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2024-09-04T02:54:59.095205 | 2020-03-11T10:12:35 | 2003.05196 | {
"authors": "Nicolas Riesterer, Daniel Brand, Marco Ragni",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26157",
"submitter": "Nicolas Riesterer",
"url": "https://arxiv.org/abs/2003.05196"
} | arxiv-papers | # Uncovering the Data-Related Limits of Human Reasoning Research:
An Analysis based on Recommender Systems
Nicolas Riesterer Cognitive Computation Lab, University of Freiburg, email:
<EMAIL_ADDRESS>Daniel Brand Cognitive Computation Lab, University
of Freiburg, email<EMAIL_ADDRESS>Marco Ragni Cognitive
Computation Lab, University of Freiburg, email<EMAIL_ADDRESS>
###### Abstract
Understanding the fundamentals of human reasoning is central to the
development of any system built to closely interact with humans. Cognitive
science pursues the goal of modeling human-like intelligence from a theory-
driven perspective with a strong focus on explainability. Syllogistic
reasoning as one of the core domains of human reasoning research has seen a
surge of computational models being developed over the last years. However,
recent analyses of models’ predictive performances revealed a stagnation in
improvement. We believe that most of the problems encountered in cognitive
science are not due to the specific models that have been developed but can be
traced back to the peculiarities of behavioral data instead.
Therefore, we investigate potential data-related reasons for the problems in
human reasoning research by comparing model performances on human and
artificially generated datasets. In particular, we apply collaborative
filtering recommenders to investigate the adversarial effects of
inconsistencies and noise in data and illustrate the potential for data-driven
methods in a field of research predominantly concerned with gaining high-level
theoretical insight into a domain.
Our work (i) provides insight into the levels of noise to be expected from
human responses in reasoning data, (ii) uncovers evidence for an upper-bound
of performance that is close to being reached urging for an extension of the
modeling task, and (iii) introduces the tools and presents initial results to
pioneer a new paradigm for investigating and modeling reasoning focusing on
predicting responses for individual human reasoners.
## 1 Introduction
The goal of human-level AI is currently approached from two directions:
solving tasks with a performance similar to or even exceeding the one of
humans [3], and understanding human cognition to a level that allows for an
application to real-world problems [18]. While the first direction has seen
major progress mainly fueled by the development of high-performant data-driven
methods over the course of the last years, the second lags behind.
Gaining insight into the processes underlying human cognition is the core
focus of cognitive science, the inter-disciplinary research area at the
junction of artificial intelligence, cognitive psychology, and neuroscience.
Currently, this field of research is focused mainly on the psychological
questions related to cognition. As a consequence, there is a distinct lack of
readily available computational models developed for use in real-world
applications such as human-like assistant systems. In this article we propose
the use of methods from information retrieval to perform data analyses for
investigating the remaining potential in modeling human cognition. In
particular, we apply models from the family of collaborative filtering
recommendation systems (for introduction see [15, 12]) to re-evaluate the
theory-focused state of the art and illustrate the potential of a more data-
driven approach to modeling human reasoning in one of its core domains:
syllogistic reasoning.
Syllogisms are one of the core domains of human reasoning research. They are
concerned with categorical assertions of the form “All A are B; All B are C”
consisting of two premises featuring a quantifier out of “All”, “Some”, “Some
not”, and “No”, and three terms, A, B, and C, two of which are uniquely tied
to their respective premise. Depending on the arrangement of terms, the
syllogism is said to be in one of four figures (_A-B;B-C_ , _B-A;C-B_ ,
_B-A;B-C_ , _A-B;C-B_).
When presented with syllogistic problems, the goal is to determine the
logically valid conclusion out of the nine possibilities constructed by
relating the two end terms of the premises via one of the four quantifiers
(eight options), or to respond with “No Valid Conclusion” (NVC) if nothing
else can be concluded. Featuring 64 distinct problems with nine conclusion
options, the domain is well-defined, small enough to gain interpretable
insight, but more detailed than most of its alternatives such as conditional
reasoning (“If it rains, then the street is wet; It rains”).
The domain of human syllogistic reasoning has seen an increase of interest in
modeling over the last years. A meta-analysis compiled a list consisting of
twelve accounts trying to provide explanations for the behavior of humans
which differs drastically from formal logics [5]. However, since cognitive
science follows a strongly theory-driven perspective on modeling, the focus of
interest often rests on analyzing and comparing specific properties of models
instead of their general predictive performance. Recent work identified a lack
of predictive accuracy of cognitive models which raises concerns about their
general expressiveness [13].
In this article, we briefly analyze the predictive accuracy of the state of
the art in modeling human syllogistic reasoning and compare the results with
data-driven models. In particular, we apply collaborative filtering-based
recommender systems which exhibit properties making them promising tools for
cognitive research. We leverage these properties to test structural
assumptions about the syllogistic domain to analyze the data’s information
content and the impact of noise on model performance. Finally, the
implications for modeling reasoning and working with human data in general are
discussed and ideas for improving the cognitive modeling problem are proposed.
## 2 Related Work
Computational modeling has become one of the prime choices for formalizing
knowledge and understanding about a domain of interest. By implementing
intuition and assumptions into computationally tractable models, competing
theories can be evaluated, progress in the understanding of a domain can be
monitored, and finally, real-world applications can be solved [9].
The field of syllogistic reasoning has seen a rise of computational models.
From initially only verbally described abstract theories [16], a recent meta-
analysis compiled a list of twelve theoretical accounts for syllogistic
reasoning, seven of which could be specified via tables relating syllogistic
problems with sets of possible conclusions [5]. While these prediction tables
still are far off fully specified implementations of the theoretical
foundations, they can serve as a starting point for conducting model
evaluation and comparison. The authors of the meta-analysis used the
prediction data in order to determine strengths and weaknesses of the
competing approaches when compared to dichotomized human response data via
classification metrics (hits, misses, and false alarms). They found that while
the approaches all exhibit distinct properties with respect to predictive
precision, no single model could be determined as an overall winner.
A recent analysis focusing on combining individual models’ strengths while
avoiding their weaknesses took the evaluation of models one step further by
avoiding the data aggregation step and focusing on the performance obtained
from querying models for individual response predictions instead [13]. Their
work revealed substantial lack of predictive performance of state-of-the-art
models for syllogistic reasoning. Simultaneously, the authors demonstrated
that data-driven modeling in form of a predictor portfolio, could be applied
successfully to increase the predictive accuracy on the task.
Information systems and machine learning as the fields concerned with data-
driven model construction and optimization have seen an astonishing increase
in popularity over the last years. Parts of this success have been due to an
integration of features related to personalities of individuals [10]. Still,
even though they share methods such as clustering, principle component
analyses or mixed models, they have yet to enter the domain of cognitive
research. Collaborative filtering as one of the default methods in the field
of recommender systems has been successfully applied to model human reasoning
before [6]. What makes this kind of memory-based collaborative filtering
approaches promising for cognitive research in general is their high
predictive capabilities paired with the similarity to the core assumption of
cognitive science, that groups of people share similar reasoning patterns.
Since recommendations are extracted from similarities between different
features of the data or the users themselves, they allow both for an analysis
of the data underlying the recommendation process, and an analysis of high-
level theoretical assumptions which can be integrated directly into the
model’s algorithmic structure (e.g., the integration of user personality [10,
7, 4]).
The following sections contrast the models from cognitive science with
collaborative filtering-based approaches in a general benchmarking setting for
syllogistic reasoning based on predictive accuracy.
## 3 Benchmarking Syllogistic Models
Figure 1: Overview over the model evaluation procedure. The benchmark selects
a task which is fed to the model in order to obtain a prediction (black
arrows). Simultaneously, by being based on experimental data, it simulates
querying a human for a response (red arrows). After obtaining the model
prediction, the true response is revealed to the model in an adaption step.
The true (human) and model conclusions are collected and ultimately evaluated
in terms of predictive accuracy.
To gain an overview over the state-of-the-art’s performance in the prediction
task, we performed a benchmark analysis using data obtained from an online
experiment conducted on Amazon Mechanical Turk consisting of $139$ reasoners
which responded to all $64$ syllogisms. Evaluations were computed relying on
leave-one-out crossvalidation, i.e., by testing one reasoner and supplying the
remaining $138$ as training data.
The model evaluation procedure is inspired by a live prediction scenario where
model predictions are retrieved simultaneously to the human reasoner selecting
a conclusion. This is illustrated by Figure 1. In particular, our benchmark
simulates this experiment by passing the tasks to a model generating
predictions (black arrows). After a prediction is obtained, the model is
supplied with the true response obtained from the human reasoner (red arrows).
This allows models to perform an adaptation to an individual’s reasoning
processes. Predictions and true responses are collected and finally compared
to compute the predictive accuracy as the average number of hits.
We included the cognitive models (matching, atmosphere, probability heuristics
model, PHM; mental models theory, MMT; PSYCOP, conversion, verbal models)
supplied with the meta-analysis on syllogistic reasoning by extracting the
prediction tables [5]. Additionally, we included two baseline models, _Random_
and _MFA_. Random represents a lower bound of predictive performance defined
by the strategy that always picks a random response out of the nine options.
MFA denotes the most-frequent answer strategy which generates predictions by
responding with the conclusion most frequently occurring in the training data.
Finally, we included two variants of memory-based collaborative filtering. The
user-based variant (UBCF) generates its prediction based on the responses of
other users weighted by the similarity computed as the number of matching
responses. The item-based variant (IBCF) compiles an item x item matrix
$\mathbf{M}$ of corresponding responses (i.e., who responded with $x$ to
syllogism $A$ also responded with $y$ to $B$) and a user vector $\mathbf{u}$
consisting of the user’s previous responses. The prediction is generated by
selecting the highest-rated response for a syllogism from the result of the
matrix-vector multiplication $\mathbf{M}\times\mathbf{u}$.
Figure 2: Accuracies of models for human syllogistic reasoning. The plot
includes cognitive models based on prediction tables reported by a recent
meta-analysis by Khemlani & Johnson-Laird (2012; Probability Heuristics Model,
PHM; Mental Models Theory, MMT; Matching, Atmosphere, PSYCOP, Conversion,
VerbalModels), baseline models (Most Frequent Answer, MFA; Random), as well as
user-based collaborative filtering (UBCF) and item-based collaborative
filtering (IBCF).
Figure 2 depicts the result of the benchmark analysis. The image highlights
the difference between cognitive models and the recommenders. This is not too
surprising since most cognitive models were not introduced with predictive
performance in mind. They were originally based on some statistical effect
(e.g., illicit conversion, a bias towards misinterpreting the direction of the
input premises [1]) or a high-level cognitive theory (e.g., PSYCOP which
assumes that reasoning is the result of interactions between different mental
rules [14]) and are analyzed with respect to their qualities in reproducing
aggregate effects of data. Still, the gap between cognitive models and data-
driven approaches calls for a re-thinking of the goals of cognitive science.
If the high-level insight cannot be integrated into successful models, their
analysis is of limited use for advancing the understanding of human cognition.
When observing the plot, special emphasis should be placed on MFA, the
baseline model responding with the most-frequent answer of the training
dataset. In terms of data-driven approaches, the MFA represents an upper bound
of performance for models which do not take inter-individual differences into
consideration. Since the cognitive models we considered for our analysis lack
computational mechansisms for handling differences between reasoners, they are
not expected to score higher than the $45\%$ achieved by MFA. In general,
models can only hope to score higher if they rely on an active adaption to
information about an individual’s reasoning processes such as previous
responses or other personality traits known to influence cognition such as
working memory capacity [17].
Being defined on an explicit database of information, collaborative filtering
is an ideal tool for data analysis and modeling. They allow researchers to
directly incorporate knowledge about the domain into the recommendation
process and thereby to directly evaluate the value of findings in rigorous
modeling scenarios. However, since this transformation of abstract findings is
out of scope for this article and remains a challenge for future research, we
do not focus on proposing an optimal recommender. We rather intend to
highlight the method’s potential for future research in the domain by
illustrating the levels of performance standard domain-agnostic
implementations can achieve.
Our benchmark shows that even domain-agnostic recommenders outperform
cognitive models. Still, they do not manage to significantly surpass MFA. This
could mean (i) that these models fail to recognize the reasoning strategies
underlying the data, or (ii) that human reasoning is too irregular, i.e., too
prone to uncontrollable noise for the approaches to succeed. In the following
section we analyze artificially generated data in order to gain further
information about the reasons behind the limited predictive performance of
syllogistic models.
## 4 Simulation Analysis
A core assumption of cognitive science is that reasoning is the result of
different processes [2]. Depending on the individual state of the reasoner
(e.g., previous experience or concentration), thorough inferences based on the
rules underlying formal logics can be conducted or simple heuristic rules can
be applied to reach a conclusion. Consequently, when assessing reasoning data,
it is usually assumed that the data at hand is the result of multiple
interleaved strategies which need to be disentangled in order to allow for an
interpretable analysis.
### 4.1 Entropy Analysis
High information content in data is essential for the success of data-driven
methods. If the data consists mostly of random effects with little structure,
patterns cannot be recognized to base future predictions on.
A common measure of information is the Shannon entropy $S$:
$S=-\sum_{i}p_{i}\log_{2}p_{i}$
Entropy can be understood as a measure of unpredictability of a state defined
via the probabilities $p_{i}$. In the case of syllogisms, entropy has
previously been applied to quantify the difficulty of the 64 problems [5].
Higher entropy results from a more uniform spread of probability mass over the
nine conclusion options and thus serves as an indicant for a more difficult
task.
Figure 3: Relationship between syllogistic problems of varying entropy and
model performances. Dotted lines represent interpolations between the data
points.
Figure 3 depicts the entropies of syllogistic problems with corresponding
model performances. It shows a distinct gap in performance between the
recommenders (IBCF, UBCF) and the remaining models. For low entropies, the
recommenders are able to leverage the information encoded in the data
resulting in high predictive accuracies. For higher entropies they are unable
to maintain their initial distance to the cognitive models which are much more
stable overall.
Entropy in reasoning data can originate from (i) inconsistencies in the
response behavior of individual human reasoners or (ii) interactions between
independent reasoning strategies. The former point is a general issue of
psychological and cognitive research since human participants are prone to
lose attention due to boredom or fatigue. As a result, inconsistent and even
conflicting data of single individuals can emerge [11]. Especially for
collaborative filtering-based models this introduces substantial problems
since users might not even be useful predictors for themselves. The latter
point is a core challenge of cognitive science. Since reasoners differ with
respect to their levels of education and experience with the task [8],
recorded datasets are likely to be the result of a large number of individual
strategies. For modeling purposes, the implications of both points differ
greatly. Since inconsistencies due to lack of attention lead to behavior
similar to guessing, it is unlikely for models to capture these effects by
relying on behavioral data alone. Interactions between different strategies,
on the other hand, are much more likely to be disentangled given additional
insight into the domains and inter-individual differences between reasoners.
Unfortunately, though, with the limited features currently contained in
reasoning datasets, i.e., the responses, it is impossible to safely attribute
the entropy of the data to either point. In the following sections, we
therefore focus on collaborative filtering to shed light on the general
capabilities of data-driven models in trying to uncover additional information
about the problems of the domain.
### 4.2 Strategy Simulation
Even though data-driven recommenders are able to achieve higher accuracies
when compared to cognitive models, they are still far from perfectly
predicting an individual reasoner. To investigate the remaining potential in
the syllogistic domain, we need to gain an understanding of potential issues
with the data.
This second analysis considers artificial data with controlled levels of
noise. Four of the cognitive models from the literature (Atmosphere, Matching,
First-Order Logic, Conversion) were implemented and assigned to one of the
four figures, respectively. By permuting the model-figure assignment and
generating the corresponding response data we obtain $256$ artificial
reasoners featuring interleaving strategies. The informativeness of this data
is reduced by additionally introducing varying levels of random noise obtained
from replacing conclusions with a random choice out of the nine conclusion
options. With increasing levels of noise, the data should be less accessible
for data-driven models.
Figure 4: Strategy reconstruction performance of models based on artificial
reasoning data with different levels of noise added by replacing a certain
proportion of responses with a random choice from the nine conclusion options.
The left and right images contrast performance with the raw noise proportions
and entropies, respectively.
Figure 4 depicts the performance of the baseline and data-driven models on the
artificial data. The left image plots the different noise levels against
predictive accuracies. It shows that a decrease in response consistency has
drastic effects on the models’ capabilities to correctly predict responses due
to the lack of information contained in the training data. The nearly linear
relationship between the levels of noise and performance suggests that the
models are stable in performance given the amount of reconstructable
information. Consequently, they allow for a data-analytic assessment of
“noise” in the data they are supplied with. In the case of syllogistic
reasoning this means that close to $50\%$ of the data would effectively be
indistinguishable from random noise. Explanations for this could be numerous
ranging from too little data with respect to the number of possible reasoning
strategies, over a lack of descriptive features, to guessing-like behavior,
i.e. strategy-less decision-making on the side of study participants. The
right image of Figure 4 presents a different perspective on the impact of
noisy data by computing corresponding entropies. Again, it shows that entropy
is tightly linked to predictive accuracies.
By comparison with the Figure 3, some interesting conclusions can be drawn. In
general, IBCF scores lower on the artificial data than on human data. Since
IBCF is based on item-item dependencies, it is unable to directly exploit
structural patterns of the data. It bases its predictions solely on
information about “reasoners responding x to problem A also respond with y to
problem B”. Higher performance on the human data therefore suggests the
existence of preferential clusters of reasoners which exhibit similar response
behavior. Since the artificial data does not feature such groups but puts more
focus on the structural information by evenly distributing the permutations of
model-figure combinations, IBCF is at a disadvantage. While we cannot formally
attribute the entropy observed in the human data to inconsistencies due to
random noise, or varying overlap between distinct reasoning strategies, the
properties of IBCF suggest the existence of key responses or groups exhibiting
similar research patterns in the data which allow the method to perform some
form of clustering to boost its accuracy. This can be interpreted as soft
evidence for the second hypothesis, that the current problem with modeling
syllogistic reasoning stems from the fact that features allowing for a
disentanglement of strategies are scarce.
A possibility to overcome these problems for the short-term progress of the
field is by explicitly integrating assumptions about the structural properties
of the data into models. If accurate enough, they should be able to boost
models’ capabilities to disentangle the overlapping strategies and allow for a
general improvement of performance. Additionaly, the converse is true: if
high-level theoretical assumptions lead to a significant improvement of the
predictor, the theory is on the right track.
### 4.3 Potential for Better Predictions
It appears as if a lack of information preventing the identification of
strategies limits the potential of modeling in the domain of syllogistic
reasoning. In general, there are two options to tackle this problem: improving
models and extending the problem domain.
There exist many possibilities to increase the predictive capabilities of
models. On the one hand, additional features known for influencing reasoning
patterns such as education [8] or working memory [17] can be integrated into
the data to boost a model’s ability to identify patterns. On the other hand,
the model can be supplied with background information about the problem
domain. Since cognitive science has a history of in-depth data analysis there
is a lot of potential for integrating theoretical findings into models. We
propose the use of collaborative filtering as an accessible tool for cognitive
scientists to transform abstract insight into testable models.
Figure 5 illustrates the potential of recommenders for insight-driven research
by contrasting item-based collaborative filtering (IBCF) and user-based
collaborative filtering (UBCF) with variants of them tuned to the structure of
the artificially generated data, i.e., the observation that syllogisms of the
same figure rely on the same inference mechanism. The plot highlights that
this additional information about the data is able to push both IBCF and UBCF
far beyond their initial performance. Especially for IBCF, the explicit
integration of the structural foundation of the data lifts its performance to
the same levels of UBCF. The gap between the domain-agnostic and tuned
variants remains clearly visible even for high levels of noise. Even though
explicit information about the structure of human data can only be
approximated from theoretical insight into the domain, this shows that
recommenders would be a useful tool for assessing the quality of assumptions.
The second option to improve modeling of human reasoning is to extend the
domain in question. If information about individuals is accumulated even
across the borders of reasoning domains, models have more data to recognize
descriptive patterns in. Additionally, it is possible to include distinctive
background information about individuals such as personality traits. This
approach has proven to boost performance in recommendation scenarios before
and is likely to generalize to the reasoning domain [4, 7]. However, since the
extension of the domain is out of scope for this work, we leave this idea open
for future research.
For research on human reasoning this final analysis shows that there exist
data-driven methods which benefit from the integration of the kind of
information that is usually uncovered in cognitive science and psychology. By
integrating correlative insight into these kinds of models, the value of the
findings can be directly assessed in benchmarking evaluations. Paired with
more informative problem domains obtained from a unification of multiple
domains of reasoning, or the addition of personality features about individual
reasoners, data-driven and theory-driven research can collaborate to overcome
the distance between the current state of the art and the goal of human-level
AI.
## 5 Conclusion
Cognitive models for human syllogistic reasoning achieve unsatisfying
accuracies when applied in a prediction setting. While the reasons for this
could be numerous, it is interesting to see that data-driven recommenders
based on collaborative filtering do not perform substantially better on an
absolute scale. This raises concerns about the data foundation of reasoning
research which is usually composed solely of reasoning problems along with the
corresponding human responses.
Our results obtained from comparison with artificially generated data suggest
that data-driven models are unable to identify and successfully exploit
patterns in the structure of human reasoning datasets when, in theory, they
should be able to. The two most likely explanations for this are noise in form
of inconsistencies in the response behavior of humans, or a lack of
distinctive features preventing data-driven approaches to identify the
patterns required for successful predictions.
Figure 5: Comparison of item-based collaborative filtering (IBCF) and user-
based collaborative filtering (UBCF) variants on artificially generated
reasoning data. Fit-versions denote implementations where structural
properties of the artificial data were actively integrated.
In order to advance the predictive performance to levels which are relevant
for applications in the era of human-level AI, reasoning research needs to
address its current shortcomings. Potential solutions include the improvement
of models by a better integration of domain-specific insight as well as an
active consideration of inter-individual differences, and the extension of the
task for example by including other domains of reasoning, recording more
comprehensive datasets, and leaving behind the current focus on data
aggregation.
For integrating insight into models, we propose collaborative filtering
recommenders as a general-purpose research method. On a technical level, they
are easy to implement and understand, and outperform the current state of the
art even in their domain-agnostic form. By integrating additional information
about the domain (even if just on the level of correlations by weighting the
dependencies between different features of the data), they allow for a
transformation of abstract hypotheses into testable assumptions for modeling.
Consequently, recommenders exhibit useful properties with respect to
comprehensibility, especially in contrast to other methods from machine
learning such as neural networks. As an example, they can naturally be applied
to clustering contexts where stereotypical users are sought after.
Generally, we see a need for an increased focus on predictive accuracies for
individual reasoners to allow more comprehensive benchmarking, to allow for a
more accessible interpretation of the results, and ultimately to enable the
models for real-world application. To facilitate this shift in perspective for
other researchers, we released the tools driving our predictive analysis as a
general benchmarking
framework111https://github.com/CognitiveComputationLab/ccobra. Only if the
different disciplines of cognitive science find together to compete in
modeling on unified informative domains using expressive and standardized
metrics such as predictive performance, will human reasoning enter a level of
progress relevant for human-level AI applications.
This paper was supported by DFG grants RA 1934/3-1, RA 1934/2-1 and RA
1934/4-1 to MR.
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|
2024-09-04T02:54:59.108757 | 2020-03-11T10:42:40 | 2003.05208 | {
"authors": "Mohammad A. Hoque, Ashwin Rao, Sasu Tarkoma",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26158",
"submitter": "Mohammad Ashraful Hoque",
"url": "https://arxiv.org/abs/2003.05208"
} | arxiv-papers | # In Situ Network and Application Performance Measurement on Android Devices
and the Imperfections
Mohammad A. Hoque University of Helsinki, Finland
<EMAIL_ADDRESS>, Ashwin Rao University of Helsinki, Finland
<EMAIL_ADDRESS>and Sasu Tarkoma University of Helsinki, Finland
<EMAIL_ADDRESS>
###### Abstract.
Understanding network and application performance are essential for debugging,
improving user experience, and performance comparison. Meanwhile, modern
mobile systems are optimized for energy-efficient computation and
communications that may limit the performance of network and applications. In
recent years, several tools have emerged that analyze network performance of
mobile applications in situ with the help of the VPN service. There is a
limited understanding of how these measurement tools and system optimizations
affect the network and application performance. In this study, we first
demonstrate that mobile systems employ energy-aware system hardware tuning,
which affects application performance and network throughput. We next show
that the VPN-based application performance measurement tools, such as Lumen,
PrivacyGuard, and Video Optimizer, aid in ambiguous network performance
measurements and degrade the application performance. Our findings suggest
that sound application and network performance measurement on Android devices
requires a good understanding of the device, networks, measurement tools, and
applications.
## 1\. Introduction
In situ Internet traffic measurement tools, such as Video Optimizer (VoP)
(Qian:2011:PRU, ), Lumen (Razaghpanah:2017, ), PrivacyGuard (PvG)
(Song:2015:PVP, ), and MopEye (Wu:2017:MOM:3154690, ), are essential for
debugging, improving user experience, and performance comparison of mobile
applications. The alternative is rooting the device and using _tcpdump_ for
offline analysis.
The above traffic measurement tools shed light on the network and application
performance. However, they may also contribute to imperfect and ambiguous
results, as we might measure something which we do not intend to measure.
Studying the sources of these imperfections is vital to calibrate the
measurement procedures and to improve the tools. At present, there is a
limited understanding of the impact of in situ mobile Internet traffic
measurement tools and how device hardware optimization affects the network and
application performance.
In this work, we quantify the performance impact of system hardware
optimization and also evaluate the impact of VoP, Lumen, and PvG on network
performance metrics, and application traffic. We focus on these three
applications, as they exemplify state-of-the-art traffic measurement and
analysis tools. These tools have similar designs and use the Android VPN
interface. However, they do not route the traffic to a remote VPN server. VoP
(vop, ), formerly known as ARO (Qian:2011:PRU, ), is a popular open-source
tool for collecting traffic from mobile devices without rooting the device,
and it also enables various diagnosis and optimization of applications,
network, CPU and GPU (vop, ) through offline analysis. In contrast, Lumen and
PvG are two online traffic analysis tool helping users to find privacy leaking
incidents. Lumen also provides insights on the TLS usage of mobile
applications (Razaghpanah:2017, ), the CDN usage by mobile applications
(8485872, ), and the DNS (Almeida2017DissectingDS, ). MopEye is another
similar application. It is currently unavailable in the Google Play Store and
also in popular source code hosting websites, such as GitHub.
This article investigates the imperfections in traffic measurements on Android
devices due to system optimization and in situ traffic measurement tools. We
demonstrate that sound Internet traffic measurement requires a thorough
understanding of the device, tools, and applications. Note that we do not aim
to establish whether a particular tool is the best or worst. Our key
observations are as follows.
_(1)_ Mobile systems employ CPU and WiFi transmit power optimization triggered
by the battery level. We observe that the CPU optimization techniques, such as
CPU hot-plugging and dynamic frequency scaling, mostly affect network I/O,
while WiFi optimization, i.e., dynamic modulation scheme, affects the uplink
throughput. These optimizations deteriorate application performance and
network throughput. Charging the device, when the battery level is below 20%,
does not improve the network performance. Therefore, one must be aware of the
adaptive performance characteristics of mobile devices while conducting
experiments (Section 2).
_(2)_ Although it is expected that VPN-based tools would provide degraded
network performance as the packets spend more time on the device
(Qian:2011:PRU, ; Razaghpanah:2017, ; Song:2015:PVP, ), we may estimate
ambiguous latency and throughput in the presence of the VPN-based tools. For
example, in the presence of PvG, SpeedCheck (speedcheck, ) estimates on-device
latency instead of the network latency. Similarly, VoP doubles the uplink
throughput estimates. The sources of these ambiguities are the implementation
of the measurement tools, as we present in Section 3. VoP also delays the
outgoing traffic, and PvG delays the incoming traffic. Therefore, to avoid
such pitfalls in network and application performance measurements, one must
have a good understanding of these applications and tools.
_(3)_ Furthermore, all these VPN-based applications fail to apply the
application intended optimization through socket options and thus affect the
application performance, as we demonstrate for the outgoing TCP traffic in
Section 3.
Finally, we summarize the sources of the above ambiguous or imperfect
measurement results (Section 4).
Figure 1. Impact of battery level. We consider two battery level (L) ranges,
L$\leq$20% & L$>$20%, on Nexus 6 over WiFi (W) and LTE (4G).
## 2\. Impact of System Optimization
Android devices may come with advanced CPU governors that save energy by hot
plugging and unplugging of CPU cores, as supported by modern Linux kernels
(cpuhotplug, ). Apart from workload characteristics, the devices may also
consider the status of the battery to employ the CPU cores. We look into the
impact of such off-the-shelf system optimization on network latency and
throughput on Nexus 6.
During our measurements with Nexus 6, we have found that two of the four cores
remain offline when the battery discharges to below 20%, and the active cores
operate at the maximum frequency of 1.73 GHz. When the battery level is above
20%, all the four cores become active, and their maximum operating frequency
increases to 2.65 GHz. Therefore, the battery level also prompts dynamic
frequency scaling.
We performed the following measurements to quantify the impact of this
optimization on the network traffic characteristics. Specifically, we used
SpeedCheck (speedcheck, ) (paid) and measured the latency and throughput on
Nexus 6 (Android 7.0) when the battery levels were above 20% and below 20%. We
performed the measurements using both WiFi and LTE. Each of the above four
scenarios was repeated ten times, and the results are presented in Figure 1.
Figure 1 shows that while hot unplugging of CPU cores on Android has a
negligible impact on the latency, its impacts on throughput is significant.
The availability of additional CPU cores, when the battery level is above 20%,
improves the I/O performance across the two access technologies, WiFi and LTE.
Furthermore, WiFi uplink throughput improves almost four times when the
battery level is above 20% compared to when it is below 20%. The closer
inspections of the MAC layer frames revealed that WiFi radio of the Nexus 6
switches from _802.11ac_ to _802.11g_ mode when the battery level drops below
20%. These performance limiting optimizations also affected the device
responsiveness for various applications, such as browsing and streaming. This
also implies that modern Android devices adapt the physical layer mechanisms
similar to the iOS
devices111https://www.forbes.com/sites/ewanspence/2017/12/20/apple-iphone-
kill-switch-ios-degrade-cripple-performance-battery/ to avoid unexpected
shutdown of the devices (8720247, ) and to improve battery life.
Figure 2. Impact of battery level on LTE modulation scheme. These snapshots
are from a single uplink and downlink throughput measurement.
Figure 1 also depicts that the downloading speed of SpeedCheck over LTE
doubles when the battery level is higher than 20%. Similar to WiFi, we further
looked into the physical layer modulation scheme used by the mobile device in
the LTE network. We rooted Nexus 6 and installed Network Signal Guru
(netsiguru, ) that samples LTE physical layer parameters after every 500 ms.
Figure 2 shows that the modulation schemes were always 16QAM (Quadrature
Amplitude Modulation) and 64QAM for uplink and downlink, respectively, during
the throughput measurements. The other attributes in the figure are discussed
in section 6. Nexus 6 employs three optimization techniques, triggered by the
battery level, which affect the network and application performance. Charging
the device, when the battery level is below 20%, does not improve the
throughput either on WiFi or LTE and application performance. The optimization
may vary from device to device.
## 3\. Impact of Measurement Tools
Figure 3. The system components of VoP, Lumen, and PvG for Android.The newly
created sockets are protected so that the Forwarder generated packets are not
in a loop. Figure 4. Impact on LTE network latency and throughput. We used
SpeedCheck and SpeedTest on Nexus 6 in the presence of Lumen (Lum.), VoP, PvG,
and Baseline, i.e., without any localhost VPN.
### 3.1. In-situ Traffic Measurement Tools
The forwarder and the packet inspector are two components of the VPN-based in
situ traffic measurement tools exemplified by VoP, Lumen, and PvG, as shown in
Figure 3.
The primary role of the forwarder is to forward (i) the packets received from
Android applications to the Internet, and (ii) the packets received from the
Internet to the Android applications. The forwarder also copies those packets
to the inspection queue to isolate traffic analysis from the path of the
packet.
The forwarder essentially creates a new TCP socket on seeing a TCP SYN packet
from the VPN interface. The forwarder in Lumen and VoP establish a socket
connection with the remote server using connect() API before sending SYN-ACK
to the application. PvG, on the other hand, establishes socket connection
after replying with SYN-ACK. Later, we demonstrate how these implementations
affect network performance measurements. The forwarder creates a new UDP
socket when it detects a new UDP flow. These newly created sockets are
protected so that packets from the newly created flows do not loop the _tun_
interface (vpnprot, ).
A packet inspector is responsible for inspecting the packets in its queue. In
the case of Lumen and PvG, the packet inspector performs the privacy analysis
on the packets, whereas the VoP’s inspector sends packets to the desktop
application.
In the later sections, we quantify the impact of VoP, Lumen, and PvG on (a)
the network performance, and (b) the network characteristics of applications.
### 3.2. Addressing Biases
We took the following steps to ensure that the measurement results presented
in the upcoming sections are not the artifacts of misconfigured tools and the
measurement setup.
(i) Battery level. For the upcoming measurements, we ensured that the devices
had more than 80% charge. This is because mobile devices might restrict
resources based on the battery level, as we have shown in section 2.
(ii) Throughput throttling. VoP also offers to throttle downlink and uplink
traffic. All the measurements in this paper were conducted without any
throughput throttling.
(iii) Software Auto Update. During the experiments, application and the auto
system updates were disabled on mobile devices.
(iv) Advertisements. We have purchased without ad subscriptions of SpeedCheck
and SpeedTest to avoid any biases caused by the free versions.
(a) Baseline
(b) VoP
(c) Lumen
Figure 5. Inter-packets gaps of the VoIP applications. Baseline refers to the
measurements without any localhost VPN.
### 3.3. Impact on Network Performance
This section explores the network performance using SpeedCheck (speedcheck, )
and SpeedTest (speedtest, ). These two applications work as the traffic load
generator without any VPN-based tools and in the presence of the listed VPN
applications. Without any VPN scenario gives the baseline performance.
SpeedCheck connects to its servers in Germany, and SpeedTest connects to the
severs in the LTE operator network within a few kilometers from the mobile
device. The measurements were repeated ten times.
_(1) Latency._ Figure 4 (left) compares the network latency reported by two
applications in the presence of the VPN-based tools. From the _tcpdump_
traces, we have identified that SpeedTest uses 10-12 requests/responses of few
bytes (less than 100 Bytes) over a TCP connection to estimate the latency.
SpeedTest estimates the baseline latency of 16-18 ms. This is expected, as the
server was located at the operator’s network. It experiences 3-5 ms additional
latency in the presence of Lumen and PvG, whereas VoP increases the latency by
three-fold. This is due to the energy optimization strategy adopted by VoP,
which we discuss in the upcoming sections.
In contrast, SpeedCheck reports the median baseline network latency of about
45 ms. From the corresponding _tcpdump_ traces, we have identified 10 empty
and consecutive TCP flows (without any data exchange) for each latency
measurements. These flows suggest that SpeedCheck uses TCP connect() API to
measure the latency. Both VoP and Lumen increase the median latency
significantly. We speculate that these two take more time to set up new TCP
flows. However, SpeedCheck underestimates the latency in the presence of PvG,
which is the consequence of the sending SYN-ACK by the PvG forwarder before
the connection is established with the remote server, as discussed in Section
3.1.
_(2) Uplink Throughput._ Figure 4 (center) depicts that SpeedTest estimates
higher uplink baseline throughput, as the server is in the LTE operator
network. It uses multiple parallel TCP connections to estimate the throughput.
Both Lumen and PvG reduce the throughput of SpeedTest/SpeedCheck by half
compared to the baseline measurements. However, Lumen severely affects the
uplink throughput measurements of the SpeedCheck. It uses a single TCP
connection and sends a large amount of data. From an exception in the debug
log, we characterized that Lumen’s forwarder cannot handle such volume.
Interestingly, VoP doubles the uplink throughput of both applications.
_(3) Downlink Throughput._ Figure 4 (right) demonstrates that SpeedTest
measures similar downlink throughput in the presence of the VPN tools to the
baseline. Lumen aids the highest throughput measurements with SpeedCheck.
However, VoP and PvG degrade the throughput of SpeedCheck significantly.
The typical network measurement tools, such as SpeedCheck and SpeedTest, can
have different methods to estimate the latency and throughput. While their
baseline estimates are reasonable, their estimates vary according to the
implementation of the VPN tools.
### 3.4. Impact on Realtime Application (UDP)
In this section, we investigate the traffic from three realtime applications;
IMO, WhatsApp, and Skype. The versions of the apps used are presented in Table
1. While these applications fall into the broad category of messaging
applications, their varying traffic characteristics help us to study the
impact of the design of VoP and Lumen. We could not use these applications in
the presence of PvG in several trials. We used a rooted Nexus 6 (Android 7.0)
and a non-rooted LG G5 (Android 8.0) for these measurements.
These apps exchange bi-directional encrypted UDP traffic. The conversations
were two minutes long over LTE, and we ran 3 iterations in each of the
following scenarios. We investigate their inter-packet gaps and bitrates. As
the baseline, we initiated conversations between Nexus 6 and LG G5 using these
apps without VoP or Lumen and captured traffic using _tcpdump_ on Nexus 6. We
then repeated the experiments with VoP running on Nexus 6 and collected
traffic from VoP. Finally, we used Lumen. Since Lumen does not store traffic,
we captured traffic with _tcpdump_ on Nexus 6.
| Baseline | VoP | Lumen
---|---|---|---
Application | (in/out) | (in/out) | (in/out)
WhatsApp (v2.18) | 21/24 kbps | 23/16 kbps | 20/22 kbps
IMO (v9.8) | 14/15 kbps | 14/13 kbps | 13/14 kbps
Skype (v8.41) | 60/50 kbps | 55/44 kbps | 48/44 kbps
Table 1. Average bitrates of UDP traffic flows from VoIP applications.
_Baseline Results._ Figure 5(a) shows that IMO has the highest inter-packet
gaps, and Skype packets have the smallest gaps. These apps also have distinct
data rates with Skype having the highest data rate, as shown in Table 1.
_Impact of VoP._ Compared to the baseline packet-gaps in Figure 5(a), VoP
significantly alters the inter-packet gaps of outgoing UDP packets, as shown
in Figure 5(b). Most of the outgoing packets across all applications have an
inter-packet gap of about 100 ms. In contrast, the incoming packets have had
similar distributions to the baseline. This delay is similar to the latency
measurements with VoP discussed earlier. Table 1 shows that the outgoing data
rates of Skype and Whatsapp reduce significantly, which we speculate to be a
consequence of the delays introduced by VoP.
_Impact of Lumen._ Figure 5(c) shows that with Lumen the inter-packet gaps of
the outgoing packets are similar to the baseline measurements. Besides, the
applications experience similar bitrates to the baseline and when using Lumen
as shown in Table 1.
### 3.5. Impact on Realtime Application (TCP)
We used Periscope (v1.24) to study the impact of VoP and Lumen on realtime TCP
flows. Periscope’s live broadcast did not work in the presence of PvG.
Periscope broadcasts over LTE across three different scenarios. We capture
traffic on Nexus 6 using _tcpdump_ for baseline and Lumen scenarios.
Similar to our observations for UDP traffic, we observed 100 ms inter-packet
gap, as shown in Figure 6 (left). From the distribution of packet size in
Figure 6 (right) (collected by VoP), we notice that more than 70% packets
captured by VoP are larger than 1500 bytes. From Traffic traces, we have
identified that VoP creates packets of a maximum of 65549 bytes for Periscope,
and the uplink throughput measurements flow from SpeedCheck.
Figure 6. Properties of uplink Periscope TCP flows.
From the source code in Github, we have identified that VoP forwarder
implements the maximum segment of 65535 bytes for the TCP flows. It
accumulates traffic from the client application, and the segments reach the
maximum size very quickly with very high bitrate traffic. This also explains
how VoP aids in higher uplink throughput measurements presented in Section
3.3. Nevertheless, these massive TCP segments are eventually fragmented once
written to the socket. Lumen has a very negligible impact on packets.
### 3.6. Analysis with Socket Options
In this section, we investigate the performance of the VPN-based tools in
processing the flows with TCP_NODELAY (Nagel’s algorithm) socket option on
Nexus 6. We specifically look into this option, as it has a direct impact on
the local delay and thus affects the performance of web browsing and other
realtime applications, such as live broadcasting, crypto/stock exchange
applications, on mobile devices. We developed a separate traffic generating
application that creates two blocking TCP sockets enabled and disabled Nagle’s
algorithm. The application sends 1300 bytes data over LTE after every 20 ms to
a remote server at the university campus. The application also receives data
from the remote server after every 20 ms in separate TCP sessions.
Figure 7. Distributions of the outgoing packet gaps observed at the network
interface. Figure 8. Distributions of incoming packet gaps observed at the
network interface and application.
_Performance of VPN-based Tools._ Figure 7(a) compares the outgoing inter-
packet gap of the application flows; having Nagel’s algorithm enabled and
disabled. When Nagel’s algorithm is enabled, more than 70% of the packets sent
from the application have more than 20 ms delays at the network layer. In the
presence of VPN applications, disabling Nagel’s algorithm by the application
does not improve the delay compared to the baseline (Figure 7(b)).
Interestingly, VoP’s packet gap reduces, as it receives packets from the local
TCP/IP stack without delay. From traffic traces, we have identified that these
VPN-based tools do not disable Nagel’s algorithm while establishing socket
connections.
Figure 8 shows the performance of the VPN applications for incoming traffic.
The application receives data at almost similar gaps observed at the network
interface. However, in the presence of PvG, the application receives 40%
packets at late. The packet-gaps patterns suggest that it uses a fixed
interval to read the VPN interface similar to VoP.
The investigations in this section reveal that the VPN-based tools do not set
the TCP/IP socket options as intended by the other user applications.
Consequently, they can misguide the developers and degrade application
performance. For example, SpeedTest disables Nagel’s algorithm or sets the
TCP_NODELAY socket option to send tiny packets to measure the network latency.
Findings in this section explain the higher latency experienced by SpeedTest
in Section 3.3.
## 4\. Sources of Imperfection
Mobile system optimizations affect downlink and uplink throughput, whereas the
VPN-based tools mostly affect the uplink throughput and latency, i.e., they
mostly affect the outgoing traffic. In this section, we summarize the sources
of such measurement results.
_Energy-Aware Optimization._ Energy-aware system optimization can affect the
network performance by limiting the network I/O and by applying adaptive
modulation schemes. Therefore, it is wise to perform such measurements when
the battery is fully charged. VoP, Lumen, and PvG rely on different sleeping
techniques to optimize their energy usage. The additional latency introduced
by VoP on outgoing packets is the artifact of using a fixed sleep interval of
100 ms in the main VPN thread. This delay further contributes to large
outgoing packets for higher bitrate uplink traffic and energy consumption for
fragmentation. PvG also introduces a fixed delay for the incoming traffic.
Regardless, these delays affect not only the quality of the measurements but
also the quality of experience when using other user applications.
_Forwarder._ In situ VPN-based measurement tools are middleboxes that tap the
packets using the VPN interface. These applications, therefore, implement a
forwarder which primarily consists of three threads: the main VPN thread, and
two-socket reader/writer threads. The reader/writer threads continuously
iterate through a list of live sockets, which contributes to the delays. The
forwarder also implements a flow state machine for each flow and
constructs/de-constructs the packets. The implementation of the forwarder
affects the latency and throughput measurements. We have also shown that the
characteristics of the newly created flows and their packet headers might not
be the same as those generated by the applications. The reason is that the
socket options must be set before the connection establishment.
## 5\. Conclusions
In this preliminary work, we investigated the challenges in measuring network
performance in the presence of system optimizations and state-of-the-art
application performance measurement tools on Android devices. System
optimizations limit the performance of the hardware components and thus the
applications, which in turn result in confusing measurement results. It can be
argued that VoP is mostly for the developers, and therefore, incurring higher
delays should not a problem. Similarly, frequent massive content uploading is
rare, and 3-4 ms additional latency is acceptable. Nevertheless, these
imperfections can significantly affect the outcome of traffic measurement
studies. An acceptable latency also depends on the application type. A user
can benefit significantly from 1-millisecond latency improvement for the
financial and other realtime applications. Therefore, there is still room for
improvement in such tools. For instance, VoP and PvG can follow Lumen’s
adaptive sleeping algorithm for reducing the gaps in the outgoing and incoming
packets, respectively. All of them can adopt some default socket options to
mitigate the performance issues with the outgoing TCP traffic. Along with the
measurement tools, it is necessary to understand the presence of various
system optimization techniques which may affect network performance.
## References
* [1] SPEEDCHECK - Speed Test. https://play.google.com/store/apps/details?id=org.speedspot.speedanalytics. [Online; accessed 7-August-2019].
* [2] Speedtest by Ookla. https://play.google.com/store/apps/details?id=org.zwanoo.android.speedtest.gworld. [Online; accessed 11-August-2019].
* [3] VPN - Android Developers. https://developer.android.com/guide/topics/connectivity/vpn. [Online; accessed 23-January-2019].
* [4] AT&T Video Optimizer. https://developer.att.com/video-optimizer, 2019. [Online; accessed 7-August-2019].
* [5] Network Signal Guru. https://play.google.com/store/apps/details?id=com.qtrun.QuickTest, 2019\.
* [6] Mario Almeida, Alessandro Finamore, Diego Perino, Narseo Vallina-Rodriguez, and Matteo Varvello. Dissecting DNS Stakeholders in Mobile Networks. In Proceedings of CoNEXT ’17, pages 28–34. ACM, 2017.
* [7] Mohammad Kawser, Nafiz Imtiaz Bin Hamid, Md Nayeemul Hasan, M Shah Alam, and M Musfiqur Rahman. Downlink snr to cqi mapping for different multiple antenna techniques in lte. International Journal of Information and Electronics Engineering, 2:756–760, 09 2012.
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* [15] Jim Zyren. Overview of the 3GPP long term evolution physical layer. 01 2007.
## 6\. LTE Radio Resource Allocation
In LTE networks, Physical Resource Block (RB) is considered as the unit of the
radio resource. With 5 MHz bandwidth, there are 25 RBs. In an RB, there are 12
sub-carriers in the frequency domain. Each of the RBs can have either $7\times
12$ or $14\times 12$ resource elements (REs), where 7 and 14 are the symbols,
in the time domain, over 0.5 and 1 ms respectively using normal cyclic prefix
(CP) [15].
Now the amount of bits an RB can carry depends on the channel quality
indicator (CQI) notification from the UE. Essentially, each CQI maps to a
modulation and coding scheme according to Table 2. CQI indicates not only the
channel quality but also a device’s capability whether the device can receive
data of a particular modulation and coding scheme or not.
The equations to compute the number bits an RB can hold for a certain CQI, and
the number of RBs is required by an eNB to transmit a packet can be expressed
as the followings.
(1) $RB_{bits}=RE_{bits}\times n\times t_{s}\\\ =C_{CQI}\times M_{bits}\times
n\times t_{s}$
In equation1, $M_{bits}$ is the bits for a modulation scheme, $n$ is the
number of usable REs, and $t_{s}$ is the duration of time slot (0.5 or 1 ms).
(2) $RB_{n}=(PacketSize_{bits}+RLC_{bits}+MAC_{bits})/RE_{bits}$
CQI | Modulation | Real Bits ($N_{m}$) | $C_{CQI}=N/1024$
---|---|---|---
1 | QPSK | 78 | 0.0762
2 | QPSK | 120 | 0.1171
3 | QPSK | 193 | 0.1884
4 | QPSK | 308 | 0.3
5 | QPSK | 449 | 0.4384
6 | QPSK | 602 | 0.5879
7 | 16QAM | 378 | 0.3691
8 | 16QAM | 490 | 0.4785
9 | 16QAM | 616 | 0.6015
10 | 64QAM | 466 | 0.4550
11 | 64QAM | 567 | 0.5537
12 | 64QAM | 666 | 0.6503
13 | 64QAM | 772 | 0.7539
14 | 64QAM | 873 | 0.8525
15 | 64QAM | 948 | 0.9258
Table 2. Channel Quality Index (CQI), Modulation Scheme, and Coding Rate
mapping [7].
Figure 9 shows the usage of the Modulation Scheme and the number of resource
blocks for a large file download on Nexus 6 with CQI11. LTE supports QPSK,
16QAM, and 64QAM, i.e., each RE can carry a maximum of 2, 4, and 6 bits
accordingly. Let us consider the duration of 1 RB is 1 ms ($t_{s}$), and there
are 168 REs. Nevertheless, mostly 120 REs ($n$) are available for carrying
data traffic. For CQI11, the modulation scheme is 64QAM and the effective code
rate $C_{CQI}=N_{m}/1024=0.55$. Therefore, an RE can hold only,
$RE_{bits}=C_{CQI}\times M_{bits}=0.55\times 6$, 3.32 bits and an RB can hold
$n\times RE_{bits}=398$ bits.
Figure 9. LTE throughput and other network parameter observed on a mobile
device using Network Signaling Guru [5].
The number of RBs required for a packet in a downlink can be computed using
equation 2 by considering the additional bits for RLC and MAC headers.
However, the network may not allocate the RBs according to the CQI. It may
have other complex resource scheduling algorithms, as it has to deal with
various types of traffic and users. The number of uplink RBs also may vary.
## 7\. Application
⬇
1int val = 1;
2// Disabling Nagel’s Algorithm
3setsockopt(sockfd,SOL_TCP,TCP_NODELAY,&one,sizeof(one));
4if (connect(sockfd, &servaddr, sizeof(servaddr)) < 0)
5 LOGE("[***Server Connect Error***");
6for (int i = 0; i < 5000; i++) {
7 usleep(20000);
8 char *daat = rand_string(1300);
9 gettimeofday(&tv, NULL);
10 times[i] = (tv.tv_sec*1000000LL+tv.tv_usec)/1000;
11 n = write(sockfd,daat, 1300);
12 if (n < 0){
13 LOGE("Error sendto %s", strerror(errno));
14 break;
15 }
16}
Listing 1: TCP sending code with/without Nagel’s algorithm.
⬇
1int BUFSIZE = 4096;
2if (connect(sockfd, &servaddr, sizeof(servaddr)) < 0)
3 LOGE("[***Server Connect Error***");
4while (true) {
5 bzero(buf, BUFSIZE);
6 n = read(sockfd, buf, BUFSIZE);
7 if (n > 0) {
8 gettimeofday(&tvo, NULL);
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Listing 2: TCP receiving code.
|
2024-09-04T02:54:59.119667 | 2020-03-11T11:40:55 | 2003.05227 | {
"authors": "Oscar Rodriguez de Rivera, Antonio L\\'opez-Qu\\'ilez, Marta Blangiardo\n and Martyna Wasilewska",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26159",
"submitter": "Oscar Rodriguez De Rivera Ortega",
"url": "https://arxiv.org/abs/2003.05227"
} | arxiv-papers |
A spatio-temporal model to understand forest fires causality in Europe
Oscar Rodriguez de Rivera1,*, Antonio López-Quílez2, Marta Blangiardo3,
Martyna Wasilewska1
1 Statistical Ecology @ Kent, National Centre for Statistical Ecology. School
of Mathematics, Statistics and Actuarial Science, University of Kent.
Canterbury, UK.
2 Dept. Estadística i Investigació Operativa; Universitat de València.
Valencia, Spain.
2 Faculty of Medicine, School of Public Health; Imperial College of London.
London, UK.
*<EMAIL_ADDRESS>
###### Abstract
Forest fires are the outcome of a complex interaction between environmental
factors, topography and socioeconomic factors (Bedia _et al_ , 2014).
Therefore, understand causality and early prediction are crucial elements for
controlling such phenomenon and saving lives.
The aim of this study is to build spatio-temporal model to understand
causality of forest fires in Europe, at NUTS2 level between 2012 and 2016,
using environmental and socioeconomic variables. We have considered a disease
mapping approach, commonly used in small area studies to assess the spatial
pattern and to identify areas characterised by unusually high or low relative
risk.
_K_ eywords Hierarchical Bayesian models $\cdot$ disease mapping $\cdot$
integrated nested laplace approximation $\cdot$ forest fires $\cdot$ causality
$\cdot$ spatio-temporal model
## 1 Introduction
Nowadays, wildfires have become one of the most significant disturbances
worldwide (Flannigan _et al_ , 2009; Pechony and Shindell, 2010; Pausas _et
al_ , 2012; Cardil _et al_ , 2014; Boer _et al_ , 2017; Molina _et al_ ,
2018). The combination of a longer drought period and a higher woody biomass
and flammability of dominant species creates an environment conducive to fire
spread (Piñol _et al_ , 1998; Millán _et al_ , 2005). Furthermore, vegetation
pattern changes with the abandonment of traditional rural activities plays a
direct role in the increase of fire severity and ecological and economic fire
impacts (Flannigan _et al_ , 2009; Chuvieco _et al_ , 2014). Fire behavior
exceeds most frequently firefighting capabilities and fire agencies have
trouble in suppressing flames while providing safety for both firefighters and
citizens (Werth _et al_ , 2016).
Many areas across the world have seen a rise in extreme fires in recent years.
Those include South America and southern and western Europe. They also include
unexpected places above the Arctic Circle, like the fires in Sweden during the
summer of 2018 (de Groot _et al_ , 2013; European Commission, 2019).
Extreme fire events, which are also referred to as “megafires”, are becoming
frequent on a global scale; recent fires in Portugal, Greece, Amazone and
other areas confirm this fact. There is not complete agreement on the term
“megafires”, which often refers to catastrophic fire events in terms of human
casualties, economic losses or both (San-Miguel-Ayanz _et al_ , 2013b).
Climate change will reduce fuel moisture levels from present values around the
Mediterranean region and the region will become drier, increasing the weather-
driven danger of forest fires. The countries in highest danger are Spain,
Portugal, Turkey, Greece, parts of central and southern Italy, Mediterranean
France and the coastal region of the Balkans, according to recent research of
the Joint Research Centre (JRC) (De Rigo _et al_ , 2017).
Most international reports on biomass burning recognize the importance of the
human factors in fire occurrence (Food, 2007). Although fire is a natural
factor in many ecosystems, human activities play a critical role in altering
natural fire conditions, either by increasing ignitions (Leone _et al_ ,
2003), or by suppressing natural fires (Johnson _et al_ , 2001; Keeley _et al_
, 1999). Both factors are contradictory, and act mainly through the mixture of
fire policy practices, on one hand, and land uses and demographic changes on
the other. Most developed countries have maintained for several decades a fire
suppression policy, which has lead to almost total fire exclusion. The long
term impact of that policy has implied an alteration of traditional fire
regimes, commonly by increasing average burn severity and size, as a result of
higher fuel accumulation (Pyne, 2001), although other authors are more
critical about the real implication of fire suppression policy (Johnson _et
al_ , 2001), or they tend to put more emphasis on the impact of climate
changes (Westerling _et al_ , 2006). For developing countries, fire is still
the most common tool for land clearing, and therefore it is strongly
associated to deforestation, especially in Tropical areas (Cochrane _et al_ ,
1999; DeFries _et al_ , 2002). The traditional use of fire in shifting
cultivation has turned in the last decades to permanent land use change, in
favour of cropland and grasslands. In addition, fire is a traditional tool to
manage permanent grasslands, which are burned annually to favour new shoots
and improve palatability (Hobbs _et al_ , 1991; Chuvieco _et al_ , 2010).
Global and local implications of changing natural fire circumstances have been
widely recognized, with major effects on air quality, greenhouse gas
emissions, soil degradation and vegetation succession (Goetz _et al_ , 2006;
Parisien _et al_ , 2006; Randerson _et al_ , 2005). The role of human
activities in changing those conditions has not been assessed at global scale.
Several local studies have identified factors that are commonly associated to
human fire ignition, such as distance to roads, forest-agricultural or forest-
urban interfaces, land use management, and social conflicts (unemployment,
rural poverty, hunting disputes,) (Leone _et al_ , 2003; Martínez _et al_ ,
2009; Vega-García _et al_ , 1995). On the other hand, humans not only cause
fires, but they suffer their consequences as well. Fire is recognized as a
major natural hazard (Food, 2007), which imply severe losses of human lives,
properties and other socio-economic values (Radeloff _et al_ , 2005; Reisen
and Brown, 2006).
Fire is no longer a significant part of the traditional systems of life;
however, it remains strongly tied to human activity (Leone _et al_ , 2009).
Knowledge of the causes of forest fires and the main driving factors of
ignition is an indispensable step toward effective fire prevention (Ganteaume
_et al_ , 2013). It is widely recognized that current fire regimes are
changing as a result of environmental and climatic changes (Pausas and Keeley
2009) with increased fire frequency in several areas in the Mediterranean
Region of Europe (Rodrigues _et al_ , 2013). In Mediterranean-type ecosystems,
several studies have indicated that these changes are mainly driven by fire
suppression policies (Minnich, 1983), climate (Pausas _et al_ , 2012), and
human activities (Bal _et al_ , 2011). Human drivers mostly have a temporal
dimension, which is why an historical/temporal perspective is often required
(Zumbrunnen _et al_ , 2011; Carmona _et al_ , 2012). In Mediterranean Europe,
increases in the number of fires have been detected in some countries,
including Portugal and Spain (San-Miguel-Ayanz _et al_ , 2013a; Rodrigues _et
al_ , 2013). In addition, a recent work by Turco _et al_ (2016) suggests huge
spatial and temporal variability in fire frequency trends especially in the
case of Spain, where increasing and decreasing trends were detected depending
on the analysis period and scale. This increase in wildfire frequency and
variability, with its associated risks to the environment and society (Moreno
_et al_ , 2011, 2014), calls for a better understanding of the processes that
control wildfire activity (Massada _et al_ , 2013). In recent decades, major
efforts have been made to determine the influence of climate change on natural
hazards, and to develop models and tools to properly characterize and quantify
changes in climatic patterns. While physical processes involved in ignition
and combustion are theoretically simple, understanding the relative influence
of human factors in determining wildfire is an ongoing task (Mann _et al_ ,
2016). It is clear that human-caused fires that occur repeatedly in a given
geographical area are not simply reducible to individual personal factors, and
thus subject to pure chance. They are usually the result of a spatial pattern,
whose origin is in the interaction of environmental and socioeconomic
conditions (Koutsias _et al_ , 2015). This is particularly true in human-
dominated landscapes such as Spain, where anthropogenic ignitions surpass
natural ignitions, and humans interact to a large degree with the landscape,
changing its flammability, and act as fire initiators or suppressors. In such
cases, human influence may cause sudden changes in fire frequency, intensity,
and burned area size (Pezzatti _et al_ , 2013).
Fire is an integral component of Mediterranean ecosystems since at least the
Miocene (Dubar _et al_ , 1995). Although humans have used fires in the region
for tens of thousands of years (Goren-Inbar _et al_ , 2004), it is only in the
last 10,000 or so that man has significantly influenced fire regime (Daniau
_et al_ , 2010). The use of fire as a management tool has persisted until
these days, although the second half of the past century saw a major change
and a regime shift due to abandonment of many unproductive lands (Moreno _et
al_ , 1988; Pausas _et al_ , 2012). Although fire still is a traditional
management tool in some rural areas for control of vegetation and enhancement
of pastures for cattle feed, most fires these days are no longer related to
the management of the land (San-Miguel-Ayanz _et al_ , 2012, 2013a, 2013b).
The European Mediterranean region is a highly populated area where nearly 200
Million people live in just 5 European Union countries, Portugal, Spain,
France, Italy and Greece. Population density varies but remains very high with
about 2500 inhabitants/km2 in the French Riviera (with peaks of up to 750,000
tourists per day during the summer) (Corteau, 2007) versus an average of 111
inhabitants/km2 in the region. The region is characterized by an extensive
wildland urban interface (WUI). Large urban areas have expanded into the
neighboring wildland areas, where expensive households are built. The WUI has
been further increased by the construction of second holiday homes in the
natural environment. Fire prone areas along the Mediterranean coast have been
extensively built up, reducing in some cases the availability of fuels, but
increasing largely the probability of fire ignition by human causes (Ganteaume
_et al_ , 2013). In other areas of the same region, abandonment of the rural
environment has lead to low utilization of forests, which are generally of
limited productivity, and the subsequent accumulation of fuel loads (San-
Miguel-Ayanz _et al_ , 2012, 2013a, 2013b; Moreira _et al_ , 2011). The
combination of the above factors converts the European Mediterranean region in
a high fire risk area (Sebastián-López _et al_ , 2008), especially during the
summer months when low precipitations and very high temperatures favor fire
ignition and spread.
About 65,000 fires take place every year in the European region, burning, on
average, around half a million ha of forest areas (European Commission, 2011).
Approximately 85% of the total burnt area occurs in the EU Mediterranean
region (San-Miguel-Ayanz _et al_ , 2010). Although fires ignite and spread
under favorable conditions of fuel availability and low moisture conditions,
ignition is generally caused by human activities. Over 95% of the fires in
Europe are due to human causes. An analysis of fire causes show that the most
common cause of fires is “agricultural practices”, followed by “negligence”
and “arson” (Vilar Del Hoyo _et al_ , 2009; Reus Dolz _et al_ , 2003).
Most fires in the region are small, as a fire exclusion (extinction) policy
prevails in Europe. Fires are thus extinguished as soon as possible, and only
a small percentage escapes the initial fire attack and the subsequent
firefighting operations. An enhanced international collaboration for
firefighting exists among countries in the European Mediterranean region. This
facilitates the provision of additional firefighting means to those in a given
country from the neighbouring countries in case of large fire events. The
trend of large fires, those larger than 500 ha, is shown quite stable in the
last decades (San-Miguel-Ayanz _et al_ , 2010). However, among these large
fires, several fire episodes caused catastrophic damages and the loss of human
lives (San-Miguel-Ayanz _et al_ , 2013a).
A first step is to identify all the factors linked to human activity,
establishing their relative importance in space and time (Martínez _et al_ ,
2009, 2013). According to Moreno _et al_ (2014), the number of fires over the
past 50 years in Spain has increased, driven by climate and land-use changes.
However, this tendency has been recently reversed due to fire prevention and
suppression policies. This highlights the influence of changes in the role of
human activities as some of the major driving forces. For instance, changes in
population density patterns—both rural and urban—and traditional activities
have been linked to an increase in intentional fires. In this sense, several
works have previously investigated the influence of human driving factors of
wildfires in Spain. These works have explored in detail a wide range of human
variables (Martínez _et al_ , 2009; Chuvieco _et al_ , 2010) and methods.
Specifically, Generalized Linear Models (Vilar Del Hoyo _et al_ , 2009;
Martínez _et al_ , 2009; Moreno _et al_ , 2014), machine learning methods (Lee
_et al_ , 1996; Rodrigues and de la Riva, 2014), and more spatial-explicit
models like Geographically Weighted Regression (Martínez _et al_ , 2013;
Rodrigues _et al_ , 2014) have previously been employed. However, all these
approaches could be considered as stationary from a temporal point of view,
since they are based on ‘static’ fire data information summarized or
aggregated for a given time span. However, the influence of human drivers
cannot be expected to be stationary (Rodrigues _et al_ , 2016). Zumbrunnen _et
al_ (2011) stress the importance of dealing with the temporal dimension of
human drivers of wildfires. Therefore, exploring temporal changes in
socioeconomic or anthropogenic drivers of wildfire will enhance our
understanding of both current and future patterns of fire ignition, and thus
help improve suppression and prevention policies (Rodrigues _et al_ , 2016).
Disease risk mapping analyses can help to better understand the spatial
variation of the disease, and allow the identification of important public
health determinants (Moraga, 2018). Spatio-temporal disease mapping models are
a popular tool to describe the pattern of disease counts and to identify
regions with an unusual incidence levels, time trend or both (Schrödle and
Held, 2011). This class of models is usually formulated within a hierarchical
Bayesian framework with latent Gaussian model (Besag _et al_ , 1991). Several
proposals have been made including a parametric (Bernardinelli _et al_ , 2014)
and nonparametric (Knorr-Held, 2000; Lagazio _et al_ , 2003; Schmid and Held,
2004) formulation of the time trend and the respective space-time
interactions.
Areal disease data often arise when disease outcomes observed at point level
locations are aggregated over subareas of study region due to constraints such
as population confidentiality. Producing disease risk estimates at area level
is complicated by the fact that raw rates can be very unstable in areas with
small population and for rare circumstances, an also by the presence of
spatial autocorrelation that may exist due to spatially correlated risk
factors (Leroux _et al_ , 2000). Thus, generalised linear mixed models are
often used to obtain disease risk estimates since they enable to improve local
estimates by accommodating spatial correlation and the effects of explanatory
variables. Bayesian inference in these models can be performed using
Integrated Nested Laplace Approximation (INLA) approach (Rue _et al_ , 2009)
which is a computational alternative to the commonly used Markov chain Monte
Carlo methods (MCMC); INLA allows to run fast approximate Bayesian inference
in latent Gaussian models.
INLA is implemented in the INLA package for the R programming language, that
provides an easy way to fit models via inla() function, which works in a
similar way as other functions to fit models, such as glm() or gam() (Palmi-
Perales _et al_ , 2019).
Statistical reporting in the European Union is done according to the
Nomenclature of Units for Territorial Statistics (NUTS) system. The NUTS is a
five-level hierarchical classification based on three regional levels and two
local levels. Each member state is divided into a number of NUTS-1 regions,
which in turn are divided into a number of NUTS-2 regions and so on. There are
78 NUTS-1 regions, 210 NUTS-2 and 1093 NUTS-3 units within the current 15 EU
countries (Eurostat, 2002).
In this paper, we explore the application of these models to understand forest
fires causality using environmental and socio-economic variables. We will work
with areal data using the number of forest fires at NUTS-2 regional level in
Europe and consider forest fires between 2012 and 2016.
## 2 Material and Methods
We extend the analysis of globalization to the NUTS-2 regions of the 27
countries of the European Union (EU-27), as not all the regions have been
included due to absence of information (forest fires or socio-economic data)
(Figure 1).
Figure 1: Study area, in grey administrative areas included in the analysis.
Our main data set comprises the number of fires in Europe at NUTS-2 level,
requested to the European Forest Fire Information System (EFFIS) (San-Miguel-
Ayanz _et al_ , 2012). We have chosen this level due to the variables that we
are interested to analyse (socioeconomic and environmental).
In order to summarise the forest fires in Europe we can see that the number of
forest fires and the area affected have decreased between 2012 and 2014.
However, the minimum was achieved in 2014, with a subsequent increase during
2015 and 2016 (Figure 2).
Figure 2: Summary of number of forest fires (left) and area affected between
2012 and 2016 (right)
The following environmental variables were obtained from the AGRI4CAST
Resources Portal: Maximum air temperature (∘C); Minimum air temperature (∘C);
Mean air temperature (∘C); Mean daily wind speed at 10 m. (m/s); Vapour
pressure (hPa); Daily precipitation (mm/day); Potential evaporation from the
water surface (mm/day); Potential evaporation from moist bare soil surface
(mm/day); Potential evapotranspiration from crop canopy (mm/day); Total global
radiation (kJ/m2/day). For each region we have the average by year. In Figure
3 we can see the average of the different variables by year for all the NUTS 2
regions.
Figure 3: Trend of average of NUTS 2 regions by year of environmental
variables between 2012 and 2015. (a) Maximum air temperature by year; (b)
Miminum air temperature;(c) Mean air temperature; (d) Mean daily wind speed;
(e) Vapour pressure; (f) Daily precipitation; (g) Potential evaporation from
the water surface; (g) Potential evaporation from moist bare soil surface; (h)
Potential evapotranspiration from crop canopy; (j)Total global radiation.
The following socio-economic variables were obtained from Eurostat: Active
population (*1000 employed persons), Woodland (*1000 hectares of Woodland in
the area), Manufactured (*1000 employed persons working in manufactured
products from woodland); Forestry (*1000 employed persons working in Forest
sector); Economic aggregates of forestry (million euro) and Unemployment (%).
In this case we have included in our model totals values by year and region.
In order to summarise the different variables we have included in Figure 4 the
total values by year for all the variables except for Unemployment where we
have done the average for all regions by year.
Figure 4: Trend the socioeconomic variables between 2012 and 2016. Totalisers
by year in the following graphs: (a) Active population; (c) Economic
aggregates of forestry; (d) Employed persons working in Forest sector; (e)
Employed persons working in manufactured products from woodland); and (b)
Average of Unemployment.
### 2.1 Spatio-Temporal model
Here we consider a disease mapping approach, commonly used in small area
studies to assess the spatial pattern of a particular outcome and to identify
areas characterised by unusually high or low relative risk (Lawson, 2013;
Pascutto _et al_ , 2000).
For the _i-th_ area, the number of forest fires $y_{i}$ is modelled as
$y_{it}\sim Poisson(\lambda_{it});\lambda_{it}=E_{it}\rho_{it}$ (1)
where the $E_{it}$ are the expected number of forest fires and $\rho_{it}$ is
the rate.
We specify a log-linear model on $\rho_{i}$ and include spatial, temporal and
a space-time interaction, which would explain differences in the time trend
for different areas. We use the following specification to explain these
differences:
$\rho_{it}=\alpha+\upsilon_{i}+\nu_{i}+\gamma_{t}+\phi_{t}+\delta_{it},$ (2)
There are several ways to define the interaction term: here, we assume that
the two unstructured effects $\nu_{i}$ and $\phi_{t}$ interact. We re-write
the precision matrix as the product of the scalar $\tau_{\nu}$ (or
$\tau_{\phi}$) and the so called structure matrix $\textbf{\emph{F}}_{\nu}$
(or $\textbf{\emph{F}}_{\phi}$), which identifies the neighboring structure;
here the structure matrix $\textbf{\emph{F}}_{\delta}$ can be factorised as
the Kronecker product of the structure matrix for $\nu$ and $\phi$ (Clayton,
1996):
$\textbf{\emph{F}}_{\phi}=\textbf{\emph{F}}_{\nu}\otimes\textbf{\emph{F}}_{\phi}=\textbf{\emph{I}}\otimes\textbf{\emph{I}}=\textbf{\emph{I}}$
(because both $\nu$ and $\phi$ are unstructured). Consequently, we assume no
spatial and/or temporal structure on the interaction and therefore
$\delta_{it}\sim Normal(0,\tau_{\phi})$ — see Knorr-Held (2000) for a detailed
description of other specifications.
In the model presented we assume the default specification of R-INLA for the
distribution of the hyper-parameters; therefore, log$\tau_{\upsilon}$ $\sim$
logGamma(1,0.0005) and log$\tau_{\nu}$ $\sim$ logGamma(1,0.0005). In addition
we specify a logGamma(1,0.0005) prior on the log-precision of the random walk
and of the two unstructured effects (Blangiardo and Cameletti, 2015).
To evaluate the fit of this model, we have applied the Watanabe-Akaike
information criterion (WAIC) (Watanabe, 2010). WAIC was suggested as an
appropriate alternative for estimating the out-of-sample expectation in a
fully Bayesian approach. This method starts with the computed log pointwise
posterior predictive density and then adds a correction for the effective
number of parameters to adjust for overfitting (Gelman and Shalizi, 2013).
Watanabe-Akaike information criterion works on predictive probability density
of detected variables rather than on model parameter; hence, it can be applied
in singular statistical models (i.e. models with non-identifiable
parameterization) (Li _et al_ , 2016).
We have used Integrated Nested Laplace Approximation (INLA) implemented in
R-INLA within the R statistical software (version 3.6.0).
## 3 Results
In this section, we show how the forest fires have evolved between 2012 and
2016.
Analysing the temporal trend, we can see graphically (Figure 5), the posterior
temporal trend for forest fires in Europe. In this graph we show how the
number of forest fires tend to be reduced over time.
Figure 5: Global linear temporal trend for number of forest fires in Europe at
NUTS2 region level. The solid line identifies the posterior mean for
$\beta_{t}$ , while the dashed lines are the 95% credibility intervals.
Analysing the posterior distribution of forest fires (Figure 6) in Europe we
can see that there is a “hot point” in western of the continent (North of
Portugal and North West of Spanish peninsula). Also, as we can see, in
general, the predicted number of forest fires is low in central Europe.
Figure 6: Map of the number of forest fires posterior distribution by region.
Comparing the different years, as we pointed previously, during 2014 the
number of forest fires decreased in all areas except in some regions of Spain
and Sicily. In addition, analysing the number of forest fires by region we can
see that the region with stronger variations is the North region from
Portugal.
In Figure 7 we can see a more detailed map focused in Mediterranean countries
(France, Greece, Italy and Spain). In this case it is clear that variability
in France is almost inexistent only with some increase in the number of forest
fires in Southern regions in 2016. The results from the data available for
Greece, show that there are not big changes during the time analysed. However,
Italy and Spain show more fluctuations during this period. The Southern part
of Italy shows great changes along the time, starting with almost 150 forest
fires in Sicily in 2012 to reduce until about 30 forest fires in 2015 and
increase again in 2016 (67 forest fires). Similarly, in Spain the Northwest
region shows several fluctuations. However, in Spain higher number of forest
fires affects more regions.
Figure 7: Detail of posterior distribution of forest fires in the
Mediterranean region.
As we can see in Table1, several variables are affecting the quantity of
forest fires. But two of them have more impact (have higher values) than the
others. Evaporation in water surface (EvaporationW) is affecting positively
the volume of forest fires at region level. On the other hand, and with
similar magnitude but negative sign, Evapotranspiration from crop canopy
(Evapotrans.) is affecting negatively the presences of forest fires.
Table 1: Posterior estimates summary (Mean, Standard deviation and 95% Credible Interval). Fixed effects and hyperparameters for spatio-temporal model. Fixed effects | | | |
---|---|---|---|---
| mean | sd | 0.025quant | 0.975quant
Active | 0.3045 | 0.1739 | -0.0375 | 0.6468
Aggregates | -0.2919 | 0.1568 | -0.6008 | 0.0152
Forestry | 0.6073 | 0.3321 | -0.058 | 1.2472
Manufactured | -0.9303 | 0.3944 | -1.717 | -0.1669
MaxTemperature | 0.1761 | 0.14 | -0.0999 | 0.4503
MinTemperature | 0.584 | 0.2153 | 0.1631 | 1.0083
AvgTemperature | -0.1396 | 0.4931 | -1.1094 | 0.8277
Wind | 0.5609 | 0.2512 | 0.0655 | 1.0522
Presion | -0.28 | 0.2978 | -0.8653 | 0.3046
Precipitation | -0.0224 | 0.0984 | -0.2158 | 0.1707
Evapotrans | -23.6247 | 9.8466 | -43.0572 | -4.3758
EvaporationW | 24.5911 | 10.907 | 3.2597 | 46.0829
EvaporationS | 0.9008 | 0.6832 | -0.4398 | 2.2435
Radiation | -0.2677 | 0.9264 | -2.0907 | 1.5486
Woodland | 0.7649 | 0.2249 | 0.3331 | 1.2172
Model hyperparameters | | | |
| mean | sd | 0.025quant | 0.975quant
Precision for AREA_ID | 2.02E-01 | 3.85E-02 | 0.1347 | 2.85E-01
Precision for Year | 1.14E-01 | 6.61E-02 | 0.0299 | 2.80E-01
Precision for AREA_ID.YEAR | 1.59E+00 | 2.66E-01 | 1.1262 | 2.17E+00
However, evapotranspiration from crop canopy is having a negative effect in
forest fires quantity. In this group we need to highlight variables having
more impact (higher values) than the others. Evaporation in water surface
(EvaporationW) is affecting positively the volume of forest fires at region
level. On the other hand, and with similar magnitude but negative sign,
Evapotranspiration from crop canopy (Evapotrans) is affecting negatively the
presences of forest fires. The rest of the variables that are affecting
positively the amount of forest fires are Minimum temperature at 10 m.
(MinTemperature) and Mean daily wind speed at 10 m (WIND). Finally,
Manufactured is affecting negatively the quantity of forest fires.
Graphical representation of estimation for the fixed effects is presented in
Figure 8. This chart presents the variables and their relationships with
forest fires. Variables distributed in a positive side contribute to higher
number of forest fires; the opposite, with variables with negative
distribution. Variables present in both areas (positive and negative) do not
have a clear relationship with answer.
Figure 8: Graphical representation of fixed effecs. Evaporation in water
surface (EvaporationW) and Evapotranspiration from crop canopy (Evapotrans)
were excluded in order to obtain more detail of the rest of the variables.
## 4 Conclusions
We have built spatio-temporal models to predict the quantity of forest fires
in Europe at NUTS-2 regional level. We have shown the relationship between the
different variables and the number of forest fires by region. We have shown
that this relationship not only is between some of the variables (fixed
effects), but also the evolution of forest fires along the time is affected
not only by time and spatial effects but also by the combination of both
(Precision for AREA_ID.YEAR).
Initially our main objective in this project was to apply these models to
Europe with more granularity, assuming that more local information will help
to understand better the causality of forest fires. However due to data
availability it was not possible to develop the project in that way. Currently
not all the socioeconomic data is available for all the NUTS-3 regions in a
continuous timestamp, being this characteristic necessary to carry a spatio-
temporal analysis.
Also, several factors can affect in different ways depending of the area. In
our case, variables have been assumed in a scale that in some of the cases
local information can help to understand cause-effect of forest fires.
Analysing the models, we believe that the use of spatio-temporal models is an
advantage for the understanding of the different dynamics, given that the
temporal and spatio-temporal perspective is not very frequent analysing forest
hazards.
Summarising, we can generalise that not only environmental factors but also
socioeconomic variables are affecting the causality of forest fires. However,
more data and more granularity in the analysis in needed in order to
understand this causality.
Landscapes became more hazardous with the time, since land abandonment led to
an increase in forest area. Treeless areas burned proportionally more than
treed ones (Urbieta _et al_ , 2019). Fires in southern Europe have more
preference shrublands than for forest types (Moreira _et al_ , 2011; Oliveira
_et al_ , 2014), but may vary along locations (Moreno _et al_ , 2011). This
could be due to a change in the ignition patterns owing to shifts in the
wildland-agricultural and wildland-urban interfaces (Rodrigues _et al_ , 2014;
Modugno _et al_ , 2016). The most vulnerable landscapes were those with
diversity of land uses, with forest-agriculture mixtures (Ortega _et al_ ,
2012). For these reasons, inclusion of vegetation to analyse causality needs
to be studied.
Fire trends can be affected by changes in ignition cause. In European
Mediterranean countries, a minor percentage of fires are caused by lightning,
and most are caused by people. Fires of these two sources tend to occur at
different locations (Vázquez and Moreno, 1998), which could affect the
vegetation they burn and the difficulty of extinction. However, no changes
between these two sources have been observed (Ganteaume _et al_ , 2013).
Regarding people-caused fires, the majority of them are voluntary, followed by
negligence (Urbieta _et al_ , 2019). In recent times, negligence fires are
increasing and voluntary ones decreasing (Ganteaume _et al_ , 2013). Whether
this is differentially affecting the number of fires trends, is something that
needs research (Urbieta _et al_ , 2019).
Spatio-temporal models and the R-INLA package appear to offer additional
benefits beyond the traditional analysis used to understand the causes of this
hazards. The combination of using a complex spatial latent field to capture
spatial processes and an underlying simple additive regression model for the
response variables relationship to the different factors, means that the fixed
effects are potentially more straightforward to interpret (Golding and Purse,
2016). R-INLA models are extremely flexible in their specifications, with
spatial autocorrelation and observer bias being straightforwardly incorporated
as random effects, while standard error distributions, such as Gaussian,
Poisson, Binomial, and a variety of zero-inflated models, can be used
interchangeably (Rue _et al_ , 2009).
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|
2024-09-04T02:54:59.132455 | 2020-03-11T11:46:07 | 2003.05230 | {
"authors": "Yang Huang, Yongtao Li, Lihua Feng, Weijun Liu",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26160",
"submitter": "Yongtao Li",
"url": "https://arxiv.org/abs/2003.05230"
} | arxiv-papers | # Inequalities for generalized matrix function and inner product
Yongtao Li Yongtao Li, College of Mathematics and Econometrics, Hunan
University, Changsha, Hunan, 410082, P.R. China<EMAIL_ADDRESS>, Yang
Huang Yang Huang, School of Mathematics and Statistics, Central South
University, Changsha, Hunan, 410083, P.R. China<EMAIL_ADDRESS>,
Lihua Feng Lihua Feng, School of Mathematics and Statistics, Central South
University, Changsha, Hunan, 410083, P.R. China<EMAIL_ADDRESS>and Weijun
Liu† Weijun Liu, School of Mathematics and Statistics, Central South
University, Changsha, Hunan, 410083, P.R. China<EMAIL_ADDRESS>
###### Abstract.
We present inequalities related to generalized matrix function for positive
semidefinite block matrices. We introduce partial generalized matrix functions
corresponding to partial traces and then provide an unified extension of the
recent inequalities due to Choi [6], Lin [14] and Zhang et al. [19, 5]. We
demonstrate the applications of a positive semidefinite $3\times 3$ block
matrix, which motivates us to give a simple alternative proof of Dragomir’s
inequality and Krein’s inequality.
###### Key words and phrases:
Block matrices; Positive semidefinite; Generalized matrix function; Partial
traces; Partial determinants; Dragomir’s inequality; Krein’s inequality.
###### 2010 Mathematics Subject Classification:
47B65, 15B42, 15A45
## 1\. Introduction
Let $G$ be a subgraph of the symmetric group $S_{n}$ on $n$ letters and let
$\chi$ be an irreducible character of $G$. For any $n\times n$ complex matrix
$A=[a_{ij}]_{i,j=1}^{n}$, the generalized matrix function of $A$ (also known
as immanant) afforded by $G$ and $\chi$ is defined as
$\mathrm{d}_{\chi}^{G}(A):=\sum\limits_{\sigma\in
G}\chi(\sigma)\prod\limits_{i=1}^{n}a_{i\sigma(i)}.$
Some specific subgroups $G$ and characters $\chi$ lead to some acquainted
functionals on the matrix space. For instance, If $G=S_{n}$ and $\chi$ is the
signum function with value $\pm 1$, then the generalized matrix function
becomes the usual matrix determinant; By setting $\chi(\sigma)\equiv 1$ for
each $\sigma\in G=S_{n}$, we get the permanent of the matrix; Setting
$G=\\{e\\}\subset S_{n}$ defines the product of the main diagonal entries of
the matrix (also known as the Hadamard matrix function).
Let $A$ and $B$ be $n\times n$ positive semidefinite matrices. It is easy to
prove by simultaneous diagonalization argument that
$\det(A+B)\geq\det(A)+\det(B).$ (1)
There are many extensions and generalizations of (1) in the literature. For
example, a remarkable extension (e.g., [17, p. 228]) says that
$\mathrm{d}_{\chi}^{G}(A+B)\geq\mathrm{d}_{\chi}^{G}(A)+\mathrm{d}_{\chi}^{G}(B).$
(2)
Recently, Paksoy, Turkmen and Zhang [19] provided a natural extension of (2)
for triple matrices by embedding the vectors of Gram matrices into a
“sufficiently large” inner product space and using tensor products. More
precisely, if $A,B$ and $C$ are positive semidefinite, they showed
$\mathrm{d}_{\chi}^{G}(A+B+C)+\mathrm{d}_{\chi}^{G}(C)\geq\mathrm{d}_{\chi}^{G}(A+C)+\mathrm{d}_{\chi}^{G}(B+C).$
(3)
Their approach to establish (3) is algebraic as well as combinatorial. Soon
after, Chang, Paksoy and Zhang [5, Theorem 3] presented a further improvement
of (3) by considering the tensor products of operators as words on certain
alphabets, which states that
$\displaystyle\mathrm{d}_{\chi}^{G}(A+B+C)+\mathrm{d}_{\chi}^{G}(A)+\mathrm{d}_{\chi}^{G}(B)+\mathrm{d}_{\chi}^{G}(C)$
(4)
$\displaystyle\quad\geq\mathrm{d}_{\chi}^{G}(A+B)+\mathrm{d}_{\chi}^{G}(A+C)+\mathrm{d}_{\chi}^{G}(B+C).$
We remark here that (4) is indeed an improvement of (3) since
$\displaystyle\mathrm{d}_{\chi}^{G}(A+B+C)+\mathrm{d}_{\chi}^{G}(C)-\mathrm{d}_{\chi}^{G}(A+C)-\mathrm{d}_{\chi}^{G}(B+C)$
$\displaystyle\quad\geq\mathrm{d}_{\chi}^{G}(A+B)-\mathrm{d}_{\chi}^{G}(A)-\mathrm{d}_{\chi}^{G}(B)\geq
0.$
We use the following standard notation. The set of $m\times n$ complex
matrices is denoted by $\mathbb{M}_{m\times n}$. If $m=n$, we use
$\mathbb{M}_{n}$ instead of $\mathbb{M}_{n\times n}$ and if $n=1$, we use
$\mathbb{C}^{m}$ instead of $\mathbb{M}_{m\times 1}$. The identity matrix of
$\mathbb{M}_{n}$ is denoted by $I_{n}$, or simply by $I$ if no confusion is
possible. We use $\mathbb{M}_{m}(\mathbb{M}_{n})$ for the set of $m\times m$
block matrices with each block being $n$-square. By convention, if
$X\in\mathbb{M}_{n}$ is positive semidefinite, we write $X\geq 0$. For two
Hermitian matrices $A$ and $B$ of the same size, $A\geq B$ means $A-B\geq 0$.
It is easy to verify that $\geq$ is a partial ordering on the set of Hermitian
matrices, referred to Löwner ordering.
On the other hand, Lin and Sra [16] gave the following extension of (1), i.e.,
if $A=[A_{ij}],B=[B_{ij}]\in\mathbb{M}_{m}(\mathbb{M}_{n})$ are block positive
semidefinite matrices, then
${\det}_{2}(A+B)\geq{\det}_{2}(A)+{\det}_{2}(B),$ (5)
where $\det_{2}(A)=[\det A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}$ and $\geq$
stands for the Löwner ordering.
The paper is organized as follows. In Section 2, we briefly review some basic
definitions and properties of tensor product in Multilinear Algebra Theory. In
Section 3, we extend the above-cited results (2), (3), (4) and (5) to block
positive semidefinte matrices (Theorem 3.5 and Corollary 3.6). As byproducts,
some new inequalities related to trace, determinant and permanent are also
included. In Section 4, we investigate the applications of a positive
semidefinite $3\times 3$ block matrix and provide a short proof of Dragomir’s
inequality (Theorem 4.4). In Section 5, we present a simple proof of Krein’s
inequality (Theorem 5.1), and then we also provide some new triangle
inequalities.
## 2\. Preliminaries
Before starting our results, we first review some basic definitions and
notations of Multilinear Algebra Theory [17]. Let $X\otimes Y$ denote the
Kronecker product (tensor product) of $X$ with $Y$, that is, if
$X=[x_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}$ and $Y\in\mathbb{M}_{n}$, then
$X\otimes Y\in\mathbb{M}_{m}(\mathbb{M}_{n})$ whose $(i,j)$-block is
$x_{ij}Y$. Let $\otimes^{r}A:=A\otimes\cdots\otimes A$ denote the $r$-fold
tensor power of $A$. We denote by $\wedge^{r}A$ the $r$th antisymmetric tensor
power (or $r$th Grassmann power) of $A$, which is the same as the $r$th
multiplicative compound matrix of $A$, and denote by $\vee^{r}A$ the $r$th
symmetric tensor power of $A$; see [1, p. 18] for more details. We denote by
$e_{r}(A),s_{r}(A)$ the $r$th elementary symmetric and $r$th complete
symmetric function of the eigenvalues of $A$ (see [11, p. 54]). Trivially,
$e_{1}(A)=s_{1}(A)=\mathrm{tr}(A)$ and $e_{n}(A)=\det(A)$ for
$A\in\mathbb{M}_{n}$.
Let $V$ be an $n$-dimensional Hilbert space and $\otimes^{n}V$ be the tensor
product space of $n$ copies of $V$. Let $G$ be a subgroup of the symmetric
group $S_{n}$ and $\chi$ be an irreducible character of $G$. The symmetrizer
induced by $\chi$ on the tensor product space $\otimes^{n}V$ is defined by its
action
$S(v_{1}\otimes\cdots\otimes v_{n}):=\frac{1}{|G|}\sum\limits_{\sigma\in
G}\chi(\sigma)v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}(n)}.$ (6)
All elements of the form (6) span a vector space, denoted by
$V_{\chi}^{n}(G)\subset\otimes^{n}V$, which is called the space of the
symmetry class of tensors associated with $G$ and $\chi$ (see [17, p. 154,
235]). It is easy to verified that $V^{n}_{\chi}(G)$ is an invariant subspace
of $\otimes^{n}V$ under the tensor operator $\otimes^{n}A$. For a linear
operator $A$ on $V$, the induced operator $K(A)$ of $A$ with respect to $G$
and $\chi$ is defined to be $K(A)=(\otimes^{n}A)\big{|}_{V^{n}_{\chi}(G)}$,
the restriction of $\otimes^{n}A$ on $V_{\chi}^{n}(G)$.
The induced operator $K(A)$ is closely related to generalized matrix function.
Let $e_{1},e_{2},\ldots,e_{n}$ be an orthonormal basis of $V$ and $P$ be a
matrix representation of the linear operator $A$ on $V$ with respect to the
basis $e_{1},\ldots,e_{n}$. Then
$\mathrm{d}_{\chi}^{G}\left(P^{T}\right)=\frac{|G|}{\mathrm{deg}(\chi)}\langle
K(A)e^{*},e^{*}\rangle,$ (7)
where $\mathrm{deg}(\chi)$ is the degree of $\chi$ and
$e^{*}:=e_{1}*e_{2}*\cdots*e_{n}$ is the decomposable symmetrized tensor of
$e_{1},\ldots,e_{n}$ (see [17, p. 227, 155]).
Now, we list some basic properties of tensor product for our latter use.
###### Proposition 2.1.
(see [1, pp. 16–20]) Let $A,B$ and $C$ be $n\times n$ matrices. Then
* (1)
$\otimes^{r}(AB)=(\otimes^{r}A)(\otimes^{r}B),\wedge^{r}(AB)=(\wedge^{r}A)(\wedge^{r}B)$
and $\vee^{r}(AB)=(\vee^{r}A)(\vee^{r}B)$.
* (2)
$\mathrm{tr}(\otimes^{r}A)=(\mathrm{tr}A)^{r}:=p_{r}(A),\mathrm{tr}(\wedge^{r}A)=e_{r}(A)$
and $\mathrm{tr}(\vee^{r}A)=s_{r}(A)$.
* (3)
$\det(\otimes^{r}A)=(\det A)^{rn^{r-1}},\det(\wedge^{r}A)=(\det
A)^{{n-r\choose r-1}}$
and $\det(\vee^{r}A)=(\det A)^{\frac{r}{n}{n+r-1\choose r}}$.
Furthermore, if $A,B$ and $C$ are positive semidefinite matrices, then
* (4)
$A\otimes B,A\wedge B$ and $A\vee B$ are positive semidefinite.
* (5)
$\otimes^{r}(A+B)\geq\otimes^{r}A+\otimes^{r}B,\wedge^{r}(A+B)\geq\wedge^{r}A+\wedge^{r}B$
and $\vee^{r}(A+B)\geq\vee^{r}A+\vee^{r}B$.
Finally, we introduce the definition of partial traces, which comes from
Quantum Information Theory [20, p. 12]. Given
$A\in\mathbb{M}_{m}(\mathbb{M}_{n})$, the first partial trace (map)
$A\mapsto\mathrm{tr}_{1}(A)\in\mathbb{M}_{n}$ is defined as the adjoint map of
the imbedding map $X\mapsto I_{m}\otimes
X\in\mathbb{M}_{m}\otimes\mathbb{M}_{n}$. Correspondingly, the second partial
trace (map) $A\mapsto\mathrm{tr}_{2}(A)\in\mathbb{M}_{m}$ is defined as the
adjoint map of the imbedding map $Y\mapsto Y\otimes
I_{n}\in\mathbb{M}_{m}\otimes\mathbb{M}_{n}$. Therefore, we have
$\langle I_{m}\otimes X,A\rangle=\langle
X,\mathrm{tr}_{1}(A)\rangle,\quad\forall X\in\mathbb{M}_{n},$
and
$\langle Y\otimes I_{n},A\rangle=\langle
Y,\mathrm{tr}_{2}(A)\rangle,\quad\forall Y\in\mathbb{M}_{m}.$
Assume that $A=[A_{ij}]_{i,j=1}^{m}$ with $A_{ij}\in\mathbb{M}_{n}$, then the
visualized forms of the partial traces are actually given in [2, Proposition
4.3.10] as
$\mathrm{tr}_{1}{(A)}=\sum\limits_{i=1}^{m}A_{ii},\quad\mathrm{tr}_{2}{(A)}=\bigl{[}\mathrm{tr}A_{ij}\bigr{]}_{i,j=1}^{m}.$
Under the above definition, it follows that both $\mathrm{tr}_{1}(A)$ and
$\mathrm{tr}_{2}(A)$ are positive semidefinite whenever $A$ is positive
semidefinite; see, e.g., [24, p. 237].
## 3\. Partial Matrix Functions
For $A=[A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$, suppose that
$A_{ij}=\bigl{[}a_{rs}^{ij}\bigr{]}_{r,s=1}^{n}$. Setting
$G_{rs}:=\bigl{[}a_{rs}^{ij}\bigr{]}_{i,j=1}^{m}\in\mathbb{M}_{m}.$
Then we can verify that
$\mathrm{tr}_{1}(A)=\sum_{i=1}^{m}A_{ii}=\sum_{i=1}^{m}\bigl{[}a_{rs}^{ii}\bigr{]}_{r,s=1}^{n}=\left[\begin{matrix}\sum\limits_{i=1}^{m}a_{rs}^{ii}\end{matrix}\right]_{r,s=1}^{n}=\bigl{[}\mathrm{tr}\,G_{rs}\bigr{]}_{r,s=1}^{n},$
Motivated by this relation, we next introduce the following definition.
###### Definition 3.1.
Let $\Gamma:\mathbb{M}_{p}\to\mathbb{M}_{q}$ be a matrix function. The first
and second partial matrix functions of $\Gamma$ are defined by
$\Gamma_{1}(A):=\bigl{[}\Gamma(G_{rs})\bigr{]}_{r,s=1}^{n}~{}~{}~{}\text{and}~{}~{}~{}\Gamma_{2}(A):=\bigl{[}\Gamma(A_{ij})\bigr{]}_{i,j=1}^{m}.$
Clearly, when $\Gamma=\mathrm{tr}$, this definition coincides with that of
partial traces; when $\Gamma=\det$, it identifies with the partial
determinants, which were introduced by Choi in [6] recently.
Let $A=[A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive
semidefinite block matrix. It is well known that both $\det_{2}(A)=[\det
A_{ij}]_{i,j=1}^{m}$ and $\mathrm{tr}_{2}(A)=[\mathrm{tr}A_{ij}]_{i,j=1}^{m}$
are positive semidefinite matrices; see, e.g., [24, p. 221, 237]. Whereafter,
Zhang [25, Theorem 3.1] extends the positivity to generalized matrix function
via generalized Cauchy-Binet formula, more precisely,
$\mathrm{d}_{\chi}^{G}{}_{2}(A)=[\mathrm{d}_{\chi}^{G}(A_{ij})]_{i,j=1}^{m}$
is also positive semidefinite.
We extend the positivity to more matrix functionals.
###### Proposition 3.2.
Let ${A}\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. If
$\Gamma$ is one of the functionals
$\mathrm{tr},\det,\mathrm{per},\mathrm{d}_{\chi}^{G},p_{r},e_{r}$ and $s_{r}$,
then $\Gamma_{1}(A)$ and $\Gamma_{2}(A)$ are positive semidefinite.
###### Proof.
We denote by
$\widetilde{A}=[G_{rs}]_{r,s=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{m})$, and
then it is easy to see that $\widetilde{\widetilde{A}}=A$ and
$\Gamma_{1}(A)=\Gamma_{2}(\widetilde{A})$. Moreover, $\widetilde{A}$ and $A$
are unitarily similar; see [6, Theorem 7] for more details. Thus, we only need
to show $\Gamma_{2}(A)$ is positive semidefinite. It is similar with the
approach in [25], we omit the details of proof. ∎
The following Lemma 3.3 plays a key step in our extension (Theorem 3.5), it
could be found in [3] or [5], we here provide a proof for convenience of
readers.
###### Lemma 3.3.
Let $A,B,C$ be positive semidefinite matrices of same size. Then for every
positive integer $r$, we have
$\displaystyle\otimes^{r}(A+B+C)+\otimes^{r}A+\otimes^{r}B+\otimes^{r}C$
$\displaystyle\quad\geq\otimes^{r}(A+B)+\otimes^{r}(A+C)+\otimes^{r}(B+C).$
The same result is true for $\wedge^{r}$ and $\vee^{r}$.
###### Proof.
The proof is by induction on $r$. The base case $r=1$ holds with equality, and
the case $r=2$ is easy to verify. Assume the required result holds for
$r=m\geq 2$, that is
$\displaystyle\otimes^{m}(A+B+C)+\otimes^{m}A+\otimes^{m}B+\otimes^{m}C$
$\displaystyle\quad\geq\otimes^{m}(A+B)+\otimes^{m}(A+C)+\otimes^{m}(B+C).$
For $r=m+1$, we get from Proposition 2.1 that
$\displaystyle\otimes^{m+1}(A+B+C)$
$\displaystyle\quad=\bigl{(}\otimes^{m}(A+B+C)\bigr{)}\otimes(A+B+C)$
$\displaystyle\quad\geq\bigl{(}\otimes^{m}(A+B)+\otimes^{m}(A+C)+\otimes^{m}(B+C)-\otimes^{m}A-\otimes^{m}B-\otimes^{m}C\bigr{)}$
$\displaystyle\quad\quad\,\otimes(A+B+C)$
$\displaystyle\quad=\otimes^{m+1}(A+B)+\otimes^{m+1}(A+C)+\otimes^{m+1}(B+C)$
$\displaystyle\quad\quad-\otimes^{m+1}A-\otimes^{m+1}B-\otimes^{m+1}C$
$\displaystyle\quad\quad+\bigl{(}\otimes^{m}(A+B)\bigr{)}\otimes
C+\bigl{(}\otimes^{m}(A+C)\bigr{)}\otimes
B+\bigl{(}\otimes^{m}(B+C)\bigr{)}\otimes A$
$\displaystyle\quad\quad-\bigl{(}\otimes^{m}A\bigr{)}\otimes(B+C)-\bigl{(}\otimes^{m}B\bigr{)}\otimes(A+C)-\bigl{(}\otimes^{m}C\bigr{)}\otimes(A+B).$
It remains to show that
$\displaystyle\bigl{(}\otimes^{m}(A+B)\bigr{)}\otimes
C+\bigl{(}\otimes^{m}(A+C)\bigr{)}\otimes
B+\bigl{(}\otimes^{m}(B+C)\bigr{)}\otimes A$
$\displaystyle\quad\geq\bigl{(}\otimes^{m}A\bigr{)}\otimes(B+C)+\bigl{(}\otimes^{m}B\bigr{)}\otimes(A+C)+\bigl{(}\otimes^{m}C\bigr{)}\otimes(A+B).$
This follows immediately by the superadditivity (5) in Proposition 2.1. ∎
We require one more lemma for our purpose.
###### Lemma 3.4.
([2, p. 93]) Let $A=[A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$.
Then $[\otimes^{r}A_{ij}]_{i,j=1}^{m}$ is a principal submatrix of
$\otimes^{r}A$ for every positive integer $r$.
Now, we present our main result, which is an unified extension of (4) and (5).
###### Theorem 3.5.
Let ${A},B,C\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. If
$\Gamma$ is one of the functionals
$\mathrm{tr},\det,\mathrm{per},\mathrm{d}_{\chi}^{G},p_{r},e_{r}$ and $s_{r}$,
then
$\displaystyle\Gamma_{1}(A+B+C)+\Gamma_{1}(A)+\Gamma_{1}(B)+\Gamma_{1}(C)$
$\displaystyle\quad\geq\Gamma_{1}(A+B)+\Gamma_{1}(A+C)+\Gamma_{1}(B+C),$
and
$\displaystyle\Gamma_{2}(A+B+C)+\Gamma_{2}(A)+\Gamma_{2}(B)+\Gamma_{2}(C)$
$\displaystyle\quad\geq\Gamma_{2}(A+B)+\Gamma_{2}(A+C)+\Gamma_{2}(B+C).$
###### Proof.
We only show that the desired result holds for $\Gamma=\mathrm{d}_{\chi}^{G}$
and $\Gamma=e_{r}$ since other case of functionals can be proved similarly. It
suffices to show the second desired result by exchanging the role of
$\widetilde{A}$ and $A$. By Lemma 3.3, we have
$\displaystyle\otimes^{r}(A+B+C)+\otimes^{r}A+\otimes^{r}B+\otimes^{r}C$
$\displaystyle\quad\geq\otimes^{r}(A+B)+\otimes^{r}(A+C)+\otimes^{r}(B+C),$
which together with Lemma 3.4 leads to the following
$\displaystyle[\otimes^{r}(A_{ij}+B_{ij}+C_{ij})]_{i,j=1}^{m}+[\otimes^{r}A_{ij}]_{i,j=1}^{m}+[\otimes^{r}B_{ij}]_{i,j=1}^{m}+[\otimes^{r}C_{ij}]_{i,j=1}^{m}$
$\displaystyle\quad\geq[\otimes^{r}(A_{ij}+B_{ij})]_{i,j=1}^{m}+[\otimes^{r}(A_{ij}+C_{ij})]_{i,j=1}^{m}+[\otimes^{r}(B_{ij}+C_{ij})]_{i,j=1}^{m}.$
By restricting above inequality to the symmetry class $V_{\chi}^{G}(V)$, we
get
$\displaystyle[K(A_{ij}+B_{ij}+C_{ij})]_{i,j=1}^{m}+[K(A_{ij})]_{i,j=1}^{m}+[K(B_{ij})]_{i,j=1}^{m}+[K(C_{ij})]_{i,j=1}^{m}$
$\displaystyle\quad\geq[K(A_{ij}+B_{ij})]_{i,j=1}^{m}+[K(A_{ij}+C_{ij})]_{i,j=1}^{m}+[K(B_{ij}+C_{ij})]_{i,j=1}^{m}.$
By combining (7), the second desired result in the case of
$\Gamma=\mathrm{d}_{\chi}^{G}$ follows.
By the same way, it follows that
$\displaystyle[\wedge^{r}(A_{ij}+B_{ij}+C_{ij})]_{i,j=1}^{m}+[\wedge^{r}A_{ij}]_{i,j=1}^{m}+[\wedge^{r}B_{ij}]_{i,j=1}^{m}+[\wedge^{r}C_{ij}]_{i,j=1}^{m}$
$\displaystyle\quad\geq[\wedge^{r}(A_{ij}+B_{ij})]_{i,j=1}^{m}+[\wedge^{r}(A_{ij}+C_{ij})]_{i,j=1}^{m}+[\wedge^{r}(B_{ij}+C_{ij})]_{i,j=1}^{m}.$
By taking trace blockwise and using Proposition 2.1, it yields the second
desired result in the case of $\Gamma=e_{r}$. ∎
From Theorem 3.5, one could get the following Corollary 3.6.
###### Corollary 3.6.
Let ${A},B,C\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. If
$\Gamma$ is one of the functionals
$\mathrm{tr},\det,\mathrm{per},\mathrm{d}_{\chi}^{G},p_{r},e_{r}$ and $s_{r}$,
then
$\displaystyle\Gamma_{1}(A+B+C)+\Gamma_{1}(C)\geq\Gamma_{1}(A+C)+\Gamma_{1}(B+C),$
and
$\displaystyle\Gamma_{2}(A+B+C)+\Gamma_{2}(C)\geq\Gamma_{2}(A+C)+\Gamma_{2}(B+C).$
In particular, by setting $m=1$ and $\Gamma=\det$ in Theorem 3.5 and Corollary
3.6, which yields the following renowned determinantal inequalities,
$\displaystyle\det(A+B+C)+\det A+\det B+\det C$
$\displaystyle\quad\geq\det(A+B)+\det(A+C)+\det(B+C),$
and
$\det(A+B+C)+\det C\geq\det(A+C)+\det(B+C).$
We remark that these two inequalities could be proved by using a majorization
approach of eigenvalues. It is more elementary and totally different from our
method. We refer to [14] and [24, p. 215] for more details.
## 4\. Positivity and Dragomir’s inequality
Recently, positive semidefinite $3\times 3$ block matrices are extensively
studied, such a partition leads to versatile and elegant theoretical
inequalities; see, e.g., [15, 9]. Assume that $X,Y,Z$ are matrices with
appropriate size, then it follows from Section 3 that the $3\times 3$ matrix
$\begin{bmatrix}\Gamma(X^{*}X)&\Gamma(X^{*}Y)&\Gamma(X^{*}Z)\\\
\Gamma(Y^{*}X)&\Gamma(Y^{*}Y)&\Gamma(Y^{*}Z)\\\
\Gamma(Z^{*}X)&\Gamma(Z^{*}Y)&\Gamma(Z^{*}Z)\end{bmatrix}$ (8)
is positive semidefinite whenever $\Gamma$ is selected for trace and
determinant. Different size of matrices in (8) will yield a large number of
interesting triangle inequalities. In particular, if $X,Y,Z$ are column
vectors, say $u,v,w\in\mathbb{C}^{n}$, it is easy to see that
$\begin{bmatrix}\mathrm{Re}(u^{*}u)&\mathrm{Re}(u^{*}v)&\mathrm{Re}(u^{*}w)\\\\[2.84544pt]
\mathrm{Re}(v^{*}u)&\mathrm{Re}(v^{*}v)&\mathrm{Re}(v^{*}w)\\\\[2.84544pt]
\mathrm{Re}(w^{*}u)&\mathrm{Re}(w^{*}v)&\mathrm{Re}(w^{*}w)\end{bmatrix}$ (9)
is positive semidefinite; see [13, 4] for more applications.
In this section, we provide two analogous results (Corollary 4.2 and
Proposition 4.3) of the above (9). Based on this result, we then give a short
proof of Dragomir’s inequality (Theorem 4.4). The following Lemma is an
Exercise in [2, p. 26], we will present a detailed proof.
###### Lemma 4.1.
Let $A=[a_{ij}]$ be a $3\times 3$ complex matrix and let
$|A|=\bigl{[}|a_{ij}|\bigr{]}$ be the matrix obtained from $A$ by taking the
absolute values of the entries of $A$. If $A$ is positive semidefinite, then
$|A|$ is positive semidefinite.
###### Proof.
We first note that the positivity of $A$ implies all diagonal entries of $A$
are nonnegative. If a diagonal entry of $A$ is zero, as $A$ is positive
semidefinite, then the entire row entries and column entries of $A$ are zero
and it is obvious that the positivity of
$\begin{bmatrix}\begin{smallmatrix}a&c\\\
\overline{c}&b\end{smallmatrix}\end{bmatrix}$ implies the positivity of
$\begin{bmatrix}\\!\begin{smallmatrix}|a|&|c|\\\
|\overline{c}|&|b|\end{smallmatrix}\\!\end{bmatrix}$. Without loss of
generality, we may assume that $a_{ii}>0$ for every $i=1,2,3$. Let
$D=\mathrm{diag}\bigl{\\{}a_{11}^{-1/2},a_{22}^{-1/2},a_{33}^{-1/2}\bigr{\\}}$
and observe that $D^{*}|A|D=|D^{*}AD|$. By scaling, we further assume that
$A=\begin{bmatrix}1&a&b\\\ \overline{a}&1&c\\\
\overline{b}&\overline{c}&1\end{bmatrix}.$
Recall that $X\geq 0$ means $X$ is positive semidefinite. Our goal is to prove
$\begin{bmatrix}1&a&b\\\ \overline{a}&1&c\\\
\overline{b}&\overline{c}&1\end{bmatrix}\geq
0\Rightarrow\begin{bmatrix}1&|a|&|b|\\\ |\overline{a}|&1&|c|\\\
|\overline{b}|&|\overline{c}|&1\end{bmatrix}\geq 0.$ (10)
Assume that $a=|a|e^{i\alpha}$ and $b=|b|e^{i\beta}$, and denote
$Q=\mathrm{diag}\left\\{1,e^{-i\alpha},e^{-i\beta}\right\\}$. By a direct
computation, we obtain
$Q^{*}AQ=\begin{bmatrix}1&|a|&|b|\\\ |a|&1&ce^{i(\alpha-\beta)}\\\
|b|&\overline{c}e^{i(\beta-\alpha)}&1\end{bmatrix}.$
Since $Q^{*}AQ\geq 0$, taking the determinant leads to the following
$1+|a||b|\left(ce^{i(\alpha-\beta)}+\overline{c}e^{i(\beta-\alpha)}\right)\geq|a|^{2}+|b|^{2}+|c|^{2}.$
Note that $2|c|\geq
2\,\mathrm{Re}\left(ce^{i(\alpha-\beta)}\right)\geq\left(ce^{i(\alpha-\beta)}+\overline{c}e^{i(\beta-\alpha)}\right)$,
then
$1+2|a||b||c|\geq|a|^{2}+|b|^{2}+|c|^{2},$
which is actually $\det|A|\geq 0$. Combining $1-|a|^{2}\geq 0$, that is, every
principal minor of $|A|$ is nonnegative, then $|A|\geq 0$. Thus, the desired
statement (10) now follows. ∎
Remark. We remark that the converse of Lemma 4.1 is not true, additionally,
the statement not holds for $4\times 4$ case. For example, setting
$B=\begin{bmatrix}1&-1&-1\\\ -1&1&-1\\\ -1&-1&1\end{bmatrix},\quad
C=\begin{bmatrix}10&3&-2&1\\\ 3&10&0&9\\\ -2&0&10&4\\\ 1&9&4&10\end{bmatrix}.$
We can see that both $|B|$ and $C$ are positive semidefinite, however, $B$ and
$|C|$ are not positive semidefinite, because $\det B=-4$ and $\det|C|=-364$.
By the positivity of Gram matrix and Lemma 4.1, we get the following
corollary.
###### Corollary 4.2.
If $u,v$ and $w$ are vectors in $\mathbb{C}^{n}$, then
$\begin{bmatrix}\bigl{|}u^{*}u\bigr{|}&\bigl{|}u^{*}v\bigr{|}&\bigl{|}u^{*}w\bigr{|}\\\\[4.26773pt]
\bigl{|}v^{*}u\bigr{|}&\bigl{|}v^{*}v\bigr{|}&\bigl{|}v^{*}w\bigr{|}\\\\[4.26773pt]
\bigl{|}{w^{*}u}\bigr{|}&\bigl{|}{w^{*}v}\bigr{|}&\bigl{|}{w^{*}w}\bigr{|}\end{bmatrix}$
is a positive semidefinite matrix.
###### Proposition 4.3.
If $u,v$ and $w$ are vectors in $\mathbb{R}^{n}$ such that $u+w=v$, then
$\begin{bmatrix}{u^{*}u}&{u^{*}v}&-{u^{*}w}\\\\[2.84544pt]
{v^{*}u}&{v^{*}v}&{v^{*}w}\\\\[2.84544pt]
-{w^{*}u}&{w^{*}v}&{w^{*}w}\end{bmatrix}$
is a positive semidefinite matrix.
###### Proof.
We choose an orthonormal basis of $\mathrm{Span}\\{u,v,w\\}$, then we may
assume that $u,v$ and $w$ are vectors in $\mathbb{R}^{3}$ and form a triangle
on a plane. We denote the angle of $u,v$ by $\alpha$, angle of $-u,w$ by
$\beta$ and angle of $-w,-v$ by $\gamma$, respectively. Note that
$\alpha+\beta+\gamma=\pi$, then we have
$\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+2\cos\alpha\cos\beta\cos\gamma=1.$
By computing the principal minor, it follows that
$R:=\begin{bmatrix}1&\cos\alpha&\cos\beta\\\ \cos\alpha&1&\cos\gamma\\\
\cos\beta&\cos\gamma&1\end{bmatrix}$
is positive semidefinite. Setting $S=\mathrm{diag}\\{\left\lVert
u\right\rVert,\left\lVert v\right\rVert,\left\lVert w\right\rVert\\}$. Thus
$S^{T}RS$ is positive semidefinite. This completes the proof. ∎
Dragomir [7] established the following inequality (Theorem 4.4) related to
inner product of three vectors, which yields some improvements of Schwarz’s
inequality; see, e.g.,[8]. We here give a short proof using Corollary 4.2.
###### Theorem 4.4.
Let $u,v$ and $w$ be vectors in an inner product space. Then
$\displaystyle\left(\left\lVert u\right\rVert^{2}\left\lVert
w\right\rVert^{2}-\bigl{|}\left\langle
u,w\right\rangle\bigr{|}^{2}\right)\left(\left\lVert
w\right\rVert^{2}\left\lVert v\right\rVert^{2}-\bigl{|}\left\langle
w,v\right\rangle\bigr{|}^{2}\right)$
$\displaystyle\quad\geq\bigl{|}\left\langle u,w\right\rangle\left\langle
w,v\right\rangle-\left\langle u,v\right\rangle\left\langle
w,w\right\rangle\bigr{|}^{2}.$
###### Proof.
Without loss of generality, by scaling, we may assume that $u,v$ and $w$ are
unit vectors. We now need to prove
$\left(1-\bigl{|}\left\langle
u,w\right\rangle\bigr{|}^{2}\right)\left(1-\bigl{|}\left\langle
w,v\right\rangle\bigr{|}^{2}\right)\geq\left(\bigl{|}\left\langle
u,w\right\rangle\bigr{|}\bigl{|}\left\langle
w,v\right\rangle\bigr{|}-\bigl{|}\left\langle
u,v\right\rangle\bigr{|}\right)^{2},$
which is equivalent to showing
$1+2\bigl{|}\left\langle u,v\right\rangle\bigr{|}\bigl{|}\left\langle
v,w\right\rangle\bigr{|}\bigl{|}\left\langle
w,u\right\rangle\bigr{|}\geq\bigl{|}\left\langle
u,v\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle
v,w\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle
w,u\right\rangle\bigr{|}^{2}.$ (11)
By Corollary 4.2, it follows that
$\begin{bmatrix}1&\bigl{|}\left\langle
u,v\right\rangle\bigr{|}&\bigl{|}\left\langle
u,w\right\rangle\bigr{|}\\\\[4.26773pt] \bigl{|}\left\langle
v,u\right\rangle\bigr{|}&1&\bigl{|}\left\langle
v,w\right\rangle\bigr{|}\\\\[4.26773pt] \bigl{|}\left\langle
w,u\right\rangle\bigr{|}&\bigl{|}\left\langle
w,v\right\rangle\bigr{|}&1\end{bmatrix}$
is positive semidefinite. Taking determinant on this matrix yields (11). ∎
Recently, Zhang gave the following inequality (see [25, Theorem 5.1] ), if
$u,v$ and $w$ are all unit vectors in an inner product space, then
$1+2\,\mathrm{Re}\left(\left\langle u,v\right\rangle\left\langle
v,w\right\rangle\left\langle w,u\right\rangle\right)\geq\bigl{|}\left\langle
u,v\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle
v,w\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle
w,u\right\rangle\bigr{|}^{2}.$ (12)
Inequality (11) seems weaker than (12). Actually, it is not difficult to prove
that (11) and (12) are mutually equivalent, we leave the details for the
interested reader.
## 5\. Some Triangle inequalities
Let $V$ be an inner product space with the inner product
$\left\langle\cdot,\cdot\right\rangle$ over the real number field $\mathbb{R}$
or the complex number field $\mathbb{C}$. For any two nonzero vectors $u,v$ in
$V$, there are two defferent ways to define the angle between the vectors $u$
and $v$ in terms of the inner product, such as,
$\displaystyle\Phi(u,v):=\arccos\frac{\mathrm{Re}\left\langle
u,v\right\rangle}{\left\lVert u\right\rVert\left\lVert v\right\rVert},$
and
$\Psi(u,v):=\arccos\frac{\bigl{|}\left\langle
u,v\right\rangle\bigr{|}}{\left\lVert u\right\rVert\left\lVert
v\right\rVert}.$
Both these definitions are frequently used in the literature, and there are
various reasons and advantages that the angles are defined in these ways; see,
e.g., [13, 4, 18] for recent studies.
The angles $\Phi$ and $\Psi$ are closely related, but not equal unless
$\left\langle u,v\right\rangle$ is a nonnegative number. We can easily see
that $0\leq\Phi\leq\pi$ and $0\leq\Psi\leq\pi/2$, and $\Phi(u,v)\geq\Psi(u,v)$
for all $u,v\in V$, since $\mathrm{Re}\left\langle
u,v\right\rangle\leq\left|\left\langle u,v\right\rangle\right|$ and
$f(x)=\arccos x$ is a decreasing function in $x\in[-1,1]$. It is easy to
verify that
$\Psi(u,v)=\min\limits_{|p|=1}\Phi(pu,v)=\min\limits_{|q|=1}\Phi(u,qv)=\min\limits_{|p|=|q|=1}\Phi(pu,qv).$
(13)
There exist two well known triangle inequalities for $\Phi$ and $\Psi$ in the
literature, we will state it as the following Theorem 5.1.
###### Theorem 5.1.
Let $u,v$ and $w$ be vectors in an inner product space. Then
$\displaystyle\Phi(u,v)$ $\displaystyle\leq\Phi(u,w)+\Phi(w,v),$ (14)
and
$\displaystyle\Psi(u,v)$ $\displaystyle\leq\Psi(u,w)+\Psi(w,v).$ (15)
The first inequality (14) is attributed to Krein who stated it without proof
in [12], and proved first by Rao [21] and [10, p. 56], whose proof boils down
to the positivity of the matrix (9). We remark that (14) on the real field
could be seen in [24, p. 31].
For the second one, Lin [13] observed that (15) can be deduced from (14)
because of the relation (13). It is noteworthy that either Corollary 4.2 or
Theorem 4.4 also guarantees (15). Indeed, by Theorem 4.4, we can obtain
$\displaystyle\left(\left\lVert u\right\rVert^{2}\left\lVert
w\right\rVert^{2}-\bigl{|}\left\langle
u,w\right\rangle\bigr{|}^{2}\right)^{1/2}\left(\left\lVert
w\right\rVert^{2}\left\lVert v\right\rVert^{2}-\bigl{|}\left\langle
w,v\right\rangle\bigr{|}^{2}\right)^{1/2}$
$\displaystyle\quad\geq\bigl{|}\left\langle u,w\right\rangle\left\langle
w,v\right\rangle\bigr{|}-\bigl{|}\left\langle u,v\right\rangle\left\langle
w,w\right\rangle\bigr{|}.$
By dividing with $\left\lVert u\right\rVert\left\lVert
v\right\rVert\left\lVert w\right\rVert^{2}$, we have
$\frac{\left|\left\langle u,v\right\rangle\right|}{\left\lVert
u\right\rVert\left\lVert v\right\rVert}\geq\frac{\left|\left\langle
u,w\right\rangle\right|}{\left\lVert u\right\rVert\left\lVert
w\right\rVert}\frac{\left|\left\langle w,v\right\rangle\right|}{\left\lVert
w\right\rVert\left\lVert v\right\rVert}-\sqrt{1-\frac{\left|\left\langle
u,w\right\rangle\right|}{\left\lVert u\right\rVert\left\lVert
w\right\rVert}}\cdot\sqrt{1-\frac{\left|\left\langle
w,v\right\rangle\right|}{\left\lVert w\right\rVert\left\lVert
v\right\rVert}},$
which is equivalent to
$\displaystyle\cos\Psi(u,v)$
$\displaystyle\geq\cos\Psi(u,w)\cos\Psi(w,v)-\sin\Psi(u,w)\sin\Psi(w,v)$
$\displaystyle=\cos(\Psi(u,w)+\Psi(w,v)).$
Thus, (15) follows by the decreasing property of cosine on $[0,\pi]$.
To end this paper, we present a new proof of inequality (14) and (15), which
can be viewed as a generalization of the method in [24, p. 31], and then we
also provide some new angle inequalities.
###### Proof of Theorem 5.1.
We here only prove (15), since (14) can be proved in a slight similar way.
Because the desireed inequality involves only three vectors $u,v$ and $w$, we
may focus on the subspace spaned by $u,v$ and $w$, which has dimension at most
$3$. We may further choose an orthonormal basis (a unit vector in the case of
dimension one) of this subspace $\mathrm{Span}\\{u,v,w\\}$. Assume that $u,v$
and $w$ have coordinate vectors $x,y$ and $z$ under this basis, respectively.
Then the desired inequality holds if and only if it holds for complex vectors
$x,y$ and $z$ with the standard product
$\left\langle
x,y\right\rangle=\overline{y_{1}}x_{1}+\overline{y_{2}}x_{2}+\cdots+\overline{y}_{n}x_{n}.$
That is to say, our mian goal is to show the following
$\Psi(x,y)\leq\Psi(x,z)+\Psi(z,y),\quad\forall\,x,y,z\in\mathbb{C}^{3}.$ (16)
We next prove the inequality (16) in two steps. If the inner product space is
a Euclidean space (i.e., an inner product space over field $\mathbb{R}$). Then
the problem is reduced to $\mathbb{R},\mathbb{R}^{2}$ or $\mathbb{R}^{3}$
depending on whether the dimension of $\mathrm{Span}\\{u,v,w\\}$ is $1,2$ or
$3$, respectively. In this real case, one can draw a simple graph to get the
result. If the inner product space is an unitary space (i.e., an inner product
space over field $\mathbb{C}$). We now do some technical tricks. We note that
the desired inequality (16) is not changed if we replace $x,y$ with $\omega
x,\delta y$ for any complex numbers $\omega,\delta$ satisfying
$|\omega|=|\delta|=1$. Therefore, we may assume further that both
$\left\langle x,z\right\rangle$ and $\left\langle z,y\right\rangle$ are real
numbers. Let $x=X_{1}+iX_{2},y=Y_{1}+iY_{2}$ and $z=Z_{1}+iZ_{2}$ for some
vectors $X_{i},Y_{i},Z_{i}\in\mathbb{R}^{3}(i=1,2)$ and denote by
$X=\begin{bmatrix}X_{1}\\\ X_{2}\end{bmatrix},\quad Y=\begin{bmatrix}Y_{1}\\\
Y_{2}\end{bmatrix},\quad Z=\begin{bmatrix}Z_{1}\\\ Z_{2}\end{bmatrix}.$
Note that $X,Y,Z\in\mathbb{R}^{6}$, then by the previous statement for
Euclidean space, we get
$\Psi(X,Y)\leq\Psi(X,Z)+\Psi(Z,Y).$ (17)
Since $\left\langle x,z\right\rangle$ and $\left\langle z,y\right\rangle$ are
real numbers, we have
$\displaystyle\left\langle x,z\right\rangle$
$\displaystyle=\mathrm{Re}\left\langle
x,z\right\rangle=Z_{1}^{T}X_{1}+Z_{2}^{T}X_{2}=\left\langle X,Z\right\rangle,$
$\displaystyle\left\langle z,y\right\rangle$
$\displaystyle=\mathrm{Re}\left\langle
z,y\right\rangle=Y_{1}^{T}Z_{1}+Y_{2}^{T}Z_{2}=\left\langle Z,Y\right\rangle,$
$\displaystyle\left\langle x,y\right\rangle$
$\displaystyle=Y_{1}^{T}X_{1}+Y_{2}^{T}X_{2}+i(Y_{1}^{T}X_{2}-Y_{2}^{T}X_{1}).$
It is easy to see that $\left\lVert x\right\rVert=\left\lVert
X\right\rVert,\left\lVert y\right\rVert=\left\lVert Y\right\rVert$ and
$\left\lVert z\right\rVert=\left\lVert Z\right\rVert$. Thus,
$\Psi(x,z)=\Psi(X,Z),\quad\Psi(z,y)=\Psi(Z,Y).$ (18)
Since $f(t)=\mathrm{arccos}\,(t)$ is a decreasing function in $t\in[-1,1]$, we
get
$\Psi(x,y)=\arccos\frac{\bigl{|}\left\langle
x,y\right\rangle\bigr{|}}{\left\lVert x\right\rVert\left\lVert
y\right\rVert}\leq\frac{\bigl{|}Y_{1}^{T}X_{1}+Y_{2}^{T}X_{2}\bigr{|}}{\left\lVert
X\right\rVert\left\lVert Y\right\rVert}=\Psi(X,Y).$ (19)
Combining (17), (18) and (19), we can get the desired inequality (16). ∎
Using the same idea of the proof of Theorem 5.1, one could also get the
following Proposition 5.2.
###### Proposition 5.2.
Let $u,v$ and $w$ be vectors in an inner product space. Then
$\displaystyle\left|\Theta(u,v)-\Theta(v,w)\right|\leq\Theta(u,w)\leq\Theta(u,v)+\Theta(v,w),$
$\displaystyle 0\leq\Theta(u,v)+\Theta(v,w)+\Theta(w,u)\leq 2\pi.$
Moreover, the above inequalities hold for $\Psi$.
The following inner product inequality is the main result in [22] and also can
be found in [24, p. 195], it is derived as a tool in showing a trace
inequality for unitary matrices. Of course, the line of proof provided here is
quite different and simple.
###### Corollary 5.3.
Let $u,v$ and $w$ be vectors in an inner product space over $\mathbb{C}$. Then
$\sqrt{1-\frac{\left|\left\langle u,v\right\rangle\right|^{2}}{\left\lVert
u\right\rVert^{2}\left\lVert
v\right\rVert^{2}}}\leq\sqrt{1-\frac{\left|\left\langle
u,w\right\rangle\right|^{2}}{\left\lVert u\right\rVert^{2}\left\lVert
w\right\rVert^{2}}}+\sqrt{1-\frac{\left|\left\langle
w,v\right\rangle\right|^{2}}{\left\lVert w\right\rVert^{2}\left\lVert
v\right\rVert^{2}}}.$
Moreover, inequality holds if we replace $|\cdot|$ with
$\mathrm{Re}\,(\cdot)$.
###### Proof.
For brevity, we denote $\alpha,\beta,\gamma$ by the angles
$\Psi(u,v),\Psi(u,w)$, $\Psi(w,v)$ or $\Phi(u,v),\Phi(u,w)$, $\Phi(w,v)$,
respectively. By Proposition 5.2, we have
$\displaystyle\frac{\alpha}{2}\leq\frac{\beta+\gamma}{2}\leq\pi-\frac{\alpha}{2},\quad
0\leq\frac{|\beta-\gamma|}{2}\leq\frac{\alpha}{2}\leq\frac{\pi}{2}.$
Then
$\displaystyle 0\leq\sin\frac{\alpha}{2}\leq\sin\frac{\beta+\gamma}{2},\quad
0\leq\cos\frac{\alpha}{2}\leq\cos\frac{\beta-\gamma}{2}.$
The required inequality can be written as
$\sin\alpha\leq
2\sin\frac{\beta+\gamma}{2}\cos\frac{\beta-\gamma}{2}=\sin\alpha+\sin\beta.$
This completes the proof. ∎
## Acknowledgments
The first author would like to expresses sincere thanks to Professor Fuzhen
Zhang for his kind help and valuable discussion [23] before its publication,
which considerably improves the presentation of our manuscript. Finally, all
authors are grateful for valuable comments and suggestions from anonymous
reviewer. This work was supported by NSFC (Grant No. 11671402, 11871479),
Hunan Provincial Natural Science Foundation (Grant No. 2016JJ 2138,
2018JJ2479) and Mathematics and Interdisciplinary Sciences Project of Central
South University.
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|
2024-09-04T02:54:59.142476 | 2020-03-11T11:49:17 | 2003.05232 | {
"authors": "Derek Reitz, Junxue Li, Wei Yuan, Jing Shi, and Yaroslav Tserkovnyak",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26161",
"submitter": "Derek Reitz",
"url": "https://arxiv.org/abs/2003.05232"
} | arxiv-papers | # Spin Seebeck Effect near the Antiferromagnetic Spin-Flop Transition
Derek Reitz Department of Physics and Astronomy, University of California,
Los Angeles, California 90095, USA Junxue Li Wei Yuan Jing Shi Department
of Physics and Astronomy, University of California, Riverside, California
92521, USA Yaroslav Tserkovnyak Department of Physics and Astronomy,
University of California, Los Angeles, California 90095, USA
###### Abstract
We develop a low-temperature, long-wavelength theory for the interfacial spin
Seebeck effect (SSE) in easy-axis antiferromagnets. The field-induced spin-
flop (SF) transition of Néel order is associated with a qualitative change in
SSE behavior: Below SF, there are two spin carriers with opposite magnetic
moments, with the carriers polarized along the field forming a majority magnon
band. Above SF, the low-energy, ferromagnetic-like mode has magnetic moment
opposite the field. This results in a sign change of the SSE across SF, which
agrees with recent measurements on Cr2O3/Pt and Cr2O3/Ta devices [Li et al.,
Nature 578, 70 (2020)]. In our theory, SSE is due to a Néel spin current below
SF and a magnetic spin current above SF. Using the ratio of the associated
Néel to magnetic spin-mixing conductances as a single constant fitting
parameter, we reproduce the field dependence of the experimental data and
partially the temperature dependence of the relative SSE jump across SF.
Introduction.—SSE involves transfer of spin angular momentum between a magnet
and a metal via thermal spin fluctuations at their interface. In a typical
experiment, a heat flux injected across the interface pumps a spin current
into the metal, which is then converted into a transverse electric voltage
$V_{\mathrm{SSE}}$ by spin-orbit interactions. This spin-current generation
can be broadly attributed to two sources: One is due to a thermal gradient
inside the magnet, which produces bulk magnon transport Adachi _et al._
(2010); Rezende _et al._ (2014, 2016a); Flebus _et al._ (2017); Prakash _et
al._ (2018); Luo _et al._ (2019) and results in interfacial spin
accumulation. The other is due to the interfacial temperature discontinuity,
which produces spin pumping directly Xiao _et al._ (2010).
SSE has been studied in ferromagnets Slachter _et al._ (2010); Uchida _et
al._ (2010a), ferrimagnets Miao _et al._ (2016); Geprägs _et al._ (2016);
Ohnuma _et al._ (2013), paramagnets Wu _et al._ (2015); Li _et al._ (2019);
Yamamoto _et al._ (2019), and recently in antiferromagnets Seki _et al._
(2015); Wu _et al._ (2016); Li _et al._ (2020); Rezende _et al._ (2016b);
Troncoso _et al._ (2020) as well as noncollinear magnets Flebus _et al._
(2019); Ma _et al._ (2020). The sign of $V_{\mathrm{SSE}}$ is determined by
the polarization of the spin current along the applied magnetic field and the
effective spin Hall angle of the metal detector. Fixing the spin Hall angle
and the gyromagnetic ratio, the observed sign of the underlying spin current
turns out to contain valuable information about the nature of spin order in
the magnet and its nonequilibrium transport properties.
Collinear ferromagnets (FMs) or noncollinear systems with weak ferromagnetic
order have their net spin ordering along the magnetic field, whereas the
elementary low-energy magnon excitations yield average spin polarization in
the opposite direction. We can also imagine another class of systems, whose
intrinsic excitations form spin-degenerate bands, with the degeneracy lifted
by Zeeman splitting. The majority species, polarized along the field, may then
determine the sign of the spin current, thus ending up opposite to the FM
case. In our formalism, uniaxial AFs fall in this latter, majority-species
scenario below SF, switching to the ferromagnetic-like SSE behavior above SF.
Unlike argued in Ref. Hirobe _et al._ (2017), therefore, the SSE with the
sign opposite to the FM case is a not a unique signature of correlated spin
liquids, but can be expected to be a rather generic low-temperature signature
of materials lacking FM order.
Theoretically, there is at present no consensus on the “correct” sign of the
SSE in antiferromagnets. Rezende et al. Rezende _et al._ (2016b) developed a
magnon transport theory for uniaxial AFs below SF and concluded it falls into
the majority-species scenario (i.e., SSE opposite to the FM case), but did not
consider the sign when comparing their theory to experiment. Yamamoto et al.
Yamamoto _et al._ (2019) used the fluctuation-dissipation theorem in a
Landau-Ginzburg theory for easy-axis AFs below SF to study SSE around the Néel
temperature $T_{N}$, concluding paramagnets and AFs below SF both have the
same sign, but that it is the same as FMs. Here, we determine the sign within
a low-temperature, long-wavelength theory for the interfacial SSE and show it
changes across SF, in agreement with recent experiments. The quantitative
aspects of the SSE over a broad range of temperatures and magnetic fields also
appear in general agreement with the data.
Spin pumping near SF transition.—In easy-axis AFs, when the Zeeman energy due
to an applied field along the easy axis exceeds the anisotropy energy, there
is a metamagnetic phase transition called spin flop (SF). Below SF (state I),
the Néel order aligns with the easy axis, and there is a small net
magnetization due to remnant longitudinal magnetic susceptibility Nordblad
_et al._ (1979); Foner (1963). Dynamically, there are two circularly-polarized
spin-wave modes with opposite handedness. When quantized, they correspond to
magnons with magnetic moment parallel or antiparallel to the order parameter,
each forming a gas (with equal and opposite chemical potentials, if driven
slightly out of equilibrium Flebus (2019)). Above SF (state II), the Néel
order reorients into the hard plane, and the spins cant giving net
magnetization along the easy axis, due to a sizeable transverse magnetic
susceptibility. There are now two distinct spin-wave modes at long
wavelengths: a ferromagnetic-like mode ($\omega\rightarrow\gamma B$ when
applied field $B\rightarrow\infty$) and a low-energy Goldstone mode associated
with the U(1)-symmetry breaking Néel orientation in the hard plane. See Fig.
1.
Figure 1: $k=0$ resonance frequencies are plotted for an easy-axis AF:
$\omega_{1}$ and $\omega_{2}$ below spin flop and $\omega_{3}$ and
$\omega_{4}$ above spin flop. $B$ is the applied magnetic field,
$B_{c}=(\gamma s)^{-1}\sqrt{K_{1}/\chi}$ is the spin-flop field (which is
about 6 Tesla for Cr2O3) according to the energy (3), and $\omega_{0}=\gamma
B_{c}$ is the gap in I. The $\omega_{1}$ mode is right-hand circularly
polarized and $\omega_{2}$ is left-hand circularly polarized in
$\delta\boldsymbol{l}$ and $\delta\boldsymbol{m}$ (however the magnitude of
$\delta\boldsymbol{m}$ is a factor $\chi K_{1}$ smaller than
$\delta\boldsymbol{l}$ below SF, so it is omitted from the Figure).
$\omega_{3}$ is linearly polarized in $\delta\boldsymbol{l}$ and
$\delta\boldsymbol{m}$ so it does not produce spin currents bey . $\omega_{4}$
is linearly polarized in $\delta\boldsymbol{l}$ and elliptically polarized in
$\delta\boldsymbol{m}$.
The spin-current density pumped across the interface consist of the Néel,
$\boldsymbol{J}_{l}$, and magnetic, $\boldsymbol{J}_{m}$, contributions:
$\boldsymbol{J}_{l}=(\hbar
g^{\uparrow\downarrow}_{l}/4\pi)\,\boldsymbol{l}\times\partial_{t}\boldsymbol{l},~{}~{}~{}\boldsymbol{J}_{m}=(\hbar
g^{\uparrow\downarrow}_{m}/4\pi)\,\boldsymbol{m}\times\partial_{t}\boldsymbol{m},$
(1)
where $g^{\uparrow\downarrow}$ is the respective (real part of the
dimensionless) interfacial spin-mixing conductance per unit area. Thermal
agitations in the metal held at temperature $T_{e}$ and in the AF at $T_{a}$
produce contributions $\boldsymbol{J}_{e}$ and $\boldsymbol{J}_{a}$ to the
spin current, respectively. The spin Seebeck coefficient $S$ can be defined as
the net spin current $J_{s}$ (projected onto the direction of the applied
field) across the interface, divided by the temperature drop $\delta
T=T_{a}-T_{e}$:
$S\equiv J_{s}/\delta T=[J_{a}(T_{a})-J_{e}(T_{e})]/\delta
T\to\partial_{T}J_{a}(T),$ (2)
in linear response, where $J_{a}=J_{l}+J_{m}$ and $J_{e}(T)=J_{a}(T)$, in
thermal equilibrium.
In this paper, we investigate the signatures of SF in the SSE. In state I,
there are two components of the Néel spin current that contribute oppositely
to the SSE. With respect to increasing field, the (anti)parallel mode
(decreases) increases in frequency. The antiparallel mode thus has greater
thermal occupation at finite field, producing a net Néel spin current
antiparallel to the field Ohnuma _et al._ (2013); Rezende _et al._ (2016b).
In state II, there is only a magnetic spin current parallel to the field from
the FM-like mode. Therefore, the SSE changes sign across SF.
Spin-wave modes.—Following standard procedure Andreev and Marchenko (1980), we
construct the low-energy long-wavelength theory for AF dynamics in terms of
the Lagrangian density
$\mathcal{L}(\boldsymbol{l},\boldsymbol{m})=s\boldsymbol{m}\cdot(\boldsymbol{l}\times\partial\boldsymbol{l}/\partial
t)-E$. The energy density is given here by
$E(\boldsymbol{l},\boldsymbol{m})=A(\gradient{\boldsymbol{l}})^{2}/2+\boldsymbol{m}^{2}/2\chi-
K_{1}l_{z}^{2}/2-b\,\boldsymbol{m}\cdot\hat{\textbf{z}},$ (3)
for a bipartite easy-axis AF subjected to a collinear magnetic field. The AF
state is parametrized by directional Néel order $\boldsymbol{l}$ and
normalized spin density $\boldsymbol{m}=\mathbf{s}/s$ ($\mathbf{s}$ being the
spin density and $s\equiv\hbar S/V$, for spin $S$ and volume $V$ per site), in
a nonlinear $\sigma$ model with constraint $\boldsymbol{l}^{2}=1$ and
$\boldsymbol{l}\cdot\boldsymbol{m}=0$. We work well below the ordering
temperature $T_{N}$, retaining the lowest-order gradient term of the Néel
order with spin stiffness $A$. $\chi$ is the transverse magnetic
susceptibility, $K_{1}$ the easy-axis anisotropy, and $b\equiv\gamma sB$, in
terms of the magnetic field $B$ applied along the easy axis in the
$\hat{\textbf{z}}$ direction (where $\gamma$ is the gyromagnetic ratio, whose
sign is lumped into the value of $B$; i.e. when $\gamma<0$, our $B$ has
opposite sign to the applied field). The Euler-Lagrange equations of motion
may be extended to include dissipative forces
$\partial\mathcal{F}/\partial\dot{\boldsymbol{m}}$ and
$\partial\mathcal{F}/\partial\dot{\boldsymbol{l}}$ from the Rayleigh
dissipation functional
$\mathcal{F}=\alpha\dot{\boldsymbol{l}}^{2}/2+\widetilde{\alpha}\dot{\boldsymbol{m}}^{2}/2$,
parametrized by Gilbert damping constants $\alpha$ and $\widetilde{\alpha}$.
The ground states I and II are
$(\boldsymbol{l}_{0},\boldsymbol{m}_{0})_{\mathrm{I}}=(\hat{\textbf{z}},0)$
and
$(\boldsymbol{l}_{0},\boldsymbol{m}_{0})_{\mathrm{II}}=(\hat{\textbf{y}},\chi
b\hat{\textbf{z}})$, with the critical field $B_{c}$ marking the jump from I
to II. Spin waves are linear excitations,
$\boldsymbol{l}=\boldsymbol{l}_{0}+\delta\boldsymbol{l}$ and
$\boldsymbol{m}=\boldsymbol{m}_{0}+\delta\boldsymbol{m}$, satisfying the
equations of motion. The dispersions are
$\displaystyle\omega_{1k},\omega_{2k}=\mp\gamma B+\sqrt{(\gamma
B_{c})^{2}+(ck)^{2}},$ (4a)
$\displaystyle\omega_{3k}=ck,~{}~{}~{}\omega_{4k}=\sqrt{\gamma^{2}B^{2}-\gamma^{2}B_{c}^{2}+(ck)^{2}},$
(4b)
where $c=s^{-1}\sqrt{A/\chi}$ is the speed of the large-$k$ AF spin waves.
The six Cartesian components of $\delta\boldsymbol{l}$ and
$\delta\boldsymbol{m}$ reduce to four independent and two slave variables,
after applying the nonlinear constraints. Correspondingly, there are four
spin-wave modes with momentum $k$, as shown in Fig. 1 (for consistency of the
gradient expansion, we require $k\ll a^{-1}$, the inverse lattice spacing).
$\omega_{1k}$ and $\omega_{2k}$ are waves with circularly precessing
$\delta\boldsymbol{l}$ and $\delta\boldsymbol{m}$ in the plane perpendicular
to $\boldsymbol{l}_{0,\mathrm{I}}$. $\omega_{3k}$ has linearly polarized
$\delta\boldsymbol{l}(t)\propto e^{i\omega_{3k}t}\hat{\textbf{x}}$ and
$\delta\boldsymbol{m}(t)\propto(\omega_{3k}/\omega_{x})e^{i(\omega_{3k}t-\pi/2)}\hat{\textbf{z}}$
bey . $\omega_{4k}$ has linearly polarized $\delta\boldsymbol{l}(t)\propto
e^{i\omega_{4k}t}\hat{\textbf{z}}$ and elliptically polarized
$\delta\boldsymbol{m}(t)\propto(\omega_{4k}/\omega_{x})e^{i\omega_{4k}t}\hat{\textbf{x}}-\chi
be^{i(\omega_{3k}t-\pi/2)}\hat{\textbf{y}}$, where $\omega_{x}\equiv 1/\chi
s$. Additional anisotropy energy $-K_{2}l_{y}^{2}/2$ within the easy plane
will slightly shift the ground states, gap $\omega_{3}$, and introduce
ellipticities in precession. When $k_{B}T\gg(\hbar/s)\sqrt{K_{2}/\chi}$,
however, these modifications are negligible k2_ .
Main results.—A thermal heat flux driven across the AF interface with a metal
is given in the bulk by $-\sigma\gradient{T}$ and at the interface by
$-\kappa\delta T$, where $\sigma$ and $\kappa$ are, respectively, the bulk and
interfacial (Kapitza) thermal conductivities. $\delta T$ here is the
temperature difference between phonons in the AF and electrons in the metal,
$\delta T=T_{p}-T_{e}$ Xiao _et al._ (2010); Adachi _et al._ (2011). The
Kapitza resistance ($\kappa^{-1}$) is large when there is poor phonon-phonon
and phonon-electron interfacial coupling. For a fixed heat flux, this results
in a larger $\delta T$, which drives the local SSE. The temperature gradient
$\gradient{T}$ inside the magnet, furthermore, generates a bulk spin current,
which flows towards the interface and contributes to the measured SSE Hoffman
_et al._ (2013). We will specialize to the limit, in which the local spin
pumping $\propto\delta T$ dominates, which corresponds to the case of an
opaque interface and/or short spin-diffusion length in the AF.
Equipped with the theory for AF dynamics, based on the Hamiltonian (3), we can
use thermodynamic fluctuation-dissipation relations in order to convert
magnetic response into thermal noise. The spin Seebeck coefficient (2) can
then be evaluated by averaging Eqs. (1) over thermal fluctuations, whose
spectral features follow the spin-wave dispersions discussed above. Carrying
out this program, we arrive at the following final results (with the details
of the derivations discussed later): Below spin flop (state I),
$S_{\mathrm{I}}=\frac{g^{\uparrow\downarrow}_{l}\hbar^{2}}{2\pi\chi
s^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\omega_{2k}\partial_{T}n_{\rm
BE}(\omega_{2k})-\omega_{1k}\partial_{T}n_{\rm
BE}(\omega_{1k})}{\omega_{1k}+\omega_{2k}},$ (5)
and above spin flop (state II),
$S_{\mathrm{II}}=\frac{g^{\uparrow\downarrow}_{m}\hbar^{2}\chi\gamma
B}{2\pi}\int\frac{d^{3}k}{(2\pi)^{3}}\omega_{4k}\partial_{T}n_{\rm
BE}(\omega_{4k}),$ (6)
where $n_{\rm BE}(\omega)=(e^{\hbar\omega/k_{B}T}-1)^{-1}$ is the Bose-
Einstein distribution function.
We may evaluate the Seebeck coefficients analytically when
$k_{B}T\gg\hbar\gamma B_{c}$. Since they are both linear in $B$, we compare
the field slopes which go as $\partial_{B}S_{\mathrm{I}}\propto
g^{\uparrow\downarrow}_{l}T$ and $\partial_{B}S_{\mathrm{II}}\propto
g^{\uparrow\downarrow}_{m}T^{3}$:
$v(T)\equiv-\frac{\partial_{B}S_{\mathrm{I}}}{\partial_{B}S_{\mathrm{II}}}\approx\frac{g^{\uparrow\downarrow}_{l}}{g^{\uparrow\downarrow}_{m}}\left(\frac{\hbar/\chi
s}{k_{B}T}\right)^{2}\sim\frac{g^{\uparrow\downarrow}_{l}}{g^{\uparrow\downarrow}_{m}}\left(\frac{T_{N}}{T}\right)^{2}.$
(7)
The ratio $v(T)$ contains the square of exchange ($\propto T_{N}$) to thermal
energy in $v(T)$ (for the complete expressions, see s_I ). Note that for the
applicability of our long-wavelength description, we require that $T\ll
T_{N}$, throughout.
Comparison to experiment.—In a conventional measurement scheme, the
(longitudinal) SSE is revealed in a Nernst geometry as a lateral voltage
induced perpendicular to the magnetic field applied in the plane of the
magnetic interface Uchida _et al._ (2010a). This voltage is understood to
arise from the inverse spin Hall effect associated with the thermally injected
spin current. Normalizing the SSE voltage by the input thermal power
$P_{\mathrm{in}}$, this gives
$\frac{V_{\mathrm{SSE}}}{P_{\mathrm{in}}}=S(B,T)\frac{2e}{\hbar}\frac{\lambda^{*}}{wt}\frac{\rho(T)}{\kappa^{*}(T)},$
(8)
where the materials-dependent interfacial spin-to-charge conversion
lengthscale $\lambda^{*}$ can be loosely broken down into a product of an
effective spin-diffusion length (a.k.a. spin-memory loss)
$\lambda_{\mathrm{sd}}$ in the (heavy) normal metal and the effective spin
Hall angle $\theta_{\mathrm{sH}}$, which converts the spin-current density
$J_{\mathrm{s}}$ injected into the normal metal into the lateral charge-
current density $J_{c}=(2e/\hbar)\theta_{\mathrm{sH}}J_{s}$. The total charge
current is $I_{c}=w\lambda_{\mathrm{sd}}J_{c}$ when $\lambda_{\mathrm{sd}}\ll
t$, the thickness of the metal film, where $w$ is the heterostructure width
transverse to the injected charge current. In the open circuit, the underlying
spin Hall motive force Uchida _et al._ (2010b) is balanced by the detectable
voltage $V_{\mathrm{SSE}}=\rho lI_{c}/wt$, along the length $l$, where $\rho$
is the normal-metal resistivity. Putting everything together and expressing
the spin current in terms of the Seebeck coefficient (2), we get the SSE
voltage (8) normalized by the input power
$P_{\mathrm{in}}=\kappa(T_{p}-T_{\mathrm{e}})lw$.
$\kappa^{*}=\kappa(T_{p}-T_{e})/(T_{a}-T_{e})$ is an effective Kapitza
conductance, which can be reduced relative to $\kappa$, if the lengthscale for
the magnon-phonon equilibration that controls the temperature mismatch
$T_{a}-T_{p}$ in the AF is long compared to $\sigma/\kappa$.
Kapitza conductances for metal-insulator interfaces have been investigated in
Refs. Stoner _et al._ (1992); Stevens _et al._ (2005); Hohensee _et al._
(2015); Lu _et al._ (2016), yielding nontrivial temperature dependences. The
parameters for Cr2O3 are: $\sqrt{A}/a=(\chi\gamma s)^{-1}\approx 500$ T,
$B_{\mathrm{c}}\approx 6$ T, $\gamma\approx\gamma_{e}$ Li _et al._ (2020)
(where $\gamma_{e}$ is the free-electron value), $K_{2}\approx 0$ Foner
(1963); for the Cr2O3/Pt and Cr2O3/Ta devices: $w=0.2$ mm, $t=5$ nm, the
resistivity of the strips are $\rho_{\mathrm{Pt}}\approx 7\times
10^{-6}~{}\Omega\cdot$m and $\rho_{\mathrm{Ta}}\approx 9\times
10^{-5}~{}\Omega\cdot$m Li _et al._ (2020) at $T=75$ K, we take $\lambda^{*}$
from spin-pumping experiments: $\lambda_{\mathrm{Pt}}^{*}\sim 0.1$ nm Sinova
_et al._ (2015) and $\lambda_{\mathrm{Ta}}^{*}\sim-0.04$ nm Hahn _et al._
(2013); Gómez _et al._ (2014); Yu _et al._ (2018), we approximate
$g^{\uparrow\downarrow}_{m}$ for Pt and Ta with YIG/Pt’s:
$g^{\uparrow\downarrow}_{m}\sim 10$ nm-2 Zhang _et al._ (2015).
Figure 2: Theoretical spin Seebeck coefficients below, Eq. (5), and above, Eq.
(6), spin flop for Cr2O3 are compared to experimental data from Li et al. Li
_et al._ (2020). (a) and (b): The ratio
$g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ is fit to the relative
slopes across SF. c) $S(T)$ is plotted until $T=80$ K; at higher temperatures,
the long-wavelength theory loses quantitative accuracy. (d) Dispersions below
SF are plotted. The majority spin carrier has magnetic moment along the field,
which determines the polarization of the spin current.
The comparison of the Seebeck coefficients (5), (6) (which may be evaluated
analytically s_I ) to the data Li _et al._ (2020) is shown in Figs. 2(a)-(b).
We use the slope of experimental $V_{\mathrm{SSE}}/P_{\mathrm{in}}$ in I to
determine $\kappa_{\mathrm{Pt}}^{*}\sim 10^{9}$ W/m${}^{2}\cdot$K and
$\kappa_{\mathrm{Ta}}^{*}\sim 10^{10}$ W/m${}^{2}\cdot$K at $T=75$ K, which
are within 1-2 orders of magnitude of Stoner et al. measurements Stoner _et
al._ (1992) of $\kappa$ in diamond$|$heavy-metal films. We also use an
independent measurement of crystalline Cr2O3’s bulk thermal conductivity
$\sigma$ Yuan _et al._ (2018), giving us an associated length scale
$\sigma/\kappa_{\mathrm{Pt}}^{*}\approx 400$ nm and
$\sigma/\kappa_{\mathrm{Ta}}^{*}\approx 60$ nm. Since the thin-film
resistivities in our samples are about ten times larger than those in Refs.
Vlaminck _et al._ (2013); Dutta _et al._ (2017) for Pt, from which we use
the values for $\lambda_{\mathrm{Pt}}^{*}$ and $g^{\uparrow\downarrow}_{m}$
which go into determining $\kappa_{\mathrm{Pt}}^{*}$, the latter can only be
taken as giving us a rough order-of-magnitude guidance.
It should be safe to suppose that $\rho$, $\kappa^{*}$, and
$g^{\uparrow\downarrow}$ are largely field independent, so that the field
dependence in $V_{\mathrm{SSE}}/P_{\mathrm{in}}$ comes from $S$. The relative
value of $S(B)$ across SF is determined theoretically up to the ratio
$g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ gl_ , which is a
property of the interfaces. Several values are chosen in plotting Fig. 2. The
best fit is determined by comparing theoretical $v(T)$ s_I , defined in Eq. 7,
to the data at $T=75$ K. Note that $S|_{B=0}=0$, as expected on symmetry
grounds. However, it is nontrivial that the $S_{\rm II}(B)$ dependence
extrapolates to zero at zero field, both experimentally and in our theory.
The temperature dependence in the calculated spin Seebeck coefficient $S$
enter through the magnon occupation number in the fluctuation-dissipation
relation (9). The overall temperature dependence of the measured SSE is,
furthermore, convoluted with thermal and charge conductivities. There are also
slower temperature dependences in various parameters, such as $\chi(T)$ Foner
(1963), which can complicate a detailed analysis. By looking at the slope
ratio $v(T)$, however, we can eliminate the common prefactor associated with
the heat-to-spin-to-charge conversions [see Eq. (8)], if the signal is
dominated by the interfacial thermal bias. The experimental $v(T)$ for a bulk
Cr2O3/Pt sample is plotted in Fig. 3 along with theoretical curves. The
experimental data points for $v(T)$ are obtained by fitting a linear-in-field
line to $V_{\mathrm{SSE}}$ in states I and II and taking the ratio of the
slopes; for the theoretical curves see s_I . At low temperatures $T<7$ K, the
theoretical slopes start becoming nonlinear [so that
$S_{\mathrm{I}},S_{\mathrm{II}}$ must be evaluated numerically using Eqs. (5),
(6)], with $S_{\mathrm{II}}(B)$ at large fields being the first portion of
$S(B)$ to become nonlinear. Nonlinearities in $V_{\mathrm{SSE}}(B)$ are also
observed experimentally above SF at $T=5$ K Li _et al._ (2020).
While we see qualitative agreement, it appears there are additional spin
Seebeck contribution(s) not captured by our formalism. The latter can stem
from a bulk SSE in state I Lebrun _et al._ (2018), since thermal magnons
polarized along the Néel order can diffuse over long distances Prakash _et
al._ (2018). In particular, an additional linear in $T$ contribution to
$S_{\mathrm{I}}$ would affect the estimate of
$g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ from the low-$T$ data,
while a cubic contribution would explain the constant offset in $v(T)$ at
larger temperatures. There may also be additional contributions in I and II
due to other types of dynamics associated with interfacial inhomogeneities and
locally uncompensated moments. In order to fit the totality of experimental
data with our interfacial SSE-based model, we would require different values
of $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ as a function of
temperature. In particular, the data shown in Fig. 2a for $T=75$ K
(corresponding to the largest temperature data point in Fig. 3) is well
reproduced by taking
$g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}\approx 15$, while the
low temperature dependence of the data follows $v(T)\approx 160/T^{2}$
corresponding to $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}\approx
300$. Although the order-of-magnitude estimate for the mixing conductance
ratio and the trend in $v(T)$ as a function of temperature are reasonably
captured by our simple model, a more complete theory (accounting for the bulk
spin transport as well as for disorder-induced mesoscopic effects at the
interface) is needed for developing a detailed quantitative understanding.
Figure 3: The ratio of the spin Seebeck coefficient field slopes $v(T)$.
Experimental data is from the same device as in Fig. 2(a) and is obtained from
the slopes of linear-in-field fit lines, as discussed in the text. Theoretical
curves are based on Eq. (7), evaluated here s_I ; plotted for various
$g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$. The dashed line shows
an approximate fit to the data.
Theoretical formalism.—We calculate the spin currents in Eqs. (1) by averaging
over thermal fluctuations of the magnetic variables. The latter can be
obtained from the symmetrized fluctuation-dissipation theorem:
$\left\langle\delta\phi_{i}\delta\phi_{j}\right\rangle=\frac{i\hbar}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\left[\chi_{ji}^{*}(\mathbf{k},\omega)-\chi_{ij}(\mathbf{k},\omega)\right]N(\omega),$
(9)
where $\delta\phi_{i}$ stands for a Cartesian component of $\boldsymbol{l}$ or
$\boldsymbol{m}$ and $\chi_{ij}$ is the corresponding linear-response
function. $N(\omega)\equiv n_{\rm BE}(\omega)+1/2$ accounts for thermal
fluctuations associated with occupied modes, according to the Bose-Einstein
distribution function $n_{\rm BE}$, with $1/2$ reflecting the zero-point
motion Landau and Lifshitz (1980). The dynamic susceptibility tensor is
defined by $\delta\phi_{i}=\chi_{ij}\xi_{j}$, for the field $\xi_{j}$
thermodynamically conjugate to $\phi_{j}$. Our system is driven according to
the energy density
$E(B,t)=E(B)-\boldsymbol{m}\cdot\boldsymbol{h}(t)-\boldsymbol{l}\cdot\boldsymbol{g}(t)$,
where $\boldsymbol{g}$ and $\boldsymbol{h}$ are conjugate to $\boldsymbol{l}$
and $\boldsymbol{m}$, respectively. The off-diagonal components of the Néel
response $\chi^{(l)}_{ij}$ thus determine the Néel pumping as
$\left\langle\boldsymbol{l}\times\partial\boldsymbol{l}/\partial
t\right\rangle_{k}\to i\omega\epsilon^{ijk}\left\langle
l_{i}l_{j}\right\rangle$ (in terms of the Levi-Civita tensor $\epsilon^{ijk}$,
and upon the Fourier transform), and similarly for the magnetic response,
$\chi^{(m)}_{ij}$.
The components contributing to spin currents in I are
$\displaystyle\chi_{xy}^{(l)}=-\frac{i}{2s^{2}\chi\omega_{0k}}\left(\frac{1}{\omega-\omega_{1k}+i\epsilon}-\frac{1}{\omega-\omega_{2k}+i\epsilon}\right),$
(10a) $\displaystyle\chi_{xy}^{(m)}=\chi^{2}K_{1}^{2}\chi_{xy}^{(l)},$ (10b)
where $\omega_{0k}=\sqrt{(\gamma B_{c})^{2}+(ck)^{2}}$ and the dispersions are
given in Eq. (4). According to Eq. (10a), the fluctuations perpendicular to
$\boldsymbol{l}_{0,\mathrm{I}}=\hat{\textbf{z}}$ at $\omega_{1k}$ and
$\omega_{2k}$ produce opposite contributions to the spin currents. The
magnetic fluctuations in I in, e.g. Cr2O3, are a factor $(\chi K_{1})^{2}\sim
10^{-7}$ smaller than the Néel fluctuations and will be neglected. In II,
$\delta\boldsymbol{l}$ is linearly polarized in the $\omega_{3k}$ and
$\omega_{4k}$ modes, so Néel fluctuations do not produce spin currents bey .
$\delta\boldsymbol{m}$ is elliptically polarized in the $\omega_{4k}$ mode,
with magnetic fluctuations producing a spin current according to
$\chi_{xy}^{(m)}=i\gamma\chi
B\left(\frac{1}{\omega-\omega_{4k}+i\epsilon}\right).$ (11)
Without dissipation, the poles $\chi_{ij}\propto
1/(\omega-\omega_{k}+i\epsilon)$ at the resonance frequencies are shifted by
positive infinitesimal $\epsilon$. With dissipation, we end up with
Lorentzians centered at these poles, whose widths are determined by bulk
Gilbert damping and the effective damping due to interfacial spin pumping
Tserkovnyak _et al._ (2002); Hoffman _et al._ (2013). When these resonance
modes’ quality factors are large, however, their spectral weight is sharp and
may be simply integrated over. We will assume this is the case, allowing us to
neglect dissipation and simply use the infinitesimal $\epsilon$.
Conclusion and outlook.—Our theory specializes to SSE from spin currents
produced by an interfacial thermal bias. The formalism may be extended to
account for bulk thermal gradients, which produce nonequilibrium interfacial
spin accumulation $\boldsymbol{\mu}$. However, determining $\boldsymbol{\mu}$
requires complimenting the interfacial transport with coupled spin and heat
transport in the bulk Prakash _et al._ (2018), which is beyond our present
scope. The purely local SSE studied here should quantitatively model SSE for
interfaces with large interfacial thermal resistances and weak interfacial
spin coupling. In this regime, SSE would provide a noninvasive probe of the
magnet’s transverse components of $\chi_{ij}$, much like scanning tunneling
microscopy is an interfacial probe of an electron density of states Tersoff
and Hamann (1983).
We have discussed two classes of systems which produce different signs for
SSE. The FM-like class involves spin excitations with magnetic moment opposite
the order parameter, such as in FMs, uniaxial AFs above SF, and DMI AFs.
Another class involves degenerate spin excitations, whose degeneracy is lifted
by magnetic field. The majority carrier, which has magnetic moment along the
magnetic field, can then dominates spin transport. In our low-temperature,
long-wavelength theory we have shown that uniaxial AFs below SF belong to this
class. However, when the bulk SSE contribution is significant, this reasoning
alone may not determine the sign. Since the majority band reaches the edge of
the BZ faster than the minority, it may suffer greater umklapp scattering at
elevated temperatures, which would lower its conductivity. A full transport
theory is then required to determine the SSE sign, as a function of
temperature.
By comparing
$v(T)\equiv-\partial_{B}S_{\mathrm{I}}/\partial_{B}S_{\mathrm{II}}$ in
experiment to our theory as a function of $T$, we see some discrepancy. Our
theory predicts $v\propto 1/T^{2}$, while the Cr2O3/Pt sample indicates
$v(T)\approx 0.7+160/T^{2}$. The constant offset could stem from a bulk
Seebeck contribution in I at higher $T$ whose coefficient goes as $T^{3}$.
Above SF, bulk contributions to SSE can be expected to be reduced, since spin
transport is then normal to the Néel order. $v(T)$ may also have contributions
from paramagnetic impurities or other extrinsic surface modes, or be
convoluted with temperature dependence in
$g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$. The magnitude of
$g^{\uparrow\downarrow}_{l}$ and $g^{\uparrow\downarrow}_{m}$ can,
furthermore, vary from one sample to another due to the amount of disorder in
the interfacial exchange coupling Takei _et al._ (2014); Troncoso _et al._
(2020). While our theory well reproduces the temperature dependence at low
$T$, a different value of
$g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ is needed to
consistently explain higher temperature data. Looking forward, a more complete
theory is called for which includes SSE contributions from both the interface
and the bulk, in addition to the dynamical effects of disorder at the
interface.
The sensitivity of the SSE to the preparation and quality of the interface may
complicate the analysis based on the measured $v(T)$ across the SF. We recall
that Seki et al. Seki _et al._ (2015) did not observe a significant SSE in I
at low temperatures in Cr2O3/Pt. Wu et al. Wu _et al._ (2016) observed SSE
with nonlinear field dependence and ferromagnetic sign signature on both sides
of SF in MnF2/Pt. Ferromagnetic sign in I was also observed in an etched-
interface Cr2O3/Pt sample by Li et al. Li _et al._ (2020). Thus the origin of
the measured sign of the signal in I, and, therefore, the physical mechanism
of SSE are unclear for these cases. We also note that both Wu et al. Wu _et
al._ (2015) in paramagnetic SSE in GGG/Pt and Li et al. Li _et al._ (2020) in
Cr2O3/Pt at $T>T_{N}$ observed the ferromagnetic sign signature, suggesting
perhaps the importance of the magnon umklapp scattering in the bulk.
The work was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences under Award No. DE-SC0012190.
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|
2024-09-04T02:54:59.151337 | 2020-03-11T11:49:26 | 2003.05234 | {
"authors": "Absos Ali Shaikh, Chandan Kumar Mondal, Prosenjit Mandal",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26162",
"submitter": "Chandan Kumar Mondal",
"url": "https://arxiv.org/abs/2003.05234"
} | arxiv-papers | # Compact gradient $\rho$-Einstein soliton is isometric to the Euclidean
sphere
Absos Ali Shaikh1, Chandan Kumar Mondal2, Prosenjit Mandal3
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
In this paper we have investigated some aspects of gradient $\rho$-Einstein
Ricci soliton in a complete Riemannian manifold. First, we have proved that
the compact gradient $\rho$-Einstein soliton is isometric to the Euclidean
sphere by showing that the scalar curvature becomes constant. Second, we have
showed that in a non-compact gradient $\rho$-Einstein soliton satisfying some
integral condition, the scalar curvature vanishes.
††footnotetext: $\mathbf{2020}$ Mathematics Subject Classification: 53C20;
53C21; 53C44.
Key words and phrases: Gradient $\rho$-Einstein Ricci soliton; scalar
curvature; Riemannian manifold.
## 1\. Introduction and preliminaries
A $1$-parameter family of metrics $\\{g(t)\\}$ on a Riemannian manifold $M$,
defined on some time interval $I\subset\mathbb{R}$ is said to satisfy Ricci
flow if it satisfies
$\frac{\partial}{\partial t}g_{ij}=-2R_{ij},$
where $R_{ij}$ is the Ricci curvature with respect to the metric $g_{ij}$.
Hamilton [9] proved that for any smooth initial metric $g(0)=g_{0}$ on a
closed manifold, there exists a unique solution $g(t)$, $t\in[0,\epsilon)$, to
the Ricci flow equation for some $\epsilon>0$. A solution $g(t)$ of the Ricci
flow of the form
$g(t)=\sigma(t)\varphi(t)^{*}g(0),$
where $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a positive function and
$\varphi(t):M\rightarrow M$ is a 1-parameter family of diffeomorphisms, is
called a Ricci soliton. It is known that if the initial metric $g_{0}$
satisfies the equation
(1) $Ric(g_{0})+\frac{1}{2}\pounds_{X}g_{0}=\lambda g_{0},$
where $\lambda$ is a constant and $X$ is a smooth vector field on $M$, then
the manifold $M$ admits Ricci soliton. Therefore, the equation (1), in
general, is known as Ricci soliton. If $X$ is the gradient of some smooth
function, then it is called gradient Ricci soliton. For more results of Ricci
soliton see [2, 7, 8]. In 1979, Bourguignon [1] introduced the notion of
Ricci-Bourguignon flow, where the metrics $g(t)$ is evolving according to the
flow equation
$\frac{\partial}{\partial t}g_{ij}=-2R_{ij}+2\rho Rg_{ij},$
where $\rho$ is a non-zero scalar constant and $R$ is the scalar curvature of
the metric $g(t)$. Following the Ricci soliton, Catino and Mazzier [4] gave
the definition of gradient $\rho$-Einstein soliton, which is the self-similar
solution of Ricci-Bourguignon flow. This soliton is also called gradient
Ricci-Bourguignon soliton by some authors.
###### Definition 1.1.
[4] Let $(M,g)$ be a Riemannian manifold of dimension $n$, $(n\geq 3)$, and
let $\rho\in\mathbb{R}$, $\rho\neq 0$. Then $M$ is called gradient
$\rho$-Einstein soliton, denoted by $(M,g,f,\rho)$, if there is a smooth
function $f:M\rightarrow\mathbb{R}$ such that
(2) $Ric+\nabla^{2}f=\lambda g+\rho Rg,$
for some constant $\lambda$.
The soliton is trivial if $\nabla f$ is a parallel vector field. The function
$f$ is known as $\rho$-Einstein potential function. If $\lambda>0$
$(\text{resp.}=0,<0)$, then the gradient $\rho$-Einstein soliton
$(M,g,f,\rho)$ is said to be shrinking (resp. steady or expanding) . On the
other hand, the $\rho$-Einstein soliton is called gradient Einstein soliton,
gradient traceless Ricci soliton or gradient Schouten soliton if
$\rho=1/2,1/n$ or $1/2(n-1)$. Later, this notion has been generalized in
various directions such as $m$-quasi Einstein manifold [11], $(m,\rho)$-quasi
Einstein manifold [12], Ricci-Bourguignon almost soliton [13].
Catino and Mazzier [4] showed that compact gradient Einstein, Schouten or
traceless Ricci soliton is trivial. They classified three-dimensional gradient
shrinking Schouten soliton and proved that it is isometric to a finite
quotient of either $\mathbb{S}^{3}$ or $\mathbb{R}^{3}$ or
$\mathbb{R}\times\mathbb{S}^{2}$. Huang [10] deduced a sufficient condition
for the compact gradient shrinking $\rho$-Einstein soliton to be isometric to
a quotient of the round sphere $\mathbb{S}^{n}$.
###### Theorem 1.1.
[10] Let $(M,g,f,\rho)$ be an $n$-dimensional $(4\leq n\leq 5)$ compact
gradient shrinking $\rho$-Einstein soliton with $\rho<0$. If the following
condition holds
$\displaystyle\Big{(}\int_{M}|W+\frac{\sqrt{2}}{\sqrt{n}(n-2)}Z\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g|^{2}\Big{)}^{\frac{2}{n}}$
$\displaystyle+\sqrt{\frac{(n-4)^{2}(n-1)}{8(n-2)}}\lambda
vol(M)^{\frac{2}{n}}$ $\displaystyle\leq\sqrt{\frac{n-2}{32(n-1)}}Y(M,[g]),$
where $Z=Ric-\frac{R}{n}g$ is the trace-less Ricci tensor, $W$ is the Weyl
tensor and $Y(M,[g])$ is the Yamabe invariant associated to $(M,g)$, then $M$
is isometric to a quotient of the round sphere $\mathbb{S}^{n}$.
In 2019, Mondal and Shaikh [14] proved the isometry theorem for gradient
$\rho$-Einstein soliton in case of conformal vector field. In particular, they
proved the following result:
###### Theorem 1.2.
[14] Let $(M,g,f,\rho)$ be a compact gradient $\rho$-Einstein soliton. If
$\nabla f$ is a non-trivial conformal vector field, then $M$ is isometric to
the Euclidean sphere $\mathbb{S}^{n}$.
Dwivedi [13] proved an isometry theorem for gradient Ricci-Bourguignon
soliton.
###### Theorem 1.3.
[13] A non-trivial compact gradient Ricci-Bourguignon soliton is isometric to
an Euclidean sphere if any one of the following holds
(1) $M$ has constant scalar curvature.
(2) $\int_{M}g(\nabla R,\nabla f)\leq 0$.
(3) $M$ is a homogeneous manifold.
We note that Catino et. al. [5] proved many results for gradient
$\rho$-Einstein soliton in non-compact manifold.
###### Theorem 1.4.
Let $(M,g,f,\rho)$ be a complete non-compact gradient shrinking
$\rho$-Einstein soliton with $0<\rho<1/2(n-1)$ bounded curvature, non-negative
radial sectional curvature, and non-negative Ricci curvature. Then the scalar
curvature is constant.
In this paper, we have showed that a non-trivial compact gradient
$\rho$-Einstein soliton is isometric to an Euclidean sphere. The main results
of this paper are as follows:
###### Theorem 1.5.
A nontrivial compact gradient $\rho$-Einstein soliton has constant scalar
curvature and therefore $M$ is isometric to an Euclidean sphere.
We have also showed that in a non-compact gradient $\rho$-Einstein soliton
satisfying some conditions the scalar curvature vanishes.
###### Theorem 1.6.
Suppose $(M,g,f,\rho)$ is a non-compact gradient non-expanding $\rho$-Einstein
soliton with non-negative scalar curvature. If $\rho>1/n$ and the
$\rho$-Einstein potential function satisfies
(3) $\int_{M-B(p,r)}d(x,p)^{-2}f<\infty,$
then the scalar curvature vanishes in $M$.
## 2\. Proof of the results
###### Proof of the Theorem 1.5.
Since the gradient $\rho$-Einstein soliton is non-trivial, it follows that
$\rho\neq 1/n$, see [4]. Taking the trace of (2) we get
(4) $R+\Delta f=\lambda n+\rho Rn.$
From the commutative equation, we obtain
(5) $\Delta\nabla_{i}f=\nabla_{i}\Delta f+R_{ij}\nabla_{j}f.$
By using contracted second Bianchi identity, we have
$\displaystyle\Delta\nabla_{i}f=\nabla_{j}\nabla_{j}\nabla_{i}f$
$\displaystyle=$ $\displaystyle\nabla_{j}(\lambda g_{ij}+\rho Rg_{ij}-R_{ij})$
$\displaystyle=$ $\displaystyle\nabla_{i}(\rho R-\frac{1}{2}R).$
and
$\nabla_{i}\Delta f=\nabla_{i}(\lambda n+\rho Rn-R)=\nabla_{i}(\rho Rn-R).$
Therefore, (5) yields
(6) $(n-1)\rho\nabla_{i}R-\frac{1}{2}\nabla_{i}R+R_{ij}\nabla_{j}f=0,$
Taking covariant derivative $\nabla_{l}$, we get
$(n-1)\rho\nabla_{l}\nabla_{i}R-\frac{1}{2}\nabla_{l}\nabla_{i}R+\nabla_{l}R_{ij}\nabla_{j}f+R_{ij}\nabla_{l}\nabla_{j}f=0.$
Taking trace in both sides, we obtain
(7) $((n-1)\rho-\frac{1}{2})\Delta R+\frac{1}{2}g(\nabla R,\nabla f)+R(\lambda
n+\rho Rn-R)=0.$
Now integrating using divergence theorem we get
$\displaystyle\int_{M}R(\lambda n+\rho Rn-R)$ $\displaystyle=$
$\displaystyle-\int_{M}((n-1)\rho-\frac{1}{2})\Delta
R-\frac{1}{2}\int_{M}g(\nabla R,\nabla f)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}R\Delta f=\frac{1}{2}\int_{M}R(\lambda n+\rho
Rn-R).$
The above equation is true only if
(8) $\int_{M}R(\lambda n+\rho Rn-R)=0,$
which implies
(9) $\int_{M}R\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}=0,$
Again integrating (4), we obtain
(10) $\int_{M}\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}=0.$
Therefore, (9) and (10) together imply that
$\int_{M}\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}^{2}=0.$
Hence, $R=\lambda n/(1-\rho n)$. Then from Theorem 1.3 we can conclude our
result. ∎
###### Proof of the Theorem 1.6.
From (4) we get
$(n\rho-1)R=\Delta f-\lambda n.$
Since $\lambda\geq 0$, the above equation implies that
(11) $(n\rho-1)R\leq\Delta f.$
Now, we consider the cut-off function, introduced in [6], $\varphi_{r}\in
C^{2}_{0}(B(p,2r))$ for $r>0$ such that
$\begin{cases}0\leq\varphi_{r}\leq 1&\text{ in }B(p,2r)\\\
\varphi_{r}=1&\text{ in }B(p,r)\\\
|\nabla\varphi_{r}|^{2}\leq\frac{C}{r^{2}}&\text{ in }B(p,2r)\\\
\Delta\varphi_{r}\leq\frac{C}{r^{2}}&\text{ in }B(p,2r),\end{cases}$
where $C>0$ is a constant. Then for $r\rightarrow\infty$, we have
$\Delta\varphi^{2}_{r}\rightarrow 0$ as
$\Delta\varphi^{2}_{r}\leq\frac{C}{r^{2}}$. Then we calculate
(12)
$\displaystyle(n\rho-1)\int_{M}R\varphi^{2}_{r}\leq\int_{M}\varphi^{2}_{r}\Delta
f$ $\displaystyle=$ $\displaystyle\int_{B(p,2r)-B(p,r)}f\Delta\varphi_{r}^{2}$
(13) $\displaystyle\leq$
$\displaystyle\int_{B(p,2r)-B(p,r)}f\frac{C}{r^{2}}\rightarrow 0,$
as $r\rightarrow\infty$. Hence, we obtain
(14) $(n\rho-1)\lim_{r\rightarrow\infty}\int_{B(p,r)}R\leq 0.$
Since $\rho>1/n$, it follows that
$\lim_{r\rightarrow\infty}\int_{B(p,r)}R\leq 0.$
But $R$ is non-negative everywhere in $M$. Therefore, $R\equiv 0$ in $M$. ∎
## 3\. acknowledgment
The third author gratefully acknowledges to the CSIR(File
No.:09/025(0282)/2019-EMR-I), Govt. of India for financial assistance.
## References
* [1] Bourguignon, J. P., Ricci curvature and Einstein metrics. Global differential geometry and global analysis, Berlin, 1979, 42–63, Lecture Notes in Math. 838, Springer, Berlin, 1981.
* [2] Cao, H. D., Recent progress on Ricci solitons, in: Recent Advances in Geometric Analysis, Adv. Lectures Math., 11 (2010), 1–38.
* [3] Cao, H. D. and Zhou, D., On complee gradient shrnking Ricci solitons. J. Diff. Geom., 85 (2010), 175–183.
* [4] Catino, G. and Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal., 132 (2016), 66–94.
* [5] Catino, G., Mazzieri, L. and Mongodi, S., Rigidity of gradient Einstein shrinkers, Commun. Contemp. Math., 17(6) (2015), 1–18.
* [6] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., 144(1) (1996), 189–237.
* [7] Chow, B. and Knopf, D., The Ricci flow: An introduction, mathematical surveys and monographs. Amer. Math. Soc., 110, 2004.
* [8] Fang, F. Q., Man, J. W. and Zhang, Z. L., Complete gradient shrinking Ricci solitons have finite topological type. C. R. Acad. Sci. Paris, Ser. I 346(1971), 653–656.
* [9] Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255–306.
* [10] Huang, G., Integral pinched gradient shrinking $\rho$-Einstein solitons, J. Math. Ann. Appl. 451(2) (2017), 1045–1055.
* [11] Hu, Z., Li, D. and Xu, J., On generalized $m$-quasi-Einstein manifolds with constant scalar curvature, J. Math. Ann. Appl. 432(2) (2015), 733–743.
* [12] Huang, G. and Wei, Y., The classification of $(m,ρ)$-quasi-Einstein manifolds, Ann. Glob. Anal. geom. 44 (2013), 269–282.
* [13] Dwivedi, S., Some results on Ricci-Bourguignon and Ricci-Bourguignon almost solitons, arXiv:1809.11103.
* [14] Mondal, C. K. and Shaikh, A. A., Some results on $\eta$-Ricci Soliton and gradient $rho$-Einstein soliton in a complete Riemannian manifold, Comm. Korean Math. Soc. 34(4) (2019), 1279–1287.
|
2024-09-04T02:54:59.159884 | 2020-03-11T12:49:09 | 2003.05263 | {
"authors": "Florian-Horia Vasilescu",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26163",
"submitter": "Florian-Horia Vasilescu",
"url": "https://arxiv.org/abs/2003.05263"
} | arxiv-papers | # Spectrum and Analytic Functional Calculus in Real and Quaternionic
Frameworks
Florian-Horia Vasilescu
Department of Mathematics, University of Lille,
59655 Villeneuve d’Ascq, France
<EMAIL_ADDRESS>
###### Abstract
We present an approach to the spectrum and analytic functional calculus for
quaternionic linear operators, following the corresponding results concerning
the real linear operators. In fact, the construction of the analytic
functional calculus for real linear operators can be refined to get a similar
construction for quaternionic linear ones, in a classical manner, using a
Riesz-Dunford-Gelfand type kernel, and considering spectra in the complex
plane. A quaternionic joint spectrum for pairs of operators is also discussed,
and an analytic functional calculus is constructed, via a Martinelli type
kernel in two variables.
Keywords: spectrum in real and quaternionic contexts; holomorphic stem
functions; analytic functional calculus for real and quaternionic operators
AMS Subject Classification: 47A10; 30G35; 47A60
Keywords: real and quaternionic operators; spectra; analytic
functionalcalculus.
Mathematics Subject Classification 2010: 47A10; 47A60; 30G35
## 1 Introduction
In this text we consider ${\mathbb{R}}$-, ${\mathbb{C}}$-, and
${\mathbb{H}}$-linear operators, that is, real, complex and quaternionic
linear operators, respectively.
While the spectrum of a linear operator is traditionally defined for complex
linear operators, it is sometimes useful to have it also for real linear
operators, as well as for quaternionic linear ones. The definition of the
spectrum for a real linear operator goes seemingly back to Kaplansky (see
[10]), and it can be stated as follows. If $T$ is a real linear operator on
the real vector space ${\mathcal{V}}$, a point $u+iv$ ($u,v\in{\mathbb{R}}$)
is in the spectrum of $T$ if the operator $(u-T)^{2}+v^{2}$ is not invertible
on ${\mathcal{V}}$, where the scalars are identified with multiples of the
identity on ${\mathcal{V}}$. Although this definition involves only operators
acting in ${\mathcal{V}}$, the spectrum is, nevertheless, a subset of the
complex plane. As a matter of fact, a motivation of this choice can be
illustrated via the complexification of the space ${\mathcal{V}}$ (see Section
2).
The spectral theory for quaternionic linear operators is largely discussed in
numerous work, in particular in the monographs [5] and [4], and in many of
their references as well. In these works, the construction of an analytic
functional calculus (called $S$-analytic functional calculus) means to
associate to each function from the class of the so-called slice
hyperholomorphic or slice regular functions a quaternionic linear operator,
using a specific noncommutative kernel.
The idea of the present work is to replace the class of slice regular
functions by a class holomorphic functions, using a commutative kernel of type
Riesz- Dunford-Gelfand. These two classes are isomorphic via a Cauchy type
transform (see [21]), and the image of the analytic functional calculus is the
same, as one might expect (see Remark 8).
