainly
encoding the contextual information of the particular word within the corpus
(collection of texts) on which we train our word2vec model.
In this project, all we did was we trained the word2vec model on our current
corpus. We did this because we felt that the corpus contained very specific
words which had a contextual meaning completely different from what is used in
general. Hence, we chose to train the corpus on the existing texts in our
corpus texts over the pre-trained word2vec models such as google models. For
training our word2vec models, we chose the minimum count as the average number
of words in each of the texts in general, since we believed that texts which
are shorter than the mean length have less context and hence we rejected those
sentences to train on. We then used the number of features as the default
number of features as 100 since we wanted to analyze on a short number of
features.
For this project, we decided on a very simple and plain approach. We obtained
the vector for each sentence by summing all the vector representations for
each word in the sentence only if the word belongs to the word2vec model. The
summed up vector is finally divided with the number of words in the sentence
since we wanted to make sure that the size of the text doesn’t affect the
vector embeddings and hence we normalized our word2vec embedding.
### 2.5 Outlier Removal
During outlier removal, the Isolation Forest algorithm isolates observations
by randomly selecting a feature and then randomly selecting a split value
between the maximum and minimum values of selected features. In Isolation
Forest, an anomaly score can be calculated as the number of conditions
required to separate given observation.
In our outlier detections and removals, Isolation Forest has been applied to
three different features. Generated from TFIDF, CV, WV. Percentages of outlier
in each feature set is calculated, bar graph of percentage of training
outliers are included.
### 2.6 Fine-tuning
Once the representations of text are pre-trained from previous unsupervised
learning, the representations are then fed into 5 different models to perform
supervised learning on the downstream task. In this case, the downstream task
is a binary classification of the fake news as either real or fake. A k-fold
prediction error is obtained from each of the 5 models, and since we have 3
different pre-training models, we have a total of 15 models to compare.
#### 2.6.1 Artificial Neural Network (ANN)
We trained simple Artificial Neural Networks which contains an input layer,
particular number of output layers (specified by a hyperparameter) in which
each hidden layer contains the same number of neurons and the same activation
function, and an output layer with just one node for the classification (real
or fake) which uses sigmoid as an activation function. We chose sigmoid as the
output layer activation and the binary_crossentropy as the loss since it is a
binary classification problem and the use of softmax normalizes the results
which is not needed for this problem and since we use only one output node to
return the activation, we applied sigmoid for the output layer activation. We
performed Grid Search strategy to find the best hyper-parameters such as
activations, optimizers, number of hidden layers and number of hidden neurons.
We had used Keras Sequential model and we used Dense Layers which contains
connections to every hidden node in the next layer.
Due to the limitation of computing resource, the grid search for Neural
Networks is divided into three sequential steps. Instead of performing grid
search on all the hyperparameters all at once, we chose to do grid search for
the activations for the hidden layers, optimizers and the number of hidden
layers and hidden neurons (done together). We coupled the number of hidden
layers and the number of neurons since we believed that each of these
hyperparameters interact with each other in improving the model training. We
also did a K-fold Split for 3 splits at each step and picked the best
hyperparameters which renders the highest accuracy.
#### 2.6.2 Long Short Term Memory networks (LSTMs)
Long Short Term Memory networks (LSTMs) is a special recurrent neural network
(RNN) introduced by Hochreiter & Schmidhuber (1997)8.
(Christopher Olah. “Understanding LSTM Networks.”)
The chain-like nature of an RNN allows information to be passed from the
beginning all the way to the end. The prediction at time step $t$ depends on
all previous predictions at time step $t\textquoteright<t$. However, when a
typical RNN is used in a larger context (i.e. a relatively large time steps),
the RNN suffers from the issue of vanishing gradient descent 9. LSTMs, a
special kind of RNN, can solve this long-term dependency problem.
(Christopher Olah. “Understanding LSTM Networks.”)
Each cell in a typical LSTMs network contains 3 gates (i.e., forget gate,
input gate, and output gate) to decide whether or not information should be
maintained in the cell state $C_{t}$.
For CountVectorizer and TfidfVectorizer, each sample of text is converted into
a 1-d feature vector of size 10000. As a result, the number of time steps
(i.e. the maximum amount of word vectors for each sample) for these two can
only be set to 1, as the pre-trained representations are done at the sample’s
level. By contrast, the number of time steps for Word2Vec can either be 1, if
we simply take an average of the word embeddings, or the length of the
sentence, where each word has an embedding and thus the pre-trained
representations are done at the word’s level. We choose the approach with 1
timestep in our model because it requires less computation power. Meanwhile,
we also do the length of the sentence, and 200 time steps are chosen as 200 is
close to the mean amount of words in each sample and it is a fairly common
choice in practice. However, since we do not have enough computation power to
fine-tune (grid search) our model, we leave it out for our model and include
it only in the final section.
In the LSTM layer, a dropout rate of 0.2, a common choice in practice10 , is
used to prevent overfitting. Grid search is performed in order to pick decent
values of hyperparameters, including the number of hidden units in the LSTM
layer, the number of hidden layers, the activation functions and the number of
nodes in the hidden layer, and the optimizer. Relatively small numbers of
hidden layers (i.e., {0, 1, 2}) and nodes (i.e., {200, 400, 600}) are selected
as the basis for grid search, because this is a simple binary classification
task and too many of them would cause overfitting.
Due to the limitation of computing resource, the grid search for LSTMs is
divided into four sequential steps. Instead of performing grid search on all
the hyperparameters all at once, the grid search is first done on the number
of hidden layers and all other hyperparameters are randomly selected from the
subset. Then, the grid search is done on the number of nodes in the hidden
layer(s), using the best number of hidden layer found in step 1. The grid
search completes when all four steps are finished. In each step we used K-fold
cross validation with $K=3$.
#### 2.6.3 Random Forest
A random forest is an ensemble classifier that estimates based on the
combination of different decision trees. So random forest will fit a number of
decision tree classifiers on various subsamples of the dataset. A random best
subsets are built by each tree in the forest. In the end, it gives the best
subset of features among all the random subsets of features.
In our project, 3 random forest algorithms have been applied with models count
vectorizer, tfidf and word-to-vector. Random forest algorithm requires 4
hyperparameters to tune, such as the number of trees in the forest (i.e.,
{200, 400, 800}); the maximum depth of the tree (i.e., {1,5,9}); the minimum
number of samples required to be at a lead node (i.e., {2, 4}); The minimum
number of samples at each leaf node has the effect of smoothing the model,
especially during regression; the minimum number of samples required to be at
a leaf node (i.e., {5, 10}). All parameters are applied to grid search and in
the end, the best set of parameters can be determined as we used K-fold cross
validation with $K=3$.
#### 2.6.4 Logistic Regression
Logistic regression is a statistical machine learning algorithm that
classifies the data by considering outcome variables on extreme ends and this
algorithm is providing a discriminatory line between classes. Compared to
another simple model, linear regression, which requires hard threshold in
classification, logistic regression can overcome threshold values for a large
dataset. Logistic regression produces a logistic curve, which is limited to
values between 0 to 1, by adding sigmoid function in the end.
In regards to our project, three logistic regressions have been applied with
models CountVectorizer, TF-IDF and Word2Vec. We did grid search on the
solvers, including newton-cg, sag, lbfgs and liblinear. Grid search is also
performed on the inverse of regularization parameter with values being {0, 4,
10}. Best parameter sets can be determined as we used K-fold cross validation
with $K=3$.
#### 2.6.5 Support Vector Machine (SVM)
SVM is a supervised machine learning algorithm in which a hyperplane is
created in order to separate and categorize features. The optimal hyperplane
is usually calculated by creating support vectors on both sides of the
hyperplane in which each vector must maximize the distance between each other.
In other words, the larger the distance between each vector around the
hyperplane, the more accurate the decision boundary will be between the
categories of features.
In regards to our project, we fit 3 support vector machines on
CountVectorizer, TfidfVectorizer, and WordToVectorizer. An SVM requires
specific parameters such as a kernel type, $C$, maximum iterations, etc. In
our case, we needed to determine the optimal $C$ as well as the optimal kernel
for each fit. We used K-fold cross validation with $K=3$. A grid search of
kernel types and $C$ was performed in order to give us the most accurate svm
model. The parameters we used for each kernel were linear and rbf while the
values we used for $C$ were 0.25 ,0.5, and 0.75. Once the grid search was
completed for these hyperparameters, the model was evaluated with the most
optimal hyperparameters using cross validation of 3 splits.
## 3 Results
Grid Search Results CountVectorizer TF-IDF Word2Vec SVM Kernel = Linear Kernel
= Linear Kernel = Linear C = 0.25 C = 0.75 C = 0.75 Logistic Regression Solver
= sag Solver = sag Solver = newton-cg C = 21.54 C = 7.74 C = 3593.81 Random
Forest Max Depth = 9 Max Depth = 9 Max Depth = 9 Min_samples_leaf = 2
Min_samples_leaf = 4 Min_samples_leaf = 2 Min_samples_split = 10
Min_samples_split = 5 Min_samples_split = 10 N_estimators = 200 N_estimators =
400 N_estimators = 400 ANN Activation = relu Activation = sigmoid Activation =
relu Optimizer = Adam Optimizer = Adam Optimizer = Adam Hidden_layers = 2
Hidden_layers = 3 Hidden_layers = 1 Num_Neurons = 600 Num_Neurons = 400
Num_Neurons = 600 LSTM Activation = sigmoid Activation = sigmoid Activation =
relu Optimizer = Adam Optimizer = Adam Optimizer = Adam Hidden_layers = 2
Hidden_layers = 2 Hidden_layers = 2 Memcells = 200 Memcells = 200 Memcells =
200 Num_Neurons = 200 Num_Neurons = 600 Num_Neurons = 600
Mean Test Scores SVM ANNs LSTMs LOGISTIC RANDOM FOREST CV 93.06% 94.29% 94.88%
94.45% 87.64% TFIDF 94.58% 93.73% 93.89% 94.79% 87.64% Word2Vec 91.17% 93.06%
92.29% 91.30% 88.60%
ANN Loss and Accuracy
LSTM Loss and Accuracy
The model is evaluated using a 3-fold of cross validation. Out of the fifteen
models, CountVectorizer with LSTMs performs the best. Word2Vec performs the
worst among the three pre-training algorithms. Random forest performs the
worst among the five fine-tuning algorithms.
## 4 Discussion
Among our three pre-training models, CountVectorizer achieves in general the
best performance comparatively and Word2Vec performs relatively poor amongst
the three models. The essential idea behind both CountVectorizer and TF-IDF is
computing a score which depends on the frequency of the word belonging to the
vocabulary. However, comparing to CountVectorizer, the TF-IDF includes an
extra inverse document frequency that “penalizes” (apparently masks) the
contextual meaning within the words that appear more frequently across
documents. They represent the importance of the word within a document. The
results may imply that even though the penalization is smoothed by a log
function, the punishment may be too high.
The results also show that in general neural networks do the best
consistently, as neural networks serve as a powerful universal approximator.
However, the loss and accuracy plots show that we are using too many epochs
and thus have the issue of overfitting. This is because our pre-training model
is already very strong so it learns a good contextual representation of text.
As a result, the epochs needed for downstream task are not much. In addition,
one thing to note is that logistic regression also performs very well. This
implies that our data are mostly linearly separable. While neural networks can
fit the data very well, but they run the risk of overfitting the data. As a
result, neural networks are not as good as SVM and Logistic Regression for TF-
IDF.
A combination of CountVectorizer and LSTMs is the best among all the models.
While LSTMs with one timestep are very similar to ANN in terms of
architecture, LSTMs have gates and a tanh activation function inside the
module. This different design may let LSTMs perform slightly better than ANN.
Word2Vec does not perform well. One reason is that we are simply taking an
average of the word embedding vectors to get a generalized vector
representation of each sample of paragraph. Taking an average fails to
represent the dependencies between words. Another reason is that we do not use
pre-trained Word2Vec embeddings available online from huge corpus but instead
build our own from the dataset. While we thought that building our own
Word2Vec would make the model specific to this task, the results show that
Word2Vec may need to be built from larger dataset.
## 5 Conclusion
This report provides a fairly simple approach to encode texts and how the
presence of words in general impacts the classification of texts as real and
fake.
We achieved high accuracy results in most of our algorithms and in particular
neural networks generally do better than the others.
What’s worth noting is that our LSTMs only use a timestep of 1 and are
essentially multi-layer perceptrons. Still, as mentioned is the LSTM’s method
section, the LSTMs with the real recurrence are performed by using Word2Vec
for representations at the word’s level. In this case, each word has its own
vector, and a sample will be a collection of vectors and thus a 2-D matrix. As
mentioned before, each vectorized word will become a timestep, and a total of
200 timesteps is used (If the paragraph has more than 200 words, only the
first 200 words will be selected). We run our model and get the following
results.
The results seem solid, but this approach is not included in our model because
it takes too much time to run and we do not have time to fine-tune the
hyperparameters. But in future work, we believe that using LSTMs with real
recurrence will give an even better results.
While we achieve great performance in this dataset, the question remains as to
whether X (to be replaced by the best model) can still perform well in tasks
that classify news into more than two categories, such as the Fake News
Challenge. In that case, a simple unidirectional LSTMs may not be so well and
may need to be replaced by a bidirectional one. In addition, it would be
interested to know how well our pre-trained model performs in other downstream
tasks, such as Spam Detection. Lastly, in our model, the pre-training is done
on the dataset given (will make the model specific to the task), instead of on
the big corpus available online, such as Google’s pre-trained Word2Vec model.
If the task were a classification of four or eight categories, pre-trained
model on large corpus may perform better as the model is pre-trained on more
words.
We can also try to improve the training by using different word embeddings.
While we only chose only 3 different types of embeddings, we could have tried
different embeddings such as GloVe and the features used are entirely
dependent only on context words. We can use different forms for encoding texts
which can be used to be trained using these algorithms to achieve a better
model. In another
State-of-the-art pre-trained models can be used if the task is no longer a
binary classification. Models like Transformer and BERT will be strong
candidates as they have learned a very strong representation that takes the
context into account when computing an embedding for a word. Unlike LSTMs
whose sequential nature prohibits parallelization, the Transformer and the
BERT can achieve parallelization by replacing recurrence with the attention
mechanism. Thus, they require less computation power and can be easily fine-
tuned in downstream tasks.
## 6 Appendix
## Github Repo
https://github.com/Sairamvinay/Fake-News-Dataset
## Author Contributions
Sairamvinay Vijayaraghavan: Project Planning, Problem Formation, DataSet
Search, POS Distribution graph, Code for CountVectorizer, Word2Vec, ANN,
Randomforest,To parse csv files (readdata), Code integration for
TextVectorizer, Grid Search model running, ROC model running, Code Base
Cleanup and management (further cleanup), PowerPoint Checking, Report Analysis
for W2V, ANN, Report editing
Zhiyuan Guo: Project Planning, DataSet Search, Polarity Graphs, Code for LSTM,
RandomForest, Adding Functionality and Readability in each of the scripts,
Code Integration, Grid Search model running, ROC model running, PowerPoint
Development, Report Analysis for TFIDF and LSTM, Report Analysis for the
Abstract, the Discussion, Conclusion, Pipeline Diagram, Report editing
Ye Wang: Project Planning, DataSet Search, Code for TFIDF, PCA, Grid Search
model running, ROC model running, Report Integration into Latex, Report
Analysis of the Results (table creations), Report Analysis for the Outlier
Removal, Random Forest, Report editing
John Voong: Word2Vec, DataCleanup (StopWord Cleanup), Grid Search model
running, ROC model running, PowerPoint Development, Report Analysis for W2V,
Pipeline Diagram, Report editing, Paper structure
Wenda Xu: Code for PCA, ROC model running, Code Base Cleanup and management,
PowerPoint Development, Report Analysis about Count Vectorizer, Report
Analysis about Logistic Regression
Armand Nasseri: Project Planning, Dataset search, Code for SVM, Data Cleanup
(StopWord Cleanup), ROC model running, PowerPoint Development, Report Analysis
about SVM
Jiaru Cai: Outlier Removal, Accuracy and Loss Plots for Neural Network,
PowerPoint Framework
Kevin Vuong: DataCleanup (remove punctuations), Code for Logistic Regression,
Grid Search model running, PowerPoint Cleanup, Report Analysis about Data
Cleanup, Introduction and Abstract
Linda Li: Unigram and Bigram analysis, Code for ROC plots, Report Analysis of
the Data Cleanup section, Graph analysis
Eshan Wadhwa: Related Work, References and Citation (Introduction and Field
research), Report Editing, PowerPoint slides,
## References
[1] Samir Bajaj, “The Pope Has a New Baby!” Fake News Detection Using Deep
Learning”, Winter 2017,
https://pdfs.semanticscholar.org/19ed/b6aa318d70cd727b3cdb006a782556ba657a.pdf
[2] Arjun Roy, Kingshuk Basak, Asif Ekbal, and Pushpak Bhattacharyya, “A Deep
Ensemble Framework for Fake News Detection and Classification”, 12 November
2018,
https://arxiv.org/pdf/1811.04670.pdf
[3] Niall J. Conroy, Victoria L. Rubin, and Yimin Chen, “Automatic Deception
Detection: Methods for Finding Fake News”, November 2015,
https://asistdl.onlinelibrary.wiley.com/doi/epdf/10.1002/pra2.2015.145052010082.
[4] Liang Wu and Huan Liu, “Tracing Fake-News Footprints: Characterizing
Social Media Messages by How They Propagate”, February 2018,
http://www.public.asu.edu/~liangwu1/WSDM18_TraceMiner.pdf
[5] Adrian Colyer, “Tracing fake news footprints: characterizing social media
messages by how they propagate”,the morning paper, February 2018,
https://blog.acolyer.org/2018/02/19/tracing-fake-news-footprints-
characterizing-social-media-messages-by-how-they-propagate/
[6] Kai Shu, Amy Sliva, Suhang Wang, Jiliang Tang and Huan Liu, “Fake News
Detection on Social Media: A Data Mining Perspective”, August 2017,
https://arxiv.org/abs/1708.01967
[7] Jiawei Zhang, Bowen Dong and Philip S. Yu, “FAKEDETECTOR: Effective Fake
News Detection with Deep Diffusive Neural Network”, August 2019,
https://arxiv.org/pdf/1805.08751.pdf
[8] Sepp Hochreiter and Jurgen Schmidhuber, “Long short-term memory”, November
1997,
http://www.bioinf.jku.at/publications/older/2604.pdf
[9] Yoshua Bengio, Patrice Simard, and Paolo Frasconi. “Learning long-term
dependencies with gradient descent is difficult”, March 1994,
http://www.comp.hkbu.edu.hk/~markus/teaching/comp7650/tnn-94-gradient.pdf
[10] Gaofeng Cheng, Vijayaditya Peddinti, Daniel Povey, et al., “An
Exploration of Dropout with LSTMs”. August 2017,
https://www.danielpovey.com/files/2017_interspeech_dropout.pdf
[11] Juan Ramos. “Using tf-idf to determine word relevance in document
queries”, December 2003,
https://www.cs.rutgers.edu/~mlittman/courses/ml03/iCML03/papers/ramos.pdf
[12] Gerard Salton and Christopher Buckley. “Term-weighting approaches in
automatic text retrieval”, January 1988,
https://www.sciencedirect.com/science/article/abs/pii/0306457388900210
[13] Jason Brownlee. “How to Prepare Text Data for Machine Learning with
scikit-learn”, August 2019,
https://machinelearningmastery.com/prepare-text-data-machine-learning-scikit-
learn/
|
2024-09-04T02:54:58.923713 | 2020-02-18T13:58:06 | 2003.04986 | {
"authors": "Vukosi Marivate, Tshephisho Sefara, Vongani Chabalala, Keamogetswe\n Makhaya, Tumisho Mokgonyane, Rethabile Mokoena, Abiodun Modupe",
"full_text_license": null,
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26147",
"submitter": "Vukosi Marivate",
"url": "https://arxiv.org/abs/2003.04986"
} | arxiv-papers | # Investigating an approach for low resource language dataset creation,
curation and classification: Setswana and Sepedi
###### Abstract
The recent advances in Natural Language Processing have been a boon for well
represented languages in terms of available curated data and research
resources. One of the challenges for low-resourced languages is clear
guidelines on the collection, curation and preparation of datasets for
different use-cases. In this work, we take on the task of creation of two
datasets that are focused on news headlines (i.e short text) for Setswana and
Sepedi and creation of a news topic classification task. We document our work
and also present baselines for classification. We investigate an approach on
data augmentation, better suited to low resource languages, to improve the
performance of the classifiers.
languageresourceLanguage Resources
Investigating an approach for low resource language dataset creation, curation
and classification: Setswana and Sepedi
Vukosi Marivate${}^{1}{}^{,}{}^{2}$, Tshephisho Sefara2, Vongani Chabalala3,
Keamogetswe Makhaya4, Tumisho Mokgonyane5, Rethabile Mokoena6, Abiodun
Modupe${}^{7}{}^{,}{}^{1}$
---
University of Pretoria1, CSIR2, University of Zululand3, University of Cape
Town4,
University of Limpopo5, North-West University6, University of the
Witwatersrand7
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract content
## 1\. Introduction
The most pressing issue with low-resource languages is of insufficient
language resources. In this study, we introduce an investigation of a low-
resource language that provides automatic formulation and customization of new
capabilities from existing ones. While there are more than six thousand
languages spoken globally, the openness of resources for each is
extraordinarily unbalanced [Nettle, 1998]. For example, if we focus on
language resources annotated on the public domain, as of November 2019, AG
corpus released about $496,835$ news articles only in English languages from
more than $200$ sources111http://groups.di.unipi.it/~gulli, Reuters News
Dataset [Lewis, 1997] comprise an roughly $10,788$ annotated texts from the
Reuters financial newswire. The New York Times Annotated Corpus [Sandhaus,
2008] holds over $1.8$ million articles, and there are no standard annotated
tokens in the low-resource language. Google Translate only supports around 100
languages [Johnson et al., 2017]. A significant number of bits of knowledge
focus on a small number of languages neglecting $17\%$ out of the world’s
language categories label as low-resource [Strassel and Tracey, 2016], which
makes it challenging to develop various mechanisms for Natural Language
Processing (NLP).
In South Africa several of the news websites (private and public) are
published in English, even though there are 11 official languages (including
English). We list the top premium newspapers by circulation as per first
Quarter 2019 [Bureau of Circulations, 2019] in Table 1. We do not have a
distinct collection of a diversity of languages with most of the reported
datasets as existed in English, Afrikaans and isiZulu. In this work, we aim to
provide a general framework that enables us to create an annotated linguistic
resource for Setswana and Sepedi news headlines. We apply data sources of the
news headlines from the South African Broadcast Corporation (SABC)
222http://www.sabc.co.za/, their social media streams and a few acoustic news.
Unfortunately, we do not have any direct access to news reports, so we hope
this study will promote collaboration between the national broadcaster and NLP
researchers.
Table 1: Top newspapers in South Africa with their languages Paper | Language | Circulation
---|---|---
Sunday Times | English | 260132
Soccer Laduma | English | 252041
Daily Sun | English | 141187
Rapport | Afrikaans | 113636
Isolezwe | isiZulu | 86342
Sowetan | English | 70120
Isolezwe ngeSonto | isiZulu | 65489
Isolezwe ngoMgqibelo | isiZulu | 64676
Son | Afrikaans | 62842
The rest of the work is organized as follows. Section 2. discusses prior work
that has gone into building local corpora in South Africa and how they have
been used. Section 3. presents the proposed approach to build a local news
corpora and annotating the corpora with categories. From here, we focus on
ways to gather data for vectorization and building word embeddings (needing an
expanded corpus). We also release and make pre-trained word embeddings for 2
local languages as part of this work [Marivate and Sefara, 2020a]. Section 4.
investigate building classification models for the Setswana and Sepedi news
and improve those classifiers using a 2 step augmentation approach inspired by
work on hierarchical language models [Yu et al., 2019]. Finally, Section 5.
concludes and proposes a path forward for this work.
## 2\. Prior Work
Creating sizeable language resources for low resource languages is important
in improving available data for study [Zoph et al., 2016] and cultural
preservation. If we focus our attention on the African continent, we note that
there are few annotated datasets that are openly available for tasks such as
classification. In South Africa, the South African Center for Digital Language
Resources (SADILAR) 333www.sadilar.org has worked to curate datasets of local
South African languages. There remain gaps such as accessing large corpora and
data from sources such as broadcasters and news organizations as they have
sizeable catalogs that are yet to make it into the public domain. In this
work, we work to fill such a gap by collecting, annotating and training
classifier models for news headlines in Setswana and Sepedi. As the data that
we do find publicly is still small, we also have to deal with the challenges
of Machine Learning on small data.
Machine learning systems perform poorly in presence of small training sets due
to overfitting. To avoid this problem, data augmentation can be used. The
technique is well known in the field of image processing [Cubuk et al., 2019].
Data augmentation refers to the augmentation of the training set with
artificial, generated, training examples. This technique is used less
frequently in NLP but a number of few studies applied data augmentation.
?) use data augmentation to counteract overfitting where recurrent neural
network (RNN) Encoder-Decoder is implemented specifically geared toward a low-
resource setting. Authors apply data augmentation by finding words that share
word stem for example fizzle and fizzling share fizzl. Then authors replace a
stem with another string.
?) apply data augmentation by using synonyms as substitute words for the
original words. However, ?) states that synonyms are very limited and the
synonym-based augmentation cannot produce numerous different patterns from the
original texts. Hence, ?) proposes contextual data augmentation by replacing
words that are predicted by a language model given the context surrounding the
original words to be augmented.
As ?) states that these techniques are valid, they are not often used in
practice because they have a high cost of implementation relative to
performance gain. They propose an easy data augmentation as techniques for
boosting performance on text classification tasks. These techniques involve
synonym replacement, random insertion, random swap, and random deletion of a
word. Authors observed good performance when using fraction of the dataset
(%):1, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, as the data size increases,
the accuracy also increases for augmented and original data. Original data
obtained highest accuracy of 88.3% at 100% data size while augmented data
obtained accuracy of 88.6% at 50% data size.
In this work, we investigate the development of a 2 step text augmentation
method in order to be improve classification models for Setswana and Sepedi.
To do this we had to first identify a suitable data source. Collect the data,
and then annotate the datasets with news categories. After the data is
collected and annotated, we then worked to create classification models from
the data as is and then use a word embedding and document embedding
augmentation approach.
## 3\. Developing news classification models for Setswana and Sepedi
Here we discuss how data was collected as well as the approach we use to build
classification models.
### 3.1. Data Collection, Cleaning and Annotation
Before we can train classification models, we first have to collect data for 2
distinct processes. First, we present our collected news dataset as well as
its annotation. We then discuss how we collected larger datasets for better
vectorization.
#### 3.1.1. News data collection and annotation
The news data we collected is from the SABC444http://www.sabc.co.za/ Facebook
pages. The SABC is the public broadcaster for South Africa. Specifically, data
was collected from the Thobela FM (An SABC Sepedi radio
station)555https://www.facebook.com/thobelafmyaka/ and Motsweding FM (An SABC
Setswana radio station)666https://www.facebook.com/MotswedingFM/. We scraped
the news headlines that are published as posts on both Facebook pages. We
claim no copyright for the content but used the data for research purposes. We
summarize the datasets in Table 2. We visualize the token distributions in
Sepedi and Setswana in Figures 1 and 2 respectively.
Table 2: News Data Sets | Setswana | Sepedi
---|---|---
Corpus Size (headlines) | 219 | 491
Number of Tokens (words) | 1561 | 3018
Figure 1: Setswana Wordcloud Figure 2: Sepedi Wordcloud
As can be seen, the datasets are relatively small and as such, we have to look
at other ways to build vectorizers that can better generalize as the word
token diversity would be very low.
We annotated the datasets by categorizing the news headlines into: _Legal_ ,
_General News_ ,_Sports_ , _Other_ , _Politics_ , _Traffic News_ , _Community
Activities_ , _Crime_ , _Business_ and _Foreign Affairs_. Annotation was done
after reading the headlines and coming up with categories that fit both
datasets. We show the distribution of the labels in both the Setswana and
Sepedi data sets in Figures 3 and 4 respectively. For this work, we only
explore single label categorization for each article. It remains future work
to look at the multi-label case. As such, there might be some noise in the
labels. Examples from the Sepedi annotated news corpus are shown next:
> _Tsela ya N1 ka Borwa kgauswi le Mantsole Weighbridge ka mo Limpopo ebe e
> tswaletswe lebakanyana ka morago ga kotsi yeo e hlagilego._ Traffic
>
> _Tona ya toka Michael Masutha,ore bahlankedi ba kgoro ya ditirelo tsa
> tshokollo ya bagolegwa bao ba tateditswego dithieeletsong tsa khomisene ya
> go nyakisisa mabarebare a go gogwa ga mmuso ka nko,ba swanetse go hlalosa
> gore ke ka lebaka la eng ba sa swanelwa go fegwa mesomong_ Legal
Figure 3: Setswana news title category distribution Figure 4: Sepedi news
title category distribution
The full dataset is made available online [Marivate and Sefara, 2020b] for
further research use and improvements to the
annotation777https://zenodo.org/record/3668495. As previously discussed, we
used larger corpora to create language vectorizers for downstream NLP tasks.
We discuss this next.
#### 3.1.2. Vectorizers
Before we get into the annotated dataset, we needed to create pre-trained
vectorizers in order to be able to build more classifiers that generalize
better later on. For this reason we collected different corpora for each
language in such as way that we could create Bag of Words, TFIDF, Word2Vec
[Mikolov et al., 2013] and FastText [Bojanowski et al., 2017] vectorizers
(Table 3). We also make these vectorizers available for other researchers to
use.
Table 3: Vectorizer Corpora Sizes in number of lines (number of tokens) Source | Setswana | Sepedi
---|---|---
Wikipedia | 478(_21924_)888https://tn.wikipedia.org/ | 300(_10190_)999https://nso.wikipedia.org/
JW300101010http://opus.nlpl.eu/JW300.php | 874464(_70251_) | 618275(_53004_)
Bible | 3110(_40497_) | 29723
Constitution111111https://www.justice.gov.za/legislation/constitution/pdf.html | 7077(_3940_) | 6564(_3819_)
SADILAR121212https://www.sadilar.org/index.php/en/resources | 33144(_61766_) | 67036(_87838_)
Total | 946264(_152027_) | 721977(_149355_)
### 3.2. News Classification Models
We explore the use of a few classification algorithms to train news
classification models. Specifically we train
* •
Logistic Regression,
* •
Support Vector Classification,
* •
XGBoost, and
* •
MLP Neural Network.
To deal with the challenge of having a small amount of data on short text, we
use data augmentation methods, specifically a word embedding based
augmentation [Wang and Yang, 2015], approach that has been shown to work well
on short text [Marivate and Sefara, 2019]. We use this approach since we are
not able to use other augmentation methods such as synonym based (requires
developed Wordnet Synsets [Kobayashi, 2018]), language models (larger corpora
needed train) and back-translation (not readily available for South African
languages). We develop and present the use of both word and document
embeddings (as an augmentation quality check) inspired by a hierarchical
approach to augmentation [Yu et al., 2019].
## 4\. Experiments and Results
This Section presents the experiments and results. As this is still work in
progress, we present some avenues explored in both training classifiers and
evaluating them for the task of news headline classification for Setswana and
Sepedi.
### 4.1. Experimental Setup
For each classification problem, we perform 5 fold cross validation. For the
bag-of-words and TFIDF vectorizers, we use a maximum token size of 20,000. For
word embeddings and language embeddings we use size 50. All vectorizers were
trained on the large corpora presented earlier.
#### 4.1.1. Baseline Experiments
We run the baseline experiments with the original data using 5-fold cross
validation. We show the performance (in terms of weighted F1 score) in the
Figures 5 & 6\. We show the baseline results as _orig_. For both the Bag-of-
Words (TF) and TFIDF, the MLP performs very well comparatively to the other
methods. In general the TFIDF performs better.
Figure 5: Baseline classification model performance for Setswana news title
categorisation
Figure 6: Baseline classification model performance for Sepedi news title
categorisation
#### 4.1.2. Augmentation
We applied augmentation in different ways. First for Sepedi and Setswana word
embeddings (word2vec), we use word embedding-based augmentation. We augment
each dataset 20 times on the training data while the validation data is left
intact so as to be comparable to the earlier baselines. We show the effect of
augmentation in Figure 5 and 6 (performance labeled with _aug_
The contextual, word2vec based, word augmentation improves the performance of
most of the classifiers. If we now introduce a quality check using doc2vec
(Algorithm 1) we also notice the impact on the performance for Sepedi (Figure
6 _aug qual_ ). We were not able to complete experiments with Setswana for the
contextual augmentation with a quality check, but will continue working to
better under stand the impact of such an algorithm in general. For example, it
remains further work to investigate the effects of different similarity
thresholds for the algorithm on the overall performance, how such an algorithm
works on highly resourced languages vs low resourced languages, how we can
make the algorithm efficient etc.
Input: $s$: a sentence, $run$: maximum number of attempts at augmentation
Output: $\hat{s}$ a sentence with words replaced
1 def _Augment(_s,run_)_:
2 Let $\vv{V}$ be a vocabulary;
3 for _ $i$ in range(_run_) _ :
4 $w_{i}\leftarrow$ randomly select a word from $s$;
5 $\vv{w}\leftarrow$ find similar words of $w_{i}$;
6 $s_{0}\leftarrow$ randomly select a word from $\vv{w}$ given weights as
distance;
7 $\hat{s}\leftarrow$replace $w_{i}$ with similar word $s_{0}$;
8 $\vv{s}\leftarrow Doc2vec(s)$;
9 $\vv{\hat{s}}\leftarrow Doc2vec(\hat{s})$;
10 $similarity$ $\leftarrow$ Cosine Similarity($\vv{s}$, $\vv{\hat{s}}$);
11 if _$similarity$ $>$ $threshold$_ :
12 return($\hat{s}$);
13
14
15
16
17
18
Algorithm 1 Contextual (Word2vec-based) augmentation algorithm with a doc2vec
quality check
Figure 7: Word2Vec feature based performance for news headline classification
Figure 8: Confusion Matrix of News headline classification models
It also interesting to look at how performance of classifiers that were only
trained with word2vec features would fair. Deep neural networks are not used
in this current work and as such we did not use recurrent neural networks, but
we can create sentence features from - word2vec by either using: the mean of
all word vectors in a sentence, the median of all word vectors in a sentence
or the concatenated power means [Rücklé et al., 2018]. We show the performance
of using this approach with the classifiers used for Bag of Words and TFIDF
earlier in Figure 8.
The performance for this approach is slightly worse with the best results for
Sepedi news headline classification being with XGBoost on the augmented data.
We hope to improve this performance using word2vec feature vectors using
recurrent neural networks but currently are of the view that increasing the
corpora sizes and the diversity of corpora for the pre-trained word embeddings
may yield even better results.
Finally, we show the confusion matrix of the best model in Sepedi on a test
set in Figure 8. The classifier categorises _General News_ , _Politics_ and
_Legal_ news headlines best. For the others there there is more error. A
larger news headline dataset is required and classification performance will
also need to be compared to models trained on full news data (with the article
body). For the Setswana classifiers, the confusion matrix shows that the data
skew results in models that mostly can categorise between categorises _General
News_ and _Other_. We need to look at re-sampling techniques to improve this
performance as well as increasing the initial dataset size.
## 5\. Conclusion and Future Work
This work introduced the collection and annotation of Setswana and Sepedi news
headline data. It remains a challenge that in South Africa, 9 of the 11
official languages have little data such as this that is available to
researchers in order to build downstream models that can be used in different
applications. Through this work we hope to provide an example of what may be
possible even when we have a limited annotated dataset. We exploit the
availability of other free text data in Setswana and Sepedi in order to build
pre-trained vectorisers for the languages (which are released as part of this
work) and then train classification models for news categories.
It remains future work to collect more local language news headlines and text
to train more models. We have identified other government news sources that
can be used. On training embedding models with the data we have collected,
further studies are needed to look at how augmentation using the embedding
models improve the quality of augmentation.
## 6\. Bibliographical References
## References
* Bojanowski et al., 2017 Bojanowski, P., Grave, E., Joulin, A., and Mikolov, T. (2017). Enriching word vectors with subword information. Transactions of the Association for Computational Linguistics, 5:135–146.
* Bureau of Circulations, 2019 Bureau of Circulations, A. (2019). Newspaper circulation statistics for the period january–march 2019 (abc q1 2019).
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|
2024-09-04T02:54:58.932182 | 2020-03-04T06:40:14 | 2003.04991 | {
"authors": "Jitin Krishnan, Hemant Purohit and Huzefa Rangwala",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26148",
"submitter": "Jitin Krishnan",
"url": "https://arxiv.org/abs/2003.04991"
} | arxiv-papers | # Unsupervised and Interpretable Domain Adaptation to Rapidly Filter Tweets
for Emergency Services
Jitin Krishnan Department of Computer Science
George Mason University
Fairfax, VA, USA
<EMAIL_ADDRESS>Hemant Purohit Department of Information
Sciences & Technology
George Mason University
Fairfax, VA, USA
<EMAIL_ADDRESS>Huzefa Rangwala Department of Computer Science
George Mason University
Fairfax, VA, USA
<EMAIL_ADDRESS>
###### Abstract
During the onset of a natural or man-made crisis event, public often share
relevant information for emergency services on social web platforms such as
Twitter. However, filtering such relevant data in real-time at scale using
social media mining is challenging due to the short noisy text, sparse
availability of relevant data, and also, practical limitations in collecting
large labeled data during an ongoing event. In this paper, we hypothesize that
unsupervised domain adaptation through multi-task learning can be a useful
framework to leverage data from past crisis events for training efficient
information filtering models during the sudden onset of a new crisis. We
present a novel method to classify relevant social posts during an ongoing
crisis without seeing any new data from this event (fully unsupervised domain
adaptation). Specifically, we construct a customized multi-task architecture
with a multi-domain discriminator for crisis analytics: multi-task domain
adversarial attention network (MT-DAAN). This model consists of dedicated
attention layers for each task to provide model interpretability; critical for
real-word applications. As deep networks struggle with sparse datasets, we
show that this can be improved by sharing a base layer for multi-task learning
and domain adversarial training. The framework is validated with the public
datasets of TREC incident streams that provide labeled Twitter posts (tweets)
with relevant classes (Priority, Factoid, Sentiment) across 10 different
crisis events such as floods and earthquakes. Evaluation of domain adaptation
for crisis events is performed by choosing one target event as the test set
and training on the rest. Our results show that the multi-task model
outperformed its single-task counterpart. For the qualitative evaluation of
interpretability, we show that the attention layer can be used as a guide to
explain the model predictions and empower emergency services for exploring
accountability of the model, by showcasing the words in a tweet that are
deemed important in the classification process. Finally, we show a practical
implication of our work by providing a use-case for the COVID-19 pandemic.
###### Index Terms:
Social Media, Crisis Analytics, Text Classification, Unsupervised Domain
Adaptation, Interpretability
Figure 1: Problem Statement: Interpretably predict labels for tweets collected
during an ongoing crisis using only the past crisis data, given a)
unavailability of labeled data in the ongoing event, and b) need for
interpretability of machine reasoning behind data filtering for emergency
managers.
## I Introduction
During the sudden onset of a crisis situation, social media platforms such as
Twitter provide valuable information to aid crisis response organizations in
gaining real-time situational awareness [1]. Effective analysis of important
information such as affected individuals, infrastructure damage, medical
emergencies, or food and shelter needs can help emergency responders make
time-critical decisions and allocate resources in the most effective manner
[2, 3, 4].
Several machine learning systems have been deployed to help towards this
humanitarian goal of converting real-time social media streams into actionable
knowledge. Classification being the most common task, researchers have
designed models [5, 6, 7, 3] that classify tweets into various crisis-
dependent categories such as priority, affected individuals, type of damage,
type of assistance needed, usefulness of the tweet, etc. Social media streams
contain short, informal, and abbreviated content; with potential linguistic
errors and sometimes contextually ambiguous. These inherently challenging
properties of tweets make their classification task and formulation less
trivial when compared to traditional text classification tasks.
In this paper, we address two practically important and underdeveloped aspects
of current research in social media mining for crisis analytics to classify
relevant social web posts: a) a fully unsupervised domain adaptation, and b)
interpretability of predictions. A fully unsupervised domain adaptation uses
no data from the ongoing crisis to train the model. Nguyen et al., 2016 [5]
showed that their convolutional neural network (CNN) model does not require
feature engineering and performed better than the state-of-the-art methods;
one of their models being completely unsupervised [5]. Similarly, Alam et al.,
2018 [6] designed a CNN architecture with adversarial training on graph
embeddings, but utilizing unlabeled target data. Our goal is to construct an
unsupervised model that does not require any unlabeled target data with the
capability of being interpretable. We specifically address the problem of data
sparsity and limited labels by designing a multi-task classification model
with domain adversarial training; which, to the best of our knowledge, is not
explored in social media mining for crisis analytics. Another crucial
component of our model is interpretability. In prior works, when a top
performing model produces an accuracy of $78\%$, for instance, it is unclear
how trustworthy it is and what features are deemed important in the model’s
decision-making process. An interpretable model like ours can present with a
convincing evidence of which words the classifier deems important when making
a certain prediction, and helps ensure reliability for domain users, e.g.,
emergency managers.
Contributions: a) To address the problems of data sparsity and limited labels,
we construct a customized multi-task learning architecture (MT-DAAN) to filter
tweets for crisis analytics by training four different classification tasks
(c.f. examples in Fig. 3) across ten different crisis events under domain
shift. This multi-task domain adversarial model consists of dedicated
attention layers for each task for interpretability and a domain classifier
branch to promote the model to be domain-agnostic. b) We demonstrate that the
attention layers provide interpretability for the predictions made by the
classifiers; with the goal to aid emergency services in a more meaningful way.
c) We empirically validate the performance of the underlying single-task
attention-based neural network architecture by comparing it to the state-of-
the-art methods, for improving generalizability and interpretability for
domain adaptation in unsupervised tweet classification tasks in general. d)
Additionally, through experiments, we show that deep networks struggle with
small datasets, and that this can be improved by sharing the base layer for
multi-task learning and domain adversarial training.
## II Related Work and Background
### II-A Domain Adaptation
Domain Adaptation in text classification tasks has a long line of fruitful
research [8, 9, 10] that try to minimize the difference between the domains so
that a model trained solely on one domain is generalizable to unseen test data
from a completely different domain. With the introduction of Domain-
Adversarial training of Neural Networks (DANN) [11], many state-of-the-art
models now utilize unlabeled target data to train classifiers that are
indiscriminate toward different domains. The speciality of this architecture
is that it consists of an extra branch, which performs domain classification
using unlabeled data from different domains. Thus, both task and domain
classifiers share some bottom layers but have separate layers towards the top.
A negative gradient (gradient reversal) from the domain classifier branch is
back-propagated to promote the features at the lower layers of the network
incapable of discriminating the domains. Recent works such as Adversarial
Memory Network (AMN) [12] utilizes attention, in addition to DANN, to bring
interpretability to capture the pivots for sentiment classification.
Hierarchical Attention Network (HATN) [13] expands upon AMN by first
extracting pivots and then jointly training networks for both pivots and non-
pivots.
For filtering social web data for crisis analytics, these models do not
suffice and need customized expansions due to the following reasons: a)
Collecting and using large unlabeled target data from the new/ongoing crisis
event may not be practically viable, thus, we aim for a fully unsupervised
modeling. b) Having access to unlabeled data from multiple crisis events can
alleviate the above problem to an extent by using it to train the domain
classifier branch to push the model to be domain independent. c) Due to the
low-resource nature of the dataset, binary classifiers may miss important
lower level features that can be potentially improved by a multi-task model
that shares the lower layers of the network for all the tasks. This is also
evident from our results in Table III and IV, which show that deep models that
perform much better than simple models on Amazon reviews do not significantly
outperform them on TREC tweet dataset for crises.
### II-B Multi-Task Learning
Multi-Task Learning (MTL) solves multiple tasks at the same time with a goal
to improve the overall generalization capability of the model [14]. Within the
context of Deep Learning, MTL is performed by sharing (or constraining) lower
level layers and using dedicated upper level layers for various tasks. A rich
overview of MTL in Deep Neural Networks is presented by Ruder (2017) [15]. MTL
has been a successful strategy over the past few years for many research
explorations such as relationship networks [16] in computer vision and Sluice
networks [17] in natural language processing. Similar problems in domain
adaptation of semantic classification and information retrieval were addressed
by jointly learning to leverage large amounts of cross-task data [18]. In low
resource datasets such as for crises, the chance of overfitting is very high.
Thus, it seems intuitively better for the model to find a shared
representation capturing different tasks and not just one, such that feature
commonalities across tasks can be exploited.
### II-C Attention Mechanism
Attention mechanism [19], originally designed for machine translation
problems, has become one of the most successful and widely used methods in
deep learning that can look at a part of a sentence at a time like humans.
This is particularly useful because of its ability to construct a context
vector by weighing on the entire input sequence unlike previous sequence-to-
sequence models [20] that used only the last hidden state of the encoder
network (typically BiLSTM [21], LSTM [22], or GRU [23]). For example, in a
sentence, the context vector is a dot product of the word activations and
weights associated with each word; thus leading to an improved contextual
memorization, especially for long sentences. Our method incorporates such
attention mechanisms to enhance interpretability of the classifier.
## III Methodology
### III-A Problem Statement: Unsupervised Domain Adaptation for Crisis Tweet
Classification
Using notations in Table I, consider a set $C$ of all crisis events such as
Guatemala Earthquake or Typhoon Yolanda. The task of unsupervised domain
adaptation for crisis analytics is to train a classifier for a specific target
crisis ($c_{t}$) using labeled ($L_{C-c_{t}}$) and unlabeled ($U_{C-c_{t}}$)
data from all other crises; where $C-c_{t}$ denotes the set of all crisis
events minus the target crisis. We assume that no data record from the target
crisis is available for training. Following the traditional domain adaptation
terminology, $X_{s}$ = $L_{C-c_{t}}$ represents the labeled data from the
source domain $S$ and $Y_{s}$ = $y_{C-c_{t}}$ represents the ground truth
labels on which the classifier is trained. And, $X_{t}$ = $L_{c_{t}}$
represents the labeled data from the target domain $T$ and $Y_{t}$ =
$y_{c_{t}}$ represents the ground truth labels; both of which are only used
for testing the classifier. $X_{d}$ = $U_{C-c_{t}}$ represents the unlabeled
data from different domains minus the target. To summarize:
Input: $X_{s}$, $Y_{s}$, $X_{d}$
Output: $Y_{t}^{pred}$ $\leftarrow$ $predict(X_{t})$
Notation | Definition
---|---
$C$ | Set of all crisis events $\\{c_{1},c_{2},...\\}$
$L_{c_{k}}$ | Set of labeled data from the event $c_{k}$
$y_{c_{k}}$ | Set of ground truth labels for $L_{c_{k}}$.
$m$ | Number of tasks (Number of bits in each label)
$U_{c_{k}}$ | Set of unlabeled data from the event $c_{k}$
$T_{x}$ | Number of words in a sentence
$x^{<k>}$ | $k$-th word of a sentence
$\alpha^{<k>}$ | attention from $k$-th word
$a^{<k>}$ | BiLSTM activation from $k$-th word
TABLE I: Notations
### III-B Overview
In the following sections, we describe three models: Single-Task Attention
Network (ST), Single-Task Domain Adversarial Attention Network (ST-DAAN), and
Multi-Task Domain Adversarial Attention Network (MT-DAAN). ST is the model we
adopt from [24] to build the single-task attention based baseline. ST-DAAN is
constructed on top of ST to make the model domain agnostic by performing
adversarial training using gradient reversal. Finally, MT-DAAN is constructed
on top of ST-DAAN with dedicated attention layers for each task on a shared
BiLSTM layer. This is shown in Figure 2.
Figure 2: Fully Unsupervised Domain Adaptation Set-up for Multi-Task Crisis
Tweet Classification.
### III-C Single-Task Attention Network (ST)
We first describe the single-task attention network [24] on top of which we
build our models. This model aligns with our goals of interpretability and
unsupervised domain adaptation. This BiLSTM based model with Attention gives
us three main advantages:
1. 1.
Unlike several existing domain adaptation methods that use unlabeled target
data to train the domain adversarial component via gradient reversal, this
method is a fully unsupervised baseline which also can be customized for
multi-task learning.
2. 2.
The method uses attention mechanism which in turn weighs each word in a
sentence based on its importance. This can be directly utilized for
interpretability.
3. 3.
The method also runs much faster (only a few minutes), i.e. highly useful in
crisis times, as compared to the top performing semi-supervised models such as
HATN [13] (hours).
This model [24] consists of a BiLSTM layer which produces $T_{x}$ activations,
each corresponding to a word in the sentence. These activations are passed
through dense and softmax layers and are combined by dot product to produce
the context vector $\sum_{k=1}^{T_{x}}\alpha^{<k>}a^{<k>}$, where $a^{<k>}$ is
the BiLSTM activation from $k$-th word and $\alpha^{<k>}$ is the attention
weight of $k$-th word. Sentences with words greater than $T_{x}$ are stripped
and those with words lower than $T_{x}$ are padded. This single-task ($m=1$)
attention network is the building block with which rest of the following
models are constructed. The single-task binary cross entropy loss function is
shown below.
$\footnotesize
L_{T}=-\frac{1}{N}\sum_{i=1}^{N}[y_{i}\log\hat{y_{i}}+(1-y_{i})\log(1-\hat{y_{i}})]$
(1)
where $T$ represents the task, $y$ is the true label, and $\hat{y}$ is the
predicted label.
### III-D Single-Task Domain Adversarial Attention Network
(ST-DAAN)
To study the specific contribution of domain adversarial training, we
construct a secondary baseline over the ST architecture by constructing an
additional branch with gradient reversal layer which is represented by the
green blocks in Figure 2. This is a single-task binary classifier with $m=1$.
Domain Adversarial Training of Neural Networks (DANN) [11] was introduced with
a goal to confuse the classifier by back-propagating a negative gradient from
a separate domain classifier branch (right-most branch, as shown in Figure 2).
This makes the classifier agnostic to difference in domains. This back-
propagation is implemented using a gradient reversal layer [11] which does
nothing during the forward pass but pushes a negative gradient
($-\lambda\frac{\partial L_{d}}{\partial\theta_{f}}$) during the backward
(gradient update) pass. $L_{d}$ is the domain classification loss, $\lambda$
is the strength of the reversal, and $f$ represents the lower level layers or
features over which the negative gradient update is performed. In our
architecture, the goal is to make the BiLSTM layer indiscriminate towards
various crisis domains such that the multi-task classification does not depend
on the domain from which the tweet/sentence is coming from. The ST-DAAN loss
function is shown below.
$\footnotesize L_{T}^{\prime}=L_{T}+w_{d}L_{d}$ (2)
where $w_{d}$ is the domain adversarial loss weight. $L_{d}$ represents the
categorical cross entropy loss for multi-domain discriminator shown below.
$\footnotesize
L_{d}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{|C-c_{t}|}[y_{ij}\log\hat{y_{ij}}]$
(3)
where $C-c_{t}$ is the set of all crisis events without the target event.
### III-E Multi-Task Domain Adversarial Attention Network
(MT-DAAN)
Building on top of ST-DAAN, we construct MT-DAAN, which is intended to
classify problems with multiple tasks or labels. For each task, a dedicated
attention layer is allocated from which it predicts binary labels. The BiLSTM
layer remains exactly the same as in the single-task model but multiple
attention blocks are added for each task along with a domain classifier. In
the architecture decision process, we first investigated a multi-label
classifier where all layers are shared with the final softmax layer making
multi-label predictions. In low resource settings, constructing a multi-label
classifier using a shared architecture is challenging for two reasons: a)
jointly balancing positive and negative samples across all classes is not
trivial and potentially challenging to make it extensible when new classes
need to be added, and b) attention layer may not always produce class-specific
insights as the weights are assigned to train for the combination of labels.
On the other hand, in the multi-task architecture with separate attention
layers, it is easy to add more classes. If some classes require more training,
it is trivial to further tune a model specific to that class. More
importantly, $context^{<t_{j}>}$ vector for $j$-th task identifies the
influential words from each sentence for that specific task. The complete
architecture is shown in Figure 2. MT-DAAN loss function is shown below:
$\footnotesize L_{MT-DAAN}=\sum_{k=1}^{m}(w_{k}L_{T_{k}})+w_{d}L_{d}$ (4)
where $m$ is the number of tasks, $w_{k}$ is the loss weight and $L_{T_{k}}$
is the loss term for each task, $w_{d}$ is the domain adversarial loss weight,
and $L_{d}$ is the domain adversarial loss term.
### III-F Model Interpretability
The output ($\alpha$) of the attention layer ($ATT$) of each task, is a
$T_{x}$-dimensional vector; $T_{x}$ being the number of words in the sentence.
The context vector ($\sum_{k=1}^{T_{x}}\alpha^{<k>}a^{<k>}$) is the product of
these attention weights and the $T_{x}$-dimensional activation ($a$) from the
$BiLSTM$ layer. $\alpha$ essentially weighs how much each word in the sentence
contributes to the classification result. Thus, $\alpha$ is the component that
is evaluated for model interpretability.
## IV DATASETS
CRISIS EVENTS | Total Tweets | Vocab | Avg #words | P | F | S | I
---|---|---|---|---|---|---|---
2012 Guatemala Earthquake | 154 | 422 | 18.74 | 104 | 108 | 12 | 15
2013 Typhoon Yolanda | 564 | 1746 | 19.47 | 249 | 46 | 119 | 51
2013 Australia Bushfire | 677 | 2102 | 20.21 | 152 | 213 | 167 | 36
2013 Boston Bombings | 535 | 1755 | 19.30 | 147 | 28 | 234 | 198
2013 Queensland Floods | 713 | 2301 | 19.08 | 293 | 54 | 173 | 215
2014 Chile Earthquake | 311 | 919 | 16.54 | 48 | 26 | 50 | 10
2014 Typhoon Hagupit | 1470 | 2893 | 15.36 | 469 | 375 | 276 | 101
2015 Nepal Earthquake | 2048 | 4026 | 13.77 | 1067 | 377 | 741 | 133
2015 Paris Attacks | 2066 | 4152 | 18.62 | 306 | 183 | 782 | 429
2018 Florida School Shooting | 1118 | 2940 | 21.40 | 329 | 64 | 206 | 70
TABLE II: TREC Dataset Statistics; Showing the number of positive samples for
each of the 4 classes. $P$=Priority, $F$=Factoid, $S$=Sentiment, and
$I$=Irrelevant.
### IV-A TREC Dataset
TREC-IS111http://dcs.gla.ac.uk/~richardm/TREC_IS/ (Text Retrieval Conference -
Incident Streams) is a program that encourages research in information
retrieval from social media posts with the goal to improve the state-of-the-
art social media based crisis analytics solutions. We use the dataset from
2018 track proposal. Statistics of this curated dataset of Twitter downloaded
from TREC is shown in Table II. The original dataset consisted of 15 crisis
events. However, due to very low data, we trimmed the events and tasks such
that there are at least 10 positive samples for each task.
The four tasks used in our experiments are shown below:
1. 1.
Priority: Different priority levels are assigned for each tweet: low, medium,
high, critical. We convert this into a binary classification problem where
$low=0$ and $\\{medium$, $high$, $critical\\}=1$.
2. 2.
Factoid: ‘Factoid’ is a categorical label that represents if a tweet is
stating a fact. Eg: ‘death toll rises ...’
3. 3.
Sentiment: ‘Sentiment’ is a categorical label that represents if a tweet
represents a sentiment. Eg: ’Worried.. Thoughts and prayers.’
4. 4.
Irrelevant: ‘Irrelevant’ is a categorical label for tweets that do not provide
any relevant information.
### IV-B Amazon Reviews Dataset
The standard benchmark
dataset222http://www.cs.jhu.edu/~mdredze/datasets/sentiment/ of Amazon reviews
[25] is widely used for cross-domain sentiment analysis. We chose four
domains: Books (B), Kitchen (K), DVD (D), and Electronics (E). The raw
data333https://github.com/hsqmlzno1/HATN/tree/master/raw_data, a part of
Blitzer’s original raw dataset, used in this work is from HATN [13]. This
dataset consists of $3000$ positive and $3000$ negative samples for each of
the $4$ domains. This dataset is used for two purposes: 1) to validate the
performance of the state-of-the-art methods including the single-task baseline
and 2) to compare and contrast the performance of deep models when trained
with rich versus sparse datasets.
### IV-C COVID-19 Tweet Dataset
For the COVID-19 use-case, we use Twitter posts collected using CitizenHelper
[26] system in March 2020, for the geo-bounding box of the Washington D.C.