As in the case of real operators, the verbatim extension of the classical
definition of the spectrum for quaternionic operators is not appropriate, and
so a different definition using the squares of operators and real numbers was
given, which can be found in [5] (see also [4]). We discuss this definition in
our framework (see Definition 1), showing later that its ”complex border“
contains the most significant information, leading to the construction of an
analytic functional calculus, equivalent to that obtained via the slice
hyperholomorphic functions.
In fact, we first consider the spectrum for real operators on real Banach
spaces, and sketch the construction of an analytic functional calculus for
them, using some classical ideas (see Theorem 2). Then we extend this
framework to a quaternionic one, showing that the approach from the real case
can be easily adapted to the new situation.
As already mentioned, and unlike in [5] or [4], our functional calculus is
obtained via a Riesz-Dunford-Gelfand formula, defined in a partially
commutatative context, rather than the non-commutative Cauchy type formula
used by previous authors. Our analytic functional calculus holds for a class
of analytic operator valued functions, whose definition extends that of stem
functions, and it applies, in particular, to a large family of quaternionic
linear operators. Moreover, we can show that the analytic functional calculus
obtained in this way is equivalent to the analytic functional calculus
obtained in [5] or [4], in the sense that the images of these functional
calculi coincide (see Remark 8).
We finally discuss the case of pairs of commuting real operators, in the
spirit of [20], showing some connections with the quaternionic case.
Specifically, we define a quaternionic spectrum for them and construct an
analytic functional calculus using a Martinelli type formula, showing that for
such a construction only a sort of ”complex border“ of the quaternionic
spectrum should be used.
This work is just an introductory one. Hopefully, more contributions on this
line will be presented in the future.
## 2 Spectrum and Conjugation
Let ${\mathcal{A}}$ be a unital real Banach algebra, not necessarily
commutative. As mentioned in the Introduction, the (complex) spectrum of an
element $a\in{\mathcal{A}}$ may be defined by the equality
$\sigma_{\mathbb{C}}(a)=\\{u+iv;(u-a)^{2}+v^{2}\,\,{\rm
is\,\,not\,\,invertible},u,v\in{\mathbb{R}}\\},$ (1)
This set is conjugate symmetric, that is $u+iv\in\sigma_{\mathbb{C}}(a)$ if
and only if $u-iv\in\sigma_{\mathbb{C}}(a)$. A known motivation of this
definition comes from the following remark.
Fixing a unital real Banach algebra ${\mathcal{A}}$, we denote by
${\mathcal{A}}_{\mathbb{C}}$ the complexification of ${\mathcal{A}}$, which is
given by $A_{\mathbb{C}}={\mathbb{C}}\otimes_{\mathbb{R}}{\mathcal{A}}$,
written simply as ${\mathcal{A}}+i{\mathcal{A}}$, where the sum is direct,
identifying the element $1\otimes a+i\otimes b$ with the element $a+ib$, for
all $a,b\in{\mathcal{A}}$.
Then ${\mathcal{A}}_{\mathbb{C}}$ is a unital complex algebra, which can be
organized as a Banach algebra, with a (not necessarily unique) convenient
norm. To fix the ideas, we recall that the product of two elements is given by
$(a+ib)(c+id)=ac-bd+i(ad+bc)$ for all $a,b,c,d\in{\mathcal{A}}$, and the norm
may be defind by $\|a+ib\|=\|a\|+\|b\|$, where $\|*\|$ is the norm of
${\mathcal{A}}$.
In the algebra ${\mathcal{A}}_{\mathbb{C}}$, the complex numbers commute with
all elements of ${\mathcal{A}}$. Moreover, we have a conjugation given by
${\mathcal{A}}_{\mathbb{C}}\ni a+ib\mapsto
a-ib\in{\mathcal{A}}_{\mathbb{C}},\,a,b\in{\mathcal{A}},$
which is a unital conjugate-linear automorphism, whose square is the identity.
In particular, an arbitrary element $a+ib$ is invertible if and only if $a-ib$
is invertible.
The usual spectrum, defined for each element $a\in{\mathcal{A}}_{\mathbb{C}}$,
will be denoted by $\sigma(a)$. Regarding the algebra ${\mathcal{A}}$ as a
real subalgebra of ${\mathcal{A}}_{\mathbb{C}}$, one has the following.
###### Lemma 1
For every $a\in{\mathcal{A}}$ we have the equality
$\sigma_{\mathbb{C}}(a)=\sigma(a)$.
Proof. The result is well known but we give a short proof, because a similar
idea will be later used.
Let $\lambda=u+iv$ with $u,v\in{\mathbb{R}}$ arbitrary. Assuming $\lambda-a$
invertible, we also have $\bar{\lambda}-a$ invertible. From the obvious
identity
$(u-a)^{2}+v^{2}=(u+iv-a)(u-iv-a),$
we deduce that the element $(u-a)^{2}+v^{2}$ is invertible, implying the
inclusion $\sigma_{\mathbb{C}}(a)\subset\sigma(a)$.
Conversely, if $(u-a)^{2}+v^{2}$ is invertible, then both $u+iv-a,u-iv-a$ are
invertible via the decomposition from above, showing that we also have
$\sigma_{\mathbb{C}}(a)\supset\sigma(a)$.
###### Remark 1
The spectrum $\sigma(a)$ with $a\in{\mathcal{A}}$ is always a conjugate
symmetric set.
We are particularly interested to apply the discussion from above to the
context of linear operators. The spectral theory for real linear operators is
well known, and it is developed actually in the framework of linear relations
(see [1]). Nevertheless, we present here a different approach, which can be
applied, with minor changes, to the case of some quaternionic operators.
For a real or complex Banach space $\mathcal{V}$, we denote by
$\mathcal{B(V)}$ the algebra of all bounded ${\mathbb{R}}$-( respectively
${\mathbb{C}}$-)linear operators on $\mathcal{V}$. As before, the multiples of
the identity will be identified with the corresponding scalars.
Let ${\mathcal{V}}$ be a real Banach space, and let
${\mathcal{V}}_{\mathbb{C}}$ be its complexification, which, as above, is
identified with the direct sum ${\mathcal{V}}+i{\mathcal{V}}$. Each operator
$T\in\mathcal{B(V)}$ has a natural extension to an operator
$T_{\mathbb{C}}\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$, given by
$T_{\mathbb{C}}(x+iy)=Tx+iTy,\,x,y\in{\mathcal{V}}$. Moreover, the map
$\mathcal{B(V)}\ni T\mapsto
T_{\mathbb{C}}\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$ is unital,
${\mathbb{R}}$-linear and multiplicative. In particular, $T\in\mathcal{B(V)}$
is invertible if and only if
$T_{\mathbb{C}}\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$ is invertible.
Fixing an operator $S\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$, we define the
operator $S^{\flat}\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ to be equal to
$CSC$, where $C:{\mathcal{V}}_{\mathbb{C}}\mapsto{\mathcal{V}}_{\mathbb{C}}$
is the conjugation $x+iy\mapsto x-iy,\,x,y\in{\mathcal{V}}$. It is easily seen
that the map $\mathcal{B}(\mathcal{V}_{\mathbb{C}})\ni S\mapsto
S^{\flat}\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ is a unital conjugate-
linear automorphism, whose square is the identity on
$\mathcal{B}(\mathcal{V}_{\mathbb{C}})$. Because
$\mathcal{V}=\\{u\in\mathcal{V}_{\mathbb{C}};Cu=u\\}$, we have $S^{\flat}=S$
if and only if $S(\mathcal{V})\subset\mathcal{V}$. In particular, we have
$T_{\mathbb{C}}^{\flat}=T_{\mathbb{C}}$. In fact, because of the
representation
$S=\frac{1}{2}(S+S^{\flat})+i\frac{1}{2i}(S-S^{\flat}),\,\,S\in\mathcal{B}(\mathcal{V}_{\mathbb{C}}),$
where
$(S+S^{\flat})({\mathcal{V}})\subset{\mathcal{V}},i(S-S^{\flat})({\mathcal{V}})\subset{\mathcal{V}}$,
the algebras $\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ and
$\mathcal{B(V)}_{\mathbb{C}}$ are isomorphic and they will be often
identified, and $\mathcal{B(V)}$ will be regarded as a (real) subalgebra of
$\mathcal{B}(\mathcal{V})_{\mathbb{C}}$. In particular, if $S=U+iV$, with
$U,V\in\mathcal{B(V)}$, we have $S^{\flat}=U-iV$, so the map $S\mapsto
S^{\flat}$ is the conjugation of the complex algebra
$\mathcal{B}(\mathcal{V})_{\mathbb{C}}$ induced by the conjugation $C$ of
${\mathcal{V}}_{\mathbb{C}}$.
For every operator $S\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$, we denote, as
before, by $\sigma(S)$ its usual spectrum. As $\mathcal{B(V)}$ is a real
algebra, the (complex) spectrum of an operator $T\in\mathcal{B(V)}$ is given
by the equality (1):
$\sigma_{\mathbb{C}}(T)=\\{u+iv;(u-T)^{2}+v^{2}\,\,{\rm
is\,\,not\,\,invertible},u,v\in{\mathbb{R}}\\}.$
###### Corollary 1
For every $T\in\mathcal{B}({\mathcal{V}})$ we have the equality
$\sigma_{\mathbb{C}}(T)=\sigma(T_{\mathbb{C}})$.
## 3 Analytic Functional Calculus for Real
Operators
Having a concept of spectrum for real operators, an important step for further
development is the construction of an analytic functional calculus. Such a
construction has been done actually in the context of real linear relations in
[1]. In what follows we shall present a similar construction for real linear
operators. Although the case of linear relations looks more general, unlike in
[1], we perform our construction using a class of operator valued analytic
functions insted of scalar valued analytic functions. Moreover, our arguments
look simpler, and the construction is a model for a more general one, to get
an analytic functional calculus for quaternionic linear operators.
If ${\mathcal{V}}$ is a real Banach space, and so each operator
$T\in\mathcal{B}({\mathcal{V}})$ has a complex spectrum
$\sigma_{\mathbb{C}}(T)$, which is compact and nonempty, one can use the
classical Riesz-Dunford functional calculus, in a slightly generalized form
(that is, replacing the scalar-valued analytic functions by operator-valued
analytic ones, which is a well known idea).
The use of vector versions of the Cauchy formula is simplified by adopting the
following definition. Let $U\subset{\mathbb{C}}$ be open. An open subset
$\Delta\subset U$ will be called a Cauchy domain (in $U$) if
$\Delta\subset\bar{\Delta}\subset U$ and the boundary of $\Delta$ consists of
a finite family of closed curves, piecewise smooth, positively oriented. Note
that a Cauchy domain is bounded but not necessarily connected.
###### Remark 2
If $\mathcal{V}$ is a real Banach space, and $T\in\mathcal{B(V})$, we have the
usual analytic functional calculus for the operator
$T_{\mathbb{C}}\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ (see [6]). That is,
in a slightly generalized form, and for later use, if
$U\supset\sigma(T_{\mathbb{C}})$ is an open set in ${\mathbb{C}}$ and
$F:U\mapsto\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ is analytic, we put
$F(T_{\mathbb{C}})=\frac{1}{2\pi i}\int_{\Gamma}F(\zeta)(\zeta-
T_{\mathbb{C}})^{-1}d\zeta,$
where $\Gamma$ is the boundary of a Cauchy domain $\Delta$ containing
$\sigma(T_{\mathbb{C}})$ in $U$. In fact, because $\sigma(T_{\mathbb{C}})$ is
conjugate symmetric, we may and shall assume that both $U$ and $\Gamma$ are
conjugate symmetric. Because the function $\zeta\mapsto F(\zeta)(\zeta-
T_{\mathbb{C}})^{-1}$ is analytic in $U\setminus\sigma(T_{\mathbb{C}})$, the
integral does not depend on the particular choice of the Cauchy domain
$\Delta$ containing $\sigma(T_{C})$.
A natural question is to find an appropriate condition to we have
$F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$, which would imply the
invariance of $\mathcal{V}$ under $F(T_{\mathbb{C}})$.
With the notation of Remark 2, we have the following.
###### Theorem 1
Let $U\subset{\mathbb{C}}$ be open and conjugate symmetric. If
$F:U\mapsto\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ is analytic and
$F(\zeta)^{\flat}=F(\bar{\zeta})$ for all $\zeta\in U$, then
$F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$ for all
$T\in\mathcal{B}(\mathcal{V})$ with $\sigma_{\mathbb{C}}(T)\subset U$.
Proof. We use the notation from Remark 2, assuming, in addition, that $\Gamma$
is conjugate symmetric as well. We put
$\Gamma_{\pm}:=\Gamma\cap{\mathbb{C}}_{\pm}$, where ${\mathbb{C}}_{+}$ (resp.
${\mathbb{C}}_{-}$) equals to $\\{\lambda\in{\mathbb{C}};\Im\lambda\geq 0\\}$
(resp. $\\{\lambda\in{\mathbb{C}};\Im\lambda\leq 0\\}$). We write
$\Gamma_{+}=\cup_{j=1}^{m}\Gamma_{j+}$, where $\Gamma_{j+}$ are the connected
components of $\Gamma_{+}$. Similarly, we write
$\Gamma_{-}=\cup_{j=1}^{m}\Gamma_{j-}$, where $\Gamma_{j-}$ are the connected
components of $\Gamma_{-}$, and $\Gamma_{j-}$ is the reflexion of
$\Gamma_{j+}$ with respect of the real axis.
As $\Gamma$ is a finite union of Jordan piecewise smooth closed curves, for
each index $j$ we have a parametrization $\phi_{j}:[0,1]\mapsto{\mathbb{C}}$,
positively oriented, such that $\phi_{j}([0,1])=\Gamma_{j+}$. Taking into
account that the function $t\mapsto\overline{\phi_{j}(t)}$ is a
parametrization of $\Gamma_{j-}$ negatively oriented, and setting
$\Gamma_{j}=\Gamma_{j+}\cup\Gamma_{j-}$, we can write
$F_{j}(T_{\mathbb{C}}):=\frac{1}{2\pi i}\int_{\Gamma_{j}}F(\zeta)(\zeta-
T_{\mathbb{C}})^{-1}d\zeta=$ $\frac{1}{2\pi
i}\int_{0}^{1}F(\phi_{j}(t))(\phi_{j}(t)-T_{\mathbb{C}})^{-1}\phi_{j}^{\prime}(t)dt$
$-\frac{1}{2\pi
i}\int_{0}^{1}F(\overline{\phi_{j}(t)})(\overline{\phi_{j}(t)}-T_{\mathbb{C}})^{-1}\overline{\phi_{j}^{\prime}(t)}dt.$
Therefore,
$F_{j}(T_{\mathbb{C}})^{\flat}=-\frac{1}{2\pi
i}\int_{0}^{1}F(\phi_{j}(t))^{\flat}(\overline{\phi_{j}(t)}-T_{\mathbb{C}})^{-1}\overline{\phi_{j}^{\prime}(t)}dt$
$+\frac{1}{2\pi
i}\int_{0}^{1}F(\overline{\phi_{j}(t)})^{\flat}(\phi_{j}(t)-T_{\mathbb{C}})^{-1}\phi_{j}^{\prime}(t)dt.$
According to our assumption on the function $F$, we obtain
$F_{j}(T_{\mathbb{C}})=F_{j}(T_{\mathbb{C}})^{\flat}$ for all $j$, and
therefore
$F(T_{\mathbb{C}})^{\flat}=\sum_{j=1}^{m}F_{j}(T_{\mathbb{C}})^{\flat}=\sum_{j=1}^{m}F_{j}(T_{\mathbb{C}})=F(T_{\mathbb{C}}),$
which concludes the proof.
###### Remark 3
If ${\mathcal{A}}$ is a unital real Banach algebra,
${\mathcal{A}}_{\mathbb{C}}$ its complexification, and $U\subset{\mathbb{C}}$
is open, we denote by $\mathcal{O}(U,{\mathcal{A}}_{\mathbb{C}})$ the algebra
of all analytic ${\mathcal{A}}_{\mathbb{C}}$-valued functions. If $U$ is
conjugate symmetric, and ${\mathcal{A}}_{\mathbb{C}}\ni
a\mapsto\bar{a}\in{\mathcal{A}}_{\mathbb{C}}$ is its natural conjugation, we
denote by $\mathcal{O}_{s}(U,{\mathcal{A}}_{\mathbb{C}})$ the real subalgebra
of $\mathcal{O}(U,{\mathcal{A}}_{\mathbb{C}})$ consisting of those functions
$F$ with the property $F(\bar{\zeta})=\overline{F(\zeta)}$ for all $\zeta\in
U$. Adapting a well known terminology, such functions will be called
(${\mathcal{A}}_{\mathbb{C}}$-valued $)$ stem functions.
When ${\mathcal{A}}={\mathbb{R}}$, so
${\mathcal{A}}_{\mathbb{C}}={\mathbb{C}}$, the space
$\mathcal{O}_{s}(U,{\mathbb{C}})$ will be denoted by $\mathcal{O}_{s}(U)$,
which is a real algebra. Note that
$\mathcal{O}_{s}(U,{\mathcal{A}}_{\mathbb{C}})$ is also a bilateral
$\mathcal{O}_{s}(U)$-module.
In the next result, we identify the algebra $\mathcal{B}(\mathcal{V})$ with a
subalgebra of $\mathcal{B}(\mathcal{V})_{\mathbb{C}}$. In ths case, when
$F\in\mathcal{O}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}})$, we shall write
$F(T)=\frac{1}{2\pi i}\int_{\Gamma}F(\zeta)(\zeta-T)^{-1}d\zeta,$
noting that the right hand side of this formula belongs to
$\mathcal{B}(\mathcal{V})$, by Theorem 1.
The properties of the map $F\mapsto F(T)$, which can be called the (left)
analytic functional calculus of $T$, are summarized by the following.
###### Theorem 2
Let ${\mathcal{V}}$ be a real Banach space, let $U\subset{\mathbb{C}}$ be a
conjugate symmetric open set, and let $T\in\mathcal{B}(\mathcal{V})$, with
$\sigma_{\mathbb{C}}(T)\subset U$. Then the assignment
${\mathcal{O}}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}})\ni F\mapsto
F(T)\in\mathcal{B}(\mathcal{V})$
is an ${\mathbb{R}}$-linear map, and the map
${\mathcal{O}}_{s}(U)\ni f\mapsto f(T)\in\mathcal{B}(\mathcal{V})$
is a unital real algebra morphism.
Moreover, the following properties are true:
(1) For all
$F\in\mathcal{O}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}}),\,f\in{\mathcal{O}}_{s}(U)$,
we have $(Ff)(T)=F(T)f(T)$.
(2) For every polynomial
$P(\zeta)=\sum_{n=0}^{m}A_{n}\zeta^{n},\,\zeta\in{\mathbb{C}}$, with
$A_{n}\in\mathcal{B}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have
$P(T)=\sum_{n=0}^{m}A_{n}T^{n}\in\mathcal{B}(\mathcal{V})$.
Proof. The arguments are more or less standard (see [6]). The
${\mathbb{R}}$-linearity of the maps
${\mathcal{O}}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}})\ni F\mapsto
F(T)\in\mathcal{B}(\mathcal{V}),\,{\mathcal{O}}_{s}(U)\ni f\mapsto
f(T)\in\mathcal{B}(\mathcal{V}),$
is clear. The second one is actually multiplicative, which follows from the
multiplicativiry of the usual analytic functional calculus of $T$.
In fact, we have a more general property, specifically $(Ff)(T)=F(T)f(T)$ for
all
$F\in\mathcal{O}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}}),\,f\in{\mathcal{O}}_{s}(U)$.
This follows from the equalities,
$(Ff)(T)=\frac{1}{2\pi
i}\int_{\Gamma_{0}}F(\zeta)f(\zeta)(\zeta-T)^{-1}d\zeta=$ $\left(\frac{1}{2\pi
i}\int_{\Gamma_{0}}F(\zeta)(\zeta-T)^{-1}d\zeta\right)\left(\frac{1}{2\pi
i}\int_{\Gamma}f(\eta)(\eta-T)^{-1}d\eta\right)=F(T)f(T),$
obtained as in the classical case (see [6], Section VII.3), which holds
because $f$ is ${\mathbb{C}}$-valued and commutes with the operators in
$\mathcal{B}(\mathcal{V})$. Here $\Gamma,\,\Gamma_{0}$ are the boundaries of
two Cauchy domains $\Delta,\,\Delta_{0}$ respectively, such that
$\Delta\supset\bar{\Delta}_{0}$, and $\Delta_{0}$ contains $\sigma(T)$.
Note that, in particular, for every polynomial
$P(\zeta)=\sum_{n=0}^{m}A_{n}\zeta^{n}$ with
$A_{n}\in\mathcal{B}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have
$P(T)=\sum_{n=0}^{m}A_{n}q^{n}\in\mathcal{B}(\mathcal{V})$ for all
$T\in\mathcal{B}(\mathcal{V})$.
###### Example 1
Let $\mathcal{V}={\mathbb{R}}^{2}$, so
$\mathcal{V}_{\mathbb{C}}={\mathbb{C}}^{2}$, endowed with its natural Hilbert
space structure. Let us first observe that we have
$S=\left(\begin{array}[]{cc}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{array}\right)\,\,\Longleftrightarrow
S^{\flat}=\left(\begin{array}[]{cc}\bar{a}_{1}&\bar{a}_{2}\\\
\bar{a}_{3}&\bar{a}_{4}\end{array}\right),$
for all $a_{1},a_{2},a_{3},a_{4}\in{\mathbb{C}}$.
Next we consider the operator $T\in\mathcal{B}({\mathbb{R}}^{2})$ given by the
matrix
$T=\left(\begin{array}[]{cc}u&v\\\ -v&u\end{array}\right),$
where $u,v\in{\mathbb{R}},v\neq 0$. The extension $T_{\mathbb{C}}$ of the
operator $T$ to ${\mathbb{C}}^{2}$, which is a normal operator, is given by
the same formula. Note that
$\sigma_{\mathbb{C}}(T)=\\{\lambda\in{\mathbb{C}};(\lambda-u)^{2}+v^{2}=0\\}=\\{u\pm
iv\\}=\sigma(T_{\mathbb{C}}).$
Note also that the vectors $\nu_{\pm}=(\sqrt{2})^{-1}(1,\pm i)$ are normalized
eigenvectors for $T_{\mathbb{C}}$ corresponding to the eigenvalues $u\pm iv$,
respectively. The spectral projections of $T_{\mathbb{C}}$ corresponding to
these eigenvalues are given by
$E_{\pm}(T_{\mathbb{C}}){\bf w}=\langle{\bf
w},\nu_{\pm}\rangle\nu_{\pm}=\frac{1}{2}\left(\begin{array}[]{cc}1&\mp i\\\
\pm i&1\end{array}\right)\left(\begin{array}[]{c}w_{1}\\\
w_{2}\end{array}\right),$
for all ${\bf w}=(w_{1},w_{2})\in{\mathbb{C}}^{2}$.
Let $U\subset{\mathbb{C}}$ be an open set with $U\supset\\{u\pm iv\\}$, and
let $F:U\mapsto\mathcal{B}({\mathbb{C}}^{2})$ be analytic. We shall compute
explicitly $F(T_{\mathbb{C}})$. Let $\Delta$ be a Cauchy domain contained in
$U$ with its boundary $\Gamma$, and containing the points $u\pm iv$. Assuming
$v>0$, we have
$F(T_{\mathbb{C}})=\frac{1}{2\pi i}\int_{\Gamma}F(\zeta)(\zeta-
T_{\mathbb{C}})^{-1}d\zeta=$
$F(u+iv)E_{+}(T_{\mathbb{C}})+F(u-iv)E_{-}(T_{\mathbb{C}})=$
$\frac{1}{2}F(u+iv)\left(\begin{array}[]{cc}1&-i\\\
i&1\end{array}\right)+\frac{1}{2}F(u-iv)\left(\begin{array}[]{cc}1&i\\\
-i&1\end{array}\right).$
Assume now that $F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$. Then we must
have
$(F(u+iv)-F(u-iv)^{\flat})\left(\begin{array}[]{cc}1&-i\\\
i&1\end{array}\right)=(F(u+iv)^{\flat}-F(u-iv))\left(\begin{array}[]{cc}1&i\\\
-i&1\end{array}\right).$
We also have the equalities
$\left(\begin{array}[]{cc}1&-i\\\
i&1\end{array}\right)\left(\begin{array}[]{c}1\\\
i\end{array}\right)=2\left(\begin{array}[]{c}1\\\
i\end{array}\right),\,\,\left(\begin{array}[]{cc}1&-i\\\
i&1\end{array}\right)\left(\begin{array}[]{c}1\\\ -i\end{array}\right)=0,$
$\left(\begin{array}[]{cc}1&i\\\
-i&1\end{array}\right)\left(\begin{array}[]{c}1\\\
-i\end{array}\right)=2\left(\begin{array}[]{c}1\\\
-i\end{array}\right),\,\,\left(\begin{array}[]{cc}1&i\\\
-i&1\end{array}\right)\left(\begin{array}[]{c}1\\\ i\end{array}\right)=0,$
Using these equalities, we finally deduce that
$(F(u+iv)-F(u-iv)^{\flat})\left(\begin{array}[]{c}1\\\ i\end{array}\right)=0,$
and
$(F(u-iv)-F(u+iv)^{\flat})\left(\begin{array}[]{c}1\\\
-i\end{array}\right)=0,$
which are necessary conditions for the equality
$F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$. As a matter of fact, this
example shows, in particular, that the condition
$F(\zeta)^{\flat}=F(\bar{\zeta})$ for all $\zeta\in U$, used in Theorem 1, is
sufficient but it might not be always necessary.
## 4 Analytic Functional Calculus for Quaternionic
Operators
### 4.1 Quaternionic Spectrum
We now recall some known definitions and elementary facts (see, for instance,
[5], Section 4.6, and/or [21]).
Let ${\mathbb{H}}$ be the abstract algebra of quaternions, which is the four-
dimensional ${\mathbb{R}}$-algebra with unit $1$, generated by the ”imaginary
units“ $\\{\bf{j,k,l}\\}$, which satisfy
${\bf jk=-kj=l,\,kl=-lk=j,\,lj=-jl=k,\,jj=kk=ll}=-1.$
We may assume that ${\mathbb{H}}\supset{\mathbb{R}}$ identifying every number
$x\in{\mathbb{R}}$ with the element $x1\in{\mathbb{H}}$.
The algebra ${\mathbb{H}}$ has a natural multiplicative norm given by
$\|{\bf x}\|=\sqrt{x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{0}^{2}},\,\,{\bf
x}=x_{0}+x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf
l},\,\,x_{0},x_{1},x_{2},x_{3}\in{\mathbb{R}},$
and a natural involution
${\mathbb{H}}\ni{\bf x}=x_{0}+x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf
l}\mapsto{\bf x}^{*}=x_{0}-x_{1}{\bf j}-x_{2}{\bf k}-x_{3}{\bf
l}\in{\mathbb{H}}.$
Note that ${\bf x}{\bf x}^{*}={\bf x}^{*}{\bf x}=\|{\bf x}\|^{2}$, implying,
in particular, that every element ${\bf x}\in{\mathbb{H}}\setminus\\{0\\}$ is
invertible, and ${\bf x}^{-1}=\|{\bf x}\|^{-2}{\bf x}^{*}$.
For an arbitrary quaternion ${\bf x}=x_{0}+x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf
l},\,\,x_{0},x_{1},x_{2},x_{3}\in{\mathbb{R}}$, we set $\Re{\bf x}=x_{0}=({\bf
x}+{\bf x}^{*})/2$, and $\Im{\bf x}=x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf
l}=({\bf x}-{\bf x}^{*})/2$, that is, the real and imaginary part of ${\bf
x}$, respectively.
We consider the complexification
${\mathbb{C}}\otimes_{\mathbb{R}}{\mathbb{H}}$ of the ${\mathbb{R}}$-algebra
${\mathbb{H}}$ (see also [8]), which will be identified with the direct sum
${\mathbb{M}}={\mathbb{H}}+i{\mathbb{H}}$. Of course, the algebra
${\mathbb{M}}$ contains the complex field ${\mathbb{C}}$. Moreover, in the
algebra ${\mathbb{M}}$, the elements of ${\mathbb{H}}$ commute with all
complex numbers. In particular, the ”imaginary units“ $\bf j,k,l$ of the
algebra ${\mathbb{H}}$ are independent of and commute with the imaginary unit
$i$ of the complex plane ${\mathbb{C}}$.
In the algebra ${\mathbb{M}}$, there also exists a natural conjugation given
by $\bar{\bf a}={\bf b}-i{\bf c}$, where ${\bf a}={\bf b}+i{\bf c}$ is
arbitrary in ${\mathbb{M}}$, with ${\bf b},{\bf c}\in{\mathbb{H}}$ (see also
[8]). Note that $\overline{\bf a+b}=\bar{\bf a}+\bar{\bf b}$, and
$\overline{\bf ab}=\bar{\bf a}\bar{\bf b}$, in particular $\overline{r\bf
a}=r\bar{\bf a}$ for all ${\bf a},{\bf b}\in{\mathbb{M}}$, and
$r\in{\mathbb{R}}$. Moreover, $\bar{{\bf a}}={\bf a}$ if and only if ${\bf
a}\in{\mathbb{H}}$, which is a useful characterization of the elements of
${\mathbb{H}}$ among those of ${\mathbb{M}}$.
###### Remark 4
In the algebra ${\mathbb{M}}$ we have the identities
$(\lambda-{\bf x}^{*})(\lambda-{\bf x})=(\lambda-{\bf x})(\lambda-{\bf
x}^{*})=\lambda^{2}-\lambda({\bf x}+{\bf x}^{*})+\|{\bf
x}\|^{2}\in{\mathbb{C}},$
for all $\lambda\in{\mathbb{C}}$ and ${\bf x}\in{\mathbb{H}}$. If the complex
number $\lambda^{2}-2\lambda\Re{\bf x}+\|{\bf x}\|^{2}$ is nonnull, then both
element $\lambda-{\bf x}^{*},\,\lambda-{\bf x}$ are invertible. Conversely, if
$\lambda-{\bf x}$ is invertible, we must have $\lambda^{2}-2\lambda\Re{\bf
x}+\|{\bf x}\|^{2}$ nonnull; otherwise we would have $\lambda={\bf
x}^{*}\in{\mathbb{R}}$, so $\lambda={\bf x}\in{\mathbb{R}}$, which is not
possible. Therefore, the element $\lambda-{\bf x}\in{\mathbb{M}}$ is
invertible if and only if the complex number $\lambda^{2}-2\lambda\Re{\bf
x}+\|{\bf x}\|^{2}$ is nonnull. Hence, the element $\lambda-{\bf
x}\in{\mathbb{M}}$ is not invertible if and only if $\lambda=\Re{\bf x}\pm
i\|\Im{\bf x}\|$. In this way, the spectrum of a quaternion ${\bf
x}\in{\mathbb{H}}$ is given by the equality $\sigma({\bf x})=\\{s_{\pm}(\bf
x)\\}$, where $s_{\pm}(\bf x)=\Re{\bf x}\pm i\|\Im{\bf x}\|$ are the
eigenvalues of $\bf x$ (see also [20, 21]).
The polynomial $P_{\bf x}(\lambda)=\lambda^{2}-2\lambda\Re{\bf x}+\|{\bf
x}\|^{2}$ is the minimal polynomial of $\bf x$. In fact, the equality
$\sigma({\bf y})=\sigma({\bf x})$ for some ${\bf x,y}\in{\mathbb{H}}$ is an
equivalence relation in the algebra ${\mathbb{H}}$, which holds if and only if
$P_{\bf x}=P_{\bf y}$. In fact, setting
$\mathbb{S}=\\{\mathfrak{\kappa}\in{\mathbb{H}};\Re\mathfrak{\kappa}=0,\|\mathfrak{\kappa}\|=1\\}$
(that is the unit sphere of purely imaginary quaternions), representig an
arbitrary quaternion $\bf x$ under the form
$x_{0}+y_{0}\mathfrak{\kappa}_{0}$, with $x_{0},y_{0}\in{\mathbb{R}}$ and
$\mathfrak{\kappa}_{0}\in\mathbb{S}$, a quaternion $\bf y$ is equivalent to
$\bf x$ if anf only if it is of the form $x_{0}+y_{0}\mathfrak{\kappa}$ for
some $\mathfrak{\kappa}\in\mathbb{S}$ (see [3] or [21] for some details).
###### Remark 5
Following [5], a right ${\mathbb{H}}$-vector space $\mathcal{V}$ is a real
vector space having a right multiplication with the elements of
${\mathbb{H}}$, such that $(x+y){\bf q}=x{\bf q}+y{\bf q},\,x({\bf q}+{\bf
s})=x{\bf q}+x{\bf s},\,x({\bf q}{\bf s})=(x{\bf q}){\bf s}$ for all
$x,y\in\mathcal{V}$ and ${\bf q},{\bf s}\in{\mathbb{H}}$.
If $\mathcal{V}$ is also a Banach space the operator $T\in\mathcal{B(V)}$ is
right ${\mathbb{H}}$-linear if $T(x{\bf q})=T(x){\bf q}$ for all
$x\in\mathcal{V}$ and ${\bf q}\in{\mathbb{H}}$. The set of right
${\mathbb{H}}$ linear operators will be denoted by $\mathcal{B^{\rm r}(V)}$,
which is, in particular, a unital real algebra.
In a similar way, one defines the concept of a left ${\mathbb{H}}$-vector
space. A real vector space $\mathcal{V}$ will be said to be an
${\mathbb{H}}$-vector space if it is simultaneously a right ${\mathbb{H}}$\-
and a left ${\mathbb{H}}$-vector space. As noticed in [5], it is the framework
of ${\mathbb{H}}$-vector spaces an appropriate one for the study of right
${\mathbb{H}}$-linear operators.
If ${\mathcal{V}}$ is ${\mathbb{H}}$-vector space which is also a Banach
space, then ${\mathcal{V}}$ is said to be a Banach ${\mathbb{H}}$-space. In
this case, we also assume that $R_{\bf q}\in\mathcal{B}({\mathcal{V}})$, and
the map ${\mathbb{H}}\ni{\bf q}\mapsto R_{\bf q}\in\mathcal{B}({\mathcal{V}})$
is norm continuous, where $R_{\bf q}$ is the right multiplication of the
elements of $\mathcal{V}$ by a given quaternion ${\bf q}\in{\mathbb{H}}$.
Similarly, if $L_{\bf q}$ is the left multiplication of the elements of
$\mathcal{V}$ by the quaternion ${\bf q}\in{\mathbb{H}}$, we assume that
$L_{\bf q}\in\mathcal{B}({\mathcal{V}})$ for all ${\bf q}\in{\mathbb{H}}$, and
that the map ${\mathbb{H}}\ni{\bf q}\mapsto L_{\bf
q}\in\mathcal{B}({\mathcal{V}})$ is norm continuous. Note also that
$\mathcal{B^{\rm r}(V)}=\\{T\in\mathcal{B(V)};TR_{\bf q}=R_{\bf q}T,\,{\bf
q}\in{\mathbb{H}}\\}.$
To adapt the discussion regarding the real algebras to this case, we first
consider the complexification ${\mathcal{V}}_{\mathbb{C}}$ of ${\mathcal{V}}$.
Because ${\mathcal{V}}$ is an ${\mathbb{H}}$-bimodule, the space
${\mathcal{V}}_{\mathbb{C}}$ is actually an ${\mathbb{M}}$-bimodule, via the
multiplications
$({\bf q}+i{\bf s})(x+iy)={\bf q}x-{\bf s}y+i({\bf q}y+{\bf s}x),(x+iy)({\bf
q}+i{\bf s})=x{\bf q}-y{\bf s}+i(y{\bf q}+x{\bf s}),$
for all ${\bf q}+i{\bf s}\in{\mathbb{M}},\,{\bf q},{\bf
s}\in{\mathbb{H}},\,x+iy\in{\mathcal{V}}_{\mathbb{C}},\,x,y\in{\mathcal{V}}$.
Moreover, the operator $T_{\mathbb{C}}$ is right ${\mathbb{M}}$-linear, that
is $T_{\mathbb{C}}((x+iy)({\bf q}+i{\bf s}))=T_{\mathbb{C}}(x+iy)({\bf
q}+i{\bf s})$ for all ${\bf q}+i{\bf
s}\in{\mathbb{M}},\,x+iy\in{\mathcal{V}}_{\mathbb{C}}$, via a direct
computation.
Let $C$ be the conjugation of ${\mathcal{V}}_{\mathbb{C}}$. As in the real
case, for every $S\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$, we put
$S^{\flat}=CSC$. The left and right multiplication with the quaternion ${\bf
q}$ on ${\mathcal{V}}_{\mathbb{C}}$ will be also denoted by $L_{\bf q},R_{\bf
q}$, respectively, as elements of $\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$.
We set
$\mathcal{B}^{\rm
r}({\mathcal{V}}_{\mathbb{C}})=\\{S\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}});SR_{\bf
q}=R_{\bf q}S,\,{\bf q}\in{\mathbb{H}}\\},$
which is a unital complex algebra containing all operators $L_{\bf q},{\bf
q}\in{\mathbb{H}}$. Note that if $S\in\mathcal{B}^{\rm
r}({\mathcal{V}}_{\mathbb{C}})$, then $S^{\flat}\in\mathcal{B}^{\rm
r}({\mathcal{V}}_{\mathbb{C}})$. Indeed, because $CR_{\bf q}=R_{\bf q}C$, we
also have $S^{\flat}R_{\bf q}=R_{\bf q}S^{\flat}$. In fact, as we have
$(S+S^{\flat})({\mathcal{V}})\subset{\mathcal{V}}$ and
$i(S-S^{\flat})({\mathcal{V}})\subset{\mathcal{V}}$, it folows that the
algebras $\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb{C}}),\,\mathcal{B}^{\rm
r}({\mathcal{V}})_{\mathbb{C}}$ are isomorphic, and they will be often
identified, where $\mathcal{B^{\rm r}(V)}_{\mathbb{C}}=\mathcal{B^{\rm
r}(V)}+i\mathcal{B^{\rm r}(V)}$ is the complexification of $\mathcal{B^{\rm
r}(V)}$, which is also a unital complex Banach algebra.
Looking at the Definition 4.8.1 from [5] (see also [4]), we give the folowing.
###### Definition 1
For a given operator $T\in\mathcal{B^{\rm r}(V)}$, the set
$\sigma_{\mathbb{H}}(T):=\\{{\bf q}\in{\mathbb{H}};T^{2}-2(\Re{\bf q})T+\|{\bf
q}\|^{2})\,\,{\rm not}\,\,{\rm invertible}\\}$
is called the quaternionic spectrum (or simply the $Q$-spectrum) of $T$.
The complement
$\rho_{\mathbb{H}}(T)={\mathbb{H}}\setminus\sigma_{\mathbb{H}}(T)$ is called
the quaternionic resolvent (or simply the $Q$-resolvent) of $T$.
Note that, if ${\bf q}\in\sigma_{\mathbb{H}}(T)$), then $\\{{\bf
s}\in{\mathbb{H}};\sigma({\bf s})=\sigma({\bf
q})\\}\subset\sigma_{\mathbb{H}}(T)$.
Assuming that ${\mathcal{V}}$ is a Banach ${\mathbb{H}}$-space, then
$\mathcal{B^{\rm r}(V)}$ is a unital real Banach ${\mathbb{H}}$-algebra (that
is, a Banach algebra which also a Banach ${\mathbb{H}}$-space), via the
algebraic operations $({\bf q}T)(x)={\bf q}T(x)$, and $(T{\bf q})(x)=T({\bf
q}x)$ for all ${\bf q}\in{\mathbb{H}}$ and $x\in{\mathcal{V}}$. Hence the
complexification $\mathcal{B^{\rm r}(V)}_{\mathbb{C}}$ is, in particular, a
unital complex Banach algebra. Also note that the complex numbers, regarded as
elements of $\mathcal{B^{\rm r}(V)}_{\mathbb{C}}$, commute with the elements
of $\mathcal{B^{\rm r}(V)}$. For this reason, for each $T\in\mathcal{B^{\rm
r}(V)}$ we have the resolvent set
$\rho_{\mathbb{C}}(T)=\\{\lambda\in{\mathbb{C}};(T^{2}-2(\Re\lambda)T+|\lambda|^{2})^{-1}\in\mathcal{B^{\rm
r}(V)}\\}=$ $\\{\lambda\in{\mathbb{C}};(\lambda-
T_{\mathbb{C}})^{-1}\in\mathcal{B^{\rm
r}(V}_{\mathbb{C}})\\}=\rho(T_{\mathbb{C}}),$
and the associated spectrum $\sigma_{\mathbb{C}}(T)=\sigma(T_{\mathbb{C}})$.
Clearly, there exists a strong connexion between $\sigma_{\mathbb{H}}(T)$ and
$\sigma_{\mathbb{C}}(T)$. In fact, the set $\sigma_{\mathbb{C}}(T)$ looks like
a ”complex border“ of the set $\sigma_{\mathbb{H}}(T)$. Specifically, we can
prove the following.
###### Lemma 2
For every $T\in\mathcal{B^{\rm r}(V)}$ we have the equalities
$\sigma_{\mathbb{H}}(T)=\\{{\bf
q}\in{\mathbb{H}};\sigma_{\mathbb{C}}(T)\cap\sigma({\bf q})\neq\emptyset\\}.$
(2)
and
$\sigma_{\mathbb{C}}(T)=\\{\lambda\in\sigma({\bf q});{\bf
q}\in\sigma_{\mathbb{H}}(T)\\}.$ (3)
Proof. Let us prove (2). If ${\bf q}\in\sigma_{\mathbb{H}}(T)$, and so the
$T^{2}-2(\Re{\bf q})T+\|{\bf q}\|^{2}$ is not invertible, choosing
$\lambda\in\\{\Re{\bf q}\pm i\|\Im{\bf q}\|\\}=\sigma({\bf q})$, we clearly
have $T^{2}-2(\Re\lambda)T+|\lambda|^{2}$ not invertible, implying
$\lambda\in\sigma_{\mathbb{C}}(T)\cap\sigma({\bf q})\neq\emptyset$.
Conversely, if for some ${\bf q}\in{\mathbb{H}}$ there exists
$\lambda\in\sigma_{\mathbb{C}}(T)\cap\sigma({\bf q})$, and so
$T^{2}-2(\Re\lambda)T+|\lambda|^{2}=T^{2}-2(\Re{\bf q})T+\|{\bf q}\|^{2}$ is
not invertible, implying ${\bf q}\in\sigma_{\mathbb{H}}(T)$.
We now prove (3). Let $\lambda\in\sigma_{\mathbb{C}}(T)$, so the operator
$T^{2}-2(\Re\lambda)T+|\lambda|^{2}$ is not invertible. Setting ${\bf
q}=\Re(\lambda)+\|\Im\lambda\|\kappa$, with $\kappa\in\mathbb{S}$, we have
$\lambda\in\sigma({\bf q})$. Moreover, $T^{2}+2(\Re{\bf q})T+\|{\bf q}\|^{2}$
is not invertible, and so ${\bf q}\in\sigma_{\mathbb{H}}(T)$.
Conversely, if $\lambda\in\sigma({\bf q})$ for some ${\bf
q}\in\sigma_{\mathbb{H}}(T)$, then $\lambda\in\\{\Re{\bf q}\pm i\|\Im({\bf
q})\|\\}$, showing that $T^{2}-2\Re(\lambda)T+|\lambda|^{2}=T^{2}+2(\Re{\bf
q})T+\|{\bf q}\|^{2}$ is not invertible.
Remark As expected, the set $\sigma_{\mathbb{H}}(T)$ is nonempty and bounded,
which follows easily from Lemma 2. It is also compact, as a consequence of
Definition 1, because the set of invertible elements in $\mathcal{B^{\rm
r}(V)}$ is open.
We recall that a subset $\Omega\subset{\mathbb{H}}$ is said to be spectrally
saturated (see [20],[21]) if whenever $\sigma({\bf h})=\sigma({\bf q})$ for
some ${\bf h}\in{\mathbb{H}}$ and ${\bf q}\in\Omega$, we also have ${\bf
h}\in\Omega$. As noticed in [20] and [21], this concept coincides with that of
axially symmetric set, introduced in [5].
Note that the subset $\sigma_{\mathbb{H}}(T)$ spectrally saturated.
### 4.2 Analytic Functional Calculus
If ${\mathcal{V}}$ is a Banach ${\mathbb{H}}$-space, because $\mathcal{B^{\rm
r}({\mathcal{V}})}$ is real Banach space, each operator $T\in\mathcal{B^{\rm
r}({\mathcal{V}})}$ has a complex spectrum $\sigma_{\mathbb{C}}(T)$.
Therefore, applying the corresponding result for real operators, we may
construct an analytic functional calculus using the classical Riesz-Dunford
functional calculus, in a slightly generalized form. In this case, our basic
complex algebra is $\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}$, endowed
with the conjugation $\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}\ni
S\mapsto S^{\flat}\in\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}$.
###### Theorem 3
Let $U\subset{\mathbb{C}}$ be open and conjugate symmetric. If
$F:U\mapsto\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb{C}})$ is analytic and
$F(\zeta)^{\flat}=F(\bar{\zeta})$ for all $\zeta\in U$, then
$F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$ for all $T\in\mathcal{B}^{\rm
r}(\mathcal{V})$ with $\sigma_{\mathbb{C}}(T)\subset U$.
Both the statement and the proof of Theorem 3 are similar to those of Theorem
1, and will be omitted.
As in the real case, we may identify the algebra $\mathcal{B}^{\rm
r}(\mathcal{V})$ with a subalgebra of $\mathcal{B}^{\rm
r}(\mathcal{V})_{\mathbb{C}}$. In ths case, when
$F\in\mathcal{O}_{s}(U,\mathcal{B}^{\rm
r}(\mathcal{V})_{\mathbb{C}})=\\{F\in{\mathcal{O}}(U,\mathcal{B}^{\rm
r}({\mathcal{V}})_{\mathbb{C}});F(\bar{\zeta})=F(\zeta)^{\flat}\,\,\forall\zeta\in
U\\}$ (see also Remark 3), we can write, via the previous Theorem,
$F(T)=\frac{1}{2\pi
i}\int_{\Gamma}F(\zeta)(\zeta-T)^{-1}d\zeta\in\mathcal{B}^{\rm
r}(\mathcal{V}),$
for a suitable choice of $\Gamma$.
The next result provides an analytic functional calculus for operators from
the real algebra $\mathcal{B}^{\rm r}(\mathcal{V})$.
###### Theorem 4
Let ${\mathcal{V}}$ be a Banach ${\mathbb{H}}$-space, let
$U\subset{\mathbb{C}}$ be a conjugate symmetric open set, and let
$T\in\mathcal{B}^{\rm r}(\mathcal{V})$, with $\sigma_{\mathbb{C}}(T)\subset
U$. Then the map
${\mathcal{O}}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}})\ni
F\mapsto F(T)\in\mathcal{B}^{\rm r}(\mathcal{V})$
is ${\mathbb{R}}$-linear, and the map
${\mathcal{O}}_{s}(U)\ni f\mapsto f(T)\in\mathcal{B}^{\rm r}(\mathcal{V})$
is a unital real algebra morphism.
Moreover, the following properties are true:
$(1)$ For all $F\in\mathcal{O}_{s}(U,\mathcal{B}^{\rm
r}(\mathcal{V})_{\mathbb{C}}),\,f\in{\mathcal{O}}_{s}(U)$, we have
$(Ff)(T)=F(T)f(T)$.
$(2)$ For every polynomial
$P(\zeta)=\sum_{n=0}^{m}A_{n}\zeta^{n},\,\zeta\in{\mathbb{C}}$, with
$A_{n}\in\mathcal{B}^{\rm r}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have
$P(T)=\sum_{n=0}^{m}A_{n}T^{n}\in\mathcal{B}^{\rm r}(\mathcal{V})$.
The proof of this result is similar to that of Theorem 2 and will be omitted.
###### Remark 6
The algebra ${\mathbb{H}}$ is, in particular, a Banach ${\mathbb{H}}$-space.
As already noticed, the left multiplications $L_{\bf q},\,{\bf
q}\in{\mathbb{H}},$ are elements of $\mathcal{B}^{\rm r}({\mathbb{H}})$. In
fact, the map ${\mathbb{H}}\ni{\bf q}\mapsto L_{\bf q}\in\mathcal{B}^{\rm
r}({\mathbb{H}})$ is a injective morphism of real algebras allowing the
identification of ${\mathbb{H}}$ with a subalgebra of $\mathcal{B}^{\rm
r}({\mathbb{H}})$.
Let $\Omega\subset{\mathbb{H}}$ be a spectrally saturated open set, and let
$U=\mathfrak{S}(\Omega):=\\{\lambda\in{\mathbb{C}},\exists{\bf
q}\in\Omega,\lambda\in\sigma({\bf q})\\}$, which is open and conjugate
symmetric (see [21]). Denotig by $f_{\mathbb{H}}$ the function $\Omega\ni{\bf
q}\mapsto f({\bf q}),{\bf q}\in\Omega$, for every $f\in\mathcal{O}_{s}(U)$, we
set
$\mathcal{R}(\Omega):=\\{f_{\mathbb{H}};f\in\mathcal{O}_{s}(U)\\},$
which is a commutative real algebra. Defining the function $F_{\mathbb{H}}$ in
a similar way for each $F\in\mathcal{O}_{s}(U,{\mathbb{M}})$, we set
$\mathcal{R}(\Omega,{\mathbb{H}}):=\\{F_{\mathbb{H}};F\in\mathcal{O}_{s}(U,{\mathbb{M}})\\},$
which, according to the next theorem, is a right $\mathcal{R}(\Omega)$-module.
The next result is an analytic functional calculus for quaternions (see [21],
Theorem 5), obtained as a particular case of Theorem 4 (see also its
predecessor in [5]).
###### Theorem 5
Let $\Omega\subset{\mathbb{H}}$ be a spectrally saturated open set, and let
$U=\mathfrak{S}(\Omega)$. The space $\mathcal{R}(\Omega)$ is a unital
commutative ${\mathbb{R}}$-algebra, the space
$\mathcal{R}(\Omega,{\mathbb{H}})$ is a right $\mathcal{R}(\Omega)$-module,
the map
${\mathcal{O}}_{s}(U,{\mathbb{M}})\ni F\mapsto
F_{\mathbb{H}}\in\mathcal{R}(\Omega,{\mathbb{H}})$
is a right module isomorphism, and its restriction
${\mathcal{O}}_{s}(U)\ni f\mapsto f_{\mathbb{H}}\in\mathcal{R}(\Omega)$
is an ${\mathbb{R}}$-algebra isomorphism.
Moreover, for every polynomial
$P(\zeta)=\sum_{n=0}^{m}a_{n}\zeta^{n},\,\zeta\in{\mathbb{C}}$, with
$a_{n}\in{\mathbb{H}}$ for all $n=0,1,\ldots,m$, we have
$P_{\mathbb{H}}(q)=\sum_{n=0}^{m}a_{n}q^{n}\in{\mathbb{H}}$ for all
$q\in{\mathbb{H}}$.
Most of the assertions of Theorem 5 can be obtained directly from Theorem 4.
The injectivity of the map ${\mathcal{O}}_{s}(U)\ni f\mapsto
f_{\mathbb{H}}\in\mathcal{R}(\Omega)$, as well as an alternative complete
proof, can be obtained as in the proof of Theorem 5 from [21].
###### Remark 7
That Theorems 3 and 4 have practically the same proof as Theorems 1 and 2
(respectively) is due to the fact that all of them can be obtained as
particular cases of more general results. Indeed, considering a unital real
Banach algebra ${\mathcal{A}}$, and its complexification
${\mathcal{A}}_{\mathbb{C}}$, identifying ${\mathcal{A}}$ with a real
subalgebra of ${\mathcal{A}}_{\mathbb{C}}$, for a function
$F\in\mathcal{O}_{s}(U,A_{\mathbb{C}})$, where $U\subset{\mathbb{C}}$ is open
and conjugate symmetric, the element $F(b)\in{\mathcal{A}}$ for each
$b\in{\mathcal{A}}$ with $\sigma_{\mathbb{C}}(b)\subset U$. The assertion
follows as in the proof of Theorem 1. The other results also have their
counterparts. We omit the details.
###### Remark 8
The space $\mathcal{R}(\Omega,{\mathbb{H}})$ can be independently defined, and
it consists of the set of all ${\mathbb{H}}$-valued functions, which are slice
regular in the sense of [5], Definition 4.1.1. They are used in [5] to define
a quaternionic functional calculus for quaternionic linear operators (see also
[4]). Roughly speaking, given a quaternionic linear operator, each regular
quaternionic-valued function defined in a neighborhood $\Omega$ of its
quaternionic spectrum is associated with another quaternionic linear operator,
replacing formally the quaternionic variable with that operator. This
constraction is largely explained in the fourth chapter of [5].
Our Theorem 4 constructs an analytic functional calculus with functions from
${\mathcal{O}}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}})$, where
$U$ is a a neighborhood of the complex spectrum of a given quaternionic linear
operator, leading to another quaternionic linear operator, replacing formally
the complex variable with that operator. We can show that those functional
calculi are equivalent. This is a consequence of the fact that the class of
regular quaternionic-valued function used by the construction in [5] is
isomorphic to the class of analytic functions used in our Theorem 5. The
advantage of our approach is its simplicity and a stronger connection with the
classical approach, using spectra defined in the complex plane, and Cauchy
type kernels partially commutative.
Let us give a direct argument concerning the equivalence of those analytic
functional calculi. For an operator $T\in\mathcal{B}^{\rm r}(\mathcal{V})$,
the so-called right $S$-resolvent is defined via the formula
$S_{R}^{-1}({\bf s},T)=-(T-{\bf s}^{*})(T^{2}-2\Re({\bf s})T+\|{\bf
s}\|)^{-1},\,\,{\bf s}\in\rho_{\mathbb{H}}(T)$ (4)
(see [5], formula (4.27)). Fixing an element $\kappa\in\mathbb{S}$, and a
spectrally saturated open set $\Omega\subset{\mathbb{H}}$, for
$\Phi\in\mathcal{R}(\Omega,{\mathbb{H}})$ one sets
$\Phi(T)=\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}\Phi({\bf s})d{\bf
s}_{\kappa}S_{R}^{-1}({\bf s},T),$ (5)
where $\Sigma\subset\Omega$ is a spectrally saturated open set containing
$\sigma_{\mathbb{H}}(T)$, such that
$\Sigma_{\kappa}=\\{u+v\kappa\in\Sigma;u,v\in{\mathbb{R}}\\}$ is a subset
whose boundary $\partial(\Sigma_{\kappa})$ consists of a finite family of
closed curves, piecewise smooth, positively oriented, and $d{\bf
s}_{\kappa}=-\kappa du\wedge dv$. Formula (5) is a (right) quaternionic
functional calculus, as defined in [5], Section 4.10.
Because the space $\mathcal{V}_{\mathbb{C}}$ is also an ${\mathbb{H}}$-space,
we may extend these formulas to the operator
$T_{\mathbb{C}}\in\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb{C}})$, extending
the operator $T$ to $T_{\mathbb{C}}$, and replacing $T$ by $T_{\mathbb{C}}$ in
formulas (4) and (5). For the function
$\Phi\in\mathcal{R}(\Omega,{\mathbb{H}})$ there exists a function
$F\in{\mathcal{O}}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb{C}}))$ such
that $F_{\mathbb{H}}=\Phi$. Denoting by $\Gamma_{\kappa}$ the boundary of a
Cauchy domain in ${\mathbb{C}}$ containing the compact set $\cup\\{\sigma({\bf
s});{\bf s}\in\overline{\Sigma_{\kappa}}\\}$, we can write
$\Phi(T_{\mathbb{C}})=\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}\left(\frac{1}{2\pi
i}\int_{\Gamma_{\kappa}}F(\zeta)(\zeta-{\bf s})^{-1}d\zeta\right)d{\bf
s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})=$ $\frac{1}{2\pi
i}\int_{\Gamma_{\kappa}}F(\zeta)\left(\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf
s})^{-1}d{\bf s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})\right)d\zeta.$
It follows from the complex linearity of $S_{R}^{-1}({\bf s},T_{\mathbb{C}})$,
and from formula (4.49) in [5], that
$(\zeta-{\bf s})S_{R}^{-1}({\bf s},T_{\mathbb{C}})=S_{R}^{-1}({\bf
s},T_{\mathbb{C}})(\zeta-T_{\mathbb{C}})-1,$
whence
$(\zeta-{\bf s})^{-1}S_{R}^{-1}({\bf s},T_{\mathbb{C}})=S_{R}^{-1}({\bf
s},T_{\mathbb{C}})(\zeta-T_{\mathbb{C}})^{-1}+(\zeta-{\bf s})^{-1}(\zeta-
T_{\mathbb{C}})^{-1},$
and therefore,
$\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf s})^{-1}d{\bf
s}_{\kappa}S_{R}^{-1}({\bf
s},T_{\mathbb{C}})=\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}d{\bf
s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})(\zeta-T_{\mathbb{C}})^{-1}+$
$\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf s})^{-1}d{\bf
s}_{\kappa}(\zeta-T_{\mathbb{C}})^{-1}=(\zeta-T_{\mathbb{C}})^{-1},$
because
$\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}d{\bf
s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})=1\,\,\,{\rm
and}\,\,\,\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf
s})^{-1}d{\bf s}_{\kappa}=0,$
as in Theorem 4.8.11 from [5], since the ${\mathbb{M}}$-valued function ${\bf
s}\mapsto(\zeta-{\bf s})^{-1}$ is analytic in a neighborhood of the set
$\overline{\Sigma_{\kappa}}\subset{\mathbb{C}}_{\kappa}$ for each
$\zeta\in\Gamma_{\kappa}$, respectively. Therefore
$\Phi(T_{\mathbb{C}})=\Phi(T)_{\mathbb{C}}=F(T_{\mathbb{C}})=F(T)_{\mathbb{C}}$,
implying $\Phi(T)=F(T)$.
## 5 Some Examples
###### Example 2
One of the simplest Banach ${\mathbb{H}}$-space is the space ${\mathbb{H}}$
itself. As already noticed (see Remark 6), taking
${\mathcal{V}}={\mathbb{H}}$, so ${\mathcal{V}}_{\mathbb{C}}={\mathbb{M}}$,
and fixing an element ${\bf q}\in{\mathbb{H}}$, we may consider the operator
$L_{\bf q}\in\mathcal{B}^{\rm r}({\mathbb{H}})$, whose complex spectrum is
given by $\sigma_{\mathbb{C}}(L_{\bf q})=\sigma({\bf q})=\\{\Re{\bf q}\pm
i\|\Im{\bf q}\|\\}$. If $U\subset{\mathbb{C}}$ is conjugate symmetric open set
containing $\sigma_{\mathbb{C}}(L_{\bf q})$, and
$F\in\mathcal{O}_{s}(U,{\mathbb{M}})$, then we have
$F({L_{\bf q}})=F(s_{+}({\bf q}))\iota_{+}(\mathfrak{s}_{\tilde{\bf
q}})+F(s_{-}({\bf q}))\iota_{-}(\mathfrak{s}_{\tilde{\bf q}})\in{\mathbb{M}},$
(6)
where $s_{\pm}({\bf q})=\Re{\bf q}\pm i\|\Im{\bf q}\|$, $\tilde{\bf q}=\Im\bf
q,\,\mathfrak{s}_{\tilde{\bf q}}=\tilde{\bf q}\|\tilde{\bf q}\|^{-1}$, and
$\iota_{\pm}(\mathfrak{s}_{\tilde{\bf q}})=2^{-1}(1\mp
i\mathfrak{s}_{\tilde{\bf q}})$ (see [21], Remark 3).
###### Example 3
Let ${\mathfrak{X}}$ be a topological compact space, and let
$C({\mathfrak{X}},{\mathbb{M}})$ be the space of ${\mathbb{M}}$-valued
continuous functions on ${\mathfrak{X}}$. Then
$C({\mathfrak{X}},{\mathbb{H}})$ is the real subspace of
$C({\mathfrak{X}},{\mathbb{M}})$ consisting of ${\mathbb{H}}$-valued
functions, which is also a Banach ${\mathbb{H}}$-space with respect to the
operations $({\bf q}F)(x)={\bf q}F(x)$ and $(F{\bf q})(x)=F(x){\bf q}$ for all
$F\in C({\mathfrak{X}},{\mathbb{H}})$ and $x\in{\mathfrak{X}}$. Moreover,
$C({\mathfrak{X}},{\mathbb{H}})_{\mathbb{C}}=C({\mathfrak{X}},{\mathbb{H}}_{\mathbb{C}})=C({\mathfrak{X}},{\mathbb{M}})$.
We fix a function $\Theta\in C({\mathfrak{X}},{\mathbb{H}})$ and define the
operator $T\in\mathcal{B}(C({\mathfrak{X}},{\mathbb{H}}))$ by the relation
$(TF)(x)=\Theta(x)F(x)$ for all $F\in C({\mathfrak{X}},{\mathbb{H}})$ and
$x\in{\mathfrak{X}}$. Note that $(T(F{\bf q}))(x)=\Theta(x)F(x){\bf
q}=((TF){\bf q})(x)$ for all $F\in C({\mathfrak{X}},{\mathbb{H}}),{\bf
q}\in{\mathbb{H}}$, and $x\in{\mathfrak{X}}$. In othe words,
$T\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb{H}}))$. Note also that the
operator $T$ is invertible if and only if the function $\Theta$ has no zero in
${\mathfrak{X}}$.
Let us compute the $Q$-spectrum of $T$. According to Definition 1, we have
$\rho_{\mathbb{H}}(T)=\\{{\bf q}\in{\mathbb{H}};(T^{2}-2\Re{\bf q}\,T+\|{\bf
q}\|^{2})^{-1}\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb{H}}))\\}.$
Consequently, ${\bf q}\in\sigma_{\mathbb{H}}(T)$ if and only if zero is in the
range of the function
$\tau({\bf q},x):=\Theta(x)^{2}-2\Re{\bf q}\,\Theta(x)+\|{\bf
q}\|^{2},\,x\in\mathfrak{X}.$
Similarly,
$\rho_{\mathbb{C}}(T)=\\{\lambda\in{\mathbb{C}};(T^{2}-2\Re\lambda\,T+\|\lambda\|^{2})^{-1}\in\mathcal{B}^{\rm
r}(C({\mathfrak{X}},{\mathbb{H}}))\\},$
and so $\lambda\in\sigma_{\mathbb{C}}(T)$ if and only if zero is in the range
of the function
$\tau(\lambda,x):=\Theta(x)^{2}-2\Re\lambda\,\Theta(x)+|\lambda|^{2},\,x\in\mathfrak{X}.$
Looking for solutions $u+iv,u,v\in{\mathbb{R}}$, of the equation
$(u-\Theta(x))^{2}+v^{2}=0$, a direct calculation shows that $u=\Re\Theta(x)$
and $v=\pm\|\Im\Theta(x)\|$. Hence
$\sigma_{\mathbb{C}}(T)=\\{\Re\Theta(x)\pm
i\|\Im\Theta(x)\|;x\in\mathfrak{X}\\}=\cup_{x\in\mathfrak{X}}\sigma(\Theta(x)).$
Of course, for every open conjugate symmetric subset $U\subset{\mathbb{C}}$
containing $\sigma_{\mathbb{C}}(T)$, and for every function
$\Phi\in\mathcal{O}_{c}(U,\mathcal{B}(C({\mathfrak{X}},{\mathbb{M}})))$, we
may construct the operator $\Phi(T)\in\mathcal{B}^{\rm
r}(C({\mathfrak{X}},{\mathbb{H}}))$, using Theorem 4.
## 6 Quaternionic Joint Spectrum of Paires
In many applications, it is more convenient to work with matrix quaternions
rather than with abstract quaternions. Specifically, one considers the
injective unital algebra morphism
${\mathbb{H}}\ni x_{1}+y_{1}{\bf j}+x_{2}{\bf k}+y_{2}{\bf
l}\mapsto\left(\begin{array}[]{cc}x_{1}+iy_{1}&x_{2}+iy_{2}\\\
-x_{2}+iy_{2}&x_{1}-iy_{1}\end{array}\right)\in{\mathbb{M}}_{2},$
with $x_{1},y_{1},x_{2},y_{2}\in{\mathbb{R}},$ where ${\mathbb{M}}_{2}$ is the
complex algebra of $2\times 2$-matrix, whose image, denoted by
${\mathbb{H}}_{2}$ is the real algebra of matrix quaternions. The elements of
${\mathbb{H}}_{2}$ can be also written as matrices of the form
$Q({\bf z})=\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
-\bar{z}_{2}&\bar{z_{1}}\end{array}\right),\,\,{\bf
z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}.$
A strong connection between the spectral theory of pairs of commuting
operators in a complex Hilbert space and the algebra of quaternions has been
firstly noticed in [17]. Another connection will be presented in this section.
If ${\mathcal{V}}$ is an arbitrary vector space, we denote by
${\mathcal{V}}^{2}$ the Cartesian product ${\mathcal{V}}\times{\mathcal{V}}$.
Let $\mathcal{V}$ be a real Banach space, and let ${\bf
T}=(T_{1},T_{2})\in\mathcal{B(V)}^{2}$ be a pair of commuting operators. The
extended pair ${\bf
T}_{\mathbb{C}}=(T_{1{\mathbb{C}}},T_{2{\mathbb{C}}})\in\mathcal{B(V_{\mathbb{C}})}^{2}$
also consists of commuting operators. For simplicity, we set
$Q({\bf
T}_{\mathbb{C}}):=\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}&T_{2{\mathbb{C}}}\\\
-T_{2{\mathbb{C}}}&T_{1{\mathbb{C}}}\end{array}\right)$
which acts on the complex Banach space $\mathcal{V}_{\mathbb{C}}^{2}$.
We now define the quaternionic resolvent set and spectrum for the case of a
pair of operators, inspired by the previous discussion concerning a single
operator.
###### Definition 2
Let $\mathcal{V}$ be a real Banach space. For a given pair ${\bf
T}=(T_{1},T_{2})\in\mathcal{B(V)}^{2}$ of commuting operators, the set of
those $Q({\bf z})\in{\mathbb{H}}_{2},\,{\bf
z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$, such that the operator
$T_{1}^{2}+T_{2}^{2}-2\Re{z_{1}}T_{1}-2\Re{z_{2}}T_{2}+|z_{1}|^{2}+|z_{2}|^{2}$
is invertible in $\mathcal{B(V)}$ is said to be the quaternionic joint
resolvent (or simply the $Q$-joint resolvent) of ${\bf T}$, and is denoted by
$\rho_{\mathbb{H}}({\bf T})$.
The complement $\sigma_{\mathbb{H}}({\bf
T})={\mathbb{H}}_{2}\setminus\rho_{\mathbb{H}}({\bf T})$ is called the
quaternionic joint spectrum (or simply the $Q$-joint spectrum) of ${\bf T}$.
For every pair ${\bf
T}_{\mathbb{C}}=(T_{1{\mathbb{C}}},T_{2{\mathbb{C}}})\in\mathcal{B(V_{\mathbb{C}})}^{2}$
we put ${\bf
T}_{\mathbb{C}}^{c}=(T_{1{\mathbb{C}}},-T_{2{\mathbb{C}}})\in\mathcal{B(V_{\mathbb{C}})}^{2}$,
and for every pair ${\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$ we put ${\bf
z}^{c}=(\bar{z}_{1},-z_{2})\in{\mathbb{C}}^{2}$
###### Lemma 3
A matrix quaternion $Q({\bf z})$ $({\bf z}\in{\mathbb{C}}^{2})$ is in the set
$\rho_{\mathbb{H}}({\bf T})$ if and only if the operators $Q({\bf
T}_{\mathbb{C}})-Q({\bf z}),\,Q({\bf T}_{\mathbb{C}}^{c})-Q({\bf z}^{c})$ are
invertible in $\mathcal{B}(\mathcal{V}_{\mathbb{C}}^{2})$.
Proof The assertion follows from the equalities
$\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-z_{1}&T_{2{\mathbb{C}}}-z_{2}\\\
-T_{2{\mathbb{C}}}+\bar{z}_{2}&T_{1{\mathbb{C}}}-\bar{z}_{1}\end{array}\right)\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-\bar{z}_{1}&-T_{2{\mathbb{C}}}+z_{2}\\\
T_{2{\mathbb{C}}}-\bar{z}_{2}&T_{1{\mathbb{C}}}-z_{1}\end{array}\right)=$
$\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-\bar{z}_{1}&-T_{2{\mathbb{C}}}+z_{2}\\\
T_{2{\mathbb{C}}}-\bar{z}_{2}&T_{1{\mathbb{C}}}-z_{1}\end{array}\right)\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-z_{1}&T_{2{\mathbb{C}}}-z_{2}\\\
-T_{2{\mathbb{C}}}+\bar{z}_{2}&T_{1{\mathbb{C}}}-\bar{z}_{1}\end{array}\right)=$
$[(T_{1{\mathbb{C}}}-z_{1})(T_{1{\mathbb{C}}}-\bar{z}_{1})+(T_{2{\mathbb{C}}}-z_{2})(T_{2{\mathbb{C}}}-\bar{z}_{2})]{\bf
I}.$
for all ${\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$, where $\bf I$ is the
identity. Consequently, the operators $Q({\bf T}_{\mathbb{C}})-Q({\bf
z}),\,Q({\bf T}_{\mathbb{C}}^{c})-Q({\bf z}^{c})$ are invertible in
$\mathcal{B}({\mathcal{V}}_{\mathbb{C}}^{2})$ if and only if the operator
$(T_{1{\mathbb{C}}}-z_{1})(T_{1{\mathbb{C}}}-\bar{z}_{1})+(T_{2{\mathbb{C}}}-z_{2})(T_{2{\mathbb{C}}}-\bar{z}_{2})$
is invertible in $\mathcal{B}(\mathcal{V}_{\mathbb{C}})$. Because we have
$T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2}=$
$[T_{1}^{2}+T_{1}^{2}-2\Re{z_{1}}T_{1}-2\Re{z_{2}}T_{2}+|z_{1}|^{2}+|z_{2}|^{2}]_{\mathbb{C}},$
the operators $Q({\bf T}_{\mathbb{C}})-Q({\bf z}),\,Q({\bf
T}_{\mathbb{C}}^{c})-Q({\bf z}^{c})$ are invertible in
$\mathcal{B}({\mathcal{V}}_{\mathbb{C}}^{2})$ if and only if the operator
$T_{1}^{2}+T_{1}^{2}-2\Re{z_{1}}T_{1}-2\Re{z_{2}}T_{2}+|z_{1}|^{2}+|z_{2}|^{2}$
is invertible in $\mathcal{B(V)}$.
Lemma 3 shows that we have the property $Q({\bf z})\in\sigma_{\mathbb{H}}({\bf
T})$ if and only if $Q(z^{c})\in\sigma_{\mathbb{H}}({\bf T}^{c})$. Putting
$\sigma_{{\mathbb{C}}^{2}}({\bf T}):=\\{{\bf z}\in{\mathbb{C}}^{2};Q({\bf
z})\in\sigma_{\mathbb{H}}({\bf T})\\},$
the set $\sigma_{{\mathbb{C}}^{2}}({\bf T})$ has a similar property,
specifically $\bf z\in\sigma_{{\mathbb{C}}^{2}}({\bf T})$ if and only if $\bf
z^{c}\in\sigma_{{\mathbb{C}}^{2}}({\bf T}^{c})$. As in the quaternionic case,
the set $\sigma_{{\mathbb{C}}^{2}}({\bf T})$ looks like a ”complex border“ of
the set $\sigma_{\mathbb{H}}({\bf T})$.
###### Remark 9
For the extended pair ${\bf
T}_{\mathbb{C}}=(T_{1{\mathbb{C}}},T_{2{\mathbb{C}}})\in{B(V_{\mathbb{C}})}^{2}$
of the commuting pair ${\bf T}=(T_{1},T_{2})\in\mathcal{B(V)}$ there is an
interesting connexion with the joint spectral theory of J. L. Taylor (see [15,
16]; see also [19]). Namely, if the operator
$T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2}$
is invertible, then the point ${\bf z}=(z_{1},z_{2})$ belongs to the joint
resolvent of ${\bf T}_{\mathbb{C}}$. Indeed, setting
$R_{j}({\bf T}_{\mathbb{C}},{\bf
z})=(T_{j{\mathbb{C}}}-\bar{z}_{j})(T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2})^{-1},$
$q=Q({\bf z})$ for $j=1,2$, we clearly have
$(T_{1{\mathbb{C}}}-z_{1})R_{1}({\bf T}_{\mathbb{C}},{\bf
z})+(T_{2{\mathbb{C}}}-z_{2})R_{2}({\bf T}_{\mathbb{C}},{\bf z})={\bf I},$
which, according to [15], implies that ${\bf z}$ is in the joint resolvent of
${\bf T}_{\mathbb{C}}$. A similar argument shows that, in this case the point
${\bf z}^{c}$ belongs to the joint resolvent of ${\bf T}_{\mathbb{C}}^{c}$. In
addition, if $\sigma(T_{\mathbb{C}})$ designates the Taylor spectrum of
$T_{\mathbb{C}}$, we have the inclusion
$\sigma(T_{\mathbb{C}})\subset\sigma_{{\mathbb{C}}^{2}}({\bf T})$. In
particular, for every complex-valued function $f$ analytic in a neighborhood
of $\sigma_{{\mathbb{C}}^{2}}({\bf T})$, the operator $f(\bf T_{\mathbb{C}})$
can be computed via Taylor’s analytic functional calculus. In fact, we have a
Martinelli type formula for the analytic functional calculus:
###### Theorem 6
Let $\mathcal{V}$ be a real Banach space, let ${\bf
T}=(T_{1},T_{2})\in\mathcal{B(V)}^{2}$ be a pair of commuting operators, let
$U\subset{\mathbb{C}}^{2}$ be an open set, let $D\subset U$ be a bounded
domain containing $\sigma_{{\mathbb{C}}^{2}}({\bf T})$, with piecewise-smooth
boundary $\Sigma$, and let $f\in\mathcal{O}(U)$. Then we have
$f({\bf T}_{\mathbb{C}})=\frac{1}{(2\pi i)^{2}}\int_{\Sigma}f({\bf z}))L({\bf
z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})d\bar{z}_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})d\bar{z}_{1}]dz_{1}dz_{2},$
where
$L({\bf
z,T_{\mathbb{C}}})=T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2}.$
Proof. Theorem III.9.9 from [19] implies that the map $\mathcal{O}(U)\ni
f\mapsto f({\bf T}_{\mathbb{C}})\in\mathcal{B(V_{\mathbb{C}})}$, defined in
terms of Taylor’s analytic functional calculus, is unital, linear,
multiplicative, and ordinary complex polynomials in ${\bf z}$ are transformed
into polynomials in ${\bf T}_{\mathbb{C}}$ by simple substitution, where
$\mathcal{O}(U)$ is the algebra of all analytic functions in the open set
$U\subset{\mathbb{C}}^{2}$, provided $U\supset\sigma({\bf T}_{\mathbb{C}})$.
The only thing to prove is that, when $U\supset\sigma_{{\mathbb{C}}^{2}}({\bf
T})$, Taylor’s functional calculus is given by the stated (canonical) formula.
In order to do that, we use an argument from the proof of Theorem III.8.1 in
[19], to make explicit the integral III(9.2) from [19] (see also [12]).
We consider the exterior algebra
$\Lambda[e_{1},e_{2},\bar{\xi_{1}},\bar{\xi_{2}},\mathcal{O}(U)\otimes\mathcal{V}_{\mathbb{C}}]=\Lambda[e_{1},e_{2},\bar{\xi_{1}},\bar{\xi_{2}}]\otimes\mathcal{O}(U)\otimes\mathcal{V}_{\mathbb{C}},$
where the indeterminates $e_{1},e_{2}$ are to be associated with the pair
${\bf T}_{\mathbb{C}}$, we put $\bar{\xi_{j}}=d\bar{z}_{j},\,j=1,2$, and
consider the operators $\delta=(z_{1}-T_{1{\mathbb{C}}})\otimes
e_{1}+(z_{2}-T_{2{\mathbb{C}}})\otimes
e_{2},\,\bar{\partial}=(\partial/\partial\bar{z_{1}})\otimes\bar{\xi_{1}}+(\partial/\partial\bar{z_{2}})\otimes\bar{\xi_{2}}$,
acting naturally on this exterior algebra, via the calculus with exterior
forms.
To simplify the computation, we omit the symbol $\otimes$, and the exterior
product will be denoted simply par juxtaposition.
We fix the exterior form $\eta=\eta_{2}=fye_{1}e_{2}$ for some
$f\in\mathcal{O}(U)$ and $y\in\mathcal{X}_{\mathbb{C}}$, which clearly satisfy
the equation $(\delta+\bar{\partial})\eta=0$, and look for a solution $\theta$
of the equation $(\delta+\bar{\partial})\theta=\eta$. We write
$\theta=\theta_{0}+\theta_{1}$, where $\theta_{0},\theta_{1}$ are of degree
$0$ and $1$ in $e_{1},e_{2}$, respectively. Then the equation
$(\delta+\bar{\partial})\theta=\eta$ can be written under the form
$\delta\theta_{1}=\eta,\,\delta\theta_{0}=-\bar{\partial}\theta_{1}$, and
$\bar{\partial}\theta_{0}=0$. Note that
$\theta_{1}=fL({\bf
z,T_{\mathbb{C}}})^{-1}[(\bar{z}_{1}-T_{1{\mathbb{C}}})ye_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})]ye_{1}$
is visibly a solution of the equation $\delta\theta_{1}=\eta$. Further, we
have
$\bar{\partial}\theta_{1}=fL({\bf
z,T_{\mathbb{C}}})^{-2}[(z_{1}-T_{1{\mathbb{C}}})(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}e_{1}-(z_{1}-T_{1{\mathbb{C}}})(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}e_{1}+$
$(z_{2}-T_{2{\mathbb{C}}})(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}e_{2}-(z_{2}-T_{2{\mathbb{C}}})(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}e_{2}]=$
$\delta[fL({\bf
z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}-fL({\bf
z,T_{\mathbb{C}}})^{-2}(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}],$
so we may define
$\theta_{0}=-fL({\bf
z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}+fL({\bf
z,T_{\mathbb{C}}})^{-2}(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}.$
Formula III(8.5) from [19] shows that
$f({\bf T}_{\mathbb{C}})y=-\frac{1}{(2\pi
i)^{2}}\int_{U}\bar{\partial}(\phi\theta_{0})dz_{1}dz_{2}=$ $\frac{1}{(2\pi
i)^{2}}\int_{\Sigma}f({\bf z}))L({\bf
z,T_{\mathbb{C}}})^{-2}[(\bar{z}_{1}-T_{1{\mathbb{C}}})yd\bar{z}_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})yd\bar{z}_{1}]dz_{1}dz_{2},$
for all $y\in\mathcal{X}_{\mathbb{C}}$, via Stokes’s formula, where $\phi$ is
a smooth function such that $\phi=0$ in a neighborhood of
$\sigma_{{\mathbb{C}}^{2}}({\bf T})$, $\phi=1$ on $\Sigma$ and the support of
$1-\phi$ is compact.
###### Remark 10
(1) We may extend the previous functional calculus to
$\mathcal{B(V}_{\mathbb{C}})$-valued analytic functions, setting, for such a
function $F$ and with the notation from above,
$F({\bf T}_{\mathbb{C}})=\frac{1}{(2\pi i)^{2}}\int_{\Sigma}F({\bf z}))L({\bf
z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})d\bar{z}_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})d\bar{z}_{1}]dz_{1}dz_{2}.$
In particular, if $F({\bf z})=\sum_{j,k\geq
0}A_{jk{\mathbb{C}}}z_{1}^{j}z_{2}^{k}$, with $A_{j,k}\in\mathcal{B(V)}$,
where the series is convergent in neighborhood of
$\sigma_{{\mathbb{C}}^{2}}({\bf T})$, we obtain
$F({\bf T}):=F({\bf T}_{\mathbb{C}})|\mathcal{V}=\sum_{j,k\geq
0}A_{jk}T_{1}^{j}T_{2}^{k}\in\mathcal{B(V)}.$
(2) The connexion of the spectral theory of pairs with the algebra of
quaternions is even stronger in the case of complex Hilbert spaces.
Specifically, if $\mathcal{H}$ is a complex Hilbert space and ${\bf
V}=(V_{1},V_{2})$ is a commuting pair of bounded linear operators on
$\mathcal{H}$, a point ${\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$ is in the
joint resolvent of ${\bf V}$ if and only if the operator $Q({\bf V})-Q({\bf
z})$ is invertible in $\mathcal{H}^{2}$, where
$Q({\bf V})=\left(\begin{array}[]{cc}V_{1}&V_{2}\\\
-V_{2}^{*}&V_{1}^{*}\end{array}\right).$
(see [17] for details). In this case, there is also a Martinelli type formula
which can be used to construct the associated analytic functional calculus
(see [18],[19]). An approach to such a construction in Banach spaces, by using
a so-called splitting joint spectrum, can be found in [14].
## References
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* [5] F. Colombo, I. Sabadini and D. C. Struppa: Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions: Progress in Mathematics, Vol. 28 Birkhäuser/Springer Basel AG, Basel, 2011.
* [6] N. Dunford and J. T. Schwartz: Linear Operators, Part I: General Theory, Interscience Publishers, New York, London, 1958.
* [7] G. Gentili and D. C. Struppa: A new theory of regular functions of a quaternionic variable, Advances in Mathematics 216 (2007) 279-301.
* [8] R. Ghiloni , V. Moretti and A. Perotti: Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013), no. 4, 1350006, 83 p.
* [9] L. Ingelstam: Real Banach algebras. Ark. Mat. 5 (1964), 239–270 (1964).
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* [11] S. H. Kulkarni: Representations of a Class of Real $B^{*}$-Algebras as Algebras of Quaternion-Valued Functions, Proceedings of the American Mathematical Society, Vol. 116, No. 1 (1992), 61-66.
* [12] R. Levi: Notes on the Taylor joint spectrum of commuting operators. Spectral theory (Warsaw, 1977), 321–332, Banach Center Publ., 8, PWN, Warsaw, 1982.
* [13] E. Martinelli: Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse, Accad. Ital. Mem. Cl. Sci. fis. mat. nat. 9 (1938), 269-283.
* [14] V. Müller and V. Kordula: Vasilescu-Martinelli formula for operators in Banach spaces, Studia Math. 113 (1995), no. 2, 127-139.
* [15] J. L. Taylor: A joint spectrum for several commuting operators. J. Functional Anal. 6 1970 172-191.
* [16] J. L. Taylor: The analytic functional calculus for several commuting operators, Acta Math. 125 (1970), 1-38.
* [17] F.-H. Vasilescu: On pairs of commuting operators, Studia Math. 62 (1978), 203-207.
* [18] F.-H. Vasilescu: A Martinelli type formula for the analytic functional calculus, Rev. Roumaine Math. Pures Appl. 23 (1978), no. 10, 1587-1605.
* [19] F.-H. Vasilescu: Analytic functional calculus and spectral decompositions, D. Reidel Publishing Co., Dordrecht and Editura Academiei R. S. R., Bucharest, 1982.
* [20] F.-H. Vasilescu: Analytic Functional Calculus in Quaternionic Framework, http://arxiv.org/abs/1902.03850
* [21] F.-H. Vasilescu: Quaternionic Regularity via Analytic Functional Calculus, Integral Equations and Operator Theory, DOI: 10.1007/s00020-020-2574-7
|
2024-09-04T02:54:59.172215 | 2020-03-11T12:54:41 | 2003.05265 | {
"authors": "D. S. Fern\\'andez, \\'A. G. L\\'opez, J. M. Seoane, and M. A. F.\n Sanju\\'an",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26164",
"submitter": "Diego S\\'anchez Fern\\'andez",
"url": "https://arxiv.org/abs/2003.05265"
} | arxiv-papers | # Transient chaos under coordinate transformations in relativistic systems
D. S. Fernández Á. G. López J. M. Seoane M. A. F. Sanjuán Nonlinear
Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad
Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
###### Abstract
We use the Hénon-Heiles system as a paradigmatic model for chaotic scattering
to study the Lorentz factor effects on its transient chaotic dynamics. In
particular, we focus on how time dilation occurs within the scattering region
by measuring the time in a clock attached to the particle. We observe that the
several events of time dilation that the particle undergoes exhibit
sensitivity to initial conditions. However, the structure of the singularities
appearing in the escape time function remains invariant under coordinate
transformations. This occurs because the singularities are closely related to
the chaotic saddle. We then demonstrate using a Cantor-like set approach that
the fractal dimension of the escape time function is relativistic invariant.
In order to verify this result, we compute by means of the uncertainty
dimension algorithm the fractal dimensions of the escape time functions as
measured with inertial and comoving with the particle frames. We conclude
that, from a mathematical point of view, chaotic transient phenomena are
equally predictable in any reference frame and that transient chaos is
coordinate invariant.
###### pacs:
05.45.Ac,05.45.Df,05.45.Pq
## I Introduction
Chaotic scattering in open Hamiltonian systems is a fundamental part of the
theoretical study of dynamical systems. There are many applications such as
the interaction between the solar wind and the magnetosphere tail seoane2013 ,
the simulation in several dimensions of the molecular dynamics lin2013 , the
modeling of chaotic advection of particles in fluid mechanics daitche2014 , or
the analysis of the escaping mechanism from a star cluster or a galaxy
zotos2017 ; navarro2019 , to name a few. A scattering phenomenon is a process
in which a particle travels freely from a remote region and encounters an
obstacle, often described in terms of a potential, which affects its
evolution. Finally, the particle leaves the interaction region and continues
its journey freely. This interaction is typically nonlinear, possibly leading
the particle to perform transient chaotic dynamics, i.e., chaotic dynamics
with a finite lifetime lai2010 ; grebogi1983 . Scattering processes are
commonly studied by means of the scattering functions, which relate the
particle states at the beginning of its evolution once the interaction with
the potential has already taken place. Thus, nonlinear interactions can make
these functions exhibit self-similar arrangements of singularities, which
hinder the system predictability aguirre2009 . Transient chaos is a
manifestation of the presence in phase space of a chaotic set called non-
attracting chaotic set, also called chaotic saddle ott1993 . This phenomenon
can be found in a wide variety of situations tel2015 , as for example the
dynamics of decision making, the doubly transient chaos of undriven autonomous
mechanical systems or even in the sedimentation of volcanic ash.
There have been numerous efforts to characterize chaos in relativistic systems
in an observer-independent manner hobill1994 . It has been rigorously
demonstrated that the sign of the Lyapunov exponents is invariant under
coordinate transformations that satisfy four minimal conditions motter2003 .
More specifically, such conditions consider that a valid coordinate
transformation has to leave the system autonomous, its phase space bounded,
the invariant measure normalizable and the domain of the new time parameter
infinite motter2003 . As a consequence, chaos is a property of relativistic
systems independent of the choice of the coordinate system in which they are
described. In other words, homoclinic and heteroclinic tangles cannot be
untangled by means of coordinate transformations. We shall utilize the Lorentz
transformations along this paper, which satisfy this set of conditions
motter2009 . Although we utilize a Hamiltonian system in its open regime, from
the point of view of Lyapunov exponents the phase space can be considered
bounded because of the presence of the chaotic saddle. This set is located in
a finite region of the system’s phase space and contains all the non-escaping
orbits in the hyperbolic regime. Hence, the Lyapunov exponents are well-
defined because these trajectories stay in the saddle forever. On the other
hand, concerning the computation of the escape time function, we shall only
consider along this work the finite part of the phase space where the escaping
orbits remain bounded, and similarly from the point of view of the finite-time
Lyapunov exponents the phase space can be considered bounded as well
vallejo2003 .
Despite the fact that the sign of the Lyapunov exponents is invariant, the
precise values of these exponents, which indicate “how chaotic” a dynamical
system is, are noninvariant. Therefore, this lack of invariance leaves some
room to explore how coordinate transformations affect the unpredictability in
dynamical systems with transient chaos. In the present work we analyze the
structure of singularities of the scattering functions under a valid
coordinate transformation. In particular, we compute the fractal dimension of
the escape time function as measured in an inertial reference frame and
another non-inertial reference frame comoving with the particle, respectively.
We then characterize the system unpredictability by calculating this fractal
dimension, since it enables to infer the dimension of the chaotic saddle
aguirre2001 . Indeed, this purely geometrical method has been proposed as an
independent-observer procedure to determine whether the system behaves
chaotically motter2001 .
Relevant works have been devoted to analyze the relationship between
relativity and chaos in recent decades barrow1982 ; chernikov1989 ; ni2012 .
More recently, the Lorentz factor effects on the dynamical properties of the
system have also been studied in relativistic chaotic scattering bernal2017 ;
bernal2018 . In this paper, we focus on how changes of the reference frame
affect typical phenomena of chaotic scattering. We describe the model in Sec.
II, which consists of a relativistic version of the Hénon-Heiles system. Two
well-known scattering functions are explored in Sec. III, such as the exit
through which the particle escapes and its escape time. In Sec. IV, we
demonstrate the fractal dimension invariance under a coordinate transformation
by using a Cantor-like set approach. Subsequently, we quantify the
unpredictability of the escape times and analyze the effect of such a
reference frame modification. We conclude with a discussion of the main
results and findings of the present work in Sec. V.
## II Model description
Figure 1: (a) The three-dimensional representation of the Hénon-Heiles
potential $V(x,y)=\frac{1}{2}(x^{2}+y^{2})+x^{2}y-\frac{1}{3}y^{3}$. (b) The
isopotential curves in the physical space show that the Hénon-Heiles system is
open and has triangular symmetry. If the energy of the particle is higher than
a threshold value, related to the potential saddle points, there exist
unbounded orbits. Following these trajectories the particle leaves the
scattering region through any of the three exits.
The Hénon-Heiles system was proposed in 1964 to study the existence of a third
integral of motion in galactic models with axial symmetry henon1964 . We
consider a single particle whose total mechanical energy can be denoted as
$E_{N}$ in the Newtonian approximation. This energy is conserved along the
trajectory described by the particle, which is launched from the interior of
the potential well, within a finite region of the phase space called the
scattering region. We have utilized a dimensionless form of the Hénon-Heiles
system, so that the potential is written as
$V(x,y)=\frac{1}{2}(x^{2}+y^{2})+x^{2}y-\frac{1}{3}y^{3},$ (1)
where $x$ and $y$ are the spatial coordinates. When the energy is above a
threshold value, the potential well exhibits three exits due to its triangular
symmetry in the physical space, i.e., the plane $(x,y)$, as visualized in Fig.
1. We call Exit 1 the exit located at the top $(y\to+\infty)$, Exit 2 the one
located downwards to the left $(x\to-\infty,y\to-\infty)$ and Exit 3 the one
at the right $(x\to+\infty,y\to-\infty)$. One of the characteristics of open
Hamiltonian systems with escapes is the existence of highly unstable periodic
orbits known as Lyapunov orbits contopoulos1990 , which are placed near the
saddle points. In fact, when a trajectory crosses through a Lyapunov orbit, it
escapes to infinity and never returns back to the scattering region.
Furthermore, we recall that the energy of the particle determines also the
dynamical regime. We can distinguish two open regimes in which escapes are
allowed. On the one hand, in the nonhyperbolic regime the KAM tori coexist
with the chaotic saddle and the phase space exhibits regions where dynamics is
regular and also chaotic sideris2006 , whereas the chaotic saddle rules the
dynamics in the hyperbolic regime, making it completely chaotic.
When the speed of the particle is comparable to the speed of light, the
relativistic effects have to be taken into account ohanian2001 . In the
present work we consider a particle which interacts in the limit of weak
external fields, and therefore we deal with a special relativistic version of
the Hénon-Heiles system, whose dynamics is governed by the conservative
Hamiltonian lan2011 ; chanda2018 ; kovacs2011 ; calura1997
$H=c\sqrt{c^{2}+p^{2}+q^{2}}+V(x,y),$ (2)
where $c$ is the value of the speed of light, and $p$ and $q$ are the momentum
coordinates. On the other hand, the Lorentz factor is defined as
$\gamma=\frac{1}{\sqrt{1-\frac{\textbf{v}^{2}}{c^{2}}}}=\frac{1}{\sqrt{1-\beta^{2}}},$
(3)
where v is the velocity vector of the particle and $\beta=|\textbf{v}|/c$ the
ratio between the speed of the particle and the speed of light. The Lorentz
factor $\gamma$ and $\beta$ are two equivalent ways to express how large is
the speed of the particle compared to the speed of light. These two factors
vary in the ranges $\gamma\in[1,+\infty)$ and $\beta\in[0,1)$, respectively.
For convenience, we shall use $\beta$ as a parameter along this work.
Hamilton’s canonical equations can be derived from Eq. (2), yielding the
equations of motion
$\displaystyle\dot{x}=$ $\displaystyle\frac{\partial H}{\partial
p}=\frac{p}{\gamma},$ $\displaystyle\dot{p}=-\frac{\partial H}{\partial
x}=-x-2xy,$ (4) $\displaystyle\dot{y}=$ $\displaystyle\frac{\partial
H}{\partial q}=\frac{q}{\gamma},$ $\displaystyle\dot{q}=-\frac{\partial
H}{\partial y}=y^{2}-x^{2}-y,$
where the Lorentz factor can be alternatively written in the momentum-
dependent form as $\gamma=\frac{1}{c}\sqrt{c^{2}+p^{2}+q^{2}}$. Although the
complete phase space is four-dimensional, the conservative Hamiltonian
constrains the dynamics to a three-dimensional manifold of the phase space,
known as the energy shell.
Some recent works aim at isolating the effects of the variation of the Lorentz
factor $\gamma$ (or $\beta$ equivalently) from the remaining variables of the
system bernal2017 ; bernal2018 . In order to accomplish this, they modify the
initial value of $\beta$ and use it as the only parameter of the dynamical
system. Since $\beta$ is a quantity that depends on $|\textbf{v}|$ and $c$,
they choose to vary the numerical value of $c$. Needless to say, the value of
the speed of light $c$ remains constant during the particle trajectory. The
fundamental reason for deciding to increase the kinetic energy of the system
by reducing the numerical value of the speed of light is simply as follows. If
we keep the Hénon-Heiles potential constant and increase the speed of the
particle to values close to the speed of light, the potential will be in a
much lower energy regime compared to the kinetic energy of the particle.
Therefore, the potential becomes negligible and the interaction between them
becomes irrelevant. Consequently, each time we select a value of the speed of
light we are scaling the system, and hence the ratio of the kinetic energy and
the potential as well. The sequence of potential wells with different values
of $\beta$ represents potential wells with the Hénon-Heiles morphology, but at
different scales in which the interaction of a relativistic particle is not
trivial. In this way, the effects of the Lorentz factor on the dynamics are
isolated from the other system variables, because the Lorentz factor is the
only parameter that differentiates all these scaled systems.
We then consider the same initial value of the particle speed
$|\textbf{v}_{0}|$ in every simulation with a different value of $\beta$,
launching the particle from the minimum potential, which is located at
$(x_{0},y_{0})=(0,0)$ and where the potential energy is null. We have
arbitrarily chosen $|\textbf{v}_{0}|\approx 0.5831$ (as in bernal2017 ;
bernal2018 ), which corresponds to the open nonhyperbolic regime with energy
$E_{N}=0.17$, close to the escape energy in the Newtonian approximation. Thus,
we analyze how the relativistic parameter $\beta$, as its value increases,
affects the dynamical properties starting from the nonhyperbolic regime. The
numerical value of $c$ varies, as shown in Fig. 2, and for instance if the
simulation is carried out for a small $\beta$, where $|\textbf{v}_{0}|\ll c$,
the initial speed of the particle only represents a very low percentage of the
speed of light. In this case, we recover the Newtonian approximation and the
classical version of the Hénon-Heiles system. On the contrary, if the
simulation takes place with a value of $\beta$ near one, the speed of the
particle represents a high percentage of the speed of light and the
relativistic effects on the dynamics become more intense.
Numerical computations reveal that the KAM tori are mostly destroyed at
$\beta\approx 0.4$, and hence the dynamics is hyperbolic for higher values of
$\beta$ bernal2018 . If some small tori survive, they certainly do not rule
the system overall dynamics. As we focus on the hyperbolic regime, the
simulations are run for values of $\beta\in[0.5,0.99]$ and by means of a fixed
step fourth-order Runge-Kutta method press1992 . We recall that the initial
values of the momentum $(p_{0},q_{0})$ depend on the chosen initial value of
$\beta$, and therefore this computational technique (to vary the value of
$\beta$ fixing $|\textbf{v}_{0}|$) is an ideal method to increase the particle
kinetic energy to the relativistic regime. For example, a particle trapped in
the KAM tori can escape if the initial value of $\beta$ is high enough, as
shown in Fig. 2.
Figure 2: The evolution of a particle launched within the scattering region
from the same initial condition for different values of $\beta$. (a) For a
very low $\beta$ (Newtonian approximation), the particle is trapped in the KAM
tori and describes a bounded trajectory. (b) The value of $\beta$ is large
enough to destroy the KAM tori and the particle leaves the scattering region
following a trajectory typical of transient chaos. (c) Finally, a larger value
of $\beta$ than in (b) makes the particle escape faster.
## III Escape times in inertial and non-inertial frames
The scattering functions enable us to represent the relation between input and
output dynamical states of the particle, i.e., how the interaction of the
particle with the potential takes place. The potential of Hénon-Heiles leads
the particle to describe chaotic trajectories before converging to a specific
exit, which makes the scattering functions exhibit a fractal structure. In
order to verify the sensitivity of the system to exits and escape times, we
launch particles from the potential minimum slightly varying the shooting
angle $\theta$ that is formed by the initial velocity vector and the positive
$x$-axis, as shown in Fig. 3(a).
The maximum value of the kinetic energy is reached at the potential minimum,
as the system is conservative. We define the value of the Lorentz factor
associated with this maximum kinetic energy as the critical Lorentz factor
$\gamma_{c}(\beta)=\frac{1}{\sqrt{1-\beta^{2}}}.$ (5)
We emphasize that the initial Lorentz factor of every particle is the critical
Lorentz factor, since every trajectory is initialized from the potential
minimum in this work. We shall monitor the Lorentz factor of the particle
along its trajectory and use the critical Lorentz factor as the criterion of
whether the particle has escaped or not. This escape criterion is based on the
fact that the value of the kinetic energy remains bounded while the particle
evolves chaotically within the potential well, bouncing back and forth against
the potential barriers before escaping. The Lorentz factor value then varies
between the unity and the critical value inside the scattering region, i.e.,
$\gamma(t)\in[1,\gamma_{c}]$. Eventually, the particle leaves the scattering
region and the value of its Lorentz factor breaks out towards infinity,
because its kinetic energy does not remain bounded anymore. In order to
prevent this asymptotic behavior of the Lorentz factor, it is convenient to
set that the escape happens at the time $t_{e}$ when
$\gamma(t_{e})>\gamma_{c}$. In this manner, we define the scattering region as
the part of the physical space where the dynamics is bounded. This escape
criterion is computationally affordable and useful to implement in any
Hamiltonian system without knowing specific information about the exits. In
addition, it includes all the escapes that take place when the Lyapunov orbit
criterion is considered.
Figure 3: (a) Each of the exits is identified with a different color, such
that Exit 1 (red), Exit 2 (green) and finally Exit 3 (blue). In order to avoid
redundant results due to the triangular symmetry of the well, we only let the
particle evolve from the angular region
$\theta_{0}\in\left[\pi/2,5\pi/6\right]$ (black dashed lines). (b) The
scattering function of the exits $(2000\times 2000)$ given the parameter map
$(\beta\in[0.5,0.99],\theta_{0}\in[\pi/2,5\pi/6])$ in the hyperbolic regime.
A particle launched with $\theta=\pi/2$ escapes directly towards the Exit 1
for every value of $\beta$ as shown in Fig. 3(b), whereas if it is launched
with $\theta=5\pi/6$ the particle bounces against the potential barrier placed
between Exit 1 and Exit 2 and escapes through the Exit 3. The whole structure
of exits in between is apparently fractal. Nonetheless, the exit function
becomes smoother when the value of $\beta$ increases, but it is never
completely smooth. On the other hand, we recall that the chaotic saddle is an
observer-independent set of points formed by the intersection of the stable
and unstable manifolds. Concretely, the stable manifold of an open Hamiltonian
system is defined as the boundary between the exit basins ott1993 . If a
particle starts from a point arbitrarily close to the stable manifold it will
spend an infinite time in converging to an exit, i.e., it never escapes. The
unstable manifold is the set along which particles lying infinitesimally close
to the chaotic saddle will eventually leave the scattering region in the
course of time tel2015 .
The escape time can be easily defined as the time the particle spends evolving
inside the scattering region before escaping to infinity. In nonrelativistic
systems, the particular clock in which the time is measured is irrelevant
since time is absolute. However, here we consider two time quantities: the
time $t$ that is measured by an inertial reference frame at rest and the
proper time $\tau$ as measured by a non-inertial reference frame comoving with
the particle. This proper time is simply the time measured by a clock attached
to the particle.
As is well known, an uniformly moving clock runs slower by a factor
$\sqrt{1-\beta^{2}}$ in comparison to another identically constructed and
synchronized clock at rest in an inertial frame. Therefore, we assume that at
any instant of time the clock of the accelerating particle advances at the
same rate as an inertial clock that momentarily had the same velocity
barton1999 . In this manner, given an infinitesimal time interval $dt$, the
particle clock will measure a time interval
$d\tau=\frac{dt}{\gamma(t)},$ (6)
where $\gamma(t)$ is the particle Lorentz factor at the instant of time $t$.
Since the Lorentz factor is greater than the unity, the proper time interval
always obeys that $d\tau\leq dt$, which is just the mathematical statement of
the twin paradox. When the particle velocities are very close to the speed of
light, the time dilation phenomenon takes place so that the time of the
particle clock runs more slowly in comparison to clocks at rest in the
potential. In the context of special relativity, it is important to bear in
mind that it is assumed that the potential does not affect the clocks rate. In
other words, all the clocks placed at rest in any point of the potential are
ticking at the same rate along this work.
Without loss of generality, Eq. 6 can be expressed as an integral in the form
$\tau_{e}=\int_{0}^{t_{e}}\frac{dt}{\gamma(t)},$ (7)
where the final time of the integration interval is the escape time in the
inertial frame. We shall solve this integral using the Simpson’s rule
jeffreys1988 . Since each evolution of the Lorentz factor is unique because
each particle describes a distinct chaotic trajectory, every particle clock
measures a different proper time at any instant of time $t$. Nonetheless, as
the dynamics is bounded in the same energetic conditions given a value of
$\beta$, the Lorentz factor of all trajectories is similar on average at any
instant of time $t$. For this reason, we assume that there exists an average
value of the Lorentz factor along the particle trajectory, and estimate it as
the arithmetic mean between the maximum and minimum values of the bounded
Lorentz factor inside the scattering region, i.e.,
$\bar{\gamma}(\beta)=\frac{1+\gamma_{c}}{2}=\frac{1+\sqrt{1-\beta^{2}}}{2\sqrt{1-\beta^{2}}}.$
(8)
Using this definition to Eq. 7, we can define an average time dilation in the
form $\bar{\tau}_{e}\equiv t_{e}/\bar{\gamma}$. This value should only be
regarded as an approximation, which shall prove of great usefulness to
interpret the numerical results obtained ahead. Accordingly, the difference
between both the average escape time and the time $t_{e}$ is also
approximately linear on average. In this manner, we can also define the
magnitude
$\delta\bar{t}_{e}\equiv
t_{e}-\bar{\tau}_{e}=\frac{1-\sqrt{1-\beta^{2}}}{1+\sqrt{1-\beta^{2}}}t_{e}.$
(9)
We emphasize that this value is again just an approximation representing the
average behavior of the system, which disregards the fluctuations of the
Lorentz factor. It reproduces qualitatively the behavior when the dynamics is
bounded in the well, as shown in Fig. 4(a).
Figure 4: (a) The Lorentz factor evolution $\gamma(t)$ of three different
trajectories: a fast escape (yellow) and two typical transient chaotic
trajectories (red and blue). The dashed guideline represents the Lorentz
factor value of $\bar{\gamma}$ (black), corresponding to $\beta=0.75$. The
time differences $\delta t(t)$ along these trajectories is also shown. (b) The
scattering function of escape times $t_{e}$ in logarithmic scale given the
parameter map $(\beta\in[0.5,0.99],\theta_{0}\in[\pi/2,5\pi/6])$. The two
black dashed lines corresponds to the subfigures (c) and (d), which show the
scattering function of escape time $t_{e}(\theta_{0})$ (blue) and
$\tau_{e}(\theta_{0})$ (red) for $\beta=0.5$ and $\beta=0.8$, respectively.
(e, f) The time difference function $\delta t_{e}(\theta_{0})$ (black) for the
same values of $\beta$.
The escape time function is similar to the exit function, as shown in Fig.
4(b); the longest escape times are located close to the the boundary of the
exit regions, i.e., the mentioned stable manifold, because these trajectories
spend long transient times before escaping. In this manner, the structure of
singularities is again associated to the stable manifold, equally that the
exit function. This is an evidence that the fractality of the escape time
function must be an observer-independent feature, since the exit through which
the particle escapes does not depend on the considered clock. Indeed, we
observe that the escape proper time function exhibits a similar structure of
singularities because of the approximated linear relation described by
$\bar{\tau}_{e}$ (see Figs. 4(c) and 4(d)). Despite being almost identical
structures, the dilation time phenomenon always makes
$\tau_{e}(\theta_{0})<t_{e}(\theta_{0})$.
Importantly, the time difference function $\delta t_{e}(\theta_{0})$ also
preserves the fractal structure as shown in Figs. 4(e) and 4(f). This occurs
because sensitivity to initial conditions is translated into sensitivity to
time dilation phenomena. The longer the time the particle spends in the well,
the more travels from the center to the potential barriers and back. If we
think of each of these travels as an example of a twin paradox journey, we get
an increasing time dilation for particles that spend more time in the well.
Since these times are sensitive to modifications in the initial conditions, so
are time dilation effects. We could then introduce what might be called the
triplet paradox. In this case an additional third sibling leaves the planet
and comes back to the starting point having a different age than their two
other siblings, because of the sensitivity to initial conditions. This
phenomenon in particular illustrates how chaotic dynamics affects typical
relativistic phenomena.
## IV Invariant fractal dimension and persistence of transient chaos
The chaotic saddle and the stable manifold are self-similar fractal sets when
the underlying dynamics is hyperbolic ott1993 . This fact is reflected in the
peaks structure of the escape time functions, which is present at any scale of
initial conditions. In this sense, the escape time functions share with the
Cantor set some properties with regard to their singularities, and therefore
to their fractal dimensions. It is possible to study the fractal dimensions of
the escape time functions in terms of a Cantor-like set lau1991 ; seoane2007 .
In this manner, we can build a Cantor-like set to schematically represent the
escape of particles launched from different initial conditions $\theta_{0}$.
We consider that a certain fraction $\eta_{t}$ of particles escapes from the
scattering region when a minimal characteristic time $t_{0}$ has elapsed. If
these particles were launched from initial conditions centered in the original
interval, two identical segments are created; the trajectories that began in
those segments do not escape at least by a time $t_{0}$. Similarly, a same
fraction of particles $\eta_{t}$ from the two surviving segments escapes by a
time $2t_{0}$. If we continue this iterative procedure for $3t_{0}$, $4t_{0}$
and so on, we obtain a Cantor-like set of Lebesgue measure zero with
associated fractal dimension $d_{t}$ that can be computed as
$d_{t}=\frac{\ln 2}{\ln 2-\ln\left(1-\eta_{t}\right)}.$ (10)
Similarly, if the escape times are measured by a non-inertial reference frame
comoving with a particle, a fraction of particles $\eta_{\tau}$ escapes every
time $\tau_{0}$, and therefore the associated fractal dimension can be defined
as $d_{\tau}$.
The behavior is governed by Poisson statistics in the hyperbolic regime.
Therefore, the average number of particles that escape follow an exponential
decay law. More specifically, the number of particles remaining in the
scattering region according to an inertial reference frame at rest in the
potential is given by
$N(t)=N_{0}e^{-\sigma t}.$ (11)
We note that this decay is homogeneous in an inertial reference frame, whereas
according to an observer describing the decay in a non-inertial reference
frame comoving with a particle, we get the decay law
$\tilde{N}(\tau)\equiv
N(t(\tau))=N_{0}e^{-\sigma\int_{0}^{\tau}\gamma(t(\tau^{\prime}))d\tau^{\prime}},$
(12)
where we have substituted the equality
$t=\int_{0}^{\tau}\gamma(t(\tau^{\prime}))d\tau^{\prime}$ from solving the Eq.
(6). In other words, for an accelerated observer the decay is still
Poissonian, but inhomogeneous. Nevertheless, if we disregard the fluctuations
in the Lorentz factor, an homogeneous statistics can be nicely approximated
once again, by defining the average constant rate
$\bar{\sigma}_{\tau}\equiv\sigma\bar{\gamma}$. We recall that $\gamma(t)$ is
the Lorentz factor along the trajectory of a certain particle, and therefore
$\tilde{N}(\tau)$ here describes the number of particles remaining in the
scattering region according to the accelerated frame co-moving with such a
particle. This particle must be sufficiently close to the chaotic saddle in
order to remain trapped in the well a sufficiently long time so as to render
useful statistics, by counting a high number of escaping test bodies.
Now we calculate, without loss of generality, the fraction of particles that
escape during an iteration according to this reference frame as
$\eta_{\tau}=\frac{\tilde{N}(\tau_{0})-\tilde{N}(\tau_{0}^{\prime})}{\tilde{N}(\tau_{0})}=\frac{N(t_{0})-N(2t_{0})}{N(t_{0})}=\eta_{t},$
(13)
where $\tau_{0}^{\prime}$ is the proper time observed by the accelerated body
when the clocks at rest in the potential mark $2t_{0}$. In this manner, we
obtain that the fraction of escaping particles is invariant under reference
frame transformations, because there exists an unequivocal relation between
the times $t$ and $\tau$ given by $\gamma(t)$. From this result we derive that
the fractal dimension of the Cantor-like set associated with the escape times
function is invariant under coordinate transformations, $d_{t}=d_{\tau}$. This
equality holds for every particle clock evolving in the well, as long as it
stays long enough. On the other hand, this result is in consonance with the
Cantor-like set nature, because its fractal dimension does not depend on how
much time an iteration lasts.
In order to compute the fractal dimensions associated with these scattering
functions, we make use of the uncertainty dimension algorithm lau1991 ;
grebogi1983_2 and the shooting method previously described. We launch a
particle from the potential minimum with a random shooting angle $\theta_{0}$
in the interval $[\pi/2,5\pi/6]$ and measure the escape times
$t_{e}(\theta_{0})$ and $\tau_{e}(\theta_{0})$, and the exit $e(\theta_{0})$
through it escapes. Then, we carry out again the same procedure from a
slightly different shooting angle $\theta_{0}+\epsilon$, where $\epsilon$ can
be considered a small perturbation, and calculate the quantities
$t_{e}(\theta_{0}+\epsilon)$, $\tau_{e}(\theta_{0}+\epsilon)$ and
$e(\theta_{0}+\epsilon)$. We then say that an initial condition $\theta_{0}$
is uncertain in measuring, e.g., the escape time $t_{e}$, if the difference
between the escape times, $|t_{e}(\theta_{0})-t_{e}(\theta_{0}+\epsilon)|$, is
higher than a given time. This time is usually associated with the integration
step $h$ of the numerical method, which is the resolution of an inertial
clock. Conveniently, we set this criterion of uncertain initial conditions as
$3h/2$, i.e., the half between the step and two times the step of the
integrator, for any clock. The reason for it is that the time differences
according to a particle clock are the result of a computation by means of Eq.
(7). Therefore, an initial condition $\theta_{0}$ is uncertain in measuring
the escape time $t_{e}$ if
$\Delta
t_{e}(\theta_{0})=|t_{e}(\theta_{0})-t_{e}(\theta_{0}+\epsilon)|>3h/2.$ (14)
Similarly, an initial condition $\theta_{0}$ is uncertain in measuring the
escape time $\tau_{e}$ if
$\Delta\tau_{e}(\theta_{0})=|\tau_{e}(\theta_{0})-\tau_{e}(\theta_{0}+\epsilon)|>3h/2.$
(15)
Finally, an initial condition is uncertain with respect to the exit through
which the particle escapes if $e(\theta_{0})\neq e(\theta_{0}+\epsilon)$.
We generally expect that the time differences holds
$\Delta\tau_{e}(\theta_{0})<\Delta t_{e}(\theta_{0})$, since we have defined
previously that $\bar{\tau}_{e}\equiv t_{e}/\bar{\gamma}$. Thus, given the
same criterion $3h/2$ in both clocks, there will be some uncertain initial
conditions $\theta_{0}$ in the inertial clock $\left(\Delta
t_{e}(\theta_{0})>3h/2\right)$ that become certain in the particle clock
$\left(\Delta\tau_{e}(\theta_{0})<3h/2\right)$. We show a scheme in Fig. 5(a)
to clarify this physical effect on the escape times unpredictability. It is
easy to see that this effect is caused by the limited resolution of the
hypothetical clocks, and becomes more intense for high values of $\beta$
because it is proportional to the Lorentz factor.
Figure 5: (a) A scheme to visualize the physical effect of a reference frame
modification on the unpredictability of the escape times, where $h=0.005$. (b)
Fractal dimensions according to exits $d_{e}$ (green), escape time $d_{t}$
(blue) and escape proper time $d_{\tau}$ (red) with standard deviations
computed by the uncertainty dimension algorithm versus twenty five equally
spaced values of $\beta\in[0.5,0.98]$.
The fraction of uncertain initial conditions behaves as
$f(\epsilon)\sim\epsilon^{1-d},$ (16)
where $d$ is the value of the fractal dimension, which enables us to quantify
the unpredictability in foreseeing the particle final dynamical state. In
particular, $d=0$ ($d=1$) implies minimum (maximum) unpredictability of the
system lau1991 . All the cases in between, $0<d<1$, imply also
unpredictability, and the closer to the unity the value of the fractal
dimension is, the more unpredictable the system is. According to our
scattering functions, it is expected that the values of their fractal
dimensions decrease as the value of $\beta$ increase, since these functions
become smoother. Taking decimal logarithms in Eq. (16), we obtain
$\log_{10}\frac{f(\epsilon)}{\epsilon}\sim-d\log_{10}\epsilon.$ (17)
This formula allows us to compute the fractal dimension of the scattering
functions from the slope of the linear relation, which obeys a representation
$\log_{10}f(\epsilon)/\epsilon$ versus $\log_{10}\epsilon$. We use an adequate
range of angular perturbations according to our shooting method and the
established criterion of uncertain initial conditions, concretely,
$\log_{10}\epsilon\in[-6,-1]$.
The computed fractal dimensions always hold $d_{e}<d_{t},d_{\tau}$ as shown in
Fig. 5(b). This occurs because it is generally more predictable to determine
the exit through which the particle escapes than exactly its escape time when
the clocks resolution is small. Therefore, there is a greater number of
uncertain conditions concerning escape times than in relation to exits. The
former ones are located outside and over the stable manifold, whereas the
uncertain conditions regarding exits can only be located on the stable
manifold by definition. We obtain computationally $d_{t}\approx d_{\tau}$ for
almost every value of $\beta$. Nonetheless, the physical effect explained
above causes a small difference between the computed fractal dimensions
regarding escape times, implying $d_{\tau}<d_{t}$ in a very energetic regime.
From a mathematical point of view, if we consider a infinitely small clock
resolution, i.e., $h\to 0$, uncertain initial conditions in any clock will be
only the ones whose associated escape time differences are also infinitely
small. Such uncertain conditions will be located on the stable manifold. In
that case, the geometric and observer-independent nature of the fractality
caused by the chaotic saddle is reflected into the values of the fractal
dimensions. It is expected that in this limit the equality
$d_{e}=d_{t}=d_{\tau}$ holds.
This equality extends the very important statement that relativistic _chaos_
is coordinate invariant to _transient chaos_ as well. The result provided in
motter2003 showing that the signs of the Lyapunov exponents of a chaotic
dynamical system are invariant under coordinate transformations can be
perfectly extended to transient chaotic dynamics. For this purpose, it is only
required to consider a chaotic trajectory on the chaotic saddle, which meets
the necessary four conditions described in motter2003 . Since the sign of the
Lyapunov exponents of a trajectory on the chaotic saddle are also invariant,
it is therefore evident that the existence of transient chaotic dynamics can
not be avoided by considering suitable changes of the reference frame. We
believe that this analytical result is at the basis of the results arising
from all the numerical explorations performed in the previous sections.
## V Conclusions
Despite the fact that the Hénon-Heiles system has been widely studied as a
paradigmatic open Hamiltonian system, we have added a convenient definition of
its scattering region. In this manner, the scattering region can be defined as
the part of the physical space where the particle dynamics is bounded, and
therefore a particle escapes when its kinetic energy is greater than the
kinetic energy value at the potential minimum.
Since relativistic chaos has been demonstrated as coordinate invariant, we
have been focused on the special relativistic version of the Hénon-Heiles
system to extend this occurrence to transient chaos. We have then analyzed the
Lorentz factor effects on the system dynamics, concretely, how the time
dilation phenomenon affects the scattering function structure. The exit and
the escape time functions exhibit a fractal structure of singularities as a
consequence of the presence of the chaotic saddle. Since the origin of the
escape time singularities is geometric, the fractality of the escape time
function must be independent of the observer. We conclude that the time
dilation phenomenon does not affect the typical structure of the singularities
of the escape times, and interestingly this phenomenon occurs chaotically.
The escape time function as measured in any clock is closely related to a
Cantor-like set of Lebesgue measure zero, since it is a self-similar set in
the hyperbolic regime. This feature allows us to demonstrate that the fractal
dimension of the escape time function is relativistic invariant. The key point
of the demonstration is that, knowing the evolution of the Lorentz factor,
there exists an unequivocal relation between the transformed times. In order
to verify this result computationally, we have used the uncertainty dimension
algorithm. Furthermore, we have pointed out that the system is more likely to
be predictable in a reference frame comoving with the particle if a limited
clock resolution is considered, even though from a mathematical point of view
the predictability of the system is independent of the reference frame.
The main conclusion of the present work is that transient chaos is coordinate
invariant from a theoretical point of view. This statement extends the
universality of occurrence of chaos and fractals under coordinate
transformations to the realm of transient chaotic phenomena as well.
## ACKNOWLEDGMENTS
We acknowledge interesting discussions with Prof. Hans C. Ohanian. This work
was supported by the Spanish State Research Agency (AEI) and the European
Regional Development Fund (ERDF) under Project No. FIS2016-76883-P.
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|
2024-09-04T02:54:59.183904 | 2020-03-09T13:49:48 | 2003.05288 | {
"authors": "Chi-Chun Zhou, Ping Zhang, and Wu-Sheng Dai",
"full_text_license": null,
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"provenance": "arxiv-papers-0000.json.gz:26165",
"submitter": "Chichun Zhou",
"url": "https://arxiv.org/abs/2003.05288"
} | arxiv-papers | 11footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# The Brownian Motion in an Ideal Quantum Qas
Chi-Chun Zhou Ping Zhang and Wu-Sheng Dai
###### Abstract
A Brownian particle in an ideal quantum gas is considered. The mean square
displacement (MSD) is derived. The Bose-Einstein or Fermi-Dirac distribution,
other than the Maxwell-Boltzmann distribution, provides a different stochastic
force compared with the classical Brownian motion. The MSD, which depends on
the thermal wavelength and the density of medium particles, reflects the
quantum effect on the Brownian particle explicitly. The result shows that the
MSD in an ideal Bose gas is shorter than that in a Fermi gas. The behavior of
the quantum Brownian particle recovers the classical Brownian particle as the
temperature raises. At low temperatures, the quantum effect becomes obvious.
For example, there is a random motion of the Brownian particle due to the
fermionic exchange interaction even the temperature is near the absolute zero.
## 1 Introduction
The Brownian motion is first observed by Robert Brown in 1827 and then
explained by Einstein (1905), Smoluchowski (1905), and Langevin (1908) in the
early 20th century [1]. The early theory of the Brownian motion not only
provides an evidence for the atomistic hypothesis of matter [2], but also
builds a bridge between the microscopic dynamics and the macroscopic
observable phenomena [2].
The classical understanding of the Brownian motion is quite well established.
However, there is an assumption in the early theory of the Brownian motion
that the medium particle obeys the Maxwell-Boltzmann distribution.
The behavior of a Brownian particle in an ideal quantum gas draws some
attentions because to study the motion of a Brownian particle in an quantum
system is now within reach of experimental tests. For example, an electron in
a black body radiation [3]. In such systems, the quantum exchange interaction,
which always leads to real difficulty in mechanics and statistical mechanics
[4, 5, 6, 7], exists and causes the medium particle obeying the Bose-Einstein
or Fermi-Dirac distribution. It is difficult to make exact or even detailed
dynamical calculations [8, 1], since a different stochastic force is provided
by the Bose-Einstein or Fermi-Dirac distribution. At high-temperature and low-
density, the classical theory serves as good approximation.
In this paper, we give an explicit expression of the mean square displacement
(MSD) of a Brownian particle in an ideal quantum gas using, e.g., the virial
expansion. Comparison with the classical Brownian motion, a correction for the
MSD, which depends on the thermal wavelength and the density of medium
particles, is deduced. The result shows that the MSD in an ideal Bose gas is
shorter than that in a Fermi gas. The behavior of the quantum Brownian
particle recovers the classical Brownian particle as the temperature raises.
At low temperature, the quantum effect becomes obvious. For example, there is
a random motion of the Brownian particle due to the fermionic exchange
interaction even the temperature is near the absolute zero.
The early studies of the Brownian motion inspired many prominent developments
in various areas such as physics, mathematics, financial markets, and biology.
In physics, exact solutions of Brownian particles in different cases, such as
in a constant field of force [1] and in a harmonically potential field [1],
are given. The Brownian motion with a time dependent diffusion coefficient is
studied in Ref. [9]. The boundary problem of Brownian motions is studied in
Refs. [10, 11]. The anomalous diffusion process, frequently described by the
scaled Brownian motion, is studied in Refs. [12, 13]. The Kramers-Klein
equation considers the Brownian particle that is in an general field of force
[1]. The generalized Langevin equations and the master equation for the
quantum Brownian motion are studied in Refs. [14, 15]. In mathematics, the
rigorous interpretation of Brownian motions based on concepts of random walks,
martingales, and stochastic processes is given [16, 8, 1]. In financial
markets, the theory of the Brownian motion is used to describe the movement of
the price of stocks and options [8, 1, 17, 18, 19, 20]. The application of the
fractional Brownian motion, which is a generalized Brownian motion, in
financial markets is studied in Refs. [21, 22, 23]. Moreover, the Brownian
motion plays a central and fundamental role in studies of soft matter and
biophysics [8], e.g., active Brownian motions, which can be used to describe
the motion of swarms of animals in fluid, are studied in Refs. [24, 25, 26,
27, 28, 8].
Among many quantities, the MSD, which is measurable, describes the Brownian
motion intuitively. There are studies focus on the MSD related problems. For
examples, the relation between the MSD and the time interval can be generally
written as $\left\langle x_{t}^{2}\right\rangle\sim t^{\alpha}$ [9]. One
distinguishes the subdiffusion with $0<\alpha<1$ and the superdiffusion with
$\alpha>1$ [29, 30, 31]. The relation between the MSD and the time interval of
the so called ultraslow Brownian motions is $\left\langle
x_{t}^{2}\right\rangle\sim\left(\ln t\right)^{\gamma}$ [32].
There are different approaches to build a quantum analog of the Brownian
motion [33, 34, 35, 36]. For examples, the method of the path integral is used
to study the quantum Brownian motion [35]. The approach of a quantum analog or
quantum generalization of the Langevin equation and the master equation, e.g.,
the quantum master equation [3] and the quantum Langevin equation [37] is used
to build a quantum Brownian motion. Among them, the method of quantum
dynamical semigroups [38] is prominent. They point it out that the quantum
equation should be casted into the Lindblad form [38, 39]. A completely
positive master equation describing quantum dissipation for a Brownian
particle is derived in Ref. [39].
This paper is organized as follows. In Sec. 2, for the sake of completeness,
we derive the brownian motion from the perspective of the particle
distribution in an ideal Boltzmann gas. In Sec. 3, we derive the MSD of a
Brownian particle in an ideal quantum gas. High-temperature and low-
temperature expansions are given. The $d$-dimensional case is considered. The
conclusion and outlook are given in Sec. 4. Some details of the calculation is
given in the Appendix.
## 2 A Brownian particle in an ideal classical gas: the Brownian motion
In this section, we consider a Brownian particle in an ideal classical gas.
For the sake of completeness, we derive, in detail, the Brownian motion from
the perspective of the particle distribution.
A brief review on the Langevin equation. For a Brownian particle with mass
$M$, the dynamic equation is given by Paul Langevin [40]
$\displaystyle dv$ $\displaystyle=-\frac{\gamma}{M}vdt+\frac{1}{M}F_{t}dt,$
(2.1) $\displaystyle dx$ $\displaystyle=vdt,$ (2.2)
where $\gamma=6\pi\eta r$ with $\eta$ the viscous coefficient and $r$ the
radius of the medium particles. $F_{t}$ is the stochastic force generated by
numerous collisions of the medium particle. It is reasonable to make the
assumption that $F_{t}$ is isotropic, i.e.,
$\left\langle F_{t}\right\rangle=0.$ (2.3)
If the collision of the medium particle is uncorrelated; that is, for $t\neq
s$, $F_{t}$ is independent of $F_{s}$:
$\left\langle F_{s}F_{t}\right\rangle\propto\delta\left(s-t\right),$ (2.4)
then, the solution of Eqs. (2.1) and (2.2) is [40]
$\displaystyle v_{t}$
$\displaystyle=v_{0}\exp\left(-\frac{\gamma}{M}t\right)+\frac{1}{M}\int_{0}^{t}\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]F_{s}ds,$
(2.5) $\displaystyle x_{t}$
$\displaystyle=x_{0}+\frac{M}{\gamma}v_{0}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]+\frac{1}{\gamma}\int_{0}^{t}\left\\{1-\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]\right\\}F_{s}ds,$
(2.6)
where $x_{0}$ and $v_{0}$ are the initial position and velocity.
The stochastic force determined by the Maxwell-Boltzmann distribution and the
MSD. In an ideal classical gas, the gas particle obeys the Maxwell-Boltzmann
distribution [41]. The number of particles possessing energy within
$\varepsilon$ to $\varepsilon+d\varepsilon$, denoted by
$\tilde{a}_{\varepsilon}$, is proportional to $e^{-\beta\varepsilon}$ [41],
i.e.,
$\tilde{a}_{\varepsilon}=\omega_{\varepsilon}e^{-\beta\varepsilon}d\varepsilon,$
(2.7)
where $\omega_{\varepsilon}$ is the degeneracy of the energy $\varepsilon$ and
$\beta=\left(kT\right)^{-1}$ with $k$ the Boltzmann constant $T$ the
temperature [41]. A collision of the medium particle with energy $\varepsilon$
gives a force of magnitude proportional to $\sqrt{2m\varepsilon}$, which is
the momentum of the particle. We have
$F=\rho\sqrt{2m\varepsilon},$ (2.8)
where $\rho$ is a coefficient and $m$ is the mass of the medium particle.
Thus, the probability of the Brownian particle subjected to a stochastic force
with magnitude within $F$ to $F+dF$ is
$P\left(F\right)dF=\sqrt{\frac{\beta}{2\pi
m\rho^{2}}}\exp\left(-\frac{F^{2}\beta}{2m\rho^{2}}\right)dF.$ (2.9)
In an ideal classical gas, there is no inter-particle interactions among
medium particles, thus, for $t\neq s$, the force $F_{s}$ and $F_{t}$ are
independent. Substituting Eq. (2.9) into Eq. (2.4) gives
$\left\langle
F_{s}F_{t}\right\rangle=\frac{m\rho^{2}}{\beta}\delta\left(s-t\right).$ (2.10)
By using Eqs (2.6), (2.9), and (2.10), a direct calculation of the MSD gives
$\displaystyle\left\langle\left(x_{t}-x_{0}\right)^{2}\right\rangle$
$\displaystyle=\frac{M^{2}}{\gamma^{2}}\left(v_{0}^{2}-\frac{1}{2m\gamma}\frac{m\rho^{2}}{\beta}\right)\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]^{2}$
$\displaystyle+\frac{1}{\gamma^{2}}\frac{m\rho^{2}}{\beta}\left\\{t-\frac{M}{\gamma}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]\right\\}.$
(2.11)
For a large-scale time, $t\gg 1$, Eq. (2.11) recovers
$\left\langle x_{t}^{2}\right\rangle=\frac{kT}{3\pi\eta r}t,$ (2.12)
where $\rho=\sqrt{12\pi\eta r/m}$ and $x_{0}$ is chosen to be $0$ without lose
of generality. Eq. (2.12) is the famous Einstein’s long-time result of the
MSD. The motion of a brownian particle in an ideal classical gas is the
Brownian motion.
## 3 The MSD of a Brownian particle in an ideal quantum gas
In this section, we give the MSD of a Brownian particle in an ideal quantum
gas. High-temperature and low-temperature expansions explain the quantum
effect intuitively.
### 3.1 The stochastic force determined by the Bose-Einstein or Fermi-Dirac
distribution
In an ideal quantum gas, the gas particle obeys Bose-Einstein or Fermi-Dirac
distribution other than the Maxwell-Boltzmann distribution. The stochastic
force is different from that in an ideal classical gas. In this section, we
discuss the properties of the stochastic force in an ideal quantum gas.
In an ideal quantum gas, the number of particles possessing energy within
$\varepsilon$ to $\varepsilon+d\varepsilon$, denoted by $a_{\varepsilon}$, is
$a_{\varepsilon}=\frac{\omega_{\varepsilon}}{\exp\left(\beta\varepsilon+\alpha\right)+g}d\varepsilon,$
(3.1)
where $\alpha$ is defined by $z=e^{-\alpha}$ with $z$ the fugacity [41]. For
Bose cases, $g=-1$, and for Fermi cases, $g=1$. Thus, the probability of the
Brownian particle subjected to a stochastic force with magnitude within $F$ to
$F+dF$ is
$p\left(F\right)dF=\sqrt{\frac{\beta}{2\rho^{2}m\pi}}\frac{1}{h_{1/2}\left(z\right)}\frac{1}{\exp\left[\beta
F^{2}/\left(2\rho^{2}m\right)+\alpha\right]+g}dF,$ (3.2)
where we $h_{\nu}\left(x\right)$ equals Bose-Einstein integral
$g_{\nu}\left(x\right)$ in Bose cases [41] and Fermi-Dirac integral
$f_{\nu}\left(x\right)$ [41] in Fermi cases.
In an ideal quantum gas, the stochastic force is also isotropic, that is, Eq.
(2.3) holds. However, the collision, due to the overlapping of the wave
package, can be correlated; that is, $\left\langle F_{s}F_{t}\right\rangle$ is
no longer a delta function but a function of $s-t$ with a peak at $s=t$.
However, as the ratio of the thermal wavelength and the average distance
between the medium particles decreases, $\left\langle
F_{s}F_{t}\right\rangle$, can be well approximated by a delta function:
$\left\langle
F_{s}F_{t}\right\rangle\sim\frac{m\rho^{2}}{\beta}\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\delta\left(s-t\right),$
(3.3)
for $n\lambda\ll 1$, where $\lambda=h/\sqrt{2\pi mkT}$ is the thermal
wavelength and $n$ is the density of the medium particle.
In this paper, we consider the case that the ratio of the thermal wavelength
and the average distance between the medium particles is small.
### 3.2 The MSD
For $n\lambda\ll 1$, by using Eqs. (3.2), (2.3), and (2.6), a direct
calculation of MSD gives
$\displaystyle\left\langle x_{t}^{2}\right\rangle$
$\displaystyle=\frac{M^{2}}{\gamma^{2}}\left\\{v_{0}^{2}-\frac{1}{2m\gamma}\frac{m\rho^{2}}{\beta}\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\right\\}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]^{2}$
$\displaystyle+\frac{1}{\gamma^{2}}\frac{m\rho^{2}}{\beta}\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\left\\{t-\frac{M}{\gamma}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]\right\\}.$
(3.4)
For a large-scale time, $t\gg 1$, Eq. (3.4) recovers
$\left\langle x_{t}^{2}\right\rangle=\frac{kT}{3\pi\eta
r}t\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)},$ (3.5)
where $h_{3/2}\left(z\right)/h_{1/2}\left(z\right)$ is a correction for the
MSD due to the Bose-Einstein or Fermi-Dirac distribution, a result of the
quantum exchange interaction among gases particles.
### 3.3 High-temperature and low-temperature expansions
In order to compare with Eq. (2.12) intuitively, we give high-temperature and
low-temperature expansions of Eq. (3.5) by using the state equation of ideal
Bose or Fermi gases [40, 41]
$\displaystyle p$ $\displaystyle=\frac{kT}{\lambda}h_{3/2}\left(z\right),$
(3.6) $\displaystyle\frac{N}{V}$
$\displaystyle=\frac{1}{\lambda}h_{1/2}\left(z\right).$ (3.7)
The high-temperature expansion. At high temperatures, the virial expansion of
Eqs. (3.6) and (3.7) directly gives [40, 41]
$\frac{pV}{N}\sim kT\left[1+ga_{1}\left(T\right)n\lambda+...\right],$ (3.8)
where $a_{1}\left(T\right)$ is the first virial coefficient [40]. For a
$1$-dimensional ideal Bose or Fermi gas, $a_{1}\left(T\right)=0.353553$ [41].
Substituting Eqs. (3.6) and (3.7) into Eq. (3.8) gives
$\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\sim\left[1+ga_{1}\left(T\right)n\lambda+...\right].$
(3.9)
Substituting Eq. (3.9) into Eq. (3.5) gives the MSD at high temperatures:
$\left\langle x_{t}^{2}\right\rangle=\frac{kT}{3\pi\eta
r}t\left[1+ga_{1}\left(T\right)n\lambda+\ldots\right].$ (3.10)
The result, Eq. (3.10), shows that the MSD in an ideal Bose gas is shorter
than that in a Fermi gas. Since $\lambda$ decreases as $T$ raises, the
behavior of the quantum Brownian particle returns the classical Brownian
particle as the temperature raises.
The low-temperature expansion for Fermi cases. At low temperatures, for Fermi
cases, $g=-1$,
$\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}=\frac{f_{3/2}\left(z\right)}{f_{1/2}\left(z\right)}.$
(3.11)
The expansion of the Fermi-Dirac integral at large $z$ gives [41]
$f_{\nu}\left(e^{\xi}\right)=\frac{\xi^{\nu}}{\Gamma\left(1+\nu\right)}\left\\{1+2\nu\sum_{j=1,3,5,...}\left[\left(\nu-1\right)....\left(\nu-j\right)\left(1-2^{-j}\right)\frac{\zeta\left(j+1\right)}{\xi^{j+1}}\right]\right\\}.$
(3.12)
Keeping only the first two terms in Eq. (3.12) gives
$f_{\nu}\left(z\right)=\frac{\left(\ln
z\right)^{\nu}}{\Gamma\left(1+\nu\right)}+2\nu\left(\nu-1\right)\frac{1}{2}\frac{\zeta\left(2\right)}{\left(\ln
z\right)^{2}}.$ (3.13)
Substituting Eq. (3.13) into Eqs. (3.6) and (3.7) gives
$\displaystyle p$ $\displaystyle=\frac{kT}{\lambda}\frac{\left(\ln
z\right)^{3/2}}{\Gamma\left(5/2\right)}\left[1+\frac{3}{4}\frac{\zeta\left(2\right)}{\left(\ln
z\right)^{2}}\right],$ (3.14) $\displaystyle\frac{N}{V}$
$\displaystyle=\frac{1}{\lambda}\frac{\left(\ln
z\right)^{1/2}}{\Gamma\left(5/2\right)}\left[1-\frac{1}{4}\frac{\zeta\left(2\right)}{\left(\ln
z\right)^{2}}\right].$ (3.15)
The fugacity $z$ can be solved from Eq. (3.15):
$\ln
z\sim\frac{\epsilon_{f}}{kT}\left[1+\frac{1}{2}\zeta\left(2\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right],$
(3.16)
where
$\epsilon_{f}=\lambda^{2}kT\left[\frac{1}{2}\Gamma\left(\frac{3}{2}\right)n\right]^{2}$
is the Fermi energy [41]. By substituting Eq. (3.13) into Eq. (3.11) with
fugacity $z$ given by Eq. (3.16), we have
$\frac{f_{3/2}\left(z\right)}{f_{1/2}\left(z\right)}=\frac{\Gamma\left(3/2\right)}{\Gamma\left(5/2\right)}\frac{\epsilon_{f}}{kT}\left\\{1+\left[\frac{\zeta\left(2\right)}{2}+\zeta\left(2\right)\right]\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right\\}.$
(3.17)
Substituting Eq. (3.17) into Eq. (3.5) gives the MSD of Fermi cases at low
temperatures:
$\left\langle x_{t}^{2}\right\rangle\sim\frac{1}{3\pi\eta
r}t\frac{\Gamma\left(3/2\right)}{\Gamma\left(5/2\right)}\epsilon_{f}\left[1+\frac{3}{2}\zeta\left(2\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right].$
(3.18)
The first term of Eq. (3.18) is independent of the temperature $T$, which
means that there is a random motion of the Brownian particle due to the
fermionic exchange interaction even the temperature is near the absolute zero.
It is a result of Pauli exclusion principle [41].
The low-temperature expansion for Bose cases. At low temperatures, for Bose
cases, $g=1$,
$\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}=\frac{g_{1+d/2}\left(z\right)}{g_{d/2}\left(z\right)}.$
(3.19)
Expanding $g_{\nu}\left(z\right)$ around $z=1$ gives [41]
$g_{\nu}\left(z\right)=\frac{\Gamma\left(1-\nu\right)}{\left(-\ln
z\right)^{1-\nu}}+\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\zeta\left(\nu-j\right)\left(-\ln
z\right)^{j}.$ (3.20)
Substituting Eq. (3.20) into Eqs. (3.6) and (3.7) gives
$\displaystyle p$
$\displaystyle=\frac{1}{\lambda^{d}}\Gamma\left(-\frac{1}{2}\right)\left(-\ln
z\right)^{1/2}+\zeta\left(\frac{3}{2}\right)-\zeta\left(\frac{1}{2}\right)\left(-\ln
z\right),$ (3.21) $\displaystyle\frac{N}{V}$
$\displaystyle=\frac{1}{\lambda^{d}}\frac{\Gamma\left(1/2\right)}{\left(-\ln
z\right)^{1/2}}+\zeta\left(\frac{1}{2}\right)-\zeta\left(-\frac{1}{2}\right)\left(-\ln
z\right).$ (3.22)
The fugacity can be solved from Eq. (3.22):
$\ln z=-\frac{\pi}{n^{2}\lambda^{2}}.$ (3.23)
By substituting Eq. (3.20) into Eq. (3.19) with fugacity $z$ given by Eq.
(3.23), we have
$\frac{g_{3/2}\left(z\right)}{g_{1/2}\left(z\right)}=\frac{\zeta\left(3/2\right)}{\sqrt{n^{2}\lambda^{2}}}-\left(2+\frac{\zeta\left(3/2\right)\zeta\left(1/2\right)}{\pi}\right)\frac{\pi}{n^{2}\lambda^{2}}.$
(3.24)
Substituting Eq. (3.24) into Eq. (3.5) gives the MSD of Bose cases at low
temperatures:
$\displaystyle\left\langle x_{t}^{2}\right\rangle$
$\displaystyle\sim\frac{kT}{3\pi\eta
r}t\left[\zeta\left(\frac{3}{2}\right)\frac{1}{n\lambda}-\left(2\pi+\zeta\left(\frac{3}{2}\right)\zeta\left(\frac{1}{2}\right)\right)\frac{1}{n^{2}\lambda^{2}}\right]$
(3.25) $\displaystyle\sim\frac{1}{3\pi\eta
r}\zeta\left(\frac{3}{2}\right)\frac{\sqrt{2\pi
m}}{h}\frac{1}{n}\left(kT\right)^{3/2}t.$ (3.26)
The MSD is proportional to $T^{3/2}$ and is reversely proportional to the
density of particle, which is also different from that of the Brownian motion.
### 3.4 The $d$-dimensional case
In this section, a similar procedure gives the MSD of a Brownian particle in a
$d$-dimensional space. For the sake of clarity, we list the result. The detail
of the calculation can be found in the Appendix.
The MSD. The MSD for a Brownian particle in an ideal quantum gas in a
$d$-dimensional space is
$\left\langle\mathbf{x}_{t}^{2}\right\rangle=\frac{kTd}{3\pi\eta
r}t\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}.$ (3.27)
The high-temperature expansion. At high temperatures, the MSD, Eq.(3.27),
becomes
$\left\langle\mathbf{x}_{t}^{2}\right\rangle=\frac{kTd}{3\pi\eta
r}t\left[1+ga_{1}\left(T\right)n\lambda^{d}+\ldots\right],$ (3.28)
where $a_{1}\left(T\right)=\frac{1}{2^{1+d/2}}$ and is the first virial
coefficient of ideal Bose or Fermi gases in a $d$-dimensional space [40, 41].
The low-temperature expansion for Fermi cases. At low temperatures, for Fermi
cases, the MSD, Eq. (3.27), becomes
$\left\langle\mathbf{x}_{t}^{2}\right\rangle\sim\frac{d}{3\pi\eta
r}\frac{\Gamma\left(1+d/2\right)}{\Gamma\left(2+d/2\right)}t\epsilon_{f}\left[1+\left(\frac{d}{2}+1\right)\zeta\left(2\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right],$
(3.29)
where $\epsilon_{f}$ is the Fermi energy in a $d$-dimensional space and
$\epsilon_{f}=\lambda^{2}kT\left[\frac{1}{2}\Gamma\left(1+\frac{d}{2}\right)n\right]^{2/d}$
[40, 41].
The low-temperature expansion for Bose cases with $d=2$. The low-temperature
expansion for a Bose gas at any dimension higher than $2$ is not given in this
section, because the Bose-Einstein condensation (BEC) occurs. Here, we only
consider the $2$-dimensional case. At low temperatures, for Bose cases, the
MSD, Eq. (3.27), becomes
$\left\langle\mathbf{x}_{t}^{2}\right\rangle=\frac{2kT}{3\pi\eta
r}t\left[-\exp\left(-n\lambda^{2}\right)-\frac{\exp\left(-n\lambda^{2}\right)}{n\lambda^{2}}+\frac{\pi^{2}}{6n\lambda^{2}}\right].$
(3.30)
## 4 Conclusions and outlooks
The difficulty in the calculation of the behavior of a Brownian particle in an
ideal quantum gas directly comes from the stochastic force caused by the Bose-
Einstein and Fermi-Dirac distribution other than the Maxwell-Boltzmann
distribution. Comparison with the classical Brownian motion, on one hand, the
distribution of the stochastic force is different; on the other hand, the
collision, due to the overlapping of the wave package, could be correlated,
that is, $\left\langle F_{s}F_{t}\right\rangle$ is no longer a delta function
but a function of $s-t$ with a peak at $s=t$. Thus, it is difficult to make
exact or even detailed dynamical calculations [8, 1].
In this paper, we consider the motion of a Brownian particle in an ideal
quantum gas. We give an explicit expression of the MSD, which depends on the
thermal wavelength and the density of medium particles. High-temperature and
low-temperature expansions explain the quantum effect intuitively. For
examples, the MSD in an ideal Bose gas is shorter than that in a Ferm gas.
There is a random motion of the Brownian particle due to the fermionic
exchange interaction even the temperature is near the absolute zero.
The result in this work can be verified by experiment test.
## 5 Acknowledgments
We are very indebted to Dr G Zeitrauman for his encouragement. This work is
supported in part by NSF of China under Grant No. 11575125 and No. 11675119.
## 6 Appendix
In the appendix, we give the detail of the calculation of Eqs (3.27), (3.28),
(3.29), and (3.30).
The detail of the calculate for the MSD, Eq. (3.27), of a Brownian particle in
a $d$-dimensional space. The Langevin equation in $d$-dimensional is
$\displaystyle M\frac{d\mathbf{v}}{dt}$
$\displaystyle=-\gamma\mathbf{v}+\mathbf{F}_{t},$ (6.1)
$\displaystyle\frac{d\mathbf{x}}{dt}$ $\displaystyle=\mathbf{v.}$ (6.2)
In a $d$-dimensional space, the stochastic force $\mathbf{F}_{t}$ is
isotropic:
$\left\langle\mathbf{F}\right\rangle=0.$ (6.3)
For different time $t$ and $s$, $\mathbf{F}_{s}$ and $\mathbf{F}_{t}$ are
almost independent when the ratio of the thermal wavelength and the average
distance between the medium particles is small, that is,
$\left\langle\mathbf{F}_{s}\cdot\mathbf{F}_{t}\right\rangle\sim\delta\left(s-t\right)$
(6.4)
holds for $n\lambda^{d}\ll 1$. The solution of Eqs. (6.1) and (6.2) is
$\displaystyle\mathbf{v}_{t}$
$\displaystyle=\mathbf{v}_{0}e^{-\frac{\gamma}{M}t}+\frac{1}{M}\int_{0}^{t}\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]\mathbf{F}_{s}ds,$
(6.5) $\displaystyle\mathbf{x}_{t}$
$\displaystyle=\mathbf{x}_{0}+\frac{M}{\gamma}v_{0}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]+\frac{1}{\gamma}\int_{0}^{t}\left\\{1-\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]\right\\}\mathbf{F}_{s}ds.$
(6.6)
In the $d$-dimensional case, the number of particle possessing momentum within
$\mathbf{P}$ to $\mathbf{P+}d\mathbf{P}$, denoted by
$a\left(\mathbf{P}\right)$, is [40, 41]
$a\left(\mathbf{P}\right)=\frac{1}{\exp\left[\beta\left(p_{x^{1}}^{2}+p_{x^{2}}^{2}+...p_{x^{d}}^{2}\right)/\left(2m\right)+\alpha\right]+g}.$
(6.7)
The force given by a collision of a particle with momentum $\mathbf{P}$ is
proportional to $\mathbf{P}$, $\mathbf{F=}\rho\mathbf{P}$. Thus the
probability of the stochastic force with magnitude within
$\left|\mathbf{F}\right|$ to $\left|\mathbf{F+}d\mathbf{F}\right|$ is
$P\left(\mathbf{F}\right)d\mathbf{F}=\left(\sqrt{\frac{\beta}{2\pi
m\rho^{2}}}\right)^{d}\frac{1}{h_{d/2}\left(z\right)}\frac{1}{\exp\left[\beta\left|\mathbf{F}\right|^{2}/\left(2m\rho^{2}\right)+\alpha\right]+g}.$
(6.8)
Substituting Eq. (6.8) into Eq. (6.4) gives
$\left\langle\mathbf{F}_{s}\cdot\mathbf{F}_{t}\right\rangle\sim\frac{dm\rho^{2}}{\beta}\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\delta\left(s-t\right).$
(6.9)
By using Eqs. (6.6), (6.8), and (6.9), a direct calculation of MSD gives
$\displaystyle\left\langle\mathbf{x}_{t}^{2}\right\rangle$
$\displaystyle=\frac{M^{2}}{\gamma^{2}}\left[\mathbf{v}_{0}^{2}-\frac{d}{2m\gamma}\frac{m\rho^{2}}{\beta}\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\right]\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]^{2}$
$\displaystyle+\frac{1}{\gamma^{2}}\frac{dm\rho^{2}}{\beta}\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\left[t-\frac{M}{\gamma}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]\right].$
(6.10)
where $\mathbf{x}_{0}$ is chosen to be the origin.
For a large-scale time, $t\gg 1$, Eq. (6.10) recovers Eq. (3.27).
The high-temperature expansion. For the $d$-dimensional case, the state
equation of an ideal quantum gas is [40, 41]
$\displaystyle p$
$\displaystyle=\frac{kT}{\lambda^{d}}h_{1+d/2}\left(z\right),$ (6.11)
$\displaystyle\frac{N}{V}$
$\displaystyle=\frac{1}{\lambda^{d}}h_{d/2}\left(z\right).$ (6.12)
The virial expansion [40, 41] directly gives
$\frac{pV}{N}\sim kT\left[1+ga_{1}\left(T\right)n\lambda^{d}+...\right]$
(6.13)
Substituting Eqs. (6.11) and (6.12) into Eq. (6.13) gives
$\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\sim\left[1+ga_{1}\left(T\right)n\lambda^{d}+...\right].$
(6.14)
Substituting Eq. (6.14) into Eq. (3.27) gives Eq. (3.28).
The low-temperature expansion for Fermi cases. For Fermi cases, $g=-1$,
$\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}=\frac{f_{1+d/2}\left(z\right)}{f_{d/2}\left(z\right)}.$
(6.15)
By the expansion of the Fermi-Dirac integral at large $z$, we have [41]
$f_{\nu}\left(e^{\xi}\right)=\frac{\xi^{\nu}}{\Gamma\left(1+\nu\right)}\left\\{1+2\nu\sum_{j=1,3,5,...}\left[\left(\nu-1\right)....\left(\nu-j\right)\left(1-2^{-j}\right)\frac{\zeta\left(j+1\right)}{\xi^{j+1}}\right]\right\\}.$
(6.16)
Keeping only the first two terms gives
$f_{\nu}\left(z\right)=\frac{\left(\ln
z\right)^{\nu}}{\Gamma\left(1+\nu\right)}+2\nu\left(\nu-1\right)\frac{1}{2}\frac{\zeta\left(2\right)}{\left(\ln
z\right)^{2}}.$ (6.17)
Substituting Eq. (6.17) into Eqs. (6.11) and (6.12) gives
$\displaystyle\frac{p}{kT}$
$\displaystyle=\frac{1}{\lambda^{d}}\frac{\left(\ln
z\right)^{1+d/2}}{\Gamma\left(2+d/2\right)}\left[1+\frac{d}{2}\left(1+\frac{d}{2}\right)\frac{\zeta\left(2\right)}{\left(\ln
z\right)^{2}}\right],$ (6.18) $\displaystyle N$
$\displaystyle=\frac{\Omega}{\lambda^{d}}\frac{\left(\ln
z\right)^{d/2}}{\Gamma\left(1+d/2\right)}\left[1+\frac{d}{2}\left(\frac{d}{2}-1\right)\frac{\zeta\left(2\right)}{\left(\ln
z\right)^{2}}\right],$ (6.19)
where $\Omega$ is the volume. The fugacity can be solved from Eq (6.19):
$\ln
z\sim\frac{\epsilon_{f}}{kT}\left[1-\zeta\left(2\right)\left(\frac{d}{2}-1\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right],$
(6.20)
where
$\epsilon_{f}=\lambda^{2}kT\left[\frac{1}{2}\Gamma\left(1+\frac{d}{2}\right)n\right]^{2/d}$
is the Fermi energy. By substituting Eq. (6.17) into Eq. (6.15) with fugacity
$z$ given by Eq. (6.20), we have
$\frac{f_{1+d/2}\left(z\right)}{f_{d/2}\left(z\right)}=\frac{\Gamma\left(1+\frac{d}{2}\right)}{\Gamma\left(2+\frac{d}{2}\right)}\frac{\epsilon_{f}}{kT}\left\\{1+\left[\frac{d\zeta\left(2\right)}{2}+\zeta\left(2\right)\right]\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right\\}.$
(6.21)
Substituting Eq. (6.21) into Eq. (3.27) gives Eq. (3.29).
The low-temperature expansion for Bose cases in the $2$-dimensional space. For
Bose cases, $g=1$,
$\frac{h_{2}\left(z\right)}{h_{1}\left(z\right)}=\frac{g_{2}\left(z\right)}{g_{1}\left(z\right)},$
(6.22)
where $d=2$. For $d=2$,
$g_{1}\left(z\right)=-\ln\left(1-z\right).$ (6.23)
Substituting Eq. (6.23) into Eq. (6.12) gives
$\frac{N}{V}=-\frac{1}{\lambda^{2}}\ln\left(1-z\right).$ (6.24)
Then, the fugacity can be solved from Eq (6.24):
$z=1-\exp\left(-n\lambda^{2}\right).$ (6.25)
Expanding $g_{2}\left(z\right)$ around $z=1$ gives
$\displaystyle g_{2}\left(z\right)$
$\displaystyle=\frac{\pi^{2}}{6}-\left(1-z\right)-\frac{\left(1-z\right)^{2}}{2^{2}}-\frac{\left(1-z\right)^{3}}{3^{2}}-\ldots$
$\displaystyle+\left(1-z\right)\ln\left(1-z\right)+\frac{\left(1-z\right)^{2}}{2}\ln\left(1-z\right)+\frac{\left(1-z\right)^{3}}{3}\ln\left(1-z\right)+\ldots$
(6.26)
Substituting Eqs. (6.26) and (6.23) with fugacity given in Eq. (6.25) into Eq.
(6.22) gives
$\frac{g_{2}\left(z\right)}{g_{1}\left(z\right)}=-\exp\left(-n\lambda^{2}\right)-\frac{\exp\left(-n\lambda^{2}\right)}{n\lambda^{2}}+\frac{\pi^{2}}{6n\lambda^{2}},$
(6.27)
Substituting Eq. (6.27) into Eq. (3.27) gives Eq. (3.30).
## Acknowledgments
We are very indebted to Dr G. Zeitrauman for his encouragement. This work is
supported in part by NSF of China under Grant No. 11575125 and No. 11675119.
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|
2024-09-04T02:54:59.199255 | 2020-03-11T14:42:58 | 2003.05336 | {
"authors": "Yoshiki Higo and Shinpei Hayashi and Shinji Kusumoto",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26166",
"submitter": "Yoshiki Higo",
"url": "https://arxiv.org/abs/2003.05336"
} | arxiv-papers | # On Tracking Java Methods with Git Mechanisms
Yoshiki Higo<EMAIL_ADDRESS>Shinpei Hayashi<EMAIL_ADDRESS>Shinji Kusumoto<EMAIL_ADDRESS>Graduate School of Information
Science and Technology, Osaka University,
Yamadaoka 1–5, Suita, Osaka 565–0871, Japan School of Computing, Tokyo
Institute of Technology,
Ookayama 2–12–1–W8–71, Ookayama, Meguro-ku, Tokyo 152–8550, Japan
###### Abstract
Method-level historical information is useful in various research on mining
software repositories such as fault-prone module detection or evolutionary
coupling identification. An existing technique named Historage converts a Git
repository of a Java project to a finer-grained one. In a finer-grained
repository, each Java method exists as a single file. Treating Java methods as
files has an advantage, which is that Java methods can be tracked with Git
mechanisms. The biggest benefit of tracking methods with Git mechanisms is
that it can easily connect with any other tools and techniques build on Git
infrastructure. However, Historage’s tracking has an issue of accuracy,
especially on small methods. More concretely, in the case that a small method
is renamed or moved to another class, Historage has a limited capability to
track the method. In this paper, we propose a new technique, FinerGit, to
improve the trackability of Java methods with Git mechanisms. We implement
FinerGit as a system and apply it to 182 open source software projects, which
include 1,768K methods in total. The experimental results show that our tool
has a higher capability of tracking methods in the case that methods are
renamed or moved to other classes.
###### keywords:
Mining software repositories , Source code analysis , Tracking Java methods
††journal: Journal of Systems and Software
## 1 Introduction
One feature of version control systems is the ability to know file-level
change information. Thus, it is easy to identify which files were changed in
given commits or counting changes for files in a given repository. However,
many approaches in mining software repositories (in short, MSR) require
information on finer-grained units such as Java methods or C functions. If we
want to count changes for Java methods, we need to parse source files to
identify method positions and then we need to match method positions with
changed code positions to identify which methods were changed. To conduct
finer-grained analyses, developers have to implement code/scripts. Besides,
incorrect analysis results will be obtained if the implemented code/scripts
include bugs.
Hata et al. proposed a technique, Historage, which enables Java methods to be
tracked with Git mechanisms [1]. Historage takes a Git repository of a Java
project as its input, and it outputs another Git repository in which each
method gets extracted as a file. Treating Java methods as files realizes that
developers/practitioners can obtain method-level historical information only
by executing Git commands such as git-log.
(a) Git repository.
(b) Historage repository.
Figure 1: Differences between Git and Historage repositories.
Figure 1 shows a simple model of Git and Historage repositories. In the Git
repository, file Person.java is managed. We can see that Person.java was
changed in two commits c100 and c101. Information for the changes on
Person.java can be retrieved by executing git-log. However, if we want to know
which methods were changed in the two commits, we have to parse Person.java to
obtain the positions of the methods and then we have to match method positions
with the positions of the changed code in the two commits. On the other hand,
in the Historage repository, each method exists as a file. Thus, just
executing git-log is sufficient to know in which commits the two methods were
changed. The command identifies that getLength() in Person.java was changed in
commit c100 and setLength(int) was changed in c101.
However, Historage has a limited capability of tracking methods in the case
that methods are renamed or moved to other classes. We explain the issue with
Figure 2, which shows refactorings on file Person.java in Figure 1. The
refactorings include the following four changes.
LABEL:Rename_Class:
Person $\rightarrow$ Engineer
LABEL:Rename_Field:
length $\rightarrow$ height
LABEL:Rename_Method (Getter):
getLength $\rightarrow$ getHeight
LABEL:Rename_Method (Setter):
setLength $\rightarrow$ setHeight
(a) Git.
(b) Historage.
Figure 2: Trackability differences between Git and Historage repositories.
In the case of the changes in Figure 2LABEL:sub@fig:GitTrackingModel, the Git
rename detection function can identify that file Person.java was renamed to
Engineer.java because the two files sufficiently share the identical lines. On
the other hand, in the Historage repository, files of Java methods get much
smaller than their original file as shown in Figure
2LABEL:sub@fig:HistorageTrackingModel. Thus, the ratio of the changed lines
against all the lines gets higher, which makes the Git function not work well.
Hata et al. addressed that changing the threshold for the Git rename function
is a way to realize a better method tracking [1]. They recommend using 30%
instead of 60%, which is a default value of Git. However, we consider that
only using a lower threshold may produce incorrect tracking results. For
example, if we use 30% instead of 60%, the Git rename function can identify
that Engineer/getHeight() is a renamed file of Person/getLength(). However, at
the same time, Person/getLength() can be tracked wrongly from
Engineer/setHeight(int) because their similarity is 1/3, which is higher than
30%.
Tracking method accurately is essential. If not, MSR approaches using
historical data gets affected. Hora et al. reported that between 10 and 21% of
changes at the method level in 15 large Java systems were untracked in the
context of refactoring detection [2]. They also found that 37% of the top-25%
most changed entities (classes and methods) have at least one untracked change
in their histories. By assessing two MSR approaches, they detected that their
results could be improved when untracked changes were resolved.
In this paper, we propose a new technique named FinerGit to improve the
trackability of Java methods. Several research areas benefit from FinerGit.
FinerGit is useful for studies in the context of assessing bug introducing
changes [3, 4, 5] or detecting code authorship [6, 7]. More broadly, any study
that compares two versions of methods can be benefited, for example, API
evolution detection [8, 9], code warning prioritization [10, 11], and many
other.
Figure 3: Tracking files with our technique.
The main contributions of this paper are the followings.
* 1.
We raise an issue on method trackability in Historage.
* 2.
We propose a new technique, FinerGit, to increase method trackability with Git
mechanisms.
* 3.
We provide a software tool based on FinerGit. The tool is open to the public
on GitHub 111https://github.com/kusumotolab/FinerGit. The tool is sufficiently
fast even for huge repositories, as shown in the evaluation.
* 4.
We show the experimental results on the tracking results of 182 open source
software (OSS) projects. These experiments have two aspects. First, they
clarify the advantage of FinerGit with an existing technique, Historage.
Second, they are the first attempt of large-scale empirical studies for the
tracking results of method-level repositories.
The remainder of this paper is organized as follows: in Section 2, we explain
our research goal and our key idea to achieve the goal; in Section 3, we
propose our new technique named FinerGit on the top of the key idea; Section 4
describes an implementation of FinerGit; then, we report the evaluation
results with the implementation in Section 5; we also describe threats to
validity on the experiments in Section 7; related work is introduced in
Section 8; lastly, we conclude this paper in Section 9.
## 2 Basic Approach
At present, there are various techniques of tracking source code entities [12,
13, 14, 15]. Those techniques utilize many types of information such as text
similarities, data dependencies, and call dependencies. On the other hand, in
this research, we utilize only line-based text similarity to track Java
methods. The reason is that our research goal is realizing accurate method
tracking with Git mechanisms.
The biggest benefit of tracking methods with Git mechanisms is that it can
easily connect with any other tools and techniques built on Git
infrastructure. For example, the following analyses can be easily performed by
using the basic commands provided by Git.
* 1.
We can know how many times each method was changed in the past by git-log.
* 2.
We can know how many developers changed a specified method in the past by
collecting author names of the commits in which the method was changed.
Git performs file comparisons by using hash values. If the size of a line is
equal to or shorter than 64 bytes, a hash value is calculated from the entire
line. If the size of a line is longer than 64 bytes, the line is chunked by 64
bytes, and a hash value is calculated from each chunk. Thus, even if just a
single token in a given line (which is shorter than 64 bytes) has been
changed, Git regards that the entire line has been changed.
Method-level tracking with Git mechanisms can be realized by treating each
method as a single file (a _method file_ hereafter). Based on this idea, Hata
et al. developed technique named Historage [16]. However, as explained with
Figure 2, simple extraction as files are inadequate for small methods. In this
research, we propose a file format that each line includes only a single
token. By using this format, each hash is calculated from a single token. In
Figure 2LABEL:sub@fig:HistorageTrackingModel, Git regards that the two red
lines of methods getLength and setLength were changed, though only the method
name and the field name were changed in methods. As a result, the ratio of
unchanged lines becomes 1/3, which is less than 60% of Git’s default value so
that the method is not tracked with Git mechanisms.
We state two restrictions for the techniques to improve method tracking with
Git mechanisms as follows.
* 1.
Since the file tracking mechanism in Git is based on line-based text
similarity, the characteristics of methods to be used in comparison must be
represented as a sequence of text lines. Based on this restriction, complex
comparison techniques of file contents such as tf/idf are not applicable.
* 2.
Since the contents of method files are visible and are utilized by developers,
they should follow a representation of source code in an understandable way by
users. Users may apply git-diff command to a method file to see how a method
was modified, and the obtained difference should represent the difference of
method contents in this case. Based on this restriction, converting method
contents to a sequence of computed numeric values used only for a comparison
purpose is not suitable.
Figure 3 shows how the changes in Figure 2LABEL:sub@fig:HistorageTrackingModel
are treated in FinerGit. The file changing mechanism in this technique
satisfies the above restrictions. The ratio of unchanged lines becomes 8/10
for getLength and 11/15 for setLength. Both values are higher than 60%, so
that both methods are tracked with Git mechanisms.
## 3 Proposed Technique
Herein, we explain our proposed technique named FinerGit to realize a better
method tracking with Git mechanisms. FinerGit is designed on the top of the
basic approach explained in Section 2. FinerGit consists of (1) naming
convention and (2) two heuristics.
### 3.1 Naming Convention
In FinerGit, a file name for a Java method includes the following information:
* 1.
a class name including the method,
* 2.
access modifiers of the method,
* 3.
a return type of the method,
* 4.
a name of the method, and
* 5.
a list of parameter types of the method.
For example, the file name for method setLength in Figure 2 becomes as
follows.
`Person#public_void_setLength(int).mjava`
Extension .mjava means that this is a method file and the file includes source
code of a Java method. Including the above information in the file name
reflects code changes around a given method as follows.
* 1.
If the name of the class including the given method is changed, the file name
of the given method gets changed, but its contents are not changed.
* 2.
If another method in the class including the given method is changed, neither
file name nor contents of the given method are changed.
* 3.
If the signature of the given method is changed, the file name of the given
method gets changed and its contents are also slightly changed since the
contents include the tokens of the method signature.
* 4.
If the contents of the given method are changed, the file name of the given
method does not get changed while its contents get changed.
We can track methods with Git mechanisms in any of the above cases if either
of them occurs alone. However, if a signature of a method is changed and its
contents are also changed broadly, it is difficult to track the method.
(a) Historage.
(b) w/o Heuristic-1.
(c) w/ Heuristic-1.
Figure 4: Tracking files w/o and w/ Heuristic-1.
### 3.2 Introducing Heuristics
It is not difficult to imagine that Git tracks wrong methods with FinerGit
because each line has only a single token and such lines will coincidentally
match with many other lines. Thus, we introduce two heuristics to reduce such
coincidental matches of unrelated lines.
Heuristic-1:
Classifying brackets, parentheses, and semicolons of termination characters in
detail.
Heuristic-2:
Removing tokens existing in all methods from the targets of similarity
calculation.
#### 3.2.1 Heuristic-1
Some termination characters such as brackets, parentheses, and semicolons are
omnipresent in Java source code. Such termination characters are used as a
part of various program elements. For example, brackets (“{” and “}”) are used
to initialize arrays in addition to code blocks such as if-statements and for-
statements. Thus, if just a bracket is placed on a line, brackets of different
roles are coincidentally matched with each other. Such accidental matchings
make the similarity between deleted and added methods inappropriately higher.
To prevent such accidental matchings, we classify termination characters in
detail. More concretely, we add a token explanation to each line. Token
explanations prevent accidental matchings of different-role characters from
being matched. In this heuristic, semicolons, brackets, and parentheses are
classified into 18, 21, and 20 categories, respectively.
Figure 4 shows how Heuristic-1 affects method tracking. Figure
4LABEL:sub@fig:Heuristic1Historage is a method file that Historage outputs.
The deleted method includes an if-statement for checking whether variable a is
null or not. The added method includes a while-statement for adding variable b
to variable total repeatedly. Those are different methods, which means a lower
similarity between them is better. In the case of Historage, the last line of
the if-statement coincidentally matches with the last line of the while-
statement so that the similarity between them becomes 1/3 (=33%). In the case
of FinerGit without Heuristic-1, the parentheses and the brackets of the if-
statement coincidentally matches with ones of the while-statement. Moreover,
the semicolon of the return-statement coincidentally matches with the one of
the expression-statement. As a result, the similarity between them becomes
5/12 (=42%). If we introduce Heuristic-1 to this example, the parentheses, the
brackets, and the semicolons get unmatched. Thus, the similarity between them
becomes 0/12 (=0%).
(a) w/o Heuristic-2.
(b) w/ Heuristic-2.
Figure 5: Tracking files w/ and w/o Heuristic-2.
#### 3.2.2 Heuristic-2
The parentheses for parameters and the brackets for method bodies are
omnipresent in compilable Java methods. The fact means that at least the four
tokens always match between any Java methods. Thus, the similarity between
non-related methods gets inappropriately higher. If methods include many
tokens, the impact of the four tokens is negligible. However, if methods are
small such as getters and setters, the impact of the four tokens become
serious. Consequently, we decided not to put the four tokens into files for
methods. By removing the four tokens, we prevent the similarity of two non-
related methods from getting higher inappropriately.
Figure 5 shows how Heuristic-2 affects tracking. This example shows a
similarity calculation between getLength (before refactoring) and setHeight
(after refactoring) in Figure 2. A lower similarity between the two methods is
better because they are different methods. In the case that we calculate a
similarity without Heuristic-2, the similarity becomes 5/10 (=50%). However,
in the case that we adopt Heuristic-2, the similarity becomes 1/6 (=17%)
because the four tokens are ignored.
## 4 Implementation
We have implemented a tool based on FinerGit. Our tool is open to the public
in GitHub, and anyone can use it freely. Our tool takes a Git repository of a
Java project, and it outputs another Git repository where each Java method
gets extracted as a file. In FinerGit repositories, method files have
extension .mjava. By executing git-log command with option \--follow for
.mjava files, we can get their histories.
The name of a method file includes the information of the signature of the
method and the class name including the method so that the file name
occasionally becomes very long. Very long file names are not compatible with
widely-used operating systems. For example, in the case of Windows 10, the
absolute path of a file must not exceed 260 characters. If a file name
violates the restriction, its file cannot be accessed with Windows’ file
manager and some other problems occur. In the case of Linux and MacOS, a file
name (not a file path) must not exceed 255 characters. For practical use in
such widely-used operating systems, if a file name becomes longer than the
restriction of operating systems, our tool cuts the file name in the middle
and then it appends a hash value that is calculated from the entire file name.
This manipulation can shorten the file name while keeping its identity.
There are three types of comments in Java source code: line comments, block
comments, and Javadoc comments. Line and block comments are removed from
.mjava files while Javadoc comments are included in .mjava files as they are
in .java files. This means that a Javadoc comment exists in the header part of
.mjava file if its original method has it.
Our tool also has a function to extract each field in Java source code as a
single file. Files for fields have extension .fjava. A field declaration
includes multiple tokens such as field name, field type, modifiers,
initializations, and annotations. Thus, fields can be tracked as well as
methods by placing a single token on a line. A file name for a Java field
include the following information:
* 1.
a class including the field,
* 2.
access modifiers of the field,
* 3.
a type of the field, and
* 4.
a name of the field.
For example, the file name for field length in Figure 2 becomes as follows.
`Person#private_int_length.fjava`
Including the above information in the file name reflects code changes around
a given field as follows.
* 1.
If the name of class including the given field is changed, the file name of
the given method gets changed, but its contents are not changed.
* 2.
If another method or field in the class including the given field is changed,
neither file name nor the contents of the given method are changed.
* 3.
If the access modifiers, type, or name of the field is changed, the file name
of the given field gets changed and its contents are also changed.
* 4.
If the annotations and/or initializations of the field are changed, the file
name of the given field does not get changed while its contents get changed.
In Historage repository, a file path of a method includes its signature
information. Historage makes a directory for each Java class. Methods included
in a class are placed in its corresponding directory. On the other hand, our
technique places files of Java methods in the same directory of their original
Java files. A reason why FinerGit does not make new directories for Java
classes is that the conversion time of Historage is long and making a large
number of directories in the conversion process is a factor of taking a long
time. Both FinerGit and Historage make a large number of files because each
Java method is extracted as a single file, but our technique does not make new
directories for Java classes. In both FinerGit and Historage, file name
collisions for extracted files do not occur as long as their source code is
compilable.
## 5 Evaluation
We evaluated FinerGit by comparing it with Historage [1]. We did not use the
published version of Historage
implementation222https://github.com/niyaton/kenja but we added Historage’s
functionality to our tool. By using the same implementation for FinerGit and
Historage, we can avoid different tracking results due to the differences in
implementation details. For example, original Historage makes directories for
each Java class while our Historage implementation outputs files of Java
methods in the same directory as their original files. The file name
convention of our Historage implementation is the same as FinerGit. Thus, in
this way, we can evaluate how much method trackability with Git mechanisms
gets improved by FinerGit.
We selected 182 Java projects in GitHub as our evaluation targets. In the
process of our target selection, we used Borges dataset [17]. This dataset
includes 2,279 popular projects in GitHub. Firstly, we extracted 202 projects
that are labeled as “Java projects”. Borges et al. classified the projects in
the dataset into six categories: Application software, System software, Web
libraries and frameworks, Non-web libraries and frameworks, Software tools,
and Documentation. Secondly, we extracted 185 projects that are other than
Documentation projects because they are repositories with documentation,
tutorials, source code examples, etc. (e.g., java-design-
patterns333https://github.com/iluwatar/java-design-patterns). Documentation
projects are outside of the scope of this evaluation. Then, we cloned the 185
repositories to our local storage on March 4th 2019. Unfortunately, we found
that three of the 185 projects did not include .java file. The three projects
(google/iosched, afollestad/material-dialogs, and googlesamples/android-
topeka) are Kotlin projects. Finally, we removed the three projects from the
185 projects.
Figure 6: Project size.
Figure 6 shows the distributions of the number of commits and LOC of the
target projects. The two largest repositories in the targets are
platform_frameworks_base444https://github.com/aosp-
mirror/platform_frameworks_base and intellij-
community555https://github.com/JetBrains/intellij-community. The two
repositories include approximate 380K and 240K commits, and their latest
revisions consist of about 3.7M and 5.0M LOC, respectively.
We generated FinerGit repositories and Historage ones from the 182 target
projects. Herein, FinerGit repositories have the file format of including a
single token per line with the two heuristics while Historage repositories
have the same line format as the original repositories.
We have evaluated FinerGit from the five viewpoints:
* 1.
tracking accuracy,
* 2.
heuristics impacts,
* 3.
project-level tracking results,
* 4.
method-size-level tracking results, and
* 5.
execution time.
Hereafter in this section, we report the results in detail.
### 5.1 Tracking Accuracy
It is not realistic to manually check whether FinerGit generates correct
tracking results for each method in the target projects. Thus, we make an
oracle for a method for each target project with the following procedure.
1. 1.
A method was randomly selected from each target project. In total, 182 methods
were selected.
2. 2.
Each of the methods in FinerGit repositories was tracked with the following
command.
`> git log --follow -U15 -M20% -C20% -p`
` -- `path/to/method`.mjava`
With the above command, a specified file is tracked even if the file was
renamed. If there is a file that has a 20% or more similarity, Git regards
that file renaming or copying occurred.
3. 3.
The tracking results were examined, and oracles of renaming and copying
history were made by two of the authors independently. Each author spent
several hours on this task. The two authors made different oracles for 34 out
of the 182 methods.
4. 4.
The two authors discussed the 34 methods so that they obtain consensus for
them. After a two-hour discussion, they got consensus oracles for the 34
methods.
With the above procedure, we obtained consensus oracles of tracking results
for the 182 methods. Finally, we obtained the resulting oracle set consisting
of 426 renaming/copying changes for the 182 methods in total.
Next, we track the methods in FinerGit’s repositories and Historage’s ones
with different thresholds. We used the following command to count how many
times Git found renaming and copying with a specified threshold.
`> git log --follow --oneline -M`t` -C`t` -p`
` -- `path/to/method`.mjava`
` | grep -e "^rename from\|^copy from"`
` | wc -l`
In the above command, t is the threshold that Git regards given two files have
a renaming or copying relationship. We tracked the target methods with 13
different thresholds (i.e., 20%, 25%, 30%, $\ldots$, 80%). If tracking results
for a method include a higher number of renaming/copying than its oracle, we
regard renaming/copying in the over-tracking part as false positives. If
tracking results for a method include a lower number of renaming/copying than
its oracle, we regard renaming/copying that are not detected as false
negatives. We calculated precision, recall, and F-measure for each threshold
by summing up the number of false positives and false negatives of all the
methods.
Figure 7: Precision, recall, and F-measure values.
Figure 7 shows how precision, recall, and F-measure changes according to given
thresholds. The graphs of Historage and FinerGit have the following features.
* 1.
Precision of Historage is very high. Historage has 93.01% of precision even in
the case of threshold 20%.
* 2.
Recall of Historage is low. Historage has only 57.04% of recall in the case of
threshold 20%.
* 3.
FinerGit has high precision in the case of high thresholds, but precision gets
rapidly decreased for lower thresholds.
* 4.
FinerGit has higher recall than Historage for all the thresholds. The recall
differences between FinerGit and Historage get bigger for lower thresholds.
Historage has a low possibility to track wrong methods while it often misses
renaming and copying. On the other hand, in FinerGit repositories, precision
gets decreased for lower thresholds while recall improves much. The highest
F-measure on FinerGit is 84.52% on threshold 50% while the highest F-measure
on Historage is 70.72% and 70.23% on thresholds 20% and 25%, respectively.
### 5.2 Heuristics Impacts
(a) Precision.
(b) Recall.
(c) F-measure.
(d) Rename count.
Figure 8: Precision, recall, F-measure, and rename count when heuristics 1 and
2 are on and/or off.
To reveal how each heuristic impacts on method tracking, we measured
precision, recall, and F-measure and we also counted found renames for the
following four types of fine-grained repositories. The target methods are the
same as Subsection 5.1. Herein, rename count means the sum of found renames
for all the target methods in a type of repositories.
H1 OFF, H2 OFF:
neither heuristics are applied to.
H1 ON, H2 OFF:
only Heuristic-1 is applied to.
H1 OFF, H2 ON:
only Heuristic-2 is applied to.
H1 ON, H2 ON:
both heuristics are applied to. This is the same repository as what we used in
Subsection 5.1.
Figure 8 shows the results. Applying only Heuristic-1 makes it possible to
find more renaming so that precision gets decreased while recall gets
increased. On the other hand, applying only Heuristic-2 slightly shorten
method tracking. As a result, precision gets increased while recall gets
decreased. The reasons why applying Heuristic-1 and Heuristic-2 have opposite
impacts on method tracking are as follows.
* 1.
Applying Heuristic-1 reduces similarities between methods. How much the
similarities are decreased depends on the contents on methods. Thus, a
different method can be tracked at a commit compared to the case that
Heuristic-1 is not applied to.
* 2.
Applying Heuristic-2 reduces similarities between all methods. Unlike
Heuristic-1, Heuristic-2 does not make a different method tracked. Thus,
Heuristic-2 just shortens method tracking.
Table 1 shows the maximum F-measure for each type of finer-grained
repositories. In this table, the maximum F-measure is the greatest F-measure
in all data. All types have almost the same maximum values. This table also
shows the maximum recall when we track methods with over 95% precision. These
results show that more method renames are found with keeping 95% precision by
applying both heuristics.
Table 1: Maximum F-measure and Maximum Recall Repository type | Max F-measure (thr.) | Max Recall (thr.)
---|---|---
H1 OFF, | H2 OFF | 82.63% (40%) | 58.45% (55%)
H1 ON, | H2 OFF | 83.77% (55%) | 56.81% (65%)
H1 OFF, | H2 ON | 83.26% (35%) | 60.09% (50%)
H1 ON, | H2 ON | 84.52% (50%) | 68.78% (55%)
### 5.3 Project-Level Tracking Results
(a) Ratio of different tracking results.
(b) Average change counts.
Figure 9: Project-level comparisons. (a) shows the ratio of methods whose
tracking results are different between FinerGit and Historage for each
project. (b) shows the average of change counts for all the methods for each
project.
In this evaluation, we measured the ratio of methods whose tracking results
are different between the two tools for each project. We compare how much the
number of detected renames is different from FinerGit and Historage under the
same precision. As shown in the previous subsection, the two tools have
different precision values for different thresholds. To realize a fair
comparison, we decided to select different thresholds for FinerGit and
Historage that satisfy the following condition: method tracking results with
the thresholds have the same precision values and the precision values are as
high as possible. Thus, we used threshold 55% for FinerGit and 25% for
Historage. The precision of FinerGit on threshold 55% is 95.73%, and Historage
on threshold 25% is 96.60%. Those precision values are almost the same and
high enough.
Figure 9 shows the comparison results. In Figure
9LABEL:sub@fig:ProjectLevelComparison:Ratio, the blue boxplot shows the ratio
of methods for which FinerGit found more renames than Historage per project
and the red boxplot shows the opposite one. FinerGit found more renames for
22.71% methods on average while the ratio of methods that Historage found more
renames than FinerGit is only 5.26%. In Figure
9LABEL:sub@fig:ProjectLevelComparison:Count, the blue boxplot shows the
average number of changes identified by FinerGit for all methods of each
project. The red one shows the average number of changes identified by
Historage. The median values of those boxplots are 3.67 and 2.86,
respectively. These results mean that FinerGit can find more renames for all
the methods on average.
Next, we show that the tracking improvement by FinerGit is effective via the
following two ways:
* 1.
considering the fact that some methods were never changed after their initial
creation, and
* 2.
conducting statistical testing for the tracking results.
#### 5.3.1 Considering Never-Changed Methods
In software development, some methods are never changed after their initial
creation. If the 182 target projects include many never-changed methods, it is
quite natural that the comparison results between FinerGit and Historage are
not so different from each other. Thus, we investigate how many never-changed
methods are included in the projects. It is not realistic to manually collect
real never-changed methods. In this experiment, we decided to regard methods
that both FinerGit and Historage were not able to detect any changes as never-
changed methods.
Figure 10 shows the relationship between the ratio of never-changed methods
and the ratio of methods for which FinerGit found more renames than Historage.
The 25 percentile, the median, and the 75 percentile of never-changed methods
are 6.88%, 15.27%, and 26.50%, respectively. The figure indicates that the
more never-changed methods there are, the fewer methods FinerGit found more
renames for. Figure 11 shows the same figures as Figure
9LABEL:sub@fig:ProjectLevelComparison:Ratio only for the projects that include
50% or more never-changed methods. As shown in Figure
11LABEL:sub@fig:ProjectLevel50on, the differences between FinerGit and
Historage are small because the majority of their methods is never-changed.
Figure 11LABEL:sub@fig:ProjectLevel50off shows the differences after we
removed never-changed methods from the projects. We can see that the
differences between the two tools get much larger. MSR approaches are
naturally applied to methods that have change histories. Never-changed methods
are exempt from MSR approaches.
We also investigated how many methods only FinerGit or Historage found at
least a change for. The former number is 97,629 and the latter one is 35,553.
They are 5.52% and 2.01% of all methods, respectively. Finding changes for
more methods means that various MSR approaches requiring past changes can be
applied more broadly.
Figure 10: Relationships between the ratio of methods for which FinerGit found
more renames than Historage and the ratio of never-changed methods.
(a) w/ never-changed methods.
(b) w/o never-changed methods.
Figure 11: The ratio of methods whose tracking results are different between
FinerGit and Historage for projects where 50% or more methods are never-
changed ones.
#### 5.3.2 Conducting Statistical Testing
We applied Paired Wilcoxson’s signed ranked test to the comparison results
between FinerGit and Historage shown in Figure 9. The test showed that the
comparison results include significant differences regarding both aspects of
the ratio ($p$-value $<$ 0.001) and average change counts ($p$-value $<$
0.001). We also applied Cliff’s Delta to the comparison results to see the
effect size. The resulting values were computed as 0.712 for the ratio and
0.221 for the average change counts, which revealed a _large_ and a _small_
effect size of the improvement achieved by using FinerGit, respectively.
Consequently, we can say that FinerGit significantly improves tracking Java
methods compared to Historage.
### 5.4 Method-Size-Level Tracking Results
We also conducted comparisons based on method size. In this comparison, we
made several method groups based on their size. Then, we compared the tracking
results for each group. Figure 12 shows the comparison results. We can see
that there are 1,036K methods whose LOC is in the range between 1 and 5.
Herein, the LOC was computed using the original format, not the single-token-
per-line one. FinerGit generated longer tracking results for 26.21% of the
1,036K methods. Our research motivation was improving the trackability for
small methods, but surprisingly FinerGit improved the trackability for methods
of any size.
This figure also shows the average rename counts that were found by FinerGit
and Historage. We can see that FinerGit found more renames for methods of any
size than Historage. Interestingly, more renames tend to be found for larger
methods by both tools.
Consequently, we conclude that the method tracking capability of FinerGit is
higher than Historage.
(a) Ratio of methods for which FinerGit or Historage found more renames than
the other tool.
(b) Average renames that were found by FinerGit or Historage.
Figure 12: Comparison based on method size.
### 5.5 Execution Time
Figure 13: Execution time of FinerGit.
We measured the time that FinerGit reconstructed the repositories of the
target projects on MacBook Pro666CPU: 2.7GHz quad-core Intel Core i7, memory
size: 16 GBytes. Figure 13 shows the measurement results. This figure shows
that FinerGit is scalable enough for large repositories. In the longest case,
FinerGit took 4,209 seconds to reconstruct the repository of intellij-
community, which includes more than 240K commits. Of course, this execution
time can be shorter if a higher specification computer is used777We also
measured execution time with our workstation whose CPU is 3.6GHz octet-core
Intel Core i9 and memory size is 32 GBytes. The execution time was
approximately 22% of MacBook Pro’s one..
Figure 13 includes the regression line for all the data. The regression line
shows that FinerGit takes around 100 seconds to process each 10K commits for
large repositories.
## 6 Comparisons with Other Techniques
We also compared FinerGit with two other techniques, AURA and RefactoringMiner
(RMiner). The first comparison target is AURA, which is a technique that takes
two versions of Java source code and generates mappings of methods between
them [15]. AURA performs call dependency and text similarity analyses to
generate mappings. The second comparison target is RMiner, which is a
technique that detects refactorings from commit history [18]. RMiner’s
refactoring detection is based on an AST-based statement matching algorithm.
RMiner defines different rules for different refactoring patterns. RMiner
checks if matching results of two ASTs before and after changes in a given
commit follow any of the rules.
We conducted this comparison on the development history of JHotDraw between
releases 5.2 and 5.3. This development history is one of the evaluation
targets in AURA’s literature [15]. Releases 5.2 and 5.3 include 1,519 and
1,981 methods, respectively. There are 19 commits between releases 5.2 and
5.3.
### 6.1 AURA
Table 2: Refactorings detected by RMiner Refactoring pattern | # of detected instances
---|---
LABEL:Change_Parameter_Type | 56
LABEL:Change_Return_Type | 10
LABEL:Move_Method | 3
LABEL:Rename_Method | 44
LABEL:Rename_Parameter | 45
Total | 158
We made FinerGit’s repository and tracked the 1,981 methods with 20% threshold
with the command shown in Subsection 5.1. The tracking results of 185 methods
included renaming and the total number of renaming was 241. Two of the authors
independently examined the tracking results to make oracles. Each author spent
several hours on this task. The two authors make different oracles for 18 out
of the 185 methods. The authors had a discussion on the 18 methods to obtain
consensus for them. After a one-hour discussion, they got consensus oracles
for the 18 methods. Our consensus oracle includes 161 renamings on 124
methods.
Next, we tracked the 1,981 methods with 50% threshold, which is the best
F-measure threshold in the evaluation in Subsection 5.1. As a result, we
obtained 161 renamings on 124 methods. By comparing the tracking results of
50% threshold with the consensus oracle, We calculated two kinds of precision
and recall: one was calculated based on renaming instances; the other was
calculated based on methods whose tracking results included at least one
renaming in the consensus oracle.
* 1.
From the viewpoint of renaming instances, precision and recall were 91.30% and
83.52%, respectively.
* 2.
From the viewpoint of methods including renames, precision and recall were
86.29% and 83.59%, respectively.
According to AURA’s literature [15], AURA generated mappings for 97 rules888A
rule is a mapping group of multiple methods. and its precision was 92.38%. By
comparing those results, we conclude that FinerGit generated mappings for more
methods with slightly-lower precision.
AURA utilizes text similarity and call dependency to generate mappings while
FinerGit utilizes only text similarity. On the other hand, AURA takes only two
versions of source code to generate mappings while FinerGit utilizes all
commits to track methods. Those are the reason why the precision values of the
two tools were not so different.
### 6.2 RefactoringMiner
We performed RMiner 999RMiner is available at
https://github.com/tsantalis/RefactoringMiner. We used the latest version of
the tool at 17th November, 2019. The commit ID is
4bb0e11550b781b61ce1c382a58ea182a2f46944. on the commit history of JHotDraw
between release 5.2 and 5.3. RMiner has a capability of detecting 38 types of
refactoring patterns and the following five refactoring patterns correspond to
renamings that FinerGit detects: LABEL:Change_Parameter_Type,
LABEL:Change_Return_Type, LABEL:Move_Method, LABEL:Rename_Method, and
LABEL:Rename_Parameter. RMiner detected 158 refactoring instances of the five
patterns. The detail numbers of refactorings detected by RMiner are shown in
Table 2. We compared the 158 refactorings with the 161 renamings detected by
FinerGit with 50% threshold. The number of common instances was 65, which was
41.14% of RMiner’s refactorings and 40.37% of FinerGit’s renamings.
Table 3: Precision and Recall of RMiner in literature [18] Refactoring pattern | Precision | Recall
---|---|---
LABEL:Move_Method | 95.17% | 76.36%
LABEL:Rename_Method | 97.78% | 83.28%
The FinerGit evaluation in Subsection 6.1 shows that FinerGit’s tracking
accuracy on JHotDraw is high (precision and recall are 91.30% and 83.52%,
respectively in 50% threshold). Table 3 shows precision and recall of RMiner
for each refactoring pattern in literature
[18]101010LABEL:Change_Parameter_Type, LABEL:Change_Return_Type, and
LABEL:Rename_Parameter were not investigated in the literature because those
refactoring patterns have been recently supported by RMiner.. According to
this table, precision and recall of RMiner are also high. However, the common
instances between FinerGit and RMiner do not occupy a large portion of all
instances detected by either of the techniques. We manually investigated
renames and refactorings that had been detected only either of the techniques
and found that the results faithfully reflected their different inheritances.
There were two major cases of renames that were detected only by FinerGit.
* 1.
New parameters were added to methods or return types of methods were changed
according to the changes in method’s bodies. Those changes were not
refactorings but functional enhancements.
* 2.
Access modifiers (public, protected, and private) were added/removed/changed.
Such changes were refactorings; however they were not supported by RMiner.
On the other hand, refactorings that were detected only by RMiner had changed
a large part of method’s bodies. Thus, line similarities of method’s bodies
between such refactorings become low, which leaded to fail to be detected as a
renaming by FinerGit.
Herein, we compared FinerGit with RMiner; however their purposes are different
from each other. The FinerGit’s purpose is tracking Java methods with high
accuracy. No matter what kinds of changes are made, FinerGit is able to track
methods if a line similarity of the method’s bodies between a change is higher
than a given threshold. On the other hand, the purpose of RMiner is detecting
refactorings in a commit history. No matter how unsimilar between method’s
bodies are between a refactoring, RMiner is able to detect the refactoring if
the refactoring is supported by RMiner.
## 7 Threats to Validity
In the experiment, we used 182 Java projects, and we investigated on tracking
results on 1,768K methods in total. Those numbers of projects and methods are
large enough so that we expect that the same results are obtained if we
conduct another experiment on different Java projects.
To measure precision, recall, and F-measure of method tracking by FinerGit and
Historage, we manually constructed oracle for 182 methods. Firstly, two of the
authors made oracle for all the 182 methods independently, and then they
discussed for which they made different oracle. This process of making oracle
is designed to avoid making mistakes and to reduce subjective view on
constructing oracle as much as possible.
One more thing about oracle is that, essentially, oracle should be made
independently from tracking results of FinerGit and Historage. However,
constructing oracle with a fully-manual work is extraordinarily difficult even
for a small number of methods. Consequently, in the experiment, we firstly
obtained high-recall tracking results with an enough low threshold, and then,
we checked how many false positives were included in the tracking results. We
consider that this construction process does not ensure 100%-correct oracle
but high enough for comparing different techniques. In other word, we made
oracle of reasonable quality with a realistic time cost.
In the manual investigation, we checked surrounding 15 lines (as shown in
Subsection 5.1) of changes in commits to judge whether method tracking by
FinerGit was correct or not. The number 15 came from our experiences with
FinerGit because we had checked tracking results of FinerGit before conducting
the experiment in this paper.
In the experiment, we discussed the comparison results by focusing on whether
FinerGit had found more renaming and copying for Java methods than Historage.
However, we also need to see the fact that there were some cases that short
tracking results by FinerGit were better than long tracking results by
Historage. Such cases mean that FinerGit was able to avoid tracking methods
incorrectly. We investigated some of such cases, and then we found that the
reason why Historage found a higher number of renames is due to the existences
of coincidentally matched lines as shown in Figure
4LABEL:sub@fig:Heuristic1Historage.
## 8 Related Work
The research that is most related to this paper is of course Historage [1].
Historage is useful in research on mining software repositories because
researchers can obtain Java method histories without implementing code/scripts
by themselves. Historage has been used in many research before now.
* 1.
Hata et al. researched predicting fault-prone Java methods by using method
histories obtained with Historage [19]. Their experimental results showed that
the method-level prediction outperformed package-level and file-level
predictions from the viewpoint of efforts for finding bugs.
* 2.
Hata et al. also used Historage to infer restructuring operations on the
logical structure of Java source code [16].
* 3.
Fujiwara et al. developed a hosting service of Historage repositories,
Kataribe111111http://sdlab.naist.jp/kataribe/ [20]. Kataribe enables
researchers/practitioners to browse method histories on the web, and they can
clone Historage repositories in Kataribe into their local storages if they
want to conduct further analyses.
* 4.
Tantithamthavorn et al. investigated the impact of granularity levels (class-
level and function-level) on a feature location technique [21]. The results
indicated that function-level feature location technique outperforms class-
level feature location technique. Moreover, function-level feature location
technique also required seven times less effort than class-level feature
location technique to localize the first relevant source code entity.
* 5.
Kashiwabara et al. proposed a technique to recommend appropriate verbs for a
method name of a given method so that developers can use various verbs
consistently [22]. Their technique recommends candidate verbs by using
association rules extracted from existing methods. They extracted renamed
methods from repositories of target projects using Historage.
* 6.
Oliveira et al. presented an approach to analyze the conceptual cohesion of
the source code associated with co-changed clusters of fine-grained entities
[23]. They obtained change histories of Java methods with Historage. By using
the change histories, they identified a set of methods that were frequently
changed together.
* 7.
Yamamori et al. proposed to use two types of logical couplings of Java methods
for recommending code changes [24]. The first type is logical couplings that
are extracted from code repositories. They used Historage and Kataribe to
obtain logical couplings of Java methods. The second type is logical couplings
that are extracted from interaction data. They used a dataset that had been
collected by Mylyn [25]. Their experimental results showed that there was a
significant improvement in the efficiency of the change recommendation
process.
* 8.
Yuzuki et al. conducted an empirical study to investigate how often change
conflicts happen in large projects and how they are resolved [26]. In their
empirical study, they used Historage to conduct method-level analysis. As a
result, they found that 44% of conflicts were caused by changing concurrently
the same positions of methods, 48% is by deleting methods, and 8% is by
renaming methods. They also found that 99% of the conflicts were resolved by
adopting one method directly.
* 9.
Suzuki et al. investigated relationships between method names and their
implementation features [27]. They showed that focusing on the gap between
method names and their implementation features is useful to predict fault-
prone methods. They used Historage to collect change histories of Java methods
in the investigation.
All the above research can be conducted with FinerGit instead of Historage.
Moreover, the experimental results may change if FinerGit is used because
there is a significant difference in the tracking results between FinerGit and
Historage.
We are not the first research group that has used single-token-per-line format
for Git repositories. To the best of our knowledge, the study by German et al.
was the first attempt to follow this approach [28]. They proposed to rearrange
source files with single-token-per-line for enabling fine-grained git-blame.
By using their technique, we can see the person who changed last for each
token of the source code. They showed that blame-by-token reports the correct
commit that adds a given source code token between 94.5% and 99.2% of the
times, while the traditional approach of blame-by-line reports the correct
commit that adds a given token between 74.8% and 90.9%. German developed a
system cregit 121212https://github.com/cregit/ based on their proposed
technique. cregit has being used in Linux development
community131313https://cregit.linuxsources.org/. cregit does not extract Java
methods as files, which is a difference between cregit and FinerGit.
Heuristic-1, which is described in Subsection 3.2, is refining symbols in
source code. On the one hand, symbol refinements are often performed in the
process of code clone detection techniques. In the context of clone detection,
some symbols are replaced with special ones prior to the matching process. For
example, in CCFinder [29] and NICAD [30], which are representative code clone
detection techniques, all variables and literals are replaced with a specific
wildcard symbol. The purpose of replacements is to detect syntactically-
similar code as code clones as much as possible. Such replacements can realize
that the matching process ignores differences in variables or literals. On the
other hand, in the context of FinerGit, we do not want to ignore differences
in variables or literals. If we ignore such differences, the similarity
between non-related methods can rise accidentally, which leads FinerGit to
make wrong method tracking. The purpose of our Heuristic-1 is to calculate
lower similarity values between non-related methods.
There are many research studies of program element matching other than
Historage [13]. Lozano et al. and Saha et al. implemented method tracking
techniques since they need to track method-level clones in their experiments
[31, 32]. Their method-level tracking techniques are line-based comparisons
and their comparisons compute numerical similarity values by comparing lines
as texts. Thus, in the case that only a small part of a line is changed, the
similarity between a before-change line and its after-change line should be
high while a simple line-based comparison like diff regards that a before-
change line is completely different from its after-change line. However, their
comparisons are still line-based ones, which include some flaws compared to
token-based ones.
* 1.
In the cases that the first token of the line is moved to the previous line or
the last token of the line is moved to the next line (e.g., left bracket (“{”)
is moved to the next line due to format change), their line-based techniques
regard that multiple lines have been changed while our technique regards that
no lines have been changed.
* 2.
The same changes have different impacts on lines of different length. For
example, variable abc is changed to def in a 10-character line, the similarity
becomes 7/10 while the same change occur in a 40-character line, the
similarity becomes 37/40.
Godfrey and Zou detected merging and splitting source code entities such as
files and functions. They extended origin analysis [33] to track source code
entities. They utilize various information for entities such as entity names,
caller/callee relationship, and code metrics values. Wu et al. proposed a
technique to identify change rules for one-replaced-by-many and many-replaced-
by-one methods [15]. Their approach is a hybrid one, which means that it uses
two kinds of data: caller/callee relationship and text similarity. Kim et al.
proposed a technique to track functions even if their names get changed [14].
Their technique computes function similarities between given two methods. They
introduced eight similarity factors such as complexity metrics and clone
existences to determine if a function is renamed from another function. Dig et
al. proposed a technique to detect refactorings performed during component
evolution [12]. Their technique can track methods even if refactorings change
their names. Their detection algorithm uses a combination of a fast syntactic
analysis to detect refactoring candidates and a more expensive semantic
analysis to refine the results. There are many other approaches for
identifying refactorings, and many of them support refactorings that changes
method names/signatures such as LABEL:Rename_Method and
LABEL:Parameterize_Method pattern [34, 35, 36, 37, 18, 38, 39]. The advantage
of the proposed technique against the above approach should be the ease to use
because it utilizes Git mechanisms to track methods. A researcher/practitioner
who wants method evolution data does not have to learn how to use new tools.
## 9 Conclusion
In this paper, we firstly discuss Historage, which is proposed in literature
[16]. Historage is a tool that converts a Git repository to a finer-grained
one. In the finer-grained repository, each Java method exists as a single
file. Thus, we can track Java method with Git commands such as git-log.
However, tracking small methods with Git mechanisms does not work well because
small methods do not have good chemistry with the Git rename detection
function. Thus, we proposed a new technique that puts only a single token of
Java methods per line. In other words, in our technique, each line includes
only a single token. We also derived two heuristics to reduce incorrect
tracking.
We implemented a software tool based on the proposed technique. We applied our
tool and Historage to 182 repositories of Java OSS projects to compare the two
tools. The 182 repositories include 1,768K methods in total, which are the
targets our comparisons. We found that FinerGit scored 84.52% as maximum
F-measure while Historage scored 70.23%. We also confirmed that the proposed
technique worked well for methods of any size in spite that our research
motivation was to realize better tracking for small methods. Furthermore, we
showed that our tool took only short time to construct finer-grained
repositories even for large repositories.
In the future, we are going to replicate some experiments of existing research
with FinerGit to check whether the better tracking of our tool changes
experimental results or not.
## Acknowledgements
This work was supported by JSPS KAKENHI Grant Number JP17H01725 and
JP18K11238.
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|
2024-09-04T02:54:59.212489 | 2020-03-11T14:58:05 | 2003.05342 | {
"authors": "Andronikos Paliathanasis",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26167",
"submitter": "Andronikos Paliathanasis",
"url": "https://arxiv.org/abs/2003.05342"
} | arxiv-papers | # Dynamics of Chiral Cosmology
Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science,
Durban University of Technology, Durban 4000, Republic of South Africa
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile,
Valdivia 5090000, Chile
###### Abstract
We perform a detailed analysis for the dynamics of Chiral cosmology in a
spatially flat Friedmann-Lemaître-Robertson-Walker universe with a mixed
potential term. The stationary points are categorized in four families.
Previous results in the literature are recovered while new phases in the
cosmological evolution are found. From our analysis we find nine different
cosmological solutions, the eight describe scaling solutions, where the one is
that of a pressureless fluid, while only one de Sitter solution is recovered.
Cosmology; Scalar field; Chiral Cosmology; Stability; Dark energy; Dynamics
###### pacs:
98.80.-k, 95.35.+d, 95.36.+x
## I Introduction
A detailed analysis of the recent cosmological observations dataacc1 ;
dataacc2 ; data1 ; data2 ; Hinshaw:2012aka ; Ade:2015xua indicates that the
universe has gone through two acceleration phases during its evolution. In
particular into a late-time acceleration phase which is attributed to dark
energy, and into an early acceleration phase known as inflation. Inflation was
proposed four decades ago guth in order to explain why in large scales the
universe appears isotropic and homogeneous. The inflationary era is described
by a scalar field known as the inflaton which when dominates drives the
dynamics of the universe such that the observations are explained.
In addition, scalar fields have been used to describe the recent acceleration
epoch of the universe, that is, they have been applied as a source of the dark
energy Ratra . In scalar field theory the gravitational field equations remain
of second-order with extra degrees of freedom as many as the scalar fields and
corresponding conservation equations hor1 ; hor2 ; hor3 . These extra degrees
of freedom can attribute the geometrodynamical degrees of freedom provided by
invariants to the modification of Einstein-Hilbert action in the context of
modified/alternative theories of gravity mod1 ; mod2 ; mod3 .
The simplest scalar field theory proposed in the literature is the
quintessence model Ratra . Quintessence is described by a minimally coupled
scalar field $\phi\left(x^{\kappa}\right)~{}$with a potential function
$V\left(\phi^{\kappa}\right)$. The scalar field satisfies the weak energy
condition, i.e. $\rho\geq 0,~{}\rho+p\geq 0,$ while the equation of state
parameters $w_{Q}=\frac{p}{\rho}$ is bounded as $\left|w_{Q}\right|\leq 1$.
For some power-law quintessence models, the gravitational field equations
provide finite-time singularities during inflation leading to chaotic dynamics
sing ; page . On the other hand, for some kind of potentials the quintessence
can describe the late-time acceleration udm .
In the cosmological scenario of a Friedmann-Lemaître-Robertson-Walker universe
(FLRW) exact and analytic solutions of the field equations for different
potentials are presented in jdbnew ; muslinov ; ellis ; barrow1 ; newref2 ;
ref001 ; ref002 and references therein. Results of similar analysis on the
dynamics of quintessence models are summarized in the recent review gen01 .
Other scalar field models which have been proposed in the literature are:
phantom fields, Galileon, scalar tensor, multi-scalar field models and others
ph1 ; ph2 ; ph3 ; ph4 ; ph5 ; ph7 ; ph8 ; ph9 ; ph10 ; ph11 . Multi-scalar
field models have been used to provide alternative models for the description
of inflation hy1 ; hy2 ; hy3 , such as hybrid inflation, double inflation,
$\alpha$-attractors hy4 ; atr1 ; atr3 and as alternative dark energy models.
Multi-scalar field models which have drawn the attention of cosmologists are,
the quintom model and the Chiral model. A common feature of these two theories
is that they are described by two-scalar fields, namely
$\phi\left(x^{\kappa}\right)$ and $\psi\left(x^{\kappa}\right)$. For the
quintom model, one of the two fields is quintessence while the second scalar
field is phantom which means that the energy density of the field can be
negative. One of the main characteristics of quintom cosmology is that the
parameter for the equation of state for the effective cosmological fluid can
cross the phantom divide line more than once qq1 ; qq2 . The general dynamics
of quintom cosmology is presented in qq3 .
In Chiral theory, the two scalar fields have a mixed kinetic term. The two
scalar fields are defined on a two-dimensional space of constant nonvanishing
curvature atr6 ; atr7 . That model is inspired by the non-linear sigma
cosmological model sigm0 . Chiral cosmology is linked with the
$\alpha-$attractor models atr3 . Exact solutions and for specific cases the
dynamics of Chiral cosmology were studied before in andimakis , while analytic
solutions in Chiral cosmology are presented in 2sfand . In the latter
reference, it was found that pressureless fluid is provided by the model,
consequently, the model can also be seen as an alternative model for the
description of the dark sector of the universe. Last but not least scaling
attractors in Chiral theory were studied in andimakis ; per1 .
In this piece of work we are interested in the evolution of the dynamics for
the gravitational field equations of Chiral cosmology in a spatially flat FLRW
background space. We consider a general scenario where an interaction term for
the two scalar fields exists in the potential term $V\left(\phi,\psi\right)$
of the two fields, that is, $V_{,\phi\psi}\neq 0$. Specifically, we determine
the stationary points of the cosmological equations and we study the stability
of these points. Each stationary point describes a solution in the
cosmological evolution. Such an analysis is important in order to understand
the general behaviour of the model and to infer about its viability. This
approach has been applied in various gravitational theories with important
results for the viability of specific theories of gravity, see for instance
dyn1 ; dyn2 ; dyn3 ; dyn4 ; dyn5 ; dyn6 ; dyn7 ; dyn8 ; dyn9 and references
therein. From such an analysis we can conclude about for which eras of the
cosmological history can be provided by the specific theory, we refer the
reader in the discussion of dyn1 . The plan of the paper is as follows.
In Section II we present the model of our consideration which is that of
Chiral cosmology in a spatially flat FLRW spacetime with a mixed potential
term. We write the field equations which are of second-order. By using the
energy density and pressure variables we observe that the interaction of the
two fields depends on the pressure term. In Section III, we rewrite the field
equations by using dimensionless variables in the $H-$normalization. We find
an algebraic-differential dynamical system consists of one algebraic
constraint and six first-order ordinary differential equations. We consider a
specific form for the potential in order to reduce dynamical system the system
by one-dimension; and with the use of the constraint equation we end with a
four-dimensional system.
The main results of this work are presented in Section IV. We find the
stationary points of the field equations which form four different families.
The stationary points of family A are those of quintessence, in family B only
the kinetic part of the second scalar field contributes to the cosmological
solutions. On the other hand, the points of family C are those where only the
dynamic part of the second field contributes. Furthermore, for the
cosmological solutions at the points of family D all the components of the
second field contributes to the cosmological fluid. For all the stationary
points we determine the physical properties which describe the corresponding
exact solutions, as also we determine the stability conditions. An application
of this analysis is presented in Section V with some numerical results.
Moreover, for completeness of our study we present an analytic solution of the
field equations by using previous results of the literature, from where we can
verify the main results of this work. In Section VII we discuss the additional
stationary points when matter source is included in the cosmological model.
Finally, in Section VIII we draw our conclusions.
## II Chiral cosmology
We consider the gravitational Action Integral to be 2sfand
$S=\int\sqrt{-g}dx^{4}R-\int\sqrt{-g}dx^{4}\left(\frac{1}{2}g^{\mu\nu}H_{AB}\left(\Phi^{C}\right)\nabla_{\mu}\Phi^{A}\nabla_{\nu}\Phi^{B}+V\left(\Phi^{C}\right)\right)$
(1)
where
$\Phi^{A}=\left(\phi\left(x^{\mu}\right),\psi\left(x^{\mu}\right)\right)$,
$H_{AB}\left(\Phi^{C}\right)$ is a second rank tensor which defines the
kinetic energy of the scalar fields, while $V\left(\Phi^{C}\right)$ is the
potential.
The Action Integral (1) describes a interacting two-scalar field cosmological
model where the interaction follows by the potential
$V\left(\Phi^{C}\right)=V\left(\phi,\psi\right),$ and the kinetic part.
In this work we assume that $H_{AB}\left(\Phi^{C}\right)$ is diagonal and
admits at least one isometry such that (1)
$S=\int\sqrt{-g}dx^{4}R-\int\sqrt{-g}dx^{4}\left(\frac{1}{2}g^{\mu\nu}\left(\phi_{;\mu}\phi_{;\nu}+M\left(\phi\right)\psi_{;\mu}\psi_{;\mu}\right)+V\left(\Phi^{C}\right)\right)$
(2)
where $M\left(\phi\right)_{,\phi}\neq 0$ and $M\left(\phi\right)\neq
M_{0}\phi^{2}$. In the latter two cases, $H_{AB}\left(\Phi^{C}\right)$
describe a two-dimensional flat space and if it is of Lorentzian signature
then it describes the quintom model. Functional of forms of
$M\left(\phi\right)$ where $H_{AB}\left(\Phi^{C}\right)$ is a maximally
symmetric space of constant curvature $R_{0}$, are given by the second-order
differential equation
$2M_{,\phi\phi}M-\left(M_{,\phi}\right)^{2}+2M^{2}R_{0}=0.$ (3)
A solution of the latter equation is $M\left(\phi\right)=M_{0}e^{\kappa\phi}$,
which can be seen as the general case since new fields can be defined under
coordinate transformations to rewrite the form of
$H_{AB}\left(\Phi^{C}\right)$. This is the case of Chiral model that we study
in this work.
Variation with respect to the metric tensor of (1) provides the gravitational
field equations
$G_{\mu\nu}=H_{AB}\left(\Phi^{C}\right)\nabla_{\mu}\Phi^{A}\nabla_{\nu}\Phi^{B}-g_{\mu\nu}\left(\frac{1}{2}g^{\mu\nu}H_{AB}\left(\Phi^{C}\right)\nabla_{\mu}\Phi^{A}\nabla_{\nu}\Phi^{B}+V\left(\Phi^{C}\right)\right),$
(4)
while variation with respect to the fields $\Phi^{A}$ give the Klein-Gordon
vector-equation
$g^{\mu\nu}\left(\nabla_{\mu}\left(H_{~{}B}^{A}\left(\Phi^{C}\right)\nabla_{\nu}\Phi^{B}\right)\right)+H_{~{}B}^{A}\left(\Phi^{C}\right)\frac{\partial
V\left(\Phi^{C}\right)}{\partial\Phi^{B}}=0.$ (5)
According to the cosmological principle, the universe in large scales is
isotropic and homogeneous described by the spatially flat FLRW spacetime with
line element
$ds^{2}=-dt^{2}+a^{2}\left(t\right)\left(dx^{2}+dy^{2}+dz^{2}\right).$ (6)
where $a\left(t\right)$ denotes the scale factor and the Hubble function is
defined as $H\left(t\right)=\frac{\dot{a}}{a}$.
For the line element (6) and the second-rank tensor
$H_{AB}\left(\Phi^{C}\right)$ of our consideration the field equations are
written as follows
$3H^{2}=\frac{1}{2}\left(\dot{\phi}^{2}+M\left(\phi\right)\dot{\psi}^{2}\right)+V\left(\phi\right)+M\left(\phi\right)U\left(\psi\right),$
(7)
$2\dot{H}+3H^{2}=-\left(\frac{1}{2}\left(\dot{\phi}^{2}+M\left(\phi\right)\dot{\psi}^{2}\right)-V\left(\phi\right)-M\left(\phi\right)U\left(\psi\right)\right),$
(8)
$\ddot{\phi}+3H\dot{\phi}-\frac{1}{2}M_{,\phi}\dot{\psi}^{2}+V_{,\phi}\left(\phi\right)+M_{,\phi}U\left(\psi\right)=0,$
(9)
$\ddot{\psi}+3H\dot{\psi}+\frac{M_{,\phi}}{M}\dot{\phi}\dot{\psi}+U_{,\psi}=0.$
(10)
where we replaced
$V\left(\phi,\psi\right)=V\left(\phi\right)+M\left(\phi\right)U\left(\psi\right)$
and we have assumed that the fields $\phi,\psi$ inherit the symmetries of the
FLRW space such that $\phi\left(x^{\mu}\right)=\phi\left(t\right)$ and
$\psi\left(x^{\mu}\right)=\psi\left(t\right)$. At this point we remark that
the field equations (8)-(10) can be produced by the variation principle of the
point-like Lagrangian
$\mathcal{L}\left(a,\dot{a},\phi,\dot{\phi},\psi,\dot{\psi}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\left(\dot{\phi}^{2}+M\left(\phi\right)\dot{\psi}^{2}\right)-a^{3}\left(V\left(\phi\right)+M\left(\phi\right)U\left(\psi\right)\right),$
(11)
while equation (7) can be seen as the Hamiltonian constraint of the time-
independent Lagrangian (11).
An equivalent way to write the field equations (7), (8) is by defining the
quantities
$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V\left(\phi\right)~{},~{}p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V\left(\phi\right),$
(12)
$\rho_{\psi}=\left(\frac{1}{2}\dot{\psi}^{2}+U\left(\psi\right)\right)M\left(\phi\right)~{},~{}p_{\psi}=\left(\frac{1}{2}\dot{\psi}^{2}-U\left(\psi\right)\right)M\left(\phi\right),$
(13)
that is,
$3H^{2}=\rho_{\phi}+\rho_{\psi},$ (14)
$2\dot{H}+3H^{2}=-\left(p_{\phi}+p_{\psi}\right),$ (15)
$\dot{\rho}_{\phi}+3H\left(\rho_{\phi}+p_{\phi}\right)=\dot{\phi}\frac{\partial}{\partial\phi}p_{\psi},$
(16)
$\dot{\rho}_{\psi}+3H\left(\rho_{\psi}+p_{\psi}\right)=-\dot{\phi}\frac{\partial}{\partial\phi}p_{\psi}.$
(17)
The latter equations give us an interesting observation, since we can write
the interacting functions of the two fields. The interaction models, with
interaction between dark matter and dark energy have been proposed as an
potential mechanism to explain the cosmic coincidence problem and provide a
varying cosmological constant. Some interaction models which have been studied
before in the literature are presented in Amendola:2006dg ; Pavon:2007gt ;
Chimento:2009hj ; Arevalo:2011hh ; an001 ; an002 while some cosmological
constraints on interacting models can be found in in1 ; in2 ; in3 ; in4 .
## III Dimensionless variables
We consider the dimensionless variables in the $H$-normalization cop
$\dot{\phi}=\sqrt{6}xH~{},~{}V\left(\phi\right)=3y^{2}H^{2}~{},~{}\dot{\psi}=\frac{\sqrt{6}}{\sqrt{M\left(\phi\right)}}zH~{},~{}U\left(\psi\right)=\frac{3}{M\left(\phi\right)}u^{2}H^{2}$
(18)
or
$x=\frac{\dot{\phi}}{\sqrt{6}H}~{},~{}y^{2}=\frac{V\left(\phi\right)}{3H^{2}}~{},~{}z=\frac{\sqrt{M\left(\phi\right)}\dot{\psi}}{\sqrt{6}H}~{},~{}u^{2}=\frac{M\left(\phi\right)U\left(\psi\right)}{3H^{2}},$
(19)
where the field equations become
$\displaystyle\frac{dx}{d\tau}$
$\displaystyle=\frac{3}{2}x\left(x^{2}-\left(1+u^{2}+y^{2}-z^{2}\right)\right)-\frac{\sqrt{6}}{2}\left(\lambda
y^{2}+\kappa\left(u^{2}-z^{2}\right)\right),$ (20)
$\displaystyle\frac{dy}{d\tau}$
$\displaystyle=\frac{3}{2}y\left(1+x^{2}+z^{2}-y^{2}-u^{2}\right)+\frac{\sqrt{6}}{2}\lambda
xy,$ (21) $\displaystyle\frac{dz}{d\tau}$
$\displaystyle=\frac{3}{2}z\left(z^{2}-\left(1+u^{2}+y^{2}-x^{2}\right)\right)-\frac{\sqrt{6}}{2}\left(\kappa
xz+\mu u^{2}\right),$ (22) $\displaystyle\frac{du}{d\tau}$
$\displaystyle=\frac{3}{2}u\left(1+x^{2}+z^{2}-y^{2}-u^{2}\right)+\frac{\sqrt{6}}{2}u\left(\kappa
x+\mu z\right),$ (23) $\displaystyle\frac{d\mu}{d\tau}$
$\displaystyle=\sqrt{\frac{3}{2}}\mu\left(2\mu
z\bar{\Gamma}\left(\mu,\lambda\right)-\kappa x-2\mu z\right),$ (24)
$\displaystyle\frac{d\lambda}{d\tau}$
$\displaystyle=\sqrt{6}\lambda^{2}x\left(\Gamma\left(\lambda\right)-1\right),$
(25)
in which
$\tau=\ln
a,~{}\lambda\left(\phi\right)=\frac{V_{,\phi}}{V}~{},~{}\kappa\left(\lambda\right)=\frac{M_{,\phi}}{M}~{},~{}\mu\left(\phi,\psi\right)=\frac{1}{\sqrt{M\left(\phi\right)}}\frac{U_{,\psi}}{U},~{}$
(26)
and functions
$\Gamma\left(\lambda\right),~{}\bar{\Gamma}\left(\mu,\lambda\right)$ are
defined as
$\Gamma\left(\lambda\right)=\frac{V_{,\phi\phi}V}{\left(V_{,\phi}\right)^{2}}~{},~{}\bar{\Gamma}\left(\mu,\lambda\right)=\frac{U_{,\psi\psi}U}{\left(U_{,\psi}\right)^{2}},$
(27)
while the constraint equation is
$1-x^{2}-y^{2}-z^{2}-u^{2}=0.$ (28)
The equation of state parameter for the effective cosmological fluid
$w_{tot},~{}$is given in terms of the dimensionless parameters as follows
$w_{tot}=-1-\frac{2}{3}\frac{\dot{H}}{H^{2}}=x^{2}+z^{2}-y^{2}-u^{2}$ (29)
while we define the variables
$\Omega_{\phi}=x^{2}+y^{2}~{},~{}\Omega_{\psi}=z^{2}+u^{2},$ (30)
with equation of state parameters
$w_{\phi}=-1+\frac{2x^{2}}{x^{2}+y^{2}}~{},~{}w_{\psi}=-1+\frac{2z^{2}}{z^{2}+u^{2}}.$
(31)
At this point it is important to mention that since the two fields interact
that is not the unique definition of the physical variables $\Omega_{\phi}$
and $\Omega_{\psi}$, $w_{\phi}$ and $w_{\psi}$. Moreover, from the constraint
equation (28) it follows that the stationary points are on the surface of a
four-dimensional unitary sphere, while the field equations remain invariant
under the transformations $\left\\{y,u\right\\}\rightarrow\left(-y,-u\right)$,
that is, the variables $\left\\{x,y,z,u\right\\}$ take values in the following
regions $\left|x\right|\leq 1~{},~{}\left|z\right|\leq 1~{},~{}0\leq y\leq
1~{}\ $and $0\leq u\leq 1$.
For the arbitrary functions $V\left(\phi\right),$ $U\left(\psi\right)$ and
$M\left(\phi\right)$, there are six dependent, namely
$\left\\{x,y,z,u,\lambda,\mu\right\\}$, where in general
$\kappa=\kappa\left(\lambda\right)$, however the dimension of the system can
be reduced by one, if we apply the constraint condition (28).
In the following Section, we determine the stationary points for the cases
where $M\left(\phi\right)=M_{0}e^{\kappa\phi},~{}\
V\left(\phi\right)=V_{0}e^{\lambda\phi},~{}$ and
$U\left(\psi\right)=U_{0}\psi^{\frac{1}{\sigma}}$. Consequently, we calculate
$\Gamma\left(\lambda\right)=1$ and
$\bar{\Gamma}\left(\mu,\lambda\right)=1-\sigma$ and $\kappa=const$. Therefore,
$\frac{d\lambda}{d\tau}=0$ is satisfied identically and the dimension of the
dynamical system is reduced by one. Therefore we end with the dynamical system
(20)-(24) with constraint (28). We remark that in Chiral model, the kinetic
parts of the two fields are defined on a two-dimensional space of constant
curvature.
## IV Dynamical behaviour
The stationary points of the dynamical system have coordinates which make the
rhs of equations (20)-(24) vanish. We categorize the stationary points into
four families. Family A, are the points with coordinates
$\left(x_{A},y_{A},z_{A},u_{A},\mu_{A}\right)=\left(x_{A},y_{A},0,0,0\right)$
and correspond to the points of the minimally coupled scalar field cosmology
cop .
The points with coordinates
$\left(x_{B},y_{B},z_{B},u_{B},\mu_{B}\right)=\left(x_{B},y_{B},z_{B},0,\mu_{B}\right)$
and $z_{B}\neq 0$ define the points of Family B. These points describe
physical solutions without any contribution of the potential
$U\left(\psi\right)$ to the energy density of the total fluid source, but only
when $\mu_{B}=0$ there is not any contribution of potential
$U\left(\psi\right)$ to the dynamics. When $\mu_{B}=0$, the stationary points
are those found before in andimakis .
Points of family $C$ have coordinates
$\left(x_{C},y_{C},z_{C},u_{C},\mu_{C}\right)=\left(x_{C},y_{C},0,u_{C},\mu_{C}\right),~{}u_{C}\neq
0$ which describe exact solutions with no contribution of the kinetic part of
the scalar fields $\psi$. Finally, the points of family D have coordinates of
the form
$\left(x_{C},y_{C},z_{C},u_{C},\mu_{C}\right)~{}$with$~{}z_{D}u_{D}\neq 0$.
Let $P$ be a stationary point of the dynamical system (20)-(24), that
is,$~{}\dot{q}^{A}=f^{A}\left(q^{B}\right)$, where $f^{A}\left(P\right)=0$. In
order to study the stability properties of the critical point $P$, we write
the linearized system which is $\delta\dot{x}^{A}=J_{B}^{A}\delta
x^{B}~{}$where $J_{B}^{A}$ is the Jacobian matrix at the point $P$,
i.e.$~{}J_{B}^{A}=\frac{\partial f^{A}\left(P\right)}{\partial x^{B}}$. The
eigenvalues $\mathbf{e}\left(P\right)$ of the Jacobian matrix determine the
stability of the station point. When all the eigenvalues have negative real
part then point $P$ is an attractor and the exact solution at the point is
stable, otherwise the exact solution at the critical point is unstable and
point $P$ is a source, when all the eigenvalues have a positive real part, or
$P$ is a saddle point.
### IV.1 Family A
There are three stationary points which describe cosmological solutions
without any contribution of the second field $\psi$. The points have
coordinates cop
$A_{1}^{\pm}=\left(\pm
1,0,0,0,0\right)~{},~{}A_{2}=\left(-\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0,0,0\right).$
(32)
Points $A_{1}^{\pm}$ describe universes dominated by the kinetic part of the
scalar field $\phi,~{}$that is by the term $\frac{1}{2}\dot{\phi}^{2}$. The
physical quantities are derived
$\left(w_{tot}\left(A_{1}^{\pm}\right),w_{\phi}\left(A_{1}^{\pm}\right),w_{\psi}\left(A_{1}^{\pm}\right),\Omega_{\phi}\left(A_{1}^{\pm}\right),\Omega_{\psi}\left(A_{1}^{\pm}\right)\right)=\left(1,1,\nexists,1,0\right).$
Point $A_{2}$ is physically accepted when $\left|\lambda\right|<\sqrt{6},$ the
physical quantities are calculated
$\left(w_{tot}\left(A_{2}\right),w_{\phi}\left(A_{2}\right),w_{\psi}\left(A_{2}\right),\Omega_{\phi}\left(A_{2}\right),\Omega_{\psi}\left(A_{2}\right)\right)=\left(-1+\frac{\lambda^{2}}{3},-1+\frac{\lambda^{2}}{3},\nexists,1,0\right).$
Therefore, point $A_{2}$ describes a scaling solution. The latter solution is
that of an accelerated universe when $\left|\lambda\right|<\sqrt{2}$.
In the case of quintessence scalar field cosmology, points $A_{1}^{\pm}$ are
always unstable, while $A_{2}$ is the unique attractor of the dynamical system
when $\left|\lambda\right|<\sqrt{3}$. However, for the model of our analysis
the stability conditions are different.
In order to conclude for the stability of the stationary points we determine
the eigenvalues of the linearized dynamical system (20)-(24) around to the
stationary points. For the points $A_{1}^{\pm}$ it follows
$\displaystyle e_{1}\left(A_{1}^{\pm}\right)$ $\displaystyle=3,$
$\displaystyle e_{2}\left(A_{1}^{\pm}\right)$
$\displaystyle=\frac{1}{2}\left(6\pm\sqrt{6}\lambda\right),$
$\displaystyle~{}e_{3}\left(A_{1}^{\pm}\right)$
$\displaystyle=\frac{1}{2}\left(6\pm\sqrt{6}\kappa\right),$
$\displaystyle~{}e_{4}\left(A_{1}^{\pm}\right)$
$\displaystyle=\mp\sqrt{\frac{3}{2}}\kappa,~{}$ $\displaystyle
e_{5}\left(A_{1}^{\pm}\right)$ $\displaystyle=\mp\sqrt{\frac{3}{2}}\kappa,$
from where we conclude that points $A_{1}^{\pm}~{}$are saddle points, while
the solutions at points $A_{1}^{\pm}$ are always unstable’ because at least
one of the eigenvalues is always positive, i.e. eigenvalue
$e_{1}\left(A_{1}^{\pm}\right)>0$.
For the stationary point $A_{2}$ the eigenvalues are derived
$\displaystyle e_{1}\left(A_{2}\right)$
$\displaystyle=\frac{1}{2}\left(\lambda^{2}-6\right),$ $\displaystyle
e_{2}\left(A_{2}\right)$ $\displaystyle=\lambda^{2}-3,$
$\displaystyle~{}e_{3}\left(A_{2}\right)$
$\displaystyle=\frac{1}{2}\kappa\lambda,$
$\displaystyle~{}e_{4}\left(A_{2}\right)$
$\displaystyle=\frac{1}{2}\left(\lambda^{2}-\kappa\lambda\right),~{}$
$\displaystyle e_{5}\left(A_{2}\right)$
$\displaystyle=\frac{1}{2}\left(\lambda^{2}-6+\kappa\lambda\right),$
that is, the exact solution at point $A_{2}$ is always unstable. However. from
the two eigenvalues $e_{1}\left(A_{2}\right),~{}e_{2}\left(A_{2}\right)$ we
can infer that in the surface $\left\\{x,y\right\\}$ of the phase space the
stationary point $A_{2}$ acts like an attractor for
$\left|\lambda\right|<\sqrt{3}$, which however becomes a saddle point for the
higher-dimensional phase space.
We remark that we determined the stability of the stationary points without
using the constant equation and reducing the dynamical system by one-
dimension. However, by replacing $z^{2}=1-x^{2}-y^{2}-u^{2}$ in the (20)-(24)
we end with a four-dimensional system, from where we find the same results,
that is, the exact solutions at the points $A_{1}^{\pm}$ and $A_{2}$ are
always unstable.
### IV.2 Family B
For $z_{B}\neq 0$ and $u_{B}=0,$ we found four stationary points which are
$\displaystyle B_{1}^{\pm}$
$\displaystyle=\left(-\frac{\sqrt{6}}{\kappa+\lambda},\sqrt{\frac{\kappa}{\kappa+\lambda}},\pm\sqrt{\frac{\lambda^{2}+\kappa\lambda-6}{\left(\kappa+\lambda\right)^{2}}},0,0\right),$
(33) $\displaystyle B_{2}^{\pm}$
$\displaystyle=\left(-\frac{\sqrt{6}}{\kappa+\lambda},\sqrt{\frac{\kappa}{\kappa+\lambda}},\pm\sqrt{\frac{\lambda^{2}+\kappa\lambda-6}{\left(\kappa+\lambda\right)^{2}}},0,\sqrt{\frac{3}{2}}\frac{\kappa}{\sqrt{\left(\lambda^{2}+\kappa\lambda-6\right)}}\right),$
(34)
which are real and are physically accepted when
$\left\\{\kappa>0,\lambda>\sqrt{6}\right\\}$ or
$\left\\{0<\lambda\leq\sqrt{6},~{}\kappa>\frac{6-\lambda^{2}}{\lambda}\right\\}$
or $\left\\{\lambda<-\sqrt{6},\kappa<0\right\\}$ or
$\left\\{-\sqrt{6}<\lambda<0,\kappa<\frac{6-\lambda^{2}}{\lambda}\right\\}$.
The latter region plots are presented in Fig. 1.
Figure 1: Region plot in the space $\left\\{\lambda,\kappa\right\\}$ where
points $\mathbf{B=}\left(B_{1}^{\pm},B_{2}^{\pm}\right)$ are real.
The stationary points have the same physical properties, that is, the points
describe universes with the same physical properties, where the physical
quantities have the following values
$w_{tot}\left(\mathbf{B}\right)=1-\frac{2\kappa}{\kappa+\lambda}~{},~{}w_{\phi}\left(\mathbf{B}\right)=-1+\frac{12}{6+\kappa\left(\kappa+\lambda\right)}~{},~{}w_{\psi}\left(\mathbf{B}\right)=1~{},$
(35)
$\Omega_{\phi}\left(\mathbf{B}\right)=1-\Omega_{\psi}\left(\mathbf{B}\right)~{},~{}\Omega_{\psi}\left(\mathbf{B}\right)=\left|\frac{\lambda\left(\kappa+\lambda\right)-6}{\left(\kappa+\lambda\right)^{2}}\right|.$
(36)
From $w_{tot}\left(\mathbf{B}\right)$ it follows that the points describe
scaling solutions and the de Sitter universe is recovered only when
$\lambda=0$, which is excluded because for $\lambda=0$, the stationary points
are not real. We continue by studying the stability of the stationary points.
In Fig. 2, we present counter plots for the physical parameters
$w_{tot}\left(\mathbf{B}\right),~{}w_{\phi}\left(\mathbf{B}\right)\,$ and
$\Omega_{\psi}\left(\mathbf{B}\right)$ in the space of variables
$\left\\{\lambda,\kappa\right\\}$.
Figure 2: Qualitative evolution of the physical variables
$w_{tot}\left(\mathbf{B}\right),~{}w_{\phi}\left(\mathbf{B}\right)\,$ and
$\Omega_{\psi}\left(\mathbf{B}\right)$ of the exact solutions at the critical
points $\mathbf{B=}\left(B_{1}^{\pm},B_{2}^{\pm}\right)$ for various values of
the free variables $\left\\{\lambda,\kappa\right\\}$.
For the stationary points $B_{1}^{\pm}$ two of the five eigenvalues are
expressed as
$e_{1}\left(B_{1}^{\pm}\right)=3\frac{\kappa}{\kappa+\lambda},~{}e_{2}\left(B_{1}^{\pm}\right)=-3\frac{\kappa-\lambda}{\kappa+\lambda},$
from where we observe that $e_{1}\left(B_{1}^{\pm}\right)>0$ in order for the
points to be real, consequently the exact solutions at the stationary points
$B_{1}^{\pm}$ are unstable.
We use the constraint $z^{2}=1-x^{2}-y^{2}-u^{2}$ such that the dynamical
system is reduced by one-dimension. Thus, for the new four-dimensional system
the eigenvalues of the linearized system around points $B_{1}^{\pm}$ are found
$\displaystyle e_{1}\left(B_{1}^{\pm}\right)$
$\displaystyle=3\frac{\kappa}{\kappa+\lambda},~{}e_{2}\left(B_{1}^{\pm}\right)=-3\frac{\kappa-\lambda}{\kappa+\lambda},$
$\displaystyle e_{3}\left(B_{1}^{\pm}\right)$
$\displaystyle=-\frac{3\kappa+i\sqrt{3\kappa\left(4\lambda^{3}+8\kappa\lambda^{2}+4\left(\kappa^{2}-6\right)\lambda-27\kappa\right)}}{2\left(\kappa+\lambda\right)},$
$\displaystyle e_{4}\left(B_{1}^{\pm}\right)$
$\displaystyle=-\frac{3\kappa-i\sqrt{3\kappa\left(4\lambda^{3}+8\kappa\lambda^{2}+4\left(\kappa^{2}-6\right)\lambda-27\kappa\right)}}{2\left(\kappa+\lambda\right)},$
from where we conclude again that the exact scaling solutions at points
$B_{1}^{\pm}$ are unstable. In particular points
Similarly, the eigenvalues of the linearized system around the points
$B_{2}^{\pm}$ are calculated
$\displaystyle e_{1}\left(B_{2}^{\pm}\right)$
$\displaystyle=-3\frac{\kappa}{\kappa+\lambda},~{}e_{2}\left(B_{2}^{\pm}\right)=-3\frac{2\sigma\left(\kappa-\lambda\right)-\kappa}{2\sigma\left(\kappa+\lambda\right)},$
$\displaystyle e_{3}\left(B_{2}^{\pm}\right)$
$\displaystyle=e_{3}\left(B_{1}^{\pm}\right),~{}e_{4}\left(B_{2}^{\pm}\right)=e_{3}\left(B_{1}^{\pm}\right),$
Hence, we infer that the stationary points $B_{2}^{\pm}$ are attractors, and
the exact solutions at the points are stable when the free parameters
$\left\\{\lambda,\kappa,\sigma\right\\}$ are constraints as follows
$\lambda\leq-\sqrt{6}:\left\\{\kappa<\lambda,\sigma<0,\sigma>\frac{\kappa}{2\left(\kappa-\lambda\right)}\right\\}\cup\left\\{\kappa=\lambda,\sigma<0\right\\}\cup\left\\{\lambda<\kappa<0,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\},$
$-\sqrt{6}<\lambda<-\sqrt{3}:\left\\{\kappa<\lambda,\sigma<0,\sigma>\frac{\kappa}{2\left(\kappa-\lambda\right)}\right\\}\cup\left\\{\kappa=\lambda,\sigma<0\right\\}\cup\left\\{\lambda<\kappa<\frac{6-\lambda^{2}}{\lambda},\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\},$
$\lambda=-\sqrt{3}:\left\\{\kappa<-\sqrt{3},~{}\sigma<0\right\\}\cup\left\\{\kappa<-\sqrt{3},~{}\frac{\kappa}{2\left(\sqrt{3}+\kappa\right)}<\sigma\right\\},$
$-\sqrt{3}<\lambda<0:\left\\{\kappa<\frac{6-\lambda^{2}}{\lambda},\sigma<0\right\\}\cup\left\\{\kappa<\frac{6-\lambda^{2}}{\lambda},\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\},$
$0<\lambda<\sqrt{3}:\left\\{\kappa>\frac{6-\lambda^{2}}{\lambda},\sigma<0\right\\}\cup\left\\{\kappa>\frac{6-\lambda^{2}}{\lambda},\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\},$
$\lambda=\sqrt{3}:\left\\{\kappa<\sqrt{3},~{}\sigma<0\right\\}\cup\left\\{\kappa<-\sqrt{3},~{}-\frac{\kappa}{2\left(\sqrt{3}-\kappa\right)}<\sigma\right\\},$
$\sqrt{3}<\lambda<\sqrt{6}:\left\\{\frac{6-\lambda^{2}}{\lambda}<\kappa<\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\}\cup\left\\{\kappa\geq\lambda,\sigma<0\right\\}\cup\left\\{\kappa>\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\},$
$\lambda\geq\sqrt{6}:\left\\{0<\kappa<\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\}\cup\left\\{\kappa\geq\lambda,\sigma<0\right\\}\cup\left\\{\kappa>\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\}.$
In Figs. 3 and 4 we plot the regions where the stationary points $B_{2}^{\pm}$
are attractors and the exact solutions on the stationary points points are
stable.
Figure 3: Region plot in the space of variabels
$\left\\{\kappa,\lambda,\sigma\right\\}$ where the points $B_{2}^{\pm}$ are
attractors. Figure 4: Region plots in the the planes
$\kappa-\sigma,~{}\lambda-\sigma$ and $\lambda-\kappa$ where points
$B_{2}^{\pm}$ are attractors. Left figures present the region in the plane
$\kappa-\sigma$ for $\lambda=-2$ and $\lambda=2$; middle figures present the
region in the plane $\lambda-\sigma$, for $\kappa=-2$ and $\kappa-2$ while
right figures are in the plane for $\lambda-\kappa$ for $\sigma=-1$ and
$\sigma=1$.
### IV.3 Family C
The stationary points of Family C are two and they have coordinates
$\displaystyle C_{1}$
$\displaystyle=\left(-\frac{\kappa}{\sqrt{6}},0,0,\sqrt{1-\frac{\kappa^{2}}{6}},0\right),$
(37) $\displaystyle C_{2}$
$\displaystyle=\left(0,\sqrt{\frac{\kappa}{\kappa-\lambda}},0,\sqrt{\frac{\lambda}{\lambda-\kappa}},0\right).$
(38)
Point $C_{1}$ is real when $\left|\kappa\right|\leq\sqrt{6}$ and the physical
quantities of the exact solution at the point are
$\left(w_{tot}\left(C_{1}\right),w_{\phi}\left(C_{1}\right),w_{\psi}\left(C_{1}\right),\Omega_{\phi}\left(C_{1}\right),\Omega_{\psi}\left(C_{1}\right)\right)=\left(-1+\frac{\kappa^{2}}{3},1,-1,\frac{\kappa^{2}}{6},1-\frac{\kappa^{2}}{6}\right).$
(39)
Thus, stationary point $C_{1}$ describes a scaling solution. The scaling
solution describes an accelerated universe when
$\left|\kappa\right|<\sqrt{2}$.
Furthermore, the exact solution at the stationary point $C_{2}$ describes a de
Sitter universe, where the two scalar fields mimic the cosmological constant,
the physical quantities are
$\left(w_{tot}\left(C_{2}\right),w_{\phi}\left(C_{2}\right),w_{\psi}\left(C_{2}\right),\Omega_{\phi}\left(C_{2}\right),\Omega_{\psi}\left(C_{2}\right)\right)=\left(-1,-1,-1,\frac{\kappa}{\kappa-\lambda},\frac{\lambda}{\lambda-\kappa}\right).$
(40)
Point $C_{2}$ is real and physically accepted when $\lambda\kappa<0$, i.e.
$\left\\{\lambda<0,\kappa>0\right\\}$ or
$\left\\{\lambda>0,\kappa<0\right\\}$.
The linearized four-dimensional system around the stationary point $C_{1}$
admits the eigenvalues
$\displaystyle e_{1}\left(C_{1}\right)$ $\displaystyle=\frac{\kappa^{2}}{2},$
$\displaystyle e_{2}\left(C_{1}\right)$
$\displaystyle=-\frac{1}{2}\left(6-\kappa^{2}\right)$ $\displaystyle
e_{3}\left(C_{1}\right)$ $\displaystyle=2\left(\kappa^{2}-3\right)$
$\displaystyle e_{4}\left(C_{1}\right)$
$\displaystyle=\frac{1}{2}\kappa\left(\kappa-\lambda\right)$
from where we infer that the exact solution at the stationary point is always
unstable. Specifically, point $C_{1}$ is a saddle point.
For the stationary point $C_{2}$, we find that one of the eigenvalues of the
linearized system around $C_{2}$ is zero. That eigenvalue corresponds to the
linearize equation (24). As far as concerns the other three eigenvalues we
plot numerically their values and we find that they have negative real parts
for all the range of parameters $\left\\{\lambda,\kappa\right\\}$ where the
point exists. In Fig. 5 we plot the real parts of the three nonzero
eigenvalues of the linearized system. Therefore, we infer that the there
exists a four-dimensional stable submanifold around the stationary point.
However, because of the eigenvalues has zero real part the center manifold
theorem (CMT) should be applied.
For simplicity on our calculations we apply the CMT for the five dimensional
system. We find that the variables with nonzero real part on their
eigenvalues, that is, variables $\left\\{x,y,z,u\right\\}$, according to the
CMT theorem are approximated as functions of variable $\mu$ as follows
$\displaystyle x$
$\displaystyle=x_{00}\mu^{2}+x_{10}\mu^{3}+x_{20}\mu^{4}+O\left(\mu^{5}\right)~{},~{}y=y_{00}\mu^{2}+y_{10}\mu^{3}+y_{20}\mu^{4}+O\left(\mu^{5}\right),~{}$
$\displaystyle z$
$\displaystyle=z_{00}\mu^{2}+z_{10}\mu^{3}+z_{20}\mu^{4}+O\left(\mu^{5}\right)~{},~{}u=u_{00}\mu^{2}+u_{10}\mu^{3}+u_{20}\mu^{4}+O\left(\mu^{5}\right)$
where
$\left\\{x_{00},y_{00},z_{00},u_{00}\right\\}=\left(0,0,z_{00},0\right)$;
$x_{10}=-\frac{z_{00}}{\kappa},~{}y_{10}=\sqrt{\frac{3}{2}}\frac{z_{00}}{\sqrt{\kappa^{3}\left(\kappa-\lambda\right)}},~{}$etc.
Hence, the fifth equation, i.e. equation (20) is written
$\frac{d\mu}{d\tau}=\alpha\mu^{4}+a_{1}\mu^{5}+O\left(\mu^{6}\right)~{}$where
$\alpha=\frac{\sqrt{6}\left(\kappa\lambda-2\left(\kappa\lambda+3\right)\sigma\right)}{2\kappa\lambda+6}z_{00}-\frac{6\kappa\left(\sqrt{\lambda\left(\lambda-\kappa\right)}\right)}{2\kappa\lambda+6}u_{10}$.
Therefore, the point is always unstable for $a\neq 0$, however from the
coefficient term $a_{1}\mu^{5}$ we find that the point can be stable.
Figure 5: Qualitative evolution for the real parts of the nonzero eigenvalues
of the linearized system around the stationary point $C_{2}$.
### IV.4 Family D
The fourth family of stationary points is consists of the following six
stationary points
$D_{1}^{\pm}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\kappa},0,\pm\frac{\sqrt{\kappa^{2}-3}}{\sqrt{2}\kappa},\frac{1}{\sqrt{2}},0\right),$
(41) $D_{2}^{\pm}=\left(x_{D_{2}},0,\pm
z_{D_{2}},\sqrt{1-\left(x_{D_{2}}\right)^{2}-\left(z_{D_{2}}\right)^{2}},\mu_{D_{2}}\right),$
(42) $D_{3}^{\pm}=\left(x_{D_{3}},0,\pm
z_{D3},\sqrt{1-\left(x_{D_{3}}\right)^{2}-\left(z_{D_{3}}\right)^{2}},\mu_{D_{3}}\right),$
(43)
with
$\displaystyle x_{D_{2}}$
$\displaystyle=-\frac{\kappa^{2}(2\sigma-1)+\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}+6\sigma}{\sqrt{6}\kappa(4\sigma-1)},$
$\displaystyle z_{D_{2}}$
$\displaystyle=\frac{\sqrt{-\kappa^{4}(1-2\sigma)^{2}+6\kappa^{2}\sigma\left(8\sigma^{2}-2\sigma+1\right)-\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}\left(\kappa^{2}(2\sigma-1)+24\sigma^{2}\right)-144\sigma^{3}}}{2\sqrt{3}\kappa\sqrt{\sigma}(4\sigma-1)},$
$\displaystyle\mu_{D_{2}}$
$\displaystyle=z_{D_{2}}\frac{\sqrt{6}\left(\kappa^{2}(1-2\sigma)^{2}+2\sigma\left(\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}-6\sigma\right)\right)}{\kappa^{2}(1-2\sigma)^{2}-24\sigma^{2}},$
$\displaystyle x_{D_{3}}$
$\displaystyle=\frac{\kappa^{2}(1-2\sigma)+\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}-6\sigma}{\sqrt{6}\kappa(4\sigma-1)},$
$\displaystyle z_{D_{3}}$
$\displaystyle=\frac{\sqrt{-\kappa^{4}(1-2\sigma)^{2}+6\kappa^{2}\sigma\left(8\sigma^{2}-2\sigma+1\right)+\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}\left(\kappa^{2}(2\sigma-1)+24\sigma^{2}\right)-144\sigma^{3}}}{2\sqrt{3}\kappa\sqrt{\sigma}(4\sigma-1)},$
$\displaystyle\mu_{D_{3}}$
$\displaystyle=z_{D_{3}}\frac{\sqrt{6}\left(2\sigma\left(\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}+6\sigma\right)-\kappa^{2}(1-2\sigma)^{2}\right)}{\kappa^{2}(1-2\sigma)^{2}-24\sigma^{2}}\text{.
}$
Points $D_{1}^{\pm}$ describe a scaling solution where the effective fluid is
pressureless, that is, it describes a dust fluid source and the scale factor
is $a\left(t\right)=a_{0}t^{\frac{2}{3}}$. The physical parameters of the
exact solution at points $D_{1}^{\pm}$ are
$w_{tot}\left(D_{1}^{\pm}\right)=0~{},~{}w_{\phi}\left(D_{1}^{\pm}\right)=1~{},~{}w_{\psi}\left(D_{1}^{\pm}\right)=\frac{3}{3-2\kappa^{2}}~{},$
(44)
$\Omega_{\phi}\left(D_{1}^{\pm}\right)=\frac{3}{2\kappa^{2}}~{},~{}\Omega_{\psi}\left(D_{1}^{\pm}\right)=1-\frac{3}{2\kappa^{2}}.$
(45)
Remark that points $D_{1}^{\pm}$ are real when $\left|\kappa\right|>\sqrt{3}$.
The eigenvalues of the four-dimensional linearized system around the
stationary points $D_{1}^{\pm}$ are derived
$\displaystyle e_{1}\left(D_{1}^{\pm}\right)$ $\displaystyle=\frac{3}{2}$
$\displaystyle e_{2}\left(D_{1}^{\pm}\right)$
$\displaystyle=\frac{3}{2}\left(\kappa-\lambda\right)$ $\displaystyle
e_{3}\left(D_{1}^{\pm}\right)$
$\displaystyle=-\frac{3+\sqrt{3\left(51-16\kappa^{2}\right)}}{4}$
$\displaystyle e_{4}\left(D_{1}^{\pm}\right)$
$\displaystyle=-\frac{3-\sqrt{3\left(51-16\kappa^{2}\right)}}{4}$
from where we infer that the stationary points $D_{1}^{\pm}$ are always
unstable. Points $D_{1}^{\pm}$ are saddle points.
Points $D_{2}^{\pm}$ are real and physically accepted when
$\left\\{\sigma\in\left(0,\frac{1}{4}\right)\cup\left(\frac{1}{4},\frac{1}{2}\right),\kappa>\frac{2\sqrt{6}\sigma}{2\sigma-1}\right\\}\cup\left\\{\frac{2\sqrt{6}\sigma}{1-2\sigma}<\kappa<-\sqrt{6}\sqrt{\frac{2\sigma^{2}+\sigma\sqrt{4\sigma-1}}{\left(1-2\sigma\right)^{2}}},\sigma>\frac{1}{2}\right\\}$
and $\left\\{\kappa<0,\sigma\,<0\right\\}$ as they are presented in Fig. 6.
The exact solution at the stationary points describe a scaling solution with
values of the equation of state parameter $w_{tot}\left(\kappa,\sigma\right)$
as they presented in Fig. 6. For the linearized four-dimensional system one of
the eigenvalues is
$e_{1}\left(D_{2}^{\pm}\right)=\frac{A\left(\kappa,\sigma\right)(2\kappa\sigma-\kappa-2\lambda\sigma)}{4\kappa\sigma(4\sigma-1)\left(2\kappa^{2}\sigma-\kappa^{2}+24\sigma^{2}\right)},$
where
$\displaystyle A\left(\kappa,\sigma\right)$
$\displaystyle=4\kappa^{4}\sigma^{2}-4\kappa^{4}\sigma+\kappa^{4}+48\kappa^{2}\sigma^{3}-12\kappa^{2}\sigma^{2}-6\kappa^{2}\sigma$
$\displaystyle+\sqrt{\left(2\kappa^{2}\sigma-\kappa^{2}+24\sigma^{2}\right)^{2}\left(4\kappa^{4}\sigma^{2}-4\kappa^{4}\sigma+\kappa^{4}-24\kappa^{2}\sigma^{2}+36\sigma^{2}\right)}+144\sigma^{3}.$
The other three eigenvalues are only functions of $\kappa,\sigma$, that is
$e_{2,3,4}\left(D_{2}^{\pm}\right)=e_{2,3,4}\left(\kappa,\sigma\right)$.
Numerically, we find that there are not any values of
$\left\\{\kappa,\sigma\right\\}$ where the points $D_{2}^{\pm}$ are defined,
such that all the eigenvalues have real part negative, consequently, the
stationary points are always sources and the exact solutions at the stationary
points $D_{2}^{\pm}$ are always unstable.
Figure 6: Left figure: Region plot in the space
$\left\\{\kappa,\sigma\right\\}$ where points $D_{2}^{\pm}$ are real and
physical accepted. Right Figure: Contour plot of the equation of state
parameter for the effective fluid $w_{tot}\left(\kappa,\sigma\right)$ at the
critical points $D_{2}^{\pm}$. Figure 7: Left figure: Region plot in the
space $\left\\{\kappa,\sigma\right\\}$ where points $D_{3}^{\pm}$ are real and
physical accepted. Right Figure: Contour plot of the equation of state
parameter for the effective fluid $w_{tot}\left(\kappa,\sigma\right)$ at the
critical points $D_{3}^{\pm}$.
Stationary points $D_{3}^{\pm}$ have similar physical properties with points
$D_{2}^{\pm}$, indeed they describe scaling solutions only. The points are
real and physically accepted in the region
$\left\\{\sigma>\frac{1}{2},\kappa<-\sqrt{\frac{6\sigma}{\sqrt{4\sigma-1}-2\sigma}}\right\\}$.
In Fig. 7 we present the region in the space $\left\\{\sigma,\kappa\right\\}$
where the points are defined as also the counter plot of the equation of state
parameter for the effective fluid source which describes the exact solution at
the points $D_{3}^{\pm}$. In a similar way with points $D_{2}^{\pm}$ we find
that there is not any range in the space $\left\\{\kappa,\sigma\right\\}$
where the points are attractors. Consequently, the stationary points
$D_{3}^{\pm}$ are sources. The main physical results of the stationary points
are summarized in Table 1.
Table 1: The physical propreties of the stationary models in chiral cosmology Point | Contribution of $\phi$ | Contribution of $\psi$ | Scaling/de Sitter | Possible $w_{tot}<-\frac{1}{3}$ | Stability
---|---|---|---|---|---
$A_{1}$ | Yes only kinetic part | No | Scaling | No | Unstable
$A_{2}$ | Yes | No | Scaling | Yes | Unstable
$B_{1}^{\pm}$ | Yes | Yes only kinetic part | Scaling | Yes | Unstable
$B_{2}^{\pm}$ | Yes | Yes only kinetic part | Scaling | Yes | Can be Stable
$C_{1}$ | Yes only kinetic | Yes only potential | Scaling | Yes | Unstable
$C_{2}$ | Yes only potential | Yes only potential | de Sitter $\left(w_{tot}=-1\right)$ | Always | CMT
$D_{1}^{\pm}$ | Yes | Yes | Scaling $\left(w_{tot}=0\right)$ | No | Unstable
$D_{2}^{\pm}$ | Yes | Yes | Scaling | Yes | Unstable
$D_{3}^{\pm}$ | Yes | Yes | Scaling | Yes | Unstable
## V Application $\left(\kappa,\sigma\right)=\left(2,\frac{1}{2}\right)$
Consider now the case where $\kappa=2$ and $\sigma=\frac{1}{2}$, while
$\lambda$ is an arbitrary constant. For that consideration, the stationary
points of the dynamical system (20)-(24) have the following coordinates
$\displaystyle\bar{A}_{1}^{\pm}$ $\displaystyle=\left(\pm
1,0,0,0,0\right),~{}$ $\displaystyle\bar{A}_{2}$
$\displaystyle=\left(-\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0,0,0\right),$
$\displaystyle\bar{B}_{1}^{\pm}$
$\displaystyle=\left(-\frac{\sqrt{6}}{\lambda+2},\sqrt{\frac{2}{\lambda+2}},\pm\sqrt{\frac{\left(\lambda+1\right)^{2}-7}{\left(\lambda+2\right)^{2}}},0,0\right),~{}$
$\displaystyle\bar{B}_{2}^{\pm}$
$\displaystyle=\left(-\frac{\sqrt{6}}{\lambda+2},\sqrt{\frac{2}{\lambda+2}},\sqrt{\frac{\left(\lambda+1\right)^{2}-7}{\left(\lambda+2\right)^{2}}},0,2\sqrt{\frac{6}{\left(\lambda+1\right)^{2}-7}}\right),$
$\displaystyle\bar{C}_{1}$
$\displaystyle=\left(-\sqrt{\frac{2}{3}},0,0,\frac{1}{\sqrt{3}},0\right),~{}$
$\displaystyle\bar{C}_{2}$
$\displaystyle=\left(0,\left(1-\frac{\lambda}{2}\right)^{-1},0,\sqrt{\frac{\lambda}{\lambda-2}}\right),$
$\displaystyle\bar{D}_{1}^{\pm}$
$\displaystyle=\left(-\frac{1}{2}\sqrt{\frac{3}{2}},0,\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2}},0\right).$
Points $\bar{A}_{1}^{\pm},~{}\bar{A}_{2}$ are sources and since they do not
depend on the parameters $\kappa,\sigma$ their physical properties are the
same as before. Recall that point $\bar{A}_{2}$ is real for
$\left|\lambda\right|<\sqrt{6}$. Stationary points
$\mathbf{B}=\left(\bar{B}_{1}^{\pm},\bar{B}_{2}^{\pm}\right)$ exist when
$\lambda>\sqrt{7}-1$. The physical parameters at the points are simplified as
follows
$w_{tot}\left(\mathbf{B}\right)=\frac{\lambda-2}{\lambda+2}~{},~{}w_{\phi}\left(\mathbf{B}\right)=-1+\frac{6}{5\lambda}~{},~{}w_{\psi}\left(\mathbf{B}\right)=1~{},$
(46)
$\Omega_{\phi}\left(\mathbf{B}\right)=\frac{2\left(\lambda+5\right)}{\left(\lambda+2\right)^{2}}~{},~{}\Omega_{\psi}\left(\mathbf{B}\right)=1-\frac{2\left(\lambda+5\right)}{\left(\lambda+2\right)^{2}}.$
(47)
The exact solutions at points $\bar{B}_{1}^{\pm}$ are always unstable.
However, for points $\bar{B}_{2}^{\pm}$ we find that
$e_{2}\left(\bar{B}_{2}^{\pm}\right)>0$ for $\lambda>\sqrt{7}-1$ which means
that points $\bar{B}_{2}^{\pm}$ are sources. The parameter for the equation of
state $w_{tot}\left(\mathbf{B}\right)$ is constraint as
$\frac{\sqrt{7}-3}{\sqrt{7+1}}<w_{tot}\left(B\right)<1$, while for
$\lambda=2$, $w_{tot}\left(\mathbf{B}\right)=0$ the exact solutions have the
scale factor $a\left(t\right)=a_{0}t^{\frac{2}{3}}$, while for $\lambda=4$,
$w_{tot}\left(\mathbf{B}\right)=\frac{1}{3}$, that is
$a\left(t\right)=a_{0}t^{\frac{1}{2}}$.
Furthermore, stationary point $\bar{C}_{1}$ is a source and describes the
radiation epoch, $w_{tot}\left(\bar{C}_{1}\right)=\frac{1}{3}$, on the other
hand, at point $\bar{C}_{2}$ the exact solution is that of de Sitter universe,
the point is real for $\lambda<0\mathbf{.}$Finally, points $\bar{D}_{1}^{\pm}$
points describe the unstable scaling solutions which describe the matter
dominated era, that is, $w_{tot}\left(\bar{D}_{1}^{\pm}\right)=0$.
In Figs. 8 and 9, the evolution of the physical variables
$\left\\{w_{tot},w_{\phi},w_{\psi},\Omega_{\phi},\Omega_{\psi}\right\\}$ is
presented for the specific model for $\lambda=-4$ and $\lambda=-2$ and for
different initial conditions for the integration of the dynamical system
(20)-(24). Recall that the de Sitter point $\bar{C}_{2}$ is a source; however,
it admits a four-dimensional stable manifold when $\mu\rightarrow 0$. We
observe that in the de Sitter point the physical parameters
$\Omega_{\phi},~{}\Omega_{\psi}$ are not zero which means that the all the
parts of the potential $V\left(\phi,\psi\right)$ contributes to the
cosmological fluid. The initial conditions have been considered such that to
describe a wide range of solutions and different behaviour. The large number
of stationary points is observed from the behaviour of $w_{tot}$, which has
various maxima before reach the de Sitter point. Similarly from the diagram of
$\left\\{\Omega_{\phi},~{}\Omega_{\psi}\right\\}$, we observe that there is a
alternation between the domination of the two fields.
Figure 8: Evolution of the physical variables
$\left\\{w_{tot},w_{\phi},w_{\psi},\Omega_{\phi},\Omega_{\psi}\right\\}$ for
numerical solutions of the field equations with
$\kappa=2,~{}\sigma=\frac{1}{2}$ and $\lambda=-4$. The plots are for different
initial conditions
$\left(x\left(0\right),y\left(0\right),z\left(0\right),u\left(0\right),\mu\left(0\right)\right)$
where $\mu\left(0\right)$ has been chosen to be near to zero, such that the de
Sitter point $\bar{C}_{2}$ to be an attractor. Figure 9: Evolution of the
physical variables
$\left\\{w_{tot},w_{\phi},w_{\psi},\Omega_{\phi},\Omega_{\psi}\right\\}$ for
numerical solutions of the field equations with
$\kappa=2,~{}\sigma=\frac{1}{2}$ and $\lambda=-2$. The plots are for different
initial conditions
$\left(x\left(0\right),y\left(0\right),z\left(0\right),u\left(0\right),\mu\left(0\right)\right)$
where $\mu\left(0\right)$ has been chosen to be near to zero, such that the de
Sitter point $\bar{C}_{2}$ to be an attractor.
Consider now the cosmographic parameters $q,~{}j$ and $s$ which are defined as
cowei
$q\left(x,y,z,u,\mu;\lambda,\kappa,\sigma\right)=-1-\frac{\dot{H}}{H^{2}}$
(48)
$j\left(x,y,z,u,\mu;\lambda,\kappa,\sigma\right)=\frac{\ddot{H}}{H^{3}}-3q-2$
(49)
$s\left(x,y,z,u,\mu;\lambda,\kappa,\sigma\right)=\frac{H^{\left(3\right)}}{H^{4}}+4j+3q(q+4)+6$
(50)
In Fig. 10 we present the evolution of the cosmographic parameters for the
application we considered in this example as also for additional values of the
free parameters $\left\\{\lambda,\kappa,\sigma\right\\}$, while all the plots
are for the same initial conditions. Here we present the qualitative evolution
of these parameters, however the cosmographic parameters as also the free
parameters of the theory can be constrained by the observations cob1 .
Figure 10: Qualitative evolution of the cosmographic parameters
$\left\\{q,j,s\right\\}$ for various values of the free parameters
$\left\\{\lambda,\kappa,\sigma\right\\}$. The plots of the first row are for
$\left\\{\lambda,\kappa,\sigma\right\\}=\left\\{-4,\pm 2,\frac{1}{2}\right\\}$
while the plots of the second row are for
$\left\\{\lambda,\kappa,\sigma\right\\}=\left\\{-2,\pm
2,\frac{1}{2}\right\\}$. From the figure we observe that in order the future
attractor to be a de Sitter point then $\kappa>0.$
In the following section we continue our analysis by presenting analytic
solutions for the model of our study.
## VI Analytic solution
We consider the point-like Lagrangian
$\mathcal{L}\left(a,\dot{a},\phi,\dot{\phi},\psi,\dot{\psi}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\left(\dot{\phi}^{2}+e^{\kappa\phi}\dot{\psi}^{2}\right)-a^{3}\left(V_{0}e^{\lambda\phi}+U_{0}\psi^{\frac{1}{\sigma}}e^{\kappa\phi}\right).$
(51)
Analytic solutions of form of Lagrangian (51) were presented before in 2sfand
. By using the results and the analysis of 2sfand we present an analytic
solutions for specific values of the parameters
$\left\\{\lambda,\kappa,\sigma\right\\}$ in order to support the results of
the previous section. Specifically for the free variables we select
$\left(\lambda,\kappa,\sigma\right)=\left(-\frac{\sqrt{6}}{2},-\frac{\sqrt{6}}{2},\frac{1}{2}\right)$.
These values are not random. In particular, from the results of 2sfand it
follows that for these specific values the field equations admit conservation
laws and they form a Liouville integrable dynamical system, such that the
field equations can be solved by quadratures.
In order to simplify the field equations and write the analytic solution by
using closed-form functions, we apply the point transformation
$a=\left(xz-\frac{3}{8}y^{2}\right)^{\frac{1}{3}}~{},~{}\phi=-2\sqrt{\frac{2}{3}}\ln\left(\frac{x}{\sqrt{\left(xz-\frac{3}{8}y^{2}\right)}}\right)~{},~{}\psi=\frac{y}{x}$
(52)
such that Lagrangian (51) is written as
$\mathcal{L}\left(x,\dot{x},y,\dot{y},z,\dot{z}\right)=-\frac{4}{3}\dot{x}\dot{z}-V_{0}x^{2}+\frac{1}{2}\dot{y}^{2}-U_{0}y^{2}.$
(53)
In the new coordinates the field equations are
$\ddot{x}=0~{},~{}\ddot{y}+2U_{0}y=0~{},~{}\ddot{z}-\frac{3}{2}V_{0}x=0,$ (54)
with constraint equation
$-\frac{4}{3}\dot{x}\dot{z}+V_{0}x^{2}+\frac{1}{2}\dot{y}^{2}+U_{0}y^{2}=0.$
(55)
Easily, we find the exact solution
$x=x_{1}t+x_{0}~{},~{}z=\frac{1}{4}V_{0}x_{1}t^{3}+\frac{3}{4}V_{0}x_{0}t^{2}+z_{1}t+z_{0}~{},$
(56)
$y\left(t\right)=y_{1}\cos\left(\sqrt{2U_{0}}t\right)+y_{2}\sin\left(\sqrt{2U_{0}}t\right)$
(57)
with constraint condition
$V_{0}x_{0}^{2}-\frac{4}{3}x_{1}z_{1}+U_{0}\left(y_{1}^{2}+y_{2}^{2}\right)$.
For $x_{0}=z_{0}=y_{1}=0$, the scale factor is written
$a\left(t\right)=\left(\frac{x_{1}}{4}V_{0}t^{4}+x_{1}z_{1}t^{2}-\frac{3}{8}\left(y_{2}\right)^{2}\sin\left(\sqrt{2U_{0}}t\right)\right)^{\frac{1}{3}}.$
It is easy to observe that the present analytic solution does not provide any
de Sitter point. That is in agreement with the result of the previous section,
since for $\lambda=\kappa$, the de Sitter point $C_{2}$ does not exist. For
more general solutions with expansion eras and de Sitter phases we refer the
reader to 2sfand .
## VII With a matter source
Let us assume now the presence of an additional pressureless matter source in
field equations with energy density $\rho_{m}$ and let us discuss the
existence of additional stationary points. For a pressureless fluid source the
dimensionless field equations (20)-(25) remain the same, while the constraint
equation (28) becomes
$\Omega_{m}=1-x^{2}-y^{2}-z^{2}-u^{2}$ (58)
where $\Omega_{m}=\frac{\rho_{m}}{3H^{2}}$, and $0\leq\Omega_{m}\leq 1$.
For this model, the stationary points found before exist and give
$\Omega_{m}=0$, while when $\Omega_{m}\neq 0$ the additional points exist
$E_{1}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\lambda},\sqrt{\frac{3}{2}}\frac{1}{\lambda},0,0,0\right)~{},~{}E_{2}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\kappa},0,0,\sqrt{\frac{3}{2}}\frac{1}{\kappa},0\right)$
(59)
$E_{3}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\kappa},0,z,\sqrt{\frac{3}{2}+\kappa^{2}z^{2}}\frac{1}{\kappa},0\right)$
(60)
Point $E_{1}$ is physically accepted when
$\left|\lambda\right|>\sqrt{\frac{3}{2}}$ and describes the tracking solution
with $\Omega_{m}\left(E_{1}\right)=1-\frac{3}{\lambda^{2}}$ where the field
$\phi$ mimics the ideal gas $\rho_{m},~{}$that is,
$w_{\phi}\left(E_{1}\right)=0,$ while the second field $\psi$ does not
contribute, i.e. $z\left(E_{1}\right)=u\left(E_{1}\right)=0$.
For $E_{2}$ we find
$\left(w_{tot}\left(E_{2}\right),w_{\phi}\left(E_{2}\right),w_{\psi}\left(E_{2}\right),\Omega_{\phi}\left(E_{2}\right),\Omega_{\psi}\left(E_{2}\right)\right)=\left(0,1,-1,\frac{3}{2\kappa^{2}},\frac{3}{2\kappa^{2}}\right)$,
which means that it is another tracking tracking solution with
$\Omega_{m}=1-\frac{3}{\kappa^{2}}$; the point is physically accepted when
$\left|\kappa\right|\geq\sqrt{\frac{3}{2}}$.
$E_{3}$ does not describe one point, but a family of points on the surface
$u\left(z\right)=$ $\sqrt{\frac{3}{2}+\kappa^{2}z^{2}}$ , for
$x\left(E_{3}\right)=-\sqrt{\frac{3}{2}}\frac{1}{\kappa},$
$y\left(E_{3}\right)=\mu\left(E_{3}\right)=0$. It describes a tracking
solution, that is $w_{tot}\left(E_{3}\right)=0$, with physical parameters
$\left(w_{tot}\left(E_{3}\right),w_{\phi}\left(E_{3}\right),w_{\psi}\left(E_{3}\right),\Omega_{\phi}\left(E_{3}\right),\Omega_{\psi}\left(E_{3}\right)\right)=\left(0,1,-\frac{3}{4+3\kappa^{2}z^{2}},\frac{3}{2\kappa^{2}},2z^{2}+\frac{3}{2\kappa^{2}}\right),$
(61)
while the point is physically accepted when
$\left|\kappa\right|\geq\sqrt{\frac{3}{2}}$ and
$\left|z\right|\leq\frac{1}{2}\sqrt{2-\frac{3}{\kappa^{2}}}$. When
$z\left(E_{3}\right)=0$, then $E_{3}$ reduces to $E_{2}$. What it is
important, to mention is that the stability analysis for all the previous
points changes, since we made use of the constraint equation (28).
## VIII Conclusions
In this work we performed a detailed study of the dynamics for a two scalar
field model with a mixed potential term known as Chiral model. The purpose of
our analysis was to study the cosmological evolution of that specific model as
also the cosmological viability of the model and which epochs of the
cosmological evolution can be described by the Chiral model.
For the scalar field potential we assumed that it is of the form
$V\left(\phi,\psi\right)=V_{0}e^{\lambda\phi}+U_{0}\psi^{\frac{1}{\sigma}}e^{\kappa\phi}$.
For this consideration and without assuming the existence of additional matter
source, we found four families of stationary points which provide nine
different cosmological solutions. Eight of the cosmological solutions are
scaling solutions which describe spacetimes with a a perfect fluid with a
constant equation of state parameter $w\left(P\right)$. One of the scaling
solutions describes a universe with a stiff matter, $w\left(P\right)=1$,
another scaling solution correspond to a universe with a pressureless fluid
source, $w\left(P\right)=0$, while for the rest six scaling solutions
$w\left(P\right)=w\left(P,\lambda,\kappa,\sigma\right)$, which can describe
accelerated eras for for specific values of the free parameters
$\left\\{\lambda,\kappa,\sigma\right\\}$. Moreover, the ninth exact
cosmological solution which was found from the analysis of the stationary
points describes a de Sitter universe.
As far as the stability of the exact solutions at the stationary points is
concerned, seven of the points are always unstable. while only the set of the
points $B_{2}^{\pm}~{}$can be stable. Point $C_{2}$ which describes the de
Sitter universe, has one eigenvalue negative while the rest of the eigenvalues
are always negative. Consequently, according to the center manifold theorem we
found the internal surface where the point $C_{2}$ is a source. Moreover, in
the presence of additional matter source only additional tracking solutions
follow, similarly to the quintessence model.
From the above results we observe that the specific Chiral cosmological model
can describe the major eras of the cosmological history, that is, the late
expansion era, an unstable matter dominated era, and two scaling solutions
describe the radiation dominated era and the early acceleration epoch,
therefore, the model in terms of dynamics it is cosmologically viable.
From this analysis it is clear that the Chiral cosmological model can be used
as dark energy candidate. In a future work we plan to apply the cosmological
observations to constrain the theory.
## References
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|
2024-09-04T02:54:59.224615 | 2020-02-25T14:44:12 | 2003.05370 | {
"authors": "Ernesto Jim\\'enez-Ruiz, Asan Agibetov, Jiaoyan Chen, Matthias Samwald,\n Valerie Cross",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26168",
"submitter": "Ernesto Jimenez-Ruiz",
"url": "https://arxiv.org/abs/2003.05370"
} | arxiv-papers | # Dividing the Ontology Alignment Task
with Semantic Embeddings and Logic-based Modules††Accepted to the 24th
European Conference on Artificial Intelligence (ECAI 2020)
Ernesto Jiménez-Ruiz City, University of London, UK, email: ernesto.jimenez-
<EMAIL_ADDRESS>and Department of Informatics, University of Oslo,
NorwaySection for Artificial Intelligence and Decision Support, Medical
University of Vienna, Vienna, AustriaDepartment of Computer Science,
University of Oxford, UKMiami University, Oxford, OH 45056, United States Asan
Agibetov Section for Artificial Intelligence and Decision Support, Medical
University of Vienna, Vienna, AustriaDepartment of Computer Science,
University of Oxford, UKMiami University, Oxford, OH 45056, United States
Jiaoyan Chen Department of Computer Science, University of Oxford, UKMiami
University, Oxford, OH 45056, United States Matthias Samwald3 Miami
University, Oxford, OH 45056, United States Valerie Cross Miami University,
Oxford, OH 45056, United States
###### Abstract
Large ontologies still pose serious challenges to state-of-the-art ontology
alignment systems. In this paper we present an approach that combines a neural
embedding model and logic-based modules to accurately divide an input ontology
matching task into smaller and more tractable matching (sub)tasks. We have
conducted a comprehensive evaluation using the datasets of the Ontology
Alignment Evaluation Initiative. The results are encouraging and suggest that
the proposed method is adequate in practice and can be integrated within the
workflow of systems unable to cope with very large ontologies.
## 1 Introduction
The problem of (semi-)automatically computing an alignment between
independently developed ontologies has been extensively studied in the last
years. As a result, a number of sophisticated ontology alignment systems
currently exist [44, 15].222Ontology matching surveys and approaches:
http://ontologymatching.org/ The Ontology Alignment Evaluation
Initiative333OAEI evaluation campaigns: http://oaei.ontologymatching.org/
(OAEI) [3, 4] has played a key role in the benchmarking of these systems by
facilitating their comparison on the same basis and the reproducibility of the
results. The OAEI includes different tracks organised by different research
groups. Each track contains one or more matching tasks involving small-size
(e.g., conference), medium-size (e.g., anatomy), large (e.g., phenotype) or
very large (e.g., largebio) ontologies. Some tracks only involve matching at
the terminological level (e.g., concepts and properties) while other tracks
also expect an alignment at the assertional level (i.e., instance data).
Large ontologies still pose serious challenges to ontology alignment systems.
For example, several systems participating in the _largebio track_ were unable
to complete the largest tasks during the latest OAEI campaigns.444Largebio
track: http://www.cs.ox.ac.uk/isg/projects/SEALS/oaei/ These systems typically
use advanced alignment methods and are able to cope with small and medium size
ontologies with competitive results, but fail to complete large tasks in a
given time frame or with the available resources such as memory.
There have been several efforts in the literature to divide the ontology
alignment task (e.g., [20, 22]). These approaches, however, have not been
successfully evaluated with very large ontologies, failing to scale or
producing partitions of the ontologies leading to information loss [42]. In
this paper we propose a novel method to accurately divide the matching task
into several independent, smaller and manageable (sub)tasks, so as to scale
systems that cannot cope with very large ontologies.555A preliminary version
of this work has been published in arXiv [25] and in the Ontology Matching
workshop [26]. Unlike state-of-the-art approaches, our method: (i) preserves
the coverage of the relevant ontology alignments while keeping manageable
matching subtasks; (ii) provides a formal notion of matching subtask and
semantic context; (iii) uses neural embeddings to compute an accurate
division by learning semantic similarities between words and ontology entities
according to the ontology alignment task at hand; (iv) computes self-
contained (logical) modules to guarantee the inclusion of the (semantically)
relevant information required by an alignment system; and (v) has been
successfully evaluated with very large ontologies.
## 2 Preliminaries
A _mapping_ (also called match) between entities666In this work we accept any
input ontology in the OWL 2 language [18]. We refer to (OWL 2) concepts,
properties and individuals as entities. of two ontologies $\mathcal{O}_{1}$
and $\mathcal{O}_{2}$ is typically represented as a 4-tuple $\langle
e_{1},\allowbreak e_{2},\allowbreak r,\allowbreak c\rangle$ where $e_{1}$ and
$e_{2}$ are entities of $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, respectively;
$r$ is a semantic relation, typically one of
$\\{\sqsubseteq,\sqsupseteq,\equiv\\}$; and $c$ is a confidence value,
usually, a real number within the interval $\left(0,1\right]$. For simplicity,
we refer to a mapping as a pair $\langle e_{1},\allowbreak e_{2}\rangle$. An
ontology _alignment_ is a set of mappings $\mathcal{M}$ between two ontologies
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$.
An ontology _matching task_ $\mathcal{M}\mathcal{T}$ is composed of a pair of
ontologies $\mathcal{O}_{1}$ (typically called source) and $\mathcal{O}_{2}$
(typically called target) and possibly an associated _reference alignment_
$\mathcal{M}^{RA}$. The objective of a matching task is to discover an
overlapping of $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ in the form of an
alignment $\mathcal{M}$. The _size_ or _search space_ of a matching task is
typically bound to the size of the Cartesian product between the entities of
the input ontologies: $\lvert Sig(\mathcal{O}_{1})\rvert\times\lvert
Sig(\mathcal{O}_{2})\rvert$, where $Sig(\mathcal{O})$ denotes the signature
(i.e., entities) of $\mathcal{O}$ and $\lvert\cdot\lvert$ denotes the size of
a set.
An ontology _matching system_ is a program that, given as input a matching
task $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$,
generates an ontology alignment $\mathcal{M}^{S}$.777Typically automatic,
although there are systems that also allow human interaction [32]. The
standard evaluation measures for an alignment $\mathcal{M}^{S}$ are
_precision_ (P), _recall_ (R) and _f-measure_ (F) computed against a reference
alignment $\mathcal{M}^{RA}$ as follows:
$P=\frac{\lvert\mathcal{M}^{S}\cap\mathcal{M}^{RA}\rvert}{\lvert\mathcal{M}^{S}\rvert},~{}R=\frac{\lvert\mathcal{M}^{S}\cap\mathcal{M}^{RA}\rvert}{\lvert\mathcal{M}^{RA}\rvert},~{}F=2\cdot\frac{P\cdot
R}{P+R}$ (1)
Figure 1: Pipeline to divide a given matching task
$\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$.
### 2.1 Problem definition and quality measures
We denote _division_ of an ontology matching task $\mathcal{M}\mathcal{T}$,
composed by the ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, as the
process of finding $n$ matching subtasks
$\mathcal{M}\mathcal{T}_{i}=\langle\mathcal{O}_{1}^{i},\mathcal{O}_{2}^{i}\rangle$
(with $i$=$1$,…,$n$), where $\mathcal{O}_{1}^{i}\subset\mathcal{O}_{1}$ and
$\mathcal{O}_{2}^{i}\subset\mathcal{O}_{2}$.
Size of the division. The size of each matching subtask is smaller than the
original task and thus reduces the search space. Let
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}_{1},\ldots,\mathcal{M}\mathcal{T}_{n}\\}$
be the division of a matching task $\mathcal{M}\mathcal{T}$ into $n$ subtasks.
The _size ratio_ of the subtasks $\mathcal{M}\mathcal{T}_{i}$ and
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ with respect to the original
matching task size is computed as follows:
$\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})=\frac{\lvert
Sig(\mathcal{O}_{1}^{i})\rvert\times\lvert
Sig(\mathcal{O}_{2}^{i})\rvert}{\lvert Sig(\mathcal{O}_{1})\rvert\times\lvert
Sig(\mathcal{O}_{2})\rvert}$ (2)
$\mathsf{SizeRatio}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M}\mathcal{T})=\sum_{i=1}^{n}\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})$
(3)
The ratio
$\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})$ is
less than $1.0$ while the aggregation
$\sum_{i=1}^{n}\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})$,
being $n$ the number of matching subtasks, can be greater than $1.0$ as
matching subtasks depend on the division technique and may overlap.
Alignment coverage. The division of the matching task aims at preserving the
target outcomes of the original matching task. The _coverage_ is calculated
with respect to a relevant alignment $\mathcal{M}$, possibly the reference
alignment $\mathcal{M}^{RA}$ of the matching task if it exists, and indicates
whether that alignment can still be (potentially) discovered with the matching
subtasks. The formal notion of coverage is given in Definitions 1 and 2.
###### Definition 1 (Coverage of a matching task)
Let $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$ be
a matching task and $\mathcal{M}$ an alignment. We say that a mapping
$m=\langle e_{1},\allowbreak e_{2}\rangle\in\mathcal{M}$ is covered by the
matching task if $e_{1}\in Sig(\mathcal{O}_{1})$ and $e_{2}\in
Sig(\mathcal{O}_{2})$. The coverage of $\mathcal{M}\mathcal{T}$ w.r.t.
$\mathcal{M}$ (denoted as
$\mathsf{Coverage}(\mathcal{M}\mathcal{T},\mathcal{M})$) represents the set of
mappings $\mathcal{M}^{\prime}\subseteq\mathcal{M}$ covered by
$\mathcal{M}\mathcal{T}$.
###### Definition 2 (Coverage of the matching task division)
Let
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}_{1},\ldots,\mathcal{M}\mathcal{T}_{n}\\}$
be the result of dividing a matching task $\mathcal{M}\mathcal{T}$ and
$\mathcal{M}$ an alignment. We say that a mapping $m\in\mathcal{M}$ is covered
by $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ if $m$ is at least covered by
one of the matching subtask $\mathcal{M}\mathcal{T}_{i}$ (with $i$=$1$,…,$n$)
as in Definition 1. The coverage of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$
w.r.t. $\mathcal{M}$ (denoted as
$\mathsf{Coverage}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M})$)
represents the set of mappings $\mathcal{M}^{\prime}\subseteq\mathcal{M}$
covered by $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$. The coverage is given
as a ratio with respect to the (covered) alignment:
$\mathsf{CoverageRatio}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M})=\frac{\lvert\mathsf{Coverage}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M})\rvert}{\lvert\mathcal{M}\rvert}$
(4)
## 3 Methods
In this section we present our approach to compute a division
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}_{1},\ldots,\mathcal{M}\mathcal{T}_{n}\\}$
given a matching task
$\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$ and the
number of target subtasks $n$. We rely on locality ontology modules to extract
self-contained modules of the input ontologies. The module extraction and task
division is tailored to the ontology alignment task at hand by embedding the
contextual semantics of a (combined) inverted index of the ontologies in the
matching task.
Figure 1 shows an overview of our approach. (i) The ontologies
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are indexed using the lexical index
LexI (see Section 3.2); (ii) LexI is divided into clusters based on the
semantic embeddings of its entries (see Section 3.4); (iii) entries in those
clusters derive potential mapping sets (see Section 3.3); and (iv) the context
of these mapping sets lead to matching subtasks (see Sections 3.1 and 3.3).
Next, we elaborate on the methods behind these steps.
### 3.1 Locality modules and context
Logic-based module extraction techniques compute ontology fragments that
capture the meaning of an input signature (e.g., set of entities) with respect
to a given ontology. That is, a module contains the context (i.e., sets of
_semantically related_ entities) of the input signature. In this paper we rely
on bottom-locality modules [13, 29], which will be referred to as locality-
modules or simply as modules. These modules include the ontology axioms
required to describe the entities in the signature. Locality-modules compute
self-contained ontologies and are tailored to tasks that require reusing a
fragment of an ontology. Please refer to [13, 29] for further details.
Locality-modules play an key role in our approach as they provide the context
for the entities in a given mapping or set of mappings as formally presented
in Definition 3.
###### Definition 3 (Context of a mapping and an alignment)
Let $m=\langle e_{1},\allowbreak e_{2}\rangle$ be a mapping between two
ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. We define the context of
$m$ (denoted as $\mathsf{Context}(m,\mathcal{O}_{1},\mathcal{O}_{2})$) as a
pair of locality modules $\mathcal{O}_{1}^{\prime}\subseteq\mathcal{O}_{1}$
and $\mathcal{O}_{2}^{\prime}\subseteq\mathcal{O}_{2}$, where
$\mathcal{O}_{1}^{\prime}$ and $\mathcal{O}_{2}^{\prime}$ include the
semantically related entities to $e_{1}$ and $e_{2}$, respectively. Similarly,
the _context_ for an alignment $\mathcal{M}$ between two ontologies
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ is denoted as
$\mathsf{Context}(\mathcal{M},\mathcal{O}_{1},\mathcal{O}_{2})=\langle\mathcal{O}_{1}^{\prime},\mathcal{O}_{2}^{\prime}\rangle$,
where $\mathcal{O}_{1}^{\prime}$ and $\mathcal{O}_{2}^{\prime}$ are modules
including the semantically related entities for the entities $e_{1}\in
Sig(\mathcal{O}_{1})$ and $e_{2}\in Sig(\mathcal{O}_{2})$ in each mapping
$m=\langle e_{1},\allowbreak e_{2}\rangle\in\mathcal{M}$.
Intuitively, as the context of an alignment (i.e.,
$\mathsf{Context}(\mathcal{M},\mathcal{O}_{1},\mathcal{O}_{2})=\langle\mathcal{O}_{1}^{\prime},\mathcal{O}_{2}^{\prime}\rangle$)
semantically characterises the entities involved in that alignment, a matching
task $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$
can be reduced to the task
$\mathcal{M}\mathcal{T}^{\mathcal{M}}_{\mathcal{O}_{1}\text{-}\mathcal{O}_{2}}=\langle\mathcal{O}_{1}^{\prime},\mathcal{O}_{2}^{\prime}\rangle$
without information loss in terms of finding $\mathcal{M}$ (i.e.,
$\mathsf{Coverage}(\mathcal{M}\mathcal{T}^{\mathcal{M}}_{\mathcal{O}_{1}\text{-}\mathcal{O}_{2}},\mathcal{M})=\mathcal{M}$).
For example, in the small OAEI _largebio_ tasks [3, 4] systems are given the
context of the reference alignment as a (reduced) matching task (e.g.,
$\mathcal{M}\mathcal{T}^{RA}_{\text{fma-
nci}}=\mathsf{Context}(\mathcal{M}^{RA}_{\text{fma-nci}},\
\mathcal{O}_{\text{FMA}},\mathcal{O}_{\text{NCI}})=\langle\mathcal{O}_{\text{FMA}}^{\prime},\mathcal{O}_{\text{NCI}}^{\prime}\rangle$),
instead of the whole FMA and NCI ontologies.
Table 1: Inverted lexical index LexI. For readability, index values have been split into elements of $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. ‘-’ indicates that the ontology does not contain entities for that entry. | # | Index key | Index value
---|---|---
Entities $\mathcal{O}_{1}$ | Entities $\mathcal{O}_{2}$
1 | $\\{$ disorder $\\}$ | $\mathcal{O}_{1}$:Disorder_of_pregnancy, $\mathcal{O}_{1}$:Disorder_of_stomach | $\mathcal{O}_{2}$:Pregnancy_Disorder
2 | $\\{$ disorder, pregnancy $\\}$ | $\mathcal{O}_{1}$:Disorder_of_pregnancy | $\mathcal{O}_{2}$:Pregnancy_Disorder
3 | $\\{$ carcinoma, basaloid $\\}$ | $\mathcal{O}_{1}$:Basaloid_carcinoma | $\mathcal{O}_{2}$:Basaloid_Carcinoma, $\mathcal{O}_{2}$:Basaloid_Lung_Carcinoma
4 | $\\{$ follicul, thyroid, carcinom $\\}$ | $\mathcal{O}_{1}$:Follicular_thyroid_carcinoma | $\mathcal{O}_{2}$:Follicular_Thyroid_carcinoma
5 | $\\{$ hamate, lunate $\\}$ | $\mathcal{O}_{1}$:Lunate_facet_of_hamate | -
### 3.2 Indexing the ontology vocabulary
We rely on a semantic inverted index (we will refer to this index as LexI).
This index maps sets of words to the entities where these words appear. LexI
encodes the labels of all entities of the input ontologies $\mathcal{O}_{1}$
and $\mathcal{O}_{2}$, including their lexical variations (e.g., preferred
labels, synonyms), in the form of _key-value_ pairs where the key is a set of
words and the value is a set of entities such that the set of words of the key
appears in (one of) the entity labels. Similar indexes are commonly used in
information retrieval applications [11], Entity Resolution systems [40], and
also exploited in ontology alignment systems (e.g., LogMap [27], ServOMap [14]
and AML [16]) to reduce the search space and enhance the matching process.
Table 1 shows a few example entries of LexI for two input ontologies.
LexI is created as follows. (i) Each label associated to an ontology entity
is split into a set of words; for example, the label “Lunate facet of hamate”
is split into the set {“lunate”, “facet”, “of”, “hamate”}. (ii) Stop-words
are removed from the set of words. (iii) Stemming techniques are applied to
each word (i.e., {“lunat”, “facet”, “hamat”}). (iv) Combinations of subsets
of words also serve as keys in LexI; for example, {“lunat”, “facet”},
{“hamat”, “lunat”} and so on.888In order to avoid a combinatorial blow-up, the
number of computed subsets of words is limited. (v) Entities leading to the
same (sub)set of words are associated to the same key in LexI, for example
{“disorder”} is associated with three entities. Finally, (vi) entries in LexI
pointing to entities of only one ontology or associated to a number of
entities larger than $\alpha$ are not considered.999In the experiments we used
$\alpha=60$. Note that a single entity label may lead to several entries in
LexI, and each entry in LexI points to one or more entities.
### 3.3 Covering matching subtasks
Each entry (i.e., a _key-value_ pair) in LexI is a source of candidate
mappings. For instance, the example in Table 1 suggests that there is a
candidate mapping
$m=\langle\mathsf{\mathcal{O}_{1}\negthickspace:\negthickspace
Disorder\\_of\\_stomach},\allowbreak\mathsf{\mathcal{O}_{2}\negthickspace:\negthickspace
Pregnancy\\_disorder}\rangle$ since these entities are associated to the _{
“disorder”}_ entry in LexI. These mappings are not necessarily correct but
will link lexically-related entities, that is, those entities sharing at least
one word among their labels (e.g., “disorder”). Given a subset of entries or
rows of LexI (i.e., $l\subseteq\textsf{LexI}$), the function
$\mathsf{Mappings}(l)=\mathcal{M}^{l}$ provides the set of mappings derived
from $l$. We refer to the set of all (potential) mappings suggested by LexI
(i.e., $\mathsf{Mappings}(\textsf{LexI})$) as $\mathcal{M}^{\textsf{LexI}}$.
$\mathcal{M}^{\textsf{LexI}}$ represents a manageable subset of the Cartesian
product between the entities of the input ontologies. For example, LexI
suggest around $2\times 10^{4}$ potential mappings for the matching task
$\mathcal{M}\mathcal{T}_{\text{fma-
nci}}=\langle\mathcal{O}_{\text{FMA}},\mathcal{O}_{\text{NCI}}\rangle$, while
the Cartesian product between $\mathcal{O}_{\text{FMA}}$ and
$\mathcal{O}_{\text{NCI}}$ involves more than $5\times 10^{9}$ mappings.
Since standard ontology alignment systems rarely discover mappings outside
$\mathcal{M}^{\textsf{LexI}}$, the context of $\mathcal{M}^{\textsf{LexI}}$
(recall Definition 3) can be seen as a reduced matching task
$\mathcal{M}\mathcal{T}^{\textsf{LexI}}=\mathsf{Context}(\mathcal{M}^{\textsf{LexI}},\mathcal{O}_{1},\mathcal{O}_{2})=\langle\mathcal{O}_{1}^{\textsf{LexI}},\mathcal{O}_{2}^{\textsf{LexI}}\rangle$
of the original task
$\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$.
However, the modules $\mathcal{O}_{1}^{\textsf{LexI}}$ and
$\mathcal{O}_{2}^{\textsf{LexI}}$, although smaller than $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$, can still be challenging for many ontology matching
systems. A solution is to divide or cluster the entries in LexI to lead to
several tasks involving smaller ontology modules.
###### Definition 4 (Matching subtasks from LexI)
Let $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$ be
a matching task, _LexI_ the inverted index of the ontologies $\mathcal{O}_{1}$
and $\mathcal{O}_{2}$, and $\\{l_{1},\ldots,l_{n}\\}$ a set of $n$ clusters of
entries in _LexI_. We denote the set of matching subtasks from _LexI_ as
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{1},\ldots,\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{n}\\}$
where each cluster $l_{i}$ leads to the matching subtask
$\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i}=\langle\mathcal{O}_{1}^{i},\mathcal{O}_{2}^{i}\rangle$,
such that $\mathsf{Mappings}(l_{i})=\mathcal{M}^{\textsf{LexI}}_{i}$ is the
set of mappings suggested by the _LexI_ entries in $l_{i}$ (i.e., _key-value_
pairs) and $\mathcal{O}_{1}^{i}$ and $\mathcal{O}_{2}^{i}$ represent the
context of $\mathcal{M}^{\textsf{LexI}}_{i}$ w.r.t. $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$.
Quality of the matching subtasks. The matching subtasks in Definition 4 rely
on LexI and the notion of context, thus it is expected that the tasks in
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ will cover most of the mappings
$\mathcal{M}^{S}$ that a matching system can compute, that is
$\mathsf{CoverageRatio}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M}^{S})$
will be close to $1.0$. Furthermore, the use of locality modules to compute
the context guarantees the extraction of matching subtasks that are suitable
to ontology alignment systems in terms of preservation of the logical
properties of the given signature.
Intuitively each cluster of LexI will lead to a smaller matching task
$\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i}$ (with respect to both
$\mathcal{M}\mathcal{T}^{\textsf{LexI}}$ and $\mathcal{M}\mathcal{T}$) in
terms of search space. Hence
$\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i},\mathcal{M}\mathcal{T})$
will be smaller than $1.0$. The overall aggregation of ratios (cf. Equation 3)
depends on the clustering strategy of the entries in LexI and it is also
expected to be smaller than $1.0$.
Reducing the search space in each matching subtask
$\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i}$ has the potential of enabling the
evaluation of systems that cannot cope with the original matching task
$\mathcal{M}\mathcal{T}$ in a given time-frame or with (limited) computational
resources.
Table 2: Matching tasks. AMA: Adult Mouse Anatomy. DOID: Human Disease Ontology. FMA: Foundational Model of Anatomy. HPO: Human Phenotype Ontology. MP: Mammalian Phenotype. NCI: National Cancer Institute Thesaurus. NCIA: Anatomy fragment of NCI. ORDO: Orphanet Rare Disease Ontology. SNOMED CT: Systematized Nomenclature of Medicine – Clinical Terms. Phenotype ontologies downloaded from BioPortal. For all tracks we use the consensus with vote=3 as system mappings $\mathcal{M}^{S}$. The Phenotype track does not have a gold standard so a consensus alignment with vote=2 is used as reference. OAEI track | Source of $\mathcal{M}^{RA}$ | Source of $\mathcal{M}^{S}$ | Task | Ontology | Version | Size (classes)
---|---|---|---|---|---|---
Anatomy | Manually created [10] | Consensus (vote=3) | AMA-NCIA | AMA | v.2007 | 2,744
NCIA | v.2007 | 3,304
Largebio | UMLS-Metathesaurus [28] | Consensus (vote=3) | FMA-NCI | FMA | v.2.0 | 78,989
FMA-SNOMED | NCI | v.08.05d | 66,724
SNOMED-NCI | SNOMED CT | v.2009 | 306,591
Phenotype | Consensus alignment (vote=2) [21] | Consensus (vote=3) | HPO-MP | HPO | v.2016 | 11,786
MP | v.2016 | 11,721
DOID-ORDO | DOID | v.2016 | 9,248
ORDO | v.2016 | 12,936
### 3.4 Semantic embeddings
We use a _semantic embedding_ approach to identify, given $n$, a set of
clusters of entries $\\{l_{1},\ldots,l_{n}\\}$ from LexI. As in Definition 4,
these clusters lead to the set of matching subtasks
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{1},\ldots,\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{n}\\}$.
The _semantic embeddings_ aim at representing into the same (vector) space the
features about the relationships among words and ontology entities that occur
in LexI. Hence, words and entities that belong to similar semantic contexts
will typically have similar vector representations.
Embedding model. Our approach currently relies on the StarSpace
toolkit101010StarSpace: https://github.com/facebookresearch/StarSpace and its
neural embedding model [49] to learn _embeddings_ for the words and ontology
entities in LexI. We adopt the _TagSpace_ [48] training setting of StarSpace.
Applied to our setting, StarSpace learns associations between a set of words
(i.e., keys in LexI) and a set of relevant ontology entities (i.e., values in
LexI). The StarSpace model is trained by assigning a $d$-dimensional vector to
each of the relevant features (e.g., the individual words and the ontology
entities in LexI). Ultimately, the look-up matrix (the matrix of embeddings -
latent vectors) is learned by minimising the loss function in Equation 5.
$\\!\sum_{\begin{subarray}{c}(w,e)\in E^{+},\\\ e^{-}\in
E^{-}\end{subarray}}L^{batch}(sim(\bm{v}_{w},\bm{v}_{e}),sim(\bm{v}_{w},\bm{v}_{e_{1}^{-}}),\ldots,\\\
sim(\bm{v}_{w},\bm{v}_{e_{j}^{-}}))$ (5)
In this loss function we compare positive samples with negative samples. Hence
we need to indicate the generator of positive pairs $(w,e)\in E^{+}$ (in our
setting those are _word-entity_ pairs from LexI) and the generator of negative
entries $e^{-}\in E^{-}$ (in our case we sample from the list of entities in
the values of LexI). StarSpace follows the strategy by Mikolov et al. [36] and
selects a random subset of $j$ negative examples for each batch update. Note
that we tailor the generators to the alignment task by sampling from LexI. The
similarity function $sim$ operates on $d$-dimensional vectors (e.g.,
$\bm{v}_{w}$, $\bm{v}_{e}$ and $\bm{v}_{e}^{-}$), in our case we use the
standard dot product in Euclidean space.
Clustering strategy. The semantic embedding of each entry
$\varepsilon=(K,V)\in$ LexI is calculated by concatenating (i) the mean
vector representation of the vectors associated to each word in the key $K$,
with (ii) the mean vector of the vectors of the ontology entities in the
value $V$, as in Equation 6, where $\oplus$ represents the concatenation of
two vectors, $\bm{v}_{w}$ and $\bm{v}_{e}$ represents $d$-dimensional vector
embeddings learnt by StarSpace, and $\bm{v}_{\varepsilon}$ is a
($2*d$)-dimension vector.
$\bm{v}_{\varepsilon}=\frac{1}{|K|}\sum_{w\in
K}\bm{v}_{w}\oplus\frac{1}{|V|}\sum_{e\in V}\bm{v}_{e}$ (6)
Based on the embeddings $\bm{v}_{\varepsilon}$ we then perform standard
clustering with the K-means algorithm to obtain the clusters of LexI entries
$\\{l_{1},\ldots,l_{n}\\}$. For example, following our approach, in the
example of Table 1 entries in rows $1$ and $2$ (respectively $3$ and $4$)
would belong to the same cluster.
Suitability of the embedding model. Although we could have followed other
embedding strategies, we advocated to learn new entity embeddings with
StarSpace for the following reasons: (i) ontologies, particularly in the
biomedical domain, may bring specialised vocabulary that is not fully covered
by precomputed word embeddings; (ii) to embed not only words but also
concepts of both ontologies; and (iii) to obtain embeddings tailored to the
ontology alignment task (i.e., to learn similarities among words and concepts
dependant on the task). StarSpace provides the required functionalities to
embed the semantics of LexI and identify accurate clusters. Precise clusters
will lead to smaller matching tasks, and thus, to a reduced global size of the
computed division of the matching task
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ (cf. Equation 3).
## 4 Evaluation
In this section we provide empirical evidence to support the suitability of
the proposed method to divide the ontology alignment task. We rely on the
datasets of the Ontology Alignment Evaluation Initiative (OAEI) [3, 4], more
specifically, on the matching tasks provided in the _anatomy_ , _largebio_ and
_phenotype_ tracks. Table 2 provides an overview of these OAEI tasks and the
related ontologies and mapping sets.
The methods have been implemented in Java111111Java codes:
https://github.com/ernestojimenezruiz/logmap-matcher and Python121212Python
codes: https://github.com/plumdeq/neuro-onto-part (neural embedding strategy),
tested on a Ubuntu Laptop with an Intel Core i9-8950HK<EMAIL_ADDRESS>and
allocating up to $25Gb$ of RAM. Datasets, matching subtasks, computed mappings
and other supporting resources are available in the _Zenodo_ repository [24].
For all of our experiments we used the following StarSpace hyperparameters:
-trainMode 0 -similarity dot --epoch 100 --dim 64.
(a) $\mathsf{CoverageRatio}$ of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$
over $\mathcal{M}^{RA}$
(b) $\mathsf{CoverageRatio}$ of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$
over $\mathcal{M}^{S}$
(c) $\mathsf{SizeRatio}$ of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$
(d) Module sizes of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ for FMA-NCI
Figure 2: Quality measures of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ with
respect to the number of matching subtasks $n$.
### 4.1 Adequacy of the division approach
We have evaluated the adequacy of our division strategy in terms of coverage
(as in Equation 4) and size (as in Equation 3) of the resulting division
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ for each of the matching task in
Table 2.
Coverage ratio. Figures 2(a) and 2(b) shows the coverage of the different
divisions $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ with respect to the
reference alignment and system computed mappings, respectively. As system
mappings we have used the consensus alignment with vote=3, that is, mappings
that have been voted by at least $3$ systems in the last OAEI campaigns. The
overall coverage results are encouraging: (i) the divisions
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ cover over $94\%$ of the reference
alignments for all tasks, with the exception of the SNOMED-NCI case where
coverage ranges from $0.94$ to $0.90$; (ii) when considering system mappings,
the coverage for all divisions is over $0.98$ with the exception of AMA-NCIA,
where it ranges from $0.956$ to $0.974$; (iii) increasing the number of
divisions $n$ tends to slightly decrease the coverage in some of the test
cases, this is an expected behaviour as the computed divisions include
different semantic contexts (i.e., locality modules) and some relevant
entities may fall out the division; finally (iv) as shown in [42], the
results in terms of coverage of state-of-the-art partitioning methods (e.g.,
[22, 20]) are very low for the OAEI _largebio_ track ($0.76$, $0.59$ and
$0.67$ as the best results for FMA-NCI, FMA-SNOMED and SNOMED-NCI,
respectively), thus, making the obtained results even more valuable.
Size ratio. The results in terms of the size (i.e., search space) of the
selected divisions $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ are presented in
Figure 2(c). The search space is improved with respect to the original
$\mathcal{M}\mathcal{T}$ for all the cases, getting as low as $5\%$ of the
original matching task size for the FMA-NCI and FMA-SNOMED cases. The gain in
the reduction of the search space gets relatively stable after a given
division size; this result is expected since the context provided by locality
modules ensures modules with the necessary semantically related entities. The
scatter plot in Figure 2(d) visualise the size of the source modules against
the size of the target modules for the FMA-NCI matching subtasks with
divisions of size $n\in\\{5,20,50,100\\}$. For instance, the (blue) circles
represent points $\big{(}\lvert Sig(\mathcal{O}_{1}^{i})\rvert,\lvert
Sig(\mathcal{O}_{2}^{i})\rvert\big{)}$ being $\mathcal{O}_{1}^{i}$ and
$\mathcal{O}_{2}^{i}$ the source and target modules (with $i$=$1$,…,$5$) in
the matching subtasks of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{5}$. It can be
noted that, on average, the size of source and target modules decreases as the
size of the division increases. For example, the largest task in
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{20}$ is represented in point
$(6754,9168)$, while the largest task in
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{100}$ is represented in point
$(2657,11842)$.
Table 3: Evaluation of systems that failed to complete OAEI tasks in the 2015-2018 campaigns. Times reported in seconds (s). Tool | Task | Matching | Performance measures | Computation times (s)
---|---|---|---|---
subtasks | P | R | F | Min | Max | Total
MAMBA (v.2015) | AMA-NCIA | 5 | 0.870 | 0.624 | 0.727 | 73 | 785 | 1,981
10 | 0.885 | 0.623 | 0.731 | 41 | 379 | 1,608
50 | 0.897 | 0.623 | 0.735 | 8 | 154 | 1,377
FCA-Map (v.2016) | FMA-NCI | 20 | 0.656 | 0.874 | 0.749 | 39 | 340 | 2,934
50 | 0.625 | 0.875 | 0.729 | 19 | 222 | 3,213
FMA-SNOMED | 50 | 0.599 | 0.251 | 0.354 | 6 | 280 | 3,455
100 | 0.569 | 0.253 | 0.350 | 5 | 191 | 3,028
SNOMED-NCI | 150 | 0.704 | 0.629 | 0.664 | 5 | 547 | 16,822
| 200 | 0.696 | 0.630 | 0.661 | 5 | 395 | 16,874
SANOM (v.2017) | FMA-NCI | 20 | 0.475 | 0.720 | 0.572 | 40 | 1,467 | 9,374
50 | 0.466 | 0.726 | 0.568 | 15 | 728 | 7,069
FMA-SNOMED | 100 | 0.145 | 0.210 | 0.172 | 3 | 1,044 | 13,073
150 | 0.143 | 0.209 | 0.170 | 3 | 799 | 10,814
POMap++ (v.2018) | FMA-NCI | 20 | 0.697 | 0.732 | 0.714 | 24 | 850 | 5,448
50 | 0.701 | 0.748 | 0.724 | 11 | 388 | 4,041
FMA-SNOMED | 50 | 0.520 | 0.209 | 0.298 | 4 | 439 | 5,879
100 | 0.522 | 0.209 | 0.298 | 3 | 327 | 4,408
ALOD2vec (v.2018) | FMA-NCI | 20 | 0.697 | 0.813 | 0.751 | 115 | 2,141 | 13,592
50 | 0.698 | 0.813 | 0.751 | 48 | 933 | 12,162
FMA-SNOMED | 100 | 0.702 | 0.183 | 0.29 | 9 | 858 | 12,688
150 | 0.708 | 0.183 | 0.291 | 7 | 581 | 10,449
Computation times. The time to compute the divisions of the matching task is
tied to the number of locality modules to extract, which can be computed in
polynomial time relative to the size of the input ontology [13]. The creation
of LexI does not add an important overhead, while the training of the neural
embedding model ranges from $21s$ in AMA-NCI to $224s$ in SNOMED-NCI. Overall,
for example, the required time to compute the division with $100$ matching
subtasks ranges from $23s$ (AMA-NCIA) to approx. $600s$ (SNOMED-NCI).
### 4.2 Evaluation of OAEI systems
In this section we show that the division of the alignment task enables
systems that, given some computational constraints, were unable to complete an
OAEI task. We have selected the following five systems from the latest OAEI
campaigns, which include novel alignment techniques but failed to scale to
very large matching tasks: MAMBA (v.2015) [35], FCA-Map (v.2016) [52], SANOM
(v.2017) [37], ALOD2vec (v.2018) [43] and POMap++ (v.2018) [30]. MAMBA failed
to complete the anatomy track, while FCA-Map, SANOM, ALOD2vec and POMap++
could not complete the largest tasks in the largebio track. MAMBA and SANOM
threw an out-of-memory exception with $25Gb$, whereas FCA-Map, ALOD2vec and
POMap++ did not complete the tasks within a $6$ hours time-frame. We have used
the SEALS infrastructure to conduct the evaluation [3, 4].
Table 3 shows the obtained results in terms of precision, recall, f-measure,
and computation times (time for the easiest and the hardest task, and total
time for all tasks) over different divisions
$\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ computed using our strategy. For
example, FCA-Map was run over divisions with 20 and 50 matching subtasks
(i.e., $n\in\\{20,50\\}$) in the FMA-NCI case. Note that for each matching
subtask a system generates a partial alignment $\mathcal{M}^{S}_{i}$, the
final alignment for the (original) matching task is computed as the union of
all partial alignments
($\mathcal{M}^{S}=\bigcup_{i=1}^{n}\mathcal{M}^{S}_{i}$). The results are
encouraging and can be summarised as follows:
1. )
We enabled several systems to produce results even for the largest OAEI test
case (e.g., FCA-Map with SNOMED-NCI).
2. )
The computation times are also very good falling under the $6$ hours time
frame, specially given that the (independent) matching subtasks have been run
sequentially without parallelization.
3. )
The size of the divisions, with the exception of FCA-Map, is beneficial in
terms of total computation time.
4. )
The increase of number of matching subtasks is positive or neutral for MAMBA,
POMap++ and ALOD2vec in terms of f-measure, while it is slightly reduced for
FCA-Map and SANOM.
5. )
Global f-measure results are lower than top OAEI systems; nevertheless, since
the above systems could not be evaluated without the divisions, these results
are obtained without any fine-tuning of their parameters.
6. )
The computation times of the hardest tasks, as $n$ increases, is also reduced.
This has a positive impact in the monitoring of alignment systems as the
hardest task is completed in a reasonable time.
## 5 Related work
Partitioning and blocking. Partitioning and modularization techniques have
been extensively used within the Semantic Web to improve the efficiency when
solving the task at hand (e.g., visualization [45, 1], reuse [29], debugging
[47], classification [7]). Partitioning or blocking has also been widely used
to reduce the complexity of the ontology alignment task [16]. In the
literature there are two major categories of partitioning techniques, namely:
_independent_ and _dependent_. Independent techniques typically use only the
structure of the ontologies and are not concerned about the ontology alignment
task when performing the partitioning. Whereas dependent partitioning methods
rely on both the structure of the ontology and the ontology alignment task at
hand. Although our approach does not compute (non-overlapping) partitions of
the ontologies, it can be considered a dependent technique.
Prominent examples of ontology alignment systems including partitioning
techniques are Falcon-AO [22], GOMMA [19], COMA++ [5] and TaxoMap [20].
Falcon-AO, GOMMA and COMA++ perform independent partitioning where the
clusters of the source and target ontologies are independently extracted. Then
pairs of similar clusters (i.e., matching subtasks) are aligned using standard
techniques. TaxoMap [20] implements a dependent technique where the
partitioning is combined with the matching process. TaxoMap proposes two
methods, namely: PAP (partition, anchor, partition) and APP (anchor,
partition, partition). The main difference of these methods is the order of
extraction of (preliminary) anchors to discover pairs of partitions to be
matched (i.e., matching subtasks). SeeCOnt [2] presents a seeding-based
clustering technique to discover independent clusters in the input ontologies.
Their approach has been evaluated with the Falcon-AO system by replacing its
native PBM (Partition-based Block Matching) module [23]. Laadhar et al. [30]
have recently integrated within the system POMap++ a hierarchical
agglomerative clustering algorithm to divide an ontology into a set of
partitions.
The above approaches, although presented interesting ideas, did not provide
guarantees about the size and coverage of the discovered partitions or
divisions. Furthermore, they have not been successfully evaluated on very
large ontologies. On the one hand, as reported by Pereira et al. [42] the
results in terms of coverage of the PBM method of Falcon-OA, and the PAP and
APP methods of TaxoMap are very low for the OAEI largebio track. On the other
hand, as discussed in Section 4, POMap++ fails to scale with the largest
largebio tasks.
Note that the recent work in [31] has borrowed from our workshop paper [26]
the quality measures presented in Section 2.1. They obtain competitive
coverage results for medium size ontologies; however, their approach, as in
POMap++, does not scale for large ontologies.
Blocking techniques are also extensively used in Entity Resolution (see [40]
for a survey). Although related, the problem of blocking in ontologies is
different as the logically related axioms for a seed signature play an
important role when computing the blocks.
Our dependent approach, unlike traditional partitioning and blocking methods,
computes overlapping self-contained modules (i.e., locality modules [13]).
Locality modules guarantee the extraction of all semantically related entities
for a given signature. This capability enhances the coverage results and
enables the inclusion of the (semantically) relevant information required by
an alignment system. It is worth mentioning that the need of self-contained
and covering modules, although not thoroughly studied, was also highlighted in
a preliminary work by Paulheim [41].
Embedding and clustering. Recently, machine learning techniques such as
semantic embedding [12] have been investigated for ontology alignment. They
often first learn vector representations of the entities and then predict the
alignment [9, 51, 46]. However, most of them focus on alignment of ontology
individuals (i.e., ABox) without considering the ontology concepts and axioms
at the terminological level (i.e., TBox). Nkisi-Orji et al. [39] predicts the
alignment between ontology concepts with Random Forest, but incorporates the
embeddings of words alone, without any other semantic components like in our
work. Furthermore, these approaches focus on predicting the alignment, while
our work aims at boosting an existing alignment system. Our framework could
potentially be adopted in systems like in [39] if facing scalability problems
for large ontologies.
Another piece of related work is the clustering of semantic components, using
the canopy clustering algorithm [33] where objects are grouped into canopies
and each object can be a member of multiple canopies. For example, Wu et al.
[50] first extracted canopies (i.e., mentions) from a knowledge base, and then
grouped the entities accordingly so as to finding out the entities with the
same semantics (i.e., canonicalization). As we focus on a totally different
task – ontology alignment, the context that can be used, such as the
embeddings for the words and ontology entities in LexI, is different from
these works, which leads to a different clustering method.
## 6 Conclusions and future work
We have developed a novel framework to split the ontology alignment task into
several matching subtasks based on a semantic inverted index, locality
modules, and a neural embedding model. We have performed a comprehensive
evaluation which suggests that the obtained divisions are suitable in practice
in terms of both coverage and size. The division of the matching task allowed
us to obtain results for five systems which failed to complete these tasks in
the past. We have focused on systems failing to complete a task, but a
suitable adoption and integration of the presented framework within the
pipeline of any ontology alignment system has the potential to improve the
results in terms of computation times.
Opportunities. Reducing the ontology matching task into smaller and more
manageable tasks may also bring opportunities to enhance (i) user interaction
[32], (ii) reasoning and repair [34], (iii) benchmarking and monitoring [3,
4], and (iv) parallelization. The computed independent matching subtasks can
potentially be run in parallel in evaluation platforms like the HOBBIT [38].
The current evaluation was conducted sequentially as (i) the SEALS instance
only allows running one task at a time, and (ii) the evaluated systems were
not designed to run several tasks in parallel; for instance, we managed to run
MAMBA outside SEALS, but it relies on a MySQL database and raised a concurrent
access exception.
Impact on the f-measure. As shown in Section 4.2, the impact of the number of
divisions on the f-measure depends on the evaluated systems. In the near
future we aim at conducting an extensive evaluation of our framework over OAEI
systems able to deal with the largest tasks in order to obtain more insights
about the impact on the f-measure. In [25] we reported a preliminary
evaluation where YAM-Bio [6] and AML [17] kept similar f-measure values, while
LogMap [27] had a reduction in f-measure, as the number of divisions
increased.
Number of subdivisions. Currently our strategy requires the size of the number
of matching subtasks or divisions as input. The (required) matching subtasks
may be known before hand if, for example, the matching tasks are to be run in
parallel in a number of available CPUs. For the cases where the resources are
limited or where a matching system is known to cope with small ontologies, we
plan to design an algorithm to estimate the number of divisions so that the
size of the matching subtasks in the computed divisions is appropriate to the
system and resource constraints.
Dealing with a limited or large lexicon. The construction of LexI shares a
limitation with state-of-the-art systems when the input ontologies are
lexically disparate or in different languages. In such cases, LexI can be
enriched with general-purpose lexicons (e.g., WordNet), more specialised
background knowledge (e.g., UMLS Metathesaurus) or with translated labels
using online services (e.g., Google). On the other hand, a large lexicon may
also have an important impact in the computation times. Our conducted
evaluation shows, however, that we can cope with very large ontologies with a
rich lexicon (e.g., NCI Thesaurus).
Notion of context. Locality-based modules are typically much smaller than the
whole ontology and they have led to very good results in terms of size and
coverage. We plan, however, to study different notions of _context_ of an
alignment (e.g., the tailored modules proposed in [8]) to further improve the
results in terms of size while keeping the same level of coverage.
This work was supported by the SIRIUS Centre for Scalable Data Access (Norges
forskningsråd), the AIDA project (Alan Turing Institute), Samsung Research UK,
Siemens AG, and the EPSRC projects AnaLOG, OASIS and UK FIRES. We would also
like to thank the anonymous reviewers that helped us improve this work.
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|
2024-09-04T02:54:59.237774 | 2020-03-11T16:40:42 | 2003.05398 | {
"authors": "Kyoungmun Lee and Siyoung Q. Choi",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26169",
"submitter": "Kyoungmun Lee",
"url": "https://arxiv.org/abs/2003.05398"
} | arxiv-papers | # Stratification of polymer-colloid mixtures via fast nonequilibrium
evaporation
Kyoungmun Lee<EMAIL_ADDRESS>Siyoung Q. Choi<EMAIL_ADDRESS>Department of Chemical and Biomolecular Engineering, Korea Advanced Institute
of Science and Technology (KAIST), Daejeon 34141, Korea
###### Abstract
In drying liquid films of polymer-colloid mixtures, the stratification in
which polymers are placed on top of larger colloids is studied. It is often
presumed that the formation of segregated polymer-colloid layers is solely due
to the proportion in size at fast evaporation as in binary colloid mixtures.
By comparing experiments with a theoretical model, we found that the
transition in viscosity near the drying interface was another important
parameter for controlling the formation of stratified layers in polymer-
colloid mixtures. At high evaporation rates, increased polymer concentrations
near the surface lead to a phase transition from semidilute to concentrated
regime, in which colloidal particles are kinetically arrested. Stratification
only occurs if the formation of a stratified layer precedes the evolution to
the concentrated regime near the drying interfaces. Otherwise, the colloids
will be trapped by the polymers in the concentrated regime before forming a
segregated layer. Also, no stratification is observed if the initial polymer
concentration is too low to form a sufficiently high polymer concentration
gradient within a short period of time. Our findings are relevant for
developing solution-cast polymer composite for painting, antifouling and
antireflective coatings.
††preprint: APS/123-QED
## I Introduction
Solution-cast polymer composite films composed of polymer matrices containing
colloidal particles have been widely studied for many applications, including
paints [1], coatings [2,3], and cosmetics [4,5] because they provide highly
improved macroscopic properties relative to the pure polymer [6], through a
simple manufacturing process. The enhanced properties of the dried films are
largely dependent on the spatial distribution of the polymer and colloid
[7-10]. In particular, stratified layers consisting of a polymer layer on a
colloidal layer have exhibited highly improved antifouling performance
[11,12], and photoactive properties [13].
Several previous studies have demonstrated ways of controlling the segregated
layers of polymer-colloid mixtures in an equilibrium state [14-16]. However,
relatively little is known about how polymer-colloid mixtures can be
stratified during the simple, fast and inexpensive nonequilibrium solvent
evaporation process. Although solvent casting is one of the simplest
manufacturing methods, from coffee ring stains [17] to many industrial
applications [1-5], the inherent nonequilibrium nature of drying has made it
difficult to clarify the underlying mechanism.
As a solvent evaporates, the spatial distribution of the solutes in liquid
films is determined by two competing factors: diffusion [18] and receding
drying interfaces. Solutes tend to distribute uniformly in drying films with a
diffusion constant _D_ , while the nonuniform concentration gradient is
developed by the downward velocity of the interface _$v_{ev}$_. Which of the
two phenomena dominates can be quantified by the dimensionless _Péclet_ number
_Pe =_ _$v_{ev}$__$z_{0}/D$_ , where _$z_{0}$_ is the initial film thickness.
If _Pe_ $>$ 1, the solutes cannot diffuse uniformly within the time of
evaporation, and they accumulate near the top of the film. On the other hand,
the drying film shows almost uniform distribution if _Pe_ $<$ 1.
In binary colloid mixtures, it was recently shown that stratifications with
smaller colloids placed on large colloids can be realized if _Pe_ is larger
than 1 [19-22]. This occurs when the concentration gradient of both the large
and smaller particles increases near the liquid/air interface. Fortini _et
al._ [20] proposed that the inverted stratification was caused by an imbalance
in the osmotic pressure between the larger and smaller colloids. Zhou _et al._
[21] suggested that the stratification phenomenon could be explained
quantitatively using a diffusion model, with cross-interaction between the
colloids. Sear and Warren [22] argued that diffusiophoretic motion induced by
the concentration gradient of the smaller components can exclude the larger
colloids from the drying interfaces.
In a way similar to binary colloid mixtures, it has been proposed that a
polymer-colloid mixture can yield the same stratified layers if the _Pe_ of
both the polymer and colloid are larger than 1 [23,24]. However, these results
have only been demonstrated by simulation and modeling studies, and few
experimental studies have been made on polymer-colloid stratification.
Although polymers and colloids can show similar behaviors at very dilute
concentrations [24,25], they might behave much differently at the high
concentrations that any drying solutions must experience for the complete
drying [26,27]. The obvious difference is viscosity. It rapidly increases at
relatively low concentrations in the polymer solution, slowing the motions of
the species [27-29]. In contrast, the viscosity of the colloidal suspension
increases relatively slowly [30]. Thus, the growth in viscosity near the
interface, which can kinetically arrest larger colloids [31-33], needs to be
considered differently for polymer and colloidal systems, but no appropriate
studies have been performed yet.
In this work, we experimentally show that the formation of stratified layers,
where a small polymer layer is placed on larger colloids, can be predicted
using two competing time scales: the time at which the colloid begins to
stratify (_$t_{s}^{*}$_) and the time the colloid is arrested by the
transitions of viscosity near the interface (_$t_{c}^{*}$_).
We consider that the colloid starts to be arrested near the drying interfaces
when the polymer concentration reaches a concentrated regime where the polymer
chains are densely packed [29]. The stratification can be observed only if
_$t_{s}^{*}$_ precedes _$t_{c}^{*}$_ , or _$t_{c}^{*}$_ /_$t_{s}^{*}$_ $>$ 1\.
Otherwise, the viscosity near the drying interface rapidly grows within a very
short time and the colloids are kinetically trapped before a sufficient
downward velocity away from the surface of large colloids is generated. In
addition, when the initial polymer concentration is too low, no stratification
can also occur because the concentration gradient of the polymer, or the
additional migration velocity of the larger colloid, is not enough until the
evaporation ends.
For the predictive analysis of _$t_{s}^{*}$_ and _$t_{c}^{*}$_ , we propose a
simple model modified from the previous work [22]. We observed quite excellent
agreement in the final film morphology of the model prediction and
experimental studies. Our comprehensive study predicts the spatial
distribution of polymers and colloids in the final dried film, based on the
experimental system and drying conditions.
## II Result and discussion
### II.1 Structure of dried films of polymer-colloid
Mixtures of aqueous polystyrene (PS) suspension with a mean diameter _$d_{c}$_
= 1 _$\mu$__m_ , and poly(ethylene glycol) (PEG) or poly(vinyl alcohol) (PVA)
were used as a model system for stratification. The molecular weights of the
polymers with PEG _$M_{n}$_ (number average molecular weight) 6,000 gmol-1,
PEG _$M_{n}$_ 20,000 gmol-1, PVA _MW_ 6,000 gmol-1, and PVA _$M_{w}$_ (weight
average molecular weight) 13,000-23,000 gmol-1 (PVA _$M_{w}$_ 18,000) were
chosen for radius of colloid (_$R_{colloid}$_) _$\gg$_ radius of polymer
(_$R_{polymer}$_). Before drying, the film solutions contained an initial
volume fraction of _$\phi_{i,p}$_ = 0.01 or 0.04 for the polymer and
_$\phi_{i,c}$_ = 0.67 _$\phi_{i,p}$_ for the colloid, respectively. The
mixture solutions were deposited on glass substrates as _$z_{0}$_ = 1.25 mm.
The evaporation was performed at ambient temperature and a relative humidity
of 23 %, resulting in an initial polymer _Péclet_ number _$Pe_{i,p}$_ $>$ 1
(See Supplemental Material). All of the experimental systems are summarized in
Table I. When the evaporation was completed, the final film morphologies were
analyzed with the help of scanning electronic microscopy (SEM) and ImageJ
analysis.
Table 1: Various polymer-colloid systems that were tested. Colloid was fixed as PS to exclude gravitational effect during drying ($\rho_{PS}$ $\approx$ $\rho_{water}$). A total of 8 systems were experimentally performed. | | | | | | | | _$Pe_{i,p}$_
---|---|---|---|---|---|---|---|---
Colloid | Polymer | | _$R_{g}$_ 111See Supplemental Material (nm) | _$\phi_{i,p}$_ | _$\phi_{i,p}:\phi_{i,c}$_ | _$h_{0}$_ (mm) | Relative humidity | _$\phi_{i,p}$_ 0.01 | _$\phi_{i,p}$_ 0.04
PS (r = 500 nm) | PEG | _$M_{n}$_ 6,000 | 3.6 | 0.01 | 3 : 2 | 1.25 | 23 % | 4 | 7
| | _$M_{n}$_ 20,000 | 7.4 | or | | | | 9 | 22
| PVA | _MW_ 6,000 | 3.5 | 0.04 | | | | 4 | 9
| | _$M_{w}$_ 18,000 | 6.8 | | | | | 8 | 24
After complete drying, the polymers were enriched at the top of the films in
PEG _$M_{n}$_ 6,000 gmol-1 (_$\phi_{i,p}$_ = 0.04) [Fig. 1(a)] and PVA _MW_
6,000 gmol-1 (_$\phi_{i,p}$_ = 0.04) [Fig. 1(c)] while other 6 dried films in
Fig. 1(b), 1(d) and Fig. 2(a) - 2(d) were not segregated, but randomly
distributed. Although the stratified layers in Fig. 1(a) and Fig. 1(c) also
showed different degrees of stratification, there was a clear boundary between
the stratified layers [Fig. 1(e)] and nonstratified layers [Fig. 1(f), Fig.
2(e), and Fig. 2(f)].
Figure 1: Cross sectional SEM images of dried films of polymer-colloid
mixtures (_$\phi_{i,p}=0.04$_ , _$\phi_{i,p}:\phi_{i,c}=3:2$_). The upper row
shows various polymer-colloid distributions according to the polymer types and
molecular weights (a) PEG _$M_{n}$_ 6,000, (b) PEG _$M_{n}$_ 20,000, (c) PVA
_MW_ 6,000, (d) PVA _$M_{w}$_ 18,000. The yellow lines represent boundary of
stratified layers. If there is no clear boundary, nothing is denoted. The
lower rows are estimated relative volume fraction of polymer _$\phi_{p}$_ (red
circles) and colloid _$\phi_{c}$_ (blue triangles) of two representatives: (e)
PEG _$M_{n}$_ 6,000 and (f) PEG _$M_{n}$_ 20,000. The colloidal volume
fractions were obtained from SEM images through the ImageJ analysis. The
remained volume fraction was considered as polymer volume fraction
_$\phi_{p}=1-\phi_{c}$_. Figure 2: SEM images of dried films formed from
polymer-colloid mixtures (_$\phi_{i,p}=0.01$_ ,
_$\phi_{i,p}:\phi_{i,c}=3:2$_). Distributions of polymer and colloid are shown
through the upper row depending on the polymer types and molecular weights (a)
PEG _$M_{n}$_ 6,000, (b) PEG _$M_{n}$_ 20,000, (c) PVA _MW_ 6,000, (d) PVA
_$M_{w}$_ 18,000. There was no clear stratified layer in all four images. The
volume fractions of polymer _$\phi_{p}$_ (red circles) and colloid
_$\phi_{c}$_ (blue triangles) of the two dried films were obtained from SEM
image analysis: (e) PEG _$M_{n}$_ 6,000 and (f) PEG _$M_{n}$_ 20,000. The
volume fractions of colloids are estimated by ImageJ analysis, and the polymer
volume fraction was determined by _$\phi_{p}=1-\phi_{c}$_.
### II.2 Modified theoretical model of dynamic stratification
As the solvent evaporated at _Pe_ $>$ 1 for both polymer and colloid, the
descending air/water interface _$z_{interface}$_ compressed the polymer and
colloid, and they accumulated near the drying interface. From previous studies
[22,34], the transition of the polymer concentration in drying film
_$\phi_{p}$_(z,t) can be written as
$\displaystyle\phi_{p}(z,t^{*})\approx\phi_{i,p}(1+Pe_{p}t^{*}exp{\bf[}-\frac{|z-z_{interface}|}{D_{p}/v_{ev}}{\bf]}),$
(1) $\displaystyle z_{interface}(t^{*})=z_{0}-v_{ev}t=(1-t^{*})z_{0}$ (2)
if _Péclet_ number of polymer _$Pe_{p}$_ $\gg$ 1, where
_$t^{*}=tv_{ev}/z_{0}$_ (0 $\leq$ _$t^{*}$_ $\leq$ 1) is the dimensionless
time. Here, _$Pe_{p}$_ and diffusion coefficient of polymer _$D_{p}$_ can be
expressed as a function of drying time when _$Pe_{p}$_ and _$D_{p}$_ vary
slowly. Since the viscosity growth derived from the increased polymer
concentration can be accompanied by the kinetic arrest of the colloidal
particles, _$t_{c}^{*}$_ can be determined by the time when the volume
fraction of polymer reaches the concentrated regime
_$\phi_{p}=\phi_{p}^{**}$_. We consider that the colloidal particles at the
drying interface (_$z=z_{interface}$_) are kinetically arrested when the
polymer fraction reaches _$\phi_{p}^{**}$_ at _$z=z_{interface}-r_{colloid}$_
$\displaystyle\phi_{p}(z_{interface}-r_{colloid},t_{c}^{*})=\phi_{p}^{**}.$
(3)
Meanwhile, increasing the concentration gradients of the small polymers can
also create the diffusiophoretic drift velocity of larger colloids
_$v_{diffusiophoresis}$_ [35,36]
$\displaystyle v_{diffusiophoresis}=-\frac{9}{4}D_{p}\nabla\phi_{p}$ (4)
under the condition of _$R_{colloid}$_ $\gg$ _$R_{polymer}$_. From the simple
1D diffusion model, the polymer concentration gradient at the interface is
_$\nabla\phi_{p}=-v_{ev}\phi_{interface}/D_{p}$_ [37]. This gives the
diffusiophoretic velocity of interfacial colloids with the combination of
_$\phi_{interface}=\phi_{i,p}(1+Pe_{p}t^{*})$_ originating from Eq. (1) at
_$z=z_{interface}$_ ,
$\displaystyle
v_{colloid,interface}\approx\frac{9}{4}v_{ev}\phi_{i,p}(1+Pe_{p}t^{*}).$ (5)
The time at which the colloid begins to stratify during the evaporation
process (_$t_{s}^{*}$_) is determined by comparing _$v_{colloid,interface}$_
and _$v_{ev}$_. Near the time when evaporation begins, the gradient of polymer
concentration is not too large and _$v_{colloid,interface}$_ does not overcome
_$v_{ev}$_. At this state, both the polymer and colloid simply accumulate at
the drying interface. If the concentration gradient of the polymer is large
enough for the formation of a higher colloidal diffusiophoretic velocity,
however, _$v_{colloid,interface}$_ is larger than _$v_{ev}$_ and it starts to
create stratified layers in the drying film. We consider the time
_$t_{s}^{*}$_ when _$v_{colloid,interface}=v_{ev}$_ , resulting in
$\displaystyle v_{colloid,interface}(t_{s}^{*})=v_{ev}.$ (6)
The final morphologies of the drying polymer-colloid mixtures are determined
by the two competing time scales _$t_{s}^{*}$_ and _$t_{c}^{*}$_. There are
three regimes for the predictive analysis of the stratification of polymer-
colloid mixtures. The first is _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, where the
downward motion of the colloidal particles appears before
_$\phi_{p}(z_{interface}-r_{colloid},t_{c}^{*})=\phi_{p}^{**}$_. The second is
_$t_{c}^{*}/t_{s}^{*}$_ $<$ 1, where the polymer volume fraction reaches
_$\phi_{p}^{**}$_ before the evolution of
_$v_{colloid,interface}(t_{s}^{*})=v_{ev}$_. The third is _$t_{s}^{*}\approx
1$_ , where _$t_{s}^{*}$_ reaches to the time at which evaporation ends
(_$t^{*}=1$_), even though _$t_{s}^{*}$_ precedes _$t_{c}^{*}$_.
### II.3 Comparison of experimental results and theoretical model
As described above, the prediction for the polymer-colloid stratification can
be estimated using the competition between _$t_{c}^{*}$_ and _$t_{s}^{*}$_.
For the time dependent volume fraction of the polymer in the drying films,
evaporation rates were determined by measuring mass reduction (Fig. SM3). To
calculate the time dependent (or concentration dependent) polymer diffusion
coefficient, the average volume fractions of polymer in the drying film were
used as _$D_{p}$_ (See Supplemental Material). The transition volume fraction
of semidilute entangled _$\phi_{e}$_ to concentrated regime _$\phi^{**}$_ in
good solvent were determined by the specific viscosity _$\eta_{sp}$_ slope
transition [27,28,38] in Fig. 3. From the slope transition of semidilute
unentangled (_$\eta_{sp}$_ $\sim$ _$\phi_{p}^{1.3}$_) to semidilute entangled
(_$\eta_{sp}$_ $\sim$ _$\phi_{p}^{3.9}$_), _$\phi_{e}$_ of the polymer in good
solvent was measured. Similarly, _$\phi^{**}$_ can be estimated using the
slope transition point between the semidilute entangled regime (_$\eta_{sp}$_
$\sim$ _$\phi_{p}^{3.9}$_) and the concentrated regime (_$\eta_{sp}$_ $\sim$
_$\phi_{p}^{\alpha}$_ , where $\alpha$ $>$ 3.9).
Figure 3: Specific viscosity of four polymer solutions as a function of
polymer volume fraction. Polymer volume fraction where it goes to concentrated
regime _$\phi^{**}$_ is estimated by the slope transition point from 3.9 to
larger than 3.9. In case of PEG _$M_{n}$_ 6,000, _$\phi^{**}$_ is considered
as max solubility ($\approx$ 630 mg/ml at 20oC). As the PEG _$M_{n}$_ 6,000
solution goes to higher than max solubility, it shows abrupt increment of
specific viscosity (empty red triangle).
In drying films of polymer-colloid mixtures, the final film morphology can be
predicted using the three regimes in the (_$t_{s}^{*}$_ , _$t_{c}^{*}$_)
plane. Regime 1 with _$t_{c}^{*}$_ /_$t_{s}^{*}$_ $>$ 1 indicates clearly
stratified layers in the dried films. Regime 2 represents nonsegregated
layers, because _$t_{c}^{*}$_ appears before _$t_{s}^{*}$_. Regime 3 also
shows nonstratified layers in the final morphology of the complete dried
polymer-colloids mixtures, since _$t_{s}^{*}$_ appears very close to 1
(_$t_{s}^{*}$_ $\approx$ 1).
The theoretical predictions based on Eq. (3), Eq. (6) and the experimental
stratification results from 8 different systems are presented in Fig. 4. There
is quite excellent agreement between the model prediction and experimental
results except for the PVA _MW_ 6,000 (_$\phi_{i,p}=0.04$_) system, which also
appears to be closest to _$t_{c}^{*}/t_{s}^{*}$_ = 1. This might be due to the
air/water interfacial activity of PVA _MW_ 6,000 (Fig. SM4), which can make
faster _$t_{s}^{*}$_ under real drying conditions, but it cannot bring
_$t_{c}^{*}$_ forward since _$t_{c}^{*}$_ is related to the
_$z=z_{interface}-r_{colloid}$_ , not _$z=z_{interface}$_. To reduce the
interfacial activity effect of PVA _MW_ 6,000 (_$\phi_{i,p}=0.04$_) on
stratification, we moved the point to deviate from _$t_{c}^{*}/t_{s}^{*}$_ $=$
1 in our theoretical model by changing _$v_{ev}$_. As it deviates from
_$t_{c}^{*}/t_{s}^{*}=1$_ , the theoretical prediction becomes consistent with
the experimental result for PVA _MW_ 6,000 (_$\phi_{i,p}=0.04$_) (Fig. 5).
Figure 4: State diagram on the (_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane. The
dotted line corresponds to _$t_{c}^{*}/t_{s}^{*}=1$_. Theoretical predictions
of 8 different systems are denoted as symbols in the diagram, and the
experimental results are represented by colors. Blue indicates regime 1
(_$t_{c}^{*}/t_{s}^{*}$_ $>$ 1) where stratified layer expected and red shows
regime 2 (_$t_{c}^{*}/t_{s}^{*}$_ $<$ 1). Orange designated regime 3
(_$t_{s}^{*}\approx 1$_) (Fig. SM5). The green indicates the intermediate
state where stratified layer is observed in experiments while it belongs to
regime 2 in model prediction. All data points show overall agreement with one
exception, filled green triangle, which also appears close to the
_$t_{c}^{*}/t_{s}^{*}=1$_. Figure 5: State diagram of PVA _MW_ 6,000
(_$\phi_{i,p}$_ = 0.04) on the (_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane. The
dotted line corresponds to _$t_{c}^{*}/t_{s}^{*}=1$_. The filled green
triangle deviated from _$t_{c}^{*}/t_{s}^{*}=1$_ in theoretical model only by
increasing _$v_{ev}$_. As it stays away from _$t_{c}^{*}/t_{s}^{*}=1$_ , the
intermediate stratified morphology where stratified layer is observed in
experiments while it belongs to regime 2 in model prediction become consistent
with model prediction. (a) SEM image of PVA _MW_ 6,000 (_$\phi_{i,p}$_ = 0.04)
at fast evaporation. (b) Top of the cross-sectional SEM image (a). The
evaporation rate was controlled by convective flow of air with a relative
humidity of 23 % at ambient temperature.
### II.4 Conditions for polymer-colloid stratification
To analyze the general conditions for polymer-colloid stratification, we
represented _$t_{c}^{*}$_ and _$t_{s}^{*}$_ in another experimental parameter.
As mentioned above, the polymer-on-top structure can be formed when the two
conditions, both _$t_{s}^{*}$_ $<$ 1 and _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, are
satisfied. From Eq. (3) and Eq. (6), _$t_{c}^{*}$_ and _$t_{s}^{*}$_ are (See
Supplemental Material)
$\displaystyle
t_{c}^{*}\approx\frac{\frac{\phi^{**}}{\phi_{i,p}}-1}{Pe_{p}(t_{c}^{*})},$ (7)
$\displaystyle
t_{s}^{*}\approx\frac{\frac{4}{9}\frac{1}{\phi_{i,p}}-1}{Pe_{p}(t_{s}^{*})},$
(8)
where _$Pe_{p}(t_{c}^{*})$_ and _$Pe_{p}(t_{s}^{*})$_ are _Pe_ of the polymer
at dimensionless time _$t^{*}=t_{c}^{*}$_ and _$t^{*}=t_{s}^{*}$_ in
respectively. The first condition for the stratification to happen,
_$t_{s}^{*}$_ $<$ 1, is
$\displaystyle Pe_{p}(t_{s}^{*})\phi_{i,p}>\frac{4}{9}-\phi_{i,p}.$ (9)
This follows a condition for similar to that for the inverted stratification
of binary colloidal mixtures [21,33].
The second requirement for stratified layers in polymer-colloid mixtures,
_$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, can be expressed as
$\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}\frac{Pe_{p}(t_{s}^{*})}{Pe_{p}(t_{c}^{*})}>1,$
(10)
$\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}\frac{\eta(t_{s}^{*})}{\eta(t_{c}^{*})}>1.$
(11)
Since _$t_{c}^{*}$_ and _$t_{s}^{*}$_ come out when the polymer solution in
the semi-dilute entangled regime (close to _$\phi_{p}=\phi^{**}$_),
_$\eta(t^{*})$_ is
$\displaystyle\eta(t^{*})=(\frac{1-t_{e}^{*}}{1-t^{*}})^{3.9}(\eta_{e}-\eta_{s})+\eta_{s}$
(12)
from Eq. (14) of Supplemental Material, where _$t_{e}^{*}$_ is the
dimensionless time when _$\eta=\eta_{e}$_ (viscosity when
_$\phi_{p}=\phi_{e}$_) from Eq. (10) of Supplemental Material. If we neglect
the last term in Eq. (12),
$\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}(\frac{1-t_{c}^{*}}{1-t_{s}^{*}})^{3.9},$
(13)
$\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}(\frac{1-t_{s}^{*}}{1-t_{c}^{*}})^{3.9}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}.$
(14)
To satisfy the condition of _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1 for polymer-colloid
stratification,
$\displaystyle\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}>1,$
(15) $\displaystyle\phi^{**}-\phi_{i,p}>\frac{4}{9}-\phi_{i,p},$ (16)
$\displaystyle\phi^{**}>\frac{4}{9}.$ (17)
It is interesting to note that the predicted stratification of the polymer-
colloid mixtures does not depend on the drying rate _$v_{ev}$_ , or _Pe_ , as
long as _$Pe\gg 1$_. This tendency also can be seen in Fig. 6, which shows the
theoretical predictions of the 8 systems above, with _$v_{ev}$_ values
changed. Ignoring the data points of _$Pe_{i,p}\leq 5$_ , failing to follow
the aforementioned assumption _$Pe\gg 1$_ , all the other points belong in
same regime once the polymer type and initial volume fraction are determined.
This is quite plausible since the increase in polymer concentration near the
drying interface accelerates both _$t_{c}^{*}$_ and _$t_{s}^{*}$_ in similar
order. Thus, it might be hard to create stratified layers in polymer-colloid
mixtures only by varying the evaporation rate _$v_{ev}$_ , or _Pe_. Altering
other properties which can increase _$t_{c}^{*}/t_{s}^{*}$_ larger than 1,
such as the interfacial activity of the polymer in Fig. 5 or the gravitational
velocity from the density difference in Fig. SM6, could be another solution to
achieve stratified layers in polymer-colloid mixtures.
Figure 6: Theoretical prediction of the stratification of 8 different systems
on the (_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane with controlled _$v_{ev}$_ (or
_$Pe_{i,p}$_) (a) PEG _$M_{n}$_ 6,000, (b) PEG _$M_{n}$_ 20,000, (c) PVA _MW_
6,000, (d) PVA _$M_{w}$_ 18,000. As _$Pe_{i,p}$_ increases, both _$t_{s}^{*}$_
and _$t_{c}^{*}$_ decrease and data points go to left bottom side on the
(_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane. Regardless of the polymer type or
molecular weight, most of the data points belong in the same regime once the
type of polymer and initial volume fraction are determined except the
relatively slow drying rate (_$Pe_{i,p}\leq 5$_ , red circles).
## III Conclusion
In summary, we demonstrated that dynamic stratification of polymer-colloid
mixtures can be achieved by controlling viscosity near the drying interface,
which results from increasing polymer concentration. When the polymer-colloid
solution evaporates, the polymer starts to increase the solution viscosity
near the air/water interface within a relatively very short time, unlike
colloidal suspensions. Since the transition in viscosity due to the polymer
can cause the kinetic arrest of colloidal particles, which hinders the
diffusiophoretic downward motion of colloids, stratified layers are only
observed if the formation of a stratified layer precedes the transition in
viscosity near the liquid/air interfaces.
Our model calculations for _$t_{c}^{*}$_ and _$t_{s}^{*}$_ , inspired by the
previous study [22], show that the segregation of polymer-colloid mixtures can
only occur under the condition of _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, unless the
solute fraction of the polymer is sufficiently low. The requirement for
stratification, _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, implies that the stratification
of polymer-colloid mixtures may not rely on drying rate if _$Pe\gg 1$_ , since
both _$t_{c}^{*}$_ and _$t_{s}^{*}$_ vary in similar order as _$v_{ev}$_
changes. Our model calculations are further supported by the consistency
between the model prediction and final experimental film morphologies
In more general terms, the consistent results of the experiments and model
prediction may shed light on methods of controlling surface enrichment in
general solution-cast polymer composites. The ability to predict morphology in
a simple nonequilibrium solvent evaporation process is highly desirable for
preparing materials whose surface properties are crucial to performance, such
as antireflective or organic photovoltaics. Our insights on how polymer
concentration affects colloidal dynamics and stratification can be exploited
to control segregated layers in solution-cast polymer-colloid mixtures.
###### Acknowledgements.
This work was supported by the Basic Science Research Program through the
National Research Foundation of Korea (Grants NRF-2015R1C1A1A01054180, and
NRF-2019R1F1A1059587).
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|
2024-09-04T02:54:59.247362 | 2020-03-11T16:42:49 | 2003.05399 | {
"authors": "Mats Vermeeren",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26170",
"submitter": "Mats Vermeeren",
"url": "https://arxiv.org/abs/2003.05399"
} | arxiv-papers | Open Communications in Nonlinear Mathematical Physics ]ocnmp[ Vol.1 (2021) pp
1–References Article
††footnotetext: © The author(s). Distributed under a Creative Commons
Attribution 4.0 International License
Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
Mats Vermeeren 1
1 School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK.
<EMAIL_ADDRESS>
Received May 18, 2021; Accepted August 31, 2021
###### Abstract
Many integrable hierarchies of differential equations allow a variational
description, called a Lagrangian multiform or a pluri-Lagrangian structure.
The fundamental object in this theory is not a Lagrange function but a
differential $d$-form that is integrated over arbitrary $d$-dimensional
submanifolds. All such action integrals must be stationary for a field to be a
solution to the pluri-Lagrangian problem. In this paper we present a procedure
to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an
integrable hierarchy of PDEs. As a prelude, we review a similar procedure for
integrable ODEs. We show that the exterior derivative of the Lagrangian
$d$-form is closely related to the Poisson brackets between the corresponding
Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples
the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian
2-form) case we present the potential and Schwarzian Korteweg-de Vries
hierarchies, as well as the Boussinesq hierarchy.
## 1 Introduction
Some of the most powerful descriptions of integrable systems use the
Hamiltonian formalism. In mechanics, Liouville-Arnold integrability means
having as many independent Hamilton functions as the system has degrees of
freedom, such that the Poisson bracket of any two of them vanishes. In the
case of integrable PDEs, which have infinitely many degrees of freedom,
integrability is often defined as having an infinite number of commuting
Hamiltonian flows, where again each two Hamilton functions have a zero Poisson
bracket. In addition, many integrable PDEs have two compatible Poisson
brackets. Such a bi-Hamiltonian structure can be used to obtain a recursion
operator, which in turn is an effective way to construct an integrable
hierarchy of PDEs.
In many cases, especially in mechanics, Hamiltonian systems have an equivalent
Lagrangian description. This raises the question whether integrability can be
described from a variational perspective too. Indeed, a Lagrangian theory of
integrable hierarchies has been developed over the last decade or so,
originating in the theory of integrable lattice equations (see for example
[14], [3], [12, Chapter 12]). It is called the theory of _Lagrangian
multiform_ systems, or, of _pluri-Lagrangian_ systems. The continuous version
of this theory, i.e. its application to differential equations, was developed
among others in [26, 28]. Recently, connections have been established between
pluri-Lagrangian systems and variational symmetries [18, 19, 22] as well as
Lax pairs [21].
Already in one of the earliest studies of continuous pluri-Lagrangian
structures [26], the pluri-Lagrangian principle for ODEs was shown to be
equivalent to the existence of commuting Hamiltonian flows (see also [24]). In
addition, the property that Hamilton functions are in involution can be
expressed in Lagrangian terms as closedness of the Lagrangian form. The main
goal of this work is to generalize this connection between pluri-Lagrangian
and Hamiltonian structures to the case of integrable PDEs.
A complementary approach to connecting pluri-Lagrangian structures to
Hamiltonian structures was recently taken in [6]. There, a generalisation of
covariant Hamiltonian field theory is proposed, under the name _Hamiltonian
multiform_ , as the Hamiltonian counterpart of Lagrangian multiform systems.
This yields a Hamiltonian framework where all independent variables are on the
same footing. In the present work we obtain classical Hamiltonian structures
where one of the independent variables is singled out as the common space
variable of all equations in a hierarchy.
We begin this paper with an introduction to pluri-Lagrangian systems in
Section 2. The exposition there relies mostly on examples, while proofs of the
main theorems can be found in Appendix A. Then we discuss how pluri-Lagrangian
systems generate Hamiltonian structures, using symplectic forms in
configuration space. In Section 3 we review this for ODEs (Lagrangian 1-form
systems) and in Section 4 we present the case of $(1+1)$-dimensional PDEs
(Lagrangian 2-form systems). In each section, we illustrate the results by
examples.
## 2 Pluri-Lagrangian systems
A hierarchy of commuting differential equations can be embedded in a higher-
dimensional space of independent variables, where each equation has its own
time variable. All equations share the same space variables (if any) and have
the same configuration manifold $Q$. We use coordinates
$t_{1},t_{2},\ldots,t_{N}$ in the _multi-time_ $M=\mathbb{R}^{N}$. In the case
of a hierarchy of $(1+1)$-dimensional PDEs, the first of these coordinates is
a common space coordinate, $t_{1}=x$, and we assume that for each $i\geq 2$
there is a PDE in the hierarchy expressing the $t_{i}$-derivative of a field
$u:M\rightarrow Q$ in terms of $u$ and its $x$-derivatives. Then the field $u$
is determined on the whole multi-time $M$ if initial values are prescribed on
the $x$-axis. In the case of ODEs, we assume that there is a differential
equation for each of the time directions. Then initial conditions at one point
in multi-time suffice to determine a solution.
We view a field $u:M\rightarrow Q$ as a smooth section of the trivial bundle
$M\times Q$, which has coordinates $(t_{1},\ldots,t_{N},u)$. The extension of
this bundle containing all partial derivatives of $u$ is called the _infinite
jet bundle_ and denoted by $M\times J^{\infty}$. Given a field $u$, we call
the corresponding section $\llbracket
u\rrbracket=(u,u_{t_{i}},u_{t_{i}t_{j}},\ldots)$ of the infinite jet bundle
the _infinite jet prolongation_ of $u$. (See e.g. [1] or [17, Sec. 3.5].)
In the pluri-Lagrangian context, the Lagrange function is replaced by a jet-
dependent $d$-form. More precisely we consider a fiber-preserving map
$\textstyle\mathcal{L}:M\times J^{\infty}\rightarrow\bigwedge^{d}(T^{*}M).$
Since a field $u:M\rightarrow Q$ defines a section of the infinite jet bundle,
$\mathcal{L}$ associates to it a section of $\bigwedge^{d}(T^{*}M)$, that is,
a $d$-form $\mathcal{L}\llbracket u\rrbracket$. We use the square brackets
$\llbracket u\rrbracket$ to denote dependence on the infinite jet prolongation
of $u$. We take $d=1$ if we are dealing with ODEs and $d=2$ if we are dealing
with $(1+1)$-dimensional PDEs. Higher-dimensional PDEs would correspond to
$d>2$, but are not considered in the present work. (An example of a Lagrangian
$3$-form system, the KP hierarchy, can be found in [22].) We write
$\mathcal{L}\llbracket u\rrbracket=\sum_{i}\mathcal{L}_{i}\llbracket
u\rrbracket\,\mbox{\rm d}t_{i}$
for 1-forms and
$\mathcal{L}\llbracket u\rrbracket=\sum_{i,j}\mathcal{L}_{ij}\llbracket
u\rrbracket\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$
for 2-forms.
###### Definition 2.1.
A field $u:M\rightarrow Q$ is a solution to the _pluri-Lagrangian problem_ for
the jet-dependent $d$-form $\mathcal{L}$, if for every $d$-dimensional
submanifold $\Gamma\subset M$ the action $\int_{\Gamma}\mathcal{L}\llbracket
u\rrbracket$ is critical with respect to variations of the field $u$, i.e.
$\frac{\mbox{\rm d}}{\mbox{\rm
d}\varepsilon}\int_{\Gamma}\mathcal{L}\llbracket u+\epsilon
v\rrbracket\bigg{|}_{\varepsilon=0}=0$
for all variations $v:M\rightarrow Q$ such that $v$ and all its partial
derivatives are zero on $\partial\Gamma$.
Some authors include in the definition that the Lagrangian $d$-form must be
closed when evaluated on solutions. That is equivalent to requiring that the
action is not just critical on every $d$-submanifold, but also takes the same
value on every $d$-submanifold (with the same boundary and topology). In this
perspective, one can take variations of the submanifold $\Gamma$ as well as of
the fields. We choose not to include the closedness in our definition, because
slightly weaker property can be obtained as a consequence Definition 2.1 (see
Proposition A.2 in the Appendix). Most of the authors that include closedness
in the definition use the term “Lagrangian multiform” (e.g. [14, 12, 32, 33,
22]), whereas those that do not tend to use “pluri-Lagrangian” (e.g. [4, 5,
27]). Whether or not it is included in the definition, closedness of the
Lagrangian $d$-form is an important property. As we will see in Sections 3.4
and 4.4, it is the Lagrangian counterpart to vanishing Poisson brackets
between Hamilton functions.
Clearly the pluri-Lagrangian principle is stronger than the usual variational
principle for the individual coefficients $\mathcal{L}_{i}$ or
$\mathcal{L}_{ij}$ of the Lagrangian form. Hence the classical Euler-Lagrange
equations are only a part of the system equations characterizing a solution to
the pluri-Lagrangian problem. This system, which we call the _multi-time
Euler-Lagrange equations_ , was derived in [28] for Lagrangian 1- and 2-forms
by approximating an arbitrary given curve or surface $\Gamma$ by _stepped_
curves or surfaces, which are piecewise flat with tangent spaces spanned by
coordinate directions. In Appendix A we give a more intrinsic proof that the
multi-time Euler-Lagrange equations imply criticality in the pluri-Lagrangian
sense. Yet another proof can be found in [23].
In order to write the multi-time Euler-Lagrange equations in a convenient
form, we will use the multi-index notation for (mixed) partial derivatives.
Let $I$ be an $N$-index, i.e. a $N$-tuple of non-negative integers. We denote
by $u_{I}$ the mixed partial derivative of $u:\mathbb{R}^{N}\rightarrow Q$,
where the number of derivatives with respect to each $t_{i}$ is given by the
entries of $I$. Note that if $I=(0,\ldots,0)$, then $u_{I}=u$. We will often
denote a multi-index suggestively by a string of $t_{i}$-variables, but it
should be noted that this representation is not always unique. For example,
$t_{1}=(1,0,\ldots,0),\qquad t_{N}=(0,\ldots,0,1),\qquad
t_{1}t_{2}=t_{2}t_{1}=(1,1,0,\ldots,0).$
In this notation we will also make use of exponents to compactify the
expressions, for example
$t_{2}^{3}=t_{2}t_{2}t_{2}=(0,3,0,\ldots,0).$
The notation $It_{j}$ should be interpreted as concatenation in the string
representation, hence it denotes the multi-index obtained from $I$ by
increasing the $j$-th entry by one. Finally, if the $j$-th entry of $I$ is
nonzero we say that $I$ contains $t_{j}$, and write $I\ni t_{j}$.
### 2.1 Lagrangian 1-forms
###### Theorem 2.2 ([28]).
Consider the Lagrangian 1-form
$\mathcal{L}\llbracket u\rrbracket=\sum_{j=1}^{N}\mathcal{L}_{j}\llbracket
u\rrbracket\,\mbox{\rm d}t_{j},$
depending on an arbitrary number of derivatives of $u$. A field $u$ is
critical in the pluri-Lagrangian sense if and only if it satisfies the multi-
time Euler-Lagrange equations
$\displaystyle\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{I}}}=0$
$\displaystyle\forall I\not\ni t_{j},$ (1)
$\displaystyle\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}-\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{u_{It_{1}}}}=0$
$\displaystyle\forall I,$ (2)
for all indices $j\in\\{1,\ldots,N\\}$, where
$\frac{\delta_{j}{}}{\delta{u_{I}}}$ denotes the variational derivative in the
direction of $t_{j}$ with respect to $u_{I}$,
$\frac{\delta_{j}{}}{\delta{u_{I}}}=\frac{\partial{}}{\partial{u_{I}}}-\partial_{j}\frac{\partial{}}{\partial{u_{It_{j}}}}+\partial_{j}^{2}\frac{\partial{}}{\partial{u_{It_{j}t_{j}}}}-\cdots,$
and $\partial_{j}=\frac{\mbox{\rm d}}{\mbox{\rm d}t_{j}}$.
Note the derivative $\partial_{j}$ equals the total derivative
$\sum_{I}u_{It_{j}}\frac{\partial{}}{\partial{u_{I}}}$ if it is applied to a
function $f\llbracket u\rrbracket$ that only depends on $t_{j}$ through $u$.
Using the total derivative has the advantage that calculations can be done on
an algebraic level, where the $u_{I}$ are formal symbols that do not
necessarily have an analytic interpretation as a derivative.
###### Example 2.3.
The Toda lattice describes $N$ particles on a line with an exponential
nearest-neighbor interaction. We denote the displacement from equilibrium of
the particles by
$u=(q^{\scriptscriptstyle[1]},\ldots,q^{\scriptscriptstyle[N]})$. We impose
either periodic boundary conditions (formally
$q^{\scriptscriptstyle[0]}=q^{\scriptscriptstyle[N]}$ and
$q^{\scriptscriptstyle[N+1]}=q^{\scriptscriptstyle[1]}$) or open-ended
boundary conditions (formally $q^{\scriptscriptstyle[0]}=\infty$ and
$q^{\scriptscriptstyle[N+1]}=-\infty$). We will use
$q^{\scriptscriptstyle[k]}_{j}$ as shorthand notation for the derivative
$q^{\scriptscriptstyle[k]}_{t_{j}}=\frac{\mbox{\rm
d}q^{\scriptscriptstyle[k]}}{\mbox{\rm d}t_{j}}$. Consider the hierarchy
consisting of the Newtonian equation for the Toda lattice,
$q^{\scriptscriptstyle[k]}_{11}=\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)-\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right),\\\
$ (3)
along with its variational symmetries,
$\begin{split}q^{\scriptscriptstyle[k]}_{2}&=\left(q^{\scriptscriptstyle[k]}_{1}\right)^{2}+\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)+\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right),\\\
q^{\scriptscriptstyle[k]}_{3}&=\left(q^{\scriptscriptstyle[k]}_{1}\right)^{3}+q^{\scriptscriptstyle[k+1]}_{1}\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)+q^{\scriptscriptstyle[k-1]}_{1}\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\\\
&\quad+2q^{\scriptscriptstyle[k]}_{1}\left(\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)+\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),\\\
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\end{split}$ (4)
The hierarchy (3)–(4) has a Lagrangian 1-form with coefficients
$\displaystyle\mathcal{L}_{1}$
$\displaystyle=\sum_{k}\left(\frac{1}{2}\left(q^{\scriptscriptstyle[k]}_{1}\right)^{2}-\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$
$\displaystyle\mathcal{L}_{2}$
$\displaystyle=\sum_{k}\left(q^{\scriptscriptstyle[k]}_{1}q^{\scriptscriptstyle[k]}_{2}-\frac{1}{3}\left(q^{\scriptscriptstyle[k]}_{1}\right)^{3}-\left(q^{\scriptscriptstyle[k]}_{1}+q^{\scriptscriptstyle[k-1]}_{1}\right)\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$
$\displaystyle\mathcal{L}_{3}$
$\displaystyle=\sum_{k}\bigg{(}-\frac{1}{4}\left(q^{\scriptscriptstyle[k]}_{1}\right)^{4}-\left(\left(q^{\scriptscriptstyle[k+1]}_{1}\right)^{2}+q^{\scriptscriptstyle[k+1]}_{1}q^{\scriptscriptstyle[k]}_{1}+\left(q^{\scriptscriptstyle[k]}_{1}\right)^{2}\right)\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)$
$\displaystyle\hskip
42.67912pt+q^{\scriptscriptstyle[k]}_{1}q^{\scriptscriptstyle[k]}_{3}-\exp\\!\left(q^{\scriptscriptstyle[k+2]}-q^{\scriptscriptstyle[k]}\right)-\frac{1}{2}\exp\\!\left(2(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]})\right)\bigg{)},$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
See [18, 29] for constructions of this pluri-Lagrangian structure. The
classical Euler-Lagrange equations of these Lagrangian coefficients are
$\displaystyle\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q^{\scriptscriptstyle[k]}}}=0\quad$
$\displaystyle\Leftrightarrow\quad
q^{\scriptscriptstyle[k]}_{11}=e^{q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}}-e^{q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}},$
$\displaystyle\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q^{\scriptscriptstyle[k]}}}=0\quad$
$\displaystyle\Leftrightarrow\quad
q^{\scriptscriptstyle[k]}_{12}=\left(q^{\scriptscriptstyle[k]}_{1}+q^{\scriptscriptstyle[k+1]}_{1}\right)e^{q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}}-\left(q^{\scriptscriptstyle[k-1]}_{1}+q^{\scriptscriptstyle[k]}_{1}\right)e^{q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}},$
$\displaystyle\mathmakebox[\widthof{{}\Rightarrow{}}][c]{\vdots}$
We recover Equation (3), but for the other equations of the hierarchy we only
get a differentiated form. However, we do get their evolutionary form, as in
Equation (4), from the multi-time Euler-Lagrange equations
$\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q^{\scriptscriptstyle[k]}_{1}}}=0,\qquad\frac{\delta_{3}{\mathcal{L}_{3}}}{\delta{q^{\scriptscriptstyle[k]}_{1}}}=0,\qquad\cdots.$
The multi-time Euler-Lagrange equations of type (2) are trivially satisfied in
this case:
$\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{q^{\scriptscriptstyle[k]}_{j}}}=q^{\scriptscriptstyle[k]}_{1}$
for all $j$.
### 2.2 Lagrangian 2-forms
###### Theorem 2.4 ([28]).
Consider the Lagrangian 2-form
$\mathcal{L}\llbracket u\rrbracket=\sum_{i<j}\mathcal{L}_{ij}\llbracket
u\rrbracket\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j},$
depending on an arbitrary number of derivatives of $u$. A field $u$ is
critical in the pluri-Lagrangian sense if and only if it satisfies the multi-
time Euler-Lagrange equations
$\displaystyle\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}=0$
$\displaystyle\forall I\not\ni t_{i},t_{j},$ (5)
$\displaystyle\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}-\frac{\delta_{ik}{\mathcal{L}_{ik}}}{\delta{u_{It_{k}}}}=0$
$\displaystyle\forall I\not\ni t_{i},$ (6)
$\displaystyle\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}+\frac{\delta_{jk}{\mathcal{L}_{jk}}}{\delta{u_{It_{j}t_{k}}}}+\frac{\delta_{ki}{\mathcal{L}_{ki}}}{\delta{u_{It_{k}t_{i}}}}=0$
$\displaystyle\forall I,$ (7)
for all triples $(i,j,k)$ of distinct indices, where
$\frac{\delta_{ij}{}}{\delta{u_{I}}}=\sum_{\alpha,\beta=0}^{\infty}(-1)^{\alpha+\beta}\partial_{i}^{\alpha}\partial_{j}^{\beta}\frac{\partial{}}{\partial{u_{It_{i}^{\alpha}t_{j}^{\beta}}}}.$
###### Example 2.5.
A Lagrangian 2-form for the potential KdV hierarchy was first given in [28].