Metro region. These tweets were annotated by volunteers of regional Community
Emergency Response Teams (CERTs), with ‘Relevant’ label denoting how relevant
a tweet is for crisis response operations. The label values range on a scale
of $1$-$4$. We convert them into binary classes by considering values $1$ and
$2$ as $-$ve ($0$) class and values $3$ and $4$ as $+$ve ($1$) class. This
dataset consists of $4911$ tweets with $-$ve ($Relevant$=$0$) and $637$ tweets
with $+$ve ($Relevant$=$1$) classes. Following unsupervised domain adaptation
criteria, the filtering models are trained using only the TREC dataset and
evaluated on the COVID-19 tweets. For each independent run of the experiment,
a balanced subset of size $637$ for both classes is selected for testing.
S $\rightarrow$ T | LR | SVM | CNN | BiLSTM | AMN | HATN | ST
---|---|---|---|---|---|---|---
B $\rightarrow$ K | 76.40 | 75.95 | 81.20 | 84.45 | 81.88 | 87.03 | 87.22
B $\rightarrow$ E | 75.53 | 74.05 | 80.44 | 84.61 | 80.55 | 85.75 | 85.51
B $\rightarrow$ D | 81.08 | 81.43 | 82.94 | 83.52 | 85.62 | 87.07 | 86.32
K $\rightarrow$ B | 76.12 | 75.78 | 78.78 | 80.67 | 79.05 | 84.88 | 81.85
K $\rightarrow$ E | 80.37 | 81.20 | 85.17 | 87.37 | 86.68 | 89.00 | 87.09
K $\rightarrow$ D | 73.32 | 74.98 | 76.41 | 78.49 | 79.50 | 84.72 | 81.13
E $\rightarrow$ B | 74.85 | 74.18 | 78.08 | 81.18 | 77.52 | 84.03 | 81.50
E $\rightarrow$ K | 81.85 | 81.85 | 86.59 | 89.00 | 87.83 | 90.08 | 89.21
E $\rightarrow$ D | 75.82 | 75.83 | 78.35 | 78.46 | 85.03 | 84.32 | 81.37
D $\rightarrow$ B | 81.17 | 82.20 | 82.26 | 84.83 | 84.53 | 87.78 | 87.02
D $\rightarrow$ K | 76.42 | 77.58 | 81.09 | 85.21 | 81.67 | 87.47 | 86.37
D $\rightarrow$ E | 72.47 | 73.68 | 79.56 | 83.66 | 80.42 | 86.32 | 85.63
AVG | 77.12 | 77.39 | 80.91 | 83.45 | 82.52 | 86.54 | 85.02
TABLE III: Performance comparison (accuracy) of various models on the standard benchmark dataset of amazon reviews. Methods in blue do not use any unlabeled target data; hence relevant in our context. Each reported score is an average of 10 independent runs of each experiment. Target | LR | SVM | CNN | BiLSTM | ST
---|---|---|---|---|---
Guatemala Earthquake | 60.14 | 56.76 | 60.47 | 65.54 | 59.97
Typhoon Yolanda | 65.39 | 65.97 | 63.05 | 65.49 | 65.53
Australia Bushfire | 65.61 | 63.23 | 62.10 | 60.10 | 62.44
Boston Bombings | 71.47 | 75.45 | 69.72 | 71.43 | 72.08
Queensland Floods | 65.56 | 64.81 | 64.13 | 66.01 | 66.21
Chile Earthquake | 43.09 | 37.94 | 43.37 | 35.45 | 39.23
Typhoon Hagupit | 49.86 | 46.22 | 49.21 | 54.13 | 52.61
Nepal Earthquake | 57.11 | 55.39 | 58.61 | 60.49 | 61.35
Paris Attacks | 71.43 | 71.72 | 72.50 | 72.14 | 71.31
Florida School Shooting | 58.79 | 63.02 | 58.82 | 59.71 | 60.55
AVG | 60.85 | 60.05 | 60.20 | 61.05 | 61.13
TABLE IV: Performance comparison (accuracy) of unsupervised models on TREC-
Priority (tweet) dataset showing that deep models are not strictly superior
than simpler models due to data sparsity. Each reported score is an average of
10 independent runs of each experiment. $Source$ = $Everything$ \- $Target$.
TARGET | Priority | Factoid
---|---|---
| ST | ST-DAAN | MT-DAAN | ST | ST-DAAN | MT-DAAN
| Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1
Guatemala Earthquake | 59.97 | 62.39 | 69.07 | 69.66 | 69.05 | 69.34 | 68.92 | 68.47 | 79.90 | 80.76 | 84.05 | 97.01
Typhoon Yolanda | 65.53 | 65.47 | 66.07 | 63.73 | 67.42 | 67.30 | 80.50 | 84.42 | 82.71 | 85.61 | 84.36 | 86.93
Australia Bushfire | 62.44 | 66.69 | 61.07 | 63.42 | 61.93 | 64.28 | 64.58 | 60.69 | 65.64 | 60.53 | 65.04 | 60.13
Boston Bombings | 72.08 | 74.29 | 72.34 | 73.37 | 73.80 | 74.74 | 83.10 | 88.51 | 81.42 | 85.90 | 85.82 | 88.82
Queensland Floods | 66.21 | 65.94 | 67.19 | 66.97 | 66.74 | 66.46 | 37.56 | 48.90 | 50.46 | 59.82 | 49.52 | 59.21
Chile Earthquake | 39.23 | 40.92 | 38.91 | 42.37 | 41.80 | 46.33 | 30.38 | 33.97 | 39.87 | 48.68 | 45.28 | 54.58
Typhoon Hagupit | 52.61 | 50.59 | 58.97 | 58.94 | 57.50 | 57.52 | 68.98 | 70.79 | 71.42 | 72.44 | 69.49 | 70.08
Nepal Earthquake | 61.35 | 59.44 | 60.18 | 57.80 | 61.65 | 59.49 | 74.04 | 76.08 | 80.72 | 81.00 | 81.04 | 81.02
Paris Attacks | 71.31 | 76.26 | 70.42 | 74.08 | 74.44 | 77.21 | 75.78 | 80.35 | 82.35 | 84.89 | 82.52 | 85.63
Florida School Shooting | 60.55 | 61.75 | 65.47 | 64.07 | 62.51 | 63.24 | 76.73 | 82.67 | 84.55 | 87.51 | 85.80 | 88.15
AVG | 61.13 | 62.37 | 62.97 | 63.44 | 63.68 | 64.59 | 66.06 | 69.49 | 71.90 | 74.71 | 73.29 | 77.16
TARGET | Sentiment | Irrelevant
---|---|---
| ST | ST-DAAN | MT-DAAN | ST | ST-DAAN | MT-DAAN
| Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1
Guatemala Earthquake | 96.96 | 97.03 | 96.45 | 96.68 | 96.76 | 92.73 | 89.36 | 89.03 | 91.22 | 91.06 | 93.11 | 92.73
Typhoon Yolanda | 75.81 | 77.62 | 77.54 | 79.01 | 76.82 | 78.35 | 76.05 | 79.77 | 78.49 | 80.59 | 80.46 | 82.31
Australia Bushfire | 75.95 | 77.58 | 78.80 | 79.12 | 78.54 | 78.92 | 35.42 | 47.164 | 53.78 | 65.11 | 51.76 | 63.36
Boston Bombings | 81.39 | 81.11 | 80.73 | 80.70 | 82.13 | 82.10 | 58.15 | 55.73 | 58.15 | 57.43 | 61.49 | 61.45
Queensland Floods | 81.69 | 80.39 | 81.05 | 81.39 | 81.53 | 81.32 | 65.68 | 65.36 | 67.26 | 65.72 | 67.88 | 67.27
Chile Earthquake | 92.69 | 92.91 | 93.10 | 93.21 | 93.62 | 93.68 | 75.16 | 84.98 | 80.46 | 86.38 | 80.64 | 86.56
Typhoon Hagupit | 84.98 | 85.86 | 85.15 | 86.14 | 85.43 | 86.38 | 63.21 | 75.04 | 71.50 | 78.25 | 70.22 | 77.27
Nepal Earthquake | 67.75 | 68.42 | 70.20 | 70.51 | 69.96 | 70.31 | 31.79 | 42.10 | 36.97 | 47.41 | 41.49 | 52.87
Paris Attacks | 76.01 | 76.63 | 73.65 | 73.98 | 74.47 | 74.60 | 33.91 | 35.25 | 44.52 | 48.32 | 47.17 | 51.32
Florida School Shooting | 68.77 | 71.77 | 67.06 | 70.03 | 68.14 | 71.05 | 32.66 | 40.90 | 44.22 | 55.27 | 47.64 | 58.65
AVG | 80.20 | 80.93 | 80.37 | 81.08 | 80.74 | 80.94 | 56.14 | 61.53 | 62.66 | 67.55 | 64.19 | 69.38
TABLE V: Unsupervised domain adaptation results on TREC dataset showing performance boost for Priority, Factoid, and Irrelevant tasks. However, Sentiment task did not show a significant improvement. See performance evaluation section for details. Each reported score is an average of 10 independent runs of each experiment. TARGET | Relevant
---|---
| ST | ST-DAAN | MT-DAAN
| Acc | F1 | Acc | F1 | Acc | F1
COVID-19 | 73.25 | 77.36 | 74.55 | 77.51 | 77.00 | 78.09
TABLE VI: Unsupervised domain adaptation results for COVID-19 tweets using
only the TREC dataset for training. Each reported score is an average of 10
independent runs of each experiment.
## V Results & Discussion
We first validate the performance of the adopted unsupervised ST model [24] by
comparing it with the following standard neural network architectures and
state-of-the-art models used for domain adaption in text. We use the standard
benchmark dataset of Amazon reviews. Following the traditional domain
adaptation experimental setup, each experiment represented as S $\rightarrow$
T consists of a source domain (S) on which the model is trained and a target
domain (T) on which the model is tested. We use Keras deep learning library
for our implementations; with $T_{x}$=$200$ for Amazon reviews and $30$ for
Tweets. We use Adam optimizer with a dropout of $0.4$, maximum epoch of $50$,
early stopping patience of $3$, batch size of $32$, and validation split of
$0.15$.
1. 1.
Simple Baselines: We construct simple baseline classifiers [27]: Logistic
Regression (LR) and Support Vector Machines (SVM). The input to these models
are constructed by aggregating the $300$-dimensional word embeddings of words
in each review.
2. 2.
CNN: A standard Convolutional Neural Network inspired by Kim, 2014 [28] is
constructed with the following architecture:
$Word\ Embeddings(T_{x},300)\rightarrow Conv1D(128,5)$
$\rightarrow MaxPooling1D(5)$ $\rightarrow Conv1D(128,5)\\\ \rightarrow
MaxPooling1D(5)$ $\rightarrow Conv1D(128,5)\\\ \rightarrow
GlobalMaxPooling1D()\rightarrow Dense(128)$
$\rightarrow Dense(2)\rightarrow y$.
This is combined with dropouts, relu activations, and ending with softmax
activation producing labels for binary classification. State-of-the-art deep
learning methods for existing social media mining approaches of crisis
analytics [6, 5] use a similar architecture.
3. 3.
BiLSTM: This is the bottom-most layer in Figure 2 with the activation
$a^{<T_{x}>}$ passed through the following: $Dense(10)\rightarrow
Dense(2)\rightarrow y$ also including dropouts, relu activation, and ending
with softmax.
4. 4.
AMN and HATN: AMN [12] and HATN [13] are attention-based methods which use
gradient reversal to perform domain adversarial training on the unlabeled data
from source and target domains. HATN is an extension to AMN by adding the
hierarchical component and jointly training pivot and non-pivot networks.
Input to all the models are word
vectors444https://code.google.com/archive/p/word2vec/ [29]. The evaluation on
amazon reviews shows how well the single-task (ST) model perform when compared
to the existing top-performing domain adaptation models on benchmark dataset.
Table III shows accuracy scores on the Amazon cross-domain sentiment analysis
dataset. HATN uses unlabeled target data, gradient reversal, explicit pivot
extraction, and joint training making it a computationally expensive method.
As shown in the experimental evaluation, we use the same Amazon dataset and
GoogleNews word vectors for our experiments. ST, being unsupervised with no
need of unlabeled target data, performed competitively with an overall
accuracy of 85.02%; thus establishing a strong fully unsupervised building
block for us to build upon.
### V-A Crisis Tweets vs Amazon Reviews
Table III and IV show that deep models struggle with small datasets such as
TREC-IS tweets. When ST model outperformed Logistic Regression by $\sim 8\%$
on the Amazon reviews dataset, the difference was only less than $1\%$ with no
statistical significance on the TREC-Priority dataset. Note that we conduct
experiments with various parameter combinations on the deep models when using
tweets. For example, $T_{x}=200$ for amazon reviews and $T_{x}=30$ for tweets
due to the difference in their average word-length. Books domain of Amazon
reviews has $182$ average number of tokens per review with a vocab size of
$105920$. On the other hand, the event with highest number of tweets in the
TREC dataset (Paris Attacks) has only 18.62 average number of tokens per tweet
with a vocab size of $4152$. This difference makes it intuitively challenging
to train deep models with several parameters that may lead the model to
memorize the entire dataset resulting in poor generalization. Multi-task
learning and domain adversarial training try to alleviate this problem by
training the shared BiLSTM layer with much more data from different tasks and
unlabeled data.
### V-B MT-DAAN Performance Evaluation
The primary purpose of the MT-DAAN model is to show that sharing the bottom
layer of the model (i.e., shared representation) for different tasks along
with domain adversarial training can help improve the generalizability of some
of the tasks that are otherwise trained alone in the single-task model. The
experiments for MT-DAAN are setup in the same unsupervised way as for single-
task. No data from the test crisis is used for training. For example, if we
are testing our model for the event ‘Typhoon Yolanda’, no data from this
crisis is used for training. Note that the domain classifier component uses
unlabeled data only from rest of the crisis; making it a fully unsupervised
domain adaptation approach. Performance scores of the four tasks (Priority,
Factoid, Sentiment, and Irrelevant) are shown in Table V. The results show
clear performance improvement for Priority, Factoid, and Irrelevant tasks.
However, Sentiment task did not show significant improvement. We speculate
that this is because other tasks do not generalize the bottom layer enough to
boost the sentiment classification performance. These results show the
usefulness of multi-task learning as well as domain adversarial training where
different tasks in multiple domains help each other when the data is sparse
and labels are limited.
Figure 3: Examples of interpretable results using attention; darker the shade,
higher the attention. Recall that no data from the crisis-event for testing is
used for training the model. Even then, relevant keywords such as ‘police
urging’, ‘death toll rises’, ‘worried’, and ‘thoughts with people’ are
correctly picked up by the attention layers of their respective tasks.
### V-C Word Vectors
We use fastText[30] as our word embeddings for tweets because of its sub-word
usage and the ability to create vectors for arbitrary and out-of-vocabulary
words. Although there exists many alternatives, picking the one that works
well for a specific dataset is not trivial. We conducted experiments using
four choices of word embeddings: fastText [30], GoogleNews [29], Glove [31],
and CrisisNLP [32]. Averaging over 10 crises, we received the following
accuracy scores (in %) respectively for the above word embeddings: {$80.20$,
$81.82$, $81.88$, $80.73$}. Unlike fastText, we fine-tune these pre-trained
vectors to create vectors for out-of-vocabulary words. Vectors for words that
are already in the vocabulary are locked while tuning for consistency in
evaluation. The tweet-based embeddings such as Glove or CrisisNLP did not
significantly outperform other models. Glove vectors are 200-dimensional while
the rest are 300-dimensional which makes the experiment favoring Glove word
vectors. This experiment shows that the problem of finding a strictly superior
word vector model for tweets still remains a challenging task.
Figure 4: Examples of interpretable results using attention for relevancy
prediction of COVID-19 tweets. With $77\%$ accuracy, although the highly
attended words in the ‘Relevant’ tweets provide some intuitive sense of
interpretability, the highlighted words in the ‘Irrelevant’ tweets are
somewhat ambiguous because it is unclear if those words are chosen due to
their specific or generic nature. This shows both the benefits and challenges
of unsupervised and interpretable domain adaptation.
### V-D Interpretability: Attention Visualization
The attention weights used to create the context vector by the dot product
operation with word activations represent the interpretable layer in our
architecture. These weights represent the importance of each word in the
classification process. Some examples are shown in Figures 3 and 4. Stronger
the color intensity stronger the word attention. In the first example, ‘boston
police urging’ is the reason why the tweet is classified as $+$ve priority.
Similarly, ‘death toll rises’ in the Factoid example, ‘worried, prayers’ in
the Sentiment example, and ‘thoughts with people’ in the Irrelevant example
are clear intuitive indicators of +ve predictions. These examples show the
importance of having interpretability as a key criterion in crisis domain
adaptation tasks for social media.
To the best of our knowledge, in social media mining for crisis analytics,
there does not exist a ground truth dataset that highlights the words that
explain the labels for tweets. Using our model as a guide, we hope to build a
robust evaluation dataset as our immediate next step so that the models can be
quantitatively evaluated using robust trust-evaluation methods such as LIME
[33]. It is also crucial to note that binary classification tasks such as
sentiment analysis of Amazon reviews has a clear class divide that produces
intuitive keywords such as ‘good’, ‘excellent’, or ‘great’ for $+$ve reviews
and ‘bad’, ‘poor’, or ‘horrible’ for $-$ve reviews. However, for short texts
such as tweets shown in Figure 4, ‘relevancy’ can depend on the context and it
is unclear which keywords truly represent the examples in the ‘irrelevant’
class.
## VI COVID-19 Use-Case
We show a practical implication of our work by applying it to COVID-19 tweets
described in Section 4.3. Our goal is to interpretably predict if a COVID-19
tweet is relevant or not; a binary classification task. The models are trained
using only the TREC dataset and evaluated on the COVID-19 tweets (a balanced
subset of size $637$ for $+$ve and $-$ve labels). We found that a combination
of ‘Priority’ and ‘Irrelevant’ labels from TREC performs better to predict
COVID-19’s ‘Relevant’ label (this can be trivially verified by constructing
two binary classifiers). We augment all three methods (ST, ST-DAAN, and MT-
DAAN) with an additional condition before label prediction:
$R_{c}=P_{t}\cap\overline{I_{t}}$, which means that a COVID-19 tweet is
‘Relevant’ only if it is predicted both ‘Priority’ = $1$ and ‘Irrelevant’ =
$0$. The scores are reported in Table VI and the attention results are shown
in Figure 4, demonstrating the effectiveness of our proposed method.
## VII Conclusion
We presented a novel approach of unsupervised domain adaptation with multi-
task learning to classify relevant information from Twitter streams for crisis
management, while addressing the problems of data sparsity and limited labels.
We showed that a multi-task learning model that shares the lower layers of the
neural network with dedicated attention layers for each task along with a
domain classifier branch can help improve generalizability and performance of
deep models in the settings of limited data. Furthermore, we showed that using
an attention-based architecture can help in interpreting the classifier’s
predictions by highlighting the important words that justify the predictions.
We also presented an in-depth empirical analysis of the state-of-the-art
models on both benchmark dataset of Amazon reviews and TREC dataset of crisis
events. The application of our generic approach for interpretable and
unsupervised domain adaptation within a multi-task learning framework can
benefit social media mining systems in diverse domains beyond crisis
management.
Reproducibility: Source code and instructions for deployment are available at
- https://github.com/jitinkrishnan/Crisis-Tweet-Multi-Task-DA.
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|
2024-09-04T02:54:58.992673 | 2020-03-11T04:29:14 | 2003.05106 | {
"authors": "Geovane Fedrecheski, Jan M. Rabaey, Laisa C. P. Costa, Pablo C.\n Calcina Ccori, William T. Pereira, Marcelo K. Zuffo",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26149",
"submitter": "Geovane Fedrecheski",
"url": "https://arxiv.org/abs/2003.05106"
} | arxiv-papers | # Self-Sovereign Identity for IoT environments:
A Perspective
Geovane Fedrecheski1, Jan M. Rabaey4, Laisa C. P. Costa1,
Pablo C. Calcina Ccori1, William T. Pereira1, Marcelo K. Zuffo1
{geovane, laisa<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>This research was partially
funded by CAPES. 1Interdisciplinary Center on Interactive Technologies,
Polytechnic School,
University of Sao Paulo, Brazil
4Berkeley Wireless Research Center, Electrical Engineering and Computer
Science Department,
University California, Berkeley, US
###### Abstract
This paper analyses the concept of Self-Sovereign Identity (SSI), an emerging
approach for establishing digital identity, in the context of the Internet of
Things (IoT). We contrast existing approaches for identity on the Internet,
such as cloud-based accounts and digital certificates, with SSI standards such
as Decentralized Identifiers (DIDs) and Verifiable Credentials (VCs). To the
best of our knowledge, this is the first thorough comparison of these
approaches. The benefits and challenges of using DIDs and VCs to identify and
authenticate IoT devices and their respective users are discussed. In the end,
we establish that SSI, with its owner-centric, privacy-aware and
decentrailized approach, provides a viable and attractive option for secure
identification of IoT devices and users.
## I Introduction
The Internet was developed as a research project to interconnect computers
[1]. Protocols like TCP/IP, developed as open standards, allowed computers to
connect in a global scale. However, even after the world-changing impacts the
Internet had on society over the last decades, it has no pervasive, privacy-
preserving, and easy to use mechanism to manage digital identities.
Where human activity is involved, a common abstraction is to use accounts,
i.e. digital records, often containing personally identifiable information
(PII), that are protected by a password and saved on a webserver. Although
this method has been working for several decades, it has many security
drawbacks, such as the use of weak passwords [2] and the potential for privacy
violation. Furthermore, the manual approach of password-protected accounts
makes it unsuitable to machine-to-machine interactions, a common scenario in
the IoT.
More automated solutions can be achieved by using Public Key Certificates
(PKCs) that bind names to public keys [3]. Widespread use of PKC, however, is
limited to organizations, due to the complexity of current methods. For
instance, while websites usually prove their identities to web browsers using
certificates, users do not use certificates in the same way, i.e. to prove
their identity to the website. Moreover, existing standards were not designed
for privacy, as evidenced by the use of real names in known certificate
formats such as PGP [4] and X.509 [5]. To aggravate the situation, the
assignment of unique names often require centralized architectures, which is
inadequate for distributed IoT applications.
A recent development towards online identification of users, organizations,
and devices has been referred to as “Self-Sovereign Identity” (SSI). The basic
premise of SSI is that subjects should own and control their own identity,
instead of having it stored and managed by a third party. This approach brings
several benefits, including enhanced privacy, control, and decentralization.
Two new standards are being proposed to realize SSI, namely, Decentralized
Identifiers (DIDs) and Verifiable Credentials (VCs) [6, 7]. While DIDs focus
on cryptographic identification, VCs provide a means for privacy-aware and
authenticated attribute disclosure.
In this paper we analyze existing approaches to identity in the Internet, such
as X.509, PGP [4], and SSI. We present a detailed comparison focusing on the
data models used to represent identity across different standards. Finally, we
discuss what are the benefits of using SSI in the Internet of Things, and
identify challenges that must be overcome.
## II Self-Sovereign Identity
Self-Sovereign Identity is an approach in which subjects are in full control
of their own digital identities [8]. SSI is analogous to offline identifiers,
which are carried by the owner (within a physical wallet), but contrasts with
current digital identity solutions, which are either based on accounts or
digital certificates, and have privacy and centralization issues.
While initially proposed by members [8] of online communities, a formal
definition of SSI was released recently [9]. Considering an identity to be
composed of an identifier associated with a set of name-value attributes, the
full self-sovereign identity of an individual is the collection of all
identities (i.e. identifiers and attributes) that span a range of
decentralized domains, such that the individual is in full control of these
identities [9]. As digital privacy concerns have been growing in recent years,
interest in SSI has intensified. This led to the definition of a set of
technical specifications to implement SSI, which we describe below.
### II-A Decentralized Identifiers
Digital identifiers so far have been either centralized or non-resolvable. For
example, Uniform Resource Locators (URLs), which can be used to resolve HTML
documents, usually depend on domains names assigned by ICANN111Internet
Corporation for Assigned Names and Numbers - https://www.icann.org/, a
centralized authority. On the other hand, unique, user-generated identifiers
such as UUIDs cannot be used to resolve associated metadata.
To address this, a new specification for Decentralized Identifiers (DIDs) is
being developed with the support of the W3C [6]. The DID has the following
syntax: did:btcr:abcdefgh12345678. The did prefix is mandatory, and colons are
used to separate a method definition and a method-specific id. A method is a
specific set of rules for working with DIDs (the example above uses the
Bitcoin method), and the format of the id depends on that method. An open
directory of different DID methods is available for public access and open for
new submissions222https://w3c-ccg.github.io/did-method-registry/.
Each DID is associated with a DID Document (DDo) that contains the DID itself
along with public keys, service endpoints, and other metadata. The public key
is used to authenticate and encrypt messages, while the endpoint provides a
way to message the entity that controls that DID. To control a specific DID, a
subject just have to own a private key associated with public keys in the DDo.
A common storage mechanism for DDos are Blockchains, from which they can be
resolved using the referred DID. On the other hand, in some cases individuals
may not want to publish their DIDs, e.g. to avoid identity correlation. In
this case, the special peer DID method can be used. Thus, DIDs are unique
identifiers that can be resolved to DID Documents, and they allow the
establishment of an end-to-end secure channel. What DIDs do not provide,
however, is a means for entities to prove claims (attributes) about
themselves.
### II-B Verifiable Credentials
Verifiable Credentials (VCs) is a W3C recommendation for portable and provable
claims about a subject. For instance, a person may claim to have the name
Alice, and a device may claim to be of type Camera. The relationship among
DIDs and VCs is shown in Figure 1. All VCs refer to the DID of the subject to
which they have been assigned (e.g. an IoT device). VCs also contain the DID
of its issuer along with a cryptographic proof. This allows a subject to
present a VC to a verifier, which can then resolve the DDo of the issuer (and
therefore its public key) from a public ledger, e.g. a Blockchain, and check
the authenticity of the VC. Figure 2 shows a use case where a user issues a VC
to a device.
A major incentive for SSI is privacy, therefore VCs are expected to be private
and stored in a personal wallet, to be shared only when necessary. To further
improve privacy, the VC specification supports zero-knowledge proofs, i.e. a
cryptographic technique “where an entity can prove to another entity that they
know a certain value without disclosing the actual value” [7].
Figure 1: A DID is the link between a DDo and a set of VCs, much like a
primary key can link different tables in a database. This allows a subject
associated with a DID to prove its identity. Figure 2: An owner-centric
scenario using SSI. Each subject generates its own DDo, while the VC is issued
by the device owner.
### II-C Decentralization, privacy, and layered authentication
Public key cryptography can be used to derive a shared secret over an insecure
channel [10]. However, a known problem is how to trust the origin of the
public key. To solve this, a signed certificate that binds a name to a public
key was proposed [3]. Two common standards for digital certificates are X.509
[5, 11] and Pretty Good Privacy (PGP) [4]. Although they differ in details,
both follow the original definition in which names are tied to public keys and
signed by a third party [3].
A crucial challenge faced by certificate-based solutions was ensuring the
uniqueness of the names. The most common solution to this was to rely on
centralized architectures. For example, the name on the subject field in X.509
must be enforced by a global authority, and the PGP id uses the name of a
person plus her email address, which ultimately depends on DNS, which is
centralized as well.
More recently, the emergence of Blockchain technology allows decentralized
consensus for choosing unique names. One problem, however, is that solutions
based on certificates put sensitive information in the identifier, which
compromises the privacy of certificate holders, and therefore might not be
suitable for storage in public, immutable ledgers.
An approach to solve this is to limit the exposure of PII on the ledger by
only writing anonymous information to it, e.g., public keys. In particular,
this approach enables public key storage and lookup, which can be used to
create a confidential and non-repudiable channel. Higher-level abstractions
can then be used to implement authentication, since the attributes necessary
to authenticate users are usually application-specific.
This is the solution that results from combining the DID and VC
specifications. Containing only pseudonymous information, such as public keys
and service endpoints, DID Documents can be used to establish a
cryptographically secure channel between two entities. After the confidential
channel is created, the entities can exchange VCs, according to the levels of
trust necessary to each application. In other words, while DIDs are lower-
level and pseudonymous, VCs are application-specific and can be used to
authenticate attributes such as name or device type. Finally, it is worth
noting that as each DID is usually a high-entropy random string, name
collisions actually stop being a concern.
## III Data models for digital identity
This section provides analysis and comparisons of existing data models for
digital identity. We start by discussing the limitations of password-based
accounts, and then proceed to compare data models based on public key
cryptography.
### III-A Accounts
The most basic method to identify subjects in computer systems is the account:
a digital record, usually composed of at least a user name and a password,
that identifies a user. Accounts are commonly stored in a server controlled by
the service provider. For example, popular IoT vendors require that a device
owner have a cloud-based account, so that she can use this virtual identity to
configure her devices.
While accounts have been used for decades in a variety of systems, they are
among the most primitive solutions for digital identities. Among the problems
related to account-based authentication are privacy and the use of passwords.
With respect to privacy, issues arise because the user is forced to store
plaintext PII in a third-party system. Regarding passwords, the literature
indicates common problems such as password reuse and difficulty to enforce
strong passwords, and points that the most widespread solution is the use of
“recommendations” [2], which depends on human factors and are difficult to
enforce.
TABLE I: Comparison of standardized data models for digital identity. | PGP | X.509 | Self-Sovereign Identity
---|---|---|---
| PGP Key | | Public Key Certificate
---
(PKC)
| Attribute Certificate
---
(AC)
| DID Document
---
(DDo)
| Verifiable Credential
---
(VC)
Goal | | Prove control of public
---
keys and identifier
(plus optional attributes)
Publish public keys
| Prove control of public
---
keys and identifier
(plus optional attributes)
Publish public keys
| Prove possession of
---
attributes
| Prove control of identifier
---
Publish public keys and
service endpoints
| Prove possession of
---
attributes
Identifier | Name and Email | Qualified Name | Same as PKC | Method-specific DID | Same as DDo
| Uniqueness
---
of identifier
| Global
---
authority (DNS)
Global authority (CA) | Same as PKC | | Ledger consensus /
---
Random number gen.
Same as DDo
Public Key(s) | 1 primary, N subkeys | 1 | n/a (points to PKC) | N | n/a (points to DDo)
Attribute(s) | Attributes field | Extensions field | Attributes field | - | subjectCredential field
Endorsement | Signature of many peers | Signature of a CA | Signature of a CA | | Self-signed (optional)
---
Indirect through VC
Signature of an Issuer
| Service
---
endpoints
- | - | n/a | Yes | n/a
| Semantic
---
schemas
- | - | - | Yes | Yes
### III-B Models based on public key cryptography
Pretty Good Privacy (PGP) [4] was created to allow individuals to prove a
binding between a public key and an identifier, the latter being composed by a
real name and an email address. This binding, along with optional attributes
and signatures, is stored in a document called a PGP Key. Conceived as a
distributed solution, individuals in the PGP scheme can sign the keys of other
individuals, so as to give an endorsement that they are who they say they are,
i.e. they are not impersonating someone or using a fake id. This scheme of
peer signatures is often referred to as the Web of Trust.
X.509 Certificates, created by the X.500 working group, defines a format for
Public Key Certificates (PKC) that binds public keys to qualified names [5].
PKCs are widely used in the Internet to authenticate domain names and protect
communications. Although technically nothing prevents peer-to-peer signature
of X.509 certificates, the vast majority of its usage is under centralized
architectures, in which a trusted authority signs the certificate to make it
trustworthy. Finally, in certain cases it is useful to have a separate
document that, instead of having public key, contains only a name associated
with signed attributes. To meet this demand, X.509 proposed a new standard
called Attribute Certificate (AC), which contains no public key, but links to
a PKC through its subject field [11].
Finally, as previously mentioned, Self-Sovereign Identity is a novel approach
that uses Decentralized Identifiers [6] and Verifiable Credentials [7] to
prove possession of identifiers and attributes, respectively.
### III-C High-level comparison
The following paragraphs compares models used by the PGP, X.509, and SSI
standards, according to Table I.
#### Goal
Both PGP Keys and PKCs are used to publish and prove control of public keys
that are tied to identifiers. Also, in these approaches, attributes can be
provided either in the same document as the public keys (PGP Key and PKC), or,
in the case of X.509, in a separate document (AC). On the other hand,
documents in the SSI paradigm have decoupled goals: DDos are be used to prove
control of an identifier and to provide a means for establishing a secure
communication; and VCs are used to prove possession of attributes.
#### Identifier (and uniqueness)
While PGP and X.509 use names and other identifiers that depend on centralized
entities, in SSI the identifiers are completely decentralized and can be auto-
generated, for example by using strong random number generators. Not only this
enables easy global uniqueness, but the pseudonymous characteristic of DIDs
also enhances privacy, when compared to previous approaches based on real
names or email addresses. Pseudonymous identifiers are also more suited for
IoT, since devices do not have names or email addresses by default.
#### Public Key(s)
PKCs are limited to only one public key, while PGP Keys and DDos can have
many. PGP still differs from DDos as the former uses a primary key that is
tied to an identifier and allows more subkeys to be included, while the latter
support multiple public keys without assumptions other than the key type,
which usually encodes its purpose, e.g. sign or encrypt.
#### Attribute(s)
Both PGP Keys and X.509 certificates support arbitrary attributes, either via
PKC extensions or dedicated ACs. In self-sovereign identity, a DDo does not
support attributes in order to stay anonymous. Instead, all PII is handled
only by VCs, which are private by default.
#### Endorsement(s)
PGP Keys can be signed by one or more peers, but X.509 certificates and VCs
can only be signed by a single issuer. DDos are not signed by external
entities, and may be self-signed. When a DDo is written to a ledger, however,
the transaction will be signed, which can be used to attest the validity of
the DDo. Another way of proving endorsement over a DID is to check the
signature of a VC associated with that DID. If the VC is signed by a trusted
issuer, the DID can be trusted. Furthermore, with respect to who can sign the
endorsements, technically it can be anyone, but there are philosophical
differences. X.509, for example, was devised to work within a centralized
architecture, where only trusted authorities can sign certificates. On the
other end of the spectrum, PGP expects peer-to-peer signatures, which
ultimately creates a Web of Trust. Finally, VCs does not make strong
assumptions on the network structure, although decentralized approaches,
especially the ones based on Blockchain, may be favorable.
#### Service endpoints
a novelty introduced by DDos is the association of a built-in mechanism to
reach the owner of a public key. This facilitates the establishment of secure
interactions between peers, from web to IoT environments.
#### Semantic schemas
only SSI-based data models allow extensibility through semantic annotations
over JSON documents. The main reason for this is that these technologies only
became popular after X.509 and PGP were developed.
### III-D Public key distribution
TABLE II: Comparison of data models for key distribution. | Raw Pub Key | PKC | DDo
---|---|---|---
| Associates key material
---
to metadata
| X | X
Privacy: no PII disclosed | X | | X
| Key rotation does not
---
requires re-signing
n/a | | X
Serialization formats | | Binary
---
Base64
| DER
---
PEM
| JSON-LD
---
JWT
Semantic schemas | | | X
| Decentralized: user
---
generates the artifact
X | | X
| Decentralized: user
---
carries the artifact
X | X | X
Service endpoint | | | X
TABLE III: Comparison of data models for attributes. | PKC | AC | VC
---|---|---|---
| Signed attributes
---
about a subject
X | X | X
| Key rotation does not
---
requires re-signing
| X | X
| Identifier differs from
---
key material
X | X | X
| Attributes decoupled
---
from key material
| X | X
Selective disclosure | | | X
Zero-knowledge proofs | | | X
Delegation | | X |
Revocation | X | X | X
Serialization formats | | DER
---
PEM
| DER
---
PEM
| JSON-LD
---
JWT
Semantic schemas | | | X
| Decentralized: user
---
carries the artifact
X | X | X
| Decentralized: Verifier
---
decoupled from Issuer
| | X
An important aspect in the design of systems based on asymmetric encryption is
the data model used to support key distribution. In the following, we compare
three approaches, as shown in Table II: Raw Public Key, Public Key
Certificates, and DID Document.
#### Raw public key
this is the simplest approach, and consists in having a public key shared as a
raw array of bytes, often encoded in some ascii-compatible format, such as
base64. Although this approach is decentralized and discloses no personal
information, it does not allow associated metadata.
#### Public Key Certificate
as previously discussed, PKCs bind a name and other attributes to a public
key, which allows subjects prove their identity. Created before privacy was a
major concern, X.509 PKCs always carry PII in the main identifier, and may
carry PII in other attributes. Finally, other drawbacks of PKCs include the
imposition of specialized serialization formats (DER and PEM), tight coupling
of keys and data (which makes key rotation more difficult), and a centralized
architecture, i.e. the artifact is not self-generated.
#### DID Document
DDos associate public keys to pseudonymous metadata, while also allowing key
rotation without re-signing any associated metadata. The latter is possible
because all signed metadata actually only lives in associated VCs. An
important difference to highlight is that DDos are not signed by third
parties, thus they cannot authenticate the origin of a public key. If this is
necessary, DDos can be composed with VCs to increase security. DDos supports
JSON-based serialization formats, which are available in most programming
languages and platforms, and can benefit from publicly available semantic
schemas. As each user auto-generates their own DIDs and DDos, the management
of the identifier is decentralized. Finally, service endpoints in DDos provide
a novel way for peers to securely establish secure channels.
### III-E Attribute distribution
Four out of the five previously described formats can be used to prove control
over attributes: PGP Keys, Public Key Certificates, Attribute Certificates,
and Verifiable Credentials. Since PGP Keys are less widely used, we only
compare the latter, as shown in Table III.
#### Public Key Certificates
the encoding of attributes in PKCs leverages the X.509 PKC extension field.
Although the reuse of an existing format may seen advantageous in terms of
compatibility, the whole certificate must be re-signed when a key is rotated,
or when selective disclosure of attributes is necessary. An important drawback
not mentioned so far is that it is impossible to disclose only a subset of the
attributes in a PKC, without contacting the issuer for a new signature.
#### Attribute Certificates
differing from PKCs, ACs contain a name and a list of attributes, but no
public key, which simplifies key rotation. Finally, while ACs support
delegation, in general they have the same drawbacks as PKCs.
#### Verifiable Credentials
similar to an AC, a VC does not contain public keys, as it focus on binding
identifiers to attributes. Among the novelties in the VC standard is the
support for selective disclosure without contacting the issuer, which is
realized using zero-knowledge cryptography. VCs also leverage JSON, a
serialization format that is both human readable and lightweight to parse. VCs
and can be further specialized into two formats: JSON Linked Data (JSON-
LD)333https://json-ld.org/, a format to serialize linked data; and JSON Web
Token (JWT), a widely used format to express security
claims444https://jwt.io/.
## IV Benefits and Challenges of SSI for IoT
As the IoT continues to evolve, new paradigms that allow spontaneous machine-
to-machine interactions started to appear [12, 13]. Necessarily decentralized,
the future IoT will require users to be the root of trust of their devices,
leading to an owner-centric IoT. As privacy concerns raise in importance,
solutions that minimize personal data sharing become paramount. Full
realization of these and other features will require novel, open, and secure
standards for identity in the IoT. The next paragraphs discuss aspects of
self-sovereign identity that are likely to improve decentralized IoT security,
while also pointing the factors that will require innovation to bring SSI to
IoT, such as support constrained devices.
### IV-A Benefits
The benefits of SSI for IoT, such as privacy and decentralization, are
discussed below.
#### Owner-Centric
The user can be the root of trust of her devices. Once a user is the owner and
controller of her identity, it is straightforward to create a network of
devices that belong to her, for example by provisioning an “owner=Alice”
credential to each device. One interesting consequence of this is that no
third party is needed to enforce security and administration of devices, as
the user herself will be able to do it. Note that in this approach devices can
have their own identity as well, and may only use the owner attribute to
facilitate the creation of trust relationships, i.e. devices that share the
same owner can automatically trust each other.
#### Privacy-preserving
Personal information is protected. By having the identity of owners and things
stored locally, sensitive data that would otherwise be stored in a service
provider will now live closer to the owner (usually in a digital wallet).
While the user can choose to backup his data for various reasons, she will be
able to do so in an encrypted way, as only she will possess the decryption
keys. Users and devices will also get to choose with whom they share their
credentials, and even be able to do so employing selective disclosure and
zero-knowledge proofs techniques, further improving privacy.
#### Decentralized
No single-point of failure. While identity providers may have been a
convenient way to authenticate users and devices so far, it is not clear what
happens when a provider stops providing, e.g. when it goes out of business. In
the self-sovereign approach, the user decides when her identity starts or
stops being valid, and she will have similar controls over her devices.
Finally, data breaches, information sharing without user consent, and other
issues are minimized when identities are not stored in a high-value data silo
that acts as a honeypot for hackers.
#### End-to-end security
Communications between two endpoints are secure. By exchanging DID Documents
and applying asymmetric cryptography, IoT devices can mutually authenticate,
derive short-lived symmetric keys, send encrypted messages, and enforce non-
repudiation. This approach can also be implemented in a transport-agnostic
way, enabling secure communication even among different protocols.
#### Layered authentication
Separates cryptographic and application-specific authentication. In the
former, two devices prove to each other that they are in possession of
specific public keys, while in the latter the devices prove different
attributes about themselves. This approach allows endpoints to always be
cryptographically protected, and leaves higher-level trust requirements to be
handled at the application layer.
#### Standardized and open approach
Fosters interoperability and robustness. Since both DIDs and VCs are being
developed as open W3C specifications, companies and researchers are free to
build solutions that are interoperable and rely on well-tested data models.
#### JSON-based encoding
Using JSON enables more applications to handle data extracted from DID
Documents and credentials, even if not originally designed to work with SSI.
### IV-B Challenges
We now discuss some challenges to apply SSI in IoT environments.
#### Constrained devices
Fully adopting SSI means that devices need to be able to run asymmetric
cryptography and cope with communication overhead of transmitting metadata,
such as DID Documents and Verifiable Credentials.
#### Asymmetric Cryptography
SSI demands execution of encryption algorithms based on asymmetric keys, which
can be challenging in devices with limited processing and energy resources.
While authors points that constrained processors such as the 32-bits Cortex M0
are well equipped to execute Elliptic Curve Cryptography (ECC) [14], the
number of operations still must be controlled to avoid battery draining. A
common tactic is to use long lived session keys that are less frequently
updated, e.g. once a day.
#### Communication overhead
Depending on the communication protocol, the size of DDos and VCs may impose a
barrier. For example, low energy protocols such as LoRA and BLE have maximum
packet sizes of 222 and 244 bytes, respectively, while DDos and VCs easily
achieve 500 bytes or more. Therefore, strategies such as compression,
fragmentation, and infrequent document transmission, will be necessary. In
extreme cases, SSI may not be possible at all, which will require proxy
approaches [15].
#### DID Resolution
Higlhy constrained devices may not be able to connect to the Internet to
download DID Documents at all. A possible solution is to create a local cache
of known DIDs, either managed by the device itself or by its gateway. On the
other hand, if both devices use peer DIDs, they can simply exchange their DIDs
directly, which shifts the problem to securely delivering the DIDs in the
first place.
#### Software availability
The SSI ecosystem is new and there is limited software available for embedded
devices. Given the foundational importance of secure cryptographic algorithms
and protocols, applications based on SSI should rely on existing libraries
that encapsulate complexity and are well tested, which reduces the chances for
vulnerabilities. Although reference implementations exist [16], they are
focused on cloud and mobile use cases. To fully incorporate SSI into IoT,
portable and lightweight libraries tailored for constrained devices must be
created and made widely available.
## V Conclusion and perspective
As the primary motivation for the development of the Internet was to remotely
connect computers, the problem of secure identification of users and devices
was left aside. While identity solutions such as accounts and certificates
were eventually developed, they feature critical issues such as weak
passwords, lack of privacy, and centralization. As it is common for systems to
mature over time, as good (and bad) practices are learned, we argue that the
Self-Sovereign Identity approach represents an important step forward in the
area of digital identity. Particularly in the context of the IoT, this paper
showed how SSI can (1) empower owners to have full control over both their
identities and their devices, (2) improve privacy by decoupling pseudonymous
and sensitive identity records, and (3) allow decentralized identity
management by reducing the dependency on third parties. As for the next steps,
the realization of SSI in the IoT will demand implementations that are
optimized for constrained devices, both for cryptographic operations and low-
power communication. Furthermore, wide adoption of SSI will depend on the
availability of open software libraries to manipulate DIDs and VCs in IoT
devices. To conclude we argue that, if adopted, SSI may significantly benefit
security and privacy of IoT applications, and potentially enable new use
cases, such as those that involve cross-owner decentralized interactions.
## References
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|
2024-09-04T02:54:59.007056 | 2020-03-11T04:51:07 | 2003.05110 | {
"authors": "Nicole F. Allard, John F. Kielkopf, Siyi Xu, Gr\\'egoire Guillon, Bilel\n Mehnen, Roberto Linguerri, Muneerah Mogren Al Mogren, Majdi Hochlaf, Ivan\n Hubeny",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26150",
"submitter": "John Kielkopf",
"url": "https://arxiv.org/abs/2003.05110"
} | arxiv-papers | # H–He collision-induced satellite in the Lyman-$\alpha$ profile of DBA white
dwarf stars
Nicole F. Allard 1,2, John F. Kielkopf 3, Siyi Xu 4, Grégoire Guillon 5, Bilel
Mehnen 6, Roberto Linguerri 6, Muneerah Mogren Al Mogren 7, Majdi Hochlaf 6,
Ivan Hubeny 8
1GEPI, Observatoire de Paris, Université PSL, UMR 8111, CNRS, 61, Avenue de
l’Observatoire, F-75014 Paris, France
2Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis
Boulevard Arago, PARIS, France
3Department of Physics and Astronomy, University of Louisville, Louisville,
Kentucky 40292 USA
4Gemini Observatory, 670 N. Aóhoku Place, Hilo, HI 96720 HI, USA
5Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR6303, CNRS, Université
de Bourgogne Franche Comté, 21078 Dijon Cedex, France
6Université Gustave Eiffel, COSYS/LISIS, 5 Bd Descartes 77454, Champs sur
Marne, France
7Chemistry Department, Faculty of Science, King Saud University, PO Box 2455,
Riyadh 11451, Kingdom of Saudi Arabia.
8Department of Astronomy, University of Arizona, 933 N Cherry Ave, Tucson, AZ
85719 USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
The spectra of helium-dominated white dwarf stars with hydrogen in their
atmosphere present a distinctive broad feature centered around 1160 Å in the
blue wing of the Lyman-$\alpha$ line. It is extremely apparent in WD 1425+540
recently observed with HST COS. With new theoretical line profiles based on ab
initio atomic interaction potentials we show that this feature is a signature
of a collision-induced satellite due to an asymptotically forbidden
transition. This quasi-molecular spectral satellite is crucial to
understanding the asymmetrical shape of Lyman-$\alpha$ seen in this and other
white dwarf spectra. Our previous work predicting this absorption feature was
limited by molecular potentials that were not adequate to follow the atomic
interactions with spectroscopic precision to the asymptotic limit of large
separation. A new set of potential energy curves and electronic dipole
transition moments for the lowest electronic states of the H–He system were
developed to account accurately for the behaviour of the atomic interactions
at all distances, from the chemical regime within 1 Å out to where the
radiating H atoms are not significantly perturbed by their neighbors. We use a
general unified theory of collision-broadened atomic spectral lines to
describe a rigorous treatment of hydrogen Lyman-$\alpha$ with these potentials
and present a new study of its broadening by radiative collisions of hydrogen
and neutral helium. These results enable ab initio modeling of radiative
transport in DBA white dwarf atmospheres.
###### keywords:
(stars:) white dwarfs < Stars - stars: atmospheres < Stars - atomic data <
Physical Data and Processes - atomic processes < Physical Data and Processes -
line: profiles < Physical Data and Processes - molecular data < Physical Data
and Processes
††pubyear: 2019††pagerange: H–He collision-induced satellite in the
Lyman-$\alpha$ profile of DBA white dwarf stars–7
## 1 Introduction
Theoretical studies of the effects of neutral atom collisions on atomic
spectral lines have often been hindered by our ignorance of the atomic
potentials. Even for systems as simple as H-H or H-He, the interactions and
the electric transition moments are quite difficult to compute with the
accuracy which is needed for evaluating a complete line profile. The
fundamental theory of calculating the spectral line profile (Allard et al.,
1999) requires knowledge of molecular potentials with high accuracy because
the shape and strength of the line profile are very sensitive to the details
of the molecular potential curves describing the atom-atom collisions. In
Allard & Christova (2009) we made an exhaustive study of the red wing of
Lyman-$\alpha$ line perturbed by H–He collisions, where we used the potentials
and electric dipole transition moments of Theodorakopoulos et al. (1984) and
Theodorakopoulos et al. (1987). We considered the high He densities met in
cool DZ white dwarfs and examined the range of validity of the one-perturber
approximation widely used to calculate the line wings. We have shown there
that the extension of the red wing of the Lyman-$\alpha$ line seen in DZ white
dwarf spectra depends strongly on the stellar temperature, while it is not
dependent on the helium density. We also predicted a blue satellite which only
very recently has been observed in Hubble Space Telescope Cosmic Origins
Spectrograph (HST COS) observations (Xu et al., 2017). The importance of a
correct determination of the blue wing of Lyman-$\alpha$ line to interpret the
asymmetrical shape of the Lyman-$\alpha$ line observed with COS is presented
in Sect. 2. An accurate prediction of the satellite and consequently the full
Lyman-$\alpha$ profile requires exacting new ab initio calculations to obtain
the ground and first excited potential energy curves and the corresponding
electric dipole transition moments for the H–He system. The new molecular data
in Sect. 3 corroborate the prediction of a line satellite in the
Lyman-$\alpha$ profile (Allard & Christova, 2009) that is described in Sect.
4. In Allard et al. (1999) we previously derived a classical path expression
for a pressure-broadened atomic spectral line shape that includes the effects
of a radiative electric dipole transition moment that is dependent on the
position of the radiating atom and its dynamic neighbors. Such a comprehensive
unified approach employing the precise molecular data is fundamentally
necessary to obtain an accurate absorption line profile that is valid over the
full breadth of spectral line for the range of densities and temperatures
found in stellar atmospheres.
Figure 1: COS observation of WD 1425+540. The broad distinctive collision-
induced satellite in the blue wing of the Lyman-$\alpha$ line about 1160 Å is
clearly visible (Xu et al., 2017). The strong emission at the center of
Lyman-$\alpha$ is from Earth’s geocoronal hydrogen above the HST orbit.
## 2 COS observation of WD 1425+540
WD 1425+540 (T=14,490 K, log g=7.95) is the prototype of DBA white dwarfs and
it is a helium-dominated white dwarf that also has a large amount of hydrogen
in its atmosphere (Bergeron et al., 2011). It was observed with HST COS under
program 13453, and the details of observation and data reduction strategy were
reported by Xu et al. (2017). Here, we focus on the spectrum of segment B of
the G130M grating, which covers 1130-1270 Å, as shown in Fig. 1. As described
in Xu et al. (2017), there are two unusual features of the Lyman-$\alpha$
profile in WD 1425+540. First, the line profile is very asymmetric exhibiting
an extend blue wing with the satellite feature as noted. Second, previous
white dwarf spectral models cannot reproduce the strength of Lyman-$\alpha$
and Balmer-$\alpha$ simultaneously. The derived hydrogen abundance is more
than a factor of 10 higher from the Lyman-$\alpha$ measurement than from
Balmer-$\alpha$. While WD 1425+540 is the most extreme case so far, these
peculiarities have been observed in other DBA white dwarfs as well, e.g. Jura
et al. (2012).
The asymmetry also could not be produced by white dwarf models of Xu et al.
(2017) because the opacity data used for the Lyman-$\alpha$ profile did not
take into account the quasi-molecular line satellite predicted in Allard &
Christova (2009). Once this feature is included, the observed asymmetry is
reproduced (Gänsicke et al., 2018). The need to have both accurate data for
Lyman-$\alpha$ and for Balmer-$\alpha$ is essential to determine the hydrogen
abundance correctly. The goal of this paper is to develop the foundation of
the atomic and molecular physics needed to compute a complete profile without
making ad hoc assumptions. We emphasize the importance of accurate potentials
and electric dipole transition moment data for this purpose, and here we
provide that data for Lyman-$\alpha$. With new potentials of H-He we also
compute a model DBA white dwarf spectrum that demonstrates their validity.
## 3 H–He diatomic potentials
### 3.1 Methodology and benchmarks
The lowest electronic excited states of hydrogen and helium are at unusually
high energies for neutral atoms (> 10 eV) with respect to their ground states,
and close to the corresponding ionization thresholds. Hydrogen with $n$= 2 or
greater is a Rydberg atom in this sense (Gallagher, 1994).
The electronic excited states of H–He diatomic system of interest in the
present work correlate adiabatically to those of these atoms. Therefore, for
the correct description of the electronic states of the H–He diatomic system
consistent with its isolated atomic fragments one needs the inclusion of
diffuse functions that can flexibly represent the states. In addition to this,
the computation of the possible interactions that may occur between these
electronic states and the subsequent mixing of their wavefunctions that
results in an apparent change in electric dipole transition moments, require
post Hartree-Fock multi-configurational approaches. More specifically, we used
the Complete Active Space Self Consistent Field (CASSCF) (Knowles & Werner,
1985; Werner & Knowles, 1985) followed by the internally contracted Multi-
Reference Configuration Interaction (MRCI) (Knowles & Werner, 1988; Werner &
Knowles, 1988; Shamasundar et al., 2011) methods as implemented in the MOLPRO
2015 package (Werner et al., 2015). In MRCI, the complete CASSCF wave
functions are used as a reference. Furthermore, the Davidson correction
(MRCI+Q) (Langhoff & Davidson, 1974) has been applied to the resulting
energies to account for the lack of size-consistency of the MRCI method. These
computations were performed in the $C_{2v}$ point group, where the $B_{1}$ and
$B_{2}$ representations were treated on equal footing.
Benchmarks on valence-Rydberg electronic states of other molecular systems
(Spelsberg & Meyer, 2001; Ndome et al., 2008; Hochlaf et al., 2010) showed the
need to use a CASSCF active space larger than the full-valence space. The
atomic basis set for the H and He atoms had to be optimized as well. Thus, we
performed a series of benchmark computations at different levels of accuracy
to find the appropriate states for convergence.
Firstly, at the lowest level of accuracy, we adopted a small active space of 3
electrons in 7 molecular orbitals in conjunction with the aug-cc-pV5Z
(Dunning, 1989; Kendall et al., 1992) basis set. With this approach, we found
inconsistencies in the calculated energies, especially in the asymptotic
region. Indeed, with this simplest choice there is a large energy gap of $\sim
0.45$ eV between the two equivalent dissociation limits H($2p\,^{2}P$) +
He($1s^{2}\,{}^{1}S$) and H($2s\,^{2}S$) + He($1s^{2}\,{}^{1}S$). Obviously,
this gap is unphysical since these two asymptotes should be strictly
degenerate because the two H ($n=2$) states have the same energy apart from
Lamb shift and negligibly small fine and hyperfine structure. Moreover, we
found a spurious second potential well ($D_{e}$ $\sim$ 660 cm-1) in the
$C\,\Sigma$ state of H–He at large internuclear separations (for
$R_{\mathrm{H-He}}$ $\sim$ 4.2 Å). Thus, at this level of accuracy, a rather
poor chemical description of the H–He molecule is obtained in spite of the
relatively large size of the MRCI computations with $\sim$ 4.3 x 104
uncontracted configuration state functions (CSFs) per $C_{2v}$ symmetry. This
may be linked to some missing correlation energy in the MRCI wavefunctions
that can be recovered by means of larger active spaces in the reference CASSCF
vector and by adopting more diffuse atomic basis sets.
Secondly, we tried an enlarged CASSCF active space of 3 electrons in 14
molecular orbitals in conjunction with the aug-cc-pV6Z (Dunning, 1989; Kendall
et al., 1992) basis set. In the subsequent MRCI treatment, the multi-
configuration wave functions included $\sim$ 2.1 x 105 uncontracted CSFs per
$C_{2v}$ symmetry. With this ansatz, the energy difference between the above
mentioned asymptotes was reduced to $\sim$ 0.33 eV but still did not vanish.
For modeling based on unified spectral line shape theory an error of this size
would be unacceptable.
Finally, using the same active space as in the second series of computations,
we added a set of diffuse functions to the aug-cc-pV6Z basis set for H and He.
Hereafter, this enlarged set will be denoted as aug-cc-pV6Z⋆. The exponents of
the added Gaussian primitives, which were left uncontracted, are listed in
Table 1 in the Appendix.