It is instructive to look at just two of the equations embedded in
$\mathbb{R}^{3}$. Then the Lagrangian 2-form has three coefficients,
$\mathcal{L}=\mathcal{L}_{12}\,\mbox{\rm d}t_{1}\wedge\mbox{\rm
d}t_{2}+\mathcal{L}_{13}\,\mbox{\rm d}t_{1}\wedge\mbox{\rm
d}t_{3}+\mathcal{L}_{23}\,\mbox{\rm d}t_{2}\wedge\mbox{\rm d}t_{3},$
where $t_{1}$ is viewed as the space variable. We can take
$\displaystyle\mathcal{L}_{12}$
$\displaystyle=-u_{1}^{3}-\frac{1}{2}u_{1}u_{111}+\frac{1}{2}u_{1}u_{2},$
$\displaystyle\mathcal{L}_{13}$
$\displaystyle=-\frac{5}{2}u_{1}^{4}+5u_{1}u_{11}^{2}-\frac{1}{2}u_{111}^{2}+\frac{1}{2}u_{1}u_{3},$
where $u_{i}$ is a shorthand notation for the partial derivative $u_{t_{i}}$,
and similar notations are used for higher derivatives. These are the classical
Lagrangians of the potential KdV hierarchy. However, their classical Euler-
Lagrange equations give the hierarchy only in a differentiated form,
$\displaystyle u_{12}$ $\displaystyle=6u_{1}u_{11}+u_{1111},$ $\displaystyle
u_{13}$
$\displaystyle=30u_{1}^{2}u_{11}+20u_{11}u_{111}+10u_{1}u_{1111}+u_{111111}.$
The Lagrangian 2-form also contains a coefficient
$\displaystyle\mathcal{L}_{23}$
$\displaystyle=3u_{1}^{5}-\frac{15}{2}u_{1}^{2}u_{11}^{2}+10u_{1}^{3}u_{111}-5u_{1}^{3}u_{2}+\frac{7}{2}u_{11}^{2}u_{111}+3u_{1}u_{111}^{2}-6u_{1}u_{11}u_{1111}$
$\displaystyle\quad+\frac{3}{2}u_{1}^{2}u_{11111}+10u_{1}u_{11}u_{12}-\frac{5}{2}u_{11}^{2}u_{2}-5u_{1}u_{111}u_{2}+\frac{3}{2}u_{1}^{2}u_{3}-\frac{1}{2}u_{1111}^{2}$
$\displaystyle\quad+\frac{1}{2}u_{111}u_{11111}-u_{111}u_{112}+\frac{1}{2}u_{1}u_{113}+u_{1111}u_{12}-\frac{1}{2}u_{11}u_{13}-\frac{1}{2}u_{11111}u_{2}$
$\displaystyle\quad+\frac{1}{2}u_{111}u_{3}$
which does not have a classical interpretation, but contributes meaningfully
in the pluri-Lagrangian formalism. In particular, the multi-time Euler-
Lagrange equations
$\frac{\delta_{12}{\mathcal{L}_{12}}}{\delta{u_{1}}}+\frac{\delta_{23}{\mathcal{L}_{23}}}{\delta{u_{3}}}=0\qquad\text{and}\qquad\frac{\delta_{13}{\mathcal{L}_{13}}}{\delta{u_{1}}}-\frac{\delta_{23}{\mathcal{L}_{23}}}{\delta{u_{3}}}=0$
yield the potential KdV equations in their evolutionary form,
$\displaystyle u_{2}$ $\displaystyle=3u_{1}^{2}+u_{111},$ $\displaystyle
u_{3}$ $\displaystyle=10u_{1}^{3}+5u_{11}^{2}+10u_{1}u_{111}+u_{11111}.$
All other multi-time Euler-Lagrange equations are consequences of these
evolutionary equations.
This example can be extended to contain an arbitrary number of equations from
the potential KdV hierarchy. The coefficients $\mathcal{L}_{1j}$ will be
Lagrangians of the individual equations, whereas the $\mathcal{L}_{ij}$ for
$i,j>1$ do not appear in the traditional Lagrangian picture.
###### Example 2.6.
The Boussinesq equation
$u_{22}=12u_{1}u_{11}-3u_{1111}$ (8)
is of second order in its time $t_{2}$, but the higher equations of its
hierarchy are of first order in their respective times, beginning with
$u_{3}=-6u_{1}u_{2}+3u_{112}.$ (9)
A Lagrangian 2-form for this system has coefficients
$\displaystyle\mathcal{L}_{12}$
$\displaystyle=\frac{1}{2}u_{2}^{2}-2u_{1}^{3}-\frac{3}{2}u_{11}^{2},$
$\displaystyle\mathcal{L}_{13}$
$\displaystyle=u_{2}u_{3}+6u_{1}^{4}+27u_{1}u_{11}^{2}-6uu_{12}u_{2}+\frac{9}{2}u_{111}^{2}+\frac{3}{2}u_{12}^{2},$
$\displaystyle\mathcal{L}_{23}$
$\displaystyle=24u_{1}^{3}u_{2}+18u_{1}u_{11}u_{12}+9u_{11}^{2}u_{2}-18u_{1}u_{111}u_{2}-2u_{2}^{3}-6uu_{2}u_{22}$
$\displaystyle\quad+6u_{1}^{2}u_{3}+9u_{111}u_{112}+3u_{11}u_{13}+3u_{12}u_{22}-3u_{111}u_{3}.$
They can be found in [30] with a different scaling of $\mathcal{L}$ and a
different numbering of the time variables. Equation (8) is equivalent to the
Euler-Lagrange equation
$\frac{\delta_{12}{\mathcal{L}_{12}}}{\delta{u}}=0$
and Equation (9) to
$\frac{\delta_{13}{\mathcal{L}_{13}}}{\delta{u_{2}}}=0.$
All other multi-time Euler-Lagrange equations are differential consequences of
Equations (8) and (9). As in the previous example, it is possible to extend
this 2-form to represent an arbitrary number of equations from the hierarchy.
Further examples of pluri-Lagrangian 2-form systems can be found in [21, 22,
29, 30].
## 3 Hamiltonian structure of Lagrangian 1-form systems
A connection between Lagrangian 1-form systems and Hamiltonian or symplectic
systems was found in [26], both in the continuous and the discrete case. Here
we specialize that result to the common case where one coefficient of the
Lagrangian 1-form is a mechanical Lagrangian and all others are linear in
their respective time-derivatives. We formulate explicitly the underlying
symplectic structures, which will provide guidance for the case of Lagrangian
2-form systems. Since some of the coefficients of the Lagrangian form will be
linear in velocities, it is helpful to first have a look at the Hamiltonian
formulation for Lagrangians of this type, independent of a pluri-Lagrangian
structure.
### 3.1 Lagrangians that are linear in velocities
Let the configuration space be a finite-dimensional real vector space
$Q=\mathbb{R}^{N}$ and consider a Lagrangian
$\mathcal{L}:TQ\rightarrow\mathbb{R}$ of the form
$\mathcal{L}(q,q_{t})=p(q)^{T}q_{t}-V(q),$ (10)
where
$\det\left(\frac{\partial{p}}{\partial{q}}-\left(\frac{\partial{p}}{\partial{q}}\right)^{T}\right)\neq
0.$ (11)
Note that $p$ denotes a function of the position $q$; later on we will use
$\pi$ to denote the momentum as an element of cotangent space. If $Q$ is a
manifold, the arguments of this subsection will still apply if there exists
local coordinates in which the Lagrangian is of the form (10). The Euler-
Lagrange equations are first order ODEs:
$\dot{q}=\left(\left(\frac{\partial{p}}{\partial{q}}\right)^{T}-\frac{\partial{p}}{\partial{q}}\right)^{-1}\nabla
V,$ (12)
where $\nabla V=\left(\frac{\partial{V}}{\partial{q}}\right)^{T}$ is the
gradient of $V$.
Note that Equation (11) implies that $Q$ is even-dimensional, hence $Q$ admits
a (local) symplectic structure. Instead of a symplectic form on $T^{*}Q$, the
Lagrangian system preserves a symplectic form on $Q$ itself [2, 20]:
$\displaystyle\omega=\sum_{i}-\mbox{\rm d}p_{i}(q)\wedge\mbox{\rm d}q_{i}$
$\displaystyle=\sum_{i,j}-\frac{\partial{p_{i}}}{\partial{q_{j}}}\,\mbox{\rm
d}q_{j}\wedge\mbox{\rm d}q_{i}$ (13)
$\displaystyle=\sum_{i<j}\left(\frac{\partial{p_{i}}}{\partial{q_{j}}}-\frac{\partial{p_{j}}}{\partial{q_{i}}}\right)\mbox{\rm
d}q_{i}\wedge\mbox{\rm d}q_{j},$
which is non-degenerate by virtue of Equation (11).
###### Proposition 3.1.
The Euler-Lagrange equation (12) of the Lagrangian (10) corresponds to a
Hamiltonian vector field with respect to the symplectic structure $\omega$,
with Hamilton function $V$.
* Proof.
The Hamiltonian vector field
$X=\sum_{i}X_{i}\frac{\partial{}}{\partial{q_{i}}}$ of the Hamilton function
$V$ with respect to $\omega$ satisfies
$\iota_{X}\omega=\mbox{\rm d}V,$
where
$\iota_{X}\omega=\sum_{i}\sum_{j\neq
i}\left(\frac{\partial{p_{j}}}{\partial{q_{i}}}-\frac{\partial{p_{i}}}{\partial{q_{j}}}\right)X_{j}\,\mbox{\rm
d}q_{i}$
and
$\mbox{\rm d}V=\sum_{i}\frac{\partial{V}}{\partial{q_{i}}}\,\mbox{\rm
d}q_{i}.$
Hence
$X=\left(\left(\frac{\partial{p}}{\partial{q}}\right)^{T}-\frac{\partial{p}}{\partial{q}}\right)^{-1}\nabla
V,$
which is the vector field corresponding to the Euler-Lagrange equation (12). ∎
### 3.2 From pluri-Lagrangian to Hamiltonian systems
On a finite-dimensional real vector space $Q$, consider a Lagrangian 1-form
$\mathcal{L}=\sum_{i}\mathcal{L}_{i}\,\mbox{\rm d}t_{i}$ consisting of a
mechanical Lagrangian
$\mathcal{L}_{1}(q,q_{1})=\frac{1}{2}|q_{1}|^{2}-V_{1}(q),$ (14)
where $|q_{1}|^{2}=q_{1}^{T}q_{1}$, and additional coefficients of the form
$\mathcal{L}_{i}(q,q_{1},q_{i})=q_{1}^{T}q_{i}-V_{i}(q,q_{1})\qquad\text{for
}i\geq 2,$ (15)
where the indices of $q$ denote partial derivatives,
$q_{i}=q_{t_{i}}=\frac{\mbox{\rm d}q}{\mbox{\rm d}t_{i}}$, whereas the indices
of $\mathcal{L}$ and $V$ are labels. We have chosen the Lagrangian
coefficients such that they share a common momentum $p=q_{1}$, which is forced
upon us by the multi-time Euler-Lagrange equation (2). Note that for each $i$,
the coefficient $\mathcal{L}_{i}$ contains derivatives of $q$ with respect to
$t_{1}$ and $t_{i}$ only. Many Lagrangian 1-forms are of this form, including
the Toda hierarchy, presented in Example 2.3.
The nontrivial multi-time Euler-Lagrange equations are
$\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q}}=0\quad\Leftrightarrow\quad
q_{11}=-\frac{\partial{V_{1}}}{\partial{q}},$
and
$\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{1}}}=0\quad\Leftrightarrow\quad
q_{i}=\frac{\partial{V_{i}}}{\partial{q_{1}}}\qquad\qquad\text{for }i\geq 2,$
with the additional condition that
$\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q}}=0\quad\Leftrightarrow\quad
q_{1i}+\frac{\partial{V_{i}}}{\partial{q}}=0.$
Hence the multi-time Euler-Lagrange equations are overdetermined. Only for
particular choices of $V_{i}$ will the last equation be a differential
consequence of the other multi-time Euler-Lagrange equations. The existence of
suitable $V_{i}$ for a given hierarchy could be taken as a definition of its
integrability.
Note that there is no multi-time Euler-Lagrange equation involving the
variational derivative
$\frac{\delta_{1}{\mathcal{L}_{i}}}{\delta{q}}=\frac{\partial{V_{i}}}{\partial{q}}-\frac{\mbox{\rm
d}}{\mbox{\rm d}t_{1}}\frac{\partial{V_{i}}}{\partial{q_{1}}}$
because of the mismatch between the time direction $t_{1}$ in which the
variational derivative acts and the index $i$ of the Lagrangian coefficient.
The multi-time Euler-Lagrange equations of the type
$\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{i}}}=\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{q_{j}}}$
all reduce to the trivial equation $q_{1}=q_{1}$, expressing the fact that all
$\mathcal{L}_{i}$ yield the same momentum.
Since $\mathcal{L}_{1}$ is regular,
$\det\left(\frac{\partial{{}^{2}\mathcal{L}_{1}}}{\partial{q_{1}^{2}}}\right)\neq
0$, we can find a canonical Hamiltonian for the first equation by Legendre
transformation,
$H_{1}(q,\pi)=\frac{1}{2}|\pi|^{2}+V_{1}(q),$
where we use $\pi$ to denote the cotangent space coordinate and
$|\pi|^{2}=\pi^{T}\pi$.
For $i\geq 2$ we consider $r=q_{1}$ as a second dependent variable. In other
words, we double the dimension of the configuration space, which is now has
coordinates $(q,r)=(q,q_{1})$. The Lagrangians
$\mathcal{L}_{i}(q,r,q_{i},r_{i})=rq_{i}-V_{i}(q,r)$ are linear in velocities.
We have $p(q,r)=r$, hence the symplectic form (13) is
$\omega=\mbox{\rm d}r\wedge\mbox{\rm d}q.$
This is the canonical symplectic form, with the momentum replaced by
$r=q_{1}$. Hence we can consider $r$ as momentum, thus identifying the
extended configuration space spanned by $q$ and $r$ with the phase space
$T^{*}Q$.
Applying Proposition 3.1, we arrive at the following result:
###### Theorem 3.2.
The multi-time Euler-Lagrange equations of a 1-form with coefficients
(14)–(15) are equivalent, under the identification $\pi=q_{1}$, to a system of
Hamiltonian equations with respect to the canonical symplectic form
$\omega=\mbox{\rm d}\pi\wedge\mbox{\rm d}q$, with Hamilton functions
$H_{1}(q,\pi)=\frac{1}{2}|\pi|^{2}+V_{1}(q)\qquad\text{and}\qquad
H_{i}(q,\pi)=V_{i}(q,\pi)\quad\text{for }i\geq 2$
###### Example 3.3.
From the Lagrangian 1-form for the Toda lattice given in Example 2.3 we find
$\displaystyle H_{1}$
$\displaystyle=\sum_{k}\left(\frac{1}{2}\left(\pi^{[k]}\right)^{2}+\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$
$\displaystyle H_{2}$
$\displaystyle=\sum_{k}\left(\frac{1}{3}\left(\pi^{[k]}\right)^{3}+\left(\pi^{[k]}+\pi^{[k-1]}\right)\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$
$\displaystyle H_{3}$
$\displaystyle=\sum_{k}\bigg{(}\frac{1}{4}\left(\pi^{[k]}\right)^{4}+\left(\left(\pi^{[k+1]}\right)^{2}+\pi^{[k+1]}\pi^{[k]}+\left(\pi^{[k]}\right)^{2}\right)\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)$
$\displaystyle\hskip
42.67912pt+\exp\\!\left(q^{\scriptscriptstyle[k+2]}-q^{\scriptscriptstyle[k]}\right)+\frac{1}{2}\exp\\!\left(2(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]})\right)\bigg{)},$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
We have limited the discussion in this section to the case where
$\mathcal{L}_{1}$ is quadratic in the velocity. There are some interesting
examples that do not fall into this category, like the Volterra lattice, which
has a Lagrangian linear in velocities, and the relativistic Toda lattice,
which has a Lagrangian with a more complicated dependence on velocities (see
e.g. [25] and the references therein). The discussion above can be adapted to
other types of Lagrangian 1-forms if one of its coefficients $\mathcal{L}_{i}$
has an invertible Legendre transform, or if they are collectively Legendre-
transformable as described in [26].
### 3.3 From Hamiltonian to Pluri-Lagrangian systems
The procedure from Section 3.2 can be reversed to construct a Lagrangian
1-form from a number of Hamiltonians.
###### Theorem 3.4.
Consider Hamilton functions $H_{i}:T^{*}Q\rightarrow\mathbb{R}$, with
$H_{1}(q,\pi)=\frac{1}{2}|\pi|^{2}+V_{1}(q)$. Then the multi-time Euler-
Lagrange equations of the Lagrangian 1-form
$\sum_{i}\mathcal{L}_{i}\,\mbox{\rm d}t_{i}$ with
$\displaystyle\mathcal{L}_{1}$ $\displaystyle=\frac{1}{2}|q_{1}|^{2}-V_{1}(q)$
$\displaystyle\mathcal{L}_{i}$
$\displaystyle=q_{1}q_{i}-H_{i}(q,q_{1})\qquad\text{for }i\geq 2$
are equivalent to the Hamiltonian equations under the identification
$\pi=q_{1}$.
* Proof.
Identifying $\pi=q_{1}$, the multi-time Euler-Lagrange equations of the type
(1) are
$\displaystyle\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q}}$
$\displaystyle=0\quad\Leftrightarrow\quad
q_{11}=-\frac{\partial{V_{1}(q)}}{\partial{q}},$
$\displaystyle\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{1}}}$
$\displaystyle=0\quad\Leftrightarrow\quad
q_{i}=\frac{\partial{H_{i}(q,\pi)}}{\partial{p}},$
$\displaystyle\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q}}$
$\displaystyle=0\quad\Leftrightarrow\quad\pi_{i}=-\frac{\partial{H_{i}(q,\pi)}}{\partial{q}}.$
The multi-time Euler-Lagrange equations of the type (2) are trivially
satisfied because
$\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{i}}}=q_{1}$
for all $i$. ∎
Note that the statement of Theorem 3.4 does not require the Hamiltonian
equations to commute, i.e. it is not imposed that the Hamiltonian vector
fields $X_{H_{i}}$ associated to the Hamilton functions $H_{i}$ satisfy
$[X_{H_{i}},X_{H_{j}}]=0$. However, if they do not commute then for a generic
initial condition $(q_{0},\pi_{0})$ there will be no solution
$(q,\pi):\mathbb{R}^{N}\rightarrow T^{*}Q$ to the equations
$\displaystyle\frac{\partial{}}{\partial{t_{i}}}(q(t_{1},\ldots
t_{N}),\pi(t_{1},\ldots t_{N}))=X_{H_{i}}(q(t_{1},\ldots
t_{N}),\pi(t_{1},\ldots t_{N}))\qquad(i=1,\ldots,N),$
$\displaystyle(q(0,\ldots,0),\pi(0,\ldots,0))=(q_{0},\pi_{0}).$
Hence the relevance of Theorem 3.4 lies almost entirely in the case of
commuting Hamiltonian equations. If they do not commute then it is an (almost)
empty statement because neither the system of Hamiltonian equations nor the
multi-time Euler-Lagrange equations will have solutions for generic initial
data.
###### Example 3.5.
The Kepler Problem, describing the motion of a point mass around a
gravitational center, is one of the classic examples of a completely
integrable system. It possesses Poisson-commuting Hamiltonians
$H_{1},H_{2},H_{3}:T^{*}\mathbb{R}^{3}\rightarrow\mathbb{R}$ given by
$\displaystyle H_{1}(q,\pi)$
$\displaystyle=\frac{1}{2}|\pi|^{2}-|q|^{-1},\quad$ the energy, Hamiltonian
for the physical motion, $\displaystyle H_{2}(q,\pi)$
$\displaystyle=(q\times\pi)\cdot\mathsf{e}_{z},$ the 3rd component of the
angular momentum, and $\displaystyle H_{3}(q,\pi)$
$\displaystyle=|q\times\pi|^{2},$ the squared magnitude of the angular
momentum,
where $q=(x,y,z)$ and $\mathsf{e}_{z}$ is the unit vector in the
$z$-direction. The corresponding coefficients of the Lagrangian 1-form are
$\displaystyle\mathcal{L}_{1}$
$\displaystyle=\frac{1}{2}|q_{1}|^{2}+|q|^{-1},$
$\displaystyle\mathcal{L}_{2}$ $\displaystyle=q_{1}\cdot q_{2}-(q\times
q_{1})\cdot\mathsf{e}_{z},$ $\displaystyle\mathcal{L}_{3}$
$\displaystyle=q_{1}\cdot q_{3}-|q\times q_{1}|^{2}.$
The multi-time Euler-Lagrange equations are
$\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q}}=0\quad\Rightarrow\quad
q_{11}=\frac{q}{|q|^{3}},$
the physical equations of motion,
$\displaystyle\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q_{1}}}=0$
$\displaystyle\quad\Rightarrow\quad q_{2}=\mathsf{e}_{z}\times q,$
$\displaystyle\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q}}=0$
$\displaystyle\quad\Rightarrow\quad q_{12}=-q_{1}\times\mathsf{e}_{z},\ $
describing a rotation around the $z$-axis, and
$\displaystyle\frac{\delta_{3}{\mathcal{L}_{3}}}{\delta{q_{1}}}=0$
$\displaystyle\quad\Rightarrow\quad q_{3}=2(q\times q_{1})\times q,$
$\displaystyle\frac{\delta_{3}{\mathcal{L}_{3}}}{\delta{q}}=0$
$\displaystyle\quad\Rightarrow\quad q_{13}=2(q\times q_{1})\times q_{1},$
describing a rotation around the angular momentum vector.
### 3.4 Closedness and involutivity
In the pluri-Lagrangian theory, the exterior derivative $\mbox{\rm
d}\mathcal{L}$ is constant on solutions (see Proposition A.2 in the Appendix).
In many cases this constant is zero, i.e. the Lagrangian 1-form is closed on
solutions. Here we relate this property to the vanishing of Poisson brackets
between the Hamilton functions.
###### Proposition 3.6 ([26, Theorem 3]).
Consider a Lagrangian 1-form $\mathcal{L}$ as in Section 3.2 and the
corresponding Hamilton functions $H_{i}$. On solutions to the multi-time
Euler-Lagrange equations, and identifying
$\pi=p(q,q_{1})=\frac{\partial{\mathcal{L}_{i}}}{\partial{q_{i}}}$, there
holds
$\begin{split}\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm
d}t_{i}}-\frac{\mbox{\rm d}\mathcal{L}_{i}}{\mbox{\rm
d}t_{j}}&=p_{j}q_{i}-p_{i}q_{j}\\\ &=\\{H_{j},H_{i}\\},\end{split}$ (16)
where $\\{\cdot,\cdot\\}$ denotes the canonical Poisson bracket and $p_{j}$
and $q_{j}$ are shorthand for $\frac{\mbox{\rm d}p}{\mbox{\rm d}t_{j}}$ and
$\frac{\mbox{\rm d}q}{\mbox{\rm d}t_{j}}$.
* Proof.
On solutions of the multi-time Euler-Lagrange equations there holds
$\displaystyle\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm d}t_{i}}$
$\displaystyle=\frac{\partial{\mathcal{L}_{j}}}{\partial{q}}q_{i}+\frac{\partial{\mathcal{L}_{j}}}{\partial{q_{1}}}q_{1i}+\frac{\partial{\mathcal{L}_{j}}}{\partial{q_{j}}}q_{ij}$
$\displaystyle=\left(\frac{\mbox{\rm d}}{\mbox{\rm
d}t_{j}}\frac{\partial{\mathcal{L}_{j}}}{\partial{q}}\right)q_{i}+\frac{\partial{\mathcal{L}_{j}}}{\partial{q_{j}}}q_{ij}$
$\displaystyle=p_{j}q_{i}+pq_{ij}.$
Hence
$\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm d}t_{i}}-\frac{\mbox{\rm
d}\mathcal{L}_{i}}{\mbox{\rm d}t_{j}}=p_{j}q_{i}-p_{i}q_{j}.$ (17)
Alternatively, we can calculate this expression using the Hamiltonian
formalism. We have
$\displaystyle\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm
d}t_{i}}-\frac{\mbox{\rm d}\mathcal{L}_{i}}{\mbox{\rm d}t_{j}}$
$\displaystyle=\frac{\mbox{\rm d}}{\mbox{\rm
d}t_{i}}(pq_{j}-H_{j})-\frac{\mbox{\rm d}}{\mbox{\rm d}t_{j}}(pq_{i}-H_{i})$
$\displaystyle=p_{i}q_{j}-p_{j}q_{i}+2\\{H_{j},H_{i}\\}.$
Combined with Equation (17), this implies Equation (16). ∎
As a corollary we have:
###### Theorem 3.7.
The Hamiltonians $H_{i}$ from Theorem 3.2 are in involution if and only if
$\mbox{\rm d}\mathcal{L}=0$ on solutions.
All examples of Lagrangian 1-forms discussed so far satisfy $\mbox{\rm
d}\mathcal{L}=0$ on solutions. This need not be the case.
###### Example 3.8.
Let us consider a system of commuting equations that is not Liouville
integrable. Fix a constant $c\neq 0$ and consider the 1-form
$\mathcal{L}=\mathcal{L}_{1}\,\mbox{\rm d}t_{1}+\mathcal{L}_{2}\,\mbox{\rm
d}t_{2}$ with
$\mathcal{L}_{1}\llbracket
r,\theta\rrbracket=\frac{1}{2}r^{2}\theta_{1}^{2}+\frac{1}{2}r_{1}^{2}-V(r)-c\theta,$
which for $c=0$ would describe a central force in the plane governed by the
potential $V$, and
$\mathcal{L}_{2}\llbracket
r,\theta\rrbracket=r^{2}\theta_{1}(\theta_{2}-1)+r_{1}r_{2}.$
Its multi-time Euler-Lagrange equations are
$\displaystyle r_{11}=-V^{\prime}(r)+r\theta_{1}^{2},$
$\displaystyle\frac{\mbox{\rm d}}{\mbox{\rm d}t_{1}}(r^{2}\theta_{1})=-c,$
$\displaystyle r_{2}=0,$ $\displaystyle\theta_{2}=1,$
and consequences thereof. Notably, we have
$\frac{\mbox{\rm d}\mathcal{L}_{2}}{\mbox{\rm d}t_{1}}-\frac{\mbox{\rm
d}\mathcal{L}_{1}}{\mbox{\rm d}t_{2}}=c$
on solutions, hence $\mbox{\rm d}\mathcal{L}$ is nonzero.
By Theorem 3.2 the multi-time Euler-Lagrange equations are equivalent to the
canonical Hamiltonian systems with
$\displaystyle H_{1}(r,\theta,\pi,\sigma)$
$\displaystyle=\frac{1}{2}\frac{\sigma^{2}}{r^{2}}+\frac{1}{2}\pi^{2}+V(r)+c\theta$
$\displaystyle H_{2}(r,\theta,\pi,\sigma)$ $\displaystyle=\sigma,$
where $\pi$ and $\sigma$ are the conjugate momenta to $r$ and $\theta$. The
Hamiltonians are not in involution, but rather
$\\{H_{2},H_{1}\\}=c=\frac{\mbox{\rm d}\mathcal{L}_{2}}{\mbox{\rm
d}t_{1}}-\frac{\mbox{\rm d}\mathcal{L}_{1}}{\mbox{\rm d}t_{2}}.$
## 4 Hamiltonian structure of Lagrangian 2-form systems
In order to generalize the results from Section 3 to the case of 2-forms, we
need to carefully examine the relevant geometric structure. A useful tool for
this is the variational bicomplex, which is also used in Appendix A to study
the multi-time Euler-Lagrange equations.
### 4.1 The variational bicomplex
To facilitate the variational calculus in the pluri-Lagrangian setting, it is
useful to consider the variation operator $\delta$ as an exterior derivative,
acting in the fiber $J^{\infty}$ of the infinite jet bundle. We call $\delta$
the _vertical exterior derivative_ and d, which acts in the base manifold $M$,
the _horizontal exterior derivative_. Together they provide a double grading
of the space $\Omega(M\times J^{\infty})$ of differential forms on the jet
bundle. The space of _$(a,b)$ -forms_ is generated by those $(a+b)$-forms
structured as
$f\llbracket u\rrbracket\,\delta u_{I_{1}}\wedge\ldots\wedge\delta
u_{I_{a}}\wedge\mbox{\rm d}t_{j_{1}}\ldots\wedge\mbox{\rm d}t_{j_{b}}.$
We denote the space of $(a,b)$-forms by
$\Omega^{(a,b)}\subset\Omega^{a+b}(M\times J^{\infty})$. We call elements of
$\Omega^{(0,b)}$ horizontal forms and elements of $\Omega^{(a,0)}$ vertical
forms. The Lagrangian is a horizontal $d$-form,
$\mathcal{L}\in\Omega^{(0,d)}$.
The horizontal and vertical exterior derivatives are characterized by the
anti-derivation property,
$\displaystyle\mbox{\rm
d}\left(\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}\right)$
$\displaystyle=\mbox{\rm
d}\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}+(-1)^{p_{1}+q_{1}}\,\omega_{1}^{p_{1},q_{1}}\wedge\mbox{\rm
d}\omega_{2}^{p_{2},q_{2}},$
$\displaystyle\delta\left(\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}\right)$
$\displaystyle=\delta\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}+(-1)^{p_{1}+q_{1}}\,\omega_{1}^{p_{1},q_{1}}\wedge\delta\omega_{2}^{p_{2},q_{2}},$
where the upper indices denote the type of the forms, and by the way they act
on $(0,0)$-forms, and basic $(1,0)$ and $(0,1)$-forms:
$\displaystyle\mbox{\rm d}f\llbracket u\rrbracket$
$\displaystyle=\sum_{j}\partial_{j}f\llbracket u\rrbracket\,\mbox{\rm
d}t_{j},$ $\displaystyle\delta f\llbracket u\rrbracket$
$\displaystyle=\sum_{I}\frac{\partial{f\llbracket
u\rrbracket}}{\partial{u_{I}}}\delta u_{I},$ $\displaystyle\mbox{\rm d}(\delta
u_{I})$ $\displaystyle=-\sum_{j}\delta u_{Ij}\wedge\mbox{\rm d}t_{j},\hskip
56.9055pt$ $\displaystyle\delta(\delta u_{I})$ $\displaystyle=0,$
$\displaystyle\mbox{\rm d}(\mbox{\rm d}t_{j})$ $\displaystyle=0,$
$\displaystyle\delta(\mbox{\rm d}t_{j})$ $\displaystyle=0.$
One can verify that $\mbox{\rm
d}+\delta:\Omega^{a+b}\rightarrow\Omega^{a+b+1}$ is the usual exterior
derivative and that
$\delta^{2}=\mbox{\rm d}^{2}=\delta\mbox{\rm d}+\mbox{\rm d}\delta=0.$
Time-derivatives $\partial_{j}$ act on vertical forms as $\partial_{j}(\delta
u_{I})=\delta u_{Ij}$, on horizontal forms as $\partial_{j}(\mbox{\rm
d}t_{k})=0$, and obey the Leibniz rule with respect to the wedge product. As a
simple but important example, note that
$\mbox{\rm d}(f\llbracket u\rrbracket\,\delta
u_{I})=\sum_{j=1}^{N}\partial_{j}f\llbracket u\rrbracket\,\mbox{\rm
d}t_{j}\wedge\delta u_{I}-f\llbracket u\rrbracket\,\delta
u_{It_{j}}\wedge\mbox{\rm d}t_{j}=\sum_{j=1}^{N}-\partial_{j}(f\llbracket
u\rrbracket\,\delta u_{I})\wedge\mbox{\rm d}t_{j}.$
The spaces $\Omega^{(a,b)}$, for $a\geq 0$ and $0\leq b\leq N$, related to
each other by the maps d and $\delta$, are collectively known as the
_variational bicomplex_ [8, Chapter 19]. A slightly different version of the
variational bicomplex, using contact 1-forms instead of vertical forms, is
presented in [1]. We will not discuss the rich algebraic structure of the
variational bicomplex here.
For a horizontal $(0,d)$-form $\mathcal{L}\llbracket u\rrbracket$, the
variational principle
$\delta\int_{\Gamma}\mathcal{L}\llbracket
u\rrbracket=\delta\int_{\Gamma}\sum_{i_{1}<\ldots<i_{d}}\mathcal{L}_{i_{1},\ldots,i_{d}}\llbracket
u\rrbracket\,\mbox{\rm d}t_{i_{1}}\wedge\ldots\wedge\mbox{\rm d}t_{i_{d}}=0$
can be understood as follows. Every vertical vector field
$V=v(t_{1},\ldots,t_{a})\frac{\partial}{\partial u}$, such that its
_prolongation_
$\operatorname{pr}V=\sum_{I}v_{I}\frac{\partial}{\partial u_{I}}$
vanishes on the boundary $\partial\Gamma$, must satisfy
$\int_{\Gamma}\iota_{\operatorname{pr}V}\delta\mathcal{L}=\int_{\Gamma}\sum_{i_{1}<\ldots<i_{d}}\iota_{\operatorname{pr}V}(\delta\mathcal{L}_{i_{1},\ldots,i_{d}}\llbracket
u\rrbracket)\,\mbox{\rm d}t_{i_{1}}\wedge\ldots\wedge\mbox{\rm d}t_{i_{d}}=0.$
Note that the integrand is a horizontal form, so the integration takes place
on $\Gamma\subset M$, independent of the bundle structure.
### 4.2 The space of functionals and its pre-symplectic structure
In the rest of our discussion, we will single out the variable $t_{1}=x$ and
view it as the space variable, as opposed to the time variables
$t_{2},\ldots,t_{N}$. For ease of presentation we will limit the discussion
here to real scalar fields, but it is easily extended to complex or vector-
valued fields. We consider functions
$u:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto u(x)$ as fields at a fixed time.
Let $J^{\infty}$ be the fiber of the corresponding infinite jet bundle, where
the prolongation of $u$ has coordinates $[u]=(u,u_{x},u_{xx},\ldots)$.
Consider the space of functions of the infinite jet of $u$,
$\mathcal{V}=\left\\{v:J^{\infty}\rightarrow\mathbb{R}\right\\}.$
Note that the domain $J^{\infty}$ is the fiber of the jet bundle, hence the
elements $v\in\mathcal{V}$ depend on $x$ only through $u$. We will be dealing
with integrals $\int v\,\mbox{\rm d}x$ of elements $v\in\mathcal{V}$. In order
to avoid convergence questions, we understand the symbol $\int v\,\mbox{\rm
d}x$ as a _formal integral_ , defined as the equivalence class of $v$ modulo
space-derivatives. In other words, we consider the space of functionals
$\mathcal{F}=\mathcal{V}\big{/}\partial_{x}\\!\mathcal{V},$
where
$\partial_{x}=\frac{\mbox{\rm d}}{\mbox{\rm
d}x}=\sum_{I}u_{Ix}\frac{\partial{}}{\partial{u_{I}}}.$
The variation of an element of $\mathcal{F}$ is computed as
$\delta\int v\,\mbox{\rm d}x=\int\frac{\delta{v}}{\delta{u}}\,\delta
u\wedge\mbox{\rm d}x,$ (18)
where
$\frac{\delta{}}{\delta{u}}=\sum_{\alpha=0}^{\infty}(-1)^{\alpha}\partial_{x}^{\alpha}\frac{\partial{}}{\partial{u_{x^{\alpha}}}}.$
Equation (18) is independent of the choice of representative $v\in\mathcal{V}$
because the variational derivative of a full $x$-derivative is zero.
Since $\mathcal{V}$ is a linear space, its tangent spaces can be identified
with $\mathcal{V}$ itself. In turn, every $v\in\mathcal{V}$ can be identified
with a vector field $v\frac{\partial}{\partial u}$. We will define Hamiltonian
vector fields in terms of $\mathcal{F}$-valued forms on $\mathcal{V}$. An
$\mathcal{F}$-valued 1-form $\theta$ can be represented as the integral of a
$(1,1)$-form in the variational bicomplex,
$\theta=\int\sum_{k}a_{k}[u]\,\delta u_{x^{k}}\wedge\mbox{\rm d}x$
and defines a map
$\mathcal{V}\rightarrow\mathcal{F}:v\mapsto\iota_{v}\theta=\int\sum_{k}a_{k}[u]\,\partial_{x}^{k}v[u]\,\mbox{\rm
d}x.$
This amounts to pairing the 1-form with the infinite jet prolongation of the
vector field $v\frac{\partial}{\partial u}$. Note that $\mathcal{F}$-valued
forms are defined modulo $x$-derivatives:
$\int\partial_{x}\theta\wedge\mbox{\rm d}x=0$ because its pairing with any
vector field in $\mathcal{V}$ will yield a full $x$-derivative, which
represents the zero functional in $\mathcal{F}$. Hence the space of
$\mathcal{F}$-valued 1-forms is $\Omega^{(1,1)}/\partial_{x}\Omega^{(1,1)}$.
An $\mathcal{F}$-valued 2-form
$\omega=\int\sum_{k,\ell}a_{k,\ell}[u]\,\delta u_{x^{k}}\wedge\delta
u_{x^{\ell}}\wedge\mbox{\rm d}x$
defines a skew-symmetric map
$\mathcal{V}\times\mathcal{V}\rightarrow\mathcal{F}:(v,w)\mapsto\iota_{w}\iota_{v}\omega=\int\sum_{k,\ell}a_{k,\ell}[u]\left(\partial_{x}^{k}v[u]\,\partial_{x}^{\ell}w[u]-\partial_{x}^{k}w[u]\,\partial_{x}^{\ell}v[u]\right)\mbox{\rm
d}x$
as well as a map from vector fields to $\mathcal{F}$-valued 1-forms
$\mathcal{V}\rightarrow\Omega^{(1,1)}/\partial_{x}\Omega^{(1,1)}:v\mapsto\iota_{v}\omega=\int\sum_{k,\ell}a_{k,\ell}[u]\left(\partial_{x}^{k}v[u]\,\delta
u_{x^{\ell}}-\partial_{x}^{\ell}v[u]\,\delta u_{x^{k}}\right)\wedge\mbox{\rm
d}x.$
###### Definition 4.1.
A closed $(2,1)$-form $\omega$ on $\mathcal{V}$ is called _pre-symplectic_.
Equivalently we can require the form to be vertically closed, i.e. closed with
respect to $\delta$. Since the horizontal space is 1-dimensional ($x$ is the
only independent variable) every $(a,1)$-form is closed with respect to the
horizontal exterior derivative d, so only vertical closedness is a nontrivial
property.
We choose to work with pre-symplectic forms instead of symplectic forms,
because the non-degeneracy required of a symplectic form is a subtle issue in
the present context. Consider for example the pre-symplectic form
$\omega=\int\delta u\wedge\delta u_{x}\wedge\mbox{\rm d}x$. It is degenerate
because
$\int\iota_{v}\omega=\int(v\,\delta u_{x}-v_{x}\,\delta u)\wedge\mbox{\rm
d}x=\int-2v_{x}\,\delta u\wedge\mbox{\rm d}x,$
which is zero whenever $v[u]$ is constant. However, if we restrict our
attention to compactly supported fields, then a constant must be zero, so the
restriction of $\omega$ to the space of compactly supported fields is non-
degenerate.
###### Definition 4.2.
A _Hamiltonian vector field_ with Hamilton functional
${\textstyle\int}H\,\mbox{\rm d}x$ is an element $v\in\mathcal{V}$ satisfying
the relation
$\int\iota_{v}\omega=\int\delta H\wedge\mbox{\rm d}x.$
Note that if $\omega$ is degenerate, we cannot guarantee existence or
uniqueness of a Hamiltonian vector field in general.
### 4.3 From pluri-Lagrangian to Hamiltonian systems
We will consider two different types of Lagrangian 2-forms. The first type are
those where for every $j$ the coefficient $\mathcal{L}_{1j}$ is linear in
$u_{t_{j}}$. This is the case for the 2-form for the potential KdV hierarchy
from Example 2.5 and for the Lagrangian 2-forms of many other hierarchies like
the AKNS hierarchy [21] and the modified KdV, Schwarzian KdV and Krichever-
Novikov hierarchies [30]. The second type satisfy the same property for $j>2$,
but have a coefficient $\mathcal{L}_{12}$ that is quadratic in $u_{t_{2}}$, as
is the case for the Boussinesq hierarchy from Example 2.6.
#### 4.3.1 When all $\mathcal{L}_{1j}$ are linear in $u_{t_{j}}$
Consider a Lagrangian 2-form $\mathcal{L}\llbracket
u\rrbracket=\sum_{i<j}\mathcal{L}_{ij}\llbracket u\rrbracket\,\mbox{\rm
d}t_{i}\wedge\mbox{\rm d}t_{j}$, where for all $j$ the variational derivative
$\frac{\delta_{1}{\mathcal{L}_{1j}}}{\delta{u_{t_{j}}}}$ does not depend on
any $t_{j}$-derivatives, hence we can write
$\frac{\delta_{1}{\mathcal{L}_{1j}}}{\delta{u_{t_{j}}}}=p[u]$
for some function $p[u]$ depending on on an arbitrary number of space
derivatives, but not on any time-derivatives. We use single square brackets
$[\cdot]$ to indicate dependence on space derivatives only. Note that $p$ does
not depend on the index $j$. This is imposed on us by the multi-time Euler-
Lagrange equation stating that
$\frac{\delta_{1}{\mathcal{L}_{1j}}}{\delta{u_{t_{j}}}}$ is independent of
$j$.
Starting from these assumptions and possibly adding a full $x$-derivative
(recall that $x=t_{1}$) we find that the coefficients $\mathcal{L}_{1j}$ are
of the form
$\mathcal{L}_{1j}\llbracket u\rrbracket=p[u]u_{j}-h_{j}[u],$ (19)
where $u_{j}$ is shorthand notation for the derivative $u_{t_{j}}$.
Coefficients of this form appear in many prominent examples, like the
potential KdV hierarchy and several hierarchies related to it [28, 29, 30] as
well as the AKNS hierarchy [21]. Their Euler-Lagrange equations are
$\mathcal{E}_{p}u_{j}-\frac{\delta_{1}{h_{j}[u]}}{\delta{u}}=0,$ (20)
where $\mathcal{E}_{p}$ is the differential operator
$\mathcal{E}_{p}=\sum_{k=0}^{\infty}\left((-1)^{k}\partial_{x}^{k}\frac{\partial{p}}{\partial{u_{x^{k}}}}-\frac{\partial{p}}{\partial{u_{x^{k}}}}\partial_{x}^{k}\right).$
We can also write $\mathcal{E}_{p}=\mathsf{D}_{p}^{*}-\mathsf{D}_{p}$, where
$\mathsf{D}_{p}$ is the Fréchet derivative of $p$ and $\mathsf{D}_{p}^{*}$ its
adjoint [17, Eqs (5.32) resp. (5.79)].
Consider the pre-symplectic form
$\begin{split}\omega&=-\delta p[u]\wedge\delta u\wedge\mbox{\rm d}x\\\
&=-\sum_{k=1}^{\infty}\frac{\partial{p}}{\partial{u_{x^{k}}}}\delta
u_{x^{k}}\wedge\delta u\wedge\mbox{\rm d}x.\end{split}$ (21)
Inserting the vector field $X=\chi\frac{\partial{}}{\partial{u}}$ we find
$\displaystyle\int\iota_{X}\omega$
$\displaystyle=\int\sum_{k=0}^{\infty}\left(\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi\,\delta
u_{x^{k}}\wedge\mbox{\rm
d}x-\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi_{x^{k}}\,\delta
u\wedge\mbox{\rm d}x\right)$
$\displaystyle=\int\sum_{k=0}^{\infty}\left((-1)^{k}\partial_{x}^{k}\\!\left(\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi\right)-\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi_{x^{k}}\right)\delta
u\wedge\mbox{\rm d}x$ $\displaystyle=\int\mathcal{E}_{p}\chi\,\delta
u\wedge\mbox{\rm d}x.$
From the Hamiltonian equation of motion
$\int\iota_{X}\omega=\int\delta h_{j}[u]\wedge\mbox{\rm d}x$
we now obtain that the Hamiltonian vector field
$X=\chi\frac{\partial{}}{\partial{u}}$ associated to $h_{j}$ satisfies
$\mathcal{E}_{p}\chi=\frac{\delta_{1}{h_{j}}}{\delta{u}},$
which corresponds the Euler-Lagrange equation (20) by identifying
$\chi=u_{t_{j}}$. This observation was made previously in the context of loop
spaces in [16, Section 1.3].
The Poisson bracket associated to the symplectic operator $\mathcal{E}_{p}$ is
formally given by
$\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm
d}x\right\\}=-\int\frac{\delta{f}}{\delta{u}}\,\mathcal{E}_{p}^{-1}\,\frac{\delta{g}}{\delta{u}}\,\mbox{\rm
d}x.$ (22)
If the pre-symplectic form is degenerate, then $\mathcal{E}_{p}$ will not be
invertible. In this case $\mathcal{E}_{p}^{-1}$ can be considered as a pseudo-
differential operator and the Poisson bracket is called _non-local_ [16, 7].
Note that $\\{\cdot,\cdot\\}$ does not satisfy the Leibniz rule because there
is no multiplication on the space $\mathcal{F}$ of formal integrals. However,
we can recover the Leibniz rule in one entry by introducing
$[f,g]=-\sum_{k=0}^{\infty}\frac{\partial{f}}{\partial{u_{x^{k}}}}\partial_{x}^{k}\,\mathcal{E}_{p}^{-1}\,\frac{\delta{g}}{\delta{u}}.$
Then we have
$\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm
d}x\right\\}=\int[f,g]\,\mbox{\rm d}x$
and
$[fg,h]=f[g,h]+[f,h]g.$
In summary, we have the following result:
###### Theorem 4.3.
Assume that $\frac{\delta_{1}{h_{j}[u]}}{\delta{u}}$ is in the image of
$\mathcal{E}_{p}$ and has a unique inverse (possibly in some equivalence
class) for each $j$. Then the evolutionary PDEs
$u_{j}=\mathcal{E}_{p}^{-1}\frac{\delta_{1}{h_{j}[u]}}{\delta{u}},$
which imply the Euler-Lagrange equations (20) of the Lagrangians (19), are
Hamiltonian with respect to the symplectic form (21) and the Poisson bracket
(22), with Hamilton functions $h_{j}$.
This theorem applies without assuming any kind of consistency of the system of
multi-time Euler-Lagrange equations. Of course we are mostly interested in the
case where the multi-time Euler-Lagrange equations are equivalent to an
integrable hierarchy. In almost all known examples (see e.g. [28, 21, 30]) the
multi-time Euler-Lagrange equations consist of an integrable hierarchy in its
evolutionary form and differential consequences thereof. Hence the general
picture suggested by these examples is that the multi-time Euler-Lagrange
equations are equivalent to the equations
$u_{j}=\mathcal{E}_{p}^{-1}\frac{\delta_{1}{h_{j}[u]}}{\delta{u}}$ form
Theorem 4.3. In light of these observations, we emphasize the following
consequence of Theorem 4.3.
###### Corollary 4.4.
If the multi-time Euler-Lagrange equations are evolutionary, then they are
Hamiltonian.
###### Example 4.5.
The pluri-Lagrangian structure for the potential KdV hierarchy, given in
Example 2.5, has $p=\frac{1}{2}u_{x}$. Hence
$\mathcal{E}_{p}=-\partial_{x}\frac{\partial{p}}{\partial{u_{x}}}-\frac{\partial{p}}{\partial{u_{x}}}\partial_{x}=-\partial_{x}$
and
$\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm
d}x\right\\}=\int\frac{\delta{f}}{\delta{u}}\partial_{x}^{-1}\frac{\delta{g}}{\delta{u}}\,\mbox{\rm
d}x.$
Here we assume that $\frac{\delta{g}}{\delta{u}}$ is in the image of
$\partial_{x}$. Then $\partial_{x}^{-1}\frac{\delta{g}}{\delta{u}}$ is
uniquely defined by the convention that it does not contain a constant term.
If $f$ and $g$ depend only on derivatives of $u$, not on $u$ itself, this
becomes the Gardner bracket [10]
$\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm
d}x\right\\}=\int\left(\partial_{x}\frac{\delta{f}}{\delta{u_{x}}}\right)\frac{\delta{g}}{\delta{u_{x}}}\,\mbox{\rm
d}x.$
The Hamilton functions are
$\displaystyle h_{2}[u]$
$\displaystyle=\frac{1}{2}u_{x}u_{t_{2}}-\mathcal{L}_{12}=u_{x}^{3}+\frac{1}{2}u_{x}u_{xxx},$
$\displaystyle h_{3}[u]$
$\displaystyle=\frac{1}{2}u_{x}u_{t_{3}}-\mathcal{L}_{13}=\frac{5}{2}u_{x}^{4}-5u_{x}u_{xx}^{2}+\frac{1}{2}u_{xxx}^{2},$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
A related derivation of the Gardner bracket from the multi-symplectic
perspective was given in [11]. It can also be obtained from the Lagrangian
structure by Dirac reduction [15].
###### Example 4.6.
The Schwarzian KdV hierarchy,
$\displaystyle u_{2}$ $\displaystyle=-\frac{3u_{11}^{2}}{2u_{1}}+u_{111},$
$\displaystyle u_{3}$
$\displaystyle=-\frac{45u_{11}^{4}}{8u_{1}^{3}}+\frac{25u_{11}^{2}u_{111}}{2u_{1}^{2}}-\frac{5u_{111}^{2}}{2u_{1}}-\frac{5u_{11}u_{1111}}{u_{1}}+u_{11111},$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
has a pluri-Lagrangian structure with coefficients [29]
$\displaystyle\mathcal{L}_{12}$
$\displaystyle=\frac{u_{3}}{2u_{1}}-\frac{u_{11}^{2}}{2u_{1}^{2}},$
$\displaystyle\mathcal{L}_{13}$
$\displaystyle=\frac{u_{5}}{2u_{1}}-\frac{3u_{11}^{4}}{8u_{1}^{4}}+\frac{u_{111}^{2}}{2u_{1}^{2}},$
$\displaystyle\mathcal{L}_{23}$
$\displaystyle=-\frac{45u_{11}^{6}}{32u_{1}^{6}}+\frac{57u_{11}^{4}u_{111}}{16u_{1}^{5}}-\frac{19u_{11}^{2}u_{111}^{2}}{8u_{1}^{4}}+\frac{7u_{111}^{3}}{4u_{1}^{3}}-\frac{3u_{11}^{3}u_{1111}}{4u_{1}^{4}}-\frac{3u_{11}u_{111}u_{1111}}{2u_{1}^{3}}$
$\displaystyle\quad+\frac{u_{1111}^{2}}{2u_{1}^{2}}+\frac{3u_{11}^{2}u_{11111}}{4u_{1}^{3}}-\frac{u_{111}u_{11111}}{2u_{1}^{2}}+\frac{u_{111}u_{113}}{u_{1}^{2}}-\frac{3u_{11}^{3}u_{13}}{2u_{1}^{4}}+\frac{2u_{11}u_{111}u_{13}}{u_{1}^{3}}$
$\displaystyle\quad-\frac{u_{1111}u_{13}}{u_{1}^{2}}+\frac{u_{11}u_{15}}{u_{1}^{2}}-\frac{27u_{11}^{4}u_{3}}{16u_{1}^{5}}+\frac{17u_{11}^{2}u_{111}u_{3}}{4u_{1}^{4}}-\frac{7u_{111}^{2}u_{3}}{4u_{1}^{3}}-\frac{3u_{11}u_{1111}u_{3}}{2u_{1}^{3}}$
$\displaystyle\quad+\frac{u_{11111}u_{3}}{2u_{1}^{2}}+\frac{u_{11}^{2}u_{5}}{4u_{1}^{3}}-\frac{u_{111}u_{5}}{2u_{1}^{2}},$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
In this example we have $p=\frac{1}{2u_{x}}$, hence
$\mathcal{E}_{p}=-\partial_{x}\frac{\partial{p}}{\partial{u_{x}}}-\frac{\partial{p}}{\partial{u_{x}}}\partial_{x}=\frac{1}{u_{x}^{2}}\partial_{x}-\frac{u_{xx}}{u_{x}^{3}}=\frac{1}{u_{x}}\partial_{x}\frac{1}{u_{x}}$
and
$\mathcal{E}_{p}^{-1}=u_{x}\partial_{x}^{-1}u_{x}.$
This nonlocal operator seems to be the simplest Hamiltonian operator for the
SKdV equation, see for example [9, 31]. The Hamilton functions for the first
two equations of the hierarchy are
$h_{2}=\frac{u_{11}^{2}}{2u_{1}^{2}}\qquad\text{and}\qquad
h_{3}=\frac{3u_{11}^{4}}{8u_{1}^{4}}-\frac{u_{111}^{2}}{2u_{1}^{2}}.$
#### 4.3.2 When $\mathcal{L}_{12}$ is quadratic in $u_{t_{2}}$
Consider a Lagrangian 2-form $\mathcal{L}\llbracket
u\rrbracket=\sum_{i<j}\mathcal{L}_{ij}\llbracket u\rrbracket\,\mbox{\rm
d}t_{i}\wedge\mbox{\rm d}t_{j}$ with
$\mathcal{L}_{12}=\frac{1}{2}\alpha[u]u_{2}^{2}-V[u],$ (23)
and, for all $j\geq 3$, $\mathcal{L}_{1j}$ of the form
$\mathcal{L}_{1j}\llbracket u\rrbracket=\alpha[u]u_{2}u_{j}-h_{j}[u,u_{2}],$
(24)
where $[u,u_{2}]=(u,u_{2},u_{1},u_{12},u_{11},u_{112},\ldots)$ since the
single bracket $[\cdot]$ denotes dependence on the fields and their
$x$-derivatives only (recall that $x=t_{1}$). To write down the full set of
multi-time Euler-Lagrange equations we need to specify all $\mathcal{L}_{ij}$,
but for the present discussion it is sufficient to consider the equations
$\frac{\delta_{12}{\mathcal{L}_{12}}}{\delta{u}}=0\quad\Leftrightarrow\quad\alpha[u]u_{22}=-\frac{\mbox{\rm
d}\alpha[u]}{\mbox{\rm
d}t_{2}}u_{2}+\frac{1}{2}\sum_{k=0}^{\infty}(-1)^{k}\partial_{x}^{k}\left(\frac{\partial{\alpha[u]}}{\partial{u_{x^{k}}}}u_{2}^{2}\right)-\frac{\delta_{1}{V[u]}}{\delta{u}}$
and
$\frac{\delta_{1j}{\mathcal{L}_{1j}}}{\delta{u_{2}}}=0\quad\Leftrightarrow\quad\alpha[u]u_{j}=\frac{\delta_{1}{h_{j}[u,u_{2}]}}{\delta{u_{2}}}.$
We assume that all other multi-time Euler-Lagrange equations are consequences
of these.
Since $\mathcal{L}_{12}$ is non-degenerate, the Legendre transform is
invertible and allows us to identify $\pi=\alpha[u]u_{2}$. Consider the
canonical symplectic form on formal integrals, where now the momentum $\pi$
enters as a second field,
$\omega=\delta\pi\wedge\delta u\wedge\mbox{\rm d}x.$
This defines the Poisson bracket
$\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm
d}x\right\\}=-\int\left(\frac{\delta{f}}{\delta{\pi}}\frac{\delta{g}}{\delta{u}}-\frac{\delta{f}}{\delta{u}}\frac{\delta{g}}{\delta{\pi}}\right)\mbox{\rm
d}x.$ (25)
The coefficients $\mathcal{L}_{1j}\llbracket
u\rrbracket=\alpha[u]u_{2}u_{j}-h_{j}[u,u_{2}]$ are linear in their velocities
$u_{j}$, hence they are Hamiltonian with respect to the pre-symplectic form
$\delta(\alpha[u]u_{2})\wedge\delta u\wedge\mbox{\rm d}x,$
which equals $\omega$ if we identify $\pi=\alpha[u]u_{2}$. Hence we find the
following result.
###### Theorem 4.7.
A hierarchy described by a Lagrangian 2-form with coefficients of the form
(23)–(24) is Hamiltonian with respect to the canonical Poisson bracket (25),
with Hamilton functions
$H_{2}[u,\pi]=\frac{1}{2}\frac{\pi^{2}}{\alpha[u]}+V[u]$
and
$H_{j}[u,\pi]=h_{j}\\!\left[u,\frac{\pi}{\alpha[u]}\right]$
for $j\geq 3$.
###### Example 4.8.
The Lagrangian 2-form for the Boussinesq hierarchy from Example 2.6 leads to
$\displaystyle H_{2}$
$\displaystyle=\frac{1}{2}\pi^{2}+2u_{1}^{3}+\frac{3}{2}u_{11}^{2},$
$\displaystyle H_{3}$
$\displaystyle=-6u_{1}^{4}-27u_{1}u_{11}^{2}+6u\pi_{1}\pi-\frac{9}{2}u_{111}^{2}-\frac{3}{2}\pi_{1}^{2},$
where the Legendre transform identifies $\pi=u_{2}$. Indeed we have
$\displaystyle\left\\{{\textstyle\int}H_{2}\,\mbox{\rm
d}x,{\textstyle\int}u\,\mbox{\rm d}x\right\\}$
$\displaystyle={\textstyle\int}\pi\,\mbox{\rm
d}x={\textstyle\int}u_{2}\,\mbox{\rm d}x,$
$\displaystyle\left\\{{\textstyle\int}H_{2}\,\mbox{\rm
d}x,{\textstyle\int}\pi\,\mbox{\rm d}x\right\\}$
$\displaystyle={\textstyle\int}(12u_{1}u_{11}-3u_{111})\,\mbox{\rm
d}x={\textstyle\int}\pi_{2}\,\mbox{\rm d}x,$
and
$\displaystyle\left\\{{\textstyle\int}H_{3}\,\mbox{\rm
d}x,{\textstyle\int}u\,\mbox{\rm d}x\right\\}$
$\displaystyle={\textstyle\int}(-6u_{1}\pi+3\pi_{11})\,\mbox{\rm
d}x={\textstyle\int}u_{3}\,\mbox{\rm d}x,$
$\displaystyle\left\\{{\textstyle\int}H_{3}\,\mbox{\rm
d}x,{\textstyle\int}\pi\,\mbox{\rm d}x\right\\}$
$\displaystyle={\textstyle\int}\left(-72u_{1}^{2}u_{11}+108u_{11}u_{111}+54u_{1}u_{1111}-6\pi\pi_{1}-9u_{111111}\right)\mbox{\rm
d}x$ $\displaystyle={\textstyle\int}\pi_{3}\,\mbox{\rm d}x.$
### 4.4 Closedness and involutivity
Let us now have a look at the relation between the closedness of the
Lagrangian 2-form and the involutivity of the corresponding Hamiltonians.
###### Proposition 4.9.
On solutions of the multi-time Euler-Lagrange equations of a Lagrangian 2-form
with coefficients $\mathcal{L}_{1j}$ given by Equation (19), there holds
$\\{h_{i},h_{j}\\}=\int\left(p_{i}u_{j}-p_{j}u_{i}\right)\mbox{\rm
d}x=\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm
d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm
d}t_{i}}\right)\mbox{\rm d}x,$ (26)
where the Poisson bracket is given by Equation (22).
* Proof.
On solutions of the Euler-Lagrange equations we have
$\displaystyle\int\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm
d}t_{j}}\,\mbox{\rm d}x$
$\displaystyle=\int\left(\frac{\delta_{1}{\mathcal{L}_{1i}}}{\delta{u}}u_{j}+\frac{\partial{\mathcal{L}_{1i}}}{\partial{u_{i}}}u_{ij}\right)\mbox{\rm
d}x$ $\displaystyle=\int\left(\left(\frac{\mbox{\rm d}}{\mbox{\rm
d}t_{i}}\frac{\delta_{1i}{\mathcal{L}_{1i}}}{\delta{u_{i}}}\right)u_{j}+\frac{\partial{\mathcal{L}_{1i}}}{\partial{u_{i}}}u_{ij}\right)\mbox{\rm
d}x$ $\displaystyle=\int\left(p_{i}u_{j}+pu_{ij}\right)\mbox{\rm d}x.$
It follows that
$\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm
d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm
d}t_{i}}\right)\mbox{\rm d}x=\int\left(p_{i}u_{j}-p_{j}u_{i}\right)\mbox{\rm
d}x.$ (27)
On the other hand we have that
$\displaystyle\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm
d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm
d}t_{i}}\right)\mbox{\rm d}x$ $\displaystyle=\int\left(\frac{\mbox{\rm
d}}{\mbox{\rm d}t_{j}}(pu_{i}-h_{i})-\frac{\mbox{\rm d}}{\mbox{\rm
d}t_{i}}(pu_{j}-h_{j})\right)\mbox{\rm d}x$
$\displaystyle=-\int\left(p_{i}u_{j}-p_{j}u_{i}\right)\mbox{\rm
d}x+2\\{h_{i},h_{j}\\}.$
Combined with Equation (27), this implies both identities in Equation (26). ∎
###### Proposition 4.10.
On solutions of the multi-time Euler-Lagrange equations of a Lagrangian 2-form
with coefficients $\mathcal{L}_{1j}$ given by Equations (23)–(24), there holds
$\\{H_{i},H_{j}\\}=\int\left(\pi_{i}u_{j}-\pi_{j}u_{i}\right)\mbox{\rm
d}x=\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm
d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm
d}t_{i}}\right)\mbox{\rm d}x,$
where the Poisson bracket is given by Equation (25) and the Hamilton functions
$H_{j}$ are given in Theorem 4.7.
* Proof.
Analogous to the proof of Proposition 4.9, with $p[u]$ replaced by the field
$\pi$. ∎
###### Theorem 4.11.
Let $\mathcal{L}$ be a Lagrangian 2-form with coefficients $\mathcal{L}_{1j}$
given by Equation (19) or by Equations (23)–(24). Consider the corresponding
Hamiltonian structures, given by $H_{1j}=h_{j}$ or $H_{1j}=H_{j}$, as in
Theorems 4.3 and 4.7 respectively. There holds $\\{H_{1i},H_{1j}\\}=0$ if and
only if
$\int\left(\frac{\mbox{\rm d}\mathcal{L}_{ij}}{\mbox{\rm
d}t_{1}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm
d}t_{i}}+\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm
d}t_{j}}\right)\mbox{\rm d}x=0$
on solutions of the multi-time Euler-Lagrange equations.
* Proof.
Recall that $t_{1}=x$, hence $\frac{\mbox{\rm d}}{\mbox{\rm
d}t_{1}}=\partial_{x}$. By definition of the formal integral as an equivalence
class, we have $\int\partial_{x}\mathcal{L}_{ij}\,\mbox{\rm d}x=0$. Hence the
claim follows from Proposition 4.9 or Proposition 4.10. ∎
It is known that $\mbox{\rm d}\mathcal{L}\llbracket u\rrbracket$ is constant
in the set of solutions $u$ to the multi-time Euler-Lagrange equations (see
Proposition A.2). In most examples, one can verify using a trivial solution
that this constant is zero.
###### Corollary 4.12.
If a Lagrangian 2-form, with coefficients $\mathcal{L}_{1j}\llbracket
u\rrbracket$ given by Equation (19) or by Equations (23)–(24), is closed for a
solution $u$ to the pluri-Lagrangian problem, then $\\{H_{1i},H_{1j}\\}=0$ for
all $i,j$.
All examples of Lagrangian 2-forms discussed so far satisfy $\mbox{\rm
d}\mathcal{L}=0$ on solutions. We now present a system where this is not the
case.
###### Example 4.13.
Consider a perturbation of the Boussinesq Lagrangian, obtained by adding $cu$
for some constant $c\in\mathbb{R}$,
$\mathcal{L}_{12}=\frac{1}{2}u_{2}^{2}-2u_{1}^{3}-\frac{3}{2}u_{11}^{2}+cu,$
combined with the Lagrangian coefficients
$\displaystyle\mathcal{L}_{13}$ $\displaystyle=u_{2}(u_{3}-1)$
$\displaystyle\mathcal{L}_{23}$
$\displaystyle=(6u_{1}^{2}-3u_{111})(u_{3}-1).$
The corresponding multi-time Euler-Lagrange equations consist of a perturbed
Boussinesq equation,
$u_{22}=12u_{1}u_{11}-3u_{1111}+c$
and
$u_{3}=1.$
We have
$\frac{\mbox{\rm d}\mathcal{L}_{12}}{\mbox{\rm d}t_{3}}-\frac{\mbox{\rm
d}\mathcal{L}_{13}}{\mbox{\rm d}t_{2}}+\frac{\mbox{\rm
d}\mathcal{L}_{23}}{\mbox{\rm d}t_{1}}=c$
on solutions, hence $\mbox{\rm d}\mathcal{L}$ is nonzero.
The multi-time Euler-Lagrange equations are equivalent to the canonical
Hamiltonian systems with
$\displaystyle H_{2}$
$\displaystyle=\frac{1}{2}\pi^{2}+2u_{1}^{3}+\frac{3}{2}u_{11}^{2}-cu$
$\displaystyle H_{3}$ $\displaystyle=\pi.$
They are not in involution, but rather
$\left\\{{\textstyle\int}H_{2}\,\mbox{\rm d}x,{\textstyle\int}H_{3}\,\mbox{\rm
d}x\right\\}=\int\left(12u_{11}u_{1}-3u_{1111}+c\right)\mbox{\rm d}x=\int
c\,\mbox{\rm d}x.$
Note that if we would allow the fields in $\mathcal{V}$ to depend explicitly
on $x$, then we would find ${\textstyle\int}c\,\mbox{\rm
d}x={\textstyle\int}\partial_{x}(cx)\,\mbox{\rm d}x=0$. Note that this is not
a property of the Lagrangian form, but of the function space we work in.
Allowing fields that depend on $x$ affects the definition of the formal
integral ${\textstyle\int}(\cdot)\,\mbox{\rm d}x$ as an equivalence class
modulo $x$-derivatives. If dependence on $x$ is allowed, then there is no such
thing as a nonzero constant functional in this equivalence class. However, in
our definition of $\mathcal{V}$, fields can only depend on $x$ through $u$,
hence $c$ is not an $x$-derivative and ${\textstyle\int}c\,\mbox{\rm d}x$ is
not the zero element of $\mathcal{F}$.
### 4.5 Additional (nonlocal) Poisson brackets
Even though the closedness property in Section 4.4 involves all coefficients
of a Lagrangian 2-form $\mathcal{L}$, so far we have only used the first row
of coefficients $\mathcal{L}_{1j}$ to construct Hamiltonian structures. A
similar procedure can be carried out for other $\mathcal{L}_{ij}$, but the
results are not entirely satisfactory. In particular, it will not lead to true
bi-Hamiltonian structures. Because of this slightly disappointing outcome, we
will make no effort to present the most general statement possible. Instead we
make some convenient assumptions on the form of the coefficients
$\mathcal{L}_{ij}$.
Consider a Lagrangian 2-form $\mathcal{L}$ such that for all $i<j$ the
coefficient $\mathcal{L}_{ij}$ only contains derivatives with respect to
$t_{1}$, $t_{i}$ and $t_{j}$ (no “alien derivatives” in the terminology of
[29]). In addition, assume that $\mathcal{L}_{ij}$ can be written as the sum
of terms that each contain at most one derivative with respect to $t_{i}$ (if
$i>1$) or $t_{j}$. In particular, $\mathcal{L}_{ij}$ does not contain higher
derivatives with respect to $t_{i}$ (if $i>1$) or $t_{j}$, but mixed
derivatives with respect to $t_{1}$ and $t_{i}$ or $t_{1}$ and $t_{j}$ are
allowed. There is no restriction on the amount of $t_{1}$-derivatives.
To get a Hamiltonian description of the evolution along the time direction
$t_{j}$ from the Lagrangian $\mathcal{L}_{ij}$, we should consider both
$t_{1}$ and $t_{i}$ as space coordinates. Hence we will work on the space
$\mathcal{V}\big{/}\left(\partial_{1}\\!\mathcal{V}+\partial_{i}\\!\mathcal{V}\right).$
For $i>1$, consider the momenta
$p^{[i]}[u]=\frac{\delta_{1i}{\mathcal{L}_{ij}}}{\delta{u_{j}}}.$
From the assumption that each term of $\mathcal{L}_{ij}$ contains at most one
time-derivative it follows that $p^{[i]}$ only depends on $u$ and its
$x$-derivatives. Note that $p^{[i]}$ is independent of $j$ because of the
multi-time Euler-Lagrange equation (6). The variational derivative in the
definition of $p^{[i]}$ is in the directions $1$ and $i$, corresponding to the
formal integral, whereas the Lagrangian coefficient has indices $i$ and $j$.
However, we can also write
$p^{[i]}[u]=\frac{\delta_{1}{\mathcal{L}_{ij}}}{\delta{u_{j}}}$
because of the assumption on the derivatives that occur in $\mathcal{L}_{ij}$,
which excludes mixed derivatives with respect to $t_{i}$ and $t_{j}$.
As Hamilton function we can take
$H_{ij}=p^{[i]}u_{j}-\mathcal{L}_{ij}.$
Its formal integral $\int H_{ij}\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}$ does
not depend on any $t_{j}$-derivatives. Since we are working with 2-dimensional
integrals, we should take a $(2,2)$-form as symplectic form. In analogy to
Equation (13) we take
$\omega_{i}=-\delta p^{[i]}\wedge\delta u\wedge\mbox{\rm d}x\wedge\mbox{\rm
d}t_{i}.$
A Hamiltonian vector field $X=\chi\frac{\partial{}}{\partial{u}}$ satisfies
$\int\iota_{X}\omega_{i}=\int\delta H_{ij}\wedge\mbox{\rm d}x\wedge\mbox{\rm
d}t_{i}$
hence
$\mathcal{E}_{p^{[i]}}\chi=\frac{\delta_{1i}{H_{ij}}}{\delta{u}},$
where $\mathcal{E}_{p^{[i]}}$ is the differential operator
$\mathcal{E}_{p^{[i]}}=\sum_{k=0}^{\infty}\left((-1)^{k}\partial_{x}^{k}\frac{\partial{p^{[i]}}}{\partial{u_{x^{k}}}}-\frac{\partial{p^{[i]}}}{\partial{u_{x^{k}}}}\partial_{x}^{k}\right)$
The corresponding (nonlocal) Poisson bracket is
$\left\\{{\textstyle\int}f\,\mbox{\rm d}x\wedge\mbox{\rm
d}t_{i},{\textstyle\int}g\,\mbox{\rm d}x\wedge\mbox{\rm
d}t_{i}\right\\}_{i}=-\int\frac{\delta_{1i}{f}}{\delta{u}}\,\mathcal{E}_{p^{[i]}}^{-1}\,\frac{\delta_{1i}{g}}{\delta{u}}\,\mbox{\rm
d}x\wedge\mbox{\rm d}t_{i}.$
Note that $H$ is not skew-symmetric, $H_{ij}\neq H_{ji}$.
The space of functionals
$\mathcal{V}\big{/}\left(\partial_{1}\\!\mathcal{V}+\partial_{i}\\!\mathcal{V}\right)$,
on which the Poisson bracket $\\{\cdot,\cdot\\}_{i}$ is defined, depends on
$i$ and is different from the space of functionals for the bracket
$\\{\cdot,\cdot\\}$ from Equation (25). Hence no pair of these brackets are
compatible with each other in the sense of a bi-Hamiltonian system.
As before, we can relate Poisson brackets between the Hamilton functionals to
coefficients of $\mbox{\rm d}\mathcal{L}$.
###### Proposition 4.14.
Assume that for all $i,j>1$, $\mathcal{L}_{ij}$ does not depend on any second
or higher derivatives with respect to $t_{i}$ and $t_{j}$. On solutions of the
Euler-Lagrange equations there holds that, for $i,j,k>1$,
$\begin{split}\int\left(\frac{\mbox{\rm d}\mathcal{L}_{ij}}{\mbox{\rm
d}t_{k}}-\frac{\mbox{\rm d}\mathcal{L}_{ik}}{\mbox{\rm
d}t_{j}}\right)\mbox{\rm d}x\wedge\mbox{\rm
d}t_{i}&=\int\left(p^{[i]}_{j}u_{k}-p^{[i]}_{k}u_{j}\right)\mbox{\rm
d}x\wedge\mbox{\rm d}t_{i}\\\ &=\left\\{{\textstyle\int}H_{ij}\,\mbox{\rm
d}x\wedge\mbox{\rm d}t_{i},{\textstyle\int}H_{ik}\,\mbox{\rm
d}x\wedge\mbox{\rm d}t_{i}\right\\}_{i}.\end{split}$ (28)
* Proof.
Analogous to the proof of Proposition 4.9. ∎
###### Example 4.15.
For the potential KdV equation (see Example 2.5) we have
$p^{[2]}=\frac{\delta_{1}{\mathcal{L}_{23}}}{\delta{u_{3}}}=\frac{3}{2}u_{111}+\frac{3}{2}u_{1}^{2},$
hence
$\displaystyle\mathcal{E}_{p^{[2]}}$
$\displaystyle=-\partial_{1}\frac{\partial{p^{[i]}}}{\partial{u_{1}}}-\partial_{1}^{3}\frac{\partial{p^{[i]}}}{\partial{u_{111}}}-\frac{\partial{p^{[i]}}}{\partial{u_{1}}}\partial_{1}-\frac{\partial{p^{[i]}}}{\partial{u_{111}}}\partial_{1}^{3}$
$\displaystyle=-3\partial_{1}u_{1}-\frac{3}{2}\partial_{1}^{3}-3u_{1}\partial_{1}-\frac{3}{2}\partial_{1}^{3}$
$\displaystyle=-3\partial_{1}^{3}-6u_{1}\partial_{1}-3u_{11}.$
We have
$\displaystyle H_{23}$ $\displaystyle=p^{[2]}u_{3}-\mathcal{L}_{23}$
$\displaystyle=-3u_{1}^{5}+\frac{15}{2}u_{1}^{2}u_{11}^{2}-10u_{1}^{3}u_{111}+5u_{1}^{3}u_{3}-\frac{7}{2}u_{11}^{2}u_{111}-3u_{1}u_{111}^{2}+6u_{1}u_{11}u_{1111}$
$\displaystyle\quad-\frac{3}{2}u_{1}^{2}u_{11111}-10u_{1}u_{11}u_{12}+\frac{5}{2}u_{11}^{2}u_{2}+5u_{1}u_{111}u_{2}+\frac{1}{2}u_{1111}^{2}-\frac{1}{2}u_{111}u_{11111}$
$\displaystyle\quad+\frac{1}{2}u_{111}u_{112}-\frac{1}{2}u_{1}u_{113}-u_{1111}u_{12}+\frac{1}{2}u_{11}u_{13}+\frac{1}{2}u_{11111}u_{2}+u_{111}u_{3},$
where the terms involving $t_{3}$-derivatives cancel out when the Hamiltonian
is integrated. Its variational derivative is
$\displaystyle\frac{\delta_{12}{H_{23}}}{\delta{u}}$
$\displaystyle=60u_{1}^{3}u_{11}+75u_{11}^{3}+300u_{1}u_{11}u_{111}+75u_{1}^{2}u_{1111}-30u_{1}^{2}u_{12}-30u_{1}u_{11}u_{2}$
$\displaystyle\quad+120u_{111}u_{1111}+72u_{11}u_{11111}+24u_{1}u_{111111}-30u_{1}u_{1112}-45u_{11}u_{112}$
$\displaystyle\quad-25u_{111}u_{12}-5u_{1111}u_{2}+2u_{11111111}-5u_{111112}.$
On solutions this simplifies to
$\displaystyle\frac{\delta_{12}{H_{23}}}{\delta{u}}$
$\displaystyle=-210u_{1}^{3}u_{11}-195u_{11}^{3}-690u_{1}u_{11}u_{111}-150u_{1}^{2}u_{1111}-210u_{111}u_{1111}$
$\displaystyle\quad-123u_{11}u_{11111}-36u_{1}u_{111111}-3u_{11111111}$
$\displaystyle=\mathcal{E}_{p^{[2]}}\left(10u_{1}^{3}+5u_{11}^{2}+10u_{1}u_{111}+u_{11111}\right)$
$\displaystyle=\mathcal{E}_{p^{[2]}}u_{3}.$
Hence
$\frac{\mbox{\rm d}}{\mbox{\rm d}t_{3}}\int u\,\mbox{\rm d}x\wedge\mbox{\rm
d}t_{2}=\int\mathcal{E}_{p^{[2]}}^{-1}\frac{\delta_{12}{H_{23}}}{\delta{u}}\,\mbox{\rm
d}x\wedge\mbox{\rm d}t_{2}=\left\\{{\textstyle\int}H_{23}\,\mbox{\rm
d}x\wedge\mbox{\rm d}t_{2},{\textstyle\int}u\,\mbox{\rm d}x\wedge\mbox{\rm
d}t_{2}\right\\}_{2}.$
### 4.6 Comparison with the covariant approach
In Section 4.5 we derived Poisson brackets $\\{\cdot,\cdot\\}_{i}$, associated
to each time variable $t_{i}$. This was somewhat cumbersome because we had a
priori assigned $x=t_{1}$ as a distinguished variable. The recent work [6]
explores the relation of pluri-Lagrangian structures to covariant Hamiltonian
structures. The meaning of “covariant” here is that all variables are on the
same footing; there is no distinguished $x$ variable. More details on
covariant field theory, and its connection to the distinguished-variable (or
“instantaneous”) perspective, can be found in [13]. The main objects in the
covariant Hamiltonian formulation of [6] are:
* •
A “symplectic multiform” $\Omega$, which can be expanded as
$\Omega=\sum_{j}\omega_{j}\wedge\mbox{\rm d}t_{j},$
where each $\omega_{j}$ is a vertical 2-form in the variational bicomplex.
* •
A “Hamiltonian multiform” $\mathcal{H}=\sum_{i<j}\mathtt{H}_{ij}\,\mbox{\rm
d}t_{i}\wedge\mbox{\rm d}t_{j}$ which gives the equations of motion through
$\delta\mathcal{H}=\sum_{j}\mbox{\rm d}t_{j}\wedge\xi_{j}\lrcorner\,\Omega,$
(29)
where $\delta$ is the vertical exterior derivative in the variational
bicomplex, $\xi_{j}$ denotes the vector field corresponding to the
$t_{j}$-flow, and $\lrcorner$ denotes the interior product. This equation
should be understood as a covariant version of the instantaneous Hamiltonian
equation $\delta H=\xi\lrcorner\,\omega$. On the equations of motion there
holds $\mbox{\rm d}\mathcal{H}=0$ if and only if $\mbox{\rm d}\mathcal{L}=0$.
Since the covariant Hamiltonian equation (29) is of a different form than the
instantaneous Hamiltonian equation we use, the coefficients $\mathtt{H}_{ij}$
of the Hamiltonian multiform $\mathcal{H}$ are also different from the
$H_{ij}$ we found in Sections 4.3–4.5. Our $H_{ij}$ are instantaneous
Hamiltonians where $t_{1}$ and $t_{i}$ are considered as space variables and
the Legendre transformation has been applied with respect to $t_{j}$.
* •
A “multi-time Poisson bracket” $\\{|\cdot,\cdot|\\}$ which defines a pairing
between functions or (a certain type of) horizontal one-forms, defined by
$\\{|F,G|\\}=(-1)^{r}\xi_{F}\delta G,$
where $\xi_{F}$ is the Hamiltonian (multi-)vector field associated to $F$, and
$r$ is the horizontal degree of $F$ (which is either 0 or 1). The equations of
motion can be written as
$\mbox{\rm d}F=\sum_{i<j}\\{|\mathtt{H}_{ij},F|\\}\,\mbox{\rm
d}t_{i}\wedge\mbox{\rm d}t_{j}.$
Single-time Poisson brackets are obtained in [6] by expanding the multi-time
Poisson bracket as
$\left\\{\left|{\textstyle\sum_{j}}F_{j}\,\mbox{\rm
d}t_{j},{\textstyle\sum_{j}}G_{j}\,\mbox{\rm
d}t_{j}\right|\right\\}=\sum_{j}\\{F_{j},G_{j}\\}_{j}\,\mbox{\rm d}t_{j}$
where
$\\{f,g\\}_{j}=-\xi_{f}^{j}\lrcorner\,\delta
g\qquad\text{and}\qquad\xi_{f}^{j}\lrcorner\,\omega_{j}=\delta f.$ (30)
These are fundamentally different from the Poisson brackets of Sections
4.3–4.5 because they act on different function spaces. Equation (30) assumes
that $\delta f$ lies in the image of $\omega_{j}$ (considered as a map from
vertical vector fields to vertical one-forms). For example, for the potential
KdV hiararchy one has $\omega_{1}=\delta v\wedge\delta v_{1}$, hence the
Poisson bracket $\\{\cdot,\cdot\\}_{1}$ can only be applied to functions of
$v$ and $v_{1}$, not to functions depending on any higher derivatives. Similar
conditions on the function space apply to the higher Poisson brackets
corresponding to $\omega_{j}$, $j\geq 2$. On the other hand, the Poisson
brackets of Sections 4.3–4.5 are defined on an equivalence class of functions
modulo certain derivatives, without further restrictions on the functions in
this class.
In summary, the single-time Poisson brackets of [6] are constructed with a
certain elegance in a covariant way, but they are defined only in a restricted
function space. They are different from our Poisson brackets of Section
4.3–4.5, which have no such restrictions, but break covariance already in the
definition of the function space as an equivalence class. It is not clear how
to pass from one picture to the other, or if their respective benefits can be
combined into a single approach.
## 5 Conclusions
We have established a connection between pluri-Lagrangian systems and
integrable Hamiltonian hierarchies. In the case of ODEs, where the pluri-
Lagrangian structure is a 1-form, this connection was already obtained in
[26]. Our main contribution is its generalization to the case of 2-dimensional
PDEs, described by Lagrangian 2-forms. Presumably, this approach extends to
Lagrangian $d$-forms of any dimension $d$, but the details of this are
postponed to future work.
A central property in the theory of pluri-Lagrangian systems is that the
Lagrangian form is (almost) closed on solutions. We showed that closedness is
equivalent to the corresponding Hamilton functions being in involution.
Although one can obtain several Poisson brackets (and corresponding Hamilton
functions) from one Lagrangian 2-form, these do not form a bi-Hamiltonian
structure and it is not clear if a recursion operator can be obtained from
them. Hence it remains an open question to find a fully variational
description of bi-Hamiltonian hierarchies.
### Acknowledgements
The author would like to thank Frank Nijhoff for his inspiring questions and
comments, Matteo Stoppato for helpful discussions about the covariant
Hamiltonian approach, Yuri Suris for his constructive criticism on early
drafts of this paper, and the anonymous referees for their thoughtful
comments.
The author is funded by DFG Research Fellowship VE 1211/1-1. Part of the work
presented here was done at TU Berlin, supported by the SFB Transregio 109
“Discretization in Geometry and Dynamics”.
## Appendix A Pluri-Lagrangian systems and the variational bicomplex
In this appendix we study the pluri-Lagrangian principle using the variational
bicomplex, described in Section 4.1. We provide proofs that the multi-time
Euler-Lagrange equations from Section 2 are sufficient conditions for
criticality. Alternative proofs of this fact can be found in [28] and [23,
Appendix A].
###### Proposition A.1.
The field $u$ is a solution to the pluri-Lagrangian problem of a $d$-form
$\mathcal{L}\llbracket u\rrbracket$ if locally there exists a $(1,d-1)$-form
$\Theta$ such that $\delta\mathcal{L}\llbracket u\rrbracket=\mbox{\rm
d}\Theta$.
* Proof.
Consider a field $u$ such that such a $(1,d-1)$-form $\Theta$ exists. Consider
any $d$-manifold $\Gamma$ and any variation $v$ that vanishes (along with all
its derivatives) on the boundary $\partial\Gamma$. Note that the horizontal
exterior derivative d anti-commutes with the interior product operator
$\iota_{V}$, where $V$ is the prolonged vertical vector field
$V=\operatorname{pr}(v\partial/\partial_{u})$ defined by the variation $v$. It
follows that
$\int_{\Gamma}\iota_{V}\delta\mathcal{L}=-\int_{\Gamma}\mbox{\rm
d}\left(\iota_{V}\Theta\right)=-\int_{\partial\Gamma}\iota_{V}\Theta=0,$
hence $u$ solves the pluri-Lagrangian problem. ∎
If we are dealing with a classical Lagrangian problem from mechanics,
$\mathcal{L}=L(u,u_{t})\,\mbox{\rm d}t$, we have
$\Theta=-\frac{\partial{L}}{\partial{u_{t}}}\delta u$, which is the pull back
to the tangent bundle of the canonical 1-form $\sum_{i}p_{i}\,\mbox{\rm
d}q_{i}$ on the cotangent bundle.
Often we want the Lagrangian form to be closed when evaluated on solutions. As
we saw in Theorems 3.7 and 4.11, this implies that the corresponding
Hamiltonians are in involution. We did not include this in the definition of a
pluri-Lagrangian system, because our definition already implies a slightly
weaker property.
###### Proposition A.2.
The horizontal exterior derivative $\mbox{\rm d}\mathcal{L}$ of a pluri-
Lagrangian form is constant on connected components of the set of critical
fields for $\mathcal{L}$.
* Proof.
Critical points satisfy locally
$\delta\mathcal{L}=\mbox{\rm d}\Theta\qquad\Rightarrow\qquad\mbox{\rm
d}\delta\mathcal{L}=0\qquad\Rightarrow\qquad\delta\mbox{\rm d}\mathcal{L}=0.$
Hence for any variation $v$ the Lie derivative of $\mbox{\rm d}\mathcal{L}$
along its prolongation $V=\operatorname{pr}(v\partial/\partial_{u})$ is
$\iota_{V}\delta(\mbox{\rm d}\mathcal{L})=0$. Therefore, if a solution $u$ can
be continuously deformed into another solution $\bar{u}$, then $\mbox{\rm
d}\mathcal{L}\llbracket u\rrbracket=\mbox{\rm
d}\mathcal{L}\llbracket\bar{u}\rrbracket$. ∎
Now let us prove the sufficiency of the multi-time Euler-Lagrange equations
for 1-forms and 2-forms, as given in Theorems 2.2 and 2.4. For different
approaches to the multi-time Euler-Lagrange equations, including proofs of
necessity, see [28] and [23].
* Proof of sufficiency in Theorem 2.2..
We calculate the vertical exterior derivative $\delta\mathcal{L}$ of the
Lagrangian 1-form, modulo the multi-time Euler-Lagrange Equations (1) and (2).
We have
$\displaystyle\delta\mathcal{L}$
$\displaystyle=\sum_{j=1}^{N}\sum_{I}\frac{\partial{\mathcal{L}_{j}}}{\partial{u_{I}}}\,\delta
u_{I}\wedge\mbox{\rm d}t_{j}$
$\displaystyle=\sum_{j=1}^{N}\sum_{I}\left(\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{I}}}+\partial_{j}\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}\right)\delta
u_{I}\wedge\mbox{\rm d}t_{j}.$
Rearranging this sum, we find
$\delta\mathcal{L}=\sum_{j=1}^{N}\left[\sum_{I\not\ni
t_{j}}\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{I}}}\delta
u_{I}\wedge\mbox{\rm
d}t_{j}+\sum_{I}\left(\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}\delta
u_{It_{j}}\wedge\mbox{\rm
d}t_{j}+\left(\partial_{j}\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}\right)\delta
u_{I}\wedge\mbox{\rm d}t_{j}\right)\right].$
On solutions of Equation (2), we can define the generalized momenta
$p^{I}=\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}.$
Using Equations (1) and (2) it follows that
$\displaystyle\delta\mathcal{L}$
$\displaystyle=\sum_{j=1}^{N}\sum_{I}\left(p^{I}\delta
u_{It_{j}}\wedge\mbox{\rm d}t_{j}+\left(\partial_{j}p^{I}\right)\delta
u_{I}\wedge\mbox{\rm d}t_{j}\right)=-d\left(\sum_{I}p^{I}\delta u_{I}\right).$
This implies by Proposition A.1 that $u$ solves the pluri-Lagrangian problem.
∎
* Proof of sufficiency in Theorem 2.4..
We calculate the vertical exterior derivative $\delta\mathcal{L}$,
$\displaystyle\delta\mathcal{L}$
$\displaystyle=\sum_{i<j}\sum_{I}\frac{\partial{\mathcal{L}_{ij}}}{\partial{u_{I}}}\,\delta
u_{I}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$
$\displaystyle=\sum_{i<j}\sum_{I}\left(\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}+\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}+\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}+\partial_{i}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\right)\delta
u_{I}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$ (31)
We will rearrange this sum according to the times occurring in the multi-index
$I$. We have
$\displaystyle\sum_{I}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}\,\delta
u_{I}=\sum_{I\not\ni
t_{i},t_{j}}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}\,\delta
u_{I}+\sum_{I\not\ni
t_{j}}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta
u_{It_{i}}$ $\displaystyle\hskip 85.35826pt+\sum_{I\not\ni
t_{i}}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta
u_{It_{j}}+\sum_{I}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta
u_{It_{i}t_{j}},$
$\displaystyle\sum_{I}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta
u_{I}=\sum_{I\not\ni
t_{j}}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta
u_{I}+\sum_{I}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta
u_{It_{j}},$
$\displaystyle\sum_{I}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta
u_{I}=\sum_{I\not\ni
t_{i}}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta
u_{I}+\sum_{I}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta
u_{It_{i}}.$
Modulo the multi-time Euler-Lagrange equations (5)–(7), we can write these
expressions as
$\displaystyle\sum_{I}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}\,\delta
u_{I}=\sum_{I\not\ni t_{j}}p_{j}^{I}\,\delta u_{It_{i}}-\sum_{I\not\ni
t_{i}}p_{i}^{I}\,\delta u_{It_{j}}+\sum_{I}(n_{j}^{I}-n_{i}^{I})\,\delta
u_{It_{i}t_{j}},$
$\displaystyle\sum_{I}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta
u_{I}=\sum_{I\not\ni t_{j}}\partial_{i}p_{j}^{I}\,\delta
u_{I}+\sum_{I}\partial_{i}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{j}},$
$\displaystyle\sum_{I}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta
u_{I}=\sum_{I\not\ni t_{i}}-\partial_{j}p_{i}^{I}\,\delta
u_{I}+\sum_{I}\partial_{j}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{i}},$
$\displaystyle\sum_{I}\partial_{i}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta
u_{I}=\sum_{I}\partial_{i}\partial_{j}(n_{j}^{I}-n_{i}^{I})\delta u_{I}.$
where
$\displaystyle
p_{j}^{I}=\frac{\delta_{1j}{\mathcal{L}_{1j}}}{\delta{u_{It_{1}}}}\qquad\text{for
}I\not\ni t_{j},$ $\displaystyle
n_{j}^{I}=\frac{\delta_{1j}{\mathcal{L}_{1j}}}{\delta{u_{It_{1}t_{j}}}}.$
Note that here the indices of $p$ and $n$ are labels, not derivatives. Hence
on solutions to equations (5)–(7), Equation (31) is equivalent to
$\displaystyle\delta\mathcal{L}$
$\displaystyle=\sum_{i<j}\Bigg{[}\sum_{I\not\ni t_{j}}\left(p_{j}^{I}\,\delta
u_{It_{i}}+\partial_{i}p_{j}^{I}\,\delta u_{I}\right)-\sum_{I\not\ni
t_{i}}\left(p_{i}^{I}\,\delta u_{It_{j}}+\partial_{j}p_{i}^{I}\,\delta
u_{I}\right)$ $\displaystyle\hskip
42.67912pt+\sum_{I}\Big{(}(n_{j}^{I}-n_{i}^{I})\,\delta
u_{It_{i}t_{j}}+\partial_{j}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{i}}$
$\displaystyle\hskip 85.35826pt+\partial_{i}(n_{j}^{I}-n_{i}^{I})\,\delta
u_{It_{j}}+\partial_{i}\partial_{j}(n_{j}^{I}-n_{i}^{I})\,\delta
u_{I}\Big{)}\Bigg{]}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}.$
Using the anti-symmetry of the wedge product, we can write this as
$\displaystyle\delta\mathcal{L}$
$\displaystyle=\sum_{i,j=1}^{N}\Bigg{[}\sum_{I\not\ni
t_{j}}\left(p_{j}^{I}\,\delta u_{It_{i}}+\partial_{i}p_{j}^{I}\,\delta
u_{I}\right)$ $\displaystyle\hskip 42.67912pt+\sum_{I}\Big{(}n_{j}^{I}\,\delta
u_{It_{i}t_{j}}+\partial_{j}n_{j}^{I}\,\delta
u_{It_{i}}+\partial_{i}n_{j}^{I}\,\delta
u_{It_{j}}+\partial_{i}\partial_{j}n_{j}^{I}\,\delta
u_{I}\Big{)}\Bigg{]}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$
$\displaystyle=\sum_{j=1}^{N}\Bigg{[}\sum_{I\not\ni t_{j}}-\mbox{\rm
d}\left(p_{j}^{I}\,\delta u_{I}\wedge\mbox{\rm
d}t_{j}\right)+\sum_{I}-d\left(n_{j}^{I}\,\delta u_{It_{j}}\wedge\mbox{\rm
d}t_{j}+\partial_{j}n_{j}^{I}\,\delta u_{I}\wedge\mbox{\rm
d}t_{j}\right)\Bigg{]}.$
It now follows by Proposition A.1 that $u$ is a critical field. ∎
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|
2024-09-04T02:54:59.265619 | 2020-03-11T16:47:22 | 2003.05402 | {
"authors": "Boxin Zhao, Y. Samuel Wang, Mladen Kolar",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26171",
"submitter": "Boxin Zhao",
"url": "https://arxiv.org/abs/2003.05402"
} | arxiv-papers | # FuDGE: A Method to Estimate a Functional Differential Graph in a High-
Dimensional Setting
Boxin Zhao<EMAIL_ADDRESS>
Booth School of Business
The University of Chicago
Chicago, IL 60637, USA
Y. Samuel Wang<EMAIL_ADDRESS>
Department of Statistics and Data Science
Cornell University
Ithaca, NY 14853, USA
Mladen Kolar<EMAIL_ADDRESS>
Booth School of Business
The University of Chicago
Chicago, IL 60637, USA
###### Abstract
We consider the problem of estimating the difference between two functional
undirected graphical models with shared structures. In many applications, data
are naturally regarded as a vector of random functions rather than a vector of
scalars. For example, electroencephalography (EEG) data are more appropriately
treated as functions of time. In such a problem, not only can the number of
functions measured per sample be large, but each function is itself an
infinite dimensional object, making estimation of model parameters
challenging. This is further complicated by the fact that the curves are
usually only observed at discrete time points. We first define a functional
differential graph that captures the differences between two functional
graphical models and formally characterize when the functional differential
graph is well defined. We then propose a method, FuDGE, that directly
estimates the functional differential graph without first estimating each
individual graph. This is particularly beneficial in settings where the
individual graphs are dense, but the differential graph is sparse. We show
that FuDGE consistently estimates the functional differential graph even in a
high-dimensional setting for both fully observed and discretely observed
function paths. We illustrate the finite sample properties of our method
through simulation studies. We also propose a competing method, the Joint
Functional Graphical Lasso, which generalizes the Joint Graphical Lasso to the
functional setting. Finally, we apply our method to EEG data to uncover
differences in functional brain connectivity between a group of individuals
with alcohol use disorder and a control group.
Keywords: differential graph estimation, functional data analysis,
multivariate functional data, probabilistic graphical models, structure
learning
## 1 Introduction
We consider a setting where we observe two samples of multivariate functional
data, $X_{i}(t)$ for $i=1,\ldots,n_{X}$ and $Y_{i}(t)$ for $i=1,\ldots,n_{Y}$.
The primary goal is to determine if and how the underlying
populations—specifically their conditional dependency structures—differ. As a
motivating example, consider electroencephalography (EEG) data where the
electrical activity of multiple regions of the brain can be measured
simultaneously across a period of time. Given samples from the general
population, fitting a graphical model to the observed measurements would allow
a researcher to determine which regions of the brain are dependent after
conditioning on all other regions. The EEG data analyzed in Section 6.2
consists of two samples: one from a control group and the other from a group
of individuals with alcohol use disorder (AUD). Using this data, researchers
may be interested in explicitly comparing the two groups and investigating the
complex question of how brain functional connectivity patterns in the AUD
group differ from those in the control group.
The conditional independence structure within multivariate data is commonly
represented by a graphical model (Lauritzen, 1996). Let $G=\\{V,E\\}$ denote
an undirected graph where $V$ is the set of vertices with $|V|=p$ and
$E\subset V^{2}$ is the set of edges. At times, we also denote $V$ as
$[p]=\\{1,2,\dots,p\\}$. When the data consist of random vectors
$X=(X_{1},\dots,X_{p})^{\top}$, we say that $X$ satisfies the pairwise Markov
property with respect to $G$ if $X_{v}\centernot\perp\\!\\!\\!\perp
X_{w}\mid\\{X_{u}\\}_{u\in V\setminus\\{v,w\\}}$ holds if and only if
$\\{v,w\\}\in E$. When $X$ follows a multivariate Gaussian distribution with
covariance $\Sigma=\Theta^{-1}$, then $\Theta_{vw}\neq 0$ if and only if
$\\{v,w\\}\in E$. Thus, recovering the structure of an undirected graph from
multivariate Gaussian data is equivalent to estimating the support of the
precision matrix, $\Theta$.
When the primary interest is in characterizing the difference between the
conditional independence structure of two populations, the object of interest
may be the _differential graph_ , $G_{\Delta}=\\{V,E_{\Delta}\\}$. When $X$
and $Y$ follow multivariate normal distributions with covariance matrices
$\Sigma^{X}$ and $\Sigma^{Y}$, let $\Delta=\Theta^{X}-\Theta^{Y}$, where
$\Theta^{X}=(\Sigma^{X})^{-1}$ and $\Theta^{Y}=(\Sigma^{Y})^{-1}$ are the
precision matrices of $X$ and $Y$ respectively. The differential graph is then
defined by letting $E_{\Delta}=\left\\{\\{v,w\\}\,:\,\Delta_{v,w}\neq
0\right\\}$. This type of differential model for vector-valued data has been
adopted in Zhao et al. (2014), Xu and Gu (2016), and Cai (2017).
In the motivating example of EEG data, the electrical activity is observed
over a period of time. When measurements smoothly vary over time, it may be
more natural to consider the observations as arising from an underlying
function. This is particularly true when data from different subjects are
observed at different time points. Furthermore, when characterizing
conditional independence, it is likely that the activity of each region
depends not only on what is occurring simultaneously in the other regions, but
also on what has previously occurred in other regions; this suggests that a
functional graphical model might be appropriate.
In this paper, we define a differential graph for functional data that we
refer to as a functional differential graphical model. Similar to differential
graphs for vector-valued data, functional differential graphical models
characterize the differences in the conditional dependence structures of two
distributions of multivariate curves. We build on the functional graphical
model developed in Qiao et al. (2019). However, while Qiao et al. (2019)
required that the observed functions lie in a finite-dimensional space in
order for the functional graphical model to be well defined, the functional
differential graphical models may be well defined even in certain cases where
the observed functions live in an infinite-dimensional space.
We propose an algorithm called FuDGE to estimate the differential graph and
show that this procedure enjoys many benefits, similar to differential graph
estimation in the vector-valued setting. Most notably, we show that under
suitable conditions, the proposed method can consistently recover the
differential graph even in the high-dimensional setting where $p$, the number
of observed variables, may be larger than $n$, the number of observed samples.
A conference version of this paper was presented at the Conference on Neural
Information Processing Systems (Zhao et al., 2019). Compared to the conference
version, this paper includes the following new results:
* •
We give a new definition for a differential graph for functional data, which
allows us to circumvent the unnatural assumption made in the previous version
and take a truly functional approach. Specifically, instead of defining the
differential graph based on the difference between conditional covariance
functions, we use the limit of the norm of the difference between finite-
dimensional precision matrices.
* •
We include new theoretical guarantees for discretely observed curves. In
practice, we can only observe the functions at discrete time points, so this
extends the theoretical guarantees to a practical estimation procedure.
Discrete observations bring an additional source of error when the estimated
curves are used in functional PCA. In Theorem 4, we give an error bound for
estimating the covariance matrix of the PCA score vectors under mild
conditions.
* •
We introduce the Joint Functional Graphical Lasso, which is a generalization
of the Joint Graphical Lasso (Danaher et al., 2014) to the functional data
setting. Empirically, we show that the procedure performs competitively in
some settings, but is generally outperformed by the FuDGE procedure.
The software implementation can be found at https://github.com/boxinz17/FuDGE.
The repository also contains the code to reproduce the simulation results.
### 1.1 Related Work
The work we develop lies at the intersection of two different lines of
literature: graphical models for functional data and direct estimation of
differential graphs.
There are many previous works studying the structure estimation of a static
undirected graphical model (Chow and Liu, 1968; Yuan and Lin, 2007; Cai et
al., 2011; Meinshausen and Bühlmann, 2006; Yu et al., 2016, 2019; Vogel and
Fried, 2011). Previous methods have also been proposed for characterizing
conditional independence for multivariate observations recorded over time. For
example, Talih and Hengartner (2005), Xuan and Murphy (2007), Ahmed and Xing
(2009), Song et al. (2009a), Song et al. (2009b), Kolar et al. (2010b), Kolar
et al. (2009), Kolar and Xing (2009), Zhou et al. (2010), Yin et al. (2010),
Kolar et al. (2010a), Kolar and Xing (2011), Kolar and Xing (2012), Wang and
Kolar (2014), Lu et al. (2018) studied methods for dynamic graphical models
that assume the data are independently sampled at different time points, but
generated by related distributions. In these works, the authors proposed
procedures to estimate a series of graphs which represent the conditional
independence structure at each time point; however, they assume the observed
data does not encode “longitudinal” dependence. In contrast, Qiao et al.
(2019); Zhu et al. (2016); Li and Solea (2018); Zhang et al. (2018) considered
the setting where the data data are multivariate random functions. Most
similar to the setting we consider, Qiao et al. (2019) assumed that the data
are distributed as a multivariate Gaussian process (MGP) and use a graphical
lasso type procedure on the estimated functional principal component scores.
Zhu et al. (2016) also assumed an MGP, but proposed a Bayesian procedure.
Crucially, however, both required that the covariance kernel can essentially
be represented by a finite dimensional object. Zapata et al. (2019) showed
that under various notions of separability—roughly when the covariance kernel
can be decomposed into covariance across time and covariance across nodes—the
conditional independence of the MGP is well defined even when the functional
data are truly infinite dimensional and that the conditional independence
graph can be recovered by the union of a (potentially infinitely) countable
number of graphs over finite dimensional objects. In a different approach, Li
and Solea (2018) did not assume that the random functions are Gaussian, and
instead used the notion of additive conditional independence to define a
graphical model for the random functions. Finally, Qiao et al. (2020) also
assumed that the data are random functions, but also allowed for the
dependency structure to change smoothly across time—similar to a dynamic
graphical model.
We also draw on recent literature which has shown that when the object of
interest is the difference between two distributions, directly estimating the
difference can provide improvements over first estimating each distribution
and then taking the difference. Most notably, when estimating the difference
in graphs in the high-dimensional setting, even if each individual graph does
not satisfy the appropriate sparsity conditions, the differential graph may
still be recovered consistently. Zhao et al. (2014) considered data drawn from
two Gaussian graphical models, and they showed that even if both underlying
graphs are dense, if the difference between the precision matrices of each
distribution is sparse, the differential graph can still be recovered in the
high-dimensional setting. Liu et al. (2014) proposed procedure based on KLIEP
(Sugiyama et al., 2008) that estimates the differential graph by directly
modeling the ratio of two densities. They did not assume Gaussianity, but
required that both distributions lie in some exponential family. Fazayeli and
Banerjee (2016) extended this idea to estimate the differences in Ising
models. Wang et al. (2018) and Ghoshal and Honorio (2019) also proposed direct
difference estimators for directed graphs when the data are generated by
linear structural equation models that share a common topological ordering.
### 1.2 Notation
Let $|\cdot|_{p}$ denote the vector $p$-norm and $\|\cdot\|_{p}$ denote the
matrix/operator $p$-norm. For example, for a $p\times 1$ vector
$a=(a_{1},a_{2},\dots,a_{p})^{\top}$, we have $|a|_{1}=\sum_{j}|a_{j}|$,
$|a|_{2}=(\sum_{j}|a^{2}_{j}|)^{1/2}$ and $|a|_{\infty}=\max_{j}|a_{j}|$. For
a $p\times{q}$ matrix $A$ with entries $a_{jk}$, $|A|_{1}=\sum_{j,k}|a_{jk}|$,
$\|A\|_{1}=\max_{k}\sum_{j}|a_{jk}|$, $|A|_{\infty}=\max_{j,k}|a_{jk}|$, and
$\|A\|_{\infty}=\max_{j}\sum_{k}|a_{jk}|$. Let $\left\lVert
A\right\rVert_{\text{F}}=(\sum_{j,k}a^{2}_{jk})^{1/2}$ be the Frobenius norm
of $A$. When $A$ is symmetric, let $\mathrm{tr}(A)=\sum_{j}a_{jj}$ denote the
trace of A. Let $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ denote the minimum
and maximum eigenvalues, respectively. Let $a_{n}\asymp{b_{n}}$ denote that
$0<C_{1}\leq{\inf_{n}|a_{n}/b_{n}|}\leq{\sup_{n}|a_{n}/b_{n}|}\leq
C_{2}<\infty$ for some positive constants $C_{1}$ and $C_{2}$.
We assume that all random functions belong to a separable Hilbert space
$\mathbb{H}$. For any two functions $f_{1},f_{2}\in\mathbb{H}$, we define
their inner product as $\langle f_{1},f_{2}\rangle=\int f_{1}(t)f_{2}(t)dt$.
The induced norm is $\|f_{1}\|=\|f_{1}\|_{\mathcal{L}^{2}}=\\{\int
f_{1}^{2}(t)dt\\}^{1/2}$.
For a function vector $f(t)=(f_{1}(t),f_{2}(t),\dots,f_{p}(t))^{\top}$, we let
$\|f\|_{\mathcal{L}^{2},2}=(\sum^{p}_{j=1}\|f_{j}\|^{2})^{1/2}$ denote its
$\mathcal{L}^{2},2$-norm. For a bivariate function $g(s,t)$, we define the
Hilbert-Schmidt norm of $g(s,t)$ as
$\|g\|_{\text{HS}}=\int\int\\{g(s,t)\\}^{2}dsdt$. Typically, we will use
$f(\cdot)$ (and similarly $g(\cdot,*)$) to denote the entire function $f$,
while we use $f(t)$ (and similarly $g(s,t)$) to mean the value of $f$
evaluated at $t$.
For a vector space $\mathbb{V}$, we use $\mathbb{V}^{\bot}$ to denote its
orthogonal complement. For $v_{1},\ldots,v_{K}\in\mathbb{V}$, and
$v=(v_{1},\ldots,v_{K})^{\top}$, we use ${\rm
Span}\left\\{v_{1},v_{2},\dots,v_{K}\right\\}={\rm Span}\left(v\right)$ to
denote the vector subspace spanned by $v_{1},\ldots,v_{K}$.
## 2 Functional Differential Graphical Models
In this section, we give a review of functional graphical models and introduce
the notion of a functional differential graphical model.
### 2.1 Functional Graphical Model
Suppose
$X_{i}(\cdot)=\left(X_{i1}(\cdot),X_{i2}(\cdot),\dots,X_{ip}(\cdot)\right)^{\top}$
is a p-dimensional _multivariate Gaussian processes (MGP)_ with mean zero and
common domain $\mathcal{T}$, where $\mathcal{T}$ is a closed interval of the
real line with length $\lvert\mathcal{T}\rvert$.111We assume mean zero and a
common domain $\mathcal{T}$ to simplify the notation, but the methodology and
theory generalize to non-zero means and different time domains. Each
observation, for $i=1,2,\ldots,n$, is i.i.d. In addition, assume that for
$j\in V$, $X_{ij}(\cdot)$ is a random element of a separable Hilbert space
$\mathbb{H}$. Qiao et al. (2019), define the conditional cross-covariance
function for $X_{i}(\cdot)$ as
${}C^{X}_{jl}(s,t)\;=\;\mathrm{Cov}\left(X_{ij}(s),X_{il}(t)\,\mid\,\\{X_{ik}(\cdot)\\}_{k\neq
j,l}\right).$ (1)
If $C^{X}_{jl}(s,t)=0$ for all $s,t\in\mathcal{T}$, then the random functions
$X_{j}(\cdot)$ and $X_{l}(\cdot)$ are conditionally independent given the
other random functions, and the graph $G_{X}=\\{V,E_{X}\\}$ represents the
pairwise Markov properties of $X_{i}(\cdot)$ if
$E_{X}=\left\\{(j,l)\,:\,j<l\text{ and }\|C^{X}_{jl}\|_{\text{HS}}\neq
0\right\\}.$ (2)
In general, we cannot directly estimate (2), since $X_{i}(\cdot)$ may be an
infinite dimensional object. Thus, before applying a statistical estimation
procedure, dimension reduction is typically required. Qiao et al. (2019) used
_functional principal component analysis_ (FPCA) to project each observed
function onto an orthonormal function basis defined by a finite number of
eigenfunctions. Their procedure then estimates the conditional independence
structure from the “projection scores” of this basis. We outline their
approach below. However, in contrast to Qiao et al. (2019), we do not restrict
ourselves to dimension reduction by projecting onto the FPCA basis, and in our
discussion we instead consider a general function subspace.
Let $\mathbb{V}^{M_{j}}_{j}\subseteq\mathbb{H}$ be a subspace of a separable
Hilbert space $\mathbb{H}$ with dimension $M_{j}\in\mathbb{N}^{+}$ for all
$j=1,2,\dots,p$. Our theory easily generalizes to the setting where $M_{j}$
may differ, but to simplify notation, we assume $M_{j}=M$ for all $j$ and
simply write $\mathbb{V}^{M}_{j}$ instead of $\mathbb{V}^{M_{j}}_{j}$. Let
$\mathbb{V}^{M}_{[p]}\coloneqq\mathbb{V}^{M}_{1}\otimes\mathbb{V}^{M}_{2}\otimes\dots\otimes\mathbb{V}^{M}_{p}$.
For any function $g(\cdot)\in\mathbb{H}$ and a subspace
$\mathbb{F}\subseteq\mathbb{H}$, let $\pi(g(\cdot);\mathbb{F})\in\mathbb{F}$
denote the projection of the function $g(\cdot)$ onto the subspace
$\mathbb{F}$, and let
$\pi(X_{i}(\cdot);\mathbb{V}^{M}_{[p]})=\left(\pi(X_{i1}(\cdot);\mathbb{V}^{M}_{1}),\pi(X_{i2}(\cdot);\mathbb{V}^{M}_{2}),\dots,\pi(X_{ip}(\cdot);\mathbb{V}^{M}_{p})\right)^{\top}.$
When the choice of the subspace is clear from the context, we will use the
following shorthand notation:
$X^{\pi}_{ij}(\cdot)=\pi(X_{ij}(\cdot);\mathbb{V}^{M}_{j})$, $j=1,2,\dots,p$,
and $X^{\pi}_{i}(\cdot)=\pi(X_{i}(\cdot);\mathbb{V}^{M}_{[p]})$.
Similar to the definitions in (1) and (2), we define the conditional
independence graph of $X^{\pi}(\cdot)$ as
$E^{\pi}_{X}=\left\\{\\{j,l\\}\,:\,j<l\text{ and
}\|C^{X,\pi}_{jl}\|_{\text{HS}}\neq 0\right\\},$ (3)
where
${}C^{X,\pi}_{jl}(s,t)\;=\;\mathrm{Cov}\left(X^{\pi}_{ij}(s),X^{\pi}_{il}(t)\,\mid\,\\{X^{\pi}_{ik}(\cdot)\\}_{k\neq
j,l}\right).$ (4)
Note that $E^{\pi}_{X}$ depends on the choice of $\mathbb{V}^{M}_{[p]}$
through the projection operator $\pi$, and as we discuss below, $E_{X}^{\pi}$
may be recovered from the observed samples.
When the data arise from an MGP, we can estimate the projected graphical
structure by studying the precision matrix of projection score vectors
(defined below) with _any_ orthonormal function basis of the subspace
$\mathbb{V}^{M}_{[p]}$. Let
$e^{M}_{j}=(e_{j1}(\cdot),e_{j2}(\cdot),\dots,e_{jM}(\cdot))^{\top}$ be any
orthonormal function basis of $\mathbb{V}^{M}_{j}$ and let
$e^{M}(\cdot)=\\{e^{M}_{j}\\}^{p}_{j=1}$ be orthonormal function basis of
$\mathbb{V}^{M}_{[p]}$. Let
$a^{X}_{ijk}=\int_{\mathcal{T}}X_{ij}(t)e_{jk}(t)dt$
denote the projection score of $X_{ij}(\cdot)$ onto $e_{jk}(\cdot)$ and let
$\displaystyle
a^{X,M}_{ij}=(a^{X}_{ij1},a^{X}_{ij2},\dots,a^{X}_{ijM})^{\top}\;\text{ and
}\;a^{X,M}_{i}=((a^{X,M}_{i1})^{\top},\ldots,(a^{X,M}_{ip})^{\top})^{\top}\in{\mathbb{R}^{pM}}.$
Since $X_{i}(\cdot)$ is a $p$-dimensional MGP, $a^{X,M}_{i}$ follows a
multivariate Gaussian distribution and we denote the covariance matrix of that
distribution as $\Sigma^{X,M}=(\Theta^{X,M})^{-1}\in\mathbb{R}^{pM\times pM}$.
Each function $X_{ij}(\cdot)$ is associated with $M$ rows and columns of
$\Sigma^{X,M}$ corresponding to $a_{ij}^{X,M}$. We use $\Theta_{jl}^{X,M}$ to
refer to the $M\times M$ sub-matrix of $\Theta^{X,M}$ corresponding to
functions $X_{ij}(\cdot)$ and $X_{il}(\cdot)$. Lemma 1, from Qiao et al.
(2019), shows that the conditional independence structure of the projected
functional data can be obtained from the block sparsity of $\Theta^{X,M}$.
###### Lemma 1
[Qiao et al. (2019)] Let $\Theta^{X,M}$ denote the inverse covariance of the
projection scores. Then, $X^{\pi}_{ij}(s)\perp\\!\\!\\!\perp
X^{\pi}_{il}(t)\mid\\{X^{\pi}_{ik}(\cdot)\\}_{k\neq j,l}$ for all222More
precisely, we only need the conditional independence to hold for all
$s,t\in{\cal T}$ except for a subset of $\mathcal{T}^{2}$ with zero measure.
$s,t\in{\cal T}$ if and only if $\Theta_{jl}^{X,M}\equiv 0$. This implies that
$E^{\pi}_{X}$—as defined in (3)—can be equivalently defined as
$E^{\pi}_{X}\;=\;\left\\{\\{j,l\\}\,:\,j<l\text{ and
}\|\Theta^{X,M}_{jl}\|_{F}\neq 0\right\\}.$ (5)
While Qiao et al. (2019) only considered projections onto the span of the FPCA
basis (that is, the eigenfunctions of $X_{ij}(\cdot)$ corresponding to $M$
largest eigenvalues), the result trivially extends to the more general case of
_any subspace_ and _any orthonormal function basis_ of that subspace.
Although $\Theta^{X,M}$ depends on the specific basis onto which
$X_{i}(\cdot)$ is projected, the edge set $E^{\pi}_{X}$ only depends on the
subspace $\mathbb{V}^{M}_{[p]}$, that is, the span of the basis onto which
$X_{i}(\cdot)$ is projected. Thus, Lemma 1 implies that although the entries
of $\Theta^{X,M}$ may change when using different orthonormal function bases
to represent $\mathbb{V}^{M}_{[p]}$, the block sparsity pattern of
$\Theta^{X,M}$ only depends on the span of the selected basis.
When $X_{i}(\cdot)\neq X^{\pi}_{i}(\cdot)$, $E^{\pi}_{X}$ may not be the same
as $E_{X}$; furthermore, it may not be the case that $E^{\pi}_{X}\subseteq
E_{X}$ or $E_{X}\subseteq E^{\pi}_{X}$. Thus, Condition 2 of Qiao et al.
(2019) requires a finite $M^{\star}<\infty$ such that $X_{ij}$ lies in
$\mathbb{V}^{M^{\star}}_{[p]}$ almost surely. When $M=M^{\star}$, then
$X_{i}(\cdot)=X_{i}^{\pi}(\cdot)$ and $E^{\pi}_{X}=E_{X}$. Under this
assumption, to estimate $E^{\pi}_{X}=E_{X}$, Qiao et al. (2019) proposed the
functional graphical lasso estimator (fglasso), which solves the following
objective:
$\hat{\Theta}^{X,M}=\operatorname*{arg\,max}_{\Theta^{X,M}}{\left\\{\log{\text{det}\left(\Theta^{X,M}\right)}-\mathrm{tr}\left(S^{X,M}\Theta^{X,M}\right)-\gamma_{n}\sum_{j\neq
l}{\left\lVert\Theta^{X,M}_{jl}\right\rVert_{\text{F}}}\right\\}}.$ (6)
In (6), $\Theta^{X,M}$ is a symmetric positive definite matrix,
$\Theta^{X,M}_{jl}\in\mathbb{R}^{M\times M}$ corresponds to the $(j,l)$ sub-
matrix of $\Theta^{X,M}$, $\gamma_{n}$ is a non-negative tuning parameter, and
$S^{X,M}$ is an estimator of $\Sigma^{X,M}$. The matrix $S^{X,M}$ is obtained
by using FPCA on the empirical covariance functions (see Section 2.3 for
details). The resulting estimated edge set for the functional graph is
$\hat{E}_{X}^{\pi}=\left\\{\\{j,l\\}\,:\,j<l\text{ and
}\left\lVert\hat{\Theta}^{X,M}_{jl}\right\rVert_{\text{F}}>0\right\\}.$ (7)
We also note that the objective in (6) was earlier used in Kolar et al. (2013)
and Kolar et al. (2014) for estimation of graphical models from multi-
attribute data.
However, the requirement that $X_{i}(\cdot)$ lies in a subspace with finite
dimension may be violated in many practical applications and negates one of
the primary benefits of considering the observations as functions.
Unfortunately, the extension to infinite-dimensional data is nontrivial, and
indeed Condition 2 in Qiao et al. (2019) requires that the observed functional
data lies within a finite-dimensional span. To see why, we first note that
$\Sigma^{X,M^{\star}}$ is always a compact operator on
$\mathbb{R}^{pM^{\star}}$. Thus, as $M^{\star}\to\infty$, the smallest
eigenvalue of $\Sigma^{X,M^{\star}}$ will go to zero. As a consequence,
$\Sigma^{X,M^{\star}}$ becomes increasingly ill-conditioned, and
$\Theta^{X,M^{\star}}$, the inverse of $\Sigma^{X,M^{\star}}$ will become ill-
defined when $M^{\star}=\infty$. This behaviour makes the estimation of a
functional graphical model —at least through the basis expansion approach
proposed by Qiao et al. (2019)—generally infeasible for truly infinite-
dimensional functional data. When the data is truly infinite-dimensional, the
best we can do is to estimate a finite-dimensional approximation and hope that
it captures the relevant information.
### 2.2 Functional Differential Graphical Models: Finite Dimensional Setting
In this paper, instead of estimating the conditional independence structure of
a single MGP, we are interested in characterizing the difference between two
MGPs, $X$ and $Y$. For brevity, we will typically only explicitly define the
notation for $X$; however, the reader should infer that all notation for $Y$
is defined analogously. As described in the introduction, Li et al. (2007) and
Zhao et al. (2014) consider the setting where $X$ and $Y$ are multivariate
Gaussian vectors, and define the differential graph
$G_{\Delta}=\\{V,E_{\Delta}\\}$ by letting
$E_{\Delta}=\left\\{(v,w)\,:\,v<w\text{ and }\Delta_{vw}\neq 0\right\\}$ (8)
where $\Delta=(\Sigma^{X})^{-1}-(\Sigma^{Y})^{-1}$ and $\Sigma^{X},\Sigma^{Y}$
are the covariance matrices of $X$ and $Y$.
We extend this definition to the functional data setting and define functional
differential graphical models. To develop the intuition, we first start by
defining the differential graph with respect to their finite-dimensional
projections, that is, with respect to $X^{\pi}_{i}(t)$ and $Y^{\pi}_{i}(t)$
for some choice of $\mathbb{V}^{M}_{[p]}$. As implied by Lemma 1, in the
functional graphical model setting, the $M\times M$ blocks of the precision
matrix of the projection scores play a similar role to the individual entries
of a precision matrix in the vector-valued Gaussian graphical model setting.
Thus, we also define a functional differential graphical model by the
difference of the precision matrices of the projection scores. Note that for
each $j\in V$, we require that both $a^{X}_{ij}$ and $a^{Y}_{ij}$ are computed
by the same function basis of $\mathbb{V}^{M}_{j}$. Let
$\Theta^{X,M}=\left(\Sigma^{X,M}\right)^{-1}$ and
$\Theta^{Y,M}=\left(\Sigma^{Y,M}\right)^{-1}$ be the precision matrices for
the projection scores for $X$ and $Y$, respectively, where the inverse should
be understood as the pseudo-inverse when $\Sigma^{X,M}$ or $\Sigma^{Y,M}$ are
not invertible. The functional differential graphical model is defined as
$\Delta^{M}=\Theta^{X,M}-\Theta^{Y,M}.$ (9)
Let $\Delta^{M}_{jl}$ be the $(j,l)$-th $M\times M$ block of $\Delta^{M}$ and
define the edges of the functional differential graph of the projected data
as:
$E^{\pi}_{\Delta}\,=\,\left\\{(j,l)\,:\,j<l\text{ and
}\,\|\Delta^{M}_{jl}\|_{F}>0\right\\}.$ (10)
While the entries of $\Delta^{M}$ depend on the choice of orthonormal function
basis, the definition of $E^{\pi}_{\Delta}$ is invariant to the particular
basis and only depends on the span. The following lemma formally states this
result.
###### Lemma 2
Suppose that ${\rm span}(e^{M}(\cdot))={\rm span}(\tilde{e}^{M}(\cdot))$ for
two orthonormal bases $e^{M}(\cdot)$ and $\tilde{e}^{M}(\cdot)$. Let
$E_{\Delta}^{\pi}$ and $E_{\Delta}^{\tilde{\pi}}$ be defined by (10) when
projecting $X$ and $Y$ onto $e^{M}(\cdot)$ and $\tilde{e}^{M}(\cdot)$,
respectively. Then, $E_{\Delta}^{\pi}=E_{\Delta}^{\tilde{\pi}}$.
Proof See Appendix B.1.
We have several comments regarding $E^{\pi}_{\Delta}$ defined in (10).
##### Projecting $X$ and $Y$ onto different subspaces:
While we project both $X$ and $Y$ onto the same subspace
$\mathbb{V}^{M}_{[p]}$, our definition can be easily generalized to a setting
where we project $X$ onto $\mathbb{V}^{X,M}_{[p]}$ and $Y$ onto
$\mathbb{V}^{Y,M}_{[p]}$, with
$\mathbb{V}^{X,M}_{[p]}\neq\mathbb{V}^{Y,M}_{[p]}$. For instance, naively
following the procedure of Qiao et al. (2019), we could perform FPCA on $X$
and $Y$ separately, and subsequently we could use the difference between the
precision matrices of projection scores to define the functional differential
graph. Although defining the functional differential graph using this
alternative approach may be suitable for some applications, it may result in
the undesirable case where $(j,l)\in E_{\Delta}^{\pi}$ even though
$C_{jl}^{X,\pi}(\cdot,*)=C_{jl}^{Y,\pi}(\cdot,*)$,
$C_{jj}^{X,\pi}(\cdot,*)=C_{ll}^{Y,\pi}(\cdot,*)$, and $C_{ll}^{\setminus
j,X,\pi}(\cdot,*)=C_{ll}^{\setminus j,Y,\pi}(\cdot,*)$. Therefore, we restrict
our discussion to the setting where both $X$ and $Y$ are projected onto the
same subspace.
##### Connection to Multi-Attribute Graphical Models:
The selection of a specific functional subspace is connected to multi-
attribute graphical models (Kolar et al., 2014). If we treat the random
function $X_{ij}(\cdot)$ as representing an infinite number of attributes,
then $X^{\pi}_{ij}(\cdot)$ will be an approximation using $M$ attributes. The
chosen attributes are given by the subspace $\mathbb{V}^{M}_{j}$. While we
allow different nodes to choose different attributes by allowing
$\mathbb{V}^{M}_{j}$ to vary across $j$, we require that the same attributes
are used to represent both $X$ and $Y$ by restricting $\mathbb{V}^{M}_{[p]}$
to be the same for $X$ and $Y$. The specific choice of $\mathbb{V}^{M}_{[p]}$,
can extract different attributes from the data. For instance, using the
subspace spanned by the Fourier basis can be viewed as extracting frequency
information, while using the subspace spanned by the eigenfunctions—as
introduced in the next section—can be viewed as extracting the dominant modes
of variation.
Given definition (10) and Lemma 2, there are two main questions to be
answered: First, how do we choose $\mathbb{V}^{M}_{[p]}$? Second, what happens
when $X$ and $Y$ are infinite-dimensional? We answer the first question in
Section 2.3 and the second question in Section 2.4.
### 2.3 Choosing Functional Subspace via FPCA
As discussed in Section 2.2, the choice of $\mathbb{V}^{M}_{[p]}$ in
Definition 10 decides—roughly speaking—the attributes or dimensions in which
we compare the conditional independence structures of $X$ and $Y$. In some
applications, we may have a very good prior knowledge about this choice.
However, in many cases we may not have strong prior knowledge. In this
section, we describe our recommended “default choice” that uses FPCA on the
combined $X$ and $Y$ observations. In particular, suppose there exist
subspaces $\\{\mathbb{V}^{M^{\star}}_{j}\\}_{j\in V}$ such that
$\mathbb{V}^{M^{\star}}_{j}$ has dimension $M^{\star}<\infty$ and
$X_{ij}(t),Y_{ij}(t)\in\mathbb{V}^{M^{\star}}_{j}$ for all $j\in V$. Then,
FPCA—when given population values—recovers this subspace.
Similar to the way principal component analysis provides the $L_{2}$ optimal
lower dimensional representation of vector-valued data, FPCA provides the
$L_{2}$ optimal finite dimensional representation of functional data. Let
$K^{X}_{jj}(t,s)=\mathrm{Cov}(X_{ij}(t),X_{ij}(s))$ denote the covariance
function for $X_{ij}$ where $j\in V$. Then, there exist orthonormal
eigenfunctions and eigenvalues
$\\{\phi^{X}_{jk}(t),\lambda^{X}_{jk}\\}_{k\in\mathbb{N}}$ such that
$\int_{\mathcal{T}}K^{X}_{jj}(s,t)\phi_{jk}^{X}(t)dt=\lambda_{jk}^{X}\phi_{jk}^{X}(s)$
for all $k\in\mathbb{N}$ (Hsing and Eubank, 2015). Since $K^{X}_{jj}(s,t)$ is
symmetric and non-negative definite, we assume, without loss of generality,
that $\\{\lambda^{X}_{js}\\}_{s\in\mathbb{N}^{+}}$ is non-negative and non-
increasing. By the Karhunen-Loève expansion (Hsing and Eubank, 2015,
Theorem7.3.5), $X_{ij}(t)$ can be expressed as
$X_{ij}(t)=\sum_{k=1}^{\infty}a^{X}_{ijk}\phi^{X}_{jk}(t)$, where the
principal component scores satisfy
$a^{X}_{ijk}=\int_{\mathcal{T}}X_{ij}(t)\phi^{X}_{jk}(t)dt$ and
$a^{X}_{ijk}\sim N(0,\lambda_{jk}^{X})$ with $E(a^{X}_{ijk}a^{X}_{ijl})=0$ if
$k\neq l$. Because the eigenfunctions are orthonormal, the $L_{2}$ projection
of $X_{ij}$ onto the span of the first $M$ eigenfunctions is
$X^{M}_{ij}(t)=\sum_{k=1}^{M}a^{X}_{ijk}\phi^{X}_{jk}(t)$. Similarly, we can
define $K^{Y}_{jj}(t,s)$,
$\\{\phi^{Y}_{jk}(t),\lambda^{Y}_{jk}\\}_{k\in\mathbb{N}}$ and
$Y^{M}_{ij}(t)$. Let $K_{jj}(s,t)=K^{X}_{jj}(s,t)+K^{Y}_{jj}(s,t)$ and let
$\\{\phi_{jk}(t),\lambda_{jk}\\}_{k\in\mathbb{N}}$ be the eigenfunction-
eigenvalue pairs of $K_{jj}(s,t)$.
Lemma 3 implies that $X_{ij}(\cdot)$ and $Y_{ij}(\cdot)$ lie within the span
of the eigenfunctions corresponding to the non-zero eigenvalues of $K_{jj}$.
Furthermore, this subspace is minimal in the sense that no subspace with
smaller dimension contains $X_{ij}(\cdot)$ and $Y_{ij}(\cdot)$ almost surely.
Thus, the FPCA basis of $K_{jj}$ provides a good default choice for dimension
reduction.
###### Lemma 3
Let $|\mathbb{V}|$ denote the dimension of a subspace $\mathbb{V}$ and suppose
$M^{\prime}_{j}=\inf\\{|\mathbb{V}|:\mathbb{V}\subseteq\mathbb{H},X_{ij}(\cdot),Y_{ij}(\cdot)\in\mathbb{V}\,\text{almost
surely}\\}.$
Let $\\{\phi_{jk}(t),\lambda_{jk}\\}_{k\in\mathbb{N}}$ be the eigenfunction-
eigenvalue pairs of $K_{jj}(s,t)$ and
$M^{\star}_{j}=\sup\\{M\in\mathbb{N}^{+}:\lambda_{jM}>0\\}.$
Then $M^{\prime}_{j}=M^{\star}_{j}$ and $X_{ij},Y_{ij}\in{\rm
Span}\\{\phi_{j1}(\cdot),\phi_{j2}(\cdot),\dots,\phi_{j,M^{\star}_{j}}(\cdot)\\}$
almost surely.
Proof See Appendix B.2.
### 2.4 Infinite Dimensional Functional Data
In Section 2.2, we defined a functional differential graph for functional data
that have finite-dimensional representation. In this section, we present a
more general definition that also allows for infinite-dimensional functional
data.
As discussed in Section 2.1, when the data are infinite-dimensional,
estimating a functional graphical model is not straightforward because the
precision matrix of the scores does not have a well-defined limit as $M$, the
dimension of the projected data, increases to $\infty$. When estimating the
differential graph, however, although $\|\Theta^{X,M}\|_{\text{F}}\to\infty$
and $\|\Theta^{Y,M}\|_{\text{F}}\to\infty$ as $M\to\infty$, it is still
possible for $\|\Theta^{X,M}-\Theta^{Y,M}\|_{\text{F}}$ to be bounded as
$M\to\infty$. For instance, $x_{n},y_{n}\in\mathbb{R}$ may both tend to
infinity, but $\lim_{n}x_{n}-y_{n}$ may still exist and be bounded.
Furthermore, even when
$\|\Theta^{X,M}-\Theta^{Y,M}\|_{\text{F}}\rightarrow\infty$, it is still
possible for the difference $\Theta^{X,M}-\Theta^{Y,M}$ to be informative.
This observation leads to Definition 1 below. To simplify notation, in the
rest of the paper, we assume that $X_{ij}(\cdot)$ and $Y_{ij}(\cdot)$ live in
an $M^{\star}$ dimensional space where $M^{\star}\leq\infty$. Recall that
$\\{\phi^{X}_{jk}(\cdot),\lambda^{X}_{jk}\\}_{k\in\mathbb{N}}$ and
$\\{\phi^{Y}_{jk}(\cdot),\lambda^{Y}_{jk}\\}_{k\in\mathbb{N}}$ denote the
eigenpairs of $K_{jj}^{X}$ and $K_{jj}^{Y}$ respectively.
###### Definition 1 (Differential Graph Matrix and Comparability)
The MGPs $X$ and $Y$ are comparable if, for all $j\in[p]$, $K_{jj}^{X}$ and
$K_{jj}^{Y}$ have $M^{\star}$ non-zero eigenvalues and
$\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)=\mathrm{span}\left(\\{\phi_{jk}^{Y}\\}_{k=1}^{M^{\star}}\right)$.
Furthermore, for every $(j,l)\in V^{2}$ and a projection subspace sequence
$\left\\{\mathbb{V}^{M}_{[p]}\right\\}_{M\geq 1}$ satisfying that $\lim_{M\to
M^{\star}}\mathbb{V}^{M}_{j}=\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$,
we have either:
$\lim_{M\to
M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}=0\qquad\text{or}\qquad\lim\inf_{M\to
M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}>0.$
In this case, we define the differential graph matrix (DGM)
$D=(D_{jl})_{(j,l)\in V^{2}}\in\mathbb{R}^{p\times p}$, where
$D_{jl}=\lim\inf_{M\to M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}.$ (11)
We say that $X$ and $Y$ are incomparable, if for some $j$, $K_{jj}^{X}$ and
$K_{jj}^{Y}$ have a different number of non-zero eigenvalues, or if
$\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)\neq\mathrm{span}\left(\\{\phi_{jk}^{Y}\\}_{k=1}^{M^{\star}}\right)$,
or if there exists some $(j,l)$ such that given
$\left\\{\mathbb{V}^{M}_{[p]}\right\\}_{M\geq 1}$ satisfying that $\lim_{M\to
M^{\star}}\mathbb{V}^{M}_{j}=\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$,
we have
$\lim\inf_{M\to
M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}=0,\qquad\text{but}\qquad\lim\sup_{M\to
M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}>0.$
In Definition 1 we say $\lim_{M\to
M^{\star}}\mathbb{V}^{M}_{j}=\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$,
to mean the following: For any $\epsilon>0$ and all
$g\in\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$, there
exists $M^{\prime}=M^{\prime}(\epsilon)<\infty$ such that
$\|g-g^{M}_{P}\|<\epsilon$ for all $M\geq M^{\prime}$, where $g^{M}_{P}$
denotes the projection of $g$ onto the subspace of $\mathbb{V}^{M}_{j}$.
When $M^{\star}<\infty$, the conditional independence structure in $X_{i}$ and
$Y_{i}$ can be completely captured by a finite dimensional representation.
When $M^{\star}=\infty$, as $M\to\infty$, $\Delta^{M}_{jl}$ approaches the
difference of two matrices with unbounded eigenvalues. Nonetheless, when $X$
and $Y$ are comparable, the limits are still informative. This would suggest
that by using a sufficiently large subspace, we can capture such a difference
arbitrarily well. On the other hand, if the MGPs are not comparable, then
using a larger subspace may not improve the approximation regardless of the
sample size. For this reason, in the rest of the paper, we only focus on the
setting where $X$ and $Y$ are comparable.
To our knowledge, there is no existing procedure to estimate a graphical model
for functional data when the functions are infinite-dimensional. Thus, it is
not straightforward to determine whether the comparability condition is
stronger or weaker than what might be required for estimating the graphs
separately and then comparing post hoc. Nonetheless, we hope to provide some
intuition for the reader.
Suppose $X$ and $Y$ are of the same dimension, $M^{\star}$. If
$M^{\star}<\infty$ and the functional graphical model for each sample could be
estimated separately (that is, $\|\Theta^{X,M}\|_{F}<\infty$ and
$\|\Theta^{Y,M}\|_{F}<\infty$), then $X$ and $Y$ are comparable when the
minimal basis which spans $X$ and $Y$ is the same. Thus, the functional
differential graph is also well defined. On the other hand, the conditions
required by Qiao et al. (2019, Condition 2) for consistent estimation are not
satisfied when $M^{\star}=\infty$, since
$\lim_{M\rightarrow\infty}\|\Theta^{X,M}\|_{F}=\infty$ due to the compactness
of the covariance operator. However, $X$ and $Y$ may still be comparable
depending on the limiting behavior of $\Theta^{X,M}$ and $\Theta^{Y,M}$. Thus,
there are settings where the differential graph may exist and be consistently
recovered even when each individual graph cannot be recovered (even when $p$
is fixed).
However, when one MGP is finite-dimensional and the other is infinite-
dimensional, then the MGPs are incomparable. To see this, without loss of
generality, we assume that MGP $X$ has infinite dimension
$M^{X}_{j}=M^{\star}_{X}=\infty$ for all $j\in V$ and MGP $Y$ has finite
dimension $M^{Y}_{j}=M^{\star}_{Y}<\infty$ for all $j\in V$. Then
$\Theta^{Y,M}$ is ill-defined when $M>M^{\star}_{Y}$ and recovering the
differential graph is not straightforward.
We now define the notion of a functional differential graph.
###### Definition 2
When two MGPs $X$ and $Y$ are comparable, we define their functional
differential graph as an undirected graph $G_{\Delta}=\\{V,E_{\Delta}\\}$,
where $E_{\Delta}$ is defined as
$E_{\Delta}=\left\\{\\{j,l\\}\,:\,j<l\text{ and }D_{jl}>0\right\\}.$ (12)
###### Remark 1
The functional graphical model defined by Qiao et al. (2019) uses the
conditional covariance function $C_{jl}^{X}(\cdot,*)$ given in (1). Thus, it
would be quite natural to use the conditional covariance functions directly to
define a differential graph where
$E_{\Delta}=\left\\{\\{j,l\\}\;:\;j<l\text{ and }C_{jl}^{X}(\cdot,*)\neq
C_{jl}^{Y}(\cdot,*)\right\\}.$ (13)
Unfortunately, this definition does not always coincide with the one we
propose in Definition 2. Nevertheless, the functional differential graph given
in Definition 2 has many nice statistical properties and retains important
features of the graph defined in (13).
The primary statistical benefit of the graph defined in Definition 2 is that
it can be directly estimated without estimating each conditional independence
function: $C^{X}_{jl}(\cdot,\cdot)$ and $C^{Y}_{jl}(\cdot,\cdot)$. Similar to
the vector-valued case considered by (Zhao et al., 2014), this allows for a
much lower sample complexity when each individual graph is dense but the
difference is sparse. In some settings, there may not be enough samples to
estimate each individual graph accurately, but the difference may still be
recovered. This result is demonstrated in Theorem 1.
The statistical advantages of our estimand unfortunately come at the cost of a
slightly less precise characterization of the difference in the conditional
covariance functions. However, many of the key characteristics are still
preserved. Suppose $X_{i}$ and $Y_{i}$ are both $M^{\star}$-dimensional with
$M^{\star}<\infty$ and further suppose that
$\\{\phi_{jm}(\cdot)\phi_{lm^{\prime}}(*)\\}_{m,m^{\prime}\in[M^{\star}]\times[M^{\star}]}$
is a linearly independent set of functions. Suppose the conditional covariance
functions for $j,l\in V$ are unchanged so that
$C_{jj}^{X}(\cdot,*)=C_{jj}^{Y}(\cdot,*)$ and $C_{ll}^{\backslash
j,X}(\cdot,*)=C_{ll}^{\backslash j,Y}(\cdot,*)$, where
$C_{ll}^{\backslash j,X}(\cdot,*)\coloneqq{\rm
Cov}(X_{l}(\cdot),X_{l}(*)\,|\,X_{k}(\cdot),k\neq j,l)$
and $C_{ll}^{\backslash j,Y}(\cdot,*)$ is defined similarly; then,
$\Delta_{jl}=0$ if and only if $C_{jl}^{X}(\cdot,*)=C_{jl}^{Y,\pi}(\cdot,*)$.
When this holds for all pairs $j,l\in V$, then the definitions of a
differential graph in Definition 2 and (13) are equivalent. When the
conditional covariance functions may change so that $C_{jj}^{X}(\cdot,*)\neq
C_{jj}^{Y}(\cdot,*)$, then we still have that $\Delta_{jl}\neq 0$ if
$C_{jl}^{X,\pi}(\cdot,*)=0$ and $C_{jl}^{Y,\pi}(\cdot,*)\neq 0$ (or vice
versa). Thus, even in this more general setting, the functional differential
graph given in Definition 2 captures all qualitative differences between the
conditional covariance functions $C_{jl}^{X}(\cdot,*)$ and
$C_{jl}^{Y}(\cdot,*)$.
Our objective is to directly estimate $E_{\Delta}$ without first estimating
$E_{X}$ or $E_{Y}$. Since the functions we consider may be infinite-
dimensional objects, in practice, what we can directly estimate is actually
$E^{\pi}_{\Delta}$ defined in (10). We will use a sieve estimator to estimate
$\Delta^{M}$, where $M$ grows with the sample size $n$. When $M^{\star}=M$,
then $E^{\pi}_{\Delta}=E_{\Delta}$. When $M<M^{\star}\leq\infty$, then this is
generally not true; however, we would expect the graphs to be similar when $M$
is large enough compared with $M^{\star}$. Thus, by constructing a suitable
estimator of $\Delta^{M}$, we can still recover $E_{\Delta}$.
### 2.5 Illustration of comparability
We provide few examples that illustrate the notion of comparability. In the
first two examples, the graphs are comparable, while in the third example, the
graphs are incomparable. First, we state a lemma that will be helpful in the
following discussions. The lemma follows directly from the properties of the
multivariate normal and the inverse of block matrices.
###### Lemma 4
Let $H^{X,M}_{jl}=\mathrm{Cov}(a^{X,M}_{ij},a^{X,M}_{il}\mid
a^{X,M}_{ik},k\neq j,l)$ and $H^{\backslash
l,X,M}_{jj}=\mathrm{Var}(a^{X,M}_{ij}\mid a^{X,M}_{ik},k\neq j,l)$. For any
$j\in V$, we have $\Theta^{X,M}_{jj}=(H^{X,M}_{jj})^{-1}$. For any $(j,l)\in
V^{2}$ and $j\neq l$, we have
$\Theta^{X,M}_{jl}=-(H^{X,M}_{jj})^{-1}H^{X,M}_{jl}(H^{\backslash
j,X,M}_{ll})^{-1}$.
Proof See Appendix B.3.
The following proposition follows directly from Lemma 4.
###### Proposition 1
Assume that for any $(j,l)\in V^{2}$ and $j\neq l$, we have
$a^{X}_{ijm}\perp\\!\\!\\!\perp a^{X}_{ijm^{\prime}}\mid a^{X,M}_{ik},k\neq
j\qquad\text{and}\qquad a^{X}_{ijm}\perp\\!\\!\\!\perp
a^{X}_{ijm^{\prime}}\mid a^{X,M}_{ik},k\neq j,l,$
for any $M$ and $1\leq m\neq m^{\prime}\leq M$. We then have
$\Theta^{X,M}_{jj}={\rm diag}\left(\frac{1}{{\rm Var}\left(a^{X}_{ij1}\mid
a^{X,M}_{ik},k\neq j\right)},\dots,\frac{1}{{\rm Var}\left(a^{X}_{ijM}\mid
a^{X,M}_{ik},k\neq j\right)}\right)$
and
$\Theta^{X,M}_{jl,mm^{\prime}}=\frac{{\rm
Cov}\left(a^{X}_{ijm},a^{X}_{ilm^{\prime}}\mid a^{X,M}_{ik},k\neq
j,l\right)}{{\rm Var}\left(a^{X}_{ijm}\mid a^{X,M}_{ik},k\neq j\right){\rm
Var}\left(a^{X}_{ilm^{\prime}}\mid a^{X,M}_{ik},k\neq
j\right)}\overset{\Delta}{=}\bar{v}^{X,jl,M}_{mm^{\prime}},$
for any $M$ and $1\leq m\neq m^{\prime}\leq M$. In addition, if
$a^{Y}_{ijm}\perp\\!\\!\\!\perp a^{Y}_{ijm^{\prime}}\mid a^{Y,M}_{ik},k\neq
j\quad\text{and}\quad a^{Y}_{ijm}\perp\\!\\!\\!\perp a^{Y}_{ijm^{\prime}}\mid
a^{Y,M}_{ik},k\neq j,l,$
for any $M$ and $1\leq m\neq m^{\prime}\leq M$, then
$\displaystyle\Theta^{X,M}_{jj}-\Theta^{Y,M}_{jj}$ $\displaystyle={\rm
diag}\left(\left\\{\frac{{\rm Var}\left(a^{Y}_{ijm}\mid a^{Y,M}_{ik},k\neq
j\right)-{\rm Var}\left(a^{X}_{ijm}\mid a^{X,M}_{ik},k\neq j\right)}{{\rm
Var}\left(a^{X}_{ijm}\mid a^{X,M}_{ik},k\neq j\right){\rm
Var}\left(a^{Y}_{ijm}\mid a^{Y,M}_{ik},k\neq
j\right)}\right\\}^{M}_{m=1}\right)$ $\displaystyle\overset{\Delta}{=}{\rm
diag}\left(\bar{w}^{j,M}_{1},\bar{w}^{j,M}_{2},\dots,\bar{w}^{j,M}_{M}\right)$
and
$\displaystyle\Theta^{X,M}_{jl,mm^{\prime}}-\Theta^{Y,M}_{jl,mm^{\prime}}$
$\displaystyle=\frac{{\rm Cov}\left(a^{X}_{ijm},a^{X}_{ilm^{\prime}}\mid
a^{X,M}_{ik},k\neq j,l\right)}{{\rm Var}\left(a^{X}_{ijm}\mid
a^{X,M}_{ik},k\neq j\right){\rm Var}\left(a^{X}_{ilm^{\prime}}\mid
a^{X,M}_{ik},k\neq j\right)}$ $\displaystyle\qquad\qquad\qquad-\frac{{\rm
Cov}\left(a^{Y}_{ijm},a^{Y}_{ilm^{\prime}}\mid a^{Y,M}_{ik},k\neq
j,l\right)}{{\rm Var}\left(a^{Y}_{ijm}\mid a^{Y,M}_{ik},k\neq j\right){\rm
Var}\left(a^{Y}_{ilm^{\prime}}\mid a^{Y,M}_{ik},k\neq j\right)}$
$\displaystyle=\bar{v}^{Y,jl,M}_{mm^{\prime}}-\bar{v}^{X,jl,M}_{mm^{\prime}}\overset{\Delta}{=}\bar{z}^{jl,M}_{mm^{\prime}},$
for any $M$ and $1\leq m\neq m^{\prime}\leq M$.
With the notation defined in Proposition 1, we have that
$\|\Delta^{M}_{jj}\|^{2}_{\text{HS}}=\sum^{M}_{m=1}\left(\bar{w}^{j,M}_{m}\right)^{2}\qquad\text{and}\qquad\|\Delta^{M}_{jl}\|^{2}_{\text{HS}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{jl,M}_{mm^{\prime}}\right)^{2}.$
(14)
As a result, we have the following condition for comparability.
###### Proposition 2
Under the assumptions in Proposition 1, assume that MGPs $X$ and $Y$ are
$M^{\star}$-dimensional, with $1\leq M^{\star}\leq\infty$, and lie in the same
space. Then they are comparable if and only if for every $(j,l)\in V\times V$,
we have either
$\lim\inf_{M\to
M^{\star}}\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{jl,M}_{mm^{\prime}}\right)^{2}>0\qquad\text{or}\qquad\lim_{M\to
M^{\star}}\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{jl,M}_{mm^{\prime}}\right)^{2}=0,$
(15)
where $\bar{z}^{jl,M}_{mm^{\prime}}$ are defined in Proposition 1.
We now give an infinite-dimensional comparable example.
###### Example 1
Assume that $\\{\epsilon^{X}_{i1k}\\}_{k\geq 1}$,
$\\{\epsilon^{X}_{i2k}\\}_{k\geq 1}$, and $\\{\epsilon^{X}_{i3k}\\}_{k\geq 1}$
are all independent mean zero Gaussian variables with ${\rm
Var}(\epsilon^{X}_{ijk})=\sigma^{2}_{X,jk}$, $j=1,2,3$, $k\geq 1$ for all $i$.
For any $k\geq 1$, let
$a^{X}_{i1k}=a^{X}_{i2k}+\epsilon^{X}_{i1k},\quad
a^{X}_{i2k}=\epsilon^{X}_{i2k},\quad
a^{X}_{i3k}=a^{X}_{i2k}+\epsilon^{X}_{i3k}.$
Let $a^{X,M}_{ij}=(a^{X}_{ij1},\cdots,a^{X}_{ijM})^{\top}$, $j=1,2,3$. We then
define $X_{ij}(t)=\sum^{\infty}_{k=1}a^{X}_{ijk}b_{k}(t)$, $j=1,2,3$, where
$\\{b_{k}(t)\\}^{\infty}_{k=1}$ is some orthonormal function basis of
$\mathbb{H}$. We define $\\{\epsilon^{Y}_{ijk}\\}_{k\geq 1}$,
$\\{a^{Y}_{ijk}\\}_{k\geq 1}$, $a^{Y,M}_{ij}$, and $Y_{ij}(t)$, $j=1,2,3$,
similarly.
The graph structure of $X$ and $Y$ is shown in Figure 1. Since $a^{X,M}_{ij}$
follows a multivariate Gaussian distribution, for any $M\geq 2$, $1\leq
m,m^{\prime}\leq M$ and $m\neq m^{\prime}$:
$\displaystyle{\rm Var}\left(a^{X}_{i1m}\mid
a^{X,M}_{i2},a^{X,M}_{i3}\right)=\sigma^{2}_{X,1m},$ $\displaystyle{\rm
Var}\left(a^{X}_{i3m}\mid a^{X,M}_{i1},a^{X,M}_{i2}\right)=\sigma^{2}_{X,3m},$
$\displaystyle{\rm Var}\left(a^{X}_{i2m}\mid
a^{X,M}_{i1},a^{X,M}_{i3}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}}{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}+\sigma^{2}_{X,1m}\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}},$
and
$\displaystyle{\rm Var}\left(a^{X}_{i1m}\mid
a^{X,M}_{i2}\right)=\sigma^{2}_{X,1m},$ $\displaystyle{\rm
Var}\left(a^{X}_{i1m}\mid
a^{X,M}_{i3}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}+\sigma^{2}_{X,1m}\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}}{\sigma^{2}_{2m}+\sigma^{2}_{3m}},$
$\displaystyle{\rm Var}\left(a^{X}_{i3m}\mid
a^{X,M}_{i2}\right)=\sigma^{2}_{X,3m},$ $\displaystyle{\rm
Var}\left(a^{X}_{i3m}\mid
a^{X,M}_{i1}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}+\sigma^{2}_{X,1m}\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}}{\sigma^{2}_{2m}+\sigma^{2}_{1m}},$
$\displaystyle{\rm Var}\left(a^{X}_{i2m}\mid
a^{X,M}_{i1}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,1m}+\sigma^{2}_{X,2m}},$
$\displaystyle{\rm Var}\left(a^{X}_{i2m}\mid
a^{X,M}_{i3}\right)=\frac{\sigma^{2}_{X,3m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}}.$
In addition, we also have
$\displaystyle{\rm Cov}(a^{X}_{i1m},a^{X}_{3m^{\prime}}\mid a^{X,M}_{i2})=0,$
$\displaystyle{\rm Cov}(a^{X}_{i1m},a^{X}_{i2m^{\prime}}\mid
a^{X,M}_{i3})=\mathbbm{1}(m=m^{\prime})\cdot\frac{\sigma^{2}_{X,3m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}},$
$\displaystyle{\rm Cov}(a^{X}_{i2m},a^{X}_{i3m^{\prime}}\mid
a^{X,M}_{i3})=\mathbbm{1}(m=m^{\prime})\cdot\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,1m}+\sigma^{2}_{X,2m}}.$
123 Figure 1: The conditional independence graph for both $X$ and $Y$ in
Example 1. The differential graph between $X$ and $Y$ has the same structure.
Similar results hold for $Y$. Suppose that
$\sigma^{2}_{X,jk},\sigma^{2}_{Y,jk}\asymp
k^{-\alpha}\quad\text{and}\quad|\sigma^{2}_{X,jk}-\sigma^{2}_{Y,jk}|\asymp
k^{-\beta},\quad j=1,2,3,$
where $\alpha,\beta>0$ and $\beta>\alpha$. Then
$\displaystyle\bar{z}^{13,M}_{mm^{\prime}}=0,$
$\displaystyle\bar{z}^{12,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\frac{\sigma^{2}_{X,1m}-\sigma^{2}_{Y,1m}}{\sigma^{2}_{X,1m}\cdot\sigma^{2}_{Y,1m}}\asymp\mathbbm{1}(m=m^{\prime})\cdot
m^{-(\beta-\alpha)},$
$\displaystyle\bar{z}^{23,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\frac{\sigma^{2}_{X,3m}-\sigma^{2}_{Y,3m}}{\sigma^{2}_{X,3m}\cdot\sigma^{2}_{Y,3m}}\asymp\mathbbm{1}(m=m^{\prime})\cdot
m^{-(\beta-\alpha)}.$
This implies that
$\displaystyle\|\Delta^{M}_{13}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{13,M}_{mm^{\prime}}\right)^{2}=0,$
(16)
$\displaystyle\|\Delta^{M}_{12}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{12,M}_{mm^{\prime}}\right)^{2}\asymp\sum^{M}_{m=1}\frac{1}{m^{\beta-\alpha}},$
$\displaystyle\|\Delta^{M}_{23}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{23,M}_{mm^{\prime}}\right)^{2}\asymp\sum^{M}_{m=1}\frac{1}{m^{\beta-\alpha}}.$
When $\beta>\alpha+1$, we have
$0<\lim_{M\to\infty}\|\Delta^{M}_{12}\|_{\text{F}}=\lim_{M\to\infty}\|\Delta^{M}_{23}\|_{\text{F}}<\infty$.
When $\beta\leq\alpha+1$, we have
$\lim_{M\to\infty}\|\Delta^{M}_{12}\|_{\text{F}}=\lim_{M\to\infty}\|\Delta^{M}_{23}\|_{\text{F}}=\infty$.
In both cases the two graphs are comparable.
The following example describes a sequence of MGPs that are comparable;
however, the differential graph is intrinsically hard to estimate.
###### Example 2
We define $\\{\epsilon^{X}_{ijk}\\}_{k\geq 1}$, $\\{a^{X}_{ijk}\\}_{k\geq 1}$,
$\\{\epsilon^{Y}_{ijk}\\}_{k\geq 1}$, and $\\{a^{Y}_{ijk}\\}_{k\geq 1}$ as in
Example 1. Let $X_{ij}(t)=\sum^{M^{\star}}_{k=1}a^{X}_{ijk}b_{k}(t)$ and
$Y_{ij}(t)=\sum^{M^{\star}}_{k=1}a^{Y}_{ijk}b_{k}(t)$, $j=1,2,3$, where
$M^{\star}$ is a positive integer. Suppose that
$\sigma^{2}_{X,jk},\sigma^{2}_{Y,jk}\asymp
k^{-\alpha}\quad\text{and}\quad|\sigma^{2}_{X,jk}-\sigma^{2}_{Y,jk}|\asymp\mathbbm{1}(k=M^{\star})k^{-\beta},\quad
j=1,2,3,$
where $\alpha,\beta>0$ and $\beta>\alpha$. Following the argument in Example
1, for any $1\leq M\leq M^{\star}$, we have
$\displaystyle\bar{z}^{13,M}_{mm^{\prime}}=0,$
$\displaystyle\bar{z}^{12,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot\frac{\sigma^{2}_{X,1m}-\sigma^{2}_{Y,1m}}{\sigma^{2}_{X,1m}\cdot\sigma^{2}_{Y,1m}}\asymp\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot
m^{-(\beta_{1}-2\alpha_{1})},$
$\displaystyle\bar{z}^{23,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot\frac{\sigma^{2}_{X,3m}-\sigma^{2}_{Y,3m}}{\sigma^{2}_{X,3m}\cdot\sigma^{2}_{Y,3m}}\asymp\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot
m^{-(\beta_{3}-2\alpha_{3})}.$
This implies that
$\displaystyle\|\Delta^{M}_{13}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{13,M}_{mm^{\prime}}\right)^{2}=0,$
(17)
$\displaystyle\|\Delta^{M}_{12}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{12,M}_{mm^{\prime}}\right)^{2}\asymp
M^{-2(\beta-2\alpha)}\mathbbm{1}(M=M^{\star}),$
$\displaystyle\|\Delta^{M}_{23}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{23,M}_{mm^{\prime}}\right)^{2}\asymp
M^{-2(\beta-2\alpha)}\mathbbm{1}(M=M^{\star}).$
Based on the calculation above, we observe that estimation of the differential
graph here is intrinsically hard. For any $M<M^{\star}$, we have
$\|\Delta^{M}_{12}\|_{\text{F}}=\|\Delta^{M}_{23}\|_{\text{F}}=0$. Thus, when
$M<M^{\star}$ is used for estimation, the resulting target graph
$E^{\pi}_{\Delta}$ would be empty. However, by Definition 1 and Definition 2,
we have $D_{12}=D_{23}\asymp(M^{\star})^{-2(\beta-2\alpha)}>0$ and
$E_{\Delta}=\\{(1,2),(2,3)\\}$.
In practice, if $M^{\star}$ is very large and we do not have enough samples to
accurately estimate $\Delta^{M}$ for a large $M$, then it is hopeless for us
to estimate the differential graph correctly. Moreover, the situation is worse
if $\beta>2\alpha$, since $D_{12}$ and $D_{23}$—the signal strength—vanish as
$M^{\star}$ increases. Figure 2 shows how the signal strength (defined as
$D_{12}$) changes as $M^{\star}$ increases for three cases: $\beta<2\alpha$,
$\beta=2\alpha$, and $\beta>2\alpha$.
This problem is intrinsically hard because the difference between two graphs
only occurs between components with the smallest positive eigenvalue. To
capture this difference, we have to use a large number of basis $M$ to
approximate the functional data, which is statistically expensive. As we
increase $M$, no useful information is captured until $M=M^{\star}$.
Furthermore, if the difference between eigenvalues decreases fast relative to
the decrease of eigenvalues, the signal strength will be very weak when the
intrinsic dimension is large. This example shows that the estimation of
functional differential graphical models is harder compared to the scalar
case.
Figure 2: Signal Strength $D_{12}\asymp(M^{\star})^{-2(\beta-2\alpha)}$ in
Example 2.
In Example 1, we characterized a pair of infinite-dimensional MGPs which are
comparable, and in Example 2 we discussed a sequence of models which are all
comparable, but increasingly difficult to recover. The following example
demonstrates that there are infinite-dimensional MGPs that may share the same
eigenspace, but are still not comparable.
###### Example 3
We construct two MGPs that are both infinite-dimensional and have the same
eigenspace, but are incomparable. As with the previous two examples, let
$V=\\{1,2,3\\}$. We assume that $X$ and $Y$ share a common set of
eigenfunctions: $\\{\phi_{m}\\}_{m=1}^{\infty}$ for $j=1,2,3$.
We construct the distribution of the scores of $X$ and $Y$ as follows. For for
any $m\in\mathbb{N}^{+}$, let $a^{X}_{i\,\cdot\,m}$ denote the vector of
scores $(a^{X}_{i1m},a^{X}_{i2m},a^{X}_{i3m})$ and define
$a^{Y}_{i\,\cdot\,m}$ analogously. For any natural number $z$, we first assume
that
$a^{X}_{i\,\cdot\,(3z-2)},a^{X}_{i\,\cdot\,(3z-1)},a^{X}_{i\,\cdot\,(3z)}\perp\\!\\!\\!\perp\\{a^{X}_{i\,\cdot\,k}\\}_{k\neq
3z,3z-1,3z-2}.$ (18)
Thus, the conditional independence graph for the individual scores is a set of
disconnected subgraphs corresponding to
$\\{a^{X}_{i\,\cdot\,(3z-2)},a^{X}_{i\,\cdot,(3z-1)},a^{X}_{i\,\cdot\,(3z)}\\}$
for $z\in\mathbb{N}^{+}$. We make the analogous assumption for the scores of
$Y$.
Within the sets
$\\{a^{X}_{i\,\cdot\,(3z-2)},a^{X}_{i\,\cdot\,(3z-1)},a^{X}_{i\,\cdot\,(3z)}\\}$
and
$\\{a^{Y}_{i\,\cdot\,(3z-2)},a^{Y}_{i\,\cdot\,(3z-1)},a^{Y}_{i\,\cdot\,(3z)}\\}$,
we assume that the conditional independence graph has the structure shown in
Figure 3. By construction, when projecting onto the span of the first $M$
functions, the edge set of individual functional graphical models for
$X^{\pi}$ and $Y^{\pi}$ is not stable as $M\rightarrow\infty$. In particular,
for both $X$ and $Y$, the edges $(1,2)$ and $(2,3)$ will persist; however, the
edge $(1,3)$ will either appear or be absent depending on $M$.
$a^{X}_{i1(3z-2)}$$a^{X}_{i2(3z-2)}$$a^{X}_{i3(3z-2)}$$a^{X}_{i1(3z-1)}$$a^{X}_{i2(3z-1)}$$a^{X}_{i3(3z-1)}$$a^{X}_{i1(3z)}$$a^{X}_{i2(3z)}$$a^{X}_{i3(3z)}$
(a) CI graph for $X$ scores
$a^{Y}_{i1(3z-2)}$$a^{Y}_{i2(3z-2)}$$a^{Y}_{i3(3z-2)}$$a^{Y}_{i1(3z-1)}$$a^{Y}_{i2(3z-1)}$$a^{Y}_{i3(3z-1)}$$a^{Y}_{i1(3z)}$$a^{Y}_{i2(3z)}$$a^{Y}_{i3(3z)}$
(b) CI graph for $Y$ scores
Figure 3: CI graph for the individual scores for two incomparable MGPs.
If $M\mod 3=1$, which corresponds to the first row in Figure 3 where $M=3z-2$
for some $z\in\mathbb{N}^{+}$, then
$\\{a^{X}_{i1k}\\}_{k<M}\perp\\!\\!\\!\perp\\{a^{X}_{i3k}\\}_{k<M}\mid\\{a^{X}_{i2k}\\}_{k\leq
M}\quad\text{ and
}\quad\\{a^{Y}_{i1k}\\}_{k<M}\perp\\!\\!\\!\perp\\{a^{Y}_{i3k}\\}_{k<M}\mid\\{a^{Y}_{i2k}\\}_{k\leq
M}.$ (19)
However, $a^{X}_{i1M}\not\perp\\!\\!\\!\perp
a^{X}_{i3M}\mid\\{a^{X}_{i2k}\\}_{k\leq M}$ since we do not condition on
$a^{X}_{i2(M+1)}$. Similarly, $a^{Y}_{i1M}\not\perp\\!\\!\\!\perp
a^{Y}_{i3M}\mid\\{a^{Y}_{i2k}\\}_{k\leq M}$ since we do not condition on
$a^{Y}_{i2(M+2)}$. Thus, the edge $(1,3)$ is in the functional graphical model
for both $X^{\pi}$ and $Y^{\pi}$; however, the specific values of
$\Theta^{X,M}$ and $\Theta^{Y,M}$ may differ.
In contrast to the previous case, when $M\mod 3=2$, which corresponds to the
second row in Figure 3 where $M=3z-1$ for some $z\in\mathbb{N}^{+}$, the
functional graphical models for $X^{\pi}$ and $Y^{\pi}$ now differ. Note that,
$\\{a^{X}_{i1k}\\}_{k\leq M}\perp\\!\\!\\!\perp\\{a^{X}_{i3k}\\}_{k\leq
M}\mid\\{a^{X}_{i2k}\\}_{k\leq M}$. Thus, the edge $(1,3)$ is absent in the
functional graphical model for $X^{\pi}$ and $\Theta^{X,M}_{1,3}=0$.
Considering $Y^{\pi}$, we have that
$\\{a^{Y}_{i1k}\\}_{k<M-1}\perp\\!\\!\\!\perp\\{a^{Y}_{i3k}\\}_{k<M-1}\mid\\{a^{Y}_{i2k}\\}_{k\leq
M}$. However, because we do not condition on $a^{Y}_{i2(M+1)}$ (the node in
the third row of Figure 3), the $(1,3)$ edge exists in the functional
graphical model for $Y^{\pi}$ since $a^{Y}_{i1(M-1)}\not\perp\\!\\!\\!\perp
a^{Y}_{i3(M-1)}\mid\\{a^{Y}_{i2k}\\}_{k\leq M}$.
In this setting where $M\mod 3=2$, for all $z\in\mathbb{N}^{+}$, we set the
covariance of the scores to be
$z^{-\beta}\times\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i1(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i1(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i1(3z)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i2(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i2(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i2(3z)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i3(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i3(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
a^{Y}_{i3(3z)}\\\a^{Y}_{i1(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
3/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i1(3z-1)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i1(3z)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i2(3z-2)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 8$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i2(3z-1)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
4$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i2(3z)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i3(3z-2)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 3/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i3(3z-1)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i3(3z)}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\\$\hfil\kern
5.0pt\crcr}}}}\right]$}},$ (20)
where $\beta>0$ is a parameter which determines the decay rate of the
eigenvalues (see Assumption 3). We then set all other elements of the
covariance to be $0$. The support of the inverse of this matrix corresponds to
the edges of the graph in Figure 3. However, when we consider the marginal
distribution of the first $M$ scores and invert the corresponding covariance,
$\Theta^{Y,M}_{1,3}$ is $0$ everywhere except for the element corresponding to
$a^{Y}_{i,1,M-1}$ and $a^{Y}_{i,3,M-1}$, that is, nodes in the top row of
Figure 3, which is equal to $-1/4\times((M+1)/3)^{\beta}$. Thus,
$\|\Delta^{M}_{1,3}\|_{F}=1/4\times((M+1)/3)^{\beta}$ and
$\lim\sup_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=\infty$.
Finally, when $M\mod 3=0$, that is, $M=3z$ for some $z\in\mathbb{N}^{+}$, the
$(1,3)$ edge is absent in both functional graphical models for $X^{\pi}$ and
$Y^{\pi}$ because
$\\{a^{X}_{i1k}\\}_{k\leq M}\perp\\!\\!\\!\perp\\{a^{X}_{i3k}\\}_{k\leq
M}\mid\\{a^{X}_{i2k}\\}_{k\leq M}\ \text{ and }\ \\{a^{Y}_{i1k}\\}_{k\leq
M}\perp\\!\\!\\!\perp\\{a^{Y}_{i3k}\\}_{k\leq M}\mid\\{a^{Y}_{i2k}\\}_{k\leq
M}.$
Thus, $\Theta^{X,M}_{1,3}=\Theta^{Y,M}_{1,3}=\Delta^{M}_{1,3}=0$. This implies
that $\lim\inf_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=0$.
Because $\lim\inf_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=0$, but
$\lim\sup_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=\infty$, $X$ and $Y$
are incomparable.
The notion of comparability illustrates the intrinsic difficulty of dealing
with functional data. However, it also illustrates when we can still hope to
estimate the differential network consistently. We have formally stated when
two infinite-dimensional functional graphical models will be comparable and
have given conditions and examples of comparability. Unfortunately, these
conditions cannot be checked using observational data. For this reason, we
mainly discuss the methodology and theoretical properties for estimation of
$E^{\pi}_{\Delta}$. Prior knowledge about the problem at hand should be used
to decide whether two infinite-dimensional functional graphs are comparable.
This is similar to other assumptions common in the graphical modeling
literature, such as “faithfulness” (Spirtes et al., 2000), that are critical
to graph recovery, but can not be verified.
## 3 Functional Differential Graph Estimation: FuDGE
In this section, we detail our methodology for estimating a functional
differential graph. Unfortunately, in most situations, there may not be prior
knowledge on which subspace to use to define the functional differential
graph. In such situations, we suggest using the principle component scores of
$K_{jj}(s,t)=K^{X}_{jj}(s,t)+K^{Y}_{jj}(s,t)$, $j\in V$ as a default choice.
In addition, each observed function is only recorded (potentially with
measurement error) at discrete time points. In Section 3.1 we consider this
practical setting. Of course, if an appropriate basis for dimension reduction
is known in advance or if the functions are fully observed at all time points,
then the estimated objects can always be replaced with their known/observed
counterparts.
### 3.1 Estimating the covariance of the scores
For each $X_{ij}$, suppose we have measurements at time points $t_{ijk}$,
$k=1,\ldots,T$,333For simplicity, we assume that all functions have the same
number of observations, however, our method and theory can be trivially
extended to allow a different number of observations for each function. and
the recorded data, $h_{ijk}$, are the function values with random noise. That
is,
$h_{ijk}=g_{ij}(t_{ijk})+\epsilon_{ijk},$ (21)
where $g_{ij}$ can denote either $X_{ij}$ or $Y_{ij}$ and the unobserved noise
$\epsilon_{ijk}$ is i.i.d. Gaussian with mean $0$ and variance
$\sigma^{2}_{0}$. Without loss of generality, we assume that
$t_{ij1}<\ldots<t_{ijT}$ for any $1\leq i\leq n$ and $1\leq j\leq p$. We do
not assume that $t_{ijk}=t_{i^{\prime}jk}$ for $i\neq i^{\prime}$, so that
each observation may be observed on a different grid.
We first use a basis expansion to estimate a least squares approximation of
the whole curve $X_{ij}(t)$ (see Section 4.2 in Ramsay and Silverman (2005)).
Specifically, given an initial basis function vector
$b(t)=(b_{1}(t),\dots,b_{L}(t))^{\top}$—for example, the B-spline or Fourier
basis—our estimated approximation for $X_{ij}(t)$ is given by:
$\displaystyle\hat{X}_{ij}(t)$ $\displaystyle=\hat{\beta}_{ij}^{\top}b(t),$
(22) $\displaystyle\hat{\beta}_{ij}$
$\displaystyle=\left(B^{\top}_{ij}B_{ij}\right)^{-1}B^{\top}_{ij}h_{ij},$
where $h_{ij}=(h_{ij1},h_{ij2},\dots,h_{ijT})^{\top}$ and $B_{ij}$ is the
design matrix for $g_{ij}$:
$B_{ij}=\left[\begin{matrix}b_{1}(t_{ij1})&\cdots&b_{L}(t_{ij1})\\\
\vdots&\ddots&\vdots\\\
b_{1}(t_{ijT})&\cdots&b_{L}(t_{ijT})\end{matrix}\right]\in\mathbb{R}^{T\times
L}.$ (23)
The computational complexity of the basis expansion procedure is
$O(npT^{3}L^{3})$, and in practice, there are many efficient package
implementations of this step; for example, fda (Ramsay et al., 2020).
We repeat this process for the observed $Y$ functions. After obtaining
$\\{\hat{X}_{ij}(t)\\}_{j\in V,i=1,2,\dots,n_{X}}$ and
$\\{\hat{Y}_{ij}(t)\\}_{j\in V,i=1,2,\dots,n_{Y}}$, we use them as inputs to
the FPCA procedure. Specifically, we first estimate the sum of the covariance
functions by
$\hat{K}_{jj}(s,t)=\hat{K}^{X}_{jj}(s,t)+\hat{K}^{Y}_{jj}(s,t)=\frac{1}{n_{X}}\sum^{n_{X}}_{i=1}\hat{X}_{ij}(s)\hat{X}_{ij}(t)+\frac{1}{n_{Y}}\sum^{n_{Y}}_{i=1}\hat{Y}_{ij}(s)\hat{Y}_{ij}(t).$
(24)
Using $\hat{K}_{jj}(s,t)$ as the input to FPCA, we can estimate the
corresponding eigenfunctions $\hat{\phi}_{jk}(t)$, $k=1,\ldots,M$,
$j=1,\ldots,p$. Given the estimated eigenfunctions, we compute the estimated
projection scores
$\displaystyle\hat{a}^{X}_{ijk}$
$\displaystyle=\int_{\mathcal{T}}\hat{X}_{ij}(t)\hat{\phi}_{jk}(t)dt\qquad\text{and}\qquad\hat{a}^{Y}_{ijk}=\int_{\mathcal{T}}Y_{ij}(t)\hat{\phi}_{jk}(t)dt,$
and collect them into vectors
$\displaystyle a^{X,M}_{ij}$
$\displaystyle=(a^{X}_{ij1},a^{X}_{ij2},\dots,a^{X}_{ijM})^{\top}\in{\mathbb{R}^{M}}\qquad\text{and}\qquad
a^{X,M}_{i}$
$\displaystyle=((a^{X,M}_{i1})^{\top},\ldots,(a^{X,M}_{ip})^{\top})^{\top}\in{\mathbb{R}^{pM}},$
$\displaystyle a^{Y,M}_{ij}$
$\displaystyle=(a^{Y}_{ij1},a^{Y}_{ij2},\dots,a^{Y}_{ijM})^{\top}\in{\mathbb{R}^{M}}\qquad\text{and}\qquad
a^{Y,M}_{i}$
$\displaystyle=((a^{Y,M}_{i1})^{\top},\ldots,(a^{Y,M}_{ip})^{\top})^{\top}\in{\mathbb{R}^{pM}}.$
Finally, we estimate the covariance matrices of the score vectors,
$\Sigma^{X,M}$ and $\Sigma^{Y,M}$, as
$\displaystyle
S^{X,M}=\frac{1}{n_{X}}\sum^{n_{X}}_{i=1}\hat{a}^{X,M}_{i}(\hat{a}^{X,M}_{i})^{\top}\qquad\text{and}\qquad
S^{Y,M}=\frac{1}{n_{Y}}\sum^{n_{Y}}_{i=1}\hat{a}^{Y,M}_{i}(\hat{a}^{Y,M}_{i})^{\top}.$
### 3.2 FuGDE: Functional Differential Graph Estimation
We now describe the FuDGE algorithm for Functional Differential Graph
Estimation. To estimate $\Delta^{M}$, we solve the following optimization
program:
$\hat{\Delta}^{M}\in\operatorname*{arg\,min}_{\Delta\in{\mathbb{R}^{pM\times
pM}}}L(\Delta)+\lambda_{n}\sum_{\\{i,j\\}\in V^{2}}\|\Delta_{ij}\|_{F},$ (25)
where
$L(\Delta)=\mathrm{tr}\left[\frac{1}{2}S^{Y,M}\Delta^{\top}{S^{X,M}}\Delta-\Delta^{\top}\left(S^{Y,M}-S^{X,M}\right)\right]$
and $S^{X,M}$ and $S^{Y,M}$ are obtained as described in Section 3.1.
We construct the loss function, $L(\Delta)$, so that the true parameter value,
$\Delta^{M}=\left(\Sigma^{X,M}\right)^{-1}-\left(\Sigma^{Y,M}\right)^{-1}$,
minimizes the population loss $\mathbb{E}\left[L(\Delta)\right]$, which for a
differentiable and convex loss function, is equivalent to selecting $L$ such
that $\mathbb{E}\left[\nabla L(\Delta^{M})\right]=0$. Since $\Delta^{M}$
satisfies
$\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}-(\Sigma^{Y,M}-\Sigma^{X,M})=0,$
a choice for $\nabla L(\Delta)$ is
$\nabla{L(\Delta^{M})}=S^{X,M}\Delta^{M}{S^{Y,M}}-\left(S^{Y,M}-S^{X,M}\right)$
(26)
so that
$\mathbb{E}\left[\nabla
L(\Delta^{M})\right]=\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}-(\Sigma^{Y,M}-\Sigma^{X,M})=0.$
Given this choice of $\nabla L(\Delta)$, $L(\Delta)$ in (25) directly follows
from properties of the differential of the trace function. The chosen loss is
quadratic (see (B.9) in appendix) and leads to an efficient algorithm. Similar
loss functions are used in Xu and Gu (2016), Yuan et al. (2017), Na et al.
(2019), and Zhao et al. (2014).
We also include the additional group lasso penalty (Yuan and Lin, 2006) to
promote blockwise sparsity in $\hat{\Delta}^{M}$. The objective in (25) can be
solved by a proximal gradient method detailed in Algorithm 1. Finally, we form
$\hat{E}_{\Delta}$ by thresholding $\hat{\Delta}^{M}$ so that:
${}\hat{E}_{\Delta}=\left\\{\\{j,l\\}\,:\,\|\hat{\Delta}^{M}_{jl}\|_{F}>\epsilon_{n}\;\;\text{or}\;\;\|\hat{\Delta}^{M}_{lj}\|_{F}>\epsilon_{n}\right\\}.$
(27)
The thresholding step in (27) is used for theoretical purposes. Specifically,
it helps correct for the bias induced by the finite-dimensional truncation and
relaxes commonly used assumptions for the graph structure recovery, such as
the irrepresentability or incoherence condition (van de Geer and Bühlmann,
2009). In practice, one can simply set $\epsilon_{n}=0$, as we do in the
simulations.
### 3.3 Optimization Algorithm for FuDGE
Algorithm 1 Functional differential graph estimation
0: $S^{X,M},S^{Y,M},\lambda_{n},\eta$.
0: $\hat{\Delta}^{M}$.
Initialize $\Delta^{(0)}=0_{pM}$.
repeat
$A=\Delta-\eta\nabla L(\Delta)=\Delta-\eta\left(S^{X,M}\Delta
S^{Y,M}-\left(S^{Y,M}-S^{X,M}\right)\right)$
for $1\leq{i,j}\leq{p}$ do
$\Delta_{jl}\leftarrow\left(\frac{\|A_{jl}\|_{F}-\lambda_{n}\eta}{\|A_{jl}\|_{F}}\right)_{+}\cdot
A_{jl}$
end for
until Converge
The optimization program (25) can be solved by a proximal gradient method
(Parikh and Boyd, 2014) summarized in Algorithm 1. Specifically, at each
iteration, we update the current value of $\Delta$, denoted as
$\Delta^{\text{old}}$, by solving the following problem:
${}\Delta^{\text{new}}=\operatorname*{arg\,min}_{\Delta}\left(\frac{1}{2}\left\|\Delta-\left(\Delta^{\text{old}}-\eta\nabla
L\left(\Delta^{\text{old}}\right)\right)\right\|_{F}^{2}+\eta\cdot\lambda_{n}\sum^{p}_{j,l=1}\|\Delta_{jl}\|_{F}\right),$
(28)
where $\nabla L(\Delta)$ is defined in (26) and $\eta$ is a user specified
step size. Note that $\nabla L(\Delta)$ is Lipschitz continuous with Lipschitz
constant $\lambda^{S}_{\max}=\|S^{Y,M}\otimes
S^{X,M}\|_{2}=\lambda_{\max}(S^{Y,M})\lambda_{\max}(S^{X,M})$. Thus, for any
step size $\eta$ such that $0<\eta\leq 1/\lambda^{S}_{\max}$, the proximal
gradient method is guaranteed to converge (Beck and Teboulle, 2009).
The update in (28) has a closed-form solution:
${}\Delta^{\text{new}}_{jl}=\left[\left(\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta\right)/\|A^{\text{old}}_{jl}\|_{F}\right]_{+}\cdot
A^{\text{old}}_{jl},\qquad 1\leq{j,l}\leq{p},$ (29)
where $A^{\text{old}}=\Delta^{\text{old}}-\eta\nabla L(\Delta^{\text{old}})$
and $x_{+}=\max\\{0,x\\},x\in{\mathbb{R}}$, represents the positive part of
$x$. Detailed derivations are given in the appendix. Note that although the
true $\Delta^{M}$ is symmetric, we do not explicitly enforce symmetry in
$\hat{\Delta}^{M}$ in Algorithm 1.
After performing FPCA, the proximal gradient descent method converges in
$O\left(\lambda^{S}_{\max}/\text{tol}\right)$ iterations, where tol is a user
specified optimization error tolerance, and each iteration takes $O((pM)^{3})$
operations; see Tibshirani (2010) for a convergence analysis of the general
proximal gradient descent algorithm.
### 3.4 Selection of Tuning Parameters
There are four tuning parameters that need to be chosen for implementing
FuDGE: $L$ (dimension of the basis used to estimate the curves from the
discretely observed data), $M$ (dimension of subspace to estimate the
projection scores), $\lambda_{n}$ (regularization parameter to tune the block
sparsity of $\Delta^{M}$), and $\epsilon_{n}$ (thresholding parameter for
$\hat{E}_{\Delta}$). While we need the thresholding parameter $\epsilon_{n}$
in (27) to establish theoretical results, in practice, we simply set
$\epsilon_{n}=0$. To select $M$, we follow the procedure in Qiao et al.
(2019). More specifically, for each discretely-observed curve, we first
estimate the underlying functions by fitting an $L$-dimensional B-spline
basis. Both $M$ and $L$ are then chosen by 5-fold cross-validation as
discussed in Qiao et al. (2019).
Finally, to choose $\lambda_{n}$, we recommend using selective cross-
validation (SCV) (She, 2012). Given a value of $\lambda_{n}$, we use the
entire data set to estimate a sparsity pattern. Then, fixing the sparsity
pattern, we use a typical cross-validation procedure to calculate the CV
error. Ultimately, we choose the value of $\lambda_{n}$ that results in the
sparsity pattern that minimizes the CV error. In addition to SCV, if we have
some prior knowledge about the number of edges in the differential graph, we
can also choose $\lambda_{n}$ that results in a desired level of sparsity of
the differential graph.
## 4 Theoretical Properties
In this section, we provide theoretical guarantees for FuDGE. We first give a
deterministic result for $\hat{E}_{\Delta}$ defined in (27) when the max-norm
of the difference between the estimates $S^{X,M},S^{Y,M}$ and their
corresponding parameters, $\Sigma^{X,M},\Sigma^{Y,M}$, is bounded by
$\delta_{n}$. We then show that when projecting the data onto either a fixed
basis or an estimated basis—under some mild conditions—$\delta_{n}$ can be
controlled and the bias of the finite-dimensional projection decreases fast
enough that $E_{\Delta}$ can be consistently recovered.
### 4.1 Deterministic Guarantees for $\hat{E}_{\Delta}$
In this section, we assume that $S^{X,M},S^{Y,M}$ are good estimates of
$\Sigma^{X,M},\Sigma^{Y,M}$ and give a deterministic result in Theorem 1. Let
$n=\min\\{n_{X},n_{Y}\\}$. We assume that the following holds.
###### Assumption 1
The matrices $S^{X,M},S^{Y,M}$ are estimates of $\Sigma^{X,M},\Sigma^{Y,M}$
that satisfy
$\max\left\\{|S^{X,M}-\Sigma^{X,M}|_{\infty},|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\right\\}\leq\delta_{n}.$
(30)
We also require that $E_{\Delta}$ is sparse. This does not preclude the case
where $E_{X}$ and $E_{Y}$ are dense, as long as there are not too many
differences in the precision matrices. This assumption is also required when
estimating a differential graph from vector-valued data; for example, see
Condition 1 in Zhao et al. (2014).
###### Assumption 2
There are $s$ edges in the differential graph; that is, $|E_{\Delta}|=s$ and
$s\ll p$.
We introduce the following three quantities that characterize the problem
instance and will be used in Theorem 1 below:
$\nu_{1}=\nu_{1}(M)=\min_{(j,l)\in
E_{\Delta}}\|\Delta^{M}_{jl}\|_{F},\quad\nu_{2}=\nu_{2}(M)=\max_{(j,l)\in
E^{C}_{\Delta}}\|\Delta^{M}_{jl}\|_{F},$
and
$\tau=\tau(M)=\nu_{1}(M)-\nu_{2}(M).$ (31)
Roughly speaking, $\nu_{1}(M)$ indicates the “signal strength” present when
using the $M$-dimensional representation and $\nu_{2}(M)$ measures the bias.
By Definition 1, when $X$ and $Y$ are comparable, we have $\lim\inf_{M\to
M^{\star}}\nu_{1}(M)>0$ and $\lim_{M\to M^{\star}}\nu_{2}(M)=0$. Therefore,
for a large enough $M$, we have $\tau>0$. However, a smaller $\tau$ implies
that the differential graph is harder to recover.
Before we give the deterministic result in Theorem 1, we first define
additional quantities that will be used in subsequent results. Let
$\displaystyle\sigma_{\max}$
$\displaystyle=\max\\{|\Sigma^{X,M}|_{\infty},|\Sigma^{Y,M}|_{\infty}\\},$
(32) $\displaystyle\lambda^{*}_{\min}$
$\displaystyle=\lambda_{\min}\left(\Sigma^{X,M}\right)\times\lambda_{\min}\left(\Sigma^{Y,M}\right),\text{
and }$ $\displaystyle\Gamma^{2}_{n}$
$\displaystyle=\frac{9\lambda^{2}_{n}s}{\kappa^{2}_{\mathcal{L}}}+\frac{2\lambda_{n}}{\kappa_{\mathcal{L}}}(\omega^{2}_{\mathcal{L}}+2p^{2}\nu_{2}),$
where
$\displaystyle\lambda_{n}$
$\displaystyle=\;2M\left[\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right)\left|\Delta^{M}\right|_{1}+2\delta_{n}\right],$
(33) $\displaystyle\kappa_{\mathcal{L}}$
$\displaystyle=\;(1/2)\lambda^{*}_{\min}-8M^{2}s\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right),$
$\displaystyle\omega_{\mathcal{L}}$
$\displaystyle=\;4Mp^{2}\nu_{2}\sqrt{\delta_{n}^{2}+2\delta_{n}\sigma_{\max}},$
and $\delta_{n}$ is defined in Assumption 1. Note that $\Gamma_{n}$—which
measures the estimation error of
$\|\hat{\Delta}^{M}-\Delta^{M}\|_{\text{F}}$—implicitly depends on
$\delta_{n}$ through $\lambda_{n}$, $\kappa_{\mathcal{L}}$, and
$\omega_{\mathcal{L}}$. We observe that $\Gamma_{n}$ decreases to zero as
$\delta_{n}$ goes to zero. The quantity $\kappa_{\mathcal{L}}$ is the maximum
restricted eigenvalue from the analysis framework of Negahban et al. (2012).
Finally, $\omega_{\mathcal{L}}$ is the tolerance parameter that comes from the
fact that $\nu_{2}$ might be larger than zero, and it will decrease to zero as
$\nu_{2}$ goes to zero.
###### Theorem 1
Given Assumptions 1 and 2, when
$\nu_{1}(M),\nu_{2}(M),\delta_{n},\lambda_{n},\sigma_{\max},M$ and $s$ satisfy
$\displaystyle
0<\Gamma_{n}<\tau/2\qquad\text{and}\qquad\delta_{n}<(1/4)\sqrt{\left(\lambda^{*}_{\min}+16M^{2}s(\sigma_{\max})^{2}\right)/\left(M^{2}s\right)}-\sigma_{\max},$
(34)
then setting
$\epsilon_{n}\in\left[\nu_{2}+\Gamma_{n},\nu_{1}-\Gamma_{n}\right)$ ensures
that $\hat{E}_{\Delta}=E_{\Delta}$.
As shown in Section 4.2, under a few additional conditions, Assumption 1 holds
for a sequence of $\delta_{n}$ that decreases to $0$ as $n$ goes to infinity.
Thus, as $M$ and $n$ both increase to infinity, we have
$\nu_{2}+\Gamma_{n}\approx 0$ and $\nu_{1}-\Gamma_{n}\approx\min_{(j,l)\in
E_{\Delta}}D_{jl}$, and we only require $0\leq\epsilon_{n}<\min_{(j,l)\in
E_{\Delta}}D_{jl}$.
### 4.2 Theoretical Guarantees for $S^{X,M}$ and $S^{Y,M}$
In this section, we prove that under some mild conditions, (30) will hold with
high probability for specific values of $\delta_{n}$. We discuss the results
in two cases: the case where the curves are fully observed and the case where
the curves are only observed at discrete time points.
#### 4.2.1 Fully Observed Curves
In this section, we discuss the case where each curve is fully observed. We
first consider the case where the basis defining the differential graph are
known in advance; that is, the exact form of $\\{e_{jk}\\}_{k\geq 1}$ for all
$j\in V$ is known. In this case, the projection score vectors $a^{X,M}_{i}$
and $a^{Y,M}_{i}$ can be exactly recovered for all $i=1,2,\dots,n$. By the
assumption that $X_{i}(t)$ and $Y_{i}(t)$ are $p$-dimensional multivariate
Gaussian processes with mean zero, we then have $a^{X,M}_{i}\sim
N(0,\Sigma^{X,M})$ and $a^{Y,M}_{i}\sim N(0,\Sigma^{Y,M})$. The following
result follows directly from standard results on the sample covariance of
multivariate Gaussian variables.
###### Theorem 2
Assume that $S^{X,M}$ and $S^{Y,M}$ are computed as in Section 3.1, except the
basis functions $\\{e_{jk}\\}_{k\geq 1}$, $j\in V$, are fixed and known in
advance. Recall that
$n=\min\\{n_{X},n_{Y}\\}\quad\text{and}\quad\sigma_{\max}=\max\\{|\Sigma^{X,M}|_{\infty},|\Sigma^{Y,M}|_{\infty}\\}.$
Fix $\iota\in(0,1]$. Suppose that $n$ is large enough so that
$\delta_{n}=\sigma_{\max}\sqrt{\frac{C_{1}}{n}\log\left(\frac{8p^{2}M^{2}}{\iota}\right)}\leq
C_{2},$
for some universal constants $C_{1},C_{2}>0$. Then (30) holds with probability
at least $1-\iota$.
Proof The proof follows directly from Lemma 1 of Ravikumar et al. (2011) and
a union bound. [2mm]
With fully observed curves and known basis functions, it follows from Theorem
2 that $\delta_{n}\asymp\sqrt{\log(p^{2}M^{2})/n}$ with high probability. As
assumed in Section 2.2 (and also in Qiao et al. (2019)), when
$\lambda^{X}_{jm^{\prime}}=\lambda^{Y}_{jm^{\prime}}=0$ for all $j$ and
$m^{\prime}>M$ (where $M$ is allowed to grow with $n$), then $\nu_{2}(M)=0$,
$\tau(M)=\nu_{1}(M)=\min_{(j,l)\in E_{\Delta}}D_{jl}>0$, and
$E_{\Delta}=E^{\pi}_{\Delta}$. We can recover $E_{\Delta}$ with high
probability even in the high-dimensional setting, as long as
$\max\left\\{\frac{sM^{2}\log(p^{2}M^{2})|\Delta^{M}|_{1}^{2}/((\lambda^{\star}_{\min})^{2}\tau^{2})}{n},\frac{sM^{2}\log(p^{2}M^{2})/\lambda^{\star}_{\min}}{n}\right\\}\rightarrow
0.$
Even with an infinite number of positive eigenvalues, high-dimensional
consistency is still possible for quickly increasing $\nu_{1}$ and quickly
decaying $\nu_{2}$.
We then consider the case where the curves are fully observed, but we do not
have any prior knowledge on which orthonormal function basis should be used.
In this case, as discussed in Section 2.3, we recommend using the
eigenfunctions of $K_{jj}(\cdot,*)=K^{X}_{jj}(\cdot,*)+K^{Y}_{jj}(\cdot,*)$ as
basis functions. We use FPCA to estimate the eigenfuctions of
$K_{jj}(\cdot,*)$ and make the following assumption.
###### Assumption 3
Let $\\{\lambda_{jk},\phi_{jk}(\cdot)\\}$ be the eigenpairs of
$K_{jj}(\cdot,*)=K^{X}_{jj}(\cdot,*)+K^{Y}_{jj}(\cdot,*)$, $j\in V$, and
suppose that $\lambda_{jk}$ are non-increasing in $k$.
1. (i)
Suppose $\max_{j\in{V}}\sum_{k=1}^{\infty}\lambda_{jk}<\infty$ and assume that
there exists a constant $\beta>1$ such that, for each $k\in\mathbb{N}$,
$\lambda_{jk}\asymp{k^{-\beta}}$ and $d_{jk}\lambda_{jk}=O(k)$ uniformly in
$j\in{V}$, where
$d_{jk}=2\sqrt{2}\max\\{(\lambda_{j(k-1)}-\lambda_{jk})^{-1},(\lambda_{jk}-\lambda_{j(k+1)})^{-1}\\}$,
$k\geq 2$, and $d_{j1}=2\sqrt{2}(\lambda_{j1}-\lambda_{j2})^{-1}$.
2. (ii)
For all $k$, $\phi_{jk}(\cdot)$’s are continuous on the compact set
$\mathcal{T}$ and satisfy
$\max_{j\in{V}}\sup_{s\in{\mathcal{T}}}\sup_{k\geq{1}}|\phi_{jk}(s)|_{\infty}=O(1).$
This assumption was used in Qiao et al. (2019, Condition 1). We have the
following result.
###### Theorem 3
Suppose Assumption 3 holds and the basis functions are estimated using FPCA of
$K_{jj}(\cdot,*)$ with fully observed curves. Fix $\iota\in(0,1]$. Suppose $n$
is large enough so that
$\delta_{n}=M^{1+\beta}\sqrt{\frac{\log\left(2C_{2}p^{2}M^{2}/\iota\right)}{n}}\leq
C_{1},$
for some universal constants $C_{1},C_{2}>0$. Then (30) holds with probability
at least $1-\iota$.
Proof The proof follows directly from Theorem 1 of Qiao et al. (2019) and the
fact that
$\|\hat{K}_{jj}(\cdot,*)-K_{jj}(\cdot,*)\|_{\text{HS}}\leq\|\hat{K}^{X}_{jj}(\cdot,*)-K^{X}_{jj}(\cdot,*)\|_{\text{HS}}+\|\hat{K}^{Y}_{jj}(\cdot,*)-K^{Y}_{jj}(\cdot,*)\|_{\text{HS}}$.
[2mm]
It follows from Theorem 3 that $\delta_{n}\asymp
M^{1+\beta}\sqrt{\log(p^{2}M^{2}/)/n}$ with high probability. Compared with
Theorem 2, there is an additional $M^{1+\beta}$ term that arises from FPCA
estimation error. Similarly, when
$\lambda^{X}_{jm^{\prime}}=\lambda^{Y}_{jm^{\prime}}=0$ for all $j$ and
$m^{\prime}>M$, we can recover $E_{\Delta}$ with high probability as long as
$\max\left\\{\frac{sM^{(4+2\beta)}\log(p^{2}M^{2})|\Delta^{M}|_{1}^{2}/((\lambda^{\star}_{\min})^{2}\tau^{2})}{n},\frac{sM^{(4+2\beta)}\log(p^{2}M^{2})/\lambda^{\star}_{\min}}{n}\right\\}\rightarrow
0.$
#### 4.2.2 Discretely-Observed Curves
Finally, we discuss the case when the curves are only observed at discrete
time points—possibly with measurement error. Following Chapter 1 of Kokoszka
and Reimherr (2017), we first estimate each curve from the available
observations by basis expansion; then we use the estimated curves to form
empirical covariance functions from which we estimate the eigenfunctions using
FPCA. The estimated eigenfunctions are then used to calculate the scores.
Recall the model for discretely observed functions given in (21):
$h_{ijk}=g_{ij}(t_{ijk})+\epsilon_{ijk},$
where $g_{ij}$ denotes either $X_{ij}$ or $Y_{ij}$, $\epsilon_{ijk}$ are
i.i.d. Gaussian noise with mean $0$ and variance $\sigma^{2}_{0}$. Assume that
$t_{ij1}<\dots<t_{ijT}$ for any $1\leq i\leq n$ and $1\leq j\leq p$. Note that
we do not need $X$ and $Y$ to be observed at the same time points and we use
$t_{ijk}$ to represent either $t^{X}_{ijk}$ or $t^{Y}_{ijk}$. Furthermore,
recall that we first compute a least squares estimator of $X_{ij}(\cdot)$ and
$Y_{ij}(\cdot)$ by projecting it onto the basis
$b(\cdot)=\left(b_{1}(\cdot),\ldots,b_{L}(\cdot)\right)$.
First, we assume that as we increase the number of basis functions, we can
approximate any function in $\mathbb{H}$ arbitrarily well.
###### Assumption 4
We assume that $\\{b_{l}\\}^{\infty}_{l=1}$ is a complete orthonormal system
(CONS) (See Definition 2.4.11 of Hsing and Eubank, 2015) of $\mathbb{H}$, that
is, $\overline{{\rm Span}\left(\\{b_{l}\\}^{\infty}_{l=1}\right)}=\mathbb{H}$.
Assumption 4 requires that the basis functions are orthonormal. When this
assumption is violated—for example, when using the B-splines basis—we can
always first use an orthonormalization process, such as Gram-Schmidt, to
convert the basis to an orthonormal one. For B-splines, there are many
algorithms that can efficiently provide orthonormalization (Liu et al., 2019).
To establish theoretical guarantees for the least squares estimator, we
require smoothness in both the curves we are trying to estimate as well as the
basis functions we use.
###### Assumption 5
We assume that the basis functions $\\{b_{l}(\cdot)\\}^{\infty}_{l=1}$ satisfy
the following conditions.
$D_{0,b}\coloneqq\sup_{l\geq 1}\sup_{t\in\mathcal{T}}\lvert
b_{l}(t)\rvert<\infty,\qquad D_{1,b}(l)\coloneqq\sup_{t\in\mathcal{T}}\lvert
b^{\prime}_{l}(t)\rvert<\infty,\qquad D_{1,b,L}\coloneqq\max_{1\leq l\leq
L}D_{1,b}(l).$ (35)
We also require that the curves $g_{ij}$ satisfy the following smoothness
condition:
$\max_{1\leq j\leq p}\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle
g_{ij},b_{m}\rangle\right)^{2}\right]D^{2}_{1,b}(m)<\infty.$ (36)
To better understand Assumption 5, we use the Fourier basis as an example. Let
$\mathcal{T}=[0,1]$ and $b_{m}(t)=\sqrt{2}\cos(2\pi mt)$, $0\leq t\leq 1$ and
$m\in\mathbb{N}$. Thus, $\\{b_{m}(t)\\}^{\infty}_{m=0}$ then constitutes an
orthonormal basis of $\mathbb{H}=\mathcal{L}^{2}[0,1]$. We then have
$b^{\prime}(t)=-2\sqrt{2}\pi m\sin(2\pi mt)$, $D_{0,b}=\sqrt{2}$,
$D_{1,b}(m)=2\sqrt{2}\pi m$ and $D_{1,b,L}=2\sqrt{2}\pi L$. In this case, (36)
is equivalent to
$\max_{1\leq j\leq p}\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle
g_{ij},b_{m}\rangle\right)^{2}\right]m^{2}<\infty.$
On the other hand, $g_{ij}(t)=\sum^{\infty}_{m=1}\langle g_{ij},b_{m}\rangle
b_{m}(t)$ and $g^{\prime}_{ij}(t)=\sum^{\infty}_{m=1}\langle
g_{ij},b_{m}\rangle b^{\prime}_{m}(t)$. Suppose that,
$\mathbb{E}\left[\|g^{\prime}_{ij}\|^{2}\right]<\infty$. Then
$\mathbb{E}\left[\|g^{\prime}_{ij}\|^{2}\right]=\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle
g_{ij},b_{m}\rangle\right)^{2}\right]\|b^{\prime}_{m}\|^{2}\asymp\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle
g_{ij},b_{m}\rangle\right)^{2}\right]m^{2}.$ (37)
Therefore, $\max_{1\leq j\leq
p}\mathbb{E}\left[\|g^{\prime}_{ij}\|^{2}\right]<\infty$, which is a commonly
used assumption in nonparameteric statistics (e.g., Section 7.2 of Wasserman
(2006)), implies (36).
Finally, we require each function to be observed at time points that are
“evenly spaced.” Formally, we require the following assumption.
###### Assumption 6
The observation time points $\\{t_{ijk}:1\leq i\leq n,1\leq j\leq p,1\leq
k\leq T\\}$ satisfy
$\max_{1\leq i\leq n}\max_{1\leq j\leq p}\max_{1\leq k\leq
T+1}\left|\frac{t_{ijk}-t_{ij(k-1)}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\leq\frac{\zeta_{0}}{T^{2}},$
(38)
where $t_{ij0}$ and $t_{ij(T+1)}$ are endpoints of $\mathcal{T}$ for any
$1\leq i\leq n$, $1\leq j\leq p$, and $\zeta_{0}$ is a positive constant that
does not depend on $i$ or $j$.
Any $g_{ij}$ can be decomposed into
$g_{ij}=g^{\shortparallel}_{ij}+g^{\bot}_{ij}$, where
$g_{ij}^{\shortparallel}\in{\rm Span}(b)$ and $g_{ij}^{\bot}\in{\rm
Span}(b)^{\bot}$. We denote the eigenvalues of the covariance operator of
$g_{ij}$ as $\\{\lambda_{jk}\\}_{k\geq 1}$ and
$\lambda_{j0}=\sum^{\infty}_{k=1}\lambda_{jk}$; and denote the eigenvalues of
the covariance operator of $g^{\bot}_{ij}$ as
$\\{\lambda^{\bot}_{jk}\\}_{k\geq 1}$ and
$\lambda^{\bot}_{j0}=\sum^{\infty}_{k=1}\lambda^{\bot}_{jk}$. Note that under
Assumption 3, we have $\max_{1\leq j\leq p}\lambda_{j0}<\infty$. Let
$1<\lambda_{0,\max}<\infty$ be a constant such that $\max_{1\leq j\leq
p}\lambda_{j0}\leq\lambda_{0,\max}$. Let $B_{ij}$ be the design matrix of
$g_{ij}$ as defined in (23) and let $\lambda^{B}_{\min}=\min_{1\leq i\leq
n,1\leq j\leq p}\left\\{\lambda_{\min}(B^{\top}_{ij}B_{ij})\right\\}$. We
define
$\displaystyle\tilde{\psi}_{1}(T,L)=\frac{\sigma_{0}L}{\sqrt{\lambda^{B}_{\min}}},\quad\tilde{\psi}_{2}(T,L)=\frac{L^{2}}{(\lambda^{B}_{\min})^{2}}\left(\lambda_{0}\left(\tilde{c}_{1}D^{2}_{1,b,L}+\tilde{c_{2}}\right)\tilde{\psi}_{3}(L)+\tilde{c}_{1}\tilde{\psi}_{4}(L)\right),$
(39) $\displaystyle\tilde{\psi}_{3}(L)\;=\;\max_{1\leq j\leq
p}\left(\lambda^{\bot}_{j0}/\lambda_{j0}\right),\quad\tilde{\psi}_{4}(L)=\max_{1\leq
j\leq p}\sum_{m>L}\mathbb{E}\left[\left(\langle
g_{ij},b_{m}\rangle\right)^{2}\right]D^{2}_{1,b}(m),$ (40)
$\displaystyle\Phi(T,L)=\min\left\\{1/\tilde{\psi}_{1}(T,L),1/\sqrt{\tilde{\psi}_{3}(L)}\right\\},$
(41)
where $\tilde{c}_{1}=18D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}$ and
$\tilde{c_{2}}=36D^{4}_{0,b}(2\zeta_{0}+1)^{2}$.
We now use superscripts or subscripts to indicate the specific quantities for
$X$ and $Y$. In this way, we define $L_{X}$, $L_{Y}$, $T_{X}$, $T_{Y}$,
$\tilde{\psi}^{X}_{1}$-$\tilde{\psi}^{X}_{4}$,
$\tilde{\psi}^{Y}_{1}$-$\tilde{\psi}^{Y}_{4}$, and $\Phi^{X},\Phi^{Y}$. In
addition, let $T=\min\\{T_{X},T_{Y}\\}$, $L=\min\\{L_{X},L_{Y}\\}$,
$\bar{\psi}_{k}=\max\\{\tilde{\psi}^{X}_{k},\tilde{\psi}^{Y}_{k}\\}$,
$k=1,\cdots,4$, $\bar{\Phi}=\min\\{\Phi^{X},\Phi^{Y}\\}$, and let $n$, $\beta$
be defined as in Section 4.1.
###### Theorem 4
Assume the observation model given in (21). Suppose Assumption 3 holds, and
Assumption 4-6 hold for both $X$ and $Y$. Suppose $T$ and $L$ are large enough
so that
$\displaystyle\bar{\psi}_{1}(T,L)\leq\gamma_{1}\frac{\delta_{n}}{M^{1+\beta}},\quad\bar{\psi}_{3}(L)\leq\gamma_{3}\frac{\delta_{n}^{2}}{M^{2+2\beta}}$
(42)
where
$\delta_{n}=\max\left\\{\frac{M^{1+\beta}\log\left(4\bar{C}_{1}np/\iota\right)}{\bar{C}_{2}\bar{\Phi}(T,L)},M^{1+\beta}\sqrt{\frac{1}{C_{6}}\bar{\psi}_{2}(T,L)\log\left(\frac{C_{5}npL}{\iota}\right)},\right.\\\
\left.M^{1+\beta}\sqrt{\frac{\log\left(4\bar{C}_{3}p^{2}M^{2}/\iota\right)}{\bar{C}_{4}n}}\right\\},$
(43)
$\bar{C}_{1}=\max\\{C^{X}_{1},C^{Y}_{1}\\}$,
$\bar{C}_{2}=\min\\{C^{X}_{2},C^{Y}_{2}\\}$,
$\bar{C}_{3}=\max\\{C^{X}_{3},C^{Y}_{3}\\}$,
$\bar{C}_{4}=\min\\{C^{X}_{4},C^{Y}_{4}\\}$,
$\bar{C}_{5}=\max\\{C^{X}_{5},C^{Y}_{6}\\}$,
$\bar{C}_{6}=\min\\{C^{X}_{6},C^{Y}_{6}\\}$. $\gamma^{X}_{k}$,
$\gamma^{Y}_{k}$, $k=1,2,3$, and $C^{X}_{k}$, $C^{Y}_{k}$, $k=1,\cdots,6$ are
constants that do not depend on $n$, $p$, and $M$. Then
$\max\left\\{|S^{X,M}-\Sigma^{X,M}|_{\infty},|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\right\\}\leq\delta_{n}$
(44)
holds with probability at least $1-\iota$.
Proof See Appendix B.5. [2mm]
The rate $\delta_{n}$ in Theorem 4 is comprised of three terms. The first two
terms correspond to the error incurred by measuring the curves at discrete
locations and are approximation errors. The third term, which also appears in
Theorem 3, is the sampling error.
We provide some intuition on how $\tilde{\psi}_{1}$, $\tilde{\psi}_{2}$,
$\tilde{\psi}_{3}$, and $\tilde{\psi}_{4}$ depend on $T$ and $L$. Note that we
choose an orthonormal basis. Then as $T\to\infty$, we have
$\displaystyle\frac{1}{T}B^{\top}_{ij}B_{ij}$
$\displaystyle=\frac{1}{T}\sum^{T}_{k=1}\left[\begin{matrix}b^{2}_{1}(t_{ijk})&b_{1}(t_{ijk})b_{2}(t_{ijk})&\cdots&b_{1}(t_{ijk})b_{L}(t_{ijk})\\\
\vdots&\vdots&\ddots&\vdots\\\
b_{L}(t_{ijk})b_{1}(t_{ijk})&b_{L}(t_{ijk})b_{2}(t_{ijk})&\cdots&b^{2}_{L}(t_{ijk})\end{matrix}\right]$
$\displaystyle\approx\left[\begin{matrix}\|b_{1}\|^{2}&\langle
b_{1},b_{2}\rangle&\cdots&\langle b_{1},b_{L}\rangle\\\
\vdots&\vdots&&\vdots\\\ \langle b_{L},b_{1}\rangle&\langle
b_{L},b_{2}\rangle&\cdots&\|b_{L}\|^{2}\end{matrix}\right]$
$\displaystyle=\left[\begin{matrix}1&0&\cdots&0\\\ \vdots&\vdots&&\vdots\\\
0&0&\cdots&1\end{matrix}\right].$
Thus, as $T$ grows, we would expect
$\lambda_{\min}(B^{\top}_{ij}B_{ij})\approx T$ for any $1\leq j\leq p$ and
$1\leq i\leq n$. This implies that $\tilde{\psi}_{1}(T,L)\approx L/\sqrt{T}$
and
$\tilde{\psi}_{2}(T,L)\approx\left(D^{2}_{1,b,L}\tilde{\psi}_{3}(L)+\tilde{\psi}_{4}(L)\right)L^{2}/T^{2}$.
Furthermore, $D^{2}_{1,b,L}\asymp L^{2}$ when we use Fourier basis.
To understand $\tilde{\psi}_{3}(L)$ and $\tilde{\psi}_{4}(L)$, note that
$\lambda^{\bot}_{j0}=\mathbb{E}[\|g^{\bot}_{ij}\|^{2}]=\mathbb{E}_{g_{ij}}[\mathbb{E}_{\epsilon}[\|g^{\bot}_{ij}\|^{2}\mid
g_{ij}]]$. Under Assumption 4, $\lambda^{\bot}_{j0}\to 0$ as $L\to\infty$;
however, the speed at which $\lambda^{\bot}_{j0}$ goes to zero will depend on
$\mathbb{H}$ and the choice of the basis functions. For example, for a fixed
$g_{ij}$, by well known approximation results (see, for example, Barron and
Sheu (1991)), if $g_{ij}$ has $r$-th continuous and square integrable
derivatives, $\|g^{\bot}_{ij}\|^{2}\approx 1/L^{r}$ for frequently used bases
such as the Legendre polynomials, B-splines, and Fourier basis. Thus, roughly
speaking, we should have $\tilde{\psi}_{3}(L)\approx 1/L^{r}$ when
$\mathbb{H}$ is a Sobolev space of order $r$. When $g_{ij}$ is an infinitely
differentiable function and all derivatives can be uniformly bounded, then
$\|g^{\bot}_{ij}\|^{2}\approx\exp(-L)$ and thus
$\tilde{\psi}_{3}(L)\approx\exp(-L)$. Similarly, we have
$\tilde{\psi}_{4}(L)\approx 1/L^{r-1}$ if $g_{ij}$ has $r$-th continuous and
square integrable derivatives; and $\tilde{\psi}_{4}(L)\approx\exp(-L)$ if
$g_{ij}$ is an infinitely differentiable function and all derivatives can be
uniformly bounded.
To roughly show how $M$, $T$, $L$ and $n$ may co-vary, we assume that $p$ and
$s$ are fixed, and all elements of $\mathbb{H}$ have $r$-th continuous and
square integrable derivatives. Then FuDGE will recover the differential graph
with high probability, if $M\ll n^{1/(2+2\beta)}$, $\sqrt{T}/L\gg
M^{1+\beta}$, $T\gg L^{2-r/2}$, and $L\gg M^{(1+\beta)/r}$.
As pointed out by a reviewer, the noise term in (21) will create a nugget
effect in the covariance, meaning that
$\text{Var}(h_{ijk})=\text{Var}(g_{ij}(t_{ijk}))+\sigma^{2}_{0}$. This nugget
effect leads to bias in the estimated eigenvalues (variances of the scores).
In our theorem, the nugget effect is reflected by $\sigma_{0}$ in
$\tilde{\psi}_{1}$. When $\sigma_{0}$ is large, adding a regularization term
when estimating the eigenvalues can improve the estimation of FPCA scores and
their covariance matrices (see Chapter 6 of Hsing and Eubank (2015)). However,
adding a regularization term increases the number of tuning parameters that
need to be chosen. An alternative approach to estimating the covariance matrix
is through local polynomial regression (Zhang and Wang, 2016). Since the focus
of the paper is on the estimation of differential functional graphical models,
we do not explore ways to improve the estimation of FPCA scores. However, we
recognize that there are alternative approaches that can perform better in
some cases.
## 5 Joint Functional Graphical Lasso
In this section, we introduce two variants of a Joint Functional Graphical
Lasso (JFGL) estimator which we compare empirically to our proposed FuDGE
procedure in Section 6.1. Danaher et al. (2014) proposed the Joint Graphical
Lasso (JGL) to estimate multiple related Gaussian graphical models from
different classes simultaneously. Given $Q\geq 2$ data sets, where the $q$-th
data set consists of $n_{q}$ independent random vectors drawn from
$N(\mu_{q},\Sigma_{q})$, JGL simultaneously estimates
$\\{\Theta\\}=\\{\Theta^{(1)},\Theta^{(2)},\dots,\Theta^{(Q)}\\}$, where
$\Theta^{(q)}=\Sigma^{-1}_{q}$ is the precision matrix of the $q$-th data set.
Specifically, JGL constructs an estimator
$\\{\hat{\Theta}\\}=\\{\hat{\Theta}^{(1)},\hat{\Theta}^{(2)},\dots,\hat{\Theta}^{(Q)}\\}$
by solving the penalized log-likelihood:
$\\{\hat{\Theta}\\}=\operatorname*{arg\,min}_{\\{\Theta\\}}\left\\{-\sum^{Q}_{q=1}n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+P(\\{\Theta\\})\right\\},$
(45)
where $S^{(q)}$ is the sample covariance of the $q$-th data set and
$P(\\{\Theta\\})$ is a penalty function. The fused graphical lasso (FGL) is
obtained by setting
$P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{i\neq
j}|\Theta^{(q)}_{ij}|+\lambda_{2}\sum_{q<q^{\prime}}\sum_{i\neq
j}|\Theta^{(q)}_{ij}-\Theta^{(q^{\prime})}_{ij}|,$ (46)
while the group graphical lasso (GGL) is obtained by setting
$P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{i\neq
j}|\Theta^{(q)}_{ij}|+\lambda_{2}\sum_{i\neq
j}\sqrt{\sum^{Q}_{q=1}\left(\Theta^{(q)}_{ij}\right)^{2}}.$ (47)
The terms $\lambda_{1}$ and $\lambda_{2}$ are non-negative tuning parameters,
while $\Theta^{(q)}_{ij}$ denotes the $(i,j)$-th entry of $\Theta^{(q)}$. For
both penalties, the first term is the lasso penalty, which encourages sparsity
for the off-diagonal entries of all precision matrices; however, FGL and GGL
differ in the second term. For FGL, the second term encourages the off-
diagonal entries of precision matrices among all classes to be similar, which
means that it encourages not only similar network structure, but also similar
edge values. For GGL, the second term is a group lasso penalty, which
encourages the support of the precision matrices to be similar, but allows the
specific values to differ.
A similar approach can be used for estimating the precision matrix of the
score vectors. In contrast to the direct estimation procedure proposed in
Section 3, we could first estimate $\hat{\Theta}^{X,M}$ and
$\hat{\Theta}^{Y,M}$ using a joint graphical lasso objective, and then take
the difference to estimate $\Delta$.
In the functional graphical model setting, we are interested in the block
sparsity, so we modify the entry-wise penalties to a block-wise penalty.
Specifically, we propose solving the objective function in (45), where
$S^{(q)}$ and $\Theta^{(q)}$ denote the sample covariance and estimated
precision of the projection scores for the $q$-th group. Note that now
$S^{(q)}$, $\Theta^{(q)}$ and $\hat{\Theta}^{(q)}$, $q=1,\ldots,Q$ are all
$pM\times pM$ matrices. Similar to the GGL and FGL procedures, we define the
Grouped Functional Graphical Lasso (GFGL) and Fused Functional Graphical Lasso
(FFGL) penalties for functional graphs. Specifically, let $\Theta^{(q)}_{jl}$
denote the $(j,l)$-th $M\times M$ block matrix, the GFGL penalty is
$P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq
l}\|\Theta^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{j\neq
l}\sqrt{\sum^{Q}_{q=1}\|\Theta^{(q)}_{jl}\|^{2}_{\text{F}}},$ (48)
where $\lambda_{1}$ and $\lambda_{2}$ are non-negative tuning parameters. The
FFGL penalty can be defined in two ways. The first way is to use the Frobenius
norm for the second term:
$P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq
l}\|\Theta^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{q<q^{\prime}}\sum_{j,l}\|\Theta^{(q)}_{jl}-\Theta^{(q^{\prime})}_{jl}\|_{\text{F}}.$
(49)
The second way is to keep the element-wise $L_{1}$ norm as in FGL:
$P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq
l}\|\Theta^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{q<q^{\prime}}\sum_{j,l}|\Theta^{(q)}_{jl}-\Theta^{(q^{\prime})}_{jl}|_{1},$
(50)
where $\lambda_{1}$ and $\lambda_{2}$ are non-negative tuning parameters.
The Joint Functional Graphical Lasso accommodates an arbitrary $Q$. However,
when estimating the functional differential graph, we set $Q=2$. We will refer
to (49) as FFGL and to (50) as FFGL2. The algorithms for solving GFGL, FFGL,
and FFGL2 are given in Appendix A.
## 6 Experiments
We examine the performance of FuDGE using both simulations and a real data
set.444Code to replicate the simulations is available at
https://github.com/boxinz17/FuDGE.
### 6.1 Simulations
Given a graph $G_{X}$, we generate samples of $X$ such that
$X_{ij}(t)=b^{\prime}(t)^{\top}\delta^{X}_{ij}$. The coefficients
$\delta^{X}_{i}=((\delta^{X}_{i1})^{\top},\ldots,(\delta^{X}_{ip})^{\top})^{\top}\in{\mathbb{R}^{mp}}$
are drawn from $N\left(0,(\Omega^{X})^{-1}\right)$ where $\Omega_{X}$ is
described below. In all cases, $b^{\prime}(t)$ is an $m$-dimensional basis
with disjoint support over $[0,1]$ such that for $k=1,\ldots m$:
$b^{\prime}_{k}(t)=\begin{cases}\cos\left(10\pi\left(x-(2k-1)/10\right)\right)+1&\text{if
}(k-1)/m\leq{x}<k/m;\\\ 0&\text{otherwise}.\end{cases}$ (51)
To generate noisy observations at discrete time points, we sample data
$h^{X}_{ijk}=X_{ij}(t_{k})+e_{ijk},\quad e_{ijk}\sim N(0,0.5^{2}),$
for $200$ evenly spaced time points $0=t_{1}\leq\ldots\leq t_{200}=1$.
$Y_{ij}(t)$ and $h^{Y}_{ijk}$ are sampled in an analogous procedure. We use
$m=5$ for the experiments below, except for the simulation where we explore
the effect of $m$ on empirical performance.
We consider three different simulation settings for constructing $G_{X}$ and
$G_{Y}$. In each setting, we let $n_{X}=n_{Y}=100$ and $p=30,60,90,120$, and
we replicate the procedure 30 times for each $p$ and model setting.
Model 1: This model is similar to the setting considered in Zhao et al.
(2014), but modified to the functional case. We generate the support of
$\Omega^{X}$ according to a graph with $p(p-1)/10$ edges and a power-law
degree distribution with an expected power parameter of 2. Although the graph
is sparse with only 20% of all possible edges present, the power-law structure
mimics certain real-world graphs by creating hub nodes with large degree
(Newman, 2003). For each nonzero block, we set
$\Omega^{X}_{jl}=\delta^{\prime}I_{5}$, where $\delta^{\prime}$ is sampled
uniformly from $\pm[0.2,0.5]$. To ensure positive definiteness, we further
scale each off-diagonal block by $1/2,1/3,1/4,1/5$ for $p=30,60,90,120$
respectively. Each diagonal element of $\Omega^{X}$ is set to $1$ and the
matrix is symmetrized by averaging it with its transpose. To get $\Omega^{Y}$,
we first select the top 2 hub nodes in $G_{X}$ (i.e., the nodes with top 2
largest degree), and for each hub node we select the top (by magnitude) 20% of
edges. For each selected edge, we set $\Omega^{Y}_{jl}=\Omega^{X}_{jl}+W$
where $W_{kk^{\prime}}=0$ for $|k-k^{\prime}|\leq{2}$, and $W_{kk^{\prime}}=c$
otherwise, where $c$ is generated in the same way as $\delta^{\prime}$. For
all other blocks, $\Omega^{Y}_{jl}=\Omega^{X}_{jl}$.
Model 2: We first generate a tridiagonal block matrix $\Omega^{*}_{X}$ with
$\Omega^{*}_{X,jj}=I_{5}$,
$\Omega^{*}_{X,j,j+1}=\Omega^{*}_{X,j+1,j}=0.6I_{5}$, and
$\Omega^{*}_{X,j,j+2}=\Omega^{*}_{X,j+2,j}=0.4I_{5}$ for $j=1,\ldots,p$. All
other blocks are set to 0. We form $G_{Y}$ by adding four edges to $G_{X}$.
Specifically, we first let $\Omega^{*}_{Y,jl}=\Omega^{*}_{X,jl}$ for all
blocks, then for $j=1,2,3,4$, we set
$\Omega^{*}_{Y,j,j+3}=\Omega^{*}_{Y,j+3,j}=W$, where $W_{kk^{\prime}}=0.1$ for
all $1\leq k,k^{\prime}\leq M$. Finally, we set
$\Omega^{X}=\Omega^{*}_{X}+\delta I$, $\Omega^{Y}=\Omega^{*}_{Y}+\delta I$,
where
$\delta=\max\left\\{|\min(\lambda_{\min}(\Omega^{*}_{X}),0)|,|\min(\lambda_{\min}(\Omega^{*}_{Y}),0)|\right\\}+0.05$.
Model 3: We generate $\Omega^{*}_{X}$ according to an Erdös-Rényi graph. We
first set $\Omega^{*}_{X,jj}=I_{5}$. With probability $.8$, we set
$\Omega^{*}_{X,jl}=\Omega^{*}_{X,lj}=0.1I_{5}$, and set it to $0$ otherwise.
Thus, we expect 80% of all possible edges to be present. Then, we form $G_{Y}$
by randomly adding $s$ new edges to $G_{X}$, where $s=3$ for $p=30$, $s=4$ for
$p=60$, $s=5$ for $p=90$, and $s=6$ for $p=120$. We set each corresponding
block $\Omega^{*}_{Y,jl}=W$, where $W_{kk^{\prime}}=0$ when
$|k-k^{\prime}|\leq{1}$ and $W_{kk^{\prime}}=c$ otherwise. We let $c=2/5$ for
$p=30$, $c=4/15$ for $p=60$, $c=1/5$ for $p=90$, and $c=4/25$ for $p=120$.
Finally, we set $\Omega^{X}=\Omega^{*}_{X}+\delta I$,
$\Omega^{Y}=\Omega^{*}_{Y}+\delta I$, where
$\delta=\max\left\\{|\min(\lambda_{\min}(\Omega^{*}_{X}),0)|,|\min(\lambda_{\min}(\Omega^{*}_{Y}),0)|\right\\}+0.05$.
Figure 4: Average ROC curves across 30 simulations. Different columns
correspond to different models, different rows correspond to different
dimensions.
We compare FuDGE with four competing methods. The first competing method
(denoted by _multiple_ in Figure 4) ignores the functional nature of the data.
We select 15 equally spaced time points, and at each time point, we implement
a direct difference estimation procedure (Zhao et al., 2014) to estimate the
graph at that time point. Specifically, for each $t$, $X_{i}(t)$ and
$Y_{i}(t)$ are simply $p$-dimensional random vectors, and we use their sample
covariances in (25) to obtain a $p\times p$ matrix $\hat{\Delta}$. This
produces 15 differential graphs, and we use a majority vote to form a single
differential graph. The ROC curve is obtained by changing the $L_{1}$ penalty,
$\lambda_{n}$, used for all time points.
The other three competing methods all estimate two functional graphical models
using either the Joint Graphical Lasso or Functional Joint Graphical Lasso
introduced in Section 5. For each method, we first estimate the sample
covariances of the FPCA scores for $X$ and $Y$. The second competing method
(denoted as _FGL_) ignores the block structure in precision matrices and
applies the fused graphical lasso method directly. The third and fourth
competing methods do account for the block structure and apply FFGL and FFGL2
defined in Section 5. To draw an ROC curve, we follow the same approach as in
Zhao et al. (2014). We first fix $\lambda_{1}=0.1$, which controls the overall
sparsity in each graph; we then form an ROC curve by varying across
$\lambda_{2}$, which controls the similarity between two graphs.
For each setting and method, the ROC curve averaged across the $30$
replications is shown in Figure 4. We see that FuDGE clearly has the best
overall performance in recovering the support of the differential graph for
all cases. We also note that the explicit consideration of block structure in
the joint graphical methods does not seem to make a substantial difference as
the performance of FGL is comparable to FFGL and FFGL2.
##### The effect of the number of basis functions:
To examine how the estimation accuracy is associated with the dimension of the
functional data, we repeat the experiment under Model 1 with $p=30$ and vary
the number of basis functions used to generate the data in (51). In each case,
the number of principal components selected by the cross-validation is $M=4$.
In Figure 5, we see that as the gap between the true dimension $m$ and the
number of dimensions used $M$ increases, the performance of FuDGE degrades
slightly, but is still relatively robust. This is because the FPCA procedure
is data adaptive and produces an eigenfunction basis that approximates the
true functions well with a relatively small number of basis functions.
Figure 5: ROC curves for Model 1 with $p=30$ and changing number of basis
functions $m$. Each curve is drawn by averaging across 30 simulations. The
number of eigenfunctions, $M$, selected by the cross-validation is 4 in each
replication.
### 6.2 Neuroscience Application
We apply our method to electroencephalogram (EEG) data obtained from a study
(Zhang et al., 1995; Ingber, 1997), which included 122 total subjects; 77
individuals with alcohol use disorder (AUD) and 45 in the control group.
Specifically, the EEG data was measured by placing $p=64$ electrodes on
various locations on the subject’s scalp and measuring voltage values across
time. We follow the preprocessing procedure in Knyazev (2007) and Zhu et al.
(2016), which filters the EEG signals at $\alpha$ frequency bands between 8
and 12.5 Hz.
Qiao et al. (2019) estimate separate functional graphs for each group, but we
directly estimate the differential graph using FuDGE. We choose $\lambda_{n}$
so that the estimated differential graph has approximately 1% of possible
edges. The estimated edges of the differential graph are shown in Figure 6.
In this setting, an edge in the differential graph suggests that the
communication pattern between two different regions of the brain may be
affected by alcohol use disorder. However, the differential graph does not
indicate exactly how the communication pattern has changed. For instance, the
edge between P4 and P6 suggests that AUD affects the communication pattern
between those two regions; however, it could be that those two regions are
associated (conditionally) in the control group, but not the AUD group or vice
versa. It could also be that the two regions are associated (conditionally) in
both groups, but the conditional covariance is different. Nonetheless, many
interesting observations can be gleaned from the results and may generate
interesting hypotheses that could be investigated more thoroughly in an
experimental setting.
Figure 6: Estimated differential graph for EEG data. The anterior region is
the top of the figure and the posterior region is the bottom of the figure.
We give two specific observations. First, edges are generally between nodes
located in the same region—either the anterior region or the posterior
region—and there is no edge that crosses between regions. This observation is
consistent with the result in Qiao et al. (2019) where there are no
connections between the anterior and posterior regions for both groups. We
also note that electrode X, lying in the middle left region has a high degree
in the estimated differential graph. While there is no direct connection
between the anterior and posterior regions, this region may play a role in
helping the two parts communicate and may be heavily affected by AUD.
Similarly, P08 in the anterior region also has a high degree and is connected
to other nodes in the anterior region, which may indicate that this region can
be an information exchange center for anterior regions, which, at the same
time, may be heavily affected by AUD.
## 7 Discussion
We proposed a method to directly estimate the differential graph for
functional graphical models. In certain settings, direct estimation allows for
the differential graph to be recovered consistently, even if each underlying
graph cannot be consistently recovered. Experiments on simulated data also
show that preserving the functional nature of the data rather than treating
the data as multivariate scalars can also result in better estimation of the
differential graph.
A key step in the procedure is first representing the functions with an
$M$-dimensional basis using FPCA. Definition 1 ensures that there exists some
$M$ large enough so that the signal, $\nu_{1}(M)$, is larger than the bias,
$\nu_{2}(M)$, due to using a finite dimensional representation. Intuitively,
$\tau=\nu_{1}(M)-\nu_{2}(M)$ is tied to the eigenvalue decay rate; however, we
defer derivation of the explicit connection for future work. In addition, we
have provided a method for direct estimation of the differential graph, but
the development of methods that allow for inference and hypothesis testing in
functional differential graphs would be fruitful avenues for future work. For
example, Kim et al. (2019) has developed inferential tools for high-
dimensional Markov networks, and future work may extend their results to the
functional graph setting.
## Acknowledgements
We thank the associate editor and reviewers for their helpful feedback which
has greatly improved the manuscript. This work is partially supported by the
William S. Fishman Faculty Research Fund at the University of Chicago Booth
School of Business. This work was completed in part with resources provided by
the University of Chicago Research Computing Center.
## Appendix A Derivation of Optimization Algorithm
In this section, we derive the key steps for the optimization algorithms.
### A.1 Optimization Algorithm for FuDGE
We derive the closed-form updates for the proximal method stated in (29). In
particular, recall that for all $1\leq{j,l}\leq{p}$, we have
$\Delta^{\text{new}}_{jl}\;=\;\left[\left(\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta\right)/\|A^{\text{old}}_{jl}\|_{F}\right]_{+}\times
A^{\text{old}}_{jl},$
where $A^{\text{old}}=\Delta^{\text{old}}-\eta\nabla L(\Delta^{\text{old}})$
and $x_{+}=\max\\{0,x\\}$, $x\in{\mathbb{R}}$ represents the positive part of
$x$.
Proof [Proof of (29)] Let $A^{\text{old}}=\Delta^{\text{old}}-\eta\nabla
L(\Delta^{\text{old}})$ and let $f_{jl}$ denote the loss decomposed over each
$j,l$ block so that
${}f_{jl}(\Delta_{jl})\;=\;\frac{1}{2\lambda_{n}\eta}\|\Delta_{jl}-A^{\text{old}}_{jl}\|^{2}_{F}+\|\Delta_{jl}\|_{F}$
(A.1)
and
$\Delta^{\text{new}}_{jl}\;=\;\operatorname*{arg\,min}_{\Delta_{jl}\in{\mathbb{R}^{M\times{M}}}}f_{jl}(\Delta_{jl}).$
(A.2)
The loss $f_{jl}(\Delta_{jl})$ is convex, so the first-order optimality
condition implies that:
${}0\in\partial f_{jl}\left(\Delta^{\text{new}}_{jl}\right),$ (A.3)
where $\partial f_{jl}\left(\Delta_{jl}\right)$ is the subdifferential of
$f_{jl}$ at $\Delta_{jl}$:
${}\partial
f_{jl}(\Delta_{jl})\;=\;\frac{1}{\lambda_{n}\eta}\left(\Delta_{jl}-A^{\text{old}}_{jl}\right)+Z_{jl},$
(A.4)
where
${}Z_{jl}\;=\;\begin{cases}\frac{\Delta_{jl}}{\|\Delta_{jl}\|_{F}}\qquad&\text{
if }\Delta_{jl}\neq{0}\\\\[10.0pt]
\left\\{Z_{jl}\in{\mathbb{R}^{M\times{M}}}\colon\|Z_{jl}\|_{F}\leq{1}\right\\}\qquad&\text{
if }\Delta_{jl}=0.\end{cases}$ (A.5)
Claim 1 If $\|A^{\text{old}}_{jl}\|_{F}>\lambda_{n}\eta>0$, then
$\Delta^{\text{new}}_{jl}\neq{0}$.
We verify this claim by proving the contrapositive. Suppose
$\Delta^{\text{new}}_{jl}={0}$. Then by (A.3) and (A.5), there exists a
$Z_{jl}\in{\mathbb{R}^{M\times{M}}}$ such that $\|Z_{jl}\|_{F}\leq{1}$ and
$0=-\frac{1}{\lambda_{n}\eta}A^{\text{old}}_{jl}+Z_{jl}.$
Thus, $\|A^{\text{old}}_{jl}\|_{F}=\|\lambda_{n}\eta\cdot
Z_{jl}\|_{F}\leq{\lambda_{n}\eta}$, so that Claim 1 holds.
Combining Claim 1 with (A.3) and (A.5), for any $j,l$ such that
$\|A^{\text{old}}_{jl}\|_{F}>\lambda_{n}\eta$, we have
$0=\frac{1}{\lambda_{n}\eta}\left(\Delta^{\text{new}}_{jl}-A^{\text{old}}_{jl}\right)+\frac{\Delta^{\text{new}}_{jl}}{\|\Delta^{\text{new}}_{jl}\|_{F}},$
which is solved by
${}\Delta^{\text{new}}_{jl}=\frac{\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta}{\|A^{\text{old}}_{jl}\|_{F}}A^{\text{old}}_{jl}.$
(A.6)
Claim 2 If $\|A^{\text{old}}_{jl}\|_{F}\leq\lambda_{n}\eta$, then
$\Delta^{\text{new}}_{jl}=0$.
Again, we verify the claim by proving the contrapositive. Suppose
$\Delta^{\text{new}}_{jl}\neq 0$. Then the first-order optimality implies the
updates in (A.6). However, taking the Frobenius norm on both sides of the
equation gives
$\|\Delta^{\text{new}}_{jl}\|_{F}=\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta$,
which implies that $\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta\geq{0}$.
The updates in (29) immediately follow by combining Claim 2 and (A.6). [2mm]
### A.2 Solving the Joint Functional Graphical Lasso
As in Danaher et al. (2014), we use the alternating directions method of
multipliers (ADMM) algorithm to solve (45); see Boyd et al. (2011) for a
detailed exposition of ADMM.
To solve (45), we first rewrite the problem as:
$\max_{\\{\Theta\\},\\{Z\\}}\left\\{-\sum^{Q}_{q=1}n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+P(\\{Z\\})\right\\},$
subject to $\Theta^{(q)}\succ 0$ and $Z^{(q)}=\Theta^{(q)}$, where
$\\{Z\\}=\\{Z^{(1)},Z^{(2)},\dots,Z^{(Q)}\\}$. The scaled augmented Lagrangian
(Boyd et al., 2011) is given by
$L_{\rho}\left(\\{\Theta\\},\\{Z\\},\\{U\\}\right)=-\sum^{Q}_{q=1}n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+P(\\{Z\\})\\\
+\frac{\rho}{2}\sum^{Q}_{q=1}\|\Theta^{(q)}-Z^{(q)}+U^{(q)}\|^{2}_{\text{F}},$
(A.7)
where $\rho>0$ is a tuning parameter and
$\\{U\\}=\\{U^{(1)},U^{(2)},\dots,U^{(Q)}\\}$ are dual variables. The ADMM
algorithm will then solve (A.7) by iterating the following three steps. At the
$i$-th iteration, they are as follows:
1. 1.
$\\{\Theta_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{\Theta\\}}L_{\rho}\left(\\{\Theta\\},\\{Z_{(i-1)}\\},\\{U_{(i-1)}\\}\right)$.
2. 2.
$\\{Z_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{Z\\}}L_{\rho}\left(\\{\Theta_{(i)}\\},\\{Z\\},\\{U_{(i-1)}\\}\right)$.
3. 3.
$\\{U_{(i)}\\}\leftarrow\\{U_{(i-1)}\\}+(\\{\Theta_{(i)}\\}-\\{Z_{(i)}\\})$.
We now give more details for the above three steps.
ADMM algorithm for solving the joint functional graphical lasso problem
(a) Initialize the variables: $\Theta^{(q)}_{(0)}=I_{pM}$,
$U^{(q)}_{(0)}=0_{pM}$, and $Z^{(q)}_{(0)}=0_{pM}$ for $q=1,\ldots,Q$.
(b) Select a scalar $\rho>0$.
(c) For $i=1,2,3,\dots$ until convergence
(i) For $q=1,\ldots,Q$, update $\Theta^{(q)}_{(i)}$ as the minimizer (with
respect to $\Theta^{(q)}$) of
$-n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+\frac{\rho}{2}\|\Theta^{(q)}-Z^{(q)}_{(i-1)}+U^{(q)}_{(i-1)}\|^{2}_{\text{F}}$
Letting $VDV^{\top}$ denote the eigendecomposition of $S^{(q)}-\rho
Z^{(q)}_{(i-1)}/n_{q}+\rho U^{(q)}_{(i-1)}/n_{q}$, then the solution is given
by $V\tilde{D}V^{\top}$ (Witten and Tibshirani, 2009), where $\tilde{D}$ is
the diagonal matrix with $j$-th diagonal element being
$\frac{n_{q}}{2\rho}\left(-D_{jj}+\sqrt{D^{2}_{jj}+4\rho/n_{q}}\right),$
where $D_{jj}$ is the $(j,j)$-th entry of $D$.
(ii) Update $\\{Z_{(i)}\\}$ as the minimizer (with respect to $\\{Z\\}$) of
$\min_{\\{Z\\}}\frac{\rho}{2}\sum^{Q}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+P(\\{Z\\}),$
(A.8)
where $A^{(q)}=\Theta^{(q)}_{(i)}+U^{(q)}_{(i-1)}$, $q=1,\ldots,Q$.
(iii) $U^{(q)}_{(i)}\leftarrow
U^{(q)}_{(i-1)}+(\Theta^{(q)}_{(i)}-Z^{(q)}_{(i)})$, $q=1,\ldots,Q$.
There are three things worth noticing. 1. The key step is to solve (A.8),
which depends on the form of penalty term $P(\cdot)$; 2. This algorithm is
guaranteed to converge to the global optimum when $P(\cdot)$ is convex (Boyd
et al., 2011); 3. The positive-definiteness constraint on $\\{\hat{\Theta}\\}$
is naturally enforced by step (c) (i).
### A.3 Solutions to (A.8) for Joint Functional Graphical Lasso
We provide solutions to (A.8) for three problems (GFGL, FFGL, FFGL2) defined
by (48), (49) and (50).
#### A.3.1 Solution to (A.8) for GFGL
Let the solution for
$\min_{\\{Z\\}}\frac{\rho}{2}\sum^{Q}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq
l}\|Z^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{j\neq
l}\left(\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}$ (A.9)
be denoted as
$\\{\hat{Z}\\}=\\{\hat{Z}^{(1)},\hat{Z}^{(2)},\dots,\hat{Z}^{(Q)}\\}$. Let
$Z^{(q)}_{jl}$, $\hat{Z}^{(q)}_{jl}$ be $(j,l)$-th $M\times M$ block of
$Z^{(q)}$ and $\hat{Z}^{(q)}$, $q=1,\ldots,Q$. Then, for $j=1,\ldots,p$, we
have
$\hat{Z}^{(q)}_{jj}=A^{(q)}_{jj},\qquad q=1,\ldots,Q,$ (A.10)
and, for $j\neq l$, we have
$\hat{Z}^{(q)}_{jl}=\left(\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{\|A^{(q)}_{jl}\|_{\text{F}}}\right)_{+}\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)_{+}A^{(q)}_{jl},$
(A.11)
where $q=1,\ldots,Q$. Details of the update are given in Appendix A.4.
#### A.3.2 Solution to (A.8) for FFGL
For FFGL, there is no simple closed form solution. When $Q=2$, (A.8) becomes
$\min_{\\{Z\\}}\;\frac{\rho}{2}\sum^{2}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+\lambda_{1}\left(\sum^{2}_{q=1}\sum_{j\neq
l}\|Z^{(q)}_{jl}\|_{\text{F}}\right)+\lambda_{2}\sum_{j,l}\|Z^{(1)}_{jl}-Z^{(2)}_{jl}\|_{\text{F}}.$
For each $1\leq j,l\leq p$, we compute $\hat{Z}^{(1)}_{jl}$,
$\hat{Z}^{(2)}_{jl}$ by solving
$\min_{\\{Z^{(1)}_{jl},Z^{(2)}_{jl}\\}}\;\frac{1}{2}\sum^{2}_{q=1}\|Z^{(q)}_{jl}-A^{(q)}_{jl}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho}\mathbbm{1}_{j\neq
l}\sum^{2}_{q=1}\|Z^{(q)}_{jl}\|_{\text{F}}+\frac{\lambda_{2}}{\rho}\|Z^{(1)}_{jl}-Z^{(2)}_{jl}\|_{\text{F}},$
(A.12)
where $\mathbbm{1}_{j\neq l}=1$ when $j\neq l$ and $0$ otherwise.
When $j=l$, by Lemma 6, we have the following closed form updates for
$\\{\hat{Z}^{(1)}_{jj},\hat{Z}^{(2)}_{jj}\\}$, $j=1,\ldots,p$. If
$\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}\leq 2\lambda_{2}/\rho$, then
$\hat{Z}^{(1)}_{jj}=\hat{Z}^{(2)}_{jj}=\frac{1}{2}\left(A^{(1)}_{jj}+A^{(2)}_{jj}\right).$
If $\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}>2\lambda_{2}/\rho$, then
$\displaystyle\hat{Z}^{(1)}_{jj}$
$\displaystyle=A^{(1)}_{jj}-\frac{\lambda_{2}/\rho}{\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}}\left(A^{(1)}_{jj}-A^{(2)}_{jj}\right),$
$\displaystyle\hat{Z}^{(2)}_{jj}$
$\displaystyle=A^{(2)}_{jj}+\frac{\lambda_{2}/\rho}{\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}}\left(A^{(1)}_{jj}-A^{(2)}_{jj}\right).$
For $j\neq l$, we get $\\{\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl}\\}$ using the
ADMM algorithm again. We construct the scaled augmented Lagrangian as:
${L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R\\},\\{V\\}\right)=\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-B^{(q)}\|_{\text{F}}+\frac{\lambda_{1}}{\rho}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}}\\\
+\frac{\lambda_{2}}{\rho}\|R^{(1)}-R^{(2)}\|_{\text{F}}+\frac{\rho^{\prime}}{2}\sum^{2}_{q=1}\|W^{(q)}-R^{(q)}+V^{(q)}\|^{2}_{\text{F}},$
where $\rho^{\prime}>0$ is a tuning parameter, $B^{(q)}=A^{(q)}_{jl}$,
$q=1,2$, and $W^{q},R^{(q)},V^{(q)}\in\mathbb{R}^{M\times M}$, $q=1,2$.
$\\{W\\}=\\{W^{(1)},W^{(2)}\\}$, $\\{R\\}=\\{R^{(1)},R^{(2)}\\}$, and
$\\{V\\}=\\{V^{(1)},V^{(2)}\\}$. The detailed ADMM algorithm is described as
below:
ADMM algorithm for solving (A.12) for $j\neq l$
(a) Initialize the variables: $W^{(q)}_{(0)}=I_{M}$, $R^{(q)}_{(0)}=0_{M}$,
and $V^{(q)}_{(0)}=0_{M}$ for $q=1,2$. Let $B^{(q)}=A^{(q)}_{jl}$, $q=1,2$.
(b) Select a scalar $\rho^{\prime}>0$.
(c) For $i=1,2,3,\dots$ until convergence
(i)
$\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}{L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R_{(i-1)}\\},\\{V_{(i-1)}\\}\right)$.
This is equivalent to
$\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-C^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho(1+\rho^{\prime})}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}},$
where
$C^{(q)}=\frac{1}{1+\rho^{\prime}}\left[B^{(q)}+\rho^{\prime}\left(R^{(q)}_{(i-1)}-V^{(q)}_{(i-1)}\right)\right].$
Similar to (28), we have
$W^{(q)}_{(i)}\leftarrow\left(\frac{\|C^{(q)}\|_{\text{F}}-\lambda_{1}/(\rho(1+\rho^{\prime}))}{\|C^{(q)}\|_{\text{F}}}\right)_{+}\cdot
C^{(q)},\qquad q=1,2.$
(ii)
$\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}{L^{\prime}}_{\rho^{\prime}}\left(\\{W_{(i)}\\},\\{R\\},\\{V_{(i-1)}\\}\right)$.
This is equivalent to
$\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}\frac{1}{2}\sum^{2}_{q=1}\|R^{(q)}-D^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{2}}{\rho\rho^{\prime}}\|R^{(1)}-R^{(2)}\|_{\text{F}},$
where $D^{(q)}=W^{(q)}_{(i)}+V^{(q)}_{(i-1)}$. By Lemma 6, if
$\|D^{(1)}-D^{(2)}\|_{\text{F}}\leq 2\lambda_{2}/(\rho\rho^{\prime})$, then
$R^{(1)}_{(i)}=R^{(2)}_{(i)}\leftarrow\frac{1}{2}\left(D^{(1)}+D^{(2)}\right),$
and if $\|D^{(1)}-D^{(2)}\|_{\text{F}}>2\lambda_{2}/(\rho\rho^{\prime})$, then
$\displaystyle R^{(1)}\leftarrow
D^{(1)}-\frac{\lambda_{2}/(\rho\rho^{\prime})}{\|D^{(1)}-D^{(2)}\|_{\text{F}}}\left(D^{(1)}-D^{(2)}\right),$
$\displaystyle R^{(2)}\leftarrow
D^{(2)}+\frac{\lambda_{2}/(\rho\rho^{\prime})}{\|D^{(1)}-D^{(2)}\|_{\text{F}}}\left(D^{(1)}-D^{(2)}\right).$
(iii) $V^{(q)}_{(i)}\leftarrow V^{(q)}_{(i-1)}+W^{(q)}_{(i)}-R^{(q)}_{(i)}$,
$q=1,2$.
#### A.3.3 Solution to (A.8) for FFGL2
For FFGL2, there is also no closed form solution. Similar to Section A.3.2, we
compute a closed form solution for
$\\{\hat{Z}^{(1)}_{jj},\hat{Z}^{(2)}_{jj}\\}$, $j=1,\ldots,p$, and use an ADMM
algorithm to compute $\\{\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl}\\}$, $1\leq
j\neq l\leq p$.
For any $1\leq j,l\leq p$, we solve:
$\min_{\\{Z^{(1)}_{jl},Z^{(2)}_{jl}\\}}\;\frac{1}{2}\sum^{2}_{q=1}\|Z^{(q)}_{jl}-A^{(q)}_{jl}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho}\mathbbm{1}_{j\neq
l}\sum^{2}_{q=1}\|Z^{(q)}_{jl}\|_{\text{F}}+\frac{\lambda_{2}}{\rho}\sum_{1\leq
a,b\leq M}|Z^{(1)}_{jl,ab}-Z^{(2)}_{jl,ab}|,$ (A.13)
where $\mathbbm{1}_{j\neq l}=1$ when $j\neq l$ and $0$ otherwise.
By Lemma 6, when $j=l$ we have
$\left(\hat{Z}^{(1)}_{jj,ab},\hat{Z}^{(2)}_{jj,ab}\right)=\left\\{\begin{aligned}
&\left(A^{(1)}_{jl,ab}-\lambda_{2}/\rho,A^{(2)}_{jl,ab}+\lambda_{2}/\rho\right)\quad\text{if}\;A^{(1)}_{jl,ab}>A^{(2)}_{jl,ab}+2\lambda_{2}/\rho\\\
&\left(A^{(1)}_{jl,ab}+\lambda_{2}/\rho,A^{(2)}_{jl,ab}-\lambda_{2}/\rho\right)\quad\text{if}\;A^{(1)}_{jl,ab}<A^{(2)}_{jl,ab}-2\lambda_{2}/\rho\\\
&\left(\left(A^{(1)}_{jl,ab}+A^{(2)}_{jl,ab}\right)/2,\left(A^{(1)}_{jl,ab}+A^{(2)}_{jl,ab}\right)/2\right)\quad\text{if}\;\left|A^{(1)}_{jl,ab}-A^{(2)}_{jl,ab}\right|\leq
2\lambda_{2}/\rho,\end{aligned}\right.$
where subscripts $(a,b)$ denote the $(a,b)$-th entry, $1\leq a,b\leq M$ and
$j=1,\ldots,p$.
For $j\neq l$, we get $\\{\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl}\\}$, $1\leq
j\neq l\leq p$ by using an ADMM algorithm. Let $B^{(q)}=A^{(q)}_{jl}$,
$q=1,2$. We first construct the scaled augmented Lagrangian:
${L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R\\},\\{V\\}\right)=\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-B^{(q)}\|_{\text{F}}+\frac{\lambda_{1}}{\rho}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}}\\\
+\frac{\lambda_{2}}{\rho}\sum_{a,b}|R^{(1)}_{a,b}-R^{(2)}_{a,b}|+\frac{\rho^{\prime}}{2}\sum^{2}_{q=1}\|W^{(q)}-R^{(q)}+V^{(q)}\|^{2}_{\text{F}},$
where $\rho^{\prime}>0$ is a tuning parameter,
$W^{q},R^{(q)},V^{(q)}\in\mathbb{R}^{M\times M}$, $q=1,2$,
$\\{W\\}=\\{W^{(1)},W^{(2)}\\}$, $\\{R\\}=\\{R^{(1)},R^{(2)}\\}$, and
$\\{V\\}=\\{V^{(1)},V^{(2)}\\}$. The detailed ADMM algorithm is described as
below:
ADMM algorithm for solving (A.13) for $j\neq l$
(a) Initialize the variables: $W^{(q)}_{(0)}=I_{M}$, $R^{(q)}_{(0)}=0_{M}$,
and $V^{(q)}_{(0)}=0_{M}$ for $q=1,2$. Let $B^{(q)}=A^{(q)}_{jl}$, $q=1,2$.
(b) Select a scalar $\rho^{\prime}>0$.
(c) For $i=1,2,3,\dots$ until convergence
(i)
$\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}.{L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R_{(i-1)}\\},\\{V_{(i-1)}\\}\right)$
This is equivalent to
$\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-C^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho(1+\rho^{\prime})}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}},$
where
$C^{(q)}=\frac{1}{1+\rho^{\prime}}\left[B^{(q)}+\rho^{\prime}\left(R^{(q)}_{(i-1)}-V^{(q)}_{(i-1)}\right)\right].$
Similar to (28), we have
$W^{(q)}_{(i)}\leftarrow\left(\frac{\|C^{(q)}\|_{\text{F}}-\lambda_{1}/(\rho(1+\rho^{\prime}))}{\|C^{(q)}\|_{\text{F}}}\right)_{+}\cdot
C^{(q)},\qquad q=1,2.$
(ii)
$\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}{L^{\prime}}_{\rho^{\prime}}\left(\\{W_{(i)}\\},\\{R\\},\\{V_{(i-1)}\\}\right)$
This is equivalent to
$\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}\frac{1}{2}\sum^{2}_{q=1}\|R^{(q)}-D^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{2}}{\rho\rho^{\prime}}\sum_{a,b}\left|R^{(1)}_{ab}-R^{(2)}_{ab}\right|,$
where $D^{(q)}=W^{(q)}_{(i)}+V^{(q)}_{(i-1)}$. Then by Lemma 6, we have
$\left(R^{(1)}_{(i),ab},R^{(2)}_{(i),ab}\right)=\left\\{\begin{aligned}
&\left(D^{(1)}_{ab}-\lambda_{2}/(\rho\rho^{\prime}),D^{(2)}_{ab}+\lambda_{2}/(\rho\rho^{\prime})\right)\quad\text{if}\;D^{(1)}_{ab}>D^{(2)}_{ab}+2\lambda_{2}/(\rho\rho^{\prime})\\\
&\left(D^{(1)}_{ab}+\lambda_{2}/(\rho\rho^{\prime}),D^{(2)}_{ab}-\lambda_{2}/(\rho\rho^{\prime})\right)\quad\text{if}\;D^{(1)}_{ab}<D^{(2)}_{ab}-2\lambda_{2}/(\rho\rho^{\prime})\\\
&\left(\left(D^{(1)}_{ab}+D^{(2)}_{ab}\right)/2,\left(D^{(1)}_{ab}+D^{(2)}_{ab}\right)/2\right)\quad\text{if}\;\left|D^{(1)}_{ab}-D^{(1)}_{ab}\right|\leq
2\lambda_{2}/(\rho\rho^{\prime}),\end{aligned}\right.$
where subscripts $(a,b)$ denote the $(a,b)$-th entry, $1\leq a,b\leq M$ and
$1\leq j,l\leq p$.
(iii) $V^{(q)}_{(i)}\leftarrow V^{(q)}_{(i-1)}+W^{(q)}_{(i)}-R^{(q)}_{(i)}$,
$q=1,2$.
### A.4 Derivation of (A.10) and (A.11)
We provide proof of (A.10) and (A.11).
Note that for any $1\leq j,l\leq p$, we can obtain
$\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl},\dots,\hat{Z}^{(Q)}_{jl}$ by solving
$\operatorname*{arg\,min}_{Z^{(1)}_{jl},Z^{(2)}_{jl},\dots,Z^{(Q)}_{jl}}\frac{\rho}{2}\sum^{Q}_{q=1}\|Z^{(q)}_{jl}-A^{(q)}_{jl}\|^{2}_{\text{F}}+\lambda_{1}\mathbbm{1}_{j\neq
l}\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\mathbbm{1}_{j\neq
l}\left(\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2},$ (A.14)
where $\mathbbm{1}_{j\neq l}=1$ when $j\neq l$ and $0$ otherwise. By (A.14),
we have that $\hat{Z}^{(q)}_{jj}=A^{(q)}_{jj}$ for any $j=1,\ldots,p$ and
$q=1,\ldots,Q$, which is (A.10). We then prove (A.11). Denote the objective
function in (A.14) as $\tilde{L}_{jl}$. Then, for $j\neq l$, the
subdifferential of $\tilde{L}_{jl}$ with respect to $Z^{(q)}_{jl}$ is
$\partial_{Z^{(q)}_{jl}}\tilde{L}_{jl}=\rho(Z^{(q)}_{jl}-A^{(q)}_{jl})+\lambda_{1}G^{(q)}_{jl}+\lambda_{2}D^{(q)}_{jl},$
where
$G^{(q)}_{jl}=\left\\{\begin{aligned}
&\frac{Z^{(q)}_{jl}}{\|Z^{(q)}_{jl}\|_{\text{F}}}\quad\text{when}\;Z^{(q)}_{jl}\neq
0\\\ &\\{G^{(q)}_{jl}\in\mathbb{R}^{M\times M}:\|G^{(q)}_{jl}\|_{\text{F}}\leq
1\\}\quad\text{otherwise}\end{aligned}\right.,$
and
$D^{(q)}_{jl}=\left\\{\begin{aligned}
&\frac{Z^{(q)}_{jl}}{\left(\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}\quad\text{when}\;\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}>0\\\
&\\{D^{(q)}_{jl}\in\mathbb{R}^{M\times
M}:\sum^{Q}_{q=1}\|D^{(q)}_{jl}\|^{2}_{\text{F}}\leq
1\\}\quad\text{otherwise}\end{aligned}\right..$
To obtain the optimum, we need
$0\in\partial_{Z^{(q)}_{jl}}\tilde{L}_{jl}(\hat{Z}^{(q)}_{jl})$
for all $q=1,\ldots,Q$. We now split our discussion into two cases.
(a) When $\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}=0$, or
equivalently, $\hat{Z}^{(q)}_{jl}=0$ for all $q=1,\ldots,Q$.
In this case, there exists $G^{(q)}_{jl}$, where
$\|G^{(q)}_{jl}\|_{\text{F}}\leq 1$, for all $q=1,\ldots,Q$; and also
$D^{(q)}_{jl}$, where $\sum^{Q}_{q=1}\|D^{(q)}_{jl}\|^{2}_{\text{F}}\leq 1$,
such that
$0=-\rho\cdot A^{(q)}_{jl}+\lambda_{1}G^{(q)}_{jl}+\lambda_{2}D^{(q)}_{jl},$
which implies that
$D^{(q)}_{jl}=\frac{\rho}{\lambda_{2}}\left(A^{(q)}_{jl}-\frac{\lambda_{1}}{\rho}G^{(q)}_{jl}\right).$
Thus, we have
$\displaystyle\|D^{(q)}_{jl}\|_{\text{F}}$
$\displaystyle=\frac{\rho}{\lambda_{2}}\left\|A^{(q)}_{jl}-\frac{\lambda_{1}}{\rho}G^{(q)}_{jl}\right\|_{\text{F}}\geq\frac{\rho}{\lambda_{2}}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\frac{\lambda_{1}}{\rho}\|G^{(q)}_{jl}\|_{\text{F}}\right)_{+}$
$\displaystyle\geq\frac{\rho}{\lambda_{2}}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\frac{\lambda_{1}}{\rho}\right)_{+},$
which implies that
$\frac{\rho^{2}}{\lambda^{2}_{2}}\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\frac{\lambda_{1}}{\rho}\right)^{2}_{+}\leq\sum^{Q}_{q=1}\|D^{(q)}_{jl}\|^{2}_{\text{F}}\leq
1,$
and then we have
$\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}\leq\lambda_{2}/\rho.$
(A.15)
(b) When $\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}>0$.
For those $q$’s such that $\hat{Z}^{(q)}_{jl}=0$, there exists $G^{(q)}_{jl}$,
where $\|G^{(q)}_{jl}\|_{\text{F}}=1$, such that
$0=-\rho A^{(q)}_{jl}+\lambda_{1}G^{(q)}_{jl}.$
Thus, we have
$\|A^{(q)}_{jl}\|_{\text{F}}=\frac{\lambda_{1}}{\rho}\|G^{(q)}_{jl}\|_{\text{F}}\leq\frac{\lambda_{1}}{\rho},$
which implies that
$\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)_{+}=0.$ (A.16)
On the other hand, for those $q$’s such that $\hat{Z}^{(q)}_{jl}\neq 0$, we
have
$0=\rho\left(\hat{Z}^{(q)}_{jl}-A^{(q)}_{jl}\right)+\lambda_{1}\frac{\hat{Z}^{(q)}_{jl}}{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}+\lambda_{2}\frac{\hat{Z}^{(q)}_{jl}}{\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}},$
which implies that
$A^{(q)}_{jl}=\hat{Z}^{(q)}_{jl}\left(1+\frac{\lambda_{1}}{\rho\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}+\frac{\lambda_{2}}{\rho\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}\right),$
(A.17)
and
$\|A^{(q)}_{jl}\|_{\text{F}}=\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}+\lambda_{1}/\rho+(\lambda_{2}/\rho)\cdot\frac{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}{\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}.$
(A.18)
By (A.18), we have
$\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)_{+}>\frac{\lambda_{2}}{\rho}\cdot\frac{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}{\sqrt{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}}>0.$
(A.19)
By (A.16) and (A.19), we have
$\displaystyle\sum^{Q}_{q=}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}$
$\displaystyle=\sum_{q:\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\neq
0}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}$ (A.20)
$\displaystyle>\frac{\lambda^{2}_{2}}{\rho^{2}}\sum_{q:\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\neq
0}\frac{\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}$
$\displaystyle>\lambda^{2}_{2}/\rho^{2}.$
We now make the following claims.
Claim 1.
$\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}=0\Leftrightarrow\sqrt{\sum^{Q}_{q=}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}\leq\lambda_{2}/\rho$.
This claim is easily shown by (A.15) and (A.20).
Claim 2. When $\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}>0$, we have
$\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}=0\Leftrightarrow\|A^{(q)}_{jl}\|_{\text{F}}\leq\lambda_{1}/\rho$.
This claim is easily shown by (A.16) and (A.19).
Claim 3. When $\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\neq 0$, then we have
$\hat{Z}^{(q)}_{jl}=\left(\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{\|A^{(q)}_{jl}\|_{\text{F}}}\right)\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q=}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)A^{(q)}_{jl}.$
To prove this claim, note that by Claim 2 and (A.18), we have
$\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)_{+}=\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\left(1+\frac{\lambda_{2}}{\rho\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}\right)$
for $q=1,\ldots,Q$. Thus,
$\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}=\sqrt{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}+\lambda_{2}/\rho,$
which implies that
$\sqrt{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}=\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}-\lambda_{2}/\rho.$
Thus, by (A.18), we have
$\displaystyle\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}$
$\displaystyle=\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{1+\frac{\lambda_{2}/\rho}{\sqrt{\sum^{Q}_{q^{\prime}=1}\left(\|A^{(q^{\prime})}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}-\lambda_{2}/\rho}}$
$\displaystyle=\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q^{\prime}=1}\left(\|A^{(q^{\prime})}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right).$
This way, combined with (A.17), we then have
$\hat{Z}^{(q)}_{jl}=\frac{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}{\|A^{(q)}_{jl}\|_{\text{F}}}A^{(q)}_{jl}=\left(\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{\|A^{(q)}_{jl}\|_{\text{F}}}\right)\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q^{\prime}=1}\left(\|A^{(q^{\prime})}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)A^{(q)}_{jl}.$
Finally, combining Claims 1-3, we obtain (A.11).
## Appendix B Main Technical Proofs
We give proofs of the results given in the main text.
### B.1 Proof of Lemma 2
We only need to prove that when we use two sets of orthonormal function basis
$e^{M}(t)=\\{e^{M}_{j}(t)\\}^{p}_{j=1}$ and
$\tilde{e}^{M}(t)=\\{\tilde{e}^{M}_{j}(t)\\}^{p}_{j=1}$ to expand the same
subspace $\mathbb{V}^{M}_{[p]}$, the definition of $E^{\pi}_{\Delta}$ will not
be changed. Since both
$e^{M}_{j}(t)=(e^{M}_{j1}(t),e^{M}_{j2}(t),\dots,e^{M}_{jM}(t))^{\top}$ and
$\tilde{e}^{M}_{j}(t)=(\tilde{e}^{M}_{j1}(t),\tilde{e}^{M}_{j2}(t),\dots,\tilde{e}^{M}_{jM}(t))^{\top}$
are orthonormal function basis of $\mathbb{V}^{M}_{j}$, there must exist an
orthonormal matrix $U_{j}\in\mathbb{R}^{M\times M}$ satisfying
$U^{\top}_{j}U_{j}=U_{j}U^{\top}_{j}=I_{M}$, such that
$\tilde{e}^{M}_{j}(t)=U_{j}e^{M}_{j}(t)$. Let $a^{X,M}_{ij}$ be the projection
score vectors of $X_{ij}(t)$ onto $e^{M}_{j}(t)$ and $\tilde{a}^{X,M}_{ij}$ be
the projection score vectors of $X_{ij}(t)$ onto $\tilde{e}^{M}_{j}(t)$. Then
$\tilde{a}^{X,M}_{ij}=U_{j}a^{X,M}_{ij}$. Denote
$U={\rm diag}\\{U_{1},U_{2},\dots,U_{p}\\}\in\mathbb{R}^{pM\times pM}.$
We then have
$\displaystyle\tilde{a}^{X,M}_{i}$
$\displaystyle=((\tilde{a}^{X,M}_{i1})^{\top},(\tilde{a}^{X,M}_{i2})^{\top},\dots,(\tilde{a}^{X,M}_{ip})^{\top})^{\top}$
$\displaystyle=((a^{X,M}_{i1})^{\top}U^{\top}_{1},(a^{X,M}_{i2})^{\top}U^{\top}_{2},\dots,(a^{X,M}_{ip})^{\top}U^{\top}_{p})^{\top}=Ua^{X,M}_{i}$
and
$\tilde{\Sigma}^{X,M}={\rm Cov}\left(\tilde{a}^{X,M}\right)=U{\rm
Cov}\left(\tilde{a}^{X,M}\right)U^{\top}=U\Sigma^{X,M}U^{\top}.$
Thus
$\tilde{\Theta}^{X,M}=\left(\tilde{\Sigma}^{X,M}\right)^{-1}=U\left(\Sigma^{X,M}\right)^{-1}U^{\top}=U\Theta^{X,M}U^{\top}.$
Therefore, $\tilde{\Theta}^{X,M}_{jl}=U_{j}\Theta^{X,M}_{jl}U^{\top}_{l}$ for
all $j,l\in V^{2}$ and, thus,
$\|\tilde{\Theta}^{X,M}_{jl}\|_{\text{F}}=\|\Theta^{X,M}_{jl}\|_{\text{F}}$
for all $j,l\in V^{2}$. This implies the final result.
### B.2 Proof of Lemma 3
We first show that $X_{ij},Y_{ij}\in{\rm
Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$ almost surely.
Let
$M^{X}_{j}=\sup\\{M\in\mathbb{N}^{+}:\lambda^{X}_{jM}>0\\}.$
By Karhunen–Loève theorem, we have $X_{ij}=\sum^{M^{X}_{j}}_{k=1}\langle
X_{ij},\phi^{X}_{jk}\rangle\phi^{X}_{jk}$ almost surely. Thus, we have
$X_{ij}\in{\rm
Span}\left\\{\phi^{X}_{j1},\dots,\phi^{X}_{j,M^{X}_{j}}\right\\}$ almost
surely. For any $1\leq k\leq M^{X}_{j}$, we have that
$\int_{\mathcal{T}}K_{jj}(s,t)\phi^{X}_{k}(s)\phi^{X}_{k}(t)dsdt\geq\int_{\mathcal{T}}K^{X}_{jj}(s,t)\phi^{X}_{k}(s)\phi^{X}_{k}(t)dsdt=\lambda^{X}_{jk}>0,$
which implies that $\phi^{X}_{k}\in{\rm
Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$. Thus, we have
${\rm
Span}\left\\{\phi^{X}_{j1},\dots,\phi^{X}_{j,M^{X}_{j}}\right\\}\subseteq{\rm
Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$ and
$X_{ij}\in{\rm Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$
almost surely. Similarly, we have that $Y_{ij}\in{\rm
Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$ almost surely.
Next, we show that $M^{\prime}_{j}=M^{\star}_{j}$ by contradiction. By the
definition of $M^{\prime}_{j}$, we have that $M^{\prime}_{j}\leq
M^{\star}_{j}$. If $M^{\prime}_{j}\neq M^{\star}_{j}$, then we have
$\mathbb{V}^{M^{\prime}_{j}}_{j}\subseteq\mathbb{H}$ such that
$M^{\prime}_{j}<M^{\star}_{j}$ and
$X_{ij},Y_{ij}\in\mathbb{V}^{M^{\prime}_{j}}_{j}$ almost surely. This implies
that there exists $\phi\in{\rm
Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}\setminus\mathbb{V}^{M^{\prime}_{j}}_{j}$
such that
$\displaystyle\mathbb{E}\left[\left(\langle\phi_{jk}(t),X_{ij}(t)\rangle\right)^{2}\right]=0\quad\text{and}\quad\mathbb{E}\left[\left(\langle\phi_{jk}(t),Y_{ij}(t)\rangle\right)^{2}\right]=0$
$\displaystyle\Rightarrow$
$\displaystyle\int_{\mathcal{T}}K^{X}_{jj}(s,t)\phi_{jk}(s)\phi_{jk}(t)dsdt=0\quad\text{and}\quad\int_{\mathcal{T}}K^{Y}_{jj}(s,t)\phi_{jk}(s)\phi_{jk}(t)dsdt=0$
$\displaystyle\Rightarrow$
$\displaystyle\int_{\mathcal{T}}K_{jj}(s,t)\phi_{jk}(s)\phi_{jk}(t)dsdt=0,$
$\displaystyle\Rightarrow$ $\displaystyle\lambda_{jk}=0,$
which contradicts the definition of $M^{\star}_{j}$. Thus, we must have
$M^{\prime}_{j}=M^{\star}_{j}$.
### B.3 Proof of Lemma 4
Let $U=V\backslash\\{j,l\\}$, and $a^{X,M}_{U}=\left((a^{X,M}_{j})^{\top},j\in
U\right)^{\top}$. Without loss of generality, assume that $\Sigma^{X,M}$ and
$\Theta^{X,M}$ take the following block structure:
$\displaystyle\Sigma^{X,M}=\left[\begin{matrix}\Sigma^{X,M}_{jj}&\Sigma^{X,M}_{jl}&\Sigma^{X,M}_{jU}\\\
\Sigma^{X,M}_{lj}&\Sigma^{X,M}_{ll}&\Sigma^{X,M}_{lU}\\\
\Sigma^{X,M}_{Uj}&\Sigma^{X,M}_{Ul}&\Sigma^{X,M}_{UU}\\\
\end{matrix}\right],\quad\Theta^{X,M}=\left[\begin{matrix}\Theta^{X,M}_{jj}&\Theta^{X,M}_{jl}&\Theta^{X,M}_{jU}\\\
\Theta^{X,M}_{lj}&\Theta^{X,M}_{ll}&\Theta^{X,M}_{lU}\\\
\Theta^{X,M}_{Uj}&\Theta^{X,M}_{Ul}&\Theta^{X,M}_{UU}\\\ \end{matrix}\right].$
Let $P$ denote the submatrix:
$P=\left[\begin{matrix}\Theta^{X,M}_{jj}&\Theta^{X,M}_{jl}\\\
\Theta^{X,M}_{lj}&\Theta^{X,M}_{ll}\end{matrix}\right].$
By standard results for the multivariate Gaussian (Heckler, 2005), we have
$\displaystyle\mathrm{Var}\left(a^{X,M}_{j}\mid a^{X,M}_{k},k\neq
j\right)=H^{X,M}_{jj}=(\Theta^{X,M}_{jj})^{-1},$
$\displaystyle\mathrm{Var}\left(\left[\begin{matrix}a^{X,M}_{j}\\\
a^{X,M}_{l}\end{matrix}\right]\mid
a^{X,M}_{U}\right)=P^{-1}=\left[\begin{matrix}(P^{-1})_{11}&(P^{-1})_{12}\\\
(P^{-1})_{21}&(P^{-1})_{22}\end{matrix}\right].$
Thus, the first statement directly follows from the first equation. To prove
the second statement, we only need to note that
$\displaystyle H^{X,M}_{jl}$
$\displaystyle=\mathrm{Cov}\left(a^{X,M}_{j},a^{X,M}_{l}\mid
a^{X,M}_{U}\right)$ $\displaystyle=(P^{-1})_{12}$
$\displaystyle=-(\Theta^{X,M}_{jj})^{-1}\Theta^{X,M}_{jl}(P^{-1})_{22}$
$\displaystyle=-H^{X,M}_{jj}\Theta_{jl}^{X,M}H^{\backslash j,X,M}_{ll},$
where the second to last equation follows from the $2\times 2$ block matrix
inverse and the last equation follows from the property of multivariate
Gaussian. This completes the proof.
### B.4 Proof of Theorem 1
We provide the proof of Theorem 1, following the framework introduced in
Negahban et al. (2012). We start by introducing some notation.
We use $\otimes$ to denote the Kronecker product. For
$\Delta\in\mathbb{R}^{pM\times pM}$, let
$\theta=\operatorname{vec}(\Delta)\in{\mathbb{R}^{p^{2}M^{2}}}$ and
$\theta^{*}=\operatorname{vec}({\Delta^{M}})$, where $\Delta^{M}$ is defined
in Section 2.2. Let $\mathcal{G}=\\{G_{t}\\}_{t=1,\ldots,N_{\mathcal{G}}}$ be
a set of indices, where $N_{\mathcal{G}}=p^{2}$ and
$G_{t}\subset\\{1,2,\cdots,p^{2}M^{2}\\}$ is the set of indices for $\theta$
that correspond to the $t$-th $M\times M$ submatrix of $\Delta^{M}$. Thus, if
$t=(j-1)p+l$, then
$\theta_{G_{t}}=\operatorname{vec}{(\Delta_{jl})}\in{\mathbb{R}^{M^{2}}}$,
where $\Delta_{jl}$ is the $(j,l)$-th $M\times{M}$ submatrix of $\Delta$.
Denote the group indices of $\theta^{*}$ that belong to blocks corresponding
to $E_{\Delta}$ as
$S_{\mathcal{G}}\subseteq{\\{1,2,\cdots,N_{\mathcal{G}}\\}}$. Note that we
define $S_{\mathcal{G}}$ using $E_{\Delta}$ and not $E_{\Delta^{M}}$.
Therefore, as stated in Assumption 2, $|S_{\mathcal{G}}|=s$. We further define
the subspace $\mathcal{M}$ as
${}\mathcal{M}\coloneqq{\\{\theta\in{\mathbb{R}^{p^{2}M^{2}}}\mid\theta_{G_{t}}=0\text{
for all }t\notin{S_{\mathcal{G}}}\\}}.$ (B.1)
Its orthogonal complement with respect to the Euclidean inner product is
$\mathcal{M}^{\bot}\coloneqq{\\{\theta\in{\mathbb{R}^{p^{2}M^{2}}}\mid\theta_{G_{t}}=0\text{
for all }t\in{S_{\mathcal{G}}}\\}}.$ (B.2)
For a vector $\theta$, let $\theta_{\mathcal{M}}$ and
$\theta_{\mathcal{M}^{\bot}}$ be the projection of $\theta$ on the subspaces
$\mathcal{M}$ and $\mathcal{M}^{\bot}$, respectively. Let
$\langle\cdot,\cdot\rangle$ represent the Euclidean inner product. Let
${}\mathcal{R}(\theta)\coloneqq{\sum_{t=1}^{N_{\mathcal{G}}}|\theta_{G_{t}}|_{2}}\triangleq{|\theta|_{1,2}}.$
(B.3)
For any $v\in{\mathbb{R}^{p^{2}M^{2}}}$, the dual norm of $\mathcal{R}$ is
given by
${}\mathcal{R}^{*}(v)\coloneqq\sup_{u\in{\mathbb{R}^{p^{2}M^{2}}\backslash{\\{0\\}}}}\frac{\langle{u},{v}\rangle}{\mathcal{R}(u)}=\sup_{\mathcal{R}(u)\leq{1}}\langle{u},{v}\rangle.$
(B.4)
The subspace compatibility constant of $\mathcal{M}$ with respect to
$\mathcal{R}$ is defined as
${}\Psi(\mathcal{M})\coloneqq{\sup_{u\in{\mathcal{M}\backslash\\{0\\}}}}\frac{\mathcal{R}(u)}{|u|_{2}}.$
(B.5)
Proof By Lemma 5 and Assumption 1, we have
$|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})|_{\infty}\leq\delta_{n}^{2}+2\delta_{n}\sigma_{\max}$
(B.6)
and
$|\operatorname{vec}{(S^{Y,M}-S^{X,M})}-\operatorname{vec}{(\Sigma^{Y,M}-\Sigma^{X,M})}|_{\infty}\leq
2\delta_{n}.$ (B.7)
Problem (25) can be written in the following form:
$\hat{\theta}_{\lambda_{n}}\in\operatorname*{arg\,min}_{\theta\in{\mathbb{R}^{p^{2}M^{2}}}}\mathcal{L}(\theta)+\lambda_{n}\mathcal{R}(\theta),$
(B.8)
where
${}\mathcal{L}(\theta)=\frac{1}{2}\theta^{\top}(S^{Y,M}\otimes{S^{X,M}})\theta-\theta^{\top}\operatorname{vec}({S^{Y,M}-S^{X,M}}).$
(B.9)
The loss $\mathcal{L}(\theta)$ is convex and differentiable with respect to
$\theta$, and it can be easily verified that $\mathcal{R}(\cdot)$ defines a
vector norm. For $h\in\mathbb{R}^{p^{2}M^{2}}$, the error of the first-order
Taylor series expansion of $\mathcal{L}$ is:
$\displaystyle\delta{\mathcal{L}}(h,\theta^{*})\coloneqq\mathcal{L}(\theta^{*}+h)-\mathcal{L}(\theta^{*})-\langle\nabla\mathcal{L}(\theta^{*}),h\rangle=\frac{1}{2}h^{\top}(S^{Y,M}\otimes{S^{X,M}})h.$
(B.10)
From (B.9), we see that
$\nabla{\mathcal{L}}(\theta)=(S^{Y,M}\otimes{S^{X,M}})\theta-\operatorname{vec}({S^{Y,M}-S^{X,M}})$.
By Lemma 9, we have
${}\mathcal{R}^{*}(\nabla{\mathcal{L}}(\theta^{*}))=\max_{t=1,2,\cdots,N_{\mathcal{G}}}\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}.$
(B.11)
We now establish an upper bound for
$\mathcal{R}^{*}(\nabla{\mathcal{L}}(\theta^{*}))$. First, note that
$(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta^{*}-\operatorname{vec}({\Sigma^{Y,M}-\Sigma^{X,M}})=\operatorname{vec}({\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}-(\Sigma^{Y,M}-\Sigma^{X,M})})=0.$
Letting $(\cdot)_{jl}$ denote the $(j,l)$-th submatrix, we have
$\displaystyle\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}$
(B.12)
$\displaystyle=\left|\left[(S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta^{*}-\operatorname{vec}{((S^{Y,M}-\Sigma^{Y,M})-(S^{X,M}-\Sigma^{X,M}))}\right]_{G_{t}}\right|_{2}$
$\displaystyle={\|(S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M})_{jl}-(S^{Y,M}-\Sigma^{Y,M})_{jl}-(S^{X,M}-\Sigma^{X,M})_{jl}\|_{F}}$
$\displaystyle\leq{\|(S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M})_{jl}\|_{F}+\|(S^{Y,M}-\Sigma^{Y,M})_{jl}\|_{F}+\|(S^{X,M}-\Sigma^{X,M})_{jl}\|_{F}}.$
For any $M\times{M}$ matrix $A$, $\|A\|_{F}\leq{M|A|_{\infty}}$, so
$\displaystyle\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}$
$\displaystyle\leq
M\left[\left|(S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M})_{jl}\right|_{\infty}+\left|(S^{Y,M}-\Sigma^{Y,M})_{jl}\right|_{\infty}+\left|(S^{X,M}-\Sigma^{X,M})_{jl}\right|_{\infty}\right]$
$\displaystyle\leq
M\left[\left|S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}\right|_{\infty}+|S^{Y,M}-\Sigma^{Y,M}|_{\infty}+|S^{X,M}-\Sigma^{X,M}|_{\infty}\right].$
For any $A\in{\mathbb{R}^{k\times{k}}}$ and $v\in{\mathbb{R}^{k}}$, we have
$|Av|_{\infty}\leq{|A|_{\infty}|v|_{1}}$. Thus, we further have
$\displaystyle|S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}|_{\infty}$
$\displaystyle=|[(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})]\operatorname{vec}{(\Delta^{M})}|_{\infty}$
$\displaystyle\leq{|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}}|\operatorname{vec}{(\Delta^{M})}|_{1}$
$\displaystyle=|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}|\Delta^{M}|_{1}.$
Combining the inequalities gives an upper bound uniform over $\mathcal{G}$
(i.e., for all $G_{t}$):
$\displaystyle\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}$
$\displaystyle\leq
M\left[|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}|\Delta^{M}|_{1}+|S^{Y,M}-\Sigma^{Y,M}|_{\infty}+|S^{X,M}-\Sigma^{X,M}|_{\infty}\right],$
which implies
$\displaystyle\mathcal{R}^{*}\left(\nabla{\mathcal{L}}(\theta^{*})\right)\leq$
$\displaystyle
M[|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}|\Delta^{M}|_{1}+$
(B.13)
$\displaystyle|S^{Y,M}-\Sigma^{Y,M}|_{\infty}+|S^{X,M}-\Sigma^{X,M}|_{\infty}].$
Assuming $|S^{X,M}-\Sigma^{X,M}|_{\infty}\leq{\delta_{n}}$ and
$|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\leq\delta_{n}$ implies
${}\mathcal{R}^{*}\left(\nabla{\mathcal{L}}(\theta^{*})\right)\leq{M[(\delta_{n}^{2}+2\delta_{n}\sigma_{\max})|\Delta^{M}|_{1}+2\delta_{n}]}.$
(B.14)
Setting
${}\lambda_{n}=2M\left[\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right)\left|\Delta^{M}\right|_{1}+2\delta_{n}\right],$
(B.15)
then implies that
$\lambda_{n}\geq{2\mathcal{R}^{*}\left(\nabla{\mathcal{L}}(\theta^{*})\right)}$.
Thus, invoking Lemma 1 in Negahban et al. (2012),
$h=\hat{\theta}_{\lambda_{n}}-\theta^{*}$ must satisfy
${}\mathcal{R}(h_{\mathcal{M}^{\bot}})\leq{3\mathcal{R}(h_{\mathcal{M}})}+4\mathcal{R}(\theta^{*}_{\mathcal{M}^{\bot}}),$
(B.16)
where $\mathcal{M}$ is defined in (B.1). Equivalently,
${}|h_{\mathcal{M}^{\bot}}|_{1,2}\leq{3|h_{\mathcal{M}}|_{1,2}}+4|\theta^{*}_{\mathcal{M}^{\bot}}|_{1,2}.$
(B.17)
By the definition of $\nu_{2}$, we have
${}|\theta^{*}_{\mathcal{M}^{\bot}}|_{1,2}=\sum_{t\notin{\mathcal{S}_{\mathcal{G}}}}|\theta^{*}_{G_{t}}|_{2}\leq\left(p(p+1)/2-s\right)\nu_{2}\leq
p^{2}\nu_{2}.$ (B.18)
Next, we show that $\delta\mathcal{L}(h,\theta^{*})$, as defined in (B.10),
satisfies the Restricted Strong Convexity property defined in definition 2 in
Negahban et al. (2012). That is, we show an inequality of the form:
$\delta\mathcal{L}(h,\theta^{*})\geq{\kappa_{\mathcal{L}}|h|^{2}_{2}}-\omega^{2}_{\mathcal{L}}\left(\theta^{*}\right)$
whenever $h$ satisfies (B.17).
By using Lemma 7, we have
$\displaystyle\theta^{\top}(S^{Y,M}\otimes{S^{X,M}})\theta$
$\displaystyle=\theta^{\top}(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta+\theta^{\top}(S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta$
$\displaystyle\geq{\theta^{\top}(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta-|\theta^{\top}(S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta|}$
$\displaystyle\geq{\lambda^{*}_{\min}}|\theta|^{2}_{2}-M^{2}|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}|\theta|^{2}_{1,2},$
where the last inequality holds because Lemma 7 and
$\lambda^{*}_{\min}=\lambda_{\min}(\Sigma^{X,M})\times{\lambda_{\min}(\Sigma^{Y,M})}=\lambda_{\min}(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})>0$.
Thus,
$\displaystyle\delta\mathcal{L}(h,\theta^{*})$
$\displaystyle=\frac{1}{2}h^{\top}(S^{Y,M}\otimes{S^{X,M}})h$
$\displaystyle\geq{\frac{1}{2}\lambda^{*}_{\min}}|h|^{2}_{2}-\frac{1}{2}M^{2}|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}|h|^{2}_{1,2}.$
By Lemma 8 and (B.17), we have
$\displaystyle|h|^{2}_{1,2}$
$\displaystyle=(|h_{\mathcal{M}}|_{1,2}+|h_{\mathcal{M}^{\bot}}|_{1,2})^{2}\leq
16({|h_{\mathcal{M}}|_{1,2}}+|\theta^{*}_{\mathcal{M}^{\bot}}|_{1,2})^{2}$
$\displaystyle\leq 16(\sqrt{s}|h|_{2}+p^{2}\nu_{2})^{2}\leq
32s|h|^{2}_{2}+32p^{4}\nu_{2}^{2}.$
Combining with the equation above, we get
$\displaystyle\delta\mathcal{L}(h,\theta^{*})$
$\displaystyle\geq{\left[\frac{1}{2}\lambda^{*}_{\min}-16M^{2}s|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}\right]}|h|^{2}_{2}$
(B.19)
$\displaystyle\qquad\qquad-16M^{2}p^{4}\nu_{2}^{2}|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}$
$\displaystyle\geq\left[\frac{1}{2}\lambda^{*}_{\min}-8M^{2}s\left(\delta^{2}_{n}+2\delta^{2}_{n}\sigma_{\max}\right)\right]|h|^{2}_{2}$
$\displaystyle\qquad\qquad-16M^{2}p^{4}\nu_{2}^{2}\left(\delta^{2}_{n}+2\delta_{n}\sigma_{\max}\right).$
Thus, appealing to (B.6), the Restricted Strong Convexity property holds with
$\displaystyle\kappa_{\mathcal{L}}$
$\displaystyle\;=\;\frac{1}{2}\lambda^{*}_{\min}-8M^{2}s\left(\delta^{2}+2\delta_{n}\sigma_{\max}\right),$
(B.20) $\displaystyle\omega_{\mathcal{L}}$
$\displaystyle\;=\;4Mp^{2}\nu_{2}\sqrt{\delta_{n}^{2}+2\delta_{n}\sigma_{\max}}.$
When
$\delta_{n}<\frac{1}{4}\sqrt{\frac{\lambda^{*}_{\min}+16M^{2}s(\sigma_{\max})^{2}}{M^{2}s}}-\sigma_{\max}$
as we assumed in the theorem, then $\kappa_{\mathcal{L}}>0$. By Theorem 1 of
Negahban et al. (2012) and Lemma 8, letting
$\lambda_{n}=2M\left[\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right)|\Delta^{M}|_{1}+2\delta_{n}\right]$,
as in (B.15), ensures
$\displaystyle\|\hat{\Delta}^{M}-\Delta^{M}\|^{2}_{F}$
$\displaystyle=|\hat{\theta}_{\lambda_{n}}-\theta^{*}|^{2}_{2}$ (B.21)
$\displaystyle\leq{9\frac{\lambda^{2}_{n}}{\kappa^{2}_{\mathcal{L}}}}\Psi^{2}(\mathcal{M})+\frac{\lambda_{n}}{\kappa_{\mathcal{L}}}\left(2\omega^{2}_{\mathcal{L}}+4\mathcal{R}(\theta^{*}_{\mathcal{M}^{\bot}})\right)$
$\displaystyle=\frac{9\lambda^{2}_{n}s}{\kappa^{2}_{\mathcal{L}}}+\frac{2\lambda_{n}}{\kappa_{\mathcal{L}}}(\omega^{2}_{\mathcal{L}}+2p^{2}\nu_{2})$
$\displaystyle=\Gamma^{2}_{n}.$
We then prove that $\hat{E}_{\Delta}=E_{\Delta}$. Recall that we have assumed
that $0<\Gamma_{n}<\tau/2=(\nu_{1}-\nu_{2})/2$ and
$\nu_{2}+\Gamma_{n}\leq\epsilon_{n}<\nu_{1}-\Gamma_{n}$. Note that we have
$\|\hat{\Delta}^{M}_{jl}-\Delta^{M}_{jl}\|_{F}\leq{\|\hat{\Delta}^{M}-\Delta^{M}\|_{F}}\leq\Gamma_{n}$
for any $(j,l)\in{V^{2}}$. Recall that
${}E_{\Delta}\;=\;\\{(j,l)\in{V^{2}}:\;j\neq{l},D_{jl}>0\\}.$ (B.22)
We first prove that $E_{\Delta}\subseteq{\hat{E}_{{\Delta}}}$. For any
$(j,l)\in{E_{\Delta}}$, by the definition of $\nu_{1}$ in Section 4.1, we have
$\displaystyle\|\hat{\Delta}^{M}_{jl}\|_{F}$
$\displaystyle\geq{\|\Delta^{M}_{jl}\|_{F}-\|\hat{\Delta}_{jl}^{M}-\Delta^{M}_{jl}\|_{F}}$
$\displaystyle\geq\nu_{1}-\Gamma_{n}$ $\displaystyle>\epsilon_{n}.$
The last inequality holds because we have assumed that
$\epsilon_{n}<\nu_{1}-\Gamma_{n}$. Thus, by definition of $\hat{E}_{{\Delta}}$
in (27), we have $(j,l)\in{\hat{E}_{{\Delta}}}$, which further implies that
$E_{\Delta}\subseteq{\hat{E}_{{\Delta}}}$.
We then show $\hat{E}_{{\Delta}}\subseteq{E_{\Delta}}$. Let
$\hat{E}^{c}_{{\Delta}}$ and $E^{c}_{\Delta}$ denote the complement set of
$\hat{E}_{{\Delta}}$ and $E_{\Delta}$. For any $(j,l)\in{E^{c}_{\Delta}}$,
which also means that $(l,j)\in{E^{c}_{\Delta}}$, by definition of $\nu_{2}$,
we have that
$\displaystyle\|\hat{\Delta}^{M}_{jl}\|_{F}$
$\displaystyle\leq{\|\Delta^{M}_{jl}\|_{F}+\|\hat{\Delta}_{jl}^{M}-\Delta^{M}_{jl}\|_{F}}$
$\displaystyle\leq\nu_{2}+\Gamma_{n}$ $\displaystyle\leq\epsilon_{n}.$
Again, the last inequality holds because because we have assumed that
$\epsilon_{n}\geq\nu_{2}+\Gamma_{n}$. Thus, by definition of
$\hat{E}_{{\Delta}}$, we have $(j,l)\notin{\hat{E}_{{\Delta}}}$ or
$(j,l)\in{\hat{E}^{c}_{{\Delta}}}$. This implies that
$E^{c}_{\Delta}\subseteq{\hat{E}^{c}_{{\Delta}}}$, or
$\hat{E}_{{\Delta}}\subseteq{E_{\Delta}}$. Combing with previous conclusion
that $E_{\Delta}\subseteq{\hat{E}_{{\Delta}}}$, the proof is complete. [2mm]
### B.5 Proof of Theorem 4
We only need to prove that
$\displaystyle P\left(\lvert S^{M}-\Sigma^{M}\rvert_{\infty}>\delta\right)$
$\displaystyle\leq C_{1}np\exp\\{-C_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}$
(B.23) $\displaystyle+C_{3}(pM)^{2}\exp\\{-C_{4}nM^{-2(1+\beta)}\delta^{2}\\}$
$\displaystyle+C_{5}npL\exp\left\\{-\frac{C_{6}M^{-2(1+\beta)}\delta^{2}}{\tilde{\psi}_{2}(T,L)}\right\\},$
where $S^{M}$ can be understood as either $S^{X,M}$ or $S^{Y,M}$ and
$\Sigma^{M}$ can be understood as either $\Sigma^{X,M}$ or $\Sigma^{Y,M}$,
with $C_{k}=C^{X}_{k}$ or $C_{k}=C^{Y}_{k}$ for $k=1,2,3,4$ accordingly. To
see that (B.23) implies (43), we first note that (B.23) implies that
$\displaystyle P\left(\lvert
S^{X,M}-\Sigma^{X,M}\rvert_{\infty}\leq\delta\,\text{and}\,\lvert
S^{Y,M}-\Sigma^{Y,M}\rvert_{\infty}\leq\delta\right)$ $\displaystyle\geq$
$\displaystyle 1-P\left(\lvert
S^{X,M}-\Sigma^{X,M}\rvert_{\infty}>\delta\right)-P\left(\lvert
S^{Y,M}-\Sigma^{Y,M}\rvert_{\infty}>\delta\right)$ $\displaystyle\geq$
$\displaystyle
1-C^{X}_{1}pM\exp\\{-C^{X}_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}-C^{X}_{3}(pM)^{2}\exp\\{-C^{X}_{4}nM^{-2(1+\beta)}\delta^{2}\\}-$
$\displaystyle
C^{Y}_{1}pM\exp\\{-C^{Y}_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}-C^{Y}_{3}(pM)^{2}\exp\\{-C^{Y}_{4}nM^{-2(1+\beta)}\delta^{2}\\}$
$\displaystyle\geq$ $\displaystyle
1-2\bar{C}_{1}pM\exp\\{-\bar{C}_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}-2\bar{C}_{3}(pM)^{2}\exp\\{-\bar{C}_{4}nM^{-2(1+\beta)}\delta^{2}\\},$
where $\bar{C}_{k}$ for $k=1,2,3,4$ are defined in Theorem 4. Thus, by letting
the last two terms in the last line of the above equation all to be $\iota/2$,
we then have (43). This way, the rest of the proof will focus on proving
(B.23).
Denote $(j,l)$-th submatrix of $S^{M}$ as $S^{M}_{jl}$, and $(k,m)$-th entry
of $S^{M}_{jl}$ as $\hat{\sigma}_{jl,km}$, thus we have
$S^{M}=(\hat{\sigma}_{jl,km})_{1\leq j,l\leq p,\leq k,m\leq M}$; similarly,
let $\Sigma^{M}=(\sigma_{jl,km})_{1\leq j,l\leq p,\leq k,m\leq M}$. Then, by
the definition of $S^{M}$ and $\Sigma^{M}$, we have
$\displaystyle\hat{\sigma}_{jl,km}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\hat{a}_{ijk}\hat{a}_{ilm}$
$\displaystyle\sigma_{jl,km}$
$\displaystyle=\mathbb{E}\left[a_{ijk}a_{ilm}\right].$
Note that
$\displaystyle\hat{a}_{ijk}$
$\displaystyle=\langle\hat{g}_{ij},\hat{\phi}_{jk}\rangle$
$\displaystyle=\langle
g_{ij}+\hat{g}_{ij}-g_{ij},\phi_{jk}+\hat{\phi}_{jk}-\phi_{jk}\rangle$
$\displaystyle=\langle g_{ij},\phi_{jk}\rangle+\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle$
$\displaystyle=a_{ijk}+\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle.$
Thus, we have
$\displaystyle\hat{\sigma}_{jl,km}-\sigma_{jl,km}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\left(\hat{a}_{ijk}\hat{a}_{ilm}-\sigma_{jl,km}\right)$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\left[a_{ijk}+\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\right]\times$
$\displaystyle\left[a_{ijk}+\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\right]-\sigma_{jl,km}$
$\displaystyle=\sum^{16}_{u=1}I_{u},$
where
$\displaystyle I_{1}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\left(a_{ijk}a_{ilm}-\mathbb{E}(a_{ijk}a_{ilm})\right),$
$\displaystyle I_{2}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$
$\displaystyle I_{3}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{4}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$
$\displaystyle I_{5}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ilm}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle,$
$\displaystyle I_{6}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$
$\displaystyle I_{7}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{8}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$
$\displaystyle I_{9}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle a_{ilm},$ $\displaystyle I_{10}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$
$\displaystyle I_{11}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{12}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$
$\displaystyle I_{13}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle
a_{ilm},$ $\displaystyle I_{14}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$
$\displaystyle I_{15}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{16}$
$\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle.$
Note that $I_{u}$, $u=1,\ldots,16$ depend on $j,l,k,m$. To simplify the
notation, we do not denote this fact explicitly. Thus, for any $0<\delta\leq
1$, when for any $1\leq j,l\leq p$ and $1\leq k,m\leq M$, if $\lvert
I_{u}\rvert\leq\delta/16$, $u=1,\ldots,16$, we will have $\lvert
S^{M}-\Sigma^{M}\rvert_{\infty}\leq\delta$. This way, for the rest of the
paper, we only need to calculate the probability of $\lvert
I_{u}\rvert\leq\delta/16$, $u=1,\ldots,16$, $1\leq j,l\leq p$ and $1\leq
k,m\leq M$.
Before we proceed to calculate the probability, we need a bit more notation.
By Assumption 3 (i), we have constants $d_{1},d_{2}>0$, such that
$\lambda_{jk}\leq d_{1}k^{-\beta}$, $d_{jk}\leq d_{2}k^{1+\beta}$ for any
$j=1,\ldots,p$ and $k\geq 1$. Let $d_{0}=\max\\{1,\sqrt{d_{1}},d_{2}\\}$, let
$\xi_{ijk}=\lambda^{-1/2}_{jk}a_{ijk}$ so that $\xi_{ijk}\sim N(0,1)$ i.i.d.
for $i=1,\ldots,n$, and denote
$\displaystyle\delta_{1}$
$\displaystyle=\frac{\delta}{144d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}},$
(B.24) $\displaystyle\delta_{2}$
$\displaystyle=9\lambda_{0,\max}\delta_{1}=\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}},$
where $\lambda_{0,\max}=\max_{j\in V}\sum^{\infty}_{k=1}\lambda_{jk}$. Recall
that $\hat{K}_{jj}$, $j=1,\ldots,p$ are defined as in (24). We define five
events $A_{1}$-$A_{5}$ as below:
$\displaystyle A_{1}$
$\displaystyle:\;\lVert\hat{g}_{ij}-g_{ij}\rVert\leq\delta_{1},\quad\forall
i=1,\ldots,n\ \forall j=1,\ldots,p,$ (B.25) $\displaystyle A_{2}$
$\displaystyle:\;\lVert\hat{K}_{jj}-K_{jj}\rVert_{\text{HS}}\leq\delta_{2}\quad\forall
j=1,\ldots,p,$ $\displaystyle A_{3}$
$\displaystyle:\;\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\leq\frac{3}{2}\quad\forall
j=1,\ldots,p\ \forall k=1,\ldots,M,$ $\displaystyle A_{4}$
$\displaystyle:\;\frac{1}{n}\sum^{n}_{i=1}\lVert g_{ij}\rVert^{2}\leq
2\lambda_{0,\max}\quad\forall j=1,\ldots,p,$ $\displaystyle A_{5}$
$\displaystyle:\;\lvert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}a_{ilm}-\sigma_{jl,km}\rvert\leq\frac{\delta}{16}\quad\forall
1\leq j,l\leq\ 1\leq k,m\leq M.$
Without loss of generality, we assume that
$\langle\hat{\phi}_{jl},\phi_{jl}\rangle\geq 0$ for any $1\leq j\leq p$ and
$1\leq k\leq M$ (If this is not true, we only need to use $-\phi_{jl}$ to
substitute $\phi_{jl}$). Then, by Lemma 10-Lemma 25, when $A_{1}$-$A_{5}$ hold
simultaneously, we have $\lvert I_{u}\rvert\leq\delta/16$ for all
$u=1,\ldots,16$, $1\leq j,l\leq p$ and $\ 1\leq k,m\leq M$. This way, we have
$\displaystyle P\left(\lvert S^{M}-\Sigma^{M}\rvert_{\infty}\leq\delta\right)$
$\displaystyle\geq P\left(\lvert I_{u}\rvert\leq\delta/16,\;\text{for
all}\;1\leq u\leq 16,1\leq j,l\leq\ 1\leq k,m\leq M\right)$ $\displaystyle\geq
P\left(\bigcap^{5}_{w=1}A_{w}\right).$
Or equivalently,
$P\left(\lvert S^{M}-\Sigma^{M}\rvert_{\infty}>\delta\right)\leq
P\left(\bigcup^{5}_{w=1}\bar{A}_{w}\right)\leq\sum^{5}_{w=1}P\left(\bar{A}_{w}\right),$
(B.26)
where the last inequality follows Boole’s inequality, and $\bar{A}$ means the
complement of $A$. This way, we then only need to give an upper bound for
$P(\bar{A}_{w})$, $w=1,\ldots,5$.
The $P(\bar{A}_{1})$ follows directly from Theorem 5. Note that by Theorem 5
and definition of $\tilde{\psi}_{1}$-$\tilde{\psi}_{4}$, we have
$\displaystyle P(\bar{A}_{1})=$ $\displaystyle
P\left(\lVert\hat{g}_{ij}-g_{ij}\rVert>\delta_{1}\;\exists 1\leq i\leq n,1\leq
j\leq p\right)$ $\displaystyle\leq$ $\displaystyle
2(np)\left\\{\exp\left(-\frac{\delta_{1}^{2}}{72\tilde{\psi}^{2}_{1}(T,L)+6\sqrt{2}\tilde{\psi}_{1}(T,L)\delta_{1}}\right)\right.$
$\displaystyle+$ $\displaystyle
L\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)$
$\displaystyle+$
$\displaystyle\left.\exp\left(-\frac{\delta_{1}^{2}}{72\lambda_{0,\max}\tilde{\psi}_{3}(L)+6\sqrt{2\lambda_{0,\max}\tilde{\psi}_{3}(L)}\delta_{1}}\right)\right\\}.$
Let $\gamma_{1}=\sqrt{2}/(12\times 144d^{2}_{0}3\sqrt{3\lambda_{0,\max}})$,
and
$\gamma_{3}=1/(72\lambda_{0,\max}\times(144d^{2}_{0}\sqrt{3\lambda_{0,\max}})^{2})$,
then when $\tilde{\psi}_{1}<\gamma_{1}\cdot\delta/M^{1+\beta}$, and
$\tilde{\psi}_{3}<\gamma_{3}\cdot\delta^{2}/M^{2+2\beta}$, we have
$72\tilde{\psi}^{2}_{1}<6\sqrt{2}\tilde{\psi}_{1}\delta_{1}$ and
$72\lambda_{0,\max}\tilde{\psi}_{3}<6\sqrt{2\lambda_{0,\max}\tilde{\psi}_{3}}\delta_{1}$,
which implies that
$\displaystyle P(\bar{A}_{1})$ (B.27) $\displaystyle\leq
2np\left\\{\exp\left(-\frac{\delta_{1}}{12\sqrt{2}\tilde{\psi}_{1}(T,L)}\right)+\exp\left(-\frac{\delta_{1}}{12\sqrt{2\lambda_{0,\max}}\sqrt{\tilde{\psi}_{3}(L)}}\right)+L\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)\right\\}$
$\displaystyle\overset{(i)}{\leq}2np\left\\{\exp\left(-\frac{\delta_{1}}{12\sqrt{2}}\Phi(T,L)\right)+\exp\left(-\frac{\delta_{1}}{12\sqrt{2\lambda_{0,\max}}}\Phi(T,L)\right)+L\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)\right\\}$
$\displaystyle\overset{(ii)}{\leq}4np\exp\left(-\frac{\delta_{1}}{12\sqrt{2\lambda_{0,\max}}}\Phi(T,L)\right)+2npL\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)$
$\displaystyle=4np\exp\left(-\frac{1}{1728\sqrt{6}\lambda_{0,\max}d^{2}_{0}}\cdot\frac{\delta}{M^{1+\beta}}\cdot\Phi(T,L)\right)$
$\displaystyle+2npL\exp\left(-\frac{\delta^{2}}{6228d^{4}_{0}\lambda_{0,\max}M^{2+2\beta}\tilde{\psi}_{2}(T,L)}\right),$
where $(i)$ follows the definition of $\Phi(T,L)$ and $(ii)$ follows the fact
that $\lambda_{0,\max}>1$.
Before we calculate $P(\bar{A}_{2})$, we first compute $P(\bar{A}_{4})$. Note
that by Jensen’s inequality, for any two real values $z_{1},z_{2}$ and any
positive integer $k$, we have
$(z_{1}+z_{2})^{k}\leq\left(|z_{1}|+|z_{2}|\right)^{k}=2^{k}\left(\frac{1}{2}|z_{1}|+\frac{1}{2}|z_{2}|\right)^{k}\leq
2^{k-1}\left(|z_{1}|+|z_{2}|\right),$
where the last line is because Jensen’s inequality with convex function
$\varphi(t)=t^{k}$, $k$ is a positive integer. Since for any $i=1,\ldots,n$
and $j=1,2\dots,p$, we have $\mathbb{E}[\|g_{ij}\|^{2}]=\lambda_{j0}$. Then,
by Jensen’s inequality and Lemma 31, for any $k\geq 2$, we have
$\displaystyle\mathbb{E}\left[\left(\|g_{ij}\|^{2}-\lambda_{j0}\right)^{k}\right]$
$\displaystyle\leq
2^{k-1}\left(\mathbb{E}\left[\|g_{ij}\|^{2k}+\lambda^{k}_{j0}\right]\right)$
$\displaystyle\leq 2^{k-1}\left((2\lambda_{j0})^{k}k!+\lambda^{k}_{j0}\right)$
$\displaystyle\leq(4\lambda_{j0})^{k}k!,$
where the second inequality is because Lemma 31. Thus,
$\sum^{n}_{i=1}\mathbb{E}\left[\left(\|g_{ij}\|^{2}-\lambda_{j0}\right)^{k}\right]\leq\frac{k!}{2}n\times(32\lambda^{2}_{j0})\times(4\lambda_{j0})^{k-2}.$
Then by Lemma 29, for any $\epsilon>0$, we have
$P\left(\left|\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}-\lambda_{j0}\right|>\epsilon\right)\leq
2\exp\left(-\frac{n\epsilon^{2}}{64\lambda^{2}_{j0}+8\lambda_{j0}\epsilon}\right).$
This way, we further get
$\displaystyle
P\left(\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}>2\lambda_{0,\max}\right)$
$\displaystyle\leq
P\left(\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}>2\lambda_{j0}\right)$
$\displaystyle\leq
P\left(\left|\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}-\lambda_{j0}\right|>\lambda_{j0}\right)$
$\displaystyle\leq 2\exp\left(-\frac{n}{72}\right).$
Since the above inequality holds for any $j=1,\ldots,p$, we then have
$P(\bar{A}_{4})=P\left(\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}>2\lambda_{0,\max},\;\exists
j=1,\ldots,p\right)\leq 2p\exp\left(-\frac{n}{72}\right).$ (B.28)
For $P(\bar{A}_{2})$, we first let
$\hat{K}^{g}_{jj}(s,t)=\frac{1}{n}\sum^{n}_{i=1}g_{ij}(s)g_{ij}(t),$
for all $j\in V$ and $K_{jj}(s,t)=\mathbb{E}[g_{ij}(s)g_{ij}(t)]$, and also
let
$A^{\prime}_{2}:\;\lVert\hat{K}^{g}_{jj}-K^{g}_{jj}\rVert_{\text{HS}}\leq\delta_{2}\quad\forall
j=1,\ldots,p.$
Note that
$\displaystyle\|\hat{K}^{g}_{jj}(s,t)-K^{g}_{jj}(s,t)\|_{\text{HS}}$
$\displaystyle=\left\|\frac{1}{n}\sum^{n}_{i=1}\left[\hat{g}_{ij}(s)-g_{ij}(s)+g_{ij}(s)\right]\left[\hat{g}_{ij}(t)-g_{ij}(t)+g_{ij}(t)\right]-K^{g}_{jj}(s,t)\right\|_{\text{HS}}$
$\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\|\hat{g}_{ij}-g_{ij}\|^{2}+\frac{2}{n}\sum^{n}_{i=1}\|\hat{g}_{ij}-g_{ij}\|\cdot\|g_{ij}\|+\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(s)g_{ij}(t)-K^{g}_{jj}(s,t)\right]\right\|_{\text{HS}}.$
Let
$\displaystyle A_{6}$
$\displaystyle:\;\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(s)g_{ij}(t)-K^{g}_{jj}(s,t)\right]\right\|_{\text{HS}}\leq
4\lambda_{0,\max}\delta_{1},\;\forall j=1,\ldots,p.$
We claim that when $A_{1}\cap A_{4}\cap A_{6}\Rightarrow A^{\prime}_{2}$. To
prove it, note that by Jensen’s inequality, we have
$\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}},$
thus, when $A_{4}$ holds, we have
$(1/n)\sum^{n}_{i=1}\left\|g_{ij}\right\|\leq\sqrt{2\lambda_{0,\max}}$ for any
$j=1,\ldots,p$. This way, when $A_{1}$, $A_{4}$ and $A_{6}$ hold
simultaneously, we have
$\|\hat{K}^{g}_{jj}(s,t)-K^{g}_{jj}(s,t)\|_{\text{HS}}\leq\delta^{2}_{1}+2\sqrt{2\lambda_{0,\max}}\delta_{1}+4\lambda_{0,\max}\delta_{1}\leq
9\lambda_{0,\max}\delta_{1},$
which is $A_{2}$. This way, we have proved $A_{1}\cap A_{4}\cap
A_{6}\Rightarrow A^{\prime}_{2}$, which implies that
$\bar{A^{\prime}}_{2}\Rightarrow\bar{A}_{1}\cup\bar{A}_{4}\cup\bar{A}_{6}$,
and thus $P(\bar{A^{\prime}}_{2})\leq
P(\bar{A}_{1})+P(\bar{A}_{4})+P(\bar{A}_{6})$. $P(\bar{A}_{1})$ has been given
by (B.27) and $P(\bar{A}_{4})$ has been given by (B.28), thus we only need to
compute $P(\bar{A}_{6})$.
By Lemma 32, for any $j=1,\ldots,p$, we have
$P\left(\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(s)g_{ij}(t)-K^{g}(s,t)\right]\right\|_{\text{HS}}>4\lambda_{0,\max}\delta_{1}\right)\leq
2\exp\left(-\frac{n\delta^{2}_{1}}{6}\right),$
thus
$P(\bar{A}_{6})\leq
2p\exp\left(-\frac{n\delta^{2}_{1}}{6}\right)=2p\exp\left(-\frac{1}{373248d^{4}_{0}\lambda^{2}_{0,\max}}\times
n\frac{\delta^{2}}{M^{2+2\beta}}\right).$ (B.29)
This way, by combining (B.27), (B.28) and (B.29), we have
$\displaystyle P(\bar{A^{\prime}}_{2})\leq$ $\displaystyle
4pM\exp\left(-\frac{1}{1728\sqrt{6}\lambda_{0,\max}d^{2}_{0}}\cdot\frac{\delta}{M^{1+\beta}}\cdot\Phi(T,L)\right)+2p\exp\left(-\frac{n}{72}\right)$
$\displaystyle+$ $\displaystyle
2p\exp\left(-\frac{1}{373248d^{4}_{0}\lambda^{2}_{0,\max}}\times
n\frac{\delta^{2}}{M^{2+2\beta}}\right).$
Finally, since
$\|\hat{K}_{j}j(s,t)-K_{jj}(s,t)\|_{\text{HS}}\leq\|\hat{K}^{X}_{j}j(s,t)-K^{X}_{jj}(s,t)\|_{\text{HS}}+\|\hat{K}^{Y}_{j}j(s,t)-K^{Y}_{jj}(s,t)\|_{\text{HS}}$,
we have $P(\bar{A}_{2})\leq
P(\bar{A^{\prime}}_{X,2})+P(\bar{A^{\prime}}_{Y,2})$, where $A^{\prime}_{X,2}$
and $A^{\prime}_{Y,2}$ are defined similarly as $A^{\prime}_{2}$ with $g$ to
be $X$ and $Y$. Thus, we have
$\displaystyle P(\bar{A}_{2})\leq$ $\displaystyle
8pM\exp\left(-\frac{1}{1728\sqrt{6}\lambda_{0,\max}d^{2}_{0}}\cdot\frac{\delta}{M^{1+\beta}}\cdot\Phi(T,L)\right)+4p\exp\left(-\frac{n}{72}\right)$
$\displaystyle+$ $\displaystyle
4p\exp\left(-\frac{1}{373248d^{4}_{0}\lambda^{2}_{0,\max}}\times
n\frac{\delta^{2}}{M^{2+2\beta}}\right).$
For $P(\bar{A}_{3})$, by Page 28-29 of Boucheron et al. (2013), and note that
$\sum^{n}_{i=1}\xi^{2}_{ijk}\sim\chi^{2}_{n}$ for any $j=1,\ldots,p$ and
$k=1,\ldots,M$, we have that for any $\epsilon>0$, we have
$P\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}-1>\epsilon\right)\leq\exp\left(-\frac{n\epsilon^{2}}{4+4\epsilon}\right).$
Thus, by letting $\epsilon=1/2$, we have
$P\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}>\frac{3}{2}\right)\leq\exp\left(-\frac{n}{24}\right),$
which implies that
$P(\bar{A}_{3})\leq pM\exp\left(-\frac{n}{24}\right).$ (B.30)
Finally, for $P(\bar{A}_{5})$, we first claim that for any $\epsilon>0$ and
$1\leq j,l\leq p$, $1\leq k,m\leq M$, we have
$P\left(\left|\frac{1}{n}\sum^{n}_{i=1}a_{ijk}a_{ilm}-\sigma_{jl,km}\right|>\epsilon\right)\leq
2\exp\left(-\frac{n\epsilon^{2}}{64d^{2}_{0}+8d_{0}\epsilon}\right).$
We now prove this claim. Note that
$\displaystyle\mathbb{E}\left[\left(a_{ijk}a_{ilm}-\mathbb{E}(a_{ijk}a_{ilm})\right)^{k}\right]$
$\displaystyle=\lambda^{k/2}_{jk}\lambda^{k/2}_{lm}\mathbb{E}\left[\left(\xi_{ijk}\xi_{ilm}-\mathbb{E}(\xi_{ijk}\xi_{ilm})\right)^{k}\right]$
$\displaystyle\leq
d^{k}_{0}\mathbb{E}\left[\left(\xi_{ijk}\xi_{ilm}-\mathbb{E}(\xi_{ijk}\xi_{ilm})\right)^{k}\right],$
and
$\displaystyle\mathbb{E}\left[\left(\xi_{ijk}\xi_{ilm}-\mathbb{E}(\xi_{ijk}\xi_{ilm})\right)^{k}\right]$
$\displaystyle\leq
2^{k-1}\left(\mathbb{E}\left[|\xi_{ijk}\xi_{ilm}|^{k}\right]+|\mathbb{E}(\xi_{ijk}\xi_{ilm})|^{k}\right)$
$\displaystyle\leq 2^{k-1}\left(\mathbb{E}[\xi^{2k}_{ij1}]+1\right)$
$\displaystyle\leq 2^{k-1}(2^{k}k!+1)$ $\displaystyle\leq 4^{k}k!,$
thus
$\mathbb{E}\left[\left(a_{ijk}a_{ilm}-\mathbb{E}(a_{ijk}a_{ilm})\right)^{k}\right]\leq(4d_{0})^{k}k!.$
The claim then follows directly from Lemma 29. By letting
$\epsilon=\delta/16$,
$P\left(\left|\frac{1}{n}\sum^{n}_{i=1}a_{ijk}a_{ilm}-\sigma_{jl,km}\right|>\frac{\delta}{16}\right)\leq
2\exp\left(-\frac{n\delta^{2}}{16^{2}\times 64\times
d^{2}_{0}+128d_{0}\delta}\right)\leq
2\exp\left(-\frac{n\delta^{2}}{16512d^{2}_{0}}\right)$
holds for any $1\leq j,l\leq p$ and $1\leq k,m\leq M$, which further implies
that
$P\left(\bar{A}_{5}\right)\leq
2(pM)^{2}\exp\left(-\frac{n\delta^{2}}{16512d^{2}_{0}}\right).$ (B.31)
Let $C_{1}=12$, $C_{2}=1/(1728\sqrt{6}\lambda_{0,\max})$, $C_{3}=9$,
$C_{4}=1/(373248d^{4}_{0}\lambda^{2}_{0,\max})$, $C_{5}=2$, and
$C_{6}=1/(6228d^{4}_{0}\lambda_{0,\max})$, then the final result follows by
combining (B.27)-(B.31).
## Appendix C More Theorems
In this section, we introduce more theorems along with their proofs.
### C.1 Theorem 5 and Its Proof
In this section, we give a non-asymptotic error bound for our basis expansion
estimated function. This theorem is used in proving Theorem 4.
For a random function $g(t)$, where $t\in\mathcal{T}$, a closed interval of
real line, and lying in a separable Hilbert space $\mathbb{H}$, we have noisy
discrete observations at time points $t_{1},t_{2},\dots,t_{T}$ generated from
the model below:
$h_{k}=g(t_{k})+\epsilon_{k},$ (C.1)
where $\epsilon_{k}\overset{\text{i.i.d.}}{\sim}N(0,\sigma^{2}_{0})$ for
$k=1,\ldots,T$. Let $b(t)=(b_{1}(t),b_{2}(t),\dots,b_{L}(t))^{\top}$ be basis
function vector. We use basis expansion to get
$\hat{g}(t)=\hat{\beta}^{\top}b(t)$, the estimator of $g(t)$, where
$\hat{\beta}\in\mathbb{R}^{L}$ is obtained by minimizing the least square
loss:
$\hat{\beta}=\operatorname*{arg\,min}_{\beta\in\mathbb{R}^{L}}\sum^{T}_{k=1}\left(\beta^{\top}b(t_{k})-h_{k}\right)^{2}.$
(C.2)
We define the design matrix $B$ as
$B=\left[\begin{matrix}b_{1}(t_{1})&\cdots&b_{L}(t_{1})\\\
\vdots&\ddots&\vdots\\\
b_{1}(t_{T})&\cdots&b_{L}(t_{T})\end{matrix}\right]\in\mathbb{R}^{T\times L},$
(C.3)
so that
$\hat{\beta}=\left(B^{\top}B\right)^{-1}B^{\top}h,$ (C.4)
where $h=(h_{1},h_{2},\dots,h_{T})^{\top}\in\mathbb{R}^{T}$.
We assume that $g(t)=\sum^{\infty}_{m=1}\beta^{*}_{m}b_{m}(t)$, and we can
decompose $g(t)$ as $g=g^{\shortparallel}+g^{\bot}$, where
$g^{\shortparallel}\in{\rm Span}(b)$ and $g^{\bot}\in{\rm Span}(b)^{\bot}$.
Let $\lambda_{0}\coloneqq\mathbb{E}[\|g\|^{2}]$ and
$\lambda^{\bot}_{0}\coloneqq\mathbb{E}[\|g^{\bot}\|^{2}]$. Then it is easy to
check that $\lambda_{0}=\sum^{\infty}_{m=1}\mathbb{E}[(\beta^{*}_{m})^{2}]$
and $\lambda^{\bot}_{0}=\sum^{\infty}_{m>L}\mathbb{E}[(\beta^{*}_{m})^{2}]$.
We assume that the basis functions $\\{b_{l}(t)\\}^{\infty}_{l=1}$ compose a
complete orthonormal system (CONS) of $\mathbb{H}$, that is, $\overline{{\rm
Span}\left(\\{b_{l}\\}^{\infty}_{l=1}\right)}=\mathbb{H}$ (see Definition
2.4.11 of Hsing and Eubank (2015)), and have continuous derivative functions
with
$D_{0,b}\coloneqq\sup_{l\geq 1}\sup_{t\in\mathcal{T}}\lvert
b_{l}(t)\rvert<\infty,\qquad D_{1,b}(l)\coloneqq\sup_{t\in\mathcal{T}}\lvert
b^{\prime}_{l}(t)\rvert<\infty,\qquad D_{1,b,L}\coloneqq\max_{1\leq l\leq
L}D_{1,b}(l).$ (C.5)
We further assume that the observation time points $\\{t_{k}:1\leq k\leq T\\}$
satisfy
$\max_{1\leq k\leq
T+1}\left|\frac{t_{k}-t_{(k-1)}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\leq\frac{\zeta_{0}}{T^{2}},$
(C.6)
where $t_{0}$ and $t_{(T+1)}$ are endpoints of $\mathcal{T}$ and $\zeta_{0}$
is a positive constant. Besides, we assume that
$\sum^{\infty}_{m=1}\mathbb{E}\left[(\beta^{*}_{m})^{2}\right]D^{2}_{1,b}(m)<\infty$,
we then define
$\psi_{4}(L)=\sum_{m>L}\mathbb{E}\left[(\beta^{*}_{m})^{2}\right]D^{2}_{1,b}(m).$
Let
$\displaystyle\psi_{1}(T,L)$
$\displaystyle=\frac{\sigma_{0}L}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}},\qquad\psi_{3}(L)=\lambda^{\bot}_{0}/\lambda_{0},$
and
$\displaystyle\psi_{2}(T,L)$
$\displaystyle=\frac{1}{(\lambda^{B}_{\min})^{2}}\left(18\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]L^{2}\psi_{3}(L)\right.$
$\displaystyle\left.\qquad\qquad+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}L^{2}\psi_{4}(L)\right),$
We then have the following theorem.
###### Theorem 5
For any $\delta>0$, we have
$P\left(\lVert g-\hat{g}\rVert>\delta\right)\leq
2\exp\left(-\frac{\delta^{2}}{72\psi^{2}_{1}(T,L)+6\sqrt{2}\psi_{1}(T,L)\delta}\right)+L\exp\left(-\frac{\delta^{2}}{\psi_{2}(T,L)}\right)\\\
+2\exp\left(-\frac{\delta^{2}}{72\lambda_{0}\psi_{3}(L)+6\sqrt{2\lambda_{0}}\sqrt{\psi_{3}(L)}\delta}\right).$
(C.7)
Proof Throughout the proof, we often use the technique to first treat $g$ as
a fixed function, that is, we consider probability conditioned on $g$, so the
only randomness comes from $\epsilon_{k}$, $k=1,\ldots,T$. We will then
include the randomness from $g$. Note that since $\epsilon_{k}$ is independent
of $g$, thus the conditional distribution of $\epsilon_{k}$ is the same with
unconditional distribution.
For a fixed $g$, since $\overline{{\rm
Span}\left(\\{b_{l}\\}^{\infty}_{l=1}\right)}=\mathbb{H}$, we can assume that
$g(t)=\sum^{\infty}_{l=1}\beta^{*}_{l}b_{l}(t)$ where $\beta^{*}_{l}=\langle
g,b_{l}\rangle=\int_{\mathcal{T}}g(t)b_{l}(t)dt$. Let
$\beta^{*}=(\beta^{*}_{1},\cdots,\beta^{*}_{L})^{\top}\in\mathbb{R}^{L}$, we
then have
$g^{\shortparallel}(t)=(\beta^{*})^{\top}b(t)=\sum^{L}_{l=1}\beta^{*}_{l}b_{l}(t)$
and $g^{\bot}(t)=\sum_{l>L}\beta^{*}_{l}b_{l}(t)$. Thus, we have
$h_{k}=g(t_{k})+\epsilon_{k}=(\beta^{*})^{\top}b(t_{k})+g^{\bot}(t_{k})+\epsilon_{k}.$
Let
$h^{\bot}=\left(g^{\bot}(t_{1}),g^{\bot}(t_{2}),\dots,g^{\bot}(t_{T})\right)^{\top}$,
$\epsilon=\left(\epsilon_{1},\epsilon_{2},\dots,\epsilon_{T}\right)^{\top}$,
we then have
$h=B\beta^{*}+h^{\bot}+\epsilon.$
Thus,
$\mathbb{E}(\mathbb{\hat{\beta}})=\beta^{*}+\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot},$
and
$\displaystyle\hat{g}(t)-g(t)$
$\displaystyle=\hat{g}(t)-g^{\shortparallel}(t)-g^{\bot}(t)$
$\displaystyle=\hat{g}(t)-(\beta^{*})^{\top}b(t)-g^{\bot}(t)$
$\displaystyle=\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)^{\top}b(t)+\left(\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot}\right)^{\top}b(t)-g^{\bot}(t).$
By Lemma 26, we then have
$\displaystyle\lVert\hat{g}-g\rVert$
$\displaystyle\leq\lVert\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)^{\top}b(t)\rVert+\lVert\left(\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot}\right)^{\top}b(t)\rVert+\lVert
g^{\bot}\rVert$
$\displaystyle\leq\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert
b\rVert_{\mathcal{L}^{2},2}+\lvert\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot}\rvert_{2}\times\lVert
b\rVert_{\mathcal{L}^{2},2}+\lVert g^{\bot}\rVert$
$\displaystyle\leq\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert
b\rVert_{\mathcal{L}^{2},2}+\frac{1}{\lambda_{\min}(B^{\top}B)}\times\left\lvert
B^{\top}h^{\bot}\right\rvert_{2}\times\lVert
b\rVert_{\mathcal{L}^{2},2}+\lVert g^{\bot}\rVert.$
Let
$\displaystyle J_{1}$
$\displaystyle=\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert_{2}b\rVert_{\mathcal{L}^{2},2}$
(C.8) $\displaystyle J_{2}$
$\displaystyle=\frac{1}{\lambda_{\min}(B^{\top}B)}\times\lvert
B^{\top}h^{\bot}\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}$
$\displaystyle J_{3}$ $\displaystyle=\lVert g^{\bot}\rVert,$
where $\lvert\mathcal{T}\rvert$ denotes the length of the interval, then
$\lVert\hat{g}-g\rVert\leq J_{1}+J_{2}+J_{3}.$ (C.9)
Since this equation holds for any $g\in\mathbb{H}$, thus when we include the
randomness from $g$, the above equation holds with probability one. We then
bound $J_{1}$, $J_{2}$ and $J_{3}$ individually.
First, for $J_{1}$, recall that $\lVert b\rVert_{\mathcal{L}^{2},2}=\sqrt{L}$
and $\psi_{1}(T,L)=\sigma_{0}\lVert
b\rVert_{\mathcal{L}^{2},2}\sqrt{L}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}$,
then for any $\delta>0$, we claim that
$P\left(J_{1}>\delta\right)\leq
2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right).$
(C.10)
To prove this result, we first treat $g$ as fixed, then note that by standard
linear regression theory, we have
$\hat{\beta}\sim
N_{L}\left(\mathbb{E}(\hat{\beta}),\sigma^{2}_{0}\left(B^{\top}B\right)^{-1}\right).$
Thus,
$\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\sim
N_{L}\left(0,I_{L}\right)$
Since
$\displaystyle J_{1}$
$\displaystyle=\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert
b\rVert_{\mathcal{L}^{2},2}$
$\displaystyle=\lvert\left(B^{\top}B\right)^{-1/2}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}\times\lVert
b\rVert_{\mathcal{L}^{2},2}$
$\displaystyle\leq\frac{1}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\lvert\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}\times\lVert
b\rVert_{\mathcal{L}^{2},2}$ $\displaystyle=\frac{\sigma_{0}\lVert
b\rVert_{\mathcal{L}^{2},2}}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\lvert\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2},$
we have
$\displaystyle P(J_{1}>\delta)$ $\displaystyle\leq
P\left(\frac{\sigma_{0}\lVert
b\rVert_{\mathcal{L}^{2},2}}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\lvert\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}>\delta\right)$
$\displaystyle=P\left(\lvert\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}>\frac{\delta}{\sigma_{0}\lVert
b\rVert_{\mathcal{L}^{2},2}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\right)$
$\displaystyle\overset{(i)}{\leq}2\exp\left(-\frac{\left(\delta/\left(\sigma_{0}\lVert
b\rVert_{\mathcal{L}^{2},2}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}\right)\right)^{2}}{8L+2\sqrt{2}\left(\left(\delta/\left(\sigma_{0}\lVert
b\rVert_{\mathcal{L}^{2},2}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}\right)\right)\right)}\right)$
$\displaystyle=2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right),$
where $(i)$ follows Lemma 28. Now if we treat $g$ as random, we only need to
note that
$\displaystyle P\left(J_{1}>\delta\right)$
$\displaystyle=\mathbb{E}_{g}\left[P\left(J_{1}>\delta_{2}|g\right)\right]$
$\displaystyle=\mathbb{E}_{g}\left[2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right)\right]$
$\displaystyle=2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right).$
Next, for $J_{2}$, we claim that for any $\delta>0$, we have
$\mathbb{P}\left(J_{2}>\delta\right)\leq
L\exp\left(-\frac{9\delta^{2}}{\psi_{2}(T,L)}\right).$ (C.11)
We use $(B^{\top}h^{\bot})_{l}$ to denote the $l$-th element of vector
$B^{\top}h^{\bot}$, then we have
$(B^{\top}h^{\bot})_{l}=\sum^{T}_{k=1}b_{l}(t_{k})g^{\bot}(t_{k})=\sum_{m>L}\beta^{*}_{m}\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k}).$
Since $g$ is a Gaussian random function with mean zero, we then have
$(B^{\top}h^{\bot})_{l}$ to be a Gaussian random variable. Besides, we have
$\mathbb{E}\left[(B^{\top}h^{\bot})_{l}\right]=0$ and
$\mathbb{E}\left[(B^{\top}h^{\bot})^{2}_{l}\right]=\sum_{m>L}\mathbb{E}\left[\beta^{*2}_{m}\right]\left(\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right)^{2}$
(C.12)
By definition of $D_{0,b}$, $D_{1,b}(\cdot)$, for any $l<m$, we have that
$\sup_{t\in\mathcal{T}}(b_{l}(t)b_{m}(t))\leq D^{2}_{0,b}$, and
$\sup_{t\in\mathcal{T}}(b_{l}(t)b_{m}(t))^{\prime}=\sup_{t\in\mathcal{T}}\\{b^{\prime}_{l}(t)b_{m}(t)+b_{l}(t)b^{\prime}_{m}(t)\\}\leq
D_{0,b}(D_{1,b}(l)+D_{1,b}(m))$. Note that
$\int_{\mathcal{T}}b_{l}(t)b_{m}(t)dt=0$ for any $l<m$, then by Lemma 30, we
have
$\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right|$
$\displaystyle=$
$\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})-\frac{1}{|\mathcal{T}|}\int_{\mathcal{T}}b_{l}(t)b_{m}(t)dt\right|$
$\displaystyle\leq$
$\displaystyle\frac{D_{0,b}(D_{1,b}(l)+D_{1,b}(m))(\zeta_{0}+1)^{2}|\mathcal{T}|/2+D^{2}_{0,b}(2\zeta_{0}+1)}{T}$
for all $1\leq l<m<\infty$, which implies that
$\left|\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right|\leq\frac{1}{2}D_{0,b}(\zeta_{0}+1)^{2}|\mathcal{T}|(D_{1,b}(l)+D_{1,b}(m))+D^{2}_{0,b}(2\zeta_{0}+1).$
Then we have
$\displaystyle\left(\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right)^{2}$
$\displaystyle\leq\left(\frac{1}{2}D_{0,b}(\zeta_{0}+1)^{2}|\mathcal{T}|(D_{1,b}(l)+D_{1,b}(m))+D^{2}_{0,b}(2\zeta_{0}+1)\right)^{2}$
$\displaystyle\leq\frac{1}{2}D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}(D_{1,b}(l)+D_{1,b}(m))^{2}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}$
$\displaystyle\leq
D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}(D^{2}_{1,b}(l)+D^{2}_{1,b}(m))+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}.$
By (C.12), we then have
$\displaystyle\mathbb{E}\left[(B^{\top}h^{\bot})^{2}_{l}\right]$
$\displaystyle\leq\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b}(l)+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\sum_{m>L}\mathbb{E}\left[\beta^{*2}_{m}\right]$
$\displaystyle+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\sum_{m>L}\mathbb{E}\left[\beta^{*2}_{m}\right]D^{2}_{1,b}(m)$
$\displaystyle\leq\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b}(l)+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\lambda^{\bot}_{0}+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)$
$\displaystyle\leq\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\lambda^{\bot}_{0}+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)$
$\displaystyle=\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\psi_{3}(L)+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)$
Thus, by tail bound of Gaussian random variable (Section 2.1.2 of Wainwright
(2019)), we have
$\displaystyle\mathbb{P}\left((B^{\top}h^{\bot})_{l}>\delta\right)\leq$
$\displaystyle\exp\left(-\frac{\delta^{2}}{2\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\psi_{3}(L)+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)}\right).$
Recall that
$\displaystyle\psi_{2}(T,L)$
$\displaystyle=\frac{1}{(\lambda^{B}_{\min})^{2}}\left(18\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]L^{2}\psi_{3}(L)\right.$
$\displaystyle\left.\qquad\qquad+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}L^{2}\psi_{4}(L)\right),$
and note that $\|b\|_{\mathcal{L}^{2},2}=\sqrt{L}$, then we have
$\displaystyle\mathbb{P}\left(J_{2}>\delta\right)$
$\displaystyle\leq\mathbb{P}\left(\lvert
B^{\top}h^{\bot}\rvert_{2}>\frac{\lambda^{B}_{\min}\delta}{\sqrt{L}}\right)\leq\mathbb{P}\left(\max_{1\leq
l\leq L}(B^{\top}h^{\bot})_{l}>\frac{\lambda^{B}_{\min}\delta}{L}\right)$
(C.13) $\displaystyle\leq
L\exp\left(-\frac{9\delta^{2}}{\psi_{2}(T,L)}\right).$
Finally, for $J_{3}$, by Lemma 31 and definition of $\psi_{3}(L)$, we have
$\mathbb{E}\left[\|g^{\bot}\|^{2k}\right]\leq(2\lambda_{0}\psi_{3}(L))^{k}k!.$
This way, by Jensesn’s inequality, we have
$\mathbb{E}\left[\|g^{\bot}\|^{k}\right]=\mathbb{E}\left[\sqrt{\|g^{\bot}\|^{2k}}\right]\leq\sqrt{\mathbb{E}\left[\|g^{\bot}\|^{2k}\right]}\leq\left(\sqrt{2\lambda_{0}\psi_{3}(L)}\right)^{k}k!.$
Thus, by Lemma 29, we have
$P\left(J_{3}>\delta\right)=P\left(\|g^{\bot}\|>\delta\right)\leq
2\exp\left(-\frac{\delta^{2}}{8\lambda_{0}\psi_{3}(L)+2\sqrt{2\lambda_{0}}\sqrt{\psi_{3}(L)}\delta}\right).$
(C.14)
The final result then follows (C.10), (C.13) and (C.14), and the fact that
$\mathbb{P}\left(J_{1}+J_{2}+J_{3}>\delta\right)\leq\mathbb{P}\left(J_{1}>\delta/3\right)+\mathbb{P}\left(J_{2}>\delta/3\right)+\mathbb{P}\left(J_{3}>\delta/3\right).$
[2mm]
## Appendix D Lemmas and their proofs
In this section, we introduce some useful lemmas along with their proofs.
###### Lemma 5
Let $\sigma_{\max}=\max\\{|\Sigma^{X,M}|_{\infty},\
|\Sigma^{Y,M}|_{\infty}\\}$. Suppose that
$|S^{X,M}-\Sigma^{X,M}|_{\infty}\leq\delta,\qquad|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\leq\delta,$
(D.1)
for some $\delta\geq 0$. Then
$\displaystyle|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})|_{\infty}\leq\delta^{2}+2\delta\sigma_{\max},$
(D.2) and
$\displaystyle|\operatorname{vec}{(S^{Y,M}-S^{X,M})}-\operatorname{vec}{(\Sigma^{Y,M}-\Sigma^{X,M})}|_{\infty}\leq
2\delta.$ (D.3)
Proof Note that for any $(j,l),(j^{\prime},l^{\prime})\in V^{2}$ and $1\leq
k,k^{\prime},m,m^{\prime}\leq M$, by (D.1), we have
$\displaystyle\left|S^{X,M}_{jl,km}S^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}-\Sigma^{X,M}_{jl,km}\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|$
$\displaystyle\leq\left|S^{X,M}_{jl,km}-\Sigma^{X,M}_{jl,km}\right|\cdot\left|S^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}-\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|+\left|\Sigma^{X,M}_{jl,km}\right|\cdot\left|S^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}-\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|$
$\displaystyle\quad+\left|\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|\cdot\left|S^{X,M}_{jl,km}-\Sigma^{X,M}_{jl,km}\right|$
$\displaystyle\leq\left|S^{X,M}-\Sigma^{X,M}\right|_{\infty}\left|S^{Y,M}-\Sigma^{Y,M}\right|_{\infty}+\sigma_{\max}\left|S^{Y,M}-\Sigma^{Y,M}\right|_{\infty}+\sigma_{\max}\left|S^{X,M}-\Sigma^{X,M}\right|_{\infty}$
$\displaystyle\leq\delta^{2}+2\delta\sigma_{\max}.$
For (D.3), note that
$\displaystyle\left|\operatorname{vec}{(S^{Y,M}-S^{X,M})}-\operatorname{vec}{(\Sigma^{Y,M}-\Sigma^{X,M})}\right|_{\infty}$
$\displaystyle=\left|(S^{X,M}-\Sigma^{X,M})-(S^{Y,M}-\Sigma^{Y,M})\right|_{\infty}$
$\displaystyle\leq|S^{X,M}-\Sigma^{X,M}|_{\infty}+|S^{Y,M}-\Sigma^{Y,M}|_{\infty}$
$\displaystyle\leq 2\delta.$
[2mm]
###### Lemma 6
For $Z^{(1)},Z^{(2)},A^{(1)},A^{(2)}\in\mathbb{R}^{M\times M}$. Denote the
solution of
$\operatorname*{arg\,min}_{\\{Z^{(1)},Z^{(2)}\\}}\;\frac{1}{2}\sum^{2}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+\lambda\|Z^{(1)}-Z^{(2)}\|_{\text{F}}$
(D.4)
as $\\{\hat{Z}^{(1)},\hat{Z}^{(2)}\\}$, where $\lambda>0$ is a constant. Then
when $\|A^{(1)}-A^{(2)}\|_{\text{F}}\leq 2\lambda$, we have
$\hat{Z}^{(1)}=\hat{Z}^{(2)}=\frac{1}{2}\left(A^{(1)}+A^{(2)}\right),$ (D.5)
and when $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$, we have
$\displaystyle\hat{Z}^{(1)}=A^{(1)}-\frac{\lambda}{\|A^{(1)}-A^{(2)}\|_{\text{F}}}\left(A^{(1)}-A^{(2)}\right)$
(D.6)
$\displaystyle\hat{Z}^{(2)}=A^{(2)}+\frac{\lambda}{\|A^{(1)}-A^{(2)}\|_{\text{F}}}\left(A^{(1)}-A^{(2)}\right).$
Proof The subdifferential of the objective function in (D.4) is
$G^{(1)}(Z^{(1)},Z^{(2)})\coloneq\partial_{Z^{(1)}}=Z^{(1)}-A^{(1)}+\lambda
T(Z^{(1)},Z^{(2)}),$ (D.7)
$G^{(2)}(Z^{(1)},Z^{(2)})\coloneq\partial_{Z^{(2)}}=Z^{(2)}-A^{(2)}-\lambda
T(Z^{(1)},Z^{(2)}),$ (D.8)
where
$T(Z^{(1)},Z^{(2)})=\left\\{\begin{aligned}
&\frac{Z^{(1)}-Z^{(2)}}{\|Z^{(1)}-Z^{(2)}\|_{\text{F}}}\quad\text{if}\;Z^{(1)}\neq
Z^{(2)}\\\ &\left\\{T\in\mathbb{R}^{M\times M}:\|T\|_{\text{F}}\leq
1\right\\}\quad\text{if}\;Z^{(1)}=Z^{(2)}\end{aligned}\right..$ (D.9)
The optimal condition is:
$0\in G^{(q)}(Z^{(1)},Z^{(2)})\quad q=1,2.$ (D.10)
Claim $\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$ if and only if
$\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$.
We first prove the necessaity, that is, when $\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$,
we prove that $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$. By (D.7)-(D.10), we
have
$\hat{Z}^{(1)}-\hat{Z}^{(2)}-\left(A^{(1)}-A^{(2)}\right)-2\lambda\frac{\hat{Z}^{(1)}-\hat{Z}^{(2)}}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}=0,$
which implies that
$\|A^{(1)}-A^{(2)}\|_{\text{F}}=2\lambda+\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}>2\lambda.$
We then prove the sufficiency, that is, when
$\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$, we prove
$\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$. Note that by (D.7)-(D.10), we have
$\hat{Z}^{(1)}+\hat{Z}^{(2)}=A^{(1)}+A^{(2)}.$
If $\hat{Z}^{(1)}=\hat{Z}^{(2)}$, we then have
$\hat{Z}^{(1)}=\hat{Z}^{(2)}=\frac{A^{(1)}+A^{(2)}}{2}.$
By (D.7) and (D.10), we have
$\|\hat{Z}^{(1)}-A^{(1)}\|_{\text{F}}=\frac{1}{2}\|A^{(1)}-A^{(2)}\|_{\text{F}}=\lambda\|T(\hat{Z}^{(1)},\hat{Z}^{(2)})\|_{\text{F}}\leq\lambda,$
which implies that
$\|A^{(1)}-A^{(2)}\|_{\text{F}}\leq 2\lambda,$
and this contradicts the assumption that
$\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$. Thus, we must have
$\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$.
Note that by this claim and the argument proving this claim, we have already
proved (D.5). We then prove (D.6). When
$\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$, by the claim above, we must have
$\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$. Then by (D.7)-(D.10), we have
$\hat{Z}^{(1)}-A^{(1)}+\frac{\lambda}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}\left(\hat{Z}^{(1)}-\hat{Z}^{(2)}\right)=0,$
(D.11)
$\hat{Z}^{(2)}-A^{(2)}-\frac{\lambda}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}\left(\hat{Z}^{(1)}-\hat{Z}^{(2)}\right)=0.$
(D.12)
(D.11) and (D.12) implies that
$\hat{Z}^{(1)}-\hat{Z}^{(2)}-\left(A^{(1)}-A^{(2)}\right)+\frac{2\lambda}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}\left(\hat{Z}^{(1)}-\hat{Z}^{(2)}\right)=0,$
which implies that
$\hat{Z}^{(1)}-\hat{Z}^{(2)}=\alpha\cdot\left(A^{(1)}-A^{(2)}\right),$ (D.13)
where $\alpha$ is a constant. We then substitue (D.13) back to (D.11) and
(D.12), we then have (D.6). [2mm]
###### Lemma 7
For a set of indices $\mathcal{G}=\\{G_{t}\\}_{t=1,\ldots,N_{\mathcal{G}}}$,
suppose $|\cdot|_{1,2}$ is defined in (B.3). Then for any matrix
$A\in{\mathbb{R}^{p^{2}M^{2}\times{p^{2}M^{2}}}}$ and
$\theta\in{\mathbb{R}^{p^{2}M^{2}}}$
$|\theta^{\top}A\theta|\leq{M^{2}|A|_{\infty}|\theta|^{2}_{1,2}}.$ (D.14)
Proof By direct calculation, we have
$\displaystyle|\theta^{\top}A\theta|$
$\displaystyle=\left|\sum_{i}\sum_{j}A_{ij}\theta_{i}\theta_{j}\right|$
$\displaystyle\leq{\sum_{i}\sum_{j}|A_{ij}\theta_{i}\theta_{j}|}$
$\displaystyle\leq{|A|_{\infty}\left(\sum_{i}|\theta_{i}|\right)^{2}}$
$\displaystyle=|A|_{\infty}\left(\sum_{t=1}^{N_{\mathcal{G}}}\sum_{k\in{G_{t}}}|\theta_{k}|\right)^{2}$
$\displaystyle=|A|_{\infty}\left(\sum_{t=1}^{N_{\mathcal{G}}}|\theta_{G_{t}}|_{1}\right)^{2}$
$\displaystyle\leq{|A|_{\infty}\left(\sum_{t=1}^{N_{\mathcal{G}}}M|\theta_{G_{t}}|_{2}\right)^{2}}$
$\displaystyle=M^{2}|A|_{\infty}|\theta|^{2}_{1,2},$
where in the penultimate line, we use that for any vector
$v\in{\mathbb{R}^{n}}$, $|v|_{1}\leq{\sqrt{n}|v|_{2}}$. [2mm]
###### Lemma 8
Suppose $\mathcal{M}$ is defined as in (B.1). For any
$\theta\in{\mathcal{M}}$, we have $|\theta|_{1,2}\leq{\sqrt{s}}|\theta|_{2}$.
Furthermore, for $\Psi(\mathcal{M})$ as defined in (B.5), we have
$\Psi(\mathcal{M})=\sqrt{s}$.
Proof By definition of $\mathcal{M}$ and $|\cdot|_{1,2}$, we have
$\displaystyle|\theta|_{1,2}$
$\displaystyle=\sum_{t\in{S_{\mathcal{G}}}}|\theta_{G_{t}}|_{2}+\sum_{t\notin{S_{\mathcal{G}}}}|\theta_{G_{t}}|_{2}$
$\displaystyle=\sum_{t\in{S_{\mathcal{G}}}}|\theta_{G_{t}}|_{2}$
$\displaystyle\leq{\sqrt{s}}\left(\sum_{t\in{S_{\mathcal{G}}}}|\theta_{G_{t}}|^{2}_{2}\right)^{\frac{1}{2}}$
$\displaystyle=\sqrt{s}|\theta|_{2}.$
In the penultimate line, we appeal to the Cauchy-Schwartz inequality. To show
$\Psi(\mathcal{M})=\sqrt{s}$, it suffices to show that the upper bound above
can be achieved. Select $\theta\in{\mathbb{R}^{p^{2}M^{2}}}$ such that
$|\theta_{G_{t}}|_{2}=c$, $\forall{t\in{S_{\mathcal{G}}}}$, where $c$ is some
positive constant. This implies that $|\theta|_{1,2}=sc$ and
$|\theta|_{2}=\sqrt{s}c$ so that $|\theta|_{1,2}=\sqrt{s}|\theta|_{2}$. Thus,
$\Psi(\mathcal{M})=\sqrt{s}$. [2mm]
###### Lemma 9
For $\mathcal{R}(\cdot)$ norm defined in (B.3), its dual norm
$\mathcal{R}^{*}(\cdot)$, defined in (B.4), is
$\mathcal{R}^{*}(v)\;=\;\max_{t=1,\ldots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}.$
(D.15)
Proof For any $u:|u|_{1,2}\leq{1}$ and $v\in{\mathbb{R}^{p^{2}M^{2}}}$, we
have
$\displaystyle\langle{v,u}\rangle$
$\displaystyle=\sum_{t=1}^{N_{\mathcal{G}}}\langle{v_{G_{t}},u_{G_{t}}}\rangle$
$\displaystyle\leq{\sum_{t=1}^{N_{\mathcal{G}}}|v_{G_{t}}|_{2}|u_{G_{t}}|_{2}}$
$\displaystyle\leq\left(\max_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}\right)\sum_{t=1}^{N_{\mathcal{G}}}|u_{G_{t}}|_{2}$
$\displaystyle=\left(\max_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}\right)|u|_{1,2}$
$\displaystyle\leq{\max_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}}.$
To complete the proof, we to show that this upper bound can be obtained. Let
$t^{*}=\operatorname*{arg\,max}_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|$,
and select $u$ such that
$\displaystyle u_{G_{t}}$ $\displaystyle=0$
$\displaystyle\qquad{\forall{t\neq{t^{*}}}},$ $\displaystyle u_{G_{t}}$
$\displaystyle=\frac{v_{G_{t^{*}}}}{|v_{G_{t^{*}}}|_{2}}$
$\displaystyle\qquad{t={t^{*}}}.$
It follows that $|u|_{1,2}=1$ and
$\langle{v,u}\rangle=|v_{G_{t^{*}}}|_{2}=\max_{t=1,\ldots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}$.
[2mm]
###### Lemma 10
Given that $A1$-$A5$ hold, we have $\lvert I_{1}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof This directly follows the assumption that $A_{5}$ holds. [2mm]
###### Lemma 11
Given that $A1$-$A5$ hold, we have $\lvert I_{2}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{2}\rvert$
$\displaystyle=\lvert\langle\frac{1}{n}\sum^{n}_{i=1}a_{ijk}(\hat{g}_{il}-g_{il}),\phi_{lm}\rangle\rvert$
$\displaystyle\leq\lVert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}(\hat{g}_{il}-g_{il})\rVert$
$\displaystyle\overset{(i)}{\leq}\sqrt{\frac{1}{n}\sum^{n}_{i=1}a^{2}_{ijk}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}$
$\displaystyle\overset{(ii)}{\leq}\delta_{1}\sqrt{\frac{1}{n}\sum^{n}_{i=1}a^{2}_{ijk}}$
$\displaystyle=\delta_{1}\lambda^{1/2}_{jk}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}}$
$\displaystyle\overset{(iii)}{\leq}\sqrt{\frac{3}{2}}\delta_{1}\lambda^{1/2}_{jk}$
$\displaystyle\leq\sqrt{\frac{3}{2}}\sqrt{d_{1}}\delta_{1}k^{-\beta/2}$
$\displaystyle\leq\sqrt{\frac{3}{2}}\sqrt{d_{1}}\delta_{1},$
where $(i)$ follows Lemma 26, $(ii)$ follows $A_{1}$, $(iii)$ follows $A_{3}$.
Note the definition of $d_{0}$, we thus have
$\lvert I_{2}\rvert\leq\sqrt{\frac{3}{2}}d_{0}\delta_{1}.$ (D.16)
Since
$\delta_{1}=\delta/\left(144d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\right)\leq\delta/(8\sqrt{6}d_{0}),$
(D.17)
we have
$\sqrt{\frac{3}{2}}d_{0}\delta_{1}\leq\sqrt{\frac{3}{2}}d_{0}\cdot\frac{\delta}{8\sqrt{6}d_{0}}=\frac{\delta}{16}.$
(D.18)
Thus,
$\lvert I_{2}\rvert\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 12
Given that $A1$-$A5$ hold, we have $\lvert I_{3}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{3}\rvert$
$\displaystyle=\lvert\langle\frac{1}{n}\sum^{n}_{i=1}a_{ijk}g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\lVert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}g_{il}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle=\lambda^{1/2}_{jk}\lVert\frac{1}{n}\sum^{n}_{i=1}\xi_{ijk}g_{il}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(i)}{\leq}\lambda^{1/2}_{jk}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert
g_{il}\rVert^{2}\right)^{1/2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(ii)}{\leq}\lambda^{1/2}_{jk}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert
g_{il}\rVert^{2}\right)^{1/2}d_{lm}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}},$
where $(i)$ follows Lemma 26, and $(ii)$ follows Lemma 27. Note that
$\lambda^{1/2}_{jk}\leq\sqrt{d_{1}}k^{-\beta/2}$, $d_{lm}\leq
d_{2}m^{1+\beta}$ and $A_{2}$-$A_{4}$ hold, thus we have
$\displaystyle\lvert I_{3}\rvert$
$\displaystyle\leq\sqrt{d_{1}}d_{2}k^{-\beta/2}m^{1+\beta}\sqrt{\frac{3}{2}}\sqrt{2\lambda_{0,\max}}\delta_{2}$
(D.19) $\displaystyle\leq
d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}.$
By definition of $\delta_{2}$, we have
$d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}\leq
d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}=\frac{\delta}{16}.$
(D.20)
Thus,
$\lvert I_{3}\rvert\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 13
Given that $A1$-$A5$ hold, we have $\lvert I_{4}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{4}\rvert$
$\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\frac{1}{n}\lVert\sum^{n}_{i=1}a_{ijk}\left(\hat{g}_{il}-g_{il}\right)\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle=\leq\lambda^{1/2}_{jk}\frac{1}{n}\lVert\sum^{n}_{i=1}\xi_{ijk}\left(\hat{g}_{il}-g_{il}\right)\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(i)}{\leq}\lambda^{1/2}_{jk}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}\right)^{1/2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(ii)}{\leq}\lambda^{1/2}_{jk}d_{lm}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}\right)^{1/2}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}},$
where $(i)$ follows Lemma 26, and $(ii)$ follows Lemma 27. Note that
$\lambda^{1/2}_{jk}\leq\sqrt{d_{1}}k^{-\beta/2}$, $d_{lm}\leq
d_{2}m^{1+\beta}$ and $A_{1}$-$A_{3}$ hold, thus we have
$\displaystyle\lvert I_{4}\rvert$
$\displaystyle\leq\sqrt{\frac{3}{2}}\sqrt{d_{1}}d_{2}k^{-\beta/2}m^{1+\beta}\delta_{1}\delta_{2}$
$\displaystyle\leq\sqrt{\frac{3}{2}}d^{2}_{0}M^{1+\beta}\delta_{1}\delta_{2}$
$\displaystyle\overset{(iii)}{\leq}\frac{\delta}{16}\times\frac{\sqrt{\frac{3}{2}}d^{2}_{0}M^{1+\beta}\delta_{1}\delta_{2}}{\sqrt{\frac{3}{2}}d_{0}\delta_{1}}$
$\displaystyle\leq\frac{\delta}{16}\times d_{0}M^{1+\beta}\delta_{2}$
$\displaystyle\leq\frac{\delta}{16}\times
d_{0}M^{1+\beta}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}$
$\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$
$\displaystyle\leq\frac{\delta}{16},$
where $(iii)$ follows (D.18). [2mm]
###### Lemma 14
Given that $A1$-$A5$ hold, we have $\lvert I_{5}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof This proof is similar to the proof of Lemma 11, thus is omitted. [2mm]
###### Lemma 15
Given that $A1$-$A5$ hold, we have $\lvert I_{6}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{6}\rvert$
$\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle\rvert$
$\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle\rvert$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle\rvert^{2}}$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}.$
By the assumption that $A_{1}$ holds, we thus have
$\lvert I_{6}\rvert\leq\delta^{2}_{1}.$ (D.21)
By (D.17),(D.18) and Lemma 11, we have
$\displaystyle\delta^{2}_{1}$
$\displaystyle\leq\frac{\delta}{16}\times\frac{\delta^{2}_{1}}{\sqrt{\frac{3}{2}}d_{0}\delta_{1}}$
(D.22)
$\displaystyle=\frac{\delta}{16}\times\frac{\delta_{1}}{\sqrt{\frac{3}{2}}d_{0}}$
(D.23)
$\displaystyle\leq\frac{\delta}{16}\times\frac{1}{\sqrt{\frac{3}{2}}d_{0}}\times\frac{\delta}{8\sqrt{6}d_{0}}$
(D.24) $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{24d^{2}_{0}}$
(D.25) $\displaystyle\leq\frac{\delta}{16},$ (D.26)
and thus
$\lvert I_{6}\rvert\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 16
Given that $A1$-$A5$ hold, we have $\lvert I_{7}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{7}\rvert$
$\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert
g_{il}\rVert^{2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert^{2}}$
$\displaystyle\overset{(i)}{\leq}\delta_{1}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert
g_{il}\rVert^{2}}$
$\displaystyle\overset{(ii)}{\leq}\delta_{1}\sqrt{2\lambda_{0,\max}}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(iii)}{\leq}\delta_{1}\sqrt{2\lambda_{0,\max}}d_{lm}\|\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$
$\displaystyle\overset{(iv)}{\leq}\delta_{1}\delta_{2}\sqrt{2\lambda_{0,\max}}d_{lm}$
$\displaystyle\leq\delta_{1}\delta_{2}\sqrt{2\lambda_{0,\max}}d_{2}m^{1+\beta}$
$\displaystyle\leq
d_{0}\sqrt{2\lambda_{0,\max}}M^{1+\beta}\delta_{1}\delta_{2},$
where $(i)$ follows the assumption that $A_{1}$ holds, $(ii)$ follows the
assumption that $A_{4}$ holds, $(iii)$ follows Lemma 27, and $(iv)$ follows
the assumption that $A_{2}$ holds. By (D.17) and (D.20), we have
$\displaystyle\lvert I_{7}\rvert$
$\displaystyle\leq\frac{\delta}{16}\times\frac{d_{0}\sqrt{2\lambda_{0,\max}}M^{1+\beta}\delta_{1}\delta_{2}}{d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}}$
$\displaystyle=\frac{\delta}{16}\times\sqrt{\frac{2}{3}}\times\frac{\delta_{1}}{d_{0}}$
$\displaystyle\leq\frac{\delta}{16}\times\sqrt{\frac{2}{3}}\times\frac{\delta}{8\sqrt{6}d^{2}_{0}}$
$\displaystyle=\frac{\delta}{16}\times\frac{\delta}{24\delta^{2}_{0}}$
$\displaystyle\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 17
Given that $A1$-$A5$ hold, we have $\lvert I_{8}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{8}\rvert$
$\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert^{2}}$
$\displaystyle\overset{(i)}{\leq}\delta^{2}_{1}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(ii)}{\leq}\delta^{2}_{1}d_{lm}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$
$\displaystyle\leq\delta^{2}_{1}d_{2}m^{1+\beta}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$
$\displaystyle\leq\delta^{2}_{1}d_{0}M^{1+\beta}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$
$\displaystyle\overset{(iii)}{\leq}d_{0}M^{1+\beta}\delta^{2}_{1}\delta_{2}$
where $(i)$ follows the assumption that $A_{1}$ holds, $(ii)$ follows the
assumption that Lemma 27 holds, and $(iii)$ follows the assumption that
$A_{2}$ holds. By (D.22), we have
$\displaystyle\lvert I_{8}\rvert$
$\displaystyle\leq\frac{\delta}{16}\times\frac{d_{0}M^{1+\beta}\delta^{2}_{1}\delta_{2}}{\delta^{2}_{1}}$
$\displaystyle=\frac{\delta}{16}\times d_{0}M^{1+\beta}\delta_{2}$
$\displaystyle=\frac{\delta}{16}\times
d_{0}M^{1+\beta}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}$
$\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$
$\displaystyle\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 18
Given that $A1$-$A5$ hold, we have $\lvert I_{9}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof This proof is similar to the proof of Lemma 12, thus is omitted. [2mm]
###### Lemma 19
Given that $A1$-$A5$ hold, we have $\lvert I_{10}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof This proof is similar to the proof of Lemma 16, thus is omitted. [2mm]
###### Lemma 20
Given that $A1$-$A5$ hold, we have $\lvert I_{11}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{11}\rvert$
$\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert\lvert\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle
g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert
g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert
g_{il}\rVert^{2}}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(i)}{\leq}2\lambda_{0,\max}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(ii)}{\leq}2\lambda_{0,\max}\delta^{2}_{2}d_{jk}d_{lm}$
$\displaystyle\leq
2\lambda_{0,\max}\delta^{2}_{2}d^{2}_{2}k^{1+\beta}m^{1+\beta},$
where $(i)$ follows because assumption $A_{4}$ holds, $(ii)$ follows Lemma 27.
Then, we have
$\lvert I_{11}\rvert\leq
2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}.$ (D.27)
Thus, by (D.20), we have
$\displaystyle 2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}$
$\displaystyle\leq\frac{\delta}{16}\times\frac{2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}}{d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}}$
(D.28)
$\displaystyle=\frac{\delta}{16}\times\frac{2}{\sqrt{3}}M^{1+\beta}\sqrt{\lambda_{0,\max}}\delta_{2}$
(D.29)
$\displaystyle=\frac{\delta}{16}\times\frac{2}{\sqrt{3}}M^{1+\beta}\sqrt{\lambda_{0,\max}}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}$
(D.30) $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{24d^{2}_{0}}$
(D.31) $\displaystyle\leq\frac{\delta}{16},$ (D.32)
which implies that
$\lvert I_{11}\rvert\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 21
Given that $A1$-$A5$ hold, we have $\lvert I_{12}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{12}\rvert$
$\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle
g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert
g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(i)}{\leq}\sqrt{2\lambda_{0,\max}}\delta_{1}\delta^{2}_{2}d_{jk}d_{lm}$
$\displaystyle\leq
d^{2}_{2}\sqrt{2\lambda_{0,\max}}k^{1+\beta}m^{1+\beta}\delta_{1}\delta^{2}_{2},$
where $(i)$ follows the assumption that $A_{1}$-$A_{3}$ hold along with Lemma
27. Then, we have
$\lvert I_{12}\rvert\leq
d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}.$ (D.33)
By (D.17) and (D.28), we have
$\displaystyle
d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}$
$\displaystyle\leq\frac{\delta}{16}\times\frac{d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}}{2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}}$
(D.34)
$\displaystyle=\frac{\delta}{16}\times\frac{\delta_{1}}{\sqrt{2\lambda_{0,\max}}}$
(D.35)
$\displaystyle\leq\frac{\delta}{16}\times\frac{1}{\sqrt{2\lambda_{0,\max}}}\times\frac{\delta}{8\sqrt{6}d_{0}}$
(D.36)
$\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$
(D.37) $\displaystyle\leq\frac{\delta}{16},$ (D.38)
which implies that
$I_{12}\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 22
Given that $A1$-$A5$ hold, we have $\lvert I_{13}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof This proof is similar to the proof of Lemma 13, thus is omitted. [2mm]
###### Lemma 23
Given that $A1$-$A5$ hold, we have $\lvert I_{14}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof This proof is similar to the proof of Lemma 17, thus is omitted. [2mm]
###### Lemma 24
Given that $A1$-$A5$ hold, we have $\lvert I_{15}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof This proof is similar to the proof of Lemma 11, thus is omitted. [2mm]
###### Lemma 25
Given that $A1$-$A5$ hold, we have $\lvert I_{16}\rvert\leq\delta/16$ for all
$1\leq j,l\leq p$, $\ 1\leq k,m\leq M$.
Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$
hold, we then have
$\displaystyle\lvert I_{16}\rvert$
$\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$
$\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$
$\displaystyle\overset{(i)}{\leq}\delta^{2}_{1}d_{jk}d_{lm}\delta^{2}_{2}$
$\displaystyle\leq
d^{2}_{2}k^{1+\beta}m^{1+\beta}\delta^{2}_{1}\delta^{2}_{2}$
$\displaystyle\leq d^{2}_{0}M^{2+2\beta}\delta^{2}_{1}\delta^{2}_{2},$
where $(i)$ follows the assumption that $A_{1}$, $A_{2}$ hold along with Lemma
27. Thus, by (D.18) and (D.34), we have
$\displaystyle\lvert I_{16}\rvert$
$\displaystyle\leq\frac{\delta}{16}\times\frac{d^{2}_{0}M^{2+2\beta}\delta^{2}_{1}\delta^{2}_{2}}{d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}}$
$\displaystyle=\frac{\delta}{16}\times\frac{\delta_{1}}{\sqrt{2\lambda_{0,\max}}}$
$\displaystyle\leq\frac{\delta}{16}\times\frac{1}{\sqrt{2\lambda_{0,\max}}}\times\frac{\delta}{8\sqrt{6}d_{0}}$
$\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$
$\displaystyle\leq\frac{\delta}{16}.$
[2mm]
###### Lemma 26
Suppose $f_{1},f_{2},\dots,f_{n}\in\mathbb{H}$ and
$v_{1},v_{2},\dots,v_{n}\in\mathbb{R}$, we have
$\lVert\sum^{n}_{i=1}v_{i}f_{i}\rVert\leq\sqrt{\sum^{n}_{i=1}v^{2}_{i}}\sqrt{\sum^{n}_{i=1}\lVert
f_{i}\rVert^{2}}$
Proof Note that
$\displaystyle\lVert\sum^{n}_{i=1}v_{i}f_{i}\rVert^{2}$
$\displaystyle=\int\left(\sum^{n}_{i=1}v_{i}f_{i}(t)\right)^{2}dt$
$\displaystyle\overset{(i)}{\leq}\int\left(\sum^{n}_{i=1}v^{2}_{i}\right)\left(\sum^{n}_{i=1}f^{2}_{i}(t)\right)dt$
$\displaystyle=\left(\sum^{n}_{i=1}v^{2}_{i}\right)\left(\sum^{n}_{i=1}\lVert
f_{i}\rVert^{2}\right),$
where $(i)$ follows Cauchy-Schwartz inequality, which directly implies the
result. [2mm]
###### Lemma 27
Suppose that Assumption 3 holds. Denote
$\tilde{\phi}_{jk}=\text{sgn}\left(\langle\hat{\phi}_{jk},\phi_{jk}\rangle\right)\phi_{jk}$,
where $\text{sgn}(t)=1$ if $t\geq 0$ and $\text{sgn}(t)=-1$ if $t<0$. Then we
have
$\lVert\hat{\phi}_{jk}-\tilde{\phi}_{jk}\rVert\leq
d_{jk}\lVert\hat{K}_{jj}-K_{jj}\rVert_{\text{HS}},$
where
$d_{jk}=2\sqrt{2}\max\\{(\lambda_{j(k-1)}-\lambda_{jk})^{-1},(\lambda_{jk}-\lambda_{j(k+1)})^{-1}\\}$
if $k\geq 2$ and $d_{j1}=2\sqrt{2}(\lambda_{j1}-\lambda_{j2})^{-1}$.