This approach, compared to the previous ones, solved all the inconsistencies
mentioned above. That is, it yielded degenerate H($2p\,^{2}P$) +
He($1s^{2}\,{}^{1}S)$ and H($2s\,^{2}S$) + He($1s^{2}\,{}^{1}S)$ dissociation
limits and no spurious potential well in the $C\,\Sigma$ state. We note that
convergence was reached at this step since a further expansion of the aug-cc-
pV6Z⋆ set by inclusion of more diffuse functions led to almost identical
results. In these calculations, the MRCI wave functions included more than
$7.5\times 10^{5}$ uncontracted CSFs per $C_{2v}$ symmetry species. These
relatively large computations for such a small molecular system were necessary
to obtain the precision needed to model the Lyman-$\alpha$ profile accurately.
Figure 2: Top: short-range part of the potential curves of the H–He molecule:
$A$ (red dotted), $B$ (green dashed line) and $C$ (blue solid). Bottom: $X$
(black solid). Note the agreement at short distance with data of
Theodorakopoulos et al. (1984) that are overplotted in dotted cyan.
Figure 3: Top: long range part of the $C\,\Sigma$ potential curve correlated
with $2s$ state. This work (full line), Theodorakopoulos et al. (1984) (dotted
line). Bottom: $\Delta V(R)$ (black) and $\tilde{d}(R)$ (blue) at 14500 K for
the $C-X$ transition. The atomic separation for the maximum in the $C-X$
difference potential is $R_{\mathrm{max}}\approx 2.2\,$Å as shown in Fig. 4
Note that the $C-X$ transition in this work is forbidden asymptotically as it
is a transition between the $2s$ and $1s$ states of the free hydrogen atom at
large $R$.
### 3.2 Potential energy curves and transition moments
The electronic states investigated in the present contribution correlate, at
large internuclear distances, to the H($1s\;^{2}S$) + He($2s^{2}\;{}^{1}S$),
H($2s\;^{2}P$) + He($2s^{2}\;{}^{1}S$), and H($2p\;^{2}P$) +
He($2s^{2}\;{}^{1}S$) dissociation limits (see Fig. 2 and Table 2 in the
Appendix). The MRCI+Q/aug-cc-pV6Z⋆ potential energy curves of the four lowest
electronic states of H–He, obtained with the largest active space and basis
set as described in the previous section, are represented in Fig. 2 as a
function of the internuclear distance, $R_{\mathrm{H-He}}$. This figure shows
that the ground state possesses a repulsive potential correlating to the
H($1s\,^{2}S$) + He($1s^{2}\,{}^{1}S$) isolated atom asymptote at large
distances.
The ground $X\,^{2}\Sigma^{+}$ state is repulsive at short range with a
shallow well at $4\,$Å. The excited $A\,^{2}\Sigma^{+}$, $B\,^{2}\Pi$ and
$C\,^{2}\Sigma^{+}$ states have rather deep potential wells in the molecular
region closer than 1 Å, and complex behavior at longer range that can affect
transition probabilities and difference potential energies in subtle ways. We
refer to these as the $X\,\Sigma$, $A\,\Sigma$, $B\,\Pi$, and $C\,\Sigma$
states, or more succinctly by the letter designations $X$, $A$, $B$, and $C$
in the following. They correlate adiabatically to the H($n=2$) +
He($1s^{2}\,{}^{1}S$) dissociation limits at large internuclear separations
(see Table 2 in the Appendix). The ordering of the assignments of labels for
the states is with $A\,\Sigma$ the lowest and $C\,\Sigma$ the highest inside
this close 1 Å region with wells in all the states of the order of
$15\,000\;\mathrm{cm}^{-1}$ deep, with minima located at $R_{\mathrm{H-He}}$ =
0.7407, 0.7686, and 0.8095 Å for the $A$, $B$ and $C$ states, respectively
(see Table 3 in the Appendix). While the $A$ and $B$ states have potentials
with a simple short-range well, the $C$ state also exhibits a potential
maximum of $\approx 0.666$ eV at $R_{\mathrm{H-He}}=2.098$ Å. Its presence
causes a related maximum in the $C-X$ transition difference potential energy
curve which affects the blue wing of Lyman-$\alpha$.
Although the $C\,\Sigma$ H-He molecular state shown in Fig. 2 is correlated
asymptotically with the $2s$ atomic state, we find that at
$R_{\mathrm{H-He}}<7\;$Å the transition probability to the $X\,\Sigma$ ground
state is not zero. Detailed electric dipole transition moments between the
$X\,\Sigma$ ground state and the $A\,\Sigma$, $B\,\Pi$ and $C\,\Sigma$ excited
states as a function of the internuclear distance have been calculated at the
MRCI/aug-cc-pV6Z⋆ level. In this calculation almost all the transition moments
are rather large, particularly for the $C\,\Sigma$ $\leftarrow$ $A\,\Sigma$
and $B\,\Pi$ $\leftarrow$ $A\,\Sigma$ transitions, where corresponding matrix
elements of around -9.2 and -7.5 debye (D or $10^{-18}$ statcoulomb-cm) are
calculated, respectively. Fig. 7 in the Appendix offers a detailed view. These
transition moments correlate to the correct atomic values at dissociation. In
particular, the $\langle X\,\Sigma|DM|C\,\Sigma\rangle$ matrix element of the
electric dipole transition moment (DM) vanishes at large $R_{\mathrm{H-He}}$
where the $1s-2s$ transition in the isolated hydrogen atom is forbidden to
one-photon electric dipole transitions by parity conservation.
## 4 Lyman-alpha opacity
The theory of spectral line shapes, especially the unified approach we
developed, determines the contributions of specific spectral lines to stellar
opacities and may be incorporated into stellar atmosphere models to make
accurate synthesis of stellar spectra possible. The line shape theory accounts
for neutral atom broadening and shift in both the centers of spectral lines
and their extreme wings with one consistent treatment without ad hoc
assumptions about the line shape or potentials. Complete details and the
derivation of the theory are provided by Allard et al. (1999). The spectrum,
$I(\Delta\omega)$, is the Fourier transform (FT) of a electric dipole
transition autocorrelation function, $\Phi(s)$. For a perturber density
$n_{p}$, we have
$\Phi(s)=e^{-n_{p}g(s)}\;,$ (1)
where the decay of the autocorrelation function with time leads to atomic line
broadening. (See Eq. (121) of Allard et al. (1999).) Our approach introduces
the concept of a modulated electric dipole transition moment
$\tilde{d}_{if}(R(t))$ into the line shape calculation.
$\tilde{d}_{if}[R(t)]=d_{if}[R(t)]e^{-\frac{V_{i}[R(t)]}{2kT}}\;\;,$ (2)
where the potential energy for the initial state is
$V_{i}(R)=E_{i}(R)-E_{i}^{\infty}\;\;.$ (3)
The difference potential energy $\Delta V(R)$ for a transition $if$ is
$\Delta V(R)=V_{if}(R)=V_{f}(R)-V_{i}(R)\;\;.$ (4)
The Boltzmann factor $e^{-\frac{V_{i}(R)}{2kT}}$ in Eq. (2) appears because
the perturbing atoms or ions are in thermal equilibrium with the radiating
atom which affects the probability of finding them initially at a given $R$.
This treatment results in Lyman series line wing profiles that exhibit a
sensitive dependence on temperature. We had to use electric dipole moments
modulated by the Boltzmann factor in the comparison of emission spectra of
Lyman-$\alpha$ (Kielkopf & Allard, 1998) and Balmer $\alpha$ (Kielkopf et al.,
2002) measured in laboratory.
### 4.1 Study of the characteristics of the line satellite
In Allard & Christova (2009) we predicted a line satellite at 1157 Å in
spectra computed for the temperature range of cool DZ white dwarfs with
potentials published in Theodorakopoulos et al. (1984). However, we noticed an
unexpected well of about 150 cm-1 (upper Fig. 3) in the potential energy of
the $C\,\Sigma$ state at $R\sim 8$ Å which may be related to the choice of
basis states and has no clear physical origin. In this work we use the new ab
initio calculations of the potentials over the full range of distances $R$
between the H and He atoms since convergence at large $R$ is now reached. The
long range well of the $C\,\Sigma$ state of Theodorakopoulos et al. (1984) and
Theodorakopoulos et al. (1987) potentials is not found in these new
calculations as we see in Fig. 3.
Figure 4: Top: variation with temperature of the line satellite. The He
density is $1\times 10^{20}$ cm-3, the temperatures are 14500 K (full black
line), $20\,000$ K (blue stars), and $5\,000$ K (red dashed line). Bottom: for
the $C-X$ transition, $\Delta V(R)$ (black solid) and $\tilde{d(R)}$ at 5000 K
(black solid), $10\,000$ K (red dotted), 14500 K (green dashed), and 20000 K
(blue solid). At the highest temperatures the He can reach the inner regions
of the lower state $X\,^{2}\Sigma$ potential and enhance the transition
probability.
The prediction of a satellite in the blue wing of the H–He line profile is
related to a potential maximum at $R=2.1$ Å (see Sect. 3.2) of the $C\,\Sigma$
state. This leads to a maximum of the potential energy difference $\Delta
V(R)$ in Eq. (4) for this transition shown in Fig. 3.
The unified theory predicts that line satellites will be centered periodically
at frequencies corresponding to integer multiples of the extrema of $\Delta
V(R)$. In the quasi-static limit the first satellite on the line would be at
$\Delta\omega=5\,000$ cm-1 corresponding to $\lambda\sim 1150$ Å on the blue
side of Lyman-$\alpha$. In this case the maximum in $\Delta V$ occurs at
rather small internuclear distance, and is quite sharp. The correspondingly
short duration of the close collision leads to a broad satellite centered at
$\lambda$ $\sim$ 1160 Å for T=$14\,500$ K (Fig. 4).
### 4.2 Temperature and density dependence
For a lower temperature, $T=5\,000$ K (Fig. 4), the duration of the collision
is longer, and the line satellite at $\lambda\sim 1153$ Å is sharper and
closer to the predicted quasi-static position than at higher temperatures. The
oscillations which appear on the red side of the quasi-molecular satellite are
due to interference effects described by Royer (1971) and Sando & Wormhoudt
(1973). They depend on the relative velocity and therefore on temperature.
Consequently velocity averaging would moderate their amplitude in observed
spectra. At temperatures below $10\,000$ K the blue wing of Lyman-$\alpha$
shortward of 1150 Å becomes significantly more transparent than at higher
temperature, an order of magnitude effect below 1120 Å. Thus this far blue
wing is a sensitive indicator of temperature in cool helium-rich WD
atmospheres.
The satellite amplitude depends on the value of the electric dipole transition
moment through the region of the potential extremum responsible for the
satellite and on the position of this extremum. The blue line wings shown in
Fig. 4 are unchanged in the range $14\,500$ to $20\,000$ K as there is no
change with $T$ of $\tilde{d}_{if}[R(t)]$ in the internuclear distance where
the potential difference goes through a maximum. $\tilde{d}_{if}[R(t)]$ at
$14\,500$ K for the $C-X$ transition is also plotted in Fig. 3. In the former
work we used electric dipole transition moments of Theodorakopoulos et al.
(1987) where the $C-X$ transition was allowed. Nevertheless the amplitude and
position of the line satellite are unchanged as they are due to a range of
internuclear distance where the potentials and the dipole moments are almost
identical as we see in Fig. 5. The main difference between the two potentials
concerns the red wing which is lowered using dipole moments of
Theodorakopoulos et al. (1987) where the $A-X$ transition was forbidden.
Figure 5: Comparison of the unified line profile using the dipole moments of
this work (black line) with the line profile using dipole moments of
Theodorakopoulos et al. (1987) (red dashed line) . The He density is $10^{20}$
cm-3 and the temperature is 14500 K.
In summary the unified line profile calculation leads to a flat blue wing due
to a line satellite. The resulting asymmetry of the Lyman-$\alpha$ line can be
easily appreciated in Fig. 5 the blue side of the line is wider than the red
side. Measured at the strength of the broad collision-induced 1160 Å
satellite, the asymmetry ratio of the width on the blue side to that on the
red is as large as 2.2. Consequently, the near wing is clearly far different
from a symmetric Lorentzian because the satellite is rather close to the
isolated atom line center. This was also the case for the Mg b triplet
perturbed by He (Allard et al., 2016). The existence of the asymmetrical shape
of these line profiles depends strongly on the maximum value of the potential
energy difference $\Delta V(R)$ which predicts the position of the line
satellite and on the atomic collision energies at the temperatures of
interest. These results enable computing atmosphere models and synthetic
spectra which we compare to an HST COS observation of WD 1425+540 in Section
5.
## 5 Model atmosphere and synthetic white dwarf spectrum
To demonstrate the importance of a proper treatment of He perturbers on
hydrogen lines, synthetic spectra of the white dwarf WD 1425+540 were computed
using the stellar atmosphere code TLUSTY (version 207) for computing the
atmospheric structure, and a companion program SYNSPEC (version 53) for
generating detailed synthetic spectra. For a description of the previous
versions (205 and 51) see the works of Hubeny & Lanz (2017) and Hubeny & Lanz
(2011a, b). This procedure allows us to study the effect of the H/He ratio on
the spectrum, and the development of line wings, though it is not fully self-
consistent with the stellar atmosphere model since that would require a
treatment of He I optical lines as well. We have computed a number of H-He
models, with the basic model parameters, $T_{\rm eff}=14,410$ K and $\log
g=7.89$, from Gänsicke et al. (2018), and with varying He/H ratio. For
treating the electron and proton broadening of the hydrogen lines we used
Tremblay & Bergeron (2009) data. The He/H ratio was adjusted to obtain a
reasonable agreement by eye with the observed spectrum, and we found a nominal
ratio of $4\times 10^{3}$ ($\log(N_{\mathrm{H}}/N_{\mathrm{He}})\approx-3.6$)
fitted the observed profile well. Liebert et al. (1979) found 3.7 from a
ground-based H$\beta$ profile, and recently Gänsicke et al. (2018) analyzed
the L$\alpha$ profile and adopted a somewhat larger
$\log(N_{\mathrm{H}}/N_{\mathrm{He}})\approx-4.0\pm 0.20$.
The potential energies for the $n=1$ and $n=2$ electronic states H-He that
were used in our models are the ones described in this paper. Stellar
opacities were computing using H-He electric dipole moments from the previous
work of Theodorakopoulos et al. (1987) in which the $A-X$ transition is
forbidden, and also using new dipole transition moments from this work in
which the $A-X$ transition is allowed. As shown in Fig. 6, the observed red
wing of Lyman-$\alpha$ is consistent with a suppressed $A-X$ transition
probability in the region of atomic separation with difference potential
energy that would contribute.
We conclude that the additional basis states used for the new ab initio
potentials improve the calculation of the potential energy curves, but may not
capture the dipole transition moments of the real H-He system correctly for
the $A-X$ transition. However the combination of this work’s potentials and
the dipole moments of Theodorakopoulos et al. (1987) achieve a remarkable fit
in Fig. 6 to the HST COS spectrum of WD 1425+540 when incorporated into the
unified line shape theory we described here.
Figure 6: The observed spectrum of WD 1425+540 (also see Fig. 1) compared with
a synthetic white dwarf spectrum in the Lyman-$\alpha$ region. The synthetic
spectrum is computed with TLUSTY and SYNSPEC for a temperature of 14 500 K and
a He/H ratio of $4\times 10^{3}$ using the unified line profile with the
potentials of this work. For the dipole moments of Theodorakopoulos et al.
(1987) (red solid line) the $A-X$ transition is forbidden and its
contributions to the opacity are suppressed. For the dipole moments of this
work (blue dashed line), the $A-X$ transition contributes in the red wing of
the model but is absent in the observed spectrum.
## 6 Conclusions
The Lyman-$\alpha$ region of the spectrum of a helium-rich white dwarf with
hydrogen in its atmosphere is determined by the changes in transition energy
and transition probability during the H-He collisions that broaden the atomic
spectral line. We developed new H-He potential energies and transition dipole
moments for the hydrogen $1s$, $2s$, and $2p$ states as input data for a
unified theory calculation of the profile of WD 1425+540 to test the
potentials and dipole moments, and to confirm the origin of the short-
wavelength “blue” satellite. We found that the spectral line profile from the
new molecular data has a satellite feature in the blue wing that agrees with
previous work. These results provide a benchmark implementation of ab initio
atomic and molecular potentials for the most basic neutral non-resonant atom-
atom pair relevant to stellar atmosphere models. The new calculations show how
the profile depends on the variation of the electric dipole transition moment
and interaction potential energy with atomic separation. A comparison with the
observed spectrum of WD 1425+540 was made by using these theoretical opacities
in a stellar atmosphere and spectrum synthesis code. While it was not our goal
to refine the stellar model based on the new theoretical data, the profiles
reproduce the observed spectrum with a reasonable He/H ratio. Further, the
absence of an extended red wing of Lyman-$\alpha$ in the observed spectrum
suggests that the states of the difference potential that could contribute to
that region have the reduced transition dipole moment that was found in
previous molecular models. The new work presented here shows clearly that
there is an opportunity to use stellar spectra to improve the atomic and
molecular physics, ultimately to yield better models for astrophysical
applications. For H–He, the $A-X$ transition dipole moment remains uncertain.
The blue wing of Lyman-$\alpha$ is sensitive to He density and the structure
and temperature of the stellar atmosphere, with a profile that for wavelengths
shortward of $1150\,$Å will have reduced opacity from regions with
temperatures under $10\,000\,$K. Profiles computed with a unified theory of
collision broadening based on accurate data from ab initio molecular physics
take into account the strong dependence of the amplitude of the electric
dipole transition moment on atom-atom separation ($R$) where the potential
energy change $\Delta V(R)$ is an extremum. Incorporated into model
atmospheres, this dependence may be used to probe white dwarf or stellar
atmospheres for density and temperature. This emphasizes the importance of the
accuracy of both the potential energies and the electric dipole transition
moments for the line shape calculations that have traditionally assumed
electric dipole transition moments are constant (Allard & Kielkopf, 1982;
Allard & Koester, 1992; Allard et al., 1994).
The effect of collision broadening is central to understanding the opacity of
stellar atmospheres, yet there have been only a few definitive comparisons
with experimental work for atomic H. (Kielkopf & Allard, 1995, 1998; Kielkopf
et al., 2004). This is because of the difficulty of creating an environment in
a laboratory experiment simulating a stellar atmosphere with accurate
diagnostics. On the theoretical side, the maturing capability of ab initio
methods now offers the possibility of accurately computing the interaction of
H with H (Drira, 1999; Spielfiedel, 2003; Spielfiedel et al., 2004) and H with
He atoms (this work). While an accurate determination of the broadening of
Balmer $\alpha$ with high density atomic hydrogen (that is H–H) has been done
by Allard et al. (2008), nothing comparable exists for H–He. Our calculations
reported in Allard et al. (2008) support the results of Barklem et al. (2000);
Barklem et al. (2002) that the Ali & Griem (1966) theory underestimates the
actual line width. Recent laboratory measurements show a similar result at
high density in environments comparable to white dwarf atmospheres (Kielkopf &
Allard, 2014). It would be possible now to similarly improve the calculation
of Balmer-$\alpha$ broadening and its contribution to the full white dwarf
opacity model. A major improvement to comprehensive theoretical models for DBA
white dwarf spectra is within reach that would determine H-He molecular data
for $n=3$ excited states, and use those to compute accurate Balmer-$\alpha$
profiles under white dwarf atmosphere conditions. Such results would help
understanding the differences in stellar parameters that are found from Balmer
and Lyman line profiles. In conclusion, complete unified line profiles based
on accurate atomic and molecular physics for both the Lyman-$\alpha$ and
Balmer-$\alpha$ lines should be incorporated into the analysis of DBA white
dwarf spectra to derive the hydrogen abundance.
## acknowledgements
The paper was based on observations made with the NASA/ESA Hubble Space
Telescope under program 13453, obtained from the data archive at the Space
Telescope Science Institute. STScI is operated by the Association of
Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.
We thank the COST Action CM1405 MOLecules in Motion (MOLIM) of the European
Community for support. The authors would like to extend their sincere
appreciation to the Deanship of Scientific Research at King Saud University
for funding the research through the Research Group Project No. RGP-333. This
work was supported by the CNRS program Physique et Chimie du Milieu
Interstellaire (PCMI) co-funded by the Centre National d’Etudes Spatiales
(CNES).
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## appendix
Parameters of the H–He molecular potentials are given in Tables 1 and 2.
Figure 7 shows the dependence on $R$ of the radiative transition moments
between the excited states and the perturbations of those states as the H and
He atoms approach from large $R$.
Table 1: Exponents of the diffuse uncontracted Gaussian primitives added to the aug-cc-pV6Z basis set to form the presently used aug-cc-pV6Z* basis sets for the H and He atoms. State | 1 | 2 | 3
---|---|---|---
H(_s_) | 0.00690204 | 0.002520537 | 0.000920468
H(_p_) | 0.026565598 | 0.010533298 | 0.004176468
H(_d_) | 0.055406537 | 0.024364162 | 0.010713761
H(_f_) | 0.106396067 | 0.046204584 | 0.020065249
H(_g_) | 0.168703345 | 0.069928301 | 0.028985598
H(_h_) | 0.175320015 | 0.045069073 | 0.011585793
He(_s_) | 0.017177900 | 0.006596920 | 0.002533450
He(_p_) | 0.050416903 | 0.019858313 | 0.007821833
He(_d_) | 0.094209988 | 0.036827891 | 0.014396494
He(_f_) | 0.151890237 | 0.056684629 | 0.021154402
He(_g_) | 0.232902520 | 0.079072280 | 0.026845675
He(_h_) | 0.248198125 | 0.060632194 | 0.014811808
Table 2: Dissociation fragments, experimental and calculated relative dissociation asymptotic energies, and molecular states for the four lowest electronic states of H–He. Experimental data are from Kramida (2010). Atomic | | Observed | Calculated | Molecular
---|---|---|---|---
H | He | (cm${}^{-}1$) | (cm${}^{-}1$) |
$1s\,^{2}S_{g}$ | $1s^{2}\,{}^{1}S_{g}$ | 0a | 0a | $X\,^{2}\Sigma^{+}$
$2p\,^{2}P_{u}$ | $1s^{2}\,{}^{1}S_{g}$ | 82259 | 82308 | $A\,^{2}\Sigma^{+}$, $B\,^{2}\Pi$
$2s\,^{2}S_{g}$ | $1s^{2}\,{}^{1}S_{g}$ | 82259 | 82308 | $C\,^{2}\Sigma^{+}$
aReference | | | |
Table 3: Spectroscopic constants and dissociation energies for the three lowest excited electronic states of H–He as deduced from the MRCI+Q /aug-cc-pV6Z* potential energy curves. $R_{e}$ corresponds to the equilibrium distance. $\omega_{e}$ and $\omega_{e}x_{e}$ are the vibrational constants. $\beta_{e}$ and $\alpha_{e}$ are the rotational constants. $D_{e}$ is the dissociation energy. State | $R_{e}$ | $\omega_{e}$ | $\omega_{e}x_{e}$ | $\beta_{e}$ | $\alpha_{e}$ | $D_{e}$
---|---|---|---|---|---|---
| Å | cm-1 | cm-1 | cm-1 | cm-1 | eV
$A\,^{2}\Sigma^{+}$ | 0.74074 | 3697.2 | 149.5 | 38.16 | 2.608 | 2.563
$B\,^{2}\Pi$ | 0.76863 | 3313.4 | 149.8 | 35.44 | 2.629 | 2.218
$C\,^{2}\Sigma^{+}$ | 0.80953 | 2906.3 | 144.0 | 31.95 | 2.551 | 1.638
Figure 7: Potential energy differences in cm-1 and electric dipole transition
moments in debye (D or $10^{-18}$ statcoulomb-cm) between the four lowest
electronic states of H–He calculated at the MRCI/aug-cc-pV6Z⋆ level. Note that
the $C\,\Sigma$ $\leftarrow$ $X\,\Sigma$ is asymptotically forbidden, while
transitions between excited states may occur. Upper panel: Energy differences
$A\Sigma-B\Sigma$ (blue) and $A\Sigma-C\Pi$ (red). Lower panel: Electric
dipole transition moments for H in the presence of He for states contributing
to H Lyman-$\alpha$.
|
2024-09-04T02:54:59.017865 | 2020-03-11T06:04:09 | 2003.05124 | {
"authors": "Yiying Yan, Zhiguo L\\\"u, JunYan Luo, Hang Zheng",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26151",
"submitter": "Yiying Yan",
"url": "https://arxiv.org/abs/2003.05124"
} | arxiv-papers | # Role of generalized parity in the symmetry of fluorescence spectrum from
two-level systems under periodic frequency modulation
Yiying Yan<EMAIL_ADDRESS>Department of Physics, School of Science,
Zhejiang University of Science and Technology, Hangzhou 310023, China Zhiguo
Lü<EMAIL_ADDRESS>Key Laboratory of Artificial Structures and Quantum
Control (Ministry of Education), Department of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center
of Advanced Microstructures, Nanjing 210093, China JunYan Luo Department of
Physics, School of Science, Zhejiang University of Science and Technology,
Hangzhou 310023, China Hang Zheng<EMAIL_ADDRESS>Key Laboratory of
Artificial Structures and Quantum Control (Ministry of Education), Department
of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240,
China Collaborative Innovation Center of Advanced Microstructures, Nanjing
210093, China
###### Abstract
We study the origin of the symmetry of the fluorescence spectrum from the two-
level system subjected to a low-frequency periodic modulation and a near-
resonant high-frequency monochromatic excitation by using the analytical and
numerical methods based on the Floquet theory. We find that the fundamental
origin of symmetry of the spectrum can be attributed to the presence of the
generalized parity of the Floquet states, which depends on the driving
parameters. The absence of the generalized parity can lead to the asymmetry of
the spectrum. Based on the generalized parity, the conditions for the symmetry
and asymmetry of the spectrum can be derived, which succeeds in predicting
symmetry and asymmetry of the spectrum for the harmonic, biharmonic, and
multiharmonic modulations. Moreover, we find that the secular approximation
widely used in the analytical calculation may lead to artifact symmetry of the
spectrum that vanishes when such approximation is avoided. The present study
provides a significant perspective on the origin of the symmetry of the
spectrum.
## I Introduction
Resonance fluorescence, arising from a quantum emitter driven by an external
field and coupled to a radiative reservoir Mollow (1969); Scully and Zubairy
(1997); Cohen-Tannoudji _et al._ (1998), is not only an important concept in
quantum optics but also has potential application in quantum information
technology, for instance, it plays an important role in realizing the single-
photon source He _et al._ (2013); Santana _et al._ (2017); Kiršanskė _et
al._ (2017). Particularly, the resonance fluorescence of two-level systems has
attracted much interest and been studied in various aspects such as spectrum
Ficek and Freedhoff (1993); Agarwal _et al._ (1991); Ficek and Freedhoff
(1996); Ficek and Rudolph (1999); Peiris _et al._ (2014); Konthasinghe _et
al._ (2014); He _et al._ (2015); Toyli _et al._ (2016), squeezing Carmichael
(1985); Grünwald and Vogel (2012, 2013), photon statistics Kimble _et al._
(1977); D’Souza _et al._ (1990); Nazir (2008); Pastukhov _et al._ (2014),
photon antibunching Itano _et al._ (1988); Ficek _et al._ (1984); Damanet
_et al._ (2018), and so on. The line shape of the spectrum is found to depend
strongly on the external field that interacts with the quantum emitters as
well as the reservoirs to which the quantum emitters are coupled. As is well-
known, for a sufficiently strong monochromatic field, the spectrum has a
symmetric three-peak structure, known as the Mollow triplet Mollow (1969).
More recently, the bi- and multi-chromatically driven quantum systems are of
interest Kryuchkyan _et al._ (2017); Antón _et al._ (2017); Yan _et al._
(2018); Saiko _et al._ (2018). In such systems, the spectrum turns out to
have a complicated multipeak structure Ficek and Freedhoff (1993); Agarwal
_et al._ (1991); Ficek and Freedhoff (1996); Ficek and Rudolph (1999); Peiris
_et al._ (2014); Konthasinghe _et al._ (2014); He _et al._ (2015), which can
be either symmetric or asymmetric. In principle, the physical origin of the
triplet and multipeak structures can be understood in terms of the transitions
between the quantum dressed states Cohen-Tannoudji _et al._ (1998) or in
terms of the transitions between the semiclassical Floquet states Breuer and
Petruccione (1997); Yan _et al._ (2016a). The studies on the resonance
fluorescence have enriched the physics concerning the light-matter
interaction.
The origin of the symmetry of the spectrum has been found in the case of the
monochromatic field. Specifically, it is the detailed balance condition that
guarantees the symmetry of the Mollow triplet Cohen-Tannoudji _et al._
(1998). As is well-known, the breakdown of such a condition leads to the
asymmetry of the spectrum, for instance, in the presence of a pure dephasing
reservoir Roy and Hughes (2012); McCutcheon and Nazir (2013) or the counter-
rotating terms of the external field under certain conditions Browne and
Keitel (2000); Yan _et al._ (2013, 2016a). The dephasing-induced asymmetric
Mollow triplet has been experimentally observed in the quantum dots (the pure
dephasing arises because of the interaction between the quantum dot and its
solid-state environment) Ulrich _et al._ (2011); Ulhaq _et al._ (2013). For
the bi- and multi-chromatic fields, the origin of the symmetry of the spectrum
is rarely discussed, owing to the fact that the physically transparent
spectrum is hardly analytically derived, and has not been comprehensively
understood.
Recent studies show that the fluorescence spectrum from a driven two-level
system with a modulated transition frequency is symmetrically multipeaked for
the vanishing detuning while asymmetrically multipeaked for the finite
detuning Yan _et al._ (2016b); Kryuchkyan _et al._ (2017); Antón _et al._
(2017); Yan _et al._ (2018). Such an exotic bichromatically driven two-level
system with coexistence of the longitudinal and transversal coupling between
the system and the applied fields has been experimentally studied in the
superconducting qubits Li _et al._ (2013); Pan _et al._ (2017), single
molecule Brunel _et al._ (1998), and nitrogen-vacancy spin qubits Rohr _et
al._ (2014). The quantum systems under frequency modulation are also of
interest in theoretical studies Kibis _et al._ (2009); Macovei and Keitel
(2014); Zhao _et al._ (2015); Silveri _et al._ (2013); Macovei _et al._
(2015), the intriguing phenomena of which were reviewed recently Silveri _et
al._ (2017). It is worthwhile to note that the bichromatically driven two-
level system with frequency modulation differs from those considered in Refs.
Agarwal _et al._ (1991); Ficek and Freedhoff (1993), where the two-level
systems are transversely driven by a bichromatic field. In such a case, the
symmetry of the fluorescence spectrum is found to depend on the average
detuning if the strengths of the two components of the bichromatic field are
the same; the pronounced asymmetry of the spectrum is revealed when the
average detuning is finite and/or the strengths of the two components of the
field are unequal Agarwal _et al._ (1991); Ficek and Freedhoff (1993). For a
bichromatically amplitude-modulated field, the spectrum is also found to be
symmetric and asymmetric for the vanishing and finite detuning, respectively
Wilkens and Rza¸ewski (1989). So far the fundamental origin of such a
detuning-dependent symmetry remains obscure.
In this work, we use both analytical and numerical methods based on the
Floquet theory to study the fundamental origin of the symmetry of the
fluorescence spectrum from the two-level system under a low-frequency periodic
modulation and a near-resonant monochromatic excitation. We address the
symmetry and asymmetry of the spectrum by considering the generalized parity
of Floquet states rather than the behaviors of the bare-state or dressed-state
populations as considered in Refs. Das and Macovei (2013); Macovei _et al._
(2015); Antón _et al._ (2017). The generalized parity is found to guarantee
the symmetry of the spectrum while the breaking of such a parity can yield
pronouncedly asymmetric spectrum even in the vanishing detuning case. Based on
the generalized parity, the conditions for the symmetric and asymmetric
spectra are derived, which are not given in the previous works and cannot be
derived from the behaviors of the bare or dressed state population. The
generalized-parity-induced symmetry of the spectrum is verified and
illustrated in the context of the biharmonic modulation by the comparison
between the analytical and numerical results. The analytical results are found
to be in agreement with the numerically exact results in the regimes where the
perturbation theory and secular approximation can be justified. In addition,
we find that the spectrum with the secular approximation may have artifact
symmetry under certain conditions, i.e., the spectrum with secular
approximation is symmetric while the numerically exact calculation shows
asymmetric spectra because of the broken parity. The present finding simply
interprets the detuning-dependent symmetry in the harmonic modulation case and
can also be extended to analyze the symmetry and asymmetry of the spectrum in
the multiharmonic modulation cases. Our results suggest that it is feasible to
control the symmetry and asymmetry of the spectrum via engineering the
generalized parity of the Floquet states.
The rest of the paper is organized as follows. In Sec. II, we first discuss
the generalized-parity-induced symmetry of the fluorescence spectrum without
the secular approximation and further elucidate the symmetry of the spectrum
with a physically transparent formal spectrum with the secular approximation.
In Sec. III, we analytically and numerically calculate the fluorescence
spectrum in the context of the biharmonic modulation to verify the symmetry
and asymmetry of the spectrum predicted based on the generalized parity. In
the last section, the conclusions are given.
## II Fluorescence spectrum and generalized parity
We consider that the transition frequency of the two-level system is modulated
periodically via a low-frequency external field $f(t)$ and the two-level
system is also excited by a near-resonant monochromatic field, which is
described by the following Hamiltonian ($\hbar=1$)
$H(t)=\frac{1}{2}[\omega_{0}+f(t)]\sigma_{z}+\frac{\Omega_{x}}{2}(\sigma_{+}e^{-i\omega_{x}t}+\sigma_{-}e^{i\omega_{x}t}),$
(1)
where $\sigma_{z(x,y)}$ is the usual Pauli matrix, $\omega_{0}+f(t)$ is the
modulated transition frequency, $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$
are the raising and lowering operators, and $\Omega_{x}$ ($\omega_{x}$) is the
strength (frequency) of the monochromatic driving. Here we choose
$f(t)=f(t+T)$ with $T$ being the fundamental period of the modulation and much
greater than $2\pi/\omega_{x}$. This is a generalized model as compared with
the previous one considered in Refs. Yan _et al._ (2016b); Kryuchkyan _et
al._ (2017); Antón _et al._ (2017).
To study the emission processes, we need to take account of the spontaneous
decay. Thus, the time evolution of the driven two-level system under study is
modeled by the Lindblad master equation. In the frame rotating at the
frequency $\omega_{x}$, the Lindblad master equation takes the form
$\frac{d}{dt}\tilde{\rho}(t)={\cal L}(t)\tilde{\rho}(t),$ (2)
where $\tilde{\rho}(t)$ is the reduced density matrix in the rotating frame
and the superoperator ${\cal L}(t)$ is given by ${\cal
L}(t)\tilde{\rho}(t)=-i[\tilde{H}(t),\tilde{\rho}(t)]-\kappa/2[\\{\sigma_{+}\sigma_{-},\tilde{\rho}(t)\\}-2\sigma_{-}\tilde{\rho}(t)\sigma_{+}]$
with $\kappa$ being the radiative decay rate. $\tilde{H}(t)$ is the effective
Hamiltonian and reads
$\tilde{H}(t)=\frac{\Omega_{x}}{2}\sigma_{x}+\frac{1}{2}[\delta+f(t)]\sigma_{z},$
(3)
with $\delta=\omega_{0}-\omega_{x}$ being the detuning between the bare
transition frequency and monochromatic excitation frequency. This master
equation is actually a set of first-order differential equations with periodic
coefficients. It can be directly solved by the so-called Floquet-Liouville
(FL) approach with a desire accuracy Ho _et al._ (1986); Yan _et al._
(2016b). Although such a Floquet-theory-based numerical method is simple and
efficient, it is not physically transparent to analyze the role of generalized
parity of Floquet states in the symmetry of the fluorescence spectrum. We use
an alternative method which is developed in our previous works Yan _et al._
(2016a, 2018) to solve the master equation and calculate the fluorescence
spectrum. We first calculate the Floquet states for $\tilde{H}(t)$ and use
them as the bases to reformulate Eq. (2) and derive its analytical formal
solutions with the aid of the secular approximation in the Floquet picture.
### II.1 The symmetry of fluorescence spectrum without secular approximation
The steady-state fluorescence spectrum is given by the Fourier transform of
the time-averaged first-order correlation function Mollow (1969); Ho _et al._
(1986)
$S(\Delta)\propto{\rm
Re}\frac{1}{T}\int_{0}^{\infty}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\left\langle\tilde{\sigma}_{+}(t^{\prime}+\tau)\tilde{\sigma}_{-}(t^{\prime})\right\rangle
e^{-i\Delta\tau}dt^{\prime}d\tau,$ (4)
where $\Delta=\omega-\omega_{x}$ and
$\left\langle\tilde{\sigma}_{+}(t^{\prime}+\tau)\tilde{\sigma}_{-}(t^{\prime})\right\rangle$
is the first-order correlation function and the tilde indicates that it is
evaluated in the rotating frame. In general, it is difficult to derive an
exact analytical spectrum. Nevertheless, we find that it is possible to show
that the spectrum is exactly symmetric about $\Delta=0$ when
$\delta+f(t)=-[\delta+f(t+T/2)]$ by realizing the fact that the driven two-
level system possesses a generalized parity symmetry, i.e.,
$\sigma_{x}\tilde{H}(t+T/2)\sigma_{x}=\tilde{H}(t).$ (5)
Here, the generalized parity transformation consists of an exchange between
the up and down states of two-level system
($\sigma_{z}\rightarrow-\sigma_{z}$) and a time shift of half period of the
modulation ($t\rightarrow t+T/2$).
We state briefly how the generalized parity guarantees the symmetry of the
spectrum. Owing to Eq. (5), we can construct a generalized parity
transformation in the Liouville space, the details of which can be found in
Appendix A. When $\delta+f(t)=-[\delta+f(t+T/2)]$, the superoperator ${\cal
L}(t)$ is similarly found to be invariant under the generalized parity
transformation. Based on this property, it can be derived from the master
equation (2) without the secular approximation that in the steady-state limit,
the time-averaged first-order correlation function is a real-valued function
in the rotating frame. As a result, the fluorescence spectrum is symmetric
about $\Delta=0$. This finding shows that the symmetry of the spectrum occurs
when $\delta+f(t)=-[\delta+f(t+T/2)]$ and results from the generalized parity.
We will numerically verify the generalized-parity-induced symmetry in Sec.
III.
### II.2 The symmetry of fluorescence spectrum with secular approximation
To further elucidate the role of the generalized parity in determining the
symmetry of the spectrum, we calculate the spectrum in the Floquet picture
which allows us to derive a physically transparent formal spectrum with the
aid of the secular approximation.
According to the Floquet theory Shirley (1965); Sambe (1973), the time-
dependent Schrödinger equation governed by $\tilde{H}(t)$ possesses a set of
formal solutions
$|\tilde{\psi}_{\alpha}(t)\rangle=|\tilde{u}_{\alpha}(t)\rangle
e^{-i\tilde{\varepsilon}_{\alpha}t}$, where
$|\tilde{u}_{\alpha}(t)\rangle=|\tilde{u}_{\alpha}(t+T)\rangle$ is Floquet
state and $\tilde{\varepsilon}_{\alpha}$ is the corresponding real-valued
quasienergy. The index $\alpha$ labels independent Floquet states.
Substituting the formal solution into the Schrödinger equation, one readily
finds that
$[\tilde{H}(t)-i\partial_{t}]|\tilde{u}_{\alpha}(t)\rangle=\tilde{\varepsilon}_{\alpha}|\tilde{u}_{\alpha}(t)\rangle.$
(6)
On solving this equation, one obtains the Floquet states and quasienergies of
the driven two-level system.
We use $|\tilde{u}_{\alpha}(t)\rangle$ ($\alpha=\pm$) as the basis to
reformulate the master equation (2) and invoke the secular approximation Yan
_et al._ (2016a, 2018), yielding
$\displaystyle\frac{d}{dt}\tilde{\rho}_{++}(t)$ $\displaystyle=$
$\displaystyle-\Gamma_{{\rm rel}}\tilde{\rho}_{++}(t)+\Gamma_{{\rm s}},$ (7)
$\displaystyle\frac{d}{dt}\tilde{\rho}_{+-}(t)$ $\displaystyle=$
$\displaystyle-(i\Delta_{+-}+\Gamma_{{\rm deph}})\tilde{\rho}_{+-}(t),$ (8)
where
$\tilde{\rho}_{\alpha\beta}(t)=\langle\tilde{u}_{\alpha}(t)|\tilde{\rho}(t)|\tilde{u}_{\beta}(t)\rangle$
is the element of density operator,
$\Delta_{+-}=\tilde{\varepsilon}_{+}-\tilde{\varepsilon}_{-}$ is the
difference of two quasienergies, and $\Gamma_{{\rm
s}}=\kappa\sum_{l}|x_{-+,l}^{(+)}|^{2}$, where $x^{(+)}_{\alpha\beta,l}$ is a
time-averaged transition matrix element defined as follows:
$x^{(\pm)}_{\alpha\beta,l}=\frac{1}{T}\int^{T}_{0}\langle\tilde{u}_{\alpha}(t)|\sigma_{\pm}|\tilde{u}_{\beta}(t)\rangle
e^{-i2\pi lt/T}dt.$ (9)
The relaxation rate $\Gamma_{{\rm rel}}$ and dephasing rate $\Gamma_{{\rm
deph}}$ are given by
$\displaystyle\Gamma_{{\rm rel}}$ $\displaystyle=$
$\displaystyle\kappa\sum_{l}(|x_{+-,l}^{(+)}|^{2}+|x_{-+,l}^{(+)}|^{2}),$ (10)
$\displaystyle\Gamma_{{\rm deph}}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}\sum_{l}(|x_{+-,l}^{(+)}|^{2}+|x_{-+,l}^{(+)}|^{2}+4|x_{++,l}^{(+)}|^{2}).$
(11)
The analytical formal solutions in the Floquet picture can be easily found as
follows:
$\displaystyle\tilde{\rho}_{++}(t)$ $\displaystyle=$
$\displaystyle\tilde{\rho}_{++}(0)e^{-\Gamma_{{\rm
rel}}t}+\tilde{\rho}_{++}^{{\rm ss}}(1-e^{-\Gamma_{{\rm rel}}t}),$ (12)
$\displaystyle\tilde{\rho}_{+-}(t)$ $\displaystyle=$
$\displaystyle\tilde{\rho}_{+-}(0)e^{-(\Gamma_{{\rm deph}}+i\Delta_{+-})t},$
(13)
where
$\tilde{\rho}_{++}^{{\rm ss}}=\frac{\Gamma_{{\rm s}}}{\Gamma_{{\rm
rel}}}=\frac{\sum_{l}|x_{-+,l}^{(+)}|^{2}}{\sum_{l}(|x_{+-,l}^{(+)}|^{2}+|x_{-+,l}^{(+)}|^{2})}$
(14)
is the steady-state population of the Floquet state. These solutions together
with the quantum regression theory enable us to derive a physically
transparent spectrum function Yan _et al._ (2016a, 2018)
$\displaystyle S(\Delta)$ $\displaystyle\propto$
$\displaystyle\sum_{l}\bigg{\\{}\pi|x_{++,l}^{(+)}|^{2}(\tilde{\rho}_{++}^{{\rm
ss}}-\tilde{\rho}_{--}^{{\rm ss}})^{2}\delta(\Delta-l\omega_{z})$
$\displaystyle+4|x_{++,l}^{(+)}|^{2}\tilde{\rho}_{++}^{{\rm
ss}}\tilde{\rho}_{--}^{{\rm ss}}\frac{\Gamma_{{\rm rel}}}{\Gamma_{{\rm
rel}}^{2}+(\Delta-l\omega_{z})^{2}}$
$\displaystyle+|x_{+-,l}^{(+)}|^{2}\tilde{\rho}_{++}^{{\rm
ss}}\frac{\Gamma_{{\rm deph}}}{\Gamma_{{\rm
deph}}^{2}+(\Delta-l\omega_{z}-\Delta_{+-})^{2}}$
$\displaystyle+|x_{-+,l}^{(+)}|^{2}\tilde{\rho}_{--}^{{\rm
ss}}\frac{\Gamma_{{\rm deph}}}{\Gamma_{{\rm
deph}}^{2}+(\Delta-l\omega_{z}+\Delta_{+-})^{2}}\bigg{\\}},$
It is evident that the accuracy of Eq. (LABEL:eq:sfunsa) is limited by the
secular approximation when the transition matrix elements
$x^{(+)}_{\alpha\beta,l}$ and quasienergies are exactly calculated. As is
well-known, the secular approximation can be justified under the strong
driving condition, i.e., $\Delta_{+-}\gg\kappa$. In general, we can calculate
the quasienergies and transition matrix elements based on both analytical and
numerical diagonalization (ND) of the Floquet Hamiltonian
$\tilde{H}(t)-i\partial_{t}$ in the Sambe space Shirley (1965); Sambe (1973) ,
yielding the analytical and semianalytical spectra, respectively.
Next, we discuss the parity phenomenon of the Floquet states resulting from
Eq. (5). We consider the behavior of the Floquet states under the generalized
parity transformation ${\cal P}_{G}$, which is defined as
${\cal
P}_{G}|\tilde{u}_{\alpha}(t)\rangle:=\sigma_{x}|\tilde{u}_{\alpha}(t+T/2)\rangle.$
(16)
By differentiating $\sigma_{x}|\tilde{u}_{\alpha}(t+T/2)\rangle$ with respect
to $t$, we readily obtain
$\left[\sigma_{x}\tilde{H}\left(t+T/2\right)\sigma_{x}-i\partial_{t}\right]\sigma_{x}\left|\tilde{u}_{\alpha}\left(t+T/2\right)\right\rangle=\tilde{\varepsilon}_{\alpha}\sigma_{x}\left|\tilde{u}_{\alpha}\left(t+T/2\right)\right\rangle.$
(17)
When $\delta+f(t)=-[\delta+f(t+T/2)]$,
$\sigma_{x}|\tilde{u}_{\alpha}(t+T/2)\rangle$ satisfies the same differential
equation as $|\tilde{u}_{\alpha}(t)\rangle$ because of Eq. (5). Recalling the
uniqueness of solutions of the differential equations, in such cases we must
have
$\sigma_{x}\left|\tilde{u}_{\alpha}\left(t+T/2\right)\right\rangle=\lambda_{\alpha}|\tilde{u}_{\alpha}(t)\rangle,$
(18)
where $\lambda_{\alpha}$ is a constant. Furthermore, we have
$\lambda_{\alpha}=\pm 1$ because of ${\cal
P}_{G}^{2}|\tilde{u}_{\alpha}(t)\rangle=\lambda_{\alpha}^{2}|\tilde{u}_{\alpha}(t)\rangle=|\tilde{u}_{\alpha}(t)\rangle.$
Specifically, when $\delta+f(t)=-[\delta+f(t+T/2)]$, the Floquet states may be
even or odd functions under the generalized parity transformation, which is
referred to as the generalized parity of the Floquet states. The generalized
parity has been previously investigated in other phenomena such as the
coherent destruction of tunneling Grossmann _et al._ (1991) and the laser-
induced electronic transport Lehmann _et al._ (2003).
Clearly, if $\delta+f(t)\neq-\left[\delta+f\left(t+T/2\right)\right]$, Eq.
(18) cannot hold as
$\sigma_{x}\tilde{H}\left(t+T/2\right)\sigma_{x}\neq\tilde{H}(t)$, i.e., the
effective Hamiltonian is no longer invariant under the generalized parity
transformation. Consequently, the Floquet states also do not have the
generalized parity.
Figure 1: The incoherent components of the fluorescence spectrum for $p=3$,
$\Omega_{x}=10\kappa$, $\delta=0$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$,
and various phase. “Ana.” and “Num.” denote the analytical and the FL
numerical results, respectively.
We show that the symmetry of the spectrum may be a consequence of the
generalized parity of the Floquet states. By using Eq. (18) and
$x_{\alpha\beta,l}^{(+)}=\left[x_{\beta\alpha,-l}^{(-)}\right]^{\ast}$, it is
straightforward to show the following identity for arbitrary integer $l$ from
the definition (9) of the transition matrix element:
$x_{\alpha\beta,l}^{(+)}=(-1)^{l}\lambda_{\alpha}\lambda_{\beta}\left[x_{\beta\alpha,-l}^{(+)}\right]^{\ast},$
(19)
provided $\delta+f(t)=-[\delta+f(t+T/2)]$. It follows that
$|x_{\alpha\beta,l}^{(+)}|=|x_{\beta\alpha,-l}^{(+)}|$ (20)
also holds for any integer $l$. We emphasize that the relation (20) can be
deduced from relation (19), however, the relation (19) cannot be derived from
relation (20). With the relation (20), it is straightforward to show that the
spectrum (LABEL:eq:sfunsa) is symmetric about $\Delta=0$ Yan _et al._ (2018).
Specifically, since $|x_{++,l}^{(+)}|=|x_{++,-l}^{(+)}|$, the emission lines
at $\Delta=\pm l\omega_{z}$ (the positions are symmetric about $\Delta=0$)
have the equal weights. Moreover, since $|x_{+-,l}^{(+)}|=|x_{-+,-l}^{(+)}|$,
we also have $\tilde{\rho}_{++}^{{\rm ss}}=\tilde{\rho}_{--}^{{\rm ss}}$
according to Eq. (14), leading to $|x_{+-,l}^{(+)}|^{2}\tilde{\rho}_{++}^{{\rm
ss}}=|x_{-+,-l}^{(+)}|^{2}\tilde{\rho}_{--}^{{\rm ss}}$. That is to say, the
emission lines at $\Delta=\pm(l\omega_{z}+\Delta_{+-})$ (the positions are
symmetric about $\Delta=0$) have the same weights. It turns out that the
symmetry of the spectrum fundamentally originates from the generalized parity
of the Floquet states when $\delta+f(t)=-[\delta+f(t+T/2)]$. Conversely, one
may expect that the symmetry of the spectrum may break when such a parity is
absent. However, it is a formidable task to analytically prove that the
spectrum is asymmetric in the absence of the generalized parity.
Let us discuss what happens to the formal spectrum if
$\delta+f(t)\neq-[\delta+f(t+T/2)]$. Under such a condition, the generalized
parity is absent, and thus we cannot have the relation (19). In principle, the
absence of the generalized parity will result in two possible situations. One
is that the spectrum becomes asymmetric about $\Delta=0$ because the relation
$|x^{(+)}_{\alpha\beta,l}|\neq|x^{(+)}_{\beta\alpha,-l}|$ can be derived at
least for a certain $l$. The other is that the spectrum is symmetric because
the equality $|x^{(+)}_{\alpha\beta,l}|=|x^{(+)}_{\beta\alpha,-l}|$ still
holds for any $l$, originating from other kinds of identities between the
transition matrix elements rather than the generalized-parity-induced identity
(19). Apparently the first situation is more trivial than the second one. Most
importantly, the present analysis suggests that the formal spectrum may be
symmetric even without the generalized parity. Consequently, we cannot
conclude from the formal spectrum (LABEL:eq:sfunsa) that the symmetry of the
spectrum breaks as long as the generalized parity is absent.
To end this section, we give some remarks on the above findings based on the
formal spectrum. First, we find that the symmetry of the spectrum may result
from the generalized parity and requires $\delta+f(t)=-[\delta+f(t+T/2)]$.
This is consistent with the analysis above without the secular approximation.
Moreover, the generalized parity is found to be an important underlying cause
of the relation (20), which was numerically found in harmonic modulation case
Yan _et al._ (2018). It turns out here that the relation (20) can be
established due to the generalized parity in the bi- and multi-harmonic cases.
Second, without the generalized parity, namely, when
$\delta+f(t)\neq-[\delta+f(t+T/2)]$, the formal spectrum can be either
trivially asymmetric or nontrivially symmetric. The symmetry requires the
relation (20) in the absence of the generalized parity, namely, Eq. (19).
Third, the formal spectrum is derived with the secular approximation and thus
the present analysis needs further verification. In what follows we consider a
concrete biharmonic modulation to verify whether the generalized parity
guarantees the symmetry of the spectrum when the secular approximation is not
invoked and we also check whether the relation (20) can be established without
the generalized parity and whether such relations lead to the symmetry of the
spectrum without the secular approximation.
Figure 2: The incoherent components of the fluorescence spectrum for $p=3$,
$\delta=5\kappa$, $\Omega_{x}=10\kappa$, $\Omega_{z}=\omega_{z}=40\kappa$,
$r=1$, and various phase.
## III Verification of symmetry and asymmetry of the spectrum
To calculate fluorescence spectrum, without loss of generality, we mainly
consider the biharmonic modulation in this work, namely, the modulation
consists of two harmonics
$f(t)=\Omega_{z}[\cos(\omega_{z}t)+r\cos(p\omega_{z}t+\phi)],$ (21)
where $\Omega_{z}$ and $\omega_{z}=2\pi/T$ are the amplitude and fundamental
frequency of the modulation, respectively, $p$ is a positive integer, $r$ is
the ratio of the amplitude of the second harmonic to that of the first one,
and $\phi$ is a relative phase. Since $\frac{1}{T}\int^{T}_{0}f(t)dt=0$, the
condition for the presence of the generalized parity
$\delta+f(t)=-[\delta+f(t+T/2)]$ is equivalent to $\delta=0$ and
$f(t)=-f(t+T/2)$. The condition for the absence of the generalized parity
$\delta+f(t)\neq-[\delta+f(t+T/2)]$ is simply divided into three cases:
$\left\\{\begin{array}[]{c}\delta\neq 0\,{\rm and}\,f(t)=-f(t+T/2);\\\
\delta=0\,{\rm and}\,f(t)\neq-f(t+T/2);\\\ \delta\neq 0\,{\rm
and}\,f(t)\neq-f(t+T/2).\end{array}\right.$ (22)
It is noted that for the biharmonic modulation (21), both $f(t)=-f(t+T/2)$ and
$f(t)\neq-f(t+T/2)$ can be realized by setting $p$ odd and even numbers,
respectively. To verify above analysis, we calculate the numerically exact
fluorescence spectrum from master equation (2) with the FL formalism Ho _et
al._ (1986); Yan _et al._ (2016b), which is compared with the analytical and
semianalytical results from Eq. (LABEL:eq:sfunsa). The analytical and
semianalytical results are obtained by using the transition matrix elements
and quasienergies calculated with the Van Vleck perturbation theory and the ND
of the Floquet Hamiltonian, respectively. The detailed analytical calculation
is presented in Appendix B. In addition, we just focus on the incoherent
components of the fluorescence spectrum, which is of interest in the
experiments. In principle, similar analysis is applicable to the coherent
components. In this work, we mainly consider the parameters regime
$\omega_{z}\sim\Omega_{z}\gg\Omega_{x}\gg\kappa$, in which case both the Van
Vleck perturbation theory (up to second order in $\Omega_{x}$) and secular
approximation can be justified. Importantly, this regime is experimentally
accessible in the artificial atoms, e.g., the transmon qubit Li _et al._
(2013). We should emphasize that if the perturbation theory is inapplicable,
we can obtain the transition matrix elements and quasienergies by the ND of
the Floquet Hamiltonian.
We first verify whether the generalized parity guarantees the symmetry of the
spectrum. In Fig. 1, we display the incoherent component of fluorescence
spectra obtained by the FL numerical method (solid line) and analytical result
(dashed line) for $p=3$, $\delta=0$, and various values of $\phi$. Apparently
the spectra are symmetric as expected. The analytical results are in agreement
with the FL results. These results also show that the spectrum depends weakly
on the relative phase $\phi$. In addition, it is straightforward to verify
that for other driving parameters, the spectrum is symmetric as well when $p$
is an odd number and $\delta=0$. In Appendix C, we show that when $\delta=0$
and $p$ is odd, the transition matrix elements indeed satisfy Eq. (19), which
guarantees the symmetry of the spectrum. The present results suggest that the
symmetry of the spectrum appears as long as $\delta=0$ and $f(t)=-f(t+T/2)$
and fundamentally originates from the generalized parity of the Floquet states
in such a situation.
Figure 3: The incoherent components of the fluorescence spectrum for $p=2$,
$\delta=0$, $\Omega_{x}=10\kappa$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$,
and various phases.
We move to examine whether the symmetry of the spectrum breaks when the
generalized parity is absent, namely, under the conditions
$\delta+f(t)\neq-[\delta+f(t+T/2)]$. We calculate the spectra with the
parameters being the same as in Fig. 1 except for the detuning
$\delta=5\kappa$, corresponding to the case of $\delta\neq 0$ and
$f(t)=-f(T+T/2)$. In Fig. 2, the analytical and FL numerical spectra agree
with each other and are found to be asymmetric for the finite detuning,
indicating that in spite of $f(t)=-f(t+T/2)$, the asymmetry of spectrum
appears when $\delta\neq 0$.