Proof This lemma can be found in Lemma 4.3 of Bosq (2000) and hence the proof
is omitted. [2mm]
###### Lemma 28
For $z\sim N_{L}\left(0,I_{L}\right)$, then for any $\delta>0$, we have
$P\left(\lVert z\rVert_{2}>\delta\right)\leq
2\exp\left(-\frac{\delta^{2}}{8L+2\sqrt{2L}\delta}\right).$
Proof Since
$\mathbb{E}\left[\lVert
z\rVert^{2k}_{2}\right]=\frac{\Gamma(\frac{L}{2}+k)}{\Gamma(\frac{L}{2})}\times
2^{k}\leq k!(2L)^{k},$
we have
$\mathbb{E}\left[\lVert z\rVert^{k}_{2}\right]\leq\sqrt{\mathbb{E}\left[\lVert
z\rVert^{2k}_{2}\right]}\leq\sqrt{k!}\left(\sqrt{2L}\right)^{k}\leq\frac{k!}{2}\cdot
4L\cdot(\sqrt{2L})^{k-2}$
for $k\geq 2$. Thus, by Lemma 29, we have proved the result. [2mm]
###### Lemma 29
Let $Z_{1},Z_{2},\dots,Z_{n}$ be independent random variables in a separable
Hilbert space with norm $\lVert\cdot\rVert$. If $\mathbb{E}[Z_{i}]=0$
($i=1,\ldots,n$) and
$\sum^{n}_{i=1}\mathbb{E}\left[\lVert
Z_{i}\rVert^{k}\right]\leq\frac{k!}{2}nL_{1}L^{k-2}_{2},k=2,3,\dots,$
for two positive constants $L_{1}$ and $L_{2}$, then for all $\delta>0$,
$P\left(\lVert\sum^{n}_{i=1}Z_{i}\rVert\geq n\delta\right)\leq
2\exp\left(-\frac{n\delta^{2}}{2L_{1}+2L_{2}\delta}\right).$
Proof This lemma can be derived directly from Theorem 2.5 (2) of Bosq (2000)
and hence its proof is omitted. [2mm]
###### Lemma 30
For a function $f(t)$ defined on $\mathcal{T}$, assuming that $f$ has
continuous derivative, and let $D_{0,f}\coloneqq\sup_{t\in{\mathcal{T}}}\lvert
f(t)\rvert$, $D_{1,f}\coloneqq\sup_{t\in{\mathcal{T}}}\lvert
f^{\prime}(t)\rvert$, assume that $D_{0,f},D_{1,f}<\infty$. Let
$\lvert\mathcal{T}\rvert$ denote the length of interval $\mathcal{T}$, and let
$u_{1}<u_{2}<\dots<u_{T}\in\mathcal{T}$, we denote endpoints of $\mathcal{T}$
as $u_{0}$ and $u_{T+1}$. Assume that there is positive constant $\zeta_{0}$
such that
$\max_{1\leq k\leq
T+1}\left|\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\leq\frac{\zeta_{0}}{T^{2}}$
(D.39)
hold. Let $\zeta_{1}=\zeta_{0}+1$, then we have
$\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|\leq\frac{D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}(\zeta_{1}+\zeta_{0})}{T}.$
Proof Since
$\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|$
$\displaystyle\leq\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})\right|$
$\displaystyle+\left|\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|,$
we will first prove the first part is smaller than $D_{0,f}\zeta_{0}/T$, and
then prove the second part is smaller than
$(D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}\zeta_{1})/T$. For
first part, we have
$\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})\right|$
$\displaystyle=\left|\sum^{T}_{k=1}f(u_{k})\left(\frac{1}{T}-\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}\right)\right|$
$\displaystyle\leq\sum^{T}_{k=1}\left|f(u_{k})\right|\left|\frac{1}{T}-\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}\right|$
$\displaystyle\leq\max_{1\leq k\leq
T}\left|\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\sum^{T}_{k=1}\left|f(u_{k})\right|$
$\displaystyle\leq\frac{\zeta_{0}}{T^{2}}\times T\times D_{0,f}$
$\displaystyle=\frac{\zeta_{0}D_{0,f}}{T}.$
To prove second part, we first note that based on (D.39), we have
$\max_{1\leq k\leq T+1}\lvert
u_{k}-u_{k-1}\rvert\leq\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}.$
Then, for any $t\in(u_{k},u_{k+1})$, by Taylor’s expansion, we have
$f(t)=f(u_{k})+f^{\prime}(\bar{t})(t-u_{k}),$
where $\bar{t}\in(u_{k},t)$. Thus,
$\lvert f(t)-f(u_{k})\rvert=\lvert f^{\prime}(\bar{t})\rvert(t-u_{k})\leq
D_{1,f}(t-u_{k}).$
This way, we have
$\displaystyle\left|\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|$
$\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}\int^{u_{k}}_{u_{k-1}}\lvert
f(u_{k})-f(t)\rvert
dt+\frac{1}{\lvert\mathcal{T}\rvert}\int^{u_{T+1}}_{u_{T}}\lvert f(t)\rvert
dt$ $\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\times T\times
D_{1,f}\times\int^{u_{k}}_{u_{k-1}}(t-u_{k})dt+\frac{1}{\lvert\mathcal{T}\rvert}\times
D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$
$\displaystyle=\frac{1}{\lvert\mathcal{T}\rvert}\times T\times
D_{1,f}\times\frac{(u_{k+1}-u_{k})^{2}}{2}+\frac{1}{\lvert\mathcal{T}\rvert}\times
D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$
$\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\times
T\times\frac{D_{1,f}}{2}\times\left(\max_{1\leq k\leq T+1}\lvert
u_{k+1}-u_{k}\rvert\right)^{2}+\frac{1}{\lvert\mathcal{T}\rvert}\times
D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$
$\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\times
T\times\frac{D_{1,f}}{2}\times\left(\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}\right)^{2}+\frac{1}{\lvert\mathcal{T}\rvert}\times
D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$
$\displaystyle=\frac{D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}\zeta_{1}}{T}.$
Thus, combining part 1 and part 2, we have
$\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|\leq\frac{D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}(\zeta_{1}+\zeta_{0})}{T}.$
[2mm]
###### Lemma 31
For Gaussian random function $g$ in Hilbert Space $\mathbb{H}$ with mean zero,
that is, $\mathbb{E}[g]=0$, we have
$\mathbb{E}\left[\|g\|^{2k}\right]\leq(2\lambda_{0})^{k}\cdot k!,$
where $\lambda_{0}=\mathbb{E}\left[\|g\|^{2}\right]$.
Proof Let $\\{\phi_{m}\\}_{m\geq 1}$ be othornormal eigenfunctions of $g$,
and $a_{m}=\langle g,\phi_{m}\rangle$, then $a_{m}\sim N(0,\lambda_{m})$ and
$\lambda_{0}=\sum_{m\geq 1}\lambda_{m}$. Let
$\xi_{m}=\lambda^{-1/2}_{m}a_{m}$, then we have $\xi_{m}\sim N(0,1)$ i.i.d..
By Karhunen–Loève theorem, we have
$g=\sum^{\infty}_{m=1}\lambda_{m}^{1/2}\xi_{m}\phi_{m}.$
Thus, $\|g\|=\left(\sum_{m\geq 1}\lambda_{m}\xi^{2}_{m}\right)^{1/2}$, and
$\|g\|^{2k}=\left(\sum_{m\geq 1}\lambda_{m}\xi^{2}_{m}\right)^{k}$.
Recall Jensen’s inequality, for convex function $\psi(\cdot)$, and real
numbers $x_{1},x_{2},\dots,x_{n}$ in its domain, and positive real numbers
$a_{1},a_{2},\dots,a_{n}$, we have
$\psi\left(\frac{\sum^{n}_{i=1}a_{i}x_{i}}{\sum^{n}_{i=1}a_{i}}\right)\leq\frac{\sum^{n}_{i=1}a_{i}\psi(x_{i})}{\sum^{n}_{i=1}a_{i}}.$
Here, let $\psi(t)=t^{k}$, and we then have
$\displaystyle\|g\|^{2k}$ $\displaystyle=\left(\sum_{m\geq
1}\lambda_{m}\right)^{k}\cdot\left(\frac{\sum_{m\geq
1}\lambda_{m}\xi^{2}_{m}}{\sum_{m\geq 1}\lambda_{m}}\right)^{k}$
$\displaystyle\leq\left(\sum_{m\geq
1}\lambda_{m}\right)^{k}\cdot\frac{\sum_{m\geq
1}\lambda_{m}\xi^{2k}_{m}}{\sum_{m\geq 1}\lambda_{m}}$
$\displaystyle=\left(\sum_{m\geq
1}\lambda_{m}\right)^{k-1}\cdot\left(\sum_{m\geq
1}\lambda_{m}\xi^{2k}_{m}\right).$
Thus,
$\displaystyle\mathbb{E}\left[\|g\|^{2k}\right]$
$\displaystyle\leq\left(\sum_{m\geq
1}\lambda_{m}\right)^{k-1}\cdot\left(\sum_{m\geq
1}\lambda_{m}\mathbb{E}\left[\xi^{2k}_{m}\right]\right)$
$\displaystyle=\left(\sum_{m\geq
1}\lambda_{m}\right)^{k}\mathbb{E}\left[\xi^{2k}_{1}\right]$
$\displaystyle=\left(\sum_{m\geq 1}\lambda_{m}\right)^{k}\cdot\pi^{-1/2}\cdot
2^{k}\cdot\Gamma(k+1/2)$ $\displaystyle\leq\left(\sum_{m\geq
1}\lambda_{m}\right)^{k}\cdot 2^{k}\cdot k!$
$\displaystyle=(2\lambda_{0})^{k}k!$
[2mm]
###### Lemma 32
For any $\delta>0$, we have
$P\left(\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(t)g_{ij}(s)-K_{jj}(s,t)\right]\right\|_{\text{HS}}>\delta\right)\leq
2\exp\left(-\frac{n\delta^{2}}{64\lambda^{2}_{0,\max}+8\lambda_{0,\max}\delta}\right)$
holding for any $j=1,\ldots,p$.
Proof Since $g_{ij}(t)=\sum_{m\geq
1}\lambda^{1/2}_{jm}\xi_{ijm}\phi_{jm}(t)$, and $\xi_{ijm}\sim N(0,1)$ i.i.d.
for $m\geq 1$, we have $g_{ij}(s)g_{ij}(t)=\sum_{m,m^{\prime}\geq
1}\lambda^{1/2}_{jm}\lambda^{1/2}_{jm^{\prime}}\xi_{ijm}\xi_{ijm^{\prime}}\phi_{jm}(s)\phi_{jm^{\prime}}(t)$,
and $K_{jj}(s,t)=\mathbb{E}[g_{ij}(s)g_{ij}(t)]=\sum_{m,m^{\prime}\geq
1}\lambda^{1/2}_{jm}\lambda^{1/2}_{jm^{\prime}}\phi_{jm}(s)\phi_{jm^{\prime}}(t)\mathbbm{1}_{mm^{\prime}}$,
where $\mathbbm{1}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})=1$ if
$m=m^{\prime}$ and $0$ if $m\neq m^{\prime}$. Thus,
$\left\|g_{ij}(s)g_{ij}(t)-K_{jj}(s,t)\right\|^{2}_{\text{HS}}=\sum_{m,m^{\prime}\geq
1}\lambda_{jm}\lambda_{jm^{\prime}}(\xi_{ijm}\xi_{ijm^{\prime}}-\mathbbm{1}_{mm^{\prime}})^{2},$
and for any $k\geq 2$, we have
$\displaystyle\mathbb{E}\left[\left\|g_{ij}(s)g_{ij}(t)-K_{jj}(s,t)\right\|^{k}_{\text{HS}}\right]$
$\displaystyle=\mathbb{E}\left[\left\\{\sum_{m,m^{\prime}\geq
1}\lambda_{jm}\lambda_{jm^{\prime}}(\xi_{ijm}\xi_{ijm^{\prime}}-\mathbbm{1}_{mm^{\prime}})^{2}\right\\}^{k/2}\right]$
$\displaystyle\overset{(i)}{\leq}\left(\sum_{m,m^{\prime}\geq
1}\lambda_{jm}\lambda_{jm^{\prime}}\right)^{k/2-1}\sum_{m,m^{\prime}\geq
1}\lambda_{jm}\lambda_{jm^{\prime}}\mathbb{E}\left[\left(\xi_{ijm}\xi_{ijm^{\prime}}-\mathbbm{1}_{mm^{\prime}}\right)^{k}\right],$
where $(i)$ follows Jensen’s inequality with convex function
$\psi(x)=x^{k/2}$. Since
$\displaystyle\mathbb{E}\left[\left(\xi_{ijm}\xi_{ijm^{\prime}}-\mathbbm{1}_{mm^{\prime}}\right)^{k}\right]$
$\displaystyle\leq
2^{k-1}\left(\mathbb{E}\left[(\xi_{ijm}\xi_{ijm^{\prime}})^{k}\right]+1\right)$
$\displaystyle\leq 2^{k-1}\left(\mathbb{E}[\xi^{2k}_{ij1}]+1\right)$
$\displaystyle\leq 2^{k-1}(2^{k}k!+1)$ $\displaystyle\leq 4^{k}k!,$
we then have
$\mathbb{E}\left[\left\|g_{ij}(s)g_{ij}(t)-K_{jj}(s,t)\right\|^{k}_{\text{HS}}\right]\leq(4\lambda_{j0})^{k}k!\leq(4\lambda_{0,\max})^{k}k!.$
The final results then follows directly from Lemma 29. [2mm]
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|
2024-09-04T02:54:59.296596 | 2020-03-11T17:21:15 | 2003.05425 | {
"authors": "Pim de Haan, Maurice Weiler, Taco Cohen and Max Welling",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26172",
"submitter": "Pim de Haan",
"url": "https://arxiv.org/abs/2003.05425"
} | arxiv-papers | # Gauge Equivariant Mesh CNNs
Anisotropic convolutions on geometric graphs
Pim de Haan
Qualcomm AI Research
University of Amsterdam
&Maurice Weiler∗
QUVA Lab
University of Amsterdam
&Taco Cohen
Qualcomm AI Research
&Max Welling
Qualcomm AI Research
University of Amsterdam
Equal ContributionQualcomm AI Research is an initiative of Qualcomm
Technologies, Inc.
###### Abstract
A common approach to define convolutions on meshes is to interpret them as a
graph and apply graph convolutional networks (GCNs). Such GCNs utilize
_isotropic_ kernels and are therefore insensitive to the relative orientation
of vertices and thus to the geometry of the mesh as a whole. We propose Gauge
Equivariant Mesh CNNs which generalize GCNs to apply _anisotropic_ gauge
equivariant kernels. Since the resulting features carry orientation
information, we introduce a geometric message passing scheme defined by
parallel transporting features over mesh edges. Our experiments validate the
significantly improved expressivity of the proposed model over conventional
GCNs and other methods.
## 1 Introduction
Convolutional neural networks (CNNs) have been established as the default
method for many machine learning tasks like speech recognition or planar and
volumetric image classification and segmentation. Most CNNs are restricted to
flat or spherical geometries, where convolutions are easily defined and
optimized implementations are available. The empirical success of CNNs on such
spaces has generated interest to generalize convolutions to more general
spaces like graphs or Riemannian manifolds, creating a field now known as
geometric deep learning (Bronstein et al., 2017).
A case of specific interest is convolution on _meshes_ , the discrete analog
of 2-dimensional embedded Riemannian manifolds. Mesh CNNs can be applied to
tasks such as detecting shapes, registering different poses of the same shape
and shape segmentation. If we forget the positions of vertices, and which
vertices form faces, a mesh $M$ can be represented by a graph ${\mathcal{G}}$.
This allows for the application of _graph convolutional networks_ (GCNs) to
processing signals on meshes.
Figure 1: Two local neighbourhoods around vertices $p$ and their
representations in the tangent planes $T_{p}M$. The distinct geometry of the
neighbourhoods is reflected in the different angles $\theta_{pq_{i}}$ of
incident edges from neighbours $q_{i}$. Graph convolutional networks apply
isotropic kernels and can therefore not distinguish both neighbourhoods. Gauge
Equivariant Mesh CNNs apply anisotropic kernels and are therefore sensitive to
orientations. The arbitrariness of reference orientations, determined by a
choice of neighbour $q_{0}$, is accounted for by the gauge equivariance of the
model.
However, when representing a mesh by a graph, we lose important geometrical
information. In particular, in a graph there is no notion of angle between or
ordering of two of a node’s incident edges (see figure 1). Hence, a GCNs
output at a node $p$ is designed to be independent of relative angles and
_invariant_ to any permutation of its neighbours $q_{i}\in\mathcal{N}(p)$. A
graph convolution on a mesh graph therefore corresponds to applying an
_isotropic_ convolution kernel. Isotropic filters are insensitive to the
orientation of input patterns, so their features are strictly less expressive
than those of orientation aware anisotropic filters.
To address this limitation of graph networks we propose Gauge Equivariant Mesh
CNNs (GEM-CNN)111Implementation at https://github.com/Qualcomm-AI-
research/gauge-equivariant-mesh-cnn, which minimally modify GCNs such that
they are able to use anisotropic filters while sharing weights across
different positions and respecting the local geometry. One obstacle in sharing
anisotropic kernels, which are functions of the angle $\theta_{pq}$ of
neighbour $q$ with respect to vertex $p$, over multiple vertices of a mesh is
that there is no unique way of selecting a reference neighbour $q_{0}$, which
has the direction $\theta_{pq_{0}}=0$. The reference neighbour, and hence the
orientation of the neighbours, needs to be chosen arbitrarily. In order to
guarantee the equivalence of the features resulting from different choices of
orientations, we adapt Gauge Equivariant (or coordinate independent) CNNs
(Cohen et al., 2019b; Weiler et al., 2021) to general meshes. The kernels of
our model are thus designed to be _equivariant under gauge transformations_ ,
that is, to guarantee that the responses for different kernel orientations are
related by a prespecified transformation law. Such features are identified as
geometric objects like scalars, vectors, tensors, etc., depending on the
specific choice of transformation law. In order to compare such geometric
features at neighbouring vertices, they need to be _parallel transported_
along the connecting edge.
In our implementation we first specify the transformation laws of the feature
spaces and compute a space of gauge equivariant kernels. Then we pick
arbitrary reference orientations at each node, relative to which we compute
neighbour orientations and compute the corresponding edge transporters. Given
these quantities, we define the forward pass as a message passing step via
edge transporters followed by a contraction with the equivariant kernels
evaluated at the neighbour orientations. Algorithmically, Gauge Equivariant
Mesh CNNs are therefore just GCNs with anisotropic, gauge equivariant kernels
and message passing via parallel transporters. Conventional GCNs are covered
in this framework for the specific choice of isotropic kernels and trivial
edge transporters, given by identity maps.
In Sec. 2, we will give an outline of our method, deferring details to Secs. 3
and 4. In Sec. 3.2, we describe how to compute general geometric quantities,
not specific to our method, used for the computation of the convolution. In
our experiments in Sec. 6.1, we find that the enhanced expressiveness of Gauge
Equivariant Mesh CNNs enables them to outperform conventional GCNs and other
prior work in a shape correspondence task.
## 2 Convolutions on Graphs with Geometry
We consider the problem of processing signals on discrete 2-dimensional
manifolds, or meshes $M$. Such meshes are described by a set ${\mathcal{V}}$
of vertices in $\mathbb{R}^{3}$ together with a set ${\mathcal{F}}$ of tuples,
each consisting of the vertices at the corners of a face. For a mesh to
describe a proper manifold, each edge needs to be connected to two faces, and
the neighbourhood of each vertex needs to be homeomorphic to a disk. Mesh $M$
induces a graph ${\mathcal{G}}$ by forgetting the coordinates of the vertices
while preserving the edges.
A conventional graph convolution between kernel $K$ and signal $f$, evaluated
at a vertex $p$, can be defined by
$\displaystyle(K\star f)_{p}\ =\
K_{\text{self}}f_{p}\,+\sum\nolimits_{q\in{\mathcal{N}}_{p}}\\!\\!K_{\textup{neigh}}f_{q},$
(1)
where ${\mathcal{N}}_{p}$ is the set of neighbours of $p$ in ${\mathcal{G}}$,
and $K_{\text{self}}\in\mathbb{R}^{C_{\textup{in}}\times C_{\textup{out}}}$
and $K_{\textup{neigh}}\in\mathbb{R}^{C_{\textup{in}}\times C_{\textup{out}}}$
are two linear maps which model a self interaction and the neighbour
contribution, respectively. Importantly, graph convolution does not
distinguish different neighbours, because each feature vector $f_{q}$ is
multiplied by the same matrix $K_{\textup{neigh}}$ and then summed. For this
reason we say the kernel is _isotropic_.
Algorithm 1 Gauge Equivariant Mesh CNN layer
Input: mesh $M$, input/output feature types
$\rho_{\textup{in}},\rho_{\textup{out}}$, reference neighbours
$(q_{0}^{p}\in{\mathcal{N}}_{p})_{p\in M}$.
Compute basis kernels $K^{i}_{\textup{self}},K^{i}_{\textup{neigh}}(\theta)$
$\rhd$ Sec. 3
Initialise weights $w_{\textup{self}}^{i}$ and $w^{i}_{\textup{neigh}}$.
For each neighbour pair, $p\in M,q\in{\mathcal{N}}_{p}$: $\rhd$ App. A.
compute neighbor angles $\theta_{pq}$ relative to reference neighbor compute
parallel transporters $g_{q\to p}$
Forward$\big{(}$input features $(f_{p})_{p\in M}$, weights
$w^{i}_{\textup{self}},w^{i}_{\textup{neigh}}$$\big{)}$:
$f^{\prime}_{p}\leftarrow\sum_{i}w_{\textup{self}}^{i}K^{i}_{\textup{self}}f_{p}+\\!\\!\\!\sum_{i,q\in{\mathcal{N}}_{p}}\\!\\!w^{i}_{\textup{neigh}}K^{i}_{\textup{neigh}}(\theta_{pq})\rho_{\textup{in}}(g_{q\to
p})f_{q}$
Consider the example in figure 1, where on the left and right, the
neighbourhood of one vertex $p$, containing neighbours
$q\in{\mathcal{N}}_{p}$, is visualized. An isotropic kernel would propagate
the signal from the neighbours to $p$ in exactly the same way in both
neighbourhoods, even though the neighbourhoods are geometrically distinct. For
this reason, our method uses direction sensitive (_anisotropic_) kernels
instead of isotropic kernels. Anisotropic kernels are inherently more
expressive than isotropic ones which is why they are used universally in
conventional planar CNNs.
We propose the Gauge Equivariant Mesh Convolution, a minimal modification of
graph convolution that allows for anisotropic kernels $K(\theta)$ whose value
depends on an orientation $\theta\in[0,2\pi)$.222 In principle, the kernel
could be made dependent on the radial distance of neighboring nodes, by
$K_{\textup{neigh}}(r,\theta)=F(r)K_{\textup{neigh}}(\theta)$, where $F(r)$ is
unconstrained and $K_{\textup{neigh}}(\theta)$ as presented in this paper. As
this dependency did not improve the performance in our empirical evaluation,
we omit it. To define the orientations $\theta_{pq}$ of neighbouring vertices
$q\in{\mathcal{N}}_{p}$ of $p$, we first map them to the tangent plane
$T_{p}M$ at $p$, as visualized in figure 1. We then pick an _arbitrary_
reference neighbour $q^{p}_{0}$ to determine a reference orientation333
Mathematically, this corresponds to a choice of _local reference frame_ or
_gauge_. $\theta_{pq^{p}_{0}}:=0$, marked orange in figure 1. This induces a
basis on the tangent plane, which, when expressed in polar coordinates,
defines the angles $\theta_{pq}$ of the other neighbours.
As we will motivate in the next section, features in a Gauge Equivariant CNN
are coefficients of geometric quantities. For example, a tangent vector at
vertex $p$ can be described either geometrically by a 3 dimensional vector
orthogonal to the normal at $p$ or by two coefficients in the basis on the
tangent plane. In order to perform convolution, geometric features at
different vertices need to be linearly combined, for which it is required to
first “parallel transport” the features to the same vertex. This is done by
applying a matrix $\rho(g_{q\to p})\in\mathbb{R}^{C_{\textup{in}}\times
C_{\textup{in}}}$ to the coefficients of the feature at $q$, in order to
obtain the coefficients of the feature vector transported to $p$, which can be
used for the convolution at $p$. The transporter depends on the geometric
_type_ (group representation) of the feature, denoted by $\rho$ and described
in more detail below. Details of how the tangent space is defined, how to
compute the map to the tangent space, angles $\theta_{pq}$, and the parallel
transporter are given in Appendix A.
In combination, this leads to the GEM-CNN convolution
$(K\star f)_{p}\ =\
K_{\textup{self}}f_{p}+\sum\nolimits_{q\in{\mathcal{N}}_{p}}K_{\textup{neigh}}(\theta_{pq})\rho(g_{q\to
p})f_{q}$ (2)
which differs from the conventional graph convolution, defined in Eq. 1 only
by the use of an anisotropic kernel and the parallel transport message
passing.
We require the outcome of the convolution to be _equivalent_ for any choice of
reference orientation. This is not the case for any anisotropic kernel but
only for those which are _equivariant under changes of reference orientations_
(gauge transformations). Equivariance imposes a linear constraint on the
kernels. We therefore solve for complete sets of “basis-kernels”
$K_{\text{self}}^{i}$ and $K_{\text{neigh}}^{i}$ satisfying this constraint
and linearly combine them with parameters $w_{\text{self}}^{i}$ and
$w_{\text{neigh}}^{i}$ such that
$K_{\text{self}}=\sum_{i}w_{\text{self}}^{i}K_{\text{self}}^{i}$ and
$K_{\text{neigh}}=\sum_{i}w_{\text{neigh}}^{i}K_{\text{neigh}}^{i}$. Details
on the computation of basis kernels are given in section 3. The full algorithm
for initialisation and forward pass, which is of time and space complexity
linear in the number of vertices, for a GEM-CNN layer are listed in algorithm
1. Gradients can be computed by automatic differentiation.
The GEM-CNN is gauge equivariant, but furthermore satisfies two important
properties. Firstly, it depends only on the intrinsic shape of the 2D mesh,
not on the embedding of the mesh in $\mathbb{R}^{3}$. Secondly, whenever a map
from the mesh to itself exists that preserves distances and orientation, the
convolution is equivariant to moving the signal along such transformations.
These properties are proven in Appendix D and empirically shown in Appendix
F.2.
$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$pick
gauge $q_{0}=q_{A}$map back to meshpick gauge $q_{0}=q_{B}$map back to
meshgeometric convconv in gauge $A$conv in gauge $B$gauge transformation
$A\\!\to\\!B$gauge transformation $A\\!\to\\!B$
(a) Convolution from scalar to scalar features.
$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$pick
gauge $q_{0}=q_{A}$map back to meshpick gauge $q_{0}=q_{B}$map back to
meshgeometric convconv in gauge $A$conv in gauge $B$gauge transfromation
$A\\!\to\\!B$gauge transfromation $A\\!\to\\!B$
(b) Convolution from scalar to vector features.
Figure 2: Visualization of the Gauge Equivariant Mesh Convolution in two
configurations, scalar to scalar and scalar to vector. The convolution
operates in a gauge, so that vectors are expressed in coefficients in a basis
and neighbours have polar coordinates, but can also be seen as a _geometric
convolution_ , a gauge-independent map from an input signal on the mesh to a
output signal on the mesh. The convolution is equivariant if this geometric
convolution does not depend on the intermediate chosen gauge, so if the
diagram commutes.
## 3 Gauge Equivariance & Geometric Features
On a general mesh, the choice of the reference neighbour, or gauge, which
defines the orientation of the kernel, can only be made arbitrarily. However,
this choice should not arbitrarily affect the outcome of the convolution, as
this would impede the generalization between different locations and different
meshes. Instead, Gauge Equivariant Mesh CNNs have the property that their
output transforms according to a known rule as the gauge changes.
Consider the left hand side of figure 2(a). Given a neighbourhood of vertex
$p$, we want to express each neighbour $q$ in terms of its polar coordinates
$(r_{q},\theta_{q})$ on the tangent plane, so that the kernel value at that
neighbour $K_{\textup{neigh}}(\theta_{q})$ is well defined. This requires
choosing a basis on the tangent plane, determined by picking a neighbour as
reference neighbour (denoted $q_{0}$), which has the zero angle
$\theta_{q_{0}}=0$. In the top path, we pick $q_{A}$ as reference neighbour.
Let us call this gauge A, in which neighbours have angles $\theta^{A}_{q}$. In
the bottom path, we instead pick neighbour $q_{B}$ as reference point and are
in gauge B. We get a different basis for the tangent plane and different
angles $\theta^{B}_{q}$ for each neighbour. Comparing the two gauges, we see
that they are related by a rotation, so that
$\theta^{B}_{q}=\theta^{A}_{q}-\theta^{A}_{q_{B}}$. This change of gauge is
called a gauge transformation of angle $g:=\theta^{A}_{q_{B}}$.
In figure 2(a), we illustrate a gauge equivariant convolution that takes input
and output features such as gray scale image values on the mesh, which are
called scalar features. The top path represents the convolution in gauge A,
the bottom path in gauge B. In either case, the convolution can be interpreted
as consisting of three steps. First, for each vertex $p$, the value of the
scalar features on the mesh at each neighbouring vertex $q$, represented by
colors, is mapped to the tangent plane at $p$ at angle $\theta_{q}$ defined by
the gauge. Subsequently, the convolutional kernel sums for each neighbour $q$,
the product of the feature at $q$ and kernel $K(\theta_{q})$. Finally the
output is mapped back to the mesh. These three steps can be composed into a
single step, which we could call a _geometric convolution_ , mapping from
input features on the mesh to output features on the mesh. The convolution is
_gauge equivariant_ if this geometric convolution does not depend on the gauge
we pick in the interim, so in figure 2(a), if the convolution in the top path
in gauge A has same result the convolution in the bottom path in gauge B,
making the diagram commute. In this case, however, we see that the convolution
output needs to be the same in both gauges, for the convolution to be
equivariant. Hence, we must have that $K(\theta_{q})=K(\theta_{q}-g)$, as the
orientations of the neighbours differ by some angle $g$, and the kernel must
be isotropic.
As we aim to design an anisotropic convolution, the output feature of the
convolution at $p$ can, instead of a scalar, be two numbers
$v\in\mathbb{R}^{2}$, which can be interpreted as coefficients of a tangent
feature vector in the tangent space at $p$, visualized in figure 2(b). As
shown on the right hand side, different gauges induce a different basis of the
tangent plane, so that the _same tangent vector_ (shown on the middle right on
the mesh), is represented by _different coefficients_ in the gauge (shown on
the top and bottom on the right). This gauge equivariant convolution must be
anisotropic: going from the top row to the bottom row, if we change
orientations of the neighbours by $-g$, the coefficients of the output vector
$v\in\mathbb{R}^{2}$ of the kernel must be also rotated by $-g$. This is
written as $R(-g)v$, where $R(-g)\in\mathbb{R}^{2\times 2}$ is the matrix that
rotates by angle $-g$.
Vectors and scalars are not the only type of geometric features that can be
inputs and outputs of a GEM-CNN layer. In general, the coefficients of a
geometric feature of $C$ dimensions changes by an invertible linear
transformation $\rho(-g)\in\mathbb{R}^{C\times C}$ if the gauge is rotated by
angle $g$. The map $\rho:[0,2\pi)\to\mathbb{R}^{C\times C}$ is called the
_type_ of the geometric quantity and is formally known as a group
representation of the planar rotation group $\operatorname{SO}(2)$. Group
representations have the property that $\rho(g+h)=\rho(g)\rho(h)$ (they are
group homomorphisms), which implies in particular that $\rho(0)=\mathbbm{1}$
and $\rho(-g)=\rho(g)^{-1}$. For more background on group representation
theory, we refer the reader to (Serre, 1977) and, specifically in the context
of equivariant deep learning, to (Lang & Weiler, 2020). From the theory of
group representations, we know that any feature type can be composed from
“irreducible representations” (irreps). For $\operatorname{SO}(2)$, these are
the one dimensional invariant scalar representation $\rho_{0}$ and for all
$n\in{\mathbb{N}}_{>0}$, a two dimensional representation $\rho_{n}$,
$\rho_{0}(g)=1,\quad\rho_{n}(g)=\begin{pmatrix}\cos ng&\shortminus\sin ng\\\
\sin ng&\phantom{\shortminus}\cos ng\end{pmatrix}.$
where we write, for example, $\rho=\rho_{0}\oplus\rho_{1}\oplus\rho_{1}$ to
denote that representation $\rho(g)$ is the direct sum (i.e. block-diagonal
stacking) of the matrices $\rho_{0}(g),\rho_{1}(g),\rho_{1}(g)$. Scalars and
tangent vector features correspond to $\rho_{0}$ and $\rho_{1}$ respectively
and we have $R(g)=\rho_{1}(g)$.
The type of the feature at each layer in the network can thus be fully
specified (up to a change of basis) by the number of copies of each irrep.
Similar to the dimensionality in a conventional CNN, the choice of type is a
hyperparameter that can be freely chosen to optimize performance.
### 3.1 Kernel Constraint
Given an input type $\rho_{\textup{in}}$ and output type $\rho_{\textup{out}}$
of dimensions $C_{\textup{in}}$ and $C_{\textup{out}}$, the kernels are
$K_{\textup{self}}\in\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$ and
$K_{\textup{neigh}}:[0,2\pi)\to\mathbb{R}^{C_{\textup{out}}\times
C_{\textup{in}}}$. However, not all such kernels are equivariant. Consider
again examples figure 2(a) and figure 2(b). If we map from a scalar to a
scalar, we get that $K_{\textup{neigh}}(\theta-g)=K_{\textup{neigh}}(\theta)$
for all angles $\theta,g$ and the convolution is isotropic. If we map from a
scalar to a vector, we get that rotating the angles $\theta_{q}$ results in
the same tangent vector as rotating the output vector coefficients, so that
$K_{\textup{neigh}}(\theta-g)=R(-g)K_{\textup{neigh}}(\theta)$.
$\rho_{\textup{in}}\to\rho_{\textup{out}}$ | linearly independent solutions for $K_{\textup{neigh}}(\theta)$
---|---
$\rho_{0}\to\rho_{0}$ | $(1)$
$\rho_{n}\to\rho_{0}$ | $\begin{pmatrix}\cos n\theta&\sin n\theta\end{pmatrix},\begin{pmatrix}\sin n\theta&\shortminus\cos n\theta\end{pmatrix}$
$\rho_{0}\to\rho_{m}$ | $\begin{pmatrix}\cos m\theta\\\ \sin m\theta\end{pmatrix},\begin{pmatrix}\phantom{\shortminus}\sin m\theta\\\ \shortminus\cos m\theta\end{pmatrix}$
$\rho_{n}\to\rho_{m}$ | $\begin{pmatrix}c&\shortminus s\\\ s&\phantom{\shortminus}c\end{pmatrix}$, $\begin{pmatrix}\phantom{\shortminus}s&c\\\ \shortminus c&s\end{pmatrix}$, $\begin{pmatrix}c_{+}&\phantom{\shortminus}s_{+}\\\ s_{+}&\shortminus c_{+}\end{pmatrix}$, $\begin{pmatrix}\shortminus s_{+}&c_{+}\\\ \phantom{-}c_{+}&s_{+}\end{pmatrix}$
$\rho_{\textup{in}}\to\rho_{\textup{out}}$ | linearly independent solutions for $K_{\textup{self}}$
$\rho_{0}\to\rho_{0}$ | $(1)$
$\rho_{n}\to\rho_{n}$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$, $\begin{pmatrix}\phantom{\shortminus}0&1\\\ \shortminus 1&0\end{pmatrix}$
Table 1: Solutions to the angular kernel constraint for kernels that map from
$\rho_{n}$ to $\rho_{m}$. We denote ${c_{\pm}=\cos((m\pm n)\theta)}$ and
${s_{\pm}=\sin((m\pm n)\theta)}$.
In general, as derived by Cohen et al. (2019b); Weiler et al. (2021) and in
appendix B, the kernels must satisfy for any gauge transformation
$g\in[0,2\pi)$ and angle $\theta\in[0,2\pi)$, that
$\displaystyle\\!\\!K_{\textup{neigh}}(\theta-g)$
$\displaystyle=\rho_{\textup{out}}(-g)K_{\textup{neigh}}(\theta)\rho_{\textup{in}}(g),$
(3) $\displaystyle K_{\textup{self}}$
$\displaystyle=\rho_{\textup{out}}(-g)\;K_{\textup{self}}\;\rho_{\textup{in}}(g).$
(4)
The kernel can be seen as consisting of multiple blocks, where each block
takes as input one irrep and outputs one irrep. For example if
$\rho_{\textup{in}}$ would be of type $\rho_{0}\oplus\rho_{1}\oplus\rho_{1}$
and $\rho_{\textup{out}}$ of type $\rho_{1}\oplus\rho_{3}$, we have $4\times
5$ matrix
$\displaystyle
K_{\textup{neigh}}(\theta)=\begin{pmatrix}K_{10}(\theta)&K_{11}(\theta)&K_{11}(\theta)\\\
K_{30}(\theta)&K_{31}(\theta)&K_{31}(\theta)\\\ \end{pmatrix}$
where e.g. $K_{31}(\theta)\in\mathbb{R}^{2\times 2}$ is a kernel that takes as
input irrep $\rho_{1}$ and as output irrep $\rho_{3}$ and needs to satisfy Eq.
3. As derived by Weiler & Cesa (2019) and in Appendix C, the kernels
$K_{\textup{neigh}}(\theta)$ and $K_{\textup{self}}$ mapping from irrep
$\rho_{n}$ to irrep $\rho_{m}$ can be written as a linear combination of the
basis kernels listed in Table 1. The table shows that equivariance requires
the self-interaction to only map from one irrep to the same irrep. Hence, we
have $K_{\textup{self}}=\begin{pmatrix}0&K_{11}&K_{11}\\\ 0&0&0\\\
\end{pmatrix}\in\mathbb{R}^{4\times 3}.$
All basis-kernels of all pairs of input irreps and output irreps can be
linearly combined to form an arbitrary equivariant kernel from feature of type
$\rho_{\textup{in}}$ to $\rho_{\textup{out}}$. In the above example, we have
$2\times 2+4\times 4=20$ basis kernels for $K_{\textup{neigh}}$ and 4 basis
kernels for $K_{\textup{self}}$. The layer thus has 24 parameters. As proven
in (Weiler & Cesa, 2019) and (Lang & Weiler, 2020), this parameterization of
the equivariant kernel space is _complete_ , that is, more general equivariant
kernels do not exist.
### 3.2 Geometry and Parallel Transport
In order to implement gauge equivariant mesh CNNs, we need to make the
abstract notion of tangent spaces, gauges and transporters concrete.
As the mesh is embedded in $\mathbb{R}^{3}$, a natural definition of the
tangent spaces $T_{p}M$ is as two dimensional subspaces that are orthogonal to
the normal vector at $p$. We follow the common definition of normal vectors at
mesh vertices as the area weighted average of the adjacent faces’ normals. The
Riemannian logarithm map $\log_{p}:{\mathcal{N}}_{p}\to T_{p}M$ represents the
one-ring neighborhood of each point $p$ on their tangent spaces as visualized
in figure 1. Specifically, neighbors $q\in{\mathcal{N}}_{p}$ are mapped to
$\log_{p}(q)\in T_{p}M$ by first projecting them to $T_{p}M$ and then
rescaling the projection such that the norm is preserved, i.e.
$|\log_{p}(q)|=|q-p|$; see Eq. 9. A choice of reference neighbor
$q_{p}\in{\mathcal{N}}$ uniquely determines a right handed, orthonormal
reference frame $(e_{p,1},\,e_{p,2})$ of $T_{p}M$ by setting
$e_{p,1}:=\log_{p}(q_{0})/|\log_{p}(q_{0})|$ and $e_{p,2}:=n\times e_{p,1}$.
The polar angle $\theta_{pq}$ of any neighbor $q\in{\mathcal{N}}$ relative to
the first frame axis is then given by $\theta_{pq}\ :=\
\operatorname{atan2}\big{(}e_{p,2}^{\top}\log_{p}(q),\
e_{p,1}^{\top}\log_{p}(q))\big{)}.$
Given the reference frame $(e_{p,1},e_{p,2})$, a 2-tuple of coefficients
$(v_{1},v_{2})\in\mathbb{R}^{2}$ specifies an (embedded) tangent vector
${v_{1}e_{p,1}+v_{2}e_{p,2}}\in T_{p}M\subset\mathbb{R}^{3}$. This assignment
is formally given by the _gauge map_ $E_{p}:\mathbb{R}^{2}\to
T_{p}M\subset\mathbb{R}^{3}$ which is a vector space isomorphism. In our case,
it can be identified with the matrix
$\displaystyle
E_{p}=\left[\begin{array}[]{cc}\rule[-2.15277pt]{0.5pt}{5.38193pt}&\rule[-2.15277pt]{0.5pt}{5.38193pt}\\\
e_{p,1}&e_{p,2}\\\
\rule[0.0pt]{0.5pt}{5.38193pt}&\rule[0.0pt]{0.5pt}{5.38193pt}\end{array}\right]\in\mathbb{R}^{3\times
2}.$ (8)
Feature vectors $f_{p}$ and $f_{q}$ at neighboring (or any other) vertices
$p\in M$ and $q\in{\mathcal{N}}_{p}\subseteq M$ live in different vector
spaces and are expressed relative to independent gauges, which makes it
invalid to sum them directly. Instead, they have to be parallel transported
along the mesh edge that connects the two vertices. As explained above, this
transport is given by group elements $g_{q\to p}\in[0,2\pi)$, which determine
the transformation of tangent vector _coefficients_ as $v_{q}\mapsto R(g_{q\to
p})v_{q}\in\mathbb{R}^{2}$ and, analogously, for feature vector coefficients
as $f_{q}\mapsto\rho(g_{q\to p})f_{q}$. Figure 4 in the appendix visualizes
the definition of edge transporters for flat spaces and meshes. On a flat
space, tangent vectors are transported by keeping them parallel in the usual
sense on Euclidean spaces. However, if the source and target frame
orientations disagree, the vector coefficients relative to the source frame
need to be transformed to the target frame. This coordinate transformation
from polar angles $\varphi_{q}$ of $v$ to $\varphi_{p}$ of $R(g_{q\to p})v$
defines the transporter $g_{q\to p}=\varphi_{p}-\varphi_{q}$. On meshes, the
source and target tangent spaces $T_{q}M$ and $T_{p}M$ are not longer
parallel. It is therefore additionally necessary to rotate the source tangent
space and its vectors parallel to the target space, before transforming
between the frames. Since transporters effectively make up for differences in
the source and target frames, the parallel transporters transform under gauge
transformations $g_{p}$ and $g_{q}$ according to $g_{q\to p}\mapsto
g_{p}+g_{q\to p}-g_{q}$. Note that this transformation law cancels with the
transformation law of the coefficients at $q$ and lets the transported
coefficients transform according to gauge transformations at $p$. It is
therefore valid to sum vectors and features that are parallel transported into
the same gauge at $p$.
A more detailed discussion of the concepts presented in this section can be
found in Appendix A.
## 4 Non-linearity
Besides convolutional layers, the GEM-CNN contains non-linear layers, which
also need to be gauge equivariant, for the entire network to be gauge
equivariant. The coefficients of features built out of irreducible
representaions, as described in section 3, do not commute with point-wise non-
linearities (Worrall et al., 2017; Thomas et al., 2018; Weiler et al., 2018a;
Kondor et al., 2018). Norm non-linearities and gated non-linearities (Weiler &
Cesa, 2019) can be used with such features, but generally perform worse in
practice compared to point-wise non-linearities (Weiler & Cesa, 2019). Hence,
we propose the _RegularNonlinearity_ , which uses point-wise non-linearities
and is approximately gauge equivariant.
This non-linearity is built on Fourier transformations. Consider a continuous
periodic signal, on which we perform a band-limited Fourier transform with
band limit $b$, obtaining $2b+1$ Fourier coefficients. If this continuous
signal is shifted by an arbitrary angle $g$, then the corresponding Fourier
components transform with linear transformation $\rho_{0:b}(-g)$, for $2b+1$
dimensional representation
$\rho_{0:b}:=\rho_{0}\oplus\rho_{1}\oplus...\oplus\rho_{b}$.
It would be exactly equivariant to take a feature of type $\rho_{0:b}$, take a
continuous inverse Fourier transform to a continuous periodic signal, then
apply a point-wise non-linearity to that signal, and take the continuous
Fourier transform, to recover a feature of type $\rho_{0:b}$. However, for
implementation, we use $N$ intermediate samples and the discrete Fourier
transform. This is exactly gauge equivariant for gauge transformation of
angles multiple of $2\pi/N$, but only approximately equivariant for other
angles. In App. G we prove that as $N\to\infty$, the non-linearity is exactly
gauge equivariant.
The run-time cost per vertex of the (inverse) Fourier transform implemented as
a simple linear transformation is $\mathcal{O}(bN)$, which is what we use in
our experiments. The pointwise non-linearity scales linearly with $N$, so the
complexity of the RegularNonLineariy is also $\mathcal{O}(bN)$. However, one
can also use a fast Fourier transform, achieving a complexity of
$\mathcal{O}(N\log N)$. Concrete memory and run-time cost of varying $N$ are
shown in appendix F.1.
## 5 Related Work
Our method can be seen as a practical implementation of coordinate independent
convolutions on triangulated surfaces, which generally rely on $G$-steerable
kernels (Weiler et al., 2021).
The irregular structure of meshes leads to a variety of approaches to define
convolutions. Closely related to our method are graph based methods which are
often based on variations of graph convolutional networks (Kipf & Welling,
2017; Defferrard et al., 2016). GCNs have been applied on spherical meshes
(Perraudin et al., 2019) and cortical surfaces (Cucurull et al., 2018; Zhao et
al., 2019a). Verma et al. (2018) augment GCNs with anisotropic kernels which
are dynamically computed via an attention mechanism over graph neighbours.
Instead of operating on the graph underlying a mesh, several approaches
leverage its geometry by treating it as a discrete manifold. Convolution
kernels can then be defined in geodesic polar coordinates which corresponds to
a projection of kernels from the tangent space to the mesh via the exponential
map. This allows for kernels that are larger than the immediate graph
neighbourhood and message passing over faces but does not resolve the issue of
ambiguous kernel orientation. Masci et al. (2015); Monti et al. (2016) and Sun
et al. (2018) address this issue by restricting the network to orientation
invariant features which are computed by applying anisotropic kernels in
several orientations and pooling over the resulting responses. The models
proposed in (Boscaini et al., 2016) and (Schonsheck et al., 2018) are
explicitly gauge dependent with preferred orientations chosen via the
principal curvature direction and the parallel transport of kernels,
respectively. Poulenard & Ovsjanikov (2018) proposed a non-trivially gauge
equivariant network based on geodesic convolutions, however, the model
parallel transports only partial information of the feature vectors,
corresponding to certain kernel orientations. In concurrent work, Wiersma et
al. (2020) also define convolutions on surfaces equivariantly to the
orientation of the kernel, but differ in that they use norm non-linearities
instead of regular ones and that they apply the convolution along longer
geodesics, which adds complexity to the geometric pre-computation - as partial
differential equations need to be solved, but may result in less
susceptibility to the particular discretisation of the manifold. A
comprehensive review on such methods can be found in Section 12 of (Weiler et
al., 2021).
Another class of approaches defines spectral convolutions on meshes. However,
as argued in (Bronstein et al., 2017), the Fourier spectrum of a mesh depends
heavily on its geometry, which makes such methods instable under deformations
and impedes the generalization between different meshes. Spectral convolutions
further correspond to isotropic kernels. Kostrikov et al. (2018) overcomes
isotropy of the Laplacian by decomposing it into two applications of the
first-order Dirac operator.
A construction based on toric covering maps of topologically spherical meshes
was presented in (Maron et al., 2017). An entirely different approach to mesh
convolutions is to apply a linear map to a spiral of neighbours (Bouritsas et
al., 2019; Gong et al., 2019), which works well only for meshes with a similar
graph structure.
The above-mentioned methods operate on the intrinsic, $2$-dimensional geometry
of the mesh. A popular alternative for embedded meshes is to define
convolutions in the embedding space $\mathbb{R}^{3}$. This can for instance be
done by voxelizing space and representing the mesh in terms of an occupancy
grid (Wu et al., 2015; Tchapmi et al., 2017; Hanocka et al., 2018). A downside
of this approach are the high memory and compute requirements of voxel
representations. If the grid occupancy is low, this can partly be addressed by
resorting to an inhomogeneous grid density (Riegler et al., 2017). Instead of
voxelizing space, one may interpret the set of mesh vertices as a point cloud
and run a convolution on those (Qi et al., 2017a; b). Point cloud based
methods can be made equivariant w.r.t. the isometries of $\mathbb{R}^{3}$
(Zhao et al., 2019b; Thomas et al., 2018), which implies in particular the
isometry equivariance on the embedded mesh. In general, geodesic distances
within the manifold differ usually substantially from the distances in the
embedding space. Which approach is more suitable depends on the particular
application.
On flat Euclidean spaces our method corresponds to Steerable CNNs (Cohen &
Welling, 2017; Weiler et al., 2018a; Weiler & Cesa, 2019; Cohen et al., 2019a;
Lang & Weiler, 2020). As our model, these networks process geometric feature
fields of types $\rho$ and are equivariant under gauge transformations,
however, due to the flat geometry, the parallel transporters become trivial.
Jenner & Weiler (2021) extended the theory of steerable CNNs to include
equivariant partial differential operators. Regular nonlinearities are on flat
spaces used in group convolutional networks (Cohen & Welling, 2016; Weiler et
al., 2018b; Hoogeboom et al., 2018; Bekkers et al., 2018; Winkels & Cohen,
2018; Worrall & Brostow, 2018; Worrall & Welling, 2019; Sosnovik et al.,
2020).
## 6 Experiments
### 6.1 Embedded MNIST
We first investigate how Gauge Equivariant Mesh CNNs perform on, and
generalize between, different mesh geometries. For this purpose we conduct
simple MNIST digit classification experiments on embedded rectangular meshes
of $28\\!\times\\!28$ vertices. As a baseline geometry we consider a flat mesh
as visualized in figure 5(a). A second type of geometry is defined as
different _isometric_ embeddings of the flat mesh, see figure 5(b). Note that
this implies that the _intrinsic_ geometry of these isometrically embedded
meshes is indistinguishable from that of the flat mesh. To generate geometries
which are intrinsically curved, we add random normal displacements to the flat
mesh. We control the amount of curvature by smoothing the resulting
displacement fields with Gaussian kernels of different widths $\sigma$ and
define the roughness of the resulting mesh as $3-\sigma$. Figures 5(c)-5(h)
show the results for roughnesses of 0.5, 1, 1.5, 2, 2.25 and 2.5. For each of
the considered settings we generate $32$ different train and $32$ test
geometries.
To test the performance on, and generalization between, different geometries,
we train equivalent GEM-CNN models on a flat mesh and meshes with a roughness
of 1, 1.5, 2, 2.25 and 2.5. Each model is tested individually on each of the
considered test geometries, which are the flat mesh, isometric embeddings and
curved embeddings with a roughness of 0.5, 1, 1.25, 1.5, 1.75, 2, 2.25 and
2.5. Figure 3 shows the test errors of the GEM-CNNs on the different train
geometries (different curves) for all test geometries (shown on the x-axis).
Since our model is purely defined in terms of the intrinsic geometry of a
mesh, it is expected to be insensitive to isometric changes in the embeddings.
This is empirically confirmed by the fact that the test performances on flat
and isometric embeddings are exactly equal. As expected, the test error
increases for most models with the surface roughness. Models trained on more
rough surfaces are hereby more robust to deformations. The models generalize
well from a rough training to smooth test geometry up to a training roughness
of 1.5. Beyond that point, the test performances on smooth meshes degrades up
to the point of random guessing at a training roughness of 2.5.
As a baseline, we build an _isotropic_ graph CNN with the same network
topology and number of parameters ($\approx 163k$). This model is insensitive
to the mesh geometry and therefore performs exactly equal on all surfaces.
While this enhances its robustness on very rough meshes, its test error of
$19.80\pm 3.43\%$ is an extremely bad result on MNIST. In contrast, the use of
anisotropic filters of GEM-CNN allows it to reach a test error of only
$0.60\pm 0.05\%$ on the flat geometry. It is therefore competitive with
conventional CNNs on pixel grids, which apply anisotropic kernels as well.
More details on the datasets, models and further experimental setup are given
in appendix E.1.
### 6.2 Shape Correspondence
As a second experiment, we perform non-rigid shape correspondence on the FAUST
dataset (Bogo et al., 2014), following Masci et al. (2015) 444These
experiments were executed on QUVA machines. . The data consists of 100 meshes
of human bodies in various positions, split into 80 train and 20 test meshes.
The vertices are registered, such that vertices on the same position on the
body, such as the tip of the left thumb, have the same identifier on all
meshes. All meshes have $6890$ vertices, making this a $6890$-class
segmentation problem.
The architecture transforms the vertices’ ${XYZ}$ coordinates (of type
$3\rho_{0}$), via 6 convolutional layers to features $64\rho_{0}$, with
intermediate features $16(\rho_{0}\oplus\rho_{1}\oplus\rho_{2})$, with
residual connections and the RegularNonlinearity with $N=5$ samples.
Afterwards, we use two $1\\!\times\\!1$ convolutions with ReLU to map first to
256 and then 6980 channels, after which a softmax predicts the registration
probabilities. The $1\\!\times\\!1$ convolutions use a dropout of 50% and 1E-4
weight decay. The network is trained with a cross entropy loss with an initial
learning rate of 0.01, which is halved when training loss reaches a plateau.
As all meshes in the FAUST dataset share the same topology, breaking the gauge
equivariance in higher layers can actually be beneficial. As shown in (Weiler
& Cesa, 2019), symmetry can be broken by treating non-invariant features as
invariant features as input to the final $1\\!\times\\!1$ convolution.
As baselines, we compare to various models, some of which use more complicated
pipelines, such as (1) the computation of geodesics over the mesh, which
requires solving partial differential equations, (2) pooling, which requires
finding a uniform sub-selection of vertices, (3) the pre-computation of SHOT
features which locally describe the geometry (Tombari et al., 2010), or (4)
post-processing refinement of the predictions. The GEM-CNN requires none of
these additional steps. In addition, we compare to SpiralNet++ (Gong et al.,
2019), which requires all inputs to be similarly meshed. Finally, we compare
to an isotropic version of the GEM-CNN, which reduces to a conventional graph
CNN, as well as a non-gauge-equivariant CNN based on SHOT frames. The results
in table 2 show that the GEM-CNN outperforms prior works and a non-gauge-
equivariant CNN, that isotropic graph CNNs are unable to solve the task and
that for this data set breaking gauge symmetry in the final layers of the
network is beneficial. More experimental details are given in appendix E.2.
Figure 3: Test errors for MNIST digit classification on embedded meshes.
Different lines denote train geometries, x-axis shows test geometries. Regions
are standard errors of the means over 6 runs.
Model | Features | Accuracy (%)
---|---|---
ACNN (Boscaini et al., 2016) | SHOT | 62.4
Geodesic CNN (Masci et al., 2015) | SHOT | 65.4
MoNet (Monti et al., 2016) | SHOT | 73.8
FeaStNet (Verma et al., 2018) | XYZ | 98.7
ZerNet (Sun et al., 2018) | XYZ | 96.9
SpiralNet++ (Gong et al., 2019) | XYZ | 99.8
Graph CNN | XYZ | 1.40$\pm$0.5
Graph CNN | SHOT | 23.80$\pm$8
Non-equiv. CNN (SHOT frames) | XYZ | 73.00$\pm$4.0
Non-equiv. CNN (SHOT frames) | SHOT | 75.11$\pm$2.4
GEM-CNN | XYZ | 99.73$\pm$0.04
GEM-CNN (broken symmetry) | XYZ | 99.89$\pm$0.02
Table 2: Results of FAUST shape correspondence. Statistics are means and
standard errors of the mean of over three runs. All cited results are from
their respective papers.
## 7 Conclusions
Convolutions on meshes are commonly performed as a convolution on their
underlying graph, by forgetting geometry, such as orientation of neighbouring
vertices. In this paper we propose Gauge Equivariant Mesh CNNs, which endow
Graph Convolutional Networks on meshes with anisotropic kernels and parallel
transport. Hence, they are sensitive to the mesh geometry, and result in
equivalent outputs regardless of the arbitrary choice of kernel orientation.
We demonstrate that the inference of GEM-CNNs is invariant under isometric
deformations of meshes and generalizes well over a range of non-isometric
deformations. On the FAUST shape correspondence task, we show that Gauge
equivariance, combined with symmetry breaking in the final layer, leads to
state of the art performance.
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## Appendix A Geometry & Parallel Transport
A gauge, or choice of reference neighbor at each vertex, fully determines the
neighbor orientations $\theta_{pq}$ and the parallel transporters $g_{q\to p}$
along edges. The following two subsections give details on how to compute
these quantities.
### A.1 Local neighborhood geometry
Neighbours $q$ of vertex $p$ can be mapped uniquely to the tangent plane at
$p$ using a map called the Riemannnian logarithmic map, visualized in figure
1. A choice of reference neighbor then determines a reference frame in the
tangent space which assigns polar coordinates to all other neighbors. The
neighbour orientations $\theta_{pq}$ are the angular components of each
neighbor in this polar coordinate system.
We define the tangent space $T_{p}M$ at vertex $p$ as that two dimensional
subspace of $\mathbb{R}^{3}$, which is determined by a normal vector $n$ given
by the area weighted average of the normal vectors of the adjacent mesh faces.
While the tangent spaces are two dimensional, we implement them as being
embedded in the ambient space $\mathbb{R}^{3}$ and therefore represent their
elements as three dimensional vectors. The reference frame corresponding to
the chosen gauge, defined below, allows to identify these 3-vectors by their
coefficient 2-vectors.
Each neighbor $q$ is represented in the tangent space by the vector
$\log_{p}(q)\in T_{p}M$ which is computed via the discrete analog of the
Riemannian logarithm map. We define this map $\log_{p}:{\mathcal{N}}_{p}\to
T_{p}M$ for neighbouring nodes as the projection of the edge vector $q-p$ on
the tangent plane, followed by a rescaling such that the norm
$|\log_{p}(q)|=|q-p|$ is preserved. Writing the projection operator on the
tangent plane as $(\mathbbm{1}-nn^{\top})$, the logarithmic map is thus given
by:
$\displaystyle\log_{p}(q)\ :=\
|q-p|\frac{(\mathbbm{1}-nn^{\top})(q-p)}{|(\mathbbm{1}-nn^{\top})(q-p)|}$ (9)
Geometrically, this map can be seen as “folding” each edge up to the tangent
plane, and therefore encodes the orientation of edges and preserves their
lengths.
The normalized reference edge vector $\log_{p}(q_{0})$ uniquely determines a
right handed, orthonormal reference frame $(e_{p,1},\,e_{p,2})$ of $T_{p}M$ by
setting $e_{p,1}:=\log_{p}(q_{0})/|\log_{p}(q_{0})|$ and $e_{p,2}:=n\times
e_{p,1}$. The angle $\theta_{pq}$ is then defined as the angle of
$\log_{p}(q)$ in polar coordinates corresponding to this reference frame.
Numerically, it can be computed by
$\theta_{pq}\ :=\ \operatorname{atan2}\big{(}e_{p,2}^{\top}\log_{p}(q),\
e_{p,1}^{\top}\log_{p}(q))\big{)}.$
Given the reference frame $(e_{p,1},e_{p,2})$, a 2-tuple of coefficients
$(v_{1},v_{2})\in\mathbb{R}^{2}$ specifies an (embedded) tangent vector
${v_{1}e_{p,1}+v_{2}e_{p,2}}\in T_{p}M\subset\mathbb{R}^{3}$. This assignment
is formally given by the _gauge map_ $E_{p}:\mathbb{R}^{2}\to
T_{p}M\subset\mathbb{R}^{3}$ which is a vector space isomorphism. In our case,
it can be identified with the matrix
$\displaystyle
E_{p}=\left[\begin{array}[]{cc}\rule[-2.15277pt]{0.5pt}{5.38193pt}&\rule[-2.15277pt]{0.5pt}{5.38193pt}\\\
e_{p,1}&e_{p,2}\\\
\rule[0.0pt]{0.5pt}{5.38193pt}&\rule[0.0pt]{0.5pt}{5.38193pt}\end{array}\right]\in\mathbb{R}^{3\times
2}.$ (13)
### A.2 Parallel edge transporters
(a) Parallel transport on a flat mesh.
(b) Parallel transport along an edge of a general mesh.
Figure 4: Parallel transport of tangent vectors $v\in T_{q}M$ at $q$ to
$R(g_{q\to p})v\in T_{p}M$ at $p$ on meshes. On a flat mesh, visualized in
figure 4(a), parallel transport moves a vector such that it stays parallel in
the usual sense on flat spaces. The parallel transporter $g_{q\to
p}=\varphi_{p}-\varphi_{q}$ corrects the transported vector _coefficients_ for
differing gauges at $q$ and $p$. When transporting along the edge of a general
mesh, the tangent spaces at $q$ and $p$ might not be aligned, see figure 4(b).
Before correcting for the relative frame orientation via $g_{q\to p}$, the
tangent space $T_{q}M$, and thus $v\in T_{q}M$, is rotated by an angle
$\alpha$ around $n_{q}\\!\times\\!n_{p}$ such that its normal $n_{q}$
coincides with that of $n_{p}$.
On curved meshes, feature vectors $f_{q}$ and $f_{p}$ at different locations
$q$ and $p$ are expressed in different gauges, which makes it geometrically
invalid to accumulate their information directly. Instead, when computing a
new feature at $p$, the neighboring feature vectors at $q\in{\mathcal{N}}_{p}$
first have to be parallel transported into the feature space at $p$ before
they can be processed. The parallel transport along the edges of a mesh is
determined by the (discrete) Levi-Civita connection corresponding to the
metric induced by the ambient space $\mathbb{R}^{3}$. This connection is given
by parallel transporters $g_{q\to p}\in[0,2\pi)$ on the mesh edges which map
tangent vectors $v_{q}\in T_{q}M$ at $q$ to tangent vectors $R(g_{q\to
p})v_{q}\in T_{p}M$ at $p$. Feature vectors $f_{q}$ of type $\rho$ are
similarly transported to $\rho(g_{q\to p})f_{q}$ by applying the corresponding
feature vector transporter $\rho(g_{q\to p})$.
In order to build some intuition, it is illustrative to first consider
transporters on a planar mesh. In this case the parallel transport can be
thought of as moving a vector along an edge without rotating it. The resulting
abstract vector is then parallel to the original vector in the usual sense on
flat spaces, see figure 4(a). However, if the (transported) source frame at
$q$ disagrees with the target frame at $p$, the _coefficients_ of the
transported vector have to be transformed to the target coordinates. This
coordinate transformation from polar angles $\varphi_{q}$ of $v$ to
$\varphi_{p}$ of $R(g_{q\to p})v$ defines the transporter $g_{q\to
p}=\varphi_{p}-\varphi_{q}$.
On general meshes one additionally has to account for the fact that the
tangent spaces $T_{q}M\subset\mathbb{R}^{3}$ and $T_{p}M\subset\mathbb{R}^{3}$
are usually not parallel in the ambient space $\mathbb{R}^{3}$. The parallel
transport therefore includes the additional step of first aligning the tangent
space at $q$ to be parallel to that at $p$, before translating a vector
between them, see figure 4(b). In particular, given the normals $n_{q}$ and
$n_{p}$ of the source and target tangent spaces $T_{q}M$ and $T_{p}M$, the
source space is being aligned by rotating it via
$R_{\alpha}\in\operatorname{SO}(3)$ by an angle
$\alpha=\arccos(n_{q}^{\top}n_{p})$ around the axis $n_{q}\\!\times\\!n_{p}$
in the ambient space. Denote the rotated source frame by
$(R_{\alpha}e_{q,1},R_{\alpha}e_{q,2})$ and the target frame by
$(e_{p,1},e_{p,2})$. The angle to account for the parallel transport between
the two frames, defining the discrete Levi-Civita connection on mesh edges, is
then found by computing
$g_{q\to p}\ =\
\operatorname{atan2}\big{(}(R_{\alpha}e_{q,2})^{\\!\top}\\!e_{p,1},\;(R_{\alpha}e_{q,1})^{\\!\top}\\!e_{p,1}\big{)}.$
(14)
In practice we precompute these connections before training a model.
Under gauge transformations by angles $g_{p}$ at $p$ and $g_{q}$ at $q$ the
parallel transporters transform according to
$g_{q\to p}\ \mapsto\ g_{p}+g_{q\to p}-g_{q}\,.$ (15)
Intuitively, this transformation states that a transporter in a transformed
gauge is given by a gauge transformation back to the original gauge via
$-g_{q}$ followed by the original transport by $g_{q\to p}$ and a
transformation back to the new gauge via $g_{p}$.
For more details on discrete connections and transporters, extending to
arbitrary paths e.g. over faces, we refer to (Lai et al., 2009; Crane et al.,
2010; 2013).
## Appendix B Deriving the Kernel Constraint
Given an input type $\rho_{\textup{in}}$, corresponding to vector space
$V_{\textup{in}}$ of dimension $C_{\textup{in}}$ and output type
$\rho_{\textup{out}}$, corresponding to vector space $V_{\textup{out}}$ of
dimension $C_{\textup{out}}$, we have kernels
$K_{\textup{self}}\in\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$ and
$K_{\textup{neigh}}:[0,2\pi)\to\mathbb{R}^{C_{\textup{out}}\times
C_{\textup{in}}}$. Following Cohen et al. (2019b); Weiler et al. (2021), we
can derive a constraint on these kernels such that the convolution is
invariant.
First, note that for vertex $p\in M$ and neighbour $q\in{\mathcal{N}}_{p}$,
the coefficients of a feature vector $f_{p}$ at $p$ of type $\rho$ transforms
under gauge transformation $f_{p}\mapsto\rho(-g)f_{p}$. The angle
$\theta_{pq}$ gauge transforms to $\theta_{pq}-g$.
Next, note that $\hat{f}_{q}:=\rho_{\textup{in}}(g_{q\to p})f_{q}$ is the
input feature at $q$ parallel transported to $p$. Hence, it transforms as a
vector at $p$. The output of the convolution $f^{\prime}_{p}$ is also a
feature at $p$, transforming as $\rho_{\textup{out}}(-g)f^{\prime}_{p}$.
The convolution then simply becomes:
$f^{\prime}_{p}=K_{\textup{self}}f_{p}+\sum_{q}K_{\textup{neigh}}(\theta_{pq})\hat{f_{q}}$
Gauge transforming the left and right hand side, and substituting the equation
in the left hand side, we obtain:
$\displaystyle\rho_{\textup{out}}(-g)f_{p}^{\prime}=$
$\displaystyle\rho_{\textup{out}}(-g)\left(K_{\textup{self}}f_{p}+\sum_{q}K_{\textup{neigh}}(\theta_{pq})\hat{f_{q}}\right)=$
$\displaystyle
K_{\textup{self}}\rho_{\textup{in}}(-g)f_{p}+\sum_{q}K_{\textup{neigh}}(\theta_{pq}-g)\rho_{\textup{in}}(-g)\hat{f_{q}}$
Which is true for any features, if $\forall g\in[0,2\pi),\theta\in[0,2\pi)$:
$\displaystyle K_{\textup{neigh}}(\theta-g)$
$\displaystyle=\rho_{\textup{out}}(-g)\;K_{\textup{neigh}}(\theta)\;\rho_{\textup{in}}(g),$
(16) $\displaystyle K_{\textup{self}}$
$\displaystyle=\rho_{\textup{out}}(-g)\;K_{\textup{self}}\;\rho_{\textup{in}}(g).$
(17)
where we used the orthogonality of the representations
$\rho(-g)=\rho(g)^{-1}$.
## Appendix C Solving the Kernel Constraint
As also derived in (Weiler & Cesa, 2019; Lang & Weiler, 2020), we find all
angle-parametrized linear maps between $C_{\textup{in}}$ dimensional feature
vector of type $\rho_{\textup{in}}$ to a $C_{\textup{out}}$ dimensional
feature vector of type $\rho_{\textup{out}}$, that is,
$K:S^{1}\to\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$, such that the
above equivariance constraint holds. We will solve for
$K_{\textup{neigh}}(\theta)$ and discuss $K_{\textup{self}}$ afterwards.
The irreducible real representations (irreps) of $\operatorname{SO}(2)$ are
the one dimensional trivial representation $\rho_{0}(g)=1$ of order zero and
$\forall n\in{\mathbb{N}}$ the two dimensional representations of order n:
$\rho_{n}:\operatorname{SO}(2)\to\operatorname{GL}(2,\mathbb{R}):g\mapsto\begin{pmatrix}\cos
ng&-\sin ng\\\ \sin ng&\phantom{-}\cos ng\end{pmatrix}.$
Any representation $\rho$ of $\operatorname{SO}(2)$ of $D$ dimensions can be
written as a direct sum of irreducible representations
$\displaystyle\rho$
$\displaystyle\cong\rho_{l_{1}}\oplus\rho_{l_{2}}\oplus...$
$\displaystyle\rho(g)$
$\displaystyle=A(\rho_{l_{1}}\oplus\rho_{l_{2}}\oplus...)(g)A^{-1}.$
where $l_{i}$ denotes the order of the irrep, $A\in\mathbb{R}^{D\times D}$ is
some invertible matrix and the direct sum $\oplus$ is the block diagonal
concatenations of the one or two dimensional irreps. Hence, if we solve the
kernel constraint for all irrep pairs for the in and out representations, the
solution for arbitrary representations, can be constructed. We let the input
representation be irrep $\rho_{n}$ and the output representation be irrep
$\rho_{m}$. Note that $K(g^{-1}\theta)=(\rho_{\textup{reg}}(g)[K])(\theta)$
for the infinite dimensional regular representation of $\operatorname{SO}(2)$,
which by the Peter-Weyl theorem is equal to the infinite direct sum
$\rho_{0}\oplus\rho_{1}\oplus...$.
Using the fact that all $\operatorname{SO}(2)$ irreps are orthogonal, and
using that we can solve for $\theta=0$ and from the kernel constraints we can
obtain $K(\theta)$, we see that Eq. 16 is equivalent to
$\hat{\rho}(g)K:=(\rho_{\textup{reg}}\otimes\rho_{n}\otimes\rho_{m})(g)K=K$
where $\otimes$ denotes the tensor product, we write $K:=K(\theta)$ and filled
in $\rho_{\textup{out}}=\rho_{m},\;\rho_{\textup{in}}=\rho_{n}$. This
constraint implies that the space of equivariant kernels is exactly the
trivial subrepresentation of $\hat{\rho}$. The representation $\hat{\rho}$ is
infinite dimensional, though, and the subspace can not be immediately
computed.
For $\operatorname{SO}(2)$, we have that for $n\geq 0$,
$\rho_{n}\otimes\rho_{0}=\rho_{n}$, and for $n,m>0$,
$\rho_{n}\otimes\rho_{m}\cong\rho_{n+m}\oplus\rho_{|n-m|}$. Hence, the trivial
subrepresentation of $\hat{\rho}$ is a subrepresentation of the finite
representation
$\tilde{\rho}:=(\rho_{n+m}\oplus\rho_{|n-m|})\otimes\rho_{n}\otimes\rho_{m}$,
itself a subrepresentation of $\hat{\rho}$.
As $\operatorname{SO}(2)$ is a connected Lie group, any
$g\in\operatorname{SO}(2)$ can be written as $g=\exp tX$ for $t\in\mathbb{R}$,
$X\in\mathfrak{so}(2)$, the Lie algebra of $\operatorname{SO}(2)$, and
$\exp:\mathfrak{so}(2)\to\operatorname{SO}(2)$ the Lie exponential map. We can
now find the trivial subrepresentation of $\tilde{\rho}$ looking
infinitesimally, finding
$\displaystyle\tilde{\rho}(\exp tX)K$ $\displaystyle=K$
$\displaystyle\Longleftrightarrow d\tilde{\rho}(X)K:=\frac{\partial}{\partial
t}\tilde{\rho}(\exp tX)|_{t=0}K$ $\displaystyle=0$
where we denote $d\tilde{\rho}$ the Lie algebra representation corresponding
to Lie group representation $\tilde{\rho}$. $\operatorname{SO}(2)$ is one
dimensional, so for any single $X\in\mathfrak{so}(2)$, $K$ is an equivariant
map from $\rho_{m}$ to $\rho_{n}$, if it is in the null space of matrix
$d\tilde{\rho}(X)$. The null space can be easily found using a computer
algebra system or numerically, leading to the results in table 1.
## Appendix D Equivariance
The GEM-CNN is by construction equivariant to gauge transformations, but
additionally satisfies two important properties. Firstly, it only depends on
the intrinsic shape of the 2D mesh, not how the mesh vertices are embedded in
$\mathbb{R}^{3}$, since the geometric quantities like angles $\theta_{pq}$ and
parallel transporters depend solely on the intrinsic properties of the mesh.
This means that a simultaneous rotation or translation of all vertex
coordinates, with the input signal _moving along_ with the vertices, will
leave the convolution output at the vertices unaffected.
The second property is that if a mesh has an orientation-preserving mesh
isometry, meaning that we can map between the vertices preserving the mesh
structure, orientations and all distances between vertices, the GEM-CNN is
equivariant with respect to moving the signal along such a transformation. An
(infinite) 2D grid graph is an example of a mesh with orientation-preserving
isometries, which are the translations and rotations by 90 degrees. Thus a
GEM-CNN applied to such a grid has the same equivariance properties a G-CNN
(Cohen & Welling, 2016) applied to the grid.
### D.1 Proof of Mesh Isometry Equivariance
Throughout this section, we denote $p^{\prime}=\phi(p),q^{\prime}=\phi(q)$. An
orientation-preserving mesh isometry is a bijection of mesh vertices
$\phi:{\mathcal{V}}\to{\mathcal{V}}$, such that:
* •
Mesh faces are one-to-one mapped to mesh faces. As an implication, edges are
one-to-one mapped to edges and neighbourhoods to neighbourhoods.
* •
For each point $p$, the differential $d\phi_{p}:T_{p}M\to T_{p^{\prime}}M$ is
orthogonal and orientation preserving, meaning that for two vectors
$v_{1},v_{2}\in T_{p}M$, the tuple $(v_{1},v_{2})$ forms a right-handed basis
of $T_{p}M$, then $(d\phi_{p}(v_{1}),d\phi_{p}(v_{2}))$ forms a right-handed
basis of $T_{p^{\prime}}M$.
###### Lemma D.1.
Given a orientation-preserving isometry $\phi$ on mesh $M$, with on each
vertex a chosen reference neighbour $q^{p}_{0}$, defining a frame on the
tangent plane, so that the log-map $\log_{p}q$ has polar angle
$\theta^{p}_{q}$ in that frame. For each vertex $p$, let
$g_{p}=\theta^{p^{\prime}}_{\phi(q_{0}^{p})}$. Then for each neighbour
$q\in{\mathcal{N}}_{p}$, we have
$\theta^{p^{\prime}}_{q^{\prime}}=\theta^{p}_{q}+g_{p}$. Furthermore, we have
for parallel transporters that $g_{q^{\prime}\to p^{\prime}}=g_{q\to
p}-g_{p}+g_{q}$.
###### Proof.
For any $v\in T_{p}M$, we have that
$\phi(\exp_{p}(v))=\exp_{p^{\prime}}(d\phi_{p}(v))$ (Tu, 2017, Theorem 15.2).
Thus
$\phi(\exp_{p}(\log_{p}q))=q^{\prime}=\exp_{p^{\prime}}(d\phi_{p}(\log_{p}q))$.
Taking the log-map at $p^{\prime}$ on the second and third expression and
expressing in polar coordinates in the gauges, we get
$(r^{p^{\prime}}_{q^{\prime}},\theta^{p^{\prime}}_{q^{\prime}})=d\phi_{p}(r^{p}_{q},\theta^{p}_{q})$.
As $\phi$ is an orientation-preserving isometry, $d\phi_{p}$ is a special
orthogonal linear map $\mathbb{R}^{2}\to\mathbb{R}^{2}$ when expressed in the
gauges. Hence $d\phi_{p}(r,\theta)=(r,\theta+z_{p})$ for some angle $z_{p}$.
Filling in $\theta^{p}_{q^{p}_{0}}=0$, we find $z_{p}=g_{p}$, proving the
first statement. The second statement follows directly from the fact that
parallel transport $q\to p$, then push-forward along $\phi$ to $p^{\prime}$
yields the same first pushing forward from $q$ to $q^{\prime}$ along $\phi$,
then parallel transporting $q^{\prime}\to p^{\prime}$ (Gallier & Quaintance,
2020, Theorem 18.3 (2)). ∎
For any feature $f$ of type $\rho$, we can define a push-forward along $\phi$
as $\phi_{*}(f)_{p^{\prime}}=\rho(-g_{p})f_{p}$.
###### Theorem D.1.
Given GEM-CNN convolution $K\star\cdot$ from a feature of type
$\rho_{\textup{in}}$ to a feature of type $\rho_{\textup{out}}$, we have that
$K\star\phi_{*}(f)=\phi_{*}(K\star f)$.
###### Proof.
$\displaystyle\phi_{*}(K\star f)_{p^{\prime}}$
$\displaystyle=\rho_{\textup{out}}(-g_{p})\left(K_{\textup{self}}f_{p}+\sum\nolimits_{q\in{\mathcal{N}}_{p}}K_{\textup{neigh}}(\theta_{pq})\rho_{\textup{in}}(g_{q\to
p})f_{q}\right)$
$\displaystyle=\rho_{\textup{out}}(-g_{p})\left(K_{\textup{self}}f_{p}+\sum\nolimits_{q^{\prime}\in{\mathcal{N}}_{p^{\prime}}}K_{\textup{neigh}}(\theta_{p^{\prime}q^{\prime}}-g_{p})\rho_{\textup{in}}(g_{q^{\prime}\to
p^{\prime}}+g_{p}-g_{q})f_{q}\right)$
$\displaystyle=\rho_{\textup{out}}(-g_{p})\left(K_{\textup{self}}f_{p}+\sum\nolimits_{q^{\prime}\in{\mathcal{N}}_{p^{\prime}}}K_{\textup{neigh}}(\theta_{p^{\prime}q^{\prime}}-g_{p})\rho_{\textup{in}}(g_{p})\rho_{\textup{in}}(g_{q^{\prime}\to
p^{\prime}})\rho_{\textup{in}}(-g_{q})f_{q}\right)$
$\displaystyle=K_{\textup{self}}\rho_{\textup{in}}(-g_{p})f_{p}+\sum\nolimits_{q^{\prime}\in{\mathcal{N}}_{p^{\prime}}}K_{\textup{neigh}}(\theta_{p^{\prime}q^{\prime}})\rho_{\textup{in}}(g_{q^{\prime}\to
p^{\prime}})\rho_{\textup{in}}(-g_{q})f_{q}$
$\displaystyle=(K\star\phi_{*}(f))_{p^{\prime}}$
where in the second line we apply lemma D.1 and the fact that $\phi$ gives a
bijection of neighbourhoods of $p$, in the third line we use the functoriality
of $\rho$ and in the fourth line we apply the kernel constraints on
$K_{\textup{self}}$ and $K_{\textup{neigh}}$. ∎
## Appendix E Additional details on the experiments
### E.1 Embedded MNIST
(a) Flat embedding
(b) Isometric embedding
(c) Curved, roughness = 0.5
(d) Curved, roughness = 1
(e) Curved, roughness = 1.5
(f) Curved, roughness = 2
(g) Curved, roughness = 2.25
(h) Curved, roughness = 2.5
Figure 5: Examples of different grid geometries on which the MNIST dataset is
evaluated. All grids have $28\\!\times\\!28$ vertices but are embedded
differently in the ambient space. Figure 5(a) shows a flat embedding,
corresponding to the usual pixel grid. The grid in Figure 5(b) is isometric to
the flat embedding, its internal geometry is indistinguishable from that of
the flat embedding. Figures 5(c)-5(h) show curved geometries which are not
isometric to the flat grid. They are produced by a random displacement of each
vertex in its normal direction, followed by a smoothing of displacements.
To create the intrinsically curved grids we start off with the flat,
rectangular grid, shown in figure 5(a), which is embedded in the $XY$-plane.
An independent displacement for each vertex in $Z$-direction is drawn from a
uniform distribution. A subsequent smoothing step of the normal displacements
with a Gaussian kernel of width $\sigma$ yields geometries with different
levels of curvature. Figures 5(c)-5(h) show the results for standard
deviations of 2.5, 2, 1.5, 1, 0.75 and 0.5 pixels, which are denoted by their
roughness $3-\sigma$ as 0.5, 1, 1.5, 2, 2.25 and 2.5. In order to facilitate
the generalization between different geometries we normalize the resulting
average edge lengths.
The same GEM-CNN is used on all geometries. It consists of seven convolution
blocks, each of which applies a convolution, followed by a RegularNonlinearity
with $N=7$ orientations, batch normalization and dropout of 0.1. This depth is
chosen since GEM-CNNs propagate information only between direct neighbors in
each layer, such that the field of view after 7 layers is $2\times 7+1=15$
pixel. The input and output types of the network are scalar fields of
multiplicity 1 and 64, respectively, which transform under the trivial
representation and ensure a gauge invariant prediction. All intermediate
layers use feature spaces of types $M\rho_{0}\oplus M\rho_{1}\oplus
M\rho_{2}\oplus M\rho_{3})$ with $M=4,\ 8,\ 12,\ 16,\ 24,\ 32$. After a
spatial max pooling, a final linear layer maps the 64 resulting features to 10
neurons, on which a softmax function is applied. The model has 163k
parameters. A baseline GCN, applying by isotropic kernels, is defined by
replacing the irreps $\rho_{i}$ of orders $i\geq 1$ with trivial irreps
$\rho_{0}$ and rescaling the width of the model such that the number of
parameters is preserved. All models are trained for 20 epochs with a weight
decay of 1E-5 and an initial learning rate of 1E-2. The learning rate is
automatically decayed by a factor of 2 when the validation loss did not
improve for 3 epochs.
The experiments were run on a single TitanX GPU.
### E.2 Shape Correspondence experiment
All experiments were ran on single RTX 2080TI GPUs, requiring 3 seconds /
epoch.
The non-gauge-equivariant CNN uses as gauges the SHOT local reference frames
(Tombari et al., 2010). For one input and output channel, it has features
$f_{p}\in\mathbb{R}$ convolution and weights $w\in\mathbb{R}^{2B+2}$, for
$B\in{\mathbb{N}}$. The convolution is:
$(K\star
f)_{p}=w_{0}f_{p}+\sum_{q\in{\mathcal{N}}_{p}}\left(w_{1}+\sum_{n=1}^{B}(w_{2n}\cos(n\theta_{pq})+w_{2n+1}\sin(n\theta_{pq})\right)f_{q}.$
(18)
This convolution kernel is an unconstranied band-limited spherical function.
This is then done for $C_{\textup{in}}$ input channels and $C_{\textup{out}}$
output channels, giving $(2B+2)C_{\textup{in}}C_{\textup{out}}$ parameters per
layer. In our experiments, we use $B=2$ and 7 layers, with ReLU non-
linearities and batch-norm, just as for the gauge equivariant convolution.
After hyperparameter search in $\\{16,32,64,128,256\\}$, we found 128 channels
to perform best.
## Appendix F Additional experiments
### F.1 RegularNonLinearity computational cost
Number of samples | Time / epoch (s) | Memory (GB)
---|---|---
none | 21.2 | 1.22
1 | 21.9 | 1.22
5 | 21.6 | 1.23
10 | 21.5 | 1.24
20 | 22.0 | 1.27
50 | 21.7 | 1.35
Table 3: Run-time of one epoch training and validation and max memory usage of
FAUST model without RegularNonLinearity of with varying number of samples used
in the non-linearity. The hyperparameters are modified to have batch size 1.
In table 3, we show the computational cost of the RegularNonLinearity,
computed by training and computing validation errors for 10 epochs. The run-
time is not significantly affected, but memory usage is.
### F.2 Equivariance Errors
In this experiment, we evaluate empirically equivariance to three kinds of
transformations: gauge transformations, transformations of the vertex
coordinates and transformations under isometries of the mesh, as introduced
above in appenndix D. We do this on two data sets: the icosahedron, a platonic
solid of 12 vertices, referred to in the plots as ’Ico’; and the deformed
icosahedron, in which the vertices have been moved away from the origin by a
factor of sampled from ${\mathcal{N}}(1,0.01)$, referred to in the plots as
’Def. Ico’. We evaluate this on the GEM-CNN (7 layers, 101 regular samples,
unless otherwise noted in the plots) and the Non-Equivariance CNN based on
SHOT frames introduced above in Eq. 18 (7 layers unless otherwise noted in the
plots). Both models have 16 channels input and 16 channels output. The
equivariance model has scalar features as input and output and intermediate
activations with band limit 2 with multiplicity 16. The non-equivariant model
has hidden activations of 16 dimensions. If not for the finite samples of the
RegularNonLinearity, the equivariant model should be exactly gauge invariant
and invariant to isometries. Both models use batchnorm, in order to evaluate
deeper models.
#### F.2.1 Gauge Equivariance
Figure 6: Equivariance error to gauge transformation.
We evaluate gauge equivariance by randomly initialising a model, randomly
sampling input features. We also sample 16 random gauge transformations at
each point. We compare the outputs of the model based on the different gauges.
As the input and output features of the equivariant model are scalars, the
outputs should coincide. This process is repeated 10 times. For the non-
equivariant model, we compute frames based on SHOT and then randomly rotate
these.
The equivariance error is quantified by as:
$\sqrt{\frac{\mathbb{E}_{\Phi,f}\mathbb{E}_{p,c}\mathrm{Var}_{g}(\Phi_{g}(f)_{p,c})}{\mathrm{Var}_{\Phi,f,p,c}(\Phi_{g_{0}}(f)_{p,c})}}$
(19)
where $\Phi_{g}(f)_{c}$ denotes the model $\Phi$ with gauge transformed by $g$
applied to input $f$ then taken the $c$-th channel, $\mathbb{E}_{\Phi,f}$
denotes the expectation over model initialisations and random inputs, for
which we take 10 samples, $\mathbb{E}_{p,c}$ denotes averaging over the 12
vertices and 16 output channels, $\mathrm{Var}_{g}$ denotes the variance over
the different gauge transformations, $\mathrm{Var}_{\Phi,f,c}$ takes the
variance over the models, inputs and channels, and $g_{0}$ denotes one of the
sampled gauge transformations. This quantity indicates how much the gauge
transformation affects the output, normalized by how much the model
initialisations and initial parameters affect the output.
Results are shown in Figure 6. As expected, the non-equivariant model is not
equivariant to gauge transformations. The equivariant model approaches gauge
equivariance as the number of samples of the Regular NonLinearity increases.
As expected, the error to gauge equivariance accumulate as the number of
layers increases. The icosahedron and deformed icosahedron behave the same.
#### F.2.2 Ambient Equivariance
Figure 7: Equivariance error to ambient transformations of the vertex
coordinates.
In this experiment, we measure whether the output is invariant to when all
vertex coordinates are jointly transformed under rotations and translations.
We perform the experiment as above, but sample as transformations $g$ 300
translations and rotations of the ambient space $\mathbb{R}^{3}$. We evaluate
again using Eq 19, where $g$ now denotes a ambient transformation.
Results are shown in Figure 7. We see that the equivariant GEM-CNN is
invariant to these ambient transformations. Somewhat unexpectedly, we see that
the non-equivariant model based on SHOT frames is not invariant. This is
because of an significant failure mode of SHOT frames in particular and
heuristically chosen gauges with a non-gauge-equivariant methods in general.
On some meshes, the heuristic is unable to select a canonical frame, because
the mesh is locally symmetric under (discrete subgroups of) planar rotations.
This is the case for the icosahedron. Hence, SHOT can not disambiguate the X
from the Y axis. The reason this happens in the SHOT local reference frame
selection (Tombari et al., 2010) is the first two singular values of the $M$
matrix are equal, making a choice between the first and second singular
vectors ambiguous. This ambiguity breaks ambient invariance. For the non-
symmetric deformed icosahedron, this problem for the non-equivariant method
disappears.
#### F.2.3 Isometry equivariance
Figure 8: Equivariance error to isometry transformation.
The icosahedron has 60 orientation-preserving isometries. We evaluate
equivariance using:
$\sqrt{\frac{\mathbb{E}_{\Phi,f}\mathbb{E}_{p,c}(\Phi(g(f)_{p,c}-\Phi(f)_{g(p),c})^{2}}{\mathrm{Var}_{\Phi,f,p,c}(\Phi(f)_{p,c})}}$
where $g:M\to M$ is an orientation-preserving isometry, sampled uniformly from
all 60 and $g(f)$ is the transformation of a scalar input feature
$f:M\to\mathbb{R}^{C_{\textup{in}}}$ by pre-composing with $g^{-1}$.
As expected, the non-equivariant model is not equivariant to isometries. The
GEM-CNN is not equivariant to the icosahedral isometries on the deformed
icosahedron, as the deformation removes the symmetry. As the number of Regular
NonLinearity samples increases, the GEM-CNN becomes more equivariant.
Interestingly, the GEM-CNN is equivariant whenever the number of samples is a
multiple of 5. This is because the stabilizer subgroup of the icosahedron at
the vertices is the cyclic group of order 5. Whenever the RegularNonLinearity
has a multiple of 5 samples, it is exactly equivariant to these
transformations.
## Appendix G Equivariance Error Bounds on Regular Non-Linearity
The regular non-linearity acts on each point on the sphere in the following
way. For simplicity, we assume that the representation is $U$ copies of
$\rho_{0}\oplus\rho_{1}\oplus...\oplus\rho_{M}$. One such copy can be treated
as the discrete Fourier modes of a circular signal with band limit $M$. We map
these Fourier modes to $N$ spatial samples with an inverse Discrete Fourier
Transform (DFT) matrix. Then apply to those samples a point-wise non-
linearity, like ReLU, and map back to the Fourier modes with a Discrete
Fourier Transform Matrix.
This procedure is exactly equivariant for gauge transformation with angles
multiple of $2\pi/N$, but approximately equivariant for small rotations in
between.
In equations, we start with Fourier modes
$x_{0},(x_{\alpha}(m),x_{\beta}(m))_{m=1}^{B}$ at some point on the sphere and
result in Fourier modes $z_{0},(z_{\alpha}(m),z_{\beta}(m))_{m=1}^{B}$. We let
$t=0,...,N-1$ index the spatial samples.
$\displaystyle x(t)$
$\displaystyle=x_{0}+\sum_{m}x_{\alpha}(m)\cos\left(\dfrac{2\pi}{N}mt\right)+\ldots$
(20)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{m}x_{\beta}(m)\sin\left(\dfrac{2\pi}{N}mt\right)$
$\displaystyle y(t)$ $\displaystyle=f(x(t))$ $\displaystyle z_{0}$
$\displaystyle=\dfrac{1}{N}\sum_{t}y(t)$ $\displaystyle z_{\alpha}(m)$
$\displaystyle=\dfrac{2}{N}\sum_{t}\cos\left(\dfrac{2\pi}{N}mt\right)y(t)$
$\displaystyle z_{\beta}(m)$
$\displaystyle=\dfrac{2}{N}\sum_{t}\sin\left(\dfrac{2\pi}{N}mt\right)y(t)$
Note that Nyquist’s sampling theorem requires us to pick $N\geq 2B+1$, as
otherwise information is always lost. The normalization is chosen so that
$z_{\alpha}(m)=x_{\alpha}(m)$ if $f$ is the identity.
Now we are interested in the equivariance error between the following two
terms, for small rotation $\delta\in[0,1)$. Any larger rotation can be
expressed in a rotation by a multiple of $2\pi/N$, which is exactly
equivariant, followed by a smaller rotation. We let $z_{\alpha}^{FT}(m)$ be
the resulting Fourier mode if first the input is gauge-transformed and then
the regular non-linearity is applied, and let $z_{\alpha}^{TF}(m)$ be the
result of first applying the regular non-linearity, followed by the gauge
transformation.
$\displaystyle z_{\alpha}^{FT}(m)$
$\displaystyle=\dfrac{2}{N}\sum_{t}\cos\left(\dfrac{2\pi}{N}mt\right)y(t+\delta)$
$\displaystyle=\dfrac{2}{N}\sum_{t}c_{m}(t)y(t+\delta)$ $\displaystyle
z_{\alpha}^{TF}(m)$
$\displaystyle=\dfrac{2}{N}\sum_{t}\cos\left(\dfrac{2\pi}{N}m(t-\delta)\right)y(t)$
$\displaystyle=\dfrac{2}{N}\sum_{t}c_{m}(t-\delta)y(t)$
where we defined for convenience $c_{m}(t)=\cos(2\pi mt/N)$. We define norms
$||x||_{1}=|x_{0}|+\sum_{m}(|x_{\alpha}(m)|+|x_{\beta}(m)|)$ and $||\partial
x||_{1}=\sum_{m}m(|x_{\alpha}(m)|+|x_{\beta}(m)|)$.
###### Theorem G.1.
If the input $x$ is band limited by $B$, the output $z$ is band limited by
$B^{\prime}$, $N$ samples are used and the non-linearity has Lipschitz
constant $L_{f}$, then the error to the gauge equivariance of the regular non-
linearity bounded by:
$\displaystyle||z^{FT}-z^{TF}||_{1}\leq\dfrac{4\pi
L_{f}}{N}\left((2B^{\prime}+\dfrac{1}{2})||\partial
x||_{1}+B^{\prime}(B^{\prime}+1)||x||_{1}\right)$
which goes to zero as $N\to\infty$.
###### Proof.
First, we note, since the Lipschitz constant of the cosine and sine is 1:
$\displaystyle|c_{m}(t-\delta)-c_{m}(t)|$ $\displaystyle\leq\dfrac{2\pi
m\delta}{N}\leq\dfrac{2\pi m}{N}$ $\displaystyle|x(t+\delta)-x(t)|$
$\displaystyle\leq\dfrac{2\pi}{N}\sum_{m}m(|x_{\alpha}(m)|+|x_{\beta}(m)|)$
$\displaystyle\leq\dfrac{2\pi}{N}||\partial x||_{1}$
$\displaystyle|y(t+\delta)-y(t)|$ $\displaystyle\leq
L_{f}\dfrac{2\pi}{N}||\partial x||_{1}$ $\displaystyle|c_{m}(t)|$
$\displaystyle\leq 1$ $\displaystyle|x(t)|$
$\displaystyle\leq|x_{0}|+\sum_{m}(|x_{\alpha}(m)|+|x_{\beta}(m)|)$
$\displaystyle\leq||x||_{1}$ $\displaystyle|y(t)|$ $\displaystyle\leq
L_{f}||x||_{1}$
Then:
$\displaystyle|c_{m}(t)y(t+\delta)-c_{m}(t-\delta)y(t)|$ $\displaystyle=$
$\displaystyle|c_{m}(t)\left[y(t+\delta)-y(t)\right]-y(t)\left[c_{m}(t-\delta)-c_{m}(t)\right]|$
$\displaystyle\leq$
$\displaystyle|c_{m}(t)||y(t+\delta)-y(t)|+|y(t)||c_{m}(t-\delta)-c_{m}(t)|$
$\displaystyle\leq$ $\displaystyle L_{f}\dfrac{2\pi}{N}||\partial
x||_{1}+L_{f}||x||_{1}\dfrac{2\pi m}{N}$ $\displaystyle=$
$\displaystyle\dfrac{2\pi L_{f}}{N}\left(||\partial x||_{1}+m||x||_{1}\right)$
So that finally:
$\displaystyle|z_{\alpha}^{FT}(m)-z_{\alpha}^{TF}(m)|$ $\displaystyle\leq$
$\displaystyle\dfrac{2}{N}\sum_{t}|c_{m}(t)y(t+\delta)-c_{m}(t-\delta)y(t)|$
$\displaystyle\leq$ $\displaystyle\dfrac{4\pi L_{f}}{N}\left(||\partial
x||_{1}+m||x||_{1}\right)$
The sinus component $|z_{\beta}^{FT}(m)-z_{\beta}^{TF}(m)|$ has the same
bound, while $|z_{0}^{FT}-z_{0}^{TF}|=|y(t+\delta)-y(t)|$, which is derived
above. So if $z$ is band-limited by $B^{\prime}$:
$\displaystyle||z^{FT}-z^{TF}||_{1}$ $\displaystyle=|z_{0}^{FT}-z_{0}^{TF}|+$
$\displaystyle\sum_{m=1}^{B^{\prime}}|z_{\alpha}^{FT}(m)-z_{\alpha}^{TF}(m)|+|z_{\beta}^{FT}(m)-z_{\beta}^{TF}(m)|$
$\displaystyle\leq\dfrac{4\pi
L_{f}}{N}\left((2B^{\prime}+\dfrac{1}{2})||\partial
x||_{1}+\sum_{m=1}^{B^{\prime}}2m||x||_{1}\right)$ $\displaystyle=\dfrac{4\pi
L_{f}}{N}\left((2B^{\prime}+\dfrac{1}{2})||\partial
x||_{1}+B^{\prime}(B^{\prime}+1)||x||_{1}\right)$
Since $||\partial x||_{1}=\mathcal{O}(B||x||_{1})$, we get
$||z^{FT}-z^{TF}||_{1}=\mathcal{O}(\frac{BB^{\prime}+{B^{\prime}}^{2}}{N}||x||_{1})$,
which obviously vanishes as $N\to\infty$. ∎
|
2024-09-04T02:54:59.310286 | 2020-03-11T17:36:44 | 2003.05428 | {
"authors": "Mitchell Kinney",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26173",
"submitter": "Mitchell Kinney",
"url": "https://arxiv.org/abs/2003.05428"
} | arxiv-papers | # Template Matching Route Classification
Mitchell Kinney
University of Minnesota - Twin Cities
###### Abstract
This paper details a route classification method for American football using a
template matching scheme that is quick and does not require manual labeling.
Pre-defined routes from a standard receiver route tree are aligned closely
with game routes in order to determine the closest match. Based on a test game
with manually labeled routes, the method achieves moderate success with an
overall accuracy of 72% of the 232 routes labeled correctly.
Keywords— Route Classification, Template Matching, Unsupervised Setting
## 1 Introduction
The world of sports is fully embracing the power of data driven decision
making and analytics. Recently, NFL players have started to wear RFID chips in
their shoulder pads to be able to track their movements on the field during
play. While there are many avenues of exploration with this new wave of data,
in this article automatic labeling of receiver routes is the focus. When
executing a passing play it is important for quarterbacks and receivers to be
coordinated so the ball can arrive on time and safely for a completed catch.
Many pass plays happen every game, and it is of interest when studying game
film to know what routes work well. Automatically labeling the types of routes
these receivers are running would be a major help in understanding what
determines success in a passing play. This paper uses a template matching
scheme to try and minimize the distance between game routes run by the players
and pre-defined routes from a typical route tree. The important aspects of
this method are scaling and translating the pre-defined routes to match
closely with the game routes. Closely matched routes will imply they are of
the same type and the label of the pre-defined route can be assigned to the
game route.
The data used in this paper comes from the NFL’s Big Data Bowl which provided
player tracking data from part of the 2017 NFL season.
## 2 Related work
Previous work in route classification has two main approaches. The first is to
manually label a portion of routes and then use those labels to train a model
to identify the remaining routes. This was done previously in [2] and [4]. In
[2] route characteristics such as the depth of the route before turning, the
direction of the turn and the length of the route after turning were recorded.
Then a training set was built from manually labeled routes and these
characteristics as well as their labels were fed to various models to train
classifiers. In [4] the author made use of a Convolutional Neural Network to
learn the hidden features of the routes after labeling approximately 1,000
routes by hand. The other approach is to use hierarchical clustering to
guarantee similar looking routes share the same label. This has been used in
[1] and [6]. In [1] the authors use a expectation-maximization (EM) algorithm
in a likelihood based approach. The authors assume each receiver trajectory
comes from a distribution with distinct parameters based on known distinct
route features. They attempt to tune the number of these distinct route types
to best separate the routes. In [6] features of the route were extracted and
hierarchical clustering was used. There were two methods shown useful in the
paper. The first used the beginning and ending of the routes as features and
the second used the length of the route before turning, the angle of the turn
and the length of the route after turning as features. The main weakness of
these methods is the amount of time that is needed to tune/ train the models
and the need in each to manually label routes at the beginning or manually
label clusters at the end. While these methods require manual labeling of
routes which the method proposed in this paper does not require.
A method which also uses pre-defined curves to compare routes was introduced
in [3]. The authors used a belief network, which is similar to a naive Bayes
classifier, to classify offensive plays. The pre-defined curves were used as
priors in the network. This method is similar to the one proposed in this
paper because it requires no manual labeling of routes. The goal of the method
in [3] is oriented more towards classifying a whole play rather than
individual routes though.
Figure 1: Empirical cumulative distribution of route cutoff times in seconds
## 3 Wrangling the routes
The data is from the Big Data Bowl competition which released tracking data
from games during the 2017 season. Each player was equipped with a tracking
device and their movements were recorded every 100 milliseconds. For this
method the positions were filtered to only include wide receivers, tight ends,
and running backs that were split out. It was found that running backs in the
backfield do not adhere to the same route tree when running routes because
they have to navigate the offensive and defensive lines some of the time. The
routes begin at the snap so no pre-snap motion was included. Once the ball was
caught by the receiver, incomplete, or intercepted the trajectory collection
was stopped. Also to avoid any creative deviations by players to possibly get
open on a broken play each route was cut off if the route was run for at least
5 seconds, (as was done in [2]). Therefore, the recorded routes ended at the
minimum time between when the pass outcome happened and 5 seconds after the
ball was snapped. A visualization of times in seconds that routes were cut off
in an example game is shown in Figure 1. There is relatively few routes that
reached the full 5 seconds, around 15%. The vast majority fell between 3 and 4
seconds. Finally, the routes were rotated and mirrored as if they were run
from the left side of the ball. There was no distinction between running from
the left or right side of the ball or the direction up or down the field.
Figure 2: Basic receiver route tree used to
classify routes
Figure 3: Game route classified as a ‘post’ run by Bennie Fowler while a
Denver Bronco
## 4 Route Tree
Routes in the NFL are differentiated based on direction changes made when
running up or down the field. Routes are an integral part of a passing play in
the NFL. Only the offense knows where a receiver will run, and these routes
help the quarterback know where a receiver will be, which allows a pass
completion to be made. In the route tree in Figure 3, the difference between
an out route and a dig route is the direction the receiver runs after running
forwards a few yards. Turning towards the center of the field will be a dig
route and turning towards the closest side line will be an out route. Another
difference is the length of the field the receiver runs before changing
direction. In a post route the receiver runs up the field before turning
slightly towards the center of the field and running towards the goal post.
While in a slant route, the receiver runs towards the center of field much
sooner, sometimes without running up the field first. As seen in Bennie
Fowler’s 20 yard post route run in the Denver Broncos versus Los Angeles
Chargers game in 2017 in Figure 3, routes run in a real NFL game have turns
that are not as crisp as seen in the route tree in Figure 3 and the length the
receivers run down the field before turning is not uniform. Therefore, a
method to classify routes must be adjustable to the angle of direction change
and the distance run down the field. The method proposed in this paper
captures this flexibility by scaling pre-defined routes to match closely to
game routes.
Figure 4: Bounding box of the post route from Figure 3
Figure 5: Aligned bounding boxes of an in-game post route (blue) and a
manually created post route (red)
Figure 6: Scaled bounding box of a manually created post route (red) from
Figure 6
## 5 Scaling Pre-Defined Routes
Scaling to match routes is a crucial first step in the method proposed,
because a distance metric is used to classify routes. Each proposed pre-
defined route should be overlaid in such a way that if a pre-defined route
label should be assigned to a game route the distance between the pre-defined
route and the game route is minimal. The only routes that are changed when
scaling/ transforming are the pre-defined routes to align closely with the
game routes. All pre-defined routes have been manually given coordinates that
match the shapes given in Figure 3. The calculations to scale any pre-defined
route only requires the bounding box of the pre-defined route and game route.
The bounding box of a set of two dimensional coordinates is the smallest
rectangle that captures all of the points. A route $R$ is defined as a set of
$(x,y)$ coordinate pairs with cardinality $|T|$ such that:
$R=\\{(x_{1},y_{1}),(x_{2},y_{2}),\dots,(x_{T},y_{T})\\},$ (1)
where $T$ is the number of timesteps the receiver’s position was recorded. The
bounding box of the set $R$ is a four dimensional tuple defined as
$\big{(}\min\limits_{x}R,\min\limits_{y}R,\max\limits_{x}R,\max\limits_{y}R\;\big{)}$.
An example of a bounding box for a route is shown in Figure 6. The scaling
approach used was to find the largest difference between the width and height
ratios of the bounding boxes of the pre-defined route and the game route and
then scale each coordinate in the pre-defined route so the largest difference
would match instead. Let the set of game route coordinates be
$R_{\text{game}}$ and the pre-defined route coordinates be $R_{\text{pre-
defined}}$. The first step in scaling is to translate all the coordinates in
both routes so the minimum $x$ and $y$ coordinates are $(0,0)$ and
$\displaystyle\min\limits_{x}R_{\text{game}}$
$\displaystyle=\min\limits_{x}R_{\text{pre-defined}}=0$ (2)
$\displaystyle\min\limits_{y}R_{\text{game}}$
$\displaystyle=\min\limits_{y}R_{\text{pre-defined}}=0$ (3)
This will align the bottom and left sides of the bounding boxes of the routes
and allow for the correct calculation of the ratio between the horizontal and
vertical direction of the bounding boxes. An example can be seen in Figure 6.
After aligning the bounding boxes the horizontal ratio $r_{h}$ and vertical
ratio $r_{v}$ can be calculated by
$\displaystyle r_{h}$ $\displaystyle=\dfrac{\max\limits_{x}R_{\text{pre-
defined}}}{\max\limits_{x}R_{\text{game}}}$ (4) $\displaystyle r_{v}$
$\displaystyle=\dfrac{\max\limits_{y}R_{\text{pre-
defined}}}{\max\limits_{y}R_{\text{game}}}.$ (5)
The larger ratio is used to scale each coordinate in the pre-defined route.
For instance, if $r_{h}>r_{v}$, then each $(x,y)$ coordinate in the pre-
defined route is multiplied by $r_{h}^{-1}$. Let $R_{\text{scaled}}$ be the
pre-defined route coordinates multiplied by the proper ratio.
$\displaystyle R_{\text{scaled}}=\min(r_{h}^{-1},r_{v}^{-1})\cdot
R_{\text{pre-defined}}.$ (6)
A Scaled dig route
B Scaled post route
Figure 7: Routes scaled using an exact match bounding box (red) over an in
game route run by Emmanuel Sanders while a Denver Bronco (blue) Figure 8: Out
route scaled using a smaller discrepancy bounding box (red) over an in game
route run by Emmanuel Sanders while a Denver Bronco (blue)
An example is shown in Figure 6. This approach was chosen because it maintains
the aspect ratio of the bounding box of the pre-defined route and reduces the
possibility of a pre-defined route not scaling “reasonably.” The aspect ratio
is the ratio between the height and width of the bounding box. Maintaining the
aspect ratio is critical to differentiating routes since direction changes are
what separates routes, for example, separating outs from corners and corners
from streaks. Allowing the aspect ratio to change could change the angles at
the direction changes in the routes so much that two types routes can become
almost indistinguishable. In Figure 7 an example of changing the aspect ratio
when scaling, is shown where a dig game route is closer to a post route than
the correct dig route because the direction change of the scaled pre-defined
post route conforms to the game route. Changing the aspect ratio occurs when
the bounding box of the pre-defined route is scaled to exactly match the game
route as in Figure 7. This angle manipulation is most problematic in routes
such as corners or posts and flats or slants. Matching bounding boxes of the
pre-defined routes and the game route would allow for “perfect” matches in the
ideal scenario but will also change the aspect ratio, sometimes drastically.
The other scaling approach is to minimize the smaller discrepancy which is
simply using the smaller ratio to scale the pre-defined route instead of the
larger ratio. The issue that arises with this approach mainly comes from how
the pre-defined routes are initially plotted. The coordinates of the pre-
defined routes were made much larger than necessary compared to game routes to
guarantee that the pre-defined routes would always scale down (both $x$ and
$y$ coordinates get smaller). This saves an additional logic step that would
be needed to possibly scale pre-defined routes up or down. The large size of
the pre-defined route in comparison to the game route is true of all pre-
defined routes. Matching the larger discrepancy implies that the pre-defined
route bounding box will be shrunk to always be contained within the game route
bounding box as in Figures 6 and 6. This way after scaling, all pre-defined
routes will be approximately similar sizes, whereas scaling to match the
smaller discrepancy might cause erratic behavior. This can be seen in Figure 8
where the scaled pre-defined route is unrealistically large compared to the
game route.
Figure 9: Example of a grid search which shifts the pre-defined route’s
bounding box
incrementally upward over an in game route run by Bennie Fowler while a Denver
Bronco
## 6 Route Classification
To classify the game routes, a simple Euclidean distance is used between the
game route and the scaled pre-defined routes after shifting the scaled pre-
defined route to align as closely as possible. Then the label of the scaled
pre-defined route that is the minimum distance from the game route is used to
also label the game route. To match up the game route and scaled route as
closely as possible, a shift is used on the scaled route. Recall from the
Section 5 that the method of scaling chosen was to minimize the largest
discrepancy. This implies that the bounding box of the scaled route will be
completely contained within the bounding box of the game route each time.
Therefore, a grid search for the optimal position of the scaled pre-defined
route can be done within the bounding box of the game route. An example of a
grid search with a scaled pre-defined out route is shown in Figure 9. Note the
scaled and game route bounding boxes will still be aligned on their bottom and
left sides. If the scaled route used $r_{h}^{-1}$ then the right side of the
bounding boxes will be aligned, and similarly if $r_{v}^{-1}$ was used the top
side of the bounding boxes will be aligned. Therefore the shift on the grid
will either be exclusively vertical or horizontal to keep the scaled route
bounding box within the game route bounding box. The chosen shift step size
was half a yard for this method. The distance that the scaled route can be
shifted is equal to $w$ defined as
$\displaystyle
w=\max\big{(}|\max\limits_{x}R_{\text{scaled}}-\max\limits_{x}R_{\text{game}}|,|\max\limits_{y}R_{\text{scaled}}-\max\limits_{y}R_{\text{game}}|\big{)}.$
(7)
Figure 10: Representation of $w$ which is the distance between the remaining
discrepancy between the bounding boxes of the in game route (blue) and scaled
pre-defined route (red)
Figure 11: Visual of how the original pre-defined route (left) gets points
added (right) to equal the number of points of the game route
One of the two elements in the max will be zero, so $w$ is equal to whichever
is positive. The number of steps taken is equal to the ceiling of
$\;\dfrac{w}{0.5}$. If the tops of the bounding boxes are aligned, then the
distances between the game and scaled routes will be measured after shifting
the $x$-coordinate of the scaled route $\\{0,0.5,\dots,w_{0.5}\\}$, where
$w_{0.5}$ is $w$ rounded down to the nearest $0.5$ increment. If the right
sides of the bounding boxes are aligned, the $y$-coordinate of the scaled
route will be shifted. A visual of $w$ is shown in Figure 11.
The distance at each step is calculated by measuring the distance between
every coordinate in the game route to the closest point on the line of the
scaled route and adding the distance between every coordinate in the scaled
route to the closest point on the line of the game route. Let $\ell_{i,i+1}$
be the line segment between coordinates $(x_{i},y_{i})$ and
$(x_{i+1},y_{i+1})$ for $i\in 1,\dots,T-1$, where $T$ is still the number of
coordinates recorded in the game route. Points are artificially added to the
scaled routes until $R_{\text{scaled}}$ has the same cardinality as
$R_{\text{game}}$; $|T|$. These points are placed evenly on the route as shown
in Figure 11. These added points do not affect the bounding box of the scaled
route. Then the collection of line segments that make up a route is defined as
$\displaystyle L=\\{\ell_{1,2},\ell_{2,3},\dots,\ell_{T-1,T}\\}.$ (8)
Let $\delta\big{(}(x,y),L\big{)}$ be defined as the minimum distance between
the point $(x,y)$ and $L$. Then the distance measurement $D_{\text{game}}$ is
found by summing the minimum distance to the line $L_{\text{scaled}}$ over the
points in $R_{\text{game}}$.
$\displaystyle D_{\text{game}}=\sum_{(x,y)\in
R_{\text{game}}}\delta\big{(}(x,y),L_{\text{scaled}}\big{)}.$ (9)
To avoid misclassification when the game route is close to only part of the
scaled route, as in Figure 13, the same measurement is taken between
coordinates of the scaled route and the line of the game route.
$\displaystyle D_{\text{scaled}}=\sum_{(x,y)\in
R_{\text{scaled}}}\delta\big{(}(x,y),L_{\text{game}}\big{)}.$ (10)
Figure 12: Partial overlap of routes
showing the necessity to calculate
the closest distances between both routes
Figure 13: Distance measured in $D_{\text{scaled}}$ with route from Figure 9
An example of the distance being measured for each coordinate in
$D_{\text{scaled}}$ can be found in Figure 13 using the routes from the
earlier grid search example in Figure 9. The total distance between the scaled
and game routes is summed with a weight $\gamma$ on $D_{\text{scaled}}$.
$\displaystyle D_{\text{route}}=D_{\text{game}}+\gamma D_{\text{scaled}}.$
(11)
Here $D_{\text{route}}$ is the measurement of distance between the game route
and one of the named routes from the route tree in Figure 3. The weight
$\gamma\leq 1$ and is designed to help balance the distance measurements since
the scaled route will necessarily be smaller than the game route because of
the scaling strategy. The weight $\gamma$ is to represent $D_{\text{game}}$
being more important since it is possible for the game route to extend further
than the scaled route while the scaled route lines up extremely closely with
only part of the game route. Weighting $D_{\text{scaled}}$ down will help with
this problem by making the distances calculated from game route coordinates
overlapping with the scaled route line more important. The route name with the
minimum distance among the entire route tree and all shifts will be assigned
to the game route.
For each classification the same pre-defined route tree is used initially.
There is no attempt made to incorporate labeled routes into future predictions
through a process such as active learning where labeling is done while
learning the coefficient space. This is done to prevent the routes used for
labeling from drifting too far from the known truth as described in [5]. This
is a phenomenon seen when the input distribution changes, especially in semi-
supervised problems. An example is in correlation matching in images. The
template being used to match within the image can start to drift away from the
truth if updated regularly. Especially in cases like route labeling when there
is little supervision, it is preferred to guarantee the templates reflect the
truth at all times rather than attempt to leverage game routes that have
already been labeled. This avoids treating a wrong label as truth.
Route | Precision | Recall | Count
---|---|---|---
Corner | 0.36 | 0.76 | 21
Dig | 0.75 | 0.27 | 45
Flat | 0.64 | 0.78 | 23
Out | 0.75 | 0.20 | 30
Post | 0.33 | 0.50 | 22
Slant | 0.67 | 0.76 | 38
Sluggo | 0.33 | 1.0 | 1
Streak | 0.54 | 0.36 | 36
Wheel | 0.33 | 0.29 | 7
Overall | 0.48 | 0.48 | 234
Route | Precision | Recall | Count
---|---|---|---
Corner | 0.54 | 0.95 | 21
Dig | 0.77 | 0.60 | 45
Flat | 0.70 | 0.91 | 23
Out | 0.95 | 0.60 | 30
Post | 0.53 | 0.86 | 22
Slant | 0.76 | 0.82 | 38
Sluggo | 0.25 | 1.0 | 1
Streak | 0.87 | 0.75 | 36
Wheel | 0.67 | 0.29 | 7
Overall | 0.72 | 0.71 | 234
Table 1: Precision and recall of labeled routes for cutoff times of 3 (left)
and 5 (right) seconds from the 2017 season game between the Denver Broncos and
Los Angeles Chargers
## 7 Performance
Overall this method performed well and was able to distinguish between routes
based on their fundamental characteristics. The method was able to handle the
direction differences (e.g. out route versus dig route) and was able to
differentiate routes based on the angle at which the receivers initially
turned. Examples of routes labeled post, corner, out, dig, slant, flat,
streak, sluggo, and wheel can be seen in the Appendix. What stands out is the
difference in when receivers are making their breaks which differentiates the
short routes such as slants and flats, the intermediate routes such as digs
and corners, and the deep routes such as posts and corners. In the flats and
slants many of these captured routes have little to no angle as a break, the
digs and outs show a very sharp turn to the center or sideline, while the
posts and corners show a more gradual turn. This method tends to over
categorize corner and post routes. It can be seen in the Appendix that there
are some dig and out routes that are mislabeled as post and corner routes
respectively because of their more gradual turns. Instead, the method should
be taking into account the ending turn which is much sharper than what would
be expected of a post or corner route.
Figure 14: Normalized confusion matrix of accuracy of routes
A more quantitative way to assess the performance of this method is similar to
the analysis performed in [2], to measure precision and recall. Precision and
recall are calculated for each route label using the number of true positives
($t_{p}$) correctly identified routes of the current label, the number of
false positives ($f_{p}$) other routes mislabeled as the current label, and
false negatives ($f_{n}$) routes of the current label that were misclassified.
They are defined as
Precision $\displaystyle=\dfrac{t_{p}}{t_{p}+f_{p}}\;,$ (12) Recall
$\displaystyle=\dfrac{t_{p}}{t_{p}+f_{n}}\;.$ (13)
Table 1 shows individual precision and recall scores for each route based on a
manually labeled game. The true labels were gathered by systematically
watching and recording a best guess for each route run during the Denver
Broncos versus the Los Angeles Chargers game in the 2017 season. The overall
score shows a moderate success at labeling. Of note is that even though there
were curl and comeback routes in the game ($\leq 5$ of both) there were no
curl or comeback routes predicted. This method will struggle with these routes
because many times when these routes are run the receiver is doubling over
onto the route which does not distinguish itself in this classification
method. Better techniques for classifying these specific routes are left for
future work. Also receivers that were labeled to be either blocking or waiting
for a bubble screen were classified as such and included in the accuracy
measurements but not shown. Blocking or waiting for a bubble screen is
indistinguishable with this method. Receivers were classified as blocking or
waiting for a bubble screen if they did not move more than 4 yards during the
play. The overall accuracy for this game was 72%. The confusion matrix in
Figure 14 shows the normalized categorization probabilities for classified
routes. The greatest mislabeling was outs erroneously labeled as flats. In
Figure 15 the distribution of routes can be seen which shows that tight end
routes were dominated by slants and flats while wide receivers seemed to run
an approximately equal amount of streaks, slants, posts, digs and corners.
These observations align closely with work done in [1].
Figure 15: Distribution of routes by position
Cutting off routes at 3 seconds was also considered because this is a more
natural amount of time a receiver would develop their routes fully. This
resulted in a sharp degradation of performance as can be seen in Table 1.
Recall in Figure 1 this cuts off many routes before the play was complete.
Another possibility would be to cut off the routes at 4 seconds but this too
showed poor performance compared to cutting off routes at 5 seconds.
## 8 Conclusion
This paper presented an unsupervised template matching method that allows for
routes to be classified using a simple distance metric. This method’s main
benefits are the overall speed and no manual labeling of routes is required.
After pre-processing the game routes, the three main parts of this method are
scaling, translating and measuring distances. Each of these operations have
$\mathcal{O}(T)$ complexity where $T$ is the number of points in the game
route. This $T$ is actually capped by the amount of maximum time allowed for
each route, which for this method is 5 seconds or 50 points. When labeling a
full game with 252 routes this method took 303 seconds or approximately 5
minutes. The other benefit is that labeling is done for each route without
having to manually label clusters afterwards or labeling routes beforehand to
use as a training set. Other methods require labeling at some time by humans,
but template matching assigns a label without any human intervention.
Converting raw coordinates of players to route labels is a step in trying to
glean more information from NFL games. The next step is to use these labels
with more standard statistical methods to understand what routes work well in
different situations: Namely how do certain route combinations work against
certain defensive coverage schemes or how does a specific player’s routes work
against various coverage types. This involves labeling individual defensive
players as zone or man, then using that information to imply an overall
coverage scheme. Producing summary statistics about these matchups will
follow. The github url github.com/kinne174 is where this project and others
are stored.
In this paper a template based search criterion was used to automatically
classify routes run by receivers. It was shown that moderate success is
achieved through an appropriate scaling method and translations of the route
to align the pre-defined routes as closely as possible with the game route.
## References
* [1] Chu, Dani, et al. “Route Identification in the National Football League.” arXiv preprint arXiv:1908.02423 (2019).
* [2] Hochstedler and Gagnon. “American Football Route Identification Using Supervised Machine Learning.” MIT Sloan Sports Analytics Conference 2017.
* [3] Intille, Stephen S., and Aaron F. Bobick. “A framework for recognizing multi-agent action from visual evidence.” AAAI/IAAI 99.518-525 (1999): 2.
* [4] Sterken, Nathan. “RouteNet: a Convolutional Nueral Network for Classifying Routes.” NFL Big Data Bowl (2019).
* [5] Quionero-Candela, Joaquin, et al. Dataset shift in machine learning. The MIT Press, 2009.
* [6] Vonder Haar, Adam. “Exploratoy Data Analysis of Passing Plays using NFL Tracking Data.” NFL Big Data bowl (2019).
## Appendix
Examples of game routes that were labeled the same in the Denver Broncos
versus Los Angeles Chargers game during the 2017 season. The magenta line
represents the median route of the group showing this method is able to
partition essential qualities of common routes.
A Corner
B Post
A Out
B Dig
A Flat
B Slant
A Wheel
B Sluggo
|
2024-09-04T02:54:59.326223 | 2020-03-11T18:06:14 | 2003.05465 | {
"authors": "Adrian Chapman and Steven T. Flammia",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26174",
"submitter": "Adrian Chapman",
"url": "https://arxiv.org/abs/2003.05465"
} | arxiv-papers | [1]Centre for Engineered Quantum Systems, School of Physics, The University of
Sydney, Sydney, Australia
# Characterization of solvable spin models via graph invariants
Adrian Chapman<EMAIL_ADDRESS>Steven T. Flammia
(May 27, 2020)
###### Abstract
Exactly solvable models are essential in physics. For many-body
spin-$\mathbf{\nicefrac{{1}}{{2}}}$ systems, an important class of such models
consists of those that can be mapped to free fermions hopping on a graph. We
provide a complete characterization of models which can be solved this way.
Specifically, we reduce the problem of recognizing such spin models to the
graph-theoretic problem of recognizing line graphs, which has been solved
optimally. A corollary of our result is a complete set of constant-sized
commutation structures that constitute the obstructions to a free-fermion
solution. We find that symmetries are tightly constrained in these models.
Pauli symmetries correspond to either: (i) cycles on the fermion hopping
graph, (ii) the fermion parity operator, or (iii) logically encoded qubits.
Clifford symmetries within one of these symmetry sectors, with three
exceptions, must be symmetries of the free-fermion model itself. We
demonstrate how several exact free-fermion solutions from the literature fit
into our formalism and give an explicit example of a new model previously
unknown to be solvable by free fermions.
## 1 Introduction
Exactly solvable models provide fundamental insight into physics without the
need for difficult numerical methods or perturbation theory. In the particular
setting of many-body spin-$\nicefrac{{1}}{{2}}$ systems, a remarkable method
for producing exact solutions involves finding an effective description of the
system by noninteracting fermions. This reduces the problem of solving the
$n$-spin system over its full $2^{n}$-dimensional Hilbert space to one of
solving a single-particle system hopping on a lattice of $O(n)$ sites. The
paradigmatic example of this method is the exact solution for the XY model
[1], where the Jordan-Wigner transformation [2] is employed to describe the
model in terms of free fermions propagating in one spatial dimension. These
fermions are resolved as nonlocal Pauli operators in the spin picture, and the
nonlocal nature of this mapping may suggest that finding generalizations to
this mapping for more complicated spin systems is a daunting task. Of the many
generalizations that have since been proposed [3, 4, 5, 6, 7, 8, 9, 10, 11], a
particularly interesting solution to this problem is demonstrated in the exact
solution of a 2-d spin model on a honeycomb lattice introduced by Kitaev [12].
For this model, the transformation to free-fermions can be made locality-
preserving over a fixed subspace through the use of local symmetries.
The dynamics of free-fermion systems are generated by Gaussian-fermionic
Hamiltonians and correspond to the class of so-called matchgate circuits. This
circuit class coincides with the group of free-fermion propagators generated
by arbitrarily time-dependent single-particle Hamiltonians [13, 14] and has an
extensive complexity-theoretic characterization. In general, matchgate
circuits can be efficiently simulated classically with arbitrary single-qubit
product-state inputs and measurement [15, 16]. However, they become universal
for quantum computation with the introduction of non-matchgates such as the
$\mathrm{SWAP}$ gate [17, 18], certain measurements and resource inputs [19,
20], and when acting on nontrivial circuit geometries [21]. Furthermore, these
circuits share an interesting connection to the problem of counting the number
of perfect matchings in a graph, which is the context in which they were first
developed [22, 23, 24, 25]. This problem is known to be very hard
computationally (it is #P-complete [26]), but is efficiently solved for planar
graphs using the so-called Fisher-Kasteleyn-Temperley algorithm [27, 28].
In this work, we develop a distinct connection between free-fermion systems
and graph theory by using tools from quantum information science. The central
object of our formalism is the _frustration graph_. This is a network
quantifying the anticommutation structure of terms in the spin Hamiltonian
when it is expanded in the basis of Pauli operators [29]. This graph has been
invoked previously in the setting of variational quantum eigensolvers [30, 31,
32, 33, 34, 35, 36, 37], commonly under the name “anti-compatibility graph".
We show that the problem of recognizing whether a given spin model admits a
free-fermion solution is equivalent to that of recognizing whether its
frustration graph is a _line graph_ , which can be performed optimally in
linear time [38, 39, 40]. From the definition of a line graph, it will be
clear that such a condition is necessary, but we will show that it is also
sufficient. When the condition is met, we provide an explicit solution to the
model.
Line graphs have recently emerged as the natural structures describing the
effective tight-binding models for superconducting waveguide networks [41, 42,
43]. In this setting, the line graph corresponds to the physical hopping graph
of photons in the network. We will see how this scenario is a kind of “inverse
problem" to the one we consider, wherein fermions are hopping on the _root_ of
the line graph. It is clear from both scenarios that the topological
connectivity structure of many-body systems plays a central role in their
behavior, and it is remarkable that this is already being observed in
experiments. We expect that further investigation of the graph structure of
many-body Hamiltonians will continue to yield important insights into their
physics.
### 1.1 Summary of Main Results
Here we give a brief summary of the main results. We first define the
frustration graph of a Hamiltonian, given in the Pauli basis, as the graph
with nonzero Pauli terms as vertices and an edge between two vertices if their
corresponding terms anticommute. A line graph $G$ of a graph $R$ is the
intersection graph of the edges of $R$ as two-element subsets of the vertices
of $R$. With these simple definitions, we can informally state our first main
result, which we call our “fundamental theorem:"
###### Result 1 (Existence of free-fermion solution; Informal version of Thm.
1).
Given an $n$-qubit Hamiltonian in the Pauli basis for which the frustration
graph $G$ is the line graph of another graph $R$, then there exists a free-
fermion description of $H$.
From this description, an exact solution for the spectrum and eigenstates of
$H$ can be constructed. This theorem illustrates a novel connection between
the physics of quantum many-body systems and graph theory with some surprising
implications. First, it gives the exact correspondence between the spatial
structure of a spin Hamiltonian and that of its effective free-fermion
description. As we will see through several examples, this relationship is not
guaranteed to be straightforward. Second, the theorem gives an exact condition
by which a spin model can _fail_ to have a free-fermion solution, the culprit
being the presence of forbidden anticommutation structures in the frustration
graph of $H$.
Some caveats to Result 1 (that are given precisely in the formal statement,
Theorem 1) involve cases in which this mapping between Pauli terms in $H$ and
fermion hopping terms is not one-to-one. In particular, if we are given a
Hamiltonian whose frustration graph is not a line graph, then a free-fermion
solution may still be possible via a non-injective mapping over a subspace
defined by fixing stabilizer degrees of freedom. Additionally, it is possible
for a given spin Hamiltonian to describe multiple free-fermion models
simultaneously, each generating dynamics over an independent stabilizer
subspace of the full Hilbert space as for the Kitaev honeycomb model [12].
These symmetries are sometimes referred to as gauge degrees of freedom, though
we will reserve this term for freedoms which cannot affect the physics of the
free-fermion model. Finally, it may be the case that the free-fermion model
contains states which are nonphysical in the spin-Hamiltonian picture, and so
these must be removed by fixing a symmetry as well. Luckily, all of these
cases manifest as structures in the frustration graph of $H$. The first,
regarding when a non-injective free-fermion solution is required, is signified
by the presence of so-called twin vertices, or vertices with the same
neighborhood. We deal with this case in our first lemma. The next two cases
are covered by our second theorem:
###### Result 2 (Graphical symmetries; Informal version of Thm. 2).
Given an $n$-qubit Hamiltonian in the Pauli basis for which the frustration
graph $G$ is the line graph of another graph $R$, then Pauli symmetries of $H$
correspond to either:
1. (i)
Cycles of $R$;
2. (ii)
A T-join of $R$, associated to the fermion-parity operator;
3. (iii)
Logically encoded qubits;
and these symmetries generate an abelian group.
We then prove that we can always fix all of the cycle symmetries by choosing
an orientation of the root graph $R$. Our results also relate the more general
class of Clifford symmetries to the symmetries of the single-particle free-
fermion Hamiltonian. We show that with exactly three exceptions, Clifford
symmetries of the spin model, in a subspace defined by fixing the symmetries
listed above, must also be symmetries of the single-particle Hamiltonian (see
Corollary 1.2 for a precise statement).
Finally, we illustrate these ideas with several examples: small systems of up
to 3 qubits, the 1-dimensional anisotropic $XY$ model in a transverse field
and its nearest-neighbor solvable generalization, the Kitaev honeycomb model,
the 3-dimensional frustrated hexagonal gauge color code [44], and the
Sierpinski-Hanoi model. To the best of our knowledge, this last model was
previously not known to be solvable.
The remainder of the paper is organized as follows. In Section 2, we will
introduce notation and give some background on the formalism of free-fermions
and frustration graphs. In Section 3, we will formally state Theorem 1 and
some general implications thereof. In Section 4, we elaborate on the structure
of symmetries which can be present in our class of solvable models. In Section
4.1, we will use the theorems of the previous two sections to outline an
explicit solution method. We close by demonstrating how the examples of free-
fermion solutions listed above fit into this formalism in Section 5.
## 2 Background
### 2.1 Frustration Graphs
The models we consider are spin-$\nicefrac{{1}}{{2}}$ (qubit) Hamiltonians
written in the Pauli basis
$\displaystyle H=\sum_{\boldsymbol{j}\in
V}h_{\boldsymbol{j}}\sigma^{\boldsymbol{j}}\mathrm{,}$ (1)
where $\boldsymbol{j}\equiv(\boldsymbol{a},\boldsymbol{b})$, with
$\boldsymbol{a}$, $\boldsymbol{b}\in\\{0,1\\}^{\times n}$ labeling an
$n$-qubit Pauli operator as
$\displaystyle\sigma^{\boldsymbol{j}}=i^{\boldsymbol{a}\cdot\boldsymbol{b}}\left(\bigotimes_{k=1}^{n}X_{k}^{a_{k}}\right)\left(\bigotimes_{k=1}^{n}Z_{k}^{b_{k}}\right)\mathrm{.}$
(2)
The exponent of the phase factor, $\boldsymbol{a}\cdot\boldsymbol{b}$, is the
_Euclidean inner product_ between $\boldsymbol{a}$ and $\boldsymbol{b}$. This
phase is chosen such that the overall operator is Hermitian, and such that
$a_{k}=b_{k}=1$ means that $\sigma^{\boldsymbol{j}}$ acts on qubit $k$ by a
Pauli-$Y$ operator. We denote the full $n$-qubit Pauli group by $\mathcal{P}$,
and $V\subseteq\mathcal{P}$ is the set of Pauli terms in $H$ (i.e.
$h_{\boldsymbol{j}}=0$ for all $\boldsymbol{j}\notin V$). Let the Pauli
subgroup generated by this set be denoted $\mathcal{P}_{H}$.
For our purposes, what is important is not the explicit Pauli description of
the Hamiltonian, but rather the commutation relations between its terms. As
Pauli operators only either commute or anticommute, a useful quantity is their
_scalar commutator_ $[\\![\cdot,\cdot]\\!]$, which we define implicitly as
$\displaystyle\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}=[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!]\sigma^{\boldsymbol{k}}\sigma^{\boldsymbol{j}}\mathrm{.}$
(3)
The scalar commutator thus only takes the values $\pm 1$. Additionally, the
scalar commutator distributes over multiplication in each argument, e.g.
$\displaystyle[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}\sigma^{\boldsymbol{l}}]\\!]=[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!][\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{l}}]\\!].$
(4)
For $n$-qubit Paulis, the scalar commutator can thus be read off from the
Pauli labels as
$\displaystyle[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!]=(-1)^{\langle\boldsymbol{j},\boldsymbol{k}\rangle}$
(5)
Here, $\langle\boldsymbol{j},\boldsymbol{k}\rangle$ is the _symplectic inner
product_
$\displaystyle\langle\boldsymbol{j},\boldsymbol{k}\rangle\equiv\begin{pmatrix}\boldsymbol{a}_{j}&\boldsymbol{b}_{j}\end{pmatrix}\begin{pmatrix}\mathbf{0}_{n}&\mathbf{I}_{n}\\\
-\mathbf{I}_{n}&\mathbf{0}_{n}\end{pmatrix}\begin{pmatrix}\boldsymbol{a}_{k}\\\
\boldsymbol{b}_{k}\end{pmatrix}\mathrm{,}$ (6)
where naturally $\boldsymbol{j}\equiv(\boldsymbol{a}_{j},\boldsymbol{b}_{j})$
and $\boldsymbol{k}\equiv(\boldsymbol{a}_{k},\boldsymbol{b}_{k})$.
$\mathbf{0}_{n}$ is the $n\times n$ all-zeros matrix, and $\mathbf{I}_{n}$ is
the $n\times n$ identity matrix. Eq. (5) captures the fact that a factor of
$-1$ is included in the scalar commutator for each qubit where the operators
$\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$ differ and neither
acts trivially. Since the inner product appears as the exponent of a sign
factor, without loss of generality, we can replace it with the _binary
symplectic inner product_
$\displaystyle\langle\boldsymbol{j},\boldsymbol{k}\rangle_{2}\equiv\langle\boldsymbol{j},\boldsymbol{k}\rangle\bmod
2.$ (7)
$H$ | $\sum\limits_{j\in\\{x,y,z\\}}h_{j}\sigma^{j}$ | $\sum\limits_{\begin{subarray}{c}\boldsymbol{j}\in\\{0,x,y,z\\}^{\times 2}\\\ \boldsymbol{j}\neq(0,0)\end{subarray}}h_{\boldsymbol{j}}\sigma^{\boldsymbol{j}}$
---|---|---
$G(H)$ $\simeq L(R)$ | |
$R$ | or |
Table 1: Example frustration graphs for general Hamiltonians on small (1- and
2-qubit) systems. (Left column) For general single-qubit Hamiltonians, the
frustration graph is the complete graph on three vertices, $K_{3}$. By the
Whitney isomorphism theorem [45], $K_{3}$ is the only graph which is not the
line graph of a unique graph, but rather is the line graph of both $K_{3}$ and
the ‘claw’ graph, $K_{1,3}$. This implies the existence of two distinct free-
fermion solutions of single-qubit Hamiltonians. (Right column) For general
two-qubit Hamiltonians, the frustration graph is the line graph of the
complete graph on six vertices $K_{6}$ [29, 46]. Colored are the size-five
cliques corresponding to the degree-five vertices of the root graph. This
mapping implies the existence of a free-fermion solution for general two-qubit
Hamiltonians by six fermions, reflecting the accidental Lie-algebra
isomorphism $\mathfrak{su}(4)\simeq\mathfrak{spin}(6)$ (see Section 5.1).
Through the binary symplectic inner product, the scalar commutator defines a
symmetric binary relation between terms in the Hamiltonian, to which we
associate the adjacency matrix of a graph. Denote the _frustration graph_ for
a Hamiltonian of the form in Eq. (1) by $G(H)\equiv(V,E)$ with vertex set
given by the Pauli terms appearing in $H$, and edge set
$\displaystyle
E\equiv\\{(\boldsymbol{j},\boldsymbol{k})|\langle\boldsymbol{j},\boldsymbol{k}\rangle_{2}=1\\}$
(8)
That is, two Pauli terms correspond to neighboring vertices in $G(H)$ if and
only if they anticommute. Without loss of generality, we can assume that
$G(H)$ is connected, as disconnected components of this graph correspond to
commuting collections of terms in the Hamiltonian and can thus be
independently treated. As such, we will further assume that $H$ has no
identity component in the expansion (1)—rendering it traceless—since this will
only contribute an overall energy shift to the system with no effect on
dynamics.
### 2.2 Majorana Fermions
A related set of Hermitian operators which only either commute or anticommute
is that of the Majorana fermion modes $\\{\gamma_{\mu}\\}_{\mu}$, which
satisfy the canonical anticommutation relations
$\displaystyle\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2\delta_{\mu\nu}I\mathrm{,}$
(9)
and for which $\gamma_{\mu}^{\dagger}=\gamma_{\mu}$. A familiar way of
realizing these operators in terms of $n$-qubit Pauli observables is through
the Jordan-Wigner transformation
$\displaystyle\gamma_{2j-1}=\bigotimes_{k=1}^{j-1}Z_{k}\otimes
X_{j}\mbox{\hskip 28.45274pt}\gamma_{2j}=\bigotimes_{k=1}^{j-1}Z_{k}\otimes
Y_{j}\mathrm{.}$ (10)
The Pauli operators on the right can easily be verified to constitute $2n$
operators satisfying Eq. (9). Of course, we will explore the full set of
generalizations to this transformation in this work. We seek to identify those
qubit Hamiltonians which can be expressed as quadratic in the Majorana modes.
Such _free-fermion_ Hamiltonians are written as
$\displaystyle\widetilde{H}=i\boldsymbol{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}}\equiv
2i\sum_{(j,k)\in\widetilde{E}}h_{jk}\gamma_{j}\gamma_{k}$ (11)
where $\boldsymbol{\gamma}$ is a row-vector of the Majorana operators, and
$\mathbf{h}$ is the _single-particle Hamiltonian_. Without loss of generality,
$\mathbf{h}$ can be taken as a real antisymmetric matrix, as we can similarly
assume $\widetilde{H}$ is traceless, and the canonical anticommutation
relations Eq. (9) guarantee that any symmetric component of $\mathbf{h}$ will
not contribute to $\widetilde{H}$. $\widetilde{E}$ is the edge-set of the
_fermion-hopping graph_ $R\equiv(\widetilde{V},\widetilde{E})$ on the fermion
modes $\widetilde{V}$. That is, $h_{jk}=0$ for those pairs
$(j,k)\notin\widetilde{E}$, and the factor of two in the rightmost expression
accounts for the fact that each edge in $\widetilde{E}$ is included only once
in the sum.
As a result of the canonical anticommutation relations (9), the individual
Majorana modes transform covariantly under the time evolution generated by
$\widetilde{H}$
$\displaystyle\mathrm{e}^{i\widetilde{H}t}\gamma_{\mu}\mathrm{e}^{-i\widetilde{H}t}=\sum_{\nu\in\widetilde{V}}\left(\mathrm{e}^{4\mathbf{h}t}\right)_{\mu\nu}\gamma_{\nu}$
(12)
since
$\displaystyle[\bm{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}},\gamma_{\mu}]=-4(\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}})_{\mu}\,.$
(13)
Since $\mathbf{h}$ is antisymmetric and real,
$\mathrm{e}^{4\mathbf{h}t}\in\mathrm{SO}(2n,\mathds{R})$. Thus, $\mathbf{h}$
can be block-diagonalized via a real orthogonal matrix,
$\mathbf{W}\in\mathrm{SO}(2n,\mathds{R})$, as
$\displaystyle\mathbf{W}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{W}=\bigoplus_{j=1}^{n}\begin{pmatrix}0&-\lambda_{j}\\\
\lambda_{j}&0\\\ \end{pmatrix}$ (14)
We can represent $\mathbf{W}$ as the exponential of a quadratic Majorana
fermion operator as well, by defining
$\displaystyle\mathbf{W}\equiv\mathrm{e}^{4\mathbf{w}}\mathrm{,}$ (15)
$\widetilde{H}$ is therefore diagonalized as
$\displaystyle\mathrm{e}^{-\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}\widetilde{H}\mathrm{e}^{\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}$
$\displaystyle=i\boldsymbol{\gamma}\cdot\left(\mathbf{W}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{W}\right)\cdot\boldsymbol{\gamma}^{\mathrm{T}}$
(16) $\displaystyle=-2i\sum_{j=1}^{n}\lambda_{j}\gamma_{2j-1}\gamma_{2j}$ (17)
$\displaystyle\mathrm{e}^{-\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}\widetilde{H}\mathrm{e}^{\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}$
$\displaystyle=2\sum_{j=1}^{n}\lambda_{j}Z_{j}$ (18)
Note that the exact diagonalization can be performed with reference to the
_quadratics_ in the Majorana fermion modes only. To completely solve the
system, it is only necessary to diagonalize $\mathbf{h}$ classically, find a
generating matrix $\mathbf{w}$, and diagonalize $\widetilde{H}$ using an
exponential of quadratics with regard to some fermionization like Eq. (10).
Eigenstates of $\widetilde{H}$ can be found by acting
$\mathrm{e}^{\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}$
on a computational basis state $\left|\mathbf{x}\right\rangle$ for
$\mathbf{x}\in\\{0,1\\}^{\times n}$. The associated eigenvalue is
$\displaystyle E_{\mathbf{x}}=2\sum_{j=1}^{n}(-1)^{x_{j}}\lambda_{j}$ (19)
Therefore, systems of the form in Eq. (11) may be considered _exactly
solvable_ classically, since their exact diagonalization is reduced to exact
diagonalization on a _poly_ $(n)$-sized matrix $\mathbf{h}$.
## 3 Fundamental Theorem
As mentioned previously, we seek to characterize the full set of Jordan-
Wigner-like transformations, generalizing Eq. (10). To be more precise, we ask
for the conditions under which there exists a mapping
$\phi:V\mapsto\widetilde{V}^{\times 2}$, for some set $\widetilde{V}$ (the
fermion modes), effecting
$\displaystyle\sigma^{\boldsymbol{j}}\mapsto
i\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}\mathrm{,}$
(20)
for $\phi_{1}(\boldsymbol{j})$, $\phi_{2}(\boldsymbol{j})\in\widetilde{V}$,
and such that
$\displaystyle[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!]=[\\![\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})},\gamma_{\phi_{1}(\boldsymbol{k})}\gamma_{\phi_{2}(\boldsymbol{k})}]\\!]$
(21)
for all pairs, $\boldsymbol{j}$ and $\boldsymbol{k}$. Such a mapping induces a
term-by-term _free-fermionization_ of the Hamiltonian (1) to one of the form
(11) such that
$\displaystyle G(H)\simeq G(\widetilde{H})\mathrm{.}$ (22)
Again, $G(H)$ is the frustration graph of $H$.
From the canonical anticommutation relations, Eq. (9), and the distribution
rule Eq. (4), we see that scalar commutators between quadratic Majorana-
fermion operators are given by
$\displaystyle[\\![\gamma_{\mu}\gamma_{\nu},\gamma_{\alpha}\gamma_{\beta}]\\!]=(-1)^{|(\mu,\nu)\cap(\alpha,\beta)|}$
(23)
Eqs. (22) and (23) can be restated graph theoretically as saying that $G(H)$
is the graph whose vertex set is the edge set of the fermion hopping graph
$R$, and vertices of $G(H)$ are neighboring if and only if the associated
edges of $R$ share exactly one vertex. Such a graph is called the _line graph_
of $R$.
###### Definition 1 (Line Graphs).
The line graph $L(R)\equiv(E,F)$ of a root graph $R\equiv(V,E)$ is the graph
whose vertex set is the edge set of $R$ and whose edge set is given by
$\displaystyle F\equiv\\{(e_{1},e_{2})\ |\ e_{1},e_{2}\in E,\ |e_{1}\cap
e_{2}|=1\\}$ (24)
That is, vertices are neighboring in $L(R)$ if the corresponding edges in $R$
are incident at a vertex.
Notice that if $L(R)$ is connected if and only if $R$ is. With these
definitions in-hand, our first main result can be stated simply as
###### Theorem 1 (Existence of free-fermion solution).
An injective map $\phi$ as defined in Eq. (20) and Eq. (21) exists for the
Hamiltonian $H$ as defined in Eq. (1) if and only if there exists a root graph
$R$ such that
$\displaystyle G(H)\simeq L(R),$ (25)
where R is the hopping graph of the free-fermion solution.
| With Twins | Without twins
---|---|---
Forbidden Graphs | (a) | (d)
Twin-Free, $L(R)$ | (b) |
Root $R$ | (c) | (e)
Table 2: A graph is a line graph if and only if it does not contain any of the
nine forbidden graphs in (a), (d), and (e) as an induced subgraph [47]. Of
these nine graphs, the three in (a) contain twin vertices, highlighted. If
these three graphs are induced subgraphs of a frustration graph such that
these highlighted vertices are twins in the larger graph, then the twins can
be removed by restricting onto a fixed mutual eigenspace of their products,
which correspond to constants of motion of the Hamiltonian. (b) The twin-free
restrictions of the graphs in (a), with all but one highlighted vertex from
(a) removed. These graphs are the line graphs of the graphs in (c). In Ref.
[48], it was shown that only five graphs contain the forbidden subgraphs in
(e) and none of those in (a) or (d). Finally, this set was further refined in
Ref. [49] to a set of three forbidden subgraphs for 3-connected line graphs of
minimum degree at least seven, though we do not display these graphs here.
###### Proof.
The proof can be found in Section 6.1. ∎
The intuition for this result is that the root graph $R$ is the graph where
the vertices are fermions and the edges are the bilinears that appear in the
Hamiltonian $H$. The result reveals a correspondence between a
characterization of line graphs and a characterization of free-fermion spin
models, as not every graph can be expressed as the line graph of some root. We
must however note that, strictly speaking, the existence of this mapping alone
does not guarantee a free-fermion solution, since the “Lie-homomorphism"
constraint, Eq. (21), does not fix the _sign_ of the terms in the free-fermion
Hamiltonian. Choosing a sign for each term is equivalent to _orienting_ the
root graph, since multiplying by a sign is equivalent to making the exchange
$\phi_{1}(\boldsymbol{j})\leftrightarrow\phi_{2}(\boldsymbol{j})$ in Eq. (20).
Different orientations may not faithfully reproduce the properties of $H$, but
we will see that such an orientation can always be chosen. The line graph
condition in Eq. (25) is therefore necessary and sufficient for a free-fermion
solution to exist. Before turning to further implications of Theorem 1, let us
first detail some properties of line graphs.
Line graphs are closely related to so-called _intersection graphs_ ,
originally studied by Erdős [50] and others (see, for example, Ref. [51]). An
intersection graph $G\equiv(V,E)$ is a graph whose vertex set, $V\subseteq
2^{S}$, consists of distinct subsets of some set $S$. Two vertices, $u$ and
$v$, are neighboring in $G$ if their intersection is nonempty ($|v\cap w|\neq
0$). A line graph is a special case of an intersection graph where every
vertex corresponds to a subset of size at most two. When we specify that
$\phi$ be injective, we are requiring that no distinct vertices have identical
subsets, and our definition of a free-fermion solution Eq. (25) identically
coincides with that of a line graph. Since terms in $H$ can thus intersect by
at most one Majorana mode, collections of terms containing a given mode are
all neighboring in $G(H)$, so this mode corresponds to a _clique_ , or
complete subgraph, of $G(H)$. This characterization of line graphs was first
given by Krausz [52] and bears stating formally.
###### Definition 2 (Krausz decomposition of line graphs).
Given a line graph $G\simeq L(R)$, there exists a partition of the edges of
$G$ into cliques such that every vertex appears in at most two cliques.
Cliques in $G(H)$ can therefore be identified with the individual Majorana
modes in a free-fermion solution of $H$. If a term belongs to only one clique,
we can ensure our resulting fermion Hamiltonian is quadratic by taking the
second clique for that term to be a clique of no edges, as we will see in
several examples below. The existence of a Krausz decomposition is utilized in
a linear-time algorithm to recognize line graphs by Roussopoulos [38], though
the earliest such algorithm for line-graph recognition was given by Lehot
[39]. A dynamic solution was later given by Degiorgi and Simon [40]. These
algorithms are optimal and constructive, and so can be applied to a given spin
model to provide an exact free-fermion solution.
We next turn to the _hereditary property_ of line graphs, for which we require
the following definition:
###### Definition 3 (Induced subgraphs).
Given a graph $G\equiv(V,E)$, an induced subgraph of $G$ by a subset of
vertices $V^{\prime}\subset V$, is a graph
$G[V^{\prime}]\equiv(V^{\prime},E^{\prime})$ such that for any pair of
vertices $u$, $v\in V^{\prime}$, $(u,v)\in E^{\prime}$ if and only if
$(u,v)\in E$ in $G$.
An induced subgraph of $G$ can be constructed by removing the subset of
vertices $V/V^{\prime}$ from $G$, together with all edges incident to any
vertex in this subset. Line graphs are a _hereditary class_ of graphs in the
sense that any induced subgraph of a line graph is also a line graph. This
coincides with our intuition that removing a term from a free-fermion
Hamiltonian does not change its free-fermion solvability. Conversely,
Hamiltonians for which no free-fermion solution exists are accompanied by
“pathological" structures in their frustration graphs, which obstruct a free-
fermion description no matter how we try to impose one. This is captured by
the forbidden subgraph characterization of Beineke [47] and later refined by
others [48, 49].
###### Corollary 1.1 (Beineke no-go theorem).
A given spin Hamiltonian $H$ has a free-fermion solution if and only if its
frustration graph $G(H)$ does not contain any of nine forbidden subgraphs,
shown in Table 2, (a) (d) (e), as an induced subgraph.
These forbidden subgraphs above can be interpreted as collections of
“frustrating" terms. At least one of the terms must be assigned to a fermion
interaction in every possible assignment from Pauli operators to fermions.
Correspondingly, ignoring these terms by removing their corresponding vertices
from the frustration graph may remove a forbidden subgraph and cause the
Hamiltonian to become solvable. The terms which we need to remove in this way
need not be unique. In the next section, we discuss one such strategy for
removing vertices such that our solution will remain faithful to the original
spin Hamiltonian by exploiting symmetries.
## 4 Symmetries
An important class of symmetries involves twin vertices in the frustration
graph.
###### Definition 4 (Twin Vertices).
Given a graph $G\equiv(V,E)$, vertices $u$, $v\in V$ are twin vertices if, for
every vertex $w\in V$, $(u,w)\in E$ if and only if $(v,w)\in E$.
Twin vertices have exactly the same neighborhood, and are thus never neighbors
in a frustration graph, which contains no self edges due to the fact that
every operator commutes with itself. Sets of twin vertices are the subject of
our first lemma.
###### Lemma 1 (Twin vertices are constants of motion).
Suppose a pair of terms $\sigma^{\boldsymbol{j}}$ and
$\sigma^{\boldsymbol{k}}$ in $H$ correspond to twin vertices in $G(H)$, then
the product $\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}$ is a nontrivial
Pauli operator commuting with every term in the Hamiltonian. Distinct such
products therefore commute with each other.
###### Proof.
The statement follows straightforwardly from the definition of twin vertices:
every term in $H$ (including $\sigma^{\boldsymbol{j}}$ and
$\sigma^{\boldsymbol{k}}$ themselves) either commutes with both
$\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$ or anticommutes with
both of these operators. Terms in $H$ therefore always commute with the
product $\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}$. This product is
furthermore a nontrivial Pauli operator, for if
$\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}=I$, then
$\boldsymbol{j}=\boldsymbol{k}$, and we would not identify these Paulis with
distinct vertices in $G(H)$. Constants of motion generated this way must
commute with one another, since they commute with every term in the
Hamiltonian and are themselves products of Hamiltonian terms. They therefore
generate an abelian subgroup of the symmetry group of the Hamiltonian. ∎
Let the symmetry subgroup generated by products of twin vertices in this way
be denoted $\mathcal{S}$. We can leverage these symmetries to remove twin
vertices from the frustration graph $G(H)$. To do this, choose a minimal
generating set $\\{\sigma^{\boldsymbol{s}}\\}$ of Pauli operators for
$\mathcal{S}$ and choose a $\pm 1$ eigenspace for each. Let
$(-1)^{x_{\boldsymbol{s}}}$ be the eigenvalue associated to the generator
$\sigma^{\boldsymbol{s}}\in\mathcal{S}$, for $x_{\boldsymbol{s}}\in\\{0,1\\}$.
We restrict to the subspace defined as the mutual $+1$ eigenspace of the
stabilizer group
$\displaystyle\mathcal{S}_{\boldsymbol{x}}=\langle(-1)^{x_{\boldsymbol{s}}}\sigma^{\boldsymbol{s}}\rangle$
(26)
For a pair of twin vertices corresponding to Hamiltonian terms
$\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$, we let
$\displaystyle\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}\equiv(-1)^{d_{\boldsymbol{j},\boldsymbol{k}}}\left[\prod_{\boldsymbol{s}\in
S_{\boldsymbol{j},\boldsymbol{k}}}(-1)^{x_{\boldsymbol{s}}}\sigma^{\boldsymbol{s}}\right]$
(27)
where $d_{\boldsymbol{j},\boldsymbol{k}}\in\\{0,1\\}$ specifies the
appropriate sign factor, and $S_{\boldsymbol{j},\boldsymbol{k}}$ is the subset
of generators of $\mathcal{S}$ such that
$\displaystyle\bigoplus_{\boldsymbol{s}\in
S_{\boldsymbol{j},\boldsymbol{k}}}\boldsymbol{s}=\boldsymbol{j}\oplus\boldsymbol{k}$
(28)
where “$\oplus$” denotes addition modulo 2 here. In the stabilizer subspace of
$\mathcal{S}_{\boldsymbol{x}}$, we can make the substitution
$\displaystyle\sigma^{\boldsymbol{k}}\rightarrow(-1)^{d_{\boldsymbol{j},\boldsymbol{k}}}\sigma^{\boldsymbol{j}}$
(29)
effectively removing the vertex $\boldsymbol{k}$ from $G(H)$.
Twin vertices capture the cases where a free-fermion solution for $H$ exists,
but is necessarily non-injective. Indeed, note that we are careful in our
statement of Theorem 1 to specify that our condition Eq. (25) is necessary and
sufficient when $\phi$ is injective. If we instead relax our requirement that
vertices of a line graph correspond to distinct subsets of size two in our
earlier discussion of intersection graphs, then we are allowing for line
graphs of graphs with multiple edges, or _multigraphs_. However, our
definition of $G(\widetilde{H})$ will differ from the line graph of a
multigraph for pairs of vertices corresponding to identical edges, which must
be adjacent in the line graph of a multigraph, but will be nonadjacent in
$G(\widetilde{H})$ from Eq. (23). Such vertices will nevertheless be twin
vertices in $G(\widetilde{H})$ due to the graph-isomorphism constraint, Eq.
(22). Therefore, if no _injective_ mapping $\phi$ satisfying Theorem 1 exists,
a many-to-one free-fermion solution exists only when twin vertices are
present. Lemma 1 allows us to deal with this non-injective case by removing
twin vertices until we obtain the line graph of a simple graph when possible.
The particular way we choose to perform this removal cannot affect the overall
solvability of the model, since the frustration graph with all twin vertices
removed is an induced subgraph of any frustration graph with only a proper
subset of such vertices removed. A model which is solvable by free fermions
this way is therefore solvable in all of its symmetry sectors.
Finally, we can see when a non-injective free-fermion solution may be possible
from the forbidden subgraph characterization, Corollary 1.1. As seen in Table
2, some of the forbidden subgraphs shown in (a) themselves contain twin
vertices. If these forbidden subgraphs are connected to the global frustration
graph such that their twin vertices remain twins in the larger graph, then
they may be removed, possibly allowing for a solution of the full Hamiltonian
by free fermions. When the twins are removed from the forbidden subgraphs,
they become line graphs as shown in Table 2 (b) and (c). An example of a model
which can be solved this way is the Heisenberg-Ising model introduced in Ref.
[1].
We next proceed to identify the remaining Pauli symmetries for a Hamiltonian
satisfying Theorem 1. For this, we invoke the natural partition of the Pauli
group $\mathcal{P}$ into the subgroup $\mathcal{P}_{H}$, again defined as that
generated by Hamiltonian terms
$\\{\sigma^{\boldsymbol{j}}\\}_{\boldsymbol{j}\in V}$, and the Pauli operators
outside this subgroup, $\mathcal{P}_{\perp}\equiv\mathcal{P}/\mathcal{P}_{H}$.
Note that the latter set does not form a group in general, as for example,
single-qubit Paulis may be outside of $\mathcal{P}_{H}$ yet may be multiplied
to operators in $\mathcal{P}_{H}$. A subgroup of the symmetries of the
Hamiltonian is the center $\mathcal{Z}(\mathcal{P}_{H})$ of $\mathcal{P}_{H}$,
the set of $n$-qubit Pauli operators in $\mathcal{P}_{H}$ which commute with
every element of $\mathcal{P}_{H}$ and therefore with every term in the
Hamiltonian. To characterize this group, we need two more definitions.
###### Definition 5 (Cycle subgroup).
A cycle of a graph $G\equiv(V,E)$ is a subset of its edges, $Y\subseteq E$,
such that every vertex contains an even number of incident edges from the
subset. If a Pauli Hamiltonian satisfies Eq. (25) for some root graph $R$, we
define its cycle subgroup $Z_{H}\subseteq\mathcal{P}_{H}$ as the abelian Pauli
subgroup generated by the cycles $\\{Y_{i}\\}_{i}$ of $R$,
$\displaystyle
Z_{H}=\bigl{\langle}\Pi_{\\{\boldsymbol{j}|\phi(\boldsymbol{j})\in
Y_{i}\\}}\sigma^{\boldsymbol{j}}\bigr{\rangle}_{i}.$ (30)
Since
$\displaystyle\prod_{\\{\boldsymbol{j}|\phi(\boldsymbol{j})\in
Y_{i}\\}}\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}=\pm
I$ (31)
we have, from Eq. (21) and the definition of $\phi$, that the elements of
$Z_{H}$ commute with every term in the Hamiltonian and thus with each other
(since they are products of Hamiltonian terms). That is,
$Z_{H}\subseteq\mathcal{Z}(\mathcal{P}_{H})$. Notice that the definition of
the generators for $Z_{H}$ in Eq. (30) may sometimes yield operators
proportional to identity.
A familiar symmetry of free-fermion Hamiltonians is the _parity operator_
$\displaystyle P\equiv
i^{\frac{1}{2}|\widetilde{V}|(|\widetilde{V}|-1)}\prod_{k\in\widetilde{V}}\gamma_{k}$
(32)
which commutes with every term in the Hamiltonian since each term is quadratic
in the Majorana modes. The phase factor is chosen such that $P$ is Hermitian.
Here, we define this operator in terms of Pauli Hamiltonian terms through a
combinatorial structure known as a T-join.
###### Definition 6 (Parity operator).
A T-join of a graph $G\equiv(V,E)$ is a subset of edges, $T\subseteq E$, such
that an odd number of edges from $T$ is incident to every vertex in $V$. If a
Pauli Hamiltonian satisfies Eq. (25) for some root graph $R$ such that the
number of vertices in $R$ is even, we define the parity operator as
$\displaystyle P\equiv i^{d}\prod_{\boldsymbol{j}\in
T}\sigma^{\boldsymbol{j}}$ (33)
where the product is taken over a T-join of $R$, and $d\in\\{0,1,2,3\\}$
specifies the phase necessary to agree with Eq. (32).
Here we have
$\displaystyle i^{d}\prod_{\boldsymbol{j}\in
T}i\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}$
$\displaystyle=i^{\frac{1}{2}|\widetilde{V}|(|\widetilde{V}|-1)}\prod_{\mu\in\widetilde{V}}\gamma_{\mu}=P\mathrm{,}$
(34)
since every fermion mode will be hit an odd number of times in the T-join.
Unlike with the cycle subgroup, $P$ is never proportional to the identity in
the fermion description, though it may still be proportional to the identity
in the Pauli description (up to stabilizer equivalences). In this case, only
solutions for the free-fermion Hamiltonian in a fixed-parity subspace will be
physical. We will see several examples of this in the next section.
When no T-join exists, we cannot form $P$ as a product of Hamiltonian terms.
In fact, $P\in\mathcal{Z}(\mathcal{P}_{H})$ only when $|\widetilde{V}|$ is
even. Now with these definitions in hand, we are ready to state our second
theorem.
###### Theorem 2 (Symmetries are cycles and parity).
Given a Hamiltonian satisfying Eq. (25) such that the number of vertices
$|\widetilde{V}|$ in the root graph is odd, then we have
$\displaystyle\mathcal{Z}(\mathcal{P}_{H})=Z_{H}.$ (35)
If the number of vertices in the root graph is even, then we have
$\displaystyle\mathcal{Z}(\mathcal{P}_{H})=\left\langle Z_{H},P\right\rangle.$
(36)
###### Proof.
The proof can be found in Section 6.2. ∎
The Pauli symmetries of the Hamiltonian outside of
$\mathcal{Z}(\mathcal{P}_{H})$ may be thought of as “logical" or “gauge"
qubits, and this characterization allows for a simple accounting of these
qubits. Suppose we express a spin Hamiltonian $H$ on $n$ qubits as a free-
fermion Hamiltonian on the hopping graph $R=(\widetilde{V},\widetilde{E})$,
and let $|\mathcal{Z}(\mathcal{P}_{H})|$ be the number of independent
generators of $\mathcal{Z}(\mathcal{P}_{H})$. The number of logical qubits
$n_{L}$ of the model is given by
$\displaystyle
n_{L}\equiv\begin{cases}n-\left[\frac{1}{2}(|\widetilde{V}|-1)+|\mathcal{Z}(\mathcal{P}_{H})|\right]&|\widetilde{V}|\
\mathrm{odd}\\\
n-\left[\frac{1}{2}(|\widetilde{V}|-2)+|\mathcal{Z}(\mathcal{P}_{H})|\right]&|\widetilde{V}|\
\mathrm{even}\end{cases}.$ (37)
This follows from the fact that the $\mathds{F}_{2}$-rank of the adjacency
matrix of $G(H)$ is twice the number of qubits spanned by the fermionic
degrees of freedom in the model, and also the number of vertices of the root
graph $R$ up to a constant shift.
$R$ | $L(R)$
---|---
|
|
|
Table 3: The Whitney isomorphism theorem [45] guarantees that the edge
automorphisms exchanging $e$ and $e^{\prime}$ in the graphs $R$ in the left
column, or corresponding vertices in their line graphs on the right, which
cannot be realized by any vertex automorphism of $R$, are the only such cases.
Finally, we note that there may be additional symmetries, such as translation
invariance, if the coefficients $h_{\boldsymbol{j}}$ themselves satisfy a
symmetry. Our characterization will allow us to say something about this
situation when the associated symmetry transformation is a Clifford
operator—that is, a unitary operator in the normalizer of the Pauli
group—commuting with the Pauli symmetries in $\mathcal{Z}(\mathcal{P}_{H})$,
such as, e.g., a spatial translation. The following statement follows from a
theorem by Whitney [45] (and extended to infinite graphs in [53]).
###### Corollary 1.2 (Clifford Symmetries and Whitney Isomorphism).
Let
$\widetilde{H}=i\boldsymbol{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}}$
be a free-fermion Hamiltonian with single-particle Hamiltonian $\mathbf{h}$ in
a fixed symmetry sector of $\mathcal{Z}(\mathcal{P}_{H})$. Then any unitary
Clifford symmetry $U$ such that $U^{\dagger}\widetilde{H}U=\widetilde{H}$
induces a signed permutation symmetry $\mathbf{u}$ such that
$\mathbf{h}=\mathbf{u}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{u}$, except for
when $U$ induces one of the three edge isomorphisms shown in Table 3.
###### Proof.
This follows from the Whitney isomorphism theorem: except for the three cases
shown in Table 3, any adjacency-preserving permutation of the vertices of
$G(\widetilde{H})$ is induced by an adjacency-preserving permutation of the
vertices of $R$. A Clifford symmetry $U$ acts as a signed permutation of the
Hamiltonian terms which preserves $\widetilde{H}$. Suppose the associated
unsigned permutation is not one of the exceptional cases, and so is induced by
a permutation $\pi$ on the vertices of $R$. This gives
$\displaystyle\widetilde{H}$ $\displaystyle=U^{\dagger}\widetilde{H}U$ (38)
$\displaystyle=i\sum_{(j,k)\in\widetilde{E}}h_{jk}\left(U^{\dagger}\gamma_{j}U\right)\left(U^{\dagger}\gamma_{k}U\right)$
(39)
$\displaystyle=i\sum_{(j,k)\in\widetilde{E}}(-1)^{x_{j}+x_{k}}h_{jk}\gamma_{\pi(j)}\gamma_{\pi(k)}$
(40)
where $x_{j}\in\\{0,1\\}$ designates the sign associated to the permutation of
vertex $j\in\widetilde{V}$. By unitarity, this sign must depend on $j$ alone,
since $U^{\dagger}\gamma_{j}U$ can only depend on $j$. Let $\mathbf{u}$ be a
single-particle transition matrix defined as
$\displaystyle u_{jk}=(-1)^{x_{j}}\delta_{k\pi(j)}.$ (41)
Then we can reinterpret Eq. (40) in the single-particle picture as
$\displaystyle
i\boldsymbol{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}}$
$\displaystyle=i\boldsymbol{\gamma}\cdot\left(\mathbf{u}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{u}\right)\cdot\boldsymbol{\gamma}^{\mathrm{T}}.$
(42)
By linear independence, Eq. (42) therefore implies
$\displaystyle\mathbf{h}=\mathbf{u}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{u}$
(43)
and the claim follows. ∎
See section 5.1 for a simple example of an exceptional Hamiltonian realizing a
frustration graph shown in Table 3. We now complete our characterization of
free-fermion solutions by choosing an orientation for every edge in the root
graph over a restricted subspace determined by the constants of motion.
### 4.1 Orientation and Full Solution
As discussed previously, the Lie-homomorphism condition Eq. (21) does not
fully constrain the free-fermion solution of a given Pauli Hamiltonian. This
is because we are free to choose a direction to each edge in the root graph by
exchanging $\phi_{1}(\boldsymbol{j})\leftrightarrow\phi_{2}(\boldsymbol{j})$,
which is equivalent to changing the sign of the term
$i\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}$ in
$\widetilde{H}$ corresponding to $\sigma^{\boldsymbol{j}}$ in $H$. A related
ambiguity corresponds to the cycle symmetry subgroup $Z_{H}$: we are free to
choose a symmetry sector over which to solve the Hamiltonian $H$ by choosing a
mutual $\pm 1$-eigenspace of independent nontrivial generators of this group.
It will turn out that both ambiguities are resolved simultaneously.
First suppose we have a Hamiltonian $H$ satisfying Eq. (25) for some root
graph $R\equiv(\widetilde{V},\widetilde{E})$. Construct a spanning tree
$\Upsilon\equiv(\widetilde{V},\widetilde{E}^{\prime})$ of $R$, defined as:
###### Definition 7 (Spanning Tree).
Given a connected graph $G\equiv(V,E)$, a spanning tree
$\Upsilon\equiv(V,E^{\prime})\subseteq G$ is a connected subgraph of $G$ such
that $E^{\prime}$ contains no cycles.
This can be performed in linear time in $|\widetilde{V}|$. Designate a
particular vertex $v\in\widetilde{V}$ as the root of this tree. Each vertex
$u\in\widetilde{V}$ has a unique path $p(u,v)\subseteq\widetilde{E}$ in
$\Upsilon$ to the root, the path $p(v,v)$ being empty. Choose an arbitrary
direction for each edge in $\widetilde{E}^{\prime}$ (we will see shortly to
what extent this choice is important).
Our choice of spanning tree determines a basis of _fundamental cycles_ for the
binary cycle space of $R$ and thus a generating set of Paulis for the cycle
subgroup $Z_{H}$. To see this, note that for each edge
$\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}$, there is a
unique cycle of $R$ given by
$\displaystyle Y_{\boldsymbol{j}}\equiv p[\phi_{1}(\boldsymbol{j}),v]\cup
p[\phi_{2}(\boldsymbol{j}),v]\cup\phi(\boldsymbol{j}).$ (44)
Let $\sigma^{\boldsymbol{y}(\boldsymbol{j})}$ be the cycle subgroup generator
associated to $Y_{\boldsymbol{j}}$, defined by
$\displaystyle\boldsymbol{y}(\boldsymbol{j})=\bigoplus_{\\{\boldsymbol{z}|\phi(\boldsymbol{z})\in
Y_{\boldsymbol{j}}\\}}\boldsymbol{z}$ (45)
such that
$\displaystyle\sigma^{\boldsymbol{y}(\boldsymbol{j})}=i^{d}\prod_{\\{\boldsymbol{z}|\phi(\boldsymbol{z})\in
Y_{\boldsymbol{j}}\\}}\sigma^{\boldsymbol{z}}$ (46)
where $d\in\\{0,1,2,3\\}$ again designates the appropriate phase. The set of
such $Y_{\boldsymbol{j}}$ contains $|\widetilde{E}|-|\widetilde{V}|+1$ cycles
and forms an independent generating set for all the cycles of $R$ under
symmetric difference. The corresponding set of
$\sigma^{\boldsymbol{y}(\boldsymbol{j})}$ is therefore an independent
generating set of the cycle subgroup up to signs, since individual Pauli
operators either commute or anticommute and square to the identity.
In a similar fashion as with twin-vertex symmetries, we restrict to a mutual
$\pm 1$ eigenspace of the cycle-subgroup generators, designated by a binary
string $\boldsymbol{x}\in\\{0,1\\}^{\times|\widetilde{E}|-|\widetilde{V}|+1}$
over the $\boldsymbol{j}$ such that
$\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}$. That is, we
restrict to the mutual $+1$ eigenspace of the stabilizer group
$\displaystyle
Z_{H,\boldsymbol{x}}\equiv\bigl{\langle}(-1)^{x_{\boldsymbol{j}}}\sigma^{\boldsymbol{y}(\boldsymbol{j})}\bigr{\rangle}.$
(47)
If Eq. (45) gives $\boldsymbol{y}(\boldsymbol{j})=\mathbf{0}$ for any
$\boldsymbol{j}$, then we take the corresponding $x_{\boldsymbol{j}}=0$. We
then simply choose the direction for the edge $\phi(\boldsymbol{j})$ such that
$\displaystyle(-1)^{x_{\boldsymbol{j}}}i^{d}\left[\prod_{\\{\boldsymbol{z}|\phi(\boldsymbol{z})\in
Y_{\boldsymbol{j}}\\}}i\gamma_{\phi_{1}(\boldsymbol{z})}\gamma_{\phi_{2}(\boldsymbol{z})}\right]=+I$
(48)
where $d$ is as defined in Eq. (46). This ensures that the product of Majorana
hopping terms around a fundamental cycle $Y_{\boldsymbol{j}}$ agrees with the
corresponding Pauli product over the restricted subspace (i.e. up to
equivalencies by stabilizers in the group $Z_{H,\boldsymbol{x}}$). By the Lie-
homomorphism constraint Eq. (21), all products of Majorana hopping terms
around a cycle of $R$ therefore agree with their corresponding Pauli products
over this subspace, and so the multiplication relations of the Paulis are
respected by their associated fermion hopping terms up to stabilizer
equivalencies. Since we have exactly as many elements $Y_{\boldsymbol{j}}$ in
our fundamental cycle basis as undirected edges $\phi(\boldsymbol{j})$, such
an orientation can always be chosen.
$\sigma^{i}_{1}$\$\sigma^{j}_{2}$ | $I$ | $X$ | $Y$ | $Z$ | 2-qubit frustration graph $L(K_{6})$
---|---|---|---|---|---
$I$ | $P\equiv i\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\gamma_{5}\gamma_{6}$ | $i\gamma_{3}\gamma_{5}$ | $i\gamma_{2}\gamma_{5}$ | $i\gamma_{2}\gamma_{3}$ |
$X$ | $i\gamma_{4}\gamma_{6}$ | $i\gamma_{1}\gamma_{2}$ | $-i\gamma_{1}\gamma_{3}$ | $i\gamma_{1}\gamma_{5}$ |
$Y$ | $-i\gamma_{1}\gamma_{6}$ | $-i\gamma_{2}\gamma_{4}$ | $i\gamma_{3}\gamma_{4}$ | $i\gamma_{4}\gamma_{5}$ |
$Z$ | $-i\gamma_{1}\gamma_{4}$ | $i\gamma_{2}\gamma_{6}$ | $-i\gamma_{3}\gamma_{6}$ | $i\gamma_{5}\gamma_{6}$ |
Table 4: (Left) Fermionization of the two-qubit Pauli algebra
$\mathcal{P}_{2}=\\{\sigma^{i}_{1}\otimes\sigma^{j}_{2}\\}_{(i,j)\neq(0,0)}$
by six fermion modes. The graph isomorphism $G(\mathcal{P}_{2})\simeq
L(K_{6})$ reflects the Lie-algebra isomorphism between $\mathfrak{su}(4)$ and
$\mathfrak{spin}(6)$. Though the scalar-commutation relations are reproduced
by quadratics in $\\{\gamma_{\mu}\\}_{\mu=1}^{6}$, the one-sided
multiplication relations are only recovered upon projecting onto the $+1$
eigenspace of $P\equiv
i\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\gamma_{5}\gamma_{6}$ on the fermion
side of the mapping. (Right) The graph $L(K_{6})$, with vertices labeled by a
particular satisfying Pauli assignment. Edges are colored to identify the six
$K_{5}$ subgraphs in the Krausz decomposition of this graph, corresponding to
the six fermion modes. Each vertex belongs to exactly two such subgraphs, as
must be the case for a line graph. This graphical correspondence was first
observed in Ref. [46] in the language of the Dirac algebra.
Why are we free then, to choose an arbitrary sign for each free-fermion term
in our original spanning tree? This choice is actually equivalent to a choice
of signs on the definitions of the individual Majorana modes themselves and so
amounts to a choice of orientation for the coordinate basis in which we write
$\mathbf{h}$. To see this, choose a fiducial orientation for $R$ satisfying
Eq. (48), and suppose our particular free-fermion solution—not necessarily
oriented this way—corresponds to the mapping
$\displaystyle\sigma^{\boldsymbol{j}}\mapsto
i(-1)^{x_{\boldsymbol{j}}}\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}$
(49)
for $\phi(\boldsymbol{j})\in\widetilde{E}^{\prime}$ and with
$x_{\boldsymbol{j}}\in\\{0,1\\}$ designating the edge-direction of
$\phi(\boldsymbol{j})$ relative to the fiducial orientation. We have
$\displaystyle(-1)^{x_{\boldsymbol{j}}}$
$\displaystyle=(-1)^{r_{\phi_{1}(\boldsymbol{j})}+r_{\phi_{2}(\boldsymbol{j})}}$
(50)
where
$\displaystyle r_{u}=\sum_{\\{\boldsymbol{k}|\phi(\boldsymbol{k})\in
p(u,v)\\}}x_{\boldsymbol{k}}.$ (51)
Since the symmetric difference of $p[\phi_{1}(\boldsymbol{j}),v]$ and
$p[\phi_{2}(\boldsymbol{j}),v]$ is the edge
$\phi(\boldsymbol{j})\in\widetilde{E}^{\prime}$, all sign factors on the right
side of Eq. (50) cancel except for $(-1)^{x_{\boldsymbol{j}}}$.
We can then absorb $(-1)^{r_{\phi_{1}(\boldsymbol{j})}}$ and
$(-1)^{r_{\phi_{2}(\boldsymbol{j})}}$ onto the definitions of
$\gamma_{\phi_{1}(\boldsymbol{j})}$ and $\gamma_{\phi_{2}(\boldsymbol{j})}$,
respectively. We furthermore see that imposing Eq. (48) gives an edge-
direction for $\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}$
that differs from that of the fiducial orientation by the associated sign
factor $(-1)^{r_{\phi_{1}(\boldsymbol{j})}+r_{\phi_{2}(\boldsymbol{j})}}$,
which remains consistent with a redefinition of the signs on the individual
Majorana modes. Letting $\mathbf{h}$ be the single-particle Hamiltonian for
the fiducial orientation, such a redefinition corresponds to conjugating
$\mathbf{h}$ by a $\pm 1$ diagonal matrix. As no scalar quantity of
$\mathbf{h}$ can depend on this choice, this redefinition corresponds to a
gauge freedom.
Proceeding this way, we can solve the effective Hamiltonian
$\displaystyle H_{\boldsymbol{x}}\equiv
H\prod_{\\{\boldsymbol{j}|\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}\\}}\left(\frac{I+(-1)^{x_{\boldsymbol{j}}}\sigma^{\boldsymbol{y}(\boldsymbol{j})}}{2}\right)$
(52)
sector-by-sector over each stabilizer eigenspace designated by
$\boldsymbol{x}$. If we also need to remove twin vertices from $G(H)$ before
it is a line graph, we project onto the mutual $+1$ eigenspace of the
stabilizer group $\mathcal{S}_{\boldsymbol{x}}$ defined previously in Section
4 as well. Finally, if the parity operator $P$ is trivial in the Pauli
description, then only a fixed-parity eigenspace in the fermion description
will be physical.
In the next section, we will see how known free-fermion solutions fit into
this characterization and demonstrate how our method can be used to find new
free-fermion solvable models, for which we give an example.
## 5 Examples
### 5.1 Small Systems
The frustration graph of single-qubit Paulis ${X,Y,Z}$ is $K_{3}$, the
complete graph on three vertices. This graph is the line graph of not one, but
two non-isomorphic graphs: the so-called ‘claw’ graph $K_{1,3}$, and $K_{3}$
itself (see Table 1). By the Whitney isomorphism theorem [45], $K_{3}$ is the
only graph which is not the line graph of a unique graph. This ambiguity
results in the existence of two distinct free-fermion solutions of a single
qubit Hamiltonian, which we will hereafter refer to as “even" (labeled “0")
and “odd" (labeled “1") fermionizations
$\displaystyle\begin{cases}X_{0}=i\gamma_{0}\gamma_{1}&X_{1}=i\gamma_{2}\gamma_{3}\\\
Y_{0}=i\gamma_{1}\gamma_{2}&Y_{1}=i\gamma_{0}\gamma_{3}\\\
Z_{0}=i\gamma_{0}\gamma_{2}&Z_{1}=-i\gamma_{1}\gamma_{3}\end{cases}\mathrm{.}$
(53)
In the even fermionization, no T-join of the root graph $K_{3}$ exists since
there are only three fermion modes $\\{\gamma_{0},\gamma_{1},\gamma_{2}\\}$.
The orientation of the root graph is constrained by the identity $XYZ=iI$. In
the odd fermionization, there are four fermion modes
$\\{\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3}\\}$, and so a T-join does
exist for the root graph $K_{1,3}$. It is the set of all edges of this graph.
The parity operator is trivial in the Pauli description however, and so the
constraint $XYZ=iI$ is enforced by restricting to the $+1$ eigenspace of
$P\equiv-\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$ in the fermion description.
We are free to choose the orientation of the root graph $K_{1,3}$ however we
like in this case, since it contains no cycles, though this choice will affect
what we call the physical eigenspace of $P$. By virtue of the line-graph
construction and our choice of orientation, both fermionizations respect the
single-qubit Pauli multiplication relations, up to stabilizer equivalencies in
some cases.
We have made the choice to label the Paulis in the two fermionizations in a
compatible way, such that
$\displaystyle\sigma^{j}_{0}=P\sigma^{j}_{1}$ (54)
This gives
$\displaystyle[\sigma^{j}_{m},\sigma^{k}_{m}]=2i\varepsilon_{jk\ell}\sigma^{\ell}_{0}\mathrm{.}$
(55)
where $m\in\\{0,1\\}$ and $\varepsilon$ is the Levi-Civita tensor. Since an
even number of parity-operator factors appear on the left side of Eq. (55),
the commutator between two Paulis in either fermionization is always a Pauli
in the even fermionization. We can additionally write multiplication relations
between the two fermionizations concisely, as
$\displaystyle\sigma^{j}_{p}\sigma^{k}_{q}=\delta_{jk}P^{p\oplus
q}+i(1-\delta_{jk})\varepsilon_{jk\ell}\sigma_{p\oplus q}^{\ell}\mathrm{.}$
(56)
Another exceptional situation arises for 2 qubits, for which the full
frustration graph is the line graph of $K_{6}$, again depicted graphically in
Table 1. This reveals a free-fermion solution for all 2-qubit Hamiltonians by
six fermion modes, listed explicitly in Table 4. We again choose our
orientation by picking a spanning tree of the root graph (for example all
terms containing the mode $\gamma_{5}$), choosing an arbitrary orientation on
this tree, and choosing the remaining orientations by enforcing the condition
in Eq. (48). Since $K_{6}$ has an even number of vertices, there exists a
T-join for this graph, e.g. the terms $\\{XX,YY,ZZ\\}$. The associated parity
operator is trivial however, as $(XX)(YY)(ZZ)=-I$, and so only the $+1$
eigenspace of $P$ in the fermion description will be physical. This solution
reflects the exceptional Lie algebra isomorphism,
$\mathfrak{su}(4)\simeq\mathfrak{spin}(6)$.
Finally, we give an example of a three-qubit Hamiltonian with an exceptional
_symmetry_ , namely
$\displaystyle H=XII+YII+ZXX+ZZZ$ (57)
This Hamiltonian has the frustration graph shown in the top right entry of
Table 3, and thus is an exceptional case to Corollary 1.2. A symmetry
transformation exchanging $e$ and $e^{\prime}$ for this Hamiltonian is the
Hadamard gate applied to the second and third qubits, which exchanges the
third and fourth terms, but cannot be realized as any permutation of the
individual Majorana modes in its free-fermion description.
Figure 1: Frustration graph for the general XY model and its root graph, shown
below. Cliques are colored to show the Krausz decomposition, which is the
image of the model under the Jordan-Wigner transform. Vertices in the root
graph are correspondingly colored, and a spanning tree is highlighted.
### 5.2 1-dimensional chains
Shown in Figure 1 is the frustration graph $G(H)$ for the most general
nearest-neighbor Pauli Hamiltonian in 1-d (on open boundary conditions) which
is mapped to a free-fermion Hamiltonian under the Jordan-Wigner
transformation,
$\displaystyle
H=\sum_{j=1}^{n-1}\sum_{\alpha,\beta\in\\{x,y\\}}\mu^{j}_{\alpha\beta}\sigma_{j}^{\alpha}\otimes\sigma_{j+1}^{\beta}+\sum_{j=1}^{n}\nu_{j}Z_{j}.$
(58)
Cliques are colored according to the Krausz decomposition of this graph, which
is easily seen by the free-fermion description. The fermion hopping graph,
$R$, is shown below. Note that the cycle symmetry subgroup $Z_{H}$ for this
model is trivial, as every product of Hamiltonian terms along a cycle in $R$
is the identity. Since the number of vertices in $R$ is even, a T-join does
exist, and the parity operator is in-fact $P=Z^{\otimes n}$. Therefore, we
have $|\mathcal{Z}(\mathcal{P}_{H})|=1$.
An example spanning tree for the root graph is highlighted, taken simply to be
the path along edges $(j,j+1)$ from $\gamma_{1}$ to $\gamma_{2n}$. Including
any additional edge in this tree will form a cycle. A natural orientation for
this tree is to direct every edge from vertex $j+1$ to vertex $j$. Note that
we can recover the Jordan-Wigner transformation from this graphical
description alone. We first adjoin a single fictitious qubit and a single
coupling term to the Hamiltonian, as
$\displaystyle H^{\prime}=\mu_{xx}^{0}X_{0}X_{1}+H.$ (59)
Since the remaining qubits only couple to qubit “0" along the $X$-direction,
all operators in $\mathcal{P}_{H}$ commute on this qubit. Furthermore, this
new term adds one vertex to the black clique at the left boundary of the chain
in Fig. 1. It thus extends the spanning tree of $R$ by one vertex – which we
label $\gamma_{0}$ – due to the fact that this new term only belongs to one
clique (so we take its additional Majorana mode to be a clique of size zero).
It can be easily verified that products of Hamiltonian terms from this new
vertex to any vertex along the chosen spanning tree have the form
$\displaystyle\begin{cases}i\gamma_{0}\gamma_{2j-1}\equiv
X_{0}\bigotimes_{k=1}^{j-1}Z_{k}\otimes X_{j}&\\\ i\gamma_{0}\gamma_{2j}\equiv
X_{0}\bigotimes_{k=1}^{j-1}Z_{k}\otimes Y_{j}&\\\ \end{cases}$ (60)
All such operators share $\gamma_{0}$, so their commutation relations are
unchanged by truncating $\gamma_{0}$. Furthermore, since all operators in
$\mathcal{P}_{H}$ commute on qubit-0, we may truncate this qubit as well
without changing the commutation relations of the operators above to obtain
the Jordan-Wigner transformation
$\displaystyle\begin{cases}\gamma_{2j-1}\equiv\bigotimes_{k=1}^{j-1}Z_{k}\otimes
X_{j}&j\ \mathrm{odd}\\\ \gamma_{2j}\equiv\bigotimes_{k=1}^{j-1}Z_{k}\otimes
Y_{j}&j\ \mathrm{odd}\\\ \end{cases}$ (61)
In principle, a similar trick would work in general, but we find it generally
simpler to define Majorana quadratic operators to avoid truncating at a
boundary. Our method is especially convenient when considering the case of
periodic boundary conditions on this model, wherein we add the boundary term
$\displaystyle
H_{\mathrm{boundary}}=\sum_{\alpha,\beta\in\\{x,y\\}}\mu^{n}_{\alpha\beta}\sigma_{1}^{\alpha}\otimes\sigma_{n}^{\beta}$
(62)
to the Hamiltonian in Eq. (58). With this term included, the model has a
nontrivial cycle symmetry given by taking the product of fermion bilinears
around the periodic boundary, and this product is also proportional to the
parity operator $P=Z^{\otimes n}$ in the spin picture. Adding a boundary term
therefore does not change $|\mathcal{Z}(\mathcal{P}_{H})|$, though it does
require that we solve the model over each of the eigenspaces of the cycle
symmetry independently by choosing the sign of the additional terms in the
fermion picture as described in Section 4.1. Let the eigenspace of $Z^{\otimes
n}$ be specified by the eigenvalue $(-1)^{p}$. For each associated free-
fermion model solution, we must then restrict to the $+1$ eigenspace of the
parity operator in the spin picture
$\displaystyle\prod_{j=1}^{n}X_{j}X_{j+1}\mapsto(-i)^{n}(-1)^{p}\prod_{k=1}^{2n}\gamma_{k}\mathrm{.}$
(63)
where index addition is taken modulo $n$. This ensures that our free-fermionic
solution respects the constraint
Figure 2: Frustration graph for the Kitaev honeycomb model (left) and its root
graph (right). Cliques are colored to show the Krausz decomposition.
Interestingly, this model’s root graph is the same as its interaction graph. A
spanning tree of the root is again highlighted.
$\displaystyle\prod_{j=1}^{n}X_{j}X_{j+1}=I\mathrm{.}$ (64)
Notice that solving the two free fermion models together (one for each
eigenspace of $Z^{\otimes n}$) gives $2^{n+1}$ eigenstates, yet restricting to
a fixed-parity sector in each keeps only $2^{n}$ of them, as required.
Finally, we see that this model contains no logical qubits via
$\displaystyle n_{L}=n-\left[\frac{1}{2}(2n-2)+1\right]=0$ (65)
as we might expect.
### 5.3 The Kitaev honeycomb model
Next we consider the Kitaev honeycomb model in two dimensions [12]. This model
has the Hamiltonian
$\displaystyle H=\sum_{\alpha\in\\{x,y,z\\}}\sum_{\alpha-\mathrm{links}\
j}J^{j}_{\alpha}\sigma^{\alpha}_{j}\sigma^{\alpha}_{j+\hat{\alpha}}$ (66)
where each of the $\alpha$ links correspond to one of the compass directions
of the edges of a honeycomb lattice. Once again, the frustration graph with
shaded cliques according to the Krausz decomposition is shown in Fig. 2.
Interestingly, the root graph of this model’s frustration graph is again the
honeycomb lattice. By going backwards, we can see that indeed, any free-
fermion model with trivalent hopping graph can be embedded in a 2-body qubit
Hamiltonian with the same interaction graph. This is because we can find a set
of Pauli operators satisfying any frustration graph whose edges can be
partitioned into triangles by assigning a different single-qubit Pauli to each
of the vertices of every triangle. A term in the Hamiltonian is then the
tensor product of all of the Pauli operators from the triangles to which its
vertex in $G(H)$ belongs.
Unlike in the one-dimensional example, the cycle subgroup of this model,
$Z_{H}$, is nontrivial. This subgroup is generated by the products of
Hamiltonian terms around a hexagonal plaquette of the honeycomb lattice,
denoted $W_{p}$ for plaquette $p$. These cycles are not independent, however,
with constraints between them depending on the boundary conditions of the
lattice. In particular, if the model is on a torus of dimension $L_{x}$ by
$L_{y}$, then the product of all Hamiltonian terms is trivial
$\displaystyle\prod_{\boldsymbol{j}\in
V}\sigma^{\boldsymbol{j}}=(-1)^{L_{x}L_{y}}I$ (67)
In this case, the cycles of the honeycomb lattice are not independent, since
they similarly multiply to the identity. There are thus $L_{x}L_{y}-1$
independent plaquettes on the lattice. There are additionally two
homotopically nontrivial cycles, which are independent as well. Notice that
the edges of the honeycomb lattice itself form a T-join, and so the above
constraint is also the statement that $P$ is furthermore trivial. Therefore,
we have
$\displaystyle|\mathcal{Z}(\mathcal{P}_{H})|=L_{x}L_{y}-1+2=L_{x}L_{y}+1$ (68)
and once again (as first computed in Ref. [54])
$\displaystyle
n_{L}=2L_{x}L_{y}-\left[\frac{1}{2}\left(2L_{x}L_{y}-2\right)+L_{x}L_{y}+1\right]=0.$
(69)
This example also illustrates that quite a large number of symmetries could be
present, and in general this will complicate finding, e.g., the symmetry
sector that contains the ground state.
Figure 3: The frustrated hexagonal gauge 3d color code, proposed in Ref. [44].
This model is based on the 3d gauge color code, whose qubits live on the
vertices of the lattice shown. Gauge generators for the 3d gauge color code
consist of Pauli-$Z$ and Pauli-$X$ operators around both the square and
hexagonal faces of the lattice. Stabilizers of the 3d gauge color code consist
of Pauli-$Z$ and Pauli-$X$ operators on both the cube and “ball" cells. The
frustrated hexagonal gauge 3D color code is given by taking the stabilizers of
the gauge color code together with the hexagonal gauge generators, which
commute with the stabilizers, but not with each other. We see from the colored
hexagonal faces above that the frustration graph of these gauge generators is
a set of disconnected path graphs. Every hexagonal plaquette term anticommutes
with exactly two others—the plaquette terms of the other Pauli type
intersecting it at exactly one qubit—and commutes with all other terms in the
Hamiltonian.
### 5.4 Frustrated Hexagonal Gauge 3D Color Code
Figure 4: The Sierpinski-Hanoi model (left) with its frustration graph,
highlighted, and its root graph (right) with a spanning tree highlighted, for
$k=5$ and local fields absent. Hamiltonian terms are 3-qubit operators acting
on qubits at the vertices of the Sierpinski sieve graph, highlighted in blue.
Cliques of the frustration graph are colored to show the graph’s Krausz
decomposition. Green and orange cells depict generators for the model’s
logical Pauli group. At the interior triangular cells of the lattice are the
3-body generators shown in green. At the interior and exterior edges of the
model are 2-body generators shown in orange. These are obtained from their
adjoining Hamiltonian terms by reflecting the action on the intersection of
their supports (so these generators act differently depending on which edge
they act on). The frustration graph of this model is the Hanoi graph
$H_{3}^{k-1}$. The vertices of this graph are in correspondence to the states
of the towers of Hanoi problem with three towers and $k-1$ discs. The root
graph of $H_{3}^{k-1}$ contains $H_{3}^{k-2}$ as a topological minor.
The frustrated hexagonal gauge 3d color code is a noncommuting Hamiltonian
whose terms consist of the stabilizer generators and a subset of the gauge
generators from the gauge color code. The gauge color code [55, 56, 57, 58]
has a Hamiltonian that is defined in terms of a natural set of gauge
generators as
$\displaystyle H=-\sum_{S\in\square,\varhexagon}\left(J_{x}\bigotimes_{j\in
S}X_{j}+J_{Z}\bigotimes_{j\in S}Z_{j}\right)$ $\displaystyle+\ \text{(boundary
terms)}$ (70)
where “$\square$" and “$\varhexagon$" denote the sets of square and hexagonal
faces on the lattice in Fig. 3, respectively (see Ref. [59] for a detailed
description of this lattice). Here the qubits live on the vertices of the
lattice. Nontrivial boundary conditions are required to restrict the logical
space of this code to a single qubit. We will ignore these boundary conditions
and consider only gauge generators in the bulk of the lattice. The stabilizers
for this model are given by products of $X$ or $Z$ around every elementary
cell, either a cube or a “ball".
We consider a model where we partially restore some of these symmetries.
Namely, we will consider the cube and balls to be “restored” symmetries of the
model, and we will remove the square generators. This leads to the following
gauge Hamiltonian that sums over only hexagonal faces, balls, and cubes,
$\displaystyle
H=-\sum_{S\in\varhexagon,\text{\mancube},{\mathchoice{\includegraphics[height=4.82224pt]{BallCell.png}}{\includegraphics[height=4.82224pt]{BallCell.png}}{\includegraphics[height=3.61664pt]{BallCell.png}}{\includegraphics[height=2.71246pt]{BallCell.png}}}}\left(J_{X}\bigotimes_{j\in
S}X_{j}+J_{Z}\bigotimes_{j\in S}Z_{j}\right).$ (71)
The cube and ball terms commute with all of the hexagon terms, and so
constitute symmetries of the model. Once we fix a sector for these terms, we
can solve the remaining model by mapping to free-fermions as follows [44].
In Figure 3, we represent a subsection of the qubit lattice of this code,
where qubits live at the tetravalent vertices. Because the cube and ball terms
commute with everything, the frustration graph depends only on the hexagonal
faces, several of which are colored in Figure 3. We see that some of these
faces intersect at exactly one vertex, and so the $X$\- and $Z$-type gauge
generators will anticommute on the associated qubit. These intersection
patterns only occur in 1D chains along the cardinal axes of the lattice. In
particular, every hexagonal face only overlaps with two other hexagonal faces
along these chains and otherwise intersects the other faces at an even number
of qubits. The frustration graph of this model thus decouples into a set of
disconnected paths, which are line graphs, and in fact they are the
frustration graph of the XY-model [1] and the 1-d Kitaev wire [60]. A free-
fermion mapping therefore exists for this model, and this demonstrates an
example of how one might construct subsystem codes with a free-fermion
solution to obtain desired spectral properties. In particular, when
$|J_{x}|\not=|J_{z}|$ the model in Eq. (71) is gapped. We note that this
observation was made previously in Ref. [44] in the context of quantum error
correcting codes.
Figure 5: Single-Particle spectrum of the Sierpinski-Hanoi model for $k=5$
with an additional local field term present in the symmetry sector for which
all cycles are $+1$. Circled are two critical points where excited bands
become degenerate.
### 5.5 Sierpinski-Hanoi model
Finally, we introduce our own example of a solvable spin model, which was
previously unknown to the best of our knowledge. This model consists of 3-body
$XYZ$-interaction terms on the shaded cells of the Sierpinski triangle, all
with the same orientation, as depicted in Fig. 4. Explicitly, the Hamiltonian
for this model is given by
$\displaystyle
H=\sum_{(i,j,k)\in\color[rgb]{0.63,0.79,0.95}\mbox{\normalsize$\blacktriangle$}}X_{i}Y_{j}Z_{k}+JH_{\text{local}}$
(72)
where $(i,j,k)$ is an ordered triple of qubits belonging to a particular
shaded cell on the lattice and we will define additional on-site terms
$H_{\text{local}}$ in Eq. (77). An instance of the model is parameterized by
$k$, the fractal recursion depth of the underlying Sierpinski lattice, where
$k=1$ is taken to be a single 3-qubit interaction.
Let us first consider the simplified model where $J=0$. Then the frustration
graph of this model is the so-called _Hanoi graph_ $H_{3}^{k-1}$. The vertices
of this graph are labeled by states of the towers of Hanoi problem with $k-1$
discs, and two vertices are neighboring if transitioning between the
corresponding states is an allowed move in the problem. Perhaps surprisingly,
this graph is a line graph, with root and highlighted spanning tree shown in
Fig. 4. Furthermore, the root graph contains $H_{3}^{k-2}$ as a topological
minor, obtained by removing the vertices of degree one and contracting the
vertices of degree two each along one of their two edges.
This model contains
$\displaystyle n=\frac{3}{2}(3^{k-1}+1)$ (73)
physical qubits, and its root graph contains
$\displaystyle|\widetilde{V}|=\begin{cases}2&k=1\\\ \frac{1}{2}\left[5\times
3^{k-2}+3\right]&k>1\end{cases}$ (74)
vertices. A T-join for this graph therefore exists only for even $k$ and
$k=1$, and the parity operator is never trivial when a T-join exists. The root
graph also contains
$\displaystyle|Z_{H}|=\sum_{j=0}^{k-3}3^{j}=\begin{cases}0&k\leq 2\\\
\frac{1}{2}\left(3^{k-2}-1\right)&k>2\end{cases}$ (75)
fundamental cycles. None of the generators of the cycle subgroup are trivial
since every qubit is acted upon by at most two anti-commuting operators. The
number of logical qubits in this model is therefore
$\displaystyle n_{L}=\begin{cases}2&k=1\\\ \frac{1}{4}\left[11\times
3^{k-2}+8+(-1)^{k}\right]&k>1,\end{cases}$ (76)
and so this Hamiltonian encodes logical qubits at a constant rate of
$\frac{11}{18}$ in the infinite $k$ limit. Perhaps unsurprisingly, these
logical qubits live on the boundaries of the fractal, and we can obtain a set
of generators for the logical Pauli group of this model as shown in Fig. 4. We
can encode logical quantum information in this model by picking symplectic
pairs of generators from this group, which anticommute with one another yet
commute with the remaining generators in the group. The remaining such
generators can be used as gauge qubits for the logical qubit we wish to
protect, and the free-fermion Hamiltonian of the model can be used for error
suppression.
We are also free to add an anisotropic local-field term to a subset of the
qubits without breaking solvability
$\displaystyle
H_{\mathrm{local}}=\sum_{i\in\color[rgb]{0.63,0.79,0.95}\mbox{\normalsize$\blacktriangle$}\mathbf{-}\color[rgb]{0.63,0.79,0.95}\mbox{\normalsize$\blacktriangle$}}\sigma_{i}^{j_{i}}$
(77)
The sum is taken over all qubits corresponding to _black_ edges connecting two
shaded cells in Fig. 4. $j_{i}$ is the third Pauli type from the two Paulis
acting on qubit $i$ by the interaction terms. The effect of these local-field
terms is to couple every black vertex in the root graph shown in Fig. 4,
except for those at the three corners, to a dedicated fermion mode. We do not
depict these additional modes to avoid cluttering the figure. These terms also
do not affect the symmetries of the model, except possibly to add the parity
operator to $\mathcal{Z}(\mathcal{P}_{H})$ when the number of vertices in the
original graph was odd, as the number of vertices in the graph with local
field terms present will always be even. The parity operator can then be
constructed as the product of all of the Hamiltonian terms, since the root
graph only has vertices of degrees 1 and 3.
In Fig. 5, we display the single-particle spectrum of the Sierpinski-Hanoi
model as a function of the local field $J$ for $k=5$ in the sector for which
all of the cycle symmetries are in their mutual $+1$ eigenspace. We highlight
two critical points where excited energy levels become degenerate to within
our numerical precision. We observe that the locations of these points are not
system-size-independent, but rather asymptotically approach $J=0$ as the
system size is increased. We conjecture that this is connected to the
emergence of scale symmetry, which the model possesses in the thermodynamic
limit, yet not for any finite size. It would be intriguing if certain physical
features of this symmetry could be realized at the critical points at finite
size, potentially opening the door to simulating scale-invariant systems on a
finite-sized quantum computer.
## 6 Proofs of Main Theorems
### 6.1 Proof of Theorem 1
We restate Theorem 1 for convenience.
###### Theorem 1, restated (Existence of free-fermion solution).
An injective map $\phi$ as defined in Eq. (20) and Eq. (21) exists for the
Hamiltonian $H$ as defined in Eq. (1) if and only if there exists a root graph
$R$ such that
$\displaystyle G(H)\simeq L(R),$ (78)
where R is the hopping graph of the free-fermion solution.
###### Proof.
If $\phi$ exists, define $R=(V,E)$, where
$E\equiv\\{(\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j}))|\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j})\in
V,\boldsymbol{j}\in E\\}$. If and only if
$|(\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j}))\cap(\phi_{1}(\boldsymbol{k}),\phi_{2}(\boldsymbol{k}))|=1$,
then the vertices corresponding to $\boldsymbol{j}$ and $\boldsymbol{k}$ are
neighboring in $G$ by Eq. (21). Thus, $G(H)\simeq L(R)$ and a mapping $\phi$
exists only if $R$ does.
If there exists a graph $R\equiv(V,E)$ such that $G\simeq L(R)$, take the
Krausz decomposition of $G(H)$. Namely, partition the edges of $G(H)$ as
$F=\\{C_{1},\dots,C_{|V|}\\}$, where each $C_{i}$ constitutes a clique in $G$
and such that every vertex in $G$ appears in at most two $C_{i}$. The cliques
in this partitioning correspond to the vertices $V$ of $R$. For each vertex
$\boldsymbol{j}$, define $\phi(\boldsymbol{j})$ to be the pair of cliques in
which $\boldsymbol{j}$ appears. Since the cliques partition the edges of $G$,
then if vertices $\boldsymbol{j}$ and $\boldsymbol{k}$ are neighboring in $G$,
they must appear in exactly one clique together, and thus
$|(\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j}))\cap(\phi_{1}(\boldsymbol{k}),\phi_{2}(\boldsymbol{k}))|=1$.
Thus, $\phi$ satisfies Eq. (21). Furthermore, $\phi$ is injective, since if
there are two vertices $\boldsymbol{j}$, $\boldsymbol{k}\in G$ such that
$\phi(\boldsymbol{j})=\phi(\boldsymbol{k})$, then $\boldsymbol{j}$ and
$\boldsymbol{k}$ appear in the same two cliques, but since the Krausz
decomposition is a partition of the edges, this would require that
$\boldsymbol{j}$ and $\boldsymbol{k}$ neighbor by two edges. However, the
definition of $G$ guarantees that pairs of vertices can only neighbor by at
most one edge, and so this is impossible. Therefore $\phi$ is injective. ∎
### 6.2 Proof of Theorem 2
Once again, we restate our theorem for convenience
###### Theorem 2, restated (Symmetries are cycles and parity).
Given a Hamiltonian satisfying Eq. (78) such that the number of vertices
$|\widetilde{V}|$ in the root graph is odd, then we have
$\displaystyle\mathcal{Z}(\mathcal{P}_{H})=Z_{H}.$ (79)
If the number of vertices in the root graph is even, then we have
$\displaystyle\mathcal{Z}(\mathcal{P}_{H})=\left\langle Z_{H},P\right\rangle.$
(80)
_Proof._ Let $G\equiv(E,F)\simeq L(R)$ be the connected line graph of a
connected root graph $R=(V,E)$, and let $G$ have adjacency matrix
$\mathbf{A}$. We will need the following well-known factorization of a line
graph adjacency matrix $\mathbf{A}$
$\displaystyle\mathbf{A}=\mathbf{B}\mathbf{B}^{\mathrm{T}}\ \mathrm{(mod\ 2)}$
(81)
where $\mathbf{B}$ is the edge-vertex incidence matrix of $R$. That is,
$\mathbf{B}$ is a $|E|\times|V|$ matrix such that
$\displaystyle B_{\boldsymbol{j}l}=\begin{cases}1&l\in\boldsymbol{j}\\\
0&\mathrm{otherwise}\end{cases}$ (82)
for all $\boldsymbol{j}\in E$ and $l\in V$. We can interpret $\mathbf{B}$ as
defining the map $\phi$ via
$\displaystyle\phi:\sigma^{\boldsymbol{j}}\mapsto\prod_{l\in
V}\gamma_{l}^{B_{\boldsymbol{j}l}}$ (83)
That is, $\phi_{1}(\boldsymbol{j})$ and $\phi_{2}(\boldsymbol{j})$ are the
indices of the nonzero elements in the row labeled by $\boldsymbol{j}$ in
$\mathbf{B}$. This then defines the adjacency matrix $\mathbf{A}$ through the
scalar commutator as
$\displaystyle[\\![\prod_{l\in V}\gamma_{l}^{B_{\boldsymbol{j}l}},\prod_{m\in
V}\gamma_{m}^{B_{\boldsymbol{k}l}}]\\!]$ $\displaystyle=\prod_{l,m\in
V}[\\![\gamma_{l}^{B_{\boldsymbol{j}l}},\gamma_{m}^{B_{\boldsymbol{k}m}}]\\!]$
(84) $\displaystyle=\prod_{l,m\in
V}(-1)^{(1-\delta_{lm})B_{\boldsymbol{j}l}B_{\boldsymbol{k}m}}$ (85)
$\displaystyle[\\![\prod_{l\in V}\gamma_{l}^{B_{\boldsymbol{j}l}},\prod_{m\in
V}\gamma_{m}^{B_{\boldsymbol{k}l}}]\\!]$
$\displaystyle=(-1)^{(\mathbf{B}\mathbf{B}^{\mathrm{T}})_{\boldsymbol{j}\boldsymbol{k}}+\left(\sum_{l}B_{\boldsymbol{j}l}\right)\left(\sum_{l}B_{\boldsymbol{k}l}\right)}$
(86) $\displaystyle(-1)^{A_{\boldsymbol{j}\boldsymbol{k}}}$
$\displaystyle=(-1)^{(\mathbf{B}\mathbf{B}^{\mathrm{T}})_{\boldsymbol{j}\boldsymbol{k}}}$
(87)
From the third to the fourth line, we replaced the left-hand side with the
definition of $\mathbf{A}$ and used the fact that the rows of $\mathbf{B}$
have exactly two nonzero elements. By the distributive property of the scalar
commutator Eq. (4), we can extend the above equation to products of
Hamiltonian terms
$\displaystyle\prod_{\boldsymbol{j}\in E}\left(\prod_{l\in
V}\gamma_{l}^{B_{\boldsymbol{j}l}}\right)^{v_{\boldsymbol{j}}}=\pm\prod_{l\in
V}\gamma_{l}^{\left(\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}\right)_{l}}\mathrm{,}$
(88)
where $\mathbf{v}\in\\{0,1\\}^{\times|E|}$, as
$\displaystyle[\\![\prod_{l\in
V}\gamma_{l}^{B_{\boldsymbol{j}l}},\prod_{\boldsymbol{k}\in
E}\left(\prod_{m\in
V}\gamma_{m}^{B_{\boldsymbol{k}m}}\right)^{v_{\boldsymbol{k}}}]\\!]=(-1)^{\left(\mathbf{B}\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}\right)_{\boldsymbol{j}}}$
(89)
since linear combinations of rows of $\mathbf{B}$ over $\mathds{F}_{2}$ will
have even-many ones. Every element of $\mathcal{P}_{H}$ is a (non-unique)
linear combination of rows of $\mathbf{B}$ over $\mathds{F}_{2}$, and so to
characterize the elements of $\mathcal{Z}(\mathcal{P}_{H})$, it is sufficient
to find a spanning set of the kernel of $\mathbf{A}$,
$\displaystyle\mathbf{A}\cdot\mathbf{v}=\mathbf{B}\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}=\mathbf{0}\
\mathrm{(mod\ 2)}$ (90)
It is again well-known that the $\mathds{F}_{2}$-kernel of
$\mathbf{B}^{\mathrm{T}}$ is the cycle space of $R$, and this specifies the
cycle subgroup $Z_{H}$ as being contained in $\mathcal{Z}(\mathcal{P}_{H})$.
All that is left is therefore to find all $\mathbf{v}$ such that
$\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}$ is in the kernel of $\mathbf{B}$.
Since we have assumed $G$ is connected, it is easy to see that the only
element in this kernel is $\mathbf{1}$, the all-ones vector. Thus,
$\mathbf{v}$ will also be in the kernel of $\mathbf{A}$ if it defines a T-join
of $G$. If $|V|$ is even, then we can construct a T-join by first pairing the
vertices along paths of $G$. We can then ensure that each edge appears at most
once in the T-join by taking the symmetric difference of all paths. If $|V|$
is odd, then no T-join exists. Indeed, assume that a T-join $T$ does exist for
$|V|$ odd, and let $\widetilde{G}=(V,T)\subseteq G$ be the subgraph of $G$
containing exactly the edges from the T-join. By construction $\widetilde{G}$
contains all the vertices of $G$ and has odd degree for every vertex, though
it may no longer be connected. Let these degrees be $\\{d_{j}\\}_{j\in V}$,
then by the handshaking lemma
$\displaystyle\sum_{j=1}^{|V|}d_{j}=2|T|\mathrm{.}$ (91)
However, the left side must be odd since we have assumed the degree of every
vertex in $\widetilde{G}$ is odd, and the number of vertices is also odd, and
so we have a contradiction. ∎
## 7 Discussion
We have seen how the tools of graph theory can be leveraged to solve a wide
class of spin models via mapping to free fermions, and given an explicit
procedure for constructing the free-fermion solution when one exists. A major
remaining open question, however, concerns the characterization of free-
fermion solutions beyond the generator-to-generator mappings we consider here.
That is, if $G(H)$ is not a line graph and no removal of twin vertices will
make it so, then it may _still_ be possible for a free-fermion solution for
$H$ to exist thanks to the continuum of locally equivalent Pauli-bases into
which $H$ may be expanded. Our fundamental theorem does not rule out the
possibility that special such bases may exist. The problem of finding such
bases is equivalent to finding specific unitary rotations of $H$ for which the
$G(H)$ again becomes a line graph. These rotations must be outside of the
Clifford group, since the frustration graph is a Clifford invariant. Their
existence may therefore depend on specific algebraic relationships between the
Pauli coefficients $h_{\boldsymbol{j}}$ in the Hamiltonian, since the
existence of a free-fermionization is a spectral invariant. We expect such
transformations will be hard to find in general, though perhaps progress can
be made for single-qubit rotations on 2-local Hamiltonians in a similar vein
as in Ref. [61] for stoquasticity. Recently, a local
spin-$\nicefrac{{1}}{{2}}$ model with a free-fermion solution – despite no
such generator-to-generator solution existing – has been found in Ref. [62].
An investigation of models which may be fermionized by these more general
transformations is therefore an interesting subject of future work.
It is natural to ask whether our results could have implications for
simulating quantum systems and quantum computation. We expect our
characterization to shed some light on the inverse problem of finding fermion-
to-qubit mappings, such as the Bravyi-Kitaev superfast encoding [63], Bravyi-
Kitaev transform [64], and generalized superfast encoding [65]. It is possible
to achieve further encodings by introducing ancillary fermion modes, as seen
in the Verstraete-Cirac mapping [7] and the contemporaneous mapping introduced
by Ball [66]. Encodings can be further improved through tailoring to specific
symmetries [67], connectivity structures [68, 69], and through the application
of Fenwick trees [70]. Recently, a treelike mapping was shown to achieve
optimal average-case Pauli-weight in Ref [71]. In their “Discussion” section,
the authors remark that an interesting future direction for their work would
involve introducing ancillary qubits to their mapping. We expect our
classification of the symmetries of free-fermion spin models to help guide
this investigation, though further work is required to fully characterize the
logical symmetry groups which can be realized by these models.
Finally, our characterization highlights the possibility of a “free-fermion
rank" for Hamiltonians as an important measure of classical simulability.
Namely, if there is no free-fermion solution for a given Hamiltonian, we can
still group terms into collections such that each collection independently has
such a solution. An interesting natural question for future work is whether
the minimal number of such collections required can be interpreted as a
quantum resource in an analogous way to the fermionic Gaussian rank [72] or
stabilizer rank [73] for states.
## Acknowledgements
We thank Samuel Elman, Ben Macintosh, Ryan Mann, Nick Menicucci, Andrew
Doherty, Stephen Bartlett, Sam Roberts, Alicia Kollár, Deniz Stiegemann,
Sayonee Ray, Chris Jackson, Jonathan Gross, and Nicholas Rubin for valuable
discussions throughout this project. This work was supported by the Australian
Research Council via EQuS project number CE170100009.
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|
2024-09-04T02:54:59.341499 | 2020-03-11T18:53:02 | 2003.05485 | {
"authors": "Riccardo Fazio and Salvatore Iacono",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26175",
"submitter": "Riccardo Fazio",
"url": "https://arxiv.org/abs/2003.05485"
} | arxiv-papers | # A Non-Iterative Transformation Method for a Class of Free Boundary Value
Problems Governed by ODEs
Riccardo Fazio111Corresponding author: e-mail<EMAIL_ADDRESS>home-page:
http://mat521.unime.it/fazio and Salvatore Iacono
Department of Mathematics and Computer Science
University of Messina
Viale F. Stagno D’Alcontres 31, 98166 Messina, Italy
###### Abstract
The aim of this work is to point out that the class of free boundary problems
governed by second order autonomous ordinary differential equations can be
transformed to initial value problems. Interest in the numerical solution of
free boundary problems arises because these are always nonlinear problems. The
theoretical content of this paper is original: results already available in
literature are related to the invariance properties of scaling or spiral
groups of point transformations but here we show how it is also possible to
use e invariance properties of a translation group. We test the proposed
algorithm by solving three problems: a problem describing a rope configuration
against an obstacle, a dynamical problem with a nonlinear force, and a problem
related to the optimal length estimate for tubular flow reactors.
Key Words. Ordinary differential equations, free boundary problems, initial
value methods, non-iterative transformation method, translation group of point
transformations.
AMS Subject Classifications. 65L10, 65L99, 34B15, 34B99.
## 1 Introduction
Free boundary value problems (BVPs) occur in all branches of applied
mathematics and science. The oldest problem of this type was formulated by
Isaac Newton, in book II of his great “Principia Mathematica” of 1687, by
considering the optimal nose-cone shape for the motion of a projectile subject
to air resistance, see Edwards [1] or Fazio [2].
In the classical numerical treatment of a free BVP a preliminary reduction to
a BVP is introduced by considering a new independent variable; see, Stoer and
Bulirsch [3, p. 468], Ascher and Russell [4], or Ascher, Mattheij and Russell
[5, p. 471]. By rewriting a free BVP as a BVP it becomes evident that the
former is always a nonlinear problem; the first to point out the nonlinearity
of free BVPs was Landau [6]. Therefore, in that way free BVPs are BVPs. In
this paper we show that free BVPs invariant with respect to a translation
group can be solved non-iteratively by solving a related initial value problem
(IVP). Therefore in this way those free BVPs are indeed IVPs. Moreover, we are
able to characterize a class of free BVPs that can be solved non-iteratively
by solving related IVPs.
The non-iterative numerical solution of BVPs is a subject of past and current
research. Several different strategies are available in literature for the
non-iterative solution of BVPs: superposition [5, pp. 135-145], chasing [7,
pp. 30-51], and adjoint operators method [7, pp. 52-69] that can be applied
only to linear models; parameter differentiation [7, pp. 233-288] and
invariant imbedding [8] can be applied also to nonlinear problems. In this
context transformation methods (TMs) are founded on group invariance theory,
see Bluman and Cole [9], Dresner [10], or Bluman and Kumei [11]. These methods
are initial value methods because they achieve the numerical solution of BVPs
through the solution of related IVPs.
The first application of a non-iterative TM was given by Töpfer in [12] for
the Blasius problem, without any consideration of group invariance theory.
Töpfer’s algorithm is quoted in several books on fluid dynamics, see, for
instance, Goldstein [13, pp. 135-136]. Acrivos, Shah and Petersen [14] first
and Klamkin [15] later extended Töpfer’s method respectively to a more general
problem and to a class of problems. Along the lines of the work of Klamkin,
for a given problem Na [16, 17] showed the relation between the invariance
properties, with respect to a linear group of transformation (the scaling
group), and the applicability of a non-iterative TM. Na and Tang [18] proposed
a non-iterative TM based on the spiral group and applied it to a non-linear
heat generation model. Belford [19] first, and Ames and Adams [20, 21] later
defined non-iterative TMs for eigenvalue problems. A review paper was written
by Klamkin [22]. Extensions of non-iterative TM, by requiring the invariance
of one and of two or more physical parameters when they are involved in the
mathematical model, were respectively proposed by Na [23] and by Scott,
Rinschler and Na [24]; see also Na [7, Chapters 8 and 9]. A survey book,
written by Na [7, Chs 7-9] on the numerical solution of BVP, devoted three
chapters to numerical TMs.
As far as free BVPs are concerned, non-iterative and iterative TMs were
proposed by Fazio and Evans [25]. Fazio [26] has shown that we can extend the
applicability of non-iterative TMs by rewriting a given free BVP using a
variables transformation obtained by linking two different invariant groups.
However, non-iterative TMs are applicable only to particular classes of BVPs
so that they have been considered as ad hoc methods, see Meyer [8, pp. 35-36],
Na [7, p. 137] or Sachdev [27, p. 218].
The transformation of BVPs to IVPs has also a theoretical relevance. In fact,
existence and uniqueness results can be obtained as a consequence of the
invariance properties. For instance, for the Blasius problem, a simple
existence and uniqueness theorem was given by J. Serrin [28] as reported by
Meyer [29, pp. 104-105] or Hastings and McLeod [30, pp. 151-153]. Moreover,
using scaling invariance properties the error analysis of the truncated
boundary formulation of the Blasius problem was developed by Rubel [31]. On
this topic a first application of a numerical test, defined within group
invariance theory, to verify the existence and uniqueness of the solution of a
free BVPs was considered by Fazio in [32]. A formal definition of the
mentioned numerical test can be found in [33].
In this paper we consider the class of free BVPs governed by second order
autonomous differential equations, and define, for these problems, a non-
iterative TM using the invariance properties of a translation group. As far as
applications of the proposed algorithm are concerned, we solve three problems.
First we test our method with a problem describing a rope configuration
against an obstacle, where we compare the obtained numerical results with the
exact solution. Then we solve a dynamical problem with a nonlinear force, and
a problem related to the optimal length estimate for tubular flow reactors,
where in both cases our results are compared to numerical data available in
literature. Finally, the last section is concerned with concluding remarks
pointing out limitations and possible extensions of the proposed approach.
## 2 The non-iterative TM
Let us consider the class of second order free BVPs given by
$\displaystyle\frac{d^{2}u}{dx^{2}}=\Omega\left(u,\frac{du}{dx}\right)\
,\qquad x\in(0,s)$ (2.1) $\displaystyle A_{1}u(0)+A_{2}\frac{du}{dx}(0)=A_{3}\
,$ (2.2) $\displaystyle u(s)=B\quad,\quad\frac{du}{dx}(s)=C\ ,$ (2.3)
where $A_{i}$, for $i=1,2,3$, $B$ and $C$ are arbitrary constants, and $s>0$
is an unknown free boundary. The differential equation (2.1) and the two free
boundary conditions (2.3) are invariant with respect to the translation group
$\displaystyle x^{*}=x+\mu\quad;\quad s^{*}=s+\mu\quad;\quad u^{*}=u\ .$ (2.4)
By using this invariance, we can define the following non-iterative algorithm
for the numerical solution of (2.1)-(2.3):
* •
we fix freely a value of $s^{*}$;
* •
we integrate backwards from $s^{*}$ to $x_{0}^{*}$ the following auxiliary IVP
$\displaystyle\frac{d^{2}u^{*}}{dx^{*2}}=\Omega\left(u^{*},\frac{du^{*}}{dx^{*}}\right)$
$\displaystyle u^{*}(s^{*})=B\ ,\qquad\frac{du^{*}}{dx^{*}}(s^{*})=C\ ,$
using an event locator in order to find $x_{0}^{*}$ such that
$A_{1}u^{*}(x_{0}^{*})+A_{2}\frac{du^{*}}{dx^{*}}(x_{0}^{*})=A_{3}\ ;$ (2.6)
* •
finally, through the invariance property, we can deduce the similarity
parameter
$\mu=x_{0}^{*}\ ,$ (2.7)
from which we get the unknown free boundary
$s=s^{*}-\mu\ .$ (2.8)
The missing initial conditions are given by
$u(0)=u^{*}(x_{0}^{*})\
,\qquad\frac{du}{dx}(0)=\frac{du^{*}}{dx^{*}}(x_{0}^{*})\ .$ (2.9)
Let us define now a simple event locator suited to the class of problems
(2.1)-(2.3). We consider first the case where
$A_{1}u^{*}(s^{*})+A_{2}\frac{du^{*}}{dx^{*}}(s^{*})<A_{3}\ .$ (2.10)
We can integrate the auxiliary IVP (• ‣ 2) with a constant step size $\Delta
x^{*}$ until at a given mesh point $x_{k}^{*}$ we get
$A_{1}u^{*}(x_{k}^{*})+A_{2}\frac{du^{*}}{dx^{*}}(x_{k}^{*})>A_{3}\ ,$ (2.11)
and repeat the last step with the smaller step size
$\Delta x_{0}^{*}=\Delta x^{*}\frac{\displaystyle
A_{3}-A_{1}u^{*}(x_{k-1}^{*})-A_{2}\frac{du^{*}}{dx^{*}}(x_{k-1}^{*})}{\displaystyle
A_{1}u^{*}(x_{k}^{*})+A_{2}\frac{du^{*}}{dx^{*}}(x_{k}^{*})-A_{1}u^{*}(x_{k-1}^{*})-A_{2}\frac{du^{*}}{dx^{*}}(x_{k-1}^{*})}\
.$ (2.12)
In defining the last step size in equation (2.12) we use a linear
interpolation. As a consequence, we have that $x_{0}^{*}\approx
x_{k}^{*}-\Delta x_{0}^{*}$. We notice that the condition imposed by this
event locator converges to the correct condition (2.6) as the step size goes
to zero, cf. the second column of table 2.
The other case
$A_{1}u^{*}(s^{*})+A_{2}\frac{du^{*}}{dx^{*}}(s^{*})>A_{3}\ ,$ (2.13)
can be treated in a similar way. Of course, also in this second case the last
step size is smaller than the previous ones.
In the next section we apply the proposed non-iterative TM to three problems.
The reported numerical results were computed by the classical fourth-order
Runge-Kutta’s method, reported by Butcher [34, p. 166], coupled with the event
locator defined above.
## 3 The obstacle problem on a string
For the obstacle problem on a string, depicted on figure 1 within the
$(x,u)$-plane where the $x$ axis is taken overlying to the obstacle, we have
to consider the following mathematical model, see Collatz [35] or Glashoff and
Werner, [36]
$\displaystyle\frac{d^{2}u}{dx^{2}}=\theta\left[1+\left(\frac{du}{dx}\right)^{2}\right]^{1/2}\
,\qquad x\in(0,s)$ $\displaystyle u(0)=u_{0}\ ,\qquad u(s)=\frac{du}{dx}(s)=0\
,$
where the positive value of $\theta$ depends on the string properties. In this
problem we have to find the position of a uniform string of finite length $L$
under the action of gravity. The string has fixed end points, say $(0,u_{0})$
and $(b,0)$, where $u_{0}>0$ and $b>0$. Furthermore, we assume that the
condition $L^{2}>\left(u_{0}^{2}+b^{2}\right)$ is fulfilled; this condition
allows us to define a free boundary $s$ for this problem, where $s$ is the
detached rope position from the obstacle.
The free BVP (3) was solved by the first author in [37] by iterative methods,
namely a shooting method and the iterative extension of the TM derived by
using the invariance with respect to a scaling group.
The exact solution of the free BVP (3) is given by
$\displaystyle
u(x)=\theta^{-1}\left[\cosh\left(\theta\left(x-s\right)\right)-1\right]\ ,$
(3.2) $\displaystyle s=\theta^{-1}\ln\left[\theta u_{0}+1+\left(\left(\theta
u_{0}+1\right)^{2}-1\right)^{1/2}\right]\ ,$
from this we easily find
$\frac{du}{dx}(0)=\sinh(-\theta s)=\frac{1}{2}\left(e^{-\theta s}-e^{\theta
s}\right)\ ,$ (3.3)
and, therefore, for $\theta=0.1$ and $u_{0}=1$ from equations (3)-(3.3) we get
the values
$s=4.356825433\ ,\qquad\frac{du}{dx}(0)=-0.458257569\ ,$ (3.4)
that are correct to the ninth decimals.
Table 1: Convergence of numerical results for $\theta=0.1$. $\Delta x$ | $\frac{du}{dx}(0)$ | $e_{r}$ | $s$ | $e_{r}$
---|---|---|---|---
$-0.1$ | $-0.458227362$ | $6.59\mbox{D}-05$ | $4.435407932$ | $6.19\mbox{D}-05$
$-0.05$ | $-0.458250809$ | $1.47\mbox{D}-05$ | $4.435621088$ | $1.39\mbox{D}-05$
$-0.025$ | $-0.458255551$ | $4.40\mbox{D}-06$ | $4.435664194$ | $4.14\mbox{D}-06$
$-0.0125$ | $-0.458257313$ | $5.59\mbox{D}-07$ | $4.435680211$ | $5.26\mbox{D}-07$
$-0.00625$ | $-0.458257463$ | $2.31\mbox{D}-07$ | $4.435681576$ | $2.18\mbox{D}-07$
$-0.003125$ | $-0.458257538$ | $6.74\mbox{D}-08$ | $4.435682258$ | $6.43\mbox{D}-08$
$-0.0015625$ | $-0.458257565$ | $8.52\mbox{D}-09$ | $4.435682504$ | $9.05\mbox{D}-09$
Let us consider a convergence numerical test for our non-iterative TM. Table 1
reports the obtained numerical results for the missing initial condition and
the free boundary value for the free BVP (3) with $\theta=0.1$ and $u_{0}=1$,
as well as the corresponding relative errors denoted by $e_{r}$. An example of
the numerical solutions is shown in figure 1.
Figure 1: Picture of the numerical solution obtained for $\theta=0.1$,
$u_{0}=1$, and $b=4.5$; the obstacle, is superimposed to the $x$ axis, and is
displayed by a black solid line.
For this numerical solution, we applied a large step size in order to show the
mesh used and to empathize how our event locator reduces the last step.
## 4 A dynamical free BVP
Suppose a particle of unitary mass is moving against a nonlinear force, given
by $-1-u-\left(\frac{du}{dx}\right)^{2}$, from the origin $u=0$ to a final
position $u=1$, our goal is to determine the duration of motion $s$ and the
initial velocity that assures that the particle is momentarily at rest at
$u=1$; see Meyer [8, pp. 97-99]. This problem can be formulated as follows
$\displaystyle\frac{d^{2}u}{dx^{2}}=-1-u-\left(\frac{du}{dx}\right)^{2}\
,\qquad x\in(0,s)$ $\displaystyle u(0)=0\ ,\qquad u(s)=1\
,\qquad\frac{du}{dx}(s)=0\ ,$
where $u$ and $x$ are the particle position and the time variable,
respectively, on the right hand side of the governing differential equation we
have the nonlinear force acting on the particle and $s$ is the free boundary.
In table 2 we propose a numerical convergence test for decreasing values of
the step size.
Table 2: Convergence of numerical results for the free BVP dynamical model. $\Delta x$ | $u(0)$ | $\frac{du}{dx}(0)$ | $s$
---|---|---|---
$-0.1$ | $1.16\mbox{D}-02$ | $3.212263787$ | $0.867662139$
$-0.05$ | $3.54\mbox{D}-03$ | $3.240676696$ | $0.870143219$
$-0.025$ | $4.61\mbox{D}-04$ | $3.2516023692$ | $0.871089372$
$-0.0125$ | $1.90\mbox{D}-04$ | $3.252564659$ | $0.871172452$
$-0.00625$ | $5.42\mbox{D}-05$ | $3.253049203$ | $0.871214290$
$-0.003125$ | $9.25\mbox{D}-06$ | $3.253209165$ | $0.871228100$
$-0.0015625$ | $3.43\mbox{D}-06$ | $3.253229900$ | $0.871229890$
$-0.00078125$ | $5.12\mbox{D}-07$ | $3.253240276$ | $0.871230785$
$-0.000390625$ | $2.01\mbox{D}-07$ | $3.253241381$ | $0.871230881$
$-0.0001953125$ | $4.62\mbox{D}-08$ | $3.253241934$ | $0.871230929$
The obtained results can be contrasted with those reported by Meyer [8, pp.
97-99] where, by using the invariant imbedding method, he found $s=1.2651$ but
a value of $u(0)=0.0163$ instead of $u(0)=0$ as prescribed by the free BVP
(4). The behaviour of the solution can be seen in the figure 2.
Figure 2: Numerical solution for the dynamical free BVP (4) obtained with
$\Delta x=0.0125$.
Again we applied a large step size in order to show how our event locator
reduces the last step.
A free BVP similar to (4) was considered by Na [7, p. 88] where the nonlinear
force was replaced by $-u\exp(-u)$. However, in this case it is possible to
prove [33], using the conservation of energy principle, that the free BVP has
countable infinite many solutions, with the missing initial conditions given
by
$\frac{du}{dx}(0)=\pm 0.726967811\ .$ (4.2)
## 5 Length estimation for tubular flow reactors
Roughly speaking, a tubular flow chemical reactor can be seen as a device
where on one side it is introduced a material A that along its passage inside
the reactor undergoes a chemical reaction so that at the exit we get a product
B plus a residual part of A; see figure 3. A $n$th order chemical reactor is
usually indicated with the notation A${}^{n}\rightarrow$ B.
Figure 3: Schematic set-up of a tubular flow reactor.
A free BVP for a tubular reactor can be formulated as
$\displaystyle\frac{d^{2}u}{dx^{2}}=N_{Pe}\left(\frac{du}{dx}+Ru^{n}\right)\
,$ $\displaystyle u(0)-\frac{1}{N_{Pe}}\frac{du}{dx}(0)=1\ ,\qquad u(s)=\tau\
,\quad\frac{du}{dx}(s)=0\ ,$
where $u(x)$ is the ratio between the concentration of the reactant A at a
distance $x$ and the concentration of it at $x=0$, $N_{Pe}$, $R$, $n$ and
$\tau$ are, the Peclet group, the reaction rate group, the order of the
chemical reaction and the residual fraction of reactant A at exit,
respectively. Moreover, $N_{Pe}$ and $R$ are both greater than zero. Finally,
for the free BVP (5), the free boundary $s$ is the length of the flow reactor
we are trying to estimate.
This is an engineering problem that consists in determining the optimal length
of a tubular flow chemical reactor with axial missing and has been already
treated by Fazio in [32], through an iterative TM, whereas Fazio in [38] made
a comparison between the results obtained with a shooting method and the upper
bound of the free boundary value obtained by a non-iterative TM.
Here, for the sake of comparing the numerical results, we fix the parameters
as follows: $N_{Pe}=6$, $R=2$, $n=2$, and $\tau=0.1$. We apply the algorithm
outlined above to the numerical solution of the free BVP (5). Table 3 shows a
numerical convergence test for decreasing values of the step size.
Table 3: Convergence of numerical results. $N_{pe}=6$, $R=2$, $n=2$, and $\tau=0.1$. $\Delta x$ | $u(0)$ | $\frac{du}{dx}(0)$ | $s$
---|---|---|---
$-0.1$ | $0.829314641$ | $-1.008175212$ | $5.117905669$
$-0.05$ | $0.830537187$ | $-1.010745699$ | $5.119104349$
$-0.025$ | $0.831147822$ | $-1.012077034$ | $5.119707352$
$-0.0125$ | $0.831227636$ | $-1.012251496$ | $5.119786158$
$-0.00625$ | $0.831267467$ | $-1.012338738$ | $5.119825502$
$-0.003125$ | $0.831271635$ | $-1.012347868$ | $5.119829619$
$-0.0015625$ | $0.831273719$ | $-1.012352436$ | $5.119831678$
$-0.00078125$ | $0.831274182$ | $-1.012353449$ | $5.119832135$
$-0.000390625$ | $0.831274327$ | $-1.012353767$ | $5.119832278$
$-0.0001953125$ | $0.831274348$ | $-1.012353814$ | $5.119832299$
The obtained numerical results are reported on table 4 and compared with
numerical results available in literature.
Table 4: Comparison of numerical results for the tubular flow reactor model. | $u(0)$ | $\frac{du}{dx}(0)$ | $s$
---|---|---|---
iterative TM [32] | $0.831280$ | $-1.012298$ | $5.121648$
shooting method [38] | $0.831274$ | $-1.012354$ | $5.119832$
non-iterative TM | $0.831274$ | $-1.012354$ | $5.119832$
As it is easily seen the computed values are in good agreement with the ones
found in [32] and [38]. The behaviour of the solution can be seen in figure 4.
Figure 4: Numerical solution for length estimation of a tubular flow reactor
obtained with $\Delta x=0.1$.
Once again, we used a large step size to make clear how our event locator
reduces the last step size.
## 6 Conclusion
In closing, we can outline some further implications coming out from this
work. First of all, the algorithm proposed in this paper can be extended to
free BVPs governed by a system of first order autonomous differential
equations belonging to the general class of problems
$\displaystyle{\displaystyle\frac{d{\bf u}}{dx}}={\bf q}\left({\bf u}\right)\
,\quad x\in[0,\infty)\ ,$ $\displaystyle u_{j}(0)=u_{j0}\ ,\qquad{\bf
u}(s)={\bf u}_{s}\ ,$
where ${\bf u}(x),{\bf u}_{s}\in\hbox{I\kern-1.99997pt\hbox{R}}^{d}$, ${\bf
q}:\hbox{I\kern-1.99997pt\hbox{R}}^{d}\rightarrow~{}\hbox{I\kern-1.99997pt\hbox{R}}^{d}$,
with $d\geq 1$, $j\in\\{1,\dots,d\\}$, $u_{j0}$ and all components of ${\bf
u}_{s}$ are given constants and $s$ is the free boundary.
Moreover, our algorithm can be applied by using an integrator from the MATLAB
ODE suite written by Samphine and Reichelt [39], and available with the latest
releases of MATLAB, with the event locator option command set in
options = odeset(’Events’,@name)
where “name” is an external, problem dependent, event function.
As mentioned in the introduction, the first application of a non-iterative TM
was defined by Töpfer in [12] more than a century ago. In this paper, by
considering the invariance with respect to a translation group, we have
investigated a possible way to solve a large class of free BVPs by a non-
iterative TM.
However, it is a simple matter to show a differential equation not admitting
any group of transformations: e.g. the differential equation considered by
Bianchi [40, pp. 470-475]. Consequently, it is easy to realize that non-
iterative TMs cannot be extended to every BVPs. Therefore, non-iterative TMs
are ad hoc methods. Their applicability depends on the invariance properties
of the governing differential equation and the given boundary conditions.
On the other hand, free BVPs governed by the most general second order
differential equation, in normal form, can be solved iteratively by extending
a scaling group via the introduction of a numerical parameter so as to recover
the original problem as the introduced parameter goes to one, see Fazio [25,
26, 33, 41]. The extension of this iterative TM to problems in boundary layer
theory has been considered in [42, 43, 44, 45]. Moreover, a further extension
to the sequence of free BVPs obtained by a semi-discretization of parabolic
moving boundary problems was repoted in [46].
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|
2024-09-04T02:54:59.350369 | 2020-03-11T19:05:24 | 2003.05489 | {
"authors": "Thomas Gerard, Christopher Parsonson, Zacharaya Shabka, Polina Bayvel,\n Domani\\c{c} Lavery, Georgios Zervas",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26176",
"submitter": "Thomas Gerard",
"url": "https://arxiv.org/abs/2003.05489"
} | arxiv-papers | SWIFT: Scalable Ultra-Wideband Sub-Nanosecond Wavelength Switching for Data
Centre Networks
Thomas Gerard*, Christopher Parsonson, Zacharaya Shabka, Polina Bayvel,
Domaniç Lavery and Georgios Zervas
Optical Networks Group, Dept. of Electronic and Electrical Engineering,
University College London, London, UK, WC1E 7JE.
<EMAIL_ADDRESS>
###### Abstract
We propose a time-multiplexed DS-DBR/SOA-gated system to deliver low-power
fast tuning across S-/C-/L-bands. Sub-ns switching is demonstrated, supporting
122$\times$50 GHz channels over 6.05 THz using AI techniques.
OCIS codes: 140.3600 Lasers, tunable, 060.6718 Switching, circuit, 060.1155
All-optical networks
## 1 Introduction
The most common data center network (DCN) packet length is $<$256 bytes which
translates to 20 ns slots in 100G links [1]. Optical circuit switching (OCS)
aims to transform data centre networks (DCNs) but needs to operate at packet
speed and granularity [2]. Recent breakthroughs have brought OCS closer to
reality. A hardware-based OCS scheduling algorithm has demonstrated
synchronous scheduling of up to 32,768 nodes within 2.3 ns [2]. A clock phase
caching method has enabled clock and data recovery in less than 625 ps,
supporting 10,000 remote nodes [1]. Yet, energy-efficient, sub-ns, many-
channel optical switching remains a challenge. Wideband fast tuneable lasers
have demonstrated switching on ns timescales [4, 5], and as low as 500 ps but
over limited bandwidths [6]. Static laser diodes (LDs) gated by semiconductor
optical amplifiers (SOAs) have achieved 912 ps 10-90% rise-fall times with
$\sim$2 ns settling time ($\pm 5\%$ of the target value) [3]; however, the
power consumption and device count limit the scalability of this approach. A
similar method used an optical comb where each wavelength was filtered then
gated by an SOA [7]; the power consumption and device count therefore also
increase linearly with number of channels, limiting scalability (see Fig.
1(b)).
In this paper, we introduce SWIFT: a modular system with Scalable Wideband
Interconnects for Fast Tuning. SWIFT combines pairs of optimised widely
tuneable lasers (TLs), multiplexing their wavelength reconfiguration on packet
timescales. The lasers are gated by pairs of fast switching SOAs, resulting in
wideband, sub-ns switching. The modular design of SWIFT (Fig. 1(a)) shows that
just two lasers and two SOAs cover each optical transmission band. SWIFT power
consumption is, therefore, practically independent of channel count; Fig. 1(b)
shows that SWIFT becomes more power efficient than alternative sub-ns
switching sources beyond 8$\times$50 GHz spaced channels.
| | |
---|---|---|---
(a) | (b) | (c) | (d)
Fig. 1: (a) Modular SWIFT architecture across S-, C- and L-bands. (b) Power
consumption comparison of laser switch designs vs. no. of channels, using data
reported in [8]. (c) PULSE DCN architecture with SWIFT modules (in red). (d)
Comparison of switching times (reported rise (solid) and estimated settling
(faded)) against no of channels for different switch systems.
The SWIFT modules can be deployed as transmitters in DCN architectures such as
PULSE [8], as shown in Fig. 1(c). In this architecture, each node has $x$
SWIFT transmitters (highlighted in red), each local pod has $N$ nodes, and
$x^{2}$ star couplers enable there to be $x$ source and $x$ destination pods.
Thus, PULSE network’s number of end-points scales with $N\times x$, where $N$
is limited by the number of wavelength channels. The large number of channels
supported by SWIFT therefore allows for significant scalability in the PULSE
DCN [2].
The concept of time-multiplexed, fast tuneable lasers was proposed in [8, 9],
but faced the challenge of optimising multiple lasers and SOAs for reliable
fast tuning. SWIFT overcomes this by applying artificial intelligence (AI)
techniques to the devices, enabling autonomous optimisation. This has allowed
us to demonstrate, for the first time, a time-multiplexed, gated laser tuning
system that can tune over 6.05 THz of bandwidth and consistently switch in 547
ps or better to support 20 ns timeslots. SWIFT outperforms other fast
switching systems in terms of rise time, settling time and channel count, as
shown in Fig 1(d).
## 2 Experimental Setup
The setup used to demonstrate SWIFT is shown in Fig. 2(a). A pair of
commercial Oclaro (now Lumentum) digital-supermode distributed Bragg reflector
(DS-DBR) lasers were driven by 250 MS/s arbitrary waveform generators (AWGs)
with 125 MHz bandwidth. Detailed IV measurements were used to map supplied
voltage to desired current. Each laser was connected to a commercial InPhenix
SOA, supporting 69 nm of bandwidth with typical characteristics of 7 dB noise
figure, 20 dB gain, and 10 dBm saturation power. Each SOA was driven with a 45
mA current source modulated by a 12 GS/s AWG with $\pm$0.5 V output and
amplified to $\pm$4 V using an electrical amplifier. All four optical devices
were held at 25∘C using temperature controllers. The SOAs were coupled
together and passed to a digital coherent receiver (50 GS/s, 22 GHz bandwidth)
and a digital sampling oscilloscope (50 GS/s, 30 GHz bandwidth), which
provided optimisation feedback to the DS-DBR lasers and to the SOAs
respectively.
\begin{overpic}[scale={0.45}]{Figures/setup_small_labels.png} \put(-5.0,55.0){(a)} \end{overpic} | \begin{overpic}[scale={0.14}]{Figures/fo.png} \put(12.0,50.0){(b)} \end{overpic} | \begin{overpic}[scale={0.14}]{Figures/dsdbr_cdf.png} \put(12.0,50.0){(c)} \end{overpic}
---|---|---
Fig. 2: (a) Experimental setup of time-multiplexed SWIFT tuneable lasers (TL)
gated by SOAs. (b) TL frequency offset (FO) of worst-case current swing w/ &
w/o optimiser. (c) CDF of all worst-case laser switch combinations w/ & w/o
optimiser.
## 3 Results and Discussion
### 3.1 Regression optimised laser switching
Fast wavelength switching can be achieved by applying ‘pre-emphasis’ to the
drive sections of an integrated semiconductor laser. Until recently, pre-
emphasis values had to be carefully tuned by hand for select samples then
extrapolated [4]. Here, we apply a linear regression optimiser to
automatically calculate the pre-emphasis values for reliable fast tuning. We
measured the output of the DS-DBR laser during a switching event using the
coherent receiver, then used the instantaneous frequency response as the error
term within a linear regression optimiser to iteratively update the applied
pre-emphasis values [5]. Fig 2(b) shows an example of the laser’s switching
response before and after application of the optimiser. We applied this
optimiser to 21 of the 122 supported channels, testing the extremes of lasing
frequency and drive current, covering 462 any-to-any switching events across
6.05 THz (1524.11-1572.48 nm). Fig. 2(c) shows the cumulative distribution of
the time taken to reach $\pm$5 GHz of the target wavelength. We measure a
worst case switch time of 14.7 ns, and a worst case frequency offset after 20
ns of $-$4.5 GHz. This indicates that SWIFT is potentially suitable for burst
mode coherent detection, as 28 GBd dual-polarisation quadrature phase shift
keying is tolerant of frequency offsets up to $\pm$7 GHz [10].
### 3.2 Particle swarm optimised SOA switching
SOA driving signals must also be optimised to approach their theoretical
rise/fall times of $\sim 100$ ps. Previous optimisation attempts did not
consider settling times nor the ability to automate the optimisation of
driving conditions for 1,000s of different SOAs in real DCNs [11, 12]. To
solve this, PSO (a population-based metaheuristic for optimising continuous
nonlinear functions by combining swarm theory with evolutionary programming)
was used in this work to optimise the SOA driving signals. PSO has previously
been applied to proportional-integral-derivative (PID) tuning in control
theory [13], but has not yet been used as an autonomous method for optical
switch control. In the optimisation, $n=160$ particles (driving signals) were
initialised in an $m=240$ (number of points in the signal) hyperdimensional
search space and iteratively ‘flown’ through the space by evaluating each
particle’s position with a fitness function $f$, defined as the mean squared
error between the drive signals’ corresponding optical outputs (recorded on
the oscilloscope) and an ideal target ‘set point’ (SP) with 0 overshoot,
settling time and rise time. As shown in Fig. 3(a) and (b), the $\pm$ 5%
settling time (effective switching time) of the SOA was reduced from 3.72 ns
(when driven by a simple square driving signal) to 547 ps, with the 10-90%
rise time also reduced from 697 ps to 454 ps. The PSO routine required no
prior knowledge of the SOA, therefore provides a flexible, automated and
scalable method for optimising SOA gating.
\begin{overpic}[scale={0.15}]{Figures/soa_outputs_annotated.png} \put(15.0,52.0){(a)} \end{overpic} | \begin{overpic}[scale={0.15}]{Figures/soa_outputs_zoomed_annotated.png} \put(12.0,52.0){(b)} \end{overpic} | \begin{overpic}[scale={0.03}]{Figures/1572p48_1524p11_1565p5_1530p72_2_channel_gating.png} \put(5.0,40.0){(c)} \end{overpic}
---|---|---
\begin{overpic}[scale={0.22}]{Figures/transitions.png} \put(3.0,45.0){(d)} \end{overpic} | \begin{overpic}[scale={0.15}]{Figures/osa_outputs_all_pairs.png} \put(12.0,50.0){(e)} \end{overpic} | \begin{overpic}[scale={0.037}]{Figures/burst_fo_data.png} \put(-1.0,45.0){(f)} \end{overpic}
Fig. 3: SOA outputs showing (a) step & (b) PSO rise & settling times, (c)
SWIFT system output, (d) $\lambda$-to-$\lambda$ 90-90% switching times, (e)
optical spectrum of the 21 worst-case channels, (f) frequency offset (FO) of
DSDBR (top) & SWIFT (bottom).
### 3.3 SWIFT module demonstration
After optimisation, the DS-DBR lasers were driven with with 12.5 MHz pre-
emphasised square waves, resulting in $\leq$40 ns bursts on each wavelength.
The lasers were driven 20 ns out of phase, so that one lased while the other
reconfigured. The SOAs were driven by 25 MHz PSO-optimised signals, resulting
in 20 ns gates, and aligned to block the first 15 ns and last 5 ns of each
laser burst, yielding four wavelength bursts of 20 ns each (see Fig. 2(a)).
Fig. 3(c) shows the oscilloscope output for the most difficult switching
instance, where DS-DBR laser 1 switched from 1572.48 nm to 1524.11 nm,
incurring a large rear current swing of 45 mA. The oscilloscope shows a flat
intensity response across each wavelength for 20 ns bursts, thereby providing
twice the granularity reported in [3]. Packet-to-packet power variations are
due to slight variations in laser wavelength power; these can be addressed by
applying slot specific SOA drive currents (not possible in our setup).
Measuring switch time by the 90-90% transition time, we report switch times
for the four transitions of 771, 812, 521, and 792 ps, respectively. These are
shown in Fig. 3(d). Furthermore, Fig. 3(f) shows the coherent receiver output
of the four wavelength slots with and without gating. The observed frequency
ripples are a result of the low sample rate of our 250 MS/s AWG that introduce
Fourier components to the driving square wave; these can be easily suppressed
by using a higher sample rate. Despite this, each slot stays within 5 GHz of
its target.
We repeated this process for each of the channels under test. Fig. 3(e) shows
the optical spectrum for all channels, all undergoing gated switching. We
measured a worst case value for the side mode suppression ratio of 35 dB,
optical power output of 0.8 dBm for a single wavelength (at 1572.48 nm) and
corresponding extinction ratio of 22 dB. The fully time-multiplexed optical
output power of SWIFT was $>$6 dBm. This represents the largest number of sub-
ns switching channels from a single sub-system ever reported, supporting
122$\times{}$50 GHz spaced channels.
In conclusion, we propose a scalable, low power, tuneable wavelength subsystem
capable of sub-ns switching. Using pairs of time-multiplexed tuneable lasers,
gated by SOAs, we have experimentally demonstrated switching times of less
than 900 ps for 122 x 50 GHz channels. Reliable and fast tuning was achieved
for each laser and SOA using regression and particle swarm optimisation AI
techniques. This enables automated, device-specific optimisation and
represents a critically important technology in OCS architectures, potentially
transforming DCN architectures.
This work is supported by EPSRC (EP/R035342/1), IPES CDT, iCASE and Microsoft
Research.
## References
* [1] K. Clark, et al., “Sub-Nanosecond Clock and Data Recovery in an Optically-Switched Data Centre Network”, ECOC, pdp, 2018.
* [2] J. Benjamin, et al., “Scaling PULSE Datacenter Network Architecture and Scheduling Optical Circuits in Sub-$\mu$seconds”, OFC, W1F.3, 2020.
* [3] K. Shi, et al., “System Demonstration of Nanosecond Wavelength Switching with Burst-mode PAM4 Transceiver,” ECOC, pdp, 2019.
* [4] J. Simsarian, et al., “Less than 5-ns wavelength switching with an SG-DBR laser”, PTL, 18(4), 2006.
* [5] T. Gerard et al., “Packet Timescale Wavelength Switching Enabled by Regression Optimisation,” arXiv:2002.11640v1 [eess.SP] 2020.
* [6] Y. Ueda, et al., “Electro-Optically Tunable Laser with $<$10-mW Tuning Power Dissipation and High-Speed $\lambda$-Switching for Coherent Networks”, ECOC, pdp, 2019.
* [7] S. Lange et al., “Sub-Nanosecond Optical Switching Using Chip-Based Soliton Microcombs,” in _OFC_ , W2A.4, 2020.
* [8] J. Benjamin, et al., “PULSE: Optical Circuit Switched Data Center Architecture Operating at Nanosecond Timescales”, arXiv:2002.04077v1 [cs.N1], 2020.
* [9] N. Ryan, et al., “A 10Gbps Optical Burst Switching Network Incorporating Ultra-fast (5ns) Wavelength Switched Tunable Laser”, ICSO, 2008.
* [10] J. Simsarian, “Fast-Tuning Coherent Burst-Mode Receiver for Metropolitan Networks”, PTL, 26(8), 2014.
* [11] C. Gallep and E. Conforti, “Reduction of semiconductor optical amplifier switching times by preimpulse step-injected current technique,”, _PTL_ , 14(7), 2002.
* [12] R. C. Figueiredo et al., “Hundred-Picoseconds Electro-Optical Switching With Semiconductor Optical Amplifiers Using Multi-Impulse Step Injection Current,”, _JLT_ , 13(1), 2015.
* [13] D. H. Kusuma et al., “The comparison of optimization for active steering control on a vehicle using PID controller based on artificial intelligence techniques,” in _International Seminar on Applications for Technology of Information and Communication_ , 2016.
|
2024-09-04T02:54:59.358682 | 2020-03-11T19:15:47 | 2003.05492 | {
"authors": "Philippe Gagnon and Florian Maire",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26177",
"submitter": "Philippe Gagnon",
"url": "https://arxiv.org/abs/2003.05492"
} | arxiv-papers | 1.87cm1.87cm1.87cm1.87cm capbtabboxtable[][]
# Lifted samplers for partially ordered discrete state-spaces
Philippe Gagnon 1, Florian Maire 1
###### Abstract
A technique called lifting is employed in practice for avoiding that the
Markov chains simulated for sampling backtrack too often. It consists in
lifting the state-space to include direction variables for guiding these
chains. Its implementation is direct when the probability mass function
targeted is defined on a totally ordered set, such as that of a univariate
random variable taking values on the integers. In this paper, we adapt this
technique to the situation where only a partial order can be established and
explore its benefits. Important applications include simulation of systems
formed from binary variables, such as those described by the Ising model, and
variable selection when the marginal model probabilities can be evaluated, up
to a normalising constant. To accommodate for the situation where one does not
have access to these marginal model probabilities, a lifted trans-dimensional
sampler for partially ordered model spaces is introduced. We show through
theoretical analyses and empirical experiments that the lifted samplers
outperform their non-lifted counterparts in some situations, and this at no
extra computational cost. The code to reproduce all experiments is available
online.111See the ArXiv page of this paper.
1Department of Mathematics and Statistics, Université de Montréal.
Keywords: Bayesian statistics; binary random variables; Ising model; Markov
chain Monte Carlo methods; Peskun-Tierney ordering; trans-dimensional
samplers; variable selection.
## 1 Introduction
### 1.1 Partially ordered state-spaces
A partially ordered set $\bm{\mathcal{X}}$ is such that there exists a
reflexive, antisymmetric, and transitive binary relation defined through a set
$\bm{\mathcal{R}}$ in which it is possible to establish that
$\mathbf{x}\leq\mathbf{y}$ and $\mathbf{y}\geq\mathbf{x}$ as a consequence of
$(\mathbf{x},\mathbf{y})\in\bm{\mathcal{R}}$ for
$\mathbf{x},\mathbf{y}\in\bm{\mathcal{X}}$. In such a set, pairs are
comparable when either $\mathbf{x}<\mathbf{y}$ or $\mathbf{y}<\mathbf{x}$, and
are incomparable when neither $\mathbf{x}<\mathbf{y}$ nor
$\mathbf{y}<\mathbf{x}$. This represents the difference with totally ordered
sets such as $\operatorname{\mathbb{R}}$ or $\mathbb{N}$ for which every pair
of different elements is comparable. We refer the reader to Trotter (1992) for
more details on partially ordered sets.
An important example of such sets is when any $\mathbf{x}\in\bm{\mathcal{X}}$
can be written as a vector $\mathbf{x}:=(x_{1},\ldots,x_{n})$ for which each
component $x_{i}$ can be of two types, say Type A or Type B, denoted by
$x_{i}\in\\{\text{A},\text{B}\\}$. In this case, $\bm{\mathcal{R}}$ can be set
to
$\bm{\mathcal{R}}:=\\{(\mathbf{x},\mathbf{y})\in\bm{\mathcal{X}}\times\bm{\mathcal{X}}:n_{\text{A}}(\mathbf{x})\leq
n_{\text{A}}(\mathbf{y})\\},$
where $n_{\text{A}}(\mathbf{x})$ is the number of components of Type A in
$\mathbf{x}$:
$n_{\text{A}}(\mathbf{x})=\sum_{i=1}^{n}\mathds{1}_{x_{i}=\text{A}}$. The
function $n_{\text{B}}(\mathbf{x})$ is defined analogously. Note that
$n=n_{\text{A}}(\mathbf{x})+n_{\text{B}}(\mathbf{x})$ for all $\mathbf{x}$ and
that the order can be symmetrically established by instead considering
$n_{\text{B}}(\mathbf{x})$ and $n_{\text{B}}(\mathbf{y})$.
Two main statistical problems fit within this framework: simulation of binary
random variables such as graphs or networks and variable selection. Indeed,
for the former, $\bm{\mathcal{X}}$ can be parameterized such that
$\bm{\mathcal{X}}=\\{-1,+1\\}^{n}$, where for example for an Ising model,
$x_{i}\in\\{-1,+1\\}$ represents the state of a spin, or, for networks,
$\bm{\mathcal{X}}=\mathcal{M}_{n}(\\{0,1\\})$ where $x_{i,j}\in\\{0,1\\}$
indicates whether nodes $i$ and $j$ are connected or not. For variable
selection, $\bm{\mathcal{X}}=\\{0,1\\}^{n}$ and $x_{i}\in\\{0,1\\}$ indicates
whether or not the $i$th covariate is included in the model characterised by
the vector $\mathbf{x}\in\bm{\mathcal{X}}$.
### 1.2 Sampling on partially ordered state-spaces
Let $\pi$ be a probability distribution defined on a measurable space
$(\bm{\mathcal{X}},\bm{\mathsf{X}})$ where $\bm{\mathcal{X}}$ is a partially
ordered set and $\bm{\mathsf{X}}$ a sigma algebra on $\bm{\mathcal{X}}$ and
consider the problem of sampling from $\pi$. We assume that it is not possible
to generate independent draws from $\pi$. Given the complex dependency
structure of modern statistical models such as graphical models and the
intractable nature of some distributions that arise, for instance, in Bayesian
statistics, this is a realistic assumption. We turn to Markov chains based
sampling methods which typically rely on an ergodic stochastic process
$\\{\mathbf{X}(m)\,:\,m\in\mathbb{N}\\}$ whose limiting distribution is $\pi$.
A typical Markov chain based sampler, such as the Glauber dynamics for
graphical models or the tie-no-tie sampler for network models, selects
uniformly at random one of the coordinates of $\mathbf{x}$, say $x_{i}$, and
proposes to flip its value from B to A (if $x_{i}=\text{A}$), and accept or
reject this move according to a prescribed probability that guarantees that
the Markov chain limiting distribution is $\pi$. Such moves are often rejected
when the mass concentrates on a manifold of the ambient space. Zanella (2019)
recently proposed a locally informed generic approach for which the
probability to select the $i$th coordinate depends on the relative mass of the
resulting proposal, i.e. $\pi(\mathbf{y})/\pi(\mathbf{x})$, aiming at
proposing less naive moves. Yet, the sampler is of reversible
Metropolis–Hastings (MH, (Metropolis et al., 1953; Hastings, 1970)) type,
implying that the chain may often go back to recently visited states. When
this is the case, the state-space is explored through a random walk, a process
exhibiting a diffusive behaviour.
The lifting technique, which can be traced back to Gustafson (1998) and even
to Horowitz (1991), aim at producing Markov chains which do not suffer from
such a behaviour. This is achieved by introducing a momentum variable
$\nu\in\\{-1,+1\\}$ which provides the process
$\\{\mathbf{X}(m)\,:\,m\in\mathbb{N}\\}$ with some memory of its past in order
to avoid backtracking. To remain Markovian, the process is thus enlarged to
$\\{(\mathbf{X},\nu)(m)\,:\,m\in\mathbb{N}\\}$ and the momentum variable is
flipped at random times according to a prescribed dynamic which guarantees
that, marginally, the process $\\{\mathbf{X}(m)\,:\,m\in\mathbb{N}\\}$ retains
its limiting distribution. Lifted Markov chains have been quantitatively
studied in Chen et al. (1999) and Diaconis et al. (2000) and have been shown
to reduce, sometimes dramatically, the mixing time of random walks on groups.
Over the past few years, there has been a renewed interest for lifted
techniques in the computational statistics community: in addition to speeding-
up the mixing time of random walk Markov chains they are also suspected to
reduce asymptotic variances of empirical averages of observables of interests,
see Andrieu and Livingstone (2019) for some precise results. We also refer to
Gagnon and Doucet (2019), Syed et al. (2019) and Neal (2020) for examples
where popular Markov chain Monte Carlo (MCMC) algorithms such as reversible
jump (Green, 1995), parallel tempering (Geyer, 1991) and slice sampling (Neal,
2003) have seen their performance improved in some situations by considering
their lifted version. Remarkably, those lifted samplers are implemented at no
additional computational cost over their non-lifted counterparts, and also
with no additional implementation difficulty.
All these successful applications of the lifting technique have been carried
out in contexts where $\bm{\mathcal{X}}$ is one-dimensional (Gustafson, 1998)
or exhibits a one-dimensional parameter which plays a central role in the
sampling scheme: the annealing parameter in Syed et al. (2019), the model
indicator reflecting the size/complexity of the nested models in Gagnon and
Doucet (2019), and the height of the level-lines
$\\{\pi(\mathbf{X}(m))\,:\,m\in\mathbb{N}\\}$ in Neal (2020). When such a one-
dimensional feature does not exist, there does not exist a straightforward way
of lifting the state-space without facing issues of reducibility or obtaining
inefficient samplers. A possibility, deemed as naive in Gustafson (1998), is
to lift each marginal component of the state-space and update them one at the
time. When the state-space is uncountable, it is possible to construct a
persistent walk by introducing bounces at random event times which change the
direction of propagation (see Vanetti et al. (2017)). However, when the state-
space is countable and partially ordered, such an approach is infeasible.
The objective of this paper is to present and analyse generic methods based on
the lifting technique to sample from a given probability mass function (PMF)
with a partially ordered countable support. In particular, we break free from
the requirement of having a one-dimensional parameter by exploiting the local
one-dimensional neighborhood structure induced by the partial order on
$\bm{\mathcal{X}}$: the neighbourhood of $\mathbf{x}$, denoted by
$\mathcal{N}(\mathbf{x})$, is separated into two parts where one comprises
states with $\mathbf{y}>\mathbf{x}$, denoted by
$\mathcal{N}_{+1}(\mathbf{x})$, and the other one comprises states with
$\mathbf{y}<\mathbf{x}$, denoted by $\mathcal{N}_{-1}(\mathbf{x})$
(considering that $\mathcal{N}(\mathbf{x})$ is only composed of states that
can be compared with $\mathbf{x}$). Looking for instance at the variable
selection example, the partial order is defined by mean of the model sizes, or
in other words, the number of covariates included in the models. If the
momentum is $\nu=+1$, the chain is forced to attempt visiting models with more
variables until a move is rejected, then $\nu$ is flipped to $\nu=-1$. As a
consequence, the momentum variable remains one-dimensional while the Markov
chain is often irreducible and efficiently explore the state-space. An
illustration showing the benefit of this approach is provided at Figure 1.
Again, we stress that the typical lifted sampler is implemented at no
additional computational cost over its non-lifted counterpart, and also with
no additional implementation difficulty.
$\begin{array}[]{cc}\textbf{Random walk behaviour}&\textbf{Persistent
movement}\cr\vspace{-1mm}\textbf{ESS = 0.12 per it.}&\textbf{ESS = 0.33 per
it.}\cr\includegraphics[width=173.44534pt]{Fig_1_a.pdf}&\includegraphics[width=173.44534pt]{Fig_1_b.pdf}\end{array}$
Figure 1: Trace plots for the statistic of number of covariates included in
the model for a MH sampler with a locally informed proposal distribution
(discussed in more details in Section 3.2) and its lifted counterpart, when
applied to solve a real variable selection problem (presented in Section 4.2);
ESS stands for effective sample size
### 1.3 Overview of our contributions
In this paper, we focus on the simulation of two-dimensional Ising models and
variable selection problems, without restricting ourselves to these examples
of applications when we present the samplers and analyse them. For these
examples, a generic sampler that we study corresponds to the discrete-time
version of a specific sampler independently developed in Power and Goldman
(2019), a paper in which the focus is rather on exploiting any structure of
$\bm{\mathcal{X}}$ identified by users. The structure identified here is, in a
sense, that $\bm{\mathcal{X}}$ exhibits a partial order. We consequently do
not claim originality for the samplers that will be presented. Our
contributions are the following:
* •
statement of the sampling problem within the specific framework of partially
ordered discrete state-spaces (so that it becomes straightforward to implement
a sampler using the lifting technique in this framework);
* •
identification of situations in which the lifted samplers are expected to
outperform their non-lifted counterparts, based on theoretical analyses and
numerical experiments;
* •
introduction of a trans-dimensional lifted sampler useful, among others, for
variable selection when it is not possible to integrate out the parameters.
### 1.4 Organisation of the paper
The generic algorithm is first presented in Section 2. We next analyse in
Section 3 two important versions with uniform proposal distributions and
locally informed ones, allowing to establish that they can outperform their
non-lifted counterparts under some assumptions. In Section 4, we show the
difference in empirical performance for a Ising model (Section 4.1) and real
variable selection problem (Section 4.2). In Section 5, we consider that
$\bm{\mathcal{X}}$ is a model space and propose a lifted trans-dimensional
sampler allowing to simultaneously achieve model selection and parameter
estimation. The paper finishes in Section 6 with retrospective comments and
possible directions for future research.
## 2 Generic algorithm
The sampler that we present is a MCMC algorithm that generates proposals
belonging to a subset of $\mathcal{N}(\mathbf{x})$ chosen according to a
“direction” $\nu\in\\{-1,+1\\}$, when the current state is
$\mathbf{x}\in\bm{\mathcal{X}}$. In particular, the proposals belong to
$\mathcal{N}_{+1}(\mathbf{x}):=\\{\mathbf{y}\in\mathcal{N}(\mathbf{x}):\mathbf{y}>\mathbf{x}\\}\subseteq\mathcal{N}(\mathbf{x})$
when $\nu=+1$ or
$\mathcal{N}_{-1}(\mathbf{x}):=\\{\mathbf{y}\in\mathcal{N}(\mathbf{x}):\mathbf{y}<\mathbf{x}\\}\subseteq\mathcal{N}(\mathbf{x})$
when $\nu=-1$, where $\mathcal{N}_{-1}(\mathbf{x})$ and
$\mathcal{N}_{+1}(\mathbf{x})$ thus denote two subsets of
$\mathcal{N}(\mathbf{x})$ such that
$\mathcal{N}_{-1}(\mathbf{x})\cup\mathcal{N}_{+1}(\mathbf{x})=\mathcal{N}(\mathbf{x})$.
More formally, assuming that the Markov chain state is
$\mathbf{x}\in\bm{\mathcal{X}}$, the proposal distribution
$q_{\mathbf{x},\nu}$ has its support restricted to
$\mathcal{N}_{\nu}(\mathbf{x})$. There exist natural candidates for such
distributions, as will be explained in the following. This makes the
implementation of the proposed sampler almost straightforward; once the
neighbourhood structure has been specified. In our framework, the partial
ordering induces a natural neighbourhood structure.
The sampler is based on the well known technique of lifting: the state-space
$\bm{\mathcal{X}}$ is lifted to an extended state-space
$\bm{\mathcal{X}}\times\\{-1,+1\\}$ such that the marginal and the conditional
distributions of the direction variable $\nu$ is the uniform distribution on
$\\{-1,+1\\}$. The algorithm, which bares a strong resemblance with the guided
walk (Gustafson, 1998), is now presented in Algorithm 1.
Algorithm 1 A lifted sampler for partially ordered discrete state-spaces
1. 1.
Generate $\mathbf{y}\sim q_{\mathbf{x},\nu}$ and $u\sim\mathcal{[}0,1]$.
2. 2.
If
$\displaystyle
u\leq\alpha_{\nu}(\mathbf{x},\mathbf{y}):=1\wedge\frac{\pi(\mathbf{y})\,q_{\mathbf{y},-\nu}(\mathbf{x})}{\pi(\mathbf{x})\,q_{\mathbf{x},\nu}(\mathbf{y})},$
(1)
set the next state of the chain to $(\mathbf{y},\nu)$. Otherwise, set it to
$(\mathbf{x},-\nu)$.
3. 3.
Go to Step 1.
If $\bm{\mathcal{X}}$ is finite, there exists a boundary, in the sense that
there may be no mass beyond a state $\mathbf{x}$ when the direction followed
is $\nu$. For instance in variable selection, the posterior probability of a
model with more covariates than the maximum number is zero. Algorithm 1 may
thus seem incomplete, in the sense that it is not explicitly specified what to
do on the boundary. We in fact consider that $q_{\mathbf{x},\nu}$ is defined
even on the boundary, and that it is defined to generate a point outside of
$\bm{\mathcal{X}}$. This point will be automatically rejected (because its
mass is 0) and the chain will remain at $\mathbf{x}$ and the direction will be
reversed. Note that this is a technical requirement. In practice, one can
simply skip Step 1 in this case and set the next state to $(\mathbf{x},-\nu)$.
It is possible to establish the correctness of the algorithm through that of a
more general version based on the lifted algorithm presented in Andrieu and
Livingstone (2019). Before presenting this more general version which has
interesting features, we introduce necessary notation. Let
$\rho_{\nu}:\bm{\mathcal{X}}\to[0,1]$, for $\nu\in\\{-1,+1\\}$, be a user-
defined function for which we require that:
$\displaystyle 0\leq\rho_{\nu}(\mathbf{x})\leq
1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}}),$ (2)
$\displaystyle\rho_{\nu}(\mathbf{x})-\rho_{-\nu}(\mathbf{x})=T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}}),$
(3)
where we have set, for all
$(\mathbf{x},\nu)\in\bm{\mathcal{X}}\times\\{-1,+1\\}$,
$\displaystyle
T_{\nu}(\mathbf{x},\bm{\mathcal{X}}):=\sum_{\mathbf{x}^{\prime}\in\bm{\mathcal{X}}}q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\,\alpha_{\nu}(\mathbf{x},\mathbf{x}^{\prime})=\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{\nu}(\mathbf{x})}q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\,\alpha_{\nu}(\mathbf{x},\mathbf{x}^{\prime}).$
(4)
These conditions make the algorithm valid and are thus considered satisfied in
the sequel. Finally, let $Q_{\mathbf{x},\nu}$ be a PMF such that
$Q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\propto
q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\,\alpha_{\nu}(\mathbf{x},\mathbf{x}^{\prime})$.
The more general algorithm is now presented in Algorithm 2.
Algorithm 2 A more general lifted sampler for partially ordered discrete
state-spaces
1. 1.
Generate $u\sim\mathcal{U}[0,1]$.
1. (i)
If $u\leq T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$, generate $\mathbf{y}\sim
Q_{\mathbf{x},\nu}$ and set the next state of the chain to $(\mathbf{y},\nu)$;
2. (ii)
if $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})<u\leq
T_{\nu}(\mathbf{x},\bm{\mathcal{X}})+\rho_{\nu}(\mathbf{x})$, set the next
state of the chain to $(\mathbf{x},-\nu)$;
3. (iii)
if $u>T_{\nu}(\mathbf{x},\bm{\mathcal{X}})+\rho_{\nu}(\mathbf{x})$, set the
next state of the chain to $(\mathbf{x},\nu)$.
2. 2.
Go to Step 1.
###### Proposition 1.
The transition kernel of the Markov chain
$\\{(\mathbf{X},\nu)(m):m\in\operatorname{\mathbb{N}}\\}$ simulated by
Algorithm 2 admits $\pi\otimes\mathcal{U}\\{-1,1\\}$ as invariant
distribution.
###### Proof.
See Section 7. ∎
It is interesting to notice that $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$
represents the probability to accept a proposal when the current state is
$(\mathbf{x},\nu)$. In Algorithm 2, we thus first decide if we accept a
proposal or not, and if it is the case, in Step 1.(i), we randomly select the
value of the proposal (using the conditional distribution). It can be readily
checked that choices for $\rho_{\nu}$ include
$\rho_{\nu}(\mathbf{x})=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ and
$\rho_{\nu}(\mathbf{x})=\max\\{0,T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\\}$.
If $\rho_{\nu}(\mathbf{x})=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$, the
condition for Case (iii) of Step 1 is never satisfied, and the algorithm
either accepts the proposal and keeps the same direction, or the proposal is
rejected and the direction is reversed. In this case, one can show that
Algorithm 2 corresponds to Algorithm 1, which is what allows ensuring the
correctness of Algorithm 1 as well. Setting $\rho_{\nu}(\mathbf{x})$ otherwise
than $\rho_{\nu}(\mathbf{x})=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ allows in
Case (iii) of Step 1 to keep following the same direction, even when the
proposal is rejected. Intuitively, this is desirable when the rejection is due
to “bad luck”, and not because there is no mass in the direction followed. The
function $\rho_{\nu}(\mathbf{x})$ aims at incorporating this possibility in
the sampler.
In the typical MCMC framework, when one wants to sample from a probability
density function, it is not possible to directly compute
$T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ as it requires computing an integral
with respect to this density function. In such a case, it is therefore usually
not possible to set $\rho_{\nu}(\mathbf{x})$ otherwise than
$1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$. This contrasts with our discrete
state-space framework in which it is often possible to directly compute
$T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$. A corollary of Theorem 3.15 in Andrieu
and Livingstone (2019) states that the best choice of function $\rho_{\nu}$ in
terms of asymptotic variance is
$\displaystyle\rho_{\nu}^{*}(\mathbf{x}):=\max(0,T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})),$
(5)
and that the worst choice is
$\rho_{\nu}^{\text{w}}(\mathbf{x}):=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$.
###### Corollary 1.
If $\bm{\mathcal{X}}$ is finite, then for any function $\rho_{\nu}$ and
$f\in\mathcal{L}_{2}^{*}(\bar{\pi}):=\left\\{f:\int
d\bar{\pi}f^{2}<\infty\text{ and }f(\mathbf{x},-1)=\right.$
$\left.f(\mathbf{x},+1)\text{ for all }\mathbf{x}\right\\}$,
$\mathrm{var}(f,P_{\rho^{*}})\leq\mathrm{var}(f,P_{\rho})\leq\mathrm{var}(f,P_{\rho^{\text{w}}}),$
where $\bar{\pi}:=\pi\otimes\mathcal{U}\\{-1,+1\\}$,
$\mathrm{var}(f,P_{\rho}):=\mathbb{V}\mathrm{ar}f(\mathbf{X},\nu)+2\sum_{k>0}\left\langle
f,P_{\rho}^{k}f\right\rangle$ and $P_{\rho}$ is the transition kernel
simulated by Algorithm 2, $\left\langle f,P_{\rho}^{k}f\right\rangle$ being
the inner product, i.e. $\left\langle f,P_{\rho}^{k}f\right\rangle:=\int
d\bar{\pi}fP_{\rho}^{k}f$.
###### Proof.
See Section 7. ∎
The price to pay for using $\rho_{\nu}^{*}$ instead of
$\rho_{\nu}^{\text{w}}$, for instance, is that the algorithm is more
complicated to implement because it is required to systematically compute
$T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ at each iteration (it is also sometimes
required to compute $T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})$). Using
$\rho_{\nu}^{*}$ also comes with an additional computation cost (which is
discussed in Section 3.2). In our numerical experiments, it is seen that the
gain in efficiency of using Algorithm 2 with $\rho_{\nu}^{*}$ over Algorithm 1
is not significant. One may thus opt for simplicity and implement Algorithm 1.
## 3 Analysis of specific samplers
In the previous section, we presented generic algorithms with conditions
ensuring that they are valid. A necessary input to implement them is the
proposal distribution $q_{\mathbf{x},\nu}$ to be used. In this section, we
explore two natural choices and analyse their asymptotic variances. We start
in Section 3.1 with the common situation where the proposal is picked
uniformly at random. As mentioned in the introduction, this choice may lead to
poor mixing. We thus go on in Section 3.2 to a distribution incorporating
information about the target. This section finishes with a brief discussion in
Section 3.3 on the computational costs associated to these choices in regular
MH samplers and their lifted counterparts.
### 3.1 Uninformed uniform proposal
In the reversible MH sampler, it is common to set the proposal distribution,
denoted by $q_{\mathbf{x}}$ for this algorithm, to
$q_{\mathbf{x}}:=\mathcal{U}\\{\mathcal{N}(\mathbf{x})\\}$. In our framework,
the analogous proposal distribution is naturally defined as
$q_{\mathbf{x},\nu}:=\mathcal{U}\\{\mathcal{N}_{\nu}(\mathbf{x})\\}$. In this
case, the acceptance probability (1) of a proposal becomes
$\alpha_{\nu}(\mathbf{x},\mathbf{y})=1\wedge\frac{\pi(\mathbf{y})\,|\mathcal{N}_{\nu}(\mathbf{x})|}{\pi(\mathbf{x})\,|\mathcal{N}_{-\nu}(\mathbf{y})|}.$
For ease of presentation of the analysis, consider again the important example
described in Section 1.1 where each component $x_{i}$ of
$\mathbf{x}=(x_{1},\ldots,x_{n})$ can be of two types. We in this section
highlight the dependency on $n$ (the dimension) of the state-space and target
because it will be relevant in our analysis. We thus write $\pi_{n}$ for the
target and $\bm{\mathcal{X}}_{n}$ for the state-space, where each state is of
the form $\mathbf{x}:=(x_{1},\ldots,x_{n})$ with
$x_{i}\in\\{\text{A},\text{B}\\}$. For now on, consider for that $\text{A}=-1$
and $\text{B}=+1$, corresponding to the case of Ising model. We note that
there is no loss of generality of considering this special case within the
important example.
In a MH sampler, one sets
$\mathcal{N}(\mathbf{x}):=\\{\mathbf{y}\in\bm{\mathcal{X}}_{n}:\sum_{i}|x_{i}-y_{i}|=2\\}=\\{\mathbf{y}\in\bm{\mathcal{X}}_{n}:\exists
j\text{ such that }y_{j}=-x_{j}\\}$, so that the algorithm proposes to flip a
single bit at each iteration. It thus chooses uniformly at random which bit to
flip. Therefore, the size of the neighbourhoods in this sampler is constant
for any $\mathbf{x}$ and is given by $n$. This implies that the acceptance
probability in this sampler, denoted by $\alpha(\mathbf{x},\mathbf{y})$,
reduces to
$\alpha(\mathbf{x},\mathbf{y})=1\wedge\pi(\mathbf{y})/\pi(\mathbf{x})$.
In the lifted case, the acceptance probability can be rewritten as
$\displaystyle\alpha_{\nu}(\mathbf{x},\mathbf{y})=1\wedge\frac{\pi(\mathbf{y})\,n_{-\nu}(\mathbf{x})}{\pi(\mathbf{x})\,n_{\nu}(\mathbf{y})}.$
(6)
Indeed,
$\mathcal{N}_{\nu}(\mathbf{x}):=\\{\mathbf{y}\in\bm{\mathcal{X}}_{n}:\exists
j\text{ such that }y_{j}=-x_{j}=\nu\\}$, which implies that
$|\mathcal{N}_{\nu}(\mathbf{x})|=n_{-\nu}(\mathbf{x})$ (with the analogous
implication for $\mathcal{N}_{-\nu}(\mathbf{y})$). The acceptance probability
$\alpha_{\nu}$ thus depends on an additional term
$n_{-\nu}(\mathbf{x})/n_{\nu}(\mathbf{y})$ compared to that in the MH sampler.
This term may have an negative impact by decreasing the acceptance
probability. This represents in fact the price to pay for using the lifted
sampler Algorithm 2 (including Algorithm 1 as a special case): the reversible
sampler is allowed to backtrack, which makes the sizes of the neighbourhoods
constant, whereas it is the opposite for Algorithm 2. The size of the
neighbourhoods diminishes in the lifted sampler as the chain moves further in
a direction (making the neighbourhoods in the reverse direction bigger and
bigger).
The impact is alleviated when $n$ is large and the mass of $\pi_{n}$
concentrates in the interior of $\bm{\mathcal{X}}_{n}$ in an area where
$|n_{-\nu}(\mathbf{x})-n/2|\leq\kappa$, where $\kappa$ is a positive integer.
Indeed, $n_{\nu}(\mathbf{y})=n_{\nu}(\mathbf{x})+1=n-n_{-\nu}(\mathbf{x})+1$,
which implies that $\alpha_{\nu}(\mathbf{x},\mathbf{y})\approx
1\wedge\pi(\mathbf{y})/\pi(\mathbf{x})=\alpha(\mathbf{x},\mathbf{y})$. In
fact, in an ideal situation where
$|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2$
for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$ (which is impossible for the
important example with types A and B), a corollary of Theorem 3.17 in Andrieu
and Livingstone (2019) establishes that Algorithm 2 with any function
$\rho_{\nu}$ dominates the MH algorithm in terms of asymptotic variances. In
particular, Algorithm 1 dominates the MH algorithm. Before presenting this
corollary, we define the transition kernel simulated by Algorithm 2 with
$q_{\mathbf{x},\nu}:=\mathcal{U}\\{\mathcal{N}_{\nu}(\mathbf{x})\\},\mathbf{x}\in\bm{\mathcal{X}}_{n},\nu\in\\{-1,+1\\}$,
as $P_{\rho,n}$ and that simulated by the MH sampler with
$q_{\mathbf{x}}:=\mathcal{U}\\{\mathcal{N}(\mathbf{x})\\},\mathbf{x}\in\bm{\mathcal{X}}_{n}$,
as $P_{\text{MH},n}$.
###### Corollary 2.
If
(a)
$\bm{\mathcal{X}}_{n}$ is finite,
(b)
$|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2=n^{*}/2$
for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$, where $n^{*}$ is a positive
integer that does not depend on $\mathbf{x}$ (but that may depend on $n$),
then for any $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and $n$,
$\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{MH},n}).$
###### Proof.
See Section 7. ∎
In light of the above, one might expect the inequality to (approximately) hold
(up to an error term) when the mass is highly concentrated on the points
$\mathbf{x}$ that are not too far from the center of the domain, where here
the notion of centrality is defined in terms of the distance between
$n_{-1}(\mathbf{x})$ or $n_{+1}(\mathbf{x})$ to $n/2$, suggesting that the
lifted sampler outperforms the reversible MH algorithm in this situation. The
rest of the section is dedicated to the introduction of conditions under which
this statement is true.
The key argument in proving Corollary 2 is to show that
$\displaystyle
q_{\mathbf{x}}(\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y})=(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})\,\alpha_{+1}(\mathbf{x},\mathbf{y})+(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})\,\alpha_{-1}(\mathbf{x},\mathbf{y}),$
(7)
for all $\mathbf{x}$ and $\mathbf{y}$. Indeed, once this is done, Theorem 3.17
in Andrieu and Livingstone (2019) can be directly applied, which yields the
result. This in fact implies that once one has designed a lifted sampler, it
is possible to identify its non-lifted counterpart through (7), and to
establish that the latter is inferior. In particular, we can establish that
the sampler that flips a coin at each iteration to next decide which PMF to
use between $q_{\mathbf{x},+1}$ and $q_{\mathbf{x},-1}$ to generate a proposal
$\mathbf{y}$ that will be subject to approval using
$\alpha_{+1}(\mathbf{x},\mathbf{y})$ or $\alpha_{-1}(\mathbf{x},\mathbf{y})$
is inferior. Denote by $P_{\text{rev.},n}$ the Markov kernel simulated by this
algorithm. Now, what we would ideally do is to show a Peskun-Tierney ordering
(Peskun, 1973; Tierney, 1998) between $P_{\text{rev.},n}$ and
$P_{\text{MH},n}$ to establish the domination of the lifted sampler over the
reversible MH algorithm. Such an ordering is difficult to obtain as one needs
to show that for any pair $(\mathbf{x},\mathbf{y})$ such that
$\mathbf{x}\neq\mathbf{y}$, $P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq
P_{\text{MH},n}(\mathbf{x},\mathbf{y})$.
A more general ordering is presented in Zanella (2019): if
$P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega
P_{\text{MH},n}(\mathbf{x},\mathbf{y})$ for all $\mathbf{x}\neq\mathbf{y}$,
then
$\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})/\omega+((1-\omega)/\omega)\mathbb{V}\mathrm{ar}f(\mathbf{X})$,
where $\omega$ is a positive constant. Note that $f$ is a function of $\nu$ as
well, but because of the restriction $f(\mathbf{x},-1)=f(\mathbf{x},+1)$,
$\nu$ can be treated as a constant. We show in this section that if
$P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega
P_{\text{MH},n}(\mathbf{x},\mathbf{y})$ with $\omega$ close to 1 for all
$\mathbf{x}\neq\mathbf{y}$ belonging to a specific set having a high
probability, then
$\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})+\text{small
error term}$ (under regularity conditions).
We start by defining this set:
$\displaystyle\bm{\mathcal{X}}_{\varphi(n)}:=\\{\mathbf{x}\in\bm{\mathcal{X}}_{n}:n/2-\varphi(n)\leq
n_{-1}(\mathbf{x}),n_{+1}(\mathbf{x})\leq n/2+\varphi(n)\\},$ (8)
where $\varphi$ is a function such that $0\leq\varphi(n)<n/2$ and, for
$\epsilon>0$, there exists $N>0$ such that for all $n\geq N$, we have that
$\varphi(n)/n<\epsilon$. For $\mathbf{x}\neq\mathbf{y}$ belonging to
$\bm{\mathcal{X}}_{\varphi(n)}$, if we consider $\omega_{n}$ now a function of
$n$, it is possible to establish that: for $\epsilon>0$, there exists a $N>0$
such that for all $n\geq N$,
$P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega_{n}P_{\text{MH},n}(\mathbf{x},\mathbf{y})$
with $1-\omega_{n}<\epsilon$. The explicit form of $\omega_{n}$ is
$\displaystyle\omega_{n}:=\left(1+\frac{\varphi(n)}{n/2}\right)^{-2}\left(1-\frac{\varphi(n)}{n/2}\right).$
(9)
One can imagine that if: the mass concentrates on
$\bm{\mathcal{X}}_{\varphi(n)}$, the chains do not often leave this set, and
when they do they do not take too much time to come back, then the asymptotic
variances should not be too different to those of chains with stationary
distribution $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$, which assigns null
mass on $\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}$ and is thus the
normalised version of $\pi_{n}$ on $\bm{\mathcal{X}}_{\varphi(n)}$:
$\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}(\mathbf{x}):=\begin{cases}\pi_{n}(\mathbf{x})\big{/}\sum_{\mathbf{x}^{\prime}\in\bm{\mathcal{X}}_{\varphi(n)}}\pi_{n}(\mathbf{x}^{\prime})&\text{if}\quad\mathbf{x}\in\bm{\mathcal{X}}_{\varphi(n)},\cr
0&\text{if}\quad\mathbf{x}\notin\bm{\mathcal{X}}_{\varphi(n)}.\end{cases}$
This is what we obtain assuming such a behaviour for the chains generated by
$P_{\text{rev.},n}$ and $P_{\text{MH},n}$. We also require that the chains
generated by the Markov kernels with the same proposal mechanisms as
$P_{\text{rev.},n}$ and $P_{\text{MH},n}$, but whose stationary distribution
is $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$, mix sufficiently well (in a
sense that will be made precise). We denote by $\tilde{P}_{\text{rev.},n}$ and
$\tilde{P}_{\text{MH},n}$ these Markov kernels.
###### Theorem 1.
Pick $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and consider that all states
$\mathbf{x}\in\bm{\mathcal{X}}_{n}$ are such that
$\mathbf{x}=(x_{1},\ldots,x_{n})$ with $x_{i}\in\\{-1,+1\\}$. If, for
$\epsilon>0$, there exists $N>0$ such that for all $n\geq N$ it is possible to
choose a constant that may depend on $n$, $\varrho(n)$, with
(a)
$\sum_{k=\varrho(n)+1}^{\infty}\left\langle f,P^{k}f\right\rangle<\epsilon$,
for all
$P\in\\{P_{\text{rev.},n},\tilde{P}_{\text{rev.},n},P_{\text{MH},n},\tilde{P}_{\text{MH},n}\\}$,
(b)
$\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cup_{m=0}^{k}A_{m}^{\mathsf{c}}(\bm{\mathcal{X}}_{\varphi(n)-1})}]<\epsilon$,
for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$, where it is considered
that the chain starts at stationarity and evolves using $P$ (and
$\mathbb{E}[f(\mathbf{X}(k))]=0$ without loss of generality) and
$A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1}):=\\{\mathbf{X}(m)\in\bm{\mathcal{X}}_{\varphi(n)-1}\\}$
(see (8) for the definitions of $\bm{\mathcal{X}}_{\varphi(n)-1}$ and
$\varphi(n)$),
(c)
$(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}]<\epsilon$,
for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$,
(d)
$(1/\omega_{n}-1)\mathrm{var}(f,\tilde{P}_{\text{MH},n})<\epsilon$ and
$((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})<\epsilon$,
where $\omega_{n}$ is defined in (9) and
$\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}$ denotes
a variance computed with respect to
$\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$,
then there exists a positive constant $\kappa$ such that
$\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})+\kappa\epsilon.$
###### Proof.
See Section 7. ∎
There are several assumptions involved in Theorem 1. But, to put this into
perspective, Assumptions (c) and (d) are automatically verified if
$\mathrm{var}(f,\tilde{P}_{\text{MH},n})$ and
$\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})$
are bounded by constants that do not depend on $n$. Assumptions (a) and (b)
are really the crucial ones.
### 3.2 Locally informed proposal
In this section, we discuss and analyse the use of locally-balanced proposal
distributions, as defined by Zanella (2019) in the reversible MH framework as
$\displaystyle
q_{\mathbf{x}}(\mathbf{y}):=g\left(\frac{\pi_{n}(\mathbf{y})}{\pi_{n}(\mathbf{x})}\right)\bigg{/}c_{n}(\mathbf{x}),\quad\mathbf{y}\in\mathcal{N}(\mathbf{x}),$
(10)
where $c_{n}(\mathbf{x})$ represents the normalising constant, i.e.
$c_{n}(\mathbf{x}):=\sum_{\mathbf{x}^{\prime}\in\mathcal{N}(\mathbf{x})}g\left(\pi_{n}(\mathbf{x}^{\prime})/\pi_{n}(\mathbf{x})\right)$,
and $g$ is a strictly positive continuous function such that $g(x)/g(1/x)=x$.
Note that we used the same notation as in Section 3.1 for the proposal
distribution; this is to simplify. We will use the same notation as in Section
3.1 for the proposal distribution of the lifted sampler and for the Markov
kernels as well. In this section, it will be implicit that the proposal
distributions are informed proposals and the Markov kernels are those induced
by these informed proposals. Note also that we again highlight the
dependencies on $n$ of some terms that will appear in our analysis.
Such a function $g$ defined in (10) implies that the acceptance probability in
the MH algorithm is given by
$\alpha(\mathbf{x},\mathbf{y})=1\wedge\frac{\pi_{n}(\mathbf{y})\,q_{\mathbf{y}}(\mathbf{x})}{\pi_{n}(\mathbf{x})\,q_{\mathbf{x}}(\mathbf{y})}=1\wedge\frac{c_{n}(\mathbf{x})}{c_{n}(\mathbf{y})}.$
Zanella (2019) shows that $c_{n}(\mathbf{x})/c_{n}(\mathbf{y})\longrightarrow
1$ as $n\longrightarrow\infty$ under some assumptions. More precisely, the
author defines $\mathbf{x}:=(x_{1},\ldots,x_{n})$ and considers that at any
given iteration, only a small fraction of the $n$ components is proposed to
change values. The result holds when the random variables
$(X_{1},\ldots,X_{n})$ exhibit a structure of conditional independence, which
implies that the normalising constants $c_{n}(\mathbf{x})$ and
$c_{n}(\mathbf{y})$ share a lot of terms. This is again a consequence of the
backtracking of the reversible sampler and is thus in contrast with what we
observe for the lifted algorithm.
Two choices for $g$ are $g(x)=\sqrt{x}$ and $g(x)=x/(1+x)$, the latter being
called the Barker proposal in reference to Barker (1965)’s acceptance
probability choice. The advantage of the latter choice is that it is a bounded
function of $x$, which stabilises the normalising constants and thus the
acceptance probability. This is shown in Zanella (2019) and in the continuous
random variable case in Livingstone and Zanella (2019). We use this function
in our numerical analyses.
The proposal distribution in the lifted sampler is given by
$q_{\mathbf{x,\nu}}(\mathbf{y}):=g\left(\frac{\pi_{n}(\mathbf{y})}{\pi_{n}(\mathbf{x})}\right)\bigg{/}c_{n,\nu}(\mathbf{x}),\quad\mathbf{y}\in\mathcal{N}_{\nu}(\mathbf{x}),$
where $c_{n,\nu}(\mathbf{x})$ is the normalising constant. In this case,
$\displaystyle\alpha_{\nu}(\mathbf{x},\mathbf{y}):=1\wedge\frac{\pi_{n}(\mathbf{y})\,q_{\mathbf{y},-\nu}(\mathbf{x})}{\pi_{n}(\mathbf{x})\,q_{\mathbf{x},\nu}(\mathbf{y})}=1\wedge\frac{c_{n,\nu}(\mathbf{x})}{c_{n,-\nu}(\mathbf{y})}.$
(11)
There are two main differences with the reversible sampler. Firstly, the
normalising constants $c_{n,\nu}(\mathbf{x})$ and $c_{n,-\nu}(\mathbf{y})$ are
sums with (in general) not the same number of terms. Consider again, for ease
of presentation of the analysis, the important example described in Section
1.1 where each component $x_{i}$ of $\mathbf{x}=(x_{1},\ldots,x_{n})$ can be
of two types and more specifically the special case of Ising model. We know
that in this case $c_{n,\nu}(\mathbf{x})$ is a sum of $n_{-\nu}(\mathbf{x})$
terms (see (6)), whereas $c_{n,-\nu}(\mathbf{y})$ is a sum of
$n_{\nu}(\mathbf{y})$ terms. The second main difference is that, in the MH
sampler, it is proposed to flip one of the $n_{-1}(\mathbf{x})$ components to
$+1$ or one of the $n_{+1}(\mathbf{x})$ components to $-1$, and
$c_{n}(\mathbf{x})$ is formed from these proposals. The constant
$c_{n}(\mathbf{y})$ is also formed from proposals to flip components to $+1$
or $-1$ with $n_{-1}(\mathbf{y})$ and $n_{+1}(\mathbf{y})$ close to
$n_{-1}(\mathbf{x})$ and $n_{+1}(\mathbf{x})$ (and this is why the ratio of
the two constants converges to 1 provided that there exists a structure of
conditional independence). In contrast, $c_{n,\nu}(\mathbf{x})$ is formed from
proposals to flip one of the $n_{-\nu}(\mathbf{x})$ components to $\nu$ and
$c_{n,-\nu}(\mathbf{y})$ from proposals to flip one of the
$n_{\nu}(\mathbf{y})=n_{\nu}(\mathbf{x})+1$ components to $-\nu$; the
compositions of these constants are thus fundamentally opposite. There is
therefore no guarantee that
$c_{n,\nu}(\mathbf{x})/c_{n,-\nu}(\mathbf{y})\longrightarrow 1$ even under the
conditions stated in Zanella (2019).
Nevertheless, there exists as in the previous section an ideal situation in
which the lifter sampler outperforms the reversible MH algorithm.
###### Corollary 3.
If
(a)
$\bm{\mathcal{X}}_{n}$ is finite,
(b)
$c_{n,-1}(\mathbf{x})=c_{n,+1}(\mathbf{x})=c_{n}(\mathbf{x})/2=c_{n}^{*}/2$
for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$, where $c_{n}^{*}$ is a positive
constant that does not depend on $\mathbf{x}$ (but that may depend on $n$),
then for any $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and $n$,
$\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{MH},n}).$
###### Proof.
Analogous to that of Corollary 2. ∎
A sufficient condition for Assumption (b) to be verified is:
$|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2=n^{*}/2$
(Assumption (b) in Corollary 2) and
$\frac{1}{n^{*}/2}\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{-1}(\mathbf{x})}g\left(\frac{\pi_{n}(\mathbf{x}^{\prime})}{\pi_{n}(\mathbf{x})}\right)=\frac{1}{n^{*}/2}\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{+1}(\mathbf{x})}g\left(\frac{\pi_{n}(\mathbf{x}^{\prime})}{\pi_{n}(\mathbf{x})}\right)=\frac{1}{n^{*}}\sum_{\mathbf{x}^{\prime}\in\mathcal{N}(\mathbf{x})}g\left(\frac{\pi_{n}(\mathbf{x}^{\prime})}{\pi_{n}(\mathbf{x})}\right)=\mu,$
for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$, where $\mu$ is a positive
constant. We thus notice that the acceptance probabilities can be directly
rewritten in terms of averages when
$|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2=n^{*}/2$
and that an additional condition to Assumption (b) in Corollary 2 is
sufficient in the locally informed case for ordering the asymptotic variances.
This thus allows establishing a connection with the uniform case.
As in the previous section, it is possible to derive conditions under which
the inequality in Corollary 3 holds approximately. They are based as before on
the definition of a set, which in this case involves states $\mathbf{x}$ that
are such that $c_{n,-1}(\mathbf{x})$ and $c_{n,+1}(\mathbf{x})$ are close to
$c_{n}^{*}/2$. These states do not have to be in an area such that
$n_{-1}(\mathbf{x})$ and $n_{+1}(\mathbf{x})$ are close to $n/2$, but in
return the mass have to be (in some sense) evenly spread out in the area to
which they belong.
We now define this set:
$\displaystyle\bm{\mathcal{X}}_{\varphi(n)}:=\\{\mathbf{x}\in\bm{\mathcal{X}}_{n}:c_{n}^{*}/2-\varphi(n)\leq
c_{n,-1}(\mathbf{x}),c_{n,+1}(\mathbf{x}),c_{n}(\mathbf{x})/2\leq
c_{n}^{*}/2+\varphi(n)\\},$ (12)
where $\varphi$ is in this section a function such that
$0\leq\varphi(n)<c_{n}^{*}/2$ and, for $\epsilon>0$, there exists $N>0$ such
that for all $n\geq N$, we have that $\varphi(n)/c_{n}^{*}<\epsilon$. We
consider to simplify that for any
$\mathbf{x},\mathbf{y}\in\bm{\mathcal{X}}_{\varphi(n)}$ there exists a
probable path from $\mathbf{x}$ to $\mathbf{y}$ generated by $P_{\text{MH},n}$
(and marginally for $P_{\rho,n}$) with all intermediate states belonging to
$\bm{\mathcal{X}}_{\varphi(n)}$ as well. We now define a restricted version of
$\bm{\mathcal{X}}_{\varphi(n)}$ for which from any state
$\mathbf{x}\in\bm{\mathcal{X}}_{\varphi(n)}$, all the possible proposals
$\mathbf{y}$ belong to $\bm{\mathcal{X}}_{\varphi(n)}$ as well; denote this
set by $\bm{\mathcal{X}}_{\varphi(n)}^{0}$. For $\mathbf{x}\neq\mathbf{y}$
belonging to $\bm{\mathcal{X}}_{\varphi(n)}$, it is possible to establish
that: for $\epsilon>0$, there exists a $N>0$ such that for all $n\geq N$,
$P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega_{n}P_{\text{MH},n}(\mathbf{x},\mathbf{y})$
with $1-\omega_{n}<\epsilon$. The explicit form of $\omega_{n}$ is
$\displaystyle\omega_{n}:=\left(1+\frac{\varphi(n)}{c_{n}^{*}/2}\right)^{-3}\left(1-\frac{\varphi(n)}{c_{n}^{*}/2}\right)^{3}.$
(13)
We are now ready to present the analogous result to Theorem 1, in which here
$\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$ is the normalised version of
$\pi_{n}$ on $\bm{\mathcal{X}}_{\varphi(n)}$ defined in (12).
###### Theorem 2.
Pick $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and consider that all states
$\mathbf{x}\in\bm{\mathcal{X}}_{n}$ are such that
$\mathbf{x}=(x_{1},\ldots,x_{n})$ with $x_{i}\in\\{-1,+1\\}$. If, for
$\epsilon>0$, there exists $N>0$ such that for all $n\geq N$ it is possible to
choose a constant that may depend on $n$, $\varrho(n)$, with
(a)
$\sum_{k=\varrho(n)+1}^{\infty}\left\langle f,P^{k}f\right\rangle<\epsilon$,
for all
$P\in\\{P_{\text{rev.},n},\tilde{P}_{\text{rev.},n},P_{\text{MH},n},\tilde{P}_{\text{MH},n}\\}$,
(b)
$\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cup_{m=0}^{k}A_{m}^{\mathsf{c}}(\bm{\mathcal{X}}_{\varphi(n)}^{0})}]<\epsilon$,
for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$, where it is considered
that the chain starts at stationarity and evolves using $P$ (and
$\mathbb{E}[f(\mathbf{X}(k))]=0$ without loss of generality) and
$A_{m}(\bm{\mathcal{X}}_{\varphi(n)}^{0}):=\\{\mathbf{X}(m)\in\bm{\mathcal{X}}_{\varphi(n)}^{0}\\}$,
(c)
$(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)}^{0})}]<\epsilon$,
for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$,
(d)
$(1/\omega_{n}-1)\mathrm{var}(f,\tilde{P}_{\text{MH},n})<\epsilon$ and
$((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})<\epsilon$,
where $\omega_{n}$ is defined in (13),
then there exists a positive constant $\kappa$ such that
$\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})+\kappa\epsilon.$
###### Proof.
Analogous to that of Theorem 1. ∎
It is possible to establish a connection with Theorem 1 as we did for
Corollary 3 with Corollary 2. Consider indeed that the set
$\bm{\mathcal{X}}_{\varphi(n)}$ is comprised of states $\mathbf{x}$ with
$n_{-1}(\mathbf{x})$ and $n_{+1}(\mathbf{x})$ close to $n/2$ and
$(1/n_{+1}(\mathbf{x}))\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{-1}(\mathbf{x})}g(\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x}))$,
$(1/n_{-1}(\mathbf{x}))\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{+1}(\mathbf{x})}g(\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x}))$
and
$(1/n)\sum_{\mathbf{x}^{\prime}\in\mathcal{N}(\mathbf{x})}g(\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x}))$
all close to a positive constant $\mu$. We thus notice that under a precise
sense of what we mean by close to, this special case fits within the
definition of $\bm{\mathcal{X}}_{\varphi(n)}$ (12), under an additional
condition compared to the definition of the set in the previous section (8).
### 3.3 Computational costs
We provide in this section an overview of the computational costs associated
to using the proposal distributions described in Sections 3.1 and 3.2. The
uniform distribution is the least expensive: at each iteration, one has to
generate from a uniform and then evaluate the acceptance probability which
requires the computation of a ratio $\pi(\mathbf{y})/\pi(\mathbf{x})$.
Consider that the cost of the latter is the important one, in the sense that
all the other costs are comparatively negligible. The approach of Zanella
(2019) thus costs twice as much, if we assume that the cost of computing any
ratio is the same and that the ratios
$\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x})$ for all $\mathbf{x}^{\prime}$ in
the neighbourhood are all computed in parallel. Indeed, these ratios are
necessary to generate the proposal $\mathbf{y}$, but once the latter has been
generated, the process has to be repeated for the reverse move. This is true
for the reversible MH sampler and Algorithm 1. Therefore, if the informed
proposal leads to a sampler at least twice as effective (in terms of ESS for
instance), then it is beneficial. It is the case in all our numerical
experiments. Note that in light of the above, implementing the reversible MH
sampler or Algorithm 1 costs essentially the same.
Algorithm 2 is more costly. When used with a uniform distribution and
$\rho:=\rho^{*}$ (5), at each iteration, $|\mathcal{N}_{\nu}(\mathbf{x})|$
ratio evaluations are required to compute
$T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ (4), and it is afterwards required to
compute $T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})$ from time to time. This makes
the implementation cost somewhere in between that of Algorithm 1 with a
uniform and Algorithm 1 using the approach of Zanella (2019). When Algorithm 2
is used with an informed proposal and $\rho:=\rho^{*}$ (5), the cost may
explode. Consider that parallel computing is used to compute any normalising
constant $c_{\nu}(\mathbf{x})$, but that the normalising constants are
computed sequentially, then at each iteration it is required to compute at
least $1+|\mathcal{N}_{\nu}(\mathbf{x})|$ normalising constants compared with
two in Algorithm 1. The computation time per iteration will thus roughly be at
least $(1+|\mathcal{N}_{\nu}(\mathbf{x})|)/2$ times larger.
## 4 Numerical experiments
We first consider in Section 4.1 simulation of an Ising model and use this as
a toy example for which we can control the dimension, the roughness of the
target and where the mass concentrates to show how the performance of the
lifted and non-lifted samplers varies when these parameters change. In Section
4.2, we evaluate their performance when employed to solve a real variable
selection problem.
### 4.1 Ising model
For the two-dimensional model, the ambiant space $(V_{\eta},E_{\eta})$ is a
$\eta\times\eta$ square lattice regarded here as a square matrix in which each
element takes either the value $-1$ or $+1$. We write each state as a vector
as before: $\mathbf{x}=(x_{1},\ldots,x_{n})$, where $n=\eta^{2}$. The states
can be encoded as for instance: the values of the components on the first line
are $x_{1},\ldots,x_{\eta}$, those on the second line
$x_{\eta+1},\ldots,x_{2\eta}$, and so on. The PMF is given by
$\pi(\mathbf{x})=\frac{1}{Z}\exp\left(\sum_{i}\alpha_{i}x_{i}+\lambda\sum_{\langle
ij\rangle}x_{i}x_{j}\right),$
where $\alpha_{i}\in\operatorname{\mathbb{R}}$ and $\lambda>0$ are known
parameters, $Z$ is the normalising constant and the notation $\langle
ij\rangle$ indicates that sites i and j are nearest neighbours.
We first consider a base target distribution for which $n=50^{2}$, the spatial
correlation is moderate and more precisely $\lambda:=0.5$, and which has the
external field (the values for the $\alpha_{i}$’s) presented in Figure 2.
Figure 2: External field of the base target
We generated the $\alpha_{i}$ independently as follows:
$\alpha_{i}=-\mu+\epsilon_{i}$ if the column index is smaller than or equal to
$\ell=\lfloor\eta/2\rfloor$ and $\alpha_{i}=\mu+\epsilon_{i}$ otherwise, where
$\mu=1$, the $\epsilon_{i}$ are independent uniform random variables on the
interval $(-0.1,+0.1)$ and $\lfloor\cdot\rfloor$ is the floor function. In the
simulation study, while keeping the other parameters fixed, we gradually
increase $\eta$ from 50 to 500 to observe the impact of dealing with larger
systems. Next, while keeping the other parameters fixed (with $\eta=50$), we
gradually increase the value of $\mu$ from 1 to 3. This has for effect of
increasing the contrast so that it is clearer which values the $x_{i}$ should
take, thus making the target rougher and concentrated on fewer configurations.
One could vary $\lambda$ as well, but this would also make the target rougher
and concentrated on fewer configurations.
We observe the impact on Algorithm 1 with uniform and informed proposal
distributions, and their non-lifted MH counterparts. For such a simulation
study, it would be simply too long to obtain the results for Algorithm 2 with
$\rho_{\nu}^{*}$ (5). We tried to vary the value of $\ell$ to observe what
happens with the acceptance rates for the uniform lifted sampler when the
ratios $n_{-\nu}(\mathbf{x})/n_{\nu}(\mathbf{y})$ (see (6)) are far from 1.
The impact in this example is however not the one expected: the acceptance
rates increase instead of deteriorating. To see why, consider for instance
that $\ell=5$. The ratios $n_{-\nu}(\mathbf{x})/n_{\nu}(\mathbf{y})$ with
$\nu=-1$ are on average around 9 (45 columns with $+1$’s and 5 with $-1$’s).
With $\nu=-1$, it is tried to flip a bit from $+1$ to $-1$ and
$\pi(\mathbf{y})/\pi(\mathbf{x})$ is thus multiplied by a factor of around 9.
It is likely that this bit is on the yellow side (see Figure 2). For such a
move, $\pi(\mathbf{y})/\pi(\mathbf{x})$ is often around $\exp(-2(1+0.5\times
4))=0.002$. Therefore, it is more likely to accept this move compared to in
the reversible MH sampler (in the MH sampler $\pi(\mathbf{y})/\pi(\mathbf{x})$
is not multiplied by 9). When $\nu=+1$, the multiplicative factor is thus
around $1/9$, but it is relatively likely that the proposal will be to flip a
bit from $-1$ to $+1$ on the yellow side, because there are some bits with the
value $-1$ on this larger side, and this move is often automatically accepted
(because $\pi(\mathbf{y})/\pi(\mathbf{x})$ is often around $1/0.002)$. Note
that these conclusions are not in contradiction with our analysis of Section
3.1, because what we observed here is specific to the Ising model. We do not
present the results because the graph is uninteresting: the performance is
essentially constant for the informed samplers and that of the uniform ones is
so low that we do not see the ESS vary.
We present the other results in Figure 3. They are based on 1,000 independent
runs of 100,000 iterations for each algorithm and each value of $\mu$ and
$\eta$, with burn-ins of 10,000. For each run, an ESS per iteration is
computed for $f(\mathbf{x},-1)=f(\mathbf{x},+1)=\sum_{i}x_{i}$ and then the
results are averaged out. This function is proportional to the magnetisation.
Monitoring such a statistic is relevant as a quicker variation of its value
(leading to a higher ESS) indicates that the whole state-space is explored
quicker.
For the base target (represented by the starting points on the left of the
graphs in Figure 3), the mass is concentrated on a manifold of several
configurations, which allows for persistent movement for informed samplers.
The lifted one indeed takes advantage of this; it is approximately 7 times
more efficient than its non-lifted counterpart. The gap widens as $\eta$
increases; it is approximately 20 and 70 times more efficient when $\eta$ is
3.2 and 10 times larger (i.e. when $n$ is 10 and 100 times larger),
respectively. We observed that the ratio of ESSs increases linearly with
$\eta$. The non-informed samplers both perform poorly (the lines are on top of
each other).
As $\mu$ increases, the target becomes rougher and concentrated on fewer
configurations. When the roughness and concentration level are too severe the
performance of the lifted informed sampler stagnates, whereas that of the non-
lifted MH sampler continues to improve. There are two reasons for this.
Firstly, the acceptance rates deteriorate more rapidly for the lifted than the
non-lifted sampler (as a consequence of the difference in the acceptance
probability, see (11)). Secondly, when the mass is concentrated on few
configurations, it leaves not much room for persistent movement for the lifted
sampler. The latter thus loses its avantage. Again, the non-informed samplers
both perform poorly (the lines are on top of each other).
$\begin{array}[]{cc}\includegraphics[width=216.81pt]{Fig_Ising_n.pdf}&\includegraphics[width=216.81pt]{Fig_Ising_mu.pdf}\cr\textbf{(a)}&\textbf{(b)}\end{array}$
Figure 3: ESS per iteration of
$f(\mathbf{x},-1)=f(\mathbf{x},+1)=\sum_{i}x_{i}$ for Algorithm 1 with uniform
and informed proposal distributions and their non-lifted MH counterparts when:
(a) $\eta$ increases from 50 to 500 and the other parameters are kept fixed
($\mu=1$, $\lambda=0.5$ and $\ell=25$); (b) $\mu$ increases from 1 to 3 and
the other parameters are kept fixed ($\eta=50$, $\lambda=0.5$ and $\ell=25$)
### 4.2 Variable selection: US crime data
A study of crime rate was first presented in Erhlich (1973) and then an
expended and corrected version appeared in Vandaele (1978) in which corrected
data were provided; the topic was more precisely on the connection between
crime rate and 15 covariates (some were added by Vandaele (1978)) such as
percentage of males of age between 14 and 23 and mean years of schooling in a
given state. The data were indeed aggregated by state and were from 47 U.S.
states in 1960. They were analysed in several statistics papers (see, e.g.,
Raftery et al. (1997)) and are available in the R package MASS.
The data are modelled using the usual linear regression with normal errors.
Here we set the prior distribution of the regression coefficients and scaling
of the errors to be, conditionally on a model, the non-informative Jeffreys
prior. It is proved in Gagnon (2019) that a simple modification to the uniform
prior on the model random variable (represented here by $\mathbf{X}$) prevents
the Jeffreys–Lindley (Lindley, 1957; Jeffreys, 1967) paradox from arising.
With the resulting likelihood function and prior density on the parameters,
the latter can be integrated out. It is thus possible to evaluate the exact
marginal posterior probability of any of the $2^{15}=$ 32,768 models, up to a
normalising constants. This allows us to implement the MH sampler with the
locally informed proposal distribution of Zanella (2019) and its lifted
counterparts (Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$ (5)). In the
previous statistical studies, it was noticed that the mass is diffused over
several models, so that we expect the lifted chains to exhibit persistent
movement (as seen in Figure 1) and to perform well. To simplify the
presentation, we do not show the performance of the uniform samplers because,
as in the previous section, it is very poor.
The performances of the algorithms are summarised in Figure 4. The results are
based on 1,000 independent runs of 10,000 iterations for each algorithm, with
burn-ins of 1,000. Each run is started from a distribution which approximates
the target. On average, Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$ are
$2.7$ and $3.3$ times more efficient than their non-lifted counterpart,
respectively. The benefits of persistent movement thus compensate for a
decrease in acceptance rates; the rate indeed decreases from 0.92 for the MH
sampler to 0.71 for Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$ (5).
Figure 4: ESS per iteration for
$f(\mathbf{x},-1)=f(\mathbf{x},+1)=\sum_{i}x_{i}$ of 1,000 independent runs
for the MH sampler with the locally informed proposal distribution and its
lifted counterparts (Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$)
## 5 Lifted trans-dimensional sampler
In this section, we introduce a trans-dimensional algorithm which is a non-
reversible version of the popular reversible jump (RJ) algorithms introduced
by Green (1995). We consider that $\bm{\mathcal{X}}$ is a model space and
$\mathbf{X}$ a model indicator. The latter indicates, for instance, with 0’s
and 1’s which covariates are included in the model in variable selection
contexts as in Section 4.2. Such an algorithm is useful when it is not
possible to integrate out the parameters, contrarily to the linear regression
with normal errors and suitable priors. Examples of such situations include
analyses based on linear regression with super heavy-tailed errors ensuring
whole robustness (Gagnon et al., 2018) and generalised linear models and
generalised linear mixed models (Forster et al., 2012).
The parameters of a given model $\mathbf{x}$ are denoted by
$\bm{\theta}_{\mathbf{x}}\in\bm{\Theta}_{\mathbf{x}}$. Trans-dimensional
algorithms are MCMC methods that one uses to sample from a target distribution
$\pi$ defined on a union of sets
$\cup_{\mathbf{x}\in\bm{\mathcal{X}}}\\{\mathbf{x}\\}\times\bm{\Theta}_{\mathbf{x}}$,
which corresponds in Bayesian statistics to the joint posterior of the model
indicator $\mathbf{X}$ and the parameters of Model $\mathbf{X}$,
$\bm{\theta}_{\mathbf{X}}$. Such a posterior allows to jointly infer about
$(\mathbf{X},\bm{\theta}_{\mathbf{X}})$, or in other words, simultaneously
achieve model selection and parameter estimation. In this section, we assume
for simplicity that the parameters of all models are continuous random
variables.
Given the current state of the Markov chain
$(\mathbf{x},\bm{\theta}_{\mathbf{x}})$, a trans-dimensional sampler generates
the next state by first proposing a model candidate $\mathbf{y}\sim
q_{\mathbf{x}}(\mathbf{y})$ and then a proposal for its corresponding
parameter values. When $\mathbf{y}=\mathbf{x}$, we say that a parameter update
is proposed, whereas we say that a model switch is proposed when
$\mathbf{y}\neq\mathbf{x}$. Note that $\mathbf{x}\in\mathcal{N}(\mathbf{x})$,
contrarily to the previous sections. This is to allow parameter updates. When
a parameter update is proposed, we allow any fixed-dimensional methods to be
used; we only require that the Markov kernels leave the conditional
distributions $\pi(\,\cdot\mid\mathbf{x})$ invariant. When a model switch is
proposed, a vector of auxiliary variables
$\mathbf{u}_{\mathbf{x}\mapsto\mathbf{y}}$ is typically generated and this is
followed by a proposal mechanism leading to
$(\bm{\theta}^{\prime}_{\mathbf{y}},\mathbf{u}_{\mathbf{y}\mapsto\mathbf{x}})$,
where $\bm{\theta}^{\prime}_{\mathbf{y}}$ is the proposal for the parameter
values in Model $\mathbf{y}$. We require the whole proposal mechanism for
$\bm{\theta}^{\prime}_{\mathbf{y}}$ to be valid in a RJ framework, in the
sense that the model switch transitions are reversible in this framework. The
non-reversibility in the lifted trans-dimensional sampler lies in the
transitions for the $\mathbf{x}$ variable during model switches. More
precisely, $\mathbf{y}$ is generated from $q_{\mathbf{x},\nu}(\mathbf{y})$
instead, but the proposal mechanism for $\bm{\theta}^{\prime}_{\mathbf{y}}$
during model switches is the same. We consider that the acceptance probability
of these model switches in RJ is given by
$\displaystyle\alpha_{\text{RJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})):=1\wedge\frac{q_{\mathbf{y}}(\mathbf{x})}{q_{\mathbf{x}}(\mathbf{y})}\,r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})),$
(14)
where the function $r$ depends on the method and may depend on several other
variables.
The algorithm is now presented in Algorithm 3. In it, we consider that the
current model $\mathbf{x}$ always belongs to the neighbourhood
$\mathcal{N}_{\nu}(\mathbf{x})$, regardless of the current direction $\nu$,
and that $q_{\mathbf{x},-1}(\mathbf{x})=q_{\mathbf{x},+1}(\mathbf{x})$, which
will typically be the case in practice.
Algorithm 3 A lifted trans-dimensional sampler for partially ordered model
spaces
1. 1.
Generate $\mathbf{y}\sim q_{\mathbf{x},\nu}$, a PMF with support restricted to
$\mathcal{N}_{\nu}(\mathbf{x})$.
2. 2.(a)
If $\mathbf{y}=\mathbf{x}$, attempt a parameter update using a MCMC kernel of
invariant distribution $\pi(\,\cdot\mid\mathbf{x})$ while keeping the current
value of the model indicator $\mathbf{x}$ and direction $\nu$ fixed.
3. 2.(b)
If $\mathbf{y}\neq\mathbf{x}$, attempt a model switch from Model $\mathbf{x}$
to Model $\mathbf{y}$. Generate $\bm{\theta}^{\prime}_{\mathbf{y}}$ using a
method which is valid in RJ and $u\sim\mathcal{U}[0,1]$. If
$\displaystyle
u\leq\alpha_{\text{NRJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})):=1\wedge\frac{q_{\mathbf{y},-\nu}(\mathbf{x})}{q_{\mathbf{x},\nu}(\mathbf{y})}\,r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})),$
(15)
set the next state of the chain to
$(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}},\nu)$. Otherwise, set it to
$(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)$.
4. 3.
Go to Step 1.
###### Proposition 2.
The transition kernel of the Markov chain
$\\{(\mathbf{X},\bm{\theta}_{\mathbf{X}},\nu)(m):m\in\operatorname{\mathbb{N}}\\}$
simulated by Algorithm 3 admits $\pi\otimes\mathcal{U}\\{-1,1\\}$ as invariant
distribution.
###### Proof.
See Section 7. ∎
The main difficulty with the implementation of trans-dimensional samplers is
the construction of efficient proposal schemes for
$(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})$ during model switches. Gagnon
(2019) discusses this in depth in the RJ framework. The author proposes a
scheme and proves that it is possible to arbitrarily approach an asymptotic
regime in which one is able to generate $\bm{\theta}^{\prime}_{\mathbf{y}}$
from $\pi(\,\cdot\mid\mathbf{y})$ (the correct conditional distribution) and
evaluate exactly the ratios of marginal probabilities
$\pi(\mathbf{y})/\pi(\mathbf{x})$ (and is therefore able to adequately
construct $q_{\mathbf{x},\nu}$). In particular, for this scheme,
$r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}}))$
is a consistent estimator of $\pi(\mathbf{y})/\pi(\mathbf{x})$. We refer the
reader to that paper for the details.
We thus conclude that the marginal behaviour of
$\\{(\mathbf{X},\nu)(m):m\in\operatorname{\mathbb{N}}\\}$ is the same as that
of the stochastic process generated by Algorithm 1 in the asymptotic regime
and considering only iterations for which model switches are proposed. All
conclusions previously drawn thus hold, at least approximatively. In
particular, one may analyse the same data as in Section 4.2, but using the
super heavy-tailed regression of Gagnon et al. (2018) for robust inference and
outlier detection. The results would be essentially the same because, as
mentioned in Raftery et al. (1997), “standard diagnostic checking (see, e.g.,
Draper and Smith (1981)) did not reveal any gross violations of the
assumptions underlying normal linear regression” and the robust method is
designed for leading to similar results in the absence of outliers. We thus
omit further analysis of Algorithm 3 and we do not illustrate how it performs
for brevity. We nevertheless highlight that it is important for
$r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}}))$
to be a low variance estimator of $\pi(\mathbf{y})/\pi(\mathbf{x})$ as
persistent movement may be interrupted otherwise, as shown in Gagnon and
Doucet (2019).
## 6 Discussion
In this paper, we presented and analysed generic algorithms allowing
straightforward sampling from any PMF $\pi$ with a support $\bm{\mathcal{X}}$
on which a partial order can be established. The algorithms rely on the
technique of lifting. We showed that these are expected to perform well when
the shape of target (the level of concentration of the mass) allows for
persistent movement. This is true even when the target concentrates on a
manifold of the ambient space in the case where the lifting technique is
combined with locally informed proposal distributions (provided that the shape
of the manifold allows for persistent movement).
The samplers are in particular useful for the simulation of binary random
variables and variable selection. Algorithm 1 can be directly employed for the
latter when the parameters of the models can be integrated out. A lifted
trans-dimensional sampler for partially ordered model spaces have been
introduced in Section 5 for, among others, variable selection when it is not
possible to integrate out the parameters. We believe it would be interesting
to continue this line of research by taking steps towards automatic generic
samplers using the technique of lifting for any discrete state-space.
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## 7 Proofs
###### Proof of Proposition 1.
It suffices to prove that the probability to reach the state
$(\mathbf{y},\nu^{\prime})$ in one step is equal to the probability of this
state under the target:
$\displaystyle\sum_{\mathbf{x},\nu}\pi(\mathbf{x})\,(1/2)\,P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))=\pi(\mathbf{y})\,(1/2).$
(16)
where $P$ is the transition kernel.
The probability to reach the state $(\mathbf{y},\nu^{\prime})$ from some
$(\mathbf{x},\nu)$ is given by:
$\displaystyle P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))$
$\displaystyle=T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\,Q_{\mathbf{x},\nu}(\mathbf{y})\,\mathds{1}(\nu=\nu^{\prime})$
$\displaystyle\qquad+\mathds{1}(\nu=-\nu^{\prime},\mathbf{x}=\mathbf{y})\left[(\rho_{\nu}(\mathbf{x})+T_{\nu}(\mathbf{x},\bm{\mathcal{X}}))-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\right]$
$\displaystyle\qquad+\mathds{1}(\nu=\nu^{\prime},\mathbf{x}=\mathbf{y})\left[1-\rho_{\nu}(\mathbf{x})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\right]$
$\displaystyle=q_{\mathbf{x},\nu}(\mathbf{y})\,\alpha_{\nu}(\mathbf{x},\mathbf{y})\,\mathds{1}(\nu=\nu)$
$\displaystyle\qquad+\mathds{1}(\nu=-\nu^{\prime},\mathbf{x}=\mathbf{y})\,\rho_{\nu}(\mathbf{x})$
$\displaystyle\qquad+\mathds{1}(\nu=\nu^{\prime},\mathbf{x}=\mathbf{y})\left[1-\rho_{\nu}(\mathbf{x})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\right].$
We have that
$\displaystyle\pi(\mathbf{x})\,(1/2)\,P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))$
$\displaystyle=(1/2)\,\pi(\mathbf{y})\,q_{\mathbf{y},-\nu^{\prime}}(\mathbf{x})\,\alpha_{-\nu^{\prime}}(\mathbf{y},\mathbf{x})\,\mathds{1}(-\nu^{\prime}=-\nu)$
$\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\,\mathds{1}(-\nu^{\prime}=\nu,\mathbf{y}=\mathbf{x})\,\rho_{-\nu^{\prime}}(\mathbf{y})$
$\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\mathds{1}(-\nu^{\prime}=-\nu,\mathbf{y}=\mathbf{x})\left[1-\rho_{-\nu^{\prime}}(\mathbf{y})-T_{-\nu^{\prime}}(\mathbf{y},\bm{\mathcal{X}})\right]$
$\displaystyle=(1/2)\,\pi(\mathbf{y})\,T_{-\nu^{\prime}}(\mathbf{y},\bm{\mathcal{X}})\,Q_{\mathbf{y},-\nu^{\prime}}(\mathbf{x})\mathds{1}(-\nu^{\prime}=-\nu)$
$\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\,\mathds{1}(-\nu^{\prime}=\nu,\mathbf{y}=\mathbf{x})\left[(\rho_{-\nu^{\prime}}(\mathbf{y})+T_{-\nu^{\prime}}(\mathbf{y},\mathcal{X}))-T_{-\nu^{\prime}}(\mathbf{y},\mathcal{X})\right]$
$\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\,\mathds{1}(-\nu^{\prime}=-\nu,\mathbf{y}=\mathbf{x})\left[1-\rho_{-\nu^{\prime}}(\mathbf{y})-T_{-\nu^{\prime}}(\mathbf{y},\mathcal{X})\right],$
where we used the definition of $\alpha$ for the first term and that
$\rho_{\nu}(\mathbf{x})-\rho_{-\nu}(\mathbf{x})=T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$
for the third term. Notice the sum on the right-hand side (RHS) is equal to
the probability to reach some $(\mathbf{x},-\nu)$, starting from
$(\mathbf{y},-\nu^{\prime})$:
$(1/2)\,\pi(\mathbf{y})\,P((\mathbf{y},-\nu^{\prime}),(\mathbf{x},-\nu))$.
Therefore,
$\displaystyle\sum_{\mathbf{x},\nu}\pi(\mathbf{x})\,(1/2)\,P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))$
$\displaystyle=\sum_{\mathbf{x},\nu}(1/2)\,\pi(\mathbf{y})\,P((\mathbf{y},-\nu^{\prime}),(\mathbf{x},-\nu))$
$\displaystyle=(1/2)\,\pi(\mathbf{y}),$
which concludes the proof. ∎
We now present a lemma that will be useful in the next proofs.
###### Lemma 1.
Let $Q$ be the Markov kernel of the Markov chain simulated by Algorithm 2, for
any valid switching function $\rho_{\nu}$, $\nu\in\\{-1,1\\}$. Assume that
$\bm{\mathcal{X}}$ is finite. Then, for any
$f\in\mathcal{L}_{2}^{*}(\bar{\pi})$,
$\lim_{\lambda\to
1}\sum_{k>0}\lambda^{k}\langle\,f,Q^{k}f\,\rangle=\sum_{k>0}\langle\,f,Q^{k}f\,\rangle\,.$
(17)
###### Proof.
For each $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$, define the sequence of
functions $S_{n}:\lambda\mapsto\sum_{0<k\leq
n}\lambda^{k}\langle\,f,Q^{k}f\,\rangle$ defined for $\lambda\in[0,1)$ and its
limit $S(\lambda)=\sum_{k>0}\lambda^{k}\langle\,f,Q^{k}f\,\rangle$ (the
dependance of $S_{n}$ and $S$ on $f$ and $Q$ is implicit). We now show that
the partial sum $S_{n}$ converges uniformly to $S$ on $[0,1)$, and since for
each $n\in\mathbb{N}$, the function
$\lambda\to\lambda^{n}\langle\,f,Q^{n}f\,\rangle$ admits a limit when
$\lambda\to 1$, we have that $S$ admits a limit when $\lambda\to 1$, given by
$\lim_{\lambda\to 1}S(\lambda)=S(1)=\sum_{k>0}\langle\,f,Q^{k}f\,\rangle\,,$
which is Eq. (17). First, note that
$\sup_{\lambda\in[0,1)}\left|S_{n}(\lambda)-S(\lambda)\right|=\sup_{\lambda\in[0,1)}\left|\sum_{k>n}\lambda^{k}\langle\,f,Q^{k}f\,\rangle\right|\leq\sup_{\lambda\in[0,1)}\sum_{k>n}\lambda^{k}\left|\langle\,f,Q^{k}f\,\rangle\right|=\sum_{k>n}\left|\langle\,f,Q^{k}f\,\rangle\right|\,.$
(18)
Thus, to prove that
$\sup_{\lambda\in[0,1)}\left|S_{n}(\lambda)-S(\lambda)\right|\to 0$, it is
sufficient to prove that the series
$\sum_{k>0}\left|\langle\,f,Q^{k}f\,\rangle\right|$ converges.
By bilinearity of the inner product and by linearity of the iterated operators
$Q,Q^{2},\ldots$, it can be checked that for any linear mapping $\phi$ on
$\mathcal{L}_{2}^{\ast}(\bar{\pi})$
$\sum_{k=1}^{\infty}\left|\left\langle
f,Q^{k}f\right\rangle\right|<\infty\Leftrightarrow\sum_{k=1}^{\infty}\left|\left\langle\phi(f),Q^{k}\phi(f)\right\rangle\right|<\infty\,.$
(19)
Since $\bm{\mathcal{X}}$ is finite, if $f\in\mathcal{L}_{2}^{\ast}(\bar{\pi})$
then $\sup|f|<\infty$. As a consequence, we may use
$\phi(f):=(f-\bar{\pi}f)/\sup|f|$ and $\bar{\pi}f:=\int f\mathrm{d}\bar{\pi}$.
In the following we denote by $\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$ the
subset of $\mathcal{L}_{2}^{\ast}(\bar{\pi})$ such that
$\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi}):=\left\\{f\in\mathcal{L}_{2}^{\ast}(\bar{\pi})\,:\,\bar{\pi}f=0\,,\;\sup|f|\leq
1\right\\}\,.$
By Eq. (19), we only need to check that the series
$\sum_{k>0}\left|\langle\,f,Q^{k}f\,\rangle\right|$ converges for each
$f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$. Since $\bm{\mathcal{X}}$ is
finite, $Q$ is uniformly ergodic and there exists constants $\varrho\in(0,1)$
and $C\in(0,\infty)$ such that for any $t\in\mathbb{N}$,
$\sup_{(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,+1\\}}\|\delta_{x,\nu}Q^{t}-\bar{\pi}\|_{\mathrm{tv}}\leq
C\varrho^{t}\,,\ $ (20)
where for any signed measure $\mu$, $\|\mu\|_{\mathrm{tv}}$ denotes its total
variation. On the one hand, note that for each
$f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$
$\langle\,f,Q^{k}f\,\rangle=\mathbb{E}f(X,\nu)Q^{k}f(X,\nu)\leq\mathbb{E}|f(X,\nu)||Q^{k}f(X,\nu)|\,,$
(21)
and on the other hand, we have that for any
$(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}$,
$|Q^{k}f(x,\nu)|=|Q^{k}f(x,\nu)-\bar{\pi}f|\leq\sup_{f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})}|Q^{k}f(x,\nu)-\bar{\pi}f|\,.$
(22)
But $\|\mu\|_{\mathrm{tv}}=(1/2)\sup_{g:\bm{\mathcal{X}}\to[-1,1]}|\mu g|$,
see for instance (Roberts and Rosenthal, 2004, Proposition 3). Since
$f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$, $|f|\leq 1$ and we have by
inclusion that for all $(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}$
$|Q^{k}f(x,\nu)|\leq\sup_{f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})}|Q^{k}f(x,\nu)-\bar{\pi}f|\leq\sup_{g:\bm{\mathcal{X}}\times\\{-1,1\\}\to[-1,1]}|Q^{k}g(x,\nu)-\bar{\pi}g|\leq
2\|\delta_{x,\nu}Q^{k}-\bar{\pi}\|_{\mathrm{tv}}\,.$ (23)
Taking the supremum over all $(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}$ in
Eq. (23), and combining with Eq. (20) yields
$\sup_{(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}}|Q^{k}f(x,\nu)|\leq
2C\varrho^{k}\,.$
Plugging this into Eq. (21), we have
$\left|\langle\,f,Q^{k}f\,\rangle\right|\leq
2C\mathbb{E}|f(X,\nu)|\varrho^{k}\,.$ (24)
which is clearly summable. As a consequence, $S_{n}$ converges uniformly to
$S$ on $[0,1)$ which concludes the proof. ∎
###### Proof of Corollary 1.
The results of Theorem 3.15 in Andrieu and Livingstone (2019) holds in our
framework, implying that
$\mathrm{var}_{\lambda}(f,P_{\rho^{*}})\leq\mathrm{var}_{\lambda}(f,P_{\rho})\leq\mathrm{var}_{\lambda}(f,P_{\rho}^{\text{w}}),$
where
$\mathrm{var}_{\lambda}(f,P_{\rho}):=\mathbb{V}\mathrm{ar}f(\mathbf{X},\nu)+2\sum_{k>0}\lambda^{k}\left\langle
f,P_{\rho}^{k}f\right\rangle$ with $\lambda\in[0,1)$. Lemma 1 allows to
conclude. ∎
###### Proof of Corollary 2.
The proof is an application of Theorem 3.17 in Andrieu and Livingstone (2019)
which will allow to establish that
$\mathrm{var}_{\lambda}(f,P_{\rho})\leq\mathrm{var}_{\lambda}(f,P_{\text{MH}}).$
We will thus be able to conclude using Lemma 1.
In order to apply Theorem 3.17, we must verify that
$q_{\mathbf{x}}(\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y})=(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})\,\alpha_{+1}(\mathbf{x},\mathbf{y})+(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})\,\alpha_{-1}(\mathbf{x},\mathbf{y}),$
for all $\mathbf{x}$ and $\mathbf{y}$. This is straightforward to verify under
the assumptions of Corollary 2:
$\displaystyle(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})\,\alpha_{+1}(\mathbf{x},\mathbf{y})+(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})\,\alpha_{-1}(\mathbf{x},\mathbf{y})$
$\displaystyle\qquad=\frac{1}{2}\frac{1}{(|\mathcal{N}(\mathbf{x})|/2)}\left(1\wedge\frac{\pi(\mathbf{y})}{\pi(\mathbf{x})}\right)\mathds{1}_{\mathbf{y}\in\mathcal{N}_{+1}(\mathbf{x})}+\frac{1}{2}\frac{1}{(|\mathcal{N}(\mathbf{x})|/2)}\left(1\wedge\frac{\pi(\mathbf{y})}{\pi(\mathbf{x})}\right)\mathds{1}_{\mathbf{y}\in\mathcal{N}_{-1}(\mathbf{x})}$
$\displaystyle\qquad=\frac{1}{|\mathcal{N}(\mathbf{x})|}\left(1\wedge\frac{\pi(\mathbf{y})}{\pi(\mathbf{x})}\right)\left(\mathds{1}_{\mathbf{y}\in\mathcal{N}_{+1}(\mathbf{x})}+\mathds{1}_{\mathbf{y}\in\mathcal{N}_{-1}(\mathbf{x})}\right)=q_{\mathbf{x}}(\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y}).$
∎
###### Proof of Theorem 1.
We first prove that
$\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{rev.},n})$. This is
done as in the proof of Corollary 2.
We now analyse $\mathrm{var}(f,P_{\text{rev.},n})$:
$\mathrm{var}(f,P_{\text{rev.},n})=\mathbb{V}\mathrm{ar}f(\mathbf{X})+2\sum_{k>0}\left\langle
f,P_{\text{rev.},n}^{k}f\right\rangle=\mathbb{V}\mathrm{ar}f(\mathbf{X})+2\sum_{k>0}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))],$
where we omitted the dependence on $\nu$ because, as we mentioned in Section
3.1, it can be treated as a constant as a consequence of the restrictions on
$f$. In the expression above, it is considered that the chain starts at
stationarity and evolves using $P_{\text{rev.},n}$. We consider without loss
of generality that $\mathbb{E}[f(\mathbf{X}(k))]=0$ (for any $k$).
We first write
$\displaystyle\mathbb{V}\mathrm{ar}f(\mathbf{X})$
$\displaystyle=\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]$
(25)
$\displaystyle=\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})^{2}]-\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]$
(26)
$\displaystyle\qquad+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}],$
(27)
where $\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}$ denotes an
expectation with respect to $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$.
Also,
$\displaystyle\sum_{k>0}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]=\sum_{k=1}^{\varrho(n)}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]+\sum_{k=\varrho(n)+1}^{\infty}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))],$
where $\varrho(n)$ is chosen according to the statement of Theorem 1;
therefore the second term on the RHS can be made as small as we want. For the
first term, we have
$\displaystyle\sum_{k=1}^{\varrho(n)}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]$
$\displaystyle=\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))]$
$\displaystyle=\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}]+\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cup_{m=0}^{k}A_{m}^{\mathsf{c}}(\bm{\mathcal{X}}_{\varphi(n)-1})}],$
where
$A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1}):=\\{\mathbf{X}(m)\in\bm{\mathcal{X}}_{\varphi(n)-1}\\}$.
By assumption, the second term on the RHS can be made as small as we want.
We have
$\displaystyle\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}]$
$\displaystyle=\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{P}_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))]$
$\displaystyle+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}],$
where $\tilde{P}_{\text{rev.},n}$ is the Markov kernel whose stationary
distribution is $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$. This equality
holds because the paths involving transition probabilities
$P_{\text{rev.},n}(\mathbf{x},\mathbf{y})$ with
$\mathbf{x},\mathbf{y}\in\bm{\mathcal{X}}_{\varphi(n)-1}$ have the same
probabilities as those of the chain with stationary distribution
$\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$. We just have to renormalise the
probabilities of the starting point by dividing by
$\pi(\bm{\mathcal{X}}_{\varphi(n)})$ to complete the argument. Note that, by
assumption, the second term on the RHS can be made as small as we want.
Now,
$\displaystyle\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{P}_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))]=\sum_{k=1}^{\varrho(n)}\text{Cov}_{\tilde{P}_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]+\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}.$
By assumption, the sum from $\varrho(n)+1$ to $\infty$ of the covariances is
small as well. If we combine this with (25), we have
$\displaystyle\mathrm{var}(f,P_{\text{rev.},n})$
$\displaystyle=\mathrm{var}(f,\tilde{P}_{\text{rev.},n})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]$
$\displaystyle\qquad+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}$
$\displaystyle\leq\mathrm{var}(f,\tilde{P}_{\text{MH},n})/\omega_{n}+((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}$
$\displaystyle\qquad+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}$
$\displaystyle=\mathrm{var}(f,\tilde{P}_{\text{MH},n})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]$
$\displaystyle\qquad+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1/\omega_{n}-1)\mathrm{var}(f,\tilde{P}_{\text{MH},n})+((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X}),$
using Theorem 2 of Zanella (2019) for the inequality and omitting the terms
that can be made as small as we want. This theorem can be used as a result of
the bound
$\tilde{P}_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega_{n}\tilde{P}_{\text{MH},n}(\mathbf{x},\mathbf{y})$
with $\omega_{n}$ defined in (9) (see Section 3.1). By assumption, the last
two terms can be made as small as we want.
Now we used the previous arguments in the reverse order to show that
$\displaystyle\mathrm{var}(f,\tilde{P}_{\text{MH},n})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]$
$\displaystyle\qquad+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}=\mathrm{var}(f,P_{\text{MH},n})+\text{small
error term},$
which yields the result. ∎
###### Proof of Proposition 2.
It suffices to prove that the probability to reach the state
$\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}\in A,\nu^{\prime}$ in one step
is equal to the probability of this state under the target:
$\displaystyle\sum_{\mathbf{x},\nu}\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime}))\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$
$\displaystyle=\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\,d\bm{\theta}_{\mathbf{y}}^{\prime},$
(28)
where $P$ is the transition kernel. Note that we abuse notation here by
denoting the measure $d\bm{\theta}_{\mathbf{y}}^{\prime}$ on the left-hand
side (LHS) given that we may in fact use vectors of auxiliary variables to
generate the proposal when switching models, which often do not have the same
dimension as $\bm{\theta}_{\mathbf{y}}^{\prime}$.
We consider two distinct events: a model switch is proposed, that we denote
$S$, and a parameter update is proposed (therefore denoted $S^{\mathsf{c}}$).
We know that the probabilities of these events are
$1-q_{\mathbf{x},\nu}(\mathbf{x})$ and $q_{\mathbf{x},\nu}(\mathbf{x})$,
respectively. We rewrite the LHS of (28) as
$\displaystyle\sum_{\mathbf{x},\nu}\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime}))\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$
(29)
$\displaystyle\quad=\sum_{\mathbf{x},\nu}\,(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid
S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ (30)
$\displaystyle\qquad+\sum_{\mathbf{x},\nu}\,q_{\mathbf{x},\nu}(\mathbf{x})\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid
S^{\mathsf{c}})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}.$
(31)
We analyse the two terms separately. We know that
$P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid
S^{\mathsf{c}})=\delta_{(\mathbf{x},\nu)}(\mathbf{y},\nu^{\prime})\,P_{S^{\mathsf{c}}}(\bm{\theta}_{\mathbf{x}},\bm{\theta}_{\mathbf{y}}^{\prime}),$
where $P_{S^{\mathsf{c}}}$ is the transition kernel associated with the method
used to update the parameters. Therefore, the second term on the RHS of (29)
is equal to
$\displaystyle\sum_{k,\nu}\,q_{\mathbf{x},\nu}(\mathbf{x})\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid
S^{\mathsf{c}})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$
$\displaystyle\quad=q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\,\pi(\mathbf{y})\times(1/2)\int\pi(\bm{\theta}_{\mathbf{y}}\mid\mathbf{y})\left(\int_{A}P_{S^{\mathsf{c}}}(\bm{\theta}_{\mathbf{y}},\bm{\theta}_{\mathbf{y}}^{\prime})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{y}}.$
We also know that $P_{S^{\mathsf{c}}}$ leaves the conditional distribution
$\pi(\,\cdot\mid\mathbf{y})$ invariant, implying that
$\displaystyle
q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\,\pi(\mathbf{y})\times(1/2)\int\pi(\bm{\theta}_{\mathbf{y}}\mid\mathbf{y})\left(\int_{A}P_{S^{\mathsf{c}}}(\bm{\theta}_{\mathbf{y}},\bm{\theta}_{\mathbf{y}}^{\prime})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{y}}$
(32)
$\displaystyle\quad=q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\,\pi(\mathbf{y})\times(1/2)\int_{A}\pi(\bm{\theta}_{\mathbf{y}}^{\prime}\mid\mathbf{y})\,d\bm{\theta}_{\mathbf{y}}^{\prime}=q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\,d\bm{\theta}_{\mathbf{y}}^{\prime}.$
(33)
For the model switching case (the first term on the RHS of (29)), we use the
fact that there is a connection between
$P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid
S)$ and the kernel associated to a specific RJ. Consider that
$q_{\mathbf{x}}(\mathbf{y})=(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})+(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})$
for all $\mathbf{x},\mathbf{y}$ and that all other proposal distributions in
RJ are the same as in Algorithm 3 during model switches. In this case,
$\alpha_{\text{RJ}}=\alpha_{\text{NRJ}}$ in the case where at the current
iteration it is chosen to use $q_{\mathbf{x},\nu}$ (which happens with
probability $1/2$) and in the reverse move it is chosen to use
$q_{\mathbf{y},-\nu}$ (which also happens with probability $1/2$).
Consider the case where Model $\mathbf{y}$ is reached from Model
$\mathbf{x}\neq\mathbf{y}$ coming from direction $\nu^{\prime}=\nu$. Given the
reversibility of RJ, the probability to go from Model $\mathbf{x}$ with
parameters in $B$ to Model $\mathbf{y}\neq\mathbf{x}$ with parameters in $A$
is
$\displaystyle\int_{B}\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\left(\int_{A}P_{\text{RJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}))\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}=\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\left(\int_{B}P_{\text{RJ}}((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}),(\mathbf{x},\bm{\theta}_{\mathbf{x}}))\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime},$
(34)
where $P_{\text{RJ}}$ is the transition kernel of the RJ. Note that
$P_{\text{RJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}))=(1/2)\,(1-q_{\mathbf{x},\nu}(\mathbf{x}))\,P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu)\mid
S),$
given that the difference between both kernels is that in RJ, it is first
decided to use $q_{\mathbf{x},\nu}$, there is thus an additional probability
of $1/2$. Analogously,
$P_{\text{RJ}}((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}),(\mathbf{x},\bm{\theta}_{\mathbf{x}}))=(1/2)\,(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\,P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid
S)$. Using this and taking $B$ equals the whole parameter (and auxiliary)
space in (34), we have
$\displaystyle(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu)\mid
S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$
$\displaystyle\qquad=(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(\int
P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid
S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}.$
The only other case to consider for model switches is where Model $\mathbf{y}$
is reached from Model $\mathbf{y}$ (because the proposal is rejected) and the
direction is $\nu^{\prime}=-\nu$. The probability of this transition is
$(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(1-\sum_{\mathbf{x}\neq\mathbf{y}}\int
P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid
S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}.$
So, the total probability of reaching
$\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}\in A,\nu^{\prime}$ through a
model switch is (recalling (29)):
$\displaystyle\sum_{\mathbf{x},\nu}\,(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid
S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$
$\displaystyle\quad=\sum_{\mathbf{x}\neq\mathbf{y}}(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu)\mid
S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$
$\displaystyle\qquad+(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(1-\sum_{\mathbf{x}\neq\mathbf{y}}\int
P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid
S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}$
$\displaystyle\quad=\sum_{\mathbf{x}\neq\mathbf{y}}(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(\int
P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid
S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}$
$\displaystyle\qquad+(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(1-\sum_{\mathbf{x}\neq\mathbf{y}}\int
P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid
S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}$
$\displaystyle\quad=(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2).$
Combining this with (32) allows to conclude the proof. ∎
|
2024-09-04T02:54:59.374374 | 2020-03-11T20:31:20 | 2003.05511 | {
"authors": "Tong Bai, Cunhua Pan, Hong Ren, Yansha Deng, Maged Elkashlan, and\n Arumugam Nallanathan",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26178",
"submitter": "Pan Cunhua",
"url": "https://arxiv.org/abs/2003.05511"
} | arxiv-papers | # Resource Allocation for Intelligent Reflecting Surface Aided Wireless
Powered Mobile Edge Computing in OFDM Systems
Tong Bai, , Cunhua Pan, ,
Hong Ren, , Yansha Deng, ,
Maged Elkashlan, , and Arumugam Nallanathan T. Bai, C. Pan, H. Ren, M.
Elkashlan and A. Nallanathan are with the School of Electronic Engineering and
Computer Science, Queen Mary University of London, London E1 4NS, U.K.
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]). Y. Deng is with the
Department of Engineering, King’s College London, London, WC2R 2LS, U.K.
(e-mail: [email protected]).
###### Abstract
Wireless powered mobile edge computing (WP-MEC) has been recognized as a
promising technique to provide both enhanced computational capability and
sustainable energy supply to massive low-power wireless devices. However, its
energy consumption becomes substantial, when the transmission link used for
wireless energy transfer (WET) and for computation offloading is hostile. To
mitigate this hindrance, we propose to employ the emerging technique of
intelligent reflecting surface (IRS) in WP-MEC systems, which is capable of
providing an additional link both for WET and for computation offloading.
Specifically, we consider a multi-user scenario where both the WET and the
computation offloading are based on orthogonal frequency-division multiplexing
(OFDM) systems. Built on this model, an innovative framework is developed to
minimize the energy consumption of the IRS-aided WP-MEC network, by optimizing
the power allocation of the WET signals, the local computing frequencies of
wireless devices, both the sub-band-device association and the power
allocation used for computation offloading, as well as the IRS reflection
coefficients. The major challenges of this optimization lie in the strong
coupling between the settings of WET and of computing as well as the unit-
modules constraint on IRS reflection coefficients. To tackle these issues, the
technique of alternative optimization is invoked for decoupling the WET and
computing designs, while two sets of locally optimal IRS reflection
coefficients are provided for WET and for computation offloading separately
relying on the successive convex approximation method. The numerical results
demonstrate that our proposed scheme is capable of monumentally outperforming
the conventional WP-MEC network without IRSs. Quantitatively, about $80\%$
energy consumption reduction is attained over the conventional MEC system in a
single cell, where $3$ wireless devices are served via $16$ sub-bands, with
the aid of an IRS comprising of $50$ elements.
## I Introduction
### I-A Motivation and Scope
In the Internet-of-Things (IoT) era, a myriad of heterogeneous devices are
envisioned to be interconnected [1]. However, due to the stringent constraints
both on device sizes and on manufacturing cost, many of them have to be
equipped with either life-limited batteries or low-performance processors.
Consequently, if only relying on their local computing, these resource-
constrained devices are incapable of accommodating the applications that
require sustainable and low-latency computation, e.g. wireless body area
networks [2] and environment monitoring [3]. Fortunately, wireless powered
mobile edge computing (WP-MEC)[4, 5, 6, 7, 8, 9, 10, 11, 12, 13], which
incorporates radio frequency (RF) based wireless energy transmission (WET)
[14, 15, 16] and mobile edge computing (MEC) [17, 18], constitutes a promising
solution of this issue. Specifically, at the time of writing, the commercial
RF-based WET has already been capable of delivering $0.05\leavevmode\nobreak\
\rm{mW}$ to a distance of $12-14\leavevmode\nobreak\ \rm{m}$ [14], which is
sufficient to charge many low-power devices, whilst the MEC technique may
provide the cloud-like computing service at the edge of mobile networks [18].
In WP-MEC systems, hybrid access points (HAP) associated with edge computing
nodes are deployed in the proximity of wireless devices, and the computation
of these devices is typically realized in two phases, namely the WET phase and
the computing phase. To elaborate, the batteries of the devices are
replenished by harvesting WET signals from the HAP in the first phase, while
in the computing phase, devices may decide whether to process their
computational tasks locally or offload them to edge computing nodes via the
HAP.
Given that these wireless devices are fully powered by WET in WP-MEC systems,
the power consumption at HAPs becomes substantial, which inevitably increases
the expenditure on energy consumption and may potentially saturate power
rectifiers. At the time of writing, the existing research contributions that
focus on reducing the power consumption mainly rely on the joint optimization
of the WET and of computing [5], as well as cooperative computation offloading
[10, 11]. However, wireless devices are still suspicious to severe channel
attenuation, which limits the performance of WP-MEC systems. To resolve this
issue, we propose to deploy the emerging intelligent reflecting surfaces (IRS)
[19, 20, 21] in the vicinity of devices, for providing an additional
transmission link both for WET and for computation offloading. Then, the power
consumption can be beneficially reduced both for the downlink and for the
uplink. To elaborate, an IRS comprises of an IRS controller and a large number
of low-cost passive reflection elements. Regulated by the IRS controller, each
IRS reflection element may adapt both the amplitude and the phase of the
incident signals reflected, for collaboratively modifying the signal
propagation environment. The gain attained by IRSs is based on the combination
of so-called the virtual array gain and the reflection-enabled beamforming
gain [19]. More explicitly, the virtual array gain is achieved by combining
the direct and IRS-reflected links, while the reflection-enabled beamforming
gain is realized by proactively adjusting the reflection coefficients of the
IRS elements. By combining these two types of gain together, IRSs are capable
of reducing the power required both for WET and for computation offloading,
thus improving the energy efficiency of WP-MEC systems. In this treatise, we
aim for providing a holistic scheme to minimize the energy consumption of WP-
MEC systems, relying on IRSs.
### I-B Related Works
The current state-of-the-art contributions are reviewed from the perspectives
of WP-MEC and of IRS-aided networks, as follows.
#### I-B1 Wireless Powered Mobile Edge Computing
This topic has attracted an increasing amount of research attention [4, 5, 6,
7, 8, 9, 10, 11, 12, 13]. Specifically, You _et al._ firstly proposed the WP-
MEC framework [4], where the probability of successfully computing was
maximized subject to the constraints both on energy harvesting and on latency.
The single-user system considered in this first trial limits its application
in large-scale scenarios. For eliminating this shortage, an energy-
minimization algorithm was proposed for the multi-user scenario [5], where the
devices’ computation offloading was realized by the time division multiple
access (TDMA) technique. Following this, Bi and Zhang maximized the weighted
sum computation rate in a similar TDMA system [6], while an orthogonal
frequency division multiple access (OFDMA) based multi-user WP-MEC system was
investigated in [7]. A holistic online optimization algorithm was proposed for
the WP-MEC in industrial IoT scenarios [8]. In the aforementioned works, the
associated optimization is commonly realized with the aid of the alternative
optimization (AO) method, because the pertinent optimization problems are
usually not jointly convex. This inevitably imposes a delay on decision
making. To mimic this issue, Huang _et al._ proposed a deep reinforcement
learning based algorithm for maximizing the computation rate of WP-MEC systems
[9], which may replace the aforementioned complicated optimization by a pre-
trained look-up table. Furthermore, as for the system where both near and far
devices have to be served, the energy consumption at the HAP has to be vastly
increased, because the farther device harvests less energy while a higher
transmit power is required for its computation offloading. Aimed for releasing
this so-called “doubly near-far” issue, the technique of user cooperation was
revisited [10, 11]. At the time of writing, the WET and computation offloading
in WP-MEC systems in the face of hostile communication environments has not
been well addressed. Against this background, we aim for tackling this issue
by invoking IRSs. Let us now continue by reviewing the relevant research
contributions on IRSs as follows.
#### I-B2 IRS-Aided Networks
In order to exploit the potential of IRSs, an upsurging number of research
efforts have been devoted in its channel modeling [22, 23], analyzing the
impact of limited-resolution phase shifts [24, 25], channel estimation [26,
27] as well as IRS reflection coefficient designs [28, 29, 30, 31]. Inspired
by these impressive research contributions, the beneficial role of IRSs was
evaluated in various application scenarios [32, 33, 34, 35, 36, 37, 38].
Specifically, an IRS was employed in multi-cell communications systems for
mitigate the severe inter-cell interference [32], where an IRS comprising of
$100$ reflection elements was shown to be capable of doubling the sum rate of
the multi-cell system. Yang _et al._ investigated an IRS-enhanced OFDMA system
[33], whose common rate was improved from around $2.75\leavevmode\nobreak\
\rm{bps/Hz}$ to $4.4\leavevmode\nobreak\ \rm{bps/Hz}$, with the aid of a
$50$-element IRS. Apart from assisting the aforementioned throughput
maximization in the conventional communications scenario, a sophisticated
design of IRSs may also eminently upgrade the performance of diverse emerging
wireless networks, e.g. protecting data transmission security [34, 35],
assisting simultaneous wireless information and power transfer (SWIPT) [36],
enhancing the user cooperation in wireless powered communications networks
[37], as well as reducing the latency in IRS-aided MEC systems [38]. These
impressive research contributions inspire us to exploit the beneficial role of
IRSs in this momentous WP-MEC scenario.
### I-C Novelty and Contributions
In this paper, an innovative IRS-aided WP-MEC framework is proposed, where we
consider orthogonal frequency-division multiplexing (OFDM) systems for its WET
and devices’ computation offloading. Under this framework, a joint WET and
computing design is conceived for minimizing its energy consumption, by
optimizing the power allocation of the WET signals over OFDM sub-bands, the
local computing frequencies of wireless devices, both the sub-band-device
association and the power allocation used for computation offloading, as well
as the pertinent IRS reflection coefficient design. Let us now detail our
contributions as follows.
* •
_Energy minimization problem formulation for the new IRS-aided WP-MEC design:_
In order to reduce the energy consumption of WP-MEC systems, we employ an IRS
in WP-MEC systems and formulate a pertinent energy minimization problem. Owing
to the coupling effects between the designs of WET and of computing, it is
difficult to find its globally optimal solution. Alternatively, we provide an
alternative optimization (AO) based solution to approach a locally optimal
solution, by iteratively optimizing settings of WET and of computing.
* •
_WET design:_ The WET setting is realized by alternatively optimizing the
power allocation of energy-carrying signals over OFDM sub-bands and the IRS
reflection coefficients. Specifically, given a set of fixed IRS reflection
coefficients, the power allocation problem can be simplified to be a linear
programming problem, which can be efficiently solved by the existing
optimization software. Given a fixed power allocation, the IRS reflection
coefficient design becomes a feasibility-check problem, the solution of which
is incapable of ensuring a rapid convergence. To tackle this issue, we
reformulate the problem by introducing a number of auxiliary variables, and
provide a locally optimal design of IRS reflection coefficients, with the aid
of several steps of mathematical manipulations and of the successive convex
approximation (SCA) method.
* •
_Computing design:_ The settings at the computing phase are specified by
alternatively optimizing the joint sub-band-device association for and the
power allocation for devices’ computation offloading, IRS reflection
coefficients at the computing phase as well as the local computing
frequencies. Specifically, as verified by [39], the duality gap vanishes when
the number of sub-bands exceeds $8$. Hence, we provide a near-optimal solution
for the joint sub-band-device association and power allocation problem,
relying on the Lagrangian duality method. The IRS reflection coefficients are
designed using the similar approach devised for that in the WET phase.
Finally, our analysis reveals that the optimal local computing frequencies can
be obtained by selecting their maximum allowable values.
* •
_Numerical validations:_ Our numerical results validates the benefits of
employing IRSs in WP-MEC systems, and quantify the energy consumption of our
proposed framework in diverse simulation environments, together with two
benchmark schemes.
The rest of the paper is organized as follows. In Section II, we describe the
system model and formulate the pertinent problem. A solution of this problem
is provided in Section III. The numerical results are presented in Section IV.
Finally, our conclusions are drawn in Section V.
Figure 1: An illustration of our IRS-aided WP-MEC system, where $K$ single-
antenna devices are served by a mobile edge computing node via a single-
antenna hybrid access point, with the aid of an $N$-element IRS.
## II System Model and Problem Formulation
As illustrated in Fig. 1, we consider an OFDM-based WP-MEC system, where $K$
single-antenna devices are served by a single-antenna hybrid access point
(HAP) associated with an edge computing node through $M$ equally-divided OFDM
sub-bands. Similar to the assumption in [5, 6, 7], we assume that these
devices do not have any embedded energy supply available, but are equipped
with energy storage devices, e.g. rechargeable batteries or super-capacitors,
for storing the energy harvested from RF signals. As shown in Fig. 2, relying
on the so-called “harvest-then-computing” mechanism [5], the system operates
in a two-phase manner in each time block. Specifically, during the WET phase,
the HAP broadcasts energy-carrying signals to all $K$ devices for replenishing
their batteries, while these $K$ devices process their computing tasks both by
local computing and by computation offloading during the computing phase. We
denote the duration of each time block by $T$, which is chosen to be no larger
than the tolerant latency of MEC applications. The duration of the WET and of
the computing phases are set as $\tau T$ and $(1-\tau)T$, respectively.
Furthermore, to assist the WET and the devices’ computation offloading in this
WP-MEC system, we place an IRS comprising of $N$ reflection elements in the
proximity of devices. The reflection coefficients of these IRS reflection
elements are controlled by an IRS controller in a real-time manner, based on
the optimization results provided by the HAP.
Figure 2: An illustration of the harvest-then-offloading protocol, where $\tau
T$ and $(1-\tau)T$ refer to the duration of the WET and the computing phases,
respectively.
Let us continue by elaborating on the equivalent baseband time-domain channel
as follows. We denote the equivalent baseband time-domain channel of the
direct link between the $k$-th device and the HAP, the equivalent baseband
time-domain channel between the $n$-th IRS element and the HAP, and the
equivalent baseband time-domain channel between the $k$-th device and the
$n$-th IRS element by
$\hat{\boldsymbol{h}}^{d}_{k}\in\mathbb{C}^{L^{d}_{k}\times 1}$,
$\hat{\boldsymbol{g}}_{n}\in\mathbb{C}^{L_{1}\times 1}$ and
$\hat{\boldsymbol{r}}_{k,n}\in\mathbb{C}^{L_{2,k}\times 1}$, respectively,
where $L^{d}_{k}$, $L_{1}$ and $L_{2,k}$ represent the respective number of
delay spread taps. Without loss of generality, we assume that the above
channels remain approximately constant over each time block. Furthermore, the
channels are assumed to be reciprocal for the downlink and the uplink.
As for the IRS, we denote the phase shift vector of and the amplitude response
of the IRS reflection elements by
$\boldsymbol{\theta}=[\theta_{1},\theta_{2},\ldots,\theta_{N}]^{T}$ and
$\boldsymbol{\beta}=[\beta_{1},\beta_{2},\ldots,\beta_{N}]^{T}$, respectively,
where we have $\theta_{n}\in[0,2\pi)$ and $\beta_{n}\in[0,1]$. Then, the
corresponding reflection coefficients of the IRS are given by
$\boldsymbol{\Theta}=[\Theta_{1},\Theta_{2},\ldots,\Theta_{N}]^{T}=[\beta_{1}e^{j\theta_{1}},\beta_{2}e^{j\theta_{2}},\ldots,\beta_{N}e^{j\theta_{N}}]^{T}$,
where $j$ represents the imaginary unit and we have $|\Theta_{n}|\leq 1$ for
$\forall n\in\mathcal{N}$. The baseband equivalent time-domain channel of the
reflection link is the convolution of the device-IRS channel, of the IRS
reflection response, and of the IRS-HAP channel. Specifically, the baseband
equivalent time-domain channel reflected by the $n$-th IRS element is
formulated as
$\hat{\boldsymbol{h}}^{r}_{k,n}=\hat{\boldsymbol{r}}_{k,n}\ast\Theta_{n}\ast\hat{\boldsymbol{g}}_{n}=\Theta_{n}\hat{\boldsymbol{r}}_{k,n}\ast\hat{\boldsymbol{g}}_{n}$.