Let us consider the case of $\delta=0$ and $f(t)\neq-f(t+T/2)$ by setting $p$
being even. We calculate the spectrum for $p=2$ and the other parameters being
the same as in Fig. 1. Figure 3 displays that the analytical and numerical
spectra are pronouncedly asymmetric even though $\delta=0$ except for
$\phi=\pi/2$ in which case the analytical spectrum is found to be strictly
symmetric (see discussion below) while the numerical spectrum is slightly
asymmetric [in particular, the intensities of emission lines at
$\Delta=\pm\omega_{z}$ are unequal as shown in Fig. 6(a)]. These results
confirm that the formal spectrum (LABEL:eq:sfunsa) may be symmetric without
the generalized parity of the Floquet states. However, the numerically exact
spectrum is asymmetric in the absence of the generalized parity. This shows
that the generalized parity plays an important role in determining the
symmetry of the exact spectrum. We will further analyze such discrepancy
between the analytical and numerical results later. In addition, we find that
in contrast with $p=3$, the spectrum is found to depend strongly on relative
phase $\phi$ when $p=2$.
Finally we calculate the spectra for $\delta\neq 0$ and $f(t)\neq-f(t+T/2)$.
Figure 4 shows the spectra obtained for the detuning $\delta=5\kappa$ and the
other parameters being the same as in Fig. 3. The spectra are still
asymmetric. In general, it is straightforward to verify the asymmetry of the
spectrum under the condition that $\delta+f(t)\neq-[\delta+f(t+T/2)]$. All in
all, it turns out that the symmetry of the spectrum breaks in the absence of
the generalized parity. Conversely, we can say that the symmetry of the
spectrum can be fully attributed to the presence of the generalized parity. In
contrast to the previous studies, we ascribe the asymmetry to the breaking of
the generalized parity rather than the unequal populations of dressed states
Antón _et al._ (2017) or the breakdown of relation (20) Yan _et al._ (2018).
Figure 4: The incoherent components of fluorescence spectrum for $p=2$,
$\delta=5\kappa$, $\Omega_{x}=10\kappa$, $\Omega_{z}=\omega_{z}=40\kappa$,
$r=1$, and various phase.
Let us explore how the analytical spectrum becomes symmetric in the absence of
the generalized parity of the Floquet states. To this end, we show that the
relation (20) can originate from the identities different from Eq. (19). Based
on the results from the Van Vleck perturbation theory, we analytically derive
the identities for the transition matrix elements in the case of vanishing
detuning and even $p$. The derivation are given in Appendix C. When $p$ is
even, $\delta=0$, and $\phi=\left(1/2+n\right)\pi$ ($n=0,\pm 1,\pm 2,\ldots$),
we find that the following relations hold for arbitrary integer $l$:
$\displaystyle x^{(+)}_{++,-l}$ $\displaystyle=$
$\displaystyle(-1)^{l}x^{(+)}_{++,l},$ (23) $\displaystyle x^{(+)}_{-+,-l}$
$\displaystyle=$ $\displaystyle-(-1)^{l}e^{-i2\theta_{0}}x^{(+)}_{+-,l},$ (24)
where $\theta_{0}$ is a phase defined in Eq. (86). Although the relations (23)
and (24) are derived based on the perturbation theory, it is straightforward
to show that they hold in the nonperturbative regimes. In Fig. 5, we calculate
$x^{(+)}_{++,l}$ $(l=\pm 1,\pm 2)$ with the variation of $\Omega_{x}$ by using
the analytical and ND methods. We see that the deviation between the
analytical and numerical results becomes larger and larger as $\Omega_{x}$
increases, which is due to the breakdown of the perturbation calculation.
Nevertheless, $x^{(+)}_{++,l}$ obtained by the ND method still satisfies Eq.
(23). This suggests that the relations (23) and (24) are not limited to the
perturbative regimes. More importantly, it follows from the identities (23)
and (24) that $|x_{\alpha\beta,l}^{(+)}|=|x_{\beta\alpha,-l}^{(+)}|$, which
leads to the symmetry of the formal spectrum (LABEL:eq:sfunsa). That is to
say, without the generalized parity of the Floquet states, the relation (20)
can also be established from other kinds of the identities for the transition
matrix elements instead of the generalized-parity-induced identity (19) under
certain conditions.
The discrepancy in the symmetry predicted by the analytical and numerical
methods shown in Fig. 3(b) indicates that the relations (23) and (24) cannot
guarantee the symmetry of the spectrum without the secular approximation. To
further verify this, in Fig. 6, we use semianalytical and FL numerical methods
to calculate the weights of the emission lines at $\Delta=\pm\omega_{z}$ as
the increasing of $\Omega_{x}$ for $p=2$, $\delta=0$, and two values of
$\phi$. It is evident that the weights calculated from the semianalytical
method (solid and dashed lines) are the same while the weights from the
numerical method (dot-dashed and dotted lines) are unequal, indicating that
the semianalytical spectrum is symmetric but the numerical spectrum is not
symmetric. The present results illustrate that that provided the relation (20)
is established in the absence of the generalized parity, the secular
approximation can induce artifact symmetry that vanishes if such approximation
is not invoked.
Figure 5: Transition matrix elements $x^{(+)}_{++,l}$ versus driving strength
$\Omega_{x}$, calculated from the analytical method and the numerical method
based on the ND of the Floquet Hamiltonian for $p=2$, $\delta=0$,
$\Omega_{z}=\omega_{z}=40\kappa$, $\phi=\pi/2$, and $r=1$.
Apart from the biharmonic modulation, we find that the conditions for the
symmetry and asymmetry of the spectrum, which are derived based on the
generalized parity, are applicable to the simple harmonic and multiharmonic
modulation cases. For the simple harmonic modulation
$f(t)=\Omega_{z}\cos(\omega_{z}t)$, $f(t)=-f(t+T/2)$ is met. Therefore, the
symmetry and asymmetry of the spectrum is uniquely controlled by the detuning
$\delta$, which simply interprets the detuning-dependent symmetry of the
spectrum. Specifically, the spectrum is expected to be symmetric when
$\delta=0$ and asymmetric when $\delta\neq 0$. This is consistent with the
findings of previous studies Yan _et al._ (2016b); Antón _et al._ (2017);
Yan _et al._ (2018). For the multiharmonic modulation
$f(t)=\sum_{p=1}^{N}\Omega_{z,p}\cos(p\omega_{z}t+\phi_{p})]$, where
$\Omega_{z,p}$ and $\phi_{p}$ are the amplitude and phase of the $p$th
harmonic, respectively, either $f(t)=-f(t+T/2)$ or $f(t)\neq-f(t+T/2)$ can be
met, similarly to the biharmonic case. We have calculated the spectrum with
the FL and semianalytical methods for the cases of $N=3$, $N=4$, and $N=5$.
The results (not shown here) further confirm that the symmetry and asymmetry
of spectrum fundamentally originate from the presence and absence of the
generalized parity of the Floquet states, respectively.
Figure 6: Weights of emission lines at $\Delta=\pm\omega_{z}$ versus driving
strength $\Omega_{x}$, calculated from the semianalytical method and the FL
method, for $p=2$, $\delta=0$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$, and
two values of $\phi$. “Semiana.” denotes the semianalytical result.
## IV Conclusions
In summary, we have studied the fundamental origin of the symmetry of the
resonance fluorescence from the two-level system subjected to a periodic
frequency modulation and a near-resonant high-frequency monochromatic
excitation by using both analytical and numerical methods based on the Floquet
theory. In such a driven two-level system, we have found that the generalized
parity of Floquet states plays a fundamental role in the symmetry of the
spectrum. Specifically, the generalized parity guarantees the symmetry of the
spectrum. On the other hand, when the generalized parity is broken, the
spectrum becomes asymmetric. This has been illustrated in the context of the
biharmonic modulation, the parameters of which can be tuned to induce or break
the generalized parity. For the biharmonic modulation, we find that when
$\delta=0$ and $f(t)=-f(t+T/2)$, the generalized parity exists and the
spectrum is symmetric. When $\delta+f(t)\neq-[\delta+f(t+T/2)]$, the
generalized parity is broken and the spectrum is found to be asymmetric.
Interestingly, we can obtain pronouncedly asymmetric spectrum by requiring the
modulation $f(t)\neq-f(t+T/2)$ even though $\delta=0$. Moreover, these
conditions for the symmetry and asymmetry of the spectrum are found to be
applicable to the simple harmonic and multiharmonic modulation cases. In
addition, we illustrated that the secular approximation may induce artifact
symmetry that vanishes if the secular approximation is avoided under certain
conditions. The present study gives a deep insight into the origin of the
symmetry of the spectrum and reveals a simple relation between the symmetry of
the spectrum and the generalized parity of the Floquet states.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
(Grants No. 11647082, No. 11774311, No. 11774226, and No. 11874260).
## Appendix A Derivation of symmetry of the spectrum without the secular
approximation
The master equation can be rewritten in a matrix form
$\frac{d}{dt}\vec{\tilde{\rho}}(t)={\cal L}(t)\vec{\tilde{\rho}}(t).$ (25)
Here the vector is defined as
$\vec{\tilde{\rho}}(t)=(\langle\tilde{\sigma}_{+}(t)\rangle,\langle\tilde{\sigma}_{-}(t)\rangle,\langle\tilde{\pi}_{+}(t)\rangle,\langle\tilde{\pi}_{-}(t)\rangle)^{{\rm
T}},$ (26)
where $\pi_{\pm}=(1\pm\sigma_{z})/2$ and
$\langle\tilde{\hat{o}}(t)\rangle\equiv{\rm Tr}[\hat{o}\tilde{\rho}(t)]$. The
superoperator ${\cal L}(t)$ in the Liouville space spanned by the matrix bases
$\\{\sigma_{\pm},\pi_{\pm}\\}$ is given by
${\cal
L}(t)=\left(\begin{array}[]{cccc}i[\delta+f(t)]-\frac{\kappa}{2}&0&-\frac{i\Omega_{x}}{2}&\frac{i\Omega_{x}}{2}\\\
0&-i[\delta+f(t)]-\frac{\kappa}{2}&\frac{i\Omega_{x}}{2}&\frac{-i\Omega_{x}}{2}\\\
\frac{-i\Omega_{x}}{2}&\frac{i\Omega_{x}}{2}&-\kappa&0\\\
\frac{i\Omega_{x}}{2}&\frac{-i\Omega_{x}}{2}&\kappa&0\end{array}\right).$ (27)
If $\delta+f(t)=-[\delta+f(t+T/2)]$, in which case the Hamiltonian is
invariant under the generalized parity transformation, one readily finds that
${\cal T}{\cal L}(t+T/2){\cal T}={\cal L}(t),$ (28)
where the transformation matrix is given by
${\cal T}=\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{array}\right),$ (29)
and ${\cal T}^{2}=I$ with $I$ being the identity matrix. Similarly to the
Hamiltonian, the matrix ${\cal L}(t)$ is invariant under the transformation
defined in Eq. (28), which can be regarded as the generalized parity
transformation in the Liouville space, similarly to that defined in Eq. (16)
of the main text.
Let us derive the specific property of the steady state in the long-time limit
[as $\det{\cal L}(t)=0$, there exists a nontrivial steady state]. It follows
from Eq. (25) that
$\frac{d}{dt}\vec{\tilde{\rho}}(t+T/2)={\cal
L}(t+T/2)\vec{\tilde{\rho}}(t+T/2),$ (30)
which leads to
$\displaystyle\frac{d}{dt}{\cal T}\vec{\tilde{\rho}}(t+T/2)$ $\displaystyle=$
$\displaystyle{\cal T}{\cal L}(t+T/2){\cal T}{\cal
T}\vec{\tilde{\rho}}(t+T/2)={\cal L}(t){\cal T}\vec{\tilde{\rho}}(t+T/2),$
(31)
which means that ${\cal T}\vec{\tilde{\rho}}(t+T/2)=c\vec{\tilde{\rho}}(t)$,
owing to the uniqueness of solutions of the differential equation. On using
the fact that $\vec{\tilde{\rho}}(t)=\vec{\tilde{\rho}}(t+T)$ as
$t\rightarrow\infty$ because of ${\cal L}(t)={\cal L}(t+T)$, we find that $c$
may be either $+1$ or $-1$. It is easy to prove by contradiction that $c=-1$.
Suppose that $c=1$, yielding
$\langle\tilde{\pi}_{+}(t+T/2)\rangle=-\langle\tilde{\pi}_{+}(t)\rangle$.
However, if one considers $\delta+f(t)=0$ in which case ${\cal L}(t)$ is time
independent while Eq. (28) still holds, the steady state becomes time
independent and one gets
$\langle\tilde{\pi}_{+}(t)\rangle=\langle\tilde{\pi}_{+}(t+T/2)\rangle$. By
contradiction, one finds that $c=-1$. Consequently, in the steady-state limit,
we have
${\cal
T}\vec{\tilde{\rho}}(t+T/2)=-\vec{\tilde{\rho}}(t)\quad(t\rightarrow\infty).$
(32)
Next, let us derive the property of the principal matrix solution
$\Pi(t,t^{\prime})$ of the master equation, which solves the differential
equation
$\frac{d}{dt}\Pi(t,t^{\prime})={\cal L}(t)\Pi(t,t^{\prime}),$ (33)
with the initial condition $\Pi(t^{\prime},t^{\prime})=I$. It is
straightforward to show that
$\displaystyle\frac{d}{dt}{\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}$
$\displaystyle=$ $\displaystyle{\cal T}{\cal L}(t+T/2){\cal T}{\cal
T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}={\cal L}(t){\cal
T}\Pi(t+T/2,t^{\prime}+T/2){\cal T},$ (34)
namely, ${\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}$ satisfies the same
differential equation and the same initial condition as $\Pi(t,t^{\prime})$.
As a result, we simply have
${\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}=\Pi(t,t^{\prime}).$ (35)
According to the quantum regression theory Mollow (1969), the two-time
correlation functions
$\vec{\tilde{g}}(t,t^{\prime})=(\langle\tilde{\sigma}_{+}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle,\langle\tilde{\sigma}_{-}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle,\langle\tilde{\pi}_{+}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle,\langle\tilde{\pi}_{-}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle)^{{\rm
T}}$ (36)
satisfy the same equation as $\vec{\tilde{\rho}}(t)$, however, with a
different initial condition
$\vec{\tilde{g}}(t^{\prime},t^{\prime})=(\langle\tilde{\pi}_{+}(t^{\prime})\rangle,0,0,\langle\tilde{\sigma}_{-}(t^{\prime})\rangle)^{{\rm
T}}.$ (37)
Similarly, another set of two-time correlation functions
$\vec{\tilde{G}}(t,t^{\prime})=(\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\sigma}_{+}(t)\rangle,\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\sigma}_{-}(t)\rangle,\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\pi}_{+}(t)\rangle,\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\pi}_{-}(t)\rangle)^{{\rm
T}}$ (38)
also satisfy the same differential equation as $\vec{\tilde{g}}(t,t^{\prime})$
but with the initial condition
$\vec{\tilde{G}}(t^{\prime},t^{\prime})=(0,\langle\tilde{\pi}_{+}(t^{\prime})\rangle,0,\langle\tilde{\sigma}_{+}(t^{\prime})\rangle)^{{\rm
T}}.$ (39)
Using Eq. (32), we have
${\cal T}\vec{\tilde{g}}(t^{\prime},t^{\prime})=\left(\begin{array}[]{c}0\\\
\langle\tilde{\pi}_{+}(t^{\prime})\rangle\\\ 0\\\
-\langle\tilde{\sigma}_{-}(t^{\prime})\rangle\end{array}\right)=\left(\begin{array}[]{c}0\\\
\langle\tilde{\pi}_{+}(t^{\prime}+T/2)\rangle\\\ 0\\\
\langle\tilde{\sigma}_{+}(t^{\prime}+T/2)\rangle\end{array}\right)=\vec{\tilde{G}}\left(t^{\prime}+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)\quad(t^{\prime}\rightarrow\infty).$
(40)
In the steady-state limit, the correlation functions are found to have the
following relation
$\displaystyle\vec{\tilde{g}}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\Pi(t,t^{\prime})\vec{\tilde{g}}(t^{\prime},t^{\prime})$ (41)
$\displaystyle=$ $\displaystyle{\cal
T}\Pi\left(t+\frac{T}{2},t^{\prime}+\frac{T}{2}\right){\cal
T}\vec{\tilde{g}}(t^{\prime},t^{\prime})$ $\displaystyle=$ $\displaystyle{\cal
T}\Pi\left(t+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)\vec{\tilde{G}}\left(t^{\prime}+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)$
$\displaystyle=$ $\displaystyle{\cal
T}\vec{\tilde{G}}\left(t+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)\quad(t^{\prime}\rightarrow\infty).$
It follows that as $t^{\prime}\rightarrow\infty$,
$\displaystyle\langle\tilde{\sigma}_{+}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\langle\tilde{\sigma}_{+}(t^{\prime}+T/2)\tilde{\sigma}_{-}(t+T/2)\rangle$
(42) $\displaystyle=$
$\displaystyle\langle\tilde{\sigma}_{+}(t+T/2)\tilde{\sigma}_{-}(t^{\prime}+T/2)\rangle^{\ast}.$
In the steady-state limit, the first-order correlation function depends
explicitly on time $t^{\prime}$, however, the $t^{\prime}$ dependence can be
eliminated by setting $t=\tau+t^{\prime}$ and integrating over $t^{\prime}$
(because the contributions of $t^{\prime}$-dependent terms are negligible to a
long-time observation), yielding the $\tau$-dependent first-order correlation
function
$\displaystyle\bar{\tilde{g}}_{1}(\tau)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{T}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle
dt^{\prime}$ (43) $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime}+T/2)\tilde{\sigma}_{-}(t^{\prime}+T/2)\rangle^{\ast}dt^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{T}\int_{T/2}^{T+T/2}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle^{\ast}dt^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle^{\ast}dt^{\prime}$
$\displaystyle=$ $\displaystyle\bar{\tilde{g}}_{1}^{\ast}(\tau),$
where we used relation (42) and the fact that
$\langle\tilde{\sigma}_{+}(\tau+t^{\prime}+T)\tilde{\sigma}_{-}(t^{\prime}+T)\rangle^{\ast}=\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle^{\ast}$
as $t^{\prime}\rightarrow\infty$. This means that the generalized parity
guarantees that the correlation function is a real-valued function of $\tau$
in the rotating frame and thus results in the symmetry of the spectrum when
$\delta+f(t)=-[\delta+f(t+T/2)]$. This is consistent with the prediction from
the spectrum (LABEL:eq:sfunsa).
In general, it is a formidable task to show that the spectrum is asymmetric
when $\delta+f(t)\neq-[\delta+f(t+T/2)]$ with or without the secular
approximation. Nevertheless, from the above derivation, one readily notes that
the generalized parity plays an important role in determining the symmetry of
the spectrum. Consequently, if such parity breaks, it is not difficult to
imagine that the symmetry of the spectrum also breaks trivially if there is no
other symmetry-inducing mechanism.
## Appendix B Analytical calculation of quasienergies and transition matrix
elements in the biharmonic modulation case
We use the Van Vleck perturbation theory Cohen-Tannoudji _et al._ (1998);
Hausinger and Grifoni (2010) to analytically calculate the quasienergies and
transition matrix elements $x_{\alpha\beta,l}^{(+)}$ for the biharmonic
modulation, which leads to the analytical fluorescence spectrum. Since we are
interested in the regime of $\Omega_{z},\,\omega_{z}\gg\Omega_{x}$, which is
accessible in the experiment Li _et al._ (2013), we use $\Omega_{x}$ as the
perturbation parameter. We first transform Eq. (6) with the unitary
transformation
$e^{S(t)}[\tilde{H}(t)-i\partial_{t}]e^{-S(t)}e^{S(t)}|\tilde{u}_{\alpha}(t)\rangle=\tilde{\varepsilon}_{\alpha}e^{S(t)}|\tilde{u}_{\alpha}(t)\rangle,$
(44)
where
$S(t)=i\frac{\Omega_{z}}{2\omega_{z}}\left\\{\sin(\omega_{z}t)+\frac{r}{p}[\sin(p\omega_{z}t+\phi)-\sin\phi]\right\\}\sigma_{z}.$
(45)
We can define the transformed Floquet states and transformed Hamiltonian as
follows:
$|u_{\alpha}^{\prime}(t)\rangle=e^{S(t)}|\tilde{u}_{\alpha}(t)\rangle,$ (46)
$\displaystyle H^{\prime}(t)$ $\displaystyle=$ $\displaystyle
e^{S(t)}[\tilde{H}(t)-i\partial_{t}]e^{-S(t)}$ (47) $\displaystyle=$
$\displaystyle\frac{1}{2}\delta\sigma_{z}+\frac{1}{2}\sum_{l}(f_{l}\sigma_{+}+f_{-l}^{\ast}\sigma_{-})e^{il\omega_{z}t},$
where
$f_{l}=\Omega_{x}F_{l},$ (48)
and
$\displaystyle F_{l}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}e^{i\frac{\Omega_{z}}{\omega_{z}}\left\\{\sin(\omega_{z}t)+\frac{r}{p}[\sin(p\omega_{z}t+\phi)-\sin\phi]\right\\}-il\omega_{z}t}dt$
(49) $\displaystyle=$ $\displaystyle
e^{-i\Theta}\sum_{k}J_{k}\left(\frac{r\Omega_{z}}{p\omega_{z}}\right)J_{l-kp}\left(\frac{\Omega_{z}}{\omega_{z}}\right)e^{ik\phi},$
with $\Theta=\frac{r\Omega_{z}}{p\omega_{z}}\sin\phi$ and $J_{k}(z)$ being the
Bessel function of the first kind. To proceed, we introduce an extended
Hilbert space in which the time-dependent Floquet Hamiltonian
$H^{\prime}(t)-i\partial_{t}$ becomes time independent Sambe (1973). One
readily introduces the Fourier basis $|l\rangle\equiv\exp(il\omega_{z}t)$ and
inner product $\langle
l|n\rangle\equiv\frac{1}{T}\int_{0}^{T}\exp[i(n-l)\omega_{z}t]dt=\delta_{l,n}$,
where $\delta_{l,n}$ is the Kronecker delta function. Denoting
$|\uparrow\rangle$ and $|\downarrow\rangle$ as the eigenstates for
$\sigma_{z}$ with the eigenvalues $+1$ and $-1$, respectively, one gets the
composite bases
$|\uparrow(\downarrow),l\rangle=|\uparrow(\downarrow)\rangle\otimes|l\rangle$.
In the extended Hilbert space spanned by such bases, we can obtain the
explicit form of the Floquet Hamiltonian, which is written as
$\displaystyle H_{{\cal F}}^{\prime}$ $\displaystyle=$ $\displaystyle
H^{\prime}(t)-i\partial_{t}$ (50) $\displaystyle=$
$\displaystyle\frac{1}{2}\delta\sigma_{z}+\sum_{n}n\omega_{z}|n\rangle\langle
n|+\frac{1}{2}\sum_{n,l}(f_{l}\sigma_{+}+f_{-l}^{\ast}\sigma_{-})$
$\displaystyle\otimes|n+l\rangle\langle n|.$
The Floquet Hamiltonian has an infinite size and is difficult to be
diagonalized exactly in analytical calculation. To carry out perturbation
calculation, we transform the Floquet Hamiltonian with a further unitary
transformation with the Hermitian generator $K$, leading to
$\displaystyle H_{{\cal F}}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
e^{iK}H_{{\cal F}}^{\prime}e^{-iK}$ (51) $\displaystyle=$ $\displaystyle
H_{\cal F}^{\prime}+[iK,H_{{\cal F}}^{\prime}]+\frac{1}{2!}[iK,[iK,H_{{\cal
F}}^{\prime}]]+\ldots,$
where the explicit form of $K$ is to be determined by requiring $H_{{\cal
F}}^{\prime\prime}$ to be block diagonal. The generator is expanded as
$K=K^{(1)}+K^{(2)}+K^{(3)}+\ldots,$ (52)
where the superscripts indicate the orders in the perturbation. We use
$H_{0}=\frac{1}{2}\delta\sigma_{z}+\sum_{n}n\omega_{z}|n\rangle\langle n|$ and
$V=\frac{1}{2}\sum_{n,l}(f_{l}\sigma_{+}+f_{-l}^{\ast}\sigma_{-})\otimes|n+l\rangle\langle
n|$ as the dominate and perturbation components, respectively. Up to the
second order in $\Omega_{x}$, we have
$\displaystyle H_{{\cal F}}^{\prime\prime}$ $\displaystyle\simeq$
$\displaystyle H_{0}+V+[iK^{(1)},H_{0}]+[iK^{(1)},V]+[iK^{(2)},H_{0}]$ (53)
$\displaystyle+\frac{1}{2}[iK^{(1)},[iK^{(1)},H_{0}]].$
Next, we discuss under which condition the transformed Hamiltonian may
reasonably be block diagonal. For the dominate component $H_{0}$, we simply
have
$H_{0}|\uparrow(\downarrow),n\rangle=[+(-)\delta/2+n\omega_{z}]|\uparrow(\downarrow),n\rangle\equiv\tilde{\varepsilon}^{(0)}_{+(-),n}|\uparrow(\downarrow),n\rangle$.
Provided that
$\tilde{\varepsilon}^{(0)}_{+,n}-\tilde{\varepsilon}^{(0)}_{-,n+m}=\delta-m\omega_{z}\approx
0$, we have a subspace spanned by two almost degenerate unperturbed states
$|\uparrow,n\rangle$ and $|\downarrow,n+m\rangle$, where $n$ is an arbitrary
integer and $m$ is the integer nearest to $\delta/\omega_{z}$. The projection
onto such a subspace is realized by the operator:
$\Pi_{n}=|\uparrow,n\rangle\langle\uparrow,n|+|\downarrow,n+m\rangle\langle\downarrow,n+m|.$
(54)
The eigenvalues of the dominate component $H_{0}$ in the $n$th subspace are
well-separated from those in the $(n+l)$th subspace as long as
$|l\omega_{z}|\gg|\delta-m\omega_{z}|$ for any $l\neq 0$. Moreover, if we
assume that
$|\langle\uparrow,n|V|\downarrow,n+l+m\rangle|\ll|\tilde{\varepsilon}^{(0)}_{+,n}-\tilde{\varepsilon}^{(0)}_{-,n+l+m}|,$
(55)
which is simply $|f_{-l-m}/2|\ll|l\omega_{z}|$, the transitions between the
states in the different subspaces can be neglected up to a certain order in
the perturbation Cohen-Tannoudji _et al._ (1998), yielding the following
condition
$\Pi_{n}H_{{\cal F}}^{\prime\prime}\Pi_{l}=0,$ (56)
for $n\neq l$. Therefore, $H_{{\cal F}}^{\prime\prime}$ is block diagonal. The
second condition that $K$ cannot have matrix elements inside each subspace of
two almost degenerate states is also assumed, i.e.,
$\Pi_{n}K\Pi_{n}=0.$ (57)
The generator can now be fully determined via Eqs. (56) and (57). The
nonvanishing elements of $K^{(1)}$ and $K^{(2)}$are given by
$\langle\uparrow,n|iK^{(1)}|\downarrow,l\rangle=\frac{1}{2}\frac{f_{n-l}}{\delta+(n-l)\omega_{z}},$
(58)
$\langle\downarrow,l|iK^{(1)}|\uparrow,n\rangle=-\frac{1}{2}\frac{f_{n-l}^{\ast}}{\delta+(n-l)\omega_{z}},$
(59)
for $n-l\neq-m$, and
$\displaystyle\langle\uparrow,n|iK^{(2)}|\uparrow,l\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4(n-l)\omega_{z}}\left\\{\sum_{k\neq
n+m,l+m}\frac{f_{n-k}f_{l-k}^{\ast}}{2}\left[\frac{1}{\delta+(n-k)\omega_{z}}+\frac{1}{\delta+(l-k)\omega_{z}}\right]\right.$
(60)
$\displaystyle\left.+\frac{f_{l-n-m}^{\ast}f_{-m}}{\delta+(l-n-m)\omega_{z}}+\frac{f_{n-l-m}f_{-m}^{\ast}}{\delta+(n-l-m)\omega_{z}}\right\\},$
$\displaystyle\langle\downarrow,n|iK^{(2)}|\downarrow,l\rangle$
$\displaystyle=$ $\displaystyle-\frac{1}{4(n-l)\omega_{z}}\left\\{\sum_{k\neq
l-m,n-m}\frac{f_{k-n}^{\ast}f_{k-l}}{2}\left[\frac{1}{\delta+(k-n)\omega_{z}}+\frac{1}{\delta+(k-l)\omega_{z}}\right]\right.$
(61)
$\displaystyle+\left.\frac{f_{l-n-m}^{\ast}f_{-m}}{\delta+(l-n-m)\omega_{z}}+\frac{f_{n-l-m}f_{-m}^{\ast}}{\delta+(n-l-m)\omega_{z}}\right\\},$
for $n\neq l$. The rest elements of $K^{(1)}$ and $K^{(2)}$ are vanishing.
The transformed Hamiltonian have the $2\times 2$ submatrix $H_{{\cal
F}}^{\prime\prime(n)}$ in the diagonal, which reads Cohen-Tannoudji _et al._
(1998)
$\displaystyle H_{{\cal F}}^{\prime\prime(n)}$ $\displaystyle=$ $\displaystyle
H_{0}\Pi_{n}+\Pi_{n}V\Pi_{n}+\frac{1}{2}\Pi_{n}[iK^{(1)},V]\Pi_{n}$ (64)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\frac{\delta}{2}+n\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{4(\delta+j\omega_{z})}&\frac{f_{-m}}{2}\\\
\frac{f_{-m}^{\ast}}{2}&-\frac{\delta}{2}+(n+m)\omega_{z}-\sum_{j\neq-m}\frac{|f_{j}|^{2}}{4(\delta+j\omega_{z})}\end{array}\right).$
One can diagonalize the submatrix $H_{{\cal F}}^{\prime\prime(n)}$
analytically. Its eigenvalues (quasienergies) are
$\tilde{\varepsilon}_{\pm,n}=\frac{1}{2}\left(m\omega_{z}\pm\Omega_{m}\right)+n\omega_{z},$
(65)
where
$\Omega_{m}=\sqrt{\left[\delta-m\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{2(\delta+j\omega_{z})}\right]^{2}+|f_{-m}|^{2}}.$
(66)
The eigenvectors are given by
$\displaystyle|\Psi_{+,n}^{\prime\prime}\rangle$ $\displaystyle=$
$\displaystyle u|\uparrow,n\rangle+v|\downarrow,n+m\rangle,$ (67)
$\displaystyle|\Psi_{-,n}^{\prime\prime}\rangle$ $\displaystyle=$
$\displaystyle v|\uparrow,n\rangle-u^{\ast}|\downarrow,n+m\rangle,$ (68)
with
$\displaystyle u$ $\displaystyle=$
$\displaystyle\frac{f_{-m}}{|f_{-m}|}\sqrt{\frac{1}{2}\left[1+\frac{1}{\Omega_{m}}\left(\delta-m\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{2(\delta+j\omega_{z})}\right)\right]},$
(69) $\displaystyle v$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}\left[1-\frac{1}{\Omega_{m}}\left(\delta-m\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{2(\delta+j\omega_{z})}\right)\right]}.$
(70)
The eigenvectors for $H_{{\cal F}}^{\prime}$ can be derived as follows:
$|\Psi_{\pm,n}^{\prime}\rangle=e^{-iK}|\Psi_{\pm,n}^{\prime\prime}\rangle\simeq\left(1-iK^{(1)}-iK^{(2)}+\frac{1}{2!}iK^{(1)}iK^{(1)}\right)|\Psi_{\pm,n}^{\prime\prime}\rangle.$
(71)
It is straightforward to derive the explicit form of the eigenvectors, which
reads
$\displaystyle|\Psi_{+,n}^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{{\cal{\cal N}}}\left\\{uB|\uparrow,n\rangle-\sum_{j\neq
0}P_{j}|\uparrow,n+j\rangle+vB|\downarrow,n+m\rangle+\sum_{j\neq
0}Q_{j}|\downarrow,n+m+j\rangle\right\\},$ (72)
$\displaystyle|\Psi_{-,n}^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{{\cal{\cal N}}}\left\\{vB|\uparrow,n\rangle+\sum_{j\neq
0}Q_{-j}^{\ast}|\uparrow,n+j\rangle-u^{\ast}B|\downarrow,n+m\rangle+\sum_{j\neq
0}P_{-j}^{\ast}|\downarrow,n+m+j\rangle\right\\},$ (73)
where
$B=1-\frac{1}{8}\sum_{l\neq-m}\frac{|f_{l}|^{2}}{(\delta+l\omega_{z})^{2}},$
(74) $\displaystyle P_{j}$ $\displaystyle=$
$\displaystyle\frac{f_{j-m}}{2[\delta+(j-m)\omega_{z}]}\left(v+\frac{uf_{-m}^{\ast}}{2j\omega_{z}}\right)+\frac{u}{4j\omega_{z}}\sum_{k\neq-m}\frac{f_{k+j}f_{k}^{\ast}}{\delta+k\omega_{z}},$
(75) $\displaystyle Q_{j}$ $\displaystyle=$
$\displaystyle\frac{f_{-j-m}^{\ast}}{2[\delta-(j+m)\omega_{z}]}\left(u+\frac{vf_{-m}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq-m}\frac{f_{k-j}^{\ast}f_{k}}{\delta+k\omega_{z}},$
(76)
and ${\cal N}=\sqrt{B^{2}+\sum_{j\neq 0}(|P_{j}|^{2}+|Q_{j}|^{2})}$ is the
normalization factor. The Floquet states $|u_{\alpha,n}^{\prime}(t)\rangle$
with the quasienergy $\tilde{\varepsilon}_{\alpha,n}$ can be derived from
$|\Psi_{\alpha,n}^{\prime}\rangle$ by replacing $|n\rangle$ with
$e^{in\omega_{z}t}$.
With above results at hand, we can analytically calculate the transition
matrix element
$\displaystyle x_{\alpha\beta,l}^{(+)}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\langle\tilde{u}_{\alpha}(t)|\sigma_{\pm}|\tilde{u}_{\beta}(t)\rangle
e^{-il\omega_{z}t}dt=\frac{1}{T}\int_{0}^{T}\langle
u_{\alpha}^{\prime}(t)|e^{S(t)}\sigma_{+}e^{-S(t)}|u_{\beta}^{\prime}(t)\rangle
e^{-il\omega_{z}t}dt$ (77) $\displaystyle=$
$\displaystyle\sum_{n}\frac{1}{T}\int_{0}^{T}F_{n}\langle
u_{\alpha}^{\prime}(t)|\sigma_{+}|u_{\beta}^{\prime}(t)\rangle
e^{i(n-l)\omega_{z}t}dt=\sum_{n}F_{n+l}\langle\Psi_{\alpha,0}^{\prime}|\sigma_{+}|\Psi_{\beta,n}^{\prime}\rangle,$
and
$\displaystyle\langle\Psi_{+,0}^{\prime}|\sigma_{+}|\Psi_{+,n}^{\prime}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{{\cal
N}^{2}}\left\\{u^{\ast}vB^{2}\delta_{n,-m}-\sum_{j\neq
0,n+m}P_{j}^{\ast}Q_{j-n-m}+(u^{\ast}Q_{-n-m}-vP_{n+m}^{\ast})B(1-\delta_{n,-m})\right\\},$
(78)
$\displaystyle\langle\Psi_{+,0}^{\prime}|\sigma_{+}|\Psi_{-,n}^{\prime}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{{\cal
N}^{2}}\left\\{-(u^{\ast})^{2}B^{2}\delta_{n,-m}-\sum_{j\neq
0,n+m}P_{j}^{\ast}P_{n+m-j}^{\ast}+2u^{\ast}P_{n+m}^{\ast}B(1-\delta_{n,-m})\right\\},$
(79)
$\displaystyle\langle\Psi_{-,0}^{\prime}|\sigma_{+}|\Psi_{+,n}^{\prime}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{{\cal
N}^{2}}\left\\{v^{2}B^{2}\delta_{n,-m}+\sum_{j\neq
0,n+m}Q_{-j}Q_{j-n-m}+2vQ_{-n-m}B(1-\delta_{n,-m})\right\\},$ (80)
$\displaystyle\langle\Psi_{-,0}^{\prime}|\sigma_{+}|\Psi_{-,n}^{\prime}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{{\cal
N}^{2}}\left\\{-u^{\ast}vB^{2}\delta_{n,-m}+\sum_{j\neq
0,n+m}P_{j}^{\ast}Q_{j-n-m}+(vP_{n+m}^{\ast}-u^{\ast}Q_{-n-m})B(1-\delta_{n,-m})\right\\},$
(81)
where $(1-\delta_{n,-m})$ indicates that the term vanishes for $n=-m$.
Clearly, the validity of the perturbation theory is limited to the condition
(55). For $\delta\approx 0$, roughly speaking, the above results can be
justified when $r\sim 1$ and $\omega_{z}\sim\Omega_{z}\gg\Omega_{x}$.
## Appendix C Equalities for transition matrix elements in the vanishing
detuning case
For the biharmonic modulation, we show the equalities that the transition
matrix elements satisfy under the vanishing detuning condition ($\delta=0$)
using the above analytical results, which helps us to understand the symmetry
of the spectrum in the main text. It follows from Eq. (49) that
$\displaystyle F_{-l}$ $\displaystyle=$ $\displaystyle
e^{-i\Theta}\sum_{k}J_{k}\left(\frac{r\Omega_{z}}{p\omega_{z}}\right)J_{-l-kp}\left(\frac{\Omega_{z}}{\omega_{z}}\right)e^{ik\phi}$
(82) $\displaystyle=$
$\displaystyle(-1)^{l}e^{-i\Theta}\sum_{k}J_{k}\left(\frac{r\Omega_{z}}{p\omega_{z}}\right)(-1)^{k(p+1)}$
$\displaystyle\times
J_{l-kp}\left(\frac{\Omega_{z}}{\omega_{z}}\right)e^{-ik\phi},$
where we used the relation $J_{-n}(z)=(-1)^{n}J_{n}(z)$. It is evident that
when $p$ is an odd number, $p+1$ is even and thus $(-1)^{k(p+1)}=1$, leading
to
$F_{-l}=(-1)^{l}e^{-i2\Theta}F_{l}^{\ast}.$ (83)
When $p$ is an even number, $(-1)^{k(p+1)}=(-1)^{k}$ may be either $+1$ or
$-1$. Nevertheless, we can obtain a simple relation between $F_{l}$ and
$F_{-l}$ by setting
$(-1)^{k}e^{-ik\phi}=e^{ik\phi},$ (84)
which yields that $\phi=\left(1/2+n\right)\pi$ $(n=0,\pm 1,\pm 2,\ldots)$.
With an even $p$ and such values of phase, we have
$F_{l}=(-1)^{l}F_{-l}.$ (85)
We should emphasize that Eqs. (83) and (85) hold under different conditions.
The former is available when $p$ is odd and regardless of $\phi$ while the
latter is established when $p$ is even and $\phi=(1/2+n)\pi$.
Provided that $\delta=0$, we get $m=\delta/\omega_{z}=0$. We define the phase
of $F_{0}$ via
$F_{0}=e^{-i\theta_{0}}|F_{0}|.$ (86)
Together with Eqs. (69) and (70), we simply have
$v=ue^{i\theta_{0}}$ (87)
with the aid of Eq. (83) or (85). Such an equality between $u$ and $v$ is
valid only for $\delta=0$ and in the valid regime of Eq. (83) or (85).
### C.1 Odd $p$
We consider that $p$ is an odd number. It follows from Eq. (49) that
$\theta_{0}=\Theta$. Using $\delta=0$ and Eqs. (83) and (87), one readily gets
from Eqs. (75) and (76) that
$\displaystyle Q_{j}$ $\displaystyle=$
$\displaystyle-\frac{f_{-j}^{\ast}}{2j\omega_{z}}\left(u+\frac{vf_{0}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq
0}\frac{f_{k-j}^{\ast}f_{k}}{k\omega_{z}}$ (88) $\displaystyle=$
$\displaystyle\frac{(-1)^{j+1}e^{i2\Theta}f_{j}}{2j\omega_{z}}\left(u+\frac{vf_{0}^{\ast}e^{-i2\Theta}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq
0}\frac{f_{-k-j}^{\ast}f_{-k}}{-k\omega_{z}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{j+1}e^{i\Theta}f_{j}}{2j\omega_{z}}\left(v+\frac{uf_{0}^{\ast}}{2j\omega_{z}}\right)+\frac{e^{i\Theta}u}{4j\omega_{z}}\sum_{k\neq
0}\frac{(-1)^{j+1}f_{k+j}f_{k}^{\ast}}{k\omega_{z}}$ $\displaystyle=$
$\displaystyle(-1)^{j+1}e^{i\Theta}P_{j}.$
From this relation and Eqs. (77)-(80), it is straightforward to show that
$\displaystyle\left[x_{-+,-l}^{(+)}\right]^{\ast}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{F_{n-l}^{\ast}}{{\cal
N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{n\neq
0,n}Q_{-j}^{\ast}Q_{j-n}^{\ast}+2vBQ_{-n}^{\ast}(1-\delta_{n,0})\right\\}$
(89) $\displaystyle=$ $\displaystyle\sum_{n}\frac{F_{-n-l}^{\ast}}{{\cal
N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{j\neq
0,-n}Q_{-j}^{\ast}Q_{j+n}^{\ast}+2vBQ_{n}^{\ast}(1-\delta_{n,0})\right\\}$
$\displaystyle=$
$\displaystyle\sum_{n}\frac{(-1)^{n+l}F_{n+l}e^{i2\Theta}}{{\cal
N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{j\neq
0,n}Q_{j}^{\ast}Q_{n-j}^{\ast}+2vBQ_{n}^{\ast}(1-\delta_{n,0})\right\\}$
$\displaystyle=$
$\displaystyle\sum_{n}\frac{(-1)^{n+l}F_{n+l}e^{i2\Theta}}{{\cal
N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{j\neq
0,n}(-1)^{n}e^{-i2\Theta}P_{j}^{\ast}P_{n-j}^{\ast}+2vB(-1)^{n+1}e^{-i\Theta}P_{n}^{\ast}(1-\delta_{n,0})\right\\}$
$\displaystyle=$ $\displaystyle(-1)^{l}\sum_{n}\frac{F_{n+l}}{{\cal
N}^{2}}\left\\{(u^{\ast})^{2}B^{2}\delta_{n,0}+\sum_{j\neq
0,n}P_{j}^{\ast}P_{n-j}^{\ast}-2u^{\ast}BP_{n}^{\ast}(1-\delta_{n,0})\right\\}$
$\displaystyle=$ $\displaystyle-(-1)^{l}x_{+-,l}^{(+)}.$
Similarly, we find that
$\left[x^{(+)}_{++,-l}\right]^{\ast}=(-1)^{l}x^{(+)}_{++,l}$. Not
surprisingly, due to the generalized parity of the Floquet states, the
transition matrix elements satisfy Eq. (19) as long as
$\delta+f(t)=-[\delta+f(t+T/2)]$. For the biharmonic modulation, such
equalities are established when $p$ is odd and $\delta=0$.
### C.2 Even $p$
We move to consider that $p$ is an even number. In such a case, the
generalized parity of the Floquet states is broken even if $\delta=0$. Thus,
we cannot expect that the transition matrix elements satisfy Eq. (19).
However, we have another type of equality. With Eqs. (85) and (87), one gets
$\displaystyle Q_{j}$ $\displaystyle=$
$\displaystyle\frac{f_{-j}^{\ast}}{-2j\omega_{z}}\left(u+\frac{vf_{0}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq
0}\frac{f_{k-j}^{\ast}f_{k}}{k\omega_{z}}$ (90) $\displaystyle=$
$\displaystyle\frac{(-1)^{j+1}f_{j}^{\ast}}{2j\omega_{z}}\left(u+\frac{vf_{0}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq
0}\frac{(-1)^{j+1}f_{j-k}^{\ast}f_{-k}}{-k\omega_{z}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{j+1}e^{-i\theta_{0}}f_{j}^{\ast}}{2j\omega_{z}}\left(v+\frac{u^{\ast}f_{0}}{2j\omega_{z}}\right)+\frac{e^{-i\theta_{0}}u^{\ast}}{4j\omega_{z}}\sum_{k\neq
0}\frac{(-1)^{j+1}f_{j+k}^{\ast}f_{k}}{k\omega_{z}}$ $\displaystyle=$
$\displaystyle(-1)^{j+1}e^{-i\theta_{0}}P_{j}^{\ast}.$
It is straightforward to derive Eqs. (23) and (24) in the main text via Eqs.
(77)-(80) and (90). We stress that the conditions for establishing such
relations require that $p$ is even, $\phi=(1/2+n)\pi$, and $\delta=0$.
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|
2024-09-04T02:54:59.030750 | 2020-03-11T06:05:47 | 2003.05126 | {
"authors": "Sergey P. Shary",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26152",
"submitter": "Sergey Shary",
"url": "https://arxiv.org/abs/2003.05126"
} | arxiv-papers | # A variability measure for estimates
of parameters in interval data fitting
Sergey P. Shary
Institute of Computational Technologies SB RAS
and Novosibirsk State University,
Novosibirk, Russia
E-mail<EMAIL_ADDRESS>
###### Abstract
The paper presents a construction of a quantitative measure of variability for
parameter estimates in the data fitting problem under interval uncertainty. It
shows the degree of variability and ambiguity of the estimate, and the need
for its introduction is dictated by non-uniqueness of answers to the problems
with interval data. A substantiation of the new variability measure is given,
its application and motivations are discussed. Several examples and a series
of numerical tests are considered, showing the features of the new
characteristic and the specifics of its use.
Keywords: data fitting problem, linear regression, interval data uncertainty,
maximum compatibility method, strong compatibility, variability measure.
MSC 2010: 65G40, 62J10, 90C90
## 1 Introduction and problem statement
The purpose of this work is to present a quantitative variability measure for
estimates of parameters of functional dependencies in the statistics of
interval data. This is a relatively young branch of modern data science that
does not rely on the probability theory, but makes extensive use of interval
analysis methods (see, e. g., the surveys in [4, 7, 10]).
Fig. 1: A variability measure can be an estimate
of the size of the set of possible solutions.
By the term “variability”, we understand the degree of variation and ambiguity
of the estimate, and the need for its introduction is dictated by the fact
that, in processing interval data, the answer is typically not unique.
Usually, we get a whole set of different estimates that are equally consistent
(compatible) with the source data and, thus, suitable as solutions to the
problem. The extent to which this set is large or small is, partly,
characterized by the term “variability”. In traditional probabilistic
statistics, estimates of parameters are known to be random variables
themselves, and the measure of their variability can be the variance of the
estimates, mean absolute difference, median absolute deviation, average
absolute deviation, and such like. What could be their analogues in the
statistics of interval data?
At first glance, the answer to this question seems quite obvious: it can be
any value that characterizes the size of the set of solutions to the problem,
if it is non-empty. We can even take an enclosure of the solution set obtained
by an interval method. A certain disadvantage of this variant is the excessive
detailing of the answer given as a box in $\mathbb{R}^{n}$, a large amount of
information that still needs to be “digested” and reduced to a compact and
expressive form. Sometimes, an interval estimate in the form of an axes-
aligned box may inadequately represent the solution set. Another disadvantage
is the complexity of finding such an estimate.
It is desirable to have a relatively simple and efficiently computable
quantity, expressed in a single number, because it would give a general
aggregate view of the subject of interest. Similarly to variance and other
probabilistic measures, it can serve as an approximate characteristic of the
quality of parameter estimation. The greater the variability of an estimate,
the less its certainty and the worse its quality, and this can serve as a
basis for conclusions about the quality of the estimate.
At the same time, the introduced variability measure should not be simply the
“size of the solution set”. If this solution set, for example, is unstable and
changes abruptly with arbitrarily small changes in the data, then its size is,
to some extent, misleading and disorienting (see example in Section 4). A
practically useful variability measure should take into account this possible
instability of the solution set to the problem and give us a robust value.
Fig. 2: An illustration for the data fitting problem under interval
uncertainty.
In our article, we are within the framework of the data fitting problem (often
called regression analysis problem): given results of measurements or
observations, it is required to construct a functional dependence of a fixed
type that “best fit” these data. Specifically, we need to determine the
parameters $x_{1}$, $x_{2}$, …, $x_{n}$ of a linear function of the form
$b=x_{1}a_{1}+\ldots+x_{n}a_{n}$ (1)
from a number of values of the independent variables $a_{1}$, $a_{2}$, …,
$a_{n}$ (also called _exogenous_ , _explanatory_ , _predictor_ or _input_
variables), and the corresponding values of the dependent variable $b$ (also
called _endogenous_ , _response_ , _criterion_ or _output_ variable). Both
$a_{1}$, $a_{2}$, …, $a_{n}$ and $b$ are not known precisely, and we only have
intervals of their possible values (see Fig. 2). To find estimates of the
coefficients $x_{1}$, $x_{2}$, …, $x_{n}$, we use the so-called maximum
compatibility method (previously called “maximum consistency method”), which
was proposed and developed in the works [6, 16, 17, 19] and others. After the
estimates for $x_{1}$, $x_{2}$, …, $x_{n}$ are found, we need to somehow
evaluate their variability. Our article presents a construction of the
variability measure in the above data fitting problem.
Note that traditional methods of data fitting and regression analysis, such as
the least squares method and its modifications, the least modulus method,
etc., cannot be applied to the solution of our problem, since they are
unsuitable for situations where the source data are intervals rather than
points.
## 2 Formulation of the main results
### 2.1 Maximum compatibility method and tolerable solution set
The initial data for our problem is a set of values of independent and
dependent variables for function (1), which are obtained as a result of $m$
measurements (observations):
$\begin{array}[]{ccccc}\text{\boldmath$a$}_{11},&\text{\boldmath$a$}_{12},&\ldots&\text{\boldmath$a$}_{1n},&\text{\boldmath$b$}_{1},\\\
\text{\boldmath$a$}_{21},&\text{\boldmath$a$}_{22},&\ldots&\text{\boldmath$a$}_{2n},&\text{\boldmath$b$}_{2},\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\text{\boldmath$a$}_{m1},&\text{\boldmath$a$}_{m2},&\ldots&\text{\boldmath$a$}_{mn},&\text{\boldmath$b$}_{m}.\end{array}$
(2)
These are intervals as we assume that these data are inaccurate and have
interval uncertainty due to measurement errors, etc. Both the data (2) and
other interval values throughout the text are highlighted in bold mathematical
font according to the informal international standard [5]. The first index of
the interval values from (2) means the measurement number, and the second one,
at $\text{\boldmath$a$}_{ij}$’s, is the number of the independent variable
that takes the corresponding value in this measurement.
To find an estimate $(\hat{x}_{1},\hat{x}_{2},\ldots,\hat{x}_{n})$ of the
parameters of the linear function (1), we “substitute” data (2) into equality
(1), thus getting an interval system of linear algebraic equations
$\left\\{\
\begin{array}[]{ccccccccc}\text{\boldmath$a$}_{11}x_{1}&+&\text{\boldmath$a$}_{12}x_{2}&+&\ldots&+&\text{\boldmath$a$}_{1n}x_{n}&=&\text{\boldmath$b$}_{1},\\\\[1.0pt]
\text{\boldmath$a$}_{21}x_{1}&+&\text{\boldmath$a$}_{22}x_{2}&+&\ldots&+&\text{\boldmath$a$}_{2n}x_{n}&=&\text{\boldmath$b$}_{2},\\\\[1.0pt]
\vdots&&\vdots&&\ddots&&\vdots&&\vdots\\\\[1.0pt]
\text{\boldmath$a$}_{m1}x_{1}&+&\text{\boldmath$a$}_{m2}x_{2}&+&\ldots&+&\text{\boldmath$a$}_{mn}x_{n}&=&\text{\boldmath$b$}_{m},\end{array}\right.$
(3)
or, briefly,
$\text{\boldmath$A$}x=\text{\boldmath$b$}$ (4)
with an interval $m\times n$-matrix
$\text{\boldmath$A$}=(\text{\boldmath$a$}_{ij})$ and interval $m$-vector
$\text{\boldmath$b$}=(\text{\boldmath$b$}_{i})$ in the right-hand side. The
sets of parameters which are compatible, in this or that sense, with the
measurement data (2) form various solution sets for the equations system (3).
The most popular of them are the _united solution set_ and _tolerable solution
set_. The united solution set, defined as
$\varXi_{uni}(\text{\boldmath$A$},\text{\boldmath$b$})=\bigl{\\{}\,x\in\mathbb{R}^{n}\mid\text{
$Ax=b\,$ for some $A\in\text{\boldmath$A$}$ and
$b\in\text{\boldmath$b$}$}\,\bigr{\\}},$
corresponds to the so-called weak compatibility between the parameters of
function (1) and data (2) (see [6, 16, 17]). The tolerable solution set,
defined as
$\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})=\bigl{\\{}\,x\in\mathbb{R}^{n}\mid\text{
$Ax\in\text{\boldmath$b$}\,$ for each matrix
$A\in\text{\boldmath$A$}$}\,\bigr{\\}},$
corresponds to the so-called strong compatibility between the parameters of
function (1) and data (2) (see [19]).
Fig. 3: An illustration of the strong compatibility between interval data and
a linear function.
Further, we assume that the solution to the data fitting problem for function
(1) is found by the maximum compatibility method (see [16, 17, 19]). As an
estimate of the parameters of function (1), it takes the maximum point of the
_recognizing functional_ , a special function that gives a quantitative
“compatibility measure” of this estimate with empirical data (2).
The maximum compatibility method has two versions, “weak” and “strong”, that
differ in understanding how exactly the interval data should be “compatible”
with the function that we construct on them. Weak and strong compatibility
reflect two different situations that may occur in data processing. In the
weak version, it is required that the graph of the constructed function just
intersects the measurement uncertainty boxes (see [16, 17]). The strong
version implies more stringent condition: it requires that the function graph
passes within the “corridors” specified by the intervals
$\text{\boldmath$b$}_{i}$, $i=1,2,\ldots,m$, for _any_ values of the
independent variables $a_{1}$, $a_{2}$, …, $a_{n}$ from the respective
intervals $\text{\boldmath$a$}_{i1}$, $\text{\boldmath$a$}_{i2}$, …,
$\text{\boldmath$a$}_{in}$ obtained in the $i$-th measurement (see [19]). This
is illustrated in Fig. 3, where the straight line of the function graph goes
through the vertical faces of the measurement uncertainty boxes. The weak
compatibility is shown in Fig. 2 by two upper straight lines. The lower line
in Fig. 2 does not satisfy compatibility condition at all, neither weak nor
strong, since it does not intersect some boxes.
The “strong version” of the maximum compatibility method has a number of
theoretical and practical advantages over the “weak version”. These are
polynomial complexity, robustness of estimates and their finite variability,
the fact that the strong compatibility partially overcomes the so-called
Demidenko paradox, etc. (see details in [19]). Hence, we consider below a
strong version of the maximum compatibility method, which corresponds to the
tolerable solution set $\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$
for the interval system of equations (4). Its recognizing functional is
usually denoted by “Tol”,
$\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})\;=\,\min_{1\leq
i\leq
m}\left\\{\,\mathrm{rad}\,\text{\boldmath$b$}_{i}-\left|\;\mathrm{mid}\,\text{\boldmath$b$}_{i}-\sum_{j=1}^{n}\,\text{\boldmath$a$}_{ij}x_{j}\,\right|\,\right\\},$
(5)
where
$\mathrm{rad}\,\text{\boldmath$b$}_{i}=\tfrac{1}{2}(\overline{\text{\boldmath$b$}}_{i}-\underline{\text{\boldmath$b$}}_{i}),\hskip
65.44133pt\mathrm{mid}\,\text{\boldmath$b$}_{i}=\tfrac{1}{2}(\overline{\text{\boldmath$b$}}_{i}+\underline{\text{\boldmath$b$}}_{i})$
are radii and midpoints of the components of the right-hand side $b$, the
arithmetic operations inside the modulus in (5) are those of the classical
interval arithmetic (see, e. g., [4, 8, 9]), and the modulus is understood as
the maximum absolute value of the points from the interval,
$|\text{\boldmath$a$}|=\max\,\\{\,|a|\mid
a\in\text{\boldmath$a$}\,\\}=\max\,\bigl{\\{}\,|\underline{\text{\boldmath$a$}}|,|\overline{\text{\boldmath$a$}}|\,\bigr{\\}}.$
Typical graphs of the functional Tol for the one-dimensional case are shown in
Fig. 4 and Fig. 5.
To solve the data fitting problem for the linear function (1) and data set
(2), it is necessary to find the unconstrained maximum, over all
$x\in\mathbb{R}^{n}$, of the functional
$\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})$,
$\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})\rightarrow\max,$
and the vector
$\hat{x}=\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})$
at which this maximum is attained provides an estimate of the parameters of
function (1).