Here, we have $\hat{\boldsymbol{h}}^{r}_{k,n}\in\mathbb{C}^{L^{r}_{k}\times
1}$ and $L^{r}_{k}=L_{1}+L_{2,k}-1$, which denotes the number of delay spread
taps of the reflection channel. Furthermore, we denote the time-domain zero-
padded concatenated device-IRS-HAP channel between the $k$-th device and the
HAP via the $n$-th IRS element by
$\boldsymbol{v}_{k,n}=[(\hat{\boldsymbol{r}}_{k,n}\ast\hat{\boldsymbol{g}}_{n})^{T},0,\ldots,0]^{T}\in\mathbb{C}^{M\times
1}$. Upon denoting
$\boldsymbol{V}_{k}=[\boldsymbol{v}_{k,1},\ldots,\boldsymbol{v}_{k,N}]\in\mathbb{C}^{M\times
N}$, we formulate the composite device-IRS-HAP channel between the $k$-th
device and the HAP as
$\boldsymbol{h}^{r}_{k}=\boldsymbol{V}_{k}\boldsymbol{\Theta}$. Similarly, we
use
$\boldsymbol{h}^{d}_{k}=[(\hat{\boldsymbol{h}}^{d}_{k})^{T},0,\ldots,0]^{T}\in\mathbb{C}^{M\times
1}$ to represent the zero-padded time-domain channel of the direct device-HAP
link. To this end, we may readily arrive at the superposed channel impulse
response (CIR) for the $k$-th device, formulated as
$\displaystyle\boldsymbol{h}_{k}=\boldsymbol{h}^{d}_{k}+\boldsymbol{h}^{r}_{k}=\boldsymbol{h}^{d}_{k}+\boldsymbol{V}_{k}\boldsymbol{\Theta},\quad\forall
k\in\mathcal{K},$ (1)
whose number of delay spread taps is given by
$L_{k}=\max(L^{d}_{k},L^{r}_{k})$. We assume that the number of cyclic
prefixes (CP) is no smaller than the maximum number of delay spread taps for
all devices, so that the inter-symbol interference (ISI) can be eliminated.
Upon denoting the $m$-th row of the $M\times M$ discrete Fourier transform
(DFT) matrix $\boldsymbol{F}_{M}$ by $\boldsymbol{f}^{H}_{m}$, we formulate
the channel frequency response (CFR) for the $k$-th device at the $m$-th sub-
band as
$\displaystyle
C_{k,m}(\boldsymbol{\Theta})=\boldsymbol{f}^{H}_{m}\boldsymbol{h}_{k}=\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta},\quad\forall
k\in\mathcal{K},\forall m\in\mathcal{M}.$ (2)
For ease of exposition, we assume that the knowledge of
$\boldsymbol{h}^{d}_{k}$ and of $\boldsymbol{V}_{k}$ is perfectly known at the
HAP. Naturally, this assumption is idealistic. Hence, the algorithm developed
in this paper can be deemed to represent the best-case bound for the energy
performance of realistic scenarios. Since different types of signals are
transmitted in the WET and computing phases, the reflection coefficients of
the IRS require separate designs in these two phases. The models of the WET
and of computing phases are detailed as follows.
### II-A Model of the Wireless Energy Transfer Phase
It is assumed that the capacity of devices’ battery is large enough so that
all the harvest energy can be saved without energy overflow. Let us use
$\boldsymbol{\Theta}^{E}=\big{\\{}\Theta^{E}_{1},\Theta^{E}_{2},\ldots,\Theta^{E}_{N}\big{\\}}$
to represent the IRS reflection-coefficient vector during the WET phase, where
we have $|\Theta^{E}_{n}|\leq 1$ for $\forall n\in\mathcal{N}$. Then, the
composite channel of the $m$-th sub-band for the $k$-th device during the WET
phase $C_{k,m}(\boldsymbol{\Theta}^{E})$ can be obtained by (2). As a benefit
of the broadcasting nature of WET, each device can harvest the energy from the
RF signals transmitted over all $M$ sub-bands. Hence, upon denoting the power
allocation for the energy-carrying RF signals at the $M$ sub-bands during the
WET phase by $\boldsymbol{p}^{E}=[p^{E}_{1},p^{E}_{2},\ldots,p^{E}_{M}]$, we
are readily to formulate the energy harvested by the $k$-th device as [5]
$\displaystyle
E_{k}(\tau,\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})=\sum^{M}_{m=1}\eta\tau
Tp^{E}_{m}\big{|}C_{k,m}(\boldsymbol{\Theta}^{E})\big{|}^{2},$ (3)
where $\eta\in[0,1]$ denotes the efficiency of the energy harvesting at the
wireless devices.
### II-B Model of the Computing Phase
We consider the data-partitioning based application [40], where a fraction of
the data can be processed locally, while the other part can be offloaded to
the edge node. For a specific time block, we use $L_{k}$ and $\ell_{k}$ to
denote the number of bits to be processed by the $k$-th device and its
computation offloading volume in terms of the number of bits, respectively.
The models of local computing, of computation offloading and of edge computing
are detailed as follows.
#### II-B1 Local Computing
We use $f_{k}$ and $c_{k}$ to represent its computing capability in terms of
the number of central processing unit (CPU) cycles per second and the number
of CPU cycles required to process a single bit, for the $k$-th device,
respectively. The number of bits processed by local computing is readily
calculated as $(1-\tau)Tf_{k}/c_{k}$, and the number of bits to be offloaded
is given by $\ell_{k}=L_{k}-(1-\tau)Tf_{k}/c_{k}$. Furthermore, we assume that
$f_{k}$ is controlled in the range of $[0,f_{max}]$ using the dynamic voltage
scaling model [40]. Upon denoting the computation energy efficiency
coefficient of the processor’s chip by $\kappa$, we formulate the power
consumption of the local computing mode as $\kappa f_{k}^{2}$ for the $k$-th
device [40].
#### II-B2 Computation offloading
In order to mitigate the co-channel interference, the devices’ computation
offloading is realized relying on the orthogonal frequency-division multiple
access (OFDMA) scheme. In this case, each sub-band is allowed to be used by at
most a single device. We use the binary vector
$\boldsymbol{\alpha}_{k}=[\alpha_{k,1},\alpha_{k,2},\ldots,\alpha_{k,M}]^{T}$
and the non-negative vector
$\boldsymbol{p}^{I}_{k}=[p^{I}_{k,1},p^{I}_{k,2},\ldots,p^{I}_{k,M}]^{T}$ to
represent the association between the sub-band and devices as well as the
power allocation of the $k$-th device to the $M$ sub-bands, respectively,
where we have
$\displaystyle\alpha_{k,m}$ $\displaystyle=\begin{cases}0,&\quad\text{if
}p^{I}_{k,m}=0,\\\ 1,&\quad\text{if }p^{I}_{k,m}>0.\end{cases}$ (4)
The power consumption of computation offloading is given by
$\sum^{M}_{m=1}\alpha_{k,m}(p_{k,m}+p_{c})$, where $p_{c}$ represents a
constant circuit power (caused by the digital-to-analog converter, filter,
etc.) [5]. Let us denote the IRS reflection-coefficient vector during the
computation offloading by
$\boldsymbol{\Theta}^{I}=\big{\\{}\Theta^{I}_{1},\Theta^{I}_{2},\ldots,\Theta^{I}_{N}\big{\\}}$,
where $|\Theta^{I}_{n}|\leq 1$ for $\forall n\in\mathcal{N}$. Then, the
composite channel of the $k$-th device at the $m$-th sub-band denoted by
$C_{k,m}(\boldsymbol{\Theta}^{I})$ can be obtained by (2). The corresponding
achievable rate of computation offloading is formulated below for the $k$-th
device
$\displaystyle
R_{k}(\boldsymbol{\alpha}_{k},\boldsymbol{p}^{I}_{k},\boldsymbol{\Theta}^{I})=\sum^{M}_{m=1}\alpha_{k,m}B\log_{2}\bigg{(}1+\frac{p_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{)},$
(5)
where $\Gamma$ is the gap between the channel capacity and a specific
modulation and coding scheme. Furthermore, in order to offload all the
$\ell_{k}$ bits within the duration of the computation phase, the achievable
offloading rate has to obey
$R_{k}(\tau,\boldsymbol{\alpha}_{k},\boldsymbol{p}^{I}_{k},\boldsymbol{\Theta}^{I})\geq\frac{\ell_{k}}{(1-\tau)T}$.
#### II-B3 Edge Computing
Invoking the simplified linear model [5], we formulate the energy consumption
at the edge node as
$\vartheta\sum^{K}_{k=1}\ell_{k}=\vartheta\sum^{K}_{k=1}\big{[}L_{k}-(1-\tau)Tf_{k}/c_{k}\big{]}$.
Furthermore, the latency imposed by edge computing comprises of two parts. The
first part is caused by processing the computational tasks. Given that edge
nodes typically possess high computational capabilities, this part can be
negligible. The second part is induced by sending back the computational
results, which are usually of a small volume. Hence, the duration of sending
the feedback is also negligible. As such, we neglect the latency induced by
edge computing.
### II-C Problem Formulation
In this paper, we aim for minimizing the total energy consumption of the OFDM-
based WP-MEC system, by optimizing the time allocation for WET and computing
phases $\tau$, both the power allocation $\boldsymbol{p}^{E}$ and the IRS
reflection coefficients $\boldsymbol{\Theta}^{E}$ at the WET phase, and the
local CPU frequency at devices $\boldsymbol{f}$, the sub-band-device
association $\\{\boldsymbol{\alpha}_{k}\\}$ and the power allocation
$\\{\boldsymbol{p}_{k}\\}$ as well as the IRS reflection coefficients
$\boldsymbol{\Theta}^{I}$ at the computing phase, subject to the energy
constraint imposed by energy harvesting, the latency requirement of
computation offloading and the sub-band-device association constraint in OFDMA
systems as well as the constraint on IRS reflection coefficients. Since the
batteries of all the wireless devices are replenished by the HAP, their energy
consumption is covered by the energy consumption at the HAP during the WET
phase. Hence, the total energy consumption of the system is formulated as the
summation of the energy consumption both of the WET at the HAP and of the edge
computing, i.e. $\tau
T\sum^{M}_{m=1}p^{E}_{m}+\vartheta\sum^{K}_{k=1}\big{[}L_{k}-(1-\tau)Tf_{k}/c_{k}\big{]}$.
To this end, the energy minimization problem is readily formulated for our
OFDM-based WP-MEC system as
$\displaystyle\mathcal{P}0\mathrel{\mathop{\mathchar
58\relax}}\mathop{\min}\limits_{\begin{subarray}{c}\tau,\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E},\boldsymbol{f},\\\
\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}\end{subarray}}\tau
T\sum^{M}_{m=1}p^{E}_{m}+\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\tau)Tf_{k}}{c_{k}}\bigg{]}$
$\displaystyle\text{s.t.}\quad 0<\tau<1,$ (6a) $\displaystyle\quad\quad
p^{E}_{m}\geq 0,\quad\forall m\in\mathcal{M},$ (6b)
$\displaystyle\quad\quad|\Theta^{E}_{n}|\leq 1,\quad\forall n\in\mathcal{N},$
(6c) $\displaystyle\quad\quad 0\leq f_{k}\leq f_{max},\quad\forall
k\in\mathcal{K},$ (6d)
$\displaystyle\quad\quad\alpha_{k,m}\in\\{0,1\\},\quad\forall
k\in\mathcal{K},\quad\forall m\in\mathcal{M},$ (6e)
$\displaystyle\quad\quad\sum^{K}_{k=1}\alpha_{k,m}\leq 1,\quad\forall
m\in\mathcal{M},$ (6f) $\displaystyle\quad\quad p^{I}_{k,m}\geq 0,\quad\forall
k\in\mathcal{K},\quad\forall m\in\mathcal{M},$ (6g)
$\displaystyle\quad\quad|\Theta^{I}_{n}|\leq 1,\quad\forall n\in\mathcal{N},$
(6h) $\displaystyle\quad\quad(1-\tau)T\bigg{[}\kappa
f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\bigg{]}\leq
E_{k}(\tau,\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E}),\quad\forall
k\in\mathcal{K},$ (6i)
$\displaystyle\quad\quad(1-\tau)TR_{k}(\boldsymbol{\alpha}_{k},\boldsymbol{p}^{I}_{k},\boldsymbol{\Theta}^{I})\geq
L_{k}-\frac{(1-\tau)Tf_{k}}{c_{k}},\quad\forall k\in\mathcal{K}.$ (6j)
Constraint (6a) restricts the time allocation for the WET and for the
computing phases. Constraint (6b) and (6c) represent the range of the power
allocation and the IRS reflection coefficients at the WET phase, respectively.
Constraint (6d) gives the range of tunable local computing frequencies.
Constraint (6e) and (6f) detail the requirement of sub-band-device association
in OFDMA systems. Constraint (6g) and (6h) restrict the range of the power
allocation and the IRS reflection coefficient at the computing phase,
respectively. Constraint (6i) indicates that the sum energy consumption of
local computing and of computation offloading should not exceed the harvested
energy for each device. Finally, Constraint (6j) implies that the
communication link between the $k$-th device and the HAP is capable of
offloading the corresponding computational tasks within the duration of the
computing phase.
## III Joint Optimization of the Settings in the WET and the Computing Phases
In this section, we propose to solve Problem $\mathcal{P}0$ in a two-step
procedure. Firstly, given a fixed $\hat{\tau}\in(0,1)$, Problem $\mathcal{P}0$
can be simplified as follows
$\displaystyle\mathcal{P}1\mathrel{\mathop{\mathchar
58\relax}}\mathop{\min}\limits_{\begin{subarray}{c}\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E},\boldsymbol{f},\\\
\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}\end{subarray}}\hat{\tau}T\sum^{M}_{m=1}p^{E}_{m}+\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\bigg{]}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_4},\eqref{eqn:P1_constraint_5},\eqref{eqn:P1_constraint_10},\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P1_constraint_9},\eqref{eqn:P1_constraint_1},\eqref{eqn:P1_constraint_2}$
(7a)
In the second step, we aim for finding the optimal $\hat{\tau}$ that is
capable of minimizing the OF of Problem $\mathcal{P}0$ using the one-
dimensional search method. In the rest of this section, we focus on solving
Problem $\mathcal{P}1$. At a glance of Problem $\mathcal{P}1$, the
optimization variables $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$ and
$\\{\boldsymbol{p}^{I}_{k}\\}$ are coupled with $\boldsymbol{p}^{E}$ and
$\boldsymbol{\Theta}^{E}$ in Constraint (6i), which makes the problem
difficult to solve. To tackle this issue, the AO technique is invoked.
Specifically, upon initializing the setting of the computing phase, we may
optimize the design of the WET phase while fixing the time allocation and the
computing phase settings. Then, the computing phase settings could be
optimized while fixing the time allocation and the design of the WET. A
suboptimal solution can be obtained by iteratively optimizing the designs of
the WET and of the computing phases. Let us detail the initialization as well
as the designs of the WET and of the computing phases, as follows.
### III-A Initialization of the Time Allocation and the Computing Phase
In order to ensure our WET design to be a feasible solution of Problem
$\mathcal{P}1$, the initial settings of the computing phase denoted by
$\boldsymbol{f}^{(0)},\big{\\{}\boldsymbol{\alpha}_{k}^{(0)}\big{\\}},\big{\\{}{\boldsymbol{p}^{I}_{k}}^{(0)}\big{\\}},{\boldsymbol{\Theta}^{I}}^{(0)}$
should satisfy Constraint (6d), (6e), (6f), (6g), (6h) and (6j). Without any
loss of generality, their initialization is set as follows.
* •
Local computing frequency $\boldsymbol{f}^{(0)}$: Obeying the uniform
distribution, each element of $\boldsymbol{f}^{(0)}$ is randomly set in the
range of $[0,f_{max}]$.
* •
IRS reflection coefficient at the computing phase
${\boldsymbol{\Theta}^{I}}^{(0)}$: Obeying the uniform distribution, the
amplitude response ${\beta^{I}_{n}}^{(0)}$ and the phase shift
${\theta^{I}_{n}}^{(0)}$ are randomly set in the range of $[0,1]$ and of
$[0,2\pi)$, respectively. Then,
${\boldsymbol{\Theta}^{I}}^{(0)}=\\{{\beta^{I}_{1}}^{(0)}e^{j{\theta^{I}_{1}}^{(0)}},\ldots,{\beta^{I}_{N}}^{(0)}e^{j{\theta^{I}_{N}}^{(0)}}\\}$
can be obtained.
* •
Sub-band-device association at the computing phase
$\big{\\{}\boldsymbol{\alpha}_{k}^{(0)}\big{\\}}$: We reserve a single sub-
band for the devices associated with the index ranging from $k=1$ to $k=K$,
sequentially. Specific to the $k$-th device, we use $k_{m}^{(0)}$ to denote
the sub-band having the maximum
$\big{|}C_{k,m}\big{(}{\boldsymbol{\Theta}^{I}}^{(0)}\big{)}\big{|}^{2}$ over
the unassigned sub-bands, and assign this sub-band to the $k$-th device.
* •
Power allocation at the computing phase
$\big{\\{}{\boldsymbol{p}^{I}_{k}}^{(0)}\big{\\}}$: For the $k$-th device, its
power allocation at the computing phase should satisfy Constraint (6j). For
minimizing the energy consumption, we assume the equivalence of two sides in
Constraint (6j). Then, its initial power allocation is given by
${p^{I^{(0)}}_{k,k_{m}^{(0)}}}=\frac{\Gamma\sigma^{2}\Big{[}2^{\frac{L_{k}}{(1-\hat{\tau})TB}-\frac{f_{k}}{c_{k}B}}-1\Big{]}}{\big{|}c_{k,k^{(0)}_{m}}\big{(}{\boldsymbol{\Theta}^{I}}^{(0)}\big{)}\big{|}^{2}}$.
For those sub-bands associated with the index $m\neq k^{(0)}_{m}$, we set
${p^{I^{(0)}}_{k,m}}=0$.
### III-B Design of the WET Phase While Fixing the Time Allocation and
Computing Settings
Given a fixed time allocation $\hat{\tau}$ and the settings of the computing
phase $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, we may simplify
Problem $\mathcal{P}1$ as
$\displaystyle\mathcal{P}2\mathrel{\mathop{\mathchar
58\relax}}\mathop{\min}\limits_{\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E}}\hat{\tau}T\sum^{M}_{m=1}p^{E}_{m}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_4},\eqref{eqn:P1_constraint_5},$
$\displaystyle\quad\quad(1-\hat{\tau})T\bigg{[}\kappa
f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\bigg{]}\leq\sum^{M}_{m=1}\eta\hat{\tau}Tp^{E}_{m}\big{|}C_{k,m}(\boldsymbol{\Theta}^{E})\big{|}^{2},\quad\forall
k\in\mathcal{K}.$ (8a)
Since Constraint (8a) is not jointly convex regarding $\boldsymbol{p}^{E}$ and
$\boldsymbol{\Theta}^{E}$, we optimize one of these two variables while fixing
the other in an iterative manner, relying on the AO technique, as follows.
#### III-B1 Optimizing the Power Allocation of the WET Phase While Fixing the
Settings of the Time Allocation, the Computing Phase and the IRS Reflection
Coefficient at the WET Phase
Given an IRS phase shift design $\boldsymbol{\Theta}^{E}$, Problem
$\mathcal{P}2$ is simplified as
$\displaystyle\mathcal{P}2\text{-}1\mathrel{\mathop{\mathchar
58\relax}}\mathop{\min}\limits_{\boldsymbol{p}^{E}}\hat{\tau}T\sum^{M}_{m=1}p^{E}_{m}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_4},\eqref{eqn:P2_constraint_1}.$
(9a)
It can be observed that Problem $\mathcal{P}2\text{-}1$ is a linear
programming problem, which can be readily solved with the aid of the general
implementation of interior-point methods, e.g. CVX [41]. The complexity is
given by $\sqrt{M+KM}M[M+KM^{3}+M(M+KM^{2})+M^{2}]$ [42], i.e.
$\mathcal{O}(K^{1.5}M^{4.5})$.
#### III-B2 Optimizing the IRS Reflection Coefficient at the WET Phase While
Fixing the Settings of the Time Allocation, the Computing Phase and the power
Allocation at the WET Phase
Given a power allocation at the WET phase $\boldsymbol{p}^{E}$, Problem
$\mathcal{P}2$ becomes a feasibility-check problem, i.e.
$\displaystyle\mathcal{P}2\text{-}2\mathrel{\mathop{\mathchar
58\relax}}\text{Find }\boldsymbol{\Theta}^{E}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},\eqref{eqn:P2_constraint_1}.$
(10a)
As verified in [28], if one of the sub-problems is a feasibility-check
problem, the iterative algorithm relying on the AO technique has a slow
convergence. Specific to the problem considered, the operation of Find in
Problem $\mathcal{P}2\text{-}2$ cannot guarantee the OF of Problem
$\mathcal{P}2$ to be further reduced in each iteration. To address this issue,
we reformulate Problem $\mathcal{P}2\text{-}2$ as follows, by introducing a
set of auxiliary variables $\\{\xi_{k}\\}$
$\displaystyle\mathcal{P}2\text{-}2^{\prime}\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{E},\\{\xi_{k}\\}}\sum^{K}_{k=1}\xi_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},$
$\displaystyle\quad\quad\xi_{k}+\kappa
f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\leq\frac{\sum^{M}_{m=1}\eta\hat{\tau}p^{E}_{m}\big{|}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{E}\big{|}^{2}}{1-\hat{\tau}},\quad\forall
k\in\mathcal{K},$ (11a) $\displaystyle\quad\quad\xi_{k}\geq 0,\quad\forall
k\in\mathcal{K}.$ (11b)
It is readily seen that the energy harvested by the wireless devices may
increase after solving Problem $\mathcal{P}2\text{-}2^{\prime}$, which implies
that the channel gain of the reflection link is enhanced. Then, a reduced
power of energy signals can be guaranteed, when we turn back to solve Problem
$\mathcal{P}2\text{-}1$. As such, a faster convergence can be obtained.
However, at a glance of Problem $\mathcal{P}2\text{-}2^{\prime}$, Constraint
(11a) is still non-convex regarding $\boldsymbol{\Theta}^{E}$. To tackle this
issue, we manipulate the optimization problem in light of [33] as follows.
Firstly, we transform Problem $\mathcal{P}2\text{-}2^{\prime}$ to its
equivalent problem below, by introducing a set of auxiliary variables
$\boldsymbol{y}^{E}$, $\boldsymbol{a}^{E}$ and $\boldsymbol{b}^{E}$
$\displaystyle\mathcal{P}2\text{-}2^{\prime}E1\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{E},\\{\xi_{k}\\},\boldsymbol{y}^{E},\boldsymbol{a}^{E},\boldsymbol{b}^{E}}\sum^{K}_{k=1}\xi_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},\eqref{eqn:P2_2_constraint_11},$
$\displaystyle\quad\quad\xi_{k}+\kappa
f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\leq\frac{\sum^{M}_{m=1}\eta\hat{\tau}p^{E}_{m}y^{E}_{k,m}}{1-\hat{\tau}},\quad\forall
k\in\mathcal{K},$ (12a) $\displaystyle\quad\quad
a^{E}_{k,m}=\Re\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{E}\big{\\}},\quad
k\in\mathcal{K},\quad m\in\mathcal{M},$ (12b) $\displaystyle\quad\quad
b^{E}_{k,m}=\Im\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{E}\big{\\}},\quad
k\in\mathcal{K},\quad m\in\mathcal{M},$ (12c) $\displaystyle\quad\quad
y^{E}_{k,m}\leq(a^{E}_{k,m})^{2}+(b^{E}_{k,m})^{2},\quad k\in\mathcal{K},\quad
m\in\mathcal{M},$ (12d)
where $\Re\\{\bullet\\}$ and $\Im\\{\bullet\\}$ represent the real and
imaginary part of $\bullet$, respectively. Following this, the successive
convex approximation (SCA) method [43] is applied for tackling the non-convex
constraint (12d). Specifically, the approximation function is constructed as
follows. The right hand side of (12d) is lower-bounded by its first-order
approximation at $(\tilde{a}^{E}_{k,m},\tilde{b}^{E}_{k,m})$, i.e.
$(a^{E}_{k,m})^{2}+(b^{E}_{k,m})^{2}\geq\tilde{a}^{E}_{k,m}(2a^{E}_{k,m}-\tilde{a}^{E}_{k,m})+\tilde{b}^{E}_{k,m}(2b^{E}_{k,m}-\tilde{b}^{E}_{k,m})$,
where the equality holds only when we have $\tilde{a}^{E}_{k,m}=a^{E}_{k,m}$
and $\tilde{b}^{E}_{k,m}=b^{E}_{k,m}$. Now we consider the following
optimization problem
$\displaystyle\mathcal{P}2\text{-}2^{\prime}E2\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{E},\\{\xi_{k}\\},\boldsymbol{y}^{E},\boldsymbol{a}^{E},\boldsymbol{b}^{E}}\sum^{K}_{k=1}\xi_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},\eqref{eqn:P2_2_constraint_11},\eqref{eqn:P2_2'E_constraint_1},\eqref{eqn:P2_2'E_constraint_6},\eqref{eqn:P2_2'E_constraint_7},$
$\displaystyle\quad\quad
y^{E}_{k,m}=\tilde{a}^{E}_{k,m}(2a^{E}_{k,m}-\tilde{a}^{E}_{k,m})+\tilde{b}^{E}_{k,m}(2b^{E}_{k,m}-\tilde{b}^{E}_{k,m}),\quad
k\in\mathcal{K},\quad m\in\mathcal{M}.$ (13a)
Both the OF and contraints in Problem $\mathcal{P}2\text{-}2^{\prime}E2$ are
affine. Hence, Problem $\mathcal{P}2\text{-}2^{\prime}E2$ is a convex
optimization problem, which can be solved by the implementation of interior-
point methods, e.g. CVX [41]. Then, a locally optimal solution of
$\mathcal{P}2\text{-}2^{\prime}$ can be approached by successively updating
$\tilde{a}^{E}_{k,m}$ and $\tilde{b}^{E}_{k,m}$ based on the optimal solution
of Problem $\mathcal{P}2\text{-}2^{\prime}E2$, whose procedure is detailed in
Algorithm 1. The computation complexity of the SCA method is analyzed as
follows. Problem $\mathcal{P}2\text{-}2^{\prime}E2$ involves $2KM$ linear
equality constraints (equivalently $4KM$ inequality constraints) of size
$2N+1$, $K$ linear inequality constraints of size $M+1$, $KM$ linear
inequality constraints of size $3$, $K$ linear inequality constraints of size
$1$, $N$ second-order cone inequality constraints of size $2$. Hence, the
total complexity of Algorithm 1 is given by
$\ln(1/\epsilon)\sqrt{4KM(2N+1)+K(M+1)+3KM+K+2N}(2N+3M+K)\\{4KM(2N+1)^{3}+K(M+1)^{3}+27KM+K+(2N+3M+K)[4KM(2N+1)^{2}+K(M+1)^{2}+9KM+K]+4N+(2N+3M+K)^{2}\\}$
[42], i.e.
$\ln(1/\epsilon)\mathcal{O}(K^{1.5}M^{1.5}N^{4.5}+K^{1.5}M^{2.5}N^{3.5}+K^{1.5}M^{2.5}N^{1.5}+K^{2.5}M^{1.5}N^{3.5}+K^{1.5}M^{4.5}+K^{2.5}M^{2.5}N^{2.5}+K^{2.5}M^{3.5}+K^{3.5}M^{1.5}N^{2.5}+K^{3.5}M^{2.5})$.
To this end, we summarize the procedure of solving Problem $\mathcal{P}2$ in
Algorithm 2.
Algorithm 1 SCA approach to design $\boldsymbol{\Theta}^{E}$, given the
settings of the time allocation, the computing phase and the power allocation
at the WET phase
0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$,
$f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$,
$\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{P}^{E}$,
$\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$, $\boldsymbol{\Theta}^{I}$ and
$\tilde{\boldsymbol{\Theta}}^{E}$
0: $\boldsymbol{\Theta}^{E}$ 1\. Initialization
Initialize $t_{1}=0$; $\epsilon_{1}=1$; $\xi_{k}=0,\forall k\in\mathcal{K}$
2\. SCA approach to design $\boldsymbol{\Theta}^{E}$
while $t_{1}<t_{\text{max}}$ $\&\&$ $\epsilon_{1}>\epsilon$ do
$\bullet$ Set
$\tilde{a}^{E}_{k,m}=\Re\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\tilde{\boldsymbol{\Theta}}^{E}\big{\\}}$
and
$\tilde{b}^{E}_{k,m}=\Im\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\tilde{\boldsymbol{\Theta}}^{E}\big{\\}},\forall
k\in\mathcal{K},\forall m\in\mathcal{M}$
$\bullet$ Obtain ${\boldsymbol{\Theta}^{E}}$ and $\\{\xi_{k}\\}$ by solving
Problem $\mathcal{P}2\text{-}2^{\prime}E2$ using CVX
$\bullet$ Set
$\epsilon_{1}=\frac{\big{|}\text{obj}\big{(}{\boldsymbol{\Theta}}^{E}\big{)}-\text{obj}\big{(}\tilde{\boldsymbol{\Theta}}^{E}\big{)}\big{|}}{\big{|}\text{obj}\big{(}\boldsymbol{\Theta}^{E}\big{)}\big{|}}$,
$\tilde{\boldsymbol{\Theta}}^{E}\leftarrow\boldsymbol{\Theta}^{E}$,
$t_{1}\leftarrow t_{1}+1$
end while3. Output optimal ${\boldsymbol{\Theta}^{E}}^{*}$
${\boldsymbol{\Theta}^{E}}^{*}\leftarrow\tilde{\boldsymbol{\Theta}}^{E}$
Algorithm 2 Alternative optimization of $\boldsymbol{p}^{E}$ and
$\boldsymbol{\Theta}^{E}$, given the settings of the time allocation and the
computing phase
0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$,
$f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$,
$\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{f}$,
$\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$,
$\boldsymbol{\Theta}^{I}$ and $\tilde{\boldsymbol{\Theta}}^{E}$
0: $\boldsymbol{P}^{E}$ and $\boldsymbol{\Theta}^{E}$ 1\. Initialization
$\bullet$ Initialize $t_{2}=0$; $\epsilon_{2}=1$;
${\boldsymbol{\Theta}^{E}}^{(0)}=\tilde{\boldsymbol{\Theta}}^{E}$
$\bullet$ Given ${\boldsymbol{\Theta}^{E}}^{(0)}$, find
${\boldsymbol{P}^{E}}^{(0)}$ by solving Problem $\mathcal{P}2\text{-}1$ via
CVX 2\. Alternative optimization of $\boldsymbol{P}^{E}$ and
$\boldsymbol{\Theta}^{E}$
while $t_{2}<t_{\text{max}}$ $\&\&$ $\epsilon_{2}>\epsilon$ do
$\bullet$ Given ${\boldsymbol{P}^{E}}^{(t_{2})}$ and
$\tilde{\boldsymbol{\Theta}}^{E}={\boldsymbol{\Theta}^{E}}^{(t_{2})}$, find
${\boldsymbol{\Theta}^{E}}^{(t_{2}+1)}$ by solving Problem
$\mathcal{P}2\text{-}2^{\prime}E1$ using Algorithm 1
$\bullet$ Given ${\boldsymbol{\Theta}^{E}}^{(t_{2}+1)}$, find
${\boldsymbol{P}^{E}}^{(t_{2}+1)}$ by solving Problem $\mathcal{P}2\text{-}1$
via CVX
$\bullet$ Set
$\epsilon_{2}=\frac{\big{|}\text{obj}\big{(}{\boldsymbol{p}^{E}}^{(t_{2}+1)},{{\boldsymbol{\Theta}}^{E}}^{(t_{2}+1)}\big{)}-\text{obj}\big{(}{\boldsymbol{p}^{E}}^{(t_{2})},{{\boldsymbol{\Theta}}^{E}}^{(t_{2})}\big{)}\big{|}}{\big{|}\text{obj}\big{(}{\boldsymbol{p}^{E}}^{(t_{2}+1)},{{\boldsymbol{\Theta}}^{E}}^{(t_{2}+1)}\big{)}\big{|}}$,
$t_{2}\leftarrow t_{2}+1$
end while3. Output optimal ${\boldsymbol{P}^{E}}^{*}$ and
${\boldsymbol{\Theta}^{E}}^{*}$
${\boldsymbol{\Theta}^{E}}^{*}\leftarrow{\boldsymbol{\Theta}^{E}}^{(t_{2})}$
and ${\boldsymbol{P}^{E}}^{*}\leftarrow{\boldsymbol{P}^{E}}^{(t_{2})}$
### III-C Design of the Computing Phase While Fixing the Time Allocation and
WET Settings
In this subsection, we aim for optimizing the design of the computing phase,
while fixing the time allocation $\hat{\tau}$ and the WET settings
$\boldsymbol{p}^{E}$ and $\boldsymbol{\Theta}^{E}$. In this case, we simplify
Problem $\mathcal{P}1$ as
$\displaystyle\mathcal{P}3\mathrel{\mathop{\mathchar
58\relax}}\mathop{\min}\limits_{\boldsymbol{f},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}}\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\bigg{]}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_10},\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P1_constraint_9},\eqref{eqn:P1_constraint_1},$
$\displaystyle\quad\quad\sum^{m}_{m=1}\alpha_{k,m}B\log_{2}\Bigg{[}1+\frac{p_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\Bigg{]}\geq\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T},\quad\forall
k\in\mathcal{K}.$ (14a)
Constraint (14a) is not jointly convex regarding
$\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and
$\boldsymbol{\Theta}^{I}$. Hence, it is difficult to find its globally optimal
solution. Alternatively, its suboptimal solution is provided by iteratively
optimizing the $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, again relying on
the AO technique, as follows.
#### III-C1 Alternative Optimization of the Sub-Band-Device Association and
the Power Allocation as well as the IRS Reflection Coefficient at the
Computing Phase
Given a fixed local CPU frequency setting $\boldsymbol{f}$, the OF of Problem
$\mathcal{P}3$ becomes deterministic. In other words, the optimization of
$\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and
$\boldsymbol{\Theta}^{I}$ seems not contributing to reducing the OF. However,
this is not always true, because if a larger feasible set of $\boldsymbol{f}$
can be obtained by optimizing $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, a reduced OF may
be achieved when we turn back to optimize $\boldsymbol{f}$. Based on this
observation, we formulate the problem below, by introducing a set of auxiliary
variables $\\{\zeta_{k}\\}$
$\displaystyle\mathcal{P}3\text{-}1\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\\{\zeta_{k}\\},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}}\sum^{K}_{k=1}\zeta_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_constraint_2}$
$\displaystyle\quad\quad\zeta_{k}\geq 0,\quad\forall k\in\mathcal{K},$ (15a)
$\displaystyle\quad\quad(1-\hat{\tau})T\bigg{[}\kappa
f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})+\zeta_{k}\bigg{]}\leq\sum^{M}_{m=1}\eta\hat{\tau}Tp^{E}_{m}\big{|}C_{k,m}(\boldsymbol{\Theta}^{E})\big{|}^{2},\quad\forall
k\in\mathcal{K}.$ (15b)
As specified in (15a), the auxiliary variables $\\{\zeta_{k}\\}$ are non-
negative, and thus a non-smaller set of $\boldsymbol{f}$ may be obtained after
solving Problem $\mathcal{P}3\text{-}1$. Given that Constraint (14a) is not
jointly convex regarding $\\{\boldsymbol{p}_{k}\\}$ and
$\boldsymbol{\Theta}^{I}$, we optimize $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ in two steps
iteratively.
In the first step, we optimize $\\{\zeta_{k}\\},\\{\boldsymbol{\alpha}_{k}\\}$
and $\\{\boldsymbol{p}^{I}_{k}\\}$, while fixing the IRS reflection
coefficient $\boldsymbol{\Theta}^{I}$. In this case, Problem
$\mathcal{P}3\text{-}1$ can be simplified as
$\displaystyle\mathcal{P}3\text{-}1a\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\\{\zeta_{k}\\},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\}}\sum^{K}_{k=1}\zeta_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P3_constraint_2},\eqref{eqn:P3_1_constraint_3},\eqref{eqn:P3_1_constraint_2}.$
(16a)
Problem $\mathcal{P}3\text{-}1a$ is a combinatorial optimization problem,
where the binary constraint (6e) is non-convex. The classic solution typically
relies on the convex relaxation method [44], where the binary constraint
imposed on $\\{\boldsymbol{\alpha}_{k}\\}$ is relaxed into a convex constraint
by introducing time-sharing variables. However, the relaxed problem is
different from the original problem, which might lead to a specific error. To
address this issue, a near-optimal solution based on the Lagrangian duality
was proposed [39], where it is verified that the duality gap vanishes in the
system equipped with more than $8$ sub-bands. Hence, in this paper, the
Lagrangian duality method [45] is invoked for solving Problem
$\mathcal{P}3\text{-}1a$. Specifically, denoting the non-negative Lagrange
multiplier vectors by
$\boldsymbol{\lambda}=[\lambda_{1},\lambda_{2},\ldots,\lambda_{K}]^{T}$ and
$\boldsymbol{\mu}=[\mu_{1},\mu_{2},\ldots,\mu_{K}]^{T}$, we formulate the
Lagrangian function of Problem $\mathcal{P}3\text{-}1a$ over the domain
$\mathcal{D}$ as
$\displaystyle\mathcal{L}\big{(}\\{\zeta_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\lambda},\boldsymbol{\mu}\big{)}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\zeta_{k}-\sum_{k=1}^{K}\lambda_{k}\bigg{[}\kappa
f_{k}^{2}+\sum^{M}_{m=1}(p^{I}_{k,m}+p_{c})+\zeta_{k}-\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}\bigg{]}$
(17)
$\displaystyle+\sum_{k=1}^{K}\mu_{k}\Bigg{[}\sum^{M}_{m=1}B\log_{2}\bigg{(}1+\frac{p^{I}_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{)}-\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}\Bigg{]},$
where the domain $\mathcal{D}$ is defined as the set of all non-negative
$p^{I}_{k,m}$ for $\forall k\in\mathcal{K}$ and for $\forall m\in\mathcal{M}$
such that for each $m$, only a single $p^{I}_{k,m}$ is positive for
$k\in\mathcal{K}$. Then, the Lagrangian dual function of Problem
$\mathcal{P}3\text{-}1a$ is given by
$\displaystyle
g(\boldsymbol{\lambda},\boldsymbol{\mu})=\mathop{\max}\limits_{\\{\zeta_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\}\in{\mathcal{D}}}\mathcal{L}\big{(}\\{\zeta_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\lambda},\boldsymbol{\mu}\big{)}.$
(18)
(18) can be reformulated as
$\displaystyle g(\boldsymbol{\lambda},\boldsymbol{\mu})$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})+\sum^{K}_{k=1}(1-\lambda_{k})\zeta_{k}-\sum^{K}_{k=1}\lambda_{k}\kappa
f_{k}^{2}$ (19)
$\displaystyle+\sum^{K}_{k=1}\lambda_{k}\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{K}_{k=1}\frac{\mu_{k}\Big{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\Big{]}}{(1-\hat{\tau})T},$
where we have
$\displaystyle\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})\triangleq\mathop{\max}\limits_{\\{\boldsymbol{p}^{I}_{k}\\}\in{\mathcal{D}}}\Bigg{\\{}-\sum_{k=1}^{K}\lambda_{k}(p^{I}_{k,m}+p_{c})+\sum_{k=1}^{K}\mu_{k}B\log_{2}\Bigg{[}1+\frac{p^{I}_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\Bigg{]}\Bigg{\\}}.$
(20)
It is readily seen that (20) is concave regarding $p^{I}_{k,m}$. Thus, upon
letting its first-order derivative regarding $p^{I}_{k,m}$ be $0$, we may give
the optimal power of the $m$-th sub-band when it is allocated to the $k$-th
device as
$\displaystyle\hat{p}^{I}_{k,m}(\lambda_{k},\mu_{k})=\bigg{[}\frac{\mu_{k}B}{\lambda_{k}\ln
2}-\frac{\Gamma\sigma^{2}}{|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}\bigg{]}^{+}.$
(21)
Then, $\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$ can be obtained, by
searching over all possible assignments of the $m$-th sub-band for all the $K$
devices, as follows
$\displaystyle\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})=\max_{k}\Bigg{\\{}-\lambda_{k}\Big{[}\hat{p}^{I}_{k,m}(\lambda_{k},\mu_{k})+p_{c}\Big{]}+\mu_{k}B\log_{2}\Bigg{[}1+\frac{\hat{p}^{I}_{k,m}(\lambda_{k},\mu_{k})|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\Bigg{]}\Bigg{\\}},$
(22)
and the suitable device is given by
$k^{*}=\arg\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$. We set
$\alpha_{k^{*},m}=1$ and $p^{I}_{k^{*},m}=\hat{p}^{I}_{k^{*},m}$ as well as
$\alpha_{k,m}=0$ and $p^{I}_{k,m}=0$ for $\forall k\neq k^{*}$. We continue by
calculating $\\{\zeta_{k}\\}$ as follows. At a glance of (21), it is observed
that $\lambda_{k}$ has to yield $\lambda_{k}>0$, $\forall k\in\mathcal{K}$,
which implies that Constraint (15b) is strictly binding for the optimal
solution of Problem $\mathcal{P}3\text{-}1a$. Therefore, $\zeta_{k}$ can be
set as
$\displaystyle\zeta_{k}=\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\kappa
f_{k}^{2}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c}).$ (23)
Once all $\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$ and $\zeta_{k}$
are obtained, $g(\boldsymbol{\lambda},\boldsymbol{\mu})$ can be calculated by
(19). Bearing in mind that the obtained
$g(\boldsymbol{\lambda},\boldsymbol{\mu})$ is not guaranteed to be optimal, we
have to find a suitable set of $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$
that minimize $g(\boldsymbol{\lambda},\boldsymbol{\mu})$, which can be
realized by the ellipsoid method [45]. More explicitly, the Lagrange
multipliers are iteratively updated following their sub-gradients towards
their optimal settings. The corresponding sub-gradients are given as follows
$\displaystyle s_{\lambda_{k}}=\kappa
f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T},$
(24) $\displaystyle
s_{\mu_{k}}=\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}-\sum_{m=1}^{M}\alpha_{k,m}B\log_{2}\bigg{(}1+\frac{p^{I}_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{)}.$
(25)
Upon denoting the iteration index by $t$, the Lagrange multipliers are updated
obeying
$\lambda_{k}(t+1)=[\lambda_{k}(t)+\delta_{\lambda}(t)s_{\lambda_{k}}]^{+}$ and
$\mu_{k}(t+1)=[\mu_{k}(t)+\delta_{\mu}(t)s_{\mu_{k}}]^{+}$, where we set
$\delta_{\lambda}(t)=\delta_{\lambda}(1)/t$ and
$\delta_{\mu}(t)=\delta_{\mu}(1)/t$ for ensuring the convergence of the OF. In
the problem considered, the ellipsoid method converges in $\mathcal{O}(K^{2})$
iterations [45, 39]. Within each iteration, the computational complexity is of
$\mathcal{O}(KM)$. Hence, the total computational complexity is given by
$\mathcal{O}(MK^{3})$. The procedure of this Lagrangian duality method is
summarized in Algorithm 3.
Algorithm 3 Design of $\\{\boldsymbol{\alpha}_{k}\\}$ and
$\\{\boldsymbol{p}^{I}_{k}\\}$, given the settings of $\hat{\tau}$,
$\boldsymbol{p}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$ and
$\boldsymbol{\Theta}^{I}$
0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$,
$f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$,
$\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{P}^{E}$,
$\boldsymbol{\Theta}^{E}$, $\boldsymbol{\Theta}^{I}$, $\boldsymbol{f}$,
$\boldsymbol{\Theta}^{I}$, $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$
0: $\\{\zeta_{k}\\}$, $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$ 1\. Initialization
Initialize $t_{3}=0$; $\epsilon_{3}=1$; Calculate $\mathcal{L}^{(0)}$ using
(17) 2\. Optimization of $\\{\zeta_{k}\\}$, $\\{\boldsymbol{\alpha}_{k}\\}$
and $\\{\boldsymbol{p}^{I}_{k}\\}$
while $t_{3}<t_{\text{max}}$ $\&\&$ $\epsilon_{3}>\epsilon$ do
for $m=1\mathrel{\mathop{\mathchar 58\relax}}M$ do
$\bullet$ Calculate $\hat{p}^{I}_{k,m}$ using (21) for each $\forall
k\in\mathcal{K}$
$\bullet$ Obtain the optimal device
$k^{*}=\arg\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$ in (22)
$\bullet$ Set $\alpha_{k^{*},m}=1$ and $p^{I}_{k^{*},m}=\hat{p}^{I}_{k^{*},m}$
as well as $\alpha_{k,m}=0$ and $p^{I}_{k,m}=0$ for $\forall k\neq k^{*}$
end for
$\bullet$ Calculate $\zeta_{k}$ using (23)
$\bullet$ Calculate $\mathcal{L}^{(t_{3}+1)}$ using (17)
$\bullet$ Update $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ using the
ellipsoid method
$\bullet$ Set
$\epsilon_{3}=\frac{\big{|}\mathcal{L}^{(t_{3}+1)}-\mathcal{L}^{(t_{3})}\big{|}}{\big{|}\mathcal{L}^{(t_{3}+1)}\big{|}}$,
$t_{3}\leftarrow t_{3}+1$
end while3. Output optimal $\\{\zeta_{k}\\}^{*}$,
$\\{\boldsymbol{\alpha}_{k}\\}^{*}$ and $\\{\boldsymbol{p}^{I}_{k}\\}^{*}$
$\\{\zeta_{k}\\}^{*}=\\{\zeta_{k}\\}$,
$\\{\boldsymbol{\alpha}_{k}\\}^{*}=\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}^{*}=\\{\boldsymbol{p}^{I}_{k}\\}$
In the second step, we optimize the IRS reflection coefficient
$\boldsymbol{\Theta}^{I}$, while fixing the settings of the resource
allocation at the computing phase $\\{\boldsymbol{\alpha}_{k}\\}$ and
$\\{\boldsymbol{p}^{I}\\}$. In this case, Problem $\mathcal{P}3\text{-}1$
becomes a feasibility-check problem below
$\displaystyle\mathcal{P}3\text{-}1b\mathrel{\mathop{\mathchar
58\relax}}\text{Find }\boldsymbol{\Theta}^{I}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_constraint_2}.$
The problem can be solved using the approach devised in Section III-B2,
detailed as follows. By introducing a set of auxiliary variables
$\\{\chi_{k}\\}$, we transform $\mathcal{P}3\text{-}2$ to the problem below
$\displaystyle\mathcal{P}3\text{-}1b\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{I},\\{\chi_{k}\\}}\sum^{K}_{k=1}\chi_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},$
$\displaystyle\quad\quad\chi_{k}\geq 0,\quad\forall k\in\mathcal{K},$ (27a)
$\displaystyle\quad\quad\sum^{m}_{m=1}\alpha_{k,m}B\log_{2}\bigg{[}1+\frac{p_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{]}\geq\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}+\chi_{k},\quad\forall
k\in\mathcal{K}.$ (27b)
Constraint (27b) is non-convex regarding $\boldsymbol{\Theta}^{I}$. To address
this issue, firstly we transform Problem $\mathcal{P}3\text{-}1b$ to its
equivalent form, by introducing a set of auxiliary variables
$\boldsymbol{y}^{I}$, $\boldsymbol{a}^{I}$ and $\boldsymbol{b}^{I}$
$\displaystyle\mathcal{P}3\text{-}1bE1\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{I},\\{\chi_{k}\\},\boldsymbol{y}^{I},\boldsymbol{a}^{I},\boldsymbol{b}^{I}}\sum^{K}_{k=1}\chi_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_2'_constraint_3},$
$\displaystyle\quad\quad\sum^{m}_{m=1}\alpha_{k,m}B\log_{2}\bigg{(}1+\frac{p_{k,m}y^{I}_{k,m}}{\Gamma\sigma^{2}}\bigg{)}\geq\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}+\chi_{k},\quad\forall
k\in\mathcal{K},$ (28a) $\displaystyle\quad\quad
a^{I}_{k,m}=\Re\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{I}\big{\\}},\quad
k\in\mathcal{K},\quad m\in\mathcal{M},$ (28b) $\displaystyle\quad\quad
b^{I}_{k,m}=\Im\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{I}\big{\\}},\quad
k\in\mathcal{K},\quad m\in\mathcal{M},$ (28c) $\displaystyle\quad\quad
y^{I}_{k,m}=(a^{I}_{k,m})^{2}+(b^{I}_{k,m})^{2},\quad k\in\mathcal{K},\quad
m\in\mathcal{M}.$ (28d)
Then, upon invoking the so-called SCA method as detailed in Section III-B2, we
approach the locally optimal solution by solving the problem below in a
successive manner
$\displaystyle\mathcal{P}3\text{-}1bE2\mathrel{\mathop{\mathchar
58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{I},\\{\chi_{k}\\}}\sum^{K}_{k=1}\chi_{k}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_2'_constraint_3},\eqref{eqn:P3_2'E1_constraint_2},\eqref{eqn:P3_2'E1_constraint_6},\eqref{eqn:P3_2'E1_constraint_7},$
$\displaystyle\quad\quad
y^{I}_{k,m}=\tilde{a}^{I}_{k,m}(2a^{I}_{k,m}-\tilde{a}^{I}_{k,m})+\tilde{b}^{I}_{k,m}(2b^{I}_{k,m}-\tilde{b}^{I}_{k,m}),\quad
k\in\mathcal{K},\quad m\in\mathcal{M}.$ (29a)
Problem $\mathcal{P}3\text{-}1bE2$ is a convex optimization problem, which can
be readily solved with the aid of the software of CVX [41]. The computational
complexity is the same as that given in Section III-B2. Note that the
optimization of $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ not only
contributes to reducing the OF of Problem $\mathcal{P}2$, but also leads to a
decreased OF of Problem $\mathcal{P}1$ by slacking its constraint (8a). Hence,
we may still reduce the OF of Problem $\mathcal{P}1$ by iteratively optimizing
the settings of the WET phase and the computing phase, even if
$\boldsymbol{f}$ reaches its maximum value.
#### III-C2 Design of CPU Frequencies
Given the settings of the sub-band-device association
$\\{\boldsymbol{\alpha}_{k}\\}$, the power allocation
$\\{\boldsymbol{p}^{I}_{k}\\}$ and the IRS reflection coefficient
$\boldsymbol{\Theta}^{I}$, Problem $\mathcal{P}3$ can be simplified as
$\displaystyle\mathcal{P}3\text{-}2\mathrel{\mathop{\mathchar
58\relax}}\mathop{\min}\limits_{\boldsymbol{f},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}}\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\bigg{]}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_10},\eqref{eqn:P3_1_constraint_2}.$
(30a)
It can be seen that the OF of Problem $\mathcal{P}3\text{-}2$ decreases upon
increasing $\boldsymbol{f}$. Hence, upon denoting
$\displaystyle\hat{f}_{k}=\sqrt{\frac{\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}}{\kappa}},$
(31)
the optimal $\boldsymbol{f}$ can be obtained as:
$\displaystyle f_{k}$ $\displaystyle=\begin{cases}0,&\quad\text{if
}\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}<0,\\\
\hat{f}_{k},&\quad\text{if
}0\leq\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}<\kappa
f_{max}^{2},\\\ f_{max},&\quad\text{if
}\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}\geq\kappa
f_{max}^{2}.\end{cases}$ (32)
The procedure of optimizing $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$, $\boldsymbol{\Theta}^{I}$ and $\boldsymbol{f}$
is summarized in Algorithm 4. To this end, it is readily to summarize the
algorithm solving Problem $\mathcal{P}1$ under a given $\hat{\tau}$ in
Algorithm 5, and an appropriate $\tau$ is found with the aid of numerical
results, as detailed in Section IV-A.
Algorithm 4 Alternative optimization of $\boldsymbol{f}$,
$\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and
$\boldsymbol{\Theta}^{I}$, given the settings of $\hat{\tau}$,
$\boldsymbol{p}^{E}$ and $\boldsymbol{\Theta}^{E}$
0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$,
$f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$,
$\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{P}^{E}$,
$\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$, and
$\tilde{\boldsymbol{\Theta}}^{I}$
0: $\\{\boldsymbol{\alpha}_{k}\\}$ $\\{\boldsymbol{p}^{I}_{k}\\}$ and
$\boldsymbol{\Theta}^{I}$ 1\. Initialization
$\bullet$ Initialize $t_{4}=0$; $\epsilon_{4}=1$;
${\boldsymbol{\Theta}^{I}}^{(0)}=\tilde{\boldsymbol{\Theta}}^{I}$
$\bullet$ Given ${\boldsymbol{\Theta}^{I}}^{(0)}$, find
$\\{\boldsymbol{\alpha}_{k}\\}^{(0)}$ and $\\{\boldsymbol{p}^{I}_{k}\\}^{(0)}$
by solving Problem $\mathcal{P}3\text{-}1a$ via Algorithm 3
$\bullet$ Obtain $\boldsymbol{f}^{(0)}$ via (32) and calculate
$\text{obj}\big{(}\boldsymbol{f}^{(0)}\big{)}$ 2\. Alternative optimization of
$\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$,
$\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$
while $t_{4}<t_{\text{max}}$ $\&\&$ $\epsilon_{4}>\epsilon$ do
$\bullet$ Given $\\{\boldsymbol{\alpha}_{k}\\}^{(t_{4})}$,
$\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{4})}$ and
$\tilde{\boldsymbol{\Theta}}^{I}={\boldsymbol{\Theta}^{I}}^{(t_{4})}$, find
${\boldsymbol{\Theta}^{I}}^{(t_{4}+1)}$ by solving Problem
$\mathcal{P}3\text{-}1bE1$ via Algorithm 1
$\bullet$ Given ${\boldsymbol{\Theta}^{I}}^{(t_{4}+1)}$, find
$\\{\boldsymbol{\alpha}_{k}\\}^{(t_{4}+1)}$ and
$\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{4}+1)}$ by solving Problem
$\mathcal{P}3\text{-}1a$ via Algorithm 3
$\bullet$ Obtain $\boldsymbol{f}^{(t_{4}+1)}$ via (32) and calculate
$\text{obj}\big{(}\boldsymbol{f}^{(t_{4}+1)}\big{)}$
$\bullet$ Set
$\epsilon_{4}=\frac{\big{|}\text{obj}\big{(}\boldsymbol{f}^{(t_{4}+1)}\big{)}-\text{obj}\big{(}\boldsymbol{f}^{(t_{4})}\big{)}\big{|}}{\big{|}\text{obj}\big{(}\boldsymbol{f}^{(t_{4}+1)}\big{)}\big{|}}$,
$t_{4}\leftarrow t_{4}+1$
end while3. Output optimal $\\{\boldsymbol{\alpha}_{k}\\}^{*}$
$\\{\boldsymbol{p}^{I}_{k}\\}^{*}$ and ${\boldsymbol{\Theta}^{I}}^{*}$
$\\{\boldsymbol{\alpha}_{k}\\}^{*}\leftarrow\\{\boldsymbol{\alpha}_{k}\\}^{(t_{4})}$,
$\\{\boldsymbol{p}^{I}_{k}\\}^{*}\leftarrow\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{4})}$
and
${\boldsymbol{\Theta}^{I}}^{*}\leftarrow{\boldsymbol{\Theta}^{I}}^{(t_{4})}$
Algorithm 5 Alternative optimization of the WET and computing phases, given
the time allocation
0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$,
$f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$,
$\\{\boldsymbol{V}_{k}\\}$ and $\hat{\tau}$
0: $\boldsymbol{P}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$,
$\\{\boldsymbol{\alpha}_{k}\\}$ $\\{\boldsymbol{p}^{I}_{k}\\}$ and
$\boldsymbol{\Theta}^{I}$ 1\. Initialization
$\bullet$ Initialize $t_{5}=0$; $\epsilon_{5}=1$;
$\tilde{\boldsymbol{\Theta}}^{E}$
$\bullet$ Initialize $\boldsymbol{f}^{(0)}$,
$\\{\boldsymbol{\alpha}_{k}\\}^{(0)}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{(0)}$
and ${\boldsymbol{\Theta}^{I}}^{(0)}$ following Section III-A
$\bullet$ Given $\boldsymbol{f}^{(0)}$, $\\{\boldsymbol{\alpha}_{k}\\}^{(0)}$,
$\\{\boldsymbol{p}^{I}_{k}\\}^{(0)}$ and ${\boldsymbol{\Theta}^{I}}^{(0)}$,
find ${\boldsymbol{P}^{E}}^{(0)}$ and ${\boldsymbol{\Theta}^{E}}^{(0)}$ by
solving Problem $\mathcal{P}2$ via Algorithm 2 2\. Alternative optimization of
$\boldsymbol{P}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$,
$\\{\boldsymbol{\alpha}_{k}\\}$ $\\{\boldsymbol{p}^{I}_{k}\\}$ and
$\boldsymbol{\Theta}^{I}$
while $t_{5}<t_{\text{max}}$ $\&\&$ $\epsilon_{5}>\epsilon$ do
$\bullet$ Given ${\boldsymbol{P}^{E}}^{(t_{5})}$,
${\boldsymbol{\Theta}^{E}}^{(t_{5})}$ and
$\tilde{\boldsymbol{\Theta}}^{I}={\boldsymbol{\Theta}^{I}}^{(t_{5})}$, find
$\boldsymbol{f}^{(t_{5}+1)}$, $\\{\boldsymbol{\alpha}_{k}\\}^{(t_{5}+1)}$,
$\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{5}+1)}$ and
${\boldsymbol{\Theta}^{I}}^{(t_{5}+1)}$ by solving Problem $\mathcal{P}3$
using Algorithm 4
$\bullet$ Given $\boldsymbol{f}^{(t_{5}+1)}$,
$\\{\boldsymbol{\alpha}_{k}\\}^{(t_{5}+1)}$,
$\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{5}+1)}$,
${\boldsymbol{\Theta}^{I}}^{(t_{5}+1)}$ and
$\tilde{\boldsymbol{\Theta}}^{E}={\boldsymbol{\Theta}^{E}}^{(t_{5})}$, find
${\boldsymbol{P}^{E}}^{(t_{5}+1)}$ and ${\boldsymbol{\Theta}^{E}}^{(t_{5}+1)}$
by solving Problem $\mathcal{P}2$ via Algorithm 2
$\bullet$ Set
$\epsilon_{5}=\frac{\big{|}\text{obj}^{(t_{5}+1)}-\text{obj}^{(t_{5})}\big{|}}{\big{|}\text{obj}^{(t_{5}+1)}\big{|}}$,
$t_{5}\leftarrow t_{5}+1$
end while3. Output optimal ${\boldsymbol{P}^{E}}^{*}$,
${\boldsymbol{\Theta}^{E}}^{*}$, $\boldsymbol{f}^{*}$,
$\\{\boldsymbol{\alpha}_{k}\\}^{*}$ $\\{\boldsymbol{p}^{I}_{k}\\}^{*}$ and
${\boldsymbol{\Theta}^{I}}^{*}$
${\boldsymbol{P}^{E}}^{*}\leftarrow{\boldsymbol{P}^{E}}^{(t_{5})}$,
${\boldsymbol{\Theta}^{E}}^{*}\leftarrow{\boldsymbol{\Theta}^{E}}^{(t_{5})}$,
$\boldsymbol{f}^{*}\leftarrow\boldsymbol{f}^{(t_{5})}$,
$\\{\boldsymbol{\alpha}_{k}\\}^{*}\leftarrow\\{\boldsymbol{\alpha}_{k}\\}^{(t_{5})}$,
$\\{\boldsymbol{p}^{I}_{k}\\}^{*}\leftarrow\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{5})}$
and
${\boldsymbol{\Theta}^{I}}^{*}\leftarrow{\boldsymbol{\Theta}^{I}}^{(t_{5})}$
## IV Numerical Results
In this section, we present the pertinent numerical results, for evaluating
the performance of our proposed IRS-aided WP-MEC design. A top view of the
HAP, of the wireless devices and of the IRS are shown in Fig. 3, where the
HAP’s coverage radius is $R$ and the IRS is deployed at the cell edge. The
locations of wireless devices are assumed to obey the uniform distribution
within a circle, whose radius and locations are specified by $r$ as well as
$d_{1}$ and $d_{2}$, respectively. Their default settings are specified in the
block of “System model” in Table I. The efficiency of the energy harvesting
$\eta$ is set as $0.5$. As for the communications channel, we consider both
the small-scale fading and the large-scale path loss. More explicitly, the
small-scale fading is assumed to be independent and identically distributed
(i.i.d.) and obey the complex Gaussian distribution associated with zero mean
and unit variance, while the path loss in $\rm{dB}$ is given by
$\displaystyle\text{PL}=\text{PL}_{0}-10\beta\log_{10}\big{(}\frac{d}{d_{0}}\big{)},$
(33)
where $\text{PL}_{0}$ is the path loss at the reference distance $d_{0}$;
$\beta$ and $d$ denote the path loss exponent of and the distance of the
communication link, respectively. Here we use $\beta_{ua}$, $\beta_{ui}$ and
$\beta_{ia}$ to represent the path loss exponent of the links between the
wireless devices and the HAP, between the wireless devices and the IRS, as
well as between the IRS and the HAP, respectively111We assume that the channel
of the direct link between the HAP and devices is hostile (due to an
obstruction), while this obstruction can be partially avoided by the IRS-
reflection link. Hence, we set a higher value for $\beta_{ua}$.. Furthermore,
the additive while Gaussian noise associated with zero mean and the variable
of $\sigma^{2}$ is imposed both on the energy signals and on the offloading
signals. The default values of the parameters are set in the block of
“Communications model” in Table I. As for the computing model, the variables
of $L_{k}$ and $c_{k}$ are assumed to obey the uniform distribution. The
offloaded tasks are assumed to be computed in parallel by a large number of
CPUs at the edge computing node, where the computing capability of each CPU is
$f_{e}=10^{9}\leavevmode\nobreak\ \rm{cycle/s}$. Then, the energy consumption
at the edge for processing the offloaded computational tasks can be calculated
as $\vartheta=c\kappa f_{e}^{2}=5\times 10^{-8}\leavevmode\nobreak\
\rm{Joule/bit}$.
Figure 3: An illustration of the locations of the HAP, of devices and of the IRS in the IRS-aided WP-MEC system. Table I: Default simulation parameter setting Description | Parameter and Value
---|---
System model [27] | $M=16$, $N=30$, $K=3$, $T=10\leavevmode\nobreak\ \rm{ms}$
$R=12\leavevmode\nobreak\ \rm{m}$, $d_{1}=11\leavevmode\nobreak\ \rm{m}$,
$d_{2}=1\leavevmode\nobreak\ \rm{m}$, $r=1\leavevmode\nobreak\ \rm{m}$
Wireless energy transfer model | $\eta=0.5$
Communication model [33] | $B=312.5\leavevmode\nobreak\ \rm{KHz}$
$\text{PL}_{0}=30\leavevmode\nobreak\ \rm{dB}$, $d_{0}=1\leavevmode\nobreak\
\rm{m}$, $\beta_{ua}=3.5$, $\beta_{ui}=2.2$, $\beta_{ia}=2.2$
$L^{d}_{k}=4$, $L_{1}=2$, $L_{2,k}=3$
$\sigma^{2}=1.24\times 10^{-12}\leavevmode\nobreak\ \rm{mW}$, $\Gamma=2$
Computing model [5] | $L_{k}=[15,20]\leavevmode\nobreak\ \rm{Kbit}$
$c_{k}=[400,500]\leavevmode\nobreak\ \rm{cycle/bit}$
$f_{max}=1\times 10^{8}\leavevmode\nobreak\ \rm{cycle/s}$
| $\kappa=10^{-28}$, $\vartheta=5\times 10^{-8}\leavevmode\nobreak\
\rm{Joule/bit}$
Convergence criterion | $\epsilon=0.001$
Apart from our algorithms developed in Section III, we also consider two
benchmark schemes for comparison. Let us describe these three schemes as
follows.
* •
_With IRS_ : In this scheme, we optimize both the power allocation
$\boldsymbol{p}^{E}$ and the IRS reflection coefficients
$\boldsymbol{\Theta}^{E}$ at the WET phase, as well as the local CPU frequency
at devices $\boldsymbol{f}$, the sub-band-device association
$\\{\boldsymbol{\alpha}_{k}\\}$, the power allocation
$\\{\boldsymbol{p}_{k}\\}$ and the IRS reflection coefficients
$\boldsymbol{\Theta}^{I}$ at the computing phase, relying on Algorithm 5.
* •
_RandPhase_ : The power allocation $\boldsymbol{p}^{E}$ at the WET phase, as
well as the local CPU frequency at devices $\boldsymbol{f}$, the sub-band-
device association $\\{\boldsymbol{\alpha}_{k}\\}$ and the power allocation
$\\{\boldsymbol{p}_{k}\\}$ at the computing phase are optimized with the aid
of Algorithm 5, while we skip the design of the IRS reflection coefficients
$\boldsymbol{\Theta}^{E}$ and $\boldsymbol{\Theta}^{I}$, whose amplitude
response is set to $1$ and phase shifts are randomly set in the range of
$[0,2\pi)$ obeying the uniform distribution.
* •
_Without IRS_ : The composite channel
$\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}$ is set to $0$
both for the WET and for the computation offloading. The power allocation
$\boldsymbol{p}^{E}$ at the WET phase, as well as the local CPU frequency at
devices $\boldsymbol{f}$, the sub-band-device association
$\\{\boldsymbol{\alpha}_{k}\\}$ and the power allocation
$\\{\boldsymbol{p}_{k}\\}$ at the computing phase are optimized with the aid
of Algorithm 5, while we skip the optimization of the IRS reflection
coefficient $\boldsymbol{\Theta}^{E}$ and $\boldsymbol{\Theta}^{I}$.
Let us continue by presenting the selection of the time allocation, sub-band
allocation in the WET and the computing phases, as well as the impact of
diverse environment settings, as follows.
### IV-A Selection of the Time Allocation
Figure 4: Simulation results of the total energy consumption versus the time
allocation $\tau$. The parameter settings are specified in Table I.
In order to find an appropriate time allocation for our WP-MEC system, we
depict the total energy consumption (the OF of Problem $\mathcal{P}1$) versus
the the time allocation $\tau$ in Fig. 4. It can be seen that the total energy
consumption becomes higher upon increasing $\tau$ for all these three schemes
considered. The reason behind it is explained as follows. For a given volume
of the computational task to be offloaded within the time duration of $T$, an
increase of $\tau$ implies a higher offloading rate required by computation
offloading, while at a glance of (5), the computation offloading rate is
formulated as a logarithmic function of the offloading power. Hence, we have
to largely increase the transmit power of computation offloading for providing
the extra offloading rate required by the increase of $\tau$, which results in
a higher energy consumption at the wireless devices. Furthermore, since the
energy required by WET is determined by the energy consumption at the wireless
devices, the total energy consumption becomes higher upon increasing $\tau$.
Based on this discussion, it seems that we should select the value of $\tau$
as small as possible. However, this may lead to an upsurge of the power
consumption for WET, which might exceed the maximum allowable transmit power
at the HAP. Therefore, as a compromise, for the environment associated with
the default settings we select $\tau=0.1$, beyond which the total energy
consumption becomes increasingly higher along with $\tau$.
### IV-B Joint Sub-Band and Power Allocation in the WET and Computing Phases
(a)
(b)
(c)
(d)
Figure 5: Joint sub-band and of power allocation for the WET and the computing
phases, relying on the Algorithm 5, where the number of bits to be processed
is set the same as $20\leavevmode\nobreak\ \rm{Kbits}$ for the three wireless
devices. (a) The channel gain at the WET phase; (b) The joint sub-band and
power allocation at the WET phase; (3) The channel gain at the computing
phase; (d) The joint sub-band and power allocation at the computing phase. The
parameter settings are specified in Table I.
Fig. 5 illustrates the channel gain as well as the joint sub-band and power
allocation both for the WET and computing phases. Our observations are as
follows. Firstly, as shown in Fig. 5b, only the $5$-th sub-band is activated
for WET. This allocation is jointly determined by the power consumption of the
computing phase and by the channel gain in the WET phase. Specifically, with
the reference of Fig. 5d, Device 3 requires the highest power consumption for
computation offloading. Given that the overall performance is dominated by the
device having the highest energy consumption, we may reduce the energy
consumption of WET, by activating the sub-band associated with the highest
channel gain of Device 3, which is the $5$-th sub-band as shown in Fig. 5a.
Secondly, with the reference of Fig. 5c, it can be observed that the power
allocation in Fig. 5d obeys the water-filling principle for each device, i.e.
allocating a higher power to the sub-band possessing a high channel gain. This
corresponds to the power allocation obtained in (21). Thirdly, comparing Fig.
5a and Fig. 5c, we can see that the channel gains in the WET and computing
phase are different for each device after we optimize the IRS reflection
coefficients, which consolidates our motivation to conceive separate IRS
designs for the WET and the computing phases.
### IV-C Performance of the Proposed Algorithms
In order to evaluate the benefits of employing an IRS in WP-MEC systems, we
compare the performance of our proposed algorithms with that of the benchmark
schemes, under various settings of the number of IRS reflection elements, of
the device location, of the path loss exponent of the IRS-related channel, and
of the energy consumption per bit at the edge, as follows.
#### IV-C1 Impact of the Number of IRS Reflection Elements
Figure 6: Simulation results of the total energy consumption versus the number
of IRS reflection elements $N$. The rest of parameters are specified in Table
I.
Figure 7: Simulation results of the total energy consumption versus the
distance between the HAP and the wireless device circle $d_{1}$. Other
parameters are set in Table I.
Fig. 6 shows the simulation results of the total energy consumption versus the
number of IRS reflection elements for the three schemes considered. We have
the following observations. Firstly, the performance gap between the scheme
“Without IRS” and the scheme “IRS RandPhase” increases along with $N$, which
implies that the IRS is capable of assisting the energy consumption reduction
in the WP-MEC system, even without carefully designing the IRS reflection
coefficients. This is due to the so-called virtual array gain induced by the
IRS, as mentioned in Section I. Secondly, the scheme “With IRS” outperforms
the scheme “IRS RandPhase”, which indicates that our sophisticated design of
IRS reflection coefficients may provide the so-called passive beamforming gain
for computation offloading. Note that different from the conventional MEC
systems [38] where WET is not employed, these two types of gain are exploited
twice in WP-MEC systems (during the WET and computing phases, respectively).
As such, IRSs are capable of efficiently reducing the energy consumption in
WP-MEC systems.
#### IV-C2 Impact of the Distance between the Device Circle and the IRS
Fig. 7 presents the simulation results of the total energy consumption versus
the distance between the HAP and the mobile wireless circles. Our observations
are as follows. Firstly, the two IRS-aided schemes do not show any visible
advantage over the scheme of “Without IRS” when we have
$d_{1}<6\leavevmode\nobreak\ \rm{m}$, which indicates that each IRS has a
limited coverage. Secondly, the benefit of deploying the IRS is becomes
visible at $d_{1}>9\leavevmode\nobreak\ \rm{m}$ in the scheme of “IRS
RandPhase”, while the advantage of the “With IRS” scheme is already notable at
$d_{1}=7\leavevmode\nobreak\ \rm{m}$. This observation implies that our
sophisticated design of IRS reflection coefficient is capable of extending the
coverage of IRS.
Figure 8: Simulation results of the total energy consumption versus the path
loss exponent of the IRS reflection link $\beta$, where we set
$\beta_{ui}=\beta_{ia}=\beta$. Other parameters are set in Table I.
Figure 9: Simulation results of the total energy consumption versus the energy
consumption per bit at the edge. Other parameters are set in Table I.
#### IV-C3 Impact of Path Loss Exponent
Fig. 8 depicts the simulation results of the total energy consumption versus
the path loss exponent of the IRS related links. It can be seen that the total
energy consumption decreases if a higher path loss exponent is encountered,
which is because a higher $\beta$ leads to a lower channel gain of the IRS-
reflected link. This observation provides an important engineering insight:
the locations of IRSs should be carefully selected for avoiding obstacles.
#### IV-C4 Impact of energy consumption at the edge
Fig. 9 shows the simulation results of the total energy consumption versus the
energy consumption per bit at the edge node. It can be observed that the
advantage of deploying IRS is eminent when we have a small value of
$\vartheta$, while the benefit becomes smaller upon increasing the value of
$\vartheta$. The reason is explained as follows. The OF of Problem
$\mathcal{P}1$ is the combination of the energy consumption of WET and of
processing the offloaded computational tasks. If the energy consumption per
bit at the edge node is of a small value, the energy consumption of WET plays
a dominant role in the total energy consumption. In this case, the benefit of
employing IRS is significant. By contrast, if $\vartheta$ becomes higher, the
total energy consumption is dominated by that at the edge. In this case,
although the energy consumption of WET can be degraded by deploying IRSs, this
reduction becomes marginal.
## V Conclusions
To reduce the energy consumption of WP-MEC systems, we have proposed an IRS-
aided WP-MEC scheme and formulate an energy minimization problem. A
sophisticated algorithm has been developed for optimizing the settings both in
the WET and the computing phases. Our numerical results reveal the following
insights. Firstly, the employment of IRSs is capable of substantially reducing
the energy consumption of the WP-MEC system, especially when the IRS is
deployed in vicinity of wireless devices. Secondly, the energy consumption
decreases upon increasing the number of IRS reflection elements. Thirdly, the
locations of IRSs should be carefully selected for avoiding obstacles. These
results inspire us to conceive a computational rate maximization design for
the IRS-aided WP-MEC system as a future work.
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|
2024-09-04T02:54:59.388750 | 2020-03-11T20:41:06 | 2003.05514 | {
"authors": "Eleftherios Kastis and Stephen Power",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26179",
"submitter": "Stephen C. Power",
"url": "https://arxiv.org/abs/2003.05514"
} | arxiv-papers | # Projective plane graphs and 3-rigidity
E. Kastis and S.C. Power Dept. Math. Stats.
Lancaster University
Lancaster LA1 4YF
U.K<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
A ${\mathcal{P}}$-graph is a simple graph $G$ which is embeddable in the real
projective plane ${\mathcal{P}}$. A $(3,6)$-tight ${\mathcal{P}}$-graph is
shown to be constructible from one of 8 uncontractible ${\mathcal{P}}$-graphs
by a sequence of vertex splitting moves. Also it is shown that a
${\mathcal{P}}$-graph is minimally generically 3-rigid if and only if it is
$(3,6)$-tight. In particular this characterisation holds for graphs that are
embeddable in the Möbius strip.
2010 Mathematics Subject Classification. 52C25 51E15
Key words and phrases: projective plane, embedded graphs, geometric rigidity
This work was supported by the Engineering and Physical Sciences Research
Council [EP/P01108X/1]
## 1\. Introduction
Let $G$ be the graph of a triangulated sphere. Then an associated bar-joint
framework $(G,p)$ in ${\mathbb{R}}^{3}$ is known to be minimally rigid if the
placements $p(v)$ of the vertices $v$ is strictly convex (Cauchy [4]) or if
the placement is generic. The latter case follows from Gluck’s result [12]
that any generic placement is in fact infinitesimally rigid. An equivalent
formulation of Gluck’s theorem asserts that if $G$ is a simple graph which is
embeddable in the sphere then $G$ is minimally 3-rigid if and only if it
satisfies a $(3,6)$-tight sparsity condition. We obtain here the exact
analogue of this for simple graphs that are embeddable in the real projective
plane ${\mathcal{P}}$. The proof rests on viewing these graphs as partial
triangulations and deriving inductive arguments based on edge contractions for
certain admissible edges. Accordingly we may state this result in the
following form. An immediate corollary is that this combinatorial
characterisation also holds for triangulated Möbius strips.
A graph $G$ is _3-rigid_ if its generic bar-joint frameworks in
${\mathbb{R}}^{3}$ are infinitesimally rigid and is _minimally 3-rigid_ if no
subgraph has this property.
###### Theorem 1.1.
Let $G$ be a simple graph associated with a partial triangulation of the real
projective plane. Then $G$ is minimally $3$-rigid if and only if $G$ is
$(3,6)$-tight.
Recall that a $(3,6)$-tight graph $G=(V,E)$ is one that satisfies the Maxwell
count $|E|=3|V|-6$ and the sparsity condition $|E^{\prime}|\leq
3|V^{\prime}|-6$ for subgraphs $G^{\prime}$ with at least 3 vertices. In
particular it follows from the Maxwell condition that such a graph falls 3
edges short of a full (possibly nonsimple) triangulation of ${\mathcal{P}}$.
The proof of Theorem 1.1 depends heavily on our main result, Theorem 6.1,
which is a purely combinatorial constructive characterisation of the
${\mathcal{P}}$-graphs which are (3,6)-tight. A key step is the identification
of edge contraction moves, for certain edges that lie in two 3-cycle faces,
such that the $(3,6)$-sparsity condition is preserved. This is done in Section
3 by exploiting the implicit topological structure of the graphs. The
associated contraction sequences must terminate and the terminal graphs are
said to in _irreducible_. They have the defining property that every
contractible edge lies on a critical $4$-, $5$\- or $6$-cycle. For the
remainder of the proof of Theorem 6.1 we show that an irreducible graph has no
contractible edges (Section 5) and we determine the uncontractible graphs
(Section 4). Determining the uncontractibles requires an extensive case-by-
case analysis leading to the 8 “base” graphs given in Figures 3, 4, 5.
The determination of construction schemes and their base graphs for various
classes of graphs is of general interest, both for embedded graph theory and
for the rigidity of bar-joint frameworks. We note, for example, that Barnette
[1] employed vertex splitting moves for the construction of triangulations of
2-manifolds and showed that there are 2 (full) triangulations of
${\mathcal{P}}$ which are uncontractible. Also, Barnette and Edelson [2], [3]
have shown that all 2-manifolds have finitely many minimal uncontractible
triangulations. Our construction theorem is in a similar spirit to this and we
expect our reduction methods, involving critical cycles and minimum hole
incidence degree, for example, to be useful for more general surface graphs
and for other sparsity classes. In particular, for $(3,6)$-tight
${\mathcal{P}}$-graphs we show that the irreducibles are the uncontractibles
and it would be interesting to determine to what extent this phenomenon is
true for other surfaces and sparsity classes.
We define a _triangulated surface graph_ associated with a classical surface
${\mathcal{M}}$, with or without boundary and we represent embeddings of these
graphs, and their connected subgraphs (${\mathcal{M}}$-graphs), in terms of
_face graphs_. A face graph is a finite connected planar graph with a
specified pairing of some or all of the edges in the outer boundary.
Identifying the paired edges gives an identification graph $G=(V,E)$ together
with a set $F$ of facial 3-cycles inherited from the finite planar graph. See
Definitions 2.1, 2.2. In Section 3 we identify the obstacles, in terms of
critical cycles of edges, which prevent edge contraction moves from preserving
the sparsity condition. The determination in Section 4 of the 8 uncontractible
${\mathcal{P}}$-graphs is given in several stages, based on the nature of the
“holes” in their partial triangulation. They may have one hole with 6-cycle
boundary, two holes with boundary cycle lengths 5 and 4, or three holes, each
with a 4-cycle boundary. Also we give a useful index for the successive
determination of these uncontractible base graphs, namely the minimum hole
incidence degree $h(G)$ (Definition 4.3).
Since Whiteley’s demonstration [14] that vertex splitting preserves generic
rigidity this construction move has become an important tool in combinatorial
rigidity theory [11]. See for example the more recent studies of generic
rigidity in the case of graphs for modified spheres [7], [8], [5], [13], and
in the case of a partially triangulated torus [6]. The proof of Theorem 1.1
given in Section 6 follows quickly from Whiteley’s theorem, Theorem 6.1, and
the 3-rigidity of the 8 base graphs.
## 2\. Graphs in Surfaces
Let ${\mathcal{M}}$ be a classical surface, possibly with boundary. Then a
_surface graph for ${\mathcal{M}}$_ is a triple $G=(V,E,F)$ where $(V,E)$ is a
simple graph, $F$ is a set of $3$-cycles of edges, called facial 3-cycles, and
where there exists a faithful embedding of $G$ in ${\mathcal{M}}$ for which
the facial 3-cycles correspond to the 3-sided faces inthe embedding. A surface
graph for ${\mathcal{M}}$, which we also refer to as an ${\mathcal{M}}$-graph,
can thus be viewed as a simple graph obtained from a full triangulation of
${\mathcal{M}}$ by discarding vertices, edges and faces. Also, $G$ is a
_triangulated surface graph for ${\mathcal{M}}$_ if the union of the embedded
faces is equal to ${\mathcal{M}}$. The following equivalent definition, based
on simplicial complexes rather than surfaces, is combinatorial and so more
elementary.
###### Definition 2.1.
A _triangulated surface graph_ is a graph $G=G(M)=(V,E,F)$ which is simple and
is determined by the $1$-skeleton and the $2$-simplexes of a finite simplicial
complex $M$ where $M$ has the following properties.
1. (i)
$M$ consists of a finite set of $2$-simplexes $\sigma_{1},\sigma_{2},\dots$
together with their $1$-simplexes and $0$-simplexes.
2. (ii)
Every $1$-simplex lies in at most two $2$-simplexes.
Condition (i) implies that each 1-simplex lies in at least one 2-simplex. It
follows that $M$ can be viewed as a _combinatorial surface_ and we define
${\mathcal{M}}={\mathcal{M}}(M)={\mathcal{M}}(G)$ to be the classical
topological surface, possibly with boundary, which is determined by $M$, the
simplicial complex [9]. Evidently, $G$ is a triangulated surface graph for
${\mathcal{M}}$.
Classical compact surfaces are classified up to homeomorphism by combinatorial
surfaces and, moreover, combinatorial surfaces arise from triangulated polygon
graphs (also called triangulated discs) by means of an identification of
certain pairs of boundary edges [9].
We now formally define such labelled triangulated discs which we refer to as
_face graphs_.
###### Definition 2.2.
A _face graph_ for a triangulated surface graph is a pair $(B,\lambda)$ where
$B$ is the planar graph of a triangulated disc and $\lambda$ is a partition of
the _boundary graph_ $\partial B$ of $B$, such that each set of the partition
has $1$ or $2$ edges, and the paired edges of the partition are directed.
A face graph $(B,\lambda)$ defines a simplicial complex $M$, with
$1$-simplexes provided by edges and identified edge pairs, and 2-simplexes
provided by the facial 3-cycles. Also, if the boundary graph of $B$ is a
3-cycle and $\lambda$ is trivial then this 3-cycle defines a 2-simplex of $M$.
If the identification graph $G=B/\lambda$ is a simple graph then $M$ is of the
type given in Definition 2.1, and so $G$ is a triangulated surface graph
$G=(V,E,F)$.
### 2.1. ${\mathcal{M}}$-graphs
We are concerned simple graphs that can be embedded in a connected classical
surface ${\mathcal{M}}$. More precisely we shall be concerned with embedded
graphs, which we refer to as ${\mathcal{M}}$-graphs, or surface graphs, and we
can define them directly in terms of more general face graphs
$(B_{0},\lambda)$, where $B_{0}\subseteq B$ with $\partial B\subset B_{0}$ and
$(B,\lambda)$ is as in the previous definition. Thus a surface graph has the
form $G=B_{0}/\lambda$ where $B_{0}$ is obtained from $B$ by the removal of
the interior edges of $k$ interior-disjoint triangulated subdiscs of $B$. We
refer to $k$ as the _number of holes_ of the embedded graph $G$. When $k=1$ we
refer to $B_{0}$ as an _annular face graph_.
$e$$f$$g$$e$$f$$g$$v_{1}$$v_{2}$$v_{3}$$v_{1}$$v_{2}$$v_{3}$
Figure 1. A face graph $(B_{0},\lambda)$ for a ${\mathcal{P}}$-graph.
Figure 1 shows the annular face graph $(B_{0},\lambda)$ for a surface graph
$G=B_{0}/\lambda$. The labelling of outer boundary edges and vertices
determines how pairs of edges are identified. A planar triangulation of the
interior of the inner 6-cycle gives a face graph $(B,\lambda)$ for the
triangulated surface graph $S=B/\lambda$, if and only if $S$ is simple. In
view of the identifications the topological surface ${\mathcal{M}}(S)$ is the
real projective plane ${\mathcal{P}}$ and $G$ is a ${\mathcal{P}}$-graph.
In this example the surface graph $G$ happens to be a (fully) triangulated
surface graph for the Möbius strip. However, in general a surface graph may
have “exposed” edges, that is, edges that belong to no facial 3-cycles, and so
in this case the surface graph will not be a triangulated surface graph for
any classical surface with boundary.
## 3\. Contraction moves and (3,6)-sparsity.
Let $G=(V,E,F)$ be a surface graph. An edge of $G$ is of type $FF$ if it is
contained in two facial $3$-cycles and an $FF$ edge is _contractible_ if it is
not contained in any non-facial $3$-cycle. For such an edge $e=uv$ there is a
natural contraction move $G\to G^{\prime}$ on the graph $G$, corresponding to
a contraction of $e$, merging $u$ and $v$ to a single vertex, leading to a
surface graph $G^{\prime}=(V^{\prime},E^{\prime},F^{\prime})$ where
$|V^{\prime}|=|V|-1,|E^{\prime}|=|E|-3,|F^{\prime}|=|F|-2$. We also say that
$G$ is _contractible_ if it has a contractible $FF$ edge.
To define formally the contracted graph $G^{\prime}$, let $e=vw$ be a
contractible $FF$ edge in $G$ and let $avw$ and $bvw$ be the two facial
3-cycles which contain $e$. Then $G^{\prime}$ is obtained from $G$ by an _edge
contraction_ on $e=vw$ if $G^{\prime}$ is obtained by (i) deleting the edges
$aw$ and $bw$, (ii) replacing all remaining edges of the form $xw$ with $xv$,
(iii) deleting the edge $e$ and the vertex $w$. That $G^{\prime}$ is simple
follows from the fact that a contractible $FF$ edge does not lie on a
nonfacial 3-cycle.
Given an edge contraction move $G\to G^{\prime}$ we note that the inverse
move, recovering $G$ from $G^{\prime}$, is a _vertex splitting move_ at $v$
which in particular introduces a new vertex $w$ and the new $FF$ edge $vw$.
Such vertex splitting move $G^{\prime}\to G$, which might be thought of as
being locally planar, creates the new surface graph $G$ for the surface
${\mathcal{M}}$ from a given surface graph $G^{\prime}$ for ${\mathcal{M}}$.
### 3.1. (3,6)-sparse ${\mathcal{P}}$-graphs.
If $G=(V,E)$ is a graph then its _freedom number_ is defined to be
$f(G)=3|V|-|E|$. A graph $G$ is _$(3,6)$ -sparse_ if $f(G^{\prime})\geq 6$ for
any subgraph $G^{\prime}$ with at least 3 vertices, and is _$(3,6)$ -tight_ if
it is $(3,6)$-sparse and $f(G)=6$. In particular a $(3,6)$-sparse graph is a
simple graph, with no loop edges and no parallel edges.
Let $B$ be a triangulated disc such that the boundary cycle $\partial B$ is of
even length $2r$. With the pairing partition $\lambda$ of opposite edges,
directed in cyclic order, the pair $(B,\lambda)$ is a face graph. If
$S=B/\lambda$ is simple then $S$ is a triangulated surface graph for the real
projective plane ${\mathcal{P}}$. Also we observe that the freedom number
$f(B)$ is equal to $6+(2r-3)$. This follows since $B$ may be viewed as a
triangulated sphere (which has freedom number $6$) with $2r-3$ edges removed.
Noting that $S$ is related to $B$ by the loss of $r$ vertices and $r$ edges it
follows that
$f(S)=(3+2r)-3r+r=3.$
Let $G$ be a surface graph for ${\mathcal{P}}$, the real projective plane,
which is determined by the annular face $(B_{0},\lambda)$ where the inner
boundary cycle of edges has length $s$. Then $f(G)=f(S)-(s-3)$ and in
particular $G$ satisfies the so-called _Maxwell count_ $f(G)=6$ if and only if
$s=6$.
###### Lemma 3.1.
Let $G$ be a triangulated surface graph for the Möbius strip. Then $G$ is a
surface graph for ${\mathcal{P}}$. Also $G$ satisfies the Maxwell count if and
only if the boundary graph $\partial G$ is the graph of a simple 6-cycle.
###### Proof.
Let $G(M)=(V,E,F)$ be a triangulated surface graph given by a finite
simplicial complex $M$ for the Möbius strip, as in Definition 2.1. Then $G(M)$
is determined by a face graph $(B,\mu)$ where $\mu$ is obtained from an
identification of two vertex-disjoint paths in the boundary of $B$, which have
the same length and orientation and which have end vertices $w_{1},w_{2}$ and
$w_{3},w_{4}$ respectively. In Figure 2 the boundary of $B$ is depicted as
rectangular.
$w_{1}$$w_{2}$$w_{3}$$w_{4}$$v_{1}$$v_{2}$$B$
Figure 2. A Möbius strip triangulated surface graph as a ${\mathcal{P}}$ with
hole graph.
Augment the planar graph $B$ to obtain a containing planar graph $B_{1}$ which
has 2 additional vertices, $v_{1}$ and $v_{2}$ say, and additional edges
$v_{1}w$ (resp. $v_{2}w$) which are incident to vertices on the boundary path
from $w_{4}$ to $w_{1}$ (resp. $w_{2}$ to $w_{3}$). This defines a
triangulated disc $B_{1}$ which is also indicated in Figure 2. Define a
partition $\lambda$ for $B_{1}$ as the augmentation of $\mu$ by the two
directed edge pairs $v_{1}w_{1},v_{2}w_{3}$ and $w_{2}v_{2},w_{3}v_{1}$ and
let $(B_{1},\lambda)$ be the resulting face graph. Then $H=B_{1}/\lambda$ is a
triangulated surface graph for ${\mathcal{P}}$. Moreover, the faces of $H$
that are incident to the vertex $v_{1}=v_{2}$ in $S$ are the faces of a
triangulated disc and $G$ is a surface graph for ${\mathcal{P}}$. ∎
Let $G$ be a ${\mathcal{P}}$-graph, with $k$ holes. If $G$ satisfies the
Maxwell count $f(G)=6$ then $k=1,2$ or $3$. For $k=1$ a representing face
graph $(B_{0},\lambda)$ for $G$ is annular with a 6-cycle inner boundary. This
inner boundary can intersect and even coincide with the outer boundary of $B$.
For $k=2$ there are two inner boundaries of length $5$ and $4$ corresponding
to the boundaries of the interior disjoint discs defining $G$, while for $k=3$
there are three inner boundaries which are 4-cycles. In particular,
$3\leq|\partial G|\leq 12.$
###### Definition 3.2.
For $k=1,2,3$ the set ${\mathfrak{P}}_{k}$ is the set of $(3,6)$-tight
${\mathcal{P}}$-graphs which have $k$ holes.
While a surface graph is a graph with extra structure we shall informally
refer to the elements of ${\mathfrak{P}}_{k}$ as graphs.
### 3.2. When contracted graphs are $(3,6)$-tight
A contraction move $G\to G^{\prime}$ on a contractible $FF$ edge $e$ of a
surface graph preserves the Maxwell count but need not preserve
$(3,6)$-tightness. We now examine this more closely in the case of a surface
graph for the real projective plane ${\mathcal{P}}$.
Suppose that $G_{1}\subseteq G$ and that both $G_{1}$ and $G$ are in
${\mathfrak{P}}_{1}$. If $e$ is a contractible $FF$ edge of $G$ which lies on
the boundary graph of $G_{1}$ then, since $G_{1}$ contains only one of the
facial 3-cycles incident to $e$, the contraction $G\to G^{\prime}$ for $e$
gives a contraction $G^{\prime}$ which is not $(3,6)$-sparse, since
$f(G_{1}^{\prime})=5$. We shall show that the failure of any contraction to
preserve $(3,6)$-sparsity is due to such a subgraph obstacle.
The following general lemma, which we refer to as the filling in lemma, is
useful for the identification of maximal $(3,6)$-tight subgraphs with specific
properties. See also [6]. In particular this lemma plays a role in the
identification of an obstacle subgraph.
###### Lemma 3.3.
Let $G\in{\mathfrak{P}}_{1}$ and let $H$ be an embedded triangulated disc
graph in G.
(i) If $K$ is a $(3,6)$-tight subgraph of $G$ with $K\cap H=\partial H$ then
$\partial H$ is a $3$-cycle graph.
(ii) If $K$ is a $(3,6)$-sparse subgraph of $G$ with $f(K)=7$ and $K\cap
H=\partial H$ then $\partial H$ is either a $3$-cycle or $4$-cycle graph.
###### Proof.
(i) Write $H^{c}$ for the subgraph of $G$ which contains the edges of
$\partial H$ and the edges of $G$ not contained in $H$. Since $G=H^{c}\cup H$
and $H^{c}\cap H=\partial H$ we have
$6=f(G)=f(H^{c})+f(H)-f(\partial H).$
Since $f(H^{c})\geq 6$ we have $f(H)-f(\partial H)\leq 0$. On the other hand,
$6\leq f(K\cup H)=f(K)+f(H)-f(\partial H)$
and $f(K)=6$ and so it follows that $f(H)-f(\partial H)=0$. It follows that
$\partial H$ is a 3-cycle.
(ii) The argument above leads to $-1\leq f(H)-f(\partial H)$ and hence to the
inequality $-1\leq f(D)-f(\partial D)$. This implies that $\partial H$ is
either a $3$-cycle or $4$-cycle graph. ∎
###### Lemma 3.4.
Let $G\in{\mathfrak{P}}_{1}$, let $e$ be a contractible $FF$ edge in $G$, and
let $G^{\prime}$ be the simple graph arising from the contraction move $G\to
G^{\prime}$ associated with $e$. Then either $G^{\prime}\in{\mathfrak{P}}_{1}$
or $e$ lies on the boundary of a subgraph $G_{1}$ of $G$ where
$G_{1}\in{\mathfrak{P}}_{1}$.
###### Proof.
Assume that $G^{\prime}\notin{\mathfrak{P}}_{1}$. It follows that $G^{\prime}$
must fail the $(3,6)$-sparsity count. Thus there exists a subgraph $K$ of $G$
containing $e$ for which the edge contraction results in a graph $K^{\prime}$
satisfying $f(K^{\prime})<6$. Let $e=vw$ and let $c$ and $d$ be the facial
$3$-cycles which contain $e$. If both $c$ and $d$ are subgraphs of $K$ then
$f(K)=f(K^{\prime})<6$, which contradicts the sparsity count for $G$. Thus $K$
must contain at most one of these facial $3$-cycles.
_Case 1_. Suppose first that $K$ is a maximal subgraph among all subgraphs of
$G$ which contain the cycle $c$, do not contain $d$, and for which contraction
of $e$ results in a simple graph $K^{\prime}$ which fails the $(3,6)$-sparsity
count. Note that $f(K)=f(K^{\prime})+1$ which implies $f(K)=6$ and
$f(K^{\prime})=5$. In particular, $K$ is $(3,6)$-tight, and is a connected
graph.
Let $(B_{0},\lambda)$ be a face graph for $G$ with an associated face graph
$(B,\lambda)$ for a triangulated surface graph for $S=(V,E,F)$ for
${\mathcal{P}}$ containing $G$. In particular $(B,\lambda)$ provides a
faithful topological embedding $\pi:S\to{\mathcal{P}}$. Let
$X(K)\subset{\mathcal{P}}$ be the closed set $\pi(E(K))$ and let
$\tilde{X}(K)$ be the union of $X(K)$ and the embeddings of the faces for the
facial $3$-cycles belonging $K$. Finally, let $U_{1},\dots,U_{n}$ be the
maximal connected open sets of the complement of $\tilde{X}(K)$ in
${\mathcal{P}}$.
Note that each such connected open set $U_{i}$ is determined by a set
${\mathcal{U}}_{i}$ of embedded faces of $S$ with the property: each pair of
embedded faces of $U_{i}$ are the endpoints of a path of edge-sharing embedded
faces in ${\mathcal{U}}_{i}$. From the topological nature of ${\mathcal{P}}$
it follows that $U_{i}$ has one of the following 3 properties.
(i) $U_{i}$ is an open disc.
(ii) $U_{i}$ is the interior of a Möbius strip.
(iii) The complement of $U_{i}$ is not connected.
The third property cannot hold since the embedding of $K$ is contained in the
complement of $U_{i}$ and contains the boundary of $U_{i}$, and yet $K$ is a
connected graph.
From the second property it follows that $K$ is a planar graph, since it can
be embedded in the complement of $U_{i}$ and this is a triangulated disc. This
is also a contradiction, since the edge contraction of a contractible $FF$
edge in a planar triangulated graph preserves $(3,6)$-sparsity.
Each set $U_{i}$ is therefore the interior of the closed set determined by an
embedding of a triangulated disc graph in $B$, say $H(U_{i})$. (Indeed, the
facial 3-cycles defining $H(U_{i})$ are those whose torus embedding have
interior set contained in $U_{i}$.) We may assume that $U_{1}$ is the open set
that contains the hole of $G$. (More precisely, $U_{1}$ contains the open set
corresponding to the embedded faces for the triangulated disc in $B$ that
determines $B_{1}$.)
For $i>1$ by the filling in lemma, Lemma 3.3, it follows that $\partial
H(U_{i})$ is a 3-cycle. By the maximality of $K$ we have $k=1$ (since adding
the edges and vertices of $S$ interior to these nonfacial 3-cycles gives a
subgraph of $G$ with the same freedom count). Thus, $K$ is a subgraph of $G$
and is equal to the surface graph for ${\mathcal{P}}$ defined by $B$ and the
embedded triangulated disc $H(U_{1})$. Thus, with $G_{1}=K$, the proof is
complete in this case.
_Case 2._ It remains to consider the case for which $K$ contains neither of
the facial $3$-cycles which contain $e$. Thus $f(K)=f(K^{\prime})+2$ and
$f(K)\in\\{6,7\\}$. Once again we assume that $K$ is a maximal subgraph of $G$
with respect to these properties and consider the complementary components
$U_{1},\dots,U_{k}$. As before each set $U_{i}$ is homeomorphic to a disc and
determines an embedded triangulated disc graph $H(U_{i})$, one of which, say
$H(U_{1})$, contains the triangulated disc which defines $G$. The filling in
lemma and maximality now implies that each boundary of $H(U_{i})$, for $i>1$,
is a $4$-cycle. By the maximality of $K$, we see once again that $k=1$ (since
adding the missing edge for such a $4$-cycle gives a subgraph of $G$ with a
lower freedom count) and the proof is completed as before. ∎
The filling in lemma holds for graphs in
${\mathfrak{P}}_{2},{\mathfrak{P}}_{3}$, with the same proof, and we may
extend Lemma 3.4 to these families of graphs.
###### Lemma 3.5.
Let $G\in{\mathfrak{P}}_{k}$, for $k=1,2$ or $3$, let $e$ be a contractible
$FF$ edge in $G$, and let $G^{\prime}$ be the simple graph arising from the
contraction move $G\to G^{\prime}$ associated with $e$. Then either
$G^{\prime}\in{\mathfrak{P}}_{k}$ or $e$ lies on the boundary of a subgraph
$G_{1}$ of $G$ where $G_{1}\in{\mathfrak{P}}_{l}$, for some $1\leq l\leq k$.
###### Proof.
The proof follows the same pattern as in the case $k=1$. Thus we assume that
$G^{\prime}\notin{\mathfrak{P}}_{k}$ and consider a subgraph $K$ of $G$ which
is maximal amongst all subgraphs which do not contain one (or both, according
to Cases 1 and 2) of the facial 3-cycles incident to $e$ and whose contraction
$K^{\prime}$ has freedom number $f(K^{\prime})=5$ (or $4$). We consider the
open set which is the complement of the embedding in ${\mathcal{P}}$ of $K$
and its facial 3-cycles. (The embedding here is denoted $\tilde{X}(K)$ in the
case $k=1$.) This open set has components $U_{1},\dots,U_{n}$ and each is the
interior of a union of an edge-connected set of ${\mathcal{P}}$-embedded
facial 3-cycles of $S$.
It follows as before, from the topological nature of ${\mathcal{P}}$, from
$(3,6)$-sparsity and from the filling in lemma, that each $U_{j}$ is an open
disc. Moreover, in the case $k=2$ each $U_{j}$ contains at least one of the 2
discs $D_{1},D_{2}$ which defines $G$ and so $n$ is 1 or 2 and it follows that
$K$ belongs to ${\mathfrak{P}}_{n}$, as desired. Similarly, for $k=3$, each
$U_{j}$ contains at least one of the 3 discs $D_{1},D_{2},D_{3}$ which defines
$G$ and so $n$ is 1, 2 or 3 and $K$ belongs to ${\mathfrak{P}}_{l}$ for some
$1\leq l\leq 3$. ∎
## 4\. The uncontractibles
Let $k=1,2$ or $3$. By the finiteness of a graph $G$ in ${\mathfrak{P}}_{k}$
it is evident that it admits a full reduction sequence
$G=G_{1}\to G_{2}\to\dots\to G_{n}$
where (i) each graph is in ${\mathfrak{P}}_{k}$, (ii) each move $G_{k}\to
G_{k+1}$ is an edge contraction for an $FF$ edge, as before, and (iii) $G_{n}$
is _irreducible_ in ${\mathfrak{P}}_{k}$ in the sense that it admits no edge
contraction to a graph in ${\mathfrak{P}}_{k}$.
Let us say that a surface graph is _uncontractible_ is every $FF$ edge lies on
a nonfacial 3-cycle. An uncontractible graph $G\in{\mathfrak{P}}_{k}$ is
certainly an irreducible graph in ${\mathfrak{P}}_{k}$ but we show in the next
section that the two classes coincide. In Section 4.2 we shall prove that
there are 8 uncontractible graphs but first we establish some useful
properties of the irreducible graphs.
### 4.1. Some properties of irreducible graphs
We say that a $k$-cycle of edges in $G$, $c$ say, is a _planar $k$-cycle_ in
$G$ if there is a face graph $(B_{0},\lambda)$ for $G$, with containing face
graph $(B,\lambda)$ for the triangulated surface graph $B/\lambda$ for
${\mathcal{P}}$, such that $c$ is determined by the boundary cycle $\hat{c}$
of a triangulated disc $D$ in $B$. Note that the holes of $G$ are defined by
embedded triangulated discs $D_{i}$ in $B_{0}$, and so we may say that a
planar cycle $c$ in $G$ _contains a hole of $G$_ if $D$ contains such an
embedded disc $D_{i}$. Also we may say that $c$ _properly contains a hole_ if
there is such an inclusion which is proper.
The following lemma shows that an irreducible (3,6)-tight
${\mathcal{P}}$-graph contains no degree 3 vertex that is incident to an $FF$
edge or lies on a planar nonfacial triangle.
###### Lemma 4.1.
Let $G$ be a graph in ${\mathfrak{P}}_{k}$, for $k=1,2$ or $3$.
(i) If $e$ is an $FF$ edge incident to a degree 3 vertex then
$G/e\in{\mathfrak{P}}_{k}$.
(ii) If $v$ is an interior vertex of $G$ and $v$ lies on a planar nonfacial
3-cycle then there is a contractible edge $vw$ with
$G/vw\in{\mathfrak{P}}_{k}$.
###### Proof.
For (i) note that since $G$ is simple $e$ is a contractible edge. Write $e=uv$
with facial 3-cycles $uvx$ and $uvy$, with $\deg v=3$. Then $e$ cannot lie on
a critical 4-, 5- or 6-cycle since one of the edges incident to $u$ would
provide an interior chord for this cycle. Also $e$ does not lie on a nonfacial
3-cycle and so (i) follows.
For (ii) let $H$ be the triangulated disc subgraph induced by the faces
incident to $v$, with vertices $v_{1},\dots,v_{n}$ in cyclic order on the
boundary of $H$. Considering the hypothesis, and relabelling, we may assume
that there is an edge $f=v_{1}v_{j}$ with $3\leq j\leq n-2$ so that the edges
$v_{3}v_{4},v_{4}v_{5},\dots,v_{j-1}v_{j},f$ are the boundary edges of a
triangulated disc. It is straightforward to show that one of the vertices
$v_{2},\dots,v_{j-1}$ has degree 3, and so (i) applies. ∎
The next lemma shows that if $G$ is irreducible then there is no critical
$m$-cycle which properly contains an $m$-hole.
###### Lemma 4.2.
Let $G$ be a graph in ${\mathfrak{P}}_{k}$, for $k=1,2$ or $3$, such that
there is a critical $m$-cycle, for $m=1,2$ or 3, which properly contains an
$m$-cycle hole, so that $G=G_{1}\cup A$ where $A$ is the annular graph
determined by the two $m$-cycles. Then $G$ is constructible from $G_{1}$ by
planar vertex splitting moves.
###### Proof.
Fix $k$ and $m\leq k$. Suppose that $|V(G)|=|V(G_{1})|+1.$ Then there is a
degree $3$ vertex on the boundary of the relevant hole of $G$. By Lemma 4.1(i)
$G$ is constructible from $G_{1}$ by a single planar vertex splitting move.
Assume next that the lemma is true whenever $|V(G)|=|V(G_{1})|+j$, for
$j=1,2,\dots,N-1$, and suppose that $|V(G)|=|V(G_{1})|+N$. Let $e$ be an
interior edge of the annular graph $A$. If the contraction $G/e$ is in
${\mathfrak{P}}_{m}$ then it follows from the induction step that $G$ is
constructible from $G_{1}$ by planar vertex splitting moves. So, by Lemma 3.5
we may assume (i), that $e$ lies on a critical $m$-cycle, with associated
graph $G^{\prime}$, or (ii), that $e$ lies on a nonfacial 3-cycle of $G$. In
the former case we may take $G_{1}^{\prime\prime}$ to be the union of $G_{1}$
and $G_{1}^{\prime}$. Then $G_{1}^{\prime\prime}$ is also in
${\mathfrak{P}}_{k}$. Since $|V(G_{1}^{\prime\prime})|-|V(G_{1})|<N$ and
$|V(G)|-|V(G_{1}^{\prime\prime})|<N$ it follows from the induction step that
the lemma holds for $G$ and $G_{1}$. So we may assume that (ii) holds, and
moreover, in view of Lemma 4.1(ii), that $e$ lies on a nonplanar nonfacial
3-cycle. To complete the proof we observe that this is not possible when $e$
is incident to a vertex on the hole which is not a vertex of the critical
$m$-cycle. ∎
### 4.2. The uncontractible graphs.
We now identify 8 uncontractible $(3,6)$-tight ${\mathcal{P}}$-graphs. Figure
3 gives two uncontractibles specified by face graphs and Figure 4 gives three
further uncontractibles as embedded graphs in ${\mathcal{P}}$. Here
${\mathcal{P}}$ is represented as a disc or a rectangle, with diagonally
opposite points of the boundary identified. The 3 remaining irreducibles are
given in Figure 5. The notation $G^{h}_{n}$ indicates that $n$ is the number
of vertices and $h=h(G)$ is the minimum _hole incidence degree_ given in the
following definition.
Figure 3. The uncontractible graphs $G^{2}_{3}\in{\mathfrak{P}}_{1}$ and
$G^{3}_{4}\in{\mathfrak{P}}_{3}$.
Figure 4. The uncontractible graphs $G^{0}_{7},$ $G^{2}_{6,\alpha}$ and
$G^{2}_{6,\beta}$ in ${\mathfrak{P}}_{3}$.
Figure 5. The uncontractible graphs $G^{1}_{5}$ in ${\mathfrak{P}}_{2}$ and
$G^{1}_{6,\alpha},$ $G^{1}_{6,\beta}$in ${\mathfrak{P}}_{3}$.
###### Definition 4.3.
Let $v$ be a vertex of $G=(V,E,F)\in{\mathfrak{P}}_{k}$ for some $k=1,2,3$.
Then (i) $\deg_{F}(v)$ is the number of facial 3-cycles incident to $v$, (ii)
$\deg_{h}(v)=\deg(v)-\deg_{F}(v)$ is the _hole incidence degree_ for $v$, and
(iii) $h(G)$ is the minimum hole incidence degree,
$h(G)=\min_{v}\operatorname{deg}_{h}(v)$.
In what follows, we shall usually consider graphs as ${\mathcal{P}}$-graphs,
with facial structure. However, let us note that as graphs: $G^{2}_{3}$ is the
triangle graph $K_{3}$; $G_{4}^{3}$ is $K_{4}$; $G_{7}^{0}$ is the cone graph
over $K_{3,3}$; $G_{5}^{1}$ is $K_{5}-e$. Also the four remaining graphs, each
with 6 vertices, are depletions of $K_{6}$ by 3 edges where these edges (i)
form a copy of $K_{3}$, (ii) are disjoint, (iii) have one vertex shared by 2
edges, (iv) have 2 vertices of degree 1. These graphs account for all possible
$(3,6)$-tight graphs on $n$ vertices for $n=3,4,5,6$, together with 1 of the
26 such graphs for $n=7$. (We remark that for $n=8,9,10$ the number of
$(3,6)$-tight graphs rises steeply, with values 375, 11495, 613092 [10].)
###### Proposition 4.4.
Let $G$ be an uncontractible graph in ${\mathfrak{P}}_{k}$, for $k=1,2$ or
$3$, which has an interior vertex. Then $k=3$ and $G$ is the hexagon graph
$G^{0}_{7}$.
###### Proof.
Let $z$ be an interior vertex in $G$. Let $X(z)$ be the subgraph of $G$
induced by $z$ and its neighbours. Assume that $z$ has degree $n$ and label
its neighbours, in cyclical order, as $v_{1},v_{2},\dots,v_{n}$. Then $X(z)$
has $n$ edges that are incident to $z$, plus $n$ perimeter edges
$v_{1}v_{2},v_{2}v_{3},\dots,v_{n}v_{1}$, and additional edges between non-
adjacent vertices $v_{1},\dots,v_{n}$. Since $G$ is uncontractible there exist
at least $\left\lceil\frac{n}{2}\right\rceil$ additional edges. It follows now
from the (3,)-sparsity that $\operatorname{deg}z=n\geq 6$. Also, since $G$ is
uncontractible there can be no vertices $v_{i}$ of degree $3$, since otherwise
the $FF$ edge $zv_{i}$ would be a contractible edge. For the same reason each
vertex $v_{i}$ has at least one additional edge $v_{i}v_{j}$ for some $j$.
Suppose that there is an additional edge $v_{i}v_{j}$ such that the
(nonfacial) 3-cycle $zv_{i}v_{j}z$ is a planar 3-cycle. Then $G$ contains a
triangulated disc $D$ with 3-cycle boundary with at least 4 vertices. Such a
graph $D$ has a contractible $FF$ edge with an interior vertex and so this
edge is also contractible in $G$, a contradiction.
Consider one of the additional edges, $v_{i}v_{j}$ with $i<j$, and let
$i^{\prime}\in\\{i+1,\dots,j-1\\}$. We claim that for every additional edge
$v_{i^{\prime}}v_{j^{\prime}}$ we have $j^{\prime}\notin\\{i+1,\dots,j-1\\}$.
Indeed, if this is not the case then there is a non-facial planar 3-cycle $c$
described by the edges
$zv_{i^{\prime}},v_{i^{\prime}}v_{j^{\prime}},v_{j^{\prime}}z$ and by the
previous paragraph this is a contradiction. Thus the additional edges have
this _non-nested_ property. It follows by a simple inductive argument that the
embedded graph $X(z)$ has faces with boundary cycles of length at most 4 since
otherwise there must be perimeter vertices of degree 3. These 4-cycles are
planar 4-cycles and so by Lemma 4.2 there are 3 holes. Thus $n=6$ and $G$ is
the hexagon graph $G^{0}_{7}$. ∎
The next lemma is key to the determination of the uncontractible graphs in
${\mathfrak{P}}_{k}$ for $k=2$ or $3$.
###### Lemma 4.5.
Let $G\in{\mathfrak{P}}_{k}$, for $k=2$ or 3, be an uncontractible (3,6)-tight
graph with no interior vertex and let $v_{1}$ be a vertex with
$\operatorname{deg}_{h}(v_{1})=1$ which lies on the boundary of a 4-cycle hole
of $G$ with edges $v_{1}v_{2},v_{2}v_{3},v_{3}v_{4},v_{4}v_{1}$. Then
$\operatorname{deg}(v_{1})=4$ if $v_{1}$ is not adjacent to $v_{3}$ and
$\operatorname{deg}(v_{1})=5$ otherwise.
###### Proof.
Let $v_{2}=w_{1},w_{2},\dots,w_{n}=v_{4}$ be the neighbours of $v_{1}$ in
cyclical order. Since $\operatorname{deg}_{h}(v_{1})=1$, we also have the
edges $w_{1}w_{2},\dots,w_{n-1}w_{n}$. Note that $\deg(v_{1})\geq 4$ since if
the degree is 3 then the edge $v_{1}w_{2}$ is contractible.
Case (a). $v_{3}\neq w_{i}$, for every $i\in\\{2,\dots,m-1\\}$.
Suppose that $n\geq 5$. It follows from the uncontractibility that for each
vertex $w_{i},2\leq i\leq n-1,$ there is an associated edge $w_{i}w_{r}$ for
some $1\leq r\leq n$ and an associated edge $w_{i+1}w_{s}$ for some $s>r$.
Since there are at most 3 holes there is an edge $w_{i}w_{i+1}$ for which the
associated cycle through $w_{i},w_{i+1},w_{s},w_{r}$ is triangulated by faces.
We claim that (i) it is a 4-cycle and (ii) it is triangulated by 2 faces. Note
that at most one of the edges $w_{i}w_{s},w_{i+1}w_{r}$ exists. Indeed,
although we can have $K_{4}\to{\mathcal{P}}$ with 3 faces this implies the
existence of a degree 3 vertex and hence a contractible edge incident to it, a
contradiction. If the face of the triangulation which contains $w_{i}w_{i+1}$
has third vertex $w$ not equal to $w_{s}$ or $w_{r}$, then at least one of the
edges $w_{i}w$, $w_{i+1}w$ is contractible, a contradiction. Since an interior
vertex $w$ does not exist the implied cycle is a 4-cycle and (i) and (ii)
hold.
Since $G$ is uncontractible $w_{i}w_{i+1}$ lies in a non-facial 3-cycle. Since
$v_{1}w_{j}$ is also an $FF$ edge for every $j\in\\{2,\dots,n-1\\}$, it
follows that there are just two candidate non-facial 3-cycles:
$w_{i-1}w_{i}w_{i+1}w_{i-1}$ or $w_{i}w_{i+1}w_{i+2}w_{i}$.
(i):
If $w_{i}w_{i+1}$ lies on the cycle $w_{i-1}w_{i}w_{i+1}w_{i-1}$, then the
4-cycle $w_{i-1}v_{1}w_{r}w_{i+1}w_{i-1}$ contains strictly the hole boundary
$v_{1}v_{2}v_{3}v_{4}v_{1}$, contradicting Lemma LABEL:l:4holelemma. Note that
this 4-cycle does contain the hole in our sense since the shading in Figure 6
indicates a triangulated disc in ${\mathcal{P}}$ with boundary equal to this
4-cycle.
$v_{1}$$w_{n}$$w_{r}$$w_{i+1}$$w_{i}$$w_{i-1}$$w_{1}$$v_{3}$ Figure 6. The
4-cycle $w_{i-1}v_{1}w_{r}w_{i+1}w_{i-1}$ contains strictly the 4-hole
$v_{1}v_{2}v_{3}v_{4}v_{1}$.
(ii):
If $w_{i}w_{i+1}$ lies on the cycle $w_{i}w_{i+1}w_{i+2}w_{i}$, then, noting
that $w_{i+2}w_{i}$ is an edge, we claim that the 5-cycle
$w_{i}v_{1}w_{r}w_{i+1}w_{i+2}w_{i}$ contains all the holes, which is a
contradiction. To see this note that by Lemma 4.2 the 4-cycle
$v_{1}w_{r}w_{i}w_{i+2}v_{1}$ contains no holes. See Figure 7.
$v_{1}$$w_{n}$$w_{r}$$w_{i+1}$$w_{i+2}$$w_{i}$$w_{1}$$v_{3}$ Figure 7. The
5-cycle $w_{i}v_{1}w_{r}w_{i+1}w_{i+2}w_{i}$ contains all the holes.
Hence none of the edges $w_{2}w_{3},w_{3}w_{4},\dots,w_{n-2}w_{n-1}$ is an
$FF$ edge. Also, the same holds for the edge $v_{2}w_{2}$, since it cannot lie
in no non-facial 3-cycle. Thus every edge of the form
$v_{2}w_{2},w_{2}w_{3},w_{3}w_{4},\dots,w_{n-1}w_{n}$ is on the boundary of a
hole. Since every edge $v_{1}w_{j}$ is an $FF$ edge, $j=2,\dots,n-1$, it
follows that $G$ contains at least $\left\lceil\frac{n}{2}+1\right\rceil$
holes. Thus, $n=4$.
Case (b). $v_{4}=w_{i_{0}}$, for some $i_{0}\in\\{2,\dots,n-1\\}$.
We have $\operatorname{deg}(v)\geq 5$ since $G$ is a simple graph. Suppose
that $n\geq 6$. As in case (a) we may assume that there exists an $FF$ edge
$w_{i}w_{i+1}$ with $i>1$ and $i+1<i_{0}$, and with vertex $w_{r}$ as before.
(See Figure 8.) Then, the only possible non-facial 3-cycle for $w_{i}w_{i+1}$
is $v_{3}w_{i}w_{i+1}v_{3}$. However, this would lead to a contradiction since
the 4-cycle $w_{i}v_{3}v_{4}v_{1}w_{i}$ strictly contains the hole
$v_{1}v_{2}v_{3}v_{4}v_{1}$.
.
$v_{1}$$v_{4}$$v_{2}$$v_{3}$$w_{i}$$w_{r}$$w_{i+1}$$v_{3}$ Figure 8. The
4-cycle $v_{4}v_{3}v_{1}w_{i}v_{4}$ contains strictly the 4-hole
$v_{1}v_{2}v_{3}v_{4}v_{1}$.
Thus, each $w_{i}w_{i+1}$ is not an $FF$ edge. Similarly, we can argue that
such a 4-cycle would be created if $w_{i_{0}-1}w_{i_{0}}$ was an $FF$ edge.
Thus again we have that $w_{1}w_{2}$ is not an $FF$ edge, since it does not
lie on a non-facial 3-cycle. Thus, the edges
$w_{1}w_{2},w_{2}w_{3},w_{3}w_{4}$ should lie on the boundaries of different
holes, which again contradicts the number of the holes of $G$. Thus
$\operatorname{deg}(v_{1})=5$. ∎
###### Proposition 4.6.
Let $G\in{\mathfrak{P}}_{k}$, for $k=1,2$ or 3, be an uncontractible
(3,6)-tight graph with no interior vertex. If there exists a vertex $v_{1}\in
V(G)$ with $\operatorname{deg}_{h}(v_{1})=1$ then $G$ is one of the graphs
$G^{1}_{6,\alpha},G^{1}_{6,\beta},G^{1}_{5}$.
###### Proof.
Case (a). Assume first that $v_{1}$ lies on the 4-cycle boundary of the hole
$H_{1}$, with vertices $v_{1},v_{2},v_{3},v_{4}$, and let
$v_{2}=w_{1},w_{2},\dots,w_{n}=v_{4}$ be all the neighbours of $v_{1}$. Since
$\operatorname{deg}_{h}(v_{1})=1$ the edges $w_{1}w_{2},\dots,w_{n-1}w_{n}$
exist. Also $\operatorname{deg}(v_{1})\geq 4$ since otherwise $v_{1}w_{2}$ is
a contractible $FF$ edge. There are two subcases.
(i):
$v_{3}\neq w_{i}$, for every $i\in\\{1,2,\dots,m\\}$.
By Lemma 4.5 we have $\operatorname{deg}(v_{1})=4$. By the uncontractibility
of the edges $v_{1}w_{2}$ and $v_{1}w_{3}$ the edges $w_{2}w_{4}$ and
$w_{1}w_{3}$ must exist. Thus $G$ contains the graph in Figure 9, except
possibly for the edge $v_{3}w_{3}$. It follows that the 4-cycle
$w_{1}w_{2}w_{4}w_{3}w_{1}$ must be the boundary of a 4-hole $H_{2}$, since
otherwise the 5-cycle $v_{1}w_{1}w_{3}w_{2}w_{4}v_{1}$ contains all the holes,
in the sense, as before, of being the boundary of an embedded disc, $B$ say,
which contains the holes. This contradicts $(3,6)$-tightness. We claim now
that the edge $v_{3}w_{2}$ or $v_{3}w_{3}$ must exist, for otherwise there is
a contractible edge in $B$. To see this check that since
$\operatorname{deg}(v_{3})\geq 3$, there exists a vertex $z$ in the interior
of the 5-cycle $v_{3}w_{4}w_{2}w_{3}w_{1}v_{3}$, such that $v_{3}z\in E(G)$.
Since $v_{3}z$ does not lie on a non facial 3-cycle, it follows that it lies
on the boundary of the third 4-hole. Thus, if $v_{3}w_{3}$ is not allowed, we
may assume by symmetry that $w_{1}z$ is an $FF$ edge in $E(G)$, so it lies on
the non-facial 3 cycle $w_{1}zw_{2}w_{1}$. Hence the third hole is described
by the 4-cycle $w_{4}v_{3}zw_{1}w_{4}$. However, this implies that $zw_{3}\in
E(G)$, which is a contractible $FF$ edge, so we have proved the claim. Hence
without loss of generality $G$ contains the subgraph $G^{1}_{6,\alpha}$ as
indicated in Figure 9. Since $G$ is uncontractible it follows that
$G=G^{1}_{6,\alpha}$.
$v_{1}$$w_{4}$$w_{3}$$w_{2}$$w_{1}$$v_{3}$ Figure 9. The uncontractible graph
$G^{1}_{6,\alpha}$.
(ii):
$v_{3}=w_{i_{0}}$ for some $i_{0}\in\\{3,\dots,n-2\\}$.
By Lemma 4.5 $\operatorname{deg}(v_{1})=5$ and so $v_{3}=w_{3}$. Since
$v_{1}w_{2}$ is an $FF$ edge, it follows that $w_{2}w_{4}\in E(G)$ and so $G$
contains the graph $G=G^{1}_{6,\beta}$ of Figure 10. Since $G$ is
uncontractible it follows that this subgraph is equal to $G$.
$v_{1}$$v_{4}$$w_{4}$$w_{2}$$v_{2}$$v_{3}$ Figure 10. The uncontractible graph
$G^{1}_{6,\beta}$.
Case (b). Let $v_{1}$ lie on the boundary of a 5-hole $H$ with boundary edges
$v_{1}v_{2}$, $v_{2}v_{3}$, $v_{3}v_{4}$, $v_{4}v_{5}$, $v_{5}v_{1}$. We may
assume that $\operatorname{deg}_{h}(v_{i})=2$, for every $i=2,3,4,5$, since
otherwise there is a vertex $v$ on a 4-hole of $G$. Since $G$ has two holes it
is straightforward to check that $\operatorname{deg}(v_{1})=4$ and that the
second hole is described by the 4-cycle $v_{2}v_{3}v_{5}v_{4}v_{2}$. Thus we
obtain that $G$ is the uncontractible (3,6)-tight given by Figure 11.
$v_{1}$$v_{5}$$v_{4}$$v_{3}$$v_{2}$ Figure 11. The uncontractible graph
$G^{1}_{5}$.
∎
Note that in the proof of the previous result we have determined the
uncontractible graphs in 2-holed case and shown that there is a unique
uncontractible graph, namely $G_{5}^{1}$. The next proposition completes the
proof that there are 8 base graphs.
###### Proposition 4.7.
Let $G\in{\mathcal{P}}$ be an uncontractible (3,6)-tight graph with
$\operatorname{deg}_{h}(v)\geq 2$ for all $v\in V(G)$. Then $G$ is one of the
four graphs $G^{2}_{6,\alpha},G^{2}_{6,\beta},G^{3}_{4},G_{3}^{2}$.
###### Proof.
Suppose first that $G$ has 2 or 3 holes. Then the hole boundaries have length
4 or 5 and it follows from the simplicity of the graph that every vertex is
common to at least 2 holes. Since there are either 2 or 3 holes it follows
that $|V|\leq 6$.
Case (a). Suppose that $G$ contains at least one $FF$ edge, say $v_{1}v_{2}$,
with non facial 3-cycle $v_{1}v_{2}v_{3}$, and associated 3-cycle faces
$v_{1}v_{2}v_{4}v_{1}$ and $v_{1}v_{2}v_{5}v_{1}$. We claim that one of the
edges $v_{3}v_{4}$ or $v_{3}v_{5}$ lies in $E(G)$. Suppose, by way of
contradiction, that neither edge exists. Then we show that the edge
$v_{4}v_{5}$ is also absent. Indeed, if $v_{4}v_{5}\in E(G)$, then we have two
planar 5-cycles; $v_{1}v_{4}v_{5}v_{2}v_{3}v_{1}$ and
$v_{1}v_{5}v_{4}v_{2}v_{3}v_{1}$, as in Figure 12.
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$
Figure 12. A subgraph with the 5-cycles $v_{1}v_{4}v_{5}v_{2}v_{3}v_{1}$ and
$v_{1}v_{5}v_{4}v_{2}v_{3}v_{1}$.
By the sparsity condition one of these has a vertex in the interior with 3
incident edges and the other has a single chordal edge in the interior and by
symmetry we may assume that the planar 5-cycle
$v_{1}v_{4}v_{5}v_{2}v_{3}v_{1}$ has the single chordal edge. However, of the
5 possibilities $v_{1}v_{2},v_{2}v_{4},v_{1}v_{5}$ are not available, by the
simplicity of $G$, and the edges $v_{3}v_{4},v_{3}v_{5}$ are absent by
assumption. This contradiction shows that $v_{4}v_{5}$ is indeed absent and
so, since $v_{4},v_{5}$ have degree at least 2, the edges
$v_{4}v_{6},v_{5}v_{6}$ must exist. Now the complement of the 2 3-cycle faces
is bounded by two 6-cycles. By the sparsity condition there are now only 2
further edges to add and so there must be a 5-cycle hole, a contradiction, and
so the claim holds.
Without loss of generality we suppose that $v_{3}v_{4}\in E(G)$. Since
$\operatorname{deg}_{h}(v_{2})\geq 2$, it follows that $v_{2}v_{6}\in E(G)$.
Moreover, the edges $v_{6}v_{2}$,$v_{2}v_{3}$ should be on the boundary of a
planar 4-hole $H_{1}$, and this implies that $v_{1}v_{6}\in E(G)$. Similarly
we obtain that the two remaining holes are determined by the cycles
$v_{1}v_{3}v_{4}v_{5}v_{1}$, and $v_{2}v_{5}v_{4}v_{6}v_{2}$. The resulting
(3,6)-tight triangulated surface graph is given in Figure 13 and is the
uncontractible graph $G_{6,\alpha}^{2}$.
$v_{5}$$v_{2}$$v_{3}$$v_{1}$$v_{4}$$v_{6}$ Figure 13. The uncontractible graph
with $h(G)=2$ and an $FF$ edge; $G_{6,\alpha}^{2}$.
Case (b). Suppose now $G$ has at least one 3-cycle face, $v_{1}v_{2}v_{3}$,
and no $FF$ edges. Then the edge $v_{1}v_{2}$ is on the boundary of a 4-hole
$H_{1}$, that is determined by the edges $v_{1}v_{2}$, $v_{2}v_{4}$,
$v_{4}v_{5}$ and $v_{5}v_{1}$.
To see that $|V|\neq 5$ note that without loss of generality the edge
$v_{3}v_{4}$ exists and $G$ contains the subgraph shown in Figure 14. Also,
since $v_{5}$ cannot have degree 2 at least one of the edges
$v_{5}v_{3},v_{5}v_{2}$ exists.
$v_{1}$$v_{2}$$v_{4}$$v_{5}$$v_{4}$$v_{5}$$v_{3}$
Figure 14. A necessary subgraph.
If $v_{5}v_{2}$ exists then the edge $v_{2}v_{3}$ is adjacent to a 4-cycle
hole and $v_{5}v_{3}$ is absent. We note next that the planar 5-cycle
$v_{3}v_{1}v_{5}v_{2}v_{4}v_{3}$ must contain a chord edge (and so provide the
third 4-cycle hole). The only available edge (by simplicity) is $v_{3}v_{5}$.
This however is inadmissible since it introduces a second 3-cycle face
$v_{3}v_{5}v_{1}$ adjacent to $v_{1}v_{2}v_{3}$.
Similarly, if $v_{5}v_{3}$ exists then we have the planar 6-cycle
$v_{3}v_{1}v_{5}v_{3}v_{2}v_{4}v_{3}$ and there must exist a diameter edge to
create the 2 additional 4-cycle holes. As there is no such edge we conclude
that $|V|=6$.
Introducing $v_{6}$ the fact that $v_{2}v_{3}$ and $v_{3}v_{1}$ lie on 4-cycle
hole boundaries leads to the graph $G_{6,\beta}^{2}$ indicated in Figure 15.
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{6}$$v_{4}$$v_{5}$ Figure 15. The
uncontractible graph $G_{6,\beta}^{2}$, with $h(G)=2$, no $FF$ edge and a
3-cycle face.
Case (c). Let now $G$ be a graph with no 3-cycle faces. Since
$\operatorname{deg}(v)\geq 3$ for each vertex it follows that
$\operatorname{deg}_{h}(v)=3$ and $\deg(v)=3$, for all $v\in V(G)$. Thus
$|V|=4$ and it follows that $G$ is the uncontractible (3,6)-tight graph
$G_{4}^{3}$ given by Figure 16.
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{2}$$v_{3}$$v_{4}$ Figure 16. The
uncontractible graph $G_{4}^{3}$ with $h(G)=3$.
Case (d). Finally, suppose that $G\in\mathfrak{P}_{1}$. We claim that the
graph has no faces and the surface graph is given by Figure 19.
Assume first that there exists an $FF$ edge, say $v_{1}v_{2}$, that lies on
the faces $v_{1}v_{2}v_{3}v_{1}$ and $v_{1}v_{2}v_{4}v_{1}$. Since the graph
is uncontractible, $v_{1}v_{2}$ lies on a non facial 3-cycle
$v_{1}v_{2}v_{5}v_{1}$. Note that $v_{3}v_{4}\notin E(G)$, since otherwise the
6-hole would lie inside a 5-cycle, either $v_{1}v_{3}v_{4}v_{2}v_{5}v_{1}$ or
$v_{1}v_{4}v_{3}v_{2}v_{5}v_{1}$, contradicting the sparsity of the graph. It
follows that we cannot have $|V(G)|\leq 5$. Indeed, in this case (see Figure
17) $v_{3}v_{5}\in E(G)$, since $\operatorname{deg}(v_{3})\geq 3$, and so
without loss of generality, in view of the symmetry, $v_{1}v_{3}$ is an $FF$
edge. But this edge does not lie on a non-facial 3-cycle, a contradiction.
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$ Figure 17. $|V(G)|\leq 5$ leads to
a contradiction.
Thus $|V(G)|=6$ and it remains to consider two subcases:
1. (i)
$v_{3}v_{5}\in E(G)$. In this case $v_{1}v_{3}$ lies on the non-facial 3-cycle
$v_{1}v_{3}v_{6}v_{1}$. However, this leads to a contradiction, since the
6-hole is contained either in the 5-cycle $v_{5}v_{3}v_{6}v_{1}v_{2}v_{5}$ or
in the 5-cycle $v_{6}v_{3}v_{2}v_{5}v_{1}v_{6}$. Hence by symmetry neither of
the edges $v_{3}v_{5},v_{4}v_{5}$ is allowed.
2. (ii)
$v_{3}v_{6},v_{4}v_{6}\in E(G)$. In this case, indicated in Figure 18, we may
assume that the hole is contained in the planar 6-cycle
$v_{1}v_{5}v_{2}v_{4}v_{6}v_{3}v_{1}$ and that the planar 6-cycle
$v_{1}v_{5}v_{2}v_{3}v_{6}v_{4}v_{1}$ is triangulated. This implies that
$v_{2}v_{3}$ is an $FF$ edge and so lies on non-facial 3-cycle. However, the
only candidate cycle is $v_{3}v_{2}v_{6}v_{3}$ and if $v_{2}v_{6}$ lies in
$E(G)$ then the hole is contained in the 5-cycle
$v_{1}v_{5}v_{2}v_{6}v_{3}v_{1}$, a contradiction.
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$$v_{6}$$v_{6}$ Figure 18. Edges
$v_{3}v_{6},v_{4}v_{6}$ in $G$ leads to a contradiction.
We have shown that no $FF$ edge is allowed. Suppose now that $G$ contains a
face, described by the vertices $v_{1},v_{2}$ and $v_{3}$. Since there are no
$FF$ edges, all edges $v_{1}v_{2},v_{2}v_{3}$ and $v_{1}v_{3}$ lie on the
boundary of the hole. Moreover, since they form a face of the graph, they
cannot form a 3-cycle path in the boundary of the hole. Only 3 edges of the
boundary cycle are left to be determined, so we may assume that the path
$v_{1}v_{2}v_{3}$ lies on the boundary. Therefore, without loss of generality,
there exists a vertex $v_{4}$ on the boundary that connects the two paths,
$v_{1}v_{2}v_{3}$and $v_{1}v_{3}$, so we obtain the 5-path
$v_{1}v_{3}v_{4}v_{1}v_{2}v_{3}$. But this implies that the remaining edge of
the 6-hole is $v_{1}v_{3}$, which would break the simplicity of the graph.
Hence the graph contains no faces and the proof is complete.
$v_{1}$$v_{2}$$v_{3}$ Figure 19. The uncontractible graph $G^{2}_{3}$.
∎
## 5\. The irreducibles
We show that an irreducible $(3,6)$-tight ${\mathcal{P}}$-graph is
uncontractible. Thus, if a graph $G$ in ${\mathfrak{P}}_{k}$, for $k=1,2$ or
$3$, has a contractible edge $e$ (so that $G/e$ is a simple graph) then there
exists a contractible edge $f$, which need not be the edge $e$, such that the
contracted graph is simple and satisfies the sparsity condition for membership
in ${\mathfrak{P}}_{k}$.
Recall that Lemma 3.5 identifies the obstacles to the preservation of
$(3,6)$-sparsity when contracting a contractible edge of
$G\in{\mathfrak{P}}_{k}$, namely that the edge lies on the boundary of a
subgraph of $G$ which is in ${\mathfrak{P}}_{l}$ for some $l\leq k$. For $k=1$
this boundary corresponds to a directed 6-cycle $c$ and we also refer to it in
subsequent proofs as a _critical 6-cycle_. Likewise for $k=2$ or $k=3$ the
edge $e$ lies on the boundary of one of the holes of a subgraph
$G\in{\mathfrak{P}}_{l}$ and we refer to the associated cycle as a _critical
5-cycle_ or _critical 4-cycle_.
###### Proposition 5.1.
Let $G\in{\mathfrak{P}}_{1}$ be irreducible. Then $G$ is uncontractible.
###### Proof.
Suppose that $G$ is irreducible with a contractible edge $e=xy$. By Lemma 3.5
there is a critical 6-cycle $c$, containing $e$, which is the boundary of a
subgraph $G_{1}\in{\mathfrak{P}}_{1}$. Since $c$ properly contains the hole of
$G$ this contradicts Lemma 4.2, completing the proof. ∎
###### Proposition 5.2.
Let $G$ be an irreducible graph in ${\mathfrak{P}}_{k}$, for $k=2$ or $3$.
Then $G$ is uncontractible.
###### Proof.
Suppose that $G$ is irreducible and $e$ is a contractible $FF$ edge in $G$. By
Lemma 3.5 there is a decomposition $G=G_{1}\cup A$ with $e\in\partial G_{1},$
and $G_{1}\in{\mathfrak{P}}_{l},$ for some $l\leq k$.
$e$$v$$v_{1}$$v_{2}$$w$$y$$x$$G_{1}$$v_{3}$$z$
Figure 20. A ${\mathcal{P}}$-diagram for a critical 6-cycle for $e$.
_Case $k=2,l=1$._ Figure 20 illustrates the planar 6-cycle boundary $c$ of
$G_{1}$ and we assume it includes the contractible edge $e$ and that it
contains the planar 4- and 5-cycle boundaries of the two holes of $G$. Since
$G$ is simple $c$ has 6 distinct vertices.
For the first part of the proof we show that $G$ contains $vv_{1}$, perhaps
after relabelling $v_{1},v_{2}$, that $yvv_{1}v_{2}v_{3}y$ is the boundary of
the 5-cycle hole, and that $xvv_{1}wx$ is the boundary of the 4-cycle hole.
Note that $G_{1}$ is a contractible graph, for otherwise, by the previous
section, $G_{1}=G_{3}^{2}$, with 3 vertices. By Proposition 5.1 $G_{1}$ is
reducible and so there is an $FF$ edge edge $h$ with
$G_{1}/h\in{\mathfrak{P}}_{1}$. If $h$ lies on a critical 6-cycle $c^{\prime}$
in $G$ then it necessarily lies on a critical 6-cycle in $G_{1}$. This is
because the subpath of $c^{\prime}$ which is interior to $c$ must have the
same length as one of boundary paths of $c$ between the corresponding
vertices. (Otherwise the 6-cycle hole is contained in a planar cycle of length
at most 5.) Thus, since $G$ is irreducible, $h$ must lie on a nonfacial
3-cycle in $G$ with some edges that are internal to $c$. To avoid sparsity
violation there must be 2 such edges, say $h_{1},h_{2}$. Moreover, since $h$
is a contractible edge in $G_{1}$ the edges $h_{1},h_{2}$ form a diameter of
the 6-cycle $c$. This diameter together with subpaths of $c$, yields two
planar 5-cycles which contain the holes of $G$. Considering the 5-cycle hole,
Lemma 4.2 implies that, perhaps after relabelling, the pair $h_{1},h_{2}$ is
equal to the pair $yv,vv_{1}$ or to a pair $wu,uv_{3}$ for some vertex $u\neq
v$ interior $c$. In the first case $yvv_{1}v_{2}v_{3}y$ is the boundary of the
5-cycle hole and, by a further application of Lemma 4.2, $xvv_{1}wx$ is the
boundary of the 4-cycle hole. The second case cannot occur, since one of the
edges $xv,yv$ must be an $FF$ edge, and one can see that it does not lie on a
nonfacial 3-cycle or a critical 4-, 5- or 6-cycle.
For the next part of the proof we show that $G_{1}$ has no interior vertices.
Let $u$ be an interior vertex of $G_{1}$ and let $f$ be one of its incident
$FF$ edges. Then since $G$ is irreducible, by the hole inclusion lemma, Lemma
4.2, $f$ does not lie on a critical 6-cycle. Also if $f$ lies on a nonfacial
3-cycle then by Lemma 4.1 it lies on a nonplanar nonfacial 3-cycle. It follows
from the $(3,6)$-sparsity of $G$ that $\deg v\geq 6$ and so there are at least
3 distinct nonplanar nonfacial 3-cycle through $u$. However this implies that
every hole of $G$ is contained in a planar 4-cycle, a contradiction.
$e$$v$$v_{1}$$v_{2}$$w$$y$$x$$G_{1}$$v_{3}$$z$$H$$H$
Figure 21. $G_{1}$ has no interior vertex and $z=v_{1}$.
Since $z$ is not an interior vertex of $G_{1}$ it is equal to $v_{1}$ (see
Figure 20). By Lemma 4.1(i) we have $\deg(v_{2})\geq 4$ and $\deg(v_{3})\geq
4$. Since $G_{1}$ is $(3,6)$-tight it follows that both vertices have degree 4
and that $G$ must have the structure indicated in Figure 21. In particular,
$v_{3}w$ does not lie on a nonfacial 3-cycle or a critical cycle and so
$G/v_{3}w$ is reducible, a contradiction.
_Case $k=2,l=2$._ We argue by contradiction and assume that $G$ is irreducible
and $e$ is a contractible $FF$ edge in $G$ which, by Lemma 3.5, lies on the
boundary of the proper subgraph $G_{1}\in{\mathfrak{P}}_{2}$. Each of the two
holes of $G_{1}$ must contain a hole of $G$, with the boundary cycles are of
the same length. By Lemma 4.2 this is a contradiction.
_Case $k=3,l=2$._ This case follows similarly. ∎
## 6\. Constructibility and 3-rigidity
Combining results of the previous sections we obtain the following
construction theorem and the proof of Theorem 1.1.
###### Theorem 6.1.
Let $G$ be a simple (3,6)-tight graph which is embeddable in the real
projective plane ${\mathcal{P}}$. Then $G$ is constructible by a finite
sequence of planar vertex splitting moves from at least one of the eight
${\mathcal{P}}$-graphs,
$G_{3}^{2},G_{4}^{3},G_{5}^{1},G_{6,\alpha}^{1},G_{6,\beta}^{1},G_{6,\alpha}^{2},G_{6,\beta}^{2},G_{7}^{0}$.
###### Proof.
As we have observed at the beginning of Section 4 it is evident that $G$ can
be reduced to an irreducible (3,6)-tight ${\mathcal{P}}$-graph, $H$ say, by a
sequence of planar edge contraction moves. By the results of Section 5 the
irreducible graph $H$ is uncontractible, and so, by the results of Section 4,
it is equal to one of the eight uncontractible ${\mathcal{P}}$-graphs. Since a
planar edge-contraction move is the inverse of a planar vertex splitting move
the proof is complete. ∎
###### Proof of Theorem 1.1.
Let $G$ be the graph of a partial triangulation of the real projective plane.
If $G$ is minimally 3-rigid then it is well-known that $G$ is necessarily
$(3,6)$-tight [11].
Suppose on the other hand that $G$ is $(3,6)$-tight. Then, by Theorem 6.1 the
graph $G$ is constructible by planar vertex splitting moves from one of the
eight uncontractible ${\mathcal{P}}$-graphs, each of which has fewer than $8$
vertices. It is well-known that all $(3,6)$-tight graphs with fewer than $8$
vertices are minimally 3-rigid. Since vertex splitting preserves minimal
3-rigidity (Whiteley [14]) it follows that $G$ is minimally 3-rigid. ∎
Acknowledgements. This research was supported by the EPSRC grant EP/P01108X/1
for the project _Infinite bond-node frameworks_ and by a visit to the Erwin
Schroedinger Institute in September 2018 in connection with the workshop on
_Rigidity and Flexibility of Geometric Structures_.
## References
* [1] D. W. Barnette, Generating the triangulations of the projective plane, J. Comb. Theory33 (1982), 222-230.
* [2] D. W. Barnette and A. Edelson, All orientable 2-manifolds have finitely many minimal triangulations, Israel. J. Math., 62 (1988), 90-98.
* [3] D.W. Barnette, and A.L. Edelson, All 2-manifolds have finitely many minimal triangulations, Israel J. Math., 67 (1989), 123-128.
* [4] A. Cauchy, Sur les polygones et polyèdres. Second Mémoir. J École Polytechn. 9 (1813) 87-99; Oeuvres. T. 1. Paris 1905, pp. 26-38.
* [5] J. Cruickshank, D. Kitson and S.C. Power, The generic rigidity of triangulated spheres with blocks and holes, J. Combin. Theory Ser. B 122 (2017), 550-577.
* [6] J. Cruickshank, D. Kitson and S.C. Power, The rigidity of a partially triangulated torus, Proc. London Math. Soc., 2018, https://doi.org/10.1112/plms.12215
* [7] W. Finbow-Singh and W. Whiteley, Isostatic block and hole frameworks, SIAM J. Discrete Math. 27 (2013) 991-1020.
* [8] W. Finbow-Singh, E. Ross and W. Whiteley, The rigidity of spherical frameworks: Swapping blocks and holes in spherical frameworks, SIAM J. Discrete Math. 26 (2012), 280-304.
* [9] N. D. Gilbert and T. Porter, Knots and Surfaces, Oxford University Press, 1994.
* [10] G. Grassegar, personal communication.
* [11] J. Graver, B. Servatius and H. Servatius, Combinatorial rigidity, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993.
* [12] H. Gluck, Almost all simply connected closed surfaces are rigid, in Geometric Topology, Lecture Notes in Math., no. 438, Springer-Verlag, Berlin, 1975, pp. 225-239.
* [13] T. Jordan and S. Tanigawa, Global rigidity of triangulations with braces, J. of Comb. Theory, Ser. B, 136 (2019), 249-288.
* [14] W. Whiteley, Vertex splitting in isostatic frameworks, Structural Topology, 16 (1990), 23-30.
|
2024-09-04T02:54:59.425039 | 2020-03-12T09:06:41 | 2003.05669 | {
"authors": "Mohammadreza Salehi, Atrin Arya, Barbod Pajoum, Mohammad Otoofi,\n Amirreza Shaeiri, Mohammad Hossein Rohban, Hamid R. Rabiee",
"full_text_license": null,
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"provenance": "arxiv-papers-0000.json.gz:26180",
"submitter": "Mohammadreza Salehi Dehnavi",
"url": "https://arxiv.org/abs/2003.05669"
} | arxiv-papers | # ARAE: Adversarially Robust Training of Autoencoders Improves
Novelty Detection
Mohammadreza Salehi,1 Atrin Arya,1 Barbod Pajoum,1 Mohammad Otoofi,1
Amirreza Shaeiri,1 Mohammad Hossein Rohban,1 Hamid R. Rabiee1
###### Abstract
Autoencoders (AE) have recently been widely employed to approach the novelty
detection problem. Trained only on the normal data, the AE is expected to
reconstruct the normal data effectively while failing to regenerate the
anomalous data. Based on this assumption, one could utilize the AE for novelty
detection. However, it is known that this assumption does not always hold.
More specifically, such an AE can often perfectly reconstruct the anomalous
data as well, due to modeling of low-level and generic features in the input.
To address this problem, we propose a novel training algorithm for the AE that
facilitates learning of more semantically meaningful features. For this
purpose, we exploit the fact that adversarial robustness promotes learning of
meaningful features. Therefore, we force the AE to learn such features by
making its bottleneck layer more stable against adversarial perturbations.
This idea is general and can be applied to other autoencoder based approaches
as well. We show that despite using a much simpler architecture in comparison
to the prior methods, the proposed AE outperforms or is competitive to state-
of-the-art on four benchmark datasets and two medical datasets.
## Introduction
---
Figure 1: Unlike DAE, ARAE that is trained on the normal class, which is the
digit $8$, reconstructs a normal instance when it is given an anomalous digit,
from the class $1$. The first row shows the input images. The second and third
rows show the DAE and ARAE reconstructions of the corresponding inputs,
respectively. ARAE is trained based on bounded $\ell_{\infty}$, $\ell_{2}$,
rotation, and translation perturbations.
In many real-world problems, it is easy to gather normal data from the
operating behavior of a system. However, collecting data from the same system
in situations where it malfunctions or is being used clumsily may be difficult
or even impossible. For instance, in a surveillance camera that captures daily
activity in an environment, almost all frames are related to the normal
behavior. This means that data associated with the anomalous behavior is
difficult to obtain from such cameras. Anomaly/novelty detection refers to the
set of solutions for such settings.
The key point in the definition of anomaly detection is the outlier notion. In
the literature, An outlier is defined as a data point that deviates from the
bulk of the remaining data (Hawkins 1980; Chalapathy and Chawla 2019).
Assuming that the normal data is generated by a distribution, the goal is to
detect whether a new unseen observation is drawn from this distribution or
not. In prior work, AE and Generative Adversarial Network (GAN) were
extensively applied for novelty detection (Sabokrou et al. 2018; Perera,
Nallapati, and Xiang 2019; Schlegl et al. 2017; Akcay, Atapour-Abarghouei, and
Breckon 2018).
In GAN-based approaches, one tries to train a model that could adversarially
generate realistic images from the normal class. This means that if the model
fails to generate a given input image, the input would probably be an
anomalous one. However, GAN-based approaches face some challenges during the
training. These include mode collapse that happens when the generator maps
several inputs to a single image in the output space. In GAN, complete mode
collapse is rare, while a partial collapse occurs more frequently (Goodfellow
2016; Kodali et al. 2017). Furthermore, high sensitivity of the training to
the choices of hyperparameters, non-convergence problem, parameter
oscillation, and non-reproducible results due to the unstable training are
counted as the other challenges in training of the GAN (Martin and Lon 2017;
Salimans et al. 2016).
On the other hand, AE is more convenient to train and gives results that are
easier to reproduce. Therefore, we propose our method based on AE-based
approaches in this paper. An AE, which has learned features that are mostly
unique to the normal class, could reconstruct the normal data perfectly, while
when given an anomalous data, it either reconstructs a corrupted or a normal
output; In the former case, the anomalous input is likely to have disjoint
features compared to the normal class, while in the latter, the input may
resemble a normal data in some aspects. Note that in both cases, unlike for
the normal data, the reconstruction Mean Squared Error (MSE) is high for the
anomalous data. This means that for such an AE, we could threshold the
reconstruction loss to distinguish the normal vs. anomalous data. One could
alternatively leverage a discriminator that is applied to the reconstructed
image to distinguish between the anomalous and normal data (Sabokrou et al.
2018; Larsen et al. 2015). In any case, as mentioned, an important premise for
the AE to work is that it learns mostly unique features to the normal class.
We call such features “semantically meaningful” or “robust”, contrasted with
generic low level features that are subject to change in presence of noise, in
the rest of the paper.
A common problem in using AE for novelty detection is its generalization
ability to reconstruct some anomaly inputs, when they share common features
with the normal class (Gong et al. 2019; Zong et al. 2018). Although this
generalization property is useful in other contexts, such as restoration (Mao,
Shen, and Yang 2016), it is considered as a drawback in novelty detection. In
other papers (Hasan et al. 2016; Zhao et al. 2017; Sultani, Chen, and Shah
2018), the main underlying assumption behind the AE-based approaches is that
the reconstruction error is high when the model is given an anomalous data,
which as mentioned does not seem to be holding perfectly.
There are two reasons why the main underlying assumption in these methods does
not hold necessarily. First, the model behavior when facing the anomalous data
is not observed and is not therefore predictable. Second, the learned latent
space may capture mostly the features that are in common between the normal
and anomalous data. When given the anomalous data, this would likely yield a
perfectly reconstructed anomalous data. To address these issues, we aimed for
a solution that learns an adversarially robust latent space, where the focus
is on learning unique or semantically meaningful features of the normal inputs
and their nuances. This could prevent the decoder from reconstructing the
anomalies.
It is shown in (Madry et al. 2017) that small imperceptible changes in the
input can easily fool a deep neural network classifier. AE’s are subject to
such attacks as well. This stems from the fact that a deep classifier or an AE
would likely learn low level or brittle non-robust features (Ilyas et al.
2019). Low level features could be exploited to reconstruct any given image
perfectly. Hence, the presence of such features seems to violate the main
underlying assumption of the earlier work for novelty detection that is based
on AE. Therefore, we propose to train an adversarially robust AE to overcome
this issue. In Figure 1, reconstructions from DAE and the proposed method are
shown. Here, the normal data is considered to be the number $8$ in the MNIST
dataset and the models are trained only on the normal category. As opposed to
the proposed ARAE, DAE generalizes and reconstructs the number $1$ perfectly.
This is not desired in the novelty detection problem. This means that the
latent space of DAE has learned features that are not necessarily meaningful.
To train a robust AE for the novelty detection task, a new objective function
based on adversarial attacks is proposed. The novel AE which is based on a
simple architecture, is evaluated on MNIST, Fashion-MNIST, COIL-100, CIFAR-10,
and two medical datasets. We will next review existing approaches in more
details, and then describe our proposed idea along with its evaluation. We
demonstrate that despite the simplicity of the underlying model, the proposed
model outperforms or stays competitive with state-of-the-art in novelty
detection. Moreover, we show that our method performs much better compared to
another state-of-the-art method in presence of adversarial examples, which is
more suitable for real-world applications.
## Related work
Figure 2: The training procedure of our method. $L_{latent}$ and $L_{rec.}$
are obtained using the MSE distance and used to form $L_{AE}$.
As explained earlier in the introduction, methods that are used in the
literature are classified into two main categories: (1) modeling the normal
behavior in the latent space; and (2) thresholding the AE reconstruction
error. Of course, a hybrid of these two approaches was also considered in the
field.
DRAE (Zhou and Paffenroth 2017), takes the second approach, i.e. it is based
on the MSE distance between the AE output and its input. An underlying
assumption in this work is that the training data may contain abnormal
samples. Therefore, the method tries to identify these samples throughout the
training process. It finally uses only the reconstruction error in the test
time.
As an extension to the AE-based methods, in OCGAN (Perera, Nallapati, and
Xiang 2019), a model is introduced in which the AE is trained by using 4 GANs,
a classifier, and the “negative sample mining” technique. Here, both the
encoder and decoder of the AE are considered as generators in the GAN. At the
inference time, the method only uses MSE between the model output and input to
make a prediction. The authors attempted to force the encoder output
distribution to be approximately uniform. They also forced the decoder output
distribution to resemble the normal input distribution in the whole latent
domain. This is expected to result in a higher MSE distance between the
decoder output and input for the abnormal data. This method achieved state-of-
the-art results at the time of presentation.
(Abati et al. 2019) and (Sabokrou et al. 2018) are the other examples in the
AE-based approaches, except that in (Abati et al. 2019), additionally, the
probability distribution over the latent space was obtained for the normal
input data. Then, in the test time, the probability of a sample being normal,
which is called the “surprise score”, is added to the reconstruction error
before the thresholding happens. In (Sabokrou et al. 2018), there is a
possibility of using the discriminator output, which is a real number between
zero and one, as an alternative to the MSE distance in order to find the
anomaly score. This is done by considering the AE as the generator in the GAN
framework.
In (Pidhorskyi, Almohsen, and Doretto 2018), a GAN is initially used to obtain
the latent space, then the probability distribution of the normal class over
the latent space is considered to be as the multiplication of two marginal
distributions, which are learned empirically. (Ruff et al. 2018) (DSVDD) tries
to model the normal latent space with the presumption that all normal data can
be compressed into a hyper-sphere. This framework can be considered as a
combination of Deep Learning and classical models such as (Chen, Zhou, and
Huang 2001) (One-class SVM), that has the advantage of extracting more
relevant features from the training data than the above-mentioned (Chen, Zhou,
and Huang 2001) because the whole network training process is done in an end-
to-end procedure. In (Schlegl et al. 2017), a GAN framework is used to model
the latent space. It is assumed that if the test data is normal, then a sample
could be found in a latent space such that the corresponding image that is
made by the generator is classified as real by the GAN discriminator.
## Method
Figure 3: Samples from the evaluation datasets. For the medical datasets, the
top row samples are anomalous and the bottom row samples are normal.
As we discussed earlier, the main problem of AE is its strong generalization
ability. We observe that DAE does not necessarily learn distinctive features
of the normal class. To remedy this problem, our approach is to force the AE
latent space implicitly to model only unique features of the normal class. To
make this happen, the framework for adversarial robustness, which is proposed
in (Madry et al. 2017; Ilyas et al. 2019), is adopted. We propose to
successively craft adversarial examples and then utilize them to train the AE.
Adversarial examples are considered as those irrelevant small changes in the
input that destabilize the latent encoding. We will next describe the details
of the proposed adversarial training in the following sections. The training
procedure is demonstrated in Figure 2.
### Adversarial Examples Crafting
In a semantically meaningful latent space, two highly perceptually similar
samples should share similar feature encodings. Therefore, searching for a
sample $X^{*}$ that is perceptually similar to a sample $X$, but has a distant
latent encoding from that of $X$, leads us to an adversarial sample. As
opposed to the normal sample $X$, the adversarial sample $X^{*}$ is very
likely to have a high reconstruction loss, thus it would be detected as
abnormal by the AE, despite being perceptually similar to a normal sample.
Therefore, based on this intuition, the following method is used to craft the
adversarial samples.
At the training epoch $i$, we craft a set of adversarial samples
$S^{i}_{(adv)}$ based on the initial training dataset $S$. For this purpose,
we slightly perturb each sample $X\in S$ to craft an adversarial sample
$X^{*}$ that has two properties: (1) $X^{*}$ is perceptually similar to $X$,
through controlling the $\ell_{\infty}$ distance of $X$ and $X^{*}$; (2)
$X^{*}$ latent encoding is as far as possible from that of $X$. This is
equivalent to solving the following optimization problem:
$\max_{\delta_{X}}L_{\text{latent}}\mbox{ s.t.
}{\|\delta_{X}\|}_{\infty}\leq\epsilon$ (1)
$L_{\text{latent}}=\|\mbox{Enc}(X+\delta_{X})-\mbox{Enc}(X)\|^{2}_{2}\ $ (2)
In this formulation, ${\|\ .\ \|}_{p}$ is the $\ell_{p}$-norm, $\epsilon$ is
the attack magnitude, and $X^{*}=X+\delta_{X}$ is the adversarial sample. We
solve this optimization problem for each sample $X\in S$ using the Projected
Gradient Descent (PGD) (Madry et al. 2017) method, to obtain $S^{i}_{(adv)}$.
### Autoencoder Adversarial Training
To train the AE using the crafted dataset $S^{i}_{(adv)}$ in the previous
section, we propose the following loss function:
$L_{\text{AE}}=L_{\text{rec.}}+\gamma L_{\text{latent}}$ (3)
where $\gamma$ is a balancing hyperparameter, $L_{\text{latent}}$ refers to
the loss function that is introduced in Eq. 2 and $L_{\text{rec.}}$
corresponds to the following loss function:
$L_{\text{rec.}}=\|X-\mbox{Dec}(\mbox{Enc}(X^{*}))\|^{2}_{2}\ $ (4)
At each step, the AE is trained one epoch on the adversarially crafted samples
using this loss function. In the training procedure, the $L_{\text{rec.}}$
term forces the AE to reconstruct the adversarial samples properly, while the
$L_{\text{latent}}$ term forces the adversarial samples to have closer
representations to that of the corresponding normal samples in the latent
space. We observe that the encoder decreases $L_{\text{latent}}$ to a limited
extent by merely encoding the whole input space into a compact latent space.
Too compact latent space results in a high $L_{\text{rec.}}$, which is not
achievable when the network is trained using $L_{\text{AE}}$. A compact latent
space causes the latent encodings of anomalous data to be close to that of
normal data. Thus for any given input, the generated image is more likely to
be a normal sample. To summarize, the whole training procedure is trying to
solve the following saddle point problem (Wald 1945):
$\begin{gathered}\delta^{*}_{X}:=\operatorname*{arg\,max}_{\|\delta_{X}\|_{\infty}\leq\epsilon}L_{\text{latent}}(X,\delta_{X},W)\\\
\min_{W}\operatorname{\mathbb{E}}_{X}\left[\gamma
L_{\text{latent}}(X,\delta^{*}_{X},W)+L_{\text{rec.}}(X,\delta^{*}_{X},W)\right]\end{gathered}$
(5)
where $W$ is denoted as the AE weights. Note that it was shown that the
adversarial training could not be solved in a single shot by the Stochastic
Gradient Descent (SGD), and one instead should try other optimization
algorithms such as the PGD. This relies on Danskin theorem to solve the inner
optimization followed by the outer optimization (Madry et al. 2017).
## Experiments
In this section, we evaluate our method, which is denoted by ARAE, and compare
it with state-of-the-art on common benchmark datasets that are used for the
unsupervised novelty detection task. Moreover, we use two medical datasets to
evaluate our method in real-world settings. We show that even though our
method is based on a simple and efficient architecture, it performs
competitively or superior compared to state-of-the-art approaches.
Furthermore, we provide insights about the robustness of our method against
adversarial attacks. The results are based on several evaluation strategies
that are used in the literature. All results that are reported in this paper
are reproducible by our publicly available implementation in the Keras
framework (Chollet 2015)111https://github.com/rohban-
lab/Salehi˙submitted˙2020.
### Experimental Setup
#### Baselines
Baseline and state-of-the-art approaches like VAE (Kingma and Welling 2013),
OCSVM (Chen, Zhou, and Huang 2001), AnoGAN (Schlegl et al. 2017), DSVDD (Ruff
et al. 2018), MTQM (Wang, Sun, and Yu 2019), OCGAN (Perera, Nallapati, and
Xiang 2019), LSA (Abati et al. 2019), DAGMM (Zong et al. 2018), DSEBM (Zhai et
al. 2016), GPND (Pidhorskyi, Almohsen, and Doretto 2018), $l_{1}$ thresholding
(Soltanolkotabi, Candes et al. 2012), DPCP (Tsakiris and Vidal 2018),
OutlierPursuit (Xu, Caramanis, and Sanghavi 2010), ALOCC (Sabokrou et al.
2018), LOF (Breunig et al. 2000), and DRAE (Xia et al. 2015) are selected to
be compared with our method. Results of some of these methods were obtained
from (Perera, Nallapati, and Xiang 2019; Wang, Sun, and Yu 2019; Pidhorskyi,
Almohsen, and Doretto 2018).
#### Datasets
We evaluate our method on MNIST (LeCun, Cortes, and Burges 2010), Fashion-
MNIST (Xiao, Rasul, and Vollgraf 2017), COIL-100 (Nene, Nayar, and Murase
1996), CIFAR-10 (Krizhevsky 2009), Head CT - hemorrhage (Kitamura 2018), and
Brain MRI - Tumor (Chakrabarty 2019) datasets. Samples from each dataset are
shown in Figure 3. These datasets differ in size, image shape, complexity and
diversity. Next, we briefly introduce each of these datasets.
* •
MNIST: This dataset contains 70,000 $28\times 28$ grayscale handwritten digits
from 0 to 9.
* •
Fashion-MNIST: A dataset similar to MNIST with 70,000 $28\times 28$ grayscale
images of 10 fashion product categories.
* •
CIFAR-10: This dataset contains 60000 $32\times 32$ color images of 10
categories.
* •
COIL-100: A dataset of 7200 color images of 100 different object classes. Each
class contains 72 images of one object captured in different poses. We
downscale the images of this dataset to the size $32\times 32$.
* •
Head CT - Hemorrhage: A dataset with 100 normal head CT slices and 100 other
with 4 different kinds of hemorrhage. Each slice comes from a different person
and the image size is $128\times 128$.
* •
Brain MRI - Tumor: A dataset with 253 brain MRI images. 155 of them contain
brain tumors and the rest 98 are normal. The image size is $256\times 256$.
#### Protocols
To carry out the training-testing procedure, we need to define the data
partitions. For MNIST, Fashion-MNIST, and CIFAR-10, one class is considered as
the normal class and samples from the other classes are assumed to be
anomalous. For COIL-100, we randomly take $n$ classes as the normal classes,
where $n\in\\{1,4,7\\}$, and use the samples from the remaining classes as the
anomalous samples. For the mentioned dataset, this process is repeated 30
times and the results are averaged. For the medical datasets, the brain images
with no damage are considered as the normal class and the rest form the
anomalous class. To form the training and testing data, there are two
protocols that are commonly used in the framework of unsupervised novelty
detection(Pidhorskyi, Almohsen, and Doretto 2018; Perera, Nallapati, and Xiang
2019; Sabokrou et al. 2018), which are as follows:
* •
Protocol 1: The original training-testing splits of the dataset are merged,
shuffled, and $80\%$ of the normal class samples are used to train the model.
The remaining $20\%$ forms some specified portion (denoted as $\tau$) of the
testing data. The other portion is formed by randomly sampling from the
anomalous classes.
* •
Protocol 2: The original training-testing splits of the dataset are used to
train and test the model. The training is carried out using the normal samples
and the entire testing data is used for evaluation.
We compare our method to other approaches using Area Under the Curve (AUC) of
the Receiver Operating Characteristics (ROC) curve, the $F_{1}$ score, and the
False Positive Rate (FPR) at $99.5\%$ True Positive Rate (TPR). Here, we let
the positive class be the anomalous one unless otherwise specified.
#### Architecture and Hyperparameters
Our AE uses a 3-layer fully connected network with layer sizes of
$(512,256,128)$, following the input-layer to encode the input. A decoder,
whose architecture is mirroring that of the encoder, is used to reconstruct
the output. Each layer of the network is followed by a sigmoid activation.
This architecture is used for all the datasets except the medical ones and
CIFAR-10. For the medical datasets and CIFAR-10, we use a convolutional AE
which is explained in (Bergmann et al. 2019). For datasets with complex and
detailed images like COIL-100, Fashion-MNIST, CIFAR-10, and the medical
datasets, the hyperparameter $\epsilon$, which is the maximum perturbation
$\ell_{\infty}$ norm as defined in Eq. 5, is set to $0.05$, while for MNIST it
is set to $0.2$. The hyperparameter $\gamma$, defined in Eq. 5, is always set
to $0.1$.
### Results
Table 1: AUC values (in percentage) for the medical datasets. The standard deviation of the last 50 epochs’ AUCs are included for the Brain MRI - Tumor dataset. Dataset | OCGAN | LSA | ARAE
---|---|---|---
Head CT - Hemorrhage | 51.2 | 81.6 | 84.8
Brain MRI - Tumor | 91.7 | 95.6 | 97.0
$\pm 3$ | $\pm 1.4$ | $\pm 0.5$
Table 2: AUC values (in percentage) on MNIST and FMNIST (Fashion-MNIST). The standard deviation of the last 50 epochs’ AUCs are included for our method on MNIST. The values were obtained for each class using protocol 2. Dataset | Method | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean
---|---|---|---|---|---|---|---|---|---|---|---|---
MNIST | VAE | 98.5 | 99.7 | 94.3 | 91.6 | 94.5 | 92.9 | 97.7 | 97.5 | 86.4 | 96.7 | 95.0
OCSVM | 99.5 | 99.9 | 92.6 | 93.6 | 96.7 | 95.5 | 98.7 | 96.6 | 90.3 | 96.2 | 96.0
AnoGAN | 96.6 | 99.2 | 85.0 | 88.7 | 89.4 | 88.3 | 94.7 | 93.5 | 84.9 | 92.4 | 91.3
DSVDD | 98.0 | 99.7 | 91.7 | 91.9 | 94.9 | 88.5 | 98.3 | 94.6 | 93.9 | 96.5 | 94.8
MTQM | 99.5 | 99.8 | 95.3 | 96.3 | 96.6 | 96.2 | 99.2 | 96.9 | 95.5 | 97.7 | 97.3
OCGAN | 99.8 | 99.9 | 94.2 | 96.3 | 97.5 | 98.0 | 99.1 | 98.1 | 93.9 | 98.1 | 97.5
LSA | 99.3 | 99.9 | 95.9 | 96.6 | 95.6 | 96.4 | 99.4 | 98.0 | 95.3 | 98.1 | 97.5
ARAE | 99.8 | 99.9 | 96.0 | 97.2 | 97.0 | 97.4 | 99.5 | 96.9 | 92.4 | 98.5 | 97.5
$\pm 0.017$ | $\pm 0.003$ | $\pm 0.2$ | $\pm 0.17$ | $\pm 0.14$ | $\pm 0.1$ | $\pm 0.03$ | $\pm 0.1$ | $\pm 0.3$ | $\pm 0.04$ | $\pm 0.04$
FMNIST | VAE | 87.4 | 97.7 | 81.6 | 91.2 | 87.2 | 91.6 | 73.8 | 97.6 | 79.5 | 96.5 | 88.4
OCSVM | 91.9 | 99.0 | 89.4 | 94.2 | 90.7 | 91.8 | 83.4 | 98.8 | 90.3 | 98.2 | 92.8
DAGMM | 30.3 | 31.1 | 47.5 | 48.1 | 49.9 | 41.3 | 42.0 | 37.4 | 51.8 | 37.8 | 41.7
DSEBM | 89.1 | 56.0 | 86.1 | 90.3 | 88.4 | 85.9 | 78.2 | 98.1 | 86.5 | 96.7 | 85.5
MTQM | 92.2 | 95.8 | 89.9 | 93.0 | 92.2 | 89.4 | 84.4 | 98.0 | 94.5 | 98.3 | 92.8
LSA | 91.6 | 98.3 | 87.8 | 92.3 | 89.7 | 90.7 | 84.1 | 97.7 | 91.0 | 98.4 | 92.2
ARAE | 93.7 | 99.1 | 91.1 | 94.4 | 92.3 | 91.4 | 83.6 | 98.9 | 93.9 | 97.9 | 93.6
Table 3: AUC and $F_{1}$ values on the COIL-100 dataset. The values were obtained using protocol 1 for $n\in\\{1,4,7\\}$ and different $\tau$s, where n and $\tau$ represent the number of normal classes and the testing data portion of the normal samples, respectively. Parameters | Metric | OutlierPursuit | DPCP | $l_{1}$ thresholding | GPND | ARAE
---|---|---|---|---|---|---
$n=1$, $\tau=50\%$ | AUC | 0.908 | 0.900 | 0.991 | 0.968 | 0.998
$F_{1}$ | 0.902 | 0.882 | 0.978 | 0.979 | 0.993
$n=4$, $\tau=75\%$ | AUC | 0.837 | 0.859 | 0.992 | 0.945 | 0.997
$F_{1}$ | 0.686 | 0.684 | 0.941 | 0.960 | 0.973
$n=7$, $\tau=85\%$ | AUC | 0.822 | 0.804 | 0.991 | 0.919 | 0.993
$F_{1}$ | 0.528 | 0.511 | 0.897 | 0.941 | 0.941
Table 4: Mean AUC values (in percentage) on CIFAR-10 using protocol 2. Metric | OCSVM | OCGAN | LSA | ARAE
---|---|---|---|---
AUC | 67.8 | 73.3 | 73.1 | 71.7
We present our AUC results for MNIST and Fashion-MNIST in Table 2. The table
contains AUC values for each class as the normal class, which were achieved
using protocol 2. Moreover, we report our results on the COIL-100 dataset in
Table 3. This table contains AUC and $F_{1}$ values for $n\in\\{1,4,7\\}$,
where $n$ is the number of normal classes. We use protocol 1 for this dataset.
For each $n\in\\{1,4,7\\}$, the percentage of the normal samples in the
testing data ($\tau$) is defined in the table. The $F_{1}$ score is reported
for the threshold value that is maximizing it. As shown in Tables 2 and 3, we
achieve state-of-the-art results in all of these datasets while using a
simpler architecture compared to other state-of-the-art methods, such as
OCGAN, LSA, and GPND. Moreover, the results in Table 3 indicate that our
method performs well when having multiple classes as normal. It also shows the
low effect of the number of normal classes on our method performance.
We also report our mean AUC results for the CIFAR-10 dataset using protocol 2,
excluding the classes with AUC near 0.5 or below, in Table 4. Consider a
classifier that labels each input as normal with probability $p$. By varying
$p$ between 0 and 1, we can plot a ROC curve and compute its AUC. We observe
that this method achieves an AUC of 0.5. So improvements below or near 0.5
aren’t valuable (see (Zhu, Zeng, and Wang 2010) for more details).
Consequently, classes 1, 3, 5, 7, and 9 which contained AUC values below 0.6
were excluded. As shown in the table, we get competitive results compared to
other state-of-the-art approaches.
The AUC values of our method on the medical datasets are reported in Table 1.
We used $90\%$ of the normal data for training and the rest in addition to the
anomalous data were used to form the testing data. Our method clearly
outperforms other state-of-the-art approaches, which shows the effectiveness
of our method on medical real-world tasks, where the dataset might be small
and complex.
To show the stability of our training procedure, we compute the standard
deviation of AUCs for the last 50 epochs of training. These values are
reported for our method on MNIST in Table 2 and for all the methods on Brain
MRI - Tumor in Table 1. From these tables, one can see the high stability of
our training procedure. Moreover, It is apparent that our method is much more
stable than other methods on the Brain MRI - Tumor dataset.
We also evaluate our method using the $F_{1}$ score on the MNIST dataset. In
this experiment, the normal class is the positive one. We use protocol 1 and
vary $\tau$ between $50\%$ and $90\%$. We use $20\%$ of the training samples
and sample from the anomalous classes to form a validation set with the same
normal samples percentage as the testing data. This validation set is used to
find the threshold that maximizes the $F_{1}$ score. As shown in Figure 4, we
achieve slightly lower $F_{1}$ scores compared to that of GPND. However, this
figure shows the low impact of the percentage of anomalous data on our method
performance.
Furthermore, FPR values at $99.5\%$ TPR on the MNIST dataset using protocol 2,
for ARAE and LSA are compared in Figure 5. One can see that despite having
equal AUCs, ARAE has lower FPR values compared to LSA and that it can reduce
the FPR value more than $50\%$ in some cases.
#### Adversarial Robustness
To show the robustness of our model against adversarial attacks, we use PGD
(Madry et al. 2017) with the $\epsilon$ parameter set to $0.05$ and $0.1$ on
the reconstruction loss, to craft adversarial samples from the normal samples
of the testing data. The normal samples of the testing data are replaced by
the adversarial ones. The AUC results for this testing data are reported in
Table 5 on the class 8 of the MNIST dataset, using protocol 2. As shown in the
table, our method is significantly more robust against adversarial samples
compared to LSA.
### Ablation
Table 5: AUC values for the attacked models. The values are reported for class 8 of MNIST using protocol 2. Parameters | LSA | ARAE
---|---|---
$\epsilon=0.05$ | 0.56 | 0.86
$\epsilon=0.1$ | 0.17 | 0.76
Table 6: AUC values (in percentage) on MNIST using protocol 2. The results are reported for both one class and two classes as the normal data. Results for other variants of our method are reported. Method | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean
---|---|---|---|---|---|---|---|---|---|---|---
DAE | 99.6 | 99.9 | 93.9 | 93.5 | 96.4 | 94.3 | 99.0 | 95.8 | 89.1 | 97.5 | 95.9
ARAE | 99.8 | 99.9 | 96.0 | 97.2 | 97.0 | 97.4 | 99.5 | 96.9 | 92.4 | 98.5 | 97.5
ARAE-A | 99.1 | 99.7 | 95.2 | 96.7 | 97.7 | 98.3 | 99.2 | 97.1 | 95.6 | 96.8 | 97.5
ARAE-R | 99.3 | 99.9 | 93.2 | 92.5 | 96.2 | 96.6 | 99.3 | 97.3 | 91.2 | 98.2 | 96.4
Method | (4, 5) | (0, 7) | (1, 3) | (2, 6) | (8, 9) | (2, 9) | (0, 8) | (0, 1) | (2, 3) | (4, 9) | Mean
DAE | 88.8 | 94.1 | 98.2 | 90.3 | 86.8 | 91.8 | 91.1 | 99.7 | 90.0 | 97.3 | 92.8
ARAE | 91.7 | 96.0 | 99.1 | 94.7 | 91.4 | 94.5 | 93.1 | 99.7 | 91.2 | 97.3 | 94.9
ARAE-A | 95.0 | 97.1 | 97.4 | 95.7 | 91.5 | 92.6 | 94.3 | 98.8 | 94.3 | 97.4 | 95.4
Figure 4: $F_{1}$ scores on the MNIST dataset using protocol 1, by taking the
normal class as the positive one. Figure 5: FPR at $99.5\%$ TPR on the MNIST
dataset using protocol 2.
We train a DAE, as a baseline method, with a random uniform noise between 0
and $0.1$ using the same network as the one that is used in our approach.
Furthermore, In addition to the $\ell_{\infty}$ perturbation set, we consider
$\ell_{2}$, and also rotation and translation perturbation sets. We need to
solve a similar optimization to the one in Eq. 5, with the only difference
being the perturbation sets (Engstrom et al. 2017). Specifically, we solve
this optimization problem on $\ell_{2}$-bounded perturbations for each sample
$X\in S$ through PGD (Madry et al. 2017) again. We next solve this
optimization on rotation and translation perturbation sets for each sample
$X\in S$ by quantizing the parameter space, and performing a grid search on
the quantized space and choosing the one with the highest latent loss. This is
the most reliable approach for solving rotation and translation perturbations
that is mentioned in (Engstrom et al. 2017). Following the approach in (Tramèr
and Boneh 2019), we use the union of these perturbation sets to make the
attack even stronger to avoid as much as brittle features that model might use
(Ilyas et al. 2019). We present our results on MNIST using protocol 2, in
Table 6. This variant of our method is denoted as ARAE-A. Notably, the AUC is
improved further in this variant in the most challenging class $8$ in MNIST
from $92.4$ based on $\ell_{\infty}$ attack to $95.6$ using the union of the
mentioned attacks. Despite this improvement, the average AUC is still the same
as in the original ARAE method.
Instead of designing the attack based on the latent layer, one could directly
use the reconstruction loss to do so. We denote this variant as ARAE-R.
However, we observed that a model that is robust to the latter attack yields a
lower improvement compared to ARAE (see Table 6). To justify this effect, we
note that an AE model that is robust based on the latter attack does not
necessarily have a stable latent layer. This stems from the fact that the
encoder and decoder are almost inverse functions by construction, and a
destabilization of the latent encoding by an attack could be repressed by the
decoder. In summary, an attack based on the latent layer is stronger than an
attack based on the reconstruction error, and hence the former promotes more
robust features.
We also report AUC values on MNIST by taking pairs of classes as the normal
ones, in Table 6. These values show the improvement yield by both of the ARAE
variants. Note that when having multiple classes as normal, one should tune
the $\epsilon$ parameter based on diversity and complexity of the training
data.
## Visualization
In the experiments section, we showed that our method improves the AE
performance and surpasses other state-of-the-art methods. In order to
demonstrate the reasons behind this improvement, we show that ARAE learns more
semantically meaningful features than DAE by interpreting these two
approaches.
MNIST | Fashion-MNIST
---|---
Input | DAE rec. | DAE map | ARAE rec. | ARAE map | Input | DAE rec. | DAE map | ARAE rec. | ARAE map
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
Figure 6: ARAE and DAE reconstructions and saliency maps for ten random inputs
from MNIST and Fashion-MNIST datasets.
ARAE | | | | | | | |
---|---|---|---|---|---|---|---|---
DAE | | | | | | | |
Figure 7: Local minima of inputs of ARAE and DAE, by initializing the input
with random noise and optimizing the reconstruction loss with respect to the
input. ARAE produces more realistic $8$ digits compared to DAE.
### Interpreting with Occlusion-1
In this method, we measure the effect of each part of the input on the output,
by occluding it and observing the difference in the output. Finally, we
visualize these differences as a saliency map (Zeiler and Fergus 2014; Ancona
et al. 2017). In the occlusion-1 method, we iteratively set each pixel to
black and then observe the reconstruction error. If it increases, we set the
corresponding pixel in the saliency map to blue, and otherwise, we set it to
red. The intensity of a pixel is determined by the amount that the
reconstruction error has changed. We compare ARAE and DAE reconstructions and
saliency maps on MNIST and Fashion-MNIST datasets, in Figure 6.
For the MNIST dataset, the model has been trained on the class $8$ and noisy
inputs are obtained by adding a uniform noise in the interval $[0,0.4]$. The
outputs and saliency maps of ARAE and DAE are shown for five random inputs in
the normal class. It is evident that DAE is focusing too much on the random
noises and has a poorer reconstruction than our model.
Similar to MNIST, we carry out the occlusion-1 method on the class dress of
the Fashion-MNIST dataset. For Fashion-MNIST, it is also obvious that random
noises have a larger effect on the output of DAE. Furthermore, DAE
reconstructions are less accurate than those of ARAE. These observations are
consistent with the known fact that adversarial robustness can increase the
model interpretability (Tsipras et al. 2018) by avoiding the learning of
brittle features (Ilyas et al. 2019).
### Local Minima Visualization
We expect from an ideal model that is trained on the MNIST class $8$, to have
a lower reconstruction error as the input gets more similar to a typical $8$.
With this motivation, we start from random noise and iteratively modify it in
order to minimize the reconstruction error using gradient descent. The results
achieved by our model and DAE are shown in Figure 7. This figure demonstrates
that inputs that lead to local minima in ARAE are much more similar to $8$,
compared to DAE.
## Conclusions
We introduced a variant of AE based on the robust adversarial training for
novelty detection. This is motivated by the goal of learning representations
of the input that are almost robust to small irrelevant adversarial changes in
the input. A series of novelty detection experiments were performed to
evaluate the proposed AE. Our experimental results of the proposed ARAE model
show state-of-the-art performance on four publicly available benchmark
datasets and two real-world medical datasets. This suggests that the benefits
of adversarial robustness indeed go beyond security. Furthermore, by
performing an ablation study, we discussed the effect of multiple perturbation
sets on the model. Future work inspired by this observation could investigate
the effect of other types of adversarial attacks in the proposed framework.
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*[AE]: Autoencoders
*[DAE]: Denoising Autoencoder
*[ARAE]: Adversarially Robust trained Autoencoder
*[GAN]: Generative Adversarial Network
*[MSE]: Mean Squared Error
*[PGD]: Projected Gradient Descent
*[SGD]: Stochastic Gradient Descent
|
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