If $\max\,\mathrm{Tol}\,\geq 0$, then the solution set
$\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$, i. e., the set of
parameters strongly compatible with the data is non-empty, and
$\hat{x}\in\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$. If
$\max\,\mathrm{Tol}\,<0$, then the solution set
$\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$ is empty and there do
not exist parameters that are strongly compatible with data (2). However, the
argument $\hat{x}$ of $\max\,\mathrm{Tol}\,$ still provides the best
compatibility of the constructed linear function with data (2) (more
precisely, the least incompatibility).
To conclude this subsection, we give a useful result on the tolerable solution
set that allows us to investigate whether it is bounded or unbounded, i. e.,
whether the tolerable solution sets is finite in size or extends infinitely.
Irene Sharaya’s boundedness criterion [13] Let the tolerable solution set to
an interval linear system $\text{\boldmath$A$}x=\text{\boldmath$b$}$ be
nonempty. It is unbounded if and only if the matrix $A$ has linearly dependent
noninterval columns.
The criterion of boundedness shows that the tolerable solution set is
unbounded, in fact, under exceptional circumstances, which are almost never
fulfilled in practice, when working with actual interval data. That is, the
tolerable solution set is mostly bounded, and the estimates obtained by the
strong version of the maximum compatibility method almost always has finite
variability.
### 2.2 Variability measures
As a quantity characterizing the variability of the estimate of the parameter
vector $\hat{x}=(\hat{x}_{1},\hat{x}_{2},\ldots,\hat{x}_{n})$ in the linear
function (1), which is obtained by the maximum compatibility method from data
(2), we propose
$\mathrm{IVE}\,(\text{\boldmath$A$},\text{\boldmath$b$})\;=\;\sqrt{n}\;\max_{\mathbb{R}^{n}}\,\mathrm{Tol}\,\cdot\Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A\,\Bigr{)}\cdot\frac{\displaystyle\bigl{\|}\,\arg\max_{\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$
(6)
In this formula,
$n$ is the dimension of the parameter vector of function (1) under
construction,
$\|\cdot\|_{2}$ is the Euclidean norm (2-norm) of vectors from
$\mathbb{R}^{n}$, defined as
$\|x\|_{2}\;=\;\left(\;\sum_{i=1}^{n}|x_{i}|^{2}\,\right)^{1/2},$
$\mathrm{cond}_{2}\,A$ is the spectral condition number of the matrix $A$,
defined as
$\mathrm{cond}_{2}\,A\;=\;\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)},$
i. e., the ratio of the maximal $\sigma_{\max}(A)$ and minimal
$\sigma_{\min}(A)$ singular values of $A$; it is an extension, to the
rectangular case, of the concept of the condition number from computational
linear algebra (see e. g. [2, 24]);
$\hat{\text{\boldmath$b$}}$ is a certain “most representative” point from the
interval vector $b$, which is taken as
$\hat{\text{\boldmath$b$}}\;=\;\tfrac{1}{2}(|\mathrm{mid}\,\text{\boldmath$b$}+\mathrm{rad}\,\text{\boldmath$b$}|+|\mathrm{mid}\,\text{\boldmath$b$}-\mathrm{rad}\,\text{\boldmath$b$}|),$
(7)
where the operations “mid” and “rad” are applied in componentwise manner.
$\varXi_{\mathit{tol}}$
Fig. 4: The maximum value of the recognizing functional
gives an idea of the size of the tolerable solution set $\varXi_{tol}$.
Despite the definite formula (7) for $\hat{\text{\boldmath$b$}}$, it should be
noted that the introduction of this point is, to a large extent, a matter of
common sense. The general approach to the definition of
$\hat{\text{\boldmath$b$}}$ is that it must be a kind of “most representative”
point from the right-hand side vector $b$, and in some situations this choice
may be different from formula (7). For example, $\hat{\text{\boldmath$b$}}$
can be a point result of the measurement, around which the uncertainty
interval is built later, based on information about the accuracy of the
measuring device.
Apart from (6), as a measure of relative variability of the parameter
estimate, the value
$n\;\Bigl{(}\,\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A\,\Bigr{)}\cdot\frac{\max_{\mathbb{R}^{n}}\,\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}},$
(8)
can have a certain significance. Both IVE and value (8) are defined for
interval linear systems (4) with nonzero right-hand sides. They can take
either positive real values or be infinite. The latter occurs in the only case
of $\,\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A=\infty$, when all the
point matrices $A\in\text{\boldmath$A$}$ have incomplete rank, i. e., when
$\sigma_{\min}(A)=0$ for every $A\in\text{\boldmath$A$}$. Then the variability
measures are set to be infinite.
The symbol IVE is built as an abbreviation of the phrase “interval variability
of the estimate”. Below, we show that the value IVE adequately characterizes
the size of non-empty tolerable solution set for a large class of practically
important situations. But it is useful to discuss informal motivations that
lead to the estimate IVE and to demonstrate that IVE has an intuitive, clear
and even visual meaning.
$\varXi_{\mathit{tol}}$$\varXi_{\mathit{tol}}$
Fig. 5: In addition to the maximum of the recognizing functional, the size
of the tolerable solution set is also affected by “steepness” of the graph.
The tolerable solution set of an interval system of linear algebraic equations
is the set of zero level of the recognizing functional Tol (see details in
[15]), or, in other words, the intersection of the hypograph of this
functional with the coordinate plane $\mathrm{Tol}\,=0$ (this is illustrated
in Fig. 4). As a consequence, the magnitude of the maximum of the recognizing
functional can, with other things being equal, be a measure of how extensive
or narrow the tolerable solution set is. The more $\max\,\mathrm{Tol}\,$, the
larger the size of the tolerable solution set, and vice versa. An additional
factor that provides “other things being equal” is the slope (steepness) of
pieces of hyperplanes of which the polyhedral graph of the functional Tol is
compiled (these are straight lines in the 1D case in Fig. 4 and Fig. 5). The
slope of the hyperplanes is determined by the coefficients of the equations
that define them, which are the endpoints of the data intervals (2). The value
of this slope is summarized in terms of the condition number of point matrices
from the interval data matrix $A$. Finally, the multiplier
$\frac{\|\arg\max\,\mathrm{Tol}\,\|_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}\
=\ \frac{\|\hat{x}\|_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}$
is a scaling coefficient that helps to provide the commensurability of the
final value with magnitudes of the solution, $\arg\max\,\mathrm{Tol}\,$, and
the right-hand side vector of the equations system. Thus, formula (6) is
obtained.
## 3 A justification of the variability measure
Considering the most general case, we should assume that the number of
measurements $m$ may not coincide with the number $n$ of unknown parameters of
the linear function (1). In this section, we consider only the case $m\geq n$.
In other words, the number of measurements (observations) made is not less
than the number of function parameters. Then the interval system of linear
equations (4) is either square or tall (overdetermined). Of course, the data
fitting problem makes sense for $m<n$ too, the maximum compatibility method
also works for this case, and the variability measure IVE is then also
applicable (see Section 4), but the latter still needs a separate
substantiation.
### 3.1 Estimates of perturbations of the solution
to rectangular linear systems
The starting point of our constructions justifying the choice of (6) exactly
in the form described above is the well-known inequality that estimates
perturbation $\Delta x$ of a nonzero solution $x$ to the system of linear
algebraic equations $Ax=b$ depending on the change $\Delta b$ of the right-
hand side $b$ (see, e. g., [2, 24]):
$\frac{\|\Delta x\|_{2}}{\|x\|_{2}}\ \leq\
\mathrm{cond}_{2}\,A\,\cdot\frac{\|\Delta b\|_{2}}{\|b\|_{2}}.$ (9)
It is usually considered for square systems of linear equations, when $m=n$,
but in the case of the Euclidean vector norm and the spectral condition number
of matrices, this inequality holds true in the more general case with $m\geq
n$. Naturally, estimate (9) makes sense only for $\sigma_{\min}(A)\neq 0$,
when $\mathrm{cond}_{2}A<\infty$, i. e., when the matrix $A$ has full column
rank. Let us briefly recall its derivation for this case.
Given
$Ax=b\quad\text{ и }\quad A(x+\Delta x)=b+\Delta b,$
we have
$A\Delta x=\Delta b.$
Further,
$\displaystyle\displaystyle\frac{\displaystyle\phantom{M}\frac{\|\Delta
x\|_{2}}{\|x\|_{2}}\phantom{M}}{\displaystyle\frac{\|\Delta
b\|_{2}}{\|b\|_{2}}}\ $ $\displaystyle=\ \frac{\|\Delta
x\|_{2}\,\|b\|_{2}\phantom{I}}{\phantom{I}\|x\|_{2}\,\|\Delta b\|_{2}}=\
\frac{\|\Delta x\|_{2}\,\|Ax\|_{2}\phantom{I}}{\phantom{I}\|x\|_{2}\,\|A\Delta
x\|_{2}}\ =\ \frac{\|\Delta x\|_{2}}{\|A\Delta
x\|_{2}}\;\frac{\|Ax\|_{2}}{\|x\|_{2}}$ $\displaystyle\leq\;\max_{\Delta x\neq
0}\frac{\|\Delta x\|_{2}}{\|A\Delta x\|_{2}}\ \max_{x\neq
0}\frac{\|Ax\|_{2}}{\|x\|_{2}}\ =\ \left(\min_{\Delta x\neq 0}\frac{\|A\Delta
x\|_{2}}{\|\Delta x\|_{2}}\right)^{-1}\ \max_{x\neq
0}\frac{\|Ax\|_{2}}{\|x\|_{2}}$ $\displaystyle=\
\bigl{(}\sigma_{\min}(A)\bigr{)}^{-1}\,\sigma_{\max}(A)\,=\
\mathrm{cond}_{2}(A)$
by virtue of the properties of the singular values (see e. g. [3, 24]). A
comparison of the beginning and the end of this calculation leads to the
inequality (9), which, as is easy to understand, is attainable for some $x$
and $\Delta x$, or, equivalently, for some right-hand sides of $b$ and their
perturbations $\Delta b$. Naturally, the above calculations and the resulting
estimate make sense only for $\sigma_{\min}(A)\neq 0$.
### 3.2 Interval systems with point matrices
Let us consider an interval system of linear algebraic equations
$Ax=\text{\boldmath$b$}$ (10)
with a point (noninterval) $m\times n$-matrix $A$, $m\geq n$, and an interval
$m$-vector $b$ in the right-hand side. We assume that $A$ has full column rank
and, therefore, $\mathrm{cond}_{2}\,A<\infty$.
Suppose also that the tolerable solution set for system (10) is non-empty, i.
e. $\varXi_{tol}(A,\text{\boldmath$b$})=\bigl{\\{}\,x\in\mathbb{R}^{n}\mid
Ax\in\text{\boldmath$b$}\,\bigr{\\}}\neq\varnothing$. We need to quickly and
with little effort estimate the size of this solution set, and our answer will
be a “radius type” estimate for $\varXi_{tol}(A,\text{\boldmath$b$})$. More
precisely, we are going to evaluate $\max\|x^{\prime}-\hat{x}\|_{2}$ over all
$x^{\prime}\in\varXi_{tol}(A,\text{\boldmath$b$})$ and for a special fixed
point $\hat{x}\in\varXi_{tol}(A,\text{\boldmath$b$})$, which is taken as
$\hat{x}\ =\
\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\text{\boldmath$b$}).$
Recall that the argument $\hat{x}$ of the maximum of the recognizing
functional for system (10) is an estimate of parameters of linear function (1)
from empirical data. Strictly speaking, this point can be determined non-
uniquely, but then let $\hat{x}$ be any one of the points at which the maximum
is reached.
Let $x^{\prime}$ be a point in the tolerable solution set
$\varXi_{tol}(A,\text{\boldmath$b$})$. How to evaluate
$\|x^{\prime}-\hat{x}\|_{2}$? It is clear that $x^{\prime}$ and $\hat{x}$ are
solutions of systems of linear algebraic equations with the matrix $A$ and
some right-hand sides $b^{\prime}$ and $\hat{b}$, respectively, from the
interval vector $b$. If $\hat{x}\neq 0$ and $\hat{b}\neq 0$, then we can apply
inequality (9), considering a perturbation of the solution $\hat{x}$ to the
system of linear algebraic equations $Ax=\hat{b}$. Then $\Delta
x=x^{\prime}-\hat{x}$, $\Delta b=b^{\prime}-\hat{b}$, and we get
$\frac{\|x^{\prime}-\hat{x}\|_{2}}{\|\hat{x}\|_{2}}\
\leq\;\mathrm{cond}_{2}\,A\cdot\frac{\|b^{\prime}-\hat{b}\|_{2}}{\|\hat{b}\|_{2}},$
from where the absolute estimate is obtained
$\|x^{\prime}-\hat{x}\|_{2}\
\leq\;\mathrm{cond}_{2}\,A\cdot\|\hat{x}\|_{2}\cdot\frac{\|b^{\prime}-\hat{b}\|_{2}}{\|\hat{b}\|_{2}}.$
(11)
The point $\hat{x}$ is found as the result of maximization of the recognizing
functional Tol, the point $\hat{b}$ coincides with $A\hat{x}$, the condition
number $\mathrm{cond}_{2}\,A$ can be computed by well-developed standard
procedures. Therefore, for practical work with inequality (11), one need
somehow evaluate $\|b^{\prime}-\hat{b}\|_{2}$.
But first, bearing in mind the further application of the deduced estimate in
a situation where the matrix $A$ may vary, we somewhat roughen (11) by taking
approximately $\|\hat{b}\|_{2}\approx\|\hat{\text{\boldmath$b$}}\|_{2}$, that
is, as the norm of the “most representative” point $\hat{\text{\boldmath$b$}}$
of the interval vector $b$, which we defined in Section 2.2:
$\|\hat{b}\|_{2}\,\approx\,\|\hat{\text{\boldmath$b$}}\|_{2},\qquad\text{
where }\
\hat{\text{\boldmath$b$}}\,=\,\tfrac{1}{2}\,\bigl{(}\,|\mathrm{mid}\,\text{\boldmath$b$}+\mathrm{rad}\,\text{\boldmath$b$}|+|\mathrm{mid}\,\text{\boldmath$b$}-\mathrm{rad}\,\text{\boldmath$b$}|\,\bigr{)}.$
In doing this, some coarsening is allowed, so instead of (11) we write
$\|x^{\prime}-\hat{x}\|_{2}\
\lessapprox\;\mathrm{cond}_{2}\,A\cdot\|\hat{x}\|_{2}\cdot\frac{\|\Delta
b\|_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$ (12)
Now it is necessary to determine the increment of the right-hand side $\Delta
b=b^{\prime}-\hat{b}$. Its obvious upper bound is
$2\,\mathrm{rad}\,\text{\boldmath$b$}$, but it is too crude. To get a more
accurate estimate of $\Delta b$, we also consider, along with system (10), a
system of linear algebraic equations
$Ax=\tilde{\text{\boldmath$b$}},$ (13)
for which the right-hand side is obtained by uniform “compressing” the
interval vector $b$:
$\tilde{\text{\boldmath$b$}}\,:=\,\bigl{[}\,\underline{\text{\boldmath$b$}}+M,\overline{\text{\boldmath$b$}}-M\,\bigr{]},$
(14)
where
$M\;:=\;\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\text{\boldmath$b$})\
\geq\ 0.$
Since the maximum $M$ is reached for a certain value of the argument,
$\hat{x}$, then
$M=\,\min_{1\leq i\leq
m}\left\\{\,\mathrm{rad}\,\text{\boldmath$b$}_{i}-\left|\;\mathrm{mid}\,\text{\boldmath$b$}_{i}-\sum_{j=1}^{n}\,\text{\boldmath$a$}_{ij}\hat{x}_{j}\,\right|\,\right\\}\
\leq\,\min_{1\leq i\leq m}\,\mathrm{rad}\,\text{\boldmath$b$}_{i}.$
As a result,
$\underline{\text{\boldmath$b$}}+M\leq\overline{\text{\boldmath$b$}}-M$ in
componentwise sense, and the endpoints in the interval vector (14) do not
“overlap” each other.
But the properties of the recognizing functional imply that, for the interval
system of linear algebraic equations (13) with the right-hand side (14), the
maximum of the recognizing functional is zero:
$\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\tilde{\text{\boldmath$b$}})\
=\ 0.$
Indeed, the values of $\mathrm{rad}\,\text{\boldmath$b$}_{i}$ are summands in
all expressions in (5), for which we take the minimum over $i=1,2,\ldots,m$.
Hence, if we simultaneously increase or decrease all
$\mathrm{rad}\,\text{\boldmath$b$}_{i}$ by the same value, keeping the
midpoints $\mathrm{mid}\,\text{\boldmath$b$}_{i}$ unchanged, then the total
value of the recognizing functional will increase or decrease by exactly same
value. In other words, if we take a constant $C\geq 0$ and the interval
$m$-vector $\text{\boldmath$e$}=([-1,1],\ldots,[-1,1])^{\top}$, then, for the
system $Ax=\text{\boldmath$b$}+C\text{\boldmath$e$}\,$ with all the right-hand
sides expanded by $[-C,C]$, we have
$\mathrm{Tol}\,(x,A,\text{\boldmath$b$}+C\text{\boldmath$e$})\ =\
\mathrm{Tol}\,(x,A,\text{\boldmath$b$})+C.$ (15)
Therefore,
$\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\text{\boldmath$b$}+C\text{\boldmath$e$})\
=\ \max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\text{\boldmath$b$})+C.$ (16)
The uniform narrowing of the right-hand side vector acts on the tolerable
solution set and the recognizing functional in a completely similar way. If we
narrow down all the components by the same value $M$, then the maximum of the
recognizing functional of the new interval system also decreases by $M$.
By virtue of the properties of the recognizing functional, the tolerable
solution set $\varXi_{tol}(A,\tilde{\text{\boldmath$b$}})$ for system (13) has
empty interior (such sets are often called “non-solid” or “meager”), which we
will consider equivalent to “having zero size”. Naturally, this is a
simplifying assumption, since in reality the tolerable solution set
corresponding to the zero maximum of the recognizing functional may be not a
single-point set. But we still accept that. This implication is also supported
by the fact that the situation with the zero maximum of the recognizing
functional is unstable: the corresponding tolerable solution set can become
empty with an arbitrarily small data perturbation (see Section 4).
Another fact concerning the auxiliary system (13) with the narrowed right-hand
side, which follows from (15)–(16), is that the point $\hat{x}$ remains to be
the argument of the maximum of the recognizing functional:
$\hat{x}\ =\
\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\tilde{\text{\boldmath$b$}}).$
For this reason, the point $\hat{b}=A\hat{x}$ lies in the interval vector
$\tilde{\text{\boldmath$b$}}$ defined by (14).
From what has been said, it follows that the solution set for the system
$Ax=\text{\boldmath$b$}$ is obtained from the solution set of the system
$Ax=\tilde{\text{\boldmath$b$}}$, which has “negligible size” and for which
$\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,(x,\text{\boldmath$A$},\tilde{\text{\boldmath$b$}})=0$,
through expanding the right-hand side vector $\tilde{\text{\boldmath$b$}}$ in
each component simultaneously by $[-M,M]$, where
$M=\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$}).$
The interval vector $\tilde{\text{\boldmath$b$}}\ni b$ may have non-zero size,
but we put
$[-\Delta b,\Delta b]=([-M,M],\ldots,[-M,M])^{\top}$
in order to make our estimate (12) attainable. Accordingly, in inequality (12)
we take
$\|\Delta
b\|=\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$}),$
if the Chebyshev norm ($\infty$-norm) is considered, or a value that differs
from it by a corrective factor from the equivalence inequality for vector
norms, if we take any other norm. As is known, for any vector
$y\in\mathbb{R}^{n}$ (see [2])
$\|y\|_{\infty}\leq\|y\|_{2}\leq\sqrt{n}\;\|y\|_{\infty}.$ (17)
Then
$\|x^{\prime}-\hat{x}\|_{2}\ \lessapprox\,\sqrt{n}\
\,\mathrm{cond}_{2}A\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$
(18)
What happens if the matrix $A$ does not have a full column rank? Then, by
virtue of the Irene Sharaya criterion, the nonempty tolerable solution set to
the system (10) is unbounded. This is completely consistent with the fact that
then $\mathrm{cond}_{2}A=\infty$ and the value of the variability measure IVE
is infinite too.
### 3.3 General interval systems
Finally, we consider a general interval system of linear equations
$\text{\boldmath$A$}x=\text{\boldmath$b$}$, with an essentially interval
matrix, i. e., when $\mathrm{rad}\,\text{\boldmath$A$}\neq 0$. In view of the
properties of the tolerable solution set (see, e. g., [15]), it can be
represented as
$\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})\ =\
\bigcap_{A\in\text{\boldmath$A$}}\;\bigl{\\{}\,x\in\mathbb{R}^{n}\mid
Ax\in\text{\boldmath$b$}\,\bigr{\\}}\ =\
\bigcap_{A\in\text{\boldmath$A$}}\varXi_{tol}(A,\text{\boldmath$b$}),$ (19)
i. e., as the intersection of the solution sets to the individual systems
$Ax=b$ with point matrices $A\in\text{\boldmath$A$}$.
For each interval linear system $Ax=\text{\boldmath$b$}$ with
$A\in\text{\boldmath$A$}$, we have estimate (18), if $A$ has full column rank.
Otherwise, if the point matrix $A$ has incomplete column rank and the
corresponding solution set $\varXi_{tol}(A,\text{\boldmath$b$})$ is unbounded,
then we do not take it into account. Consequently, for the tolerable solution
set of the system $\text{\boldmath$A$}x=\text{\boldmath$b$}$, which is the
intersection of the solution sets $\varXi_{tol}(A,\text{\boldmath$b$})$ for
all $A\in\text{\boldmath$A$}$, the following should be true:
$\|x^{\prime}-\hat{x}\|_{2}\ \lessapprox\
\min_{A\in\text{\boldmath$A$}}\,\left\\{\;\sqrt{n}\;\,\mathrm{cond}_{2}A\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}\,\right\\}.$
(20)
The transition from representation (19) to inequality (20) can be both very
accurate and rather crude (as can be seen from considering the intersection of
two 1D intervals). It all depends on the size of the intersection of the
solution sets of individual systems $Ax=\text{\boldmath$b$}$. On the other
hand, the amount of this intersection is indirectly characterized by the
magnitude of $\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,$.
Taking the above facts into account, we perform approximate estimation of the
right-hand side of inequality (20) by moving the minimum over
$A\in\text{\boldmath$A$}$ through the curly brackets. First of all, we
evaluate the factor $\|\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\|_{2}$,
which changes to the smallest extent, by the constant available to us after
the numerical solution of the data fitting problem:
$\bigl{\|}\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\text{\boldmath$b$})\bigr{\|}_{2}\,\approx\;\mathrm{const}\;=\;\bigl{\|}\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})\bigr{\|}_{2}.$
(21)
Next, the minimum of $\mathrm{cond}_{2}A$ naturally turns to
$\min\mathrm{cond}_{2}A$, and the most important factor
$\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\text{\boldmath$b$})$ will be
changed to
$\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})$.
This choice (as well as (21)) is rather rigidly determined by the following
reasons.
The expression for our variability measure should preserve its simplicity and
be uniform for all cases and situations. In particular, if the interval matrix
$A$ squeezes to a point matrix $A$, then our measure should turn to the
estimate (18) for the point case. Finally, if $\max\,\mathrm{Tol}\,=0$, then
our measure must be zero too, since the size of the (stable) tolerable
solution set is also zero, and our variability measure should reliably detect
such situations. All this taken together leads to the estimate
$\|x^{\prime}-\hat{x}\|_{2}\;\,\lessapprox\ \sqrt{n}\
\Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A\,\Bigr{)}\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$
(22)
The same estimate as (22), by virtue of the equivalence inequality (17), is
also true for the Chebyshev norm:
$\max_{x^{\prime}\in\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})}\|x^{\prime}-\hat{x}\|_{\infty}\
\lessapprox\
\sqrt{n}\;\,\Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A\,\Bigr{)}\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$
This completes the rationale for (6).
If we want to evaluate the relative size of the tolerable solution set,
expressing it in ratio to the norm of its points, then it is reasonable to
take $\hat{x}=\arg\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,$ as the “most
typical” point from the tolerable solution set
$\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$. Using (17) again, we
obtain
$\frac{\max_{x^{\prime}\in\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})}\|x^{\prime}-x^{\prime\prime}\|_{\infty}}{\|\hat{x}\|_{\infty}}\
\lessapprox\
n\;\Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A\,\Bigr{)}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$
This gives value (8).
## 4 Numerical examples and some tests
First of all, we consider an example of unstable tolerable solution set that
changes abruptly with small perturbations in the system of equations. For all
interval $2\times 2$-systems of linear algebraic equations of the form
$\begin{pmatrix}[-1,1]&[-1,1]\\\\[2.0pt] 1&-1\\\\[2.0pt]
\end{pmatrix}\begin{pmatrix}x_{1}\\\\[2.0pt]
x_{2}\end{pmatrix}=\begin{pmatrix}{[-1,1]}\\\\[2.0pt]
{[1,1+\eta]}\end{pmatrix},\qquad\eta\geq 0,$ (23)
the tolerable solution sets are the same: this is the straight line segment
joining the points $(0,-1)$ and $(1,0)$ and depicted in Fig. 6. The diameter
of the solution set is essentially non-zero (namely, $\sqrt{2}$), but the
unconstrained maximum of the recognizing functional Tol for all such systems
is zero, and it is attained at the point $(0.5,-0.5)$.
$\varXi_{\mathit{tol}}$ Fig. 6: The tolerable solution set for the interval
equations systems (23).
At the same time, any arbitrarily small increase in the lower endpoint of the
interval $[1,1+\eta]$ in the right-hand side of the second equation makes the
tolerable solution set empty. An arbitrarily small reduction of the upper
endpoint of the interval $[-1,1]$, located in the first component of the
right-hand side vector, produces a similar effect. It turns out that the
maximum value of the recognizing functional Tol characterizes very precisely
the instability of the original solution set and the zero size of the solution
sets of perturbed systems.
As the second example, we consider the problem of constructing a linear
function of two variables $a_{1}$ and $a_{2}$,
$b=x_{1}a_{1}+x_{2}a_{2},$ (24)
from the interval data obtained in 3 measurements:
$\begin{array}[]{c|ccc}&\text{\boldmath$a$}_{1}&\text{\boldmath$a$}_{2}&\text{\boldmath$b$}\\\
\hline\cr\\\\[-8.53581pt] 1&[98,100]&[99,101]&[190,210]\\\\[3.0pt]
2&[97,99]&[98,100]&[200,220]\\\\[3.0pt]
3&[96,98]&[97,99]&[190,210]\end{array}$
Note that in these data the three-dimensional uncertainty boxes of
measurements 1 and 2, as well as 2 and 3, substantially “overlap” each other:
their intersections are boxes with non-empty interiors, the sizes of which are
comparable to the sizes of the original data boxes.
$x_{1}$$x_{2}$$\hat{\text{\boldmath$x$}}$
Fig. 7: The tolerable solution set of the system of equations (25)
in comparison with the box constructed by using the estimate IVE.
To determine the coefficients $x_{1}$ and $x_{2}$, we compose an interval
$3\times 2$-system of linear algebraic equations
$\begin{pmatrix}[98,100]&[99,101]\\\\[2.0pt] [97,99]&[98,100]\\\\[2.0pt]
[96,98]&[97,99]\end{pmatrix}\begin{pmatrix}x_{1}\\\\[2.0pt]
x_{2}\end{pmatrix}=\begin{pmatrix}{[190,210]}\\\\[2.0pt]
{[200,220]}\\\\[2.0pt] {[190,210]}\end{pmatrix}.$ (25)
Its matrix has incomplete rank, since its member is a point matrix with the
rank 1:
$\begin{pmatrix}98&99\\\ 98&99\\\ 98&99\end{pmatrix}.$ (26)
The united solution set for system (25) is unbounded, therefore it is hardly
possible to determine, with certainty, the coefficients of the linear function
(24) satisfying the weak compatibility between parameters and data (see
Section 2). However, the interval matrix of system (25) does not contain
linearly dependent point columns, and therefore, according to the Irene
Sharaya criterion [13] (see Section 2.1), the tolerable solution set is
bounded. It is depicted in Fig. 7, which is drawn by the procedure EqnTol2D
from the package IntLinInc2D [14]. The minimum spectral condition number of
the point matrices contained in the interval matrix of (25) is $103.83$, and
it is reached on the matrix
$\begin{pmatrix}100&99\\\ 97&100\\\ 96&99\end{pmatrix}.$
This result can be obtained, for example, using the simulated annealing
algorithm on the set of point matrices contained in the interval matrix of
(25).
Numerical solution of the maximization problem for the recognizing functional
Tol can be carried out within MATLAB environment, using the free program
tolsolvty2.m (available from the author’s page at ResearchGate [21]). It
implements a modified version of the so-called $r$-algorithms for non-
differentiable optimization proposed and developed by N.Z. Shor and N.G.
Zhurbenko [20]. Using the precision settings specified in it “by default”, we
get $\max\,\mathrm{Tol}\,=1.9095$, which is reached at $\hat{x}=(5.1857\cdot
10^{-7},2.0603)^{\top}$. Then,
$\mathrm{IVE}\,=\sqrt{2}\cdot 1.9095\cdot
103.83\cdot\frac{\|\hat{x}\|_{2}}{\sqrt{200^{2}+210^{2}+200^{2}}}=1.6399.$
Interval hull of the tolerable solution set for system (25) (that is, its
optimal interval enclosure) is the box
$\begin{pmatrix}[-0.9620,3.0227]\\\\[2.0pt] [-0.9320,3.0127]\end{pmatrix},$
which can also be found by the procedure EqnTol2D. We see that the value of
IVE is in satisfactory agreement with the radii of the components of the
optimal estimate of the solution set, equal to $1.9924$ and $1.9724$
respectively.
In the maximum compatibility method, the argument
$\hat{x}=\arg\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,$ of the unconstrained
maximum of the recognizing functional plays a crucial role, and, in fact, our
variability estimate IVE relies heavily on it. This is why it makes sense to
look at the box $\hat{\text{\boldmath$x$}}$ with the components
$[\hat{x}_{i}-\mathrm{IVE}\,,\hat{x}_{i}+\mathrm{IVE}\,]$, $i=1,2$. It is also
depicted in Fig. 7, and the substantial asymmetry of its location relative to
the solution set is, of course, explained by the specific position of the
center, the point $\hat{x}$, as well as the ill-conditioning of the point
systems from (25). With other data, the box $\hat{\text{\boldmath$x$}}$
estimates the tolerable solution sets significantly better (see further).
Next, we give an example of the opposite type (in a sense, dual to the
previous example), where a linear function of three variables
$b=x_{1}a_{1}+x_{2}a_{2}+x_{2}a_{3}$ (27)
is to be constructed from the data of two experiments summarized below:
$\begin{array}[]{c|cccc}&\text{\boldmath$a$}_{1}&\text{\boldmath$a$}_{2}&\text{\boldmath$a$}_{3}&\text{\boldmath$b$}\\\
\hline\cr\\\\[-8.53581pt] 1&[98,100]&[97,99]&[96,98]&[190,210]\\\\[3.0pt]
2&[99,101]&[98,100]&[97,99]&[200,220]\end{array}$
To find the parameters of function (27), we come to an underdetermined
interval system of linear algebraic equations
$\begin{pmatrix}[98,100]&[97,99]&[96,98]\\\\[2.0pt]
[99,101]&[98,100]&[97,99]\end{pmatrix}\begin{pmatrix}x_{1}\\\\[1.0pt]
x_{2}\\\\[1.0pt] x_{3}\end{pmatrix}=\begin{pmatrix}{[190,210]}\\\\[2.0pt]
{[200,220]}\end{pmatrix}.$ (28)
Its matrix is the transposed matrix of system (25), so
$\min_{A\in\text{\boldmath$A$}}\mathrm{cond}_{2}\,A$ is the same for it. Also,
the matrix of system (28) contains a point matrix of the incomplete rank 1,
which is transposed for (26) (and many more such matrices).
$x_{1}$$x_{2}$$x_{3}$ Fig. 8: The tolerable solution set for the interval
equations system (28).
Again, the united solution set for system (28) is unbounded, and it is
difficult (if at all possible) to determine the coefficients of the linear
function (27), relying on the weak compatibility between parameters and data,
due to “infinite variability” of the resulting estimate. Nevertheless, in
these adverse conditions, the nonempty tolerable solution set to the interval
system of equations (28) is bounded by virtue of the Irene Sharaya criterion
[13] (see Section 2.1). In Fig. 8, the tolerable solution set is depicted as a
thin hexagonal plate. Computation of the maximum of the recognizing functional
for this system using the code tolsolvty2.m gives the value
$\max\mathrm{Tol}\,=3.9698$, which is reached at the point
$\hat{x}=\arg\max\mathrm{Tol}\,=\,\bigl{(}\,2.0603,3\cdot 10^{-6},2.1\cdot
10^{-6}\,\bigr{)}^{\top}.$
It can be taken as an estimate of the coefficients in (27). Then the
varibility measure of the above estimate is
$\mathrm{IVE}\,=\sqrt{2}\cdot 3.9698\cdot
103.83\cdot\frac{\|\hat{x}\|_{2}}{\sqrt{200^{2}+210^{2}}}=4.1413.$
Interval hull (optimal interval enclosure) of the tolerable solution set for
system (28) is the box
$\begin{pmatrix}[-1.9747,4.0302]\\\\[2.0pt] [-1.9899,4.0759]\\\\[2.0pt]
[-1.9949,4.1071]\end{pmatrix},$
which can also be computed by the procedure EqnTolR3. The radii of the
components of this interval vector are $3.0024$, $3.0329$, $3.0510$
respectively, which is also not very different from the value of IVE. The
example shows that the value IVE works even in the case of $m<n$, when the
number of measurements is less than the number of parameters to be determined.
But a rigorous justification of this fact is waiting for its study.
To conclude the section, we present, in Table 1, the results of numerical
tests for the interval linear $n\times n$-system
$\left(\begin{array}[]{cccc}\theta&{[0,2]}&\cdots&{[0,2]}\\\\[1.0pt]
{[0,2]}&\theta&\cdots&{[0,2]}\\\\[1.0pt]
\vdots&\vdots&\ddots&\vdots\\\\[1.0pt]
{[0,2]}&{[0,2]}&\cdots&\theta\end{array}\right)\;\left(\begin{array}[]{@{\,}c@{\,}}x_{1}\\\\[1.0pt]
x_{2}\\\\[1.0pt] \vdots\\\\[1.0pt]
x_{n}\end{array}\right)=\left(\begin{array}[]{@{\;}c@{\;}}{[1,K]}\\\\[1.0pt]
{[1,K]}\\\\[1.0pt] \vdots\\\\[1.0pt] {[1,K]}\end{array}\right),$ (29)
with various $n$ and $K$. System (29) resembles that proposed in [9], having
exactly same matrix. But the right-hand sides were taken as positive intervals
$[1,K]$, since the original balanced intervals $[-1,1]$ in the system from [9]
make the tolerable solution set “too symmetric”.
Table 1: Results of the computational tests with system (29) $\theta\rule[-8.53581pt]{0.0pt}{22.76219pt}$ | $\mathrm{IVE}\,$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{\infty}$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{2}$ | $\theta\rule[-8.53581pt]{0.0pt}{22.76219pt}$ | $\mathrm{IVE}\,$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{\infty}$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{2}$
---|---|---|---|---|---|---|---
$n=5$, $K=10$ | $n=10$, $K=10$
2.0 | 1.019 | 1.25 | 2.795 | 6.0 | 0.894 | 0.5 | 1.581
4.0 | 1.081 | 0.875 | 1.957 | 9.0 | 1.491 | 0.389 | 1.230
6.0 | 0.786 | 0.639 | 1.429 | 12.0 | 0.582 | 0.313 | 0.988
8.0 | 0.681 | 0.5 | 1.118 | 15.0 | 0.495 | 0.26 | 0.822
10.0 | 0.534 | 0.41 | 0.917 | 20.0 | 0.396 | 0.203 | 0.640
$n=5$, $K=20$ | $n=10$, $K=20$
2.0 | 2.953 | 3.75 | 8.385 | 6.0 | 2.489 | 1.333 | 4.216
4.0 | 2.698 | 2.125 | 4.752 | 9.0 | 1.831 | 0.944 | 2.987
6.0 | 2.015 | 1.472 | 3.292 | 12.0 | 1.432 | 0.729 | 2.306
8.0 | 1.591 | 1.125 | 2.516 | 15.0 | 1.255 | 0.593 | 1.876
10.0 | 1.378 | 0.91 | 2.035 | 20.0 | 0.985 | 0.453 | 1.431
The interval matrix of system (29) is known to be regular if and only if
$\theta>n$ for even $n$ and $\theta>\sqrt{n^{2}-1}$ for odd $n$ [9].
Consequently, in Table 1, the first two rows that correspond to each separate
case of $n$ and $K$ refer to systems with singular matrices. As the parameter
$\theta$ grows, the matrix of the system becomes regular and better
conditioned.
The values of IVE are compared with
$\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip
1.0pt}\varXi_{tol})\|_{\infty}$ and
$\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip
1.0pt}\varXi_{tol})\|_{2}$, that is, with the Chebyshev norm (max-norm) and
the Euclidean norm of the radius of the interval hull of the tolerable
solution set (denoted as $\scalebox{0.67}[0.87]{$\Box$\hskip
1.0pt}\varXi_{tol}$). We can see that, with the exception of two cases
corresponding to $n=5$ and $K=10,20$, the values of IVE are always within the
two-sided bounds given by
$\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip
1.0pt}\varXi_{tol})\|_{\infty}$ (lower bound) and
$\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip
1.0pt}\varXi_{tol})\|_{2}$ (upper bound). And that is reasonable. Overall, our
numerical experiments confirm the adequacy of the new measure of variability,
which gives quite satisfactory approximate estimates of the size of the
tolerable solution sets in various situations.
## 5 Discussion
IVE is the first measure of variability proposed in the statistics of interval
data, for estimation using the maximum compatibility method, and we can not
compare IVE with similar other measures, since they simply do not exist.
However, it is useful to correlate the estimate IVE with the ideal
mathematical characteristics of the solution set, such as its diameter, in
terms of computational convenience and laboriousness.
First of all, IVE reflects instabilities in the solution set better than the
diameter (see the first example in Section 4). An instability of the tolerable
solution set for an interval linear system arises in the case when the maximum
value of the recognizing functional is zero, $\max\mathrm{Tol}\,=0$. Then the
tolerable solution set can be either a single-point or an extended set with
non-zero diameter and empty interior [15]. After an arbitrarily small
perturbation of data, the latter situation can abruptly turn into the empty
solution set. In any case, this phenomenon is signaled by the zero value of
the maximum of the recognizing functional. The corresponding variability
measure IVE is also zero, which is quite natural: it makes sense to show only
“stable size” of the solution set. The equality of IVE to zero or “almost
zero” thus allows us to diagnose unstable cases.
Next, the problem of computing the diameter, in 2-norm, of the tolerable
solution set to an interval linear system of equations is NP-hard in general.
This follows from its reducibility to the quadratic programming problem with
linear constraints (see [22]). Indeed, the membership of a point in the
tolerable solution set to an interval $m\times n$-system of equations is
determined by a system of linear inequalities of the size $2m\times 2n$ (the
Rohn theorem [12]). To compute the diameter of the tolerable solution set in
2-norm, we have to maximize the quadratic objective function
$\|x^{\prime}-x^{\prime\prime}\|^{2}_{2}$ over all pairs of points
$x^{\prime}$, $x^{\prime\prime}$ from the tolerable solution set, i. e.
satisfying $2m\times 2n$-systems of linear inequalities. So, computing the
diameter of the tolerable solution set is not easy.
The diameter of the interval hull of the tolerable solution set can be
computed more simply, but it is not better than IVE in any case, since the
interval hull is not the solution set itself, and the coarsening resulted from
such a replacement may be large.
Calculation of IVE by formula (6) requires solving the data fitting problem,
that is, finding $\max\,\mathrm{Tol}\,$ and $\arg\max\,\mathrm{Tol}\,$, and
then we need to evaluate the minimum of the condition number of the matrices
from the interval data matrix. In turn, the recognizing functional Tol is a
concave piecewise linear function [15], so computing its maximum is
polynomially complex. The author efficiently solves it by nonsmooth
optimization methods developed in recent decades, in particular, using
$r$-algorithms proposed by N.Z. Shor [20], or using separating plane
algorithms (see, e. g., [11, 23]). The most difficult part in calculating IVE
is thus evaluating the minimum condition number of point matrices from a given
interval matrix.
Computing the exact minimum of the condition number is not simple, but to
address practical problems which will apply the value IVE, it is sufficient to
have an approximate estimate for
$\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A$ from above. This
follows from our considerations in Section 3.3. Sometimes, it is not necessary
to compute $\min\,\mathrm{cond}_{2}\,A$ at all, if we have to compare, with
each other, the variability of the estimates obtained for the same data matrix
$A$.
If the interval matrix is “sufficiently narrow”, being not very different from
a point matrix, then we can approximate
$\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A\;\approx\;\mathrm{cond}_{2}(\mathrm{mid}\,\text{\boldmath$A$}).$
(30)
But in general, this recipe may work poorly, since the left and right sides of
the approximate equality (30) can be quite different. In the examples with
systems (25) and (28) from Section 4, the condition number of the midpoint
matrix is $2.38\cdot 10^{4}$, and, using the simplified formula (30), we are
mistaken in evaluating the variability measure IVE by more than 20 times.
In the general case, for a more accurate evaluation of
$\min\,\mathrm{cond}_{2}\,A$, we can use popular evolutionary optimization
methods, such as a genetic algorithm, simulated annealing, particle swarm
optimization, etc., within the interval matrix $A$. In the numerical
experiments from Section 4, the minimum of the condition number was found
using the standard program of simulated annealing from free computer math
system Scilab.
Note that there is a fundamental difference between computing the variability
measure IVE and computing the diameter of the tolerable solution set: in the
first case, we calculate a minimum, while in the second we have to find a
maximum. Using traditional optimization methods and various heuristics, in the
first case we compute an approximation to the minimum from above, and in the
second case we find an approximation to the maximum from below. If we want to
get, with our variability measure, a guaranteed outer estimation of the
solution set, then the upper estimate, which is obtained by calculating the
minimum in IVE, is more preferable.
There exists one more viewpoint at the variability measure IVE.
In traditional probabilistic statistics, the phenomenon of collinearity of
data (also called “multicollinearity”) plays a large role. It is the presence
of a linear dependence between the input (predictor) variables of the
regression model. The $k$ variables of the model in question are usually
called _collinear_ if the vectors representing them lie in a linear space of
dimension less than $k$ [1], so that one of these vectors is a linear
combination of the others. In practice, such exact collinearity of data is
rare, but real computational problems in data fitting often starts when the
data vectors are “almost linearly dependent”. Then the parameter estimates are
unstable, which leads to increased statistical uncertainty, i. e., an increase
in the variance of the estimates.
According to modern views, the collinearity of data is most adequately
described by the condition number of the matrix made up of these data (see, e.
g., [1], Chapter 3). In this sense, our IVE is, in fact, a measure of the
collinearity of the data, corrected with the help of the actual value of the
estimate and compatibility of this estimate with the data (which is indicated
by the maximal value of the recognizing functional). The minimum over all
$\mathrm{cond}_{2}A$ for $A\in\text{\boldmath$A$}$ is taken due to the
specifics of the strong compatibility of parameters and data, and it agrees
well with the regularizing role of the tolerable solution set (see [18]).
With this interpretation, IVE makes sense even with a negative maximum of the
recognizing functional, max Tol, when the tolerable solution set is empty and
the parameters of the linear function (1), which are strongly compatible with
the data, do not exist. The absolute value of IVE still shows, up to a certain
scaling coefficient, a measure of the collinearity of the data (a measure of
their ill-conditioning), and the negative sign indicates the status of the
solution to the problem, i. e., that the parameter vector computed is not
strongly compatible with the data, but only provides the best possible
approximation for the input data of the problem.
The immediate goal of further research is to justify the use of IVE for
underdetermined situations, where the number $m$ of observations is less than
the number $n$ of parameters to be determined. The maximum compatibility
method works well in this case too, we get parameter estimates and we can
calculate their values of IVE, but its application needs to be justified, at
least at the same level of rigor as was done in this work for $m\geq n$.
#### Aknowledgements
The author is grateful to Alexander Bazhenov, Sergey Kumkov, and Sergei Zhilin
for stimulating discussions and valuable comments on the work. Also, the
author thanks the anonymous reviewers for their constructive criticism and
good suggestions.
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|
2024-09-04T02:54:59.042857 | 2020-03-11T06:31:10 | 2003.05133 | {
"authors": "B\\\"ulent Karas\\\"ozen",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26153",
"submitter": "Bulent Karas\\\"ozen",
"url": "https://arxiv.org/abs/2003.05133"
} | arxiv-papers | # Model Order Reduction in Neuroscience
Bülent Karasözen Institute of Applied Mathematics & Department of Mathematics,
Middle East Technical University, Ankara-Turkey<EMAIL_ADDRESS>
###### Abstract
Human brain contains approximately $10^{9}$ neurons, each with approximately
$10^{3}$ connections, synapses, with other neurons. Most sensory, cognitive
and motor functions of our brains depend on the interaction of a large
population of neurons. In recent years, many technologies are developed for
recording large numbers of neurons either sequentially or simultaneously.
Increase in computational power and algorithmic developments have enabled
advanced analyses of neuronal population parallel to the rapid growth of
quantity and complexity of the recorded neuronal activity. Recent studies made
use of dimensionality and model order reduction techniques to extract coherent
features which are not apparent at the level of individual neurons. It has
been observed that the neuronal activity evolves on low-dimensional subspaces.
The aim of model reduction of large-scale neuronal networks is accurate and
fast prediction of patterns and their propagation in different areas of brain.
Spatiotemporal features of the brain activity are identified on low
dimensional subspaces with methods such as dynamic mode decomposition (DMD),
proper orthogonal decomposition (POD), discrete empirical interpolation (DEIM)
and combined parameter and state reduction. In this paper, we give an overview
about the currently used dimensionality reduction and model order reduction
techniques in neuroscience.
Keywords:neuroscience, dimensionality reduction, proper orthogonal
decomposition, discrete empirical interpolation, dynamic mode decomposition,
state and parameter estimation.
Classification[MSC 2010]: 93A15,92C55, 37M10,37M99,37N40,65R32.
## 1 Introduction
Due to the advances in recording and imaging technologies, the number of
recorded signals from brain cells increased significantly in the last few
years. The recorded spatio-temporal neural activity give rise to networks with
complex dynamics. In neuroscience, molecular and cellular level details are
incorporated in large-scale models of the brain in order to reproduce
phenomena such as learning and behavior. The rapid growth of simultaneous
neuronal recordings in scale and resolution brings challenges with the
analysis of the neuronal population activity. New computational approaches
have to be developed to analyze, visualize, and understand large-scale
recordings of neural activity. While algorithmic developments and the
availability of significantly more computing power have enabled analysis of
larger neuronal networks, these advances cannot keep pace with increasing size
and complexity of recorded activity. The activity of complex networks of
neurons can often be described by relatively few distinct patterns. Model
order reduction techniques enable us to identify the coherent spatial–temporal
patterns.
The presence or absence of a neural mechanism can be analyzed for neuronal
populations. Dimensionality reduction methods [6] are data-driven statistical
techniques for forming and evaluating hypotheses about population activity
structure, which are summarized in Section 2. One of the goals of
neurosciences is fast and accurate predictions of the potential propagation in
neurons. The differential equations describing the propagation of potential in
neurons were developed and are described by Hodgkin and Huxley equations [12].
They consists of a coupled system of ordinary and partial differential
equations (ODEs and PDEs). The dimension of the associated discretized systems
is very large for accurately simulating neurons with realistic morphological
structure and synaptic inputs. In Section 3 we present two model order
reduction approaches based on POD and DEIM [5] which can predict accurately
the potential propagation in large scale neuronal networks leading to
important speedups [17, 16, 2]. Using the functional neuroimagining data from
electroencephalography (EEG) or functional magnetic resonance imaging (fMRI),
different regions of the brain can be inferred by dynamic causal modeling
(DCM) [7]. Effective connectivity is parameterised in terms of coupling among
unobserved brain states and neuronal activity in different regions. In Section
4 we describe a combined state and parameter reduction for parameter
estimation and identification [10] to extract effective connectivity in
neuronal networks from measured data, such as data from electroencephalography
(EEG) or functional magnetic resonance imaging (fMRI). In Section 5 the data-
driven, equation free, model order reduction method dynamic mode decomposition
(DMD) is described for identifying sleep spindle networks [3]. Reduced order
models with POD and DEIM and four variants of them are presented for neuronal
synaptic plasticity and neuronal spiking networks in Section 6.
## 2 Dimensionality reduction methods
Coordination of responses across neurons exists only at the level of the
population and not at the level of single neurons. The presence or absence of
a neural mechanism can be analyzed for neuronal populations. Dimensionality
reduction methods are data-driven statistical techniques for forming and
evaluating hypotheses about population activity structure. They produce low-
dimensional representations of high-dimensional data with the aim to extract
coherent patterns which preserve or highlight some feature of interest in the
data [6]. The recorded neurons of dimension $D$ are likely not independent of
each other, because they belong to a common network of neuronal populations.
From the high-dimensional data of neuronal recordings, a smaller number of
explanatory variables $K$ ( $K<D$) are extracted with the help of
dimensionality reduction methods. The explanatory variables are not directly
observed, therefore they are referred to as latent variables. The latent
variables define a $K$-dimensional space representing coherent patterns of the
noisy neural activity of $D$ neurons.
There exists several dimensionality reduction methods which differ in the
statistical interpretation of the preserved and discarded features of the
neuronal populations. We summarize the commonly used statistical methods of
dimensionality reduction methods in [6], where further references about the
methods can be found.
Principal component and factor analysis; The most widely known technique to
extract coherent patterns from high dimensional data is the modal
decomposition. A particularly popular modal decomposition technique is
principal component analysis (PCA), which derives modes ordered by their
ability to account for energy or variance in the data. In particular, PCA is a
static technique and does not model temporal dynamics of time-series data
explicitly, so it often performs poorly in reproducing dynamic data, such as
recordings of neural activity. The low-dimensional space identified by PCA
captures variance of all types, including firing rate variability and spiking
variability, whereas factor analysis (FA) discards the independent variance
for each neuron. and preserves variance that is shared across neurons.
Time series methods: The temporal dynamics of the population activity can be
identified if the data comes from a time series. The most commonly used time
series methods for dimensionality reduction neural recordings are: hidden
Markov models (HMM) [18], kernel smoothing followed by a static dimensionality
reduction method, Gaussian process factor analysis (GPFA) [35], latent linear
dynamical systems (LDS) [4] and latent nonlinear dynamical systems (NLDS)
[26]. They produce latent neural trajectories that capture the shared
variability across neurons. The HMM is applied when a jump between discrete
states of neurons exists, other methods identify smooth changes in firing
rates over time.
Methods with dependent variables: On many neuronal recordings the high-
dimensional firing rate space is associated with labels of one or more
dependent variables, like stimulus identity, decision identity or a time
index. The dimensionality reduction aims in this case to project the data such
that differences in these dependent variables are preserved. The linear
discriminant analysis (LDA) can be used to find a low-dimensional projection
in which the $G$ groups to which the data points belong are well separated.
Nonlinear dimensionality reduction methods: All the previous methods assume a
linear relationship between the latent and observed variables. When the data
lies on a low-dimensional, nonlinear manifold in the high-dimensional space, a
linear method may require more latent variables than the number of true
dimensions of the data. The most frequently used methods to identify nonlinear
manifolds are Isomap79 [31] and locally linear embedding (LLE) [28]. Because
the nonlinear methods use local neighborhoods to estimate the structure of the
manifold, population responses may not evenly explore the high-dimensional
space. Therefore theses methods should be used with care.
## 3 Proper orthogonal decomposition (POD) and discrete empirical
interpolation (DEIM) for Hodgin-Huxley model
One of the goals of neurosciences is fast and accurate predictions of the
potential propagation in neurons. The differential equations describing
propagation of potential in neurons are described by Hodgkin and Huxley (HH)
cable equations [12]. They consists of a coupled system of ordinary (ODEs) and
partial differential equations PDEs. Accurate simulation of morphology,
kinetics and synaptic inputs of neurons requires solution of large systems of
nonlinear ODEs. The complexity of the models are determined by the synapse
density of the dentritic length ($1\mu$ one micron). In simulations, for one
synapse per micron on a cell $5$ mm dendrite at $5,000$ compartments each with
$10$ variables are needed, which results in $50,000$ coupled nonlinear system
of ODEs [17, 16]. To recover complex dynamics, efficient reduced order
neuronal methods are developed using the POD and DEIM from the snapshots of
the in space and time discretized coupled PDEs and ODEs [17, 16, 2]. In this
section we describe two of them. They differ in the formulation of the HH
cable equation and of the equations for the gating variables.
### 3.1 Morphologically accurate reduced order modeling
The neuronal full order models (FOMs) in [17, 16] consists of $D$ branched
dendritic neurons $B=\sum_{d=1}^{D}B_{d}$ meeting at the soma, where the
$d^{th}$ has $B_{d}$ branches. It is assumed that the branch $b$ carries $C$
distinct ionic currents with associated densities and $G_{bc}(x)$ and reversal
potentials $E_{c},c=1,\ldots,C$. The kinetics of current $c$ on branch $b$ is
governed by the $F_{c}$ gating variables, $w_{bcf},f=1,\ldots,F_{c}$. When
subjected to input at $S_{b}$ synapses, the nonlinear HH cable equation for
the transmembrane potential $v_{b}(x,t)$ with the equation for the gating
variables $w_{bcf}$ is given by (see [2] Fig.1, model network with three
cables)
$\displaystyle a_{b}C_{m}\frac{\partial v_{b}}{\partial t}=$
$\displaystyle\frac{1}{2R_{i}}\frac{\partial}{x}\left(a_{b}^{2}\frac{\partial
v_{b}}{\partial x}\right)$ (1) $\displaystyle-
a_{b}\sum_{c=1}^{C}G_{bc}(x)(v_{b}-E_{c})\prod_{f=1}^{F_{c}}w_{bcf}^{q_{cf}}$
$\displaystyle\frac{1}{2\pi}\sum_{s=1}^{S_{b}}g_{bs}(t)\delta(x-x_{bs})(v_{b}-E_{bs})$
$\displaystyle\frac{\partial w_{bcf}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{w_{cf,\infty}(v_{b})-w_{bcf}}{\tau_{cf}(v_{b})},\quad
0<x<l_{b},\;t>0,$ (2)
where $g_{bs}(nS)$ is the time course, $x_{bs}$ is the spatial location and
$E_{bs}$ is the reversal potential of the $s$th synapse on branch $b$. The
variables and parameters in (1) are described in [17, 16].
These branch potentials interact at $J$ junction points, where junction $J$
denotes the soma. The $D$ dendrites join at soma. Continuity of the potential
at the soma leads to a common value at current balance denoted by
$v_{\sigma}(t)$. Then the networked form of (1) becomes
$\displaystyle a_{b}C_{m}\frac{\partial v_{\sigma}}{\partial t}=$
$\displaystyle\frac{\pi}{A_{\sigma}R_{i}}\sum_{d=1}^{D}\frac{\partial}{\partial
x}\left(a_{b_{J}^{d}}^{2}(l_{b_{J}^{d}})\frac{\partial
v_{b_{J^{d}}}(l_{b_{J^{d}}},t)}{\partial x}\right)$ (3) $\displaystyle-
a_{b}\sum_{c=1}^{C}G_{\sigma
c}(x)(v_{\sigma}-E_{c})\prod_{f=1}^{F_{c}}w_{\sigma cf}^{q_{cf}}(t)$
$\displaystyle\frac{1}{A_{\sigma}}\sum_{s=1}^{S_{b}}g_{\sigma
s}(t)(v_{\sigma}(t)-E_{\sigma s})$ $\displaystyle\frac{\partial w_{\sigma
cf}(t)}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{w_{cf,\infty}(v_{\sigma}(t))-w_{\sigma
cf}(t)}{\tau_{cf}(v_{\sigma})(t)},\quad 0<x<l_{b},\;t>0.$ (4)
When the cell is partitioned into $N$ compartments, with $C$ distinct ionic
currents per compartment and with $F$ gating variables per current, the
following nonlinear ODEs are obtained
$\displaystyle v^{\prime}(t)=$ $\displaystyle
Hv(t)-(\Phi(w(t)e).v(t)+\Phi(w(t))E_{i}$ (5)
$\displaystyle+G(t).(v(t)-E_{s}),\quad v(t)\in\mathbb{R}^{N}$ $\displaystyle
w^{\prime}(t)=$ $\displaystyle(A(v(t))-w(t))./B(v(t)),\quad
w(t)\in\mathbb{R}^{N\times C\times F}$ (6)
where $H\in\mathbb{R}^{N\times N}$ is the Hines matrix [11], $e=[1\;1\cdots
1]^{T}\in\mathbb{R}^{C}$ and the ‘dot’ operator, $a.b$, denotes element-wise
multiplication. $E_{i}$ and $E_{s}$ are respectively the vector of channel
reversal potentials and the vector of synaptic reversal potentials,
respectively Eq. (5) is discretized in time by the second order discretized
Euler scheme [11].
In [16] using the snapshots of $v(t)$ and of the nonlinear term
$N(v(t),w(t))\equiv(\Phi)w(t))e).v(t)-\Phi(w(t)))E_{i}$ at times
$t_{1},t_{2},\ldots,t_{n}$ the POD and DEIM modes are constructed.
The reduced membrane potential $v_{r}$ are constructed using the POD basis,
the reduced gating variables $w_{r}$ are obtained after applying the DEIM to
the nonlinear terms. The reduced order model in [16] preserves the spatial
precision of synaptic input, captures accurately the subthreshold and spiking
behaviors.
In [17] a linearized quasi active reduced neuronal model is constructed using
balanced truncation and ${\mathcal{H}}_{2}$ approximation of transfer
functions in time. ROMs preserve the input-output relationship and reproduce
only subthreshold dynamics.
### 3.2 Energy stable neuronal reduced order modeling
In [1, 2] a different form of the HH cable equation and ODEs for gating
variables is considered. The intracellular potential $v(x,t)$ and three gating
variables $m(x,t),\;h(x,t)$, and $n(x,t)$ describe the activation and
decativation of the ion channels, of the sodium channels and of the potassium
channels, respectively. A single cable in the computational domain
$(x,t)\in[0,L]\times(0,T]$ describing the distribution of the potential
$u(x,t)$ is given by [1, 2]
$\frac{\partial u}{\partial
t}=\frac{\mu}{a(x)}\left(a(x)^{2}u_{x}\right)_{x}-\frac{1}{C_{m}}g(m,h,n)u+\frac{1}{C_{m}}f(m,h,n,x,t),$
(7)
where $a(x)$ the radius of the neurons and $C_{m}$ is specific membrane
capacitance, $\mu=\frac{1}{2C_{m}R_{i}}>0$ the ratio with $R_{i}$ the axial
resistivity. The conductance $g(x,t)$ is a polynomial of the gating variables
$g(x,t)=g_{1}m^{3}h+g_{2}n^{4}+g_{3}>0,$ (8)
with the source term
$f(m,h,n,x,t)=g_{1}E_{1}m^{2}h+g_{2}E_{2}n^{4}+g_{3}E_{3}-i(x,t),$ (9)
where $E_{l},\;l=1,2,3$ are equilibrium potentials and $i(x,t)$ input current
at $x$
$i(x,t)=\sum_{s=1}^{N_{s}}i_{s}(x,t),\quad x\in[0,L].$ (10)
The nonlinear ODEs for the gating variables are given by
$\displaystyle\frac{\partial m}{\partial t}$ $\displaystyle=$
$\displaystyle\alpha_{m}(v(x,t)(1-m(x,t))-\beta_{m}v(x,t))m(x,t),$
$\displaystyle\frac{\partial h}{\partial t}$ $\displaystyle=$
$\displaystyle\alpha_{h}(v(x,t))(1-h(x,t))-\beta_{h}v(x,t))h(x,t),$ (11)
$\displaystyle\frac{\partial n}{\partial t}$ $\displaystyle=$
$\displaystyle\alpha_{n}(v(x,t))(1-n(x,t))-\beta_{n}v(x,t))n(x,t),$
Expression for
$\alpha_{m},\;\alpha_{h},\;\alpha_{n},\;\beta_{m},\;\beta_{h},\;\beta_{n}$ and
boundary conditions can be found in [2].
In [1, 2], a model network with three cables connected to a soma is used. The
equations governing the potential propagation in the network $N_{c}$ neuron
cables-dentrites and /or axons with the superscript ${}^{(c)},\;c=1,\ldots
N_{c}$, are given as
$\displaystyle\frac{\partial v^{(c)}}{\partial t}=$
$\displaystyle\frac{\mu}{a^{(c)}(x^{(c)})}\left(\left(a^{(c)}\left(x^{(c)}\right)^{2}\right)v^{(c)}_{x}\right)_{x}-\frac{1}{C_{m}}g\left(m^{(c)},h^{(c)},n^{(c)}\right)u^{(c)}$
$\displaystyle+$
$\displaystyle\frac{1}{C_{m}}f\left(m^{(c)},h^{(c)},n^{(c)},x^{(c)},t\right)$
(12)
$\displaystyle\frac{\partial m^{(c)}}{\partial t}$ $\displaystyle=$
$\displaystyle\alpha_{m}(v^{(c)}(1-m^{(c)})-\beta_{m}v^{(c)})m^{(c)},$
$\displaystyle\frac{\partial h^{(c)}}{\partial t}$ $\displaystyle=$
$\displaystyle\alpha_{h}(v^{(c)})(1-h^{(c)})-\beta_{h}v{(c)})h^{(c)},$ (13)
$\displaystyle\frac{\partial n^{(c)}}{\partial t}$ $\displaystyle=$
$\displaystyle\alpha_{n}(v^{(c)}))(1-n^{(c)})-\beta_{n}v^{(c)})n^{(c)},$
for $x^{(c)}\in\Omega^{(c)}=[0,L^{(c)}]$ together with the boundary
conditions.
The semi-discrete form of these equations are approximated using energy stable
summation by parts [1, 2] for the model network. The reduced order bases (ROB)
for multiple cables of identical lengths are assembled into a network in block
form. The block structure of the ROB allows a flexible structure-preserving
model reduction approach with an independent approximation in each cable and
energy stability and accuracy properties follow from this block structure.
Computation of the time varying reduced variables in the gating equations at
every time $t$ is costly because they scale with dimension of FOM. A
nonnegative variant of the discrete empirical interpolation method (NNDEIM) is
developed in [2] to preserve the structure and energy stability properties of
the equations.
The capability of the greedy-based approach to generate accurate predictions
in large-scale neuronal networks is demonstrated for system with more than
$15,000$ degree of freedoms (dofs). The state variable ROB of dimension $l=15$
with POD modes together with the nonnegative ROBs of dimension $p=60$ with
NNDEIM modes are constructed using a greedy approach to predict the potential
variation at the soma. The speedup of simulations is about $20$ larger than
Galerkin projection only is $1.3$ without using the NDEIM.
## 4 Combined state and parameter reduction for dynamic causal modelling
In neuroscience different regions of the brain are inferred using
neuroimagining data from EEG or fMRI recordings using the method od dynamic
causal modeling (DCM) [7]. Effective connectivity is parameterised in terms of
coupling among unobserved brain states and neuronal activity in different
regions. In DCM the neuronal activity is of the observed brain region is
represented as a SISO (single input single output) linear state-space system
$\dot{x}=A_{\mathrm{d}yn}(\mu)x+B_{\mathrm{d}yn}u$ (14)
with the parameterized connectivity $A_{\mathrm{d}yn}(\mu)$ and external input
matrices $B_{\mathrm{d}yn}$.
Linearization of the nonlinear DCM hemodynamic forward sub-model (balloon
model) [7] transforms the neuronal activity to the measured BOLD (blood oxygen
level dependent) response. Linearization around the equilibrium results in the
following single input, single output (SISO) system:
$\displaystyle B_{obs}$ $\displaystyle:=$ $\displaystyle(1\;0\;0\;0)^{T},\quad
C_{obs}=(0\;0\;V_{0}k_{1}\;V_{0}k_{2}),$ (15) $\displaystyle\dot{z}_{i}$
$\displaystyle=$ $\displaystyle A_{obs}z_{i}+B_{obs}x_{i},$ (16)
$\displaystyle y_{i}$ $\displaystyle=$ $\displaystyle C_{obs}z_{i},$ (17)
$\displaystyle z_{0}$ $\displaystyle=$ $\displaystyle(0\;0\;0\;0)^{T},$ (18)
$A_{\mathrm{o}bs}:=\left(\begin{array}[]{cccc}\frac{1}{\tau_{s}}&\frac{1}{\tau_{f}}&0&0\\\
1&0&0&0\\\
0&\frac{1}{\tau_{0}E_{0}}(1-(1-E_{0})(1-\ln(1-E_{0})))&\frac{1}{\tau_{0}}&\frac{1-\alpha}{\tau_{0}\alpha}\\\
0&\frac{1}{\tau_{0}}&0&\frac{1}{\tau_{0}\alpha}\end{array}\right).$ (19)
The fMRI measurements at the $i^{th}$ brain region are reflected by the output
variables $y_{i}$. For the meaning of the variables and parameters in (15) and
(19) we refer to [10, 9]. The linearized forward sub-models are embedded into
the fMRI connectivity model
$\left(\begin{array}[]{c}\dot{x}\\\ \dot{z}_{1}\\\ \dot{z}_{2}\\\ \vdots\\\
z_{N_{dyn}}\end{array}\right)=\left(\begin{array}[]{ccccc}A_{dyn}(\mu)&0&0&\cdots&0\\\
\delta_{1,1}&A_{obs}&0&&\\\ \delta_{2,1}&0&A_{obs}&&\\\ \vdots&&\ddots&\\\
\delta_{1,N_{dyn}}&&&A_{obs}\end{array}\right)\left(\begin{array}[]{c}x\\\
z_{1}\\\ z_{2}\\\ \vdots\\\
z_{N_{dyn}}\end{array}\right)+\left(\begin{array}[]{c}B_{dyn}\\\ 0\\\ 0\\\
\vdots\\\ 0\end{array}\right)v,$ (20)
$y=\left(0\left(\begin{array}[]{ccc}C_{obs}&&\\\ &\ddots&\\\
&&C_{obs}\end{array}\right)\right)\left(\begin{array}[]{c}x\\\ z_{1}\\\
z_{2}\\\ \vdots\\\ z_{N_{dyn}}\end{array}\right),$ (21)
where $\delta_{ij}\in\mathbb{R}^{4\times N_{\mathrm{d}yn}}$ denotes the
Kronecker matrix with non-zero elements located at the $(i,j)^{th}$ component.
The linearized state-space forward model (20) and (21) corresponds to a
multiple input, multiple output (MIMO) system
$\dot{x}(t)=A(\mu)x(t)+Bu(t),\qquad y(t)=Cx(t),$ (22)
where $x\in\mathbb{R}^{N}$ is the internal state, $u\in\mathbb{R}^{J}$ the
external input, $y\in\mathbb{R}^{O}$ the observed output and $\mu$ are the
parameters describing different conditions.
For large number of $M:=N^{2}$ parameters, the computational cost for
inferring the parameters and states is very high. In [10, 8] a combined state
and parameter model order reduction is developed for parameter estimation and
identification to extract effective connectivity. The inversion procedure
consists of two phases, the offline and online phases. In the offline phase,
the underlying parameterized model is reduced jointly in states and
parameters. In online phase, the reduced order model’s parameters are
estimated to fit the observed experimental data. Using state and parameter
reduction, the computational cost is reduced in the offline phase. The
simultaneous reduction of state and parameter space is based on Galerkin
projections with the orthogonal matrices for the state
$V\in\mathbb{R}^{N\times n}$ and for the parameters $P\in\mathbb{R}^{M\times
m}$. The reduced model is of lower order $n<<N,\;m<<M$ than the original full
order model. The reduced states $x_{r}(t)\in\mathbb{R}^{n}$ and the reduced
parameters $\mu\in\mathbb{R}^{m}$ are computed as
$\dot{x}_{r}(t)=A_{r}(\mu_{r})x_{r}(t)+B_{r}u(t),\qquad y_{r}(t)=C_{r}x(t)$
(23)
with a reduced initial condition $x_{r,0}=V^{T}x_{0}$ and the reduced
components
$\displaystyle\mu_{r}$ $\displaystyle=$ $\displaystyle
P^{T}\mu\in\mathbb{R}^{m},$ $\displaystyle A_{r}(\mu_{r})$ $\displaystyle=$
$\displaystyle V^{T}A(P\mu_{r})V\in\mathbb{R}^{n\times n},$ $\displaystyle
B_{r}$ $\displaystyle=$ $\displaystyle V^{T}B\in\mathbb{R}^{n\times J},$
$\displaystyle C_{r}$ $\displaystyle=$ $\displaystyle CV\in\mathbb{R}^{O\times
m}.$
In the online phase, the optimization based inverse problem is combined with
the reduction of state and parameter space. The inversion is based on
generalized data-driven optimization approach to construct the ROMs in [23]
and enhanced with the Monte-Carlo method to speed up the computations. The
state projection $V\in\mathbb{R}^{N\times n}$ and parameter projection
$P\in\mathbb{R}^{m\times m}$ are determined iteratively based on a greedy
algorithm by maximizing the error between the high-fidelity original and the
low-dimensional reduced model in the Bayesian setting.
Numerical experiments with the DCM model in [23] show highly dimensional
neuronal network system is efficiently inverted due to the short offline
durations. In the offline phase, Monte-Carlo enhanced methods require more
than an order of magnitude less offline time compared to the original and
data-misfit enhanced methods. In the online phase the reduced order model has
a speedup factor about an order of magnitude more compared to the full-order
inversion. The output error of the data-misfit enhanced method is close to
full order method. The output-errors of Monte-Carlo decrease with increasing
number of simulation but does not reach the output error of the full order
system. The source code is available in MATLAB [8].
## 5 Dynamic mode decomposition
Dynamic mode decomposition (DMD) is a data-driven, equation free ROM technique
[20]. It was initially developed to reduce the high dimensional dynamic data
obtained from experiments and simulations in the fluid mechanics into a small
number of coupled spatial–temporal modes [29, 30]. DMD was applied to explore
spatial–temporal patterns in large-scale neuronal recordings in [3]. DMD can
be interpreted as combination of discrete Fourier transform (DFT) in time and
principal component analysis (PCA) [15] in space. Both PCA and independent
component analyses (ICA) [13] are static techniques, which perform poorly in
reproducing dynamic data, such as recordings of neural activity.
The data is taken from electrocorticography (ECoG) recordings. Voltages from
$n$ channels of an electrode array sampled every $\Delta t$. These
measurements are arranged at snapshot $k$ to the column vectors
${\mathbf{x}}_{k}$. The $m$ snapshots in time construct to data matrices
shifted in time with $\Delta t$
${\mathbf{X}}=\left[\begin{array}[]{cccc}|&|&&|\\\
{\mathbf{x}}_{1}&{\mathbf{x}}_{2}&\cdots&{\mathbf{x}}_{m-1}\\\
|&|&&|\end{array}\right],\quad{\mathbf{X}}^{\prime}=\left[\begin{array}[]{cccc}|&|&&|\\\
{\mathbf{x}}_{2}&{\mathbf{x}}_{3}&\cdots&{\mathbf{x}}_{m}\\\
|&|&&|\end{array}\right]$ (24)
These matrices are assumed to be related linearly in time
${\mathbf{X}}^{\prime}={\mathbf{A}}{\mathbf{X}}.$ (25)
The DMD of the data matrix pair ${\mathbf{X}}$ and ${\mathbf{X}}^{\prime}$ is
given by the eigendecomposition of ${\mathbf{A}}$ using the singular value
decomposition (SVD) of the data matrix ${\mathbf{X}}=U\Sigma V^{*}$ by
computing the pseudoinverse
${\mathbf{A}}\approx{\mathbf{X}}^{\prime}{\mathbf{X}}^{\dagger}\equiv{\mathbf{X}}^{\prime}{\mathbf{V}}{\mathbf{\Sigma}}^{-1}{\mathbf{U}}^{*}.$
The spatio-temporal modes are computed by the exact DMD algorithm [32].
Because DMD does not contain explicit spatial relationship between neighboring
measurements, traveling waves occurring in neuronal networks can not be
captured well with a few coherent modes. DMD was also used as a windowed
technique with a temporal window size constrained by lower bound as for the
discrete Fourier transformation (DFT). In contrast to the fluid dynamics where
$n>>m$, in neuroscience the electrode arrays that have tens of channels $n$ in
the recordings with $m$ number of snapshots in the windows data per second, so
that $n<m$. The number of singular values $v$ of ${\mathbf{X}}$ are less than
$n$ and $m-1$, which restricts the maximum number of DMD modes and eigenvalues
to $n$. Because of this the dynamics can be captured over $m$ snapshots. The
rank mismatch is resolved by appending to the snapshot measurements with $h-1$
time-shifted versions of the data matrices. The augmented data matrix
${\mathbf{X}}_{\mathrm{a}ug}$ is given as
${\mathbf{X}}_{\mathrm{a}ug}=\left[\begin{array}[]{cccc}|&|&&|\\\
{\mathbf{x}}_{1}&{\mathbf{x}}_{2}&\cdots&{\mathbf{x}}_{m-h}\\\ |&|&&|\\\
|&|&&|\\\ {\mathbf{x}}_{2}&{\mathbf{x}}_{3}&\cdots&{\mathbf{x}}_{m-h-1}\\\
|&|&&|\\\ &&\cdots&\\\ |&|&&|\\\
{\mathbf{x}}_{h}&{\mathbf{x}}_{h+1}&\cdots&{\mathbf{x}}_{m-1}\\\ |&|&&|\\\
\end{array}\right].$ (26)
The augmented matrices ${\mathbf{X}}_{{\mathrm{a}ug}}$ and
${\mathbf{X}}^{\prime}_{{\mathrm{a}ug}}$ are Hankel matrices, which are
constant along the skew diagonal, as in the Eigenvalue Realization Algorithm
(ERA) [14]. The number of the stacks $h$ is chosen such that $hn>2m$. A
measure to determined the optimal number of stacks $h$ is the approximation
error
$E=\frac{||{\mathbf{X}}-\hat{\mathbf{X}}||_{F}}{||{\mathbf{X}}||_{F}}$
where $||\cdot||_{F}$ is the Frobenius norm. The approximation error $E$ is
decreasing with increasing number of stacks $h$ and reaches a plateau, so that
the DMD accuracy does not significantly increases.
DMD is applied in [3] as an automated approach to detect and analyze reliably
spatial localization and frequencies of sleep spindle networks from human ECoG
recordings. A MATLAB implementation is available at github.com/bwbrunton/dmd-
neuro/.
## 6 Reduced order modeling of biophysical neuronal networks
Recently reduced order models for ODEs
$\dot{x}(t)=A(t)x(t)+f(x(t))+Bu(t)$ (27)
are constructed using POD and DEIM to investigate input-output behavior of the
neuronal networks in the brain [22, 21], where $x(t)$ are state, and $u(t)$
are input variables, respectively.
The model in [22] is based on the chemical reactions of molecules in synapses,
that are the intercellular information transfer points of neurons. The
signaling pathways in striatal synaptic plasticity are modeled in [19]. This
model describes how certain molecules, which are a prerequisite for learning
in the brain, act in synapses. The stoichiometric equations obey the law of
mass action, which leads to a deterministic system of $44$ ODEs of the form
(27) . The state $x(t)$ of the control system (27) is a collection of ions,
molecules, and proteins that act in neuronal synapses. The linear part of (27)
is sparse, the nonlinearities are quadratic. The time dependent stimulus
$u(t)$ consists of molecules that are important for neuronal excitability and
plasticity, calcium and glutamate.
In [21], a nonlinear biophysical network model is considered, describing
synchronized population bursting behavior of heterogeneous pyramidal neurons
in the brain [27]. Neurons communicate by changing their membrane voltage to
create action potentials (spikes), propagating from cell to cell. Spiking is
the fundamental method of sensory information processing in the brain, and
synchronized spiking is an emergent property of biological neuronal networks.
The ODE system (27) in [21] consists of the states $x(t)$ as a collection of
$50$ neurons, each modeled with $10$ ODEs, leading totally to a system of ODEs
of dimension $500$. Each cell is modeled with Hodgkin-Huxley equations, where
each cell has only two compartments (dendrites and soma) and these
compartments have different ion channels. The state variables $x(t)$ include
the voltages of somatic and dendritic compartments, dendritic calcium
concentration, synaptic and ion channel gating variables and the input $u(t)$
is an injected current. Additionally, the soma compartment voltages are
coupled to dentritic compartments of randomly chosen cells. This networking of
the output of cells as input to other cells is key for producing synchronized
population behavior. The nonlinearities are Hodgkin-Huxley type, i.e.
exponential functions as well as cubic and quartic polynomials.
In [22], POD+DEIM was applied to a data-driven biological model of plasticity
in the brain (27). The ROMs with POD-DEIM reduce significantly the simulation
time and error between the original model and reduced order solutions can be
tuned by adjusting the number POD and DEIM bases independently. When the ROMs
are trained in a matching time interval of $10000$ seconds, accurate results
are obtained. However, generalizing the reduced model to longer time intervals
is challenging, which is characteristic for all nonlinear models.
In [21], the network model (27) is reduced with localized DEIM (LDEIM) [24],
discrete adaptive POD (DAPOD) [33, 34], and adaptive DEIM [25]. DEIM and the
variations are used here in combination with POD. ROMs require large number
POD and DEIM bases, to accurately simulate the input-output behavior in the
ROMs. In this model, every cell is heterogeneous in parameters and there are
also jump/reset conditions, which are factors that pose additional challenges
to the reduced order methods. However, the ROMs in [21] were able to replicate
the emergent synchronized population activity in the original network model.
DAPOD and ADEIM perform best in preserving the spiking activity of the
original network model. ADEIM is too slow and does not allow low enough
dimensions to offset the computational costs of online adaptivity. DAPOD is
able to find a lower dimensional POD basis online than the other methods find
offline, but the runtime is close to the original model.
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|
2024-09-04T02:54:59.054047 | 2020-03-11T08:03:25 | 2003.05150 | {
"authors": "G. Hasinger",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26154",
"submitter": "Guenther Hasinger",
"url": "https://arxiv.org/abs/2003.05150"
} | arxiv-papers | # Illuminating the dark ages: Cosmic backgrounds from accretion onto
primordial black hole dark matter
G. Hasinger
###### Abstract
The recent interpretation of cold dark matter as the sum of contributions of
different mass Primordial Black Hole (PBH) families [1] could explain a number
of so far unsolved astrophysical mysteries. Here I assume a realistic
$10^{-8}$–$10^{10}$ M⊙ PBH mass distribution providing the bulk of the dark
matter, consistent with all observational constraints. I estimate the
contribution of baryon accretion onto this PBH population to various cosmic
background radiations, concentrating first on the cross-correlation signal
between the Cosmic X–ray and the Cosmic infrared background fluctuations
discovered in deep Chandra and Spitzer surveys. I assume Bondi capture and
advection dominated disk accretion with reasonable parameters like baryon
density and effective relative velocity between baryons and PBH, as well as
appropriate accretion and radiation efficiencies, and integrate these over the
PBH mass spectrum and cosmic time. The prediction of the PBH contribution to
the X–ray background is indeed consistent with the residual X–ray background
signal and the X–ray/infrared fluctuations. The predicted flux peaks at
redshifts z$\approx$17–30, consistent with other constraints requiring the
signal to come from such high redshifts. The PBH contribution to the 2–5
$\mu$m cosmic infrared background fluctuations is only about 1%, so that these
likely come from star formation processes in regions associated with the PBH.
I discuss a number of other phenomena, which could be significantly affected
by the PBH accretion. Magnetic fields are an essential ingredient in the Bondi
capture process, and I argue that the PBH can play an important role in
amplifying magnetic seed fields in the early universe and maintaining them
until the galactic dynamo processes set in. Next I study the contribution of
the assumed PBH population to the re-ionization history of the universe and
find that they do not conflict with the stringent ionization limits set by the
most recent Planck measurements. X–ray heating from the PBH population can
provide a contribution to the entropy floor observed in groups of galaxies.
The tantalizing redshifted 21-cm absorption line feature observed by EDGES
could well be connected to the radio emission contributed by PBH to the cosmic
background radiation. Finally, the number of intermediate-mass black holes and
the diffuse X–ray emission in the Galactic Center region are not violated by
the assumed PBH dark matter, on the contrary, some of the discrete sources
resolved in the deepest Chandra observations of the Galactic Ridge could
indeed be accreting PBH.
## 1 Introduction
Recent years saw a revival of the idea originally put forward by S. Hawking
[2], that Primordial Black Holes (PBH) could make up the so far elusive Dark
Matter. LIGOs first detection of gravitational waves from merging binary black
holes of approximately equal masses in the range 10–30 M⊙ [3, 4] led to the
suggestion that these could be a signature of dark matter stellar mass PBH [5,
6, 7] in a mass window not yet excluded by other astrophysical constraints. A
recent review about the rich literature constraining the possible
contributions of PBH to the dark matter is e.g. given in [8].
In a recently published theoretical prediction [1, 9] PBH are created in the
QCD phase transitions (around 100 MeV) of different particle families freezing
out of the primordial Quark-gluon plasma within the first two seconds after
the inflationary phase. When W+/-, Z bosons, baryons, pions are created, and
e+e- pairs annihilate, they leave an imprint in form of a significant
reduction of the sound speed at the corresponding phase transitions, and allow
regions of high curvature to collapse and form PBH [see also 10]. The typical
mass scale of these PBH is defined by the size of the horizon at the time of
the corresponding phase transition. In this model four distinct populations of
PBH in a wide mass range are formed: planetary mass black holes at the W+/-, Z
transition, PBH of around the Chandrasekhar mass when the baryons (protons and
neutrons) are formed from 3 quarks, PBH of masses of order 30 M⊙ (these
correspond to the LIGO black holes), when pions are formed from two quarks,
and finally supermassive black holes (SMBH) at the e+e- annihilation [see also
11]. Another remarkable aspect of this theory is, that the gravitational
energy released at at the PBH collapse locally reheats regions (hot spots)
around the black holes to the electroweak transition scale (around 100 GeV),
where chiral sphaleron selection effects can introduce the matter/antimatter
asymmetry. The PBH in this picture would therefore also be responsible for the
baryogenesis and fix the ratio of dark matter to baryons. Clustering of the
PBH in a very wide mass distribution could alleviate some of the more
stringent observational constraints on the allowed contribution of PBH to the
dark matter [7, 12]. The interpretation of cold dark matter as the sum of
contributions of different mass PBH families could explain a number of so far
unsolved mysteries, like e.g. the massive seed black holes required to create
the supermassive black holes in the earliest QSOs [13], the ubiquitous massive
LIGO/VIRGO massive binary black holes [e.g. 6], or even the putative "Planet
X" PBH in our own Solar System [14].
The most abundant family of PBH should be around the Chandrasekhar mass (1.4
M⊙). This prediction may already have been vindicated by the recent OGLE/GAIA
discovery of a sizeable population of putative black holes in the mass range
1–10 M⊙ [15]. The microlensing survey OGLE has detected $\sim$60 long-duration
microlensing events. About 20 of these have GAIA DR2 parallax distances of a
few kpc, which break the microlensing mass–distance degeneracy and allow the
determination of masses in the few solar mass range, implying that these
objects are most likely black holes, since stars at those distances would be
directly visible by OGLE.
Important fingerprints of a population of PBH may be hidden in the Cosmic
infrared and X–ray background radiation (see [16] for a comprehensive review).
Indeed, [6] argues, that the near-infrared Cosmic background (CIB)
anisotropies detected in deep Spitzer [17, 18, 19, 20] and Akari [21] images,
which cannot be accounted for by known galaxy populations [22], could be
connected to PBH. Similar fluctuations were discovered in the Cosmic X–ray
background (CXB) observed in a deep Chandra survey, which are correlated with
the CIB anisotropies in the same field [23]. Later studies of wider/deeper
fields covered by both Chandra and Spitzer [24, 25, 26] have substantially
improved the detection significance of the observed signal. The X–ray
fluctuations contribute about 20% to the CIB signal, indicating that black
hole accretion should be responsible for such highly efficient X–ray emission.
Similar studies of deep fields observed with the Hubble Space Telescope in the
optical range do not show such a cross-correlation signal down to mAB$\sim$28
[see 16]. The angular scale of the fluctuation power spectra of the CIB and
CXB reach values >1000", much larger than expected for the known galaxy
populations [27]. All of these findings can be understood, if the fluctuation
signal comes from a high-redshift (z$\gtrsim$12) population of black holes.
The spectral shape of the CXB fluctuations determined from a combination of
the deepest/widest fields [26] can be fit either with a very high redshift
population of obscured black holes, or with completely unobscured black hole
accretion. Original models [28] invoked highly obscured Direct Collapse Black
Holes formed in metal-free halos at z>12 to explain the observed CIB and CXB
signal. However, accreting massive black holes have recently been firmly ruled
out as the source of these fluctuations [29], because they would require an
unfeasible amount of black hole accretion at z>6, locking up a larger amount
of mass in massive black holes at high redshift, than the known black hole
mass function at z=0. These authors also ruled out local diffuse emission as
the source of the X–ray fluctuations. The CXB has been largely resolved into
discrete sources in deep X–ray images, either directly [see 30, 31], or by
crosscorrelating with the deepest Hubble galaxy catalogues [32, 33]. However,
[32] show that some marginally significant diffuse CXB still remains after
accounting for all discrete contributions. This is consistent with the
independent determination of [34]. The residual unresolved flux is about 3
times larger than the X-ray flux associated with the above CXB/CIB
fluctuations.
Given the difficulties in explaining the CIB/CXB correlation with known
classes of sources, and motivated by the notion that the dark matter could be
dominated by an extended mass distribution of PBH, I constructed a toy model
to explore the potential contribution to the cosmic backgrounds by the
accretion of baryons throughout cosmic history onto such a population of early
black holes. Assuming a combination of Bondi-Hoyle-Lyttleton quasi-spherical
capture at large distances from the PBH, and advection-dominated disk
accretion flows (ADAF) in the vicinity of the central object, I can explain
the observed residual CXB flux and the CXB/CIB crosscorrelation with minimal
tuning of the input parameters, and find a maximum contribution to the
extragalactic background light in the redshift range 15<z<30\. I further
estimate that this accretion onto PBH can produce enough flux to significantly
contribute to the pre-ionization of the intergalactic medium with UV photons
by a redshift z$\gtrsim$15 and to the pre-heating of the baryons with X–ray
photons, observed as an "entropy floor" in the X–ray emission of galaxy
groups.
In section 2 the assumed PBH mass distribution is introduced and contrasted
with recent observational limits on the PBH contribution to the dark matter.
The basic ingredients of the toy model for the accretion onto PBH are
presented in section 3. The assumed radiation mechanism and efficiency is
discussed in section 4. The contribution of the PBH emission to the different
bands is compared with the observational constraints in section 5. Other
potential diagnostics of this putative dark matter black hole population are
discussed in section 6, and conclusions are presented in section 7. Throughout
this work a $\Lambda$CDM cosmology with $\Omega_{M}$=0.315,
$\Omega_{\Lambda}$=0.685, and H0=67.4 km s-1 Mpc-1 [35] is used. These
parameters define the baryon density $\Omega_{bar}$=0.049, the dark matter
density $\Omega_{DM}$=0.264, and the critical mass density of the universe
$\rho_{crit}$=1.26$\times 10^{20}M_{\odot}~{}{\rm Gpc}^{-3}$. All logarithms
in this paper are taken to the base 10.
## 2 The assumed PBH mass distribution
The theoretical predictions in [1, 9, 11, 36] yield a broad distribution of
PBH masses with a number of peaks corresponding to the particle families
freezing out from the Big Bang. Depending on the spectral index $n_{s}$ of the
primordial curvature fluctuation power spectrum, the PBH mass distribution has
a different overall slope. [36] find consistency of these predictions with a
number of recent observational limits on the PBH contribution to the dark
matter, but there is a tension of their models with the Cosmic Microwave
Background (CMB) constraints from accretion at large PBH masses [37, 38].
Recent limits from gravitational lensing of type Ia supernovae on a maximum
contribution of stellar-mass compact objects to the dark matter of around 35%
[39], and from the LIGO OI gravitational wave merger rate of black holes in
the mass range 10–300 M⊙ [40] are also in tension with these models. An
additional important constraint comes from a comparison of the predicted PBH
fraction with the measured local mass function of supermassive black holes
(SMBH) in the centers of nearby galaxies. Integrating the local SMBH mass
function of [41] (see figure 1) in the range $10^{6}$–$10^{10}$ M⊙ yields a
local SMBH mass density of $\rho_{SMBH}$=6.3$\times$105 M⊙ Mpc-3,
corresponding to a dark matter fraction of fSMBH=1.89$\times$10-5, which is
about a factor of 10–100 lower than the fPBH predictions in [1, 36].
Figure 1: The PBH mass spectrum (thick red line) assumed for this work
(García-Bellido, 2020, priv. comm.), compared to a number of observational
constraints. Microlensing limits from SNe [39], EROS [42], and the Subaru M31
survey [43] are shown as solid, dashed and dotted green lines, respectively.
LIGO limits from gravitational merger event rates are shown as blue solid line
for subsolar masses [44], and as blue dashed line for 10-300 M⊙ [40]. The CMB
accretion limits from [37] are shown as orange dashed line. Multiwavelength
limits from the Galactic Center [45] are shown in magenta for X-ray (solid)
and radio (dashed) observations. Finally, the local SMBH mass function [41] is
shown as black line at 106-10 M⊙.
For these reasons, García-Bellido et al. (2020 in prep.) are revising their
model parameters in order to predict a steeper PBH mass function at large MPBH
and shared one of their new models, shown as red curve in figure 1. Here a
value of ns=0.987 is assumed for the spectral index of the primordial
fluctuation power spectrum, as well as a running curvature of dns=$-$0.0006.
The integral of this PBH distribution over the whole mass range yields fPBH=1.
On the other hand, the distribution yields only $\sim$40% of the dark matter
in the peak mass range [0.1,10] M⊙, and is thus fully consistent with the
microlensing constraints in figure 1. In the mass range of the LIGO black hole
binaries it predicts just the right amount of dark matter to explain the
gravitational wave merger rates, and in the SMBH range it is consistent with
the local black hole mass function (taking into account the accretion onto
supermassive PBH over cosmic time producing the bulk of the X-ray background
[46]). Apart from small sections, the new PBH mass function is thus fully
consistent with the most recent observational constraints.
## 3 Baryon accretion onto the PBH
In the following I use the PBH mass spectrum presented in section 2 to
calculate the accretion of baryons onto PBH over cosmic time, and to predict
the electromagnetic emission from this process. As we will see, for most of
the cosmic history these black holes move at supersonic speeds among the
baryons and will therefore undergo Bondi-Hoyle-Lyttleton quasi-spherical
capture [47, 48, 49, 50]. In the Bondi-Hoyle picture of a black hole moving
supersonically through a homogeneous gas, the capture happens in the wake of
the moving object. Behind the object, material moves in from a wide cone, and
needs to lose angular momentum before it can fall towards the black hole. The
gas is in principle collisionless, so that only the magnetic field trapped in
the plasma allows particles to lose angular momentum and start to behave like
a fluid. This gas forms the accretion flow, in which it is adiabatically
heated. The accreting gas is ionized and embedded in the magnetic field. Any
plasma drawn in by the gravitational field will carry along the magnetic
field. Shvartsman [51] argues that in the black hole tail, where the matter
flow stops, the gravitational and magnetic energy densities become nearly
equal. This equipartition is preserved in the infalling flow and thus the
magnetic field grows towards the black hole. Like the heat has to be
ultimately radiated away, the magnetic field needs a way to dissipate energy
on its way inward. [52] discuss that the most likely dissipation mechanism for
the magnetic field is reconnection of field lines in narrow current sheets,
similar to the processes we observe in solar flares and active galactic
nuclei. Magnetic reconnection causes the acceleration and non-thermal heating
of a small fraction of the infalling electrons. In parallel, decoupled
magnetic field lines can carry some of the amplified magnetic field outward
and eject plasma [52].
An important question is, whether the accretion flow is spherically symmetric
close to the black hole, or whether an accretion disk is formed. Originally
most researchers assumed spherical accretion for PBH [e.g. 53, 54, 38].
However, [37] argues, that the accreted angular momentum is large enough, that
an accretion disk is formed, at least close to the black hole. According to
these authors, the granularity of the PBH distribution and the formation of
PBH binaries at the scale of the Bondi radius will imprint density and
velocity gradients into the relative distribution of baryons and PBH, such
that ultimately an accretion disk and an advection-dominated accretion flow
(ADAF) will form [55]. The formation of an ADAF disk significantly reduces the
accretion rate and the radiative efficiency [56], compared to spherical
accretion. But to first order the Bondi-Hoyle-Lyttleton mechanism can be used
to estimate the accretion rate $\dot{M}$ onto the PBH [37, 8].
Bondi [49] discusses two different approximations to the spherical gas
accretion problem, (i) the velocity-limited case, where the motion of the
accreting object through the gas is dominant and an accretion column is formed
in the wake of the moving object, and (ii) the temperature-limited case, where
the sound speed of the gas is dominant and a spherical accretion flow forms.
In the velocity-limited case (i) the mass accretion rate is given as
$\dot{M}=2.5\pi\rho(GM)^{2}v_{rel}^{-3},$ (3.1)
where $\rho$ is the gas density, $M$ is the PBH mass, and $v_{rel}$ is the
relative velocity between object and gas. In the temperature-limited case (ii)
with negligible relative velocity, the thermal velocity of the gas particles
is dominant and the corresponding accretion rate is given by
$\dot{M}=2.5\pi\rho(GM)^{2}c_{s}^{-3},$ (3.2)
where $c_{s}$ is the sound speed. For intermediate cases, [49] introduces an
effective velocity
$v_{eff}=\sqrt{v_{rel}^{2}+c_{s}^{2}}$ (3.3)
and the corresponding mass accretion rates becomes
$\dot{M}=2\lambda\pi\rho(GM)^{2}v_{eff}^{-3},$ (3.4)
where the so called accretion eigenvalue $\lambda$ is is a fudge factor of
order unity, dependent on non-gravitational aspects of the problem, like e.g.
the gas equation of state or outflows from feedback effects. Different authors
have discussed this parameter for the particular application of gas accretion
onto PBH in the early universe. [53] find values of $1.12>\lambda>10^{-3}$,
depending e.g. on the PBH mass. For masses of order 1 M⊙ they find
$\lambda=1.12$. [5] discriminate between isothermal and adiabatic gas with
accretion eigenvalues of $\lambda$=1.12, and 0.12, respectively. In this paper
I assume an eigenvalue $\lambda$=0.05. The motivation for this choice is
discussed in section 4, while section 5 and 6 show that this choice fits the
observational constraints quite well.
Figure 2: Left: Baryon temperature as a function of redshift. Right: Mean
relative velocity $\langle v_{rel}\rangle$ between dark matter and baryons,
sound speed $c_{s}$ and the effective velocity $v_{eff}$ (eq. 3.8) as a
function of redshift.
Let us first look at the thermal history and thus the sound speed of the gas
over cosmic history. A nice summary is given in figure 15 of [57]. Despite
having decoupled from the CMB at z$\approx$1089, the gas temperature continues
to follow the temperature evolution T$\propto$(1+z) of the background photons
due to Compton scattering off residual ionized electrons from the
recombination era. Below redshifts z$\approx$200 the residual ionization in
the gas is low enough, that it decouples from the background radiation and
cools adiabatically following the relation T$\propto$(1+z)2. When the first
objects form and reionization starts around z$\lesssim$20, the gas is heated
up to temperatures $\sim$104 K. The details of re-ionization are still
uncertain and will be discussed below. I have deliberately chosen a redshift
of z$\approx$20 for re-ionization to become dominant, with full ionization
occurring around z$\approx$7\. Finally, at z<3, when the bulk of the cosmic
baryons are falling into increasingly larger dark matter halos and become
virialized, they are further heated up to form the warm/hot intergalactic
medium at temperatures $10^{5-7}$K [58]. Using figure 2b in this paper I
estimate average temperatures for the IGM of 5$\times 10^{4}$, 1.5$\times
10^{5}$, and 8$\times 10^{5}$ K at z=2, 1, 0, respectively. The baryon
temperature as a function of redshift assumed in this work is shown in figure
2 (left). The sound speed of the gas is given by
$c_{s}=\sqrt{\frac{\gamma kT}{\mu m_{H}}},$ (3.5)
where $\gamma$=5/3 for an ideal monoatomic gas, $\mu$=1.22 is the mean
molecular weight including a helium mass fraction of 0.24, $m_{H}$ is the mass
of the hydrogen atom, and $T$ is the temperature of the baryons as a function
of cosmic history discussed above [59]. The sound speed as a function of
redshift is the dotted curve shown in figure 2 (right).
I now discuss the relative velocity $v_{rel}$ between the dark matter PBH and
the baryons throughout cosmic history. In the radiation-dominated phase of the
universe at z>1089, the dark matter is already hierarchically clustering under
the influence of its own gravity. The sound speed of the photon-baryon fluid
is very high, of order one third of the velocity of light, and thus the normal
matter undergoes baryonic acoustic oscillations [60, 61]. This leads to a
spatial separation between baryons and dark matter and thus to a Gaussian
distribution of relative velocities with an average around $\langle
v_{rel}\rangle$$\approx$30 km/s [see 59, 62]. At z$\approx$1089, when
electrons and protons combine and the universe becomes transparent, the sound
speed of the gas dramatically drops to $\sim$6 km/s. The dark matter PBH
kinematically decouple from the baryons and their relative velocities become
highly supersonic. In the linear growth phase of the universe, at scales
larger than the gas Jeans-length, the dark matter and the baryons fall in the
same gravitational potentials of the cosmic web and thus their relative
velocity decreases with the cosmic expansion:
$\langle v_{rel}\rangle_{linear}\approx 30~{}{\frac{1+z}{1000}}~{}{\rm
km~{}s}^{-1}.$ (3.6)
This relation is shown as the right branch of the dashed line in figure 2
(right), above redshifts $z\gtrsim 20$. From this figure it becomes apparent,
that between recombination and re-ionization the PBH move with highly
supersonic, but decreasing velocities through the gas, due to the decaying
sound waves. As we will see below, in this branch of the velocity curve the
contribution of PBH to the cosmic backgrounds has a maximum. At lower
redshifts, at scales smaller than the gas Jeans-length, the hierarchical
clustering becomes non-linear and baryons falling into growing dark matter
halos are virialized. As a consequence, the velocity dispersion between dark
matter and gas increases again towards lower redshifts, scaling as
$M_{Halo}^{1/3}$, where $M_{Halo}$ is the mass of the dark matter halo
becoming non-linear. I used two different methods to estimate the average
virial velocity for redshifts z$\lesssim$20\. First, the Millenium Simulation
run described in [63] gives the mass function of dark matter halos with halo
masses $M_{Halo}$>$10^{10}M_{\odot}$ for five different redshifts between z=10
and z=0. I extrapolated these simulated mass functions to lower masses
($M_{Halo}$>10${}^{3}M_{\odot}$) using the empirical universal halo mass
function shape found through simulations by [64]. For every mass bin I
determined the virial velocity according to the calibration of the velocity
dispersion as a function of halo mass described in [65], and then calculated
the average for each epoch. These virial velocities are shown as crosses in
figure 2 (right). The extrapolation to halo masses as small as
$M_{Halo}>10^{3}M_{\odot}$ is rather uncertain, both for the mass function and
the velocity dispersion, because the cosmological simulations do not have a
fine enough mass resolution at this scale. Also, the velocity dispersion
relevant for Bondi capture onto PBH is determined by the smallest mass scales
becoming non-linear at any redshift. A second possibility to calculate the
relative velocities in the non-linear phase is therefore to determine the
velocity dispersion directly from the dark matter power spectrum and integrate
this over the smallest non-linear scales. This calculation has been performed
by M. Bartelmann (2020, priv. comm.), adopting the normalization of the
primordial power spectrum of $\sigma_{8}$=0.8. The relative velocity in the
non-linear regime can be approximated by
$\langle v_{rel}\rangle_{nonlinear}\approx 620~{}(1+z)^{-2.3}~{}{\rm
km~{}s}^{-1},$ (3.7)
and is shown as the left branch ($z\lesssim 20$) of the dashed line in figure
2 (right). At z=2 the cluster velocity dispersion agrees with this estimate,
but it systematically overestimates the small-scale velocity dispersion at
larger redshifts.
Since we are interested in the total contribution of PBH to the
electromagnetic radiation of the universe, we have to average over the whole
Gaussian distribution of relative velocities. The Bondi accretion rate is
proportional to $v_{rel}^{-3}$ (see above), and therefore smaller velocities
dominate. For this particular case [38] propose to replace the quadratic
average of relative velocity and sound speed in Bondi’s formula (3.3) above
with their harmonic mean:
$v_{eff}=\sqrt{\langle v_{rel}\rangle~{}c_{s}.}$ (3.8)
This is the assumption I adopt here, and the resulting effective velocity
$v_{eff}$ is shown as solid red curve in figure 2 (right). With equation (3.8)
the accretion rate becomes
$\dot{M}=2\lambda\pi\rho(GM)^{2}~{}(\langle v_{rel}\rangle~{}c_{s})^{-3/2}$
(3.9)
It is interesting to note that in the range 200<z<20 both relative velocity
and sound speed decrease linearly with (1+z). Therefore the mass accretion
rate is expected to be constant in this era. It is important to understand
that the redshift, at which both the sound speed and the relative velocity of
the gas turn around due to re-ionization and virialization, respectively, and
rapidly increase towards lower redshift, is crucial for our analysis. The
redshift, where the minimum velocity occurs, ultimately determines the maximum
flux contribution of PBH accretion to the cosmic backgrounds.
The calculation of the Bondi accretion rate in equation (3.9) requires the
density $\rho$ as a function of redshift. With $\Omega_{bar}$=0.049 and
$\rho$=n$\cdot$mH, where $n$ is the number density of particles, I find
$n=250~{}\left(\frac{1+z}{1000}\right)^{3}~{}{\rm cm}^{-3}.$ (3.10)
I define $\dot{m}$ as the normalized mass accretion rate
$\dot{m}=\dot{M}/\dot{M}_{Edd}$, with the Eddington accretion rate
$\dot{M}_{Edd}$=1.44$\times 10^{17}M/M_{\odot}$ g s-1. Then I can rewrite
equation (3.9) into normalized quantities
$\dot{m}=\lambda\cdot
0.321\left(\frac{1+z}{1000}\right)^{3}~{}\left(\frac{M}{M_{\odot}}\right)\left(\frac{v_{eff}}{1~{}{\rm
km~{}s}^{-1}}\right)^{-3}$ (3.11)
With a very broad PBH mass spectrum, including intermediate-mass and
supermassive black holes (MPBH>1000 M⊙), it is important to include the
effective viscosity due to the Hubble expansion in the early universe [53].
The Bondi radius determines the amount mass captured by the PBH:
$r_{B}={\frac{G~{}M}{v_{eff}^{2}}}\approx 1.34\cdot
10^{16}\left(\frac{M}{M_{\odot}}\right)\left(\frac{v_{eff}}{1~{}{\rm
km~{}s}^{-1}}\right)^{-2}cm.$ (3.12)
This is shown for two different PBH masses in figure 8 (left). The
characteristic time scale for accretion is the Bondi crossing time
$t_{cr}=r_{B}/v_{eff}$, which can be compared to the Hubble time $t_{H}$ at
the corresponding redshift. If $t_{cr}<t_{H}$ there will be stationary
accretion, while for Bondi crossing times larger than the Hubble time the
accretion is suppressed. For every redshift we can calculate a critical PBH
mass Mcr, below which the steady-state Bondi assumption can be applied. For
redshifts z=1000, 200, 20 this critical mass corresponds to
$log(M_{cr}/M_{\odot})$=5.3, 4.8, 3.4, respectively. At redshifts below z=20
Mcr rapidly increases to values above 106 M⊙. For PBH masses close to and
above Mcr the Bondi accretion rate can be scaled by the Hubble viscosity loss
given in the dashed curve in figure 3 (left) of [53].
Inserting $v_{eff}$ from equation (3.8) and figure 2 (right) into equation
(3.11), assuming an accretion eigenvalue $\lambda$=0.05 and applying the above
Hubble viscosity correction, I can finally calculate the normalized accretion
rate as a function of redshift and PBH mass. The results are shown in figure 3
(left). For PBH masses smaller than $\sim$1000 M⊙ the normalized accretion
rate is roughly constant in the redshift range 20<z<200 due to the fact that
the density and velocity dependence on redshift in equation (3.9) roughly
cancel out (see also the lower panel of figure 4 in [38]). At z<20 $\dot{m}$
drops dramatically because of the effective velocity increase. PBH masses
larger than $\sim 10^{4}$ M⊙ reach accretion rates close to the Eddingon limit
at z$\gtrsim$100, but are significantly affected by the Hubble viscosity at
z$\gtrsim$20\. For all PBH masses the accretion rate is small enough, that the
growth of the PBH population can be neglected over cosmic time (PBH with
masses in the range 105-7 M⊙ accrete about 0.5–2% of their mass until z>20).
Figure 3: Left: Normalized accretion rate onto PBH with masses in the range
0.1–107 M⊙ as a function of redshift. Right: Radiative efficiencies derived
from the accretion rates, assuming the hot accretion flow model of [56] with a
viscous heating parameter $\delta$=0.5.
Figure 4: Spectra of the hot disk accretion flow (ADAF) from [55] with a
viscous heating parameter $\delta$=0.5, divided by the normalized accretion
rate. Left: accretion onto a 10 M⊙ black hole for different accretion rates,
as indicated. Right: same for an accretion rate of $log(\dot{m})$=-1.6 but
different black hole masses (as indicated) .
## 4 Accretion spectrum and radiative efficiency
For the accretion flow and the electromagnetic emission mechanism I follow
[37, 8] and assume the formation of an accretion disk. Accretion rates close
to the Eddington limit will lead to the formation of a standard Shakura-
Sunyaev disk [66], which has a canonical radiative efficiency $\eta\approx
0.1$. For much lower accretion rates $\dot{m}$$\ll$1 an advection-dominated
hot accretion flow [55] is expected, with a significantly lower radiation
efficiency [56], roughly scaling according to $\eta\propto\dot{m}$. Figure 4
shows hot accretion flow spectra from [55] with a viscous heating parameter
$\delta$=0.5 for black holes, normalized by Eddington luminosity and mass
accretion rate. The left graph shows radiation from a 10 M⊙ black hole at
different mass accretion rates. The right graph shows the spectrum from black
holes with different masses in the range 10-109 M⊙ and a mass accretion rate
$log(\dot{m})$=–1.6.
It is important to understand, that for advection dominated accretion flows
not all the matter entering the Bondi radius will actually reach the black
hole. This is due to feed-back mechanisms acting on the accreted gas, e.g.
producing outflows or jets. The advection dominated flow models of [56, 55]
therefore find a radial dependence of mass accretion rate $\dot{M}\propto
R^{\alpha}$, typically with $\alpha\sim 0.4$. Within a radius of about 10 RS,
where $R_{S}=2GM/c^{2}$ is the Schwarzschild radius, the accretion flow more
closely follows the standard Shakura-Sunyaev description of a constant
accretion rate with radius down to the last stable orbit ($\sim 3R_{S}$). In
terms of the classical Bondi description of quasi-spherical capture, the loss
of accreted matter can be associated with the accretion eigenvalue:
$\lambda\approx\left(\frac{10R_{S}}{R_{D}}\right)^{\alpha},$ (4.1)
where $R_{D}$ is the outer radius of the accretion disk formed. For
$\alpha$=0.4, the value of $\lambda$=0.05 chosen for the analysis in this
paper therefore corresponds to an outer disk radius of $R_{D}\sim$2$\times$104
$R_{S}$, about 8 orders of magnitude smaller than the Bondi radius. In this
picture the accretion flow on large scales follows the Bondi quasi-spherical
flow for most of the radial distance, until the advection-dominated accretion
disk is formed.
The radiative efficiency for the ADAF spectra in figure 4 is the integral over
these curves and has been calculated through numerical simulations by [56].
For this work I use a digitized version of the highest efficiency curve in
their figure 1, with a viscous heating parameter $\delta$=0.5111Please take
note that the definition of $\dot{m}$ between these authors and the analysis
presented here differs by a factor of 10.. A maximum radiative efficiency of
$\eta$$\sim$0.08 is achieved for log($\dot{m}$)>–1.6. We can now calculate the
radiative efficiency for every mass and redshift bin from the normalized
accretion rate in figure 3 (left). The result is shown in figure 3 (right). It
turns out that above redshifts z$\gtrsim$20 and black hole masses MPBH>100 M⊙,
which dominate the contribution to the extragalactic background light, the
radiative efficiencies are relatively large (typically >3%).
Figure 5: Density-weighted bolometric luminosity of single PBH as a function
of mass for different redshifts indicated (left), and as a function of
redshift for different mass bins indicated (right).
Figure 6: Density-weighted bolometric flux of single PBH as a function of mass
for different redshifts indicated (left), and as a function of redshift for
different mass bins indicated (right).
We now have the ingredients to calculate the bolometric luminosity and flux
expected from the baryon accretion onto the assumed PBH mass spectrum over
cosmic time. For every black hole of mass MPBH I calculate the expected
bolometric luminosity L${}_{bol}=\dot{m}~{}\eta~{}L_{Edd}$, where
LEdd=1.26$\times$10${}^{38}~{}M_{PBH}/M_{\odot}$ erg/s is the Eddington
luminosity, and the normalized mass accretion rate $\dot{m}$ as well as the
radiation efficiency $\eta$ are taken from the data in figure 3. In every mass
bin, the relative number density of PBH compared to those of 1 M⊙ is
nPBH=fPBH/MPBH, where fPBH is the PBH mass function from figure 1. For every
mass and redshift bin I thus multiply the bolometric luminosity with this
relative number density in order to obtain the density-weighted luminosity
$\langle L_{bol}\rangle^{*}$ for an equivalent PBH of 1 M⊙. This quantity is
shown in figure 5 as a function of PBH mass (left) and redshift (right). It
shows that the largest contribution to the PBH luminosity over cosmic time
comes from PBH in the mass range MPBH=103-7 at redshifts z>100\. The
Chandrasekhar PBH mass peak is subdominant in this representation. The total
PBH luminosity deposited in the baryonic gas at high redshifts is important
for the pre-ionization and X–ray heating of the intergalactic medium discussed
in section 6.
To calculate the contribution of PBH accretion to the extragalactic background
light we need to convert the density-weighted luminosities in Figure 5 to
bolometric fluxes using the luminosity distance $D_{L}$ at the corresponding
redshift: $\langle F_{bol}\rangle^{*}$=$\langle
L_{bol}\rangle^{*}$/(4$\pi~{}D_{L}^{2}$). This quantity is shown in figure 6
as a function of PBH mass (left) and redshift (right). It shows that the
largest contribution to the extragalactic surface brightness is produced at a
redshift z$\approx$20 from PBH in the mass range MPBH=102-5, and a similar
contribution from the Chandrasekhar mass peak. SMBH at M${}_{PBH}\sim
10^{6.5}$ have a notable contribution around z$\sim$10.
## 5 The contribution of PBH to the extragalactic background light
To calculate the surface brightness per redshift shell in a particular
observed frequency band [$\nu_{1}$;$\nu_{2}$] of the electromagnetic spectrum,
I take into account the spectral shape and the fraction of the radiation
falling into the rest frame frequency band [$\nu_{1}$/(1+z);$\nu_{2}$/(1+z)].
The exact spectral shape is not so important for this derivation, it is mainly
used to calculate the bolometric corrections, i.e. the fraction of the total
luminosity falling into the various frequency bands as a function of redshift.
The ADAF spectra in figure 4, in particular those at high $\dot{m}$ values,
can be approximated by power laws with an exponential cutoff at $\sim$200 keV.
Following [37] and [8], I assume a power law slope of –1 (corresponding to a
flat line in figure 4). Below a critical frequency $\nu_{c}$ the power law
spectrum is cut off by synchrotron self-absorption into a power law with a
steep slope of approximately +1.86. As can be seen in figure 4 (right),
$\nu_{c}$ depends on MPBH and can be approximated by
log($\nu_{c}$)$\approx$14.82–0.4log(MPBH/M⊙). The bolometric corrections are
then obtained by integrating the analytic normalized spectra over the observed
frequency bands. For the 2–5$\mu$m band we have to consider in addition the
Lyman-$\alpha$ break, which produces a sharp cutoff at z$\gtrsim$30 (see e.g.
[28, 67]). These bolometric corrections are shown in figure 7 (left) for the
2–5$\mu$m NIR band, the 0.5–2 keV and the 2–10 keV X–ray bands, respectively.
To predict the surface brightness of all PBH across cosmic time in these
observed frequency bands, the total flux per PBH in figure 6 (right) has to be
multiplied with the bolometric correction and the PBH surface density in a
particular redshift shell. Using the critical mass density of the universe
$\rho_{crit}$=1.26$\times 10^{20}M_{\odot}~{}{\rm Gpc}^{-3}$ and the Dark
Matter density $\Omega_{DM}$=0.264, as well as the reference mass 1M⊙, a
comoving PBH space density of $n_{PBH}=3.32\times 10^{19}(1+z)^{3}~{}{\rm
Gpc}^{-3}$ is obtained. For every redshift shell [z+$\Delta$z] the PBH space
density is multiplied with the volume of the shell [V(z+$\Delta$z)–V(z)] and
divided by 4$\pi$ deg2 to obtain the number of PBH per deg2. Figure 7 (right)
shows the derived surface brightness as a function of redshift (per
$\Delta$z=1 interval) for the three spectral bands considered here. The
emission in all three bands peaks around z$\approx$20 with a FWHM of
$\Delta$z$\approx$[-3;+6].
Figure 7: Left: The bolometric correction, i.e. the fraction of the total
luminosity falling into the respective observed frequency band as a function
of redshift, for the 2–5$\mu$m NIR band, as well as the 0.5–2 and 2–10 keV
X–ray bands. Right: Predicted surface brightness of the PBH in the same
observed bands as a function of redshift (per $\Delta$z=1).
The curves in figure 7 (right) can now be integrated to predict the total PBH
contribution to the extragalactic background light as SB2-5μ$\approx$10-13,
SB0.5-2keV$\approx$1.9$\times$10-13, and SB2-10keV$\approx$1.3$\times$10-13
erg cm-2 s-1 deg-2, respectively. The minimum amount of X–ray surface
brightness necessary to explain the CXB/CIB cross-correlation signal observed
by [23] in the 0.5–2 keV band has been discussed by [29]. This is $9\times
10^{-14}$ erg cm-2 s-1 deg-2, corresponding to roughly 1% of the total CXB
signal in this band. The 0.5–2 keV PBH contribution predicted for an accretion
eigenvalue of $\lambda$=0.05 in equation (3.11) is thus about a factor of 2
larger than the observed CXB fluctuation signal, which could well be
consistent, given the coherence between the CXB and CIB signals. As discussed
above, there is a marginally significant diffuse CXB remaining after
accounting for all discrete source contributions [31, 34]. Extrapolating into
the X–ray bands considered here, this residual flux corresponds to
$\approx$(7$\pm$3) and $\approx$(9$\pm$20)$\times 10^{-13}$ erg cm-2 s-1 deg-2
in the 0.5–2 keV and 2–10 keV band, respectively. Assuming the $\lambda$=0.05
value, the predicted PBH contribution is therefore well below the upper limit
(15–25%) of any unresolved component left in the CXB. The main result of this
paper is therefore, that the assumed PBH population for the dark matter can
indeed explain the X–ray fluctuation signal, with a Bondi accretion eigenvalue
of $\lambda$=0.05.
The flux measured in the 2–5$\mu$m CIB fluctuations at angular scales >100" is
about 1 nW m-2 sr-1 [68], or 3$\times 10^{-10}$ erg cm-2 s-1. The cross-
correlated CIB/CXB fluctuations contribute about 10% to the total CIB
fluctuations [23], i.e. 3$\times 10^{-11}$ erg cm-2 s-1. Therefore the
predicted PBH contribution to these CIB fluctuations is only about 0.5% for
$\lambda$=0.05. It is argued in [6] that PBH in the early universe could
amplify the cosmic power spectrum at small spatial scales (see below).
Together with the pre-ionization of the intergalactic medium discussed below,
the PBH can therefore significantly increase the associated star formation.
The NIR emission in this picture would then be dominated by early star
formation associated with PBH instead of direct PBH emission.
## 6 Discussion
### 6.1 Linear versus post-linear growth
In this simplified treatment I only consider the linear evolution of the power
spectrum above the virialization redshift around z$\approx$20 (see figure 2
right). On sufficiently large scales the initial power spectrum has been very
precisely determined as nearly scale invariant with overdensities of 10-4
[35], and the PBH density field is expected to follow the standard adiabatic
perturbations. On small scales the power spectrum is only poorly constrained
and could be significantly amplified by the discrete nature of the PBH
population itself [6, 69, 70]. Poisson variations in the density of PBH will
introduce non-linear growth of density fluctuations and the corresponding
velocity dispersion already well before the virialization redshift z$\sim$20
discussed above. However, from numerical simulations [70] conclude that the
nonlinear velocity perturbations introduced by >20 M⊙ PBH are too small to
dominate the relative velocities between baryons and PBH at z$\gtrsim$100 [see
also 71]. However, non-linear effects definitely become more important at
lower redshifts (see above) and could effectively reduce the Bondi capture
rate.
### 6.2 Magnetic fields in the early universe
The accretion mechanism assumed in the Bondi capture model only works, if
there is a rough equipartition between the kinetic energy and magnetic fields
in the accreted gas, as it is the case in the turbulent interstellar medium of
our Galaxy. It is therefore justified to ask, whether this mechanism can also
work at high redshifts, where the existence and magnitude of magnetic fields
is still unclear. Magnetic fields are present at almost every scale of the low
redshift universe, from stars and planets to galaxies and clusters of galaxies
and possibly even in the intergalactic medium in voids of the cosmic web, as
well as in high-redshift galaxies. [72] and [73] review the observations and
possible origin of magnetic fields. There is a surprising similarity between
the relatively strong magnetic fields measured in our own Galaxy (0.3–0.4 nT)
and other nearby galaxies ($\sim$1 nT) with magnetic fields measured in
clusters of galaxies (0.1–1 nT), as well as in high redshift galaxies ($\sim$1
nT), when the universe was only about 1/3 of its current age. There are even
indications of magnetic fields of order $\gtrsim$10-20 T in cosmic voids
derived from the gamma ray emission of blazars [74].
One can come to the conclusion that the origin of cosmic magnetism on the
largest scales of galaxies, galaxy clusters and the general intergalactic
medium is still an open problem [75]. It is usually assumed that primordial or
cosmic seed fields are amplified over time through the galactic dynamo effect
to produce the rather strong fields observed in local galaxies. In this
picture it is, however, unclear how similar fields can be created in such
different settings (e.g. clusters) and different cosmic times (high-redshift
galaxies). An interesting possibility is therefore that cosmic magnetic fields
could be remnants from the early universe, or created in a process without
galactic dynamos. Assuming equipartition, the energy density in the CMB
photons would correspond to a magnetic field of about 0.3 nT. Magnetic fields
of $10^{-20}$ T, as observed in cosmic voids today, would only require a
minute fraction of $10^{-10}$ of this energy density in the early universe to
be channeled into magnetic fields.
Figure 8: Left: The Bondi radius for a 104 M⊙ (thin blue) and 1 M⊙ (thick
blue) PBH compared to the proton (red) and electron (green) Larmor radius,
assuming a magnetic field of B=$10^{-20}$ T, as observed in local galaxy
voids. Right: Baryon ionization/heating fraction $\chi_{e}$ as a function of
redshift. The thin dash-dotted line shows the residual ionization left over
from the radiation dominated era [76]. The red curve shows the ionization
fraction from UV photons produced by accreting PBH. The blue curve shows the
corresponding heating fraction by >1 keV X–ray photons. The thick dashed black
line shows one of the models consistent with the Planck satellite data [35]
(see text). The green hatched areas shows the range of high–redshift
ionization fractions considered in [16].
Here I argue, that PBH could play a role in amplifying early magnetic seed
fields and sustaining them until the epoch of galaxy formation. I compare the
Bondi radius in eq. (3.12) and figure 8 (left) with the Larmor radius
$r_{L}={\frac{m~{}v_{\bot}}{|q|~{}B}},$ (6.1)
which determines the gyro motion of particles moving through a magnetic field.
Here $m$ is the mass of the particle (either proton or electron), $v_{\bot}$
is the velocity component of the particle perpendicular to the magnetic field,
$|q|$ is the absolute electric charge of the particle, and $B$ is the magnetic
field. Assuming a seed field of $B$=10-20 T and approximating the velocity
with the sound speed $v_{\bot}$$\approx$$c_{s}$ yields the gyro radius for
both protons and electrons. The proton gyro radius is about a factor of 2000
larger than the electron gyro radius.
Figure 8 (left) shows the Bondi radius as well as the proton and electron
Larmor radii as a function of redshift. If the gyro radius is smaller than the
Bondi radius, the respective particle is easily accreted by the PBH. If,
however, the gyro radius is larger than the Bondi radius, the particle will
first not be easily accreted, but rather spiral around the PBH. From 8 we see,
that at redshifts z$\gtrsim 70$ and PBH masses in the range
MPBH$\approx$0.3–500 for our assumed magnetic field strength the proton Larmor
radius is larger than the Bondi radius, while the electron Larmor radius is
smaller than the Bondi radius. There is still a substantial fraction of
residual electrons and protons/helium ions from the era before recombination
(see the dash-dotted curve in 8 right from [76]). These electrons have
therefore no problem being accreted, while for certain PBH masses protons
resist the accretion. This will create a net electric current, which in turn
will increase the average magnetic field strength until the proton gyro radius
becomes smaller than the Bondi radius. This way the PBH can amplify the
average magnetic field. The supersonic motion between baryon gas and PBH
discussed above is expected to be coherent over large scales (of the order of
Mpc) and can therefore induce large-scale ordered magnetic fields. A further
magnetic field amplification occurs, as discussed above, in the accretion
funnel, when magnetic fields are dissipated through reconnection and ejected
with the plasma. In a sense, the ubiquitous PBH can possibly create their own
magnetic field and distribute it throughout the universe. It is, however,
plausible to assume, that magnetic fields in the early universe should be
smaller than today, and the fractions of ionized baryons is less. This could
also explain a rather small Bondi accretion eigenvalue required to match the
observations.
### 6.3 Re-Ionization
Next I turn to the contribution of PBH accretion to the re-ionization and re-
heating history of the universe. At z$\approx$1089, when the photons decoupled
from the baryons and formed the CMB radiation, the universe became
predominantly neutral. Afterwards the universe entered a long period of
“darkness”, in which the residual ionization left over from the primordial
photon-baryon fluid diminished (see figure 8 right), the background photons
cooled down, and any higher-frequency emission was quickly absorbed in the
atomic gas. In the model described here the first sources to illuminate the
“dark ages” would be the PBH accreting from the surrounding gas. Their
ultraviolet emission, above the Hydrogen ionization energy of 13.6 eV, would
start to re-ionize small regions around each PBH. However, in the beginning
the density of the surrounding gas is still so high that the ionized matter
quickly recombines. As long as the re-combination time is much shorter than
the Hubble time at the corresponding epoch, UV photons from the PBH cannot
penetrate the surrounding medium, but instead produce an ionized bubble
growing with time. In this type of ionization equilibrium the number of
ionizing photons $N_{ion}$ required to overcome recombination is given as the
ratio between the Hubble time $t_{H}(z)$ and the recombination time
$t_{rec}(z)$ at the particular epoch, and can be derived from equations (2)
and (3) in [77] as
$N_{ion}=t_{H}/t_{rec}=max[1,0.59~{}\left(\frac{1+z}{7}\right)^{1.5}].$ (6.2)
At a redshift z=1000, $N_{ion}$ is about 1000, and reaches a level of unity at
z$\lesssim$10 for the assumed set of parameters. For this calculation I ignore
clumping of the ionized gas. In reality the effective clumping factor is
relatively large for reionization at high redshift because the ionizing
sources are more numerous in the filaments of the cosmic web, but must also
ionize a correspondingly larger fraction of dense gas in the filaments, and
thus ionization is slowed down. At lower redshift, when molecular gas and
stars have already formed, not all UV photons will escape the dense regions.
The effective escape fraction is one of the largest uncertainties in our
current understanding of re-ionization. For simplicity, I assume an escape
fraction $f_{esc}$=0.1 for UV photons, and $f_{esc}$=1 for X–ray photons,
independent of redshift.
To calculate the history of pre-ionization by PBH I integrate the above
normalized ADAF model for frequencies log($\nu$)>15.52 Hz, corresponding to
the hydrogen ionization energy of 13.6 eV. To calculate the number of ionizing
photons per PBH of reference mass 1 M⊙ I take the density-weighted luminosity
$\langle L_{bol}\rangle^{*}$ from figure 5 (right). To determine the average
space density of ionizing photons I multiply with the average comoving space
density of PBH (assuming the reference mass 1 M⊙):
$n_{PBH}=1.06\times 10^{-54}~{}\left(\frac{1+z}{1000}\right)^{3}~{}{\rm
cm}^{-3},$ (6.3)
and with the escape fraction $f_{esc}$ and finally divide by $N_{ion}$ from
eq. (6.3) and the average density of baryons given in equation (2.10) to
determine the ionization rate of baryons over cosmic time.
The red curve in figure 8 (right) shows the cumulative ionization fraction
$\chi_{e}$ as a function of redshift for the accretion eigenvalue
$\lambda$=0.05. A maximum cumulative ionization fraction of $\sim$2.8%, is
reached at a redshift z$\approx$10\. This can be compared to one of the recent
models determined from the Planck satellite data [35]. Here the 1$\sigma$
upper percentile of the FlexKnot model in their figure 45, which is consistent
with the ionization optical depth determined from the most recent Planck data,
is shown as dashed curve. A high-redshift contribution to the ionization
history of the universe has also been discussed by [78] and [16]. The range of
$\chi_{e}$ values assumed in the latter work is shown as green hatched region
in figure 8 (right). For the choice of $\lambda$=0.05, the UV emission from
the PBH population assumed in the toy model is therefore fully consistent with
the observational constraints from Planck.
### 6.4 X–ray heating
The role of X–ray heating in shaping the early universe has been discussed by
[79]. Compared to UV photons, X–ray photons have a much larger mean free path
and can therefore ionize regions far away from the source. In addition, most
of the X–ray energy gets deposited into heating up the gas. In order to
estimate the amount of X–ray heating of the gas I applied the same mechanism
as for the UV photons above, but integrating the above ADAF model for
frequencies log($\nu$)>17.68 Hz, corresponding to 2 keV. I assume an escape
fraction of $f_{esc}$=1 and $N_{ion}$=1. The blue curve in figure 8 (right)
shows the cumulative 2 keV heating fraction per baryon as a function of
redshift for the assumed accretion eigenvalue of $\lambda$=0.05. The maximum
cumulative heating fraction is $\sim$1.6%. X–rays from PBH therefore have only
a small contribution to the pre-ionization of the universe as a whole, but can
be responsible for a part of the pre-heating of gas observed in the "entropy
floor" of groups of galaxies. [80] reviewed the energetics of groups and
clusters of galaxies, which cannot be reproduced by simple models, where the
gas density is proportional to dark matter density. [81] and [82] argued that
the gas must have been pre-heated before falling into the cluster potential.
X–ray observations of groups of galaxies with ROSAT by [83] confirmed the need
for a non-gravitational entropy injection in the group gas. These authors
coined the term "entropy floor", which amounts to an energy of about 2 keV per
baryon injected into the group gas. The pre-heating of the gas by PBH, albeit
only contributing to a small fraction of the total baryon content of the
universe, could have played an important role in heating the high-density
regions, which first formed groups and clusters.
### 6.5 Cosmological 21-cm signal
Figure 9: Density-weighted 1.4 GHz (observed) luminosity of a single PBH as a
function of mass for different redshifts indicated.
The red-shifted 21-cm line can provide important new constraints on the
physical processes in the early universe [see e.g. 84, 8]. The Experiment to
Detect the Global EoR Signature (EDGES) has measured a strong, sky-averaged
21-cm absorption line profile after subtracting the Galactic synchrotron
emission [85]. The signal is centered at a frequency around 78 MHz and covers
a broad range in redshift z=15–20. The signal may be due to ultraviolet light
from first objects in the universe altering the emission of the 21-cm line by
lowering the spin temperature of neutral hydrogen relative to the CMB.
However, the signal is about three times larger than that expected from the
standard $\Lambda$CDM cosmology, which led some authors to suggest new dark
matter physics [e.g. 86]. Instead of new physics, an increased 21-cm radio
background contribution above the CMB at the epoch around z=15–20 could also
explain the EDGES signal. Indeed, [87] estimate the additional 21-cm radio
background from accretion onto putative radio-loud intermediate-mass black
holes (IMBH) forming in first molecular cooling halos at redshifts z=16–30.
This could be sufficient to explain the EDGES feature, however, it requires
extreme assumptions about the radio loudness of the IMBH population. Instead
of assuming an interpretation in terms of mini-QSOs from IMBH grown through
accretion, I estimate here, whether PBH accretion could have a significant
contribution to the EDGES signal. A full treatment of this effect for the PBH
toy model is beyond the scope of this paper, but similar to the treatment of
the PBH contribution to the CXB and CIB derived in section 5, I can estimate
the PBH contribution to the observed low-frequency cosmic radio background,
and thus to the EDGES signal.
The balloon-borne double-nulled Absolute Radiometer for Cosmology,
Astrophysics and Diffuse Emission (ARCADE2) instrument has measured the
absolute temperature of the sky at frequencies 3, 8, 10, 30, and 90 GHz, and
detected a significant excess over the CMB blackbody spectrum at a temperature
of 2.731K [88]. Combining the ARCADE2 measurements with lower frequency data
from the literature, the excess brightness temperature can be characterized as
a power law TR=1.19 ($\nu$/1 GHz)-2.62 K, which translates into a radio
spectrum with a slope of -0.62 and a normalization of 3$\times$10-22 W m-2
sr-1 at 1.4 GHz. This cosmic radio synchrotron background is substantially
larger than that expected from an extrapolation of the observed radio counts
[89], and thus presents a major new challenge in astrophysics. [90] found that
the global 21cm signal can be significantly amplified by an excess background
radiation compared to the standard $\Lambda$CDM models, especially in
absorption. Assuming that only 10% of the radio synchrotron background
originates at large redshifts, they predict a 21cm feature almost an order of
magnitude stronger than that expected purely from the CMB. Interpolating
between their calculations for 0.1% and 10% excess background I find, that an
excess high-redshift radiation field of about 5% of the observed radio
synchrotron background is sufficient to explain the EDGES findings.
In order to calculate the expected PBH contribution to the radio background I
assume that each black hole has a radio emission following the fundamental
plane relation between X-ray luminosity, radio luminosity and black hole mass
found by [91]. I use the parameterization for radio-quiet AGN from [92]:
log(LR)=0.85 log(LX)+0.12 log(MPBH), where LR is the 1.4 GHz radio luminosity
in units of 1040 erg/s, LX is the 0.1–2.4 keV X–ray luminosity in units of
1044 erg/s, and MPBH is the PBH mass in units of 108 M⊙. The X–ray luminosity
is calculated from the bolometric luminosity shown in figure 5 (right).
Assuming the ADAF radiation spectrum above, the fraction of the bolometric
luminosity radiated in the 0.1-2.4 keV band is 0.23. For the radio spectrum I
assume a power law with spectral index -0.62. This means that the bolometric
correction is $\propto$(1+z)0.38. The radio luminosities derived this way as a
function of PBH mass and redshift are shown in figure 9. Multiplying these
luminosities with the PBH density over cosmic time, converting into observed
fluxes and integrating over mass and redshift I obtain a contribution of
radio-quiet PBH to the observed radio background of $\sim$3$\times$10-25 W m-2
sr-1 at 1.4 GHz, i.e. a fraction of 0.1% of the observed synchrotron radio
background. Most of this additional radiation field is accumulated at
redshifts z$\gtrsim$20\. Following [90], this excess radio flux would increase
the depth of the 21cm absorption line only by about 30%. If, however, some
fraction of the PBH would be radio-loud (e.g. 5% with 1000 times higher
fluxes), like observed in the AGN population, the 5% excess high-redshift
radio background flux necessary to explain the EDGES feature could be easily
achieved by PBH.
### 6.6 Primordial Black Holes in the Galactic Center
Next I discuss some observational effects of the putative PBH population
residing in the Galactic Center region. First, assuming a Milky Way dark
matter halo of $\sim$10${}^{12}M_{\odot}$, the PBH mass spectrum from section
2 (figure 1) indeed predicts about one supermassive PBH with a mass $\gtrsim
10^{6.5}M_{\odot}$, consistent with the Sgr A∗ SMBH in the center of our
Galaxy [93]. To estimate the density of dark matter and baryons in the
Galactic bulge region itself, I refer to dynamical models of the Milky Way’s
center, using the density of red clump giant stars stars measured in infrared
photometric surveys, as well kinematic radial velocity measurements of M-giant
stars in the Galactic bulge/bar constructed in [94]. From N–body simulations
of stellar populations for barred spiral discs in different dark matter halos
these authors were able to determine with high precision the mass in a volume
of ($\pm 2.2\times\pm 1.4\times\pm 1.2$ kpc3) centered on the Galactic
Bulge/Bar. The total mass is (1.84$\pm$0.07)$\times$1010 M⊙. Depending on the
assumed model, about 9–30% consists of dark matter, i.e. 1.7–5.5$\times$109
M⊙. Applying the above PBH mass spectrum, we thus expect 5–10 intermediate-
mass PBH with MPBH>104 M⊙ in the Galactic bulge region, but zero with MPBH>105
M⊙.
Recent high-resolution observations of high-velocity compact clouds (HVCC) in
the central molecular zone of our Milky Way with the Atacama Large
Millimeter/submillimeter Array (ALMA) have indeed identified five promising
IMBH candidates, wandering through the patchy ISM in the Galactic Center [see
95]. The most compelling case is HCN–0.044–0.009, which shows two dense
molecular gas streams in Keplerian orbits around a dark object with a mass
MIMBH=(3.2$\pm$0.6)$\times$104 M⊙ [96]. The semimajor axis of these Keplerian
streams are around 2 and 5$\times$1017 cm. Another interesting case is the
infrared and radio object IRS13E, a star cluster close to the Galactic Center
potentially hosting an IMBH [97]. ALMA observations identified a rotating,
ionized gas ring around IRS13E [98], with an orbit radius of 6$\times$1015 cm
and a rotation velocity of $\sim$230 km/s. This is thus another promising IMBH
candidate with a mass of MIMBH=2.4$\times$104 M⊙.
Two of the five IMBH candidate sources in [95] are possibly associated with
X–ray sources detected in the deep Chandra images of the Galactic Center [99].
IRS13E has the X–ray counterpart CXOGC 174539.7-290029 with an X–ray
luminosity L2-10keV$\approx$3$\times$1030 erg/s, and CO–0.31+0.11 has the
possible X–ray counterpart CXOGC 174426.3-290816 with an X–ray luminosity
L2-10keV$\approx$4$\times$1029 erg/s. The other three sources have X–ray upper
limits in the range of several 1030 erg/s. Assuming a bolometric correction
factor of 1/30 for the 2–10 keV range, the combination of the mass accretion
eigenvalue $\lambda$ and the radiative efficiency $\eta$ therefore has to be
extremely small, on the order of 3$\times$10-11. This is about a factor of 100
lower than the 2$\times$10-9 LEdd derived for the Galactic Center black hole
Sgr A∗ [55]. Even assuming a very low efficiency ADAF model, a steady-state
accretion solution is unlikely for these objects. The solution of this puzzle
may come from the fact, that the velocity and density gradients of the gas in
the Galactic Center region are so large, that the angular momentum forces any
accreted matter into Keplerian motion well outside the classical Bondi radius
[see 37]. Indeed, the orbital periods and lifetimes of the Keplerian streams
around HVCCs are in the range 104-5 years, and thus accretion is expected to
be highly variable on very long time scales. Another possibility to halt
accretion for a considerable time is the feedback created by outflows during
efficient accretion events. Numerical simulations of the gas dynamics in the
center of the Galaxy [100] show that the outflows significantly perturb the
gas dynamics near the Bondi radius and thus substantially reduce the capture
rate. The net result of both these effects would be a highly variable, low
duty cycle bursty accretion onto the IMBH and SMBH in the Galactic Center,
consistent with the extremely low accretion efficiencies observed. The
accretion limits for black holes in the mass range MPBH=20–100 M⊙ derived from
deep Chandra and radio observations of the Galactic Center [45], are already
shown in figure 1 to be consistent with the assumed PBH mass spectrum. Recent
NuSTAR observations of the Galactic Center, including the effects of gas
turbulence and the uncertainties related to the dark matter density profile
even further weaken these constraints [101]. At any rate, the assumed PBH mass
distribution of section 2 is fully consistent with the observational
constraints for all PBH masses >20 M⊙ in the Galactic Center.
Finally, I check the PBH predictions for lower masses against the Galactic
ridge X–ray emission (GRXE), an unresolved X–ray glow at energies above a few
keV discovered almost 40 years ago and found to be coincident with the
Galactic disk. The GRXE in the 2–10 keV band has a background-corrected
surface brightness of (7.1$\pm$0.5)$\times 10^{-11}$ erg cm-2 s-1 deg-2 which
was largely resolved into discrete sources [102], with the brightest source
having an X–ray luminosity of about 1032 erg s-1, and the minimum detectable
luminosity around 1030 erg s-1. The integrated emission has a strong iron line
from hot plasma at 6.7 keV, and the authors interpret the X–ray emission as
coming from a large population of cataclysmic variables and coronally active
stars. Using the mass determination in the Galactic bulge/bar above I find
that the average baryon density in this region is in the range 17–22 cm-3.
However, most of these baryons are locked up in stars. In order to estimate
the physical conditions of the gas in the Galactic Bulge/Bar region I follow
[103]. According to these authors, there are four phases of the interstellar
medium in the Galactic center region: (1) a cold molecular phase in Giant
Molecular Clouds with temperatures around 50 K and gas densities n=103.5-4
cm-3 covering a volume fraction around 1%; (2) a warm molecular phase with
temperatures around 150 K and gas density n=102.5 cm-3, covering a volume
fraction of $\sim$10%; (3) an atomic phase with temperatures around 500-1000 K
and density $\sim$10 cm-3, covering a volume fraction around 70%, and (4)
ionized gas with temperatures 104-8 K and an unknown density. Depending on the
temperature of the interstellar medium, the sound speeds are in the range
$c_{s}$=1–100 km/s. The stellar velocity dispersion in the central region of
our Galaxy is in the range 100–140 km/s [104], while the dark matter velocity
dispersion is about 110 km/s [105]. In the spirit of the discussion leading up
to equation 3.9 and figure 2 (right) above, I assume an effective velocity for
Bondi accretion $v_{eff}\approx$50 km/s and $\lambda$=0.1. As shown in figures
5 and 6, the PBH emissivity for the assumed mass spectrum is typically
dominated by objects with MPBH>100, which already are discussed above. Indeed,
calculating the Bondi accretion rates and radiative efficiencies for objects
with MPBH<100 for the four ISM phases in the Galactic Center I obtain
negligible PBH contributions to the total GRXE brightness. Some individual
MPBH$\sim$100 M⊙ objects in high density regions could in principle have X–ray
luminosities up to $L_{2-10keV}$=1033 erg/s, more luminous than the brightest
X–ray source detected in the Galactic Ridge survey [102], but taking into
account the strong variability and small duty cycle expected for this class of
objects, their absence in the surveys is understandable. Some of the fainter
unidentified sources in the current deep X–ray surveys could indeed be
accreting PBH and future large X–ray observatories like ATHENA [106] or LYNX
[107] should be able to identify more. See also [108] for future searches in
the IR and sub-mm region.
## 7 Conclusions and Outlook
The interpretation of cold dark matter as the sum of contributions of
different mass PBH families [1] could explain a number of so far unsolved
mysteries, like e.g. the massive seed black holes required to create the
supermassive black holes in the earliest QSOs [13], the ubiquitous massive
LIGO/VIRGO massive binary black holes [e.g. 6], or even the putative "Planet
X" PBH in our own Solar System [14]. The most abundant family of PBH should be
around the Chandrasekhar mass (1.4 M⊙). This prediction may already have been
vindicated by the recent OGLE/GAIA discovery of a sizeable population of
putative black holes in the mass range 1-10 M⊙ [15]. Here I estimate the
contribution of baryon accretion onto the overall PBH population to various
cosmic background radiations, concentrating first on the crosscorrelation
signal between the CXB and the CIB fluctuations discovered in deep Chandra and
Spitzer surveys [23]. Assuming Bondi capture and advection dominsted disk
accretion with reasonable parameters like baryon density and the effective
relative velocity between baryons and PBH over cosmic time, as well as
appropriate accretion and radiation efficiencies, I indeed predict a
contribution of PBH consistent with the residual X–ray fluctuation signal.
This signal peaks at redshifts z$\approx$17–30. The PBH contribution to the
2–5 $\mu$m CIB fluctuations, however, is only about 1%, so that these would
have to come from star formation processes associated with the PBH.
I discuss a number of other phenomena, which could be significantly affected
by the PBH accretion. Magnetic fields are an essential ingredient in the Bondi
accretion process, and I argue that the PBH can play an important role in
amplifying magnetic seed fields in the early universe and maintaining them
until the galactic dynamo processes set in. Next I study the contribution of
the assumed PBH population to the re-ionization history of the universe and
find that they do not conflict with the stringent ionization limits set by the
most recent Planck measurements [35]. X–ray heating from the PBH population
can provide a contribution to the entropy floor observed in groups of galaxies
[83]. The tantalizing redshifted 21-cm absorption line feature observed by
EDGES [85] could well be connected to the radio emission contributed by PBH to
the cosmic background radiation. Finally, the number of IMBH and the diffuse
X–ray emission in the Galactic Center region are not violated by the PBH dark
matter, on the contrary, some of the discrete sources in the resolved GRXE
could be accreting PBH.
It is obvious, that our simple PBH toy model for the dark matter requires
significantly more work to turn it into quantitative predictions. Real
magnetohydrodynamic simulations of the whole PBH mass spectrum including their
own hierarchical clustering would be required to obtain the full history of
their contribution to the cosmic backgrounds. The exciting EDGES discovery
definitely requires a full-blown analysis of the radio contribution of PBH to
the cosmic background. Future X–ray observations with eROSITA and ATHENA,
infrared wide field surveys with Euclid and WFIRST, and microlensing
observations with WFIRST will provide important additional diagnostics in this
exciting and dramatically developing PBH field (see [109, 110]).
## Acknowledgments
I am thankful to Juan García-Bellido for sharing a digital copy of the new
running spectral index PBH mass distribution model in figure 1 in advance of
publication, as well as many very useful discussions about PBH. I am indebted
to Matthias Bartelmann for computing the small-scale non-linear relative
velocity dispersion (figure 2 right) and providing very valuable comments and
corrections to the manuscript. I would like to thank Sergey Karpov for very
helpful discussions and inputs about their spherical accretion model. I would
also like to thank my colleagues Nico Cappelluti, Sasha Kashlinsky and
Alexander Knebe for very helpful discussions and contributions. Finally, I
thank an anonymous referee for pointing out a substantial flaw in the first
version of the paper, which has been corrected here and led to significant
improvements. Throughout this work I made use of Ned Wright’s cosmology
calculator [111] and the NASA Astrophysics Data System (ADS), operated by the
Smithsonian Astrophysical Observatory under NASA Cooperative Agreement
NNX16AC86A.
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|
2024-09-04T02:54:59.069368 | 2020-03-11T08:05:02 | 2003.05151 | {
"authors": "Jukka Ruohonen and Kalle Hjerppe",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26155",
"submitter": "Jukka Ruohonen",
"url": "https://arxiv.org/abs/2003.05151"
} | arxiv-papers | 11institutetext: Department of Future Technologies, University of Turku,
Turku, Finland
# Predicting the Amount of GDPR Fines
Jukka Ruohonen Kalle Hjerppe
{juanruo<EMAIL_ADDRESS>
###### Abstract
The General Data Protection Regulation (GDPR) was enforced in 2018. After this
enforcement, many fines have already been imposed by national data protection
authorities in the European Union (EU). This paper examines the individual
GDPR articles referenced in the enforcement decisions, as well as predicts the
amount of enforcement fines with available meta-data and text mining features
extracted from the enforcement decision documents. According to the results,
articles related to the general principles, lawfulness, and information
security have been the most frequently referenced ones. Although the amount of
fines imposed vary across the articles referenced, these three particular
articles do not stand out. Furthermore, good predictions are attainable even
with simple machine learning techniques for regression analysis. Basic meta-
data (such as the articles referenced and the country of origin) yields
slightly better performance compared to the text mining features.
###### Keywords:
T
ext mining Legal mining Data protection Law enforcement
## 1 Introduction
Data protection has a long history in the EU. In particular, the GDPR repealed
the earlier Directive 95/46/EC. Although this directive laid down much of the
legal groundwork for EU-wide data protection and privacy, its national
adaptations, legal interpretations, and enforcement varied both across the
member states and different EU institutions [10]. In short: it was a paper
tiger. In contrast, Regulation (EU) 2016/679, the GDPR, is a regulation; it is
binding throughout the EU with only a minimal space for national adaptations.
In practice, only a few Articles (A) in the GDPR provide some but limited room
for national maneuvering; these include A6 with respect to relaxation in terms
of other legal obligations or public interests, A9 in terms of sensitive data,
and A10 regarding criminal matters. Thus, in general, this particular
legislation should be interpreted and enforced uniformly through the European
Union by national data protection authorities whose formal powers are defined
in A58. In practice, however, already the resources and thus the actual power
for enforcement vary across the member states [1, 7]. Coupled with a lack of
previous research on the enforcement of the GDPR, this variance provides a
motivation for the present work to examine the recent enforcement fines
imposed according to the conditions specified in A83. In addition, the work is
motivated by a tangential question; is it also possible to predict these fines
by machine learning methods?
To answer to the question, the paper uses meta-data and text miming features
extracted from the decision documents released by the national authorities. As
such, only black-box predictions are sought; the goal is not to make any legal
interpretations whatsoever. Nevertheless, the answer provided still
establishes a solid contribution—especially when considering that the paper is
presumably the very first to even examine the GDPR fines. As is discussed in
Section 2, the black-box approach also places the paper into a specific branch
of existing research dealing with legal documents. This section also refines
the question into two more specific research questions. Afterwards, the
structure is straightforward: the dataset and methods are elaborated in
Sections 3 and 4, results are presented in Section 5, and conclusions follow
in Section 6. As will be noted in the final section, there are also some
lessons that should not be learned from this work.
## 2 Background
Legal mining—in lack of a better term—has emerged in recent years as a
promising but at times highly contested interdisciplinary field that uses
machine learning techniques to analyze various aspects related to law [8].
Although the concrete application domains vary, case law and court cases are
the prime examples already because these constitute the traditional kernel of
legal scholarship. Within this kernel, existing machine learning applications
range from the classification of judges’ ideological positions [12], which may
be illegal in some European countries [3], to the prediction of decisions of
the European Court of Human Rights [16, 17]. These examples convey the
traditional functions of applied machine learning; exploratory data mining and
the prediction of the future.
There is also another closely related application domain. Again in lack of a
better term, data extraction could be a label for this domain: by exploiting
the nature of law as an art of persuasion [8], the domain uses distinct
information retrieval techniques to extract and quantify textual data from
legal documents into structured collections with a predefined logic and
semantics [2, 24, 28]. To gain a hint about the extraction, one might consider
a legal document to contain some facts, rights, obligations, and prohibitions,
statements and modalities about these, and so forth. Although the two
application domains are complementary in many respects, the underlying
rationales exhibit some notable differences.
Oftentimes, the legal mining domain is motivated by a traditional rationale
for empirical social science research: to better understand trends and
patterns in lawmaking and law enforcement; to contrast these with legal
philosophies and theories; and so forth. This rationale extends to public
administration: machine learning may ease the systematic archiving of legal
documents and the finding of relevant documents, and, therefore, it may also
reduce administrative costs [4]. These administrative aspects reflect the goal
of building “systems that assist in decision-making”, whereas the predictive
legal mining applications seek to build “systems that make decision” [21].
Although the data extraction domain can be motivated by the same
administrative rationale, providing data to predictive systems is seldom the
intention behind the extraction. Instead, there is a further rationale in this
domain: to extract requirements for software and systems in order to comply
with the laws from which a given extraction is done [24]. Driven by the
genuine interest to facilitate collaboration between lawyers and engineers in
order to build law-compliant software and systems [26], this rationale has
been particularly prevalent in the contexts of data protection and privacy.
For instance, previous work has been done to extract requirements from the
Health Insurance Portability and Accountability Act in the United States [2].
Against this backdrop, it is no real surprise that data extraction has been
applied also for laws enacted in the EU. While there is previous work for
identifying requirements from the GDPR manually [13], there indeed exists also
more systematic data extraction approaches [25]. However, neither domain has
addressed the enforcement of this EU-wide regulation. In fact, a reasonably
comprehensive literature search indicates no previous empirical research on
the GDPR’s enforcement. Given this pronounced gap in the existing literature,
this paper sets to examine the following two Questions (Q) regarding the
enforcement fines:
$\textmd{Q}_{1}$: (i) Which GDPR articles have been most often referenced in
the recent enforcement cases, (ii) and do the enforcement fines vary across
these articles?
$\textmd{Q}_{2}$: How well the recent GDPR fines can be predicted in terms of
basic available (i) meta-data and (ii) textual traits derived from the
enforcement decisions?
These two questions place the present work into the legal mining domain. Also
the underlying rationales are transferable. For instance, an answer to
$\textmd{Q}_{1}$ helps to understand which aspects of the GDPR have been
actively enforced during the early roll out of the regulation. Also
$\textmd{Q}_{2}$ carries a practical motivation: by knowing whether the
penalties are predictable by machine learning techniques, a starting point is
available for providing further insights in different practical scenarios.
These scenarios range from the automated archival of enforcement decisions and
the designation of preventive measures to litigation preparations. However, it
is important to remark that the GDPR’s enforcement is done by national data
protection authorities. Although the focus on public administration is
maintained nevertheless, documents about the enforcement decisions reached by
these authorities should not be strictly equated to law-like legal documents.
This point provides an impetus to move forward by elaborating the dataset
used.
## 3 Data
The dataset is based on a GDPR enforcement tracker that archives the fines and
penalties imposed by the European data protection authorities [5]. This
tracker is maintained by an international law firm for archiving many of the
known enforcement cases. Each case is accompanied by meta-data supplied by the
firm as well as a link to the corresponding decision from a national
authority. In addition to potentially missing cases due to the lack of
publicly available information, the archival material is unfortunately
incomplete in many respects. The reason originates from the incoherent
reporting practices of the European data protection authorities. Therefore,
all cases were obtained from the tracker, but the following four steps were
followed to construct a sample for the analysis:
1. 1.
To maintain coherence between $\textmd{Q}_{1}$ and $\textmd{Q}_{2}$, only
those cases were included that had both meta-data and links to the decisions
available. In terms of the former, some cases lacked meta-data about the fines
imposed, the particular GDPR articles referenced in the decisions, and even
links to the decisions.
2. 2.
To increase the quality of the sample, only those cases were included that
were accompanied with more or less formal documents supplied on the official
websites of the data protection authorities. Thus, those cases are excluded
whose archival material is based online media articles, excerpts collected
from annual reports released by the authorities, and related informal sources.
3. 3.
If two or more cases were referenced with the same decision, only one decision
document was included but the associated meta-data was unified into a single
case by merging the articles references and totaling the fines imposed.
4. 4.
All national decisions written in languages other than English were translated
to English with Google Translate. In general, such machine translation is
necessary due to the EU-wide focus of the forthcoming empirical analysis.
Given these restrictions, the sample amounts to about 72% of all cases
archived to the tracker at the time of data collection. Even with these
precautions, it should be stressed that the quality of the sample is hardly
optimal. While the accuracy of the meta-data supplied by the firm is taken for
granted, there are also some issues with the quality of the publicly available
decisions. The authorities in some countries (e.g., Hungary and Spain) have
released highly detailed and rigorous documents about their decisions, while
some other authorities (e.g., in Germany) have opted for short press releases.
Although most of the documents were supplied in the portable document format
(PDF) and informally signed by the authorities, it should be thus stressed
that the data quality is not consistent across the European countries
observed. In addition, it is worth remarking the detail that scanned PDF
documents (as used, e.g., in Portugal) had to be excluded due to the automatic
data processing. While these data quality issues underline the paper’s
exploratory approach, these carry also political and administrative
ramifications that are briefly discussed later on in Section 6.
## 4 Methods
Descriptive statistics and regression analysis are used for answering to the
two questions asked. In terms of Question $\textmd{Q}_{1}$, dummy variables
for the GDPR articles referenced are simply regressed against the logarithm of
the fines imposed by using the conventional analysis-of-variance (ANOVA). As
many of the cases reference multiple articles, it should be remarked that
these dummy variables are not so-called fixed effects. The methods for
answering to the second Question $\textmd{Q}_{2}$ require a more thorough
elaboration. In addition to (i) the GDPR articles, the meta-data aspects
include dummy variables for the following features: (ii) the year of a given
enforcement case; (iii) the country in which the given fine was imposed; and
(iv) the sector of the violating organization. The last feature was
constructed manually by using five categories: individuals, public sector
(including associations), telecommunications, private sector (excluding
telecommunications), and unknown sector due to the lack of meta-data supplied
in the enforcement tracker. In total, these features amount to $49$ dummy
variables.
The textual aspects for $\textmd{Q}_{2}$ are derived from the translated
decisions. Seven steps were used for pre-processing: (a) all translated
decision documents were lower-cased and (b) tokenized according to white space
and punctuation characters; (c) only alphabetical tokens recognized as English
words were included; (d) common and custom stopwords were excluded; (e) tokens
with lengths less than three characters or more than twenty characters were
excluded; (f) all tokens were lemmatized into their common English dictionary
forms; and, finally, (g) those lemmatized tokens were excluded that occurred
in the whole decision corpus in less than three times. A common natural
language processing library [22] was used for this processing together with a
common English dictionary [20]. In addition to the stopwords supplied in the
library, the twelve most frequent tokens were used as custom excluded
stopwords: data, article, personal, protection, processing, company,
authority, regulation, information, case, art, and page. After this pre-
processing, the token-based term frequency (TF) and term frequency inverse
document frequency (TF-IDF) were calculated from the whole corpus constructed
(for the exact formulas used see, e.g., [23]). These common information
retrieval statistics are used for evaluating the other part in
$\textmd{Q}_{2}$. In general, TF-IDF is often preferred as it penalizes
frequently occurring terms.
Sparsity is the biggest issue for prediction. There are only $154$
observations but already the meta-data amounts to $49$ independent
variables—and the TF and TF-IDF each to $4189$ independent variables.
Fortunately, the problem is not uncommon, and well-known solutions exist for
addressing it. Genomics is a good example about the application domains
riddled with the problem; within this domain, it is not uncommon to operate
with datasets containing a few thousand observations and tens of thousands of
predictors [6]. Dimension reduction is the generic solution in this domain and
other domains with similar problems. Thus, three common dimension reduction
methods for regression analysis are used: principal component regression
(PCR), partial least squares (PLS), and ridge regression (for a concise
overview of these methods see, e.g., [11]). In essence, PCR uses uncorrelated
linear combinations as the independent variables; PLS is otherwise similar but
also the dependent variable is used for constructing the combinations. Ridge
regression is based on a different principle: the dimensionality is reduced by
shrinking some of the regression coefficients to zero. In general, all three
methods are known to yield relatively similar results in applied work.
In terms of practical computation, the number of components for the PCR and
PLS models, and the shrinkage parameter for the ridge regression, is optimized
during the training while the results are reported with respect to a test set
containing 20% of the enforcement cases. Centering (but not scaling) is used
prior to the training with a $5$-fold cross-validation. Computation is carried
out with the caret package [14] in conjunction with the pls [18] and foba [30]
packages. Although root-mean-square errors (RMSEs) are used for optimizing the
training, the results are summarized with mean absolute errors (MAEs) due to
their straightforward interpretability. These are defined as the arithmetic
means of the absolute differences between the observed and predicted fines in
the test set.
## 5 Results
The GDPR fines imposed vary greatly. As can be seen from Fig. 1, a range from
about $e^{6}$ euros to $e^{12}$ euros capture the majority of the enforcement
fines observed. This range amounts roughly from about four hundred to $163$
thousand euros. That said, the distribution has a fairly long tail; also a few
large, multi-million euro fines are present in the sample. Therefore, the
sample cannot be considered biased even though the restrictions discussed in
Section 3 exclude some of the largest enforcement cases, including the
announcements about the intention to fine the British Airways and Marriott
International by the Information Commissioner’s Office in the United Kingdom.
Although these two excluded cases are—at least at the time of
writing—preliminary announcements, they are still illuminating in the sense
that both were about large-scale data breaches.
Figure 1: Enforcement Fines in the Sample
However, the GDPR’s corresponding A32 for information security has not been
the most frequently referenced article in the recent enforcement cases.
Instead, A5 and A6, which address the general principles and lawfulness of
personal data processing, have clearly been the most referenced individual
articles, as can be seen from Fig. 2. These two articles account for as much
as 87% of all $252$ references made in the $154$ enforcement cases. More than
six references have been made to A13 (informing obligations to data subjects),
A15 (right to access), A21 (right to object), and A17 (right to erasure).
These references indicate that enforcement has been active also with respect
to the rights granted by the GDPR for individual data subjects. Furthermore,
less frequent references have been made in the decisions to numerous other
articles. These include the obligations to designate data protection officers
(A37), conduct impact assessments (A35), and consult supervisory authorities
(A36), to name three examples. While the principles, lawfulness, and
information security account for the majority, the less frequent but still
visible references to more specific articles hint that the regulation’s whole
scope is slowly being enforced by the European authorities.
Figure 2: Referenced GDPR Articles in the Enforcement Cases
Turning to the second part of $\textmd{Q}_{1}$, the regression coefficients
from the log-linear ANOVA model are visualized in Fig. 3 (the intercept is
present in the model but not shown in the figure, and A36 is omitted as the
single reference made to the article corresponds with the single reference
made to A35 in the same decision; the dummy variable for A35 thus captures the
effect of both articles). As can be seen, the confidence intervals (CIs) are
quite wide for the articles referenced only infrequently, and only six
coefficients are statistically significant at the conventional threshold.
Thus, some care is required for interpretation.
Figure 3: Enforcement Fines Across Articles (logarithm, ANOVA, 95% CIs)
When looking at the coefficients with relatively tight CIs, it is evident that
variation is present but the magnitude of this variation is not substantial.
Most of the coefficients remain in the range $[-5,5]$. However, together all
the references do yield a decent model; an $F$-test is statistically
significant and the coefficient of determination is large ($R^{2}\simeq
0.44$). To put aside the statistical insignificance, it is also interesting to
observe that some of the coefficients have negative signs, meaning that some
references indicate smaller fines compared to the average. Among these are the
conditions for consent (A7), sensitive data (A9), transparency (A12), and
informing (A13), as well as the already noted right to access (A15), proper
notifications about data breaches (A33), and the powers granted for the
supervisory authorities (A58). Finally, the magnitude of the coefficient
($1.52$) for the information security article (A32) is significant but does
not stand out in terms of magnitude. When compared to cases without a
reference to this article, only about $1.5\%$ higher fines have been imposed
in cases referencing A32.
Figure 4: Prediction Performance (logarithm, MAEs) Figure 5: Observed and
Predicted Values in the Test Set
The results regarding $\textmd{Q}_{2}$ are summarized in Fig. 4 (the MAEs for
the training refer to the best cross-validated models). Three noteworthy
observations can be drawn from this summary. First and foremost, the
prediction performance is generally decent: the best-performing cases all
yield MAEs roughly between $1.3$ and $1.5$ for the log-transformed fines.
These average prediction errors seem also reasonable when taking a closer look
at the actual predictions—except for the outlying large fines. Take Fig. 5 as
a brief example; the figure displays the observed fines and the predicted
fines based on the PLS and ridge regression estimators for the first meta-data
model. Even though most of the predicted observations are fairly close to the
observed fines, the test set also contains one five million euro fine that is
quite severely underestimated by both regression estimators. The
underestimations amount to over $246$ thousand euros. Though, when a magnitude
is measured in millions, it is a matter of interpretation whether an error
measured in hundreds of thousands is large, small, or something else.
Second, there are some interesting differences between the regression
estimators. In particular, PLS and ridge regression exhibit relatively large
differences between training and testing. The explanation relates to the RMSE-
based optimization during training. For instance, PCR was estimated with only
one component for the first meta-data model and three components for the
remaining three models, whereas two components were picked for all four PLS
models.
Last but not least, the smallest MAE for the test set is outputted by ridge
regression using only the $49$ meta-data variables. The second and third
models containing the TF and TF-IDF variables both perform worse. Furthermore,
the fourth model, which contains the meta-data and TF-IDF variables, indicates
that the text mining features tend to slightly weaken the predictions. It is
also worth remarking that some redundancy is present among the meta-data
variables; comparable performance is obtained with only $17$ meta-data
variables that are left after prior pre-processing with the caret’s
nearZeroVar function. All this said, the overall interpretation should be less
explicit when considering the practical motivation for $\textmd{Q}_{2}$ noted
in Section 2. If only the decision documents are available without any prior
work to manually construct the meta-data from these, even the simple text
mining features could be used for black-box predictions.
## 6 Conclusion
This paper explored two questions. The answers to these can be summarized as
follows. First: regarding $\textmd{Q}_{1}$, the articles related to the
general principles (A5), lawfulness (A6), and information security (A32) have
been most frequently referenced by the national data protection authorities
during the early enforcement period observed in this paper. Although also the
enforcement fines vary across the various GDPR articles referenced in the
authorities’ decisions, the effects of these three articles do not stand out
in particular. A good corollary question for further work would be to examine
the future evolution of these references; a hypothesis is that the
regulation’s enforcement is slowly moving from the principles and lawfulness
conditions to more specific elements. Then: regarding $\textmd{Q}_{2}$, it is
possible to obtain decent predictions even with standard machine learning
techniques for regression analysis. Basic meta-data (i.e., articles
referenced, year of enforcement, country or origin, and industry sector) seems
to provide slightly better predictive performance compared to basic text
mining features (i.e., TF and TF-IDF) extracted from the decision documents.
Yet, even the text mining features seem sufficient for blind black-box
predictions. There are also many potential ways to improve the predictions
reported, including those related regression analysis (such as using specific
sparse-PLS estimators) and text mining (such as using word embeddings). Data
mining techniques (such as topic modeling) could be used also for better
understanding the nuances behind the decisions. An alternative path forward
would be to extend the specific data extraction approaches discussed in
Section 2 to the enforcement decisions. However, the motivation to move
forward is undermined by practical problems. As was remarked in Section 3,
already the quality of data is a problem of its own.
Recently, the enforcement of the GDPR has been fiercely criticized by some
public authorities and pundits alike. The reasons are many: a lack of
transparency and cooperation between national data protection authorities,
diverging legal interpretations, cultural conflicts, the so-called “one-stop-
shop” system, old-fashioned information systems and poor data exchange
practices, and so on and so forth [27]. The data collection used for the
present work testifies on behalf of the criticism: the decision documents
released by the national authorities have varied wildly in terms of quality
and rigor. Some national authorities have even hidden their decisions from
public scrutiny. A paradox is present: although A15 grants a right for data
subjects to access their personal data, the same subjects may need to exercise
their separate freedom of information rights to obtain cues about decisions
reached by national authorities. Four legs good, two legs bad.
Finally, it is necessary to briefly point out the bigger issues affecting the
legal mining and data extraction domains—and, therefore, also the present
work. For one thing, the practical usefulness of legal expert systems has been
questioned for a long time. The artificial intelligence hype has not silenced
the criticism [15]. Like with the “code is law” notion, which has never
existed in reality [19], there are also many philosophical counterarguments
against the legal mining and data extraction domains [8, 9, 21]. It is
problematic at best to codify the methodology of a scholarly discipline into
rigid schemas in order to nurse the methodological requirements of another
discipline; legal reasoning is distinct from other types of reasoning
exercised in empirical sciences; and so forth. Law is not code. But code is
increasingly used to predict law enforcement decisions. The legal mining
domain, in particular, is frequently involved with a motivation to build “a
system that could predict judicial decisions automatically” but with a
provision that there is “no intention of creating a system that could replace
judges” [17]. Such system-building leads to another delicate paradox. Namely,
the GDPR and related laws (such as Directive 2016/680 for data protection in
criminal matters) were also designed to provide certain guards against legal
mining and the resulting automated decision-making involving human beings
[29]. This paper is not immune to criticism originating from this fundamental
paradox. If it is seen as undesirable to build systems for making law
enforcement decisions, it should be also seen as undesirable to build systems
for automatically fining companies.
### Acknowledgements
This research was funded by the Strategic Research Council at the Academy of
Finland (grant no. 327391).
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|
2024-09-04T02:54:59.080131 | 2020-03-11T09:12:44 | 2003.05174 | {
"authors": "Jose Blanchet, Renyuan Xu and Zhengyuan Zhou",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26156",
"submitter": "Renyuan Xu",
"url": "https://arxiv.org/abs/2003.05174"
} | arxiv-papers | # Delay-Adaptive Learning in
Generalized Linear Contextual Bandits
Jose Blanchet Department of Management Science and Engineering, Stanford
University, USA. Email<EMAIL_ADDRESS>Renyuan Xu Mathematical
Institute, University of Oxford, UK. Email<EMAIL_ADDRESS>Zhengyuan Zhou
Stern School of Business, New York University, USA. Email<EMAIL_ADDRESS>
###### Abstract
In this paper, we consider online learning in generalized linear contextual
bandits where rewards are not immediately observed. Instead, rewards are
available to the decision maker only after some delay, which is unknown and
stochastic. We study the performance of two well-known algorithms adapted to
this delayed setting: one based on upper confidence bounds, and the other
based on Thompson sampling. We describe modifications on how these two
algorithms should be adapted to handle delays and give regret
characterizations for both algorithms. Our results contribute to the broad
landscape of contextual bandits literature by establishing that both
algorithms can be made to be robust to delays, thereby helping clarify and
reaffirm the empirical success of these two algorithms, which are widely
deployed in modern recommendation engines.
## 1 Introduction
The growing availability of user-specific data has welcomed the exciting era
of personalized recommendation, a paradigm that uncovers the heterogeneity
across individuals and provides tailored service decisions that lead to
improved outcomes. Such heterogeneity is ubiquitous across a variety of
application domains (including online advertising, medical treatment
assignment, product/news recommendation ([LCLS2010],
[BCN2012],[chapelle2014],[bastani2015online],[SBF2017])) and manifests itself
as different individuals responding differently to the recommended items.
Rising to this opportunity, contextual bandits ([besbes2009dynamic,
rigollet2010nonparametric, goldenshluger2011note, hsu2014taming,
agrawal2016efficient]) have emerged to be the predominant mathematical
formalism that provides an elegant and powerful formulation: its three core
components, the features (representing individual characteristics), the
actions (representing the recommendation), and the rewards (representing the
observed feedback), capture the salient aspects of the problem and provide
fertile ground for developing algorithms that balance exploring and exploiting
users’ heterogeneity.
As such, the last decade has witnessed extensive research efforts in
developing effective and efficient contextual bandits algorithms. In
particular, two types of algorithms–upper confidence bounds (UCB) based
algorithms ([LCLS2010, FCGS2010, chu2011contextual, JBNW2017, LLZ2017]) and
Thompson sampling (TS) based algorithms ([AG2013a, AG2013b, RV2014,
russo2016information, agrawal2017thompson])–stand out from this flourishing
and fruitful line of work: their theoretical guarantees have been analyzed in
many settings, often yielding (near-)optimal regret bounds; their empirical
performance have been thoroughly validated, often providing insights into
their practical efficacy (including the consensus that TS based algorithms,
although sometimes suffering from intensive computation for posterior updates,
are generally more effective than their UCB counterparts, whose performance
can be sensitive to hyper-parameter tuning). To a large extent, these two
family of algorithms have been widely deployed in many modern recommendation
engines.
However, a key assumption therein–both the algorithm design and their
analyses–is that the reward is immediately available after an action is taken.
Although useful as a first-step abstraction, this is a stringent requirement
that is rarely satisfied in practice, particularly in large-scale systems
where the time-scale of a single recommendation is significantly smaller than
the time-scale of a user’s feedback. For instance, in E-commerce, a
recommendation is typically made by the engine in milliseconds, whereas a
user’s response time (i.e. to buy a product or conversion) is typically much
larger, ranging from hours to days, sometimes even to weeks. For instance, a
thorough empirical study in [chapelle2014] found that more than 10% of the
conversions in Criteo (a real-time bidding company) were at least 2 weeks old.
Furthermore, [chapelle2014] found that the delay distribution from the
company’s data follows the exponential distribution closely and hence does
have heavy tails. Similarly, in clinical trials, it is infeasible to
immediately observe and hence take into account the medical outcome after
applying a treatment to a patient–collecting medical feedback can be a time-
consuming and often random process; and in general, it is common to have
applied trial treatments to a large number of patients, with individual
medical outcomes only available much later at different, random points in
time. In both the E-commerce ([KCW2001, chapelle2014])and the clinical trials
cases ([CC2011]), a random and often significantly delayed reward is present.
Further, such delays empirically often follow a heavy tail distribution, and
hence a priori can have substantially negative impact on the learning
performance. Consequently, to understand such impact of delays, adjustments in
classical formulations must be made, both at the algorithmic level and at the
analysis level.
### 1.1 Related Work
In the past five years or so, the problem of learning on bandits with delays
has received increasing attention and has been studied in several different
settings in the existing literature, where most of the efforts have
concentrated on the multi-armed bandits setting, including both the stochastic
multi-armed bandits and the adversarial multi-armed bandits.
For stochastic multi-armed bandits with delays, [JGS2013] show a regret bound
$O(\log T+\mathbb{E}[\tau]+\sqrt{\log T\mathbb{E}[\tau]})$ where
$\mathbb{E}[\tau]$ is the mean of the iid delays. [DKVB2014] consider Gaussian
Process bandits with a bounded stochastic delay. [MLBP2015] follow the work of
[JGS2013] and propose a queue-based multi-armed bandit algorithm to handle
delays. [PASG2017] match the same regret bound as in [JGS2013] when feedback
is not only delayed but also anonymous.
For adversarial multi-armed bandits with delays, [NAGS2010] establish the
regret bound of $\mathbb{E}[R_{T}]\leq
O(\tau_{\text{const}})\times\mathbb{E}[R^{\prime}_{T}(\frac{T}{\tau_{\text{const}}})]$
for Markov decision process, where $\tau_{\text{const}}$ is the constant delay
and $R^{\prime}_{T}$ is the regret without delays. [CGM2019] consider
adversarial bandits with fixed constant delays on the network graph, with a
minimax regret of the order $\tilde{O}\sqrt{(K+\tau_{\text{const}})T}$, where
$K$ is the number of arms. Another related line of work to adversarial multi-
armed bandits is adversarial learning with full information, where the rewards
for all arms are observed. Different variants of this problems in the delayed
setting have been studied by [WO2002], [mesterharm2005], [QK2015] and
[GST2016].
On the other hand, learning in contextual bandits with delays are much less
explored. [JGS2013] consider learning on adversarial contextual bandits with
delays and establish an expected regret bound
$\mathbb{E}\left[R_{T}\right]\leq(1+\mathbb{E}[M_{T}^{*}])\times\mathbb{E}\left[R^{\prime}_{T}\left(\frac{T}{1+\mathbb{E}[M_{T}^{*}]}\right)\right]$
by using a black-box algorithm, where $M_{T}^{*}$ is the running maximum
number of delays up to round $T$. [DHKKLRZ2011] consider stochastic contextual
bandits with a fixed constant delay. The reward model they consider is general
(i.e. not necessarily parametric); however, they require the policy class to
be finite. In particular, they obtain the regret bound $O(\sqrt{K\log
N}(\tau_{\text{const}}+\sqrt{T}))$, where $N$ is the number of policies and
$\tau_{\text{const}}$ is again the fixed constant delay.
Finally, we also note that there is a growing literature on offline contextual
bandits (for a highly incomplete list, see [dudik2011doubly,
swaminathan2015batch, athey2017efficient, zhou2018offline, kitagawa2018should,
off-policy-evaluation-slate-recommendation, deep-learning-logged-bandit-
feedback]). This is a setting where all the data has been collected upfront
and a policy needs to be learned from this batch data at once. Although
sharing the same primitives (contexts, actions and rewards), this problem has
important differences from the online setting. In particular, the exploration
part is missing in this problem and a separate set of challenges exist in the
offline case. In this setting, delays would have no impact since all the
rewards will have been collected at the end (except perhaps at the tail of the
batch).
### 1.2 Our Contributions
In this paper, we consider learning on generalized linear (stochastic)
contextual bandits with stochastic unbounded delays. Our contributions are
two-fold. First, we design two delay-adaptive algorithms for generalized
linear contextual bandits, one based on UCB, the other based on TS. We refer
to the two variants as Delayed UCB (DUCB, as given in Algorithm 1) and Delayed
TS (DTS, as given in Algorithm 2) respectively. DUCB requires a carefully
designed delay-adaptive confidence parameter, which depends on how many
rewards are missing up to the current time step. In contrast, DTS is a
straightforward adaptation that incorporates the delayed rewards as they
become available.
Second, we give regret characterizations of both DUCB and DTS under (1)
independent stochastic, unbounded delays that can have heavy tails, (2)
unbounded Markov delays that can have near-heavy tails (tails that are
arbitrarily close to exponential tails), and (3) unbounded delays with any
dependency structure that have light (sub-Gaussian) tails. In particular, as a
special case of our results, when the delays are iid with mean $\mu_{I}$, we
have a high-probability regret bound of
$\tilde{O}\left(\left(\sigma_{G}\sqrt{d}+\mu_{I}d+d\right)\sqrt{T}\right)$ on
DUCB, where $\sigma_{G}$ is a parameter characterizing the tail bound of the
delays and $d$ is the feature dimension. For comparison, the state-of-the-art
regret bound of UCB on generalized linear contextual bandits without delays is
$\tilde{O}\left(d\sqrt{T}\right)$ ([FCGS2010, LLZ2017]). For DTS, we have the
Bayesian regret bound of
$\tilde{O}\left(\left(\sigma_{G}\sqrt{d}+\mu_{I}\sqrt{d}+d\right)\sqrt{T}\right)$.
For comparison, the state-of-the-art Bayesian regret bound of TS on
generalized linear contextual bandits without delays is
$\tilde{O}\left(d\sqrt{T}\right)$ ([RV2014, russo2016information]). The regret
bounds we have obtained highlight the dependence on the delays in two ways:
one is how much delay is present on average, the other is how heavy the tail
of the distribution is. Both factors contribute to the degradation of the
regret bounds: that the average delay enlarges regret is intuitive; that the
tail influences regret is because a more likely large delay (at the far right
end of a tail) can delay the learning for that context significantly,
particularly in the early stages when the decision maker is unsure about the
underlying parameter is.
To the best of our knowledge, these regret bounds provide the first
theoretical characterizations in generalized linear contextual bandits with
large delays. Our results contribute to the broad landscape of contextual
bandits literature by establishing that both algorithms are robust to delays,
thereby helping clarify and reaffirm the empirical success of these two
algorithms, which are widely deployed in modern recommendation engines.
Some of the initial results have appeared in the conference version
[zhou2019]. Our work here provides a comprehensive treatment of learning in
generalized linear contextual bandits with large delays that incorporates
substantially more in-depth inquiries on several fronts. First, we consider
the heavier-tailed delays that include exponential distributions whereas
[zhou2019] only dealt with light-tailed delays that are either sub-Gaussian or
have ($1+q$)-th moment (for some $q>0$). This relaxation is important both
from an empirical standpoint and from a theoretical standpoint. Empirically,
as mentioned earlier, the field study in [chapelle2014] found that the delay
distribution from the company’s data follows the exponential distribution
closely, rather than a sub-Gaussian distribution that is commonly assumed in
the bandits literature. Theoretically, establishing guarantees in this larger-
delay regime requires us to develop a new (and arguably more elegant) argument
from that in [zhou2019], which is not applicable here. We explain the
technical difficulty in more detail in Section 3.3. Second, the sole focus of
[zhou2019] is on adapting and analyzing UCB-based algorithms. However, as
mentioned earlier, it is known that Thompson sampling often achieves superior
empirical performance, despite the fact that their theoretical bounds (when no
delays are present) may not match exactly those of the UCB algorithms.
Furthermore, TS-based algorithms do not suffer from hyper-parameter tuning and
can effectively incorporate prior and can therefore significantly outperform
(when priors are available and correct). Consequently, in this paper, in
addition to adapting and analyzing the UCB-based algorithms, we also discuss
(in Section 4) the adaptation of TS-based algorithms in the delayed feedback
setting and obtain regret bounds that characterize the corresponding
performance. Finally, we move beyond the regime of the independent delay
setting studied in [zhou2019], and instead consider (in Section 5) the much
more general and realistic history-dependent delays setting. We give regret
bounds of both UCB-based algorithms and TS-based algorithms, under both the
Markov delays assumption and the general stationary delays assumption. We also
highlight, in this unified presentation, the comparison of the various regret
bounds as the assumption on delays get progressively weakened.
## 2 Problem Setup
In this section, we describe the formulation for learning in generalized
linear contextual bandits (GLCB) in the presence of delays. We start by
reviewing the basics of generalized linear contextual bandits, followed by a
description of the delay model. Before proceeding, we first fix some notation.
For a vector $x\in\mathbb{R}^{d}$, we use $\|x\|$ to denote its $l_{2}$-norm
and $x^{\prime}$ its transpose.
$\mathbb{B}^{d}:=\\{x\in\mathbb{R}^{d}:\|x\|\leq 1\\}$ is the unit ball
centered at the origin. The weighted $l_{2}$-norm associated with a positive-
definite matrix $A$ is defined by $\|x\|_{A}:=\sqrt{x^{\prime}Ax}$. The
minimum and maximum singular values of a matrix $A$ are written as
$\lambda_{\min}(A)$ and $\|A\|$ respectively. For two symmetric matrices $A$
and $B$ the same dimensions, $A\succeq B$ means that A-B is positive semi-
definite. For a real-valued function f, we use $\dot{f}$ and $\ddot{f}$ to
denote its first and second derivatives. Finally, $[n]:=\\{1,2,\cdots,n\\}$.
### 2.1 Generalized Linear Contextual Bandits
#### Decision procedure.
We consider the generalized linear contextual bandits problem with $K$
actions. At each round $t$, the agent observes a context consisting of a set
of $K$ feature vectors $x_{t}:=\\{x_{t,a}\in\mathbb{R}^{d}|a\in[K]\\}$, which
is drawn iid from an unknown distribution $\gamma$ with $\|x_{t,a}\|\leq 1$.
Each feature vector $x_{t,a}$ is associated with an unknown stochastic reward
$y_{t,a}\in[0,1]$. If the agent selects one action $a_{t}$, there is a
resulting reward $y_{t,a_{t}}\in[0,1]$ associated. In the standard contextual
bandits setting, the reward is immediately observed after the decision is made
and the observed reward can be utilized to make decision in the next round.
Although it is generally understood in the contextual bandits literature, for
completeness, here we briefly discuss the meaning of the above quantities, as
well as where they come from. In general, at each round $t$, an individual
characterized by $v_{t}$ (a list of characteristics associated with that
individual) is drawn from a population and becomes available. When the
decision maker decides to apply action $a_{t}$ (one of the available $K$
actions) to this individual, then a reward $y_{t}(v_{t},a_{t})$ is obtained:
this reward can depend stochastically on both the individual characteristics
$v_{t}$ and the selected action $a_{t}$. However, in practice, for both
modelling and computational reasons, one often first featurizes the individual
characteristics and the actions. In particular, with sufficient generality,
one assumes $\mathbf{E}[y_{t}(v_{t},a_{t})\mid
v_{t},a_{t}]=g_{\theta}(\phi(v_{t},a_{t}))$, where $g_{\theta}(\cdot)$ is the
parametrized mean reward function and $\phi(v_{t},a_{t})$ extracts the
features from the given raw individual characteristics $v_{t}$ and action
$a_{t}$. In the above formulation, as is standard in the contextual bandits
literature, we assume the feature map $\phi(\cdot)$ is known and given and
$x_{t,a}=\phi(v_{t},a)$. If $V_{t}$ is already a vector in Euclidean space,
then a common choice for the feature extractor is
$\phi(v_{t},a)=[\mathbf{0},\dots,\mathbf{0},v_{t},\mathbf{0},\dots,\mathbf{0}]$:
that is, a $Kd$-dimensional vector with all zeros except at the $a$-th block.
#### Relationship between reward $Y$ and context $X$.
In terms of the relationship between $Y_{t,a}$ and $X_{t,a}$, we follow the
standard generalized linear contextual bandits literature ([FCGS2010,
LLZ2017]). Define $\mathcal{H}^{0}_{t}=\\{(s,x_{s},a_{s},y_{s,a_{s}}),s\leq
t-1\\}\cup\\{x_{t}\\}$ as the information available at the beginning of round
$t$. The agent maximizes the cumulative expected rewards over $T$ rounds with
information $\mathcal{H}^{0}_{t}$ at each round $t$ ($t\geq 1$). Suppose the
agent takes action $a_{t}$ at round $t$. Denote by $X_{t}=x_{t,a_{t}}$,
$Y_{t}=y_{t,a_{t}}$ and we assume the conditional distribution of $Y_{t}$
given $X_{t}$ is from the exponential family. Therefore its density is given
by
$\displaystyle\mathbb{P}_{\theta^{*}}(Y_{t}|X_{t})=\exp\left(\frac{Y_{t}X_{t}^{\prime}\theta^{*}-m(X_{t}^{\prime}\theta^{*})}{h(\eta)}+A(Y_{t},\eta)\right).$
(1)
Here, $\theta^{*}$ is an unknown number under the frequentist setting;
$\eta\in\mathbb{R}^{+}$ is a given parameter; $A$, $m$ and $h$ are three
normalization functions mapping from $\mathbb{R}$ to $\mathbb{R}$.
For exponential families, $m$ is infinitely differentiable,
$\dot{m}(X^{\prime}\theta^{*})=\mathbb{E}[Y|X]$, and
$\ddot{m}(X^{\prime}\theta^{*})=\mathbb{V}(Y|X)$. Denote
$g(X^{\prime}\theta^{*})=\mathbb{E}[Y|X]$ , one can easily verify that
$g(x^{\prime}\theta)=x^{\prime}\theta$ for linear model,
$g(x^{\prime}\theta)=\frac{1}{1+\exp(-x^{\prime}\theta)}$ for logistic model
and $g(x^{\prime}\theta)=\exp(x^{\prime}\theta)$ for Poisson model. In the
generalized linear model (GLM) literature ([NW1972, McCullagh2018]), $g$ is
often referred to as the inverse link function.
Note that (1) can be rewritten as the GLCB form,
$\displaystyle Y_{t}=g(X_{t}^{\prime}\theta^{*})+\epsilon_{t},$ (2)
where $\\{\epsilon_{t},t\in[T]\\}$ are independent zero-mean noise,
$\mathcal{H}^{0}_{t}$-measurable with
$\mathbb{E}[\epsilon_{t}|{\mathcal{H}^{0}_{t}}]=0$. Data generated from (1)
automatically satisfies the sub-Gaussian condition:
$\displaystyle\mathbb{E}\left[\exp({\lambda\epsilon_{t}})|{\mathcal{H}^{0}_{t}}\right]\leq\exp\left({\frac{\lambda^{2}\hat{\sigma}^{2}}{2}}\right).$
(3)
Throughout the paper, we denote $\hat{\sigma}>0$ as the sub-Gaussian parameter
of the noise $\epsilon_{t}$.
###### Remark 1
In this paper, we focus on the GLM with exponential family (1). In general,
one can work with model (2) under the sub-Gaussian assumption (3). Our
analysis will still hold by considering maximum quasi-likelihood estimator for
(2). See more explanations in Section 3.1.
### 2.2 The Delay Model
Unlike the traditional setting where each reward is immediately observed, here
we consider the case where stochastic and unbounded delays are present in
revealing the rewards. Let $T$ be the number of total rounds. At round $t$,
after the agent takes action $a_{t}$, the reward $y_{t,a_{t}}$ may not be
available immediately. Instead, it will be observed at the end of round
$t+D_{t}$ where $D_{t}$ is the delay at time $t$. We assume $D_{t}$ is a non-
negative random number which is independent of $\\{D_{s}\\}_{s\leq t-1}$ and
$\\{x_{s},y_{s,a_{s}},a_{s}\\}_{s\leq t}$. First, we define the available
information for the agent at each round.
#### Information structure under delays.
At any round $t$, if $D_{s}+s\leq t-1$ (reward occurred in round $s$ is
available at the beginning of round $t$), then we call
$(s,x_{s},y_{s,a_{s}},a_{s})$ the complete information tuple at round $t$. If
$D_{s}+s\geq t$, we call $(s,x_{s},a_{s})$ the incomplete information tuple at
the beginning of round $t$. Define
$\mathcal{H}_{t}=\left\\{(s,x_{s},y_{s,a_{s}},a_{s})\,\,|\,\,s+D_{s}\leq
t-1\right\\}\cup\left\\{(s,x_{s},a_{s})\,\,|\,\,s\leq t-1,s+D_{s}\geq
t\right\\}\cup\left\\{x_{t}\right\\},$
then $\mathcal{H}_{t}$ is the information (filtration) available at the
beginning of round $t$ for the agent to choose action $a_{t}$. In other words,
$\mathcal{H}_{t}$ contains all the incomplete and complete information tuples
up to round $t-1$ and the content vector $x_{t}$ at round $t$.
Moreover define
$\displaystyle\mathcal{F}_{t}=\\{(s,x_{s},a_{s},y_{s,a_{s}})\,\,|\,\,s+D_{s}\leq
t\\}.$ (4)
Then $\mathcal{F}_{t}$ contains all the complete information tuples
$(s,x_{s},a_{s},y_{s,a_{s}})$ up to the end of round $t$. Denote
$\mathcal{I}_{t}=\mathcal{F}_{t}-\mathcal{F}_{t-1}$, $\mathcal{I}_{t}$ is the
new complete information tuples revealed at the end of round $t$.
#### Performance criterion.
Under the frequentist setting, assume there exists an unknown true parameter
$\theta^{*}\in\mathbb{R}^{d}$. The agent’s strategy can be evaluated by
comparing her rewards to the best reward. To do so, define the optimal action
at round $t$ by $a_{t}^{*}=\arg\max_{a\in[K]}g(x_{t,a}^{\prime}\theta^{*})$.
Then, the agent’s total regret of following strategy $\pi$ can be expressed as
follows
$R_{T}(\pi):=\sum_{t=1}^{T}\left(g\left(x_{t,a^{*}_{t}}^{\prime}\theta^{*}\right)-g\left(x_{t,a_{t}}^{\prime}\theta^{*}\right)\right),$
where $a_{t}\sim\pi_{t}$ and policy $\pi_{t}$ maps $\mathcal{H}_{t}$ to the
probability simplex
$\Delta^{K}:=\\{(p_{1},\cdots,p_{K})\,\,|\,\,\sum_{i=1}^{K}p_{i}=1,p_{i}\geq
0\\}$. Note that $R_{T}(\pi)$ is in general a random variable due to the
possible randomness in $\pi$.
#### Assumptions.
Throughout the paper, we assume the following assumption on distribution
$\gamma$ and function $g$, which is standard in the generalized linear bandit
literature ([FCGS2010, LLZ2017, JBNW2017]).
###### Assumption 1 (GLCB)
* •
$\lambda_{\min}(\mathbb{E}[\frac{1}{K}\sum_{a\in[K]}x_{t,a}x_{t,a}^{\prime}])\geq\sigma_{0}^{2}$
for all $t\in[T]$.
* •
$\kappa:=\inf_{\\{\|x\|\leq 1,\|\theta-\theta^{*}\|\leq
1\\}}\dot{g}(x^{\prime}\theta)>0$.
* •
$g$ is twice differentiable. $\dot{g}$ and $\ddot{g}$ are upper bounded by
$L_{g}$ and $M_{g}$, respectively.
In addition, we assume the delay sequence $\\{D_{t}\\}_{t=1}^{T}$ satisfies
the following assumption.
###### Assumption 2 (Delay)
Assume $\\{D_{t}\\}_{t=1}^{T}$ are independent non-negative random variables
with tail-envelope distribution $(\xi,\mu,M)$. That is, there exists a
constant $M>0$ and a distribution $\xi$ with mean $\mu<\infty$ such that for
any $m\geq M$ and $t\in[T]$,
$\mathbb{P}(D_{t}\geq m)\leq\mathbb{P}(D\geq m),$
where $D\sim\xi$. Furthermore, assume there exists $q\geq 0$ such that
$\mathbb{P}(D-\mu\geq x)\leq\exp\left(\frac{-x^{1+q}}{2\sigma^{2}}\right),$
where $\mathbb{E}[D]=\mu$.
Assumption 2 includes the most common delay patterns in real-world
applications. $D$ is sub-Gaussian when $q=1$ and $D$ has exponential delays
when $q=0$. When $D_{t}$’s are iid, the following condition guarantees
Assumption 2:
$\mathbb{P}(D_{t}-\mathbb{E}[D_{t}]\geq
x)\leq\exp\left(\frac{-x^{1+q}}{2\tilde{\sigma}^{2}}\right),$
with some $\tilde{\sigma}>0$ and $q\geq 0$. We summarize the parameter
definition in Table LABEL:tab:parameters. (See Section LABEL:app:table.)
Note that with Assumption 2, we do not need to assume all delays have
identical distributions, as long as they are independent over time. Since
there exists an envelop distribution $\xi$ uniformly dominating the tail
probability of all delays, we can get a handle on the tail of all the delay
distributions. This can be viewed as the regularity condition on the delays.
## 3 Delayed Upper Confidence Bound (DUCB) for GLCB
In this section, we propose a UCB type of algorithm for GLCB adapting the
delay information in an online version. Let us first introduce the maximum
likelihood estimator we adopt and then state the main algorithm.
### 3.1 Maximum Likelihood Estimators (MLEs).
Denote $T_{t}=\\{s:s\leq t-1,D_{s}+s\leq t-1\\}$ as the set containing
timestamps with complete information tuples at the beginning of round $t$. We
use data with timestamps in $T_{t}$ to construct the MLE. Suppose we have
independent samples of $\\{Y_{s}:s\in T_{t}\\}$ condition on $\\{X_{s}:s\in
T_{t}\\}$. The log-likelihood function of $\theta$ under (1) is
$\displaystyle\log l\left(\theta\,\,|\,\,T_{t}\right)$ $\displaystyle=$
$\displaystyle\sum_{s\in
T_{t}}\left[\frac{Y_{s}X_{s}^{\prime}\theta-m(X_{s}^{\prime}\theta)}{v(\eta)}+B(Y_{s},\eta)\right]$
$\displaystyle=$ $\displaystyle\frac{1}{v(\eta)}\sum_{s\in
T_{t}}\left[Y_{s}X_{s}^{\prime}\theta-m(X_{s}^{\prime}\theta)\right]+\text{constant}.$
Therefore, the MLE can be defined as
$\hat{\theta}_{t}\in\arg\max_{\theta\in\Theta}\sum_{s\in
T_{t}}\left[Y_{s}X_{s}^{\prime}\theta-m(X_{s}^{\prime}\theta)\right].$
Since $m$ is differentiable with $\ddot{m}\geq 0$, the MLE can be written as
the solution of the following equation
$\displaystyle\sum_{s\in T_{t}}(Y_{s}-g(X_{s}^{\prime}\theta))X_{s}=0,$ (5)
which is the estimator we use in Step 4 of Algorithm 1.
Note that, the general GLCB, a semi-parametric version of the GLM, is obtained
by assuming only that $\mathbb{E}[Y|X]=g(X^{\prime}\theta^{*})$ (see (2))
without further assumptions on the conditional distribution of $Y$ given $X$.
In this case, the estimator obtained by solving (5) is referred to as the
maximum quasi-likelihood estimator. It is well-documented that this estimator
is consistent under very general assumptions as long as matrix $\sum_{s\in
T_{t}}X_{s}X_{s}^{\prime}$ tends to infinity as $t\rightarrow\infty$
([CHY1999, FCGS2010]).
### 3.2 Algorithm: DUCB-GLCB
Denote $G_{t}=\sum_{s=1}^{t-1}\mathbb{I}\\{s+D_{s}\geq t\\}$ as the number of
missing reward when the agent is making a prediction at round $t$. Further
denote $W_{t}=\sum_{s\in T_{t}}X_{s}X_{s}^{\prime}$ as the matrix consisting
feature information with timestamps in $T_{t}$ and
$V_{t}=\sum_{s=1}^{t-1}X_{s}X_{s}^{\prime}$ as the matrix consisting all
available features at the end of round $t-1$. Then the main algorithm is
defined as follows.
Algorithm 1 DUCB-GLCB
1: Input: the total rounds $T$ , model parameters $d$ and $\kappa$, and tuning
parameters $\tau$ and $\delta$.
2: Initialization: randomly choose $\alpha_{t}\in[K]$ for $t\in[\tau]$, set
$V_{\tau+1}=\sum_{i=1}^{\tau}X_{s}X_{s}^{\prime}$,
$T_{\tau+1}:=\\{s\,:\,s\leq\tau,s+D_{s}\leq\tau\\}$,
$G_{\tau+1}=\tau-|T_{\tau+1}|$ and $W_{\tau+1}=\sum_{s\in
T_{\tau+1}}X_{s}X_{s}^{\prime}$
3: for $t=\tau+1,\tau+2,\cdots,T$ do
4: Update Statistics: calculate the MLE $\hat{\theta}_{t}$ by solving
$\sum_{s\in T_{t}}(Y_{s}-g(X_{s}^{\prime}\theta))X_{s}=0$
5: Update Parameter:
$\beta_{t}=\frac{\hat{\sigma}}{\kappa}\sqrt{\frac{d}{2}\log\left(1+\frac{2(t-G_{t})}{d}\right)+\log(\frac{1}{\delta})}+{\sqrt{G_{t}}}$
6: Select Action: choose
$a_{t}=\arg\max_{a\in[K]}\left(x_{t,a}^{\prime}\hat{\theta}_{t}+\beta_{t}\|x_{t,a}\|_{V_{t}^{-1}}\right)$
7: Update Observations: $X_{t}\leftarrow x_{t,a_{t}}$, $V_{t+1}\leftarrow
V_{t}+X_{t}X_{t}^{\prime}$ and $T_{t+1}\leftarrow
T_{t}\cup\\{s\,:\,s+D_{s}=t\\}$, $G_{t+1}=t-|T_{t+1}|$, and
$W_{\tau+1}=W_{\tau}+\sum_{s:s+D_{s}=t}X_{s}X_{s}^{\prime}$
8: end for
###### Remark 2 (Comparison to UCB-GLM Algorithm in [LLZ2017])
We make several adjustments to the UCB-GLM Algorithm in [LLZ2017]. First, in
step 4 (statistics update), we only use data with timestamps in $T_{t}$ to
calculate the estimator using MLE. In this step, using data without reward
will cause bias in the estimation. Second, when selecting the action in step
5, parameter $\beta_{t}$ is updated adaptively at each round whereas in
[LLZ2017], the corresponding parameter is constant over time. Moreover, in
step 4, we choose to use $V_{t}$ to normalize the context vector $X_{t,a}$
instead of $W_{t}$.
### 3.3 Preliminary Analysis
Denote $G_{t}^{*}=\max_{1\leq s\leq t}G_{s}$ as the running maximum number of
missing reward up to round $t$. The property of $G_{t}$ and $G^{*}_{t}$ is the
key to analyze the regret bound for both UCB and Thompson sampling algorithms.
We next characterize the tail behavior of $G_{t}$ and $G^{*}_{t}$.
###### Proposition 1 (Properties of $G_{t}$ and $G_{t}^{\star}$)
Assume Assumption 2. Denote $\sigma_{G}=\sigma\sqrt{2+q}$. Then,
1. 1.
$G_{t}$ is sub-Gaussian. Moreover, for all $t\geq 1$, with probability
$1-\delta$
$\displaystyle{G}_{t}\leq
2(\mu+M)+\sigma_{G}\sqrt{2\log\left(\frac{1}{\delta}\right)}+2\sigma_{G}^{2}\log
C_{3}+1,$ (6)
where $C_{3}=2\sigma^{2}+1$.
2. 2.
With probability $1-\delta$,
$\displaystyle{G}_{T}^{*}$ $\displaystyle\leq$ $\displaystyle
2(\mu+M)+\sigma_{G}\sqrt{2\log T}+2\sigma_{G}^{2}\log C_{3}$ (7)
$\displaystyle+\sigma_{G}\sqrt{2\log\left(\frac{1}{\delta}\right)+2\log
C_{3}\sigma_{G}\sqrt{2\log T}+2\log C_{3}}+1,$
where $G_{T}^{*}=\max_{1\leq s\leq T}G_{s}$.
3. 3.
Define $W_{t}=\sum_{s\in T_{t}}X_{s}X_{s}^{\prime}$ where $X_{t}$ is drawn
iid. from some distribution $\gamma$ with support in the unit ball
$\mathbb{B}_{d}$. Furthermore, let $\Sigma:=\mathbb{E}[X_{t}X_{t}^{\prime}]$
be the second moment matrix, and $B$ and $\delta>0$ be two positive constants.
Then there exist positive, universal constants $C_{1}$ and $C_{2}$ such that
$\lambda_{\min}(W_{t})\geq B$ with probability at least $1-2\delta$, as long
as
$\displaystyle
t\geq\left(\frac{C_{1}\sqrt{d}+C_{2}\sqrt{\log(\frac{1}{\delta})}}{\lambda_{\min}(\Sigma)}\right)^{2}+\frac{2B}{\lambda_{\min}(\Sigma)}+2(\mu+M)+\sigma_{G}\sqrt{2\log\left(\frac{1}{\delta}\right)}+2\sigma_{G}^{2}\log
C_{3}+1.$ (8)
A special case of Proposition 1-1 is when $D_{i}$’s are iid and $q=0$. Now
assume $D_{i}\sim D$ are iid with exponential-decays:
$\displaystyle\mathbb{P}(D-\mu_{I}\geq
t)\leq\exp(-\frac{t}{2\sigma_{I}^{2}}),$ (9)
and $\mu_{I}=\mathbb{E}D$. Then with probability $1-\delta$, we have
$\displaystyle G_{t}-\mu_{I}\leq
2\sigma_{I}\sqrt{\log\left(\frac{1}{\delta}\right)}+1+4\sigma_{I}^{2}\log(2\sigma_{I}^{2}).$
(10)
At a high level, the proof utilizes the fact that, with high probability,
there will be a lot of zero terms in the summation
$G_{t}=\sum_{s=1}^{t-1}\mathbb{I}(s+D_{s}\geq s)$ when $t$ is large. This is
done by designing a sequence of stopping times for the successes. We highlight
the idea by showing result (10) for the special case when $D_{t}$’s are iid
and $q=0$. The full version of the proof is deferred to Appendix LABEL:proof.
###### Sketch of the proof..
Define $V=\sum_{i=1}^{\infty}\mathbb{I}(D_{i}-\mu_{I}\geq i)$ where $D_{i}\sim
D$ are iid that satisfies (9).
Now let us define the following sequence of stopping times, $(k\geq 1)$,
$T(k)=\inf\\{t>T(k-1):D_{t}\geq t\\},$
where $T(k)$ is the time of the $k^{\text{t}h}$ success. Therefore,
$\displaystyle\mathbb{P}(V\geq j)$ $\displaystyle=$
$\displaystyle\mathbb{P}(T(1)<\infty,T(2)<\infty,\cdots,T(j-1)<\infty,T(j)<\infty)$
(11) $\displaystyle=$
$\displaystyle\Pi_{k=1}^{j}\mathbb{P}\left(T(k)<\infty|T(i)<\infty\,\,\text{
for }\,\,i\leq k-1\right)$ $\displaystyle=$
$\displaystyle\Pi_{k=2}^{j}\mathbb{P}\left(T(k)<\infty|T(k-1)<\infty\right)\mathbb{P}\left(T(1)<\infty\right)$
(12) $\displaystyle\leq$
$\displaystyle\Pi_{k=1}^{j}\left(\sum_{i=k}^{\infty}\exp\left(-\frac{i}{2\sigma_{I}^{2}}\right)\right)$
(13) $\displaystyle\leq$
$\displaystyle\Pi_{k=1}^{j}\left(2\sigma_{I}^{2}\exp\left(-\frac{k-1}{2\sigma_{I}^{2}}\right)\right)$
(14) $\displaystyle=$
$\displaystyle(2\sigma_{I}^{2})^{j}\exp\left(-\frac{(j-1)j}{4\sigma_{I}^{2}}\right)$
(15)
(11) holds by tower property. (12) holds since event
$\\{T(k)<\infty|T(k-1)<\infty\\}$ is equivalent to event
$\\{T(k)<\infty|T(j)<\infty\,\,{\text{f}or}\,\,j\leq k-1\\}$. Condition on
$T(k-1)<\infty$, we have
$\mathbb{P}\left(T(k)<\infty|T(k-1)<\infty\right)\leq\mathbb{P}(\cup_{j\geq
k}\mathbb{I}(D_{j}\geq
j))\leq\sum_{i=k}^{\infty}\exp\left(-\frac{i}{2\sigma_{I}^{2}}\right)$. The
last inequality holds by the union bound. Therefore (13) holds. Finally, 14
holds by integration.
Given (14), $V$ is sub-Gaussian and with probability $1-\delta$,
$V\leq
2\sigma_{I}\sqrt{\log\left(\frac{1}{\delta}\right)}+1+4\sigma_{I}^{2}\log(2\sigma_{I}^{2}).$
Similarly, we can show that, for any $t\geq 1$, $G_{t}$ is sub-Gaussian. With
probability $1-\delta$, we have
$G_{t}-\mu_{I}\leq
2\sigma_{I}\sqrt{\log\left(\frac{1}{\delta}\right)}+1+4\sigma_{I}^{2}\log(2\sigma_{I}^{2}).$
∎
Note that $G_{t}$ is sub-Gaussian even when $D$ has near-heavy-tail
distribution ($p\in[0,1)$).
###### Remark 3
The proof of Proposition 1 is simple but essential. It fully utilizes the
property that the sequence in $V$ has a lot of zero terms (with high
probability). In particular, one will not be able to fully obtain the result
if one uses the standard approach and directly works at the level of “the-sum-
of-sub-Guassians-is-sub-Gaussian" and thereafter analyzing sum of sub-Gaussian
constants, which is the method used in [zhou2019]. In order to drive this
point home, we provide an approach in this direction using Hoeffding bound
(Theorem LABEL:thm9). See Appendix LABEL:further_G. With such a approach, one
can only handle the case when $q>0$, which excludes the most difficult
scenario with exponential delays. With Hoeffding bound, the sub-Gaussian
parameter for $V$ is of the form
$\sigma=\sqrt{\sum_{i=1}^{\infty}\sigma_{i}^{2}}$ where $\sigma$ is the sub-
Gaussian parameter for indicator function $\mathbb{I}(G_{i}\geq i)$.
Intuitively speaking, this Hoeffding bound does not take into consideration of
the sparsity in the sequence. Therefore, the argument cannot reach the limit
for $q=0$.
### 3.4 Regret Bounds
###### Theorem 1
Assume Assumptions 1-2. Fix any $\delta$. There exists a universal constant
$C:=C(C_{1},C_{2},M,\mu,\sigma_{0},\hat{\sigma},\,\sigma,\kappa)>0,$
such that if we run DUCB-GLCB with
$\tau:=C\left(d+\log(\frac{1}{\delta})\right)$ and
$\beta_{t}=\frac{\hat{\sigma}}{\kappa}\sqrt{\frac{d}{2}\log\left(1+\frac{2(t-G_{t})}{d}\right)+\log(\frac{1}{\delta})}+G_{t}$,
then, with probability at least $1-5\delta$, the regret of the algorithm is
upper bounded by
$\displaystyle R_{T}$ $\displaystyle\leq$
$\displaystyle\tau+L_{g}\left[4\sqrt{\mu+M}\sqrt{Td\log\left(\frac{T}{d}\right)}+2^{7/4}\sqrt{\sigma_{G}}(\log
T)^{1/4}\sqrt{d\log\left(\frac{T}{d}\right)T}+\frac{2d\hat{\sigma}}{\kappa}\log\left(\frac{T}{d\delta}\right)\sqrt{T}\right.$
(16)
$\displaystyle\,\,+\left.2\sqrt{2Td\log\left(\frac{T}{d}\right)}\left(\sqrt{\sigma_{G}}\left({2\log\left(\frac{1}{\delta}\right)+2\log
C_{3}\sigma_{G}\sqrt{2\log T}+2\log C_{3}}\right)^{1/4}\right.\right.$
$\displaystyle\,\,\left.\left.+\sqrt{1+2\sigma_{G}^{2}\log
C_{3}}\right)\right]$
For parameter definition, we refer to Table LABEL:tab:parameters in Section
LABEL:app:table.
The proof of Theorem 1 consists of three steps. The first step is to construct
a confidence ball associated with the adaptive parameter $\beta_{t}$ and show
that the true parameter falls into the confidence ball with high probability.
The second step is to upper bound the normalized context sequence
$\sum_{t=\tau+1}^{\tau+n}\|X_{t}\|_{V_{t}^{-1}}$. And the last step is to
utilize the property of $G_{t}$ and $G^{*}_{t}$ proved in Proposition 1. The
details is deferred to Appendix LABEL:proof.
Given the high probability bound in Theorem 1, one can show the expected
regret bound without much of work.
###### Corollary 1 (Expected regret)
Assume Assumptions 1-2. The expected regret is bounded by
$\displaystyle\mathbb{E}[R_{T}]={O\left(d\sqrt{T}\log(T)+\sqrt{\sigma_{G}}\sqrt{Td}(\log(T))^{3/4}+(\sqrt{\mu+M}+\sigma_{G})\sqrt{Td\log\left({T}\right)}\right).}$
(17)
Given the result in (16), (17) holds by choosing $\delta=\frac{1}{T}$ and
using the fact that $R_{T}\leq T$.
The highest order term $O(d\sqrt{T}\log(T))$ does not depend on delays. This
result is in line with the non-contextual stochastic bandit literature
([JGS2013]). Delay impacts the expected regret bound in two folds. First, the
sub-Gaussian parameter $\sigma_{G}$ and the mean-related parameter ${\mu+M}$
appears in the second-highest order term. Second, the sub-Gaussian parameter
${\sigma_{G}}$ appears in the third-order term. Note that here we include the
log factors in deciding the highest order term, the second highest order term
and so on. If we exclude the log terms, then both delay parameters impact the
regret bound multiplicatively.
### 3.5 Tighter Regret Bounds for Special Cases
When the sequence $\\{D_{s}\\}_{s=1}^{T}$ satisfies some specific assumptions,
we are able to provide tighter high probability bounds on the regret.
###### Proposition 2
Given Assumptions 1-2, we have the following results.
1. 1.
If there exists a constant $D_{\max}>0$ such that $\mathbb{P}(D_{s}\leq
D_{\max})=1$ for all $s\in[T]$. Fix $\delta$. There exists a universal
constant $C>0$ such that by taking
$\tau=D_{\max}+C(d+\log(\frac{1}{\delta}))$, with probability $1-3\delta$, the
regret of the algorithm is upper bounded by
$\displaystyle
R_{T}\leq\^{A}~\tau+L_{g}\left(2{\sqrt{D_{\max}}}\sqrt{2Td\log\left(\frac{T}{d}\right)}+\frac{2d\hat{\sigma}}{\kappa}\log\left(\frac{T}{d\delta}\right)\sqrt{T}\right).$
(18)
2. 2.
Assume $D_{1},\cdots,D_{T}$ are iid non-negative random variables with mean
$\mu_{I}$. There exists $C>0$ such that by taking
$\tau:=C\left(d+\log(\frac{1}{\delta})\right)$, with probability $1-5\delta$,
the regret of the algorithm is upper bounded by
$\displaystyle R_{T}\leq$
$\displaystyle\tau+L_{g}\left[{4\sqrt{\mu_{I}}}\sqrt{Td\log\left(\frac{T}{d}\right)}+{2^{7/4}\sqrt{\sigma_{G}}(\log
T)^{1/4}}\sqrt{d\log\left(\frac{T}{d}\right)T}+\frac{2d\hat{\sigma}}{\kappa}\log\left(\frac{T}{d\delta}\right)\sqrt{T}\right.$
$\displaystyle\,\,+\left.2\sqrt{2Td\log\left(\frac{T}{d}\right)}\left(\sqrt{\sigma_{G}}\left({2\log\left(\frac{1}{\delta}\right)+2\log
C_{3}\sigma_{G}\sqrt{2\log T}+2\log C_{3}}\right)^{1/4}\right.\right.$
$\displaystyle\,\,\left.\left.+\sqrt{1+2\sigma_{G}^{2}\log
C_{3}}\right)\right]$
When delays $\\{D_{s}\\}_{s=1}^{T}$ are bounded by $D_{\max}$, the delay
paramter $D_{\max}$ only appears in the term $\sqrt{Td\log{{T}}}$ and does not
affect the highest order term $d\log(\frac{T}{d\delta})\sqrt{T}$. Compared to
(17), there is no regret term on the order of
$O(\sqrt{Td}\left(\log(T)\right)^{3/4})$ in (18). This is because we can
provide a smaller number on the right hand side of (8) when delays are
bounded. When delays are iid, $\mu+M$ is replaced by $\mu_{I}$, which is the
common expectation of all the random delays.
We refer to Appendix LABEL:proof for the proof of Proposition 2.
## 4 Delayed Thompson Sampling (DTS) for GLCB
In section 3, under the frequentist set-up, we assume there exists a true
parameter $\theta^{*}$ and use UCB to encourage exploration and construct the
confidence interval for $\theta^{*}$. On the contrary, posterior sampling does
not make use of upper confidence bounds to encourage exploration and instead
relies on randomization. In this section, we operate in the Bayesian decision
making setting and assume the decision maker is equipped with a prior
distribution on $\theta^{*}$. In this setting, the standard performance metric
is Bayesian regret, defined as follows:
$R^{B}_{T}(\pi)=\mathbb{E}_{\theta^{*},x}[R_{T}(\pi,\theta^{*})]=\sum_{t=1}^{T}\mathbb{E}_{\theta^{*},x}\left[g\left(x_{t,a^{*}_{t}(\theta^{*})}^{\prime}\theta^{*}\right)-g\left(\
x_{t,a_{t}}^{\prime}\theta^{*}\right)\right],$
where $a_{t}\sim\pi_{t}$. Next, we present the Thompson sampling algorithm
when adapted to the delayed setting. Algorithm 2 provides a formal
description.
Algorithm 2 DTS-GLCB
1: Input: the total rounds $T$ , tuning parameter $\tau$, prior $Q_{0}$
2: Initialization: randomly choose $\alpha_{t}\in[K]$ for $t\in[\tau]$
3: Update information: $\mathcal{F}_{\tau}$ according to (4).
4: if $\mathcal{I}_{\tau}=\emptyset$ then
5: $Q_{1}(\theta)=Q_{0}(\theta)$
6: else
7: $Q_{1}(\theta)\propto
Q_{0}(\theta|\mathcal{F}_{\tau})\Pi_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{\tau}}\mathbb{P}(y_{s,a_{s}}|\theta,x_{s,a_{s}})$
8: end if
9: for $t=1,2,\cdots,T-\tau$ do
10: Sample Model: $\hat{\theta}_{t+\tau}\sim Q_{t}$
11: Select Action: $\bar{a}_{t+\tau}\in\arg\max_{a\in[K]}\left\langle
x_{t+\tau,a},\hat{\theta}_{t+\tau}\right\rangle$
12: Update information: $\mathcal{F}_{t+\tau}$ according to (4). Define
$\mathcal{I}_{t+\tau}:=\mathcal{F}_{t+\tau}-\mathcal{F}_{t+\tau-1}$ as the new
information at round $t+\tau$
13: if $\mathcal{I}_{t+\tau+1}=\emptyset$ then
14: $Q_{t+1}(\theta)=Q_{t}(\theta)$
15: else
16: $Q_{t+1}(\theta)\propto
Q_{t}(\theta|\mathcal{F}_{\tau+t})\Pi_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t+\tau+1}}\mathbb{P}(y_{s,a_{s}}|\theta,x_{s,a_{s}})$
17: end if
18: end for
###### Remark 4
Note that in Algorithm 2, there is an exploration period of length $\tau$. The
posterior distribution employed at round $\tau+1$ is conditioned on
observations made over the first $\tau$ time rounds. Another point to note is
that Algorithm 2 is kept at an abstract level. The exact computation depends
on the prior chosen and the exponential family. Note that every exponential
family has a conjugate prior ([DY1979]), which admits efficient posterior
update. Section 4.1 provides a concrete example on linear contextual bandits,
which is a simple special case. We use this special case to illustrate how one
can perform efficient incremental update in the presence of delays.
### 4.1 Delayed Thompson Sampling For Linear Contextual Bandits
When $g(x)=x$ and $m(x)=\frac{x^{2}}{2}$, (1) reduces to
$\displaystyle\mathbb{P}(Y|X)=\exp\left(\frac{YX^{\prime}\theta^{*}-(X^{\prime}\theta^{*})^{2}/2}{h(\eta)}+A(Y,\eta)\right).$
(19)
Recall, from Bayes’ theorem, the posterior distribution is equal to the
product of the likelihood function $\theta\rightarrow\mathbb{P}(y|\theta)$ and
prior $\mathbb{P}(\theta)$, normalized by the probability of the data
$\mathbb{P}(y)$:
$\mathbb{P}(\theta|y)=\frac{\mathbb{P}(y|\theta)\mathbb{P}(\theta)}{\int\mathbb{P}(y|\theta)\mathbb{P}(\theta^{\prime})d\theta^{\prime}}$
Different choices of the prior distribution $\mathbb{P}(\theta)$ may make the
integral more or less difficult to calculate. Moreover, the product
$\mathbb{P}(y|\theta)\mathbb{P}(\theta)$ may take one form or another. But for
certain choices of the prior, the posterior will have the same form as the
prior, with possibly different parameter values. Such a choice is a conjugate
prior. The conjugate prior, giving a closed-form expression for the posterior,
makes Thompson sampling efficient to update. Further notice that, every
exponential family has a conjugate prior ([DY1979]).
Now we consider the normal conjugate prior for the linear model (19). Let
$B_{t}=aI_{d}+\sum_{s\in T_{t}}x_{s,a_{s}}x_{s,a_{s}}^{\prime}$ and
$\theta_{t}=B_{t}^{-1}\left(\sum_{s\in T_{t}}x_{s,a_{s}}y_{s,a_{s}}\right)$.
Given the linear model (19), suppose we have $Y|X$ is Gaussian with
$\mathcal{N}(X^{\prime}{\theta},v^{2})$. If the prior for $\theta$ at round
$t$ is given by $\mathcal{N}(\theta_{t},v^{2}B_{t}^{-1})$, then it is easy to
verify that the posterior distribution at around $t+1$ is
$\mathcal{N}(\theta_{t+1},v^{2}B_{t+1}^{-1})$. Then Algorithm 2 becomes
Algorithm 3 DTS-LCB
1: Input: the total rounds $T$ , constant $v>0$ and $a\geq 0$, tuning
parameter $\tau$, conjugate prior $\mathcal{N}(\theta_{0},v^{2}B_{0}^{-1})$
with $\theta_{0}=0$ and $B_{0}=aI_{d}$, $f_{0}=0$
2: Initialization: randomly choose $\alpha_{t}\in[K]$ for $t\in[\tau]$
3: Update information: $\mathcal{F}_{\tau}$ according to (4).
4: if $\mathcal{I}_{t}=\emptyset$ then
5: $\theta_{1}=\theta_{0}$, $B_{1}=B_{0}$
6: else
7:
$B_{1}=\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t}}x_{s,a_{s}}x_{s,a_{s}}^{T}$,
$f_{1}=\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t}}x_{s,a_{s}}y_{s,a_{s}}$
and $\theta_{1}=B_{1}^{-1}f_{1}$
8: end if
9: for $t=1,2,\cdots,T-\tau$ do
10: Sample Model:
$\hat{\theta}_{t+\tau}\sim\mathcal{N}(\theta_{t},v^{2}B_{t}^{-1})$
11: Select Action: $\bar{a}_{t+\tau}\in\arg\max_{a\in[K]}\left\langle
x_{t+\tau,a},\hat{\theta}_{t+\tau}\right\rangle$
12: Update information: $\mathcal{F}_{t+\tau}$ according to (4). Define
$\mathcal{I}_{t+\tau}:=\mathcal{F}_{t+\tau}-\mathcal{F}_{t+\tau-1}$ as the new
information at round $t+\tau$
13: if $\mathcal{I}_{t+\tau+1}=\emptyset$ then
14: $B_{t+1}=B_{t}$
15: $\theta_{t+1}=\theta_{t}$
16: else
17:
$B_{t+1}=B_{t}+\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t+\tau}}x_{s,a_{s}}x_{s,a_{s}}^{T}$,$f_{t+1}=f_{t}+\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t+\tau}}x_{s,a_{s}}y_{s,a_{s}}$,
and $\theta_{t+1}=B_{t+1}^{-1}f_{t+1}$
18: end if
19: end for
Note that the update (line 17) is on the incremental form which is practically
efficient.
### 4.2 Regret Bounds
Denote $\pi_{\tau}^{\text{PS}}$ as the posterior sampling policy described in
Algorithm 2 with an exploration period $\tau$. We have the following result.
###### Theorem 2
Assume Assumptions 1-2. There exists a universal constant
$C:=C(C_{1},C_{2},M,\mu,\sigma_{0},\sigma_{G},\,\sigma,\kappa)>0$, such that
if we run exploration with $\tau:=C\left(d+\log(\frac{1}{\delta})\right)$,
$\displaystyle R^{B}_{T}(\pi_{\tau}^{\text{PS}})={O\left(d\log
T\sqrt{T}+\sqrt{\sigma_{G}}\sqrt{Td}(\log(T))^{3/4}+(\sqrt{\mu+M}+\sigma_{G})\sqrt{dT\log\left(T\right)}\right).}$
(20)
For parameter definition, we refer to Table LABEL:tab:parameters in Section
LABEL:app:table.
We follow the steps in [RV2014] to prove the Bayesian regret bound in Theorem
2. The idea is the follows. We first decompose the Bayesian regret and the UCB
the regret and build a connection between them. We then provide the Bayesian
regret bound by utilizing a sequence of upper confidence bounds. We defer the
details to Appendix LABEL:proof.
When $\\{D_{s}\\}_{s=1}^{T}$ satisfies some specific assumptions, we are able
to provide tighter Bayesian regret bounds.
###### Corollary 2
Assume Assumptions 1-2, we have the following result:
1. 1.
If there exists a constant $D_{\max}>0$ such that $\mathbb{P}(D_{s}\leq
D_{\max})=1$ for all $s\in[T]$. Then,
$R^{B}_{T}(\pi_{\tau}^{\text{PS}})=O\left(d\log
T\sqrt{T}+{\sqrt{D_{\max}}}\sqrt{dT\log T}\right).$
2. 2.
Assume $D_{1},\cdots,D_{T}$ are iid non-negative random variables with mean
$\mu_{I}$. Then,
$R^{B}_{T}(\pi_{\tau}^{\text{PS}})={O\left(d\log
T\sqrt{T}+\sqrt{\sigma_{G}}\sqrt{Td}(\log(T))^{3/4}+(\sqrt{\mu_{I}}+\sigma_{G})\sqrt{dT\log\left(T\right)}\right).}$
We defer the proof of Corollary 2 to Appendix LABEL:proof. The results in
Theorem 2 and Corollary 2 are comparable to the results in Section 3.
## 5 Extensions: History-dependent Delays
In previous sections, we have analyzed the regret bounds for both DUCB-GLCB
and DTS-GLCB when delays are independent. In practice, such independence
assumption may not hold and current delays may depend on historical delays. In
this section, we explore two types of dependency structures for the delays. In
section 5.1, we discuss Markov delays where the stationary distribution is
near-heavy-tail. In section LABEL:sec:random_delay, we discuss delays with
random dependency structures but under a stronger assumption on the stationary
distribution, which is lighter-than-sub-Gaussian.
### 5.1 Markov Delays
###### Assumption 3 (Markov Delay)
Let $\\{D_{t}\\}_{t=1}^{T}$ be a stationary Markov chain on the general state
space $\mathcal{X}=\mathbb{N}^{+}$ with invariant distribution $\pi$. Given
$D\sim\pi$ with $\mu_{M}=\mathbb{E}[D]$, we further assume that
$\mathbb{P}(D-\mu_{M}\geq
x)\leq\exp\left(\frac{-x^{1+q}}{2\sigma_{M}^{2}}\right),$
for some $q>0$ and $\sigma_{M}>0$.
Under Assumption 3, the stationary distribution $\pi$ can have near-heavy-tail
property when $q$ is small.
Recall that $G_{t}=\sum_{s=1}^{t-1}\mathbb{I}\\{s+D_{s}\geq t\\}$ is the
number of missing reward and $G_{t}^{*}=\max_{1\leq s\leq t}G_{t}$ is the
running maximum number of missing reward. Under Assumption 3, $G_{t}$ and
$G_{t}^{*}$ has the following properties and again this is the key to analyze
regret bounds for both DUCB and DTS.
###### Proposition 3 (Properties of $G_{t}$ and $G_{t}^{\star}$ under Markov
delays)
Assume Assumption 3 and $l_{2}$-spectral gap $1-\lambda\in(0,1]$. Then,
1. 1.
For any $0<\delta<1$ and any $t$ we have, with probability at least
$1-\delta$,
$\displaystyle G_{t}-\mu_{M}\leq
A_{2}(\lambda)\log\left(\frac{1}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{1}{\delta}\right)},$
(21)
where $A_{1}(\lambda)=\frac{1+\lambda}{1-\lambda}$ and
$A_{2}(\lambda)=\frac{1}{3}\mathbb{I}(\lambda=0)+\frac{5}{1-\lambda}\mathbb{I}(\lambda>0)$.
2. 2.
With probability at least $1-\delta$,
$\displaystyle
G_{T}^{*}\leq\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)},$
(22)
where $G_{T}^{*}=\max_{1\leq t\leq T}G_{t}$.
3. 3.
Define $W_{t}=\sum_{s\in T_{t}}X_{s}X_{s}^{\prime}$ where $X_{t}$ is drawn
iid. from some distribution $\gamma$ with support in the unit ball
$\mathbb{B}_{d}$. Furthermore, let $\Sigma:=\mathbb{E}[X_{t}X_{t}^{\prime}]$
be the second moment matrix, and $B$ and $\delta>0$ be two positive constants.
Then there exist positive, universal constants $C_{1}$ and $C_{2}$ such that
$\lambda_{\min}(W_{t})\geq B$ with probability at least $1-2\delta$, as long
as
$\displaystyle
t\geq\left(\frac{C_{1}\sqrt{d}+C_{2}\sqrt{\log(\frac{1}{\delta})}}{\lambda_{\min}(\Sigma)}\right)^{2}+\frac{2B}{\lambda_{\min}(\Sigma)}+\mu_{M}+A_{2}(\lambda)\log\left(\frac{1}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{1}{\delta}\right)}.$
(23)
$\lambda$ is the $l_{2}$-spectral gap of the transition probability. We refer
the formal concepts and the definition of $l_{2}$-spectral gap to [JSF2018,
Section 2.2]. Proposition 3-1 is proved by utilizing the Berstein’s inequality
for general Markov chains ([JSF2018, Theorem 1.1]) and Proposition 3-2 is
proved by applying union bound.
###### Proof.
Proof of Proposition 3. Recall $G_{t}=\sum_{s=1}^{t-1}\mathbb{I}\\{D_{s}\geq
t-s\\}$. Define $f_{i}(D_{i})=\mathbb{I}\\{D_{s}\geq t-s\\}-p_{i}$ with
$p_{i}=\mathbb{P}(D_{s}\geq t-s)$. Then $\mathbb{E}[f_{i}(D_{i})]=0$,
$\mathbb{V}[f_{i}(D_{i})]=p_{i}(1-p_{i})\leq p_{i}$, and
$\sum_{s=1}^{t-1}\mathbb{V}[f_{i}(D_{i})]\leq\sum_{s=1}^{t-1}p_{s}<\mu_{M}$.
From [JSF2018, Theorem 1.1], we have
$\displaystyle\mathbb{P}\left(\sum_{s=1}^{t-1}f_{i}(D_{i})>x\right)\leq\exp\left(-\frac{x^{2}}{2(A_{1}(\lambda)\mu_{M}+A_{2}(\lambda)x)}\right).$
(24)
Note that the right hand side in (24) is independent of $t$. Technically
speaking, this is because the summation of the variance
$\sum_{s=1}^{t-1}\mathbb{V}[f_{i}(D_{i})]$ is upper bounded by $\mu_{D}$ which
is independent of $t$. Therefore, Property 1 in Proposition 3 holds for any
$t\geq 1$.
Property 2 holds by the union bound and Property 1,
$\displaystyle\mathbb{P}\left(\max_{1\leq t\leq
T}G_{t}>\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)}\right)$
$\displaystyle\leq\sum_{t=1}^{T}\mathbb{P}\left(G_{t}>\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)}\right)\leq
T\times\frac{\delta}{T}.$
Therefore, the following holds with probability no smaller than $1-\delta$,
$\mathbb{P}\left(\max_{1\leq t\leq
T}G_{t}<\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)}\right).$
∎
